^{1}, François A. Detcheverry

^{1}, Marcus Müller

^{2}and Juan J. de Pablo

^{1,a)}

### Abstract

A Monte Carlo formalism for the study of polymeric melts is described. The model is particle-based, but the interaction is derived from a local density functional that appears in the field-based model. The method enables Monte Carlo simulations in the , , semigrandcanonical and Gibbs ensembles, and direct calculation of free energies. The approach is illustrated in the context of two examples. In the first, we consider the phase separation of a binary homopolymer blend and present results for the phase diagram and the critical point. In the second, we address the microphase separation of a symmetric diblock copolymer, examine the distribution of local stresses in lamellae, and determine the order-disorder transition temperature.

I. INTRODUCTION

II. METHOD

III. RESULTS

IV. CONCLUSION

### Key Topics

- Block copolymers
- 17.0
- Chemical potential
- 11.0
- Monte Carlo methods
- 9.0
- Polymers
- 9.0
- Mean field theory
- 7.0

## Figures

The ratio as a function of , the number of particles each particle interacts with. Here, the contour discretization is . The line is a guide to the eye. All other results in this work use unless otherwise mentioned.

The ratio as a function of , the number of particles each particle interacts with. Here, the contour discretization is . The line is a guide to the eye. All other results in this work use unless otherwise mentioned.

(a) Coexistence curves computed in the semigrandcanonical ensemble for binary mixtures of homopolymers with (◼), (◆), (▲), and (●). The curve obtained with simulation in the Gibbs ensemble for is also shown . Note that the vertical axis corresponds to , not . The black line represents the mean-field prediction for an incompressible blend. (b) The probability distribution of the order parameter ; the black line shows the result for the Ising universality class,^{32} and the colored curves show results for different blends at our estimate of . (c) as a function of . The line is a linear fit to the data.

(a) Coexistence curves computed in the semigrandcanonical ensemble for binary mixtures of homopolymers with (◼), (◆), (▲), and (●). The curve obtained with simulation in the Gibbs ensemble for is also shown . Note that the vertical axis corresponds to , not . The black line represents the mean-field prediction for an incompressible blend. (b) The probability distribution of the order parameter ; the black line shows the result for the Ising universality class,^{32} and the colored curves show results for different blends at our estimate of . (c) as a function of . The line is a linear fit to the data.

Density profiles in a lamellar phase of a symmetric diblock, computed with Monte Carlo (MC) simulations and SCFT, in the ensemble, for . The coordinates are normalized by the lamellar spacing . The top curves correspond to the total density. The system size for the MC simulation is .

Density profiles in a lamellar phase of a symmetric diblock, computed with Monte Carlo (MC) simulations and SCFT, in the ensemble, for . The coordinates are normalized by the lamellar spacing . The top curves correspond to the total density. The system size for the MC simulation is .

Local stress in the lamellar phase of a symmetric diblock. The vertical dashed line corresponds to the interface. Each component is divided by the total pressure of . The long-dashed curve is the total density of the melt.

Local stress in the lamellar phase of a symmetric diblock. The vertical dashed line corresponds to the interface. Each component is divided by the total pressure of . The long-dashed curve is the total density of the melt.

(a) Snapshot of the disordered phase at . (b) Snapshot of the lamellar phase at .

(a) Snapshot of the disordered phase at . (b) Snapshot of the lamellar phase at .

(a) Particle density as a function of . Lines are a guide to the eye. (b) Excess chemical potential as a function of . The curves have been shifted for clarity. Empty and filled symbols correspond to the disordered and lamellar phases, respectively. Lines are linear fits to the data. The star denotes a metastable lamellar state. The error in is comparable to the symbol size. (c) Properties of single chain conformation. (dashed lines) and (solid lines) are defined in the text. Both quantities are normalized by their value for an ideal chain . Lines are a guide to the eye. Note that graph (c) is plotted as a function of while graphs (a) and (b) are plotted as a function of .

(a) Particle density as a function of . Lines are a guide to the eye. (b) Excess chemical potential as a function of . The curves have been shifted for clarity. Empty and filled symbols correspond to the disordered and lamellar phases, respectively. Lines are linear fits to the data. The star denotes a metastable lamellar state. The error in is comparable to the symbol size. (c) Properties of single chain conformation. (dashed lines) and (solid lines) are defined in the text. Both quantities are normalized by their value for an ideal chain . Lines are a guide to the eye. Note that graph (c) is plotted as a function of while graphs (a) and (b) are plotted as a function of .

Partial structure factors and . Here, , , , and . From Eq. (13), one obtains .

Partial structure factors and . Here, , , , and . From Eq. (13), one obtains .

How the ODT deviates from the SCFT value in various studies. MC, SCMF, and FTS refer to Refs. 43, 24, and 46, respectively. The lines are given by the equation , with (dashed line) and (solid line). Note that the different studies employ different definitions for the Flory-Huggins parameter.

How the ODT deviates from the SCFT value in various studies. MC, SCMF, and FTS refer to Refs. 43, 24, and 46, respectively. The lines are given by the equation , with (dashed line) and (solid line). Note that the different studies employ different definitions for the Flory-Huggins parameter.

The segmental chemical potential for a symmetric diblock copolymer system with , , , and . Each segment pair is taken as an and particles, with the first pair being the particles in the middle and the last pair being the two end particles.

The segmental chemical potential for a symmetric diblock copolymer system with , , , and . Each segment pair is taken as an and particles, with the first pair being the particles in the middle and the last pair being the two end particles.

Excess pressure of the monomeric fluid as a function of the particle density for ranging from 50 to (see text for details). The simulation box is a cube of size .

Excess pressure of the monomeric fluid as a function of the particle density for ranging from 50 to (see text for details). The simulation box is a cube of size .

Dimensionless compressibility of the monomeric fluid as a function of .

Dimensionless compressibility of the monomeric fluid as a function of .

RDF for the monomeric fluid. The interaction range is .

RDF for the monomeric fluid. The interaction range is .

Root-mean-square displacement of particles in the monomeric fluid for , 280, 500, 900, 1600, 2800, 5000, 9000, , , , , , , and (from top to bottom). The acceptance ratio of particle displacement is 50% in all cases. The system includes 33000 particles.

Root-mean-square displacement of particles in the monomeric fluid for , 280, 500, 900, 1600, 2800, 5000, 9000, , , , , , , and (from top to bottom). The acceptance ratio of particle displacement is 50% in all cases. The system includes 33000 particles.

## Tables

Summary of parameters and ODT values.

Summary of parameters and ODT values.

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