Interaction energy between flat surfaces

Surface forces measurement directly determines interaction forces between two surfaces as a function of the surface separation (D) using a simple spring balance. Instruments employed are a surface forces apparatus (SFA), developed by Israelachivili and Tabor [17], and a colloidal probe atomic force microscope introduced by Ducker et al. [18] (Fig. 1). The former utilizes crossed cylinder geometry, and the latter uses the sphere-plate geometry. For both geometries, the measured force (F) normalized by the mean radius (R) of cylinders or a sphere, F/R, is known to be proportional to the interaction energy, Gf, between flat plates (Derjaguin approximation). 

The behaviour of 77(h), already encountered in the previous section, is the most central theme of colloid stability and this issue will be dealt with extensively in Volume rv. Here, we shall emticipate this discussion by reviewing the main elements. The disjoining pressure was already introduced in sec. 1.4.2. For two colloidal particles, or macroscopic phases, a distamce h apart, let the Gibbs energy of interaction be G(h). For the present purpose only the interaction between flat surfaces has to be considered. The macroscopic phases are the (mostly solid, but in some cases liquid) support S (or Lj) cind the vapour G, interacting across the liquid... 

The force measured between crossed cylinders (F ), as in the SEA, and between spheres (Fg), as in the MASIF, a distance D apart is normalized by the local geometric mean radius (R). This quantity is related to die fiee energy of interaction per unit area between flat surfaces (W) according to the Deijaguin approximation (30) ... 

First, after minimizing the free energy one obtains the order parameter profile across the nematic layer. Then, the free-energy is calculated and the force between the surfaces is just a derivative of the free energy with respect to surface separation D. However, in the AFM force experiments, the forces are conveniently measured between a sphere and a flat surface. In this case, the so-called Derjaguin approximation is used [2], which relates the forces between curved surfaces and the interaction energy per unit surface area J D) between two flat surfaces. The pre-nematic force F D) between a sphere of radius R and a flat surface is F D) = 2nR F D) — F oo)) ... 

For small surface charge densities the free energy of interaction per unit area, between flat surfaces is given by the following (107) ... 

This equation is useful in that it is applicable to any type of force law so long as the range of interaction and the separation are much less than the radius of the sphere. Thus the force to overcome the work of adhesion between a rigid sphere and a flat surface written in terms of the surface energy Ay is ... 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

A force microscope actually measures the forces between two macroscopic bodies. The finite size and the macroscopic surface of the tip and the surface spot lead to a number of fundamental consequences in their interaction (Fig. 2). First, the net force is stronger than the intermolecular forces and it acts at much larger distances. Even in the 10-100 nm range, the interaction energy, which is proportional to the size of the tip, can exceed kBT. Secondly, the force between a spherical tip and a flat surface decays with the separation as F D 2 (Fig. 2b) compared to f r 7 for the attraction between two atoms (Fig. 2a). In combination with the finite tip size, the low force gradient increases the effective interaction area and limits the resolution (see Sect. 2.3.3). Third, the surrounding me-... 

This variation of AAm/Bm with separation comes from the "relativistic screening function" R (l), which is subsequently elaborated. This factor becomes important at large distances when we must be concerned with the finite velocity of the electromagnetic wave. At short distances, Rn(l) = 1 the energy of interaction between two flat surfaces varies with the square of separation. At large distances any effective power-law variation of the interaction depends on the particular separation of materials and wavelengths of the operative electromagnetic waves between them. 

Solution The free energy of interaction between a perfect sphere and a flat surface goes as [-(AHam/6)](R//) whereas the interaction free energy between two flats goes as -( Ham/127r/2) per area. What if the sphere flattens slightly ... 

The SFA consists of a hermetically closed stainless steel chamber that can be filled with any transparent liquid or gas of choice. Mica is a preferred substrate in the SFA, though other surfaces, such as single-crystal sapphire plates have also been used [10]. In the SFA, the force acting between the surfaces, mounted in a crossed cylinder configuration, as a function of surface separation is measured. The data obtained are normally plotted as force, FC(D), normalised by the undeformed geometric mean radius of the surfaces, R This quantity is related to the free energy of interaction per unit area, G, between two flat surfaces at the same separation [11] ... 

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