## Abstract

Using simple exactly solvable models, we show that event-dependent time delays may lead to significant non-Poisson effects in the statistics of polymer chain growth. The results are confirmed by stochastic simulation of various growth scenarios. Our interest in mathematical aspects of non-Markovian growth arises from recent successful application of delayed probability density functions in stochastic modelling of controlled radical polymerization.

## 1. Introduction

A substantial part of the polymer market is produced by free radical polymerization (FRP). This is a chain-growth process that proceeds by addition of monomer units to a growing polymer chain. The addition of monomer units is called propagation. Chain growth may terminate by bimolecular termination (reaction of two growing chains) and by chain transfer of the radical to monomer and chain transfer agent. In the polymerization of acrylic monomers transfer to polymer is frequent [1]. Intramolecular transfer, called backbiting, is the most common transfer to polymer process [2], where the radical is transferred to a monomer unit located two positions behind the unit where the radical was. It has been reported that the rate of backbiting may be affected by the presence of polar solvents [3]. The midchain radical formed by backbiting may (i) react with a monomer, thus continuing the chain growth [2,4], (ii) migrate along the chain [5], (iii) undergo beta-scission [6], and (iv) react with other species present in the medium, such as a chain transfer agent [7], and solvents. The most likely event is propagation. Therefore, the chain does not stop growing, but simply continues from a different monomer unit, and hence a branch is formed.

In the last two decades, controlled radical polymerization (CRP) has revolutionized FRP allowing an unprecedented control of the polymer microstructure. In CRP of acrylic monomers, an unexplained decrease of the branching level has been observed and has prompted a lively debate on the causes of such findings [8–11]. A recent publication [11] has shown that these and other kinetic anomalies can be explained by assuming a non-Markovian kinetics. With this in mind, we analyse below some simple exactly solvable non-Markovian delayed growth models.

The rest of the paper is organized as follows. In §2, we define the probabilities for different growth scenarios. In §3, we briefly review growth without a delay, leading to Markovian master equations for the probabilities of interest. In §4, we consider linear growth with a fixed ‘downtime’ introduced after each attachment of a monomer. We show that such a delay leads to non-Poisson distribution of the polymer length, and a set of time-delayed differential equations for the relevant probabilities. In §5, we study the short- and the long-time limits of the mean length of the grown polymer. In §6, we consider a branching process, whose delay is determined by the polymer's growth rate. In §7, we confirm and extend our results by employing a numerical stochastic algorithm similar to the one pioneered by Gillespie *et al*. [12]. Section 8 contains our conclusion.

## 2. Linear growth

We start with the usual mathematical prerequisite. Consider the growth of a linear polymer which proceeds by attaching monomers to, say, its right end at discreet times *t*_{j}=*jdt*.

We begin with a single monomer. At each *t*_{j}, an extra monomer is added with the probability *p*_{j}, or else nothing happens with the probability 1−*p*_{j}. Thus, the probability to add a monomer after *J*−1 unsuccessful attempts is *t*_{J} is *t*_{i} and let the probabilities depend on both *t*_{j} and *t*_{i}, *p*=*p*_{j,i}. In the continuum limit, we should send d*t*→0 and introduce the growth rate *c*(*t*_{j},*t*_{i}) d*t*≡*p*_{j,i}, assuming *c* to be a slowly varying function. The probability to add nothing for *t*′≤*t*′′≤*t* is then given by
*t*′ and add the first monomer in the interval [*t*,*t*+d*t*], we have
*t*, the monomer is either attached or not, so the two corresponding probabilities add up to 1,

With many monomers able to join the polymer chain between *t*' and *t*, we are looking for the probability *P*(*n*,*t*,*t*′) to have *n* new additions by the time *t*. This is just the probability for adding monomers at *t*′≤*t*_{1}≤*t*_{2}≤⋯≤*t*_{n}≤*t*, multiplied by the probability that no more monomers are added between *t*_{n} and *t*, and summed over all *t*_{i}, *i*=1,2,…*n*,
*P*(*n*,*t*,*t*′),

Another useful quantity is the probability density *W*(*n*,*t*,*t*′) for *n* monomers to be attached in the interval [*t*′,*t*], with the last of them added in [*t*,*t*+d*t*],
*P*(*n*,*t*,*t*′) is expressed as
*W*(*n*,*t*,*t*′) have the advantage that they satisfy the simple evolution equations (EEs),

At least three cases need to be distinguished.

*A.* The growth rate depends only on the current time, and not on the previous history of the chain,
*Markovian* case we will briefly review in §3.

*B.* The growth rate depends only on the chain's past and is not manipulated externally.
*non-Markovian* case is the main subject of this paper.

*C.* Finally, the growth rate, which depends on the polymer's history in the sense outlined above, may also be manipulated externally. In this case, *c* is a function of both *t* and *t*', and the process is also non-Markovian.

## 3. Markovian growth

Consider the case when there is an unlimited supply of monomers to be added to the chain, and the probability to add one at a given time is modified externally, e.g. by varying the temperature at which the process takes place. The growth begins at some *t*', and we are interested in the length of a polymer at a time *t*. The probability for adding a monomer in [*t*,*t*+d*t*] is *c*(*t*) d*t*, and the function *g* and the PDF *f* in equations (2.1) and (2.2) are of the form
*c* over the growth period. If the external conditions remain unchanged, *c*(*t*)=const, the growth is linear with time, 〈*n*(*t*)〉=*c*(*t*−*t*′). In the special case (3.1), differentiating equation (2.6) [or, directly, equation (3.2)] yields a closed master equation for the probabilities *P*_{M}(*n*,*t*,*t*′)
*δ*_{nm} is the Kronecker delta. Equations (3.5) are obviously Markovian, as the rate at which a *P*_{M}(*n*,*t*,*t*′) changes depends only on the current state of the system, {*P*(*n*,*t*,*t*′)}, *n*=0,1,2….

## 4. Non-Markovian growth with delays

Suppose next that, as in §3, there is an unlimited supply of monomers, and the external conditions remain unchanged. But each added monomer, except the first, now needs a time *τ* to properly settle into the chain structure, only after which the chain is ready to attach again, with the same constant growth rate *c*. The process is now explicitly non-Markovian: to check whether a monomer can be added, one needs to know the history of the chain. Accordingly, the probability *c*(*t*) depends not on the time elapsed since *t*', but on the time elapsed since the last monomer was added. In equation (2.4), we therefore have *c*(*t*,*t*′)=*c*(*t*−*t*′). Explicitly, we obtain (NM stands for ‘non-Markovian’)
*θ*(*z*)=1 for *z*≥0 and 0 otherwise. The physical background of equations (4.1)–(4.3) is as follows: in classical kinetics, the pseudo-first-order processes are considered purely stochastic, i.e. described by an exponential probability distribution function. However, this is intuitively difficult to justify because the probability density that a reaction occurs at *t*=0 should be 0, as instantaneous reactions do not occur. This concept has been used to analyse the competitive processes occurring in CRP and to explain the reduction of branching in CRP of acrylic monomers [11]. Equation (4.2) can be considered a simplification of the linear-exponential (LE) equation used in [11]. This simplification allows us to obtain solutions analytically.

It is easy to see that the model described by equations (4.1)–(4.3) has a simple exact solution. Indeed, returning to equation (2.5) and putting *t*′=0, we note that the probability *W*(*n*,*t*,*t*′=0)≡*W*(*n*,*t*) is the same as for growth with a constant *c*, but for a shorter time. The effective time of growth, *t*_{eff}, is, therefore, the elapsed time *t* minus the total time the growth was shut down due to adding *n*−1 monomers, i.e. *t*_{eff}=*t*−(*n*−1)*τ*. Should (*n*−1)*τ* exceed *t*, the process is not possible, and the corresponding probability is zero. From (2.5) and (4.3), we easily find
*P*(*n*,*t*,*t*′=0)≡*P*(*n*,*t*) are no longer given by a Poisson distribution, but can be obtained as quadratures using equations (2.6), (4.2) and (4.4)
*P*_{NM}s in equations (4.5). There are, however, EEs (2.7) which, as ∂_{t}*f*(*t*)=*cδ*(*t*−*τ*)−*cf*(*t*), read
*W*_{NM}(*n*−1,*t*−*τ*|*τ*). As for any suitable function *F*(*t*), _{t}, and their properties may differ significantly from those of (3.5), as will be illustrated in §5.

## 5. Mean chain length for a delayed growth

One quantity of practical interest is the mean length of the chain grown in the presence of a delay,
*cτ*. We note that *cτ*<1 implies that the delay is shorter than the average reaction time. Similarly, *cτ*=1, and *cτ*>1 indicate that the delay is equal, and longer than the average reaction time, respectively. Figure 2 shows the standard deviation of the length for *cτ*=10. As *cτ*→0, we recover the Poisson distribution (3.2), and for *ct*≪1, 〈*n*(*t*,*τ*)〉 grows linearly at the rate close to *c*. In the opposite limit, *ct*≫1, the behaviour is more interesting with the curves showing a steplike variation at short times before settling into a linear behaviour later. This has a simple physical explanation. The time it takes the chain to add one monomer at a constant growth rate *c*, *t*_{add}, is approximately 1/*c*. If *τ*/*t*_{eff}=*cτ*≫1, a monomer is added quickly, but then the system has to wait long until another one can be attached. Thus, for *t*_{add}<*t*<*τ*, it behaves as if the delay were infinite, i.e. as if the only two possible outcomes were one or none monomers added, with the probabilities *t*≈*τ*, when the system ‘recalls’ that the delay is not infinite after all. A second monomer is added quickly, and 〈*n*(*t*,*τ*)〉_{NM} remains flat and close to 2 until *t*≈2*τ*, and so on. This behaviour can be expected from the way we have constructed our model. Equivalently, it can be seen as an illustration of the ability of time-delayed differential equations to produce rapid variations in their solutions after they seem to have reached an asymptotic limit [14].

As *K* is a constant),
*n*(*t*)〉 monomers is accompanied by switching off the growth for a duration of approximately [〈*n*(*t*)〉−1]*τ*−*τ*/2. (The last addition may occur close to *t*, so its delay is, on average, shorter.) Thus, the growth is similar to the growth without delay over a time 〈*n*(*t*)〉*τ*−*τ*/2 at the rate *c*. Equating 〈*n*(*t*)〉 to *c*[*t*−〈*n*(*t*)〉*τ*+*τ*/2] yields

## 6. Linear growth with branching: growth-induced delay

Next, we consider the delay in backbiting, caused by the fact that in order to form the six-membered ring required for the backbiting reaction, a minimum of three monomer units is needed in a linear segment of the chain. It is assumed that the polymer chain grows at a constant rate *c*_{add}, without any delay, its full length at a time *t* being *c*_{add}*t*. It can, however, form branches [15,16] as illustrated in figure 4. The branching occurs with a rate *c*_{branch}, but can only happen after a linear segment contains at least three monomers [17]. In other words, a branching event should be preceded by at least *n*_{0}=3 attachments of monomers.^{1} It is the number of branchings, *n*_{branch}, we are interested in. To make the problem tractable, we assume that the growth is deterministic, i.e. that exactly one monomer is added to the chain at *t*_{add}=1/*c*_{add}. This makes branching a delayed reaction of the type considered in the §§4 and 5, and the delay time is now given by
*n*_{0}, as a branching event can occur just before or just after attachment of an extra monomer, so that the actual delay may lie between (*n*_{0}−1)*t*_{add} and *n*_{0}*t*_{add}. We will address this issue shortly. For now, the mean number of branchings in the chain, 〈*n*_{branch}(*t*)〉 is given by a formula similar to equation (5.2), but shifted by *τ*, as already the first branching may occur only after at least *n*_{0} growth events,

For *γ*≫1 and *n*_{0}/*c*_{add}+1/*c*_{branch}<*t*<2*n*_{0}/*c*_{add}, there is exactly one branching, the second one appearing approximately after *t*=2*n*_{0}/*c*_{add}+1/*c*_{branch}.

For *γ*≫1 and *c*_{branch}≫*c*_{add}, a branching event occurs just after a linear segment has grown to contain *n*_{0} monomers, so that *N*(*t*)〉=*n*_{0}. In the opposite limit, *γ*≪1 growth outpaces branching. After an initial delay of *n*_{0}/*c*_{add}, branching proceeds at a rate ≈*c*_{branch}, and the mean length of the linear segment, 〈*N*(*t*)〉, is approximately *c*_{add}/*c*_{branch}.

Figure 5 shows the dependence of 〈*N*(*t*)〉 on *t*, for *n*_{0}=3 and different values of *γ* in equation (6.4). Figure 6 shows a related quantity, the ratio of the mean number of the branchings to the mean total length of the chain, 〈*n*_{branch}(*t*)〉/*c*_{add}*t*. We note that for *γ*≪1, the curved part of the graph in figure 6 is due to the fact that branching is delayed relative to growth by about *n*_{0}/*c*_{add}, after which it proceeds at a constant rate of *c*_{add}*t*. For the ratio we, therefore, have
*t*≈5*n*_{0}/*c*_{add}, as can be seen from figure 7. This approximates the range of times in which the initial delay in building a branchable chain leads to non-constant behaviour of the ratio number of branches/chain length.

## 7. Stochastic simulation of delayed growth

If the growth of the chain does not occur at regular times, but is itself a Poisson process, we need to take into account the small probability that three monomers can be added also within a short time. For growth occurring at a constant rate, *c*_{add} the probability to add *n*_{0} monomers within a time *τ* is
*n*_{0} are shown in figure 8. Then we replace *f* in equations (2.5) with *et al*. [12]. There are two ways to simulate delayed branching of a polymer chain using the method similar to that of [12,18]. A rigorous proof of their equivalence is given in appendix A.

### (a) Stochastic simulation of a single delayed process, with the delay built into the corresponding probability density function

A random number generator is prepared, so that it ‘draws’ a random number *t*_{j}, *j*=1,2,….*K*, with a probability
*f*(*t*) is the PDF given by equation (7.2). The number of branches is set to zero, and then the first value *t*_{1} is drawn. If it lies between 0 and the time *t* at which the growth is stopped, the number of branches is increased by 1, and *t*_{1} becomes the new starting time. This step is repeated until the *k*th step yields *t*_{k}>*t*, at which point the drawing stops, and the vector *N*, one obtains a collection of histories, from which the probability of any particular property can be obtained as the relative frequency with which the property occurs. For example, the probability to have *n* branches by a time *t* is given by
*N*_{n} the number of realizations with exactly *n* events.

The results of a stochastic simulation with the PDF (4.3), shown in figure 3, are in full agreement with the analytical results (4.4) and (4.5). The results for the PDF (7.2) are presented in figure 9.

### (b) Stochastic simulation of two Poisson processes with an additional constraint

Alternatively, one can perform a simulation of two simultaneous Poisson processes with their respective PDFs defined as follows:
*n*_{0} monomers have been added previously. (Note that without such a constraint the processes are independent, and the ratio of the mean number of branchings to the mean length of a polymer in figure 6 is a constant equal to *c*_{branch}/*c*_{add} at all times.)

Now in each step of the simulation, one draws random values of *f*_{add} and *f*_{branch} in equations (7.5), respectively. If *t*_{k} is found to be greater than *t*, and a history consisting of branching events interspersed among acts of growth is stored. With many histories collected, average values of observables are evaluated as in the previous subsection.

A comparison demonstrates a good agreement between the single- and two-processes simulations of branching events. The results are shown in figure 9 for various values of *γ*. For growth events distributed in time, the sharp features present in the time dependence of observables in the model of §4 are smoothed over, yet there remain significant non-Markovian effects.

Note also that computationally the approach of §7a is more efficient than the straightforward algorithm described in §7b (figure 10), as it reduces a number of simulated processes to one.

### (c) A simpler choice of the delayed probability density function

The algorithm outlined in §7a suggests an efficient way of simulating delayed processes if the amount of delay is known *a priori*. In practice, however, it is often not the case. If so, one may use a set of experimental data to tune parameters of PDFs taking an appropriate optimization route [11]. Optimization is most easily achieved for a PDF having a simple analytical representation and depending only on a small number of parameters. A simple two-parametric analytical form broadly similar to equation (7.2) is the LE one, used in [11]
*t*=*b* corresponds to the maximum of *τ*=(1−*kb*^{2}/2)/*kb*. Figure 11 compares the PDF (7.2) with its LE approximation (7.6), while figure 12 illustrates the level of accuracy achieved by the approximate PDF in computation of a branching fraction (6.7). The LE approximation accurately reproduces the short-time behaviour of the branching fraction curve, while in the long-time limit it underestimates its value by less than 8%. This justifies the use of approximation (7.6) as an initial guess for a delayed PDF used to fit the experimental data [11]. Finally, the comparison of computational efficiency of simulated approaches suggested in §§7a, 7b and 7c is presented in figure 13. The approach of §7c outperforms the methods §§7a and b by up to two orders of magnitude, which will become more dramatic with increasing size and complexity of simulated processes.

## 8. Conclusion and discussion

In summary, introduction of a delay after each growth event significantly changes the statistics of polymer chain growth in CRP of acrylic monomers. The process is governed by the ratio *τ*/*t*_{add}, where *τ* is the length of the delay, and *t*_{add} is the time it takes, on average, to add a monomer to the chain. For *τ*/*t*_{add}≪1, the growth remains essentially Markovian, with the mean chain length growing as *t*/*t*_{add}, except at very short times. For *τ*/*t*_{add}≫1, the growth at short times proceeds by rapid attachments of single monomers, separated by long ‘waiting periods’ in which nothing happens. As time progresses, the step-like variations of the mean polymer chain become less pronounced, and a growth with a renormalized constant growth rate

In the simple model of §6, formation of branches in the chain is a delayed process, whose delay is determined by the growth rate of the polymer chain. The growth itself is unaltered by the branching rate *c*_{branch} which, in turn, depends on the structural properties of the chain and not on the rate at which the monomers are added. The behaviour of the mean number of branches is, therefore, similar to that of the mean chain length in a linear growth with delay, described in the preceding paragraph. The ratio of the mean number of branchings to the mean length of the chain, 〈*n*_{branch}(*t*)〉/*c*_{add}*t*, has a constant value *c*_{branch}/*c*_{add} in the Markovian case. In the presence of a delay, at short times *c*_{branch}/(*c*_{add}+*n*_{0}*c*_{branch}) as shown in figure 6. In general, the presence of a delay is best visible for

The growth rate *c*_{add}=1/*t*_{add} is proportional to the concentration of monomers and is therefore variable. Our analysis suggests a way of checking whether there are possible time delays of order *τ* in the growth of a particular polymer. One way is to examine the mean number of branches occurring at growth times *c*_{add}. If for

Apart from time delays, there may be various complex processes accompanying growth of a polymer. Yet, the delayed nature of the growth is one possible reason for observed deviations from the predictions of the Poisson law.

Finally, we note that computer simulation of the phenomenon can be achieved by different methods whose accuracy and efficiency have been discussed in detail.

## Ethics

This work did not involve animal experimentation, human studies, conservation issues or potential risk of misuse or maltreatment of animals.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

D.S., E.A., S.R. and J.M.A conceived the mathematical models, and interpreted the computational results and wrote the paper. S.R. implemented and performed most of the simulations. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the Basque Government (grant nos. IT-472-10, IT-373-10, and Etortek Nanoiker IE11-304), University of the Basque Country UPV/EHU (UFI 11/55 and 11/56), and the Ministry of Science and Innovation of Spain (grant nos. FIS2009-12773-C02-0, CTQ 2011-25572 and MTM2013-46553-C3-1-P).

## Acknowledgements

S.R. and E.A. also acknowledge support of the Basque Government through the BERC 2014–2017 programme and of the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and grant SVP-2014-068451. The SGI/IZO-SGIker UPV/EHU is acknowledged for providing computational resources. E.A. acknowledges support from MINECO project MTM2013-46553-C3-1-P.

## Appendix A. Proof of the equivalence between the methods of §sec7a,b

The aim of this section is to prove that stochastic simulation of two explicitly constrained Poisson processes (§7b) is equivalent to simulating stochastically a single delayed process (§7a), with the delay built into the correspondent PDF given by equation (7.2).

Let us consider two competing Poisson processes, an addition and a branching, which occur at constant rates *c*_{add} and *c*_{branch}, respectively, with a branching event possible only after at least *n*_{0} additions. We are interested in the time *n*_{0} initial additions required, *n*_{0}th addition, respectively, and *T*_{branch} is a time for the next branching event to occur if no delay is imposed.

First, we derive the corresponding PDFs for each term of the r.h.s of (A 1) and then find the resulting PDF for the sum of random variables in equation (A 1). As addition is a Poisson process, its PDF is
*α*,*β*), *Γ*(*α*) is the gamma function (*Γ*(*α*)=(*α*−1)! if *t*≥0, and 0 otherwise. From the appendix B, it follows that the PDF for

Once the required minimal length of *n*_{0} monomers is achieved, branching is also a Poisson process, and *T*_{branch} is distributed with the PDF

The PDF of

## Appendix B. A useful relation

Consider *m* independent random variables *T*_{i}, *i*=1,…*m*, which are sampled from the Gamma distribution defined in appendix A, *T*_{1},…,*T*_{m}∼Gamma(*α*_{i},*β*). Then their sum, *T*_{i},
_{1}=λ_{2}=⋯=λ_{m}=λ yields the mgf for the sum *T*, *ψ*_{T}(λ). By (B 1), we have

## Footnotes

↵1 Note that

*n*_{0}=3 is the*minimal*number of added monomers, for the chain to be able to branch. The actual value of*n*_{0}depends on the properties of the chain, and on the precise mechanism by which branching occurs. It is, therefore, an adjustable parameter of our simple model.

- Received November 19, 2014.
- Accepted June 3, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.