Multimerizations, Aggregation, and Transfer Reactions of Small Numbers of Molecules

Abstract

Chemical equilibria of multimerizations in systems with small numbers of particles exhibit a behavior seemingly at odds with that observed macroscopically. In this paper, we apply the recently proposed expression of equilibrium constant for binding, which includes cross-correlations in reactants’ concentrations, to write an equilibrium constant for the formation of clusters larger than two (e.g., trimer, tetramer, and pentamer) as series of two-body reactions. Results obtained by molecular dynamics simulations demonstrate that the value of this expression is constant for all concentrations and system sizes, as well as at an onset of a phase transition to an aggregated state, where densities in the system change discontinuously. In contrast, the value of the commonly utilized expression of equilibrium constant, which ignores correlations, is not constant and its variations can reach few orders of magnitude. Considering different paths for the same multimer formation, with elementary reactions of any order, yields different expressions for the equilibrium constant, yet, with exactly the same value. This is also true for routes with essentially zero probability to occur. Existence of different expressions for the same equilibrium constant imposes equalities between averages of correlated, along with uncorrelated, concentrations of participating species. Moreover, a relation between an average particle number and relative fluctuations derived for two-body reactions is found to be obeyed here as well despite couplings to additional equilibrium reactions in the system. Analyses of transfer reactions, where association and dissociation events take place on both sides of the chemical equation, further indicate the necessity to include cross-correlations in the expression of the equilibrium constant. However, in this case, the magnitudes of discrepancies of the uncorrelated expression are smaller, likely because of partial cancellation of correlations, which exist on both the reactant and product sides.

Breakdown of Langmuir Adsorption Isotherm in Small Closed Systems

“…Yet, Polak and Rubinovich argued that adsorption under nanoconfinement exhibits equilibrium properties deviating from those predicted by Langmuir’s model due to an entropic effect, and Ramaswamy et al argued that rate equations are qualitatively incorrect in subcritical volumes . Furthermore, single-molecule experiments of small-sized systems undergoing association reactions (where both reactants are mobile in space) find that concentrations of bound complexes do not agree with predictions of the conventional chemical equilibrium theory. − Similar behavior was also reported by computational studies. − In light of these findings, we recently demonstrated that for bimolecular reactions, averages of quantities observed at small (finite) systems are different from those observed at large or macroscopic systems. − This inhomogeneous character of the functions describing the system’s properties is applicable for closed systems, that is, for systems in which the total numbers of particles are fixed, such as the canonical ensemble. Then, by definition, as time or configurations are propagated, the particle numbers of all components are subjected to fluctuations with relative magnitudes that increase as system’s size decreases.…”

“…What is the difference then between small and large systems? Because we are dealing with bimolecular reactions, which necessarily proceed via two-body interactions, cross-correlations in particle numbers (or concentrations) must be taken into account when describing mass-actions at equilibrium. − The importance of these cross-correlations is augmented as particle numbers and/or volume decrease, as well as, for lower temperatures or larger binding energies, and the amplitude of their effect can reach few orders of magnitude. On the other hand, when the system is large enough (hereafter, will be used interchangeably with the term macroscopic), these cross-correlations are negligible and can be completely ignored.…”

“… 41 − 56 In light of these findings, we recently demonstrated that for bimolecular reactions, averages of quantities observed at small (finite) systems are different from those observed at large or macroscopic systems. 57 − 59 This inhomogeneous character of the functions describing the system’s properties is applicable for closed systems, that is, for systems in which the total numbers of particles are fixed, such as the canonical ensemble. Then, by definition, as time or configurations are propagated, the particle numbers of all components are subjected to fluctuations with relative magnitudes that increase as system’s size decreases.…”