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Reconciling lattice and continuum models for polymers at interfaces

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10.1063/1.3693515

### Abstract

It is well known that lattice and continuum descriptions for polymers at interfaces are, in principle, equivalent. In order to compare the two models quantitatively, one needs a relation between the inverse extrapolation length *c* as used in continuum theories and the lattice adsorption parameter Δχ_{s} (defined with respect to the critical point). So far, this has been done only for ideal chains with zero segment volume in extremely dilute solutions. The relation Δχ_{s}(*c*) is obtained by matching the boundary conditions in the two models. For depletion (positive *c* and Δχ_{s}) the result is very simple: Δχ_{s} = ln(1 + *c*/5). For adsorption (negative *c* and Δχ_{s}) the ideal-chain treatment leads to an unrealistic divergence for strong adsorption:*c* decreases without bounds and the train volume fraction exceeds unity. This due to the fact that for ideal chains the volume filling cannot be accounted for. We extend the treatment to real chains with finite segment volume at finite concentrations, for both good and theta solvents. For depletion the volume filling is not important and the ideal-chain result Δχ_{s} = ln(1 + *c*/5) is generally valid also for non-ideal chains, at any concentration, chain length, or solvency. Depletion profiles can be accurately described in terms of two length scales: ρ = tanh^{2}[(*z* + *p*)/δ], where the depletion thickness (distal length) δ is a known function of chain length and polymer concentration, and the proximal length *p* is a known function of *c* (or Δχ_{s}) and δ. For strong repulsion *p* = 1/*c* (then the proximal length equals the extrapolation length), for weaker repulsion *p* depends also on chain length and polymer concentration (then *p* is smaller than 1/*c*). In very dilute solutions we find quantitative agreement with previous analytical results for ideal chains, for any chain length, down to oligomers. In more concentrated solutions there is excellent agreement with numerical self-consistent depletion profiles, for both weak and strong repulsion, for any chain length, and for any solvency. For adsorption the volume filling dominates. As a result *c* now reaches a lower limit *c* ≈ −0.5 (depending slightly on solvency). This limit follows immediately from the condition of a fully occupied train layer. Comparison with numerical SCF calculations corroborates that our analytical result is a good approximation. We suggest some simple methods to determine the interaction parameter (either *c* or Δχ_{s}) from experiments. The relation Δχ_{s}(*c*) provides a quantitative connection between continuum and lattice theories, and enables the use of analytical continuum results to describe the adsorption (and stretching) of lattice chains of any chain length. For example, a fully analytical treatment of mechanical desorption of a polymer chain (including the temperature dependence and the phase transitions) is now feasible.

© 2012 American Institute of Physics

Received 06 December 2011
Accepted 22 February 2012
Published online 05 April 2012

Acknowledgments: A.M.S. acknowledges financial support through the Russian-German Grant No. N09-03091344 NNIO-a.

Article outline:

I. INTRODUCTION

II. LATTICE MODEL

A. Field

B. Propagator

C. Composition law

III. CONTINUUM MODEL

A. Edwards equation

B. Exact solution of the Edwards equation

C. Approximations for depletion

IV. GROUND-STATE APPROXIMATION AND THE RELATION *c*(χ_{s})

A. GS equation

B. The GSA boundary condition

C. The lattice boundary condition

D. The relation *c*(χ_{s})

V. DEPLETION RESULTS

A. Explicit expressions for *g*(*z*)

B. GSA and SCF depletion profiles

C. Extension to swollen chains

VI. ADSORPTION RESULTS

A. Adsorption from semidilute solutions

B. Adsorption from dilute solutions

C. The relations c(χ_{s}) and φ_{0}(χ_{s}) for adsorption

D. GSA and SCF adsorption profiles

E. Determination of the adsorption energy parameter from experiment

VII. CONCLUDING REMARKS

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2012-04-05

2014-04-20

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