Equilibrium Polymerization as a Critical Phenomenon

The equilibrium polymerization of sulfur is investigated by Monte Carlo simulations. The potential energy model is based on density functional results for the cohesive energy, structural, and vibrational properties as well as reactivity of sulfur rings and chains ͓Part I, J. Chem. Phys. 118, 9257 ͑2003͔͒. Liquid samples of 2048 atoms are simulated at temperatures 450рTр850 K and Pϭ0 starting from monodisperse S 8 molecular compositions. Thermally activated bond breaking processes lead to an equilibrium population of unsaturated atoms that can change the local pattern of covalent bonds and allow the system to approach equilibrium. The concentration of unsaturated atoms and the kinetics of bond interchanges is determined by the energy ⌬E b required to break a covalent bond. Equilibrium with respect to the bond distribution is achieved for 15р⌬E b р21 kcal/mol over a wide temperature range (Tу450 K), within which polymerization occurs readily, with entropy from the bond distribution overcompensating the increase in enthalpy. There is a maximum in the polymerized fraction at temperature T max that depends on ⌬E b . This fraction decreases at higher temperature because broken bonds and short chains proliferate and, for T рT max , because entropy is less important than enthalpy. The molecular size distribution is described well by a Zimm-Schulz function, plus an isolated peak for S 8 . Large molecules are almost exclusively open chains. Rings tend to have fewer than 24 atoms, and only S 8 is present in significant concentrations at all T. The T dependence of the density and the dependence of polymerization fraction and degree on ⌬E b give estimates of the polymerization temperature T f ϭ450Ϯ20 K.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Density functional and Monte Carlo studies of sulfur. II. Equilibrium polymerization of the liquid phase

Ballone

Jones

2003

“…We also note that the introduction of constraints into the self-assembly process also tends to make the thermodynamic transition sharper ͑i.e., less rounded͒. 12,57 Taking these constraints to extreme limits can make the selfassembly transition a true second-order phase transition. 58 This is an aspect of the assembly process that merits investigation in future work.…”

Section: B Basic Thermodynamic and Cluster Propertiesmentioning

confidence: 99%

Model for reversible nanoparticle assembly in a polymer matrix

Rahedi

Douglas

Starr

2008

The clustering of nanoparticles ͑NPs͒ in solutions and polymer melts depends sensitively on the strength and directionality of the NP interactions involved, as well as the molecular geometry and interactions of the dispersing fluids. Since clustering can strongly influence the properties of polymer-NP materials, we aim to better elucidate the mechanism of reversible self-assembly of highly symmetric NPs into clusters under equilibrium conditions. Our results are based on molecular dynamics simulations of icosahedral NP with a long-ranged interaction intended to mimic the polymer-mediated interactions of a polymer-melt matrix. To distinguish effects of polymer-mediated interactions from bare NP interactions, we compare the NP assembly in our coarse-grained model to the case where the NP interactions are purely short ranged. For the "control" case of NPs with short-ranged interactions and no polymer matrix, we find that the particles exhibit ordinary phase separation. By incorporating physically plausible long-ranged interactions, we suppress phase separation and qualitatively reproduce the thermally reversible cluster formation found previously in computations for NPs with short-ranged interactions in an explicit polymer-melt matrix. We further characterize the assembly process by evaluating the cluster properties and the location of the self-assembly transition. Our findings are consistent with a theoretical model for equilibrium clustering when the particle association is subject to a constraint. In particular, the density dependence of the average cluster mass exhibits a linear concentration dependence, in contrast to the square root dependence found in freely associating systems. The coarse-grained model we use to simulate NP in a polymer matrix shares many features of potentials used to model colloidal systems. The model should be practically valuable for exploring factors that control the dispersion of NP in polymer matrices where explicit simulation of the polymer matrix is too time consuming.

“…A large number of studies on the equilibrium properties of equilibrium polymers have been published, both theoretical 3,4 and experimental; 5,6 for recent reviews on this subject see Refs. 5 and 7.…”

Section: Introductionmentioning

confidence: 99%

End-evaporation dynamics revisited

Dubbeldam

Schoot

2005

We present analytical results on the so-called end-evaporation kinetics in equilibrium polymeric systems following a temperature jump ͑T jump͒. A T jump prepares the system with a nonequilibrium length distribution, after which it relaxes back to its equilibrium state. Starting from a master equation, we develop a mean-field analytical theory based on a generating function approach, which allows explicit approximate expressions for the monomer and dimer concentrations to be derived in a discrete setting; the concentrations of the other chains as well as the average chain length were shown to be entirely expressible in terms of the monomer and dimer concentrations. We find that the calculated monomer and dimer concentrations as well as the average chain length are in good agreement with numerical simulation results and do not suffer from some of the defects of earlier continuum theories. Furthermore, the relaxation was shown to take place in three different stages. The first stage comprises the very fast relaxation of the monomers to almost their equilibrium concentration; the other polymer chains have hardly relaxed. During the second stage, which is highly nonlinear, a redistribution of material at practically constant monomer density takes place. Only in the final stage of the relaxation process the chain concentrations approach their true equilibrium values. In this stage there are only very small shifts in the concentrations of chains, which are governed by extremely slow "indirect" monomer-mediated processes.