^{1,a)}and John G. Curro

^{2}

### Abstract

A recently developed density functional theory(DFT) for tethered bead-spring chains is used to investigate colloidal forces for the good solvent case. A planar surface of tethered chains is opposed to a bare, hard wall and the force exerted on the bare wall is calculated by way of the contact density. Previously, the case of large wall separation was investigated. The density profiles of the unperturbed chains, in that case, were found to be neither stepfunctions nor parabolas and were shown to accurately predict computer simulation results. In the present paper, the surface forces that result from the distortion of these density profiles at finite wall separation is studied. The resulting force function is analyzed for varying surface coverages, wall separations, and chain lengths. The results are found to be in near quantitative agreement with the scaling predictions of Alexander [S. Alexander, J. Phys. (Paris)38, 983 (1977)] when the layer thickness is “correctly” defined. Finally, a hybrid Alexander–DFT theory is suggested for the analysis of experimental results.

Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION

II. MOLECULAR MODEL AND GENERAL BACKGROUND

III. SCALING THEORY

IV. RESULTS

V. DISCUSSION

### Key Topics

- Density functional theory
- 59.0
- Solvents
- 22.0
- Free energy
- 14.0
- Polymers
- 8.0
- Computer simulation
- 7.0

## Figures

The reduced surface energy, , as a function of wall separation, . The squares are the simulation results of Murat and Grest^{6} for chains of and surface coverage ; the circles, experimental results of Taunton *et al.* ^{23} on toluene-solvated polystyrene tethered to mica surfaces; the triangles, the result of DFT theory ; the solid line is the Alexander–DFT prediction. The theory and simulation curves are shifted in the vertical direction by an arbitrary amount so as to match the experiment. In the inset, the unshifted DFT and simulation results are shown along with the (shifted) Alexander–DFT prediction.

The reduced surface energy, , as a function of wall separation, . The squares are the simulation results of Murat and Grest^{6} for chains of and surface coverage ; the circles, experimental results of Taunton *et al.* ^{23} on toluene-solvated polystyrene tethered to mica surfaces; the triangles, the result of DFT theory ; the solid line is the Alexander–DFT prediction. The theory and simulation curves are shifted in the vertical direction by an arbitrary amount so as to match the experiment. In the inset, the unshifted DFT and simulation results are shown along with the (shifted) Alexander–DFT prediction.

The average layer thickness as a function of the cube root of the surface coverage . Chain lengths of , 20, 30, 40, 50, and 100 are shown. In the scaling regime, a linear relationship exists as predicted by scaling theory. In the inset, the slopes of vs are plotted vs. chain length. A linear relation is found—also in agreement with scaling theory. The solid points are the result of DFT calculations while the open points are computer simulation results of Grest *et al.* ^{13} The dashed curve is the boundary between the mushroom and scaling regimes.

The average layer thickness as a function of the cube root of the surface coverage . Chain lengths of , 20, 30, 40, 50, and 100 are shown. In the scaling regime, a linear relationship exists as predicted by scaling theory. In the inset, the slopes of vs are plotted vs. chain length. A linear relation is found—also in agreement with scaling theory. The solid points are the result of DFT calculations while the open points are computer simulation results of Grest *et al.* ^{13} The dashed curve is the boundary between the mushroom and scaling regimes.

The surface force as a function of the wall separation. . The symbols are the results of DFT theory. The lines are the Alexander–DFT predictions.

The surface force as a function of the wall separation. . The symbols are the results of DFT theory. The lines are the Alexander–DFT predictions.

The reduced surface force as a function of reduced layer thickness. All the systems indicated as being in the scaling regime in Fig. 2 are shown. The different chains lengths have been shifted vertically for clarity with, from lowest to highest, , 20, 30, 40, 50, and 100. The lines are the Alexander predictions. In the inset, the unshifted points are shown.

The reduced surface force as a function of reduced layer thickness. All the systems indicated as being in the scaling regime in Fig. 2 are shown. The different chains lengths have been shifted vertically for clarity with, from lowest to highest, , 20, 30, 40, 50, and 100. The lines are the Alexander predictions. In the inset, the unshifted points are shown.

The reduced layer thickness vs. reduced wall separation. The symbols are results of DFT theory have been shifted vertically for clarity with, from lowest to highest, , 20, 30, 40, 50, and 100. The lines are the curve fit reported Eq. (4.2) and is combined with the Alexander surface force expression in the Alexander–DFT predictions of Figs. 1 and 3. In the inset, the unshifted points are shown.

The reduced layer thickness vs. reduced wall separation. The symbols are results of DFT theory have been shifted vertically for clarity with, from lowest to highest, , 20, 30, 40, 50, and 100. The lines are the curve fit reported Eq. (4.2) and is combined with the Alexander surface force expression in the Alexander–DFT predictions of Figs. 1 and 3. In the inset, the unshifted points are shown.

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