Viscoelastic Properties of Polymer Melts from Equilibrium Molecular Dynamics Simulations
Viscoelastic Properties of Polymer Melts from Equilibrium Molecular Dynamics Simulations
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DOI：
10.1021/ma035487l
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发表时间：
2005-01
期刊：
影响因子：
5.5
通讯作者：
S. Sen;Sanat K. Kumar;P. Keblinski
中科院分区：
文献类型：
--
作者：
S. Sen;Sanat K. Kumar;P. Keblinski
介绍。聚合物链的动力学现在普遍很好地理解了短链的Rouse模型和长链的重复模型的先驱框架。虽然这些理论给出了众所周知的扩散率D和粘度η随链长N变化的表达式，但一个悬而未决的问题是，从劳斯行为过渡到重复行为的链长是多少。这个长度，称为纠缠长度Ne，在实验上比较容易估计，但在模拟中很难获得，因为直到最近，还没有令人满意的纠缠定义在数值研究中，大多数对纠缠长度的估计涉及进行模拟和检查动态量，例如频率相关的存储模量。克雷默和格雷斯特已经做了大量的工作来确定链缠结的起源，结果有些复杂。对链扩散的链长依赖性的研究得出了在长度为35的链附近从劳斯标度（N-1）到重复标度（N-2）行为的交叉。相比之下，从非平衡分子动力学模拟中估计的存储模量产生的纠缠链长度在80附近。我们用平衡分子动力学（MD）的工具来解决这个问题根据Smith等人的建议，从这些模拟中自然产生的平衡应力波动直接通过Green-Kubo方程提供了粘度的估计。10存储和损失模量也从应力自相关函数中计算出来，并且在存储模量中可以看到平台的迹象。平台期是复制动力学开始的最强指示，并且在链长度为80或更高时可见。对纠缠长度Ne的估计，提供了一个接近30的数字，与Kremer和Grest提供的第一个估计一致。仿真模型与方法。MD仿真采用标准链模型。非成键单体之间的相互作用由一个位移的纯排斥性Lennard-Jones （LJ）势来描述：U(r)) 4 [(σ/r)12 (σ/r)6] +， r为21/6σ。相邻的键合单体通过刚性的FENE电位相互作用，其形式为VFENE) k(R0/2) ln(1 (r/R0))，这将相邻单体之间的距离限制在约1σ（我们使用与参考文献5和8中的Grest和Kremer相同的参数来计算FENE电位）。我们对不同链长的单分散聚合物熔体进行了定体积模拟，并将重点关注交叉区，分别是N) 20、40、80和120。在一个典型的模拟中，我们在周期模拟盒中共嵌入了2400个单体，每个方向的尺寸为14.1σ，对应于缩减的段密度F*) 0.85。在一些模拟中，我们将单体的数量增加了一倍，对于最长的链系统N) 120，这相当于将链的数量从20增加到40。由于从模拟中推断出的特性与系统尺寸无关，即使对于这些相对较小的系统，我们得出结论，我们的结果仅具有较小的有限尺寸效应。我们将报告所有数量的减少单位，这是在论文的最后定义。我们在δt) 0.001t*的微正则系综中使用五阶Gear算法进行了MD模拟，确保在整个恒定能量模拟运行过程中能量守恒在0.5%以内。我们运行的起始结构是通过在降低温度T*) 1下，通过~ 1000万MD步骤逐渐挤压半稀溶液至最终密度0.85来构建的。这些结构进一步在恒温下运行，额外的1000万步以获得恒定能量运行的起始配置。在结构制备和平衡之后，我们进行了恒定能量模拟，对于N) 20，我们进行了至少5000万MD步长，对于N) 120，我们进行了高达3亿MD步长。由于我们没有将模拟与热浴相结合，因此降低的温度在单位周围表现出很小（几个百分点）的波动。检查有限尺寸效应的模拟结果是很重要的。克雷默和格雷斯特建议，对于这些链长度，至少需要20个链才能避免这种影响。我们的体系包含2400个单体，即20个链对应N) 120。在下面的表1中，我们列出了所有链长度的各种量，如端到端矢量和扩散率。特别是，当体系规模从2400个单体增加到4800个单体时，扩散系数D从0.0012变化到0.0011。由于D通常对有限尺寸效应最敏感，并且由于这些D值密切跟踪Kremer和Grest的结果，5我们得出结论，我们的系统尺寸足够大。为了检查平衡，我们使用链的端到端矢量的自相关函数，4 < rend(t)rend(0) > / < rend >，其中rend(t)表示时刻t的端到端距离矢量。也可以为旋转的均方根半径定义类似的自相关函数。这两个自相关函数都表现出指数衰减，伦斯勒理工学院化学与生物工程系的Isermann。伦斯勒理工学院材料科学与工程系。表1。a N) 20 N) 40 N) 80 N) 120 < Re > (σ2) 29.2 62.6 127.5 195.1 D (σ2τ-1) 0.02 0.0074 0.0023 0.0012 a Re为端到端平均矢量星等。D是扩散系数。BATCH: ma2a46 USER: rjb69 DIV: @xyv04/data1/CLS_pj/GRP_ma/JOB_i03/DIV_ma035487l日期：2004年12月28日
Introduction. The dynamics of polymer chains are now generally very well understood in the pioneering frameworks of the Rouse model for short chains, and the reptation model for longer chains.1-5 While these theories yield well-known expressions for the variation of the diffusivity D and viscosity η with the chain length N, an open question is the chain length at which one crosses over from Rouse-like behavior to reptation. This length, termed the entanglement length, Ne, is relatively easy to estimate experimentally, but has been hard to access in a simulation due to the fact that, up until recently, there was no satisfactory definition of an entanglement.6,7 In numerical studies, most estimates of the entanglement length involve conducting simulations and examining dynamic quantities, such as the frequency dependent storage modulus. Extensive work has been done by Kremer and Grest8 to identify the onset of chain entanglement, with somewhat mixed results. Examination of the chain length dependence of chain diffusion yields a cross over from the Rouse scaling (N-1) to reptation scaling (N-2) behavior in the vicinity of chains of length 35. In contrast, estimates of storage modulus from nonequilibrium molecular dynamics simulations yield an entanglement chain length in the vicinity of 80. We approach this problem with the tools of equilibrium molecular dynamics (MD).9 Equilibrium stress fluctuations which result naturally from these simulations directly provide estimates of the viscosity through the Green-Kubo equation, as suggested by Smith et al.10 The storage and loss modulus are also computed from the stress autocorrelation function, and the hints of a plateau are seen in the storage modulus. The plateau is the strongest indication of onset of reptation dynamics, and is seen for chain lengths of 80 and higher. Estimates of the entanglement length Ne, provide a number closer to 30, consistent with the first estimate provided by Kremer and Grest. Simulation Model and Methods. The MD simulation employs a standard chain model. Interaction between nonbonded monomers are described by a shifted, purely repulsive Lennard-Jones (LJ) potential: U(r) ) 4 [(σ/r)12 (σ/r)6] + for r 21/6σ. Adjacent bonded monomers interact via a stiff FENE potential, in the form of VFENE ) k(R0/2) ln(1 (r/R0)), which constrains the distance between adjacent monomers to about 1σ (we use the same parameters for the FENE potential as Grest and Kremer in refs 5 and 8). We performed constant volume simulations of monodisperse polymer melts of varying chain lengths, and will focus our attention on the crossover regime, specifically N ) 20, 40, 80 and 120, respectively. In a typical simulation we use a total of 2400 monomers embedded in the periodic simulation box, of size 14.1σ in each direction, which corresponds to the reduced segment density of F* ) 0.85. In a few simulations we have doubled the number of monomers, and for the longest chain system, N ) 120, this corresponds to increasing the number of chains from 20 to 40. Since the properties deduced from the simulation were independent of system size even for these relatively small systems, we conclude that our results only have minor finite size effects. We will report all quantities in terms of reduced units, which are defined at the end of the paper. We conduct MD simulations using a fifth order Gear algorithm in the microcanonical ensemble with a δt ) 0.001t* ensuring energy conservation to within 0.5% over the whole constant energy simulation run. The starting structures for our runs were constructed by gradually squeezing semidilute solutions to a final density of 0.85 over ∼10 million MD steps at the reduced temperature, T* ) 1. These structures were further run at constant temperature, for an additional 10 million steps to obtain a starting configuration for the constant energy run. After the structure preparation and equilibration, we followed with the constant energy simulation for a minimum of 50 million MD steps for N ) 20, up to 300 million MD steps for N ) 120. The reduced temperature exhibited small (several percent) fluctuations around unity since we did not couple our simulations to a heat bath. It is important to check our simulation results for finite-size effects. Kremer and Grest suggest that for these chain lengths a minimum number of 20 chains is required to avoid such effects. Our systems contain 2400 monomers, i.e., 20 chains for N ) 120. In Table 1 below we have tabulated various quantities like the end-toend vector and diffusivity, for all the chain lengths. In particular, the diffusivity, D, changed from 0.0012 to 0.0011, when the system size was doubled from 2400 to 4800 monomers. Since D is typically the most sensitive to finite size effects, and since these D values closely track the results of Kremer and Grest,5 we conclude that our system sizes are large enough. To check for equilibration we use the autocorrelation function of the end-to-end vector of the chains,4 〈rend(t)rend(0)〉/〈rend〉, where rend(t) denotes the end-to-end distance vector at time t. A similar autocorrelation function can also be defined for the root-mean-square radius of gyration. Both of these autocorrelation functions showed an exponential decay, from which the † Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute. ‡ Department of Materials Science and Engineering, Rensselaer Polytechnic Institute. Table 1. a N ) 20 N ) 40 N ) 80 N ) 120 〈Re〉 (σ2) 29.2 62.6 127.5 195.1 D (σ2τ-1) 0.02 0.0074 0.0023 0.0012 a Re is the average end-to-end vector magnitude. D is the diffusivity. BATCH: ma2a46 USER: rjb69 DIV: @xyv04/data1/CLS_pj/GRP_ma/JOB_i03/DIV_ma035487l DATE: December 28, 2004