Photoisomerization Kinetics and Mechanical Stress in Azobenzene

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Letter

Photoisomerization Kinetics and Mechanical Stress in Azobenzene-Containing Materials Vladimir Petrovich Toshchevikov, Jaroslav M Ilnytskyi, and Marina Grenzer Saphiannikova J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b00173 • Publication Date (Web): 17 Feb 2017 Downloaded from http://pubs.acs.org on February 19, 2017

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The Journal of Physical Chemistry Letters

Photoisomerization Kinetics and Mechanical Stress in Azobenzene-Containing Materials

Vladimir Toshchevikov,1,2,* Jaroslav Ilnytskyi,1,3 Marina Saphiannikova1

1

Leibniz-Institut für Polymerforschung, Hohe Str. 6, 01069 Dresden, Germany;

2

Institute of Macromolecular Compounds, Bolshoi pr. 31, 199004 Saint-Petersburg, Russia;

3

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii

Str. 1, 79011 Lviv, Ukraine

* The author to whom correspondence should be addressed. E-mail: [email protected], [email protected]

1 ACS Paragon Plus Environment


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ABSTRACT

Kinetics of photoisomerization and time evolution of ordering in azobenzene-containing materials are studied theoretically and by using computer simulations. Starting from kinetic equations of photoisomerization, we show that the influence of light is equivalent to the action of the effective potential which reorients chromophores perpendicularly to polarization direction. The strength of the potential is defined by optical and viscous characteristics of the material. The potential generates photomechanical stress of giant values ~GPa, in accordance with recent experimental findings for azobenzene materials deep in a glassy state. The proposed approach has a great predictive strength for deeper understanding and further development of the photocontrollable smart compounds.

TABLE OF CONTENTS GRAPHIC


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Stimuli-responsive smart materials found already much application in nanotechnology1 and medicine.2 One of the most promising stimuli is light, due to its clean nature and possibility to apply its action with an ultra-high speed and spatial precision. As a result, photocontrollable nano- and micro-templates,3-6 sensors, artificial muscles7-12 and actuators can be produced. To make a polymer-based material photosensitive it is modified by inclusion of special chromophoric groups, the azobenzene being the most widely used.3-12 Its reaction to the light via photoisomerisation from the trans- to cis-state and the following conversion of absorbed energy into mechanical work are of great interest for understanding how the photosensitive devices work and how to tune their performance for each particular application.

The key quantity, which defines the deformation, is the mechanical stress. In many cases, the light-induced deformation occurs at the light intensities as low as I = 0.1 W / cm 2 and below the glass-transition temperature of the polymer. This state is characterized by the values of the elastic modulus as high as E ~ 1 GPa and viscosity13 of η ~ 10 3 GPa ⋅ s and remains nearly unaffected under illumination.14 Interestingly, recent experiments5,15,16 showed that light-induced stress can reach a giant value of 2 GPa and is able to break the metallic layer on the surface of a glassy azo-polymer as well as to deform covalent bonds. There is no yet a theoretical explanation for the light-induced stress of such a large magnitude.

In this Letter we present a theory that provides such an explanation. We start from the kinetic equations for the photoisomerization dynamics between trans- and cis-isomers of azobenzene. Using theoretical solutions of these kinetic equations, as well as computer simulation technique, we show that the angular selectivity of the trans-isomer photoisomerization is equivalent to the application of the effective orientation potential, which acts on each chromophore:



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U eff (θ ) = V 0 cos 2 θ .

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(1)

Here V0 > 0 is the potential strength and angle θ defines orientation of azobenzene with respect to polarization vector of the light E. Under potential (1) the trans-isomers reorient perpendicularly to E, the effect is known from experiments as “angular hole burning” or Weigert effect.17,18 This effective potential was introduced phenomenologically earlier19,20 and was used recently to describe static (steady) photodeformation of azobenzene polymers of various structures.21-25

Similar ideas to reduce description of complicated phenomena to the action of effective potentials are widely used in physics. The examples include such well-known cases as: the Lennard-Jones potential for intermolecular forces; the mean-field potential in liquid crystals;26 effective potential of quantum particle in rapidly oscillating field;27 potential of entropic elasticity for polymer chains.28 The meaning of these potentials is quite clear and has a strong physical background. On the other hand, the potential (1) has never been justified from the first principles. In this Letter, by studying the full time-dependent process of reorientation of chromophores under illumination with the polarized light, we relate for the first time the effective potential (1) to optical and viscous characteristics of the azobenzene material.

Now, we focus on the description of the photoisomerization kinetics of azobenzene chromophores. In contrast to previous works29-31 which considered a simplified, angular independent, population dynamics of cis-isomers, we describe in detail the light-induced orientation ordering of both isomers taking into account the angular selectivity of photoisomerization with respect to the polarization vector E. The photoisomerization kinetics is studied in a thin sample under constant light intensity. Our aim is to solve angular-dependent


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Figure 1. Azobenzene chromophore under photo-isomerization process and a unit vector k

related to its orientation in space.

kinetic equations and to derive the mechanical stress, the task which is still missing in the literature.

The reorientation of chromophores under illumination with the polarized light can be described in terms of orientation kinetics for a unit vector k directed along the chemical bond at the end of the phenyl ring, see Fig. 1. Typically, the azobezene chromophores are attached to polymer chains through this chemical bond. Reorientation of the vector k defines reorientation of chain fragments and, as a result, induces the deformation of the material. The orientation of the vector k is defined by two Euler angles Ω ≡ (θ , ϕ ) . Here θ is the angle between the vector k and the polarization vector E; ϕ is the azimuthal angle in respect to E. The kinetic equations for the orientation distribution functions nT ,C (Ω) for trans- and cis-isomers can be written as follows (cf. with ref. 18):

 ∂nT (Ω) 2 2  ∂t = − PT cos θ nT (Ω) + PC ∫ dΩ ′nC (Ω′) f CT (Ω′ → Ω) + DT ∇ θ nT  ∂n (Ω)  C = PT ∫ dΩ′ cos 2 θ ′ nT (Ω ′) f TC (Ω ′ → Ω) − PC nC (Ω) + DC ∇ θ2 nC .  ∂t

(2)



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The first terms in the r.h.s. of both equations describe the contribution from the trans-cis isomerization process. Its probability is angular dependent via cos 2 θ term,18 resulting from the “hole burning” effect. Strictly speaking, the trans-cis isomerization is determined by the transition dipole moment for light absorption.18 However, the vector of the transition dipole moment is directed mainly along the long axis of the trans-isomer and its difference to the vector k will give only a small correction to the value PT defined here for k. The function

f TC (Ω ′ → Ω) in Eqs. (2) determines the probability of stochastic reorientation of the vector k ′(Ω ′) to a new vector k (Ω) as the result of the photoisomerization. We assume that the

azimuthal redistribution of the vector k is random and f TC is a function solely of the angle χ between the vectors k′ and k: f TC = f TC ( χ ) .

The second terms in the r.h.s. of Eqs. (2) describe the contribution from the cis-trans isomerization process. In a good approximation this process can be considered as an isotropic with respect to the polarization vector E due to isotropic polarizability tensor of bent cisisomers.32 The probabilities of the trans-cis, PT , and cis-trans, PC , photoisomerizations are related to the light intensity, I : PT = k TC ⋅ I and PC = k CT ⋅ I + γ . Here kTC and k CT are the corresponding rate constants33 and the parameter γ describes the cis-trans thermal relaxation. In contrast to previous works,29-31 which neglected the light-induced cis-trans isomerization assuming k CT = 0 , we consider explicitly this effect in our theory. Since the rate constants k TC and k CT depend strongly on the wavelength,33 their variation allows us to describe the photoisomerization kinetics in a wide range of wavelengths from UV light ( k TC > k CT ) to visible light ( kTC < k CT ).


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Similarly to the trans-cis isomerization process, it is assumed that f CT = f CT ( χ ) . Integrations over the angles in Eqs. (2) are defined via the volume element: dΩ = sin θ dθ dϕ , θ ∈ [0, π ] and ϕ ∈ [0, 2π ] .

The last terms in the r.h.s. of Eqs. (2) describe the orientation diffusion of the trans- and cisisomers. The theory, in its general form, contains two independent rotational diffusion coefficients DT and DC . However, as it was shown in ref. 34, these values are very close to each other. Thus, below we will use an approximation DT ≈ DC ≡ D . Due to the axial symmetry of the system with respect to the polarization vector E, the distribution functions nT ,C should be independent of the azimuthal angle ϕ : nT ,C = nT ,C (θ , t ) . Thus, the Laplace operator, ∇ θ2 , in Eqs. (2) includes derivatives only with respect to the angle θ . Note that we do not consider the translation diffusion of chromophores, which can be strongly hindered in stiff glassy materials, but their rotational mobility is taken explicitly into account.

The reorientation kinetics is studied starting from the initial (dark) state, in which all chromophores are in the ground trans-state and distributed isotropically:

nT (t = 0) = 1 / 4π and nC (t = 0) = 0 .

(3)

The total number of chromophores is constant:

∫ dΩ

n(θ ) = 1 ,

(4)

where n(θ ) = nT (θ ) + nC (θ ) is the angular distribution function for all isomers. Next, we analyze the photoisomerization kinetics at short and at long times.



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At short times, t ⋅ PT 1 . One can see from Eqs. (5) and (6) that the initial stage at short times, t PT−1 the back cis-trans isomerization process comes into play. This leads to the change of the potential strength with time. To study this effect, we introduce orientation order parameters:

S T ,C =

3〈cos 2 θ 〉 T ,C − 1 2

,

(10)

where the brackets 〈...〉 T ,C mean the averaging over the orientation distribution functions for trans- and cis-isomers. The kinetic equations for S T ,C and for the number fractions of the transisomers, Φ T ≡ ∫ dΩ nT (θ ) , and cis-isomers, Φ C = 1 − Φ T , can be derived from Eqs. (2), see Supporting Information. For example, we obtain the following kinetic equation for S T :  3〈 cos 4 θ 〉 T 2 S T + 1 3〈 cos 2 χ 〉 CT − 1 ∂ (Φ T S T ) = − PT Φ T  − + P Φ S − 6 DT Φ T S T ,  C C C ∂t 2 6 2  

(11)

which includes the higher moment 〈 cos 4 θ 〉 T of the distribution function nT (θ ) . To find S T ,C and Φ T ,C at any time we need a closure approximation which would relate 〈 cos 4 θ 〉 T with S T


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Figure 2. Time evolution of the angular distribution function nT for the trans-isomers.

defined by 〈 cos 2 θ 〉 T . Analyzing the numerical solution of Eq. (2), we found that time evolution of nT (θ ) can be well described by the exponential distribution function: nT (θ , t ) = Φ T Z exp[− βVT (t ) cos 2 θ ] .

(12)

Here β = 1 / kT , Z is the normalization constant and the time-dependent strength VT (t ) is defined from the condition (10). One can see from Fig. 2 that exact numerical solution for

nT (θ ) , obtained from Eqs. (2), is approximated at all t extremely well by the analytical distribution (12). Thus, we found an accurate closure approximation to the problem: varying the value of VT , the dependence of 〈cos 4 θ 〉 T on ST can be obtained, see Supporting Information. Using this closure dependence, one can calculate the order parameters ST , S C and the fraction of trans-isomers Φ T as functions of time. The results for the order parameters obtained in the framework of the closure approximation agree very well with those obtained by explicit numerical solutions of kinetic equations (2), as can be seen in Fig. 3.



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Figure 3. The order parameters S T and S C as functions of time: comparison between exact

numerical calculations and the closure approximation based on the effective potential.

Using the closure approximation we analyze the time dependence of the strength of the potential, V (t ) , as a function of ratio PC / PT . As shown in Supporting Information, the magnitude of V (t ) deviates from that of V0 less than two times. Thus, when comparing to experiment, it is safe to consider the effective potential having the strength up to 10 −18 J order of magnitude as predicted by Eq. (9).

To prove the correctness of the theory developed above, we compare our theoretical results with computer simulations performed recently.36 There we considered the ensemble of soft spherocylinder beads each representing a fragment of the azo-polymer. Mixed deterministicstochastic approach was applied, where the Newtonian equations of motion were complemented by stochastic photoisomerisation events between the trans-like and cis-like beads. As the result, both the photostationary state and system evolution towards it were analyzed. In Figure 4 we compare the time dependence of the number fraction of cis-isomers obtained in simulations36 and in the current study. A good agreement between the theory and computer simulations demonstrates that the closure approximation based on orientation potential (1) provides a very


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good approach to study the photoisomerization kinetics in azobenzene-containing materials. Note that the closure approximation simplifies the problem tremendously since it reduces the photoisomerization kinetics to three scalar parameters S T , S C , Φ C instead of complete calculation of angular distribution functions nT ,C (θ ) which satisfy Eqs. (2).

The rest of our Letter is devoted to estimation of light-induced mechanical stress. The mechanical stress produced in the system of rod-like particles upon application of an external field is discussed in detail in the monograph.28 The stress tensor for an isotropic material under potential U eff (θ ) can be written as, see Eq. (9.52) of ref. 28:

σ αβ = − n 0 〈 (u × RU eff ) α u β 〉 ,

(13)

where n 0 is the number density of particles, u is the unit vector along the rod-like particle and R = [u × ∂ / ∂u] is the rotational operator. Substituting the potential (1) in Eq. (13), one can see

that the components of the stress tensor are proportional to n0V0 . The calculation of the prefactors given in Supporting Information leads to the following estimation of the magnitude of the tensile normal stress: σ ≅ 0.4n 0V0 . In agreement with experiments11,37 the sign of deformation (expansion / contraction) is defined by the chemical structure of the material, as shown in Supporting Information. Using typical value21 of n 0 ≈ 1.5 ⋅10 21 cm −3 and V 0 ≤ 7 × 10 −18 J estimated above, the characteristic magnitude of the tensile stress for the material near T g is given by

σ ≅ 0.4n 0V0 ≤ 4 GPa at the light intensity I = 0.1 W / cm 2 . It has the same order of magnitude as σ ≈ 2 GPa found in experimental works.15,16



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Figure 4. The number fraction of cis-isomers Φ C as a function of time. The time t′ is given in

′ = 1 / 2 D ′ = 80 . 〈sin 2 χ 〉 TC = 〈sin 2 χ 〉 CT = 0.16 . units of the computer simulations: τ rot

The stress can be lower than 4 GPa due to the factor 〈sin 2 χ 〉 < 1 . For instance, even at small redistribution angles 10 o ≤ χ ≤ 20 o , expected for chromophores in a stiff glassy material, we obtain 100 MPa ≤ σ ≤ 500 MPa at I = 0.1 W / cm 2 . The last values are still higher than typical values of the yield stress, σ Y , for glassy materials (e.g. σ Y ≈ 50 MPa for PMMA). At σ > σ Y the glassy material deforms irreversibly. Thus, the developed theory provides for the first time the physical background of the light-induced mechanical stress of such a giant magnitude. We conclude this Letter by noting, that as was shown by us previously,21-25,38,39 the orientation approach based on the effective potential (1) provides results in a good agreement with experimental data for photodeformation of azobenzene-containing polymers. Here we justify this approach by explicit theoretical study of photoisomerization kinetics. Thus, the orientation approach receives now a strong physical background from the first theoretical principles, from computer simulations and from experimental data.


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The effective orientation potential can be applied to study the reorientation kinetics, deformation and mass transport of a wide spectrum of azo-materials including amorphous, liquid-crystalline11 and cross-linked12 polymers, azobenzene-functionalized dendrimers and brushes,5 azobenzene-decorated plasmonic particles40 and many others. Essential generalization of the theory can be description of bending deformation due to attenuation of the light intensity across a thick sample due to absorption.29-31,41 The established structure-property relationships can be used for designing specific photo-deformable materials with target technologically important properties.

Acknowledgements. The authors thank the Deutsche Forschungsgemeinschaft for support of

this work through the grant GR 3725/2-2.

Supporting Information. Derivation of the effective potential; Closure approximation;

Kinetics of photoisomerization at high viscosities; Stress tensor.

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Azobenzene-Containing Polymers with Different Molecular Architecture: Molecular Dynamics Study. J. Chem. Phys. 2011, 135, 044901. (39) Ilnytskyi, J. M.; Saphiannikova, M.; Neher, D.; Allen, M. P. Computer Simulation of Side-

Chain Liquid Crystal Polymer Melts and Elastomers. In: Liquid Crystalline Polymers, Vol. 1, p. 93, Springer, 2016. (40) Raimondo, C.; Reinders, F.; Soydaner, U.; Mayor, M.; Samorì, P. Light-Responsive

Reversible Solvation and Precipitation of Gold Nanoparticles. Chem. Commun. 2010, 46, 1147 – 1149. (41) Geue, T.; Saphiannikova, M.; Henneberg, O.; Pietsch, U. Formation Mechanism and

Dynamics in Polymer Surface Gratings. Phys. Rev. E 2002, 65, 052801.


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Figure 1. Azobenzene chromophore under photo-isomerization process and a unit vector k related to its orientation in space. 145x103mm (150 x 150 DPI)

ACS Paragon Plus Environment


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Figure 2. Time evolution of the angular distribution function nT for the trans-isomers. 35x24mm (300 x 300 DPI)


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Figure 3. The order parameters ST and SC as functions of time: comparison between exact numerical calculations and the closure approximation based on the effective potential. 35x24mm (300 x 300 DPI)



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Figure 4. The number fraction of cis-isomers as functions of time. The time t' is given in units of the computer simulations. 35x24mm (300 x 300 DPI)


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TOC-graphic. 51x54mm (150 x 150 DPI)


Photoisomerization Kinetics and Mechanical Stress in Azobenzene

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