Monte Carlo Technique

The model system3738 in this study consists of 2504 chains of length N=10 confined to a cubic lattice. To simulate an infinite set of chains, the system is approximated as a set of infinitely many identical cells of length L with periodic boundary conditions in all three orthogonal directions (x, y, z). In this study, the only interaction energy is a nearest neighbor monomer-monomer interaction, £A.B, £A.B is zero if two neighboring monomers are of different type (A-B) and is negative otherwise. £A.B applies to any two adjacent monomers, whether a bond connects them or not. Steric interactions are included as excluded volume; simultaneous occupation of a given lattice site by more than one monomer is prohibited. The density of the system is calculated as the fraction of occupied lattice sites, p = NpN/L3, where Np = 2504 is the number ofpolymer chains present, N=10 is the length of all chains, homopolymers and copolymers, and L=31 is the size of the cubic lattice. In the present study, p = 84.05%.

tension between the two phases. The copolymer also overcomes an important shortcoming of a phase-separated blend as it strengthens the interface. In a non-compatibilized blend, the biphasic interface is sharp and there is very little entanglement between the two phases, which leads to a very weak interface and poor transfer of stress. The copolymer, however, can strengthen that interface by entangling with both homopolymers. This is visualized most easily for a diblock copolymer where the A block unfurls into the homopolymer A phase and entangles with the homopolymer and the B block does the same thing with homopolymer B. It is also thought that the copolymer acts as a buffer at the biphasic interface to inhibit droplet coalescence, which in turn results in a finer dispersion of the minor phase in the major phase.

Thus, to act as a useful compatibilizer in a phase separated system, the copolymer must expand at the interface as it entangles with the homopolymer phases. The copolymer must also take up substantial volume to act as a buffer and be able to inhibit droplet coalescence. Thus, the temperature dependence of the volume of the copolymer at the biphasic interface can be utilized as an assessment of the ability of a copolymer to act as a compatibilizer. If the copolymer entangles with the homopolymers, the volume will swell as the temperature is lowered and the system becomes more phase-separated. However, if the copolymer does not entangle with the homopolymers at the interface, it will be trapped between the two homopolymer phases and collapse as the temperature is lowered.

Figure 1. Plot ofthe volume of copolymers with different architectures as a function oftemperature

The volume of the copolymers examined in this study as a function of temperature is shown in Figure 1 for a system with 7.5% copolymer. Blends with other percent copolymers show similar trends. The volumes are calculated from the three radii of gyration as V = 53/2x rgX x.rgy x rgZ. the 53/2 factor is a result of the relationship between the radius of gyration, moment of inertia and principal axes of an ellipse, which the chain approximates in these conditions.

Inspection of this figure shows that both the alternating and diblock copolymers exhibit a steady and significant increase in the volume of the copolymer as the system becomes more phase-separated. One explanation for this response is that the copolymer is being swollen by its interaction with the homopolymer, suggesting that the copolymer is entangled with the homopolymers in these systems, and will, thus, be an effective compatibilizer. However, examination of the 'random' copolymers (Px = 0.5, 1.0, 1.5) shows an interesting trend. The volume of the alt-ran copolymer (Px = 0.5) does not change much as the system undergoes phase-separation, and actually decreases in volume slightly as the system begins to phase separate. The statistically random copolymer (Px = 0) is slightly better in that it shows a constant increase in size with phase separation, suggesting some interaction with the homopolymers, however the increase in volume (and therefore interaction with homopolymer) is modest. The blocky random copolymer, however shows a steady, significant increase in volume as the system becomes more phase separated.

Thus, from this data, it appears that both the alternating and diblock copolymers are the most efficient at compatibilizing polymer blends. Unfortunately, these two copolymers are also the most difficult to realize and therefore, this is of little use from a commercial standpoint. It is interesting that the alternating copolymer may rival the diblock copolymer as a compatibilizer and this possibility is currently under investigation in our laboratory. Within the random copolymers (Px = 0.5, 1 .0, and 1.5), these results suggest that the blocky structure is much better at interacting with the homopolymers than the alt-ran structure, and thus should be a more effective interfacial modifier.

Figure 2. Plot of compatibilizing efficiency as a function of sequence distribution for a linear copolymer.

Figure 2. Plot of compatibilizing efficiency as a function of sequence distribution for a linear copolymer.

The change in volume of the copolymer as the system goes from a miscible to an immiscible state has been utilized as a qualitative measure of its ability to compatibilize a biphasic blend. This can be quantified by using the difference between the volume of the copolymer in the miscible system and the volume of the copolymer in the phase separated system at its deepest quench as a measure of the effectiveness of the copolymer as a compatibilizer. This value is plotted vs. the sequence distribution, Px in Figure 2. This data quantifies the trend that is described above; the alternating and diblock copolymers are the best compatibilizers, however within the random structures, the more blocky structures is a more effective interfacial modifier than the statistically random copolymer which is more effective than an alternating-random structure.

Comparison to Experimental Results

It is interesting to compare these calculations to recent experimental results that have examined the ability of 'random' copolymers to strengthen a biphasic interface. To make this comparison, an experimental parameter must be found that can correlate the sequence distribution of the experimental copolymers to the simulation parameter Px. Px is essentially a measure of the percentage of neighboring monomers along the copolymer chain that are of different type. The reactivity ratios of the two monomers can also be utilized to ascertain the propensity of two different monomers to be bonded together along the copolymer chain. Recall that rb the reactivity ratio of monomer A, equals kn / ; the ratio of reactivity of an A monomer with another A monomer to it's reactivity with a B monomer. Therefore, if ri < 1 , there is more probability that an A monomer will react with a B monomer to continue chain growth, whereas if ri > 1, there is a greater probability that an A monomer will react with another A monomer than it will react with a B monomer. Thus the copolymer that is formed when r1 < 1 will be more alternating in nature, whereas, if ri > 1 , the copolymer will be more blocky in nature. Similar arguments can be made regarding r2. Therefore, conceptually, the reactivity ratios define similar characteristics regarding the sequence distribution of a copolymer to the simulation parameter Px; they both relate the propensity to form bonds between dissimilar monomers along the copolymer chain. Therefore, the reactivity ratios of copolymers used in the experimental studies will be utilized to determine the corresponding Px value of the copolymers.

Modeling the copolymerization on a computer completes this correlation between Px and the reactivity ratio. In this model, one million chains that are 2,000 monomers long are synthesized. The composition of the monomer pool is initially set to 50% of each monomer. 1000 chains are grown simultaneously and then repeated until 1 x 1 06 chains are created. This is meant to account for the broad initiation times that may occur in free radical polymerizations. The reactivity ratios of the monomer pairs as well as the composition of the remaining monomer pool control the evolution of the sequence and composition distributions of the copolymer chain. The probability, PAAthat a monomer A will add to the growing chain that ends in an A. monomer radical is where r is the reactivity ratio for the A monomer, [A] is the concentration of monomer A in the monomer pool, and [B] is the concentration of monomer B in the monomer pool. Similar equations for PAB(= 1- PAA),PBB,and PBA(= 1 - PBB)can be derived. As a copolymer chain grows, these probabilities are utilized to statistically determine the identity of the next monomer that is added to the growing polymer chain. The final copolymers are then analyzed to determine the corresponding Px values.

The experimental results that will be examined consist of studies that look at the ability of a 'random' copolymer to improve the properties of mixtures of the two homopolymers relative to the ability of a diblock copolymer. The three different systems that are examined include copolymers of poly(styrene-co-methyl methacrylate) (S/MMA), poly(styrene-co-2-vinyl pyridine) (S/2VP), and poly(styrene-co-ethylene) (S/E) in mixtures of the two homopolymers. The experiments that have been utilized to examine the ability of the copolymer to strengthen a polymer blend include the examination of the tensile properties of the compatibilized blend and the determination of the interfacial strength between the two homopolymers using asymmetric double cantilever beam (ADCB) experiments.

The first set of experiments that will be considered has examined the ability of random copolymers of styrene and methyl methacrylate to improve the interfacial strength between polystyrene and poly(methyl methacrylate). 121318 Using the asymmetric double cantilever beam technique, the researchers have found that a diblock copolymer (50/50 composition, Mw = 282,000) creates an interface with strength of400 J/m2. When utilizing a random copolymer however, it was found that the strongest interface (70% styrene, Mw = 250,000) had strength of ca. 80 J/m2. The bare interface between the two homopolymers in the absence of a copolymer had strength of approximately 6 J/m2. Thus, the random copolymer does strengthen the interface, however not as well as the diblock copolymer. It is interesting to note that Macosko and co-workers have found that random copolymers of styrene and methyl methacrylate do not inhibit static droplet coalescence an applied shear flow, and thus do not function as an effective compatibilizer in that respect.

However, for a copolymer of styrene and 2-vinyl pyridine, similar experiments show a slightly different pattern.11,25Using ADCB, Kramer and coworkers have found that a diblock copolymer (50/50 composition, Mw = 170,000) creates an interface with strength of 70 J/m2, whereas a random copolymer (50/50 composition, Mw = 700,000) was able to produce an interface with strength of 140 J/m2. Thus, for this system, the random copolymer is a better interfacial modifier than the diblock copolymer.

The last system that will be compared is that of polystyrene and polyethylene. In these studies,35the tensile properties of the blend with an added copolymer were examined. These studies showed that the addition of 10% diblock copolymer (50/50 composition, Mw = 80,000) to a blend that is 80% polystyrene produces an ultimate strength of 33 MPa, whereas the addition of the same amount of a random copolymer (67% polyethylene, Mw = 80,000) produces a blend with an ultimate strength of 41 MPa. Thus, in this system, the random copolymer again creates a superior system to the diblock copolymer.

Interestingly, these results may be explained by a careful examination of the sequence distributions of experimental copolymers. Due to the reactivity ratios of the monomers, the PS/PMMA random copolymer (r1 = 0.46, r2= 0.52) produces a copolymer with a Px value of is 1.28 and is thus alternating in nature, whereas, both PS/P2VP (r1=0.5, r2= 1.3; Px = 0.92) and PS/PE (^=0.78, r2= 1.39, Px = 0.89) random copolymers are more blocky in nature. Thus, the 'blocky' random copolymer strengthens the interface while the alternating-random copolymer does not. Unfortunately, in all of the experimental studies mentioned above there exist other parameters besides sequence distribution that could influence the correlation of these experimental results to the Monte Carlo work. Thus, further experiments are currently underway in our laboratory to more carefully correlate the influence of copolymer sequence distribution to its ability to modify the biphasic interface in a polymer blend.

Nonetheless, Monte Carlo results suggest that in order for a copolymer to be an effective compatibilizer in polymer blends, the copolymer must be blocky in nature. This interpretation of the simulation results are in agreement with the experimental results which show that the "random" copolymers that are better than the diblock copolymer at compatibilizing a blend are those that are blockier than a statistical random copolymer. The copolymers that do not behave as effectively as a compatibilizer are more alternating than a statistically random copolymer.

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