# Analytic results on the polymerisation random graph model

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## Abstract

The step-growth polymerisation of a mixture of arbitrary-functional monomers is viewed as a time-continuos random graph process with degree bounds that are not necessarily the same for different vertices. The sequence of degree bounds acts as the only input parameter of the model. This parameter entirely defines the timing of the phase transition. Moreover, the size distribution of connected components features a rich temporal dynamics that includes: switching between exponential and algebraic asymptotes and acquiring oscillations. The results regarding the phase transition and the expected size of a connected component are obtained in a closed form. An exact expression for the size distribution is resolved up to the convolution power and is computable in subquadratic time. The theoretical results are illustrated on a few special cases, including a comparison with Monte Carlo simulations.

## Keywords

Random graph Connected components Polymerisation Molecular network## Mathematics Subject Classification

05C80 82D60## 1 Introduction

The chemical graph theory is the branch of mathematical chemistry that applies graph theory to mathematical modelling of chemical processes. This theory centres its attention on the concept of a molecular graph, which identifies atoms (or monomers) as vertices and chemical bonds as edges. This structure, finite or infinite, is usually defined a priori, e.g. molecular graphs describing structural isomers or Euclidian graphs describing crystal nets [5, 27]. The graph-theoretical invariants of such chemical objects are known to be strongly correlated with physical properties of the resulting materials. These invariants include but are not restricted to: Wiener index, average shortest path, shape index, centric index, and connectivity index [7, 21, 22, 26]. Not all molecular topologies can be described by a single graph, but rather by a probability measure over graphs [17, 19]. This scope covers (hyper-)branched polymers, cross-linked polymers, molecular networks, and gels to name a few. A branch of graph theory that operates with probability distributions over graphs—random graph theory—has little documented applications to chemistry at present.

Consider a chemical system where each monomer has a predefined functionality, that is the maximum number of neighbours in the network. If the spatial positioning of the monomers is disregarded, the monomers can be represented as vertices in a graph model. From this perspective, the polymerisation process is a random graph process that respects the limitations induced by the chemistry, for instance, the bound on the vertex degree. The fact that this chemical system can be well described by graph theory is already hinted by a broad range of analogues to graph-theoretical terminology that exists in polymer chemistry: vertex (*monomer*), degree bound (*functionality*), graph (*polymer network*), tree (*branched polymer*), connected component (*polymer molecule*), giant component (*gel*), density (*conversion*), etc.

In this paper a random graph process is introduced to model an evolving molecular network. The degree distribution of this random graph is defined by a time-continuous evolution equation that mimics the chemistry of the step-growth polymerisation process. This process starts with disconnected vertices and progresses up to the point where no new edge can be placed. The degree of each vertex is bounded, but different bounds may be defined for distinct vertices. Therefore, we distinguish between the degree—actual number of incident edges, and the functionality—pre-imposed bound on the number of incident edges. At each time step, the probability that a vertex receives an edge is proportional to the difference between the vertex’s functionality and degree. The share of vertices in each functionality class is pre-defined, and constitutes the only input parameter for the random graph model.

Most of the available studies target narrow special cases of this system and pursue results with a distinct reasoning from the graph-theoretical one. Important contributions include: Hamilton–Jacobi formalism as applied to dynamic graphs with globally bounded degrees [1], results on the grabbing-particle system [4], open-form analytical results for non-phase-transiting systems [11], combinatorial analysis for monomers bearing identical groups [6], closed-form analytical [31] and numerical [17] results for trifunctional vertices in a directed topology, analytical results for mixture of bi- and trifunctional vertices [12], analytical results on phase transition in evolving directed graphs [14], and stochastic simulations on molecular networks [16]. The random graph model is also related to many processes outside polymer chemistry. For instance, Smoluchowski coagulation equation with a multiplicative kernel governs the dynamics of component-size distribution of the polymerisation random graph with trifunctional vertices. Only in this special case, the analytical expression for component sizes is available also after the phase transition, for a review on Smoluchowski coagulation see Refs. [3, 29]. In probability theory, the gambler’s ruin problem for infinite number of games is equivalent to finding criteria for the phase transition in the polymerisation random graph with vertices not exceeding degree three [10].

The rest of the paper is organised as follows. First, a differential-difference equation describing evolution of the degree distribution due to the step-growth polymerisation process is formulated and solved in time. Then, given the time dependent degree distribution, the emergence of the giant component is analysed. This includes results on the edge density at which the giant component appears and the criterion on the functionality distribution that admit emergence of the giant component at finite time. Furthermore, the size distribution of connected components is resolved and expressions for the expected component size are given. Finally, the theoretical results are discussed for a few special cases. The theory is also compared against the size distributions that were generated by a Monte Carlo simulation.

## 2 Evolution process for the degree distribution

*n*adjacent edges [24]. Since a degree of a vertex cannot be arbitrary large in a chemical system, each vertex is assigned a bound on its degree, \(m=0,1,2,\dots \). To copy the chemical terminology, we refer to this bound as the functionality [28]. So that one may speak of a two-variate distribution \(\text {u}(n,m), \;n,m=0,1,2,\dots \) as the probability to sample a vertex with degree

*n*and functionality

*m*, such that \(\text {u}(n,m)=0\) for \(n>m\). We will now construct an evolutionary process for \(\text {u}(k)\) that mimics the step-growth polymerisation of multifunctional monomers. This linking process starts with disconnected vertexes, that is the probability to sample a vertex of degree zero is \(d(0,k)=1,\) and the process ends when one samples a vertex with \(n=m\) with probability one. The precise rule of assigning a new edge is the following conceptualisation of the step-growth polymerisation process: on each time step, one samples two candidate vertices with probability proportional to \((m-n)\text {u}(n,m)\) and connects them with an edge. So that

*m*) as introduced in Ref. [1], Eq. (3). An alternative way of introducing (1) is by writing the corresponding reaction mechanism for monomer species \(M_{n,m}\):

*m*is constant over time, \(\sum \limits _{n=0}^{\infty } u(n,m,t) = f_m,\; \sum \limits _{m=1}^\infty f_m=1\), and \(f_m\) is treated as the only parameter of the model. The sum written in the second line of Eq. (3) represents the expected number of unused but potentially available edges and can be viewed as a difference of two partial moments, \(\mu _{01}(t)-\mu _{10}(t),\) where

*t*to the expected number of edges at the end of the process:

*c*(

*t*) as an alternative measure of the progress. The differential equation (3) falls into the class of linear population balance equations. This class of equations frequently appears as a model for many chemical and biological problems where it is usually approached numerically [18, 20]. In the current case, it is possible to find an analytical solution of (3) by transforming the equation to the domain of generating functions, solving the corresponding partial differential equation, and applying the inverse transform.

*c*(

*t*) instead of time. To do this, it is enough to realise that \(c(t)=\frac{\mu _{10}(t)}{\mu _{01}} = \frac{\mu _{01} t}{1 + \mu _{01} t}\) and \((\mu _{01} t)^n (1+\mu _{01} t)^{-m}=(\mu _{01} t)^n(1+\mu _{01} t)^{-n} (1+\mu _{01} t)^n (1+\mu _{01} t)^{-m}= \left( \frac{\mu _{01} t }{1+\mu _{01} t}\right) ^n \left( \frac{1}{1+\mu _{01} t}\right) ^{m-n}= \left( \frac{\mu _{01} t }{1+\mu _{01} t}\right) ^n \left( 1-\frac{\mu _{01} t}{1+\mu _{01} t}\right) ^{m-n}\) so that Eq. (11) transforms to

*m*,

*u*(

*n*,

*t*) evolves form the Kronecker’s delta function, \(\delta _n\) at \(t=0\) to \(f_m\) in the limit of \(t\rightarrow \infty .\) The moments of the degree distribution, \(\mu _i=\sum \limits _{n=0}^\infty n^i u(n,t) = \mu _{i0}\) can be directly found from summation of Eq. (13). For instance the expressions for the first three moments read,

## 3 Global properties of the network, the giant component

### Corollary

*M*monomer species of functionalities \(m=1,\dots ,M\) and fractions \(f_1,f_2,\dots ,f_M,\; \sum \limits _{m=1}^M f_m=1\) react at constant rate \(k_p\), then the system features the phase transition in a finite time if and only if

- 1.
If all monomers have the same functionality

*m*, then the phase transition is reached in a finite time only if \(m\ge 3\) (i.e.*m*is the smallest positive integer satisfying \(m^2-2m>0\)). - 2.
Adding (or removing) monomers of functionality two does not affect phase transition time \(t_g\), whereas it does alter the edge density at the phase transition, \(c_g\).

- 3.
Adding sufficient amount of \(f_1\) to any system will prevent the phase transition.

- 4.Consider a system that consists of two species: monomers with functionality
*m*that are present at fraction \(f_m\) and monomers with functionality one, that are present at fraction \(f_1=1-f_m\). The system does not go through the phase transition in finite time if,$$\begin{aligned} f_1 > \frac{m^2-2 m}{m^2-2m+1}. \end{aligned}$$(20) - 5.When all monomers have functionality
*m*, the polymerisation leads to an infinite network at edge density$$\begin{aligned} c_g=\frac{m}{m^2-m}=\frac{1}{m-1}. \end{aligned}$$

## 4 Size distribution of connected components

*t*where it leads to no confusion, and refer to the degree distribution, as given in Eq. (13), by simply \(\text {u}(n)\) or by its generating function,

*an edge*at random, so that every edge has equal probability to be sampled. Then, one of incident to this edge vertices is chosen and the edge itself is removed. We will refer to this vertex as the biased vertex. Let \(\text {u}_1(n)\) denotes the probability that a biased vertex has

*n*incident edges. Then,

*w*(

*n*) denotes the probability that a randomly sampled node belongs to a connected component of size

*n*. Similarly to definition of \(\text {u}_1(n)\), let \(w_1(n)\) denotes the probability that a biased vertex belongs to a connected component of size

*n*. Newman et al. [24] noticed that the generating functions for \(u_1(n)\) and \(w_1(n)\) are related by a functional equation

*w*(

*n*) and \(U_1(x)\) generates \(u_1(n)\). This equation has a straightforward interpretation: the equation unfolds the generating function for \(w_1(n)\) as a sum over all configurations of a biased vertex. Each configuration occurs with probability \(\text {u}_1(n)\) and involves

*n*biased sub-components of size \(w_1(n).\) Furthermore, the sum in Eq. (23) can be in itself viewed as the definition of the generating function. So that one may write,

*w*(

*n*) reads

*w*(

*n*) can be written out in terms of convolution powers[15],

*m*, \(f_m=1\), then

*w*(

*n*) is simply given by,

*w*(

*n*) at large \(n\gg 1,\) see Ref. [15]. Namely,

One may see that at the phase transition, when \(t=t_g\), the coefficient in the exponential function in (29) vanishes and the asymptote switches to the power law decay.

## 5 Expected size of connected components

*w*(

*n*) describes only finite components. Before the phase transition, a randomly sampled node belongs to a finite component with probability one, therefore \(\sum \limits _{n=1}w(n)=W(1)=1\). After the phase transition, when \(t>t_{\text {g}},\) the probability that a randomly sampled node belongs to a finite component is smaller than one and

*w*(

*n*) fails to be normalised:

## 6 Interpretation of the results and examples

The present paper introduces a model for studying polymer networks composed of multifunctional monomers that polymerise according to the step-growth mechanism (2). This model associates a vertex with a monomer and an edge with a chemical bond between two such monomers in the network. A resulting topology of the polymer network is viewed as a random graph defined by its degree distribution. Initial fractions of monomers of different functionalities \(f_m\) are directly related to molar concentrations of monomer species. The reaction kinetics is formalised by the master equation (3) and yields an analytical expression for the degree distribution at any point of time (11). Although the master equation (3) has a unit rate, an arbitrary reaction rate can be modelled by simply scaling time variable *t* in a linear fashion. An example of a degree distribution evolving in time is given in Fig. 1. In this example, the initial condition of the kinetic model is chosen to be \(f_{10}=1,\) that corresponds to pure 10-functional monomers. In the given context, both, initial and terminal degree distributions are Kronecker’s delta functions positioned correspondingly at \(m=0\) and \(m=10\).

A deeper analysis reveals that when initial concentrations of monomers satisfy condition (18), the random graph develops a giant component at time \(t_g\) that is given by Eq. (16). This event is related to the fact that the molecular network undergoes a phase transition. Such phase transition is called gelation, and is a well-documented chemical phenomenon that signifies transition from liquid-like to solid-like state in soft matter[30, 32]. Figure 2 presents two examples showing how \(t_g\) is influenced by varying \(f_m.\) The figure illustrates the fact that addition of one- and two- functional vertices may be used to control the timing of the phase transition: addition of two-functional vertices postpones the emergence of the giant component in terms of \(c_g,\) whereas \(t_g\) remains invariant; addition of one-functional vertices may entirely prevent it.

The size distribution of connected components, as given in Eq. 26, is interpreted as the molecular weight distribution, whereas the asymptote (29) might serve as a good way to approximate the latter if rapid computations are required. Evolution of the expected number of this distribution, also known in the chemical literature as number-average molecular weight, is given by Eqs. (32), (34).

### Example 1

We consider vertices with at most degree 2, that is \(f_2=1\). Graphs generated by such a process are always linear and, according to (18), the giant component is reachable only asymptotically at \(t\rightarrow \infty \). Furthermore, a small perturbation, \(f_1=\varepsilon ,\; f_2=1-\varepsilon ,\) prevents emergence of the giant component even at infinite time (see points A at barycentric plot of configurations, Fig. 2). The component-size distribution is illustrated in Fig. 3. One may notice the constant “drift” (as indicated with an arrow) of the distribution towards larger values of components sizes. The distribution features the exponential asymptote at any \(t>0\).

### Example 2

As shown in Fig. 2, one may postpone the phase transition so it occurs anywhere between 0.5 and 1 by adding vertices of functionality 2 to the system. For instance, a mixture of vertices with functionalities two and three having fractions \(f_2=\frac{49}{50}\) and \(f_3=\frac{1}{50},\) as denoted by point B in Fig. 2, postpones the phase transition to \(c_g=\frac{101}{104}\approx 0.97.\) The evolution of the size distribution for this case is depicted in Fig. 4b.

*termination agents*within the chemical context. Depending on what is the degree of the other species, the probability of randomly selecting a component may feature regular oscillations. For instance, in a dense, \(c=1\), mixture of

*m*-functional and one-functional vertices, connected components can take their sizes only from

### Example 3

We consider a mixture of one- and six-functional vertices present with fractions \(f_1=\frac{24}{25},\;f_6=\frac{1}{25}\). This distribution of functionalities features the phase transition at \(c=1.\) As illustrated in Fig. 6a, the size distribution decays monotonically at low edge densities, but switches to oscillations as *c* approaches 1. The switch itself is gradual as can be seen in Fig. 6b. In Fig. 7, the theoretical results are compared to component-size distribution generated by Monte Carlo (MC) computations. The theory and MC data are in a perfect agreement; however, despite extensive size of MC computations (100 ensembles of size \(10^6\) vertices), the MC resolution in the tail of the distributions remains poor.

## Notes

### Acknowledgements

This work is part of the Project Number 639.071.511, which is financed by the Netherlands Organisation for Scientific Research (NWO) VENI. Some of the results published in this paper were obtained during work under PAinT (Paint Alterations in Time) project as part of the NWO Science4Arts Program.

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