170 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 02 03 o Figure 4.--Potential energy of repulsion between two spherical particles for different values of a. (From Overbeek (Kruyt), "Colloid Science.") repel each other as indicated in Fig. 4. The a in this figure is our r, the droplet radius, and H0 is the distance between the droplets (so the hori- zontal axis is in units of double layer thickness). /zR is the repulsive elec- trical potential between the two droplets, and when •b0 is small r _•2e_KH ø (7) /7n = 4.62 X 10 -6 • where v is the valence of the counter ions. 3' is a not too simple function of Z(= Ve•o/kT) where e is the electron charge. eZ/2 -- 1 (= 0 when Z is small) 3' -- eZ/2 q- 1 (= 1 when Z is large) Thus it is seen that the solution of the electrostatic problem gives a poten- tial which is always repulsive and which is exponential in H0, the distance between the particles. Observation has shown that emulsions flocculate and coalesce and, probably even earlier in our experience, that gases can be condensed to liquids. For these to occur there must be attractive forces acting between neutral molecules. These universal long range attractive forces, responsible for deviations from ideal gas laws are called van der Waals' forces. Fritz London, in 1932, gave a quantum mechanical explanation of the forces of attraction between nonpolar molecules. One interpretation of this quan- tum mechanical theory is as follows:

THEORY OF EMULSION STABILITY 171 If one were to have an instantaneous picture of the nuclei and the elec- trons in a nonpolar molecule, say carbon tetrachloride, one would observe a spacial distribution of the electrons and the positive nuclei such that the center of gravity of the positive and negative charges were separate. Thus, in this instantaneous picture the molecule is not nonpolar but has a dipole-- a positive and a negative end. Since electrons move very rapidly this picture changes rapidly and with it the magnitude and orientation of the dipole. In the case of a nonpolar molecule the superposition of the series of pictures for any appreciable time gives a zero dipole. However, to a neighbor molecule, which is quite as shifty an entity itself, this instantane- ous nonzero dipole is observable. The momentary dipole in the first mole- cule induces in the second a dipole in the opposite direction which is thus attracted to the first dipole. In this way a force of attraction is engendered in the entire system. The magnitude of the force between the molecules depends on the polarizability, or ease with which a dipole is induced, of the molecules and some natural frequencies of the molecules. Upon the latter depends the ability of an individual pair to "jitter" together and thus main- tain this temporary bond for a long time. When two droplets are in collision or close together, that is when the distance between them is small compared to their radii the potential energy of the London-van der Waals attraction is given approximately by 1 r •'• = - i5 • •o (8) in which .4 is a constant taking into account the polarizability and jitter factors. It is to be noted that the attraction potential is, under these circumstances, inversely proportional to the first power of the distance between the droplets. It is only when dealing with very large droplets, when compared with molecules, that the dependence on distance is of so low an order. This comes about because of cooperation among the induced dipoles near the quite fiat, compared to molecular radii of curva- ture, collision regions of the droplets. The total potential energy of the interaction is given by the sum •' = •,,• + •,• (9) This summation is shown diagrammatically in Fig. 5 for one set of values of the various parameters. A small set of summation curves for different values of the surface potential •0 for a 1 micron radius particle and assum- ing reasonable values for the other parameters is given in Fig. 6. Inspection of Fig. 6 shows that in the system involved one can have two types of interactions. Where •b0 is below a certain value, 23 millivolts in this case, the barrier in the potential rs. i•aterparticle distance curve always lies below kT. There is then enough thermal energy present to allow drop- lets to come right together at a rapid rate. This is exemplified by the two