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A depletion force is an entropic force between colloidal particles suspended in a solution of smaller nonadsorbing polymers, micelles, or hard spheres ]. The force arises when the separation between the colloidal particles is less than the diamater of the smaller depletant particles. Depletion forces are generally attractive, but can be repulsive at larger distances and higher depletant concentration[physica A].

Causes

Entropy and Hemholtz Energy

An entropic force arises when there is overlap of restricted volumes of the (large) spheres or plates in solution. The overlap increases the volume accessible to solute which increases the systems entropy. This process happens spontaneously at constant temperature since the Hemholtz energy decreases.

Sterics

When two parallel plates are in a suspension of small particles at a distance less than two times the radius of the small particles, the small particles cannot fit in between the plates dude to spacial restrictions. For spherical particles in a suspension of small particles, there is a region surrounding large spherical particles that small spherical particles in the suspension are excluded from. When spheres get close enough (i.e. h< (Dlarge+dsmall)/2), the regions overlap which reduces the volume unavailable to small spheres.

Osmotic Pressure

When small spheres are excluded from between two parallel plates, the difference in chemical potential between the region between the plates and outside the plates results in osmotic pressure that acts on the outer sides of the plates, resulting in aggregation.

Asakura-Oosawa model

Asakura-Oosawa Model Earliest model (1954) force is always attractive force is proportional to osmotic pressure p0=kTN/V Theory Distance Between Plates (general case)[1] *two plates in a solution of rigid spherical macromolecules

			*If distance between two plates, a, is smaller than the diameter of solute molecules, d, no solute can enter between the plates.

*Pure solvent between plates *Force=osmotic pressure acts on plates * if medium is a very dilute and monodisperse solution, p=kTN ∂lnQ/ ∂a, force p, N is the total number of solute molecules *force causes entropy of macromolecules to increase [2] * force attractive for a < d (or z<2r) [2] Other Cases Plates in a solution of rod like macromolecules *Macromolecules of length (l), where (l^2<<A) A= area of plates Plates in a solution of polymers * The configurational entropy of chain molecules is decreased in the neighborhood of the plates * approximate as diffusion in a vessel with walls which absorb diffusing particles

                               * P = -Apo{(1-f)- a∂f/∂a}

1-f = attraction from osmotic effect a∂f/∂ repulsion due to chain molecules confined between plates P is on order of <r>, the mean end-to-end distance of chain molecules in free space Large hard spheres in a solution of small hard spheres * Large Spheres D = 2RB * Small spheres D= 2 Rs * if distances between center of spheres, h < (D + d)/2 . Then the small spheres are excluded from the space between the large spheres [3]

Derjaguin Approximation

Theory *The Derjaguin approximation is valid for any type of a force law [3]| *for high concentrations, structural correlation effects in the macromolecular liquid become important [2] *the repulsive interaction strength strongly increases for large values of R/r. (large radius/small radius) [2] *Force between two spheres can be related to the force between two plates *Force is integrated between small regions on one surface and the opposite surface, which is assumed to be locally flat [3] Equations * For two sphere radii R1, R2 on the z axis h+R1+R2 distance apart, force F in Z direction if h<<R1,R2 is: [3] F(h) ≈ 2π (R1R2/R1+R2) W(h) where W(h) =∫_h^∞▒f(z)dz and f(z) is the normal force per unit area between two flat surfaces distance z apart. [3] *Applied to Depletion Forces, and taking 0<h<2Rs , the depletion force given by the Derjaguin approximation is [3] F(h)= - π ε(RB+RS)[p(ρ)(2Rs-h)+ γ (ρ,∞)] ε is the geometrical factor, which is set to 1 γ (ρ,∞) = 2 γ (ρ), the interfacial tension at the wall-fluid interface

Density functional theory

• When any fluid is exposed to external potential V(R) all equilibrium quantities become functions of number density profile ρ(R), minimizing free energy. [4]

Ω([ρ(R)]; µ,T)=F([ρ(R)]; T)-∫▒〖d^3 R[µ-V(R)]ρ(R)〗 µ= chemical potential T=Temperature F[ρ] = hemholtz free energy *For force in terms of the equilibrium βFz(h)=∫▒〖d^3 R∂(|R|-(Rb+Rs))(-z/R)xρ(R+(2Rb+h) e_z)〗 β=(kbT)-1

Measurement and Experimentation

Atomic Force Microscopy

o Atomic Force Microscopy  Attractive/repulsive depletion forces deflects cantilever  Known deflection of cantilever associated with a force and used to find depletion force. o Video Microscopy  Relate position and movement of particles to attractive/repulsive depletion forces. o Optical Tweezer Experiments  Attractive or repulsive force is applied to particles using a focused laser

Applications

Colloidal Stability

o Colloidal Stability  Depletion attraction • Depletion attraction between particles in a solvent can result in flocculation. • Destabilizes colloids Flocculation results in destabilization of the colloid • Attraction is observed for short distances between particles.  Depletion repulsion • Data suggest depletion forces are not always attractive but can also repel particles (Mao). • Repulsion can be found with greater distances between particles • Depletion repulsion can be used as a method of stabilizing colloids and preventing flocculation.  Oscillatory depletion interactions • Depletion force oscillates between attraction and repulsion as a function of particle distance.  Packing fraction effects • Increased packing fraction of particles in the solvent amplifies depletion potential.

Biological Systems

 Depletion forces between membranes • Depletion forces exist between adjacent membranes in a solvent of large molecules • May influence vesicle transport and adsorption. • Measurement of these forces is difficult due to complexity of biological systems and presence of other forces. • Curvature effects of membrane on depletion force.

References

1. Mao, Y., M. Cates, and H. Lekkerkerker. "Depletion Force in Colloidal Systems." Physica A: Statistical and Theoretical Physics 222.1-4 (1995): 10-24. Print.