# Spreading of Nonvolatile Liquids in a Continuum Picture

## Paper Details

Title: Spreading of Nonvolatile Liquids in a Continuum Picture

Authors: F. Brochard-Wyart, J.M. di Meglio, D. Quere, P.G. de Gennes

Journal: Langmuir, 7 (1991), pgs. 335-338

## Introduction and Fundamental Parameters

This paper focuses on the wetting criteria for solid-liquid interfaces where the long-range interaction is not oscillating and is described by a Hamaker constant (A). The two key parameters are the Hamaker constant and the spreading coefficient (S) which contains contributions from short range interactions. Three fundamental regimes are considered:

1) Complete wetting - a small droplet spreads to become a flat "pancake" surrounded by a dry solid

2) Pseudo-partial wetting - a droplet forms a spherical cap with a finite contact angle but - the surrounding solid is "wet" - i.e. the drop is in equilibrium with a molecular film

3) Partial wetting - the contact angle is nonzero and the solid around the drop is "dry"

Spreading of liquids on ideal smooth solid surfaces can be expressed in terms of free energy per unit area of a film of thickness e:

$\gamma$ sl and $\gamma$ are the solid/liquid and liquid/air interfacial tensions respectively

When e is larger than the molecular size, P(e) is controlled by long range van der Waals forces

When thickness is very large, P(e) goes to zero because the van der Waals forces only act over a certain distance.

A is the difference of the Hamaker constants i.e. A(sl) - A(ll)

The Hamaker constant is the measure of the strength of a Van der Waals particle-particle interaction.

For small thicknesses:

S is the spreading coefficient on a dry solid surface and $\gamma$ so is the surface energy of the dry solid.

P(e) is not just a function of van der Waals forces - other interactions (dipole-dipole and H bonds for example) may also be important at small thicknesses

A and S are independent of each other and can be any sign - therefore - there are 4 cases to consider.

It is important to keep in mind the fact that the energy function P(e) must be convex for the film to be thermodynamically stable.

## Complete Wetting

In this case - both A and S are positive and the disjoining pressure (which is the slope of the P(e) versus e function) is monotonically decreasing).

A small droplet in contact with a flat, solid surface spreads out and becomes a pancake (depicted below). How thick the film is is determined by the competition long-range forces, which tend to thicken the film, and S, which favors a large wet region. The equilibrium thickness corresponds to the minimum free energy.

## Pseudo Partial Wetting

When S>0 and A<0 we call this "pseudo" partial wetting

In this case, the free energy must have a minimum at a certain value of film thickness. Depending on the volume of the drop, two regimes may arise:

a) microscopic droplet with two possible modes of spreading:

i) a "pancake" of finite thickness (see figure 3a) ii) a very dilute gas of molecules expanding indefinitely on the solid (thickness of film =0) (see figure 3b)

## Partial Wetting

The situation in which S<0 and A>0 is referred to as partial wetting. In this case, the liquid droplet makes a non-zero finite contact angle (defined by the Young equation) on a dry surface.

Young Equation