Kinetics of reversible polymerization

Dongen, P. G. J. van; Ernst, M. H.

doi:10.1007/bf01011836

Cited by 124 publications

(135 citation statements)

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“…Such equations play an important role in reversible polymerization processes and in related aggregation and fragmentation processes. [30][31][32][33][34][35] In our model, the variable window sizes for the chain fusion/fragmentation processes and for the irreversible mechanical breakage of chains imposes somewhat intricate constraints on the corresponding fusion/fragmentation and breakage kernels which must be accounted for correctly as many detailed features, revealed by the subsequent numerical simulations, depend non-linearly on these window sizes themselves. The details of how these window size constraints are handled are explained in the Appendix.…”

Section: Modelmentioning

confidence: 99%

Mechanically Induced Homochirality in Nucleated Enantioselective Polymerization

2017

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Section: Modelmentioning

confidence: 99%

Mechanically Induced Homochirality in Nucleated Enantioselective Polymerization

2017

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“…These equations differ from the Becker-Döring equations in allowing clusters arbitrary size to coalesce together whereas the Becker-Döring equations only allow cluster-monomer interactions, but allow both aggregation and fragmentation. The Smoluchowski coagulation-fragmentation problem is much more complex but has been analysed by van Dongen & Ernst [20] and Carr & da Costa [8,9,10]. The scaling behaviour of the Becker-Döring equations has been studied by several authors, for example, Brilliantov & Krapivsky [6] and Blackman & Marshall [2].…”

Section: Introductionmentioning

confidence: 99%

Renormalization-theoretic analysis of non-equilibrium phase transitions: I. The Becker-Döring equations with power law rate coefficients

Wattis¹,

Coveney²

2001

J. Phys. A: Math. Gen.

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We study in detail the application of renormalisation theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes. We investigate the Becker-Döring equations, originally formulated to describe and analyse non-equilibrium phase transitions, and more recently generalised to describe a wide range of physicochemical problems. In the present paper we analyse how the systematic coarse-graining renormalisation of the Becker-Döring system of equations affects the aggregation and fragmentation rate coefficients. We consider the case of power-law size-dependent cluster rate coefficients which we show lead to only three classes of system that require analysis: coagulationdominated systems, fragmentation-dominated systems and those where coagulation and fragmentation are exactly balanced. We analyse the late-time asymptotics associated with each class. PACS

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“…The equality in the fraction of formed bonds p (the extent of reaction in chemical language) provides the connection between t during reversible or irreversible aggregation and T in equilibrium. Van Dongen and Ernst [18] also provide an analytic expression for the t-dependence of p following a sudden change in the external control parameters, offering the first soluble example of reversible self-assembly of loopless branched structures. Interestingly, for functionality two (chain assembly) the solution coincides with the mean-field analytic expressions later on derived for equilibrium polymerization kinetics [20].…”

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“…The study of the chemical version of this model [21] showed that formation of closed bonding loops in finite size clusters is disfavored, possibly due to the non-spherical particle shape and the location of the reactive sites. Therefore, it offers to us the possibility to carefully check the Stockmayer [14] and Van Dongen and Ernst [18] predictions. We can assess how closely -in the absence of loop formation -the chemical gelation process in a system of functionalized or patchy units, as well as the reversible evolution from a monomeric to an equilibrium bonded state, can be envisaged as a progressive sequence of equilibrium states, closely connecting t with T .…”

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Connecting Irreversible to Reversible Aggregation: Time and Temperature

et al. 2009

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We report molecular dynamics simulations of a gel-forming mixture of ellipsoidal patchy particles with different functionality. We show that in this model, which disfavors the formation of bondloops, elapsed time during irreversible aggregation -leading to the formation of an extended network -can be formally correlated with equilibrium temperature in reversible aggregation. We also show that it is possible to develop a parameter-free description of the self-assembly kinetics, bringing reversible and irreversible aggregation of loopless branched systems to the same level of understanding as equilibrium polymerization.Several natural and synthetic materials, as well as biological structures, result from the self-assembly of elementary units into branched aggregates and networks [1,2,3,4,5]. This self-assembly process is receiving considerable attention in two fast-growing fields: supramolecular chemistry [1, 2, 3] and collective behavior of patchy and functionalized particles [6,7,8], among the most promising building blocks of new materials [9,10]. The process of formation of an extended three-dimensional network of bonds connecting independent molecules, proteins or colloidal particles, is named gelation and the resulting material a gel [11,12,13]. The ratio between the bond energy u 0 and the thermal energy k B T (where k B is the Boltzmann constant and T is the temperature) can be used to classify aggregation into two broad categories: for strong attraction strength (chemical case [11,14,15,16]), bond formation is irreversible and the number of bonds continuously grows with time. In the case of weak attraction strength (physical case [17]), bonds break and reform while the number of bonds progressively reaches its equilibrium value. In the latter case, the final structure of the system can in principle be predicted with equilibrium statistical mechanics methods.The value of the ratio u 0 /k B T separates the two classes. Conceptually, any model of physical aggregation may be turned into a chemical model by studying its properties following a quench to k B T ≪ u 0 . Similarly, applying temperatures comparable to the bond energy turns an irreversible aggregation model into a physical one. The idea of a close connection between irreversible and reversible aggregation is already contained in the early mean-field theoretical work of Stockmayer [14], considering the Smoluchowski's kinetic equations solved in the limit of absence of closed bonding loops. In Stockmayer's calculations, at any time t during chemical aggregation, the distribution of clusters of finite size k (N k (t)) is identical to that found following equilibrium statistical mechanics prescriptions, i.e., by maximizing the entropy with the constraint of a fixed number of bonds. The N k are commonly referred to as Flory-Stockmayer (FS) distributions. Later on, Van Dongen and Ernst [18] confirmed that the FS distributions are also solutions of the Smoluchowski's equations when bond-breaking processes are accounted for. According to these theoretical wo...

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Kinetics of reversible polymerization

Cited by 124 publications

References 28 publications

Mechanically Induced Homochirality in Nucleated Enantioselective Polymerization

Mechanically Induced Homochirality in Nucleated Enantioselective Polymerization

Renormalization-theoretic analysis of non-equilibrium phase transitions: I. The Becker-Döring equations with power law rate coefficients

Connecting Irreversible to Reversible Aggregation: Time and Temperature

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