7106
J. Phys. Chem. 1994,98, 7106-7115
FEATURE ARTICLE Energetics of Organic Solid-State Reactions: Piezomodulated Raman Spectroscopy, Modal Motion, and Anharmonicity in the 2,S-Distyrylpyrazine (DSP) Photoreaction N. M. Peachey and C. J. Eckhardt' Department of Chemistry, University of Nebraska-Lincoln, Lincoln, Nebraska 68588 Received: December 17, 1993; In Final Form: May 16, 1994'
The concepts of phonon assistance and chemical pressure have been advanced to better understand the collective nature of the energetics and dynamics of solid-state reactions. With the 2,5-distyrylpyrazine photoreaction, these hypotheses were examined by employing lattice dynamical calculations and Raman and piezomodulated Raman spectroscopy in the lattice mode region. Since modal motion and anharmonicity are important in substantiating these models, the analysis of this solid-state reaction focused on these aspects. The motions calculated by lattice dynamical methods are consistent with the results from the piezomodulated Raman spectra. The anharmonicity of the phonon modes was found to increase significantly following the initiation of the reaction. This was particularly evident for those modes having motions along the reaction coordinate. Chemical pressure is shown to be of great utility in explaining this solid-state reaction while phonon-phonon interactions are demonstrated to have no significant role.
I. Introduction The growth of interest in the chemistry of materials has given impetus to an area of the solid state that has been, until recently, not as prominent as other branches. Although solid-state chemistry is assumed by many to involve only the study of inorganic substances, an assertion supported by the inclusion of the solid-state chemistry subdivision in the Inorganic Division of the American Chemical Society, the area of organic solid-state chemistry has been steadily growing and demonstrating its importance during the last half of this century. This may be attributed to the potential offered by the vast pallet of organic chemical reactions and the exquisite tunability of organic molecules. Whether the concern is with nonlinear optical materials, organic metals and magnets, production of polymers of extended chain conformation, stability and biological activity of drugs or extension to problems in ordered films, liquid crystals, inclusion compounds, or aggregates and clusters, organic solidstate chemistry is proving its relevance. Although threads of study of reactions of organic crystals are traceable well into the nineteenth century, the first concerted investigationof organic solid-state reactions was initiated by G . M. J. Schmidt at the beginning of the postwar period. This work led to the formulation of the topochemical postulate' that "reaction in the solid state occurs with a minimum amount of atomic or molecular movement." This generated extensive studies on photoaddition chemistry in crystals, and not surprisingly, strong emphasis was placed on the structural aspects of solid-state reactions. The successes were many2 although, with further investigation, significant exceptions to the topochemical postulate were found. The topochemicalpostulate was expanded to include the idea of a reaction cavity,3 which is taken to be the volume in the lattice occupied by the molecules directly involved in the reaction. The identification of cases which did not conform to the topochemical postulate led to the notion of steric compressi~n.~ A .force ~ in the vicinity of the reaction site is posited such that it affects, not necessarily directly, the reacting species' atomic Author to whom correspondence should be addressed. published in Aduance ACS Abstracts, June 15, 1994.
a Abstract
0022-365419412098-7 106304.50/0
displacements in the direction of the reaction coordinate. While presenting a causal relationship, the concept was not formulated sufficiently precisely to permit analytical formulation. With the positive advances made through application of the topochemical postulate, topochemistry, the study of reactions in organized media wherein geometric factors define the reaction coordinate, became dominated by structural investigations. To a large degree, the crystal has been viewed only as a matrix that provides geometrical constraints to the reaction. Solid-state aspects, especially collective properties, have been largely neglected. However, Cohen and Schmidt' had warned that "Topochemical influences are expected to be dominant only in certain types of reaction: they are probably of minor importance when the reaction mechanism involves long-distance migration of electrons or of excitation energy..." Others, however, have investigated other aspects of reactions in organic crystals. Some of these were still largely structural as in the case of study of the role of defects on solid-state organic reactions6 of which more recent note, especially of impurities, has been made.' The role of collective states, perhaps the most distinguishing characteristic of solids, has until recently largely been the province of physicists.* However, more detailed consideration of the role of the lattice, particularly manifested as strain, has also been inve~tigated.~,~ Nevertheless, thedominant theme in the study of organic solid-state reactions has been structural. The phonon assistance modello was the first to consider the dynamics of the solid-state reaction and the role of collective excitations. There are two scenarios. The first is that of excitonphonon assistance. Here the electronic excitation causes a displacement of the molecule from its equilibriumposition, thereby creating a local lattice distortion. This, in effect, localizes the electronicexcitation. Both the distortion, which may be regarded as a local strain, and the localized excitation can assist in the photochemical reaction either through formation of an excimer intermediate or by deformation along the reaction coordinate. This process does not affect the identity of the individual molecules but is a purely lattice effect. Further, if the excited state is 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 29, 1994 7107
Feature Article
n
(c)
Figwe 1. (a) DSP molecule, (b) repeat unit of the oligomer and polymer,
and (c) two-stage photoreaction. sufficiently long-lived, the excitation can propagate through the crystal together with its accompanying local strain. An example of the electron-phonon mechanism that manifests itself in an unusual way is found in the crystal-to-crystal oligomerization of 2,5-distyrylpyrazine (DSP), which was originally used to make the case for such a mechanism." The monomer crystal photoreacts at excitation frequencies less than 25 000 cm-l to form oligomers, but at higher frequencies high polymer is formed12 (Figure 1). This wavelength dependence cannot be addressed by the topochemical postulate. We have shown13 that collective electronic interactions (excitons) cause an exciton branch of the lowest P* P transition to lie in the same energy region as the much more localized P* n transition. The latter causes a distortion which acts as a trap for the P* r exciton which thereby funnels energy into the localized region of the distortion and prevents growth of the high polymer. However, at higher energiesthe mechanism is no longer biexcitonic since the traps are not created and the "delocalized" *-excitons can continually provide energy to feed the growth of the polymer. The second process is the phonon-phonon assistancemechanism where certain lattice vibrations (phonons) present in the reacting crystal possess motions along the reaction coordinate and can assist in bringing the moieties or their reacting centers together to form product. These phonon modes are, by definition,1°J4 anharmonic. Like soft modes in crystals which cause a structural instability in the lattice to initiate a phase transition, the phononassisting modes should have large amplitudes of motion due to anharmonicoverdamping. It has been suggested that these modes serve the same function as molecular collisions do in the gas phase and that large amplitudes of vibrational motion of these modes assist in initiating the reaction.14 As thereaction proceeds,
-
-
+-
the phonon-assisting modes should shift to lower energy or "softenn as they are frozen out by the formation of product. Shifts in the lattice phonon frequencies have been observed in solid-state reactions of organic crystals.lSJ6 Recently, the measured and calculated vibrational motions for those modes whose frequencies undergo the greatest change in the photoreaction in DSP have been shown to be consistent with the motions expected for phonon assistance.15 In these experiments the changes of lattice frequencies in DSP crystals were measured as a function of the extent of reaction. Some frequencies were observed to begin changing at low conversion while others resisted any variance until approximately 60%of the crystal had reacted. Lattice dynamical calculations were used to obtain the modal motions associated with the frequencies showing the greatest "softening". These movements were found to be consistent with displacements that would enhance interaction of the reacting centers of the DSP molecules. However, the mere existence of vibrational motions and frequency shifts which satisfy the description of phonon-assistingmodes is not sufficientto establish the model. The observed frequency shifts may be only a consequence of the reaction arising, for example, because the motions are associated with the direction in which the anisotropic potential distorts most due to developing polymer, rather than a validation of the phonon-phonon assistance mechanism. In fact, Prasad et al.10,11.14 have viewed the DSP system as displaying only exciton-phonon assistance. The phonon assistance model was important for introducing dynamical considerations of the lattice into the vocabulary of organic solid-state reactions. It demonstrated that complete relianceon geometricconsiderationswas insufficienttounderstand mechanisms in reacting crystals. The hypothesis requires a quantitative formulation that permits some predictive capability. Further, the clear associationof modal motion along the reaction coordinate with mode "softening" has yet to be established. Despite this intense interest in organic solid-state reactions and the formulation of many useful qualitative models, quantitative descriptions have been essentially nonexistent. This may be attributed to a paucity of investigations on the role of collective states in the reactions of crystals and the difficulty of inventing more precise constructions of the above models. To better grasp the fundamentals of these processes, structural and moleculecentered information must be augmented by a thorough depiction of the crystal energetics and dynamics. Achieving this requires developmentand refinement of quantitative theories that account for both local perturbations caused by product molecules forming in the lattice and collective properties that determine the overall behavior of the solid. Since any solid-statereaction has properties of both a chemical reaction and a structural phase transition,17 the process may be thought of as resulting from a convolution of thereactionenergysurfacewith theoverall latticepotentialenergy. Any theoretical construct which attempts to adequately describe such processesmust account for theeffectsofthese twointerrelated entities. Such a model has been recently formulated by Luty and Fouret.18 Developed within a generalizedsusceptibilityformalism, the concept of "chemical pressure" is employed to describe the effect of the product guest molecules upon the host reactant lattice potential. The model quantitatively describes the static strain that interacts with the lattice to influence the continuingcreation of product within the locally perturbed crystal. Initially, the elasticity of the lattice is able to accommodate the local strains originating from the guest product molecules, and the reactant lattice potential, although perturbed, is largely unchanged. As the product grows within the lattice, however, the localized strain fields coalesce as they begin to interact, directly and indirectly. At this point, the original lattice potential is no longer able to accommodatethe growing strain and evolution into a new potential reflecting the nature of the product molecule interactions is
Peachey and Eckhardt
7108 The Journal of Physical Chemistry, Vol. 98, No. 29, 1994
observed. The indirect interaction is mediated by phonons, and the crystal energy decreasesby an amount equivalent to the lattice deformation energy. Depending upon the magnitude of the difference between the product and lattice potentials, the change can be either catastrophic, leading to phase separation, or continuouswhere solid solutionsexist in the transformation from the pure reactant to pure product phase. In this manner, the chemical pressure model quantitatively addresses both the local characteristics of the reaction and the resultant collective consequences of the local perturbations. Recent investigations have demonstrated its value in describing the prototypical DSP crystal-to-crystal reaction.16 The earlier models mentioned previously have not lent themselves easily to quantitative formulation although they have provided invaluable conceptual frameworks for considering reactions in the solid state. Albeit these ideas historically preceeded the chemical pressure model, these earlier concepts can be regarded as specific extensions of the chemical pressure hypothesis. Since the phonon assistance model addresses the collective nature of the crystalline solid, it can also be related to the chemical pressure hypothesis. This hypothesis is useful for understanding solid-state reactions since it is predictivein nature, is amenableto experimentalverification, and unifies many earlier models. Indeed, by extending the chemical pressure hypothesis to the more general case where the lattice potential is directionally dependent and the resulting elasticity is anisotropic, phonon modes would be expected to be differently affected by the reaction at a given fraction of conversion. Thme modes having displacements along the reaction coordinate should be more directly influenced by the growth of product in the lattice and thus display significant frequency shifts. Furthermore, these shifts should occur quite early in the reaction. To investigate the role that the phonon modes play in solidstate photoreactions, the anharmonicity and, most importantly, the motions of those modes that fit the description of phonon assistance must be probed. Recently, this has become experimentally possible with the development of piezomodulated Raman spectroscopy (PRS).19,20 This spectroscopic technique probes the coupling of acoustic phonons with the optical phonons observed in the conventional Raman spectrum and allows the quantification of such coupling by measuring the anharmonic coupling constant. Since coupling can only occur for anharmonic phonons, the magnitude of the measured coupling constant is a gauge of the anharmonicity of the phonon mode being investigated. Should the modes that display the greatest frequency shifts early in the reaction be truly phonon-assisting, they should be very anharmonic even in the unreacted crystal. Alternatively, if these shifts are only consequences of the chemical pressure affecting the anisotropic lattice potential, the modes in question would be expected to display no significant anharmonicity until after the reaction has begun to occur. As these questions are addressed, an important feature of piezomodulated Raman spectroscopy is that it provides an independent experimental method of verifying the motions calculated by lattice dynamical calculations. Performing PRS measurements with the uniaxial stress along appropriatedirections allows experimental confirmation of the vibrational motion obtained through a lattice dynamical calculation. Since these motions are integral to phonon-phonon assistance, the information gleaned using PRS is important in understanding phonon assistanceand related mechanismsinvolved in solid-state reactions. The solid-state reaction used here to investigate the details of these two models is the DSP photopolymerization. This investigation focuses on the oligomerizationreaction since it is singlecrystal to single-crystal, thereby allowing the continuous monitoring of the phonon modes throughout the courseof the reaction.
Following this introduction, a brief theoretical treatment of the thermodynamicsof solid-statereactions, the chemical pressure model, latticedynamicalcalculations, and piezomodulated Raman spectroscopy will be presented. Subsequently, the experimental aspects of piezomodulated Raman spectroscopy will be discussed, followed by an outlineof experimentalprocedures for purification, crystallization, and oligomerization of the DSP crystal. In section IV, the results of the piezomodulated Raman spectroscopy of both the monomer and a partially oligomerized crystal will be considered. In section V the focus is on the analysis of the experimental data and its importance in understanding and differentiating between the chemical pressure and phonon assistance models. 11. Theory
A. Homogeneity and Heterogeneity. The formation of the product in the reactant lattice is an obvious case of mixing, and the degree of heterogeneity or homogeneity of the reaction is of importance. For the former, the reaction would be expected to proceed preferentiallyfrom the initiation sites while, for the latter, the site of formation of the product is random and the reactant and product molecules form a solid solution. The situation may be treated in the same fashion as a crystal-to-crystal phase transition. For a solid, the Helmholtz free energy of two phases, F Iand Fz, must be equal at the transition temperature. This means that
where, for the transition to a high-temperature phase, the entropy change will, with rare exception, be positive, thereby requiring a positive change in the internal energy. A phase transition may be regarded as always being associatedwith a solid-state reaction, but additional energy changes must also be considered. Chief of these is the free energy of the reaction which will be an important, if not dominant, contribution. Generally, a product crystal that has been formed by reaction will be left in a metastable state. Since the product has been formedin a latticewithan incompatible potential, the process is often kinetically instead of thermodynamically driven. The total free energy of the solid-state reaction, which is the sum of the intermolecular and intramolecular contributions, determines whether the reaction will be a single-phaseprocess or will undergo phase separation. Since the reaction itself will necessarily lead to a lowering of the free energy regardless of which process prevails, the determinative interactions will be intermolecular. A homogeneous reaction will occur when the reactant-product interactions are lower in energy than found in the pure reactant or product lattices; otherwise the reaction will be heterogeneous. If, however, the difference in energy between the homogeneous and heterogeneousphase behaviors is small, it is possible that the reaction may display single-phase behavior because of kinetic control. B. Cbemical Pressure Model. Basic considerationsof the free energy changes associated with a solid-state reaction provide the basis for formulation of a quantitative model for the chemistry of crystals. As noted above, the isothermal reaction of a solid must necessarily involve a change in the Helmholtz free energy, F,and this will normally be accompanied by a volume change. This may be represented as (tlF/tlV)~,which is clearly a pressure. The local perturbations developing in the crystal lattice may be regarded as generating a “pressure” which affects the stability of the solid. These perturbations may be chemical and may cause local symmetry breaking in the lattice, leading to local strain fields.IE As the reaction proceeds in the crystal, the local strain fields with eventually couple to effect a global or collective transformation. The nature of the transformation, and indeed
The Journal of Physical Chemistry, Vol. 98, No. 29, 1994 7109
Feature Article the course of the reaction itself, is markedly influenced by the accumulated strain within the lattice. The growth and propagation of the local strain fields are the defining characteristics of the chemical pressure which brings about the solid-state transformation. The change in the elastic constants of the lattice, which themselves have the dimensions of energy/volume, reflects this evolution from the reactant to the product lattice. The chemical pressure model assumes a linear response of the local elastic field to the forming product molecules and considers only pairwise interactionsbetween the different sites in the crystal. In this case, the energy of the pure crystal comprised of reactant molecules (M) positioned at sites n throughout the crystal can be writtenla
Upon the introduction of product molecules ( P )in the lattice, the energy of the crystal becomes
[ PM(nn?
+ vPP(nn? - 2 P P ( n n 9 ] [ 1 - .(n)]
[ 1 - u(n?]] (2.2)
where u(n) is assigned a value of 1 if the site is occupied by a reactant molecule and 0 if occupied by a product molecule. The excess energy due to the introduction of product into the lattice is the difference between the energies of the mixed and pure reactant lattices, 1 AV(u) = - T { Z [ l 2,
- ~ ( n ’ ) ] [ F ~ ( n n-’ P) M ( n n ’ ) ] + [l - ~(n)]AvP~(nn’)[l - ~ ( n ? ] )(2.3)
where the substitution APp(nn? = pp(nn?
+ P‘(nn’) - 2 P P ( n n 9
(2.4)
has been employed. Thedifferencein theenergies YMP(nn’)and VMM(nn’) isderived from the dissimilarity of the chemical properties of the product and reactant molecules and is the source of the local chemical pressure. The resultant force acting on a particular site can be calculatedby obtainingthe first derivativeof the energy difference with respect to displacement w’(n),
a[PP(nn? - P‘(nn?] afl(n)
Within the perturbed lattice, each reactant molecule will experience a local elastic field resulting from the local strains. This local elastic field is
where 4(n,n’) is the 6 X 6 force constant tensor describing the interactions between the pairs of molecules. Since the force constant tensor is in general anisotropic, the local elastic field will be directionally dependent. With the assumption that the response of the molecule to the field will be linear and that the susceptibility of individual
molecules x 0 is largely unchanged by the perturbations in the lattice, the displacements become
~ ( n=) xo*h(n)
(2.7)
This equation is a mechanical analogue of the expression for a dipole, i.r, induced in a medium of polarizability, CY,by an external effective field, 6: p
= cu.6
where the molecular polarizability is replaced by
and the thermal average is taken with the local on-site Hamiltonian. The effects of chemical pressure and elastic screening can be best described by considering the Fourier transform of the local field,ls
where p(q,u) are the concentrationfluctuationsI8in Fourier space. The denominator describes the elastic screening of the chemical pressure. Since the elastic screening will be directionally dependent, its effects are represented by a tensor. In certain directions, the lattice potential may be less able to screen the chemical pressure and it will begin deforming in the region of these coordinates much earlier in the reaction. This should be evidenced in the phonon spectrum as frequency shifts of certain phonons early in the reaction. The phonons resulting from the portion of the lattice potential that has the least amount of screening are expected to display more dramatic frequency shifts than those derived from the region of the lattice potential energy surface in which the elastic screening is more effective. The numerator in eq 2.8 describes the chemical pressure as a consequenceof the lattice perturbations resulting from formation of the product molecules. Initially, the elastic screening sufficiently compensates for the strain introduced by the product perturbations, and the overall lattice potential is relatively unchanged. At this stage in the reaction, few shifts in phonon frequencies are expected. However, as the chemical pressure continues to increase in the lattice, the reactant lattice potential reaches a point at which it can no longer accommodate the growing strain. At this juncture, it radically evolves into the new product lattice potential. The remaining reactant molecules in the lattice now become the local guest perturbations, and the new lattice screening diminishestheir effect upon the overall lattice potential. The point at which the old reactant lattice evolves into the new product lattice can be observed in the lattice phonon mode frequencies. When this occurs, the vibrational frequencies will show dramaticshifts mirroring thechangesof thelattice potential. C. Lattice D y ~ m i c dCalculations. Since investigation of vibrational motions of the lattice phonons is important in the understanding of the proposed phonon assistance mechanism, the librations associated with the lattice phonon modes of DSP need to be evaluated. To investigate the vibrational frequencies and the motions associated with each of the lattice phonon modes of DSP,lattice dynamical calculations were performed2*in the harmonic and rigid-molecule approximations.22 The molecular interactions are accounted for by a summation of the pairwise atom-atom interactionsusing a modified Buckingham potentialz3 of the form
where ‘0 is the distance between atom i and a t o m j of different
7110 The Journal of Physical Chemistry, Vol. 98, No. 29, 1994
Peachey and Eckhardt
molecules. The potential was parameterized using "universal" parameters,*4 and a minimum potential energy crystal structure was calculated using a gradient minimization routine. The calculated lattice constants tend to be shorter than those observed experimentally since the calculations assume no vibrational motion. The forceconstantsat themolecular equilibrium positions were calculated by
The harmonic approximation of the internal energy of the crystal, (2.1 1) is used to construct the lattice dynamical problem where c$,@ is an element of the force constant matrix and u, is the displacement of the molecule a. The dynamical matrix is then obtained by solving the equations of motion. From the dynamical matrix, the eigenvalues giving the frequencies of vibrations and the eigenvectors which describe the symmetries and motions of the lattice phonon modes are obtained from the equation
where D(9) is the dynamical matrix, M is the mass-inertial tensor, U(q) are the displacement vectors, and w(q) are the angular frequencies of the phonons. D. PiezomodulatedRaman Spectroscopy. The piezomodulated Raman spectrum results from the response of the Raman-active optical phonon modes to a periodic stress applied to the crystal. The symmetries of the strain and the stress that generate it are identical. Since macroscopic strains can be related to the displacements and propagation directions of acoustic phonons in the limit of zero wave vector, the piezomodulation can be regarded as exciting sets of acoustic phonons of zero wave vector but the same symmetry. The coupling of the optical phonons with the acoustic phonons generated by piezomodulation causes the optical mode frequencies to shift relative to those in the unperturbed crystal. The spectrum obtained is a difference spectrum with an intensity determined by19
7
,
I
,
I
,
,
I
,
,
I
v
, , , , , , ,
ENERGY Figure 2. (a) Unperturbed Raman band (- - -) and the bands at compression(wwmp) and at extension (weXt) of the crystal. (b) Derivative-
like piezomodulated Raman band
(wat
- wcomp).
Q(oJ) = Q"" + Qj
(2.15)
where QCIj represents the elastic displacements of the molecules for a given strain and Qj is the instantaneous vibrational equilibrium position of the molecule. The perturbed Hamiltonian for the unit cell of volume u can then be written,
(2.13) where ZeXt(w) is the band shape of the mode when the crystal is extended and Zmmp(w) is its band shape when compressed. Should the uniaxial compression cause the Raman band to shift to higher frequency, the expected signal, which is detected synchronously with theapplied stress,takes the formof thederivative-like reponse shown in Figure 2. This is defined to be a positive phase responsive. Alternatively, upon compression, a band may shift to lower frequency. In this case, the observed response is similar to Figure 2 except that it is 180° out-of-phase, giving a negative phase response. To better understand the origin of the coupling of the acoustic with the phonon modes and the subsequent piezomodulated Raman spectrum, it is necessary to consider the Hamiltonian for the strained crystal. Since the stress applied to the crystal is relatively small, only changing the energy by about l%,25 the perturbation can be represented by 7f=7fo+%f
(2.14)
where 7f0 is the Hamiltonian of the undeformed lattice. The Hamiltonian of the perturbed lattice, 7ff,can be renormalized after the normal coordinates Q(0J) are separated into two parts,
where C,B,~is the effective elastic constant, woj is the angular frequency of modej , and the coupling between theoptical phonons, Qj, and the strain, e@,is described by 4,s~. The first two terms in eq 2.16 represent largely harmonic contributions to the crystal energy by the strain and the lattice vibrations, respectively. The final term describes the strain-induced coupling of the lattice phonons and is dependent upon the magnitude of the coupling constant c $ a B j j j which is a measure of the anharmonicity of the phonon mode.26 For a Raman resonance at 00 and a small strain, the PRS response may be represented as
where it is seen that a piezomodulation spectrum can only be obtained for those modes which exhibit anharmonic coupling. Since the c$,# is a measure of the anharmonicity, it is clear that the more harmonic the mode, the smaller the piezomodulation response will be. Because of this, the magnitude of the
The Journal of Physical Chemistry, Vol. 98, No.29, 1994 7111
Feature Article piezomodulated response need bear no necessary relationship to the intensity of the Raman band for the mode. The coupling constants 4 4 may be evaluated using a curvefitting procedure. The Raman response can be expressed as a Lorentzian line shape, 07
I(,) = A [(Wi
- ,2)*
+ ,2y2]2
(2.18)
where y is the full width at half-height of the Raman band and A is a constant which can be determined experimentally from the Raman band of frequency 00. Determining y from the experimental Raman band and fitting the calculated to the experimental peak toobtain A provide the values needed to evaluatethecoupling constant. Using eq 2.17 gives the corresponding expression for the piezomodulated Raman response as
where e,b is the strain in the crystal resulting from the applied stress. The piezomodulated Raman bands can be fit to the experimental bands using eq 2.18, from which values for Ae,p r$,# were obtained. 111. Experiment
A. Piezomodulation Spectroscopy. For the piezomodulation experiment the crystal is glued between a stainless steel and a piezoceramic (PZT) bar to which an oscillating voltage of 1OL 103 Hz is applied. This provides periodic, uniaxial, tensile, and compressive stresses to the crystal. Polarized excitation light obtained from either an argon ion or HeNe laser passes through a polarization rotator and impinges on a properly oriented crystal. The inelastically scattered light is gathered by a lens, passed through a polarization scrambler, and synchronously detected at the driving frequency of the piezomodulator. The photomultiplier photocurrent is accepted by a current-to-voltage follower which feeds the amplified signal to a lock-in amplifier. The angle between the incident and scattered light is 90'. Piezomodulated Raman spectra were obtained from DSP monomer and partially oligomerized crystals. The partially oligomerized crystals were irradiated until about 15% of the monomers had reacted. The monomer crystals were uniaxially stress-modulated in twodirections. First, stress was applied along the crystallographic c-axis while scattered light was obtained from the naturally occurring (010) face. Next, the crystal was stressed along the a-axis with the scattering direction normal to the (001) face which was obtained by cutting. The partially oligomerized crystals were only stressed along the c-axis since these results were only used for comparison with those from the monomer crystal to establish the change in anharmonicity in the reacting crystals. In order to accurately analyze the piezomodulated response, both Raman and piezomodulated Raman spectra were obtained from all the crystals. Since the crystals become increasingly brittle upon reaction, it was not possible to piezomodulate crystals for which the reaction had proceeded much beyond 15%. The PRS response was analyzed using a digital lock-in amplifier (SR850, Stanford Research) which was triggered by theoscillating voltage modulating the piezoceramic. The SRSSOhas two phasesensitive detectors (PSDs) with which to detect and analyze the signal response. The Y output of the first PSD is related to the signal voltage Vsi, by
Y = v,, COS e
(3.1)
where B is the phase difference between the signal and the lock-in reference. A second PSD is utilized which samples the signal
90' out-of-phase from the first, and its X output is
X = Vs&sin B Thus, Y is the in-phase component of the signal, and X is the quadrature component. These two PSDs can be used to remove the phase dependence of the signal to the lock-in reference by obtaining R from R = /X2
+ Y2
(3.3)
Thus R measures only signal amplitude and is insensitive to the phase difference. Since the PRS response is dependent upon the phase shift between the Raman band in the unstrained crystal and that of the strained, the spectrum is normally rccorded as the response from a single PSD. On occasion, however, the noise level of the PRS signal is too high to obtain a measurable spectrum in this fashion. Then the phaseinsensitive response, R, may be obtained. While this does not provide the phase-sensitive data required for the full analysis of the piezomodulated response, it does allow determination of the magnitude of the anharmonic coupling constant. Since previous experiments used lock-in amplifiers which contained only a single PSD,Io analysis of R for very weak PRS responses was not possible. However, implementation of a digital lock-in amplifier with dual PSDs makes the investigation of coupling constants at low-level signals feasible. B. DSPSynthesis,Purification,andCrystalCrowth. Synthesis of the DSP employed the procedures described by Hasegawa et al.27 The compound was recrystallized from xylene and further purified using column chromatography. Both sublimation and zone-refining were found to be inadequate as they reintroduced impurities from decomposition of the compound. Details of the purification are published e1~ewhere.l~ Crystals of sufficient size and quality for spectroscopic studies were grown by slow evaporation from saturated tetrahyrofuran or acetone solutions. Crystals were typically 2 X 2 X 1.5 mm in size. The photoreactive DSP crystal (aform) packs into the orthorhombic space group Pbca with four molecules in the unit cell.2* The crystals grow such that (100) is the large plate face with smaller (010) edge faces. The crystals were easily cut using a LKB-Produkter ultramicrotome (Bromma, Sweden) to enlarge the (010) face or to obtain the non-naturally occurring (001) face. Monomer crystals were oligomerized on a rotating stage using light from a 1OOO-W quartz halogen source. The light was rendered monochromatic by a (2-nm bandpass) Jarrel-Ash 82410 monochromator. Crystals being oligomerizedwere continuously rotated about an axis perpendicularto thea crystallographic axis toaverage inhomogeneities in the light sourceand polarization effectsof both the crystal optics and the monochromator. Greater average penetration of the light was achieved by illumination at a wavelength which was slightly lower in energy than the first absorption band in the crystal. The extent of oligomerization in the crystal was established using both mass spectrometry and comparativeintensitiesof the ethylene and cyclobutanevibrations in the Raman vibrational spectrum. Details of the method used to assess extent of reaction are published elsewhere.'3J6
IV. Results The factor group symmetry for the DSP unit cell is Du,and 12 Raman-active modes are expected: three each of A, [b(cc)a], B1, [b(ab)c], B 4 [b(ac)a], and B3, [a(bc)b] symmetry, where Port0 notation29 for associated experimental geometry is noted in brackets.30 Lattice dynamical calculations were performed for the DSP monomer crystal, and the results are tabulated in Table 1. The displacement vectors in Table 1 are axial vectors which represent rotational vibrations about an axis. Since the
7112 The Journal of Physical Chemistry, Vol. 98, No. 29, 1994
Peachey and Eckhardt
TABLE 1: Calculated Rotational Displacement Vectors and Calculated and Exwrimental Mode Freauencies for Monomer
0.3
DSP.
mode symm A, Big
Bz, B3,
0.78
0.2
calcd displacement vectors6 x-axis y-axis z-axis 0.027 503 0.011 509 0.004 170 -0.003 970 0.014 815 0.026 107 -0.026 105 0.009 622 0.011 580 -0.001 674 0.009 943 0.028 365
0.000 131 0.001 553 -0.005 515 -0.002 203 0.004 833 -0.002 866 0.002 884 0.002 787 0.004 136 -0.002 229 0.004 964 -0.001 885
-0.002 402 0.004 941 0.001 386 0.009 622 0.001 263 -0.000 368 0.000 363 0.004 594 -0.003 297 0.005 208 0.002 161 -0.000 572
calcd exptl freq (cm-l) freq (cm-1) 46.5 16 66.505 100.529 39.075 101.402 143.765 50.453 57.339 77.066 22.434 90.616 134.851
0.08
ar
47.0 66.0 78.0 39.0 96.0 135.0 41.0 54.0 84.0 30.0 90.5 136.0
rug
8
$
0
O
.a I
g
42
.0.08
D
30
40
50
0.4
0.08
In the coordinatesystemused for the axialvectors the DSP molecular long axis is the x-axis and that normal to the plane of the molecule is the z-axis (see Figure 3). Rotational displacementvectorsgiven in inertiallyweighted coordinates.
60
a
0.2
0.04
0
O
y-axis
82 8 g
u, .O.M
42
.0.08
-0.4
g
x-axis
20
30 40 ENERGY(cm")
50
0
ENERGY(cm")
Figure 4. Raman band (- - -) and the piezomodulated Raman band (-) of (a) the 47.0 cm-I A, mode, (b) the 39.0 cm-I B1, mode, (c) the 41.0 cm-1 Bzg mode, and (d) the 30.0 cm-' B3, mode. Raman and PRS intensities are on the same relative scale. Left ordinates show scales for Raman intensities and right ordinates for PRS response.
y-axis
x-axis
Figure 3. (a) Approximatemotion of the 47.0 cm-l A,and the 41.0 cm-l Bz, modes. (b) Approximate motion of the 39.0 cm-l B1, and the 30.0 cm-1 B3, modes. The relative size of the arrows depicts the relative magnitude of the motion about the particular axis.
DSP molecule is situated on an inversion site, only these librons are Raman-active. Examination of the calculated motions for the lattice modes allows several observations to be made. The axial displacement vectors for the lowest frequency A, and Bz, modes have similar magnitudes, and the same is true for the low-frequency B1, and B3, modes. The largest components of rotational motion for the low-frequency A, and Bz, modes are out-of-plane librations about the molecular long axis while the BI, and Ba motions are mostly in-plane librations about the axis perpendicular to the molecule (Figure 3). Two other modes whose vibrations are important to the analysis of the DSP reactions are the A, 66.0 cm-1 and Bzr 84.0 cm-1 modes. These have previously been shown to fit the description of phonon-assisting modes. l 5 The magnitudes and directions of their calculated motions are also strikingly similar with most, but not all, of their rotational motion being out-ofplane oscillations about the long axis of the molecule. Polarized PRS was used to analyze lattice phonon modes in each symmetry direction. These results can be used to confirm the motions obtained from the lattice dynamical calculations as well as provide a probe of the anharmonicity of the lattice modes.
Unfortunately, only the intense, low-frequency modes of each symmetrydirection yielded unambiguous piezomodulation results. The Raman and piezomodulated Raman spectra of these modes are shown in Figure 4. Since the anharmonicity of the 66.0 cm-l A, and 84.0 cm-1 Bzs modes are crucial to the understanding of the mechanism of the DSP photoreaction, it is important to assess their coupling constants. Although it was not possible to obtain phase-sensitive data for these two modes due to degraded signals, an evaluation of the magnitude of the coupling constant could be made using the phase-insensitive response (R). These spectra are shown in Figure 5 . Since the elastic tensor for DSP has not been determined, the strain in the crystal could not be evaluated. Nevertheless, since the largest components of the stiffness tensor are generally the diagonal elements, it was assumed that this is true for the DSP crystal. This allowed thedeterminationof thecoupling constants, which permitted comparison of the anharmonicity of the important modes in this crystal. The Raman-active modes for which only phaseinsensitive data were obtained were also analyzed in the manner described above. For these bands, the absolute value of eq 3.2 was used to calculate the magnitude of the coupling constant. This prevented the assessment of the motions of the modes but did allow analysis of the anharmonicity. The results of this analysis are tabulated in Table 2. V. Discussion
Even a casual consideration of the results of the piezomodulated Raman experiment for the low-frequency monomer crystal Raman bands (Table 2) shows that they agree qualitatively with the lattice dynamical results. The47 cm-' A, and 41 cm-1 Bz, modes, which calculations indicate have similar vibrational motions, mainly rotation about the x-axis, show the same phase behavior when stressed along both the a and c crystallographic axes, and
The Journal of Physical Chemistry, Vol. 98, No. 29, 1994 7113
Feature Article
.............................................
u
50
40
60
60
ENERQY
70
w
W
l
W
(m-9
Figure 5. Lattice spectrum including insets of the phase-insensitive(R) spectra over the 60-1 00 cm-I energy region of the monomer and the 15% oligomer of (a) the A, modes and (b) the B g modes. Raman intensities are relative. The 4 R spectra have been amplified 40-fold and the BzII R spectra amplified 30-fold relative to their respective Raman bands.
the same is true for 39 cm-I B1, and 30 cm-I B3, modes. The following detailed analysis confirms this observation. The outof-planelibrations indicated by the lattice dynamical calculations for the low-frequency A, and B2, modes would cause greater repulsive interactionswith neighboring molecules when the crystal is compressed along both the a and c axes (Figure 6). This explains the positive phase of the piezomodulated Raman spectra of the crystal stressed in both these directions since Z”p(W) > Zcxt(w). Calculationsindicate that the low-frequency BI, and BJ, modes are largely in-plane librations about the axis perpendicular to the plane of the molecule with little out-of-plane motion. Thus, compression along the c-axis, considering that other effects are less important, would have negligible influence on the intermolecular interactions. However, compressingthe crystal along the c-axis will cause elastic expansion along the other crystallographic axes. Such expansion will lessen the interactions between neighboring molecules for those vibrational motions that involve in-plane interactions, thereby allowing the mode frequencies to shift to lower energies so that Zcxt(o)> Zcomp(w). This is confirmed in the PRS spectrum by a negative phase of these two bands when the crystal is stressed along the c-axis. Compression along the a-axis, however, will increase the interactions for the molecules undergoing both out-of-plane and in-plane librations. In this case, all four modes are expected to display a positive-phase piezomodulated Raman band. This is observed. Consideration of the mode anharmonicity also supports the calculational assignmentsof librational motion. To compare the anharmonicity of the various modes calculated from the coupling constants, the constants must be divided by the square of the mode frequency. This normalization accounts for the greater coupling of a low-frequency mode with the oscillating stress.31 These values are included in Table 2, where error is expressed by the least significant figure. The increase in anharmonicity in the partially oligomerized crystal for the B1, and B3, modes is greater than that for the A, and Bz, modes. Sinceoligomerization involves an increase of the molecular length, librations about the molecular long axis should show less increase in anharmonicity
than librations about the axis perpendicular to the molecular plane. This further confirms that the motions indicated by the lattice dynamical calculations are in agreement with those of the molecules in the crystal lattice. Establishment of the credibility of the calculated librational motions permits elaboration of the role of phonon assistance in the DSP photoreaction. The phonon assistance hypothesis supposes that the modes involved in such assistance are necessarily anharmonic to allow for the large oscillations due to overdamping. The large anharmonicity of the phonon-assisting m d e can be a feature of the mode in the unreacted crystal or it may develop in the early stagesof the reaction. However, if the mode becomes anharmonic as the crystal reacts, the reaction would be expected to show autocatalytic behavior in order for phonon assistanceto be present. Should a phonon-assistingmode be operative in an autocatalytic reaction, it would imply that the reaction proceeds slowly until sufficient anharmonicity has developed in the lattice potential for the enabling overdampedoscillations to occur. Autocatalytic behavior is known for some of the diacetylene solid-state reactions.32 For reactions which are not autocatalytic, however, the phonon-assisting mode must be largely anharmonic in the unreacted crystal. The coupling constants for both the 66 cm-I A, and 84 cm-’ Bz, modes are listed in Table 2. In each case, these modes are quite harmonic, i.e. weakly coupled to acoustic phonons, in the monomer crystal. The values of 14aB/wiI for both these modes are considerably less than that for any of the other bands investigated in the monomer. However, this changes significantly in the partially reacted crystal where the 66cm-I A, mode increases in anharmonicity by an order of magnitude while the 84 cm-I B2, modeincreasesapproximately5-fold (Table 2). Thisis the largest increase of the coupling constant observed for any of the investigated modes, indicating that, by this stage in the reaction, both these modes have become quite anharmonic. The above analysisof anharmonicity relates the roles of phonon assistance and chemical pressure in the DSP photoreaction. We had previously observed that some phonons in the DSP crystal appeared to match the description of phonon-assisting modes.15 However, since the 66 cm-’ A, and 84 cm-1Bz, modes are relatively harmonic in the monomer crystal and the DSP reaction is not autocatalytic, these phonon modes cannot be phonon-assisting. Rather, the anharmonicitywhich developsas the reaction proceeds is a consequence of the reaction. The developing anharmonicity and frequency shifts observed for these modes result from the perturbations of the product molecules in the lattice. The conclusion is that, regardless of the lattice vibrational motion, phonon assistancedue to phonon-phonon coupling is not operative in this reactivesystem. That thesemodesbecomevery anharmonic upon reaction is not surprising from the viewpoint of chemical pressure. Since the librations of the molecules for the modes in question involve motions along the reaction coordinate, these modes should be affected more directly by the growing product in the lattice. The elastic screeningprovided by the lattice, being anisotropic, is expected to be least effective in this direction, and the lattice potential energy surface in this region is more directly affected by the influence of the molecular reaction. Contrasting the phonon assistancehypothesis with the chemical pressure theory clarifies the importance of the latter. Clearly, phonon assistance due to phonon motions along the reaction coordinates is not responsible for the reactivity of the DSP molecular crystal. Should it be found to be operative in a particular reactive system,it would not besufficiently established as a generally applicable mechanism. Indeed, what may appear to be phonon-phonon assistance could be a manifestation of the effects of chemical pressure. Since the lattice potential energy surface is expected to deform more dramatically in the region in which the reaction occurs, the lattice phonons with displacements
7114 The Journal of Physical Chemistry, Vol. 98, No. 29, 1994
TABLE 2
Piezomodulnted Ramnn Spectroscopy Results response phase
mode
freq (cm-1)
A,
47.0 66.0 39.0 41.0
BI, B2g
B3,
Peachey and Eckhardt
84.0 30.0
e-axis
a-axis
+
+ + + +
-
+
-
15% oligomer
monomer
b d (cm-2/103) 3.2
1.4 -2.3 3.0 2.9 -2.0
1bd4/w;I 1.5 0.3 1.7 2.1
0.4 2.5
‘$a#
(~m-2/103) 14d/wil 3.6 13.0
-4.0 4.1 13.0 -3.8
1.7 3.1 3.0 2.9 1.9 5.4
Ab,, (cm-2/103)
A[~,/o$
0.4 12.6 1.7
0.2 2.8 1.3
1.1
0.8
10.1 1.8
1.5 2.9
associated motions along the reaction coordinates, these modes were found to be quite harmonic in the monomer crystal. They shift to lower energy early in the reaction, as suggested by phonon assistance, but become anharmonic only after the reaction begins. Since the DSP reaction is not autocatalytic, this suggeststhat the observed shifts to lower frequency and increased anharmonicity are consequentialrather than causal and result from the chemical pressure of the developing product. This study demonstrates the broad applicabilityof the concept of chemical pressure and shows how it subsumesthe concept of phonon assistance which, because it is more restrictive, does not have general applicability. The combination of theoretical models that accurately rationalize solid-state reactions with experimental results is expected to yield new paradigms for understanding these processes. As important as knowledge of the geometrical structure and local perturbation behavior is for a solid-state reaction, it must be accompanied by an equally lucid understanding of the collective energetics and lattice dynamics associated with chemical transformationsin crystals. Such studiescan assist in opening avenues for better control, development, and utilization of solid-state reactions. Acknowledgment. N.M.P. thanks the University of Nebraska-Lincoln and the University of Nebraska Foundation for fellowship support. This research would have been impossible to execute without support from the Center for Materials Research and Analysis of the University of Nebraska-Lincoln, which is gratefully acknowledged. Figure 6. (a) Stress along the c crystallographic axis. (b) Stress along the a crystallographic axis.
along the reaction coordinates will experience frequency shifts early in the reaction.
VI. Conclusions The elucidationof solid-state reactions requires the refinement of models accurately describing the crystal’s energetic and dynamic interactions. The rudiments of such a model are presented in the chemical pressure hypothesis which, beginning from a local, microscopic level, defines how.perturbations in the lattice can lead to global transformations. This hypothesis has been tested by observing the Raman-active lattice vibrational modes throughout the DSP oligomerization reaction, and the experimental results were found to supportthe model qualitatively. Strengthened by experimental evidence, the chemical pressure hypothesis must be extended to quantitatively mirror the shifts in modal frequenciesobserved in the Raman spectrum. This will require a treatment which more completely accounts for the anisotropic nature of the lattice potential and elastic screening. Furthermore, the model needs to address the changing molecular susceptibilities during the rapid evolution of the product lattice potential after the elastic screening of the reactant lattice is no longer effective. Analysis of the development of anharmonicity in the lattice modes has allowed a more complete appraisal of the phonon assistancemechanism. Although phonons exist in DSP that have
References and Notes (1) Cohen, M.D.; Schmidt, G. M.J. J. Chem. Soc. 1964, 1996. (2) Schmidt, G. M.J.; et al. In Solid Srare Photochemistry; Ginsburg, D., Ed.; Verlag Chemie: Weinheim, Germany, 1976. (3) Cohen, M. D. Angew. Chem., Int. Ed. Engl. 1975, 14, 386. (4) McBride, J. M.Acc. Chem. Res. 1983, 16, 304. (5) Ariel, S.;Askari, S.; Scheffer, J. R.;Trotter, J.; Walsh, L. In Organic Phototransformations in Nonhomogeneous Media; Fox, M. A., Ed.; ACS Symposium Series 278; American Chemical Society: Washington, DC, 1985; p 243. (6) Thomas, J. M.Phil. Trans. R. Soc. 1971,277, 251. (7) Sandman, D. J.; Haahma, R. A.; Foxman, B. M.Chem. Mater. 1991, 3, 471 and references therein. (8) Davydov, A. S. Theory of Molecular Excironr; Plenum Press: New York, 1971. Kitaigorodsky,A. I. Molecular Ctystals and Molecules: Academic Press: New York, 1973. Califano, S.;Schettino, V.; Netto, N. Lorrice Dynamics of Molecular Crysrals; Springer-Verlag: New York, 1981, (9) Baughman, R. H. J. Chem. Phys. 1978,68, 3110. Hollingsworth, M. D.; McBride, J. M. J. Am. Chem. Soc. 1985, 107, 1792. (10) Prasad, P. N.; Dwarakanath, K.J. Am. Chem. Soc. 1980,102,4254. (11) Swiatkiewicz, J.; Prasad, P. N. J. Polym. Sci. Polym. Phys. Ed. 1984, 22, 1417. (12) Tamaki, T.; Suzuki, Y.; Hasegawa, M.Bull. Chem. Soc. Jpn. 1972, 45, 1988. (13) Peachey,N. M.; Eckhardt,C. J.J. Am. Chem.Soc. 1993,115,3519. (14) Prasad, P. N.; Swiatkiewicz, J. Mol. Crysr. Liq. Crysr. 1983,93,25. Prasad, P. N.; Swiatkiewicz, J.; Eisenhardt, G. Appl. Specrrosc. t e r r . 1982, 18,59. Prasad, P. N. In Crystallographically Ordered Polymers; Sandman, D. J., Ed.; ACS Symposium Series 337, ACS: Washington, DC, 1987. (15) Stezowski, J. J.; Peachey, N. M.;Goebel, P.; Eckhardt, C. J. J . Am. Chem. Soc. 1993, 115,6499. (16) Peachey, N. M.; Eckhardt, C. J. J . Phys. Chem. 1993, 97, 10489. (17) Dunitz, J. D. Pure Appl. Chem. 1991, 63, 177. (18) Luty, T.; Fouret, R.J . Phys. Chem. 1989, 90, 5696. (19) Luty, T.; Eckhardt, C. J. J. Chem. Phys. 1985, 82, 1515. (20) White, K.M.; Eckhardt, C. J. J. Chem. Phys. 1990, 92, 2214.
Feature Article (21) Born, M.;von Umh, Th.Z . Phys. 1912,13,297. (22) Vankataraman, G.; Sahni, V. C. Rw. Mod.Phys. 1970, 42, 409. (23) White, K. M.; Eckhardt, C. J. J. Chem. Phys. 1989,90,4709,1990, 92,2214. Kulver, R.; Eckhardt, C. J. J. Chem. Phys. 1988,90,4709. (24) Pertsin, A. J.; Kitaigorodsky,A. I. TheAtom-AtomPotentialMefhod; Springer-Verlag: Berlin, 1968; p 89. (25) White, K. M.; Eckhardt, C. J. Phys. Rw. Lcrt. 1987,59, 574. (26) Cowley, R. A. In The R u m Effcct; Anderson, A., Ed.; Marcel Dekker: New York, 1971; p 95, vol. 1. (27) Hasegawa, M.;Suzuki,Y.;Suzuki,F.; Nahnishi, H. J. Polym. Sci.
The Journal of Physical Chemistry, Vol. 98, No. 29, 1994 7115 A1 1%9,7,743. Nahnishi, H.; Suzuki,Y.;Suzuki,F.; Hasegawa, M.Ibid. 753. (28) Sasada, Y.; Shimanouchi, H.; Nakanishi, H.; Hasegawa, M.Bull. Chem. SOC.Jpn. 1971,44, 1262. (29) Damen, T. C.; Porto, S.P. S.;Tell, B. S.Phys. Reo. 1966,142,570. (30) Hayes, W.; Loudon, R. Scattering of Light by Crystals; Wiley-Interscience: New York, 1978. (31) White, K. M.Ph.D. Thesis, University of Nebrash, Lincoln, NE, 1987. (32) Baughman, R. H.; Chance, R. R. J. Chem. Phys. 1980, 73,4113.
