Complex Dynamics of Photo-Switchable Guest Molecules in All

Jan 15, 2018 - We study theoretically the kinetics of noninteracting photoswitchable guest molecules (model azo-dye) dispersed at low concentration in...
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Article Cite This: J. Phys. Chem. B 2018, 122, 1756−1765

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Complex Dynamics of Photo-Switchable Guest Molecules in AllOptical Poling Close to the Glass Transition: Kinetic Monte Carlo Modeling W. Radosz, G. Pawlik, and A. C. Mitus* Department of Theoretical Physics, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego 27, 50−370 Wroclaw, Poland ABSTRACT: We study theoretically the kinetics of noninteracting photoswitchable guest molecules (model azo-dye) dispersed at low concentration in host (model polymer matrix) in the all-optical poling process close to the glass transition temperature Tg. We modify kinetic Monte Carlo model used in our previous studies of nonlinear optical processes in host−guest systems. The polymer matrix is simulated using the bond-fluctuation model. The kinetics of multiple trans−cis−trans cycles is formulated in terms of transition probabilities which depend on local free volume in the matrix and its dynamics. Close to Tg, the buildup of polar order, monitored in terms of angular probability density functions, follows a power-law in time while the evolution of the nonlinear susceptibilities related to second harmonic generation effect follows the stretched-exponential law. This complex dynamics of guest molecules implies the presence of dynamic heterogeneities of the matrix in space and time which spread the complexity from the matrix to the otherwise simple dynamics of noninteracting guest molecules. A qualitative physical picture of mosaic-like states intertwined areas of free- and hindered angular motion of guest moleculesis proposed and the role of related short and longer scales in space for the promotion of complex dynamics of guest molecules is discussed. A brief comparison of the theory to available experimental data is given. forming a constant electric field acting on the dipolar moments of the choromphores.4 The former effect promotes axial symmetry, the latter−polar symmetry. This simple physical picture of light-matter interaction-driven buildup of polar order is fundamentally modified by the equilibrium thermal fluctuations of the local structure of the host matrix in the nanoscale, typical for the size of, e.g., Disperse Red 1 (DR1) azobenzene molecule.4 Unluckilly, little is known, on the quantitative level, about this kind of local structure of a polymer matrix and about its spatial and temporal correlations. The main challenge is the characterization of static local inhomogeneities in the distribution of the monomers and their dynamic evolution, driven by local jamming/clustering effects, which modify the dynamics of the guest molecules. The quantification of those topics is in its very early stage.6,7 Local jamming/clustering events constitute the microscopic origin of complex macroscopic polymer dynamics, particularly well expressed in the close vicinity of the glass transition (GT) temperature Tg. The slowing down of polymer dynamics8,9 is usually ascribed to an appearance of long-tailed (power-law) distributions of waiting times for local structural changes, triggered by jamming effects which ultimately cause dynamic heterogeneities10−12 in space and time (this scenario occurs for

1. INTRODUCTION: PHYSICAL PICTURE Host−guest systems composed of a host matrix (polymer, biopolymer or liquid crystal) with chemically or physically attached guest molecules (chromophores, dipoles, octupoles, coated spheres, and others) are commonly used in nonlinear optical (NLO) applications and techniques like, e.g., second (SHG) or third harmonic generation, sum and difference frequency generation, all-optical poling, four-wave mixing, selffocusing, self-phase modulation, and many others.1−3 A wide spectrum of those effects, which offer numerous applications in modern material sciences, makes both experimental and theoretical studies of host−guests system of key importance. In this paper we model theoretically one of those effects − SHG in a system of photoswitchable chromophores (azo-dyes) dispersed in a polymer matrix. The necessary condition for this effect is a buildup of a nonvanishing global polar order for the distribution of long axes of azo-dyes in the space, which manifests itself in nonzero macroscopic second-order nonlinear susceptibilities. Polar order can be promoted either by an external electric field poling, photoassisted electric field poling or by all-optical poling techniques.4 While the electric-field driven build-up was studied theoretically in our paper,5 current study is related to the latter method. In the all-optical poling experiment the chromophores interact with two linearly polarized light beams (with frequencies ω and 2ω) which result, first, in multiple one- and two-photon-driven photoisomerization cycles trans−cis−trans of azo-dyes and, second, in © 2018 American Chemical Society

Received: December 4, 2017 Revised: January 12, 2018 Published: January 15, 2018 1756

DOI: 10.1021/acs.jpcb.7b11949 J. Phys. Chem. B 2018, 122, 1756−1765

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The Journal of Physical Chemistry B two-dimensional Lennard-Jones liquid13). Thus, the dynamics of nonlinear optical phenomena in host−guest systems close to Tg belongs to the class of complex systems’ dynamics and its theoretical analysis, which encounters methodological difficulties, constitutes a challenge. The objective of this paper is to study theoretically the influence of glass transition in the host polymer matrix on the dynamics of guest (azo-dye) molecules dispersed at low concentration, in a model host−guest system mimicking alloptical poling experiment. Host−guest systems can be studied theoretically using kinetic Monte Carlo (MC) modeling, combining standard MC bond fluctuation simulations of a polymer matrix with suitable kinetic models which account for the interaction of guest molecules with the matrix and with external fields as well as for their intermolecular interactions. An advantage of this kind of modeling is that it grants a deeper insight into ”microscopic” processes promoting macroscopic effects. More specifically, it makes possible a detailed characterization of ”microscopic” orientational order of guest molecules and its temporal evolution which is hard (if at all possible!) to infer indirectly from experimental data. This approach was used by some of us to study chosen NLO phenomena in host−guest systems with photochromic and dipolar molecules serving as guest molecules: inscription and erasure of diffraction gratings,14−16 diffraction gratings in DNA based biomaterials,17 SHG in electrically poled systems,5 inscription of surface relief gratings,18 and photoinduced reorientation of nematic liquid crystals;19,20 see also a review paper.21 Those studies were performed for temperatures below Tg; moreover, no systematic analysis of temperature effects was done. In this study we use the same approach with a suitably generalized MC kinetic model onto the case of all-optical poling. Quite recently, kinetic Monte Carlo simulations22 were used to study the dynamics of needle-like photoswitchable molecules on a surface, motivated by a power-law decay of initial birefringence in a dense monolayer of photoswitchable dye methyl-red molecules demonstrated experimentally.23 The authors report the complex dynamics of the molecules in low-temperature and high density limit, traced to dynamic heterogeneity originating from intermolecular interactions; the polymer matrix is not present in their model. The paper is organized as follows. In the next section, we present the basic features of MC modeling of all-optical poling in a host−guest system. The third section is devoted to the characterization of the process of buildup of polar order in this system in a wide temperature interval around Tg. The discussion of complex kinetics of guest molecules close to Tg constitutes the main part of the fourth section in which the topics of the localization of the glass transition temperature and the relation of our results to experimental data are also briefly discussed.

2.1. Polar Order and Nonlinear Susceptibilities. Efficiency of the SHG in poled thin films consisting of quasi one-dimensional (1D) charge transfer chromophores in a polymer matrix depends on the degree of polar order. For such quasi 1D rod-like molecule the orientation of its long axis is described by angle Ω = (θ, ϕ), where θ and ϕ denote, respectively, the polar and azimuthal angle defined wrt. an external direction Z being the poling field direction (linearly polarized poling/probing optical field perpendicular to the thin film surface), see Figure 1. The distribution of orientations of

Figure 1. Schema of SHG model setup.

long axes of the molecules is given by the normalized probability density function (PDF) ρ(Ω). This PDF makes possible the calculation of the statistical averages ⟨...⟩ of an arbitrary function f(θ, ϕ): ⟨f (θ , ϕ)⟩ = =

∫ f (θ , ϕ)ρ(Ω) dΩ ∫0

2π

1

dϕ

∫−1 f (θ , ϕ)ρ(θ , ϕ) d[cos θ]

(1)

The second order NLO susceptibility χ(2) for a set of rod-like molecules is characterized by two tensor components:24,25 the diagonal (2) χZZZ ( −2ω; ω , ω) = NFβzzz ⟨cos3 θ ⟩

(2)

and the off diagonal one (2) χXXZ ( −2ω; ω , ω) =

1 NFβzzz ⟨sin 2 θ cos θ ⟩ 2

(3)

Here, X denotes a direction perpendicular to Z, N the number density of azo-dyes in trans state, βzzz the the dominant component of hyperpolarizability tensor for a rod-like molecule in trans state (molecular first hyperpolarizability), and F the local field factor. Monte Carlo modeling yields the PDF ρ(θ, ϕ; t), where t denotes the Monte Carlo “time”number of Monte Carlo steps, see belowwhich fully characterizes the dynamics of (2) buildup of polar order and of NLO susceptibilities χ(2) ZZZ, χXXZ. 2.2. MC Modeling of Polymer Host. We have used the same standard MC lattice bond-fluctuation model of the equilibrium polymer matrix26 as in our previous studies of host−guest systems (section 1). The polymer chain consisted of a set of beads (monomers) located on a simple cubic (s.c.) lattice, connected by bonds with lengths 2, √5, √6, 3, 3, and √10 (in units of lattice constant); six nonequivalent bond orientations were used. The dimensionless energies ϵ of the bonds are ϵ = 1 for the first three cases, and ϵ = 0 for the last three lengths. N0 = 24000 polymer chains, each consisting of L = 20 monomers, were placed on a Vp = 200 × 200 × 200 lattice; the reduced density ρ = 8 N0 L/Vp = 0.48 corresponds

2. MC MODELING OF ALL-OPTICAL POLING Monte Carlo modeling of all optical poling in host−guest systems addresses three topics: (i) characterization of orientational order of guest azo-dyes molecules and its relation to macroscopic SHG susceptibilities, (ii) MC simulation of equilibrium polymer matrix and an analysis of local parameters which condition the local dynamics of azo-dyes, and (iii) the kinetic model for the interaction of azo-dyes with optical field and with the matrix. Those topics are discussed in the next subsections. 1757

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Figure 2. Snapshot of a polymer system (10% of all chains are displayed) (a) and scheme of calculation of parameter V (b).

Figure 3. Instantaneous 2D cross-section of the polymer system: map of V (large map) and part of this map scaled up (small map).

to the concentrated polymer solution.26 In spite of its simplicity, this model plays one of central roles in modeling of polymer systems.27 Simulations were made at constant reduced temperature T (its unit is related to the dimensionless energy scale ϵ). A single Monte Carlo Step (MCS) consisted of trial moves of unit length of all the monomers along one of the three axes x, y, z of the s.c. lattice. Each move was accepted if (a) the trial location (lattice node) was empty, (b) the lengths of emerging new bonds were acceptable, and (c) the standard Metropolis acceptance criterion26 was satisfied. The starting configuration consisted of isotropically oriented bonds. Periodic boundary conditions were used. A typical instantaneous configuration of the polymer system is shown in Figure 2a. Local static and dynamic features of the polymer matrix which affect the dynamics of the guest molecules are characterized by two parameters, V(r)⃗ and C(r)⃗ , introduced in our earlier papers,15,16 see also in ref 21. Those quantities enter the expressions for the transition rates in the kinetic model described in section 2.4. The first one, V(r)⃗ , describes the local inhomogeneities (local voids) of the distribution of the monomers around lattice point r.⃗ The value of V(r)⃗ is calculated using a 3 × 3 × 3 cube, centered around r ⃗ (Figure 2b). If the distance of a monomer from the lattice node r ⃗ is 0 or 1, then V = 0. In other cases V = 7−k, where 0 ≤ k ≤ 7 stands for the number of monomers in the cube. High values of V indicate that there are few monomers in a close vicinity of point r;⃗ low values imply that its closest environment is crowded with

monomers. An eye-inspection of instantaneous maps of parameter V(r)⃗ (Figure 3) reveals a spatial heterogeneity in its distribution. Clearly, some complex spatial correlations are present, nevertheless the pattern is too complicated to draw, at this point, any far-reaching quantitative conclusions. The characterization of local static and dynamic heterogeneity constitutes one of the challenges for the statistical physics of model polymer systems. It amounts to the characterization of the random spatiotemporal fields V(r,⃗ t) and C(r,⃗ t) (see below), which requires the introduction of various kinds of probability density functions and correlation functions. This task goes far beyond the scope of current paper. Some time independent correlation functions which quantify the concept of voids were calculated in refs 6 and 7. The second parameter, C(r)⃗ , describes a short-time reorganization of the monomers in a local neighborhood of point r ⃗ and affects the rotational dynamics (rotational mobility) of a guest molecule at this point, see section 2.4. Its value is equal15,16,21 to the total number of changes δni (in a single MCS) of the occupation state ni of a node by the monomers (ni = 1 for site i occupied by a monomer; ni = 0 when site i is monomer-free): C(r)⃗ = ∑i |δ ni|. The summation extends over the nodes of a 9 × 9 × 9 cube centered around the investigated cell r.⃗ The large size of the cube is dictated by the fact that monomers are relatively sparse - in our simulations one monomer is localized, on the average, in a volume composed of 16 lattice sites. 1758

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The Journal of Physical Chemistry B 2.3. Glass Transition Temperature Tg. For further analysis an estimation of the glass transition temperature Tg is necessary. Its localization is more ambiguous than for equilibrium I- or II-order phase transitions, where thermodynamics plays an important role.28,29 Some parameters, mostly geometric in nature but also of thermodynamic origin, like mean bond energy ϵ, Flory parameter, specific heat, or internal temperature, were proposed and studied for the bondfluctuation model in the seminal paper.30 Those parameters clearly mark the glass transition but the values of Tg are not localized in a narrow temperature interval. The geometric parameters studied were: mean squared bond length, radius of gyration or the mean free volume (MFV) parameter30 v, originally introduced for 2D systems.31 Similar in spirit, approach using dynamically available volume,32 was used in ref 33. MFV parameter v (0 ≤ v ≤ 1) is equal to the normalized average (over polymer chains and configurations) number of lattice sites available for monomers in a single MC step. The lattice site is available if the displacement of a monomer is not blocked and the emerging new bond length is acceptable. Figure 4 shows the temperature dependence of v.

model. Namely, both static (excluded volume) and dynamic (photoisomerization cycles) effects of the model azodyes on the polymer matrix are neglected. In other words, we study a system of noninteracting azo molecules in dynamical external field (due to optical field and the matrix) which is independent of the dynamics of those molecules. The motivation for this approximation is as follows. We are interested in the ”transfer” of complexity from the matrix close to glass transition onto a system of noninteracting azo-dyes which, in the absence of the matrix, display a simple (biexponential) dynamics.14 In this context, the concentration of the azodyes has to be sufficiently low to avoid steric and electrostatic dye−dye interactions. Those interactions become important for sufficiently high concentration of the dyes and lead to their complex dynamics, even in the absence of the matrix, as shown experimentally23 and through simulations.22 So, in a general case, there are two very different origins/mechanisms of complex dynamics of the guest molecules. While the ultimate goal might be to study a general case, where both mechanisms contribute, in various ways, to the overall complex dynamics, in this paper we study only the first one (matrix induced)the second mechanism (interaction induced in the absence of the matrix), was studied theoretically.22 The validity of the assumedly weak impact of azo molecules on the matrix in the low concentration case follows from an analysis of SHG experiments in host−guest systems. Consider, e.g., SGH in doped PMMA/DCNP system34 with 2 wt % concentration of DCNP molecules. The molar masses of DCNP and of PMMA repeat unit (r.u.) are, respectively, 222.25 g/mol and 100.12 g/mol. Two wt % concentration corresponds to 1 DCNP molecule per 109 PMMA r.u. PMMA has 8.8 r.u. per Kuhn segment35 and thus a single DCNP molecule corresponds to 12.4 Kuhn segments. In our simulations the chain consists of 20 Kuhn segments which yields approximately 1.6 azo molecules per modeled chain. Actually, SHG signal can be measured at 1 wt % or even lower concentrations, which corresponds in our simulations to less than a single azo molecule per chain. Those data give a sound ground to assume that the effect of the azo molecules on the matrix can be weak/negligible while SHG signal can be still monitored. The kinetic MC modeling in the case of pure light-matter interaction was studied by us some time ago.14 The transition rates in the presence of the matrix, in the low concentration limit when dye−dye intermolecular interactions and the impact of the isomerization dynamics on the matrix are negligible were proposed in our papers.15,16 Some applications of MC kinetic modeling to other physical systems are reviewed in ref 36. In this paper we generalize the transition rates15,16 onto the case of all−optical poling in low concentration limit. The light-driven trans−cis−trans cycle of azo-dye molecules consists of three events: transitions trans →cis, cis →trans and angular diffusion (see Figure 5). The transition rate for onephoton trans →cis transition p(θ, ϕ) ∝ cos2 θ is independent of angle ϕ. In the case of all-optical poling additional transition rates have to be accounted for,4 which correspond to (i) twophoton absorption, where p(θ) ∝ cos4 θ, and (ii) the presence of electric field which breaks the centrosymmetry: p(θ) ∝ cos3 θ. An overall transition rate is a weighted sum of those rates. Upon optimization requirement (equal excitation probabilities through one- and two-photon absorptions) the rates p(trans → cis) and p(cis →trans) read:4

Figure 4. Plot of MFV parameter v (black points, left axis) and of mean bond energy ϵ (blue circles, right axis) against reduced temperature T. Inset: plot of mean squared radius of gyration of polymer chain against T. Straight lines represent linear fits.

Three regimes are present: linear at low (T ≤ 0.15) and linear at high (T ≥ 0.35) temperatures and a transient regime 0.15 < T < 0.35. Linear regimes correspond, respectively, to glassy and melt polymer phases; the crossover between them takes place in the transient regime. Clearly, no precise estimation of Tg is possible. Some estimation can be obtained using the linear fits; their intersection yields Tg ≈ 0.25. Similar conclusions are drawn from the plot of average bond energy ϵ (T) ; we find Tg ≈ 0.23. On the other hand, the temperature dependence of the radius of gyration, Figure 4 (inset), displays a well-defined crossover between linear regimes at T = 0.26. To conclude, we estimate that Tg lies in the interval 0.23 ≤ Tg ≤ 0.26. 2.4. Kinetic Model for All-Optical Poling in Low Concentration Limit. The kinetic MC modeling uses analytical expressions for transition probabilities p in unit time (transition rates) for various constituent processes. For host−guest systems the kinetic model accounts for the interaction of azo-dyes with an optical field (light-matter interaction), with the matrix as well as for intermolecular dye− dye interactions. This study concerns the case of a low concentration of the azo molecules, leading to a simplification of the general kinetic 1759

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molecule after cis → trans transition can be chosen at random. All-optical poling process consists of two periods: the writing period of length 2 × 105 to 5 × 105 MCS when the intensity of linearly polarized light is constant and the decay period performed in darkness (I = 0, 106 MCS), where the dynamics is due only to the angular diffusion. The detailed structure and size of the model azo molecules are not relevant because in the low concentration limit they do not influence the dynamics of the matrix (and, correspondingly, no interactions are included in the Metropolis conditions). The azo molecules react to the local fields (optical and matrixinduced), are fixed in space and occupy in a random way the nodes of the same s.c. lattice on which the monomers are located. This choice follows from the following argumentation. Since in the low concentration limit azo molecules do not influence the dynamics of the matrix, the statistics of the data for dynamics of the azo molecules can be increased by a simple trick: for a single instantaneous configuration of the matrix the azo molecules can be safely placed in all nodes of the lattice. Consequently, in the limit of low density, the only effect of the translation in the space would be to probe another node. Since all nodes are probed, as explained above, the translation becomes inessential in this case.

Figure 5. Illustration of trans−cis−trans cycle and angular diffusion process. 2

4

3

p(trans → cis) = VIptc (cos θ + cos θ + 2cos θ)

(4)

p(cis → trans) = VIpct

(5)

3. RESULTS 3.1. Onset and Evolution of Polar Order. The transition rates discussed in the previous section are independent of angle ϕ and the PDF ρ(θ, ϕ; t) reduces to ρ(θ; t). In Figure 6a, we present the polar plot of ρ(θ; t)/sin θ for t = 0, 104 and 2 × 105 MCS. Here, the factor sin θ ensures that an isotropic distribution of orientations of dyes is represented as a circle on the polar plot of ρ(θ) /sin θ. Clearly, the symmetry breaking from isotropic to polar order occurs: the plot, initially (t = 0) consisting of a circle, develops a maximum around θ = 180° and a minimum at θ = 0°. The latter, so-called angular hole-burning (AHB) effect, has a double origin: axial symmetry promoted by cos2 θ and cos4 θ terms and polar symmetry promoted by the cos3 θ term in eq 4. We conclude that our kinetic model correctly describes the effect of centrosymmetry breaking and an onset of polar order in all-optical poling phenomenon. The AHB effect depends on the temperature. Figure 6b shows the plots of ρ(θ)/sin θ at the end of the inscription phase for three temperatures: well below Tg (T = 0.1), close to

where I denotes light intensity and ptc and pct denote the probabilities of photoisomerization in a single act of interaction with light. The parameter R = ptc/pct has a strong impact on the kinetics of host−guest systems.17,22,37 The transition rate for rotational diffusion reads:15,16 ⎧ p VC ,ΔΩ < ΔΩ 0 ⎪ diff p(Ω → Ω + ΔΩ) = ⎨ ⎪ ,ΔΩ > ΔΩ 0 , ⎩ 0

(6)

where pdif f is a constant. Equations 4−6 imply that the azo-dye molecules couple to the polymer matrix by the factors V and C. In the simulations we have used the values pct = 0.0001, ptc = 0.1 pct, and pdiff = 0.005 (as in our previous papers). We assume14,38 that the orientation of a long axis of a trans

Figure 6. (a) Polar plot of angular distribution ρ(θ; t)/sin θ for t = 0 (thick solid line, black), t = 104 MCS (thin solid line, blue), and t = 2 × 105 MCS (dashed line, red) for T = 0.3. (b) Angular distribution ρ(θ)/sin θ at the end of the inscription phase for T = 0.1 (thin solid line, red), T = 0.275 (thick solid line, black), and T = 0.5 (dashed line, blue). Excerpt: Part of the plot close to θ = 0°. 1760

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Figure 7. (a) Temperature dependence of normalized average number N0−20/Nmax of molecules in trans state with 0° ≤ θ ≤ 20° at the end of inscription phase. (b) The log−log plots of N0−20(t) for a few temperatures.

Tg (T = 0.275) and well above Tg (T = 0.5). The inset shows the plot for angles from θ = 0° to θ = 60°. In each case a welldeveloped polar order is present. An eye-inspection shows that the order of appearance of the plots does not show any systematic dependence on T. To quantify this statement we plot in Figure 7a, the temperature dependence of the normalized average number N0−20/Nmax of particles in trans state for which 0° ≤ θ ≤ 20°, at the end of the inscription phase. Here, Nmax = N0−20(T = 0.075). A well-defined minimum which corresponds to the strongest AHB effect occurs between T = 0.275 and T = 0.3. At higher temperatures a linear regime sets in. Moreover, the temporal evolution of the number N0−20 close to this interval of temperatures has a very different character than away from it, see Figure 7b. Namely, the double logarithmic plot is linear for T = 0.25 (and nearly linear for T = 0.275) which indicates a power-law behavior, while for two other temperatures away from Tg, T = 0.2 and 0.4, the power-law is absent. 3.2. Nonlinear Susceptibilities. The nonlinear suscepti(2) bilities χ(2) ZZZ, χXXZ depend on two parameters calculated in MC modeling: number density N of trans molecules and averages of trigonometric functions of angle θ. We have found that the former was practically constant during the simulation: |N(t) − 3 N(0)|/N(0) ≈ 5 × 10−3. Susceptibility χ(2) ZZZ becomes then ⟨cos θ⟩ scaled by a constant factor. For this reason in what follows we analyze ⟨cos3 θ⟩ factor only. The inset in Figure 8 shows the time dependence of ⟨cos3 θ⟩ for temperatures: T = 0.1, 0.25, 0.3, and 0.5. A qualitative conclusion is that different types of kinetics are present close to Tg and away from it: in the first case the plot does not saturate after t = 2 ·105 MCS while it does for T = 0.5 and shows a weak growth effect for T = 0.1. Moreover, the maximum value of ⟨cos3 θ⟩ is larger close to Tg than away from it. The buildup of susceptibility χ(2) XXZ follows the same pattern and in the remaining part of the paper we present the results for χ(2) ZZZ only. The curves ⟨cos3 θ⟩(t) were fitted (Origin 9.5) using single exponential fit (three parameters), biexponential fit (four parameters), and a stretched exponential fit (three parameters):

Figure 8. Plot of parameter ⟨cos3 θ⟩(t) in the inscription phase. Inset: temperatures T = 0.1 (thin red solid line), 0.25 (thick black line), 0.3 (dashed blue line), and 0.5 (dotted green line). Main plot: MC results (solid line, black) and stretched exponential fit (dashed line, red). T = 0.15.

simulation interval (from 2 × 105 MCS to 5 × 105 MCS) while the biexponential fit has failed in this respect. We conclude that the stretched exponential fit represents a statistical model for the simulated data while the biexponential fit does not. This conclusion holds for both inscription and erasure phases. For this reason, we interpret the simulated data using stretched exponential fit; see Figure 8. The results of fitting ⟨cos3 θ⟩(t) for different temperatures are shown in Figure 9. The plot of characteristic time τ(T) (panel a) in the inscription phase is not monotonic and displays a distinct maximum around T = 0.25, in the temperature interval where the glass transition occurs. There is a substantial increasee.g., 2 timesof relaxation time τ close to T = 0.25 as compared to temperatures away from it. In the decay phase, on the contrary, there is no maximum close to the glass transition temperature and the characteristic times decrease nearly linearly with the temperature (a maximum occurs at low temperatures but it is not related to the glass transition - this effect is not of interest for the present study). Panel b shows the plot of exponent d(T). As in the case of τ(T), in the inscription phase the plot is not monotonic and displays a minimum in the temperature range 0.2 < T < 0.225, slightly below the assumed glass transition temperature. In the decay phase the parameter d has higher values than in the inscription phase (around 0.8− 0.9) and displays larger fluctuations (not shown in Figure 9).

d

⟨cos3 θ ⟩(t ) = A[1 − e−(t / τ) ]

(7)

where τ denotes the characteristic time. The visual inspection showed that the single exponential fit failed while both stretched and biexponential fits satisfactorily reproduced the data. However, the parameters of the stretched exponential fit were with good accuracy independent of the length of 1761

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Figure 9. Parameters of stretched exponential fits, eq 7, of ⟨cos3 θ⟩(t): (a) τ(T) and (b) d(T). Inscription phase: black circles. Decay phase: red squares.

The complex kinetics of guest molecules close to the glass transition temperature is a direct consequence of complex dynamics of polymer matrix. Namely, (i) the intermolecular interactions are negligible due to the low concentration of the dyes, and (ii) in the absence of polymer matrix, noninteracting dyes display biexponential kinetics.14 Our studies based on local analysis of the structure of the matrix offer a deeper insight into its local dynamics close to Tg. Namely, the origin of stretched exponential complexity is usually ascribed39−43 to a broad (power-law) distribution of trapping periods of time when the dynamics of the guest molecules is blocked or hindered. In the case of a polymer matrix the trapping events are governed by local voids characterized by the time-dependent field V(r,⃗ t). A typical instantaneous configuration of V(r,⃗ t) (Figure 3) has a form of an intertwining mosaic (speculated on a long time ago44) of blocked (red) and free (green) spaces. In the former, the dynamics are strongly hindered; in the latter, it is free from steric limitations. The temporal evolution of the matrix changes the field V(r,⃗ t) and modifies the spatial distribution of trapping events, leading to complex behavior of guest molecules. Let us point out an analogy to this hypothetical scenario. For the 2D Lennard-Jones liquid close to melting, the temporal correlation functions describing local structure display stretch-exponential behavior while away from it simple exponential dynamics sets in. The origin of this behavior was ascribed to observed powerlaw distribution of trapping times related to a change of some topological characteristics of mosaic-like local arrangements of the atoms.13 Characterization of the mosaic-like states V(r,⃗ t) is a challenge and constitutes one of the most important topics in modeling of polymer systems. First quantitative results, related to static characterization of the mosaic, were presented in refs 6,7. Present study yields some qualitative conclusions concerning the local scales and local dynamics of the mosaic-like states, originating from an analysis of a very different temperature dependence of time τ in the inscription and decay phases (Figure 9a). In the former case, the maximum of τ reflects, as expected, the slowing-down effect due to the matrix close to GT. Surprisingly, the slowing-down has no visible impact on relaxation time τ in the decay phase. The origin of this effect is bound up with the type of the motions of the guest molecules, in particular with the way they probe the angular phase space. In the decay phase there are no photoisomerization transitions (I = 0) and the molecules perform a diffusive-like orientational motion with a small amplitude; the transition rate depends both on V and C, see eq 6. The angular kinetics is weakly complex close to glass transition (exponent d = 0.8−0.9 is close to d = 1). In the d = 1 limit, the normal angular diffusion is the result

The emerging physical picture of complex dynamics of guest molecules is discussed in the next section.

4. DISCUSSION 4.1. Glass Transition, Complex Dynamics, and MosaicLike States. The results presented above clearly show an influence of the glass transition on the kinetics of azo-dyes, expressed in a number of anomalous physical effects in temperature and time. The angular hole burning effect is the strongest for the temperatures close to the GT and becomes weaker for higher and lower temperatures (Figure 6b and Figure 7a). The process of the buildup of the polar phase and emergence of nonlinear susceptibilities is clearly marked by the closeness to the GT. First, the characteristic time τ displays a maximum close to the GT (Figure 9a). This effect can be also observed in the time dependence of the polar order parameter which saturates more slowly close to the GT than away from it (Figure 8). Second, the exponent d displays a minimum close to the GT (Figure 9b). Still more important is the fact that the kinetics of guest molecules close to the glass transition displays features typical for complex systems: power laws and nonexponential relaxation. The former appears in the angular hole burning effect, Figure 7b, where the number of trans molecules N0−20(t) displays a power law at T = 0.25: N0 − 20(t ) ∼ t α

(8)

with the exponent α ≈ 0.3. The power law, inherently related to a lack of a characteristic time scale, breaks down upon a small departure from T = 0.25. Complex relaxation is present in the process of buildup of the polar phase (nonlinear susceptibilities): the stretched exponential function, eq 7, constitutes a reliable model for the simulated data, in contrast with single- or double-exponential functions. The exponent d close to the GT has a value around d = 0.55, well away from d = 1 which corresponds to a simple kinetics. For temperatures close to the GT exponent d keeps its low value. At larger departures from GT exponent d increases but in different ways for lower and higher temperatures. In the former case it keeps its value well below d = 1, indicating a complex dynamics. With increasing temperatures d displays a strong systematic increase toward d = 1 (e.g., d ≈ 0.85 for T = 0.5), approaching a simple, noncomplex dynamics. We conclude that while in the interval of temperatures studied in this paper the kinetics of guest molecules is complex, this effect is the strongest close to the GT. 1762

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The Journal of Physical Chemistry B of random, uncorrelated in time and space, collisions with the molecules of the environment, and has a thermal origin. Thus, the observed kinetics of dye molecules is mainly due to the dynamics of parameter C, while the weak complexity results from a correlated dynamics of local voids. Since the amplitude of trial diffusive steps is small, the molecules probe only some small parts of the angular phase space and the contribution of spatiotemporal correlations of the field V(r,⃗ t) to the kinetics is weak. The situation in the inscription phase is very different. The guest molecules in the trans state probe the whole angular phase space because of the random orientation of a trans molecule after the cis → trans transition. The probability of this transition depends on the local structure of the mosaic at some elapsed time after the trans →cis transition took place. In this case the spatiotemporal correlations of the field V(r,⃗ t) become important; in particular, they govern the probability of trapping events. In turn, trapping of a guest molecule in a cis state leads to a wide spectrum of waiting times and well-developed complex dynamics. This qualitative physical picture does not suffice to explain the origin of a rather counterintuitive effect: the largest efficiency of the AHB close to the glass transition (Figure 7a). Intuitively one expects that the AHB process should be the least effective in this case because of the slowing down effect. This process is governed by the spatiotemporal correlations of V(r,⃗ t) but its kinetics depends on the time scales related to trans → cis and cis → trans transitions. Preliminary results show that the minimum in the plot N0−20(T) gradually vanishes as R increases. Nevertheless, parameter N0−20 always differentiates between glassy and melt phases of the matrix. The quantitative analysis of the origin of complex dynamics of guest molecules has to take into account an interplay of time scales related to pct, ptc and of the characteristic times for the spatiotemporal correlations of the field V(r,⃗ t). This analysis is beyond the scope of this paper. 4.2. Localization of Glass Transition via All-Optical Poling. Our study provides tools for the localization of glass transition on the basis of some anomalies of directly or indirectly measurable physical quantities and not of geometrical (like MFV) parameters. The strongest anomaly corresponds to the order parameter ⟨cos3 θ⟩ directly related to the intensity of SHG. The maximum of characteristic time τ in the inscription phase (Figure 9a) is at T = 0.25. The well-defined minimum of exponent d is localized in the interval T = 0.225−0.25 (Figure 9b). The AHB effect, indirectly measured in SHG experiments, is the strongest in an interval T = 0.275−0.3 (Figure 6b, Figure 7a). The power law is present at T = 0.25 and vanishes already at 0.275 (Figure 7b). Those anomalies lie in a rather wide interval of temperatures: 0.225 ≤ T ≤ 0.3. We ascribe the origin of the scatter of those estimates for Tg obtained on the basis of the anomalies of physical parameters to the fact that the glass transition manifests itself through singularities of various physical quantities in different ways which reflect their specific dynamics. 4.3. Prediction of MC Model vs Experiment. Experimental study of the photoinduced SHG in all-optical poling on thin polymer films of PMMA grafted copolymer with DR1 guest molecules was reported in ref 45 for three temperatures: T = 50, 90, and 110 °C. The authors have analyzed only the decay process. We have analyzed the time evolution of χ(2) in the inscription phase using the data retrieved from original data in ref 45. The plots of χ(2)(t) and their fits are shown in Figure 10. The results

Figure 10. χ(2) amplitude for PMMA grafted copolymer with DR1: experimental data for 50 (○), 90 (△) and 110 °C (□). Red solid line and green dashed line represent, correspondingly, exponential and stretched exponential fits. Inset shows the corresponding relaxation times τ (black circles and empty circles) obtained from fitting procedures; the size of the error bars are of the size of graphical symbols. Based on data from ref 45.

of the fitting procedure were inconclusive−both single exponential and stretched exponential fits reproduced the data in a satisfactory way and gave approximately the same values of characteristic times τ (inset in Figure 10). However, in the experiment the ratio of the time of measurement to maximal value of τ was approximately equal 3. We have shown in section 3.2 that reliable conclusions concerning the functional form of χ(2)(t) can be drawn when this ratio is at least 1 order of magnitude larger. Experimental data clearly show a nonmonotonic behavior of τ on the temperature (inset). At maximum, for T = 90 °C, those times are approximately twice larger (τ = 2000 s) than at lower (T = 50 °C) and higher (T = 110 °C) temperatures (τ = 1100 s). This observation remains in a close analogy to slowingdown behavior found in the simulations, see Figure 9a. While in the simulations the maximum was located close to the glass temperature Tg, the situation in the experiment is not clear: while the authors of ref 45 state that glass temperature of the investigated system was Tg = 130 °C, in the literature46 a wide range of possible glass temperature values for such systems is reported, from 85 up to 165 °C. All optical poling of DR1 molecules dispersed in PMMA was studied at room temperature in ref 47. The statistics of the data for the inscription phase was not as good as in the above case and no reliable conclusions could be drawn. We conclude that much longer all-optical poling experiments are necessary to reveal the complex nature of underlying physical processes.

5. SUMMARY We have proposed and studied a simple kinetic Monte Carlo model for the theoretical analysis of the kinetics/dynamics of all-optical poling process in a host (polymer matrix) − guest (azo-dye molecules) system in the limit of low concentration of guest molecules. We have found that the kinetics/dynamics of guest molecules close to the glass transition temperature Tg belongs to the class of complex systems. The “increase” of complexity close to Tg constitutes an analogy to the increase of complexity in a coexistence phase for 2-dimensional Lennard-Jones liquid13 and sheds some light on the structure/dynamics of microscopic/ mesoscopic inhomogeneities in mosaic-like states of the matrix, 1763

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scientific research of young scientists and Ph.D. students. We thank A. Miniewicz for helpful discussions.

which (hypothetically) constitute an origin of complexity in locally ordered but globally disordered systems. We suggest that the large-scale correlations in mosaic-like states play an important role in the inscription phase, leading to ”strong” complexity while the decay phase is mostly driven by short scale collisions of kinetic origin, leading to “weak”complexity. The analysis of both spatial and temporal correlations in mosaic-like states is under progress now and the results will be published elsewhere. Finally, let us discuss briefly some perspectives for a modification of the model onto the case of higher concentration of guest molecules and some related physical effects. The influence of the dynamics of azo molecules on the chain, neglected in this paper, becomes important for sufficiently high concentration. The generalization of the model onto that case is possible but not straightforward. The dye−dye interaction can be studied following the lines of ref 22. The impact of photoisomerization on the dynamics/diffusion of the chain48 can be incorporated into the model following the lines of our paper,18 where a high concentration of azo molecules chemically attached to the chain was studied. In this case a photoisomerization act grants an additional nonthermal trial MC move of neighboring monomer/s. Preliminary results in the two-dimensional case published recently49 and previous studies50 clearly indicate that isomerization changes the character of the diffusion of the chain as a whole. Next, for high concentration of the azo molecules their translation has to be accounted for because it would dynamically modify the local environment of polymer chains and thus would have an impact on their dynamics. However, the modeling of the translation induced by isomerization is a nontrivial problema sound solution to it was proposed in ref 51. An interesting topic which could be studied using the extended model is the modification of the glass transition temperature due to the photoisomerization.23,48 Qualitatively, two main competing effects emerge. First, isomerization increases locally the mobility of the chain, thus decreasing the Tg. On the other hand, the steric effects may hinder this mobility and increase Tg. Let us point out that in the case of high concentration of the azo molecules the mosaic-like states will undergo a change as compared to the case of the matrix without the dyes and the evaluation of additional correlation functions will be necessary. Finally, a replacement in the extended model of azo molecules by dipolar molecules oriented using corona poling method would make possible a theoretical analysis of recent measurements.52

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AUTHOR INFORMATION

Corresponding Author

*(A.C.M.) E-mail: [email protected]. ORCID

W. Radosz: 0000-0002-5641-3344 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This paper is partially supported in the form of a scholarship from funds given to the Faculty of Fundamentals Problems of Technology, Wrocaw University of Science and Technology, by the Ministry of Science and Higher Education in 2016 for the 1764

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