Diffusion of copolymers composed of monomers with drastically different friction factors in copolymer/homopolymer blends

Blending multicomponent miscible polymer systems has proven to be a cost effective method for developing unique materials that feature a mixture of chemical and physical properties. However, many polymer mixtures that are predicted to have desirable properties are not miscible in one another and, as a result, blending these materials requires a copolymer compatibilizer composed of the molecular components of each individual homopolymer. This compatibilizer selectively segregates to the interface between the two immiscible homopolymers, thereby creating a defined interface and therefore promoting dispersion among the two materials.^1,2 The speed of this segregation process plays a crucial role in the effectiveness of a chosen copolymer as a compatibilizer; the faster the copolymer diffuses through the homopolymer matrix, the higher the probability it reaches this interface. The copolymer must be able to reach the interface between the immiscible homopolymers within an appropriate time period, as defined by the processing conditions, in order to ensure compatibilization.

The dynamics of polymer chains within multicomponent homopolymer/copolymer systems are not easily predicted because the role of connectivity on the motion of the different components within an individual polymer chain is not well understood. Moreover, the impact of connectivity on polymer chain dynamics affects the bulk properties of the blends and therefore the potential uses for the system. Recent work studying the impact of connectivity of segments in partially miscible homopolymer blends^3–5 and multi-block copolymers^3,4 suggests that a potential driving force that controls the dynamics of individual chains in a polymer blend is the local concentration of the segments around a given polymer chain. However, many of these studies only account for the connectivity between polymer segments and bulk composition fluctuations in the blend, while neutron reflectivity experiments⁵ suggest that the sequence distribution of the segments within an individual copolymeric chain plays a crucial role in the local concentration around the polymer. Simulation studies of homopolymer/copolymer systems have shown that the thermodynamic interactions between interacting monomer segments have a critical impact on the local structure of the copolymer and the composition of its local environment and thus dynamics of the copolymer.^6,7

In viscosity and tracer diffusion studies on styrene-MMA block copolymers,⁸ the effective chain friction factor $ζeff$ was experimentally measured and compared with the predictions of two different mixing rules: the first assumes a composition weighted average of the friction factor of pure PMMA and PS melts; in the second, local motion of a given monomer is dependent on its friction factor attenuated by the composition of its local environment. For example, using the first set of mixing rules, a given styrene monomer will move at the same speed regardless of the composition of the volume around it, while using the second set of rules, the presence of a MMA monomer in the moving styrene’s local volume will slow the styrene down.

While neither of these two models exactly describes the resulting experimental data, the data appear to exhibit a nonlinear dependence of $ζeff$ on copolymer composition, in general agreement with the neutron reflectivity experiments mentioned above. The combined results of these studies demonstrate that both the connectivity and thermodynamic interactions between the different monomer components impact the local concentration fluctuations around a copolymer. Therefore, connectivity and thermodynamic interactions become important driving forces that govern the dynamics of copolymers, including copolymer diffusion, in homopolymer blends.

In this work, we have developed a Monte Carlo simulation that monitors the diffusion of an AB copolymer containing two different monomers and examined the importance of their thermodynamic interactions on the diffusion of the copolymer in an A homopolymer matrix. In this study, the variation of the effective friction factor, $ζeff$⁠, of each monomer is also accounted for. These features allow us to adjust the monomer connectivity, the thermodynamic interactions between the monomers which manifest within a chain, and between copolymer and homopolymer matrix and define the chain friction coefficient of each component independently, allowing for the measurement of the combined effect of these parameters on copolymer chain diffusion. The system chosen is designed to mimic an experimental polystyrene/poly(methyl methacrylate) copolymer, thus the connectivity of the chains is defined by experimentally determined reactivity ratios of MMA and styrene. Further studies to determine the effect of varying this connectivity are of interest for future studies. Using this technique, we have developed a method to implement computer simulations in a single experiment to probe the combined effect of both the connectivity among different monomers in the copolymer chain along with changes in local composition fluctuations on the dynamics of each component in multicomponent polymer blends.

The model used in this work is the Bond Fluctuation Model (BFM) which has been successfully used in previous studies to simulate dynamics in polymer blends.^9–14 In the three-dimensional version of the bond-fluctuation model,^11,12,15,16 each monomer represents a Kuhn segment and occupies eight sites on a simple cubic lattice. Monomers on a chain are connected with one another by bonds whose lengths vary in the range 2 ≤ l ≤ $10$⁠. Only six possible bond classes are allowed: [2,0,0], [2,1,0], [2,1,1], [2,2,1], [3,0,0], and [3,1,0], where [ ] implies all permutations of the coordinates with either positive or negative signs. This restriction leads to 108 possible bond vectors and prevents chain crossing. As a result of these restraints, the bond fluctuation model incorporates some of the flexibility associated with an off-lattice model while maintaining the advantages of working on a lattice. Previous studies indicate that the model can effectively capture the dynamics of polymer melts, as well as blends.

The modified model used in this study varies from the standard bond fluctuation model by allowing the monomer segments to overlap, but this overlap incurs a thermodynamic penalty. This model has been used to simulate finite excluded volume effects in simulations of polymer melts on a lattice.^16,17 To incorporate this into the model, an overlap penalty is assigned to the monomer segment attempting to move and varies depending on the degree of overlap. Overlapping one corner of another monomer incurs a penalty of $ϵ8$ where ϵ is the penalty for complete overlap, which is not allowed. This energetic penalty is added on a move-by-move basis to the energy of the respective monomer during the calculation of the total change in energy used in the Metropolis sampling of that Monte Carlo step (MCS). A result of this condition is that the minimum bond length between monomers in the simulation is lowered to 1, a distance that corresponds to an overlap penalty of $ϵ4$⁠, while also permitting the monomer bonds to cross in a few statistically insignificant cases. Through this overlapping process, the system of polymer chains in this modified model better simulate the excluded volume effects and allow for correlation between these effects and their consequences seen in experimental work.⁵ 

The systems in each of the simulations in this study consist of 1728 polymer chains, where 10% are tagged as copolymers, of chain length N = 32 in a cubic lattice of 96 units in length, resulting in an average monomer density of ρ = 0.5. Thermodynamic interactions between monomers have been implemented by tagging each monomer as type A or type B, which represent MMA and styrene, respectively, in this study. The thermodynamic interactions between monomers are represented by the pairwise interactions $ϵAA=ϵBB=−1$ and $ϵAB=1$⁠. Non-bonded interactions were applied to pairs of monomers that have bond vectors of [2,0,0], [2,1,0], and [2,1,1], which results in a $6$ radius containing 54 possible neighboring lattice sites.^6,18 The temperature in these simulations was defined as $T*=defkBTϵ$ and the simulations were carried out using a standard Metropolis algorithm. These parameters and conditions are consistent with previous studies of polymer melts using the unmodified bond fluctuation model.^6,7,12

The initial configurations were generated by filling the simulation box with homopolymer chains that are parallel to the z-axis, where all bond lengths are equal to 2, which is the minimum bond distance allowed using the unmodified bond fluctuation model. An appropriate number of the initial homopolymer chains were then tagged as copolymer chains depending on the total copolymer chain loading. The monomer sequence of each copolymer chain was determined by an algorithm using the reactivity ratios of MMA and styrene monomers in a free radical polymerization.¹⁹ This algorithm utilizes the reactivity ratio of MMA to estimate the probability that an MMA monomer would bond to a styrene molecule, while the reactivity ratio of styrene provides insight into the probability that a styrene molecule will bond to an MMA monomer. The monomer segments of the chosen copolymer chain were then retagged using this generated copolymer sequence.

The monomer friction factor $ζeff$ of MMA is much larger than that of styrene, where this variation is included in the model by incorporating an additional barrier to monomer motion, which was employed on a step-by-step basis. Two methods to implement the barrier to motion were used to explore how the detailed model of monomer motion attenuation alters the global diffusion of the copolymer chains. The first method is analogous to a mean field approximation where the probability of movement of a given monomer is proportional to the friction factor of the monomer that is moving and is weighted by the global blend composition. In the second method, the probability of monomeric motion is determined based on the friction factor of that monomer weighted by the instantaneous local composition that exists around the moving monomer. The first method, which we term the A-type method, determines the probability of movement of a monomer solely on the type of monomer moving, where the local friction factor of MMA is ∼3077 times greater than that of styrene, a value determined from reported rheological experiments,⁸ as shown in Equation (1),

In Equation (1), $B∼0A$ is the activation energy of the movement of an MMA monomer and $B∼0B$ is the activation energy for the movement of a styrene monomer. Thus, when an MMA monomer is chosen to move, the move is allowed with the probability given in Equation (2),

Due to computational limitations, this barrier to motion was reduced such that the ratio of the friction factors was 30 in the simulations performed during this course of this work. However, when a styrene monomer is moved, the move’s probability is always unity. The movement probability for a given monomer is calculated on a move-by-move basis and is in addition to the Metropolis condition determined by the configuration’s total change in energy. This results in a pseudo-mean-field approximation with respect to the local friction factors of each monomer type.

The second method, which we term R-type motions, determines the probability of motion of a given monomer as a product of the friction factor of the type of monomer attempting to move and the composition of the immediate neighborhood around it. The probability of MMA movement in this method is similar to the probability of MMA movement in the A-type method, but the barrier to motion is offset proportionally by the number of styrene monomers within a local volume surrounding the moving MMA monomer, as shown in Equation (3). Similarly the styrene movement probability is unity but is also offset by the number of MMA monomers nearby, as described in Equation (4),

In the above equations, the B_AB term is the difference in the activation energy of movement of the two monomers and is equal to $B∼0A−B∼0B$⁠. In Equation (4), z_nn represents the total number of neighboring monomers in the local configuration, and N_AB(c) and N_AB(c′) represent the number of styrene or MMA monomers that are in contact with the moving monomer in the pre-move configuration and the post-move configuration, respectively.

Once all of the chains were placed in the simulation box and their monomer segments were properly tagged, the simulation was carried out at a reduced temperature $T*=10$ where the mean-squared center-of-mass, $⟨rcm2⟩$⁠, and the radius of gyration, R_g, for each chain were recorded as the simulation was allowed to equilibrate. The point where the system has equilibrated was defined as the point in time where the homopolymer and copolymer chain motion was diffusive, $⟨rcm2⟩∼t1$⁠, and the R_g converges to a constant value. The structure of the resultant blend was also monitored by calculating the radial pair distribution functions (rdfs) for both blend components. System dynamics were quantified by calculating the diffusion coefficient D of the various polymers in the system as the $limt→∞⟨rcm2⟩6t$⁠, where t is the time defined in the number of Monte Carlo steps (MCSs). Finally, an effective chain friction factor, $ζeff$⁠, of the polymer or the copolymer was calculated from the diffusion coefficient using the Stokes-Einstein relationship.

The structure of the copolymer chains contained within the simulation box was probed by measuring the radial distribution profile of the following pairs of monomers: the distance between an MMA monomer that is in a copolymer and surrounding MMA monomers that are in a homopolymer; the distance between a styrene monomer (that must be in a copolymer) and surrounding MMA monomers that are in a copolymer, and the distance between two styrene monomers, which both must reside in a copolymer. The resultant radial pair distribution functions, (rdfs), provide insight into the structure and packing of the monomers within a local volume. Figure 1 plots the rdf of a blend that contains a ∼60% MMA copolymer chain for systems that are equilibrated using the R-type or A-type models. These figures demonstrate that there is very little difference between the structural packing around each of the three possible monomer pairs between the two movement protocols, the A-type and R-type. This is not unexpected since there is no explicit difference in structural limitations between the two models and this parallelism is reinforced by the similarity of the rdfs in Figure 2, which shows the rdfs of the systems with a variation in the copolymer composition. Additionally, Figures 1 and 2 demonstrate that there is no evidence of a tendency towards coalescence of the constituent polymer species and indicate that the copolymer/homopolymer blend remains miscible throughout our simulations, in agreement with similar previous studies performed using the unmodified bond fluctuation method.⁷ The small differences between the curves in Figure 2 are most likely due to insufficient statistics that, given more computational time, would average out. Moreover, all chains in each simulation exhibit an average R_g of ∼6 lattice units, which is in agreement with previous bond-fluctuation studies.^9,10

For this study, a simulation box contains a total of 1555 homopolymer chains of MMA type and 173 copolymer chains, where the distribution of MMA and styrene monomers within the copolymer was determined using the method described above. The average effective chain friction factors of the copolymer chains that are calculated from the diffusion coefficients listed in Table I are presented in Figure 3. The black data points in this figure represent the average effective chain friction factor of a styrene homopolymer and MMA homopolymer melt of the same degree of polymerization. The line fit between these two points predicts the effective friction factor assuming an exponential dependence on the composition of MMA in the copolymer with this representation of the data. The data gathered for both the A-type and R-type motion models indicate that these copolymer chains are both faster (i.e., have lower effective friction factors) than this linear approximation. The dependence of $ζeff$ on copolymer composition for both the A- and R-type motions is non-linear, with the A-type model demonstrating a very strong non-linearity. The dependence of the chain friction factor on copolymer composition exhibits an inflection point between 0.5% and 0.6% MMA.

Copolymer compatibilizers are important components of polymer blends of immiscible homopolymers and in order to act as compatibilizers, these copolymers must readily diffuse throughout the target homopolymer matrices towards a biphasic interface. A primary goal of this work is to study how the connectivity of the copolymer chain, a significant difference in the friction factor of the two monomers in the copolymer, and thermodynamic interactions between the two monomers impact the global motion of the copolymer chains. This global motion is parameterized by the effective friction factor of the copolymer chain, which is derived from the diffusion coefficient of the copolymer chains.

Our simulation results indicate that the conformation and radii of gyration of both the homopolymer and copolymer chains are independent of the method by which the variation in friction factors is incorporated in the simulation model. The rdf curves that describe the copolymer chain configurations also verify that the packing of the final copolymer monomer is very similar for the A- and R-type movement models. There is an increase in styrene clustering in copolymers that are rich in styrene (i.e., lower MMA composition copolymers), implying that the thermodynamic environment resulting from the repulsive interactions between the styrene and MMA does alter the local structure of the copolymer in some systems. Another significant factor that impacts the rdf curves is the compressibility of the monomers in the melt, which reduces the average excluded volume. These data confirm that the method by which the variation in monomer friction factor is incorporated into the simulation technique modeling does not impact the final structure and packing of the copolymer chains in the homopolymer matrix. This, however, does not necessarily imply that the composition fluctuations that arise from the variation in friction factors do not affect local monomer packing during the simulation convergence.

As Figure 3 shows, the effective local friction that is experienced by the copolymer chain, $ζeff$⁠, differs between the two movement types and both exhibit a non-exponential dependence on copolymer composition. This non-linear log-linear dependence generally follows the copolymer composition dependence of the experimentally determined viscosity and copolymer tracer diffusion coefficient of PS-b-PMMA in a PMMA matrix as reported by Milhaupt et al. and similar non-linearities in the copolymer composition dependence of the diffusion coefficient of PS-ran-MMA copolymers in a PMMA matrix reported by our group.^5,8 The inflection point of the curve around 0.55 MMA loading also matches these data. While the magnitude of the nonlinear change in friction factor presented in Figure 3 is not as severe as the shift that is observed in the experimental data, this may be due to the fact that the ratio of friction factors in Equation (2) is 30, rather than 3077 due to computational limitations. However, it should be emphasized that the impact of altering this ratio on the diffusion of the copolymer chains was studied and found to be of limited effect as the ratio exceeds ∼15. Thus, while the ratio of ζ(MMA)/ζ(styrene) examined in these simulations is less than that which is found experimentally, these data indicate that a value of ζ(MMA)/ζ(styrene) = 30 reasonably documents the impact of the variation in effective friction factor of the two monomers on the diffusive behavior of the resultant copolymer that is made up of those two monomers.

We believe that this non-linear logarithmic dependence on composition is a result of the connectivity between dissimilar styrene and MMA monomers within the copolymer chain and the resulting thermodynamic environment within the copolymer’s local volume, which results in composition concentration fluctuations. In a solution of MMA and styrene small molecules, where there is no connectivity between the individual particles, these local composition fluctuations are dependent almost entirely on the thermodynamic interaction potential between the styrene and MMA monomers. In such an untethered system, the average monomer diffusion coefficient in the blend can be represented as a mean-field approximation determined by the compositionally weighted average of the diffusion coefficients of the MMA and styrene monomers. Since the diffusing movement is not tethered, the movement of fast-moving styrene monomers allows for concentration fluctuations that form and dissipate on a very fast time scale around a particular local volume, which corresponds to the expected linear log-scale trend in $ζeff.$ However, the segments in a copolymer are connected, and the faster moving styrene monomers are tethered to the slower moving MMA monomers. This connectivity, in turn, dampens the dissipation (and formation) of local concentration fluctuations. The results presented above can thus be explained based on this connectivity, indicating that the fast-moving styrene monomers dominate the motion of the copolymer chain, dragging the slow moving MMA monomers along with it. This outcome has the effect of decreasing the effective friction factor, $ζeff,$ of the copolymer below the weighted average of the copolymer composition and as a result the copolymer chain moves faster than the mean field average based on the copolymer composition.

This comparison of the dynamics of copolymers to that of a free particle system also provides insight as to why $ζeff$ of the copolymers that are moved by the A-type motion is much lower than the expected mean field average. The increased mobility of these chains can be ascribed to the fact that the A-type motion only depends on the composition of the copolymer but does not take into account the composition of the local volume around the polymer. Because movement in the A-type system is not dependent on the local environment surrounding the copolymer, the motion of the faster moving styrene monomers dominates the motion of the copolymer chains. Consequentially, the diffusion of the copolymer chains in the A-type system becomes a kinetic effect. This interpretation is also consistent with the fact that the A-type copolymer chains are faster than the R-type copolymers with the same styrene loading in Figure 3.

The implementation of the R-type model introduces an extra layer of complexity that is not present in the A-type system. In the R-type model, the motion of any given monomer depends on the local composition surrounding the moving monomer. Thus, in the R-type system, the probability that a styrene monomer will move can be attenuated by the statistically probable presence of a neighboring MMA monomer. Similarly, the presence of neighboring styrene monomers can speed up the motion of MMA monomers. As a result, styrene monomers that move by the R-type model do not move as fast as those that move by the A-type model, while the motion of R-type MMA monomers will be faster than their A-type counterparts. However, since the monomeric friction factor for styrene monomers is an order of magnitude less than that of the MMA monomers and given the abundance of MMA monomers within the system, attenuation of the rate of styrene monomer motion greatly dominates the copolymer chain movement.

The decrease in styrene motion slows the formation and dissipation of concentration fluctuations within the copolymer’s local volume, which further dampens the effect of the thermodynamic interactions on copolymer chain diffusion relative to the copolymer in the A-type system, resulting in an increase in the copolymer chain effective friction factor $ζeff$⁠. Therefore, the R-type copolymers exhibit a weaker log-log nonlinearity in chain $ζeff$ than in the A-type model in Figure 3 due to the decreased movement of the copolymer styrene monomers, which in turn increases the local concentration of the styrene in the volume around the copolymer chain. This interpretation is also consistent with the increased styrene-styrene contacts at smaller distances in the rdf data in Figures 1 and 2.

Given the known importance of the local composition of a moving monomer on its motion, one explanation for the decrease in the effective friction factor of the copolymers as shown in Figure 3 is that the local composition of the copolymer is richer in styrene than the mean field average. To test this hypothesis, we correlate the results of our simulations with predictions from Lodge-McLeish (LM) theory on dynamics in multicomponent systems.²⁰ The LM theory stipulates that the local environment around a polymer chain in a blend controls the dynamics of the chain, where the connectivity of the polymers creates a local composition that differs from that of the average composition of the blend. Therefore LM theory states that the local environment around the two blend components will differ from the average composition even for very miscible blends, with the effect of this local composition on the chain dynamics increasing if the two blend components exhibit distinctly different friction factors. The determination of the local composition in the LM theory is ascribed solely to the chain connectivity in polymer chains and completely disregards the thermodynamic interactions between the components.

To evaluate the impact of the thermodynamic interaction on the local composition of the copolymer in our simulation, the local concentration is determined for the simulated copolymer/homopolymer blends and compared to predictions of the LM theory. According to the LM theory, the effective local concentration $ϕeff$ around a monomer is given by Equation (5) where $ϕs$⁠, the self-concentration of the minor components, describes its excess concentration within a local volume relative to the average matrix composition, and φ is the bulk composition for the two-component blend. The self-concentration $ϕs$ in Equation (6) is calculated from the characteristic ratio C_∞, monomer molecular mass M₀, monomer density ρ, the number of backbone bonds per Kuhn segment, κ, the average number of monomers per repeat unit N_av, and the volume around a Kuhn segment V. We have calculated values for $ϕeff$ and $ϕs$ for a monomer on the copolymer chains in our simulations assuming values of these parameters for the copolymer chains that are compositional averages of the corresponding values for PS and PMMA. The $ϕeff$ values calculated using the LM theory for the copolymer chains in our simulations are presented in Figure 4 as the solid black line. Since our simulations provide the positions of the monomers of the copolymer chains and their respective types, we can directly measure the number of styrene monomers in the same volume as the local volume in Equation (6) in the simulation. These values, which we term $ϕeffs$⁠, are presented for the A- and R-type simulations in Figure 4 as the green squares and brown circles, respectively. According to the results in Figure 4, the Lodge-McLeish model consistently under predicts the local concentration of the styrene component near the copolymer,

This result is not entirely surprising since the Lodge-McLeish theory does not explicitly account for the repulsion or attraction between monomers due to thermodynamic interactions in estimating the local composition around a copolymer. The LM theory assumes the correlation-hole effect, which states that each monomer is essentially guaranteed to be surrounded by a certain number of identical monomers as a result of the polymer chain connectivity. While this assumption is valid for some homopolymer blends, our results clearly indicate that this assumption is of questionable validity in a copolymer/homopolymer blend, where that local composition in blends containing copolymers depends on both monomer connectivity and the thermodynamic potential between the connected monomers. These concentration fluctuations, which are impacted by the kinetic variation of the speed of styrene monomer diffusion relative to that of MMA, as well as the thermodynamic interactions between MMA and styrene monomers that are inherent to both the A- and R-type methods are used in our simulations. Thus, the copolymers exhibit a higher local concentration of the faster styrene monomers within the copolymer local volume than the LM theory predicts. This increased concentration of styrene in the local volume increases the probability of motion for the copolymer chain, increases their speed, and decreases the average effective chain friction factor for copolymers in both A- and R-type from the expected linear power law dependence with composition.

The effect of connectivity and thermodynamic interactions between components on chain copolymer dynamics in partially miscible homopolymer-copolymer blends was studied using a Monte Carlo simulation featuring a modified bond-fluctuation model. Our studies have confirmed that copolymer connectivity, composition distribution, and heterogeneity of the monomer friction factors in the blend impact concentration fluctuations within the copolymer local volume, which in turn effects the diffusion of the copolymer within the homopolymer matrix. These effects can be quantified by measuring the effective chain friction factor of the copolymers thereby measuring the diffusion of the copolymers throughout the homopolymer melt. Our results demonstrate that copolymer diffusion does not have a linear power law dependence on the copolymer composition but is kinetically affected by the dominant motion of the faster monomer and thermodynamically affected by the concentration of the minority monomer component within a given monomer’s local volume.

The configuration of our simulations reveals that this effective local concentration of styrene is larger than what the Lodge-McLeish theory predicts. Since the minority component has an experimentally measured monomeric friction coefficient that is three orders of magnitude lower than that of the majority component, this process results in the development of local volumes that are richer in the minority component throughout the blend, which in turn results in faster copolymer diffusion. This kinetically driven motion is attenuated by the aforementioned composition fluctuations, leading to increased thermodynamic interactions between the major and minor components within the local volume around a copolymer chain, which therefore slows the diffusion of the copolymer. These simulation results expand on our group’s previous copolymer dynamics simulations and agree with our experimental neutron reflectivity studies of P(S-ran-MMA) copolymer diffusion in a PMMA matrix.

This research is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.