# Difference between revisions of "Two-dimensional nanometric confinement of entangled polymer melts"

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== Keywords == | == Keywords == | ||

− | entangled polymer melts, two-dimensional confinement, Monte Carlo simulation | + | [[entangled polymer melts]], [[two-dimensional confinement]], [[Monte Carlo simulation]] |

== Introduction == | == Introduction == | ||

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In the simulation, the polymer film of thickness <math>h</math> is represented by a simple cubic lattice with periodic boundary condition along the lateral direction. A dense melt at high thickness <math>h>R_g</math> is first generated, with chains of length <math>N</math> and a small number of vacancies placed on the lattice. The chains are then equilibrated by elementary rotations and exchanges of segments between adjacent chains. It is guaranteed that the process obeys detailed balance, a necessary condition for thermodynamic equilibrium. Full equilibrium is considered to be achieved when the average radius of gyration for the chains and the eigenvalues of their gyration tensor approach steady state values. Next, the film thickness is slowly decreased by randomly picking a small number of chains and converting them into vacancies so that the density remains the same. A new equilibrium can be achieved. The whole process is repeated until a sufficiently small <math>h</math> is attained, as is illustrated in Fig. 1. | In the simulation, the polymer film of thickness <math>h</math> is represented by a simple cubic lattice with periodic boundary condition along the lateral direction. A dense melt at high thickness <math>h>R_g</math> is first generated, with chains of length <math>N</math> and a small number of vacancies placed on the lattice. The chains are then equilibrated by elementary rotations and exchanges of segments between adjacent chains. It is guaranteed that the process obeys detailed balance, a necessary condition for thermodynamic equilibrium. Full equilibrium is considered to be achieved when the average radius of gyration for the chains and the eigenvalues of their gyration tensor approach steady state values. Next, the film thickness is slowly decreased by randomly picking a small number of chains and converting them into vacancies so that the density remains the same. A new equilibrium can be achieved. The whole process is repeated until a sufficiently small <math>h</math> is attained, as is illustrated in Fig. 1. | ||

+ | |||

+ | == Results == | ||

+ | |||

+ | [[Image:polymer_2.jpg|800px|right]] | ||

+ | |||

+ | Polymer chains appear as flattened ellipsoids with their large semiaxis being <math>\sqrt{\lambda_1}</math> where <math>\lambda_1</math> is the largest eigenvalue of the gyration tensor '''G'''. Renormalized quantities are defined: <math>h^*=h/R_{g,bulk}</math> and <math>R_g^*=R_g/R_{g,bulk}</math>. The dependence of <math>\lambda_1/\lambda_3</math> and <math>\lambda_2/\lambda_3</math> on <math>h^*</math> is studied and depicted in Fig. 2. At large <math>h^*</math>, the curves converge towards their limiting values for Gaussian chains. When <math>h^*</math> is below 1, the two curves go up dramatically. However, <math>\lambda_1/\lambda_2</math> remains close to the value expected for Gaussian chains, as is seen in insert. The dependence of <math>R_g^*</math> on <math>h^*</math> is plotted in Fig. 3. A minimum of <math>R_g^*</math> is found around the critical value <math>h^*</math> ~ 1. This minimum value increases as the chain length <math>N</math> decreases. These trends are in agreement with previous results. | ||

+ | |||

+ | |||

+ | [[Image:polymer_3.jpg|800px|right]] | ||

+ | |||

+ | The dependence of the number of chain segments between entanglement points <math>N_e</math> on <math>h^*</math> is also studied and presented in Fig. 4. The data reveal an onset of an increase in <math>N_e</math> below the critical value <math>h^*</math> ~ 1. This is also consistent with previous experimental measurements. | ||

+ | |||

+ | Details of the internal structure of the chains are shown in Fig. 5, in which <math>r_{\parallel}</math> is the in-plane projection of the end-to-end vector length of a strand of n segments along the chain contour, averaged over all the chains of length <math>n</math>. It can be revealed from this study that at short length scales, chains appear to organize into compact disc-like domains, called globules, whose size and density increase with the extent of confinement. A decrease in the overall chain length also increases the globules size but it has no effect on their density. | ||

+ | |||

+ | The model findings are considered relevant to the molecular origin of the exceptional stiffness of nacre, a composite of parallel plates of calcite which confine protein chains to small distances. |

## Latest revision as of 03:55, 3 December 2011

Entry by Yuhang Jin, AP225 Fall 2011

## Contents

## Reference

Yves Termonia, *Polymer*, 2011, **52**, 5193.

## Keywords

entangled polymer melts, two-dimensional confinement, Monte Carlo simulation

## Introduction

Behavior of polymer chains under a two-dimensional confinement is of interest and importance to a variety of engineering applications, and has significant relevance to biological macromolecules as well. Both experimental studies and numerical approaches have been devoted to this topic, with a focus on thin polymer films whose thickness is smaller than the radius of gyration <math>R_g</math>. One important result of these efforts is that for film thickness <math>h<R_g</math> the polymer chains retain their Gaussian behavior.

In this paper, simulation at extreme confinements, <math>h<<R_g</math>, is performed for the first time. A Monte-Carlo approach based on previous work is employed.

## Model

In the simulation, the polymer film of thickness <math>h</math> is represented by a simple cubic lattice with periodic boundary condition along the lateral direction. A dense melt at high thickness <math>h>R_g</math> is first generated, with chains of length <math>N</math> and a small number of vacancies placed on the lattice. The chains are then equilibrated by elementary rotations and exchanges of segments between adjacent chains. It is guaranteed that the process obeys detailed balance, a necessary condition for thermodynamic equilibrium. Full equilibrium is considered to be achieved when the average radius of gyration for the chains and the eigenvalues of their gyration tensor approach steady state values. Next, the film thickness is slowly decreased by randomly picking a small number of chains and converting them into vacancies so that the density remains the same. A new equilibrium can be achieved. The whole process is repeated until a sufficiently small <math>h</math> is attained, as is illustrated in Fig. 1.

## Results

Polymer chains appear as flattened ellipsoids with their large semiaxis being <math>\sqrt{\lambda_1}</math> where <math>\lambda_1</math> is the largest eigenvalue of the gyration tensor **G**. Renormalized quantities are defined: <math>h^*=h/R_{g,bulk}</math> and <math>R_g^*=R_g/R_{g,bulk}</math>. The dependence of <math>\lambda_1/\lambda_3</math> and <math>\lambda_2/\lambda_3</math> on <math>h^*</math> is studied and depicted in Fig. 2. At large <math>h^*</math>, the curves converge towards their limiting values for Gaussian chains. When <math>h^*</math> is below 1, the two curves go up dramatically. However, <math>\lambda_1/\lambda_2</math> remains close to the value expected for Gaussian chains, as is seen in insert. The dependence of <math>R_g^*</math> on <math>h^*</math> is plotted in Fig. 3. A minimum of <math>R_g^*</math> is found around the critical value <math>h^*</math> ~ 1. This minimum value increases as the chain length <math>N</math> decreases. These trends are in agreement with previous results.

The dependence of the number of chain segments between entanglement points <math>N_e</math> on <math>h^*</math> is also studied and presented in Fig. 4. The data reveal an onset of an increase in <math>N_e</math> below the critical value <math>h^*</math> ~ 1. This is also consistent with previous experimental measurements.

Details of the internal structure of the chains are shown in Fig. 5, in which <math>r_{\parallel}</math> is the in-plane projection of the end-to-end vector length of a strand of n segments along the chain contour, averaged over all the chains of length <math>n</math>. It can be revealed from this study that at short length scales, chains appear to organize into compact disc-like domains, called globules, whose size and density increase with the extent of confinement. A decrease in the overall chain length also increases the globules size but it has no effect on their density.

The model findings are considered relevant to the molecular origin of the exceptional stiffness of nacre, a composite of parallel plates of calcite which confine protein chains to small distances.