The Logistic Function in Glass Transition Models of Amorphous Polymers: A Theoretical Framework for Isobaric Cooling Processes

IF 1.8 4区 工程技术 Q3 POLYMER SCIENCE
Claudio Corbisieri
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Abstract

Studying the macroscopic-phenomenological behavior of amorphous polymers at glass transition is often subject to limitations because the ordinary differential equations that describe the material behavior require numerical solution. To avoid these limitations, ad-hoc-formulated models of the glass transition have been proposed. However, their scope of application is expected to be limited due to insufficient theoretical foundation. This work establishes a theoretical framework for models that use the logistic function to approximate the macroscopic-phenomenological behavior of amorphous polymers at glass transition. For this purpose, an exactly-solvable Riccati equation is derived within thermodynamics with internal state variables. A closed-form expression in terms of mathematical functions for the temperature derivative of a single internal state variable is the result. This closed-form expression contains the logistic function thus featuring a continuous transition region centered around a pressure and cooling-rate dependent transition temperature. Based on comparison of existing models with the exact solution derived from the Riccati equation, generalized models that approximate the thermal expansion coefficient and heat capacity at glass transition are proposed. This work thus demonstrates the validity of the logistic function in glass transition models of amorphous polymers and provides suggestions as to how existing models can be extended in their applicability.

Abstract Image

无定形聚合物玻璃转化模型中的对数函数:等压冷却过程的理论框架
由于描述材料行为的常微分方程需要数值解法,因此研究无定形聚合物在玻璃化转变过程中的宏观现象学行为往往受到限制。为了避免这些限制,人们提出了玻璃化转变的临时模型。然而,由于理论基础不足,这些模型的应用范围预计会受到限制。本研究建立了一个理论框架,利用对数函数来逼近无定形聚合物在玻璃化转变过程中的宏观现象行为。为此,在热力学中导出了一个可精确求解的 Riccati 方程,并带有内部状态变量。以数学函数的形式对单一内部状态变量的温度导数进行了闭式表达。这个闭式表达式包含逻辑函数,因此以压力和冷却速率相关的过渡温度为中心,形成了一个连续的过渡区域。在将现有模型与里卡提方程得出的精确解进行比较的基础上,提出了近似玻璃化转变时热膨胀系数和热容量的广义模型。因此,这项研究证明了逻辑函数在无定形聚合物玻璃化转变模型中的有效性,并就如何扩展现有模型的适用性提出了建议。
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来源期刊
Macromolecular Theory and Simulations
Macromolecular Theory and Simulations 工程技术-高分子科学
CiteScore
3.00
自引率
14.30%
发文量
45
审稿时长
2 months
期刊介绍: Macromolecular Theory and Simulations is the only high-quality polymer science journal dedicated exclusively to theory and simulations, covering all aspects from macromolecular theory to advanced computer simulation techniques.
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