Diffusiophoresis and diffusioosmosis

Diffusiophoresis is the spontaneous motion of colloidal particles or molecules in a fluid, induced by a concentration gradient of a different substance.

For example, concentration gradients in ethanol solutions in water move 1 μm diameter colloidal particles with diffusiophoretic velocities

Diffusioosmosis, also referred to as capillary osmosis, is flow of a solution relative to a fixed wall or pore surface, where the flow is driven by a concentration gradient in the solution.

This is distinct from flow relative to a surface driven by a gradient in the hydrostatic pressure in the fluid.

In diffusioosmosis the hydrostatic pressure is uniform and the flow is due to a concentration gradient.

The term diffusioosmosis is used when the surface is viewed as static, and the solution flows.

A well studied example of diffusiophoresis is the motion of colloidal particles in an aqueous solution of an electrolyte solution, where a gradient in the concentration of the electrolyte causes motion of the colloidal particles.

[6][7] Colloidal particles may be hundred of nanometres or larger in diameter, while the interfacial double layer region at the surface of the colloidal particle will be of order the Debye length wide, and this is typically only nanometres.

So here, the interfacial width is much smaller than the size of the particle, and then the gradient in the smaller species drives diffusiophoretic motion of the colloidal particles largely through motion in the interfacial double layer.

Simple diffusion of colloids is fast on length scales of a few micrometres, and so diffusiophoresis would not be useful, whereas on length scales larger than millimetres, diffusiophoresis may be slow as its speed decreases with decreasing size of the solute concentration gradient.

Thus, typically diffusiophoresis is employed on length scales approximately in the range a micrometre to a millimetre.

This was studied by Prieve[10] in the context of latex particles being pulled towards, and coating, a dissolving steel surface.

Diffusiophoresis is an analogous phenomenon to thermophoresis, where a species A moves in response to a temperature gradient.

Simply speaking, a gradient in any thermodynamic quantity, such as the concentration of any species, or temperature, will drive motion of all thermodynamic quantities, i.e., motion of all species present, and a temperature flux.

Multicomponent diffusion is diffusion in mixtures, and diffusiophoresis is the special case where we are interested in the movement of one species that is usually a colloidal particle, in a gradient of a much smaller species, such as dissolved salt such as sodium chloride in water.

Thus diffusiophoresis always occurs in a mixture, typically a three-component mixture of water, salt and a colloidal species, and we are interested in the cross-interaction between the salt and the colloidal particle.

It is the very large difference in size between the colloidal particle, which may be 1μm across, and the size of the ions or molecules, which are less than 1 nm across, that makes diffusiophoresis closely related to diffusioosomosis at a flat surface.

In both cases the forces that drive the motion are largely localised to the interfacial region, which is a few molecules across and so typically of order a nanometer across.

Both diffusioosmosis and the Marangoni effect are driven by gradients in the interfacial free energy, i.e., in both cases the induced velocities are zero if the interfacial free energy is uniform in space, and in both cases if there are gradients the velocities are directed along the direction of increasing interfacial free energy.

To understand these expressions better, we can consider a very simple model, where the surface simply excludes an ideal solute from an interface of width

So diffusiophoresis moves particles towards lower solute concentrations, in this case.

So the only non-zero component of the Stokes' equation is In diffusioosmosis, in the bulk of the fluid (i.e., outside the interface) the hydrostatic pressure is assumed to be uniform (as we expect any gradients to relax away by fluid flow) and so in bulk[13][12] for

This result is for a specific and very simple model, but it does illustrate general features of diffusioosmoisis: 1) the hydrostatic pressure is, by definition (flow induced by pressure gradients in the bulk is a common but separate physical phenomenon) uniform in the bulk, but there is a gradient in the pressure in the interface, 2) this pressure gradient in the interface causes the velocity to vary in the direction perpendicular to the surface, and this results in a slip velocity, i.e., for the bulk of the fluid to move relative to the surface, 3) away from the interface the velocity is constant, this type of flow is sometimes called plug flow.

This charged surface of the colloidal particle interacts with a gradient in salt concentration, and this gives rise to diffusiophoretic velocity

is the reduced difference between the diffusion constant of the positively charged ion,

Note that there are two contributions to diffusiophoresis of a charged particle in a salt gradient, which give rise to the two terms in the above equation for

The first is due to the fact that whenever there is a salt concentration gradient, then unless the diffusion constants of the positive and negative ions are exactly equal to each other, there is an electric field, i.e., the gradient acts a little like a capacitor.

The second part is due to the surface free energy of the surface of a charged particle, decreasing with increasing salt concentration, this is a similar mechanism to that found in diffusiophoresis in gradients of neutrial substances.

, and if the positively charged ions diffuse faster than the negatively charged ones, then this term will push particles down a salt gradient, but if it is the negatively charged ions that diffuse faster, then this term pushes the particles up the salt gradient.

A group from Princeton University[14] reported the application of diffusiophoresis to water purification.

Schematic illustrating diffusiophoretic motion of a colloidal particle (blue) in a concentration gradient of a solute (red). Note that there is also a concentration gradient of the solvent (green). The particle is moving a diffusiophoretic velocity , in a fluid that is stationary far from the particle. The fluid velocity decays from for fluid in contact with the surface of the particle, to near zero, within the interface at the particle's surface.
This schematic illustrates diffusioosmotic flow above a surface in contact with a solution that has a concentration gradient of a solute (red). The flow as a function of height above the surface is shown as black arrows of length proportional to the flow velocity at that height. The flow is left-to-right as this solute is repelled by the surface, and its concentration increases from left-to-right. Therefore, the surface free energy increases right-to-left, which drives flow from right-to-left.