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Mechanically Responsive Molecular Crystals Panče Naumov,*,† Stanislav Chizhik,‡,§ Manas K. Panda,† Naba K. Nath,† and Elena Boldyreva*,‡,§ †
New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates Institute of Solid State Chemistry and Mechanochemistry, Siberian Branch of Russian Academy of Sciences, ul. Kutateladze, 18, Novosibirsk 630128, Russia § Novosibirsk State University, ul. Pirogova, 2, Novosibirsk 630090, Russia ‡
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S Supporting Information *
2.9. Thin Isometric Crystals 2.10. Symmetry Restrictions and Photoinduced Structural Transformations 3. A Model for Bending of Crystalline Needles 3.1. General Considerations 3.2. Distribution of the Yield Normal to the Irradiated Face 3.3. Effect of the Reverse Thermal Reaction on the Concentration Profile 3.4. Bending, Twisting, and Coiling of a Needle Irradiated from One Side 3.4.1. Bending and Torsion Moments 3.4.2. Bending 3.4.3. Twisting 3.4.4. Coiling 3.4.5. Time Profile of the Crystal Curvature 3.4.6. Dependence on the Geometric Parameters of the Crystal 3.4.7. Residual Stresses in the Crystal 3.5. Strain−Stress Relations in Crystals Stressed by an Internal Transformation 4. Relaxation of the Mechanical Stresses Related to Photoinduced Transformations 4.1. Relaxation by Fracturing 4.1.1. Fracture of Bulky Macroscopic Objects 4.1.2. Fracture of Thin Films and Fine Particles 4.1.3. Fractures Resulting from Photoinduced Transformations 4.2. Estimation of the Critical Conditions 4.2.1. Stochastic Nature of Crack Formation 4.2.2. Thick Crystals: Tensile Stresses in the Product Layer 4.2.3. Thick Crystals: Compressive Stresses in the Product Layer 4.2.4. Thin Needle-Shaped Crystals or Ribbons: The Structure Shrinks along the Long Crystal Axis 4.2.5. Thin Crystal Needles or Ribbons: The Structure Expands along the Long Crystal Axis 4.2.6. Thin Crystal Needles or Ribbons: Shear Strain in the Plane of the Irradiated Face
CONTENTS 1. Introduction 1.1. Materials for Conversion of Energy into Mechanical Motion 1.2. Actuation at Different Levels of Structural Hierarchy 1.3. Advantages of Single Crystals for Actuation 1.4. Scope of This Review 2. A General Model for Mechanical Effects in Crystals 2.1. General Considerations of Photomechanical Effects 2.2. Main Types of Mechanical Effects 2.3. Factors That Determine the Mechanical Response 2.3.1. Regular and Random Factors 2.3.2. Tensor of Strain Induced by Photochemical Transformation 2.3.3. Distribution of the Product through the Crystal Bulk 2.3.4. Boundary Conditions in the Elastic Strain Problem 2.3.5. Crystal Size and Shape 2.4. Large, Thick Crystals 2.5. Thin Plates 2.6. Needle-Shaped Crystals 2.6.1. Modes of Deformation 2.6.2. Isotropic Material 2.6.3. Anisotropic Material 2.7. Other Factors That Determine the Mechanical Effect 2.8. Very Thin (Nanosize) Needles © 2015 American Chemical Society
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Chemical Reviews 4.3. Estimation and Quantification of the Critical Conditions 4.3.1. Fracture at the Surface of a Thick Crystal 4.3.2. Fracture of Thin Crystals 4.3.3. Transformation of the Excess Strain Energy to Kinetic Energy of the Fragments 4.4. Role of Plastic Deformation in Photomechanical Effects 4.4.1. General Considerations 4.4.2. A Mathematical Model 5. Overview of Experimentally Observed Mechanical Effects in Crystals 5.1. Introductory Notes 5.2. Relaxation by Bending and Twisting 5.2.1. Crystals of Inorganic Compounds 5.2.2. Crystals of Coordination Compounds 5.2.3. Bending Organic Crystals 5.3. Relaxation by Rapid Deformation and Disintegration (Photo- and Thermosalient Effects) 5.3.1. General Considerations 5.3.2. Thermosalient Effect 5.3.3. Photosalient Effect 5.3.4. Other Mechanical Effects 6. Conclusions and Future Perspectives Associated Content Supporting Information Author Information Corresponding Authors Notes Biographies Acknowledgments References
Review
electric or magnetic fields. In the past, photorestrictive effects in inorganic ceramic materials have been commonly employed to control actuators, which are known to be robust, stiff, and highly temperature resistant. More recently, thin strips or films of soft materials, such as organic polymers, elastomers, or liquid crystalline materials, were used for the same purpose. These materials are characterized by relatively low melting or glass transition temperatures, and they are sufficiently elastic to readily absorb the macroscopic mechanical stress that builds up during mechanical reconfiguration. Traditionally, similar processes have been considered detrimental to the physical integrity of single crystals of organic, metal−organic, or organometallic materials. Research on mechanically responsive single crystals had already been developed by the 1980s,1−3 during which time it was very active.4 Dating just back over a decade, this field was revived alongside the growing realization that the mechanical effects in crystals can be utilized as dynamic elements in macroscopic devices, and several research groups have been particularly prolific in this research field.1 More recent research on actuating materials has brought a paradigm shift in the perception that single crystals are stiff and brittle entities. The renewed interest in materials with superb mechanical properties comes from the need to utilize such materials for the design of new fast and efficient energy-storage materials or actuators. Along these lines of pursuit, there are intermittent reports on piezochromic, sonochemical, and triboluminescent reactions and processes in molecular crystals that relate their mechanical properties to their molecular and crystalline structure. The favorable elastic constants of organic crystals are expected to result in faster responses and to relax more rapidly for recovery of the initial state. Indeed, as we discuss later in this Review, some of these materials are capable of colossal leaps (distances 105−106 times their own size) or bending over a right angle, and represent extreme, visually impressive cases of the response to strains that can accumulate in the crystal interior. These rare organic crystals are self-actuators par excellence and could also be utilized as convenient models in solid-state chemistry research to examine the relation between the collective force of weak intermolecular interactions and the macroscopic response of a crystal up to the limit of susceptibility of ordered matter to strains. Although the number of mechanically responsive crystals is still limited, and the quantification and systematic comparison of the performance of these phenomena is statistically impractical, if taken together they set the path to establish more quantitative and systematic correlations. Indeed, the increasing number of reported examples of mechanically active single crystals promises a bright future for both basic research and practical applications of these extraordinary materials.
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1. INTRODUCTION 1.1. Materials for Conversion of Energy into Mechanical Motion
Transformation of energy into motion (work) is one of the basic processes of energy conversion in nature. The processes of actuation underlying the transduction of light, thermal, or chemical energy into work by mechanically responsive materials are of special interest to materials science (throughout this Review, “mechanically responsive” refers to materials that respond with a mechanical effect, regardless of whether this effect has been induced by light, heat, mechanical force, or other stimulus). Materials that are capable of stimuli-responsive mechanical effects are invaluable for the fabrication of mechanically tunable elements for actuation and energy harvesting, including flexible electronics, displays, artificial muscles, microfluidic valves and gates, dynamic components in the soft robotics, switchable reflector units for projective displays, and tunable components for contact printing. The advanced materials that will qualify for these applications in the future must fulfill an extended list of requirements including reversibility, rapid and controllable mechanical response that is proportional to the applied stimulus, and extended lifetime with insignificant fatigue. Their properties hinge on the capability for preservation of physical integrity and endurance to defects that normally accumulate to cause fatigue during prolonged exposure to mechanical stress, temperature, light, pH, and
1.2. Actuation at Different Levels of Structural Hierarchy
The mechanical profile of structurally ordered materials is directly related to their molecular structure and crystal packing. As such, an understanding of the molecular processes that underlie generation of mechanical response by external force is critically important for target-oriented design of solid materials that will meet the stringent requirement of enhanced mechanical robustness. It follows that the fundamental relationships between structure organization at various hierarchical levels (molecular, supramolecular, macroscopic) and the mechanical response are instrumental for the 12441
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on polymeric or liquid-crystalline materials,59−62 as exemplified by the design of a macroscopic motor60 and a high-frequency oscillator.63 Although still with limited efficiency, at least in principle these materials can be fabricated into dynamic elements and implemented into devices for direct practical applications, as was demonstrated by a cantilever that contains photoactive acetylacetone groups made of a polymeric hydrogel that operates on the basis of keto−enol tautomerism.64 More recently, humidity (in fact, spatial gradient in relative aerial humidity) is being considered as a stimulus to drive mechanical deformations in hydrogels and polymers in the absence of light.65−73
occurrence (or absence) of actuating phenomena. Understanding of their mechanistic subtleties could also facilitate the efforts to quantify and, in the future, to possibly even predict their magnitude and intensity, as well as their spatial and temporal characteristics and performance. At a molecular level, artificial molecular machines5−7 are molecule-scale chemical replicas of simple macroscopic mechanical devices, such as gyroscopes, compasses, tweezers, or ratchets that can convert energy into molecular work. They were originally designed to mimic active biological supramolecular assemblies, the myosin−actin system8 being only one of the many examples. In living systems, operation of such molecular motors normally requires high-energy molecules as secondary or tertiary components, while it remains a pronouncedly stochastic, externally uncontrollable process. Downscaling macroscopic photomechanical systems to the molecular level has brought the development of molecules or molecular assemblies that are molecular analogues of macroscopic objects and could reproduce their motions at the molecular level.9 Fascinating organic (photo)chemistry has been developed around similar molecular machines10,11 that are capable of converting kinetic energy or light into mechanical motion at a molecular level. As only some of the most appealing examples, selected to illustrate the variety of such machines reported to date, one can highlight molecular rotors composed of weak assemblies of molecules12−14 and single molecules that oscillate around a single axis.15,16 In parallel with the development of molecular-scale supramolecular machinery, photoactive elements for energy conversion at the macroscopic level based on polymers (mainly, elastomeric, liquid-crystalline, and hybrid materials) were also developed.17−25 Much of the work in that regard has been done within the realm of bioinspired and biomimetic materials that mimic some natural, highly organized structures with specific function.26−32 The photomechanical response in photoactive polymers appears as an outcome of interplay between the polymer network morphology, thermomechanical properties, absorption coefficient, and sample geometry, in addition to other factors.33 Substantial efforts have been made to decipher the factors that determine the behavior of such dynamic polymers in response to external effectors, including light irradiation, and their effects on the properties,34−42 and several book chapters34−36,41 and reviews37,38,43−46 elaborate the tremendous advances on this topic. The common approach to transforming light into mechanical energy by using polymers is inclusion of light-sensitive chromophores (azobenzenes, stilbenes, spyropyrans, diarylethenes, complexes that undergo linkage isomerism, etc.) that are chemically coupled47−50 or embedded in polymeric materials or liquid crystal blends.51 Fascinating mechanical or actuating responses have been documented for some of these materials, some of them carrying the potential to directly harness sunlight.52 In addition to surface gratings, which are now routinely fabricated by photoirradiation of azopolymer or spiropyran films, coatings, and microarrays,53−56 photoactive polymer fibers, for instance, can undergo sizable photoinduced stretching or shrinking.57 Some azobenzene-doped elastomers and polymer films exhibit remarkable photomechanical effects,58 contracting as much as 22% of their length.17 In some instances, this bending can be controlled by modulating the light polarization38 or by changing the location of the azobenzene moieties from the cross-links to side chains.25 These properties have recently been utilized for the fabrication of photomechanical elements based
1.3. Advantages of Single Crystals for Actuation
Some of the primary requirements to be met by the nextgeneration actuating materials are rapid, reversible, and fatigueless mechanical deformations with conversion efficiency coefficients and response speeds that supersede those of polymeric materials. While highly elastic and reversible deformations are common for liquid crystalline materials and elastomers, where reversible and directional deformations have already been demonstrated, they are far less intuitive for single crystals, which are considered brittle and devoid of the elasticity required for perpetual deformation.74,75 Nevertheless, although still limited in number relative to polymer materials, there is an emerging number of instances of elastic and mechanically compliant organic crystals with remarkable mechanical response.4,45,76,77 Within this line of pursuit, it recently has been shown78−80 that under mechanically applied force, certain organic molecular crystals can respond with impressive mechanical properties;81 for instance, they can be bent elastically or plastically over 180°82 or to 360°83,84 or can even be superelastic,85,86 much like superelastic inorganic materials. The mechanical properties of single crystals under external mechanical force were summarized in a recent review article80 and will not be a part of the focus in this Review. For all practical purposes, the elastic properties of organicbased molecular crystals can be considered intermediate between those of hard (inorganic materials, such as ceramics) and soft (polymers and polymer-like materials) actuating materials, this property unquestionably being their greatest asset. Among the advantages of organic-based crystals over polymers and liquid crystals for the fabrication of photomechanical actuators are the occasionally superior elastic moduli, which in practical terms translate into stronger resulting force for identical deflection. Moreover, although single crystals may be expected to exhibit smaller deformations, the local stresses induced by photoreaction are comparably higher than in polymers. Unlike dye-doped elastomers, the dense and regularly ordered molecular environment in single crystals provides a platform for rapid energy transfer with less energy dissipation, which translates into comparably more efficient coupling between the light and mechanical energy. Last, the coupling between the incident force and the mechanical energy inherent to the ordered structure of organic single crystals is more efficient relative to that of elastomers and is reflected in their favorable elastic constants. Consequently, faster response and relaxation with shorter recovery times can be anticipated with actuators made of organic-based single crystals, these properties having strong implications for their future applications. In addition to the immediate advantages described above, at least three important properties warrant attention amidst 12442
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discussion of the mechanical effects in single crystals, as opposed to similar effects in polymeric films or other amorphous materials. First, the enhanced degree of cooperativity between constituent molecules that is inherent to the dense and regular packing is particularly efficient in facilitating energy transport throughout the medium. In turn, when acting collectively, even weak intermolecular interactions could lead to sizable responses. These interactions effectively assist the progression of the local perturbation throughout the crystal. The facile transfer of mechanical strain that develops inside the crystal spreads quickly (occasionally through domino-like effects that might develop in such ordered setting) to affect larger domains, before it ultimately manifests itself as apparent shape reconfiguration or motility. Being unique to the crystalline state, the enhanced cooperative action is of imminent relevance to elicit rapid response. Thus, superior kinetics can also be expected. Second, in contrast to polymers, the ordered solid state provides additional convenience, with an opportunity to access the related processes by X-ray diffraction methods. Single-crystal diffraction as well as powder X-ray and neutron diffraction under varying temperature and pressure are paramount for elucidation and direct observation of the detailed mechanism of such processes. On the contrary, insight into the mechanism in amorphous materials is limited and often indirect; it is based on (polarized) spectroscopic or low-angle diffraction data, which oftentimes rely on assignments and fitting procedures. A third asset of organic-based single crystals is that they provide synthetically accessible supramolecular diversity (salts, mixed crystals, cocrystals, or solvates) that could be purposefully generated to quantify, at a molecular level, the effects of statistical dilution on the response of such materials. Secondary or tertiary molecular components have important effects on energy transfer in these materials, especially when they interact strongly with the photoactive components, and occasionally exert remarkable effects on their performance. Crystalline actuators that can be powered and/or driven by light provide further advantage over their mechanically, thermally, or chemically driven counterparts, as they do not require secondary or tertiary chemical species for their operation. Moreover, light-driven crystalline actuators are devoid of problems with spatial control encountered by thermally fueled devices, and can be conveniently controlled by modulating the properties of the excitation light. These photoactive machineries are also devoid of physical contacts with the primary stimulus, while providing more efficient response to the stimulus relative to the diffusion-controlled action that is inherent to stimulation by chemical species.
responsive single crystals. We provide an extensive summary of earlier, oftentimes serendipitous, sporadically reported observations of single crystals that can bend, curl, twist, hop, leap, spin, explode, split, roll, or respond otherwise to external stimuli, such as light or heat. All of these various phenomena represent different relaxation modes of the mechanical stress and structural strain that has been accumulated over the course of photochemical transformation, and can be considered on the basis of the fundamental properties of the mechanics of solids: elasticity and plasticity. We also establish a general model for the mechanical effects in molecular crystals. The proposed approach offers means to describe and to explain mechanical and photomechanical phenomena at semiquantitative and quantitative level, and can be used to design new experiments both by controlling the internal structure via chemical substitutions and crystal engineering, as well as by selection of macroscopic characteristics of crystals (their size and shape). Additional design factors may include the radiation wavelength, intensity, polarization, temperature, external mechanical load, and defects in the solid, among other parameters.
2. A GENERAL MODEL FOR MECHANICAL EFFECTS IN CRYSTALS 2.1. General Considerations of Photomechanical Effects
Quantification of the thermodynamics, kinetics, and kinematics of mechanically responsive systems beyond the qualitative observations of such effects requires elucidation and modeling of the kinetics, comprehension of the kinematics, and quantification of the ensuing reaction yields. Determination of the kinetics and understanding the mechanisms underlying the conversion of other types of energy into mechanical work in the solid state is far from being a trivial task. Universal parameters for single crystalline actuators have not yet been set despite the fact that they are becoming increasingly important for quantitative benchmarking of performance in the development of technologically important materials. For example, a concept that requires further clarification surrounds the nature of the mechanism that underpins the photochemical response; both homogeneous and heterogeneous reactions could be considered. To attain a general understanding, the models should be devoid of parameters that are characteristic for particular molecular structure or crystal packing. Here, we set an overarching model of photomechanical effects that includes processes that lead to preservation of crystal integrity as well as those that are destructive to the crystal. The heuristic approach that was adopted for the characterization of photomechanical effects in substituted anthracenes, azobenzenes, diarylethenes, and cobalt(III) coordination compounds by several very prolific research groups in the past4 was instrumental in the efforts to establish a set of common measurable parameters to compare the performance of various single crystals. The study of the interrelation between photoexcitation and photoreactions in molecular crystals and their structural instability was very active in the 1970s and 1980s.87−92 These early burgeoning research efforts revolved around establishing molecular-level mechanisms of solid-state transformations and deciphering the feedback between solid-state reactions and mechanical strain/stress.93−95 Macroscopic manifestations of the structural strain induced by photoinduced transformations were documented,1−3,96 but remained out of research focus. Dating back just over a decade, research on photomechanical effects was revived.4 This line of inquiry brought up new
1.4. Scope of This Review
Although some photoactive, thermally active, and mechanically active single crystals that respond with mechanical effects have already been characterized, substantial understanding of the complete quantitative mechanistic profile of these materials is yet to be accomplished. Oftentimes, the difficulties with complete characterization of such mechanically active single crystalline materials are related to practical reasons surrounding very large structural changes and subsequent deterioration of the quality of single crystals (disintegration, explosion, surface effects, sublimation, twinning, fatigue, etc.). In many cases, however, the mere observation of mechanical effects in crystals is ignored, remaining unreported. Our aim with this Review is to provide a detailed overview on the current state of the emerging field of mechanically 12443
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can also be quite long. There can be a significant time delay between stages 2 and 3. In the context of this Review, which focuses on the macroscopic mechanical response of a crystal to irradiation, the delay between stages 2 and 3 (not the delay between stages 1 and 2) will be termed induction period. In general, elastic strain occurs simultaneously with structural transformations (there is no induction period). However, relaxation of mechanical stresses via plastic deformation or fracture requires energy to be accumulated and therefore occurs after an induction period. All mechanical effects of crystals can be classified into two major groups (Figure 2). The first group comprises phenomena
reports on a variety of macroscopic motions of crystals induced by light, including bending, twisting, creeping, jumping, exploding, rotation, and other effects (for an overview of these effects, see section 5). From a chemist’s perspective, the solid−solid transformations that are accompanied by photomechanical effects are very diverse. They can be homogeneous or heterogeneous and can include intra- and intermolecular reactions, isomerizations, photodimerizations, polymerizations, as well as both single- and two-photon processes. Mechanical effects range from “simple” dilation or compression to more complicated effects such as bending, twisting, coiling (spontaneous or after an induction period), as well as spontaneous fragmentation, jumping or rotation (after an induction period), creeping, and trembling. Irreversible plastic bending and formation of surface cracks without violent separation of fragments have also been documented (see Table S1). Generally, all of these phenomena represent different modes of relaxation of the mechanical stress and structural strain that has been accumulated over the course of a photochemical transformation. Despite the diversity, these processes have common features, and can thus be treated within the same general model. A common characteristic for all reactions is that the absorption of light accounts for the nonuniform distribution of transformation within the crystal in the direction normal to the irradiated surface. In the discussion that follows, we will first consider the case where the whole crystal surface is irradiated uniformly and the transformation is homogeneous within the layers normal to the light flux. Later in this Review, we will also discuss the special case where the irradiation is localized at a small spot on the surface. 2.2. Main Types of Mechanical Effects
A photochemical transformation in a solid can be considered as following the sequence: (1) light absorption, (2) a series of steps resulting in a structural transformation, and (3) macroscopic mechanical response (Figure 1). The second stage takes some time and can be very short (as compared to the total duration of irradiation and subsequent processes), but
Figure 2. Schematic (a) and examples (b) of the main types of photomechanical effects in crystals. The strain accumulated in a crystal by structural change in response to photoexcitation can be released slowly, which leads to (reversible or irreversible) bending or twisting. If the relaxation does not occur immediately, the strain is accumulated during the induction period and is released suddenly, whereby the crystals splinter, explode, move, jump, or creep. Some images of the crystals were recorded in our laboratory, and others were adapted from the literature with permission from refs 225 and 115. Copyright 2007 Nature Publishing Group and 2013 Wiley-VCH.
where crystals relax by mechanical reconfiguration (reshaping) without fragmentation. Figure 3 depicts a schematic of some exemplary deformations of crystals that occur with preservation of macroscopic integrity. The mechanical reconfiguration is slow and commences starting with the onset of the photoinduced transformation, without an induction period, although the peak in the rate of reshaping is normally observed with some delay after the photodeexcitation. These effects typically occur over seconds or minutes, and the mechanical reconfiguration of the crystal generally follows the kinetics of the photoinduced (chemical and structural) transformation
Figure 1. General mechanistic scheme of the photomechanical effects in crystals. The plot shows the relation between the time for accumulation of stress and the type of mechanical effect. 12444
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product through the crystal bulk, and the boundary conditions. The random factors are related to stochastic irregularities (defects) within crystals. The necessity to account for the random factors stems from the mechanisms underlying plastic deformation and fragmentation of crystals. These processes depend on the presence, concentration, and distribution of defects in a crystal, and, consequently, on what is usually referred to as “sample history”, a collective and general term that sometimes refers to multiple factors such as the method of sample preparation, storage, and preliminary treatment, among other factors. Defects normally are nonuniformly distributed in the bulk and on the surface of crystals, and they affect the threshold conditions. Together with the inhomogeneity of loading due to the random distribution of crystal shapes, this renders the sites that act as onset points for relaxation highly unpredictable. Another consequence of the stochastic nature of the relaxation processes relates to the fact that different processes compete for relaxation by channeling the strain energy. As these relaxation processes occur, they also affect the distribution of defects as well as the boundary conditions of the deformation. Therefore, every act of relaxation of stress/strain, regardless of its nature, adds a new (stochastic) event to the sample history. By knowing the structural data, defect distribution, crystal size and shape, and the elasticity tensor, one can, at least in principle, obtain information on the distribution of local strain and stresses in the crystal bulk, although such exercise is normally quite arduous. 2.3.2. Tensor of Strain Induced by Photochemical Transformation. Local perturbations in the structure that result from a photochemical transformation can be considered dilation centers that act as local point defects similar to defects introduced by chemical substitution in solid solutions. This is a rather oversimplified approximation that can strictly be applied only to intramolecular isomerizations. If the reaction is homogeneous and accompanied by expansion that is proportional to the degree of transformation, this approach can also be applied to photoinduced dimerizations, charge transfer processes, and polymerizations. If the solid medium is approximated as a continuum, the cumulative effect of these dilation centers manifests itself at the macroscopic level as an averaged lattice strain, which can be described by a strain tensor. This tensor can be calculated if the difference between the crystal structures of the reactant and product of the photochemical transformation is known. In a selected coordinate system, this tensor can be represented as a diagonal matrix, where the diagonal elements define the strain along the three orthogonal axes (the principal axes of a strain ellipsoid). Under the effect of the strain tensor, a sphere made up of the crystal material would transform into such an ellipsoid (Figure 4). Strain ellipsoids allow visualization of the directions in which a structure expands or shrinks over the course of the transformation, and also contain information on the magnitude of the related strain.97 This approach has been successfully applied to describe the structural perturbations of photochemical reactions in molecular crystals.98−104 2.3.3. Distribution of the Product through the Crystal Bulk. In setting a universal model for photomechanical effects in crystals, additional useful assumptions are (1) homogeneity of the photochemical reaction in layers normal to the irradiated surface, (2) absence of interfaces between the reactant and product phases, (3) displacive nature of the transformation, (4) nonreconstructive nature of the transformation, (5) that
Figure 3. Schematic of (plastic or elastic) mechanical reconfiguration of crystal without disintegration. When exposed to light, straight crystals (a) can bend (b and c), twist (d and e), or coil (f and g).
although with some delay. Being very slow processes (as compared to the speed of sound), in principle the system of this type could be considered to be in a state of mechanical equilibrium during the whole process. Thus, similar to the thermoelastic materials, these phenomena can be treated as standard quasistatic problems within the elasticity theory by considering the strain induced over the course of the transformation. Phenomena from the second group are fast and accompanied by an induction period. The (fast or slow) photoinduced transformation initially proceeds without apparent mechanical effect. Although a critical state for mechanical reconfiguration is reached as a prerequisite for relaxation, this state does not lead to immediate relaxation; by exceeding the critical conditions, the system is merely advanced to a metastable state. Long lifetime of the metastable state increases the probability of relaxation, which occurs suddenly, by very rapid structural transformation on the order of microseconds or less. This group of mechanical effects consists of spontaneous but sudden motions, such as jumping, splintering, explosive fragmentation, and creeping of crystals (note that creeping is included in this group as it could be considered a series of small, rapid plastic deformations). On the basis of the estimated kinetic energies associated with these processes, their rates are comparable to the speed of sound. Such high rates are characteristic for macroscopic effects such as dynamic fragmentation, plastic deformation by gliding of dislocations, and martensitic transformations. Quantification of these complex spontaneous motions is a challenging task. Unlike the phenomena from the first group, these effects cannot be treated within the static model of mechanical equilibrium. However, it is possible to at least estimate the portion of the total energy accumulated before the motion commences, which is transformed to kinetic energy during the relaxation. For most practical applications, this estimation is usually sufficient. 2.3. Factors That Determine the Mechanical Response
2.3.1. Regular and Random Factors. The type of response, its intensity, as well as the duration of the induction period (the time lapse between the onset of irradiation and the mechanical response) depend on multiple factors including regular factors that are intrinsic to the system, as well as random factors. The regular factors are deterministic and are related to parameters that characterize the distribution of stress in the crystal, that is, the strain tensors, the distribution of the 12445
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obtain the distribution of local strain and stress in the crystal interior, and this problem can only be solved numerically. In real experiments, however, crystals come with very different sizes and shapes, and individual solutions for each case are impractical. Relatively simple solutions are viable in two representative cases: (1) a very large and thick (relative to the characteristic absorption depth) crystal, which can formally be considered as an infinite body with a single plane surface, or (2) a needle-shaped crystal of finite size.
Figure 4. Schematic representation of the strain ellipsoid. Under the effect of strain, a sphere defined by three unit vectors is distorted to ellipsoid with principal component strains ε1, ε2, and ε3.
2.4. Large, Thick Crystals
In the case of a large crystal, one has to solve a simple problem involving a planar stress state; the stresses that develop normal and tangential to the irradiated surface are equal to zero (Figure 6). In this case, the strain (with respect to the initial state) in
reactions proceed via formation of a solid solution of product in the reactant, and (6) the absence of feedback phenomena (see below). Nonuniformity of product distribution is then observed exclusively in the direction orthogonal to the irradiated surface and accounts for the light absorption (Figure 5; the case of
Figure 6. Schematic representation of the planar stress state. In elasticity theory, the planar stress state is the case when the stressed state of a bulk physical body can be described as a system of parallel 2D planes. Each of these planes is stressed when subjected to forces applied only within the plane. There is no force applied normal to the layers, and there are no shear forces between the layers.
each layer parallel to the surface will be null. The strain with respect to the local, partly transformed state in the coordinates of the irradiated crystal face is equal to the local transformation strain, but is of opposite sign (as if the structure of the local, partly transformed state were strained to the size of the initial structure). The strain can be represented by a deformation ellipse, calculated as the projection of the strain ellipsoid onto the irradiated crystal face (a parallelogram can be conveniently used for a better graphical representation, as we have also done in this Review). This solution means that the large thickness of the crystal excludes any other effects but the parallel shift of layers along the surface (i.e., the crystal cannot bend, twist, etc.). It is apparent that in this way strained local states with the maximum possible stress can be achieved. These states are essentially local because no macroscopic strain or change in crystal shape occurs. This means that the critical threshold states for the relaxation processes in large crystals will be reached earlier relative to finite-size (small) crystals. In this limiting case, one can only study fast photomechanical effects and elucidate the threshold conditions of plastic deformation and fragmentation. A special case of this situation, indeed, more involved from the view of its mathematical description, yet relatively simple for experimental study, is the local irradiation of small areas at the surfaces of large crystals. This would make it possible to study, under well-reproducible conditions, identical crystal and identical crystal face of that crystal, the dependence of the photomechanical response on various parameters such as the excitation wavelength, light intensity, temperature, size of the irradiated spot, Miller indices of the irradiated face, etc.
Figure 5. Bending and twisting of a needle-shaped crystal induced by a photochemical transformation. (a) Light absorption generates a gradient of the transformation degree normal to the irradiated surface. (b) Expansion of the surface layers along the needle causes bending of the crystal (a cross-section of the strain ellipsoid is shown by a dashed line). (c) Stretching of the surface layers along the diagonal of the needle axis (left) or application of shear stress (right) causes twisting of the crystal.
nonuniform radiation will be considered in section 2.8). The local degree of transformation defines the local structural deformation. In a first approximation, the local deformation induced by the transformation is proportional to the degree of transformation and the deformation tensor. The transformation gradient throughout the crystal volume is determined by kinetic factors (flux of light quanta, quantum yield, characteristic absorption depth, rate of the reverse reaction) and geometrical factors (crystal shape, relative disposition of the direction of irradiation and particular crystal faces, that is, the boundary conditions of the transformation). 2.3.4. Boundary Conditions in the Elastic Strain Problem. The faces of the crystal act as boundaries to which the zero-force conditions apply. These boundary conditions include, in particular, the orientation of the strain tensor with respect to the crystal faces (Figure 5). 2.3.5. Crystal Size and Shape. For the general case of a crystal with arbitrary shape and size, it is rather arduous to 12446
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Figure 7. Deformation of beams and deformation tensors in the surface plane represented by a hypothetical isotropic material that undergoes anisotropic deformation. (a) A model for crystal as undistorted beam. The sampled area is represented by a square with ay = az = a. The coordinate system is defined with the crystal surface (y,z) and a normal to the surface (x). (b) Bending by stretching. By bending outward, the sampled square stretched along the z axis turns into a rectangle with ay = a and az = a(1 + εzz). (c) Bending by compression. By bending inward, the sampled square compressed along the z axis turns into a rectangle with ay = a and az = a(1 − εzz). (d) Right-handed twisting. By right-handed torsion, the square turns into a rhombus with ay = az = a, long diagonal = √2a(1 + εyz), and short diagonal = √2a(1 − εyz). (e) Left-handed twisting. By left-handed twisting, the square turns into a rhombus with ay = az = a and change in the long and short diagonals. (f) Right-handed coiling. The square turns into a parallelogram with ay = a, az = a(1 + εzz), long diagonal = √2a(1 + εyz + εzz/2), and short diagonal = √2a(1 − εyz + εzz/2). (g) Left-handed coiling. The square turns into a parallelogram with ay = a, az = a(1 − εzz), long diagonal = √2a(1 + εyz − εzz/2), and short diagonal = √2a(1 − εyz − εzz/2). In (f) and (g), the sign of εzz does not affect the twisting direction (right or left); only εyz relevant. The sign of εzz, related to the type of deformation (tension or compression), is defined by the side of the helix (outer or inner, respectively). Note that in a real (anisotropic) crystal, the symmetry-dependent deformation can be a combination of bending and torsion.
2.5. Thin Plates
absent or is significantly smaller than in the other direction, or the width of the plate in one dimension is much smaller than the curvature radius in the other direction. The latter condition holds for needle-like (acicular) crystals, which are very convenient for the study of photomechanical effects.
Detection of macroscopic deformation of a nearly isometric crystal (where the three dimensions are comparable in size) is an involved exercise: relative to the dimensions of the crystal, different points of the surface shift very little with respect to each other. This burden can be avoided when working with thin plates. Provided that the strain is nonuniform normal to the surface, the crystal plate can bend. The amplification of this effect is roughly proportional to the ratio of the longitudinal and the transverse dimension of the plate-like crystal. However, if the curvature radii become comparable with the longitudinal dimension of the plate, the simultaneous bending motions of the plate in two directions are coupled to each other due to the significant strain that arises over the plate perimeter,105 and the resulting bending is small. Bending of the plate can become free and consequently more pronounced only if the bending in one direction does not hinder the bending in the other direction. The latter becomes possible in one of two cases: either the initial deformation that accounts for bending in one direction is
2.6. Needle-Shaped Crystals
2.6.1. Modes of Deformation. A needle-like crystal can also be characterized by a critical threshold value for free bending, that is, the range of curvature radius within which it behaves as a classical elastic beam. When the curvature radius approaches the thickness of the needle, one has to use the socalled “approximation of large deformations”.106 However, for most crystals that are studied in real experiments, these corrections can be neglected. Elastic beams can undergo two main types of deformations, bending and twisting. Some of us have recently set the first quantitative model for bending of crystals that accounts for the kinetics of both forward (photochemical) and reverse (thermal) reaction;106 however, the model represents a special case that is 12447
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limited to bending of slender crystals. Experimentally, it has been observed that while thin needles bend, ribbons of the same material tend to twist.107 Bending and twisting can be combined to elicit more complex deformations. When a needleshaped crystal is irradiated on one face, it is generally observed that a combination of uniform bending and uniform twisting strains results in coiling of the needle into a helix. The cause for this effect is rooted in the stress that is induced by the photochemical transformation and distributed nonuniformly through the depth of the needle. The collective stress exerts two effects on the needle: (1) expansion (or compression) stresses along the needle axis result in a bending moment, and (2) shear stresses in the planes parallel to the surface generate a torsion moment that acts on the needle axis. A mathematical treatment has been developed to analytically predict the (equilibrium) deformed shape of an initially flat, straight ribbon subjected to arbitrary surface stressing and/or undergoing buildup of residual strain, both of which afford the magnitudes and orientations of the principal curvatures. The basis of this approach stems from the standard methods that have been developed for bending and torsion of beams within the realm of mechanical engineering.108−111 A detailed mathematical description of spontaneous bending and twisting of ribbons has been suggested, for example, by Chen et al.112 The processes described in this reference are driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between the layers of a laminated composite. In most of the recent literature, in analogy with bending beams, crystals irradiated on one face have been considered “layered composites” because of the gradient of product that results from nonuniform transformation normal to the irradiated surface. This nonuniformity is due to light absorption. 2.6.2. Isotropic Material. In the special case of mechanically isotropic material, the relative contributions of bending and torsion to the overall deformation are determined by the orientation of the needle axis with respect to the principal axes of the strain ellipsoid (Figure 7). If one axis of the strain ellipsoid, projected onto the irradiated surface, coincides with the needle axis, only longitudinal stresses and bending moments will arise during deformation. In effect, the crystal only undergoes bending. If that is not the case, torque will arise, and the needle will coil into a helix. The outcome of the deformation (bending vs twisting) can be understood by considering a thin sheet of a material that can only be deformed in one direction along the irradiated surface where the photoinduced transformation occurs (Figure 8). When exposed to light, such sheets will roll up into a cylinder. If a thin strip (corresponding to a needle crystal) is cut out of this object at some angle to the circular line of the cylinder, a coiled strip is obtained. If the two axes (the axes of the projection of the strain ellipsoid onto the irradiated surface and the needle axis) do not coincide, the deformation strain can be considered in coordinates that coincide with the needle axis and divided into isotropic and shear components. The bending and torsion moments that correspond to these two components result in the two types of deformation, bending and twisting, respectively, which are independent from each other. If the deformation is isotropic within a layer parallel to the surface, the gradient of the degree of transformation normal to the irradiated surface results in a bending moment only, and the crystal will bend. It is quite unlikely that the strain within the layer parallel to the surface of an irradiated isotropic crystal will be anisotropic. However, such anisotropic strain can be
Figure 8. Mechanism of formation of helix from isotropic material. (a−c) An anisotropic strain is generated by adhesion of anisotropically extended or compressed layers. (d) Definition of parameters that define the coiling of a helix, the radius ρ, helical step (pitch) L, and the helical angle tg(φ) = L/(2πρ). Adapted with permission from ref 112. Copyright 2011 American Institute of Physics.
artificially generated by gluing anisotropically expanded or compressed layers of a material together.112 In this case, one can observe all possible combinations of bending and torsion. The bending contribution is unambiguously determined by the bending moment of the longitudinal strain component, whereas the twisting contribution is determined by the torsion moment of the shear strain component in the plane of the strained layer. If one of the principal axes of the strain ellipsoid coincides with the axis of the needle, only bending is observed. Otherwise, bending is combined with torsion (Figure 7). 2.6.3. Anisotropic Material. The situation is more complex for the more common case of anisotropic materials, which includes nearly all molecular crystals. With anisotropic materials, either of the evolving moments, bending or torsion, can concurrently cause both bending and twisting.109−111 In addition to the ratio of the moments, the relative contribution of each deformation mode depends on the ratio of the components of the elasticity tensor. If any of the strain ellipsoid axes (more precisely, their projection onto the irradiated surface) coincides with the needle axis, and the cross-section of the needle coincides with the symmetry plane of the elasticity tensor, then only the bending moment acts on the needle and only bending (i.e., without twisting) will be observed. This scenario is applicable to structures with higher symmetry than triclinic, and is very likely for naturally grown, well-shaped crystals. An example of this type of bending was observed for [Co(NH3)5NO2]Cl(NO3).2,98,113,114 In all other cases, in principle one should observe a combination of bending and twisting that results in coiling, unless these two components accidentally cancel out. It can be inferred from the above discussion that in contrast to isotropic materials, the twisting in anisotropic crystals can arise without a torsion moment, and bending can occur without a bending moment. Instead, the inherent anisotropy in the elastic properties of the material accounts for generation of shear strain when the material is stretched or compressed. In return, extension or compression results from shear stresses. Exclusive bending or exclusive twisting can be observed only under certain symmetry restrictions. Thus, an additional condition to avoid twisting is that the needle cross-section coincides with the symmetry plane of the elastic tensor. 12448
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gradient of the radiation intensity absorbed by the crystal reaches its asymptotic function with time, independent of the crystal thickness. Therefore, the gradient in the degree of transformation will also approach some asymptotic function of time, which does not depend on the crystal thickness. After the onset of the transformation, the product gradient first increases, but it decreases when the reaction finishes. In this case, irradiated crystals should go through the same sequence of bending and unbending (twisting and untwisting), which will end with uniform stretching or compression in accordance with the strain tensor. We shall further demonstrate that situations are possible in which the gradient of the degree of transformation normal to the crystal surface does not disappear, even after a phototransformation in a thin crystal has been completed (see sections 3.3 and 3.4.5). This happens, for example, if there is a reverse thermal transformation with a rate comparable to that of the forward phototransformation. In this case, the gradient in the degree of transformation and the corresponding parameters that characterize strain (bending or torsion) decrease by the end of the phototransformation to fixed nonzero values. By the same means, the gradient in the transformation degree normal to the irradiated surface is constant through the bulk at all times. This enables bending or twisting without development of stresses. The curvature of the crystal can reach its maximum value independent of the crystal thickness. Because no mechanical stresses arise in thin crystals, relaxation that leads to irreversible structural changes, that is, defect accumulation and fracture, does not occur. Therefore, multiple bending/ unbending cycles are possible without damage or fatigue of the crystal. In this case, it is most convenient to study slow changes of the crystal shape, allowing one to obtain information on the transformation strain tensor, as well as on the characteristic temporal and spatial parameters of the phototransformation. An important condition, however, is that the transformation is homogeneous. If the transformation product is amorphous or the photochemical reaction results in a reconstructive phase transformation that yields an interface between reacted and nonreacted crystal segments, changes to the crystal shape will be irreversible. Strong irreversible twisting of nanosized crystals, accompanied by amorphization, has been reported.118 Another interesting case of the mechanical response of thin nanosize crystals to photochemical transformations corresponds to their local irradiation by a light spot with diameter smaller than the linear dimension of the crystal. In this case, a longitudinal nonuniformity near the irradiated spot arises. The border of the irradiated area coincides with the border between the initial and the transformed structures. If the cross-section of the crystal is small, the difference between the structures can be compensated for by rotation of the lattice preserving the interface between the two phases. Dislocations and disclinations can play an important role in forming the interface. As a result, the crystal can bend in the irradiated region. Because the perimeter of the interface is small, the bending can be reversible and the crystal structure can be completely restored during the reverse transformation without permanent damage to the crystal. Two main factors account for this reversibility. First, the limits of strength and plasticity increase drastically with decreasing size of the particles (crystals). Second, if the glide of dislocations is involved in the formation of the interface, then the dislocations migrate from one surface to the other without being accumulated in the crystal bulk near the interface. If the dislocations do remain at the interface, a
Otherwise, the bending moment will cause additional torsion due to the anisotropy of the elastic moduli. 2.7. Other Factors That Determine the Mechanical Effect
Crystal deformation is a dynamic process. If the crystal shape changes significantly during irradiation, the treatment becomes progressively more complex because the irradiation conditions also change during the process. In the case of sole bending, the only parameter that varies is the incident angle of radiation with respect to the crystal surface. This results in a smooth and continuous variation of irradiation intensity along the needle axis. If a nonzero torsion is added and the needle twists, different faces of the crystal will be exposed to the light source at different angles for different time intervals. These evolving conditions bring about a very complex time profile of the crystal shape. In general, one should account for the variation of local curvature, plane bending, direction and value of the torsion moment per unit length, as well as the rate by which these parameters change along the needle axis. If polarized light is used, the light absorption by different sites of the crystal may also change as the crystal twists and different faces are exposed. Some of these effects can be controlled by polarized light. For example, recently a crystal has been described that twists on irradiation so that opposite faces are exposed to the light flux, one after the other.115 It can be anticipated that this irradiation regime will have a balanced effect on the two faces of the crystal, and that bending will cease after the first turnover. However, because opposite faces have different propensities to absorb the polarized radiation, the bending does not cease. There are also cases when even a significant change in the crystal shape does not affect the reaction as a result of the changing irradiation conditions. This is, for example, the case when the photoinduced transformation is heterogeneous and the nuclei of the product phase are formed at the irradiated surface. If subsequent transformation results from the growth of product nuclei that are already present, the effect of changes in crystal shape on irradiation conditions during crystal bending will not be very pronounced. Another situation in which crystal shape exhibits negligible influence on the reaction is related to the existence of relatively long intermediate stages between the absorption of light and subsequent formation of the reaction product. One can find examples of published work describing macroscopic strain of crystals that starts at the moment of irradiation, but continues long after the light is switched off.116,117 For the sake of simplicity, the analysis in this Review does not take into account these complex effects, which are difficult to strictly control experimentally. Instead, the model described in this Review assumes constant irradiation intensity of the whole surface. In the first approximation, for relatively small macroscopic strains, these considerations are applicable to the whole acicular (needle-like) crystal. The conclusions can be used to develop more sophisticated models that account for variation in the irradiation conditions along the needle axis over the course of a transformation. 2.8. Very Thin (Nanosize) Needles
Macroscopic changes of the crystal shape in response to photoinduced transformation can also be observed in very thin crystals for which the thickness is much smaller than the characteristic absorption depth. At first sight, one could expect that uniform irradiation of such thin crystals will cause practically uniform transformation throughout the crystal bulk. This would result in uniform strain, either stretching or compression, without bending or twisting. However, the 12449
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regular dislocation structure is formed, similar to small-angle boundaries. The reverse transformation also generates stresses at the interface with comparable amplitude. This results in reversible deformation that can completely restore the initial structure. An example of this reversible bending and unbending of nanosized needle-like crystals has been described by Bardeen et al.117
Even if these solid solutions are metastable, they can be preserved without phase separation for a long time because of poor diffusivity in molecular crystals. The symmetry of the two pure phases can differ arbitrarily. From the mechanistic point of view, the formation of a solid solution of one phase in another can be considered an arbitrary deformation of the initial phase, without any restrictions imposed by the initial symmetry. This description is complicated by the fact that changes of composition over the course of a phototransformation are accompanied by changes in elastic properties. If these changes are small, their contribution to the evolving stress and to the values characterizing bending and torsion will be the next order of magnitude smaller relative to the degree of transformation. However, if the crystal thickness is comparable to the characteristic absorption depth, then the change in elastic properties over the course of the transformation can have pronounced effects on the crystal deformation. This will be the case, for example, if some of the elastic coefficients and compliance tensors become nonzero as a result of symmetry lowering. In such cases, mixed deformation modes will arise instead of pure torsion and pure bending, and the “admixtures” of additional modes will increase as the reaction progresses and the degree of transformation increases. In a general case, this development will eventually result in deviation from the constant relation between bending and torsion that would otherwise be expected if the elastic properties remained constant. As an approximation, one could neglect the effects related to changes in elastic properties. Significant variations in the experimentally determined relation between the torsion and bending are indicative of changes in the elastic properties over the course of reaction, and should be taken into consideration. If the symmetry does not change even after transformation to the product has been completed, the strain can be formally considered as if it were similar to thermal deformation of the same phase, even that a solid solution is formed. In this case, the symmetry of the strain tensor is unambiguously related to the symmetry of the elastic moduli. If the product has a symmetry different from that of the reactant, the process can still be considered as if it had occurred in the same phase, which is defined as the phase with the lowest symmetry. The identity of the lower symmetry phase (the initial or the final one) is irrelevant; the state with higher symmetry can be considered a threshold variant of the changing structure where additional symmetry elements can appear and disappear over the course of the reaction. In such cases, the symmetry properties of the strain induced by the transformation will be invariably related to the symmetry of the less symmetric phase. However, when considering the reaction within the approximation of constant elastic properties of the crystal throughout the transformation, one usually resorts to use of the elastic moduli of the initial phase, even if that phase is not the least symmetric one. Thus, if the crystal symmetry is preserved over the course of the transformation, or if the structure becomes more symmetric, the problem can be treated similar to that of the thermal strain. Otherwise, the symmetry of the strain tensor does not depend on the symmetry of the elastic properties of the initial phase.
2.9. Thin Isometric Crystals
The theoretical treatment of a crystal with size comparable to the characteristic absorption depth, and with nearly isometric shape, presents the most complicated case. While a shape change does occur, and leads to a decrease in the elastic strain energy, the main burden with the treatment of these crystals is the difficulty in reliably measuring macroscopic changes in their shape. In turn, although an induction period must exist prior to stress relaxation, it is difficult to predict it as well as to detect it. That said, the visual effect exhibited by systems in this category is very impressive; crystals start shaking, vibrating, or jumping. If larger crystals of the same materials are irradiated, these effects are hardly noticeable. Time-dependent studies of such systems can be used to obtain empirical statistics of induction periods for different types of mechanical response. Such results can be useful to understand the mechanisms of the structural response to irradiation. For example, the lower limit of irradiation dose required to induce an observable effect can be estimated for a range of crystal sizes. One can also determine this parameter as a function of other factors, including crystal size, shape, light intensity, temperature, etc. It is also a worthwhile exercise to estimate the fraction of accumulated energy that is released as kinetic energy during the mechanical response. One can expect that the farther the system departs from the critical threshold conditions, the larger the portion of accumulated energy that will be transformed to kinetic energy. 2.10. Symmetry Restrictions and Photoinduced Structural Transformations
Generally, a phototransformation results in the formation of a new phase. As a first approximation, lattice strain induced by a photochemical transformation can be compared to thermal expansion.114 Because temperature is a scalar variable, the anisotropy of thermal expansion is determined by the symmetry of the initial crystal structure and is not altered by expansion. In contrast to this, some phototransformations do not follow this simple rule. The space group, and even the crystal system, may change during the transformation, for example, from monoclinic to orthorhombic.99,119 The symmetry of the product could be (although it is not necessarily) related to the symmetry of the initial structure. An intramolecular photochemical transformation (e.g., isomerization) or a photochemical reaction between two molecules (e.g., dimerization) is coupled to change in the phase. In that respect, the photochemical transformation is distinct from a thermal expansion, which involves only lattice strain within the initial phase. The strain tensor of a thermal expansion is unambiguously related to the symmetry of the crystal structure and the symmetry of other physical properties, as is the elasticity tensor.120 A peculiar feature of some photochemical transformations is that the difference in structures of the reactant and product (mutual juxtaposition of molecules and the cell parameters) is rather small. In such cases, a solid solution of product and reactant molecules can be formed across a full range of stoichiometries, from product/reactant ratio of 0:1 to 1:0.121
3. A MODEL FOR BENDING OF CRYSTALLINE NEEDLES 3.1. General Considerations
As discussed in section 2, needle-shaped crystals are the most suitable as probes for establishing quantitative models of the 12450
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transfer and formation of heterogeneous product phase are not considered; however, these factors should be considered in more specific models. (2) The intensity of light received by the crystal is spatially homogeneous over the exposed crystal face and does not depend on a change of crystal shape during the reaction. The light absorption, quantum yield of transformation, reaction rate constant, and mechanical properties of the crystal remain constant throughout the transformation. The possible feedback phenomena linking the evolving structure and reaction rate can be included in a more sophisticated model in the future. (3) The photochemical transformation results in structural strain that varies linearly with the degree of transformation. The ensuing mechanical stresses are considered within the linear elasticity approximation. We consider a general anisotropic case where the elastic modulus and the structural strain tensor that results from the transformation are both anisotropic.
macroscopic manifestation of photomechanical effects. The mechanical strain in needle-shaped crystals can be described quantitatively on the basis of the approaches that were developed in the 19th century by J.-M. K. Duhamel when considering the stresses induced by nonuniform temperature distribution.122 A classical solution to the problem of stresses in a plate experiencing a temperature gradient is described in ref 123. Stresses resulting from nonuniform distribution of temperature can be similar in many respects to stresses induced by nonuniform composition. The basics of the theory of stress concentration also date back to the 19th century.124 The theory was further developed by Shternberg,125 who introduced the concepts of composition and temperature heterometry, the nonuniformity of cell parameters resulting from the nonuniformity of chemical composition or temperature. The term was used recently to explain photoinduced bending and twisting of microneedles and microribbons.107 To apply these approaches to quantification of photomechanical effects, one needs to account for composition nonuniformity that results from the light absorption by the crystal substance, so that the degree of transformation decreases as one moves from the irradiated surface to the crystal bulk. Several models have been proposed to quantify the macroscopic deformation of needle-shaped crystals on irradiation (Table S2). In all models, the crystals have been assumed to be mechanically isotropic. As a result of the crystal shape, this nonobvious approximation can be applied, even if the crystal structure is in fact anisotropic: if the needle axis coincides with a principal axis of the strain ellipsoid, the symmetry of the mechanical properties of crystals makes it possible to treat needle-shaped crystals as isotropic. Generally speaking, this approximation can be used to describe bending, but not twisting. To describe torsion strains, one must consider the anisotropy of mechanical properties or at least the anisotropy of strain induced by transformation. In the latter case, one can consider two-layered structures composed of isotropic materials that have been preliminarily subjected to two-dimensional (plane) strain. These considerations allow one to describe the anisotropic tensor of thermal/composition expansion. Exact quantification of bending and torsion requires explicit treatment of the anisotropy of the mechanical properties of the material and the strain tensor generated by the transformation. The need to account for both anisotropy terms results from the interrelation of bending and torsion components. More complex phenomena can add to the direct effect of the composition heterometry on the strain experienced by a crystal on irradiation. For example, irradiation can result in charge separation in space, leading to the formation of bulk and surface charges (exited electrons can migrate through the crystal until they are trapped at some sites; for example, molecules at the crystal surface can act as such traps for electrons or holes). This can induce an electric field and account for piezoelectric and electrostrictive phenomena. Regardless of the origin of strain, macroscopic crystal deformation can be described within the same or very similar theoretical models. Here, we present the most general model to date to describe the photoinduced bending/twisting phenomena. This overarching treatment is based on the following assumptions: (1) The absorption of light by the crystal follows the Lambert−Beer−Bouguer law. The light induces a local, homogeneous transformation similar to a chemical substitution in a solid solution. Secondary effects such as excited-energy
3.2. Distribution of the Yield Normal to the Irradiated Face
A description of the deformation of a needle over the course of a photoinduced transformation requires information on the profile of transformation yield through the crystal, normal to the irradiated surface. Before the product of a photoinduced transformation is formed as a distinct phase, a multistep process takes place. This process includes photon absorption, various electronic rearrangement processes, transfer of electronic excitation through the crystal bulk until it is trapped at a defect site or dissipates, formation of intermediates, and diffusion of chemical species. In the solid state, these processes are typically accompanied by generation of mechanical stresses and their relaxation. The stress generation and relaxation can affect the process at any stage, either by stimulation or by retardation. It also has a strong effect on the spatial propagation of the process and accounts for spatial self-organization of solidstate transformations. This phenomenon, termed feedback, has been considered in several studies of solid-state reactivity.94,126−133 The feedback effect makes it extremely difficult to suggest a universal and exhaustive description of the kinetics of solid-state phototransformations. That said, the main features of photomechanical effects can be understood well if a simple model of a photoreaction is considered. The model assumes that the degree of transformation in the solid is homogeneous in a layer parallel to the irradiated surface and decreases as a function of penetration depth, normal to the exposed surface. This inhomogeneity of the photoinduced transformation results in photomechanical effects. A similar reasoning holds, for example, for (1) a heterogeneous process during which a layer of the product phase is formed and grows normal to the surface, with product phase conversion proportional to the dose of absorbed radiation, and (2) a homogeneous process where the absorbed photons cause local chemical transformation and a solid solution of the product in the starting reactant is formed. Here, we will consider the case of a homogeneous transformation with accompanying formation of a solid solution, and we will also take into account the possibility of reverse, thermally activated transformations. If in the first approximation we assume that the light absorption and quantum yield depend on neither the degree of transformation nor the stress induced by the transformation (i.e., there is no feedback), the kinetics of the photoinduced transformation can be described by the following equation: 12451
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In this case, the maximum degree of transformation can only reach a value equal to the ratio kph/kth. The final width of the concentration profile is also different and depends on the rate of the reverse thermal reaction. In general, the width of the profile, corresponding to one-half of the maximum degree of transformation, is equal to
(1)
where C is the local degree of transformation, I is the local density of photon flux, α is the quantum (conversion) yield of the transformation, v0 is the crystal volume subjected to photoinduced transformation normalized by the total number of chemical species, and kth is the rate of the reverse reaction (thermal isomerization). The first term on the right side of eq 1 is the rate of phototransformation, which is proportional to the rate of photon absorption (the divergence of flux density), the quantum yield, and the concentration of the reactant that remains untransformed at corresponding time t. If the characteristics of the absorption remain constant during the exposure, and the irradiation flux with intensity I0 is normal to the flat exposed surface, then the intensity decreases with penetration depth x as I = I0 exp( −μx)
x0.5 =
(2)
Ct = k ph exp(−μx)(1 − C) − k thC (3)
(Note that the subscripts in italic font in the equations throughout this Review are the corresponding partial derivatives.) The solution of this equation is C(x , t ) = C0(x)[1 − exp(−t[k th + k ph exp(−μx)])] C0(x) =
k ph exp( −μx) k th + k ph exp( −μx) (4)
The equation is a product of the function C0(x) that describes the stationary profile of the degree of transformation that is −1 attainable within a characteristic time kth , and a time−1 dependent part. At times t ≪ (kth + kph) , the profile can be approximated by an exponential function that increases linearly with time: C(x , t ) ≈ tk ph exp( −μx)
(5)
3.3. Effect of the Reverse Thermal Reaction on the Concentration Profile
Depending on the relative rates of the direct phototransformation and the reverse thermal reaction, the evolution of the concentration profile in the early stage can be different from that in the late stages of the process. If the rate of the reverse thermal reaction is significantly smaller than the rate of the direct phototransformation (as is the case with many reactions), complete transformation is nearly achieved on the surface of the crystal within time on the order of k−1 ph . The profile of the product concentration propagates deep into the crystal as a slowing wave with product layer width following a logarithmic law as a function of time (approximately as μ−1 ln(kpht) according to eq 4). If the reverse reaction is faster (e.g., at high temperatures), the exponential law given by eq 5 describing the concentration profile is preserved, but its growth slows exponentially with time: C(x , t ) ≈
k ph k th
exp( −μx)[1 − exp(−k tht )]
(7)
At high rates of the reverse thermal reaction, the characteristic width of the profile is determined only by the characteristic depth of light penetration into the crystal, μ−1. If the reverse reaction is slow, this width becomes approximately 1.4 ln(kph/ kth) times larger (i.e., roughly an order of magnitude larger). This is a rather significant difference, although it takes a very long time to be reached. During the initial period up to k−1 ph , the width of μ−1 is reached quickly. Thereafter, the expansion of the concentration profile slows, and the final width is reached only after time on the order of k−1 th . The ratio of the reaction rate constants, kph and kth, has a pronounced effect on the final stationary concentration profile C0(x) in the crystals with thickness much less than x0.5. According to eq 4, if the rate of the reverse reaction is low (kth ≪ kph), the phototransformation in thin crystals can reach completion with practically no gradient in the degree of transformation through the crystal bulk. In cases where the two reaction rates are of comparable values or if the rate of the reverse reaction is higher, the stationary concentration profile C0(x) becomes approximately exponential; that is, at the end of the phototransformation, a nonzero gradient in the degree of transformation normal to the crystal surface exists. As mentioned above, a structural distortion alone is not sufficient to cause a photomechanical effect; an important additional condition is that the transformation is nonuniform through the crystal bulk. Only nonuniform structural strain resulting from a nonuniform transformation can account for significant changes in the shapes of crystals, the generation of stresses, and the accumulation of strain energy that can eventually induce rapid, spontaneous motions. In thick crystals (with thickness on the order of x0.5 or larger), the nonuniformity of the transformation through the crystal bulk is preserved during the entire transformation, whereas for thin crystals, the nonuniformity reaches its maximum and then starts decreasing to a level dependent on kth/kph. For thin crystals, eq 4 can be used to calculate the ratio between the final gradient in concentration that is reached after the phototransformation is complete, Cx,fin, and the maximum gradient reached at the intermediate time moment when kpht = 1, Cx,max. The expression can be written as
where 1/μ is the characteristic penetration depth. Consequently, the kinetic equation can be written as
k ph = I0αv0μ
k ph ⎞ 1 ⎛ ln⎜2 + ⎟ k th ⎠ μ ⎝
Cx ,fin Cx ,max
=
k th /k ph k th /k ph + exp[−(1 + k th /k ph)]
(8)
This ratio is equal to 0.5 when kth/kph ≈ 0.28. At this ratio, the final gradient of the transformation degree is one-half of the maximum value of the gradient that has been achieved at some intermediate moment. If kth/kph ≈ 0.004, the ratio 8 decreases to 0.01. At smaller values of kth/kph, one can neglect any nonuniformity in the product distribution through the bulk of a thin crystal at the end of the phototransformation. For completeness, the situation in which kth/kph = 10−3 (slow reverse thermal transformation) will be considered in section 3. In this case, the maximum achievable width of the product
(6) 12452
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the resulting stressed state is not the classically simple case for which stresses are linear functions of the coordinates. To obtain a more precise solution, one has to take into account all of the stress tensor components, not all of which are necessarily zero. This is a complex problem, which can only be solved by using numerical methods. However, these additional complications (the nonzero tensor components) would bring only minor corrections in the values of the moments, whereas the qualitative description of the phenomena remains unaffected. The main correction can be expected to be related to the σ2 component (i.e., stresses in the transverse direction). Depending on the sign of this correction, the corrected values of the moments can either be increased or be reduced. However, these corrections will have a tangible effect only when the thickness of the transformed layer is much smaller than the thickness of the needle, that is, in cases where there is a significantly nonlinear distribution of stress through the depth of the needle. In reality, the boundary effects become significant even when the thickness of the transformed layer reaches approximately one-tenth of the total needle thickness; the transverse stresses start to decrease significantly and their influence also decreases. Therefore, for simplicity, we shall consider these stresses to be zero. To solve this problem, let us subdivide it into two parts. First, let us calculate the mechanical stresses that would arise in the needle if it could not bend, twist, or be strained along its longest axis. We will use the calculated values of stresses to find the values of bending and torsion moments and the average longitudinal force. We will then allow the needle to be strained freely so that all of the moments and the longitudinal force cancel. The solutions of the two separate problems make it possible to find the resulting stresses and the strain energy. The nonzero stress components in the first part of the problem can be calculated on the basis of the zero-deformation condition of the needle:
concentration profile will be approximately 7 times larger than the characteristic penetration depth (eq 7), and this must be taken into account when analyzing the plots illustrating the results of corresponding calculations. 3.4. Bending, Twisting, and Coiling of a Needle Irradiated from One Side
3.4.1. Bending and Torsion Moments. By knowing the concentration profile (the distribution of the degree of transformation throughout the crystal), one can solve the problem of a bending needle that is irradiated from one side. Let us define the z axis of the coordinate system as the longest needle axis (Figure 9). The x coordinate, as defined above,
Figure 9. Coordinate system and labels of crystal dimensions used in the mathematical model for bending of thin needles.
coincides with the axis normal to the irradiated surface (yz). The y coordinate is within the plane of the irradiated surface, normal to the needle axis. Let us also label the thickness of the needle b and its width a. As in the classical engineering problem of bending and torsion of a beam with known geometry, we assume that all components of the stress tensor are zero except for σiz. We will consider the general case where the object has anisotropic mechanical properties; the tensor components are considered in the coordinate system of the needle, which in this general case does not coincide with the system related to the tensor principal axes. The needle bends and twists in such a way that the evolving bending and torsion moments compensate the moments that result from the strain (nonuniform through the depth of the crystal) caused by the phototransformation. In addition, the needle experiences uniform strain that compensates the average general forces at its free surfaces; that is, the needle changes its length and thickness. In the following, we will use Voight’s matrix notation120 for this tensor (ei instead of eij) as well as for all tensors (sij instead of sijkl, σi instead of σij, and εi instead of εij). The local strain caused by the transformation is proportional to the local degree of transformation, C(x,t), and the strain at complete transformation, and thus εti (x,t) = eiC(x,t). Further, to make notations shorter, we will omit the dependence on the coordinate (x) and time (t), but develop the model within an understanding that these dependencies do exist. In the approximation within which the problem of bending and twisting of beams is considered, the layer of the transformed compound at the surface of the needle can be characterized by two nonzero stress tensor components, σ3 and σ4, which generate bending and torsion moments. In general, because the strain resulting from transformation is not uniform,
s33σ3t + s34σ4t + ε3t = 0 s44σ4t + s34σ3t + ε4t = 0
(9)
where sij are the components of the elastic compliance tensor (inverse of the elastic stiffness tensor) in the coordinate system of the needle, and σti are the stresses resulting from the transformation. From these relations, one can obtain:
σ3t = − σ4t = −
s44ε3t − s34ε4t s33s44 − s34 2 s33ε4t − s34ε3t s33s44 − s34 2
(10)
A special choice of the needle axis and “suitable” symmetry will give s34 = 0 and εt4 = 0, which would further give σt4 = 0 and σt3 = −εt3/s33, a situation that implies that only bending will occur. The bending moment caused by the transformation is ⎛ b⎞ t ⎜x − ⎟σ d x 3 ⎝ 2⎠ b⎛ s e − s34e4 b⎞ ⎜x − ⎟C(x , t ) dx = −a 44 3 2 2⎠ s33s44 − s34 0 ⎝
M=a
∫0
b
∫
(11)
The torsion moment is equal to 12453
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⎛ b⎞ t ⎜x − ⎟σ d x 4 ⎝ 2⎠ s e − s34e3 b ⎛ b⎞ ⎜x − ⎟C(x , t ) dx = −a 33 4 2 2⎠ s33s44 − s34 0 ⎝
MT = a
∫0
b
∫
twisting always occurs along the needle axis, the result is very easy to predict: the rate of twist ϑ (the rate at which the torsion angle changes along the z axis) is equal to the sum of torsions resulting from the two moments. The expression describing the torsion is134
(12)
As can be inferred from these equations, the ratio of these two moments does not change with time as both have identical dependencies on the concentration profile function. Consequently, the ratio between twisting and bending of a needle will also be constant with time; that is, the needle will have the shape of a helix with a constant helical angle that depends only on the transformation strain ellipsoid and the elastic moduli of the substance. 3.4.2. Bending. As mentioned above, for compounds with anisotropic mechanical properties, even if only one of the bending or torsion moments arises, bending and twisting will occur simultaneously. Because of the linearity of elasticity, these effects are additive. To describe bending of a needle in a general case, we should consider bending deformations caused by two actions, a bending moment M and a torsion moment MT. The bending caused by the moment M will occur in-plane (xz), whereas the torsion moment MT will result in bending in both planes (xz and yz) to varying degrees. Therefore, the combined bending will be observed in some intermediate plane. The moment M can be calculated as M = I/R, where I is the moment of inertia (I = Eab3/12 for rectangular beams, E is Young’s modulus, equal to 1/s33 in the present case, and R is the curvature radius). The bending component resulting from the moment M then will be bending in the plane (xz) with a curvature radius: 12s33 1 = M RM ab3
ϑ=
∫0
⎛ b⎞ ⎜x − ⎟C(x , t ) dx ⎝ 2⎠
ab3
M+
MT WT
(17)
3W0 s33
(
2 s35
ba
3
+
2 s34
ab3
)
(18)
The value of W0 can be calculated as the sum of an infinite series dependent on the parameter d = (a/b)(s44/s55)1/2: W0 =
⎡ (ab)2 1 ⎢ 1 64 − 5 s44s55 d ⎢⎣ 3 πd
⎤ 1 ⎛ m π d ⎞⎥ ⎜ ⎟ th m5 ⎝ 2 ⎠⎥⎦
∞
∑ m = 1,3,5...
(19)
If d ≥ 1, the second and subsequent terms within the infinite series of this expression can be neglected, and only the first term considered. The error in this case will not exceed 1% in a given range of d. Hence, provided d ≥ 1, W0 can be approximated as W0 ≈
(ab)2 1 ⎡ 1 64 ⎛ πd ⎞⎤ ⎢ − 5 th⎜⎝ ⎟⎠⎥ s44s55 d ⎣ 3 2 ⎦ πd
(20)
If d < 1, then the approximate expression given by eq 20 can still be used, if d−1 is substituted for d. If the normal crosssections of the beam coincide with the mirror planes of the symmetry in the mechanical properties, then according to eq 18 WT = W0 (because s35 and s34 are zero in this case). Yet to obtain a twisting at such symmetry conditions, the transformation should distort the initial crystal symmetry so that, at minimum, e4 ≠ 0. This situation is possible, for instance, if an orthorhombic structure with its axes oriented along the needle axes undergoes a monoclinic distortion, with its 2-fold axis oriented along the x axis. As it follows from eq 17, twisting can be absent in only two cases: either accidently, when the two terms in the sum compensate each other, or if the aforementioned restrictions on both the crystal symmetry and the mutual orientation of the needle and strain ellipsoid axes are fulfilled (see section 2.6.3 and section 3.4.1), so that both terms are zero (two conditions are sufficient in this case, s34 = 0 and e4 = 0). 3.4.4. Coiling. The combination of bending and twisting will result in coiling of the needle into a helix. The characteristics of a helical shape are the helix radius ρ, the step L, and the helix angle tg(φ) = L/(2πρ) (for definition of the parameters, see Figure 8d). From the differential geometry, it is known135 that
(14)
∂φy 6M 1 = − 3T s34 = RT2 ∂z ab (15)
The curvature radius of the resulting combined bending can be determined from the relation: 2 ⎛ 1 1 1 ⎞ 1 = + + 2 ⎟ ⎜ RT2 ⎠ R2 RT1 ⎝ RM
6s34
W0 1+
In this case, the curvature radius does not depend on the elastic moduli of the substance, but is determined solely by the thickness of the needle, the strain of the completely transformed substance along the needle axis, and the distribution of the degree of transformation normal to the surface. The bending caused by moment MT will be a combination of bending in the planes xz and yz, and can be characterized by the curvature radii:
∂φ 6M 1 = x = − 3 T s35 RT1 ∂z ab
= ϑM + ϑT = −
WT =
(13)
b
∂z
where WT is the torsional stiffness of the beam. The torsional stiffness of the beams with rectangular cross-section can be described in the general case of anisotropic elastic moduli by the expression:134
In fact, this value alone determines the curvature of a needle for “correctly” selected symmetry conditions. So: 12e 1 = − 33 RM b
∂φz
(16)
3.4.3. Twisting. Similar to bending, twisting of a needle will also be caused by both bending and torsion moments. Because
ρ=
R 1 + (R ϑ)2
L=
2πϑR2 1 + (R ϑ)2
tg(φ) = R ϑ 12454
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finite nonzero value. The same is true for the cases of crystal bending and torsion. For example, the final curvature of a thin crystal will be equal to
A remarkable consequence of these relations is that the helical angle does not depend on C(x,t), but is determined only by mechanical properties, the geometry of the needle, and the tensor of the transformation strain (note that when the rate of twist is multiplied by the curvature radius, the dependences on C(x,t) are eliminated). This follows from the equivalent dependences of the torsion and bending moments on the distribution of the product concentration normal to the crystal surface (see eqs 11 and 12). The helical radius ρ and the helical step L (pitch) are changed simultaneously in complete accordance: both values are proportional to the curvature radius R. 3.4.5. Time Profile of the Crystal Curvature. It follows from the relations obtained above that the reciprocal curvature radius 1/R (as well as the rate of twist ϑ) is always proportional to the dimensionless value: ψr = −
12 b2
∫0
b
⎛ b⎞ ⎜x − ⎟C(x , t ) dx ⎝ 2⎠
k th /k ph ⎛1⎞ ⎜ ⎟ = e3μ ⎝ R ⎠fin (1 + k th /k ph)2
(24)
This value reaches its maximum at e3μ/4 when the rates of direct photoinduced transformation and the reverse thermal transformation are equal. The ratio of the final and maximum curvatures is defined by eq 8. It can be seen that in the case of a very slow reverse reaction (kth ≪ kph), the curvature first reaches a peak and then the crystal unbends almost completely (Figure 10). However, as the rate of the reverse reaction approaches that of the forward reaction, the unbending becomes incomplete; the final curvature increases with kth/kph, and the peak in curvature becomes less pronounced. If the reverse reaction goes faster than the forward one, the peak in curvature almost disappears. The curvature simply plateaus with the level of the final curvature decreasing on further increase of kth/kph. These effects can be observed in experiments if a reverse thermal reaction is possible and a phototransformation is studied over a range of temperatures. 3.4.6. Dependence on the Geometric Parameters of the Crystal. Let us now consider the effect of crystal thickness on the crystal curvature as the photoinduced transformation proceeds. As shown in Figure 11a, as the thickness of the needles increases, the value of the maximum curvature decreases, and simultaneously the overall duration of the bending−unbending cycle increases. Further increase in
(22)
which fully describes its time-dependence (other coefficients depend only on the geometry, mechanical properties, and strain tensor corresponding to complete transformation). This quantity can be termed normalized reciprocal curvature radius. The evolution with time, ψr(t), varies as a function of the ratio between the characteristic width of the profile of the degree of transformation and the thickness of the needle. If the crystal thickness is much less than μ−1, eq 22 can be simplified to ψr = −bCx(x = 0,t). Figure 10 shows the dependence of the curvature, expressed as 1/(e3μR) = ψr/(μb) with time (kpht). The maximum
Figure 10. Time-dependence of the curvature of a bending crystal for different rates of the forward and reverse reactions. The normalized curvature, expressed as 1/(e3μR), is shown for different ratios of the rate constants of the forward photochemical (kph) and the reverse thermal (kth) reactions. The curves were calculated by using eq 22.
curvature corresponds to the maximum product gradient through the crystal bulk. According to eq 4, this maximum is achieved when kpht = 1. The maximum curvature corresponding to eq 22 reaches the value of k th /k ph + exp[−(1 + k th /k ph)] ⎛1⎞ ⎜ ⎟ = e3μ ⎝ R ⎠max (1 + k th /k ph)2
Figure 11. Variation of the curvature of a bending crystal with time and its dependence on the crystal thickness. The normalized curvature, expressed as 1/(e3μR), is shown for different thicknesses of the crystal expressed as μb where μ is the characteristic absorption depth and b is the thickness of the crystal. (a) Time-dependence of the curvature for crystals with varying thickness. (b) Dependence of the stationary crystal curvature after long irradiation (kpht ≫ 1) on the crystal thickness and the ratio between the rate constants of the forward and reverse reactions. The curves were calculated by using eq 22.
(23)
If kth/kph ≪ 1, then thin needles first bend until they reach the maximum curvature. The original shape of these needles is subsequently restored, corresponding to near completion of the transformation throughout the entire bulk of the needle. If one cannot neglect the reverse thermal reaction, then as shown in section 3.3, the concentration gradient will be asymptotic at a 12455
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total forces along the needle axis, as well as the total shear forces acting on the side faces of the needle, are zero:
thickness results in complete disappearance of the peak in the curvature radius, and typically in a decrease in the asymptotically reached curvature values. By approximating the experimental curves with calculated functions, one can deduce the parameters that can be employed to characterize the photoinduced transformation. The result can be qualitatively interpreted as follows. When the needles are thin, the transformation essentially occurs throughout the entirety of the crystal bulk. Because of the nature of light absorption, a gradient of photon flux is established normal to the irradiated surface. Initially this results in an increased gradient of the transformation degree normal to the surface. The crystal bends, and this free bending allows for near complete relaxation of the growing stresses. The gradient of the degree of transformation reaches its maximum, corresponding to maximal curvature of the bent crystal, but starts to decrease afterward, corresponding to near completion of the transformation throughout the crystal. Correspondingly, the curvature starts decreasing and ultimately disappears (Figure 11a). In thicker crystals, the transformation cannot occur in the crystal interior because of light absorption in areas near the crystal surface. The thicker is the needle, the smaller is the portion of the crystal in which the transformation can take place. The maximum bending moment imposed on the needle by the product layer grows linearly with needle thickness. However, when a needle bends, a restoring moment evolves that is proportional to the thickness cubed and the reverse curvature radius, b3/R. Therefore, the maximum attainable curvature decreases as b−2. Additionally, using eq 22, one can calculate the dependence of the curvature on the crystal thickness with time for the case of infinitely long irradiation. Figure 11b, for example, shows the final curvature calculated for different kth/kph ratios. It is apparent from there that the change of curvature is not monotonous when kth/kph < 1, and the thickness at which a maximal curvature is achieved depends on the ratio kth/kph. Another interesting result is that the final curvature of thin needles (μb ≈ 1) may be a rising or falling function of the thickness, depending on the ratio kth/kph. These results demonstrate that there is no simple and obvious dependence of the curvature on the crystal thickness that is defined only by the crystal stiffness and the energy of chemical transformation as suggested in a model proposed previously.136 Fitting of the bending dynamics (curvature or its change with time) with Timoshenko’s model,123,137,138 which adds the effects of shearing to the Euler−Bernoulli theory for bending of slender beams139 as it is commonly done recently,140 is clearly an oversimplification. Instead, the curvature should be considered by including all factors that are relevant for solution of a quasielastostatic problem, the nonuniformity of the structure deformation in the crystal (defined mostly by the properties related to light absorption and mechanisms of the ensuing chemical reactions, including possibility of heterogeneous process), as well as the crystal geometry, which is required for the determination of the crystal torsion/bending stiffness and the corresponding moments for a given inhomogeneity of the transformation. 3.4.7. Residual Stresses in the Crystal. To calculate the stress that remains in the needle after bending, the possibility of additional uniform deformation of the whole needle should be considered along with the bending. The conditions that are required to find this uniform component of strain are that the
∫0
b
σ3 dx =
∫0
b
σ4 dx = 0
(25)
From eq 10, it is easy to find that the corresponding uniform strain values are equal to ε30 = ⟨C⟩e3 ε40 = ⟨C⟩e4
(26)
where ⟨C⟩ =
1 b
∫0
b
C(x , t ) dx
(27)
is the average degree of transformation as a function of time. The resulting stresses will be determined by the sum of all contributions to strain: those from the transformation, bending, twisting, and uniform deformation. For example, in case of “favorable” symmetry conditions, only the longitudinal stress σ3 remains: σ3 =
⎤ e3 ⎡ b − 2x − C(x , t )⎥ ⎢⟨C⟩ + ψr ⎦ s33 ⎣ 2b
(28)
The spatial distribution of stresses, which is determined by the factor in square brackets in eq 28, and its evolution with time are also qualitatively different for thick and thin needles. The profiles for thin needles (Figure 12a) are roughly symmetric with respect to the center, and thus the stresses in the central part of the needle cross-section and at its edges have opposite signs. It is worthwhile to note that the amplitude of the profile first grows with time and subsequently starts to decrease. Later, the profile inverts, for example, at approximately kpht ≈ 1.6 in Figure 12a. Ultimately, having passed the second extremum, the amplitude falls to near zero. In other words, at different distances from the surface of the needle, the material is subjected to alternating tensile and compression stresses, and eventually returns to a nearly nonstressed state. In thicker needles, the stress profiles become asymmetric (Figure 12b). The profile evolution resembles propagation of a wave that results in inversion of the sign of stresses at the irradiated surface. Because of the large thickness of the needle, the transformation cannot be completed throughout the needle bulk; thus, the stresses do not disappear with time, but instead approach some finite profile. If the needles are very thick, one can distinguish the regions in which the reaction has already occurred from regions where it will never occur due to the absorption of light by the surface layers. This situation is similar to formation of an epitaxial layer at a substrate with different lattice parameters (Figure 12c). As a result of summing the contributions of all deformations in the needle, in contrast to simple bending, the number of neutral surfaces (planes) is always greater than one, and in some cases it can be up to three. The most important points to be considered are the values of stresses attainable in different cases, and the stored strain energy. These parameters are related to triggering of different relaxation processes, such as plastic deformation and fragmentation, and they also determine the magnitude of these processes. Considering the strain with respect to the maximum possible values (Figure 12), one can see that in the case of thin needles, the change in shape 12456
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Figure 13. Time dependence of the specific strain energy accumulated per unit of irradiated surface for different thicknesses of the needle. (a) Thick needles, (b) thin needles. The curves were calculated by using eq 29. The quantity on the ordinate is dimensionless (W/[(e3)2/ 2μs33]).
3.5. Strain−Stress Relations in Crystals Stressed by an Internal Transformation
In general, one can expect that regions with positive strain (stretched) should experience positive (tensile) stresses and vice versa. However, if the origin of strain is not an external load, but a transformation inside the crystal, the situation becomes more complex. If the reaction proceeds nonuniformly through the crystal bulk, with different degrees of transformation as a function of distance into the bulk of the crystal, nonuniform stress is also induced through the crystal bulk. The sign of stresses in layers at different depths in the crystal will depend on the sign of strain and on crystal thickness. The simplest case is when a crystal is thick as compared to the light penetration depth and the phototransformation product is formed in a thin surface layer. The crystal remains practically unbent in this case. If the crystal structure expands as a result of the phototransformation, the surface layer will experience compressive stresses. This might seem counterintuitive; the reason for this is that the surface layer is “attached” to the nontransformed crystal bulk, which does not allow the surface layer to expand as much as it would like if the surface layer were “free”. Therefore, the substance in the surface layer is stretched as compared to its initial state (before the transformation), but compressed as compared to its natural nonstressed state after the transformation. Similarly, if the crystal structure compresses as a result of phototransformation, the transformed surface layer will experience tensile stresses. Now let us decrease the crystal thickness so that the crystal starts to bend on irradiation. If the structure expands, the irradiated surface will be convex, but the surface product layer will experience compressive stresses. Expansion due to bending is insufficient to fully compensate for compressive stresses and change the sign of the stresses. If the phototransformation causes the crystal structure to shrink, the crystal will bend and the irradiated surface will become concave, but the transformed surface layer will experience tensile stresses. A special
Figure 12. Variation of the profile of the longitudinal stresses. (a) Thin needle (bμ = 1), (b) thick crystal (bμ = 10), and (c) very thick crystal (bμ = 50). The curves were calculated by using eq 28.
(bending) releases the stresses almost completely. One can demonstrate that in the case of very thin needles, the amplitude of the residual stress is proportional to the square of needle thickness. In thick needles, the stresses in the transformed part of the needle approach the maximum possible values, which are determined by the complete deformation that results from the transformation. For clarity, in Figure 13 we present the time dependences of the strain energy accumulated per unit irradiated surface, which can be calculated as W (t ) =
∫0
b
s33σ32(x , t ) dx 2
(29)
(Note that in Figure 13 the data are normalized to dimensionless units by division with the coefficient (e3)2/ (2μs33).) It can be inferred from Figure 13 that in the case of a needle with thickness of b = 1/μ, the maximum value of energy that can be accumulated is 4 orders of magnitude smaller than in the case of thick needles. If the needles are even thinner, the energy decreases no slower than ∼b5. Thus, increasing the thickness of a needle causes a very fast increase in the residual stresses. Because the lattice strain that results from the phototransformation is often equal to several percent, the potential stresses that are generated can be very high, and irreversible relaxation processes such as plastic deformation or fragmentation become inevitable. 12457
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sufficient to support the irreversible growth of these nuclei. Although in general both conditions are necessary, there are several special cases where one of the conditions becomes more important than the other. In situations that are commonly encountered in mechanical engineering, fracture of a macroscopic object is induced by external loading. The stress fields are extended, and their size is comparable to the characteristic dimensions of the body. In this case, the first of the two conditions above becomes more important. One can distinguish between short- and long-term strength of the material. The short-term strength is related to the existence of various structural peculiarities or defects (scratches, discontinuities, cavities, incisions, etc.) that can act as stress concentrators. Fracture starts when critical conditions are met at these sites. According to the force criterion of fracture mechanics, a crack can start irreversibly growing at a stress concentrator when the stress intensity factor exceeds a certain critical value Kc, a characteristic property of the material. The stress intensity factor (K) depends on the characteristic size of the stress concentrator as
intermediate case is when the strain gradient induced by phototransformation is constant through the crystal bulk. In this case, the contributions of all force momenta will result in zero net stress throughout the crystal bulk, independent of the sign of strain induced by phototransformation. For the inner, untransformed side of the crystal that is opposite to the irradiated surface, the relation between the signs of strain and stresses is as anticipated, and stresses arise because the opposite side of the crystal experiences a phototransformation. This is intuitive based on experience with the external loading of beams, as for these layers the origin of strain is “external”. Therefore, the convex side is stretched as compared to its normal state, experiencing tensile stresses, and the concave side is compressed as compared to its normal state and experiences compressive stresses. The stress distribution in a crystal that bends because of the phototransformation is very different from that resulting from an applied external force. In a crystal bent by external force, the convex side experiences tensile stresses, and the concave side experiences compressive stresses. The line of zero stress is inside the crystal, strictly in the middle for a uniform crystal. If a crystal is bent because of the phototransformation, both the concave and the convex surfaces experience stresses of the same sign. These are either tensile stresses if the structure shrinks after phototransformation, or compressive stresses if the structure expands. The two opposite external surfaces can only experience stresses of the same sign if somewhere in the crystal bulk there appear regions with stresses of the opposite sign. As a consequence, there should be more than one line of zero stress in the crystal bulk. Under certain conditions, more complicated cases are also possible, when the nonuniform strain causes bending sufficient to change the sign of stress once more in the very vicinity of the irradiated surface, generating one additional neutral surface (see Figure 12b, line tkph = 50). The intricacies of this complex picture depend on the detailed distribution of the transformation product through the crystal bulk and on the structural strain induced by the transformation. The formulas and plots given above illustrate different possibilities of stresses distribution. This information will be of further importance in section 4 when analyzing photoinduced crystal fracture and the existence of different types of photomechanical responses: jumping, creeping, exploding, vibrating, etc.
K≈σ h
(30)
where h is the characteristic size of the stress concentrator (e.g., scratch depth), and σ is the tensile stress normal to the surface of the fracture (for fractures of the first type) or shear stresses in the surface plane (for fractures of the second and third types; see Figure 14 for explanations of the classification of fracture types).
Figure 14. Types of fractures and cracks induced by deformation. (a− c) The three types of fractures. (d−f) Cracks induced by compression (d), stretching (e), and twisting (f).
4. RELAXATION OF THE MECHANICAL STRESSES RELATED TO PHOTOINDUCED TRANSFORMATIONS
The fracture is possible only if K exceeds a critical value Kc, which is a characteristic of the material: K ≥ Kc (31)
4.1. Relaxation by Fracturing
4.1.1. Fracture of Bulky Macroscopic Objects. Mechanical reconfiguration (change in crystal size and shape) is not the sole channel by which mechanical stresses can relax; other options are plastic deformation and fragmentation. Of the mechanisms of plastic deformation, only those based on the dislocation glide can account for the fast relaxation processes that can induce a macroscopic effect such as crystal jumping or creeping. The plastic deformation or fragmentation that occurs during a phototransformation is subject to critical conditions. The fracture requires two conditions to be fulfilled. First, inhomogeneities that act as stress concentrators must be present in the material before the reaction, or they should appear as a result of structural fluctuations; these are referred to as crack nuclei. Second, the elastic stress energy must be
The critical stress values in this case are determined solely by the mechanical properties of the material and the size of the defects (σc ≈ Kc/√h). If these values are not reached (i.e., there are no concentrators of sufficiently large size present), the critical crack nuclei form as a result of structural fluctuations mentioned in section 4.1.141 As a result of the atomic oscillation in crystals, temporary atomic configurations appear and disappear in which interatomic distances are above some critical value, analogous to small cracks. Another option is crack nucleation via plastic deformation. This is the case of long-term strength in which the characteristic time period that precedes fracture increases exponentially with the decrease of stresses. 12458
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4.1.2. Fracture of Thin Films and Fine Particles. When the strength of thin films and fine particles is considered, the second of the above conditions becomes more important. In small particles, the number of intrinsic defects is negligible, and thus the fracture starts with nucleation of critical stress concentrators. At the same time, according to eq 30, the level of critical stresses increases as the size of the object, and thus the threshold size of the stress concentrators decreases.142,143 For example, a fracture on the micrometer scale requires stress values that are equivalent to strains of one-hundredth the order of magnitude, that is, about 0.01 times the elastic modulus.130−132 These strain values are typical for photoinduced transformations in crystals. Nucleation of critical stress concentrators under such conditions will occur sufficiently fast. The limiting step will therefore be achieving a sufficient amount of stored energy to induce irreversible growth of fractures. 4.1.3. Fractures Resulting from Photoinduced Transformations. A peculiar feature of the stress that arises in crystals during a photoinduced transformation is that the origin of stress is not an external force, but rather an inhomogeneous distribution of strain that evolves throughout the crystal bulk. In this case, as discussed in section 2, the stressed state can be characterized by two measurables: a characteristic degree of inhomogeneity in the extent of transformation and a characteristic crystal size. The main portion of the strain energy is concentrated in an area that is the smallest of the two sizes. Because of the absorption of radiation by a crystal, a photoinduced stress field will be of a distinctive size, corresponding to the characteristic absorption depth of the material. If the excited state can migrate, the stress field can be larger or smaller than the absorption depth, but will be limited by the rate at which the excited state relaxes. Finally, the extent of stressed area appearing as the result of a phototransformation is always limited in size by the product thickness, which is typically of micrometers size. Consequently, the size most “dangerous” for cracking stress concentrators h in eq 30 should be the same as the size of stressed area, which is in fact the thickness of the product (or the whole crystal thickness if the reaction occurs in most of a thin crystal bulk). Because of the natural limit on the size of stressed regions that arise during a phototransformation, a critical state given by eq 31 can be only achieved under certain conditions. According to eq 30, fracture is possible when the product of stresses and √h (where h should be considered as the size of the stressed region as outlined above) exceeds a certain critical value. Several different cases can be considered. If K does not reach the critical value Kc at any radiation dose, then in principle fracture is not possible. This is the case when the size of the stressed region (the crystal thickness or the characteristic penetration depth, whichever is smaller) is too small for stresses induced by phototransformation to reach the critical level necessary for fracture. Alternatively, the evolving stresses are insufficient to induce fracture in the region on the order of magnitude of the penetration depth. Consequently, for some transformations fracture may not be possible, and for others the possibility of fracturing is limited by the minimum crystal thickness bc, which is necessary for crack nucleation. If both sufficient thickness of the crystal and the critical level of stresses can be achieved, fracture will be observed after sufficient exposure to light. The amount of phototransformed product is sufficiently large to generate a critical stress. In general, similar conclusions can be reached in the case of plastic deformation where the efficient plasticity limit (yield point)
also depends, albeit inversely, on the size of the stressed region.144 4.2. Estimation of the Critical Conditions
4.2.1. Stochastic Nature of Crack Formation. Given the highly stochastic nature of fracturing process, it is practically impossible to derive a single mathematical expression to predict the exact point at which critical conditions will be achieved. Apart from some idealized cases, a critical condition is subject to probability and thus can only be estimated, rather than calculated exactly. Moreover, because critical crack nuclei appear as a result of fluctuations (lattice vibrations), the fracture itself always occurs after the critical conditions are exceeded. The farther the system has been brought from the critical condition, the higher is the probability of fracture. In all cases, energy is required to form the surface of a crack. In general, the value of the specific surface energy depends on the orientation of the crack in the crystal and on the velocity of crack propagation. For simplicity, in the first approximation, it can be considered that the specific energy depends solely on the material. Because in most cases a fracture on the small scale is initiated by structural fluctuations, the sign of corresponding stresses is very important for the process of fracturing. A crack nucleus can be generated more easily if the stress tensor has stretching components, regardless of the coordinate system. When testing a material, one always obtains lower values of the critical stress for extension relative to compression, composite materials being the only exception. In fact, a fracture that results from compressive stresses always starts at sites where inhomogeneity of the stress field is most pronounced, a manifestation of edge effects forming local regions of tensile stresses. In photoinduced transformations where the transformation results in positive structural strain in two or more directions, pure compressive stresses can only arise in crystals with thickness much greater than that of the photoinduced product layer. In all other cases, regions with tensile stresses are always present in the crystal because of bending or twisting. In continuation, the five typical cases of critical stressed states will be described. 4.2.2. Thick Crystals: Tensile Stresses in the Product Layer. If the critical conditions required for fracture can in principle be achieved (see section 4.1.1), then fracture (Figure 14) can start when the critical conditions are slightly exceeded. This results in formation of a network of type I surface cracks no deeper than the thickness of the product layer (Figure 14d). The small excess over the critical stresses accounts for the quasistationary regime of the fracture. It is in this regime that nearly all of the elastic strain energy that is released forms cracks; no significant energy can be transferred into kinetic energy. Therefore, no crystal motions or violent fragmentation with crystal debris projected over large distances are possible. In fact, this is the only case in which a quantitative description of the process is possible. The results can be considered a description of the threshold critical state. Under other conditions (smaller crystals, compressive stresses), the critical state will be achieved at a higher irradiation dose. 4.2.3. Thick Crystals: Compressive Stresses in the Product Layer. This case is possible if the crystal structure expands (cell parameters increase) in the plane parallel to the irradiated face so that stresses are compressive (see section 3.5; Figure 14e). Crack formation under compressive stresses is difficult and requires significantly higher stresses than fracture on extension. First, nucleation of cracks in stretched regions is 12459
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formation of type I surface cracks normal to the crystal axis (i.e., in the direction normal to the stretching forces). For thin crystals, stress relaxation via crystal bending becomes important; as a result, the critical state for fracture is achieved at a higher degree of transformation. As in section 3.4.2, if the crystals are sufficiently thin, the characteristic absorption depth exceeds the crystal thickness and the strain energy decreases steadily to zero. At a certain value of crystal thickness, fragmentation becomes impossible. In other words, for any crystal there exists a threshold value for the crystal thickness below which the crystals will not crack if strain is induced by the transformation. This critical thickness must be several times larger than the thickness of the layer in which fracture is observed for thick crystals (section 4.2.2), because the stresses are alleviated by bending. As long as b > bc, the process of fragmentation depends on crystal thickness. In relatively thick crystals, a network of surface cracks normally forms, similar to the case described in section 4.2.2. Because of the large mass of the crystals, the motions, if they occur, will be slow and accompanied by sudden local unbending that results from alleviation of the surface stresses when a crack is formed. As the crystal becomes thinner, the motions accompanying this unbending become faster (because of smaller crystal weight). As the crystals approach the threshold thickness value, the stressed area occupies an increasingly larger part of the crystals’ bulk. This will result in the crystal splitting into fragments, which will subsequently fly apart. The velocity of the fragments will first increase with decreasing crystal thickness (weight). For even thinner crystals, the maximum possible value of accumulated strain energy will decrease, and the velocity of the flying fragments will also decrease. Finally, fragmentation will no longer be possible for crystals with insufficient thickness. 4.2.5. Thin Crystal Needles or Ribbons: The Structure Expands along the Long Crystal Axis. In this case, the observed phenomena are qualitatively similar to those considered in section 4.2.4. The differences are related to the fact that surface layers experience compressive stresses. Because of this, the values of critical degree of transformation at which fracture will be observed are comparatively higher, similar to the case described in section 4.2.3. Here, type II cracks, as opposed to type I, should appear at the surface (Figure 14e). An additional phenomenon should be observed because of crystal bending. In crystal needles, tensile stresses arise in the inner layers to compensate for the compressive stresses at the crystal surface (Figure 12). Because tensile stresses are more favorable for crack nucleation, under certain conditions fracture can start in these inner layers, nucleating cracks of type I. The thickness of the inner stretched layer is roughly equal to onehalf of the total thickness of the bent crystal. Cracks initiated at the inner site will propagate through the crystal to the irradiated surface layers. Because a large strain energy is accumulated in the latter, crack propagation through the whole crystal is likely. As compared to the case in which the transformation reduces the volume and cell parameters (i.e., tensile stresses arise at the irradiated surface), significant motion of crystal fragments will be observed over a much larger range of crystal thickness, and the kinetic energy of splitting fragments will be higher. Still, similar to the previously discussed case, there exists a minimum crystal thickness below which no fracture is possible. 4.2.6. Thin Crystal Needles or Ribbons: Shear Strain in the Plane of the Irradiated Face. In this case, photo-
easier than in compressed ones. In the case under consideration, such extended regions can only be present at the crystal edges where the level of mechanical stresses is significantly lower due to edge effects. This is because the stress-free surface (i.e., the adjacent face) is near. Second, only type II cracks (Figure 14b,e) can arise in the compressed surface layer far from the edges. These cracks will penetrate the crystal along the direction of the maximum shear stresses (at 45° to the surface in an isotropic material). It is very difficult to generate such a crack under the conditions where crack edges cannot move freely with respect to each other because of compression stresses. Plastic deformation along the entire surface of the future crack is a necessary condition to make such a crack appear. Even if such a crack does nucleate, the corresponding stress intensity factor is significantly lower than the corresponding value for a type I crack that arises in a layer of the same thickness at the comparable level of tensile stresses. According to Murakami,145 the stress intensity factor for a type II crack is about 2.5 times less than that for a type I crack in the stretched layer. Therefore, to initiate fracture in a compressed layer, one needs a stress of at least 2.5 times the magnitude of that required to generate a crack in a stretched layer of the same thickness. Because of this, the energy accumulates in the compressed layer and at the moment of fracture can be an order of magnitude higher than the maximum energy preceding fracture in a stretched layer (the energy is proportional to the square of stresses). A possible consequence is that surface fracture can then be followed by a dynamic fracture in which a substantial portion of the accumulated energy transforms into kinetic energy, propelling small crystal fragments. The dimensions of the splitting fragments will be always comparable to the thickness of the product layer. Once initiated, the progression of the fracture cannot be terminated as long as the front of the crack remains in the supercritical stressed state, continuing along the already transformed, stressed part of the crystals. The crack cannot extend far into the originally nonstressed part of the crystal (the untransformed region), where the stress level will be below the critical value required for crack propagation. The starting crystal will remain practically immobile because of its large mass; if it moves because of reactive forces, it will move at a velocity inversely proportional to its mass due to the law of the conservation of momentum. This reactive crystal movement is rather complex and subject to the effect of several factors: the reactive force due to fragments splitting, the law of the conservation of momentum, the law of the conservation of energy, friction from the support, and the reaction itself. When fragments split from the crystal, a reactive force arises. Initially, the crystal momentum is equal and opposite to that of the fragment. The crystal may jump in various directions, depending on the fragment momentum, along the support, in the upward direction, or it may even try to jump downward, subsequently springing out of the support. In all cases, the crystal will slow down, be it because of friction, work against gravity, or because of energy dissipation during bouncing on the support. However, if fragments continue to split off at some frequency, the crystal continues jumping and shifting. 4.2.4. Thin Needle-Shaped Crystals or Ribbons: The Structure Shrinks along the Long Crystal Axis. As in the case of thick crystals (section 4.2.2), a layer of substance experiencing tensile stresses will arise at the irradiated side of the crystal. Because of this, fracture will be possible via 12460
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of K will change over the course of a transformation from zero to its maximum value:
transformation is expected to induce sole torsion as a result of shear strain in the surface layer. As a consequence, tensile stress will arise at some angle relative to the crystal axis, enabling the initiation of a fracture. The occurrence of the fracture and the phenomena ensuing thereafter are similar to those described in section 4.2.4. Additionally, in this case, type III cracks may form (Figure 14f); these cracks are responsible for direct relaxation of the shear stresses. Which of the fracture types will dominate in twisted crystals depends on the relation between the critical conditions for each of the types. However, any observed macroscopic photomechanical effects will not be significantly different.
Klim ≈
∫0
σ (x , t ) l2 − x2
K max(t ) = Kc
To estimate when the latter condition holds, it is practical to reformulate it in terms of the energy criterion for fracture. According to this, the strain energy release rate must reach the critical value:
G = Gc
(39)
The strain energy release rate G is equal to the derivative of the change in energy of elastic strains required to increase the crack surface. The ratio given by eq 39 means that the energy released as the crack grows and the stresses decrease is used to form new crack surfaces. Because two surfaces are formed when a crack grows, it is common to consider Gc = 2γ, where γ is the specific fracture work required to form a new surface. It is postulated in fracture theory that criteria given by eqs 36 and 39 are equivalent and are related by
dx (32)
(33)
where E is the elastic (Young’s) modulus characterizing extension along the surface, and ε is the tensile strain within the plane that corresponds to the shrinkage resulting from complete transformation. The dependence of K on the crack length l (eq 32) is plotted for a range of time points in Figure 15. These functions have a
K 2 = GE
(40)
Consequently, the condition of the possibility of fracture, eq 38, can be approximated as 2γ ε2 > μ E
(41)
For compounds with relatively low plasticity, the ratio in the right part of eq 41 is approximately equal to the length of an interatomic bond, that is, ∼ 0.1 nm. If the characteristic depth of light penetration is about 10 μm, and strain is typically several percent, then eq 41 holds. As an example of a transformation when fracture is impossible, one can compare the case where the depth of light absorption is 1 μm and the strain is less than 1%. If the left part of the inequality is much larger than the right side, the critical condition given by eq 37 will be true already at the early stages of the reaction. Using eq 5, one can estimate the time needed to achieve the critical state:
Figure 15. Reduced stress intensity coefficient of an edge crack at the surface of a thick crystal as a function of its length l at different time points (t·kph). The curves were calculated by using eq 32.
k phtc ≈
maximum at the crack length that corresponds to the characteristic width of the C(x,t) profile. The dependence of the maximum K value versus time can be qualitatively expressed as K max(t ) ≈ EεC(x = 0, t ) htr(t )
(37)
For fracture to become possible, at least in principle, it is necessary that Klim > Kc (38)
where the function σ(x,t) defines tensile stresses that act in the product layer due to its shrinkage. The function can be written as σ(x , t ) = EεC(x , t )
(35)
that expresses the force criterion for fracture. Because the function given by eq 32 increases monotonously with time, a critical state always corresponds to the maximum value of K:
The conditions for achieving critical stressed states can be considered on the basis of the classification of the typical situations described in section 4.2. To facilitate the analysis, we will neglect the special features of fracture that are related to the anisotropy of mechanical properties of a crystal and consider the case of uniaxial strain of a crystal with isotropic properties. 4.3.1. Fracture at the Surface of a Thick Crystal. The stress intensity factor of the edge crack propagating normal to the crystal surface at the depth h can be calculated as145 l
1 + k th /k ph
A critical stressed state after which a fracture becomes possible arises when K reaches its critical value: K = Kc (36)
4.3. Estimation and Quantification of the Critical Conditions
l K (l , t ) = 2.24 π
Eε x0.5
2γμ Eε 2
(42) −1
For example, for the case when μ ≈ 10 μm and ε ≈ 0.03, kphtc ≈ 0.1; that is, the fracture will become possible when the local degree of transformation at the surface reaches 10%. If a transformation does not generate tensile stresses at the surface of the crystal (i.e., only compressive components are present), then as mentioned in section 4.2.2 the fracture can be initiated either by nucleation of type I cracks at the crystal edges (strain nonuniformity causes stretched regions) or by
(34)
The characteristic thickness of the product layer htr changes from μ−1 to x0.5, as defined by eq 7. Thus, the maximum value 12461
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of the crystal. The critical fracture conditions for different reactions (initially inducing tensile or compressive stress) become more similar. For reactions that induce compressive stresses in the surface layer of the product, the fracture can now start not only at the crystal surface or edges, but also in the internal layer that is stretched as a result of the tensile stress (see section 4.2.4). It is possible to estimate the conditions under which fracture will preferentially start in the inner stretched layer of the crystal. If the surface layer experiences compressive stresses, two options exist: fracture can start with a surface crack of type II, or with an inner crack of type I. Let us assume that a crack will nucleate in the region of the crystal where the accumulated strain energy will be sufficient to trigger fracture. For fracture to start in the compressed surface region, the critical stresses must be at least 2.5 times higher, and consequently the accumulated energy must be 6.25 higher than what would be required if stresses in the same surface layer were tensile (see section 4.2.2). The ratio between the amount of energy accumulated in the surface and in the inner layers is plotted as a function of crystal thickness in Figure 16a. Wneg and Wpos are calculated
type II cracks formed at the irradiated surface. Qualitatively, the criteria for fracture remain the same; however, the critical deformation ε and degree of transformation (defined by kphtc) will be several times higher. 4.3.2. Fracture of Thin Crystals. As the thickness of the crystals decreases, macroscopic change in crystal shape (bending, twisting, coiling) becomes possible, resulting in a general decrease in the stresses and elastic stress energy (Figures 12 and 13). Obviously, in this case, the stress intensity factors and the strain energy release rates of the potential cracks will be smaller than the corresponding values for thick crystals under similar conditions. As the thickness of the crystal approaches the characteristic depth of the product layer, these values approach zero. It follows that, as the crystals get thinner, the critical state shifts toward higher transformation yield (longer irradiation time, higher radiation doses) until the crystal becomes so thin that fracture becomes impossible. Stress distribution over the crystal bulk becomes very complex in this case (Figure 12). If crystals are sufficiently thick, the most highly stressed regions are still localized near the layer of the transformed part of the crystal. However, as the crystal cross-section decreases, stresses become comparable throughout the crystal bulk, and more than one neutral (zero stress) surface can arise. This effect renders establishment of a universal criterion for fracture very difficult, because the criterion should include all possible variants and configurations of potential critical cracks. Nevertheless, the rate of maximum strain energy release will be limited by a value proportional to the density of strain energy localized in the region of the potential fracture and the size of that region. As shown above, the major fraction of the energy and the largest stresses are always concentrated in a region with size comparable to the thickness of the product layer. If the crystal is sufficiently thin, this region spreads throughout the whole crystal. The maximum strain energy release rate will always correspond to cracks that have the same length as the size of the region with the highest density of strain energy. Therefore, to estimate the maximum possible strain energy release rate, one can simply take the integral of the strain energy density over the crystal depth, that is, the value of the accumulated specific strain energy per unit of the irradiated surface. This value was calculated by using eq 29 and is plotted in Figure 13. For thick crystals, the release rate grows steadily to some maximum with the degree of transformation. As the crystal thickness approaches the characteristic depth of the product layer, the change in specific strain energy versus time deviates from monotonic behavior and shows two maxima. The first maximum is related to the initial stages of the reaction where the largest stress is localized at the irradiated crystal surface. The second maximum is observed when the reaction is close to completion. The largest stresses in this case are localized mainly at the side of the crystal that is opposite to the irradiated face. The ratio of maxima given by this expression reflects the concentration profile of the product that follows from the model. A general trend suggests that as crystals become thicker, the second maximum disappears initially as the reaction is incapable of spreading through the crystal bulk. With further growth in thickness, the first maximum disappears as the reaction becomes unable to reach even the middle of the crystal. In turn, the accumulation of energy becomes monotonous as is the case with very thick crystals. Interestingly, as the crystal thickness decreases, both tensile and compressive stresses of comparable value arise in the bulk
Figure 16. Relations between energies accumulated in inner and outer layers of crystal (layers with opposite signs of stress) as a function of crystal thickness. (a) Analysis of the cracking origination possibilities (crack I in the inner layer or crack II in the surface layer) for the case of structure expansion during reaction. (b) Analysis of possibility of cracking into parts or termination of the cracks (structure expansion). (c) The same analysis performed for structure compression. These curves were calculated by using eq 29 and specially selected integration limits (see section 4.3.2). 12462
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as a result of the photochemical transformation as a function of the crystal thickness. The expression for this value (made dimensionless by dividing by Eε2/μ) is
according to eq 29 but by using limits of integration that correspond to regions where compressive (negative) or tensile (positive) stresses are produced by the transformation. For simplicity, the ratio is calculated assuming that fracture occurs at early stages of the phototransformation when the concentration profile is exponential (eq 5). If the crystal thickness is smaller than b ≈ 40 μ−1, cracks should nucleate in the inner layer of the crystal as the energy that can be spent for fracture in the inner layer is higher than that for the outer in this case. Otherwise, if the crystal is thicker, fracture will start from the surface via type II crack generation. This plot allows one to predict if the fracture, after it had started, can propagate through the whole crystal bulk, and to estimate the excess energy that can be released after fragmentation. A type I crack formed inside the crystal can also grow in the surface layer using accumulated strain energy stored in this layer. This can be up to 6 times larger than that in the inner parts of the crystal (Figure 16a). Fracture will split the crystal into fragments that will fly apart with high velocities because the “excess” energy released after the fragmentation is 1−6 times higher than the work required for crack generation. If a crystal is thicker than b ≈ 40 μ−1, the majority of the strain energy is spent generating type II cracks in the surface layer. The energy accumulated in the irradiated part of the crystal may also be sufficient to continue fracture and complete crystal fragmentation. An estimate of the ratio of the energy values in the inner and surface parts of a crystal (Figure 16b, where Wneg2 is calculated for the compressed region near opposite to the irradiated side of the crystal) suggests that even for rather thick crystals, complete fragmentation is possible. However, in thicker crystals, the kinetic energy of the fragments will be rather low, not exceeding a few tenths of the energy required to generate the cracks. For crystals thicker than b ≈ 45 μ−1, the energy remaining after initial crack generation will no longer be sufficient to let the crack propagate through the whole crystal and fracture will be limited to the compressed surface layer. It follows that in the opposite case, when a reaction induces tensile stresses in the surface layer, fracture will always start from the surface before sufficient energy can be accumulated to nucleate an internal type II crack. The ratio of energy in the bulk to the energy in the stretched surface layer is plotted as a function of the crystal thickness in Figure 16c (Wpos2 refers to the stretched region near the opposite crystal side). Complete crystal fragmentation will be possible if crystal thickness does not exceed b ≈ 5 μ−1. In this case, the excess energy released as kinetic energy of the flying fragments will be equivalent to approximately one-half of the energy spent to generate the fragments by fracture. In summary, one can expect that if a photoreaction results in structure expansion, fracture is likely to break crystals into fragments with a significant portion of the accumulated energy transformed into kinetic energy. In the opposite case, it will be less likely that supercritical conditions will be achieved, and the fracturing will be significantly less violent; either splitting will occur with low kinetic energy of the leaving fragments, or surface cracks will be generated to decrease the stresses in the crystal bulk, unbending the crystal that was bent as a result of irradiation. Refraining from a detailed analysis of the conditions required to achieve the critical state in all possible cases, let us estimate the conditions under which crystal fracture will not be possible. To this end, one must consider the dependence of the maximum value of the specific strain energy that is accumulated
gmax (b) =
⎡ μ max⎢ 2 ⎣ Eε
∫0
b
σ 2(x , t ) ⎤ dx ⎥ 2E ⎦
(43)
where the stress function is calculated using eq 28 and accounts for crystal bending and the maximum value of the integral as a function of time. The results of the calculations based on these assumptions are plotted in Figure 17. The calculated value
Figure 17. Maximum possible specific strain energy as a function of crystal thickness for kth/kph = 0.001. The specific energy is given as gmax in eq 43; the reduced needle thickness is μb (similar to Figure 16). (a) Thick needles, (b) thin needles. The curves were calculated by using eq 43.
makes it possible to qualitatively estimate the dependence of the maximal strain energy release rate on the crystal thickness according to
gmax (b) ≈
Gmax μ
(44) Eε 2 Thus, fracture is possible if 2γμ gmax (b) > A 2 (45) Eε where A is a coefficient close to unity. The crystal thickness at which the equality is achieved in the eq 45 is the critical thickness bc mentioned in section 4.1.3. In this case, fracturing becomes impossible when the crystal thickness decreases to less than several times the characteristic depth of radiation absorption. As one can see from the dependence in Figure 17, fracture becomes almost impossible when b < μ−1. 4.3.3. Transformation of the Excess Strain Energy to Kinetic Energy of the Fragments. To estimate maximum values for kinetic energy, let us consider a hypothetical case in which the entire strain energy that is released as a result of mechanical stress relaxation is transformed to kinetic energy, fueling the motion of the crystal or its fragments. If initially the bulk of the crystal was uniformly stressed, the specific kinetic 12463
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energy, ρV2max/2 (here, ρ is the density, Vmax is the velocity of the crystal fragments), will be equal to the initial density of the strain energy Eε2/2. Because the value of (E/ρ)1/2 is equal to the velocity of sound c in the solid, then Vmax = εc and is of the order of tens of meters per second. Experimentally measured velocities of flying fragments splitting from a crystal as a result of a phototransformation are at least an order of magnitude lower.146,147 This result implies that at most several percent of the strain energy is transformed to kinetic energy. The main reason for this discrepancy is the energy required for fracturing. If fracturing causes complete separation of the fragments from each other, their kinetic energy T can be calculated as T = ΔW − Ccr
result from the geometry of the fracture with respect to the crystal geometry. If we consider, for example, a case when strain energy is sufficient to break a needle-like crystal into two parts, the velocity of the flying crystal fragments can be estimated on the basis of a number of considerations. The kinetic energy of the fragments is T ≈ δγScr
Assuming that a crack splits a crystal with cross-section Scr and length L, the kinetic energy of the particles is ρV 2 ≈
δγ T ≈ ScrL L
(49)
Taking into account that γ ≈ λE (λ ≈ 0.1 nm, see the comment to eq 41), we obtain
(46)
where ΔW is the change in the total strain energy that results from a decrease in the stress of the split fragment, and Ccr is the work spent on fracture. The latter is usually related to formation of new crack surfaces Scr and can be formally expressed as Ccr = 2γScr. The critical state at which fracture becomes possible is achieved when the difference between the two terms in the right half of eq 46 becomes zero. This corresponds to the stationary state of a crack, the tip of which starts propagating as soon as the right part of eq 46 becomes positive. The growth of a crack at a small positive excess of ΔW over Ccr is termed a quasistationary fracture and can only occur if a crack already exists as an intrinsic defect in the crystal before external stress was applied. When the crack must nucleate prior to growth, the critical state must be exceeded by a certain value. Immediately after the critical state is reached, the probability of fracture nucleation is zero and increases as this critical state is exceeded. Fracture is a stochastic process, and as such, even for the same system, it may start when the critical state is exceeded by different amounts. It appears reasonable to assume that the amount by which the system exceeds the critical state is equal to a fraction of Ccr. When tensile stresses prevail (the transformation results in compression of the material), this fraction must be relatively small as the strain values usually reach several percent and the evolving stresses are rather high. As such, crack nucleation is fast and takes place as soon as the critical state is exceeded. If compressive stresses dominate, fracture will preferentially start at the location where tensile stresses are localized, even for small excesses over the critical state. However, if the growing crack reaches a region containing a high density of compressive strain energy, the state of the crack will largely exceed the critical state for this region. A dynamic fracture is triggered whereupon a large portion of the energy is transformed to kinetic energy. In this case, the total kinetic energy can amount to a significant part of Ccr, but cannot exceed the total fracture energy. In summary, the kinetic energy can be expressed as T ≈ δCcr
(48)
V≈c
δλ L
(50)
This expression gives the velocity of the fragments obtained after fracture from a millimeter-sized crystal on the order of tens of centimeters per second. This result is in good agreement with experimental observations.146,147 4.4. Role of Plastic Deformation in Photomechanical Effects
4.4.1. General Considerations. Plastic deformation is an alternative channel through which mechanical stresses may relax, and can either be complementary or an alternative to fracture. There are several possible mechanisms of plastic deformation, and the temperature (relative to the melting temperature of the substance) determines which of the various mechanisms will dominate. For a complete description of photomechanical effects, it is necessary to consider all possible processes by which accumulated elastic strain can dissipate. These relaxation processes may contribute new “records” to the history of the crystal that is undergoing photochemical transformation. These records include all information concerning generated defects and the corresponding changes in mechanical properties. In the context of this Review, our main focus is on photomechanical effects, and we shall thus limit discussion to plastic deformations related to dislocation glide. Similar to the case of fracture discussed in section 4.3, two conditions must be met for plastic deformation to become possiblea dislocation can both generate and subsequently move under the action of shear stresses. Qualitatively, these conditions can be expressed as reaching a certain level of shear stress in the glide planes of the dislocations. The level of shear stress that is sufficient to trigger a glide of dislocation within a crystal that has a low concentration of defects (i.e., noninterfering defects) is known as the Peierls−Nabarro barrier.148 For a given dislocation type, this value is a characteristic of the material and can vary broadly; it depends strongly on the nature of the chemical bonds and intermolecular interactions in the crystal. Depending on the value of the Peierls−Nabarro barrier, compounds can be classified as plastic and brittle. To allow dislocation glide, the average values of the macroscopic shear stresses must be no lower than the Peierls−Nabarro barrier. If the crystal contains impurities, heterogeneous inclusions, or the density of dislocations is initially high, perhaps as a result of the phototransformation, free dislocation glide is hindered and slows or, in some cases, completely stops. The critical stress required to initiate the dislocation glide will
(47)
where the coefficient δ can vary from small values (0.01−0.1) where tensile stresses dominate (or for thin crystals when the contributions of tensile and compressive stresses to the total energy are comparable) to some value close to unity (section 4.3.2) for compressive stresses in relatively thick crystals. It is difficult to estimate this value more precisely as it is highly stochastic and depends on a multitude of factors. We have seen how changes in crystal thickness and stress regime can affect the kinetic energy of flying crystal fragments. Observed decreases in the kinetic energy of fragments may also 12464
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be many times higher than the Peierls−Nabarro barrier for an ideal crystal, but this will also be a characteristic of the material in its nonideal state. A detailed description of the dynamics of dislocation glide is a very complex problem that requires consideration of multiple factors including characteristics of the material averaged over the bulk, the nature of the stress field, as well as local interaction between dislocations and with internal (defects) and external (surface) crystal inhomogeneities. Qualitatively, one can define the condition of reaching the level of shear stresses τs required to trigger dislocation glide as τs ≥ τp
By this point, we have seen that the conditions enabling plastic deformation within the photoinduced product layer are described by eqs 51 and 53. In a general case, it is not clear which of the two conditions will be more important, because in both cases the characteristic values of critical shear stresses fall in the range from 10−5Gs to 10−2Gs, depending on the crystal structure, the nature of the chemical bonds, and the radiation penetration depth. However, it is worth noting that when eq 53 applies, plastic deformation of a crystal can occur when externally applied mechanical stresses reach values comparable to those generated by a photoinduced transformation (or even lower). However, this cannot occur simultaneously to the phototransformation as dislocations simply cannot nucleate in stressed areas as small as those resulting from irradiation. The probability of plastic deformation can therefore change dramatically depending on the presence of surface defects in the crystals, that is, on the procedures of crystals preparation, storage, and manipulation. For plastic deformation, it is typical that the critical stress level is exceeded in the initial stage of the deformation (sharp “yield strength (yield point)”). Similar to the homogeneous nucleation of cracks (or any other homogeneous nucleation), the critical conditions must be exceeded. According to Malygin,144 the typical value of stress required in excess of the critical level for the deformation of thin (90° upon [2+2] dimerization. Adapted with permission from ref 213. Copyright 2010 American Chemical Society. (b) Microribbons of 9-anthracene carboxylic acid undergoing [4+4] photodimerization twist around the longest axis. Reprinted with permission from ref 116. Copyright 2011 American Chemical Society. (c) Bending of 200 nm nanorods of 9-anthracene carboxylic acid upon local UV excitation in the area denoted with a circle. Reprinted with permission from ref 211. Copyright 2007 Wiley-VCH. The lengths of the bars in panels b and c are 20 and 20.7 μm, respectively.
× 10−3 to 9.5 × 10−3 s−1 (kunbending = 9.0 × 10−3 s−1). When exposed to spatially uniform radiation, microribbons of the same compound reversibly twist and thermally relax back to the initial shape after several minutes (Figure 22b).116 Prominent crystal bending was observed with a tetramorphic molecule that models the Green Fluorescent Protein (GFP) lumophore (20b, Scheme 1).213 Polymorph C of this molecule undergoes a τ-one bond flip upon exposure to unfocused UV light, whereas micrometer to millimeter-size crystals of the form A bend along their longest axis to angles >90° without breaking. After the crystal bends, it starts to twist until a recess appears normal to the long axis (Figure 22a). The bending angle can be controlled with exposure time. The photomechanical effect is due to a [2+2] photodimerization. The change in the surface morphology was followed by AFM. The structure of the unreacted crystal of form A is composed of layered two-dimensional sheets of all-syn hydrogen-bonded strings parallel to the c-axis, alternating and perpendicular to allanti zigzag strings that run parallel to the b-axis. Upon photodimerization, the pairs of dimerized layers slide by nearly shear-free motion atop each other, accounting for the unusual elasticity of the crystals. 5.2.3.3. Bending by Trans−Cis Isomerization. By utilizing trans-to-cis isomerization to drive a photochemical reaction, nanosized crystals of azobenzene can undergo reversible size changes by alternatively exposing the sample to 313 and 435 nm light.214 Macroscopic plate-like crystals and cocrystals215 of 12471
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substituted azobenzenes also bend upon excitation (1b−16b, Scheme 1). The bending of thin crystals of azobenzenes is a general phenomenon, and it is fast and reversible.216 For instance, crystals of trans-4-(dimethylamino)azobenzene (2b, Scheme 1) exposed on their (001) surface to UV light bend along the b axis and to a smaller extent along the a axis (Figure 23).217 By alternating exposure to UV light (2 s) and thermal
Figure 23. Bending of (001) face of the microcrystal of trans-4(dimethylamino)azobenzene induced by UV light (scale bar is 200 μm). Reprinted with permission from ref 217. Copyright 2009 American Chemical Society. Figure 24. Bending of salicylideneaniline crystal. (a and b) Bending of a slender crystal by alternating irradiation with UV light (a → b) and visible light (b → a), and (c) cyclability of the process monitored by the deflection of the tip of the crystal during alternative irradiation with UV and visible light. The length of the bar is 10 μm. Reprinted with permission from ref 221. Copyright 2011 The Royal Society of Chemistry.
back-isomerization (5 s), the crystal can be switched more than 100 times despite the very low quantum yield of the isomerization, estimated as ∼1%. The displacement of the tip of the crystal is dependent on the radiation intensity.218 The bending appears to be a common property of azobenzenecontaning crystals, as crystals of trans-4-aminoazobenzene (1b, Scheme 1) can also bend reversibly. The identity of the phase of the slender crystals, however, which is normally assumed to be identical to the bulk material in this and other cases, requires additional verification. Control experiments have confirmed that the photomechanical effect is not due to thermal effects,216 although the photothermal effects in azobenzene crystals can be significant.219 A convenient model for heating effects in thin films of azobenzene polymers developed by Yager and Barrett220 can be used to estimate these effects. Another instance of mechanical response in photochromic crystalline materials based on cis−trans isomerization was provided with the report on bending salicylideneaniline crystals.221 If irradiated with a UV LED light, platelike crystals of N-3,5-di-tert-butylsalicylidene-3-nitroaniline (18b, Scheme 1) undergo intramolecular proton transfer followed by cis−trans isomerization, whereupon they bend 45° away from the incident light within 1 s (Figure 24). The original shape of the crystals can be recovered within 10 s by bleaching with visible light. It was reported that the crystal can be alternatively bent and straightened over 200 times without obvious deterioration. The photomechanical effect of R- and Senantiomers of photochromic salicylidenephenylethylamine (an analogue of the salicylideneanilines, 17b in Scheme 1) was compared to that of the crystals of the racemic compound.222 While both chiral and racemic crystals bent, the reversibility of the racemic crystal was superior to that of the chiral crystal, indicating that subtle details of the structure can have a pronounced effect on the photomechanical response. 5.2.3.4. Bending by Cyclization and Ring Opening. Some photochromic diarylethenes in which the molecules have poor cycloreversion efficiency and thus can be retained in the cyclic form react with conversion yields >90%. There are also several reports on plate-like crystals of diarylethenes that can not only bend, but can also contract by excitation with UV light, depending on the crystal size and shape.223 It was recently demonstrated that such crystals can perform mechanical
work.223,224 Reversible change in the shape of crystals from square to lozenge was observed with single crystals of photochromic diarylethenes,225 similar to those reported earlier for the thermal phase transition of terephthalic acid.226 The internal crystal angles of the parallelepiped crystal of 1,2-bis(2ethyl-5-phenyl-3-thienyl)perfluorocyclopentane change from 88° and 92° to 82° and 98°, respectively. Thin crystalline rods of the methyl derivative can also bend slightly, and perform work by shifting small metal objects.225 Mechanical work can be also performed with mixed diarylethene crystals (36b, Scheme 1).224 A rotation of a small ratchet wheel (diameter 3.2 mm) can be driven by alternative exposure of a 1.3 mm long bending crystal to UV and visible light. Figures 25 and 26 depict crystals of two other photochromic materials, dithienylhexafluorocyclopentane and furylfulgide (38b and 39b/40b, Scheme 1).227,228 Slender crystals of some diarylethenes can exhibit remarkable flexibility. Upon UV irradiation, plate-like crystals of symmetrically substituted
Figure 25. Single crystal of dithienylhexafluorocyclopentene rolled under UV light. Reprinted with permission from ref 228. Copyright 2008 The Royal Society of Chemistry. 12472
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Figure 26. Bending of a crystal of photochromic furylfulgide irradiated in the direction of the blue arrow with UV light for 0 s (left), 1 s (middle), and 2 s (right). Reprinted with permission from ref 227. Copright 2012 Chemical Society of Japan.
Scheme 2. Molecular Structures of Reported Thermosalient Crystalsa
a
Ligand abbreviations: 3-CNpy = 3-cyanopyridine.
5.3. Relaxation by Rapid Deformation and Disintegration (Photo- and Thermosalient Effects)
dithienylhexafluorocyclopentene with chiral (R)-N-phenylethylamide groups (40b, Scheme 1) bend concavely before they curl up (Figure 25).228 The rolled crystals can then be straightened by exposure to visible light. Similar reversible bending of other photochromic crystals was also reported recently,227 indicating that the deformation of photochromic crystals that can be prepared as thin plates is not exceptional, but a common property that has not previously been noticed. In addition to bending photochromic salicylideneaniline described above, Koshima et al. have also reported that platelike microcrystals of the open E-form of the furylfulgide (38b, Scheme 1) irradiated with LED light at 365 nm change color from yellow to red. Simultaneously, these crystals bend approximately 9° in 1 s due to a ring closure to its C form (Figure 26).227 The original shape of the crystal is restored by excitation with light >390 nm, although the surface modification observed after the irradiation is irreversible.
5.3.1. General Considerations. Some crystals can jump over large distances during solid-state transformations. As has been discussed in the previous sections, this type of mechanical response is possible when there is an induction period preceding relaxation of mechanical stresses that accumulates over the course of phototransformation (Table S1). In many cases, the jumping results from a process without any gas evolution, such as a phase transition, or a photochemical reaction. Such effect can be induced either by heat or by light; in the former case, the phenomenon is termed thermosalient effect, while in the latter case it is known as photosalient effect (“salient” means leaping; see section 5.3.2). These effects where the motion is due to structural transformation should not be confused with chemical reactions where the explosion or motion of the crystals is induced by evolution of gas, such as the thermal decomposition of KMnO4 or (NH4)2Cr2O7. 12473
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Figure 27. Examples of single crystals that display thermosalient effect. (a,b) (Phenylazophenyl)palladium(II) hexafluoroacetylacetonate, (c,d) pyroglutamic acid, and (e,f) 1,2,4,5-tetrabromobenzene. Upon heating, the crystals jump off the base or splinter.
Recently, similar kinematic effects due to decomposition (evolution of oxygen) were reported in heterospin hexafluoroacetylacetonate complexes with nitronyl nitroxide.229 The displacement of crystals, accompanied by cracking and disintegration, continues for several weeks. 5.3.2. Thermosalient Effect. When heated or cooled, some crystalline materials can jump many times their own size. There is only a handful of reported instances of such jumping (hopping) solids (Scheme 2).230−232 In 1987, Gigg and collaborators coined the less colloquial term thermosalient crystals, standing for thermally induced jumping or leaping crystals.231 Figure 27 shows selected examples of thermosalient crystals. The first report that explicitly mentions jumping of organic crystals is a communication by Etter and Siedle in 1983.230 In her last lecture, recorded in 1992 (retrieved by the kind courtesy of Joel Bernstein from his private collection), Etter described the jumping effect using crystals of (phenylazophenyl)palladium hexafluoroacetylacetonate (1t, Scheme 2). She also proposed a mechanism based on the structures of the two phases (see video in the Supporting Information of ref 4). Between 85 and 95 °C, acicular yellow crystals of 1 undergo expansion of nearly 10% along their longest axis, whereupon they were reported to “fly off the hot stage”. The expansion to an intermediate “expanded yellow form” is followed by a color change between 90−100 °C from yellow to red, and the progression of the front of the phase transition along the crystal can be readily detected by visual means (Figure 28). The structure of the high-temperature phase determined from a recrystallized sample indicated that, during the phase transition, the neighboring stacks in the low temperature phase slip together in the direction of the long molecular axis ([011]). This was considered an indication of a mechanism similar to that underlying martensitic phase transitions. The details of the mechanism of the thermosalient effect in this prototypical compound were clarified recently (Figure 29).233 The compound has five forms and four phase transitions at ambient pressure, of which one is thermosalient. During this transition, the head-to-head stacked molecules separate from each other. The rate of advancement of the habit plane (the phase boundary) recorded with a high-speed camera (Figure 28) was 0.54 m s−1, corresponding to a turnover number of 1.272 × 109 molecules per second. This rate is 4 orders of magnitude faster than that of the second (nonthermosalient) high-temperature transition and also supersedes greatly the rate of spin-crossover transitons (typically on the order of 10−5 m s−1). The rapid spatial phase progression is the key factor driving the mechanical effect.
Figure 28. Manifestation of the thermosalient effect in crystals of (phenylazophenyl)palladium(II) hexafluoroacetylacetonate. (a−e) Upon heating, the crystal on the left expands along its longest axis (a,b), followed by expansion of the smaller crystal on the right (b,c), whereby they are transformed to (intermediate “expanded yellow form”). The expansion is succeeded by a transformation to the red phase, accompanied by a spatially resolved gradual change of color from yellow to red (d,e). These snapshots were extracted and adapted from the original video recording of Margaret Etter’s lecture in 1992 (the video material was kindly contributed by Joel Bernstein). The pink and violet colors are artifacts from the video recording. (f−m) Snapshots from high-speed recordings showing the progression of the phase front in a crystal which was restrained from motion with a small amount of oil. The phase transition starts from one of the corners shown to the right and rapidly progresses to the left with a straight front between two phases. The images in panels f−m were adapted with permission from ref 233. Copyright 2014 The Nature Publishing Group.
In 1960, it was reported that “a single crystal of ferrocene [15t in Scheme 2] cooled to 77 K undergoes a violent disintegration” (a private communication with Wilkinson cited in ref 234; a similar effect was described for nickelocene, 14t). This description resembles the visual effect observed during a thermosalient transition. Bodenheimer and Low provided evidence that the phase transition and crystal disintegration are separate phenomena.235 However, the effect was later studied by calorimetry,236 and it was shown that the disintegration is a result of the energy transfer from the accumulated strain to the kinetic energy of the debris; thus, the mechanical effect is caused by the latent strain accumulated during the λ-transition at 163.9 K. More examples have been reported in the literature since these pioneering publications. When heated above room 12474
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related through two phase transitions, I ↔ II at 30 °C (11 °C on cooling) and II ↔ III at 70 °C (40 °C on cooling).231,242−245 The crystals of this material hop when they undergo a phase change between forms II and III. The rapidly moving front was observed with needle-shaped crystals that were fixed at one end. On irradiation a sharp kink of several degrees developed along the needle axis, and the crystal was observed to shrink by over 10%. The molecular packing in the two forms is similar, and the molecules are only slightly offset with respect to each other. Expectedly, the largest conformational differences are observed with the acetyl and benzyl groups, which rotate to accommodate the new lattice. As the very complex conformational changes precluded drawing direct mechanistic conclusions, the effect was attributed to cooperative all-substituent motions. Much deeper insight into the mechanism of the thermosalient phenomenon was accomplished recently with the anticholinergic medicine oxitropium bromide (2t, Scheme 2).246,247 Depending on the crystal habit of the roomtemperature phase (blocky or prismatic), heating to 45 °C as well as during subsequent cooling to 27 °C, crystals of 2t can jump to roughly 2 cm, disintegrate, or simply expand. The thermosalient effect is due to a first-order phase transition between two phases that have identical crystal symmetry.246 Although the transition on heating is accompanied by a highly anisotropic cell expansion of 4%, where the b axis increases by 11% and the c axis is shortened by 7%, the molecular conformations in the two phases are surprisingly similar.247 The terminal groups are slightly offset, and the phenyl ring is more disordered in the high temperature phase. Having the two rigid fragments (the tricyclic moiety and phenyl ring) connected by a flexible linker (ester group), the thermosalient effect of 2t was qualitatively explained by drawing analogy with a shuttle (Figure 31). The heating induces significant strain in the structure that parallels compression of a spring. At the point of the phase transition, the stress is suddenly released by anisotropic expansion of the cell, and the packing switches to that of the high temperature phase. The sudden release of stress
Figure 29. Mechanism of the phase transitions and the thermosalient effect in (phenylazophenyl)palladium(II) hexafluoroacetylacetonate. The thermosalient transition occurs between forms α and γ. Reprinted with permission from ref 233. Copyright 2014 The Nature Publishing Group.
temperature, long prismatic crystals of 1,2,4,5-tetrabromobenzene (10t, Scheme 2) undergo hopping of several centimeters due to a phase transition from the room temperature β-form to the high-temperature γ-form.237 Single crystals undergo a transition at 45.0 °C, whereas twinned crystals transform at 45.5 °C, presumably due to stabilization of the β structure by twinning. If heated from one side with a point-type heater, the two components of the twinned crystals separate as a result of the bending that occurs during the phase transition of one of the crystal components (Figure 30).238 Both of these phases are
Figure 30. Twinned acicular crystal of 1,2,4,5-tetrabromobenzene splits along the twinning plane as a result of bending caused by the phase transition of the twin components. Adapted with permission from ref 238. Copyright 2013 American Chemical Society.
layered structures, and the hopping phenomenon is entirely governed by intermolecular Br···Br and C−H···Br interactions. Comparison of the two structures showed238−240 that the angle between the neighboring rings decreases over the transition from 22.6° in the β phase to 13.7° in the γ phase. This difference is indicative of slight tilting of the molecules and flattening of the molecular sheets, ultimately accumulating strain and causing the crystal to hop. The recent study with nanoindentation showed that these crystals are very soft (elastic recovery of 60%), which could be the key to susceptibility of the structure to internal pressure required for accumulation of strain for the thermosalient effect.238 The temperature dependence of the sound velocity and the elastic constant were measures by Brillouin light scattering.241 The results showed that the jumping is driven by the elastic instability and that the large intermolecular anharmonic interaction is associated with the molecular motions in the (110) plane. Three polymorphs (I, II, and III) of inositol (±)-3,4-di-Oacetyl-1,2,5,6-tetra-O-benzyl-myo-inositol (3t, Scheme 2) are
Figure 31. Structure of oxitropium bromide and mechanism of the thermosalient effect. (a,b) Molecule structures of oxitropium bromide in the low-temperature (a) and high-temperature (b) phase. (c,d) Mechanistic analogy between the molecule of the cation (d) and a mechanical spring (c).247 12475
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Figure 32. Schematic of the morphological and packing relationship between forms I and II of terephthalic acid. Adapted with permission from ref 226. Copyright 1994 Taylor & Francis.
to a reconstructive, first-order phase transition with increased disorder in the γ form and is reflected as the typical sawtooth profile in the DSC curve. The molecules pack in layers in both phases, where each molecule is surrounded by six other molecules. The number of the hydrogen bonds remains the same; however, in contrast to the other thermosalient transitions, there is no significant weakening of the hydrogen bonds. The difference is in the shape of the layers; they are slightly corrugated in phase β, but more planar in phase γ. There are a few additional examples of thermally induced jumping crystals where the phenomenon was noted, but not investigated further.254−257 A thermosalient effect was mentioned for 4,5-bis(fluorodinitromethyl)-2-methoxy-l,3-dioxolane (8t)254 and [36]-superphane (5t, Scheme 2).255 Crystals of the dioxolane 8t were reported to jump at 40 °C up to 1 cm high, and the effect was accompanied by development of cracks on their surface.254 The only significant intermolecular interactions in the structure before the transition are the O··· O contacts from the nitro groups, which connect the molecules within layers parallel to the bc plane, as well as pairs between these layers. Colorless crystals of the multibridged cyclophane 5t heated above 300 °C first start to “blink” and jump several times between 340 and 380 °C. The structure of the hightemperature phase and the mechanism of the jumping effect remain unresolved.255 The thermosalient effect of pyroglutamic acid (11t, Scheme 2) is particularly robust; the crystals can be propelled at least 15 times without apparent disintegration.258,259 Surprisingly, while crystals of both enantiomerically pure forms undergo the thermosalient phase transition and are motile, the racemate is not active, a result that indicates the long-term order that is required for accumulation of strain. The effect was correlated with the crystal faces and occurs only when the crystal expands perpendicular to its resting surface.258 On the basis of the above examples, some general characteristics of the thermosalient effect can be outlined. In general, the true thermosalient effect appearing as a result of a phase transition is a rapid, nearly instantaneous release of
from the crystals results in a forceful jump. As the new packing of the high temperature phase does not stabilize the new molecular conformation, the molecules relax back to a conformation similar to that of the low-temperature phase, with increased disorder due to the looser packing. When heated above 71.5 °C or subsequently cooled below 65.5 °C, the colorless needle-shaped crystals of trans,trans,anti,trans,trans-perhydropyrene (hexadecahydropyrene) (4t, Scheme 2) show strong movements and hop about 6 cm high, accompanied by a change of their polarization orientation.244,248 In the room temperature phase, the cyclohexane rings are in a chair conformation. Thus, the perhydropyrene molecules are packed in corrugated layers parallel to the ac plane that interact with each other through van der Waals interactions.248 The thermally excited molecules shift within the layers, thus contributing to the buildup of strain.249 At the transition point the internal stress is relieved, whereby the layers slide relative to each other and the crystal jumps. Crystals of polymorph II of terephthalic acid (9t, Scheme 2) heated to 70−100 °C (typically 90−95 °C) undergo a phase transition whereby they reorganize from a rhomboid shape into nearly rectangular plates (Figure 32).226 The morphological change occurs in two stages, where one-half of the crystal straightens up first, followed by the other half. Some crystals were observed to jump during the phase transition, although this phenomenon has not been described in detail.226 This transformation is generally reversible, and reoccurs on cooling to 30 °C. Fast reshaping of the crystals that did not transform can be also triggered mechanically by poking them with a metal point, thus indicating that local stress is required to trigger this transformation. Supposedly, the hydrogen-bonded organization of the molecules in parallel sheets is preserved on phase transition,250,251 and the sheets are displaced relative to each other. A thermosalient phase transition was recently reported between the high-temperature forms β and γ for the high energy material FOX-7 (12t, Scheme 2).252,253 The effect is due 12476
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pyroglutamic acid (11t), and the molecules 12t and 13t are enlisted as class III; they do not fit into either of these classes, and further development in this field is necessary to discover if there are other compounds with similar packing and characteristics. The hydrogen-bonding potential of these molecules is saturated by formation of hydrogen-bonded polymers or dimers. The available data on the structures (see Table 2 in ref 147) and the morphological changes that occur during the thermosalient transition indicate that the thermosalient effect in these three classes is associated with slightly different driving mechanisms. Accordingly, different consequences on the crystal integrity and morphology are observed. A common feature of all molecules is the presence of potential gliding planes devoid of strong interactions, either due to an absence of strong hydrogen-bond donors or due to an engagement of such functionalities in strong intramolecular hydrogen bonding. Most of the molecules of class I thermosalient crystals do not contain such groups, while the class II crystals have potential hydrogen-bond donors or acceptors that are “buried” by bulky groups. Therefore, they are effectively shielded from strong intermolecular interactions. At first sight, 2t, 9t, and 11t might appear as anomalies. Although the cation of 2t contains a hydroxyl group, the acidic proton of this functionality is engaged in a strong intramolecular hydrogen bond, and thus it is unavailable for intermolecular interactions.51 Crystals of 2t are thermosalient because the charges of the oxitropium cation and the bromide cancel, and thus the ion pairs effectively act as integral neutral units that are separated from each other during the phase transition. In effect, there is little change in the distance between the two opposite ions, despite the significant change in the distance between the ion pairs.247 The carboxyl groups in 9t (form II) and 11t form the typical symmetric hydrogen-bonded dimer,226 and therefore their hydrogen-bonding potential for additional intermolecular interactions is alleviated. These considerations indicate that an all-weak interaction environment at least along one direction in the crystal is a necessary prerequisite for large changes in the molecular disposition during the transition. Extended threedimensional hydrogen-bonding networks or strong intermolecular interactions are expected to act as a damper and absorb the strain in the crystal caused by heating. As a consequence, crystal motion will be suppressed. It should be noted that crystals of 9t, unless mechanically stimulated with a sharp object, tend to deform rather than jump.226 This correlates well to the different nature of the effect in this particular case. In accordance with the above classification, two main molecular mechanisms of the thermosalient phenomenon can be proposed. Heating thermosalient class I and class III crystals induces a small rearrangement between the layers of the flat molecules that are connected by very weak interactions; this cooperative effect results in an accumulation of strain between the layers. The strain outweights the forces between the layers and triggers sliding of the layers with respect to each other. Because of the pronounced structural anisotropy in these crystals, the sliding is observed as an anisotropic change of the unit cell. The anisotropic distribution of the local pressure during the transition and the limited molecular conformational flexibility render such structures susceptible to twinning. The transition in the thermosalient class II crystals, on the other hand, is mainly related to torsions of the flexible substituent groups. Heating to the point of the phase transition initially causes conformational changes, mainly in the flexible groups, leading to an accumulation of strain. When the packing
sizable mechanical strain that has accumulated in the crystal interior during the structural change. In effect, when micrometer- or millimeter-sized crystals are taken over the thermosalient phase transition, they suddenly hop to distances that can range from several millimeters up to a meter. This process is normally accompanied by anisotropic thermal expansion in the unit cell. The high-temperature phase usually has a cell volume up to several percent larger than the lowtemperature phase, and is thus less dense. The thermosalient effect usually (but not always) is accompanied by fracture, the development of cracks, and other surface imperfections. On the basis of the scarce data that exist on this effect and the limited number of characterized structures, it is now commonly accepted that the true thermosalient phenomenon is associated with sharp phase transitions and markedly large, anisotropic changes of the unit cell volume, tentatively related to the martensitic family of phase transitions. A martensitic phase transition is a first-order displacive solid-to-solid transition that proceeds with homogeneous lattice deformation and without atom diffusion.260−262 These transitions proceed by movement of the habit plane (the plane between the parent and product phase) induced by small cooperative movement of the atoms, while the overall chemical composition and atomic order are retained. Several examples of thermosalient-like effects were reported where the crystals burst, shift, or jump in response to desolvation or gas evolution, but the origin of the locomotion in these cases is different from the “true” thermosalient effect (vide supra). The displacement of jumping crystals from their original positions depends on the particular system, and can range from several millimeters to several centimeters. Occasionally, the movements can be colossal; recently, it was reported that small crystals of N′-2-propylidene4-hydroxybenzohydrazide (13t, Scheme 2) can jump distances of nearly 1 m, over 1000 times their own size!263 The processes related to plastic deformation and fracture that can account for crystal jumping with or without explosion have been considered in section 4. In the literature, one can also find attempts to systematically consider molecular and crystal structures of the compounds that exhibit thermosalient properties. The basic structural data pertaining to the known examples of thermosalient crystals are listed in Table 2 in ref 147. The majority of the structural information available is scattered and incomplete. The existing data have been obtained for chemically very different species, and oftentimes, due to technical obstacles, these materials have only been partially characterized. There are practical difficulties related to crystal degradation that render the structure determination of the high-temperature phase very difficult. Nonetheless, there are attributes common to all thermosalient materials that allow one to at least establish qualitative structure−kinematic inferences. With respect to molecular shape and intermolecular interactions, the known jumping crystals (Scheme 2) were classified4,147 into three main groups: rigid flat molecules and flexible molecules. In the former are molecules that are devoid of strong hydrogen-bond donors and pack in layers, mainly by π−π interactions, such as 1t, 4t, 5t, 6t (columnar packing with π−π interactions), 7t, 10t, 14t, 15t, 16t, 17t, and 18t (class I). The latter molecules contain a central fragment that is decorated with multiple bulky functional groups of low hydrogen-bonding potential such as 2t, 3t, and 8t (the hydroxyl group in 2t forms a strong intramolecular hydrogen bond and is thus unavailable for intermolecular hydrogen bonding; see below). Two small organic acids, terephthalic acid (9t) and 12477
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transitions underlying the thermosalient effect are not reconstructive and the locomotion is accompanied by crystal splitting, disintegration into smaller fragments, crystal deformation, explosion, twinning, and sublimation at high temperature. These obstacles render the detailed structural analysis of the thermosalient phase transition very difficult or practically impossible. Consequently, the existing experimental evidence on the mechanisms of related phase transitions is rather limited, and systematic correlations of the phenomenon with the structure have not yet been developed to allow prediction of this effect. The jumping effect is also known for metal−organic and for purely inorganic materials. The effect was recently reported for the complex [ZnBr2(2,2′-bipyridine)] (18t, Scheme 2), which on cooling from room temperature undergoes a transition from its monoclinic to a triclinic form.264 The original structural report of Mn0.98CoGe by Jeitschko,265 which undergoes a displacive phase transition around 67 and 32 °C on heating and cooling, respectively (the respective values for the stoichiometric compound MnCoGe are 125 and 185 °C), did not report any mechanical effects. Private communication with the author cited in ref 243, however, describes colossal jumps of the crystals of up to 30 cm. For inorganic systems, the phenomenon was studied in the spinels of nickel(II) (NiCr2O4) and copper(II) (CuCr2O4).266 These compounds are tetragonal (I41/amd) at room temperature,267,268 but undergo a transition to a cubic phase (Fd3̅m) at high temperature.65,66 The temperatures of the first-order phase transition are 47 °C (NiCr2O4) and 581 °C (CuCr2O4). Because the transition temperature of NiCr2O4 is close to room temperature, its crystals can be seen jumping several millimeters during illumination under a microscope.266 These two spinels undergo discontinuous variation of the cell parameters, although the relative cell change (4.5% and −1.36% for NiCr2O4 and CuCr2O4, respectively) is considerably smaller than the values observed for organic compounds. The thermosalient effect was related to the Jahn−Teller effect (Figure 33). As a recent example, one can mention a paper on first-order phase transitions in a series of AMoO4 compounds (A = Co, Mn, Fe, Ni, Cu, or Zn) studied by neutron diffraction.269 The phase transition is due to the higher compressibility coefficient, despite the presence of shorter bonds for the high-temperature form. The cell volume difference of 13% between the high- and low-temperature
subsequently switches to the high-temperature phase, the strain is released and the molecules relax back to their most energetically stable conformation. Because of the lower density of this relaxed structure, molecular conformations are similar to their original form. Accordingly, the high-temperature phase typically shows increased disorder in the bulky groups to compensate for the cell expansion. Because of the increased conformational freedom, it is more difficult to envisage the mechanism of class II phase transitions on the basis of the initial structure. As can be inferred from the preceding discussion, the two key characteristics of these transitions are their dynamics and the mechanical work that they are capable of generating. The mechanical work can be estimated by considering the volume change during the phase transition. There is no accurate information on the time scales at which these processes occur. In a complex process such as the thermosalient effect, the kinematics will vary according to the event (accrual of mechanical strain, jump, release of the strain, and structural relaxation). The spatial evolution of the phase transition front in 3t was observed within 0.1 s,243 while hopping of the crystals of 10t occurs in 0.04 s.237 By approximating the molecular flexibility to a single degree of freedom, the estimated duration of the phase transition step in 2t is on the order of 10−7 s.247 The above discussion can provide very qualitative guidelines in the search for new thermosalient solids. Regardless of whether the molecules are flat, rigid, and packed in weakly interacting layers (classes I and III) or they are bulky, flexible, and shielded from strong intermolecular interactions by bulky substituents (class II), they should be devoid of strong intermolecular bonds and be capable of only very weak, preferably nondirectional interactions. The phase transition of such molecules should be accompanied by an anisotropic change in the cell volume. Along these lines of reasoning, it can be anticipated that there are many more thermosalient systems in addition to the few known examples. One of the reasons for the small number of reported cases may be the lack of interest among solid-state researchers to use optical microscopy (e.g., supplemented with a hot stage) to visually observe the changes in color and morphology that sometimes accompany solid− solid phase transitions. Indeed, phase transitions are becoming routinely characterized by “bulk” analytical techniques, such as DSC and PXRD, which render such effects unobservable. The second reason may be that despite the occasional observation of the jumping effect, such information is not always reported, either due to lack of more thorough characterization of the underlying process, or because the mechanical effect was considered irrelevant. The anomalous expansion (of thermosalient solids) is accommodated by shrinkage along some of the axes. The potential practical applications of the thermosalient effect for direct conversion of thermal energy into mechanical work in organic-based actuators turn these solids into important materials for future technologies. Jumping crystals represent the most visually impressive demonstration of the macroscopic effects caused by the collective action of relatively weak intermolecular interactions at the molecular level. Indeed, these materials are capable of exerting mechanical force that can develop as a result of accumulating strain within the interior of single crystals. The few examples that are documented are precious cases and provide the opportunity to acquire a detailed understanding of the expression of molecular structural changes at the macroscopic level. Regrettably, in most cases, the phase
Figure 33. Geometrical changes in the NiCr2O4 octahedra during the phase transition. Thin lines, cubic phase; thick lines, tetragonal phase. Reprinted with permission from ref 266. Copyright 1997 The Royal Society of Chemistry. 12478
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Scheme 3. Molecular Structures of Photosalient Crystalsa
a
Ligand abbreviations: 4-spy = 4-styrylpyridine; 3F-4spy = 3′-fluoro-4-styrylpyridine.
macroscopic effects will depend on the orientation of the interface in the crystal, on the interaction between the crystal and the support, and on many other factors. It will not, however, require a temperature gradient in the crystal. The first reported example of the few reports that exist on the photosalient effect seems to be that of α-santonin (1p, Scheme 3).270−272 When exposed to sunlight, crystals of this compound undergo a complex multistep intramolecular rearrangement whereupon they were reported to “burst”. The high-speed recording of the effect showed that it is due to splitting of the crystals orthogonal to the b axis of the unit cell, presumably due to formation of a cage dimer.271 A similar effect was observed with a model meta-GFP chromophore (3p, Scheme 3).213 At low UV intensity, nonfixed crystals of this material bend >90°. However, when flashed or continuously irradiated with more intense UV light, the crystals spring off and jump a few millimeters as a result of acute lattice distortion. Because of the delayed release of the structural stress accumulated in the crystal lattice, the photomechanical effect is latent, and the motions may continue even after the excitation has terminated. A photosalient effect induced by cyclization was also observed with diarylethene crystals (4p, Scheme 3).273,274 Continuously UV-irradiated crystals initially underwent a photochromic change, but when the energy reached ∼10 μJ they typically jumped 1−2 mm nearly perpendicular to the long molecular axis. However, larger crystals did not jump and merely cracked under the accumulated strain. On the basis of
forms leads to the thermosalient effect that is associated with fracture along a certain crystallographic plane. 5.3.3. Photosalient Effect. By analogy with thermally induced crystal jumping, the term photosalient effect has been proposed to demarcate the sudden hopping of crystals when they are exposed to UV or visible light. As in the case with thermosalient effects, the origin of this phenomenon is related to the generation of mechanical stresses in the crystal. These stresses accumulate during an induction period due to structural strain before critical conditions of their relaxation via dislocation glide, twinning, or/and fracture are achieved (see section 4). The jumping occurs as a result of instantaneous release of the colossal strain that has accumulated in the crystal interior. Some of the immediately apparent requirements for the occurrence of the photosalient effect are anisotropic changes in the lattice and a very short time interval for response. Despite a large similarity in the manifestations of the thermo- and photosalient effects, there is an important difference in the underlying mechanisms. For photosalient phenomena, the nonuniform distribution of the reaction product in the crystal bulk that results from light absorption is the main source of strain, and is of the utmost importance for the interpretation of the observed macroscopic phenomena. In contrast to that, thermosalient effects can also be observed on uniform heating. The product phase nucleates, and the reactant−product interface propagates rapidly through the crystal. A rapid structural transformation is related to large shear strain and shear stresses. Under these conditions, 12479
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Figure 34. Photosalient effect of four prismatic crystals of [Co(NH3)5(NO2)]Cl(NO3) undergoing linkage photoisomerization from the nitro to the nitrito form under strong UV light. The five basic kinematic effects are shown. The snapshots were recorded at 0.25 ms intervals. Reprinted with permission from ref 146. Copyright 2013 Wiley-VCH.
crystals of this compound irradiated with strong UV light jump when they transition to the yellow-emitting polymorph. The transition is accompanied by shortening of the intermolecular aurophilic bonds. More recent studies on a series of paddlewheel complexes [Zn2(benzoate)4(L)2] having substituted fluorostyrylpyridine ligands, which can undergo photodimerization (8p, 9p in Scheme 3), showed that the photosalient effect could be a more common phenomenon than it has been thought in the past.278,279 When crystals of these materials are exposed to even weak UV light, they explode and pop violently.278 An analysis of partially polymerized crystal showed that even after ∼21% polymerization, a significant strain is accumulated and the volume expands so that the crystals pop out (in air) or crack (in oil). In an extensive study of another series of silver complexes AgL2X2 (L = 4-styrylpyridine, 2′-fluoro-4-styrylpyridine, and 3′-fluoro-4-styrylpyridine, X = BF4−, ClO4−, and NO3− ; 10p−15p in Scheme 3), six out of nine compounds were photosalient.279 The photosalient behavior, which is due to rapid volume expansion during the photoreaction, is determined by the ligand and the crystal packing. 5.3.4. Other Mechanical Effects. Other various mechanical effects, such as creeping or trembling, jumping with
the development of irreversible surface cracks, the effect was related to Grinfeld surface instability.275,276 We have recently observed a strong photosalient effect with linkage isomerization in crystals of [Co(NH3)5(NO2)]Cl(NO3) (5p, Scheme 3).146 The miniscule change in the coordination of the nitro ligand (nitrogen to oxygen) triggers a vast macroscopic mechanical response. If subjected to UV light, blocky crystals of this compound leap distances of over 105− 106 times their own size to release the colossal amount of strain that accumulates in their interior during the photochemical reaction. Individual kinematic analysis of over 200 crystal blocks from snapshots recorded at 250 ms showed that the photosalient effect in this material results in separation of debris, splitting, and explosion (Figure 34). The debris separates in less than 0.1 ms, and this event often propels the crystal into a ballistic spinning motion around its barycenter in the direction of the longest crystal axis. The movement is preceded by a period of latency of 0.6−4.5 s during which the strain is accumulating. It was found that this accrual period is determined by the volume of the crystals, and that the probability of crystal explosion increases with the stress accrual time. A photosalient effect was also recently reported for a gold(I) isocyanide complex (6p, Scheme 3).277 Blue-emitting 12480
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of these materials as mechanical actuators for rapid and efficient conversion of light or thermal energy into mechanical work, the emerging concept of single-crystal-based machinery is posed to continue to develop and expand. One of the ultimate goals in this emerging field is to build crystals from molecules that have been structurally “preprogrammed” to collectively respond to mechanical, electric, magnetic, or photonic stimuli in a predictable and determinable manner, fulfilling specific functions. Organic-based actuators that act as prototypic biomimetic and technomimetic crystals also pose a major challenge to fundamental studies in the field of crystal engineering. The thermal technomimetic or biomimetic single crystal actuator, in which a solid−solid phase transition occurs with significant reshaping of the crystal habit as the system is heated or cooled, can be incorporated in a number of devices. Such devices may include thermally actuated valves, switches, latches, clamps, and other dynamic elements in motion-control devices. Similarly, piezoelectric actuators can utilize an electric field coupled to a lever system to amplify the spatial displacements to technologically useful magnitudes. Thermal actuators rely on a temperature gradient or temporal variation in temperature to generate mechanical force to push/pull and open/close objects, or to move a load. Photochemically active counterparts provide additional advantages with spatially resolved, tunable, and contact-free operation. They are driven by nonuniform excitation and/or anisotropic response of the structure to the excitation. Light-induced changes in the shape of crystals are qualitatively and quantitatively comparable to, or superior than, those known for polymers and gels. These changes in crystal morphology appear as contraction, bending, or volume changes and are increasingly being reported, even for common photochromic compounds that have been known to chemists for centuries. Because of their high power density, better phase transfer accuracy, reliability, and instant thermal/photo-to-mechanical energy transfer, crystalline thermosalient and photosalient solids that undergo sudden structure changes that induce visually impressive movements certainly pave the way to a new generation of actuating materials. The crystals that reversibly bend upon exposure to light form yet another family of mechanically responsive solids that promises exciting endeavors. As some of the future prospects in this field, single crystalline materials that show a shape-memory effect could be designed to be deformed and fixed into a temporary shape and then recover their original, permanent shape only after exposure to an external stimulus. Considering the scientific and technological relevance of such materials in sensing applications, single crystals that behave similar to shapememory polymers are likely to become a major breakthrough in this research field and are likely to replace them in the future. An increase in temperature, heating on exposure to an electric current, or light illumination of a single crystal could activate a thermally induced shape-memory effect. Although counterintuitive, there is an increasing body of evidence that single crystals can exhibit impressive elastic properties. The elasticity of such single crystals could be complemented with the ultrafast energy transfer in the solid state to arrive at rapidly responsive actuators. Single crystals can be also cast into biomaterials to afford a rubber-like material with high resilience at low strain and that would behave similar to shock-absorber-like materials at high strain, by effectively dissipating the mechanical energy. Properties comparable to the passive elastic properties of the
rotation, violent fragmentation, or surface cracking, etc., which are reported in the literature as curiosities, can usually be reduced to the previously considered cases of elastic deformation (changing crystal shape) and plastic relaxation or fracture (Table S1). The mechanics of these phenomena have been described in detail in section 4. Expansion upon mechanical stimulation (without hopping),280 deformation,281 delamination,83 exfoliation, and layering282 can also occur upon irradiation, even in the absence of chemical reaction. Microribbons from asymmetric perylene diimide, for example, exhibit lateral and longitudinal macroscopic change under laser irradiation (Figure 19c,d).282 When excited at 488 nm, the stacked layers in these crystals undergo impressive lateral sliding, which is observed as an apparent increase in thickness of the crystals. The reverse sliding can be induced by exposure to an electron beam in the SEM, whereupon the original shape of the photodeformed crystal is restored. Single crystals of the charge transfer complex [Cu(TCNQ)] (TCNQ = 7,7,8,8-tetracyanoquinodimethane) can expand along the stacking axis by irradiating it normal to the axis due to partial charge transfer [Cu+(TCNQ−•)] ↔ Cu0 + TCNQ0.283,284 Observation of this process by ultrafast electron microscopy285 shows that by expanding the two portions of a split micrometer-long crystal, the width of a “gate” between two portions of the same crystal that was previously created by applying a shock with a pulsed laser can be reversibly modulated. At low laser fluence, the “gate” can be opened and closed by increasing the lattice spacing in the two crystalline pieces by excitation with femtosecond pulses. Higher fluence results in loss of crystallinity and phase separation of the domains of metallic copper and strands of neutral TCNQ. By measuring the frequency of motion modes by time-resolved electron microscopy and using the Young’s modulus, the force and energy were determined for cantilevers fabricated from this material.286
6. CONCLUSIONS AND FUTURE PERSPECTIVES The target-oriented design and preparation of molecules that mimic mechanical motors capable of movement has already been accomplished within the realm of supramolecular chemistry. Solution-based molecular motors that convert light or chemical energy into directional rotary or linear motions have been prepared. However, to perform useful work at a macroscopic scale, the design and fabrication of functional solid-state supramolecular systems that are active as macroscopic objects (on a surface or in the bulk) is essential. In the past, this has mainly been accomplished with elastomer-based multistable materials, which are normally endowed with the advantage of facile preparation and good processability. Although the understanding of how elastomeric or liquidcrystalline actuating materials operate has already reached maturity, the most recent attempts to quantify the performance of these materials have unraveled limitations with the inherently low coupling between the light/thermal energy and mechanical responses and associated long response times. It has become evident that a substantially new approach has to be taken to push the limits of performance toward efficient, controllable, rapid, and fatigueless mechanical responses, maintaining very high coefficient-of-performance characteristics. The material presented in this Review illustrates the growing interest of the solid-state chemistry research community in the mechanical properties and corresponding macroscopic effects of molecular crystals. In light of the prospects for applications 12481
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human muscles are anticipated, and such composites could serve as substitutes for muscle-mimetic biomaterials. Their mechanical properties could be tuned by adjusting the composition of the elastomeric proteins, thus leading to versatile biomaterials that mimic different types of muscles. Among other things, these biomaterials could be utilized as scaffolds in tissue engineering and as a matrix for artificial muscles.
Solid State, the Advisory Committee of CrystEngComm, and Associated Editor for RSC Advances. Dr. Naumov is a recipient of the Friedrich Wilhelm Bessel Research Award from the Alexander von Humboldt Foundation, the Asian and Oceanian Photochemistry Association Prize for Young Scientists, and a Global Center of Excellence Fellowship.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemrev.5b00398. Table S1, main types and features of mechanical response observed with photoinduced solid-state transformations; Table S2, selected theoretical models describing strain induced by phototransformations in thin needle-shaped crystals; and additional references (PDF)
AUTHOR INFORMATION Dr. Stanislav Chizhik earned his B.S. from the Department of Physics of Novosibirsk State University in 1993. He then began working at the Institute of Solid State Chemistry and Mechanochemistry, Siberian Branch of the Russian Academy of Sciences, where he obtained his Ph.D. degree in solid-state chemistry in 2001. He continues to work at the Siberian Branch Russian Academy of Sciences and is currently employed as a Senior Researcher in the Laboratory of NonEquilibrium Solid-Phase Systems. His research interests are focused on the theoretical studies of mechanisms and kinetics of solid-state transformations.
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest. Biographies
After acquiring his Ph.D. in chemistry and materials science from Tokyo Institute of Technology in 2004, Dr. Naumov, who is originally from Macedonia, continued his research in the National Institute for Materials Science in Japan where he established a laboratory for the study of solid phenomena. In 2007, he was appointed as an Associate Professor at Osaka University, where he led a small but very active research group, and after a short stint at Kyoto University, in 2012 he became an Associate Professor at New York University’s new campus in Abu Dhabi (NYUAD). He also has an external appointment as an Associate Professor with the Sts. Cyril and Methodius University in Macedonia. The research in Naumov’s group at NYUAD is in the domain of structural and solid-state chemistry, photochemistry, and materials science, especially focusing on smart materials. His publication portfolio includes over 150 publications that have been cited more than 2200 times, with an h-index of 22. He has been a member on the review panels with the NSF, ERC, ACS-PRF, he is an active reviewer for more than 45 journals published by the NPG, ACS, Wiley-VCH, RSC, and Elsevier, and is a member of the Advisory Committee of International Conference on the Chemistry of Organic
Dr. Manas K. Panda is from West Bengal, India, and he received his preliminary education there. He obtained his Ph.D. in Chemistry (2010) from the Indian Institute of Technology Bombay, India. After a short stay at Dow Chemical International R&D lab in Pune, India, he moved to the University of Crete, Greece (2011−2012) for a postdoctoral research. He joined New York University Abu Dhabi as a Postdoctoral Associate in Panče Naumov’s laboratory in 2013, and in 2014 he was promoted to Research Scientist. His research projects in Prof. Naumov’s group are in the domain of solid-state chemistry and smart materials. Specifically, his research has been focused on solidstate behavior of organic/inorganic molecules upon exposure to various external stimuli, such as light, heat, pressure, sound, and humidity. 12482
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effects, mechanochemistry, and high-pressure research to crystal growth and design, polymorphism, and physical pharmacy.
ACKNOWLEDGMENTS This work was supported by New York University Abu Dhabi (P.N., M.K.P., and N.K.N.) and a grant 14-03-00902 from RFBR (S.C. and E.B.). The authors from Novosibirsk thank Dr. A. P. Chupakhin, Dr. A. A. Sidelnikov, and Dr. A. A. Matvienko, with whom they have been collaborating on this topic since the 1980s, for the valuable discussions on this topic. We thank Dr. Patrick Commins and Adam Michalchuk for reading the manuscript and language correction. The Chemical Abstract codes of the older publications published in Russian language are available from the authors upon request.
Dr. Naba K. Nath obtained his Ph.D. degree from the University of Hyderabad in India (2012). He worked as Postdoctoral Associate with Panče Naumov at Kyoto University Japan (2012−2013) and New York University Abu Dhabi (2013−2015). Currently he is working as an Assistant Professor in the Department of Chemistry, National Institute of Technology Meghalaya, India. His research interests include the investigation of solid-state forms of pharmaceuticals, stimuli-responsive crystalline materials, and X-ray diffraction.
REFERENCES (1) Abakumov, G. A.; Nevodchikov, V. I. Thermomechanical and Photomechanical Effects in the Crystals of Complexes with Free Radicals. Dokl. Phys. Chem. 1982, 266, 1407−1410. (2) Boldyreva, E. V.; Sidelnikov, A. A.; Chupakhin, A. P.; Lyakhov, N. Z.; Boldyrev, V. V. Deformation and Mechanical Fragmentation of the Crystals [Co(NH3)5NO2]X2 (X = Cl, Br, NO3) in the Course of Linkage Photoisomerization. Dokl. Phys. Chem. 1984, 277, 893−896. (3) Ivanov, F. I.; Urban, N. A. Mechanism of Photomechanical Deformation of β-Lead Azide Whisker Crystals. React. Solids 1986, 1, 165−170. (4) Nath, N. K.; Panda, M. K.; Sahoo, S. C.; Naumov, P. Thermally Induced and Photoinduced Mechanical Effects in Molecular Single CrystalsA Revival. CrystEngComm 2014, 16, 1850−1858. (5) Kinbara, K.; Aida, T. Toward Intelligent Molecular Machines: Directed Motions of Biological and Artificial Molecules and Assemblies. Chem. Rev. 2005, 105, 1377−1400. (6) Balzani, V.; Clemente-León, M.; Credi, A.; Ferrer, B.; Venturi, M.; Flood, A. H.; Stoddart, J. F. Autonomous Artificial Nanomotor Powered by Sunlight. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 1178− 1183. (7) Feringa, B. L.; van Delden, R. A.; Koumura, N.; Geertsema, E. M. Chiroptical Molecular Switches. Chem. Rev. 2000, 100, 1789−1816. (8) Yildiz, A.; Forkey, J. N.; McKinney, S. A.; Ha, T.; Goldman, Y. E.; Selvin, P. R. Myosin V Walks Hand-Over-Hand: Single Fluorophore Imaging with 1.5-nm Localization. Science 2003, 300, 2061−2065. (9) Carella, A.; Coudret, C.; Guirado, G.; Rapenne, G.; Vives, G.; Launay, J.-P. Electron-Triggered Motions in Technomimetic Molecules. Dalton Trans. 2007, 177−186. (10) Khuong, T.-A. V.; Nuñez, J. E.; Godinez, C. E.; Garcia-Garibay, M. A. Crystalline Molecular Machines: A Quest Toward Solid-State Dynamics and Function. Acc. Chem. Res. 2006, 39, 413−422. (11) A special issue of the Accounts of Chemical Research with a collection of impressive contributions was devoted to the subject: Acc. Chem. Res. 2001, 34, 409−522, 10.1021/ar0100881. (12) Kottas, G. S.; Clarke, L. I.; Horinek, D.; Michl, J. Artificial Molecular Rotors. Chem. Rev. 2005, 105, 1281−1376. (13) Badjić, J. D.; Balzani, V.; Credi, A.; Silvi, S.; Stoddart, J. F. A Molecular Elevator. Science 2004, 303, 1845−1849. (14) Muraoka, T.; Kinbara, K.; Aida, T. Mechanical Twisting of a Guest by a Photoresponsive Host. Nature 2006, 440, 512−515. (15) Gimzewski, J. K.; Joachim, C.; Schlittler, R. R.; Langlais, V.; Tang, H.; Johannsen, I. Rotation of a Single Molecule within a Supramolecular Bearing. Science 1998, 281, 531−533. (16) Zheng, X.; Mulcahy, M. E.; Horinek, D.; Galeotti, F.; Magnera, T. F.; Michl, J. Dipolar and Nonpolar Altitudinal Molecular Rotors Mounted on an Au(111) Surface. J. Am. Chem. Soc. 2004, 126, 4540− 4542. (17) Finkelmann, H.; Nishikawa, E.; Pereira, G. G.; Warner, M. A New Opto-Mechanical Effect in Solids. Phys. Rev. Lett. 2001, 87, 015501−1−015501−4.
Dr. Elena V. Boldyreva graduated from Novosibirsk State University in 1982. She received her Ph.D. degree in physical chemistry in 1988 and Dr. Sci. degree in solid-state chemistry in 2000 from the Institute of Solid State Chemistry and Mechanochemistry at the Siberian Branch of the Russian Academy of Sciences. Since 1982 she has been employed simultaneously by this institute and Novosibirsk State University. She is now Head of a research group and of the Chair of Solid State Chemistry. From 1990 until 1999, she spent several research terms abroad with grants from the DFG, BMBF, and fellowships from the Humboldt Foundation (Germany), from CNR (Italy), from the Royal Society (UK), and as a visiting professor (France). She is the author and coauthor of over 200 research papers in peer-reviewed journals that have been cited more than 3000 times and of several monographs, including “Reactivity of Molecular Solids” (Wiley, 1999) (with V. Boldyrev) and “High-Pressure Crystallography: From Novel Experimental Approaches to Applications in CuttingEdge Technologies” (Springer, 2010) (with P. Dera). In 2007, she was awarded a Prize from the European Society for Applied Physical Chemistry (Eurostar-Science) for profound research of structure and reactivity of organic and inorganic solids. She served as an elected member of the Executive Committee of the International Union of Crystallography (2008−2014) and is a member of the Advisory Committee of International Conference on the Chemistry of the Organic Solid State. Her research interests span from photomechanical 12483
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