THEORY OF EMULSION STABILITY 167 nc = concentration at bottom of tube g -= gravitation constant k -- Boltzmann's constant T -= absolute temperature This distribution is illustrated for mono disperse particles in Fig. 1. The total concentration of disperse phase at height h can be obtained by sum- mation over all of the particle sizes. If the particle size distribution is known this summation can probably be performed analytically. It is quite clear, however, that the smaller the density difference between the two phases and the smaller the particle size the less steep is the composition gradient in the emulsion. III. DOUBLE LAYER. THEORY Ideas developed by the early part of this century have dominated our analysis up until this point. In discussing flocculation and coalescence the work of Verwey, Overbeek and others since about 1935 will provide the principal guides (3). From the comparatively simple questions of the mechanical properties of the droplets and the medium we go to the complex ß. ß .:.'....'. : ""'" .: "5'":'.".' : o . : ".•: . '." '.:" . • '.r ..' },:'" '. A". '.' ::':'" ß ß .-..•.. ½•.: ...... :': :ø .o • .... ß . Figure 1 .--Dia- grammatic repre- sentation of the dis- tribution of uni- formly sized particles in a gamboge sol (Pertin). (From Wieser, "Colloid Chemistry," p. 202.) Figure 2.--The electrokinetic potential (D and the electrical double layer. (From Alexander and Johnson, "Colloid Science," p. 297.)

168 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS chemical and electrical interactions. In this we will indulge in a bit of double talk for the discussion will be about double diffuse--double layer-- double particle interactions. In general the positive and negative ions present in an emulsion will dis- tribute themselves between the oil phase and the water phase so as to put a charge on the oil droplets. That is, there will be a higher concentration of ions of one charge than of the other in the oil. This is illustrated dia- grammatically in Fig. 2. In this particular case the negative ions are in excess in the oil phase. Unfortunately the scale is too small to show a distribution of the negative charges of decreasing concentration toward the center of the droplet. The first layer around the droplet, the "solrating layer," consists of strongly bound water molecules, hydrogen atoms pointed toward the oil since the water dipoles will orient in the charge field, and excess positive ions strongly held by the negative charge on the oil. This layer remains bound to the droplet as the oil moves through the water and having an excess of positive ions it reduces the effective net charge of the drop. It is this net charge which is effective in electrokinetic experiments such as electrophoresis measurements. Continuing out toward the body of the water phase distant from the droplet, it is seen that the concentration of the positive ions decreases and that of the nega- tive ions increases much like in Fig. 1. This ion atmosphere is the "diffuse double layer" in the water phase. It is often convenient to consider instead of the double layer itself a shell of charges which would have the same effect as the double layer. What is referred to as the thickness of the double layer is the distance d = 1/K from the droplet to this shell as illus- trated (K2 is proportional to the ionic strength of the solution divided by its dielectric constant). It should be noted that the same type of diffuse double layer is present in the oil phase. Below the diagram in Fig. 2 is a plot of electrical potential, the energy required to bring an electron from infinity to the given point, as a function of distance from the center of the droplet for the region outside the surface of shear. The shape of the potential curve is determined by the ion distribution discussed above. The difference in potential between the surface of shear and a distant point in the water phase is known as the zeta potential or electrokinetic potential. Figure 3 is a plot of the entire potential-distance curve, with the direction of increasing potential reversed when compared with Fig. 2. The difference in potential between the in- teriors of the water and oil layers is denoted by •/• and that portion of the potential difference due to dipole orientation and charges at the interface by chi. It should be noted that most of the potential drop occurs in the oil phase. Implied above was that the water was pure and the only ions present were those from the ionization of water. Although the relative distances