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 ${\displaystyle \sigma }$, the interaction potential as a function of interparticle separation is:

${\displaystyle V(h)=\left\{{\begin{matrix}0&{\mbox{if}}\quad h\geq \sigma \\\infty &{\mbox{if}}\quad h<\sigma \end{matrix}}\right.}$

called the hard-sphere potential where ${\displaystyle h}$ is the center-to-center distance between the spheres.^[8]

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.^[7] Again, approximating the particles to be hard-spheres, the interaction potential between colloids of diameter ${\displaystyle D}$ and depletant sols of diameter ${\displaystyle d}$ is:

${\displaystyle V(h)=\left\{{\begin{matrix}0&{\mbox{if}}\quad h\geq {\Big (}{\frac {D+d}{2}}{\Big )}\\\infty &{\mbox{if}}\quad h<{\Big (}{\frac {D+d}{2}}{\Big )}\end{matrix}}\right.}$

where ${\displaystyle h}$ is the center-to-center distance between the spheres. Typically, depletant particles are very small compared to the colloids so ${\displaystyle d\ll D}$

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

Excluded volumes of hard spheres overlap resulting in an increase in the total volume available to depletants. This increases the entropy of the system and lowers the Helmholtz free energy

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.^[7][8] The volume of the excluded region is

${\displaystyle V_{\mathrm {E} }={\frac {\pi {\big (}D+d{\big )}^{3}}{3}}}$

where ${\displaystyle D}$ is the diameter of the large spheres and ${\displaystyle d}$ 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.^[1][4] The reduced excluded volume, ${\displaystyle V'_{\mathrm {E} }}$ can be written

${\displaystyle V'_{\mathrm {E} }=V_{\mathrm {E} }-{\frac {2\pi l^{2}}{3}}{\bigg [}{\frac {3\left(D+d\right)}{2}}-l{\bigg ]}}$

where ${\displaystyle l=(D+d)/2-h/2}$ is half the width of the lens-shaped region of overlap volume formed by spherical caps. The volume available ${\displaystyle V_{\mathrm {A} }}$ 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.^[7] For each case, the available volume for small spheres is given by

${\displaystyle V_{\mathrm {A} }=\left\{{\begin{matrix}V-V_{\mathrm {E} }&{\mbox{if}}\quad h\geq D+d\\V-V'_{\mathrm {E} }&{\mbox{if}}\quad h<D+d\end{matrix}}\right.}$

In the latter case small spheres are depleted from the interparticle region between large spheres and a depletion force ensues.

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.^[1] Because the canonical ensemble is an athermal system at a constant volume the Helmholtz free energy is written

${\displaystyle A=-TS}$

where ${\displaystyle A}$ is the Helmholtz free energy, ${\displaystyle S}$ is the entropy and ${\displaystyle T}$ 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

${\displaystyle A=-k_{\mathrm {B} }T\ln Q}$

where ${\displaystyle Q}$ 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.^[7] If hard-spheres are assumed, the partition function ${\displaystyle Q}$ is

${\displaystyle Q={\frac {V_{\mathrm {A} }^{N}}{N!\Lambda ^{3N}}}}$

The volume available for small spheres,${\displaystyle V_{\mathrm {A} }}$ was calculated above.${\displaystyle N}$ is the number of small spheres and ${\displaystyle \Lambda }$ is the de Broglie wavelength. Substituting ${\displaystyle Q}$ into the statistical definition, the Helmholtz free energy now reads

${\displaystyle A=-k_{\mathrm {B} }T\ln {\bigg (}{\frac {V_{\mathrm {A} }^{N}}{N!\Lambda ^{3N}}}{\bigg )}}$

The magnitude of the depletion force, ${\displaystyle {\mathcal {F}}}$ is equal to the change in Helmholtz free energy with distance between two large spheres and is given by^[7]

${\displaystyle {\mathcal {F}}=-{\bigg (}{\frac {\partial A}{\partial h}}{\bigg )}_{T}}$

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.^[14][15][16] Other examples include many systems with hard-core only interactions.^[17]

Theory

Asakura and Oosawa assumed low concentrations of macromolecules. However, at high concentrations of macromolecules, structural correlation effects in the macromolecular liquid become important. Additionally, the repulsive interaction strength strongly increases for large values of ${\displaystyle R/r}$ (large radius/small radius).^[18] In order to account for these issues, the Derjaguin approximation, which is valid for any type of force law, has been applied to depletion forces. The Derjaguin approximation relates the force between two spheres to the force between two plates. The force is then integrated between small regions on one surface and the opposite surface, which is assumed to be locally flat.^[7]

The Derjaguin Approximation relates the force between two spheres to the force between two plates.

Equations

If there are two spheres of radii ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ on the ${\displaystyle Z}$ axis, and the spheres are ${\displaystyle h+R_{1}+R_{2}}$ distance apart, where ${\displaystyle h}$ is much smaller than ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$, then the force, ${\displaystyle F}$, in the ${\displaystyle z}$ direction is

${\displaystyle F(h)\approx 2\pi \left({\frac {R_{1}R_{2}}{R_{1}+R_{2}}}\right)W(h)}$

In this equation, ${\displaystyle W(h)=\textstyle \int _{h}^{\infty }f(z)dz}$, and ${\displaystyle f(z)}$ is the normal force per unit area between two flat surfaces distance ${\displaystyle z}$ apart.

When the Derjaguin approximation is applied to depletion forces, and 0<h<2Rs, then the depletion force given by the Derjaguin approximation is

${\displaystyle F(h)=-\pi \epsilon \left(R_{B}+R_{S}\right){\big [}p(\rho )(2R_{S}-h)+\gamma (\rho ,\infty ){\big ]}}$

In this equation, ${\displaystyle \epsilon }$ is the geometrical factor, which is set to 1, and ${\displaystyle \gamma (\rho ,\infty )=2\gamma (\rho )}$, the interfacial tension at the wall-fluid interface.^[7]

Theory

Asakura and Oosawa assumed a uniform particle density, which is true in a homogeneous solution. However, if an external potential is applied to a solution, then the uniform particle density is disrupted, making Asakura and Oosawa's assumption invalid. Density functional theory accounts for variations in particle density by using the grand canonical potential. The grand canonical potential, which is a state function for the grand canonical ensemble, is used to calculate the probability density for microscopic states in macroscopic state. When applied to depletion forces, the grand canonical potential calculates the local particle densities in a solution.^[7]

Equations

Density functional theory states that when any fluid is exposed to an external potential, ${\displaystyle V(R)}$, then all equilibrium quantities become functions of number density profile, ${\displaystyle \rho (R)}$. As a result, the total free energy is minimized. The Grand canonical potential, ${\displaystyle \Omega \left({\big [}\rho (R){\big ]};\mu ,T\right)}$, is then written

${\displaystyle \Omega \left({\big [}\rho (R){\big ]};\mu ,T\right)=A\left({\big [}\rho (R){\big ]};T\right)-\int d^{3}R{\big [}\mu -V(R){\big ]}\rho (R),}$

where ${\displaystyle \mu }$ is the chemical potential, ${\displaystyle T}$ is the temperature, and ${\displaystyle A[\rho ]}$ is the helmholtz free energy.^[23]