THE LAWS OF BROWNIAN MOVEMENT Displacement in a Given Time.—It is owing to Brownian movement that an equilibrium distribution is established in an emulsion, the more vigorous the movement the more rapid being the establishment of equilibrium. Variations in Brownian activity, however, do not affect the final distribution, which is always the same for particles of the same size and apparent density. There fore the ultimate conditions can be studied without consideration of the mechanism by which they have been established. Detailed analysis of this mechanism was made by Einstein in his brilliant theoretical studies. Besides this, the tentative, but very sugges tive, analysis of this same problem due to Smoluchowski cer tainly deserves mention, although published subsequently, because of the different method of attack. Einstein and Smoluchowski both defined the activity of Brownian movement in the same manner. Previous to this, a "mean speed of agitation" had to be determined by following the path of a particle as closely as possible. The values thus found were always a few microns per second for par ticles of the order of a micron in diameter (one micron is , of a millimetre). Such values are seriously in error. The irregu larities of the path are so numerous and so rapid that it is im possible to follow them, and the recorded path is far simpler and shorter than the true path. Further, the apparent mean speed of a particle during a given period varies enormously in magnitude and direction, with no tendency towards a limit as the time of observation diminishes; this may readily be seen by noting the successive positions of a particle in the camera lucida, first every minute, then, say, every five seconds, and better still, by photo graphing them every a of a second.
Dismissing, therefore, the true velocity as immeasurable, and neglecting the extremely tangled path described by a particle in a given time, Einstein and Smoluchowski chose to define the magni tude of the movements in terms of a rectilinear segment connect ing the starting and finishing points, which, on the average, will evidently be greater the more vigorous is the motion. This seg ment will represent the displacement of the particle during the time taken. The projection to a horizontal plane, observed directly in the microscope under ordinary experimental conditions, with the microscope vertical, will be the horizontal displacement.
The same applies, consequently, to one-half of this square, i.e., to the mean square of the projection of the horizontal dis placement on an arbitrary horizontal axis. Expressed in another way, is constant for a given particle immersed in a given liquid. This ratio, which is obviously greater the more violently the particle moves, defines the activity of the Brownian movement of the particle.
Diffusion of Emulsions.—It is to be expected that if pure water were in contact with an aqueous emulsion of uniform par ticles, Brownian movement would cause diffusion of the particles in the water by a process analogous to that causing the true diffusion of dissolved substances. Further, such diffusion would be more rapid the more active the Brownian movement of the particles. Einstein's calculations, made entirely on the assump tion that Brownian motion is completely irregular, show that an emulsion actually does diffuse like a solution, the coefficient of a diffusion D being just equal to half the number which expresses the activity of Brownian movement:— In a vertical column of emulsion, the final permanent condition is created and maintained by the interplay of two opposing processes, gravity which is constantly attracting the particles downwards, and Brownian movement which is continually dis persing them. This idea can be more precisely stated in the fol lowing form : for each stratum, the loss through diffusion towards regions of low concentration balances the gain through gravitation towards regions of high concentration.
In the special case when the particles are spheres of radius a (to which an attempt can be made to apply Stokes' law which has been actually verified for spherules of microscopic dimen sions), and assuming also that at equal concentrations particles and molecules produce the same osmotic pressure, it is found that: I N 6Irart where ri is the viscosity of the liquid, T its absolute temperature, and N is Avogadro's number. Seeing that the coefficient of dif fusion is one-half of the activity of Brownian movement, this equation can be put in the form: RT I t N 31w7 Thus, the activity of Brownian movement (and the rapidity of diffusion) must be proportional to the absolute temperature, and inversely proportional to the viscosity of the liquid and to the radius of the particles.
Einstein immediately perceived that the order of magnitude of movement seemed to fit in with these conclusions. Smoluchow ski reached the same result in a learned discussion of the data then available, which consisted of the nature and the density of the particles being immaterial ; the qualitative observations of the increase of movement with rise of temperature and with diminu tion of particle size; and a rough evaluation of the displacements of particles of the order of a micron in size. Thence it became possible to state definitely that Brownian movement is certainly not more than five times as active as the predicted motion, nor less than one-fifth of it. This approximate agreement in the order of magnitude and in the qualitative properties of the phenomenon at once lent considerable force to the kinetic theory, and this was clearly indicated by the originators of that theory.
No support to these observations of Einstein and Smoluchowski was published until 1908, when an interesting, but only partial, verification appeared, by Seddig, who compared, at various tem peratures, the displacements experienced by ultramicroscopic par ticles of cinnabar estimated to be nearly uniform in size. If Ein stein's formula is correct, the mean displacements d and d at 17° C and 90° C must be in the ratio 2.05, allowing for the change viscosity. Experiment gave 2.2. The discrepancy was much smaller than the possible experimental error.