Appendix:  Patterns in Drying Water Films

 

 

Box 1:

 

 

 

תיבת טקסט: Two-phase behaviour of the film. Analytically, we describe the interaction between the film (thickness h) and the substrate by free energy per unit area. The Van der Waals term is proportional to h-2 and the polar interaction to -exp(-h/d0). Then we can write: Now the chemical potential of the fluid is , since adding unit volume of fluid to the film adds ρ molecules. In equilibrium, , which corresponds to the minimum of the thermodynamic potential Ψ: Looking at Ψ(h) in Fig. B1 for various values of the vapour pressure p, we find that when p is below psat there are two minima, representing two phases. Depending on the value of p, one phase might be stable and the other meta-stable. At the extremes, only one phase exists. There is a particular value of p at which both phases are in equilibrium with the vapour; this is the triple point. This model represents very well the two-phase behaviour of the water films throughout the whole pressure range p<psat. Box 2:

 

תיבת טקסט: An evolution equation for film dynamics. The dynamics of the film can be described by an evolution equation for h(t). When the curvature is approximated by and the hydrodynamic flow is driven by gradients in the chemical potential. There are two sources for this: one is and the other is the Gibbs-Thomson pressure . Material is lost at a rate proportional to the difference between the chemical potentials of the film and the vapour, i.e. Together, taking into account viscosity of the fluid η and the usual condition of no slip at the interface with the substrate, we get a dynamic equation: This equation captures most of the film behaviour, at least qualitatively, as a function of film thickness and vapour pressure, and has many similarities to the equations for solidification derived using the phase-field model.

Box 3:

 

תיבת טקסט: Three-beam interferometry. When the two light waves of equal amplitude from the top of the film and the substrate interfere, with phase difference φ, the intensity observed is proportional to . This is very insensitive to changes in φ when it is close to zero, and most sensitive when φ has value around an odd multiple of π/2. In three-beam interferometry, in order to measure very small values of φ, a third wave is added which has a constant phase relation to the substrate reflection, so that its vector sum with that reflection creates a wave differing in phase by an odd multiple of π/2 from the expected phase of the top reflection. This is shown as a vector diagram in Fig. B2. In this way, the reflection from the film behaves as if it were thicker by an odd number of λ/8, for which thickness the sensitivity is highest. The phase of the third wave (bottom reflection) is tuned by carefully cleaving the mica to get the right thickness. Then, a measurement accuracy of about 5Ǻ was achieved, limited by the sensitivity (one gray level) of the 8-bit camera used.

 

Figure B1:

The thermodynamic potential Ψ(h) for various vapour pressures: (a) p=psat  (b) where there is only a thick stable film, (c) where there is a thick stable film and a thin metastable film, (d) where thick and thin films are in equilibrium (triple point), (e) where there is a thick metastable film and a thin stable film, (f) where only the thin film exists.

 

 

 

 

 

Figure B2:

 

Vector diagram showing the reflections from the top surface of the film, the interface with the substrate and the bottom of the substrate (blue). The resultant vector of the latter two reflections (red) has to be normal to the top reflection for maximum sensitivity to changes in  φ.