Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA.
J Chem Phys. 2011 Feb 28;134(8):084902. doi: 10.1063/1.3548888.
Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.
两种密切相关的场论方法已被用于先前的工作中，以构建用于嵌段共聚物熔体和混溶性聚合物共混物相关的随机相位近似的粗粒化理论的校正。“辅助场”（AF）方法基于粗粒化模型的配分函数 Z 的严格表达式，该表达式是辅助化学势场的泛函积分。“有效哈密顿量”（EH）方法则基于 Z 作为可观测量子场的泛函积分的表达式。EH 方法中的精确有效哈密顿量 H(eff) 被定义为具有约束有序参数场的系统的自由能。然而，在实践中，H(eff) 通常由平均场自由能泛函近似，从而产生我们所谓的平均场有效哈密顿量（MFEH）近似。这种近似是 Fredrickson-Helfand 对嵌段共聚物中涨落效应的分析以及更早的聚合物共混物中 Ginzburg 判据的工作的起点。Holyst 和 Vilgis 的更严格的 EH 方法使用精确 H(eff) 的辅助场表示，并允许该场的高斯涨落。迄今为止，AF 和 EH 方法的所有应用都依赖于化学势和/或有序参数场的高斯或“单圈”涨落的某种形式，以平均场鞍点为中心。Holyst 和 Vilgis 的单圈 EH 近似和单圈 AF 理论彼此等效，但与单圈 MFEH 理论不等效。单圈 AF 和 MFEH 理论用于预测逆结构因子 S(-1)(q)，（在没有对这两种理论的进一步近似的情况下）通过与 Flory-Huggins 相互作用参数 χ 无关的函数来区分。因此，这些理论用于预测峰散射强度的预测在旋节线附近表现出相似的 χ 依赖性。无序嵌段共聚物熔体中结构因子的 Fredrickson-Helfand 理论是 MFEH 单圈理论的渐近近似，该理论捕获了非常接近有序-无序转变的非常长的对称共聚物的主导渐近行为。
