# Disjoining pressure and the energy of thin films

## Contents

## The energy of a thin film

Here we consider a thin, liquid film (gray) between a solid (black), and a vapor (white). The thickness of the film is \epsilon, and the \sigma coefficients give the energy density of the surfaces. However, the total energy density also depends on some function P of the thickness of the film:

This function P comes from the fact that we are considering a liquid with some sort of structure. For example, we could have molecular ordering at a surface, or a liquid with large constituents (i.e. a polymer) whose configurations become important as the film thickness approaches that size scale.

We can better understand P by considering some limits. If the film thickness is taken to zero, we get the energy of the bare surface \sigma_s0, minus the energy densities of the solid/liquid and liquid/vapor interfaces. This is equal to the spreading coefficient S; we will see more about S in the future, but S measures the tendency of a liquid placed on the surface to spread.

In the other limit where the film is extremely thick, the thickness of the film should not matter, and the only energy will come from the interfaces. In other words, inside a large bulk of fluid, there can be no energy gained from the surface interactions, so adding more of this bulk shouldn't change anything.

There are of course intermediate ranges, for example:

- Between surfaces: P(e) ~ 1/e^2. Recall that the energy between two surfaces goes as the inverse of their separation squared, so this is just a surface-surface interaction energy.
- For polymers: P(e) should vary on the length scale of the polymer; this makes sense because as the film becomes thin, the configuration of the individual polymers becomes important and we have to actually squish them to make the film thinner
- Electrostatics: The electromagnetic interaction is important for molecules on short scales, typically 10's of nm in water, 100's of nm in oil

## Disjoining pressure - liquid surfaces

A disjoining pressure (<math>\Pi</math>) describes the supplementary pressure that arises when the hydrostatic pressure of a thin layer differs from the pressure of a bulk phase. This pressure may have either positive or negative values.

A decreasing P(e) gives rise to thicker films that are more stable. A positive disjoining pressure <math>\Pi</math> corresponds to spreading of the liquid. One physical example of a disjoining pressure helps make this clearer is the case of a surface sandwiched between two parallel plates. Suppose one had stacked two parallel plates separated by a distance h apart filled with a thin material. An equalibrium between the plates will be reached if the material between the plates satisfies the following properties: <math>\partial</math><math>\Pi</math>(h)/<math>\partial</math>h < 0 ie. the disjoining pressure increases as the seperation of the plates h goes to zero. In the reverse case, the interlayer will be unstable. Measurement of the disjoining pressure is only feasible if <math>\partial</math><math>\Pi</math>/<math>\partial</math>h < <math>\partial</math>N/<math>\partial</math>h. In words, one can only make a reasonable measurement of the disjoining pressure if small changes in the disjoining pressure result in large changes in an external pressure. One method for measuring disjoining pressures is featured in figure 4.1.

In the Sheludko method, the film being studied is placed inside a porous ring filled with the same bulk material with density <math>\rho</math>. When hydrostatic pressure is applied to this ring <math>p_h</math> = <math>\rho </math> gH, the chemical potentials of the film and bulk liquid must be equal. This results in a correlation between the disjoining pressure and external pressure given below.

Curves of the disjoining pressure as a function of film thickness may be constructed by measuring the film's thickness for a range of applied pressures. Reference: Derjaguin (ch2). De Gennes (pg. 88-89)

## Disjoining pressure - across films

The usual method used to analyze the thermodynamics of interfaces is by excess properties (an idea developed by Gibbs precisely) (See a). However when two interfaces interpentrate the idea of an excess is imprecise (See b) and the thermodynamics better done with disjoining pressures. In figure a, Gibb's theory assumes that middle layer between the two shaded bounded regions has the intensive properties of the bulk phase. Once the boundary layers overlap (figure b), a surface no longer follows the laws of hydrostatics.

Similar considerations arise when the interfacial energy is calculated by Bakker's equation:

The graph below shows the distribution of the pressure tensor components <math>P_n</math> (normal) and <math>P_t</math> (tangential) in thin liquid interlayers. It is bounded by two phases that are exactly the same, either liquid or gaseous. When in equilibrium we have <math>P_t=P_n=P</math> only **outside** the boundary <math>h_0</math>, where P is some pressure of the phase contiguous to the interlayer. The normal component does not change at the boundary, whereas the tangential component changes in a way that depends on the nature of the contiguous phases and outside forces. The conclusion we can draw from the fact that <math>P_n</math> is constant is that resultant of the volume forces acting in the interlayer is *zero*!

**Where do these interpenetrations occur? At any interface or are there certain requirements of the materials involved?**

## Profile of a free film

The force of gravity acting on an elementary segment of the film between the levels, H and dH is balanced by a difference dσ between film tensions at the two levels due to a different thickness at the two levels. The tension is higher in the thinner regions. All extensions of the film perturb the equilibrium and modify the values of adsorbtion at liquid/gas interface. The effect of "film elasticity" was first recognized by Gibbs.

## Disjoining Pressure for Non-uniform Thin Films

It is known that the concept of the disjoining pressure between two parallel interfaces due to van der Waals interactions cannot generally be extended to films of non-uniform thickness. Below is a formula for the disjoining pressure for a film of non-uniform thickness by minimizing the total Helmholtz free energy for a thin film residing on a solid substrate. In the derivation, it is chosen to utilize a definition of the disjoining pressure based on Yeh’s thermal equilibrium condition for a liquid thin film residing on a flat solid substrate. Assuming a van der Waals hard sphere interaction, scientists integrate the intermolecular potential throughout the whole heterogeneous system and obtain the excess energy associated with the disjoining pressure by subtracting the interfacial tension potential from the total intermolecular potential. In the limit of a small slope, the disjoining pressure given by this approach takes the relatively simple form:

- <math>Pressure=-A_{123}(4-3h_{x}^{2}+3hh_{xx})/24 \pi h^{3}</math>

where <math>A_{123}</math> is the Hamaker constant for a phase 1 and 2 interacting through a phase 3; h, <math>h_{x}</math>, and <math>h_{xx}</math> are the local film thickness, slope, and second order derivative, respectively. A key point is that the form for the Hamaker constant depends on the properties of all three phases, and is completely consistent with Lifshitz theory for non-retarded van der Waals forces.