5 Special Fluid Medium Lubrication
5.4 Electric Double Layer Effect in Water Lubrication
5.4.1 Electric Double Layer Hydrodynamic Lubrication Theory
The electric double layer occurs in the contact interface of an ion fluid and a solid. Figure 5.21 shows the diagram of the electric double layer structure. The electric double layer is composed of the Stern layer and the diffusion layer. The Stern layer is an absorbing layer of water on a solid surface, and it has a potential𝛹. The macro-movement of fluid is believed to occur in the intersecting plane, with a few molecules of the Stern layer and the diffusion layer. The plane is called the slip plane. The electric potential on the slip plane is called the electric poten- tial, 𝜁, which can be determined theoretically according to the characteristics of water and ceramic.
5.4.1.2 Hydrodynamic Lubrication Theory of Electric Double Layer
Figure 5.22 shows the hydrodynamic lubrication model of the electric double layer, based on an electric double layer existing on the two solid surfaces. The lower surfacez=0 has a velocityU
Figure 5.21 Schematic diagram of electric double layer.
Figure 5.22 Electric double layer hydrodynamic lubrication model.
k k along the x direction, but the upper surface is static. The potential distribution of the electric
double layer along the z direction is determined by the following formula 𝜓=
{𝜁exp(−𝜒z) 0<z<h∕2
𝜁exp(−𝜒(h−z)) h∕2<z<h, (5.34)
wherehis the lubricant film thickness;𝜒−1is the Debye double layer thickness;𝜁is the electric potential of the slip plane.
In classical lubrication theory, only the viscous force and pressure are considered. However, due to the electric double layer, we must consider the electric field. The viscous forces on a micro-element along the x and y directions are
dFx= 𝜕𝜏zx
𝜕z dxdydz dFy = 𝜕𝜏zy
𝜕z dxdydz. (5.35)
According to Newtonian viscosity law, the shearing stresses in the x and y directions are equal to
𝜏zx=𝜂𝜕ux
𝜕z 𝜏zy=𝜂𝜕uy
𝜕z , (5.36)
whereuxanduyare the velocities of the fluid along the x and y directions;𝜂is the fluid viscosity.
Substituting the above equations into Equation 5.35, we have dFx=𝜂𝜕2ux
𝜕z2 dxdydz dFy =𝜂𝜕2uy
𝜕z2 dxdydz. (5.37)
The pressure differencesdPxanddPycaused bydxanddyalong the x and y directions are dPx= −𝜕p
𝜕xdxdydz dPy= −𝜕p
𝜕ydxdydz. (5.38)
Along the x and y directions, the electric forcesdRxanddRyare dRx=Ex𝜌dxdydz
dRy =Ey𝜌dxdydz, (5.39)
whereExandEyare the fluid electric potentials inside the electric double layer along the x and y directions respectively, which are generated by the flow;𝜌is the density.
k k The relationships between the flow potential Ex and Ey, and pressure pare given by the
Helmholtz–Smoluchowski formula:
Ex= − 𝜁𝜀 4𝜋𝜂a𝜆
𝜕p
𝜕x Ey= − 𝜁𝜀
4𝜋𝜂a𝜆
𝜕p
𝜕y, (5.40)
where𝜂ais the apparent viscosity of the fluid;𝜆is the conductivity;𝜀is the dielectric constant.
Equation 5.40 is derived under the assumption of capillarity. Because the capillary diameter is much larger than the thickness of electric double layer, the macroscopic viscosity𝜂in the original formula uses the micro-apparent𝜂ato take account of the electric double layer effect.
The force balance conditions for a micro-element are dFx+dPx+dRx=0
dFy+dPy+dRy =0. (5.41)
Substituting Equations 5.37–5.39 into Equation 5.41, we have 𝜂𝜕2ux
𝜕z2 −𝜕p
𝜕x +Ex𝜌=0 𝜂𝜕2uy
𝜕z2 −𝜕p
𝜕y +Ey𝜌=0. (5.42)
In the electric double layer, the electric potential𝛹is given by
∇2𝜓= −4𝜋𝜌
𝜀 . (5.43)
Substitute Equation 5.43 into 5.42 and consider that the sizes of electric double layer in the x and y directions are much larger than that in the z direction, that is,
𝜕2∕𝜕x2≪ 𝜕2∕𝜕z2 and𝜕2∕𝜕y2≪ 𝜕2∕𝜕z2; after simplification we obtain 𝜂𝜕2ux
𝜕z2 −𝜕p
𝜕x −Ex𝜀𝜕2𝜓 4𝜋𝜕z2 =0 𝜂𝜕2uy
𝜕z2 −𝜕p
𝜕y −Ey𝜀𝜕2𝜓
4𝜋𝜕z2 =0. (5.44)
Suppose the pressure across the thickness is constant, integrating the first formula of Equation 5.44 toztwice, we have
𝜂ux−Ex𝜀 4𝜋𝜓 = 1
2
𝜕p
𝜕xz2+Az+B, (5.45)
whereAandBare the integral constants to be determined by the boundary conditions:
{
ux|z=0= −U, 𝜓|z=0=𝜁
ux|z=h=0, 𝜓|z=h=𝜁. (5.46)
k k Substituting the boundary conditions into Equation 5.33 gives
A= −h 2
𝜕p
𝜕x +U𝜂 h B= −Ex𝜀
4𝜋𝜁−U𝜂. (5.47)
Substituting the above expressions forAandBinto Equation 5.45, it becomes 𝜂ux= z2
2
𝜕p
𝜕x− hz 2
𝜕p
𝜕x +Ex𝜀
4𝜋(𝜓−𝜁) −𝜂( 1− z
h )
U. (5.48)
With the same method to the y direction, we have 𝜂uy= z2
2
𝜕p
𝜕y−hz 2
𝜕p
𝜕y+ Ey𝜀
4𝜋(𝜓−𝜁). (5.49)
In the derivation of Equation 5.49, the following boundary conditions are used:
{ uy=0
𝜓=𝜁 forz=0 andz=h. (5.50)
The flow along the x direction is equal to Qx=∫
h 0
uxdz. (5.51)
Then, substituting Equation 5.48 into the above equation, we have Qx= 1
𝜂 {
−h3 12
𝜕p
𝜕x −Ex𝜀𝜁 4𝜋
[ h− 2
𝜒(1−e−𝜒h∕2) ]}
− hU
2 . (5.52)
We can also get the flow along the y direction as Qy= 1
𝜂 {
−h3 12
𝜕p
𝜕y− Ey𝜀𝜁 4𝜋
[ h− 2
𝜒(1−e−𝜒h∕2) ]}
. (5.53)
Under the incompressible assumption, the integral continuity equation becomes
𝜕Qx
𝜕x + 𝜕Qy
𝜕y =0. (5.54)
Substituting Equations 5.52 and 5.53 into Equation 5.54, we have
𝜕
𝜕x ( h3
12𝜂
𝜕p
𝜕x )
+ 𝜕
𝜕y ( h3
12𝜂
𝜕p
𝜕y )
= U 2
𝜕h
𝜕x − 𝜕
𝜕x {Ex𝜀𝜁
4𝜋𝜂 [
h− 2
𝜒(1−e−𝜒h∕2) ]}
− 𝜕
𝜕y {Ey𝜀𝜁
4𝜋𝜂 [
h− 2
𝜒(1−e−𝜒h∕2) ]}
. (5.55)
k k SubstitutingExandEyin Equation 5.40 into Equation 5.55, and modifying it into the form of
Reynolds equation gives
𝜕
𝜕x ( h3
12𝜂a
𝜕p
𝜕x )
+ 𝜕
𝜕y ( h3
12𝜂a
𝜕p
𝜕y )
= U 2
𝜕h
𝜕x. (5.56)
In Equation 5.56,𝜂ais the apparent viscosity:
𝜂a=𝜂+ 3𝜀2𝜁2
{ h− 2
𝜒(1−e−𝜒h∕2) }
4𝜋2𝜆h3 . (5.57)
Equation 5.56 is called the Reynolds equation considering the electric double layer effect.
The apparent viscosity expression Equation 5.57 shows that due to the existence of the electric double layer, the increase of𝜂ais proportional to the square of𝜁, and is inversely proportional to the cube ofh. In other words, the film thickness will have a significant influence on the electric double layer hydrodynamic lubrication. In addition,𝜁increases the load-carrying capacity of the lubrication film. This conclusion may help us choose surface materials and lubricants of thin film lubrication.