Dynamics Of Activated Processes Involving Chain Molecules
This thesis presents our recent study of few interesting problems involving activated processes. This chapter gives an overview of the thesis. It is now possible to do single molecule experiments involving enzyme molecules. The kinetics of such reactions exhibits dynamic disorder associated with conformational changes of the enzyme-substrate complex. The static disorder and dynamic disorder of reaction rates, which are essentially indistinguishable in ensemble-averaged experiments, can be determined separately by the real-time single-molecule approach. In our present work we have given a theoretical description of how rate of reactions involving dynamic disorder is studied using path integral approach. It is possible to write the survival probability and the rate of the process as path integrals and then use variational approaches to get bounds for both. Though the method is of general validity, we illustrate it in the case of electronic relaxation in stochastic environment modeled by a particle experiencing diﬀusive motion in harmonic potential in presence of delta function sink. The exact solution of corresponding Smoluchowski equation was found earlier analytically in Laplace domain with sink having arbitrary strength and position. Exact evaluation of path integral calculation to survival probability is not possible analytically. Wolynes et al. have done an approximate calculation to get bounds to the survival probability in the Laplace domain. A bound in the Laplace domain is not as useful as a bound in the time domain and hence we use the direct approximate variational path integral technique to calculate both lower and upper bound of survival probability in time domain. We mimic the delta function sink by quadratic sink for which the path integral can be solved exactly. The strength of the quadratic sink is treated as variational parameter and using the optimized value for it, one can estimate the optimized lower as well as upper bound of survival probability. We have also calculated a lower bound to the rate. The variational results are compared with the exact ones, and it is found that the results for the two parameter case are better than those of one parameter case. To understand how good our approximation is, we calculate the bounds in survival time and found them to be in good agreement with exact results. Our approach is valid for any arbitrary initial distribution that one may start with. We consider the Kramers problem for a long chain polymer trapped in a biased double well potential. Initially the polymer is in the less stable well and it can escape from this well to the other well by the motion of its N beads across the barrier to attain the configuration having lower free energy. In one dimension we simulate the crossing and show that the results are in agreement with the kink mechanism suggested earlier. In three dimensions, it has not been possible to get analytical “kink solution”for an arbitrary po-tential; however, one can assume the form of the solution of the non-linear equation as a kink solution and then find a double well potential in three dimensions. To verify the kink mechanism, simulations of the dynamics of a discrete Rouse polymer model in a double well in three dimensions were done. We find that the time of crossing is proportional to the chain length which is in agreement with the results of kink mechanism. The shape of the kink solution is also in agreement with the analytical solution in both one and three dimensions. We then consider the dynamics of a short chain polymer crossing over a free energy barrier in space. Adopting the continuum version of the Rouse model, we find exact expressions for the activation energy and the rate of crossing. For this model, the analysis of barrier crossing is analogous to semiclassical treatment of quantum tunneling. Finding the saddle point for the process requires solving a Newton-like equation of motion for a fictitious particle. The analysis shows that short chains would cross the barrier as a globule. The activation free energy for this would increase linearly with the number of units N in the polymer. The saddle point for longer chains is an extended conformation, in which the chain is stretched out. The stretching out lowers the energy and hence the activation free energy is no longer linear in N . The rates in both the cases are calculated using a multidimensional approach and analytical expressions are derived using a new formula for evaluating the infinite products. However, due to the harmonic approximation made in the derivation, the rates are found to diverge at the point where the saddle point changes over from the globule to the stretched out conformation. The reason for this is identified to be the bifurcation of the saddle to give two new saddles. A correction formula is derived for the rate in the vicinity of this point. Numerical results using the formulae are presented. It is possible for the rate to have a minimum as a function of N . This is due to the confinement effects in the initial state. We analyze the dynamics of a star polymer of F arms confined to a double well potential. Initially the molecule is confined to one of the minima and can cross over the barrier to the other side. We use the continuum version of Rouse-Ham model. The rate of crossing is calculated using the multidimensional approach due to Langer.Finding the transition state for the process is shown to be equivalent to the solution of Newton’s equations for F independent particles, moving in an inverted potential. For each star polymer, there is a critical total length N Tc below which the polymer crosses over as a globule. The value of NTc depends on the curvature at the top of the barrier as well as the individual arm lengths. So we keep the lengths of (F -1) arms fixed and increase the length of the F th arm to get the minimum total length NTc. Below NTc the activation energy is proportional to the total arm length of the star. Above N Tc the star crosses the barrier in a stretched state. Thus, there is a multifurcation of the transition state at NTc. Above NTc, the activation energy at first increases and then decreases with increasing arm length. This particular variation of activation energy results from the fact that in the stretched state, only one arm of the polymer is stretched across the top of the barrier, while others need not to be. We calculate the rate by expanding the energy around the saddle upto second order in the fluctuations. As we use the continuum model, there are infinite modes for the polymer and consequently, the prefactor has infinite products. We show that these infinite products can be reduced to a simple expression, and evaluated easily. However, the rate diverges near N Tc due to the multifurcation, which results in more than one unstable mode. The cure for this divergence is to keep terms upto fourth order in the expansion of energy for these modes. Performing this, we have calculated the rate as a function of the length of the star. It is found that the rate has a nonmonotonic dependence on the length, suggesting that longer stars may actually cross over faster.
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