Role of Distributions in Binders and Curatives and Their Effect on Network Evolution and Structure

Abstract

Precursors of cross-linked polymer systems are blends of many compounds, some of them forming series (distributions) of compounds of increasing molecular weights, type and number of functional groups, or some other property. The distributions may arise from impurities in the raw materials, or a result of side reactions. In most cases, the distributions are generated intentionally; functional copolymers, hyperbranched polymers, off-stoichiometric highly branched polymers, chain extended systems, or precursors prepared in several stages can serve as examples. The distributions affect processing and materials properties. We show ways to generate these distributions from bond formation kinetics and reaction mechanisms. Statistical branching theories based on assemblage of branched molecules and gel structure from building units in different reaction states are used to model the evolution of the cross-linked system.

Appendix

Definitions of Distributions and Their Transform Using the Formalism of Probability Generating Functions

In the following tables, the degree of polymerization, molecular weight, and functionality distributions and their averages are defined and expressed in the form of probability generating functions (pgf). The transform between various distributions and their averages by manipulation with the pgfs is also shown.

Probability Generating Functions

This communication deals with various distributions related to properties of the precursors, and often they are multivariate. To keep track of what property we are dealing with, the distributions are post-signed by labels and presented in the form of probability generating functions. They have been used for description of branching and cross-linking systems since a long time ago, and their formulation and use have been repeatedly explained (cf., e.g., Refs. [1, 14, 45]). Nevertheless, some explaining notes about their use are summarized below.

As examples, the following distributions in the form probability generating functions are discussed:

$$ \begin{array}{l}A(Z)={\sum}_i{a}_i{Z}^i,\kern1em {\sum}_i{a}_i=1\\ {}N(Z)={\sum}_i{n}_i{Z}^i,\kern1em W(Z)={\sum}_i{w}_i{Z}^i,\kern1em {\sum}_i{n}_i={\sum}_i{w}_i=1\\ {}Q(\mathbf{Z},\mathbf{z})={\sum}_{i,j}{q}_{ij}{Z}_{\mathrm{A}}^{X_{\mathrm{A}}}{Z}_{\mathrm{B}}^{X_{\mathrm{B}}}{z}_{\mathrm{A}\mathrm{B}}^i{z}_{\mathrm{B}\mathrm{A}}^j,\kern1em Z={Z}_{\mathrm{A}},{Z}_{\mathrm{B}},\kern1em z={z}_{\mathrm{A}\mathrm{B}}{z}_{\mathrm{B}\mathrm{A}},\kern1em {\sum}_{i,j}{q}_{ij}=1\end{array} $$

The function A(Z) describes a set of probabilities a = a ₁, a ₂, …,  a _i ,…,  a _n of finding the object A in all its possible states distinguished by the value i of a property. In the function N(Z), the property is degree of polymerization, and its value is characterized by the exponent i, related is the set of number fractions n; for the function W(Z), the property is also the degree of polymerization, and the related probability is the set of weight fractions w. Boldface quantities are vectors like a above. The function Q(Z, z) describes the probabilities of finding the building unit Q in states characterized by the properties X _A and X _B and by number of bonds i of AB bonds and j of BA bonds. Z and z are auxiliary variables of the pgfs; by manipulation with them, one can transform them from one to another and to get average values of the property, as will be shown below. The average value is obtained by differentiation and putting the auxiliary variable equal to 1; for example,

$$ \frac{\partial N(Z)}{\partial Z}={\sum}_i{ i n}_i{Z}^i,\kern1em {\left[\frac{\partial N(Z)}{\partial Z}\right]}_{Z=1}={\sum}_i{ i n}_i={P}_{\mathrm{n}} $$

Degree of Polymerization and Molecular Weight Distributions

Averages	Probability generating function formulation
Number-average degree of polymerization, P xn, and molecular weight, M xn, of homopolymer X	$\begin{array}{l} N(Z)={\sum}_i{n}_i{Z}^i\\ {}\frac{\partial N(Z)}{\partial Z}={\sum}_i{ i n}_i{Z}^{i-1}\kern1em {\left[\frac{\partial N(Z)}{\partial Z}\right]}_{Z=1}\equiv N^{\prime }(1)={\sum}_i{ i n}_i={P}_{\mathrm{Xn}}\\ {}{N}_{\mathrm{M}}(Z)={\sum}_i{n}_i{Z}^{i M}\\ {}\frac{\partial {N}_{\mathrm{M}}(Z)}{\partial Z}={\sum}_i{ i n}_i{ M Z}^{i M-1}\\ {}{\left[\frac{\partial {N}_{\mathrm{M}}(Z)}{\partial Z}\right]}_{Z=1}\equiv N{\prime}_M(1)={\sum}_i{ i n}_i M={P}_{\mathrm{Xn}}\end{array}$
$\begin{array}{l}{P}_{\mathrm{X}\mathrm{n}}={\sum}_i{ i n}_i\\ {}{M}_{\mathrm{X}\mathrm{n}}={\sum}_i{ i n}_i{M}_{\mathrm{x}}={M}_{\mathrm{X}}{\sum}_i{ i n}_i\end{array}$
w-Average (weight-average) degree of polymerization P Xw and molecular weight M xw of homopolymer X	$\begin{array}{l} W(Z)={\sum}_i{w}_i Z^{\prime }= Z\frac{\partial N(Z)}{\partial Z}/{\left[\frac{\partial N(Z)}{\partial Z}\right]}_{Z=1}\\ {}\frac{\partial W(Z)}{\partial Z}={\sum}_i{ i w}_i{Z}^{i-1};\kern1em {\left[\frac{\partial W(Z)}{\partial Z}\right]}_{Z=1}\equiv W^{\prime }(1)={\sum}_i{ i w}_i={P}_{\mathrm{X}\mathrm{w}}\\ {}{W}_{\mathrm{M}}(Z)={\sum}_i{w}_i{Z}^{i M}= Z\frac{\partial {N}_{\mathrm{M}}(Z)}{\partial Z}/{\left[\frac{\partial {N}_{\mathrm{M}}(Z)}{\partial Z}\right]}_{Z=1}\\ {}\frac{\partial {W}_{\mathrm{M}}(Z)}{\partial Z}={\sum}_i{ i w}_i{M}_{\mathrm{X}}{Z}^{i M-1}\\ {}{\left[\frac{\partial {N}_{\mathrm{M}}(Z)}{\partial Z}\right]}_{Z=1}\equiv N{\prime}_{\mathrm{M}}(1)={\sum}_i{ i n}_i{M}_{\mathrm{x}}={M}_{\mathrm{X}}{P}_{\mathrm{X}\mathrm{n}}\end{array}$
$\begin{array}{l}{P}_{\mathrm{X}\mathrm{w}}={\sum}_i{i w}_i={\sum}_i{i}^2{n}_i/{\sum}_i{i n}_i\\ {}{M}_{\mathrm{X}\mathrm{w}}={\sum}_i{i w}_i{M}_{\mathrm{x}}={M}_{\mathrm{X}}{\sum}_i{i}^2{n}_i/{\sum}_i{i n}_i\end{array}$
Number-average degree of polymerization P ABn and molecular weight M ABn of copolymer AB	$\begin{array}{l} N\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)={\sum}_k{n}_k\sum_{i=0}^k{x}_{i, k- i}{Z}_{\mathrm{A}}^i{Z}_{\mathrm{B}}^{k- i}\overset{\mathrm{random}}{=}{\sum}_k{n}_k{\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}+\left(1-{x}_{\mathrm{A}}\right){Z}_{\mathrm{B}}\right)}^k\\ {}\mathrm{DP}:{Z}_{\mathrm{A}}={Z}_{\mathrm{B}}= Z,\kern1em {\left[\frac{\partial N(Z)}{\partial Z}\right]}_{Z=1}={\sum}_k{ k n}_k={P}_{\mathrm{A}\mathrm{Bn}}\\ {}{N}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)={\sum}_k{n}_k\sum_{i=0}^k{n}_{i, k- i}{Z}_{\mathrm{A}}^{{i M}_{\mathrm{A}}}{Z}_{\mathrm{B}}^{\left( k- i\right){M}_{\mathrm{B}}}\overset{\mathrm{random}}{=}\\ {}{\sum}_k{n}_k{\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}^{M_{\mathrm{A}}}+\left(1-{x}_{\mathrm{A}}\right){Z}_{\mathrm{B}}^{M_{\mathrm{B}}}\right)}^k\\ {}{\left[\frac{\partial {N}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)}{\partial {Z}_{\mathrm{A}}}\right]}_{Z_{\mathrm{A}}={Z}_{\mathrm{B}}=1}+{\left[\frac{\partial {N}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)}{\partial {Z}_{\mathrm{B}}}\right]}_{Z_{\mathrm{A}}={Z}_{\mathrm{B}}=1}=\\ {}{\sum}_k{n}_k\sum_{i=0}^k\left({ i n}_{i, k- i}{M}_{\mathrm{A}}+\left( k- i\right){n}_{i, k- i}{M}_{\mathrm{B}}\right)\overset{\mathrm{random}}{=}\\ {}{\sum}_k{ k n}_k\left({x}_{\mathrm{A}}{M}_{\mathrm{A}}+\left(1-{x}_{\mathrm{A}}\right){M}_{\mathrm{B}}\right)={M}_{\mathrm{A}\mathrm{Bn}}\end{array}$
$\begin{array}{l}{P}_{\mathrm{A}\mathrm{Bn}}={\sum}_i{ i n}_i={\sum}_i\sum_{j=0}^i\left({ j n}_{\mathrm{A} j}+\left( i- j\right){n}_{\mathrm{B}\left( i- j\right)}\right)\\ {}{M}_{\mathrm{A}\mathrm{Bn}}={\sum}_i\sum_{j=0}^i\left({ j n}_{\mathrm{A} j}{M}_{\mathrm{A}}+\left( i- j\right){n}_{\mathrm{B}\left( i- j\right)}{M}_{\mathrm{B}}\right)\end{array}$
w-Average (weight-average) degree of polymerization P Xw and molecular weight M Xw of copolymer AB	$\begin{array}{l} W\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)=\sum_k{w}_k\sum_{i=0}^k{x}_{i, k- i}{Z}_{\mathrm{A}}^i{Z}_{\mathrm{B}}^{k- i}\overset{\mathrm{random}}{=}\sum_k{w}_k\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}+\right(1-{x}_{\mathrm{A}}\left){Z}_{\mathrm{B}}\right){}^k\\ {}\mathrm{DP}:{Z}_{\mathrm{A}}={Z}_{\mathrm{B}}= Z\kern1em \mathsf{as}\kern0.5em \mathsf{homopolymer}\\ {}{\left[\frac{\partial W(Z)}{\partial Z}\right]}_{Z=1}=\sum_k{k w}_k={P}_{\mathrm{A}\mathrm{Bw}}\\ {}{W}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)=\sum_k{w}_k\sum_{i=0}^k{n}_{i, k- i}{Z}_{\mathrm{A}}^{i{M}_{\mathrm{A}}}{Z}_{\mathrm{B}}^{\left( k- i\right){M}_{\mathrm{B}}}\overset{\mathrm{random}}{=}\sum_k{w}_k\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}^{M_{\mathrm{A}}}+\right(1-{x}_{\mathrm{A}}\left){Z}_{\mathrm{B}}^{M_{\mathrm{B}}}\right){}^k\\ {}{W}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)=\sum_k{n}_k\sum_{i=0}^k{x}_{i, k- i}{Z}_{\mathrm{A}}^{i{M}_{\mathrm{A}}}{Z}_{\mathrm{B}}^{\left( k- i\right){M}_{\mathrm{B}}}\overset{\mathrm{random}}{=}\sum_k{n}_k\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}^{M_{\mathrm{A}}}+\right(1-{x}_{\mathrm{A}}\left){Z}_{\mathrm{B}}^{M_{\mathrm{B}}}\right){}^k\\ {}{W}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)=\Big(1/{M}_{\mathrm{A}\mathrm{Bn}}\Big)\left({Z}_{\mathrm{A}}\frac{\partial {N}_{\mathrm{M}}}{\partial {Z}_{\mathrm{A}}}+{Z}_{\mathrm{B}}\frac{\partial {N}_{\mathrm{M}}}{\partial {Z}_{\mathrm{B}}}\right)=\sum_k{n}_k\sum_{i=0}^k{x}_{i, k- i}{Z}_{\mathrm{A}}^{i{M}_{\mathrm{A}}}{Z}_{\mathrm{B}}^{\left( k- i\right){M}_{\mathrm{B}}}\\ {}\kern3pc =\left(1/{M}_{\mathrm{A}\mathrm{Bn}}\right)\sum_k{n}_k\sum_{i=0}^k\left({ i M}_{\mathrm{A}}+\right( k- i\left){M}_{\mathrm{B}}\right){x}_{i, k- i}{Z}_{\mathrm{A}}^{i{M}_{\mathrm{A}}}{Z}_{\mathrm{B}}^{\left( k- i\right){M}_{\mathrm{B}}}\\ {}{M}_{\mathrm{A}\mathrm{Bw}}={\left[\frac{\partial {W}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)}{\partial {Z}_{\mathrm{A}}}\right]}_{Z_{\mathrm{A}}={Z}_{\mathrm{B}}=1}+\left[\frac{\partial {W}_{\mathrm{M}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}}\right)}{\partial {Z}_{\mathrm{B}}}\right]{}_{Z_{\mathrm{A}}={Z}_{\mathrm{B}}=1}\\ {}\kern1.5pc =\left(1/{M}_{\mathrm{A}\mathrm{Bn}}\right)\sum_k{n}_k\sum_{i=0}^k\left({ i M}_{\mathrm{A}}+\right( k- i\left){M}_{\mathrm{B}}\right){}^2{x}_{i, k- i}\overset{\mathrm{random}}{=}\Big(1/{M}_{\mathrm{A}\mathrm{Bn}}\Big)\\ {}\sum_k{k}^2\left({x}_{\mathrm{A}}{M}_{\mathrm{A}}+\left(1-{x}_{\mathrm{A}}\right){M}_{\mathrm{B}}\right){}^2{n}_k=\left({x}_{\mathrm{A}}{M}_{\mathrm{A}}+\right(1-{x}_{\mathrm{A}}\left){M}_{\mathrm{B}}\right){P}_{\mathrm{A}\mathrm{Bw}}\end{array}$
$\begin{array}{l}{P}_{\mathrm{A}\mathrm{B}\mathrm{w}}\equiv {P}_{\mathrm{A}\mathrm{B}2}=\sum_i\sum_i{i}^2{n}_i/\sum_i{i n}_i\\ {}{M}_{\mathrm{A}\mathrm{B}\mathrm{w}}=\sum_i\sum_{j=0}^i\left({ j w}_{\mathrm{A}\mathrm{j}}{M}_{\mathrm{A}}+\left( i- j\right){w}_{\mathrm{B}\left( i- j\right)}{M}_{\mathrm{B}}\right)\\ {}{w}_{\mathrm{A}\mathrm{j}}=\frac{{ j n}_{\mathrm{A}\mathrm{j}}{M}_{\mathrm{A}}}{\sum_{j=0}^i\left({ j n}_{\mathrm{A}\mathrm{j}}{M}_{\mathrm{A}}+\left( i- j\right){n}_{\mathrm{B}\left( i- j\right)}{M}_{\mathrm{B}}\right)}\end{array}$

Functionality Distributions

Definition	Probability generating function formulation
Number-average and w-average (second moment) functionality average—single functional group	$\begin{array}{l}{F}_{\mathrm{f}\mathrm{n}}\left({Z}_{\mathrm{f}}\right)=\sum_f{n}_f{Z}_{\mathrm{f}}^f\\ {}{f}_{\mathrm{n}}={\left[\frac{\partial {F}_{\mathrm{f}\mathrm{n}}\left({Z}_{\mathrm{f}}\right)}{\partial {Z}_{\mathrm{f}}}\right]}_{Z_{\mathrm{f}}=1}\equiv F{\prime}_{\mathrm{f}\mathrm{n}}(1)\\ {}{F}_{\mathrm{f}2}\left({Z}_{\mathrm{f}}\right)=\sum_f{f n}_f{Z}_{\mathrm{f}}^f/{f}_{\mathrm{n}}={Z}_{\mathrm{f}}\frac{\partial {F}_{\mathrm{f}\mathrm{n}}\left({Z}_{\mathrm{f}}\right)}{\partial {Z}_{\mathrm{f}}}/{f}_{\mathrm{n}}\\ {}{f}_{\mathrm{w}}\equiv {f}_2={\left[\frac{\partial {F}_{\mathrm{f}2}\left({Z}_{\mathrm{f}}\right)}{\partial {Z}_{\mathrm{f}}}\right]}_{Z_{\mathrm{f}}=1}\end{array}$
$\begin{array}{l}{f}_{\mathrm{n}}=\sum_f{f n}_f\\ {}{f}_{\mathrm{w}}\equiv {f}_2=\frac{\sum_f{f}^2{n}_f}{\sum_f{f n}_f}\end{array}$
Number-average and w-average (second moment) functionality average—two types of functional groups (R, S)	$\begin{array}{l}{F}_{\mathrm{fn}}({Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{Bf}})=\sum_{i,j}{n}_{ij}{Z}_{\mathrm{A}\mathrm{f}}^i{Z}_{\mathrm{Bf}}^j\\ {}{f}_{\mathrm{A}\mathrm{n}}={[\frac{\mathrm{\partial}{F}_{\mathrm{fn}}({Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{Bf}})}{\mathrm{\partial}{Z}_{\mathrm{A}\mathrm{f}}}]}_{Z_{\mathrm{A}\mathrm{f}}={Z}_{\mathrm{Bf}}=1}\\ {}{F}_{\mathrm{A}\mathrm{f}2}({Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{Bf}})=\sum_i{in}_{ij}{Z}_{\mathrm{A}\mathrm{f}}^i{Z}_{\mathrm{Bf}}^j/{f}_{\mathrm{A}\mathrm{n}}=\\ {}{Z}_{\mathrm{A}\mathrm{f}}\frac{\mathrm{\partial}{F}_{\mathrm{fn}}({Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{Bf}})}{\mathrm{\partial}{Z}_{\mathrm{A}\mathrm{f}}}/{f}_{\mathrm{A}\mathrm{n}}\\ {}{f}_{\mathrm{A}\mathrm{w}}\equiv {f}_{\mathrm{A}2}={[\frac{\mathrm{\partial}{F}_{\mathrm{A}\mathrm{f}2}({Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{Bf}})}{\mathrm{\partial}{Z}_{\mathrm{A}\mathrm{f}}}]}_{Z_{\mathrm{A}\mathrm{f}}={Z}_{\mathrm{Bf}}=1}=\sum_{i,j}{i}^2{n}_{ij}/{f}_{\mathrm{A}\mathrm{n}}\end{array}$
$\begin{array}{l}{f}_{\mathrm{A}\mathrm{n}}=\sum_f{f}_{\mathrm{A}}{n}_{f\mathrm{A}};\kern1em {f}_{\mathrm{B}\mathrm{n}}=\sum_f{f}_{\mathrm{B}}{n}_{f\mathrm{B}}\\ {}{f}_{\mathrm{A}\mathrm{w}}\equiv {f}_{\mathrm{A}2}=\frac{\sum_f{f}_{\mathrm{A}}^2{n}_{f\mathrm{A}}}{\sum_f{f}_{\mathrm{A}}{n}_{f\mathrm{A}}}\end{array}$Other second-moment averages possible (depends on reaction paths)

Distribution of Reaction States of Building Units in the Branching Process

Cross-linking of f A-functional precursor with f B-functional cross-linker; groups of equal and independent reactivity	States in the form of a pgf: units, unreacted groups Af and Bf, and groups A reacted with B groups and B groups reacted with C groups of cross-linker marked; type of bond (A → B) indicated
$\begin{array}{l}{\left(1-{\alpha}_{\mathrm{A}}+{\alpha}_{\mathrm{A}}\right)}^{f_{\mathrm{A}}}=\\ {}\sum_{i=0}^f\frac{f_{\mathrm{A}}!}{i!\left({f}_{\mathrm{A}}- i\right)!}{\alpha}_{\mathrm{A}}^i{\left(1-{\alpha}_{\mathrm{A}}\right)}^{f_{\mathrm{A}}- i}=\\ {}{\left(1-{\alpha}_{\mathrm{A}}\right)}^{f_{\mathrm{A}}}+ f\left(1-{\alpha}_{\mathrm{A}}\right){}^{f_{\mathrm{A}}-1}{\alpha}_{\mathrm{A}}+\dots \\ {}+{f}_{\mathrm{A}}\left(1-{\alpha}_{\mathrm{A}}\right){\alpha}_{\mathrm{A}}^{f_{\mathrm{A}}-1}+{\alpha}_{\mathrm{A}}^{f_{\mathrm{A}}}\end{array}$	n A, n B mole fractions of precursors A and B, respectively
	$\begin{array}{l}{F}_0\left({Z}_{\mathrm{A}},{Z}_{\mathrm{A}\mathrm{f}},{Z}_{\mathrm{B}},{Z}_{\mathrm{B}\mathrm{f}},{z}_{\mathrm{A}\mathrm{B}},{z}_{\mathrm{B}\mathrm{A}}\right)\equiv {F}_0\left(\mathbf{Z},\mathbf{z}\right)=\\ {}{n}_{\mathrm{A}}{F}_{0\mathrm{A}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{A}\mathrm{f}},{z}_{\mathrm{A}\mathrm{B}}\right)+{n}_{\mathrm{A}}{F}_{0\mathrm{B}}\left({Z}_{\mathrm{B}},{Z}_{\mathrm{B}\mathrm{f}},{z}_{\mathrm{B}\mathrm{A}}\right);\\ {}{F}_{0\mathrm{A}}\left({Z}_{\mathrm{A}},{Z}_{\mathrm{A}\mathrm{f}},{z}_{\mathrm{A}\mathrm{B}}\right)={Z}_{\mathrm{A}}{\left(\left(1-{\alpha}_{\mathrm{A}}\right){Z}_{\mathrm{A}\mathrm{f}}+{\alpha}_{\mathrm{A}}{z}_{\mathrm{A}\mathrm{B}}\right)}^{f_{\mathrm{A}}}=\\ {}{Z}_{\mathrm{A}}{\left(1-{\alpha}_{\mathrm{A}}\right)}^{f_{\mathrm{A}}}{Z}_{\mathrm{A}\mathrm{f}}^{f_{\mathrm{A}}}+{Z}_{\mathrm{A}}{\left(1-{\alpha}_{\mathrm{A}}\right)}^{f_{\mathrm{A}}-1}{Z}_{\mathrm{A}\mathrm{f}}^{f_{\mathrm{A}}-1}{\alpha}_{\mathrm{A}}{z}_{\mathrm{A}\mathrm{B}}+\dots +\\ {}{Z}_{\mathrm{A}}{f}_{\mathrm{A}}\left(1-\alpha \right){Z}_{\mathrm{A}\mathrm{f}}{\alpha}_{\mathrm{A}}^{f_{\mathrm{A}}-1}+{Z}_{\mathrm{A}}{\alpha}_{\mathrm{A}}^{f_{\mathrm{A}}}{z}_{\mathrm{A}\mathrm{B}}^{f_{\mathrm{A}}}\end{array}$
Cross-linking of functional copolymer with cross-linker C:	$\begin{array}{l}{F}_0\left(\mathbf{Z},\mathbf{z}\right)={n}_{\mathrm{A}\mathrm{B}}{F}_{0\mathrm{AB}}+{n}_{\mathrm{C}}{F}_{0\mathrm{C}}\\ {}{F}_{0\mathrm{AB}}\left(\mathbf{Z},\mathbf{z}\right)={N}_{\mathrm{A}\mathrm{B}}\left({Z}_{\mathrm{A}},{F}_{0\mathrm{B}}\left({Z}_{\mathrm{B}},{z}_{\mathrm{B}\mathrm{C}}\right)\right)=\\ {}\sum_k{n}_k{\left({x}_{\mathrm{A}}{Z}_{\mathrm{A}}+\left(1-{x}_{\mathrm{A}}\right){Z}_{\mathrm{B}}\left(1-{\alpha}_{\mathrm{B}}+{\alpha}_{\mathrm{B}}{z}_{\mathrm{B}\mathrm{C}}\right)\right)}^k\end{array}$
Distribution of states of AB copolymer;
B-unit carries a reactive group which reacts with C-group of C-cross-linker	$\begin{array}{l}{F}_{0\mathrm{AB}}\left(\mathbf{Z},\mathbf{z}\right)={N}_{\mathrm{A}\mathrm{B}}\left(\xi \right)=\sum_k{n}_k{\xi}^k\\ {}\xi \left({Z}_{\mathrm{A}},{Z}_{\mathrm{B}},{z}_{\mathrm{B}\mathrm{C}}\right)={x}_{\mathrm{A}}{Z}_{\mathrm{A}}+\left(1-{x}_{\mathrm{A}}\right){Z}_{\mathrm{B}}\left(1-{\alpha}_{\mathrm{B}}+{\alpha}_{\mathrm{B}}{z}_{\mathrm{B}\mathrm{C}}\right)\end{array}$