Simulations of Morphology Evolution in Polymer Blends during Light

Article pubs.acs.org/JPCC

Simulations of Morphology Evolution in Polymer Blends during Light Self-Trapping Saeid Biria and Ian D. Hosein* Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse, New York 13244, United States S Supporting Information *

ABSTRACT: Simulations are presented for binary phase morphologies prepared via coupling the self-trapping properties of light with photopolymerization induced phase separation in blends of reactive monomer and inert linear chain polymer. The morphology forming process is simulated based on a spatially varying photopolymerization rate, dictated by self-trapped light, coupled with the Cahn−Hilliard equation that incorporates the free energy of polymer mixing, degree of polymerization, and polymer mobility. Binary phase morphologies form with a structure that spatially correlates to the profile of the self-trapped beam. Attaining this spatial correlation emerges through a balance between the competitive processes entailed in photopolymerization-induced decreases in diffusion mobility and the drive for the blend components to phase separate. The simulations demonstrate the ability for a self-trapped optical beam to direct binary phase morphology along its propagation path. Such studies are important for controlling the structure of polymer blends, whereby physical properties and critical physical and chemical phenomena may be enhanced.

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INTRODUCTION Polymer blends are attractive systems for applications in transport,1−4 thermoelectrics,5 solid-state lighting,6,7 and composites.8,9 An important aspect to enhance their critical physical phenomenon is to establish novel synthetic routes to the rational design of their morphology, in order to leverage structure−property relationships. To this end, control over polymer blend morphology remains an important area of investigation. One attractive approach to prepare different morphologies is through irradiation of mixtures consisting of one or more photopolymerizable components. The increase in molecular weight associated with photopolymerization renders the components immiscible, and spontaneous phase separation occurs via a spinodal decomposition mechanism, in what is referred to as photopolymerization-induced phase separation (PIPS). Currently, PIPS approaches have fallen under either one of two categories: uniform irradiation of either constant or temporally varying intensity10 or holographic polymerization using a constructive interference field.11 Uniform irradiation is attractive for its scalability; however, precise control over the morphology is allusive owing to the stochastic nature of phase separation. Holographic polymerization provides precise structures; however, there are scalability and complexity issues associated with establishing 3D constructive fields over large scales and depths. Hence, a new approach to direct polymer blend organization is desirable. It is well-known that photopolymerizable media display nonlinear optical behavior, via photopolymerization-induced changes in refractive index, that enable nonlinear optical waveforms to propagate through the media.12−14 For example, a microscale optical beam will experience a self-focusing © 2017 American Chemical Society

nonlinearity that counters its natural divergence in space over its propagation path. The dynamic balance between divergence and self-focusing results in a “self-trapped” beam, a nonlinear waveform characterized by divergence-free propagation in its own self-induced waveguide. As light self-trapping is facilitated by an irreversible, spatially local photopolymerization reaction, the beam imprints a permanent, microscale waveguide structure of cylindrical geometry over its propagation path.14,15 This dielectric structure is characterized by a higher degree of polymerization (i.e., molecular weight, cross-linking) which establishes the appropriate core-cladding refractive index profile to guide light. Self-focusing and self-trapping of optical beams in photopolymerizable media have been employed to fabricate microstructured materials, in a process referred to as “light induced self-writing”. An enticing area of inquiry is in eliciting nonlinear light propagation in photoreactive polymer blends. Along with the phenomenon of self-trapping, the consequent increase in polymerization degree in the region of self-trapping can result in the occurrence of spatially localized PIPS. We recently observed in a irradiated photoreactive blend the formation of binary phase morphologies with structures that closely replicate an arrangement of spontaneously emerging self-trapped beams. 16 The morphology consisted of one polymer component in the regions of self-trapped light, and the other component phase separated into the surroundings. Phase separation has also been observed in epoxide−solvent mixtures, Received: April 10, 2017 Revised: May 10, 2017 Published: May 19, 2017 11717

DOI: 10.1021/acs.jpcc.7b03348 J. Phys. Chem. C 2017, 121, 11717−11726

Article

The Journal of Physical Chemistry C

we explored a larger (20) and smaller (5) beam size. We started with a uniformly distributed mixture of polymer 1 (reactive monomer) and polymer 2 (inert linear chain) of equal volume fractions, and we define herein φ as the volume fraction of polymer 1. The refractive index, n, of the blend was calculated as an average refractive index based on the weighted mixture rule:28

in which self-trapped beams produced cylindrical, cross-linked epoxy waveguides, with solvent in the surrounding regions.17 Light-induced self-writing of waveguides has been employed in a blend of two monomers, whose phase separation establishes the core-cladding refractive index profile to enable waveguiding.14,18,19 To attain spatially correlated morphologies over a range of binary polymer mixtures, it is critical to understand the effects of several determinative factors on concurrent light selftrapping and phase separation and how to control blend morphology with self-trapped beams. To this end, a theoretical framework by which generalized understanding may be established would be highly instructive. PIPS in polymer blends has been theoretically examined for the case of uniform irradiation20 and holographic polymerization.21,22 Jisha and coworkers theoretically examined self-writing and phase separation in liquid crystal monomer mixtures.23 However, there have been no theoretical studies on the evolution of morphology in photoreactive polymer blends during the self-trapping of a transmitted optical beam, especially for a range of blends characterized by different polymerization rates, molecular weights, and miscibility. For the self-trapping of light to be a feasible process to control the organization of a wide range of polymer blend compositions, theoretical insight into morphology evolution is necessary. Here, we present a theoretical study of the evolution of polymer blend morphology over the course of irradiation with a microscale transmitted optical beam that undergoes selftrapping. Our blend consists of a photoreactive monomer and an inert (nonreactive) linear chain polymer. Under appropriate blend conditions, PIPS is induced along the path length of the self-trapped beam, and high spatial correlation between the light profile and the binary phase morphology is achieved. Morphology is mapped over parameters that vary the photoreactivity and blend miscibility, and we explain the resultant morphologies in terms of the dynamic and temporal balance between increased polymerization degree and diffusion, both of which are induced by self-trapped light. To the best of our knowledge, this is the first comprehensive study that provides theoretical insight into the determinative factors at play in morphology evolution of a polymer blend irradiated with self-trapped light. Furthermore, our approach is a new computational methodology that combines light propagation, photopolymerization, and phase separation of polymer blend systems and is of general interest for their light directed synthesis and organization.

n = ϕn1 + (1 − ϕ)n2

where n1 is the refractive index of the polymer 1 and n2 is the refractive index of polymer 2. We selected refractive index values for polymer 1 and 2 to emulate those for trimethylolpropane triacrylate (n1 = 1.474) and dimethylsiloxane oligomer (n2 = 1.446), respectively, for which we have observed light self-trapping in our previous work.16,29 Photopolymerization. The change in the degree of polymerization of polymer 1, Xn, was expressed as a function of the light intensity (I) and the polymerization rate constant (kp):30

dX n = k pI1/2X n dt

⎛ 1 ⎞ n1 = n1,m + Δn⎜1 − ⎟ Xn ⎠ ⎝

(4)

where n1,m is the refractive index of the monomer of polymer 1 (1.474), and Δn is the refractive index of the polymer 1 at saturation, which we estimate as 0.007 for simulations herein, which is typical for acrylate systems. The exception is for two simulations for which we changed Δns to 10× and 0.1× this value. n1 is maximum when fully polymerized (i.e., n1 = n1,m + Δn, as Xn → ∞). Equations 2, 3, and 4 provide n in eq 2 with the intensity dependence that establishes optical nonlinearity, whereby nonlinear light propagation may be simulated with eq 1. Specifically, such a photoreactive system possesses an integrating nonlinearity,33 whose refractive index change is the sum of all changes over time. Kewitsch et al. related the refractive index change directly to the light intensity.12 Whereas, herein we relate refractive index change to intensity through Xn (i.e., eq 3 → 4 → 2), because the calculation of Xn itself is also required to perform calculations of free energy and for phase separation. Phase Separation. In order to simulate diffusion during phase separation, we employed the Cahn−Hilliard equation:34

EXPERIMENTAL SECTION Theoretical Framework. Light Propagation. Propagation of a microscale optical beam (Gaussian profile) of diameter d through the mixture was computed via the beam propagation method (BPM)24−26 to solve the Helmholtz equation in the paraxial approximation:27 ⎞ ∂ψ 1 1 ⎛ n2 + ∇⊥ ψ + k 0⎜ 2 − 1⎟ = 0 ∂z 2k 0 2 ⎝ n̅ ⎠

(3)

Equation 3 can be derived from with photopolymerization rate equations under the steady state approximation,31 and the relation Xn = [M]0/[M],32 where [M]0 and [M] are the monomer concentrations at the beginning of the reaction and at some time thereafter, respectively. Equation 3 can model the polymerization of monofunctional or polyfunctional monomers, which would form linear or cross-linked structures, respectively. Absorbance losses as well as aspects of photoinitiation efficiency were excluded from consideration in the simulations. To incorporate the degree of polymerization in the refractive index of polymer 1, we calculated n1 as

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j

(2)

(1)

where ψ is the light field, z is the propagation direction, k0 is the wave vector, n is the refractive index, n̅ is the average refractive index of the medium, and ∇⊥ is the transverse Laplacian (∂2/ ∂x2 + ∂2/∂y2). Light simulated herein represents propagation of a coherent source (i.e., a laser beam). The diameter of the beam for all simulations was 10, except for two simulations for which

⎡ ⎤⎤ ⎡ ∂f (ϕ) ∂ϕ = ∇·⎢M ∇⎢ − 2κ ∇2 ϕ⎥⎥ ⎦⎦ ⎣ ∂ϕ ∂t ⎣

(5)

where M is the mobility, κ is the gradient energy coefficient, and f(φ) is the Flory−Huggins (FH) free energy equation 11718


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and (5) the updated refractive index profile. This new refractive index profile is used as input for the beginning of the next loop. Iterating through these steps, the evolution of both the transmitted optical beam and the polymer morphology were determined. The time step for each loop was considered as constituting a 100 ms duration (i.e., irradiation with the beam), which is an approximate time to yield a polymerization event in photoinitiated acrylate-based monomers under laser irradiation.12 The time step for diffusion was 10 ms, resulting in 10 time steps per iteration (i.e., 10 ms × 10 = 100 ms). This time step was sufficient to provide stable solutions for the phase evolution. The blends are thermodynamically stable at the onset of the simulation, namely their temperature−composition coordinate exists in the single phase region and then enters under the spinodal line over the course of photopolymerization (see Supporting Information). We explored different polymerization rates by varying the values in the polymerization rate constant (kp), different miscibilities via the interaction parameter (χ), and different polymerization degrees for polymer 2 (N2). All simulations were run for a total simulated duration of 60 s, which was sufficient for the phase morphology to evolve to its fullest extent, except for kp = 0.01, which was run for a duration of 120 s. In this study we did not vary light intensity; we aimed herein to examine in particular changes in the parameters of the mixture itself (i.e., reactivity and miscibility). Varying intensity will have a similar effect as increasing the polymerization rate constant, as both are comprised in the constant factor (I1/2 or kp) in the polymerization rate equation (eq 3). Table 1 summarizes the parameters explored in this study.

based on a combinatorial lattice theory for a binary polymer blend, expressed as follows:34 f (ϕ) =

⎤ (1 − ϕ) RT ⎡ ϕ ln φ + ln(1 − ϕ) + χϕ(1 − ϕ)⎥ ⎢ ν ⎣ N1 N2 ⎦ (6)

where R is the ideal gas constant, T is the temperature in kelvin (298 K), ν is the volume of a lattice site, N1 and N2 are the degrees of polymerization of the two polymers, and χ is the interaction parameter. As polymer 1 is photoreactive (N1 = N(t) = Xn), its degree of polymerization increases over the course of irradiation. Polymer 2 has a fixed degree of polymerization (N2). We employed a polymerization-dependent mobility to simulate an expected decrease in diffusion over time. The mobility was calculated as34

M0 X̅ n

M=

(7)

where M0 is the mobility of the monomer estimated as M0 =

35

Dν RT

(8) −13

In eq 8, D is the diffusivity, which is estimated as 5 × 10 m2/ s for the polymers,36−40 and ν is chosen as the molar volume of water, 1.8 × 10−5 m−3/mol. The gradient energy coefficient was estimated using the random phase approximation:41 κ=

RTa 2 3ν

(9)

where a is the monomer size, estimated to be 6.3 × 10−8 m. Chan et al. employed in simulations of uniform irradiation a gradient energy coefficient that was dependent on polymerization degree (i.e., κ = κ0N).34 In our case, polymerization is highly localized to the region of the optical beam, and the surrounding mixture remains unreacted. To account for this, the gradient energy coefficient was kept constant. Simulation Methods. Calculations were carried out in a 3D volume (x, y, z) spatially discretized into 33 × 33 × 1001 points. The spacing between points in all dimensions was normalized to represent a 1 μm unit distance, in order to simulate the physical scenario of a microscale optical beam and morphology evolution at the microscale. Light propagation was along the z-axis (consisting of 1001 points, 1 mm total length); we also refer to the z-dimension as the “depth”. We selected a total depth of 1 mm, as this is more than sufficient for most thin-film applications of polymer blends. Herein, all 3D distributions for both polymer concentration and also light intensity are visually displayed with xz- and yz-plane slices that cut through the center of the simulation cell. Beam propagation was carried out using commercial software (BeamPROP, Synopsys). The Cahn−Hilliard equation was solved numerically, in MATLAB, through spatial discretization in both space and time, and using spectral methods as described elsewhere.42,43 Simulation Procedure. Simulation of optical beam propagation and the evolution of polymerization and phase separation was achieved through a temporal loop, for which each iteration sequentially calculates the effect of the following phenomena: (1) propagating a beam through the mixture; (2) the intensity- and position-dependent increase in polymerization degree of polymer 1; (3) the updated FH free energy of mixing; (4) time-stepping the diffusion of the two components;

Table 1. Parameters Explored in This Simulation Study parameters

values

Δn φ χ N2 kp d I λ n1 n2

0.0007, 0.007, 0.07 0.25, 0.50, 0.75 0.01, 0.1, 0.2, 0.5, 1.0 1, 5, 50, 500, 5000 0.01, 0.1, 1, 10 5, 10, 20 0.01 0.532 1.446, 1.474 1.446, 1.474

We selected beam diameters particularly for their significant divergence over a path length of 1 mm. This was confirmed through calculation of the beams’ Rayleigh ranges (zR), namely, the distance along the propagation direction at which the crosssectional area is doubled: πr 2 (10) λ where r is the initial beam radius and λ is the wavelength of light. The calculated zR values were 36.9, 147.6, and 590.5 μm for beam diameters of 5, 10, and 20 μm, respectively. Hence, a 1 mm path length exceeds this range, and appreciable divergence is expected whereby self-trapping can be examined. We also selected these beam diameters as it is a common range examined in experimental work on light-induced self-writing.12,14 Based on experiments on curing kinetics with lasers,44 the dimensionless values of kp used herein would correspond to realistic steady-state polymerization rate constants of 0.1−10 zR =

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Figure 1. Time series of the evolution of a transmitted optical beam and polymer morphology, illustrating the coupling of light self-trapping to phase separation. Parameters for data shown are φ = 0.5, χ = 0.5, kp = 1, and N2 = 50.

mol L−1 s−1, and the intensity I would correspond to a realistic value of ∼100 mW/cm2.

cylindrically symmetric. As a result, the phase retains its cylindrical symmetry over the depth of the blend. The binary phase structure forms in a relatively short time (∼15 s), which is similar to the duration over which light-induced self-writing forms cured structures in photopolymerizable resins.12,14 The evolution also reveals that the overall dynamics of phase separation are predominantly coaxial to the propagation axis of the self-trapped beam, although there is a “frontal kinetic” aspect to phase evolution, as phase separation can be observed to begin at the front of the simulated volume. Figure 2 maps the final morphology (φ = 0.5) over a range of polymerization rate constants and χ values, particularly for N2 = 50 (see Supporting Information for other values of N2). Figure 2 reveals the dependence of the morphology on both photopolymerization rate and blend miscibility. For certain values of kp and χ, a phase region of cylindrical geometry rich in polymer 1 is attained over the entire depth (1 mm) of the blend and that spatially correlates to the self-trapped beam profile (see Supporting Information for complementary optical beam profiles for data shown in Figure 2 as well as for all other molecular weights N2), whereas a spatially correlated structure is lacking for other parameters. Namely, high values of χ show low correlation, except for when kp = 1. Also, blends with very low (kp = 0.01) and very high (kp = 10) rate constants show lower correlation over the range of χ values explored. The degree of phase separation, i.e., richness of the cylindrical phase, also increases over the range of kp from 0.01 to 1 and for χ from 0.01 to 1; however, for a high kp of 10, this trend does not persist. Furthermore, in conjunction with phase richness, with increasing χ, the width of the cylindrical phase becomes smaller. For very low values of kp and χ (i.e., kp = 0.01 and χ < 0.5), the blends do not undergo phase separation, which is expected according to their corresponding temperature−composition coordinates not entering the two-phase coexistence region (i.e., under the spinodal curve) in the phase diagram (see Supporting Information).

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RESULTS Figure 1 reveals the coupling of optical beam propagation to morphology evolution in the photoreactive polymer blend. The consequent increase in refractive index associated with photopolymerization results in the initially divergent beam to undergo self-focusing, which dynamically balances to form a self-trapped beam. Concurrently, the mixture undergoes phase separation with the concentration of polymer 1 increasing in the region of self-trapping and polymer 2 expelled into the surrounding regions. This leads to a stable phase-separated morphology whose 3D spatial profile is commensurate with the self-trapped beam: polymer 1 in the region of self-trapping as a phase region with cylindrical geometry and polymer 2 in the surrounding dark regions. Owing to conservation of mass (C1 + C2 = 1), the distribution of polymer 2 is complementary to the distribution of polymer 1 (see Supporting Information for polymer 2 distributions corresponding to Figure 1). Hence, the concentration distributions of polymer 1 (C1) are shown hereon. Figure 1 shows an exemplary parameter set for which the polymer blend morphology spatially correlates to the profile of the self-trapped beam. The multimodal nature of the transmitted light is observable in the spatial profiles in Figure 1 and is expected from the waveguide characteristics of the selfwritten structure.45 The modes appear as oscillations in beam diameter over the propagation path, owing to the sinusoidal profiles of the modes expected for the graded index profile46 induced by the optical beam. Such oscillations were also observed by Kewitsch et al. in single photopolymer systems12 and are characteristic of self-trapped beams.47 The cylindrical phase rich in polymer 1 showed no replication of any multimodal profile, nor any oscillatory shape, most likely due to it responding to the time-average intensity distribution of the self-trapped beam, for which the multimodal optical structure is averaged into a relatively Gaussian-shaped beam, which is 11720


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Figure 2. Resultant distribution of polymer 1 (C1) mapped over polymer rate constant (kp) and interaction parameter (χ). Parameters for data shown are φ = 0.5 and N2 = 50.

formation kinetics that attain three general morphologies: (1) spatially correlated phase separation in the region of selftrapping (Figure 3a−d), (2) spatially correlated phase separation with enrichment of both the cylindrical phase and the surroundings (Figure 3e−h), and (3) phase evolution not spatially correlated to the self-trapped beam (Figure 3i−l). These plots reveal the kinetics for their corresponding morphologies shown in Figure 2. For the first observed morphology, the refractive index shows a gradual rise to saturation over the course of irradiation (plots over the first 15 s was sufficient to show changes, as the system was static thereafter). This refractive index profile over time is typically observed in experiments on light self-trapping.12,14 Selftrapping occurs rapidly within the first second of irradiation. The illuminated region shows a gradual enrichment of polymer 1 due to phase separation, which eventually ceases in accordance to a decrease in the mobility of the components

We tracked four key parameters during the process in the region of illumination (nominal diameter of d = 10) of the optical beam to reveal the underlying kinetics of formation: the refractive index, the degree of self-trapping, the average concentration of polymer 1 (C̅ 1), and 1/N1 which is proportional to the mobility. The degree of self-trapping was quantitatively determined by examining the power output at the end of the simulation cell, over an circular area (diameter d) centered on the beam’s propagation path. Self-trapping is indicated by the rise in the ratio of the power at the end of the simulation cell to the power at input (P/P0). Lower values, particularly at the beginning of the simulation, indicate beam divergence and over the course of irradiation self-trapping can result in light perfectly confined to the self-written high refractive index region over its entire propagation path (i.e., P/ P0 = 1). Figure 3 shows plots of the four key parameters over time for three different blend conditions that exemplify the 11721


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Figure 3. Kinetics associated with self-trapping and consequent phase separation over the course of irradiation for parameters: (a−d) kp = 1, N2 = 50, and χ = 0.01; (e−h) kp = 1, N2 = 50, and χ = 0.5; (i−l) kp = 10, N2 = 50, and χ = 1. The plateau apparent in (l) is an artifact from the limit to very large values for N1 that are storable in MATLAB.

by 3 orders of magnitude, such that morphology evolution is arrested. This morphology has the width of the cylindrical phase approximately equal to the diameter of the beam. For the second observed morphology, a rise in the refractive index occurs at the beginning, along with rapid self-trapping of the optical beam and a concurrent enrichment of the region with polymer 1. In contrast to the first morphology, the average concentration of polymer 1 subsequently decreases over time, until the dynamics are arrested. This corresponds to a decrease in the width of the cylindrical phase, due to enrichment, such that it becomes smaller than the diameter analyzed. This artificially yields a lower C1 because it now includes the surroundings where polymer 1 concentration is low. For the third morphology, a sudden rise in refractive index is followed by a sudden drop. Self-trapping occurs; however, light confinement over the course of irradiation is not maintained. C1 rapidly rises at the beginning of irradiation, but then suddenly drops, owing to the random phase morphology bringing both polymer 1 rich and polymer 2 rich phases into the region of self-trapped light. We also considered the effect of different volume fractions on resultant phase morphology. Figure 4 shows morphologies for blends with φ = 0.25 and 0.75, for kp = 1, over a range of χ values at N 2 = 50 (see Supporting Information for morphologies at other N2 values). For φ = 0.75, spatially correlated morphologies can be observed with higher richness as compared to φ = 0.5, at the same values of χ and N2. Whereas for φ = 0.25, the phase-separated morphologies show polymer 2 positioned in the center and polymer 1 in the surroundings. This inverted morphology also replicates the profile of a divergent beam, which is corroborated by that fact that no self-trapping was observed (see Supporting Information).

Figure 4. Spatial distribution of polymer component (C1) for volume fractions of φ = 0.25 and φ = 0.75, mapped over χ, for kp = 1 and N2 = 50.

In the last part of our study, we examined the effect of other conditions on the morphology, namely, conditions of different saturable refractive index (Δn) and beam size. We also switched the refractive index values of polymers 1 and 2, such that phase separation would result in a low-index core, higher-index cladding type waveguide. Morphologies that spatially correlate with the self-trapped beam were observed under all such conditions (Figure 5). Specifically, when the refractive index values were switched, the polymer 1 rich phase does not extend over the entire depth. The variation in beam size and the resultant cylindrical phases are of similar diameter, which shows 11722


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correlation to the self-trapped beam profile. If the χ value is significantly high, then even at high polymerization rates the initial growth will induce rapid phase separation before any correlated structure can be captured. Hence, a temporal balance between the thermodynamic drive to phase separate and the decrease in mobility must be achieved in order to capture spatially correlated structures. These arguments explain the variation in morphology observed (Figure 2) as resulting from variation in this competition over the examined range of rate constants, χ values, and polymerization degrees (N2) and also identify the optimal conditions at which spatially correlated morphologies are achieved. Hence, while a self-trapped beam is easily attained in the blends studied, the polymerization and consequent phase separation that it induces, depending on the reactivity and miscibility of the blend, dictate the subsequent evolution and whether or not spatially correlated morphologies are obtained. Change in the volume fraction shifts the temperature− composition coordinate in the phase diagram, whereby the mixture enters under the spinodal curve at different times over the course of irradiation. This leads to phase separation proceeding under different conditions of mobility and drive to phase separate. Namely, owing to the asymmetry of the spinodal curves, especially for higher N2 values, higher volume fractions of polymer 1 (φ > 0.5) will enter into the spinodal regime at earlier times when N1 is lower (i.e., higher mobility), relative to φ = 0.5, whereas lower volume fractions (φ < 0.5) will enter into the spinodal regime when N1 is higher. As a result, phase separation can proceed for a longer duration for φ > 0.5 and thus attain richer phases, as opposed to lower volume fractions of φ < 0.5 for which under the same conditions low richness is achieved. The inversion of the spatial locations of the phases for volume fractions of φ = 0.25 can be understood as an effect of the “uphill” diffusion during phase separation; namely, when φ ≥ 0.5, local regions of higher free energy will become rich in polymer 1, whereas when φ < 0.5, local regions of higher free energy will become depleted of polymer 1. Consequently, for φ = 0.25, polymer 1 diffuses out of and polymer 2 into the region of illumination where the blend first becomes unstable. Consequently, the refractive index profile is inverted, and the consequent low-index core, high-index cladding is unable to elicit self-trapping of the optical beam. Spatial correlation of the morphology prefers that the photoreactive polymer component have the higher refractive index, such that a high-index core, low-index cladding is established upon phase separation, to maintain the capability to confine and waveguide light. A lower index core, higher index cladding results in an efflux of light intensity, which can inhibit the phase separation from extending over the entire depth of the blend. Initial self-trapping is still possible in this case because in an initially homogeneous mixture the refractive index is an average of the two polymer components, and any polymerization yields a rise in refractive index relative to this homogeneous medium, which creates a high-index-core, lowindex cladding. However, when phase separation occurs, the surroundings can gain a higher refractive index value, such that intensity is lost from the self-trapping region. This is shown in the intensity profiles of the light, which become splayed over the propagation path into the surroundings, away from the center of the beam. We confirmed that the irradiated region must attain a higher refractive index by showing for blends with φ = 0.25 that switching the refractive index values of polymers 1 and 2 enables self-trapping (see Supporting Information). In

Figure 5. Morphology for different refractive index values, beam diameter, and saturable refractive index. Results are shown for parameters φ = 0.5, kp = 1, χ = 0.5, and N2 = 50.

that phase separation occurs predominately in the region of the self-trapped light. This indicates the precision of the process, whereby the width of the cylindrical phase region is defined by the beam width. This also explains why the concentration profiles remain relatively the same as the initial concentration profile for regions further away from irradiated region. Also observed, the richness of the cylindrical phase increases with either an increase in the beam width or a decrease in the saturable refractive index. The ability to attain spatially correlated morphologies for a saturable refractive index value as low as 0.0007 indicates that monomers with even very small rises in refractive index can enable self-trapping and spatially correlated morphologies; in other words, a wide range of polymer systems may be patterned.

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DISCUSSION The morphology of the blends simulated herein are determined by two competitive processes: the mixture’s drive to phase separate into two components and polymerization-induced decreases in mobility which keeps to two components together.10 Yet, polymerization can be either cooperative or antagonistic to phase separation, as it initially induces the onset of phase separation but later inhibits it by decreasing diffusion. Physically, the decrease in mobility is associated with the increase in viscosity of the system owing to greater molecular weight chains as well as cross-linking. Hence, while selftrapping of light is easily achieved in such photopolymerizationdependent refractive index media, the ability to attain binary phase structures that spatially correlate to the self-trapped beam depends on the balance of these two processes. Herein, the reactivity of the blend and miscibility of its components are critical. At very low polymerization rates, no appreciable phase separation occurs because the very slow polymerization cannot drive phase separation over the duration explored. On the other hand, very high polymerization rates rapidly decrease mobility such that phase separation proceeds under highly inhibitive conditions, which can keep the system in a homogeneous phase or yield only low richness phases. Generally, higher polymerization rates are necessary to counter the stronger drive to phase separate in blends with higher molecular weight (N2) or χ. It appears that for higher polymerization rates spatial correlation begins to emerge at the front of the blend, but the blend “freezes” too quickly for spatially correlated morphologies to extend over the entire depth. If the drive to phase separate is too strong, and a sufficient polymerization rate is not induced so that the mobility is reduced at an appropriate rate, the phase separation proceeds uncontrolled and loses its spatial 11723


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weight initially reduces the drive to phase separate (i.e., lower free energy), but its growth will contribute to the decrease in mobility of the system. This was our approach to achieve phaseseparated morphologies for polymer blends with calculated χ values close to 2.16 The results reported herein are corroborated by our recent experimental in situ study on structure evolution in binary polymer blends,48 wherein morphology was mapped over exposure intensity (to change kp) and weight fraction. Spatially correlated phase-separated structures were achieved for relatively lower intensities (i.e., lower kp), whereas spatial correlation was lost at higher ones. However, spatially correlated morphologies were found for φ < 0.5, whereas for φ ≥ 0.5 significant evolution in phase separation resulted in random structures. This is in contrast to the spatially correlated structures observed herein for φ = 0.5. This may be explained by the significantly lower polymerization rates realized in our experimental work, as compared to what was simulated herein, which cannot capture this morphology for φ ≥ 0.5. Nevertheless, attaining a spatially correlated morphology for large values of φ is possible, if higher intensities are used. Shoji and co-workers employed resin mixtures with φ ∼ 0.7 and were able to attain uniform cylindrical phases over 20 mm in length by irradiating with 380 mW blue laser light for 10 s.49 Clearly in this case the kinetics are fast enough to freeze in the structure. Therefore, the conclusions from this simulation work are supported by experimental results, indicating that the principles established herein may be used to attain experimental control over polymer blend morphology. Our simulations specifically reveal phenomenon for coherent light (i.e., lasers), yet self-writing is possible with spatially and temporally incoherent light, both UV and visible. Incoherent optical beams are more divergent than coherent beams, owing to the former’s highly “speckled” and multimodal nature. Nevertheless, even modest index values have been shown to trap incoherent light.13 The inherent femtoscale fluctuations in amplitude and phase in incoherent light are not deleterious to its self-trapping, as the slow chemical reaction of polymerization responds to the time-average intensity of the light, in which such fluctuations wash out. Hence, our findings herein can apply to incoherent light. Indeed, as previously mentioned, our recent experimental work48 has demonstrated the ability for self-trapped incoherent light to be employed to control polymer blend morphology in the same way as revealed by this study. While kp and I have the same effect on polymerization rate, nonlinear wave propagation has experimentally shown an intensity dependence. At higher intensities the transmitted optical beams undergo nonlinear effects such as filamentation, self-phase modulation, diffraction rings, and excitation of high order modes.50−53 Self-trapping occurs at the lowest intensities over this intensity range, and our selected intensity was suitable to observed and study this phenomenon. Our future studies on intensity dependence will examine coupling of such complex nonlinear waveforms to phase separation.

this case, spatially correlated morphologies are attained because the influx of polymer 2 (which now has a higher refractive index) establishes the high-index core, low-index cladding to trap light. Our results show that the mechanisms (self-trapping, onset of mixing instability, phase separation, and arrested evolution of structure) entailed in morphology evolution are sequential; namely, self-trapping is rapidly established, and this self-trapped light drives polymerization, which then induces the system to become unstable and to phase separate, until reduction in mobility freezes in the structure. Under conditions that lead to spatially correlated morphologies, the predominant direction of diffusion for the components is coaxial to the beam path. The slight frontal kinetic aspect of phase separation can be explained by the initially divergent beam being most intense at the front of the simulation volume, and hence polymerization degree is higher than further along the depth. This results in the local onset of phase separation beginning at the front and then rapidly “propagating” over the depth of the blend. The rapid self-trapping of light enables the process to be unaffected by any possible early onset of phase separation, such that selftrapping is always established; however, depending on morphology evolution, the self-trapped beam can later become disrupted from its well-confined, wave-guiding state. We found no instance of the initial onset of light self-trapping being disrupted or inhibited by the blend, owing to its need for only a modest rise in refractive index, which can occur before any significant phase separation. The smaller size of the cylindrical phase with increase degree of polymerization is in accordance with conservation of mass: as the cylindrical phase includes more polymer 1, polymer 2 must be expelled into the surroundings, and narrowing in width of the cylindrical phase is a consequence. The increased richness in the cylindrical phase with either an increase in beam diameter or a decrease in Δn is due to the light intensity being distributed over a larger volume (total beam power is fixed), which yields lower polymerization rates and thus slower changes in mobility that allows diffusion to persist for longer. Herein, spatially correlated morphologies were achieved over a depth of 1 mm, yet light self-trapping can be expected to pattern blends with greater thicknesses. From the perspective of materials synthesis, the implications of the results are instructive. First, the interaction parameter for a blend may be determined from the solubility parameters (δ) of the two components, according to the following equation:32 χ=

Vr [δ1 − δ2]2 RT

(11)

This value for χ will inform on the conditions needed for controlling morphology. Experimentally, photoinitiator would be varied to control the polymerization rate (i.e., increasing kp) in order to tune morphology to attain spatially correlated morphologies. Higher photoinitiator concentrations will be necessary for blends with higher χ and longer linear chains in polymer 2. Below optimal values phase separation will not spatially correlate to the beam, nor even be induced, and above optimal values the richness of the phases will be lower or phase separation significantly inhibited. Our results also indicate that for blends with very high values of χ, attaining correlated morphologies might pose a challenge. One way to mitigate this experimentally is to employ a lower molecular weight linear chain that can either be cured after phase separation or one that reacts at a slower rate during irradiation. Its lower molecular

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CONCLUSION In summary, we have theoretically shown self-trapping of optical beams and consequent phase separation in binary blends of photoreactive monomer and linear chain polymer. Self-trapping is attained over all blend conditions, as it depends only on refractive index changes. Under suitable conditions, the optical beam remains trapped and binary phase structures form 11724


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that spatially correlate to the optical profile. The wide range of miscibilities over which spatially correlated morphologies were observed implies potential applicability of this approach to pattern a broad range of polymer blend compositions. The cylindrical phase introduces strong directionality and anisotropy of spatial composition in the blend, which can be favorable for critical physical phenomenon, particularly for thermal, proton, or ion transport, as well as for anisotropic mechanical properties. If the cylindrical phase can be chemical removed (i.e., a porogen), then cylindrical pores may be attained to create membranes that enable solvent flux or particle flow. Hence, we envisage that this work will provide guiding principles to direct the morphology of photoreactive systems using transmitted optical beams. Studies on ternary blends and other blend compositions as well as multiple beam propagation are currently in progress and will be reported in the future.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b03348. Final morphologies for other values of volume fraction, χ, and N2 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. ORCID

Ian D. Hosein: 0000-0003-0317-2644 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support from the College of Engineering and Computer Science at Syracuse University. REFERENCES

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Simulations of Morphology Evolution in Polymer Blends during Light

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