以布朗動力學模擬探討流體動力作用影響之下高斯鏈的行為

Abstract

在不考慮排外體積效應的q溫度下，可在理論分析及實驗上來研究在受到流體動力作用(HI)影響之下的稀薄溶液的鏈動力學行為。在加入流體動力作用後的Langevin方程式會變成非線性，而因此無法得其理論分析解。齊姆(Zimm)對流體動力作用(the Oseen tensor)採取了預先平均(preaveraged)的做法。在這樣的近似作法下，可以用數值計算來求得其正規運動模式的特徵值(eigenvalues)。而對於那些無法得到理論分析解或者是數值解的情形下，布朗動力學模擬即可在鏈動力學的研究上成為有用的方法 首先，藉由計算鬆弛模數(relaxation modulus)及末端至末端向量時間相關函數並與由正規運動模式(齊姆理論)的特徵值所計算出之結果的方式，從模擬結果與數值解的一致性更證實了以布朗動力學模擬方式來研究一有限珠球數下的高斯鏈動力學的有效性。 當使用Rotne-Prager-Yamakawa tensor (RPYT)時，在本研究中，我們對應力鬆弛的非線性行為加以研究。此外，第二正向應力差異函數不再如齊姆理論所預估的零值，而有一負的峰值且其發生點，不管珠球的數目，皆在時間t/Dt~3或4間。 模擬的另一個應用則是計算在不同鏈長及流體動力作用強度h*下的旋轉半徑與流體動力作用半徑之比值(r=Rg/RH)。結果顯示由preaveraged Oseen tensor模擬出的結果與高斯鏈假設下用Kirkwood方程式算得之值有很好的一致性。但是由Kirkwood理論結果與實驗上的數據來比較，約有10~15%的不一致性存在，此一不一致性無法以preaveraged Oseen tensor來加以解釋。而一般相信此一不一致性起因於高分子鏈型態即使是在q條件下，也並非全然為理論分析中的高斯鏈。 本研究中的最後一個主題是動態結構因子，此一物理量可以藉由光子相關譜儀來測得。動態結構因子有兩個極端的區域(qRg<<1與qRg>>1)，它們反應出兩種不同的分子鏈運動：一是整條高分子鏈的擴散運動(qRg<<1)，而另一則是分子內部的運動(qRg>>1)。由模擬結果與由正規運動模式的特徵值所計算出的數值解的吻合，我們可指出以preaveraged Oseen tensor與RPYT兩種方法來模擬是沒有差別的。此外，我們也比較了數值分析解與實驗數據。根據在不同散射角度與不同的流體動力作用強度的結果，我們發現對一在q溶劑(環己烷)內的幾近於單一分子量分佈的聚苯乙烯樣品F2000( )而言，最適合的流體動力作用參數 h*約為0.20~0.25。我們也使用由Dubois-Violette與de Gennes所推導出的一連續的積分方程式來與實驗結果做比較。在高角度時(90。與120。)，其qRg值>>1，此時可以觀察到q3的依附關係。但在低角度時，如30。時，其qRg值只稍大於在1，而在這種qRg值下，內部運動及擴散運動皆會影響此一動態結構因子。 在本研究中，我們展示了布朗運動模擬的可行性及多方的應用。由模擬中，我們呈現出令人洞察鏈動力學的諸多細節。

At the q temperature, where the excluded volume effect is absent, the chain dynamic behavior under the influence of hydrodynamic interaction(HI) in a dilute solution has been studied theoretically and experimentally by many. The inclusion of the HI into the Langevin equation renders it to become non-linear and unsolvable analytically. Zimm preaveraged the HI(the Oseen tensor). Under such an approximation, the eigenvalues for the normal modes of motion can be obtained numerically. In the case where an analytic solution or numerical solution is not possible, the Brownian dynamic simulation becomes very valuable in studying the chain dynamics. First, the validity of the Brownian dynamics simulation for studying the dynamics of a Gaussian chain with a finite number of beads, N, is established by showing the close agreement between the simulation results and the values for the relaxation modulus and the time-correlation function of the end-to-end vector numerically calculated from the eigenvalues of the normal modes of motion(the Zimm theory). With the Rotne-Prager-Yamakawa tensor(RPYT) for HI, the non-linear stress-relaxation behavior is studied in this report. It is found from simulation that the second normal stress difference function is no longer zero as predicted by the Zimm theory but reveal a negative peak occurring at the same time t/Dt~3 or 4 for different values of N. Another simulation topic is the ratio of the radius of gyration to the hydrodynamic radius(r=Rg/RH) as a function of chain length under different hydrodynamic interaction strengths, h*. The result shows the close agreement between the simulation of using preaveraged Oseen tensor and the calculated values of the Gaussian chain using the Kirkwood equation. It has been noticed that there exists a discrepancy of 10~15% between Kirkwood's values and experimental data, which cannot be explained when the preaveraged Oseen tensor is used. It is believed that this disagreement is due to the conformation of the polymer chain being not entirely Gaussian even under the q condition. The final topic of this study is the dynamic structural factor, which can be measured by the photon-correlation spectroscopy. The dynamic structural factor has two limiting regimes(qRg<<1 and qRg>>1), which are characterized by different kinds of chain motions: one is the diffusion motion of the whole chain (qRg<<1) and the other is the internal motions(qRg>>1). The close agreement between simulation results and values numerically calculated from the eigenvalues of normal modes of motion indicates that there is no difference between the results of using the preaveraged Oseen tensor and RPYT. The comparison of numerically calculated values and experimental data is also made. According to the comparison at different scattering angles and under different hydrodynamic strengths, the optimum hydro-dynamic interaction parameter h* for the nearly monodisperse polystyrene sample F2000( ) in the q solvent, (cyclohexane)should be in the range 0.20~0.25. The continuous integral equation of Dubois-Violette and de Gennes is also compared with experimental results. At higher scattering angles(90。and 120。) where the condition qRg>>1 holds, the q3 dependence is observed. But at a low angle,30。, where the qRg value is not much larger than 1,not only internal motions but also the diffusion motion affect the dynamic structural factor. In this study, we demonstrate the feasibility and applications of Brownian dynamics simulation. From the simulation, insightful details of chain dynamics are revealed.