Fu Henry C, Shenoy Vivek B, Powers Thomas R
Division of Engineering, Brown University, Providence, Rhode Island 02912, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061503. doi: 10.1103/PhysRevE.78.061503. Epub 2008 Dec 16.
In the technique of microrheology, macroscopic rheological parameters as well as information about local structure are deduced from the behavior of microscopic probe particles under thermal or active forcing. Microrheology requires knowledge of the relation between macroscopic parameters and the force felt by a particle in response to displacements. We investigate this response function for a spherical particle using the two-fluid model, in which the gel is represented by a polymer network coupled to a surrounding solvent via a drag force. We obtain an analytic solution for the response function in the limit of small volume fraction of the polymer network, and neglecting inertial effects. We use no-slip boundary conditions for the solvent at the surface of the sphere. The boundary condition for the network at the surface of the sphere is a kinetic friction law, for which the tangential stress of the network is proportional to relative velocity of the network and the sphere. This boundary condition encompasses both no-slip and frictionless boundary conditions as limits. Far from the sphere there is no relative motion between the solvent and network due to the coupling between them. However, the different boundary conditions on the solvent and network tend to produce different far-field motions. We show that the far-field motion and the force on the sphere are controlled by the solvent boundary conditions at high frequency and by the network boundary conditions at low frequency. At low frequencies compression of the network can also affect the force on the sphere. We find the crossover frequencies at which the effects of sliding of the sphere past the polymer network and compression of the gel become important. The effects of sliding alone can lead to an underestimation of moduli by up to 33%, while the effects of compression alone can lead to an underestimation of moduli by up to 20%, and the effects of sliding and compression combined can lead to an underestimation of moduli by up to 43%.
在微观流变学技术中，宏观流变参数以及关于局部结构的信息是从微观探针粒子在热或主动驱动下的行为推导出来的。微观流变学需要了解宏观参数与粒子因位移而感受到的力之间的关系。我们使用双流体模型研究球形粒子的这种响应函数，其中凝胶由通过拖曳力与周围溶剂耦合的聚合物网络表示。我们在聚合物网络的小体积分数极限下并忽略惯性效应，得到了响应函数的解析解。我们对球体表面的溶剂使用无滑移边界条件。球体表面网络的边界条件是一个动摩擦定律，对于该定律，网络的切向应力与网络和球体的相对速度成正比。这个边界条件包含了无滑移和无摩擦边界条件这两个极限情况。远离球体时，由于溶剂和网络之间的耦合，它们之间没有相对运动。然而，溶剂和网络上不同的边界条件往往会产生不同的远场运动。我们表明，远场运动和球体上的力在高频时由溶剂边界条件控制，在低频时由网络边界条件控制。在低频时，网络的压缩也会影响球体上的力。我们找到了球体滑过聚合物网络的滑动效应和凝胶压缩效应变得重要的交叉频率。仅滑动效应可能导致模量低估高达33%，而仅压缩效应可能导致模量低估高达20%，滑动和压缩联合效应可能导致模量低估高达43%。
