Laboratory of Physical and Colloid Science, Wageningen University, 6703 HB Wageningen, The Netherlands.
J Chem Phys. 2012 Apr 7;136(13):134707. doi: 10.1063/1.3693515.
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.
众所周知，聚合物界面的格子和连续体描述在原则上是等效的。为了定量比较这两种模型，需要建立连续体理论中使用的逆外推长度 c 与格子吸附参数Δχ(s)（相对于临界点定义）之间的关系。到目前为止，这种关系仅在非常稀的溶液中具有零段体积的理想链中得到了研究。通过匹配两种模型的边界条件，可以得到Δχ(s)(c)关系。对于耗尽（正 c 和Δχ(s)），结果非常简单：Δχ(s) = ln(1 + c/5)。对于吸附（负 c 和Δχ(s)），理想链处理导致强烈吸附的不现实发散：c 无限制地减小，而链体积分数超过 1。这是因为对于理想链，无法考虑体积填充。我们将处理扩展到具有有限段体积的真实链，适用于良好和 theta 溶剂。对于耗尽，体积填充并不重要，理想链的结果Δχ(s) = ln(1 + c/5)通常也适用于任何浓度、链长或溶剂化的非理想链。可以根据两个长度尺度准确描述耗尽轮廓：ρ = tanh(2)[(z + p)/δ]，其中耗尽厚度（远端长度）δ是已知的链长和聚合物浓度函数，近端长度 p 是 c（或Δχ(s)）和δ的已知函数。对于强排斥，p = 1/c（此时近端长度等于外推长度），对于较弱的排斥，p 还取决于链长和聚合物浓度（此时 p 小于 1/c）。在非常稀的溶液中，我们发现与理想链的先前分析结果在任何链长下都具有定量一致性，直至低聚物。在更浓的溶液中，与弱和强排斥的数值自洽耗尽轮廓具有极好的一致性，适用于任何链长和任何溶剂化。对于吸附，体积填充占主导地位。因此，c 现在达到下限 c ≈ -0.5（稍微取决于溶剂化）。这一下限直接来自完全占据的链层条件。与数值 SCF 计算的比较证实，我们的分析结果是一个很好的近似。我们建议了一些从实验确定相互作用参数（无论是 c 还是Δχ(s)）的简单方法。Δχ(s)(c)关系提供了连续体和格子理论之间的定量联系，并使分析连续体结果能够用于描述任何链长的格子链的吸附（和拉伸）。例如，现在可以对聚合物链的机械解吸（包括温度依赖性和相变）进行完全分析处理。
