Extracting Aggregation Free Energies of Mixed Clusters from Simulations of Small Systems: Application to Ionic Surfactant Micelles

Abstract

Micelle cluster distributions from molecular dynamics simulations of a solvent-free coarse-grained model of sodium octyl sulfate (SOS) were analyzed using an improved method to extract equilibrium association constants from small-system simulations containing one or two micelle clusters at equilibrium with free surfactants and counterions. The statistical-thermodynamic and mathematical foundations of this partition-enabled analysis of cluster histograms (PEACH) approach are presented. A dramatic reduction in computational time for analysis was achieved through a strategy similar to the selector variable method to circumvent the need for exhaustive enumeration of the possible partitions of surfactants and counterions into clusters. Using statistics from a set of small-system (up to 60 SOS molecules) simulations as input, equilibrium association constants for micelle clusters were obtained as a function of both number of surfactants and number of associated counterions through a global fitting procedure. The resulting free energies were able to accurately predict micelle size and charge distributions in a large (560 molecule) system. The evolution of micelle size and charge with SOS concentration as predicted by the PEACH-derived free energies and by a phenomenological four-parameter model fit, along with the sensitivity of these predictions to variations in cluster definitions, are analyzed and discussed.

Simulations of NaCl Aggregation from Solution: Solvent Determines Topography of Free Energy Landscape

“…The partition‐enabled analysis of cluster histograms (PEACH) method uses simulation cluster size distributions from a set of simulations over a range of concentration and/or system sizes to generate equilibrium association constants K i , j for each cluster with i and j representing the numbers of Na + and Cl − ions in the cluster . Using the exact relationship between the single‐cluster partition functions q i , j (= q ° i , j V / V ° , with q° the partition function of a single cluster in a standard volume V° = 1/c°) and the canonical partition functions Q ( N A , N B , V , T ) for a system with a total of N A and N B parts of each molecule type, the average cluster frequency is, …”

“…A number of approaches have been used to account for this depletion effect in simulations of nucleation at fixed number of particles . Here, we apply the recently developed partition enabled analysis of cluster histograms (PEACH) method, which avoids this problem as it accounts for the discrete nature of state partitions explicitly . It allows extraction of free energies of formation from cluster distributions obtained through unbiased simulations of small systems, even when a large fraction of the monomers are involved in cluster formation, and has been applied to the free energy of assembly of anionic surfactant micelles .…”

“…[39] It allows extraction of free energies of formation from cluster distributions obtained through unbiased simulations of small systems, even when a large fraction of the monomers are involved in cluster formation, and has been applied to the free energy of assembly of anionic surfactant micelles. [39] Cluster definitions will be designed to encompass all stable cluster structures (i.e., those that occur with significantly greater probability than would be predicted from randomly diffusing ions), whether or not they lie on the pathway to crystal nucleation. A definition that excludes linear or branched structures, for instance, would fail to consistently account for the reduced availability of ions when those structures are formed.…”

“…Here, we apply the recently developed partition enabled analysis of cluster histograms (PEACH) method, which avoids this problem as it accounts for the discrete nature of state partitions explicitly . It allows extraction of free energies of formation from cluster distributions obtained through unbiased simulations of small systems, even when a large fraction of the monomers are involved in cluster formation, and has been applied to the free energy of assembly of anionic surfactant micelles . Cluster definitions will be designed to encompass all stable cluster structures (i.e., those that occur with significantly greater probability than would be predicted from randomly diffusing ions), whether or not they lie on the pathway to crystal nucleation.…”

Free energy of micellization of dodecyl phosphocholine (DPC) from molecular simulation: Hybrid PEACH‐BAR method

“…3 Statistical thermodynamics shows that applying the law of mass action directly to simulations of small systems results in distortions to the size distribution and unimer concentration, 4 as has been confirmed recently in coarse-grained simulations of surfactants. 5 We have reported a new method to extract cluster free energy profiles and cluster size distributions from MD simulations of few clusters for several systems [6][7][8][9] including the zwitterionic surfactant octyl phosphocholine (OPC). 6 This method, dubbed "PEACH" standing for "Partition-enabled Analysis of Cluster Histograms," finds the globally optimized set of equilibrium association constants for all cluster sizes through an iterative fitting for the cluster size distributions observed in simulations.…”