In our previous report on resonance energy transfer from a dye molecule to graphene [J. Chem. Phys.129, 054703 (2008)], we had derived an expression for the rate of energy transfer from a dye to graphene. An integral in the expression for the rate was evaluated approximately. We found a Yuwaka-type dependence of the rate on the distance. We now present an exact evaluation of the integral involved, leading to very interesting results. For short distances (z<20 A), the present rate and the previous rate are in good agreement. For larger distances, the rate is found to have a z(-4) dependence on the distance, exactly. Thus we predict that for the case of pyrene on graphene, it is possible to observe fluorescence quenching up to a distance of 300 A. This is in sharp contrast to the traditional fluorescence resonance energy transfer where the quenching is observable only up to 100 A.

We study the distance dependence of the rate of resonance energy transfer from the excited state of a dye to the pi system of graphene. Using the tight-binding model for the pi system and the Dirac cone approximation, we obtain the analytic expression for the rate of energy transfer from an electronically excited dye to graphene. While in traditional fluorescence resonance energy transfer, the rate has a (distance)(-6) dependence, we find that the distance dependence in this case is quite different. Our calculation of rate in the case of the two dyes, pyrene and nile blue, shows that the distance dependence is Yukawa type. We have also studied the effect of doping on energy transfer to graphene. Doping does not modify the rate for electronic excitation energy transfer significantly. However, in the case of vibrational transfer, the rate is found to be increased by an order of magnitude due to doping. This can be attributed to the nonzero density of states at the Fermi level that results from doping.

It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))(2)⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))(2)⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.

We consider a long chain molecule, initially confined to the metastable side of a biased double well potential. It can escape from this side to the other by the motion of its N segments across the barrier. We assume that the length of the molecule is much larger than the width w of the barrier. The width w is taken to be sufficiently large that a continuum description is applicable to even the portion over the barrier. We use the Rouse model and analyze the mechanism of crossing the barrier. There can be two dominant mechanisms: end crossing and hairpin crossing. We find the free energy of activation for the hairpin crossing to be two times that for end crossing. The pre-exponential factor for hairpin crossing is proportional to N, while it is independent of N for end crossing. In both cases, the activation energy has a square root dependence on the temperature T, leading to a non-Arrhenius form for the rate. We also show that there is a special time dependent solution of the model, which corresponds to a kink in the chain, confined to the region of the barrier. The movement of the polymer from one side to the other is equivalent to the motion of the kink on the chain in the reverse direction. If there is no free energy difference between the two sides of the barrier, then the kink moves by diffusion and the time of crossing t(cross) approximately N(2)/T(3/2). If there is a free energy difference, then the kink moves with a nonzero velocity from the lower free energy side to the other, leading to t(cross) approximately N/sqrt[T]. We also discuss the applicability of the mechanism to the recent experiments of Kasianowicz [Proc. Natl. Acad. Sci. USA 93, 13 770 (1996)], where DNA molecules were driven through a nanopore by the application of an electric field. The prediction that t(cross) approximately N is in agreement with these experiments. Our results are in agreement with the recent experimental observations of Han, Turner, and Craighead [Phys. Rev. Lett. 83, 1688 (1999)]. We also consider the translocation of hydrophilic polypeptides across hydrophobic pores, a process that is quite common in biological systems. Biological systems accomplish this by having a hydrophobic signal sequence at the end that goes in first. We find that for such a molecule, the transition state resembles a hook, and this is in agreement with presently accepted view in cell biology.

Deviations from the usual R -6 dependence of the rate of fluorescence resonance energy transfer (FRET) on the distance between the donor and the acceptor have been a common scenario in the recent times. In this paper, we present a critical analysis of the distance dependence of FRET, and try to illustrate the non-R -6 type behaviour of the rate for the case of transfer from a localized electronic excitation on the donor, a dye molecule to three different energy acceptors with delocalized electronic excitations namely, graphene, a two-dimensional semiconducting sheet and the case of such a semiconducting sheet rolled to obtain a nanotube. We use simple analytic models to understand the distance dependence in each case.