通过分数阶粘弹性和Mittag-Leffler-Prabhakar函数分析凝胶化转变
Analyzing the Gelation Transition through Fractional Viscoelasticity and the Mittag-Leffler-Prabhakar Function
AI总结：
本文利用分数阶粘弹性模型和基于三参数Mittag-Leffler-Prabhakar函数的模型，研究凝胶化转变中临界凝胶态的标度无关幂律粘弹性行为，并验证了临界指数间的超标度关系。
摘要：
凝胶化转变是将可流动液体转变为弹性固体的过程，存在于从胶体到聚合物的各种系统中。在凝胶化转变过程中，系统经历一个临界凝胶态，其特征是无标度幂律粘弹性。有趣的是，分数阶微积分为这种幂律粘弹性提供了自然的数学语言。在这项工作中，我们开发了物理约束的分数阶粘弹性模型以及基于三参数Mittag-Leffler-Prabhakar函数的模型，用于前凝胶态和后凝胶态区域，确保与每个区域中的传统标度关系一致。虽然观察到分数阶前凝胶模型仅在有限的参数值子集内有效，但基于Prabhakar函数的模型严格消除了这一限制。我们强制动态模量及其导数在临界凝胶点处连续，这普遍地施加了临界凝胶态两侧松弛动力学的对称性。这种强制进一步验证了连接临界指数的超标度关系，使其成为理论必然性而非经验巧合。我们针对时域和频域实验数据验证了所提出的模型。还识别了一个与模型无关、频率无关的临界凝胶态流变指纹，该指纹由两个临界指数唯一确定。
英文摘要：
The gelation transition, a process that transforms a flowable liquid into an elastic solid, occurs in a variety of systems, ranging from colloidal to polymeric. During the gelation transition, a system passes through a critical gel state characterized by scale-free power-law viscoelasticity. Interestingly, the fractional calculus provides a natural mathematical language for such power-law viscoelasticity. In this work, we develop physically constrained fractional viscoelastic models, as well as those based on the three-parameter Mittag-Leffler-Prabhakar function, for both the pre-gel and post-gel regimes, ensuring consistency with the conventional scaling relations in each regime. While the fractional pre-gel model is observed to be valid only for a restricted subset of parameter values, the Prabhakar function-based model for the relaxation modulus, which translates into a Havriliak-Negami function-based complex modulus, rigorously removes this limitation. We enforce continuity of the dynamic moduli and their derivatives across the critical gel point, which universally imposes a symmetry in the relaxation dynamics on either side of the critical gel state. Such enforcement further validates the hyperscaling relation connecting the critical exponents, making it a theoretical necessity rather than an empirical coincidence. We validate the proposed models against time- and frequency-domain experimental data. A model-agnostic, frequency-independent rheological fingerprint of the critical gel state, uniquely determined by two critical exponents, is also identified.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
