Diffusion-limited growth of polymer chains (original) (raw)
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Continuous Self-Avoiding Walk with Application to the Description of Polymer Chains
The Journal of Physical Chemistry B, 2006
We develop the continuous self-avoiding walk (CSAW) methodology for investigating temperature dependent thermodynamic properties of finite polymer chains without imposing a lattice. This leads to a new concept: the free energy theta temperature, T θF , at which the free energy is proportional to chain length. Above T θF , the polymer chain-solvent mixture leads to a single phase, whereas below T θF the polymer solvent system has a positive surface tension with a tendency to phase separation to form a globular phase. For finite chains this coil-globule transition lies above the geometric theta temperature at which the distribution describes a Gaussian coil. CSAW provides the basis for a new approach to predict globular properties of real polymers.
2009
Essential physics associated with the conformational behavior of a linear semiflexible homopolymer chain have been derived from a model of directed self avoiding walk (DSAW) on a two dimensional rectangular lattice. The DSAW model has been solved analytically to study phase transitions occurring in the polymer chain and exact values of conformational properties and transition points have been reported. We have analyzed the variation of critical value of step fugacity and persistent length with bending energy of the semiflexible polymer chain for a case when the chain is in the bulk. In presence of an attractive impenetrable surface, variation of critical value of monomer-surface attraction with bending energy of the polymer chain shows that adsorption of a stiff polymer chain takes place at a smaller value of monomer surface attraction than a flexible polymer chain. We have compared the results obtained for a two dimensional rectangular lattice case to the corresponding results obtained using square lattice and found that qualitative nature of phase diagrams are similar in the case of both the lattices. [
AIP Advances, 2018
Recent calculations on the change in radial dimensions of reacting (growing) polyethylene in the gas phase experiencing Lennard Jones and Kihara type potentials revealed that a single reacting polyethylene molecule does not experience polymer collapse. This implies that a transition that is the converse of what happens when molten polyethylene crystallizes, i.e. it transforms from random coil like structure to folded rigid rod type structure, occurs. The predicted behaviour of growing polyethylene was explained by treating the head of the growing polymer chain as myopic whereas as the whole chain (i.e. when under equilibrium conditions) being treated as having normal vision, i.e. the growing chain does not see the attractive part of the LJ or Kihara Potentials. In this paper we provide further proof for this argument in two ways. Firstly we carry forward the exact enumeration calculations on growing self avoiding walks reported in that paper to larger values of number of steps by using Monte Carlo type calculations. We thereby assign physical significance to the connective constant of self avoiding walks, which until now was treated as a purely abstract mathematical entity. Secondly since a reacting polymer molecule that grows by addition polymerisation sees only one step ahead at a time, we extend this calculation by estimating the average atmosphere for molecules, with repulsive potential only (growing self avoiding walks in two dimensions), that look at two, three, four, five ...steps ahead. Our calculation shows that the arguments used in the previous work are correct.
Modified Self-Avoiding Walk in a Polymerization Process
2005
The modified SAW (mSAW) is defined as a statistical method to treat a polymerization process in a manner similar to that used to treat chains with excluded volume statistics where no two monomers can occupy the same site in space. Unlike the chains with excluded volume statistics, the walk algorithm does not terminate when the next walk is an occupied site. Instead the walker continues along a different direction. Monte Carlo simulations of the random walk are carried out on both 2D and 3D lattices. Universality classes different from those of the chain with excluded volume statistics are found. The critical exponents of the mean-square end-to-end distance are found to be 1.437 (0.005) for 2D and 1.007 (0.004) for 3D, where the figures in the parentheses are the uncertainties of the last digit. The universality classes are determined from rigorous computer simulations.
Excluded-volume effects in linear polymers: Universality of generalized self-avoiding walks
Physical Review B, 1985
A random walk which can visit each lattice site at most twice is considered. The universality of selfavoiding-walk critical behavior with respect to variations of a fugacity for self-intersections is predicted on the basis of general renormalization-group arguments and explicitly tested in two dimensions, both by exact enumeration analysis and by cluster scaling calculations. The meaning of the above universality and its consequences, as far as a correct formulation of Flory approximations is concerned, are briefly discussed.
Diffusion of a polymer chain in random media
Macromolecules, 1989
Dynamic properties of a self-avoiding walk chain, which performs Brownian motion between randomly distributed impenetrable fixed obstacles, have been investigated by Monte Carlo simulations. Analogous to the case of a random walk chain in random media, the chain dynamics is found to be slower than even reptation demonstrated by a stronger inverse dependence of the chain diffusion coefficient on chain length. This phenomenon is attributed to the slowing down of the chain due to the presence of bottlenecks in the random medium. The bottlenecks squeeze the chain and reduce the chain entropy setting up entropic barriers at random locations. A scaling analysis is adopted to account for the effects of such entropic barriers on chain diffusion. The simulation data are consistent with the predictions of the scaling arguments demonstrating that chain diffusion in random media is controlled by the entropic barriers of the media.
Existence of four-dimensional polymer collapse I. Kinetic growth trails
Physica A: Statistical Mechanics and its Applications, 1998
We present the results of simulations of kinetic growth trails (KGT) (bond-avoiding walks) in four dimensions. We use a mapping from a kinetic growth model to a static model of selfinteracting trails (ISAT) at a particular temperature to argue that this temperature is precisely the collapse temperature of four-dimensional interacting trails. To do this we show that the kinetic growth trails behave neither like static non-interacting trails, which should behave as excludedvolume-dominated four-dimensional polymers (that is self-avoiding walks), or collapsed fourdimensional polymers, but rather show an intermediate behaviour. This is the ÿrst indication of collapse in any four-dimensional lattice polymer model and so may be helpful in deciding which of the competing models of polymers is a good model in lower dimensions. We have calculated various exponents of the KGT model and identiÿed them with certain critical exponents of the static ISAT problem.
Anomalous diffusion of ideal polymer networks
Physical Review E, 1997
Internal dynamics of swollen polymer arrays were investigated with Brownian dynamic techniques applied to regular Rouse networks. In all cases local or self-diffusion decayed as a power law with a power proportional to the given topological dimension. This behavior allows for the classification of three dynamic regimes: subcritical topologies accommodate power law anomalous diffusion; logarithmic anomalous diffusion occurs within the critical topological dimension d t ϭ2; and upper-critical topologies siege bounded anomalous diffusion. ͓S1063-651X͑97͒13206-8͔
Dynamical Transitions in a Dragged Growing Polymer Chain
2016
We extend the Rouse model of polymer dynamics to situations of non-stationary chain growth. For a dragged polymer chain of length N(t) = t^α, we find two transitions in conformational dynamics. At α= 1/2, the propagation of tension and the average shape of the chain change qualitatively, while at α = 1 the average center-of-mass motion stops. These transitions are due to a simple physical mechanism: a race duel between tension propagation and polymer growth. Therefore they should also appear for growing semi-flexible or stiff polymers. The generalized Rouse model inherits much of the versatility of the original Rouse model: it can be efficiently simulated and it is amenable to analytical treatment.
A directed walk model of a long chain polymer in a slit with attractive walls
Journal of Physics A: Mathematical and General, 2005
We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit. Apart from the general interest in the effect of geometrical confinement, this can be viewed as a two-dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions. We demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half-plane geometry, even when the polymer is constrained to be fixed at both ends on one wall. We are not able to find a closed form solution for the free energy for finite width, at all values of the interaction parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane. This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur.