高分子鏈型態與運動的蒙地卡羅模擬

Abstract

摘 要 本研究嘗試經由蒙地卡羅模擬(Monte Carlo simulation)研究一有限珠球(beads)數勞斯鏈(Rouse chain)的鏈動力學，藉由末端至末端向量的時間相關函數的模擬結果和分析解之比較，直接證明了蒙地卡羅模擬研究高分子運動的有效性。然後我們以此法計算鏈中單一勞斯節(Rouse segment)和彈性啞鈴(elastic dumbbell)之鍵向量(bond vector)b(t)和方向向量u(t)=b(t)/|b(t)|的動態相關函數，觀察鏈連接效應對單一勞斯節運動的影響，特別是對 的觀察。此函數主要的動態範圍與去偏極光子相關譜（depolarized photo-correlation）所測得動態函數的動態範圍相同，在此範圍內只要珠球數夠大（ ）則此函數是獨立於分子量的。在 條件下聚苯乙烯-環己烷濃溶液( )的去偏極光子相關譜與模擬所得 圖譜形狀的一致性，可完全由包含鏈連接效應之各單一勞斯節運動說明，並由此求得在濃溶液中聚苯乙烯的一勞斯節分子量大約是1100；這比由其他方法估算的熔融態聚苯乙烯勞斯節分子量(780-900)稍為大些。 本研究亦在原本的勞斯鏈上對非相鄰兩節間加入雷努-瓊斯位能(Lennard-Jones potential)來模擬排外體積效應(excluded volume effect)，模擬顯示高分子在高溫（相當於優良溶劑條件下）呈現膨脹鏈圈狀態；在低溫（相當於不良溶劑（poor solvent）條件下）呈現收縮鏈圈狀態，在高溫及低溫兩極的模擬結果與理論預測的指數律 （對膨脹鏈圈而言 ；對收縮鏈圈 ）相近。 另一方面由上述模型模擬尋找出的 條件下計算旋轉半徑與流體動力半徑之比 ，所得無限鏈長下的 值和實驗測量值很一致；實驗值和Kirwood理論值及純粹勞斯模型的模擬結果之間的偏差，導因於實驗系統中 條件下的高分子鏈型態並非真正理想的高斯鏈。

Abstract The validity of the Monte Carlo simulation for studying the dynamics of a Rouse chain with finite number of beads, N, is established by showing the close agreement between the simulation results and the analytical solutions for the time-correlation function of the end-to-end vector. Then, the Monte Carlo simulation is used to calculate the dynamic functions associated with the bond vector b(t) or direction u(t)=b(t)/|b(t)| of an elastic dumbbell and a Rouse segment in a chain. The effect of chain connectivity on the motions of a single Rouse segment is studied. In particular, it is shown that the dynamic function over a wide dynamic range, which is the main region probed by the depolarized photo-correlation spectroscopy, is basically independent of the value of N 8 in agreement with the experimental results. Furthermore, the line shape of the depolarized photo-correlation functions of the concentrated solutions ( ) of polystyrene in cyclohexane at theta point can be fully accounted for by including the effect of chain connectivity regardless of the crudeness of the Rouse segment relative to the chemical structure. From this study, the molecular weight for a Rouse segment of polystyrene in concentrated solutions should be around 1100, which is slightly larger than the values m=780~900 obtained for polystyrene in the melt state by other methods. Another simulation topic is the excluded volume effect, which is introduced by adding the Lennard-Jones interaction potential between all the beads not directly linked by a spring (i.e. beyond the nearest neighbor) to the Rouse model. Our aim has been to describe the change of dimensions with temperature , reaching the globular state at low temperature and the swollen state at high temperature. In the high and low temperature limits, the simulation results are in close agreement with theoretical scaling laws , for the swollen state and for the globular state, respectively. From this simulation, we calculate the ratio of the radius of gyration to the hydrodynamic radius （ ） as a function of chain length under the theta condition. The ratio obtained from extrapolation for infinitely long chain is in much better agreement with the experimental result than the theoretical value of the Gaussian chain and the value obtained from the simulation based on the Rouse chain model (without involving the LJ potential). It is shown that the discrepancy between theory and experiments is due to the conformation of the polymer chain being not totally Gaussian, even under the theta condition.