A depletion force is an effective attractive force that arises between large colloidal particles that are suspended in a dilute solution of depletants, which are smaller solutes that are preferentially excluded from the vicinity of the large particles. One of the earliest reports of depletion forces that lead to particle coagulation is that of Bondy, who observed the separation or "creaming" of rubber latex upon addition of polymer depletant molecules (sodium alginate) to solution. More generally, depletants can include polymers, micelles, osmolytes, ink, mud, or paint dispersed in a continuous phase.
Depletion forces are often regarded as entropic forces, as was first explained by the established Asakura–Oosawa model. In this theory the depletion force arises from an increase in osmotic pressure of the surrounding solution when colloidal particles get close enough such that the excluded cosolutes (depletants) cannot fit in between them. Because the particles were considered as hard-core (completely rigid) particles, the emerging picture of the underlying mechanism inducing the force was necessarily entropic.
Causes
Sterics The system of colloids and depletants in solution is typically modeled by treating the large colloids and small depletants as dissimilarly sized hard spheres. Hard spheres are characterized as non-interacting and impenetrable spheres. These two fundamental properties of hard spheres are described mathematically by the hard-sphere potential. The hard-sphere potential imposes steric constraint around large spheres which in turn gives rise to excluded volume, that is, volume that is unavailable for small spheres to occupy.
Hard-sphere potential In a colloidal dispersion, the colloid-colloid interaction potential is approximated as the interaction potential between two hard spheres. For two hard spheres of diameter of , the interaction potential as a function of interparticle separation is:
called the hard-sphere potential where is the center-to-center distance between the spheres.
If both colloids and depletants are in a dispersion, there is interaction potential between colloidal particles and depletant particles that is described similarly by the hard-sphere potential. Again, approximating the particles to be hard-spheres, the interaction potential between colloids of diameter and depletant sols of diameter is:
where is the center-to-center distance between the spheres. Typically, depletant particles are very small compared to the colloids so
The underlying consequence of the hard-sphere potential is that dispersed colloids cannot penetrate each other and have no mutual attraction or repulsion.
Excluded volume
When both large colloidal particles and small depletants are in a suspension, there is a region which surrounds every large colloidal particle that is unavailable for the centers of the depletants to occupy. This steric restriction is due to the colloid-depletant hard-sphere potential. The total volume of the excluded region for two spheres is
where is the diameter of the large spheres and is the diameter of the small spheres.
When the large spheres get close enough, the excluded volumes surrounding the spheres intersect. The overlapping volumes result in a reduced excluded volume, that is, an increase in the total free volume available to small spheres. The reduced excluded volume, can be written
where is half the width of the lens-shaped region of overlap volume formed by spherical caps. The volume available for small spheres is the difference between the total volume of the system and the excluded volume. To determine the available volume for small spheres, there are two distinguishable cases: first, the separation of the large spheres is big enough so small spheres can penetrate in between them; second, the large spheres are close enough so that small spheres cannot penetrate between them. For each case, the available volume for small spheres is given by
In the latter case small spheres are depleted from the interparticle region between large spheres and a depletion force ensues.
Thermodynamics The depletion force is described as an entropic force because it is fundamentally a manifestation of the second law of thermodynamics, which states that a system tends to increase its entropy. The gain in translational entropy of the depletants, owing to the increased available volume, is much greater than the loss of entropy from flocculation of the colloids. The positive change in entropy lowers the Helmholtz free energy and causes colloidal flocculation to happen spontaneously. The system of colloids and depletants in a solution is modeled as a canonical ensemble of hard spheres for statistical determinations of thermodynamic quantities.
However, recent experiments and theoretical models found that depletion forces can be enthalpically driven. In these instances, the intricate balance of interactions between the solution components results in the net exclusion of cosolute from macromolecule. This exclusion results in an effective stabilization of the macromolecule self-association, which can be not only enthalpically dominated, but also entropically unfavorable.
Entropy and Helmholtz energy The total volume available for small spheres increases when the excluded volumes around large spheres overlap. The increased volume allotted for small spheres allows them greater translational freedom which increases their entropy. Because the canonical ensemble is an athermal system at a constant volume the Helmholtz free energy is written
where is the Helmholtz free energy, is the entropy and is the temperature. The system's net gain in entropy is positive from increased volume, thus the Helmholtz free energy is negative and depletion flocculation happens spontaneously.
The free energy of the system is obtained from a statistical definition of Helmholtz free energy
where is the partition function for the canonical ensemble. The partition function contains statistical information that describes the canonical ensemble including its total volume, the total number of small spheres, the volume available for small spheres to occupy, and the de Broglie wavelength. If hard-spheres are assumed, the partition function is
The volume available for small spheres, was calculated above. is the number of small spheres and is the de Broglie wavelength. Substituting into the statistical definition, the Helmholtz free energy now reads
The magnitude of the depletion force, is equal to the change in Helmholtz free energy with distance between two large spheres and is given by
The entropic nature of depletion forces was proven experimentally in some cases. For example, some polymeric crowders induce entropic depletion forces that stabilize proteins in their native state. Other examples include many systems with hard-core only interactions.
Osmotic pressure The depletion force is an effect of increased osmotic pressure in the surrounding solution. When colloids get sufficiently close, that is when their excluded volumes overlap, depletants are expelled from the interparticle region. This region between colloids then becomes a phase of pure solvent. When this occurs, there is a higher depletant concentration in the surrounding solution than in the interparticle region. The resulting density gradient gives rise to an osmotic pressure that is anisotropic in nature, acting on the outer sides of the colloids and promoting flocculation. If the hard-sphere approximation is employed, the osmotic pressure is:
where is osmotic pressure and is number density of small spheres and is the Boltzmann constant.
Asakura–Oosawa model Depletion forces were first described by Sho Asakura and Fumio Oosawa in 1954. In their model, the force is always considered to be attractive. Additionally, the force is considered to be proportional to the osmotic pressure. The Asakura–Oosawa model assumes low macromolecule densities and that the density distribution, , of the macromolecules is constant. Asakura and Oosawa described four cases in which depletion forces would occur. They first described the most general case as two solid plates in a solution of macromolecules. The principles for the first case were then extended to three additional cases.
Free energy change due to the depletion force In the Asakura–Oosawa model for depletion forces, the change in free-energy imposed by an excluded cosolute, , is:
where is the osmotic pressure, and is the change in excluded volume (which is related to molecular size and shape). The very same result can be derived using the Kirkwood-Buff solution theory.
Solid plates in a solution of macromolecules In the first case, two solid plates are placed in a solution of rigid spherical macromolecules. If the distance between two plates, , is smaller than the diameter of solute molecules, , then no solute can enter between the plates. This results in pure solvent existing between the plates. The difference in concentration of macromolecules in the solution between the plates and the bulk solution causes a force equal to the osmotic pressure to act on the plates. In a very dilute and monodisperse solution the force is defined by
where is the force, and is the total number of solute molecules. The force causes the entropy of the macromolecules to increase and is attractive when
Rod-like macromolecules Asakura and Oosawa described the second case as consisting of two plates in a solution of rod like macromolecules. The rod like macromolecules are described as having a length, , where , the area of the plates. As the length of the rods increases, the concentration of the rods between the plates is decreased as it becomes more difficult for the rods to enter between the plates due to steric hindrances. As a result, the force acting on the plates increases with the length of the rods until it becomes equal to the osmotic pressure. In this context, it is worth mentioning that even the isotropic-nematic transition of lyotropic liquid crystals, as first explained in Onsager's theory, can in itself be considered a special case of depletion forces.
Plates in a solution of polymers The third case described by Asakura and Oosawa is two plates in a solution of polymers. Due to the size of the polymers, the concentration of polymers in the neighborhood of the plates is reduced, which result the conformational entropy of the polymers being decreased. The case can be approximated by modeling it as diffusion in a vessel with walls which absorb diffusing particles. The force, , can then be calculated according to:
In this equation is the attraction from the osmotic effect. is the repulsion due to chain molecules confined between plates. is on order of , the mean end-to-end distance of chain molecules in free space.
Large hard spheres in a solution of small hard spheres The final case described by Asakura and Oosawa describes two large, hard spheres of diameter , in a solution of small, hard spheres of diameter . If the distance between the center of the spheres, , is less than , then the small spheres are excluded from the space between the large spheres. This results in the area between the large spheres having a reduced concentration of small spheres and therefore reduced entropy. This reduced entropy causes a force to act upon the large spheres pushing them together. This effect was convincingly demonstrated in experiments with vibrofluidized granular materials where attraction can be directly visualized.
Improvements upon Asakura–Oosawa model
Derjaguin approximation
Theory