高分子熔液內高度糾纏單獨鍊的動態結構因子之Rouse-Mooney模式

Abstract

我們以Rouse-Mooney model的Langevin equation推導得到高度糾纏高分子熔液之單獨鍊中子散射動態結構因子的連續與不連續函數表示式與Monte Carlo模擬和實驗結果相比較。模擬的結果與理論曲線相當一致，因此模擬的過程可以解說Rouse-Mooney model所內含的物理意義。就實用的目的來看，只要在可應用於以Rouse segemt為基礎所得理論的“安全“ q範圍(q代表散射波向量(q)的大小)內，使用連續的理論式便能充分地分析實驗結果。此動態結構因子函數表示式與自由 Rouse鍊的結果，在(qa)**2或(qR)**2→∞時( a與 R分別代表entanglement distance 與end-to-end distance平方的平均值)，可轉化為同一極限式，而這個轉化驗證了過去以物理概念所做的預測。此動態結構因子函數表示式可以很自然地轉化至極限式，使得其全範圍的曲線皆能夠以reduced Rouse variable (q**2(Zd*t)**0.5 )一致地表示出來。這樣所展現出的全範圍成了一個架構或圖像，使得不同 範圍之中子散射曲線所造成的效應可在一致性下尋找其方位並加以研究。其中一個明顯的效應就是，此動態結構因子函數表示式的曲線，在qa約為1~5範圍，當 ~0.1<q**2(Zd*t)**0.5 <~4時(以時間來說在t<te1範圍)，會比極限式的曲線偏向較快的一側。這個現象在糾纏PVE樣品的實驗結果與理論曲線的比較中得到支持。另外，此動態結構因子函數表示式預測到的隨波向量q變化之平台高度，也就是隨q而分開之平台，與實驗上 qa在~1到~7之間，在時間大於Rouse-Mooney model中最低mode之鬆弛時間(te1)後，所觀察到的平台現象一致。隨q分開之平台分布對a值的高敏感性使得我們可以讓理論值與實驗結果配合以求出a值。高度糾纏高分子樣品PEB-2由此方式得到的a值與流變值很接近。此研究顯示出Rouse-Mooney model所描述的兩端固定的邊界條件是讓我們能夠正確預測隨q分開之平台的分布的主要原因。

Dynamic structure factor (DSF) functions of single (labeled) chains well-entangled in polymer melts derived from the Langevin equation of the Rouse–Mooney model in both discrete and continuous forms have been compared with the results of Monte Carlo simulations and of experiments. The procedure involved in carrying out the simulations, whose results are in close agreement with the theoretical curves, illustrates the physical meaning of the Rouse–Mooney model. It is shown that for all practical purposes, it is sufficient to use the continuous form to analyze experimental results in the “safe” q region (q being the magnitude of the scattering wave vector q) where the Rouse-segment-based theories are applicable. The DSF function reduces to the same limiting form as that of the free Rouse chain as(qa)**2 or(qR)**2→∞ (a and R being the entanglement distance and the root mean square end-to-end distance, respectively), confirming what has bee expected physically. The natural reduction to the limiting form allows the full range of DSF curves to be displayed in terms of the reduced Rouse variable(q**2(Zd*t)**0.5 ) in a unified way. The displayed full range represents a framework or “map”, with respect to which effects occurring in different regions of the DSF may be located and studied in a consistent manner. One effect is the significant or noticeable deviations of the theoretical DSF curves from the limiting curve in the region ~0.1<q**2(Zd*t)**0.5 <~4(a time region where t<te1 ) to the faster side as is in the range 1~5.This is supported by the comparison of the experimental results of an entangled PVE (poly(vinylethylene)) sample with the theoretical curves. The DSF functional forms predict plateaus with heights depending on the scattering wave vector q-q-split plateaus―as can be experimentally observed in the time region greater than the relaxation time of the lowest Rouse-Mooney mode, ( te1), when qa ( a being the entanglement distance) falls between ~1 and ~7. High sensitivity of the distribution of the q-split plateaus to enables its value to be extracted from matching the calculated with experimental results.The thus obtained value for a well-entangled PEB-2 (poly(ethylene-co-butene)) polymer is close agreement with the rheological result. It is shown that the fixed-end boundary conditions in the Rouse-Mooney model are responsible for the correct prediction of the distribution of the q-split plateaus.