Effective interface potential
The stability of a thin liquid film on a solid surface is determined by the balance of short and long range interactions between the liquid and the solid. Short range forces arise due to the repulsion of overlapping electron orbitals and vary as a function of r−12,r being the distance between the molecules. Considering two planar surfaces, this repulsion yields an interaction energy per area varying as h−8, h being the distance between the two surfaces. The long range interactions are caused by correlations in the fluctuating polarizations of nearby molecules and vary as a function of h−6. For large distances (>100nm) retardation effects have to be considered and the interaction falls as h−7.
1.1. Fundamentals of wetting and dewetting 13
Considering two planar surfaces, the non-retarded interaction yields [19, 20]:
W(h) =− A
12πh2 (1.7)
whereW is the energy per unit area and A is known as the Hamaker constant [21] and is equal to :
A=π2ρ1ρ2C (1.8)
ρ1 and ρ2 being the molecular number densities of the two surfaces involved. C is a coefficient in the particle-particle interaction and it depends on molecular parameters such as polarizability, but, not on the geometry of the bodies. The definition of the Hamaker constant in Equation 1.8 ignores the influence that neighboring atoms can have on the interaction between any pair of molecules. Even though this approximation is definitely not exact for condensed phase, it turns out that the expression of Equation 1.7 remains valid even within the framework of continuum theories. The only change is in the way the Hamaker constant is calculated [19, 22], namely by the optical properties of the materials involved. For A < 0, the system can gain energy by increasing the distanceh between the surfaces, while for A >0, the system gains energy by reducing the distanceh.
The effective interface potential Φ is the result of the combination of the short and long range interactions and is defined as the excess free energy per unit area which is nec-essary to bring two interfaces from infinity to a certain distance and can be characterized by the following expression [23, 24]
Φ(h) =C∗ 1
h8 +W(h) (1.9)
whereC∗ is a constant characterizing the interaction strength.
Symmetry breaking mechanisms
The distinction between stable, metastable and unstable films can be made in terms of Φ as seen in Figure 1.7.
The black curve characterizes a film that is stable on the substrate, since energy would be necessary to thin the film. The equilibrium film thickness is infinite. The two other curves exhibit a global minimum of Φ(h) ath=h∗ and the system can gain energy
0
film thickness Φ
stable
unstable metastable
Φ min
h*
{
Figure 1.7: Effective interface potential Φ as a function of film thickness h. The black curve corresponds to the stable case, the red one to the metastable and the blue curve corresponds to the unstable case.
by changing its present film thickness h toh∗. The blue curve characterizes a film that is unstable on the substrate. It is readily shown [25–28] that, if the second derivative of Φ(h) with respect to h is negative, unstable modes exist, whose amplitudes grow exponentially according toexp(t/τ), whereτ is the growth time that is characteristic for the respective mode. Furthermore, there is a characteristic wavelengthλsof these modes, the amplitude of which grows the fastest and will therefore dominate the emerging dewetting pattern. This process is analogous to spinodal decomposition of a blend of incompatible liquids, which occurs if the second derivative of the free energy with respect to the composition is negative. There, as well, a certain wavelength exists, the amplitude of which is amplified the strongest. Following this analogy, dewetting via unstable surface waves has been termed spinodal dewetting.
The red curve characterizes a film that is metastable on the substrate. For film
thick-1.1. Fundamentals of wetting and dewetting 15 nesses where the second derivative Φ00(h) is positive, the film is considered metastable, since the system has to overcome a potential barrier in order to reach its state of lowest energy at h =h∗. Some kind of nucleus, e.g. a dust particle, is required to lower Φ(h) and can therefore induce dewetting. This rupture mechanism is termedheterogeneous nucleation. For film thicknesses where Φ00(h) is negative and |Φ00(h)|is increasing with decreasing film thickness, no nucleus is necessary in order for the film to lower its lo-cal thickness, as the thermal ”activation” is sufficient to overcome the energy barrier (homogeneous nucleation). Finally, for film thicknesses where Φ00(h) is negative and
|Φ00(h)| is decreasing with decreasing film thickness, the film becomes unstable and its decay will proceed via spinodal dewetting.
Linking the effective interface potential to macroscopic properties
Already in 1938, A. Frumkin had stressed that there is a link between the effective interface potential and the macroscopic contact angle of Young [29], which is given in the following equation:
Φ(h∗)
γLV = 1−cos(θ) (1.10)
Thus, knowing the surface tension γLV and the macroscopic contact angle θ, we can use Equation 1.10 to determine the global minimum of the effective interface potential Φ(h∗).
Another ”macroscopically” accessible quantity that is linked to the effective interface potential is the spinodal wavelengthλs [26, 27]:
λs(h) =
s−8π2γ
Φ00(h) (1.11)
The above equation illustrates that only if Φ00(h)<0 (spinodal dewetting), is λs(h) real. For Φ00(h) = 0, λs(h) diverges to infinity. By determining the spinodal wavelength as a function of the film thickness, one can gain insight into the course of Φ00(h). By measuring the equilibrium layer thickness h∗ and the contact angle θ, it is possible to reconstruct the complete effective interface potential. An offspring of the reconstruction of the effective interface potential Φ is that the Hamaker constantAof the system can be determined. The values forA can be compared with the values calculated from optical properties of the media involved.