The excluded volume effect is added to a fractional viscoelastic model for modeling fractal polymers. This reveals a physical connection between the fractional time derivative, fractal geometry, and excluded volume effect. This derivation is a general theoretical framework based on the Scott-Blair fractional model of viscoelasticity when the excluded volume and the hydrodynamic interaction are explicitly taken into account to derive the microscopic stress within the molecular theory of Rouse and Zimm. The methodology extends the generalized molecular theory of Zimm by adding the effect of excluded volume where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. The modified distribution of the end-to-end vector of a monomer contained within a polymer network is used for pre-averaging approximations of the mobility matrix in the Zimm model. The pre-averaging approximation is important since the mobility matrix is a nonlinear function and it is difficult to explicitly calculate. Through application of thermodynamic laws, we derive the linear fractional model of viscoelasticity based on its spectral dimension, fractal dimension, and the excluded volume parameter for fractal media. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is strongly correlated with fractal structure and excluded volume effects.

排除的体积效应被添加到分数粘弹性模型中，用于对分形聚合物进行建模。这揭示了分数时间导数、分形几何和排除体积效应之间的物理联系。该推导是基于粘弹性的 Scott-Blair 分数模型的一般理论框架，当明确考虑排除体积和流体动力相互作用以推导 Rouse 和 Zimm 的分子理论内的微观应力时。该方法通过添加排除体积的影响扩展了 Zimm 的广义分子理论，其中新的松弛公式包含内部状态变量，这些变量自然取决于变形的分数时间导数。包含在聚合物网络中的单体端到端向量的修正分布用于在 Zimm 模型中对迁移率矩阵进行预平均近似。预平均近似很重要，因为迁移率矩阵是一个非线性函数，很难明确计算。通过应用热力学定律，我们根据粘弹性的谱维数、分形维数和分形介质的排除体积参数推导出了粘弹性的线性分数模型。该推导显示了粘弹性线性分数模型中分数导数的阶数与分形结构和排除的体积效应密切相关。预平均近似很重要，因为迁移率矩阵是一个非线性函数，很难明确计算。通过应用热力学定律，我们根据粘弹性的谱维数、分形维数和分形介质的排除体积参数推导出了粘弹性的线性分数模型。该推导显示了粘弹性线性分数模型中分数导数的阶数与分形结构和排除的体积效应密切相关。预平均近似很重要，因为迁移率矩阵是一个非线性函数，很难明确计算。通过应用热力学定律，我们根据粘弹性的谱维数、分形维数和分形介质的排除体积参数推导出了粘弹性的线性分数模型。该推导显示了粘弹性线性分数模型中分数导数的阶数如何与分形结构和排除的体积效应密切相关。