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Center for the Study of Systems Biology

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Phenomenological theory of the dynamics of polymer melts. I. Analytic treatment of self-diffusion

(Georgia Institute of Technology, 1988-01-15) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

In the context of dynamic Monte Carlo (MC) simulations on dense collections of polymer chains confined to a cubic lattice, the nature of the dynamic entanglements giving rise to the degree of polymerization n, dependence of the self-diffusion constant D~n[superscript −2] is examined. Consistent with our previous simulation results, which failed to find evidence for reptation as the dominant mechanism of polymer melt motion [J. Chem. Phys. 86, 1567, 7164, 7174 (1987)], long-lived dynamic entanglement contacts between pairs of segments belonging to different chains are extremely rare and are mobile with respect to the laboratory fixed frame. It is suggested that dynamic entanglements involve the dragging of one chain by another through the melt for times on the order of the terminal relaxation time of the end-to-end vector. Employing the physical description provided by the MC simulation, the general expression of Hess [Macromolecules 19, 1395 (1986)] for the friction constant increment experienced by a polymer due to the other polymers forms the basis of a phenomenological derivation of D~n[superscript −2] for monodisperse melts that does not require the existence of reptation. Rather, such behavior is dependent on the relatively benign assumptions that the long distance global motions of the chains are uncorrelated, that the dynamic contacts can be truncated at the pair level, and that the propagator describing the evolution between dynamic contacts contains a free Rouse chain component. The mean distance between dynamic entanglements is predicted to depend inversely on concentration, in agreement with experiment. Moreover, as the free Rouse component is frozen out, for chains greater than an entanglement length ne, a molecular weight independent glass transition is predicted. Extension to bidisperse melts predicts that the probe diffusion coefficient Dp depends on the matrix degree of polymerization, nm, as n. Finally, comparison is made between the theoretical expressions and MC results for mono- and bidisperse melts
Phenomenological theory of the dynamics of polymer melts. II. Viscoelastic properties

(Georgia Institute of Technology, 1988-01-15) Skolnick, Jeffrey ; Yaris, Robert

A phenomenological theory of the nonmechanical and viscoelastic properties of polymer melts is developed. Consistent with computer simulation results [A. Kolinski, J. Skolnick, and R. Yaris, J. Chem. Phys. 86, 1567, 7164, 7174 (1987)], that fail to find evidence for reptation as the dominant mechanism of long distance motion in a melt, we assume that the long-time behavior of a chain is that of a Rouse-like chain having a number of slow moving points, each with a friction constant proportional to the degree of polymerization n. Coupled with the assumption of rubber like behavior at short times made previously by Doi and Edwards [J. Chem. Soc., Faraday Trans. 2 74, 1802 (1978)], the theory predicts that over a broad molecular weight range the shear viscosity scales with n as approximately the 3.4 power of the molecular weight, and that ~n³ in the infinite molecular weight limit. Furthermore, the theory rationalizes the origin of the different crossover molecular weights for the shear viscosity and the self-diffusion coefficient, D. It also accounts for the origin of the intermediate time coupling of the center-of-mass motion into the internal coordinates and for the time dependence of the single bead positional autocorrelation functions seen in previous simulations. Proceeding by analogy to Graessley [J. Poly. Sci. Poly. Phys. Ed. 18, 27 (1980)], in the large molecular weight limit, phenomenological expressions for D and are derived and comparison is made with experiment.
Monte Carlo studies on the long time dynamic properties of dense cubic lattice multichain systems. II. Probe polymer in a matrix of different degrees of polymerization

(Georgia Institute of Technology, 1987-06-15) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

The dynamics of a probe chain consisting of n[subscript P]=100 segments in a matrix of chains of length of n[subscript M]=50 up to nM=800 at a total volume fraction of polymer φ =0.5 have been simulated by means of cubic lattice Monte Carlo dynamics. The diffusion coefficient of the probe chain over the range of n[subscript M]=under consideration decreases by about 30%, a behavior rather similar to that seen in real melts of very long chains. Furthermore, the analysis of the probe chain motion shows that the mechanism of motion is not reptation-like and that the cage effect of the matrix is negligible. That is, the local fluctuations of the topological constraints imposed by the long matrix chains (even for n[subscript M]=800) are sufficiently large to provide for essentially isotropic, but somewhat slowed down, motion of the probe, n[subscript P] =100, chains relative to the homopolymer melt. The results of these MC experiments are discussed in the context of theoretical predictions and experimental findings for related systems.
Monte Carlo studies on the long time dynamic properties of dense cubic lattice multichain systems. I. The homopolymeric melt

(Georgia Institute of Technology, 1987-06-15) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

Dynamic Monte Carlo simulations of long chains confined to a cubic lattice system at a polymer volume fraction of φ =0.5 were employed to investigate the dynamics of polymer melts. It is shown that in the range of chain lengths n, from n=64 to n=800 there is a crossover from a weaker dependence of the diffusion coefficient on chain length to a much stronger one, consistent with D~n⁻². Since the n⁻² scaling relation signals the onset of highly constrained dynamics, an analysis of the character of the chain contour motion was performed. We found no evidence for the well-defined tube required by the reptation model of polymer melt dynamics. The lateral motions of the chain contour are still large even in the case when n=800, and the motion of the chain is essentially isotropic in the local coordinates. Hence, the crossover to the D~ n⁻² regime with increasing chain length of this monodisperse model melt is not accompanied by the onset of reptation dynamics.
Does reptation describe the dynamics of entangled, finite length polymer systems? A model simulation

(Georgia Institute of Technology, 1987-02-01) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

In order to examine the validity of the reptation model of motion in a dense collection of polymers, dynamic Monte Carlo (MC) simulations of polymer chains composed of n beads confined to a diamond lattice were undertaken as a function of polymer concentration and degree of polymerization n. We demonstrate that over a wide density range these systems exhibit the experimentally required molecular weight dependence of the center-of-mass self-diffusion coefficient D~n−[superscript 2.1] and the terminal relaxation time of the end-to-end vector R~n[superscript 3.4]. Thus, these systems should represent a highly entangled collection of polymers appropriate to look for the existence of reptation. The time dependence of the average single bead mean-square displacement, as well as the dependence of the single bead displacement on position in the chain were examined, along with the time dependence of the center-of-mass displacement. Furthermore, to determine where in fact a well-defined tube exists, the mean-square displacements of a polymer chain down and perpendicular to its primitive path defined at zero time were calculated, and snapshots of the primitive path as a function of time are presented. For an environment where all the chains move, no evidence of a tube, whose existence is central to the validity of the reptation model, was found. However, if a single chain is allowed to move in a partially frozen matrix of chains (where all chains but one are pinned every ne beads, and where between pin points the other chains are free to move), reptation with tube leakage is recovered for the single mobile chain. The dynamics of these chains possesses aspects of Rouse-like motion; however, unlike a Rouse chain, these chains undergo highly cooperative motion that appears to involve a backflow between chains to conserve constant average density. While these simulations cannot preclude the onset of reptation at higher molecular weight, they strongly argue at a minimum for the existence with increasing n of a crossover regime from simple Rouse dynamics in which reptation plays a minor role at best.
Possible role of helix-coil transitions in the microscopic mechanism of muscle contraction

(Georgia Institute of Technology, 1987-02) Skolnick, Jeffrey

Local helix-coil transitions in the coiled coil portion of myosin have long been implicated as a possible origin of tension generation in muscle. From a statistical mechanical theory of conformational transitions in coiled coils, the free energy required to form a randomly coiled bubble in the hinge region of myosin of the type conjectured by Harrington (Harrington, W. F., 1979, Proc. Natl. A cad. Sci. USA, 76:5066-5070) is estimated to be -25 kcal/mol. Unfortunately this is far more than the free energy available from ATP hydrolysis if the crossbridges operate independently. Thus, in solution such bubbles are predicted to be absent, and the theory requires that the rod portion of myosin be a hingeless, continuously deforming rod. While such bubble formation in vivo cannot be entirely ruled out, it appears to be unlikely. We further conjecture that in solution the swivel located between myosin subfragments 1 and 2 (S-2 and S-I) is due to a locally random conformation of the chains caused by the presence of a proline residue at the point that physically separates the coiled coil from the globular portion of myosin. On attachment of S-I to actin in the strong binding state, the configurational entropy of the random coil in the swivel region is greatly reduced relative to the case where the ends are free. This produces a spontaneous coil-to-helix transition in the swivel region that causes rotation of S-I and the translation of actin. Thus, the model predicts that the actin filaments are pushed rather than pulled past the thick filaments by the crossbridges. The specific mechanism of force generation is examined in detail, and a simple statistical mechanical realization of the model is proposed. We find that the model gives a substantial number of qualitative and at times quantitative predictions in accord with experiment, and is particularly appealing in that it provides a simple means of free energy transduction-the well known fact that topological constraints shift the equilibrium between helical and random coil states.
The collapse transition of semiflexible polymers. A Monte Carlo simulation of a model system

(Georgia Institute of Technology, 1986-09-15) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

Monte Carlo simulations have been performed on a diamond lattice model of semiflexible polymers for a range of flexibilities and a range of chain lengths from 50 to 800 segments. The model includes both repulsive (excluded volume) and attractive segment–segment interactions. It is shown that the polymers group into two classes, "flexible" and "stiff." The flexible polymers exhibit decreasing chain dimensions as the temperature decreases with a gradual collapse from a loose random coil, high temperature state to a dense random coil, low temperature state. The stiffer polymers, on the other hand, exhibit increasing chain dimensions with decreasing temperature until at a critical temperature there is a sudden collapse to an ordered high density, low temperature state. This difference is due to the relative strength of the segment–segment attractive interactions compared to the energetic preference for a trans conformational state over a gauche state. When the attractive interaction is relatively strong (flexible case) the polymer starts to collapse before rotational degrees of freedom freeze out, leading to a disordered dense state. When the attractive interaction is relatively weak (stiff case) the polymer starts to freeze out rotational degrees of freedom before it finally collapses to a highly ordered dense state.
On the short time dynamics of dense polymeric systems and the origin of the glass transition: A model system

(Georgia Institute of Technology, 1986-02-01) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

In order to model the short time (and distance) scale motions for dense polymeric systems, we have performed dynamic Monte Carlo simulations of chains on a diamond lattice at considerably greater densities than those done previously. Chain dynamics were simulated by a random sequence of three- and four-bond kink motions and end moves. For times shorter than the chain diffusion time, the single bead autocorrelation function g(t) exhibits three distinct regimes: a short time Rouse-like regime where g(t)~t[superscript ½]; a mid-region where g(t)~t β, followed by a longer time, Rouse-like regime where g(t)~t1/2. There is a smooth crossover from Rouse-like dynamics, β =1/2, at low density to smaller values of β at higher density, and β =0 at the glass transition density (Φ[subscript G] =0.92±0.01). It is shown that the major motion of the chains is transverse to the chain contour rather than along the chain. The observed motion is successfully analyzed in terms of the motion of defects (holes) through the sample. It is shown that the glass transition at Φ[subscript G] ±0.92 is caused by the shutting down of the orientation changing four-bond motions.
Monte Carlo dynamics of diamond-lattice multichain systems

(Georgia Institute of Technology, 1986-01-30) Kolinski, Andrzej ; Skolnick, Jeffrey ; Yaris, Robert

We present preliminary results of Monte Carlo studies on the dynamics of multichain diamond-lattice systems at considerably greater densities than those done previously. Chain dynamics were simulated by a random sequence of three or four bond kink motions. The single bead autocorrelation function exhibits "slow" mode relaxation behavior with a g(t)∝ tβ. There is a smooth crossover from Rouse-like dynamics, β=1/2, at low density to smaller values of β at higher density and β=0 at the glass transition density (φG≅0.92). The simulation provides a self-diffusion coefficient D ∝ n-2, with n the number of beads, in agreement with experiment. A phenomenological model, different from the widely accepted reptation picture, is proposed.
Effects of topological solitons on autocorrelation functions for chains of coupled torsional oscillators

(Georgia Institute of Technology, 1983-06-01) Perchak, Dennis ; Yaris, Robert ; Skolnick, Jeffrey

Brownian dynamics computer simulations were performed on chains of coupled torsional oscillators. The purpose was to observe the changes in autocorrelation functions, related to typical experimental measurements, caused by the introduction of topological solitons or kinks into the system. We considered three model systems: a chain of coupled torsional oscillators, a chain of coupled torsional oscillators with additional onefold rotational potentials acting on each oscillator, and a chain of coupled torsional oscillators with additional threefold rotational potentials. These models are of interest because of their application to torsional motions in polymeric systems, and, in particular, the system with onefold rotational potentials has been studied extensively as the sine–Gordon chain. We present simulation results for three autocorrelation functions of these three systems both with and without topological solitons.