(a) Illustration of our model system. Solute particles are confined within a cubic solution compartment located in the centre of the simulation box. (b) Simulation snapshot; solute and solvent particles are coloured green and blue, respectively. For clarity only particles located in the solution compartment are shown.
Local density profiles ρ(x) (in units of σ−3) measured across the middle of the simulation box, for solute concentration c s = 0.254σ−3. Panel (a) shows the total particle density, ρtotal(x); panel (b) shows the local solvent density ρv(x).
Spatially averaged solvent density ρv as a function of solute concentration c s (both in units of σ−3), in the solution and solvent compartments (circles and squares, respectively). The dashed lines show theoretical predictions based on the Carnahan-Starling equation of state (see Appendix B).
Osmotic pressure ΔP (in units of k B Tσ−3) as a function of solute concentration c s (in units of σ−3). In the main plot, the symbols show simulation results. The circles show direct measurements of ΔP in our simulations, computed using the method of planes (see Appendix A). The squares show ΔP computed from our simulations using Eq. (1). The statistical errors on the symbols are ±3% for c s (circles and squares) and ±3% for ΔP (squares only)—i.e., approximately the size of the symbols. These errors arise mainly from uncertainty in the position of the solution boundary, as discussed in Appendix A. The lines show theoretical predictions. The dotted line shows the van‘t Hoff relation, ΔP = k B Tc s. The dashed line shows the pressure of a system of hard spheres at density c s, computed using the Carnahan-Starling equation of state. The dot-dashed line shows the osmotic pressure predicted by a full thermodynamic calculation, including solute and solvent, using the Carnahan-Starling equation of state. In the inset, the circles are the same as in the main plot, while the triangles show the prediction of our simple hopping model (Eq. (2)) (where and are the average solvent densities in the solution and solvent compartments, respectively).
Net force per particle, normal to the membrane (in units of k B Tσ−1) acting on the solvent particles, as a function of position x, in a slab taken through the middle of the simulation box, for solute concentration c s = 0.254σ−3. Solvent particles close to the boundaries of the solution compartment tend to be pushed out of the solution.
Probability distributions for the distance d to the nearest solute particle (a) and the perpendicular velocity v of the nearest solute particle (b), for solvent particles located less than 0.001 particle diameters outside the solution compartment, which do (black lines) or do not (red lines) enter the solution compartment in the next simulation time step. In these simulations, c s = 0.05σ−3. The boundary of the solution compartment is defined as described in Appendix A.
(a) θ is defined as the angle between the instantaneous velocity of a solvent particle i as it enters the solution compartment and the vector joining it to particle j within the solution compartment. (b) Probability distribution P(θ) when particle j is defined to be the nearest solute particle (black line) and when particle j is the nearest particle of either solute or solvent (red line). In these simulations c s = 0.05σ−3. The boundary of the solution compartment is defined as described in Appendix A.
(a) Method of planes with a plane of finite extent in the y and z directions. Solid unidirectional arrows denote contributions to the kinetic part of Eq. (A1) (ϕ); solid bidirectional arrows denote contributions to the interaction part of Eq. (A1). All interactions in which the line connecting the two particles crosses the plane are included (e.g., the bottom pair of particles); however, the interaction between the top pair of particles (dotted arrow) is excluded. (b) Pressure-density relation of a “gas” of solute particles, confined in the solution compartment in the absence of solvent (closed circles), compared with that of a periodic box of equivalent particles (open circles). The pressure is in units of k B Tσ−3; the density is in units of σ−3. For the closed circles, the density of the solute gas is defined using an effective boundary of the solution compartment positioned 0.91σ inside the locus of divergence of the confining potential.
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