TY - JOUR

T1 - A statistical mechanical theory of proton transport kinetics in hydrogen-bonded networks based on population correlation functions with applications to acids and bases

AU - Tuckerman, Mark E.

AU - Chandra, Amalendu

AU - Marx, Dominik

N1 - Funding Information:
Partial financial support is gratefully acknowledged as follows. M.E.T.: NSF Grant No. CHE-0704036 and the Alexander von Humboldt Foundation (AvH) for the Bessel Award 2005 that allowed stays as a Visiting Professor in Bochum; A.C.: Department of Science and Technology (DST), Department of Atomic Energy (DAE), and Council of Scientific and Industrial Research (CSIR), all Government of India, and Alexander von Humboldt Foundation (AvH) for a Fellowship that allowed stays as a Visiting Professor in Bochum; D.M.: Ruhr-Universität Bochum (RUB), Deutsche Forschungsgemeinschaft (DFG), and Fonds der Chemischen Industrie (FCI). The calculations were performed on Linux Clusters granted to A.C. by DST, DAE, CSIR, and AvH Foundation as well as at Bovilab@RUB (Bochum) and Rechnerverbund-NRW. In addition, we are most grateful to Rudolf Hilfer for providing us with numerical data of the Mittag–Leffler functions used to compute the intermittent correlation function depicted in Fig. of .

PY - 2010/9/28

Y1 - 2010/9/28

N2 - Extraction of relaxation times, lifetimes, and rates associated with the transport of topological charge defects in hydrogen-bonded networks from molecular dynamics simulations is a challenge because proton transfer reactions continually change the identity of the defect core. In this paper, we present a statistical mechanical theory that allows these quantities to be computed in an unbiased manner. The theory employs a set of suitably defined indicator or population functions for locating a defect structure and their associated correlation functions. These functions are then used to develop a chemical master equation framework from which the rates and lifetimes can be determined. Furthermore, we develop an integral equation formalism for connecting various types of population correlation functions and derive an iterative solution to the equation, which is given a graphical interpretation. The chemical master equation framework is applied to the problems of both hydronium and hydroxide transport in bulk water. For each case it is shown that the theory establishes direct links between the defect's dominant solvation structures, the kinetics of charge transfer, and the mechanism of structural diffusion. A detailed analysis is presented for aqueous hydroxide, examining both reorientational time scales and relaxation of the rotational anisotropy, which is correlated with recent experimental results for these quantities. Finally, for OH- (aq) it is demonstrated that the "dynamical hypercoordination mechanism" is consistent with available experimental data while other mechanistic proposals are shown to fail. As a means of going beyond the linear rate theory valid from short up to intermediate time scales, a fractional kinetic model is introduced in the Appendix in order to describe the nonexponential long-time behavior of time-correlation functions. Within the mathematical framework of fractional calculus the power law decay ∼ t-σ, where σ is a parameter of the model and depends on the dimensionality of the system, is obtained from Mittag-Leffler functions due to their long-time asymptotics, whereas (stretched) exponential behavior is found for short times.

AB - Extraction of relaxation times, lifetimes, and rates associated with the transport of topological charge defects in hydrogen-bonded networks from molecular dynamics simulations is a challenge because proton transfer reactions continually change the identity of the defect core. In this paper, we present a statistical mechanical theory that allows these quantities to be computed in an unbiased manner. The theory employs a set of suitably defined indicator or population functions for locating a defect structure and their associated correlation functions. These functions are then used to develop a chemical master equation framework from which the rates and lifetimes can be determined. Furthermore, we develop an integral equation formalism for connecting various types of population correlation functions and derive an iterative solution to the equation, which is given a graphical interpretation. The chemical master equation framework is applied to the problems of both hydronium and hydroxide transport in bulk water. For each case it is shown that the theory establishes direct links between the defect's dominant solvation structures, the kinetics of charge transfer, and the mechanism of structural diffusion. A detailed analysis is presented for aqueous hydroxide, examining both reorientational time scales and relaxation of the rotational anisotropy, which is correlated with recent experimental results for these quantities. Finally, for OH- (aq) it is demonstrated that the "dynamical hypercoordination mechanism" is consistent with available experimental data while other mechanistic proposals are shown to fail. As a means of going beyond the linear rate theory valid from short up to intermediate time scales, a fractional kinetic model is introduced in the Appendix in order to describe the nonexponential long-time behavior of time-correlation functions. Within the mathematical framework of fractional calculus the power law decay ∼ t-σ, where σ is a parameter of the model and depends on the dimensionality of the system, is obtained from Mittag-Leffler functions due to their long-time asymptotics, whereas (stretched) exponential behavior is found for short times.

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U2 - 10.1063/1.3474625

DO - 10.1063/1.3474625

M3 - Article

C2 - 20886925

AN - SCOPUS:77957730975

SN - 0021-9606

VL - 133

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

IS - 12

M1 - 124108

ER -