Transition into the gel regime for crosslinking radical polymerisation in a continuously stirred tank reactor

Macro Theory & Simulations

Iedema

2017

Self Cite

A novel technique is developed to predict the evolving topology of a diacrylate polymer network under photocuring conditions, covering the low‐viscous initial state to full transition into polymer gel. The model is based on a new graph theoretical concept being introduced in the framework of population balance equations (PBEs) for monomer states (mPBEs). A trivariate degree distribution that describes the topology of the network locally is obtained from the mPBE, which serves as an input for a directional random graph model. Thus, access is granted to global properties of the acrylate network which include molecular size distribution, distributions of molecules with a specific number of crosslinks/radicals, gelation time/conversion, and gel/sol weight fraction. Furthermore, an analytic criterion for gelation is derived. This criterion connects weight fractions of converted monomers and the transition into the gel regime. Valid results in both sol and gel regimes are obtained by the new model, which is confirmed by a comparison with a “classical” macromolecular PBE model. The model predicts full transition of polymer into gel at very low vinyl conversion (<2%). Typically, this low‐conversion network is very sparse, as becomes apparent from the predicted crosslink distribution.

Section: Introductionmentioning

confidence: 99%

Acrylate Network Formation by Free‐Radical Polymerization Modeled Using Random Graphs

Schamboeck

Macro Theory & Simulations

Iedema

2017

Self Cite

“…Now, when we have an explicit expression for the degree distribution, it is possible to apply the existence criterion (13) to this expression, and in this way we find the critical parameters. Let c n (t) and c k (t) denote the fraction of in-spots and out-spots that were converted into in-edges,…”

Section: B Phase Transitionmentioning

confidence: 99%

“…Plugging (34) into (13) and realizing that c k (t) = ν 10 ν 01 c n (t) yields a criterion for the existence of the giant weak component as a quadratic function of c n (t) with the coefficients involving exclusively initial moments, Out-degree, 01 . Both roots of (35) are real, and the smallest root is always negative.…”

Section: B Phase Transitionmentioning

confidence: 99%

“…Some notable examples of the kinetic perspective on random graphs include, but are not limited to, analytic results obtained by Ben-Naim [4][5][6], Lushnikov [7][8][9], Buffet [10], Gordon [11], and their coauthors; numerical studies of stepgrowth and cross-linking polymerization by Kryven et al [12][13][14]; and the algorithmic method introduced by Hillegers and Slot [15,16]. In the above enlisted cases (with the exception of Refs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

2016

Phys. Rev. E

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions Kryven, I. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition, we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step-polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.

“…In polymer chemistry, for example, the infinite configuration network is used as a toy model for hyper-branched and cross-linked polymers. In this context, the component size distribution predicts viscoelastic properties of the material while the emergence of the giant component is interpreted as a phase transition from liquid to solid state of the soft matter [4,12]. Since connected components are closely related to clusters in bond percolation processes, the distribution of component sizes can be used to model outbreaks for SIR epidemiological processes [2].…”

Section: Introductionmentioning

confidence: 99%

General expression for the component size distribution in infinite configuration networks

2017

Phys. Rev. E

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale, and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.