Washburn's equation

Powder wettability measurement with the Washburn method.

In its most general form the Lucas Washburn equation describes the penetration length (${\displaystyle L}$) of a liquid into a capillary pore or tube with time ${\displaystyle t}$ as ${\displaystyle L=(Dt)^{\frac {1}{2}}}$, where ${\displaystyle D}$ is a simplified diffusion coefficient.[4] This relationship, which holds true for a variety of situations, captures the essence of Lucas and Washburn's equation and shows that capillary penetration and fluid transport through porous structures exhibit diffusive behaviour akin to that which occurs in numerous physical and chemical systems. The diffusion coefficient ${\displaystyle D}$ is governed by the geometry of the capillary as well as the properties of the penetrating fluid. A liquid having a dynamic viscosity ${\displaystyle \eta }$ and surface tension ${\displaystyle \gamma }$ will penetrate a distance ${\displaystyle L}$ into the capillary whose pore radius is ${\displaystyle r}$ following the relationship:

${\displaystyle L={\sqrt {\frac {\gamma rt\cos(\phi )}{2\eta }}}}$

Where ${\displaystyle \phi }$ is the contact angle between the penetrating liquid and the solid (tube wall).

Washburn's equation is also used commonly to determine the contact angle of a liquid to a powder using a force tensiometer.[5]

In the case of porous materials, many issues have been raised both about the physical meaning of the calculated pore radius ${\displaystyle r}$[6] and the real possibility to use this equation for the calculation of the contact angle of the solid.[7] The equation is derived for capillary flow in a cylindrical tube in the absence of a gravitational field, but is sufficiently accurate in many cases when the capillary force is still significantly greater than the gravitational force.

In his paper from 1921 Washburn applies Poiseuille's Law for fluid motion in a circular tube. Inserting the expression for the differential volume in terms of the length ${\displaystyle l}$ of fluid in the tube ${\displaystyle dV=\pi r^{2}dl}$, one obtains

${\displaystyle {\frac {\delta l}{\delta t}}={\frac {\sum P}{8r^{2}\eta l}}(r^{4}+4\epsilon r^{3})}$

where ${\displaystyle \sum P}$ is the sum over the participating pressures, such as the atmospheric pressure ${\displaystyle P_{A}}$, the hydrostatic pressure ${\displaystyle P_{h}}$ and the equivalent pressure due to capillary forces ${\displaystyle P_{c}}$. ${\displaystyle \eta }$ is the viscosity of the liquid, and ${\displaystyle \epsilon }$ is the coefficient of slip, which is assumed to be 0 for wetting materials. ${\displaystyle r}$ is the radius of the capillary. The pressures in turn can be written as

${\displaystyle P_{h}=hg\rho -lg\rho \sin \psi }$

${\displaystyle P_{c}={\frac {2\gamma }{r}}\cos \phi }$

where ${\displaystyle \rho }$ is the density of the liquid and ${\displaystyle \gamma }$ its surface tension. ${\displaystyle \psi }$ is the angle of the tube with respect to the horizontal axis. ${\displaystyle \phi }$ is the contact angle of the liquid on the capillary material. Substituting these expressions leads to the first-order differential equation for the distance the fluid penetrates into the tube ${\displaystyle l}$:

${\displaystyle {\frac {\delta l}{\delta t}}={\frac {[P_{A}+g\rho (h-l\sin \psi )+{\frac {2\gamma }{r}}\cos \phi ](r^{4}+4\epsilon r^{3})}{8r^{2}\eta l}}}$

• Bosanquet equation
• Mercury intrusion porosimetry (MIP)

• Powder wettability measurement with the Washburn method