Self-consistent field theory (SCFT) is used to evaluate the excess free energy per unit area (i.e... more Self-consistent field theory (SCFT) is used to evaluate the excess free energy per unit area (i.e., tension) of different boundaries (or interfaces) between the coexisting phases of a block copolymer blend. In this first study of its kind, we focus on the boundaries separating the short- and long-period lamellar phases that form in mixtures of small and large symmetric diblock copolymers of polymerizations Ns and Nl, respectively, when Ns ≪ Nl. According to strong-segregation theory (SST), the tension is minimized when both sets of lamellae orient parallel to the boundary, but experiments [Hashimoto, , Macromolecules 1994, 27, 1562] tend to observe kink boundaries instead, where the domains of one lamellar phase evolve continuously into those of the other lamellar phase. Our more refined SCFT calculations, on the other hand, do predict a lower tension for the kink boundary consistent with the experimental observations. For completeness, we also examine the boundaries that form when the short-period lamellar phase disorders, and again the SCFT results are in agreement with experiment.
The interactions between mesophase-forming copolymers and nanoscopic particles can lead to highly... more The interactions between mesophase-forming copolymers and nanoscopic particles can lead to highly organized hybrid materials. The morphology of such composites depends not only on the characteristics of the copolymers, but also on the ...
Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends. Frank S. Bates * , Wayne Maure... more Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends. Frank S. Bates * , Wayne Maurer, Timothy P. Lodge, Mark F. Schulz, and Mark W. Matsen Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455. ...
Entropic segregation of chain ends to the surface of a monodisperse polymer melt and its effect o... more Entropic segregation of chain ends to the surface of a monodisperse polymer melt and its effect on surface tension are examined using self-consistent field theory (SCFT). In order to assess the dependence on chain stiffness, the SCFT is solved for worm-like chains. Our focus is still on relatively flexible polymers, where the persistence length of the polymer, ℓp, is comparable to the width of the surface profile, ξ, but still much smaller than the total contour length of the polymer, ℓc. Even this small degree of rigidity causes a substantial increase in the level of segregation, relative to that of totally flexible Gaussian chains. Nevertheless, the long-range depletion that balances the surface excess still exhibits the same universal shape derived for Gaussian chains. Furthermore, the excess continues to reduce the surface tension by one unit of kBT per chain end, which results in the usual N−1 reduction in surface tension observed by experiments. This enhanced segregation will also extend to polydisperse melts, causing the molecular-weight distribution at the surface to shift towards smaller Nn relative to the bulk. This provides a partial explanation for recent quantitative differences between experiments and SCFT calculations for flexible polymers.
Monte Carlo field-theoretic simulations (MC-FTS) are performed on structurally symmetric binary h... more Monte Carlo field-theoretic simulations (MC-FTS) are performed on structurally symmetric binary homopolymer blends for invariant polymerization indexes of N ≥ 103. It is shown that the ultraviolet (UV) divergence that plagues MC-FTS at small N can be removed by an appropriate renormalization of the Flory–Huggins interaction parameter, χ, allowing one to extract meaningful results that are independent of the wavevector cutoff. Once the divergence is taken care of, the fluctuation corrections to mean-field theory are found to be exceptionally small. In particular, the disordered-state structure function, S(k), is virtually indistinguishable from the RPA prediction, and there is a slight shift in the critical point, (χN)c, that roughly scales as N–1/2. An implication of the small corrections is that previous experimental determinations of χ based on homopolymer blends should be relatively accurate.
The performance of thermoplastic elastomers composed of block copolymers is dependent upon the mo... more The performance of thermoplastic elastomers composed of block copolymers is dependent upon the molecular bridges linking together the discrete minority domains. Here we devise a strategy for calculating the bridging statistics for complex block copolymer architectures, using self-consistent field theory. The method is demonstrated on (AB)M stars with M identical diblock arms. The fraction of molecules forming bridges, νb, is found to increase rapidly with M to values well beyond that of conventional ABA triblock copolymers. Once M is of order 10, virtually all molecules form bridges, and furthermore their arms tend to be distributed equally among neighboring minority domains. These high bridging fractions combined with the tendency of single molecules to bridge multiple domains make diblock-arm stars an excellent candidate for improved thermoplastic elastomers.
This work explores the use of continuous thermodynamic integration in field‐theoretic simulations... more This work explores the use of continuous thermodynamic integration in field‐theoretic simulations of a symmetric diblock copolymer melt. Free energies of the lamellar and disorder phases are evaluated by thermodynamic integration from a reference state (an Einstein crystal, λ = 0) to a diblock copolymer (λ = 1). This is followed by integration over the interaction parameter, χb, to locate the order–disorder transition (ODT). Then the equilibrium lamellar spacing and free energy cost of stretching and compressing lamellae are examined. The ODT, lamellar spacing, and compression modulus are consistent with previous calculations, though found faster and more precisely. The above quantities do not depend on simulation box size, suggesting that finite‐size effects are small and simulating two lamellar periods is sufficient to accurately evaluate bulk behavior. Furthermore, the statistical uncertainty in the ODT increases quickly with system size, suggesting that small systems may lead to more precise results.
Field-theoretic simulations (FTSs) are performed on ternary blends of A- and B-type homopolymers ... more Field-theoretic simulations (FTSs) are performed on ternary blends of A- and B-type homopolymers of polymerization Nh and symmetric AB diblock copolymers of polymerization Nc. Unlike previous studies, our FTSs are conducted in three-dimensional space, with the help of two new semi-grand canonical ensembles. Motivated by the first experiment to discover bicontinuous microemulsion (BμE) in the polyethylene-polyethylene propylene system, we consider molecules of high molecular weight with size ratios of α ≡ Nh/Nc = 0.1, 0.2, and 0.4. Our focus is on the A + B coexistence between the two homopolymer-rich phases in the low-copolymer region of the phase diagram. The Scott line, at which the A + B phases mix to form a disordered melt with increasing temperature (or decreasing χ), is accurately determined using finite-size scaling techniques. We also examine how the copolymer affects the interface between the A + B phases, reducing the interfacial tension toward zero. Although comparisons with self-consistent field theory (SCFT) illustrate that fluctuation effects are relatively small, fluctuations do nevertheless produce the observed BμE that is absent in the SCFT phase diagram. Furthermore, we find evidence of three-phase A + B + BμE coexistence, which may have been missed in the original as well as subsequent experiments.
Traditional particle-based simulations struggle with large bottlebrush copolymers, consisting of ... more Traditional particle-based simulations struggle with large bottlebrush copolymers, consisting of many side chains grafted to a backbone. Field-theoretical simulations (FTS) allow us to overcome the computational demands in order to calculate their equilibrium behavior. We consider bottlebrushes where all grafts are symmetric diblock copolymers, focusing on the order-disorder transition (ODT) and the size of ordered domains. Increasing the number of grafts and decreasing the spacing between them both raise the transition temperature. The ODT and lamellar period asymptotically approach constants as the number of grafts increases. As the spacing between grafts becomes large, the bottlebrushes behave like diblock copolymers, and as it becomes small, they behave like starblock copolymers. In between, the period increases, reaching a maximum when the spacing is approximately 0.35 times the length of the grafts. A comparison of FTS with mean-field calculations allows us to assess the effect of compositional fluctuations. Fluctuations suppress ordering, while having little effect on the period, as is the case for diblock copolymers.
Self-consistent field theory (SCFT) is used to evaluate the excess free energy per unit area (i.e... more Self-consistent field theory (SCFT) is used to evaluate the excess free energy per unit area (i.e., tension) of different boundaries (or interfaces) between the coexisting phases of a block copolymer blend. In this first study of its kind, we focus on the boundaries separating the short- and long-period lamellar phases that form in mixtures of small and large symmetric diblock copolymers of polymerizations Ns and Nl, respectively, when Ns ≪ Nl. According to strong-segregation theory (SST), the tension is minimized when both sets of lamellae orient parallel to the boundary, but experiments [Hashimoto, , Macromolecules 1994, 27, 1562] tend to observe kink boundaries instead, where the domains of one lamellar phase evolve continuously into those of the other lamellar phase. Our more refined SCFT calculations, on the other hand, do predict a lower tension for the kink boundary consistent with the experimental observations. For completeness, we also examine the boundaries that form when the short-period lamellar phase disorders, and again the SCFT results are in agreement with experiment.
The interactions between mesophase-forming copolymers and nanoscopic particles can lead to highly... more The interactions between mesophase-forming copolymers and nanoscopic particles can lead to highly organized hybrid materials. The morphology of such composites depends not only on the characteristics of the copolymers, but also on the ...
Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends. Frank S. Bates * , Wayne Maure... more Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends. Frank S. Bates * , Wayne Maurer, Timothy P. Lodge, Mark F. Schulz, and Mark W. Matsen Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455. ...
Entropic segregation of chain ends to the surface of a monodisperse polymer melt and its effect o... more Entropic segregation of chain ends to the surface of a monodisperse polymer melt and its effect on surface tension are examined using self-consistent field theory (SCFT). In order to assess the dependence on chain stiffness, the SCFT is solved for worm-like chains. Our focus is still on relatively flexible polymers, where the persistence length of the polymer, ℓp, is comparable to the width of the surface profile, ξ, but still much smaller than the total contour length of the polymer, ℓc. Even this small degree of rigidity causes a substantial increase in the level of segregation, relative to that of totally flexible Gaussian chains. Nevertheless, the long-range depletion that balances the surface excess still exhibits the same universal shape derived for Gaussian chains. Furthermore, the excess continues to reduce the surface tension by one unit of kBT per chain end, which results in the usual N−1 reduction in surface tension observed by experiments. This enhanced segregation will also extend to polydisperse melts, causing the molecular-weight distribution at the surface to shift towards smaller Nn relative to the bulk. This provides a partial explanation for recent quantitative differences between experiments and SCFT calculations for flexible polymers.
Monte Carlo field-theoretic simulations (MC-FTS) are performed on structurally symmetric binary h... more Monte Carlo field-theoretic simulations (MC-FTS) are performed on structurally symmetric binary homopolymer blends for invariant polymerization indexes of N ≥ 103. It is shown that the ultraviolet (UV) divergence that plagues MC-FTS at small N can be removed by an appropriate renormalization of the Flory–Huggins interaction parameter, χ, allowing one to extract meaningful results that are independent of the wavevector cutoff. Once the divergence is taken care of, the fluctuation corrections to mean-field theory are found to be exceptionally small. In particular, the disordered-state structure function, S(k), is virtually indistinguishable from the RPA prediction, and there is a slight shift in the critical point, (χN)c, that roughly scales as N–1/2. An implication of the small corrections is that previous experimental determinations of χ based on homopolymer blends should be relatively accurate.
The performance of thermoplastic elastomers composed of block copolymers is dependent upon the mo... more The performance of thermoplastic elastomers composed of block copolymers is dependent upon the molecular bridges linking together the discrete minority domains. Here we devise a strategy for calculating the bridging statistics for complex block copolymer architectures, using self-consistent field theory. The method is demonstrated on (AB)M stars with M identical diblock arms. The fraction of molecules forming bridges, νb, is found to increase rapidly with M to values well beyond that of conventional ABA triblock copolymers. Once M is of order 10, virtually all molecules form bridges, and furthermore their arms tend to be distributed equally among neighboring minority domains. These high bridging fractions combined with the tendency of single molecules to bridge multiple domains make diblock-arm stars an excellent candidate for improved thermoplastic elastomers.
This work explores the use of continuous thermodynamic integration in field‐theoretic simulations... more This work explores the use of continuous thermodynamic integration in field‐theoretic simulations of a symmetric diblock copolymer melt. Free energies of the lamellar and disorder phases are evaluated by thermodynamic integration from a reference state (an Einstein crystal, λ = 0) to a diblock copolymer (λ = 1). This is followed by integration over the interaction parameter, χb, to locate the order–disorder transition (ODT). Then the equilibrium lamellar spacing and free energy cost of stretching and compressing lamellae are examined. The ODT, lamellar spacing, and compression modulus are consistent with previous calculations, though found faster and more precisely. The above quantities do not depend on simulation box size, suggesting that finite‐size effects are small and simulating two lamellar periods is sufficient to accurately evaluate bulk behavior. Furthermore, the statistical uncertainty in the ODT increases quickly with system size, suggesting that small systems may lead to more precise results.
Field-theoretic simulations (FTSs) are performed on ternary blends of A- and B-type homopolymers ... more Field-theoretic simulations (FTSs) are performed on ternary blends of A- and B-type homopolymers of polymerization Nh and symmetric AB diblock copolymers of polymerization Nc. Unlike previous studies, our FTSs are conducted in three-dimensional space, with the help of two new semi-grand canonical ensembles. Motivated by the first experiment to discover bicontinuous microemulsion (BμE) in the polyethylene-polyethylene propylene system, we consider molecules of high molecular weight with size ratios of α ≡ Nh/Nc = 0.1, 0.2, and 0.4. Our focus is on the A + B coexistence between the two homopolymer-rich phases in the low-copolymer region of the phase diagram. The Scott line, at which the A + B phases mix to form a disordered melt with increasing temperature (or decreasing χ), is accurately determined using finite-size scaling techniques. We also examine how the copolymer affects the interface between the A + B phases, reducing the interfacial tension toward zero. Although comparisons with self-consistent field theory (SCFT) illustrate that fluctuation effects are relatively small, fluctuations do nevertheless produce the observed BμE that is absent in the SCFT phase diagram. Furthermore, we find evidence of three-phase A + B + BμE coexistence, which may have been missed in the original as well as subsequent experiments.
Traditional particle-based simulations struggle with large bottlebrush copolymers, consisting of ... more Traditional particle-based simulations struggle with large bottlebrush copolymers, consisting of many side chains grafted to a backbone. Field-theoretical simulations (FTS) allow us to overcome the computational demands in order to calculate their equilibrium behavior. We consider bottlebrushes where all grafts are symmetric diblock copolymers, focusing on the order-disorder transition (ODT) and the size of ordered domains. Increasing the number of grafts and decreasing the spacing between them both raise the transition temperature. The ODT and lamellar period asymptotically approach constants as the number of grafts increases. As the spacing between grafts becomes large, the bottlebrushes behave like diblock copolymers, and as it becomes small, they behave like starblock copolymers. In between, the period increases, reaching a maximum when the spacing is approximately 0.35 times the length of the grafts. A comparison of FTS with mean-field calculations allows us to assess the effect of compositional fluctuations. Fluctuations suppress ordering, while having little effect on the period, as is the case for diblock copolymers.
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