^{1}and Marcus Müller

^{1,a)}

### Abstract

The description of fluctuations by single chain in mean field (SCMF) simulations is discussed and the results of this particle-based self-consistent field technique are quantitatively compared to Monte Carlo simulations of the same discretized Edwards-Hamiltonian providing exact reference data. In SCMF simulations one studies a large ensemble of noninteracting molecules subjected to real, external fields by Monte Carlo simulations. The external fields approximate nonbonded, instantaneous interactions between molecules. In the self-consistent mean field theory the external fields are static and fluctuation effects are ignored. In SCMF simulations, the external fieldsfluctuate since they are frequently recalculated from the instantaneous density distribution of the ensemble of molecules. In the limit of infinitely high density or instantaneous update of the external fields, the SCMF simulation method accurately describes long-wavelength fluctuations. At high but finite updating frequency the accuracy depends on the discretization of the model. The accuracy is illustrated by studying the single chain structure and intermolecular correlations in polymer melts, and fluctuation effects on the order-disorder transition of symmetric diblock copolymers.

It is a pleasure to thank F. Detcheverry, G. H. Fredrickson, P. F. Nealey, J. J. de Pablo, M. Schick, M. P. Stoykovich, and J. P. Wittmer for enjoyable and fruitful discussions. Financial support was provided by the Volkswagen Foundation and the calculations were performed at the John von Neumann Institute for Computing, Jülich, Germany.

I. INTRODUCTION

II. A MINIMAL MODEL FOR COMPRESSIBLE DIBLOCK COPOLYMER MELTS

A. Coarse-grained description and discretization

B. Representation through pairwise interactions and direct MC simulations

C. Field-theoretic representation and SCF theory

D. SCMF simulations

III. QUANTITATIVE COMPARISON BETWEEN SCMF SIMULATIONS AND DIRECT MC SIMULATIONS

A. Homopolymer melts

1. Molecular conformations

2. Density fluctuations

3. Long-ranged inter- and intramolecular correlations

B. Diblock copolymer melts

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Monte Carlo methods
- 82.0
- External field
- 42.0
- Mean field theory
- 31.0
- Statistical mechanics models
- 22.0
- Block copolymers
- 20.0

## Figures

Deviation between SCMF simulations using the single move updating scheme and direct MC simulations as a function of the control parameter for , and various values of , , and . The inset compares the single chain structure factor, , obtained from direct MC (open circles) and SCMF (solid circle) simulations using random local segment displacement and updating frequency for system parameters , , , and .

Deviation between SCMF simulations using the single move updating scheme and direct MC simulations as a function of the control parameter for , and various values of , , and . The inset compares the single chain structure factor, , obtained from direct MC (open circles) and SCMF (solid circle) simulations using random local segment displacement and updating frequency for system parameters , , , and .

(a) Error of (left scale) as a function of the updating frequency . The right scale displays the characteristic ratio, . System parameters: , , , and . The ratios , are compiled in Table I. (b) Error of (left scale) as a function of the mean square average change of the number of segments per cell, , between updates of the fields. The right scale shows how this number depends on the updating frequency, .

(a) Error of (left scale) as a function of the updating frequency . The right scale displays the characteristic ratio, . System parameters: , , , and . The ratios , are compiled in Table I. (b) Error of (left scale) as a function of the mean square average change of the number of segments per cell, , between updates of the fields. The right scale shows how this number depends on the updating frequency, .

Density distributions obtained from direct MC (open circles) and SCMF (solid circles) utilizing the single move updating scheme. The comparison in the inset refers to the system of Fig. 1 (i.e., , , , , and ).

Density distributions obtained from direct MC (open circles) and SCMF (solid circles) utilizing the single move updating scheme. The comparison in the inset refers to the system of Fig. 1 (i.e., , , , , and ).

Error of the second moment of the density distribution for the single move updating scheme as a function of the inverse control parameter .

Error of the second moment of the density distribution for the single move updating scheme as a function of the inverse control parameter .

The error of the second moment of the density distribution as a function of the updating frequency . , , , and . Density fluctuations are analyzed on different length scales as indicated in the key. The arrow marks the commonly used updating frequency .

The error of the second moment of the density distribution as a function of the updating frequency . , , , and . Density fluctuations are analyzed on different length scales as indicated in the key. The arrow marks the commonly used updating frequency .

Kratky plot of the single chain structure factor for a homopolymer melt obtained by SCMF and direct MC simulations (solid and open circles, respectively). , , , , and . The Kratky plot for an ideal chain with (thick solid line) is also shown for comparison. The dotted line is the plateau value of in the range of the self-similar structure of the chain.

Kratky plot of the single chain structure factor for a homopolymer melt obtained by SCMF and direct MC simulations (solid and open circles, respectively). , , , , and . The Kratky plot for an ideal chain with (thick solid line) is also shown for comparison. The dotted line is the plateau value of in the range of the self-similar structure of the chain.

Density profiles in the vicinity of an impenetrable surface for a homopolymer melt with , , and and various values of . The ground state approximation for the Gaussian chain model is also shown for comparison. The inset shows the profiles for and two spatial discretizations and .

Density profiles in the vicinity of an impenetrable surface for a homopolymer melt with , , and and various values of . The ground state approximation for the Gaussian chain model is also shown for comparison. The inset shows the profiles for and two spatial discretizations and .

Total density structure factor calculated for the homopolymer melt of Fig. 6. The dashed line is the RPA prediction, Eq. (28), while open circles are the direct MC simulation data. The solid circles show the SCMF simulation results for . The inset presents the inverse structure factor for small wave vectors. The shift between the RPA and simulation results indicates the renormalization of the compressibility by local fluctuations. The arrow marks the limiting RPA behavior, .

Total density structure factor calculated for the homopolymer melt of Fig. 6. The dashed line is the RPA prediction, Eq. (28), while open circles are the direct MC simulation data. The solid circles show the SCMF simulation results for . The inset presents the inverse structure factor for small wave vectors. The shift between the RPA and simulation results indicates the renormalization of the compressibility by local fluctuations. The arrow marks the limiting RPA behavior, .

Segment-segment radial distribution functions, , for the system of Fig. 6. The RPA, MC, and SCMF results for the total pair correlation function, , are shown with open circles, solid lines, and dashed lines, respectively. The intermolecular part, , is shown for direct MC and SCMF with open and solid triangles.

Segment-segment radial distribution functions, , for the system of Fig. 6. The RPA, MC, and SCMF results for the total pair correlation function, , are shown with open circles, solid lines, and dashed lines, respectively. The intermolecular part, , is shown for direct MC and SCMF with open and solid triangles.

Logarithmic plot of decay of bond-bond correlations along the chain contour for the homopolymer melt of Fig. 6. The direct MC and SCMF results are represented by open triangles and circles, respectively. The dashed line is the asymptotic power law , while the solid line corresponds to Eq. (31).

Logarithmic plot of decay of bond-bond correlations along the chain contour for the homopolymer melt of Fig. 6. The direct MC and SCMF results are represented by open triangles and circles, respectively. The dashed line is the asymptotic power law , while the solid line corresponds to Eq. (31).

Structure factor, , of composition fluctuations calculated for a diblock copolymer melt in the disordered phase at various values of . The open and solid symbols show the direct MC and SCMF simulation results, respectively, for systems with , , , and . The size of the simulation cell was units and was utilized. The lines show Leibler’s RPA predictions.

Structure factor, , of composition fluctuations calculated for a diblock copolymer melt in the disordered phase at various values of . The open and solid symbols show the direct MC and SCMF simulation results, respectively, for systems with , , , and . The size of the simulation cell was units and was utilized. The lines show Leibler’s RPA predictions.

The reciprocal peak height of the structure factor, , for , , , and as a function of . The size of the simulation cell was . The symbols denote the SCMF results, while the thick dashed line shows the Leibler RPA prediction. The solid line is the Fredrickson-Helfand theory result [cf. Eq. (33)].

The reciprocal peak height of the structure factor, , for , , , and as a function of . The size of the simulation cell was . The symbols denote the SCMF results, while the thick dashed line shows the Leibler RPA prediction. The solid line is the Fredrickson-Helfand theory result [cf. Eq. (33)].

Hysteresis loop for the interaction energy density, normalized by [see Eq. (34)]. Snapshots show the morphology at various stages of “cooling down.” The inset shows the hysteresis loop for the largest eigenvalue, , of the Saupe matrix defined via the unit vectors along the line connecting the centers of mass of and blocks, Eq. (35). , , , , and cell size .

Hysteresis loop for the interaction energy density, normalized by [see Eq. (34)]. Snapshots show the morphology at various stages of “cooling down.” The inset shows the hysteresis loop for the largest eigenvalue, , of the Saupe matrix defined via the unit vectors along the line connecting the centers of mass of and blocks, Eq. (35). , , , , and cell size .

Stress anisotropy in lamellar morphology at and , , , and . The main panel shows the fluctuation of the difference of stress components and , with axis and axis being parallel and normal to the lamellae (the lower right inset shows a snapshot of the setup) as a function of SCMF steps measured in units of Rouse time, . The solid line shows the fluctuation of for the lamella of the hysteresis morphology, while solid and open circles correspond to lamellae with 10% larger and 10% smaller spacing, respectively. The top right inset shows the average value of as a function of strain . The equilibrium lamellar spacing is .

Stress anisotropy in lamellar morphology at and , , , and . The main panel shows the fluctuation of the difference of stress components and , with axis and axis being parallel and normal to the lamellae (the lower right inset shows a snapshot of the setup) as a function of SCMF steps measured in units of Rouse time, . The solid line shows the fluctuation of for the lamella of the hysteresis morphology, while solid and open circles correspond to lamellae with 10% larger and 10% smaller spacing, respectively. The top right inset shows the average value of as a function of strain . The equilibrium lamellar spacing is .

## Tables

Comparison between the bare chain extension which is a model parameter and the measured mean squared end-to-end distance , observed in direct MC simulations for different discretizations and densities. is the invariant degree of polymerization.

Comparison between the bare chain extension which is a model parameter and the measured mean squared end-to-end distance , observed in direct MC simulations for different discretizations and densities. is the invariant degree of polymerization.

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