arXiv 2103.161812021-03-30cond-mat.soft

含有限环的Polymer模型网络的弹性

On the Elasticity of Polymer Model Networks Containing Finite Loops

Michael Lang

发表信息：: Macromolecules 52 (2019) 6266-6273
DOI：: 10.1021/acs.macromol.9b00996

摘要：

基于电阻类比和使用的理想回路气体近似（ILGA），它示出，只有挂起的循环减少模量，否则完美的网络由单分散链和路口相同的功能。因此，如果可以采用电阻器类比，则移除挂起结构（环状和分支）的网络的循环秩足以表征模量。它进一步表明，它是不可能的，将有限的周期到一个聚合物网络，使个别网络链处于平衡构象，同时保持在交界处的力平衡。因此，电阻类比仅提供了包含有限环路的网络的幻影模的近似值。改进的方法，幻影模量可以构建从考虑在交界处的力平衡，这需要在不完善的网络中的交联波动的分布的知识。假设回路具有平衡构象和所有回路连接处的力平衡，则在末端连接模型网络的ILGA内以及在$L_{1}\llξ$的极限内获得体模模量$G_{\text{ph}}\approx\left(ξ-c_{\text{f}}L_{1}\right)kT/V$的下限估计。这里，$L_{1}$是主（“待决”）循环的数量，$ξ$是网络的循环秩，$k$是玻尔兹曼常数，$V$是样品的体积，$T$是绝对温度。$c_{\text{f}}$是一个官能度依赖系数，对于连接官能度$f=3$，它是$\approx2.56$，对于$f=4$，它是$\approx3.06$，而在大$f$的限制下，它快速地向$\approx4.2$收敛。进一步修正幻影模量超出有限循环简要地解决。

英文摘要：

Based upon the resistor analogy and using the ideal loop gas approximation(ILGA) it is shown that only pending loops reduce the modulus of an otherwise perfect network made of monodisperse strands and junctions of identical functionality. Thus, the cycle rank of the network with pending structures removed (cyclic and branched) is sufficient to characterize modulus, if the resistor analogy can be employed. It is further shown that it is impossible to incorporate finite cycles into a polymer network such that individual network strands are at equilibrium conformations while maintaining simultaneously a force balance at the junctions. Therefore, the resistor analogy provides only an approximation for the phantom modulus of networks containing finite loops. Improved approaches to phantom modulus can be constructed from considering a force balance at the junctions, which requires knowledge of the distribution of cross-link fluctuations in imperfect networks. Assuming loops with equilibrium conformations and a force balance at all loop junctions, a lower bound estimate for the phantom modulus, $G_{\text{ph}}\approx\left(ξ-c_{\text{f}}L_{1}\right)kT/V$ is obtained within the ILGA for end-linked model networks and in the limit of $L_{1}\llξ$. Here, $L_{1}$ is the number of primary ("pending") loops, $ξ$ the cycle rank of the network, $k$ the Boltzmann constant, $V$ the volume of the sample, and $T$ the absolute temperature. $c_{\text{f}}$ is a functionality dependent coefficient that is $\approx2.56$ for junction functionality $f=3$ and $\approx3.06$ for $f=4$, while it converges quickly towards $\approx4.2$ in the limit of large $f$. Further corrections to phantom modulus beyond finite loops are addressed briefly.

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英文摘要：

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