This paper considers the evolution of a polydisperse polymerizing system comprising g₁,g₂…$g_1\text{,}{g}_2\dots$ – mers carrying ϕ₁,ϕ₂…${\varphi }_1\text{,}{\varphi }_2\dots$ functional groups reacting with one another and binding the g–mers together. In addition, the g–mers are assumed to be added at random by one at a time with a known rate depending on their mass g   and functionality ϕ$\varphi$. Assuming that the rate of binding of two g–mers is proportional to the product of the numbers of nonreacted functional groups the kinetic equation for the distribution of clusters (g–mers) over their mass and functionalities is formulated and then solved by applying the generating function method. In contrast to existing approaches this kinetic equation operates with the efficiencies proportional to the product of the numbers of active functional groups in the clusters rather than to the product of their masses. The evolution process is shown to reveal a phase transition: the emergence of a giant linked cluster (the gel) whose mass is comparable to the total mass of the whole polymerizing system. The time dependence of the moments of the distribution of linked components over their masses and functionalities is investigated. The polymerization process terminates by forming a residual spectrum of sol particles in addition to the gel.

中文翻译：

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精确可解的聚合模型

本文考虑了包含g ₁，g ₂ …的多分散聚合体系的发展。$G_{\mathrm{1个}}\text{，}{G}_{\mathrm{2个}}\dots$–携带ϕ ₁，ϕ ₂ …的海员${\varphi }_{\mathrm{1个}}\text{，}{\varphi }_{\mathrm{2个}}\dots$官能团相互反应并将g- mers结合在一起。另外，克假定聚体被随机通过一次一个地添加具有取决于它们的质量已知的速率克  和功能φ$\varphi$。假设两个g- mers的结合速率与未反应的官能团数量的乘积成正比，则为簇分布的动力学方程（g-mers）在其质量和功能上被公式化，然后通过应用生成函数方法进行求解。与现有方法相反，该动力学方程的效率与簇中活性官能团数量的乘积成正比，而不与其质量的乘积成正比。演化过程显示出一个相变：一个巨大的团簇（凝胶）的出现，其质量与整个聚合体系的总质量相当。研究了链接组件的分布力矩在其质量和功能上的时间依赖性。除了形成凝胶外，聚合过程还通过形成溶胶颗粒的残留光谱而终止。