^{1,a)}and Rosalind J. Allen

^{1}

### Abstract

Osmosis is one of the most important physical phenomena in living and soft matter systems. While the thermodynamics of osmosis is well understood, the underlying microscopic dynamical mechanisms remain the subject of discussion. Unravelling these mechanisms is a prerequisite for understanding osmosis in non-equilibrium systems. Here, we investigate the microscopic basis of osmosis, in a system at equilibrium, using molecular dynamics simulations of a minimal model in which repulsive solute and solvent particles differ only in their interactions with an external potential. For this system, we can derive a simple virial-like relation for the osmotic pressure. Our simulations support an intuitive picture in which the solvent concentration gradient, at osmotic equilibrium, arises from the balance between an outward force, caused by the increased total density in the solution, and an inward diffusive flux caused by the decreased solvent density in the solution. While more complex effects may occur in other osmotic systems, our results suggest that they are not required for a minimal picture of the dynamic mechanisms underlying osmosis.

The authors thank Daan Frenkel for drawing this topic to our attention, Leo Lue for assistance with the calculations in Appendix B, Mike Cates, Davide Marenduzzo, and Aron Yoffe for useful discussions and Mike Cates and Patrick Warren for helpful comments on the manuscript. T.W.L. was supported by an EPSRC DTA studentship and R.J.A. by a Royal Society University Research Fellowship and by EPSRC under Grant No. EP/J007404. This work has made use of the Edinburgh Compute and Data Facility, which is partially supported by the eDIKT initiative.

I. INTRODUCTION

II. A MINIMAL MODEL FOR OSMOSIS

III. OSMOSIS IN THE MODEL SYSTEM

A. Density imbalance

B. Osmotic pressure

IV. WHAT MAINTAINS THE SOLVENT DENSITY GRADIENT?

A. Balance between outward and inward fluxes

B. Solvent-solute correlations

V. DISCUSSION

### Key Topics

- Solvents
- 117.0
- Osmosis
- 27.0
- Colloidal systems
- 15.0
- Solution processes
- 11.0
- Chemical solutions
- 9.0

## Figures

(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.

(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, ρ_{total}(*x*); panel (b) shows the local solvent density ρ_{v}(*x*).

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, ρ_{total}(*x*); panel (b) shows the local solvent density ρ_{v}(*x*).

Spatially averaged solvent density ρ_{v} 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).

Spatially averaged solvent density ρ_{v} 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 Δ*P* (in units of *k* _{ B } *T*σ^{−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 Δ*P* in our simulations, computed using the method of planes (see Appendix A). The squares show Δ*P* computed from our simulations using Eq. (1). The statistical errors on the symbols are ±3% for *c* _{ s } (circles and squares) and ±3% for Δ*P* (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, Δ*P* = *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).

Osmotic pressure Δ*P* (in units of *k* _{ B } *T*σ^{−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 Δ*P* in our simulations, computed using the method of planes (see Appendix A). The squares show Δ*P* computed from our simulations using Eq. (1). The statistical errors on the symbols are ±3% for *c* _{ s } (circles and squares) and ±3% for Δ*P* (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, Δ*P* = *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 } *T*σ^{−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.

Net force per particle, normal to the membrane (in units of *k* _{ B } *T*σ^{−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.

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) 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 } *T*σ^{−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.

(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 } *T*σ^{−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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