November 1949
INDUSTRIAL AND ENGINEERING CHEMISTRY
applications requiring lubricants that function satisfactorily at both high and low temperatures. On the other hand, in applications where low temperatures are encountered ( -75 to 200 F.) and there are severe conditions conducive to boundary friction, the diester greases would be preferred. O
O
CONCLUSIONS
.”
4
The silicone-base lithium stearate greases which have been developed offer valuable advantages for the lubrication of antifriction bearings over the wide temperature range of -60” to 325” F. They appear to be uniquely suitable for lubrication at elevated temperatures. In applications requiring a wide range of operating temperatures with emphasis on extremely low temperatures ( -130” to 250” F.) the silicone-diester base lithium atearate greases are believed more suitable. Satisfactory methods have been developed for preparing both types of grease by modification of conventional methods. ACKNOWLEDGMENT
The authors wish to acknowledge the early and valuable work of George M. Hain in the development of the first silicone-base greases. The bearing and wear test results reported herein were made available by J. E. Brophy and J. Larson of the Naval Research Laboratory. The spectroscopic data on the silicone fluids were kindly supplied by D. C. Smith of the Naval Research Laboratory. It is a pleasure to acknowledge the cooperation of S. M. Collegeman of the Engineering Experiment Station, Annapolis, Md., who made available the results of his tests on lithium stearate, silicone-diester base greases,
2551
LITERATURE CITED
(1) Atkins, D. C., Baker, H. R., Murphy, C. M., and Zisrnan, W. A, IND. ENG.CHEM.,39,491 (1947). (2) Atkins, D. C., Murphy, C. M., and Saunders, C. E., Zhid., p. 1395. (3) Bried, E. M., Kidder, H. F.,Murphy, C. M., and Zismqn, UT.A,, Zbid., p. 485. (4) Brophy, J. E., Militz, R. O., and Zisman, W. A., Trans. A m . SOC.Mech. Engrs., 68,355 (1946). ( 5 ) Denison, G. H. (to California Research Corporation), U. S. Patents 2,398,415-6 (April 16, 1946). (6) Dow Corning Corp., catalog, “Dow Corning Silicones” (1945). (7) Fitzsimmons, V. G., Pickett, D. L., Militz, R. O., and Zisman, W. A., Trans. Am. Soc. Mech. Engrs., 68,361 (1946). (8) Hain, G. M., A.S.T.M. Bull., 147,86 (August 1947). (9) Hain, G. M., Jones, D. T., Merker, R. L., and Zisman, W. A., IND.ENG.CHEM.,39,500 (1947). (10) Hain, G. M., and Stone, E. E., Zbid., p. 506. (11) Hain, G. M., and Zisman, W. A., U. S. Patent 2,448,567 (Sept. 7, 1948). (12) Zbid., 2,446,177 (Aug. 3, 1948). (13) Kauppi, T. A., Institute Spokesmen, X , No. 3, pp. 1-4 (1946). (14) Kauppi, T.A., and Pederson, W., Machine Design, 18, No. 7, pp. 109-13 (1946). (15) McGregor, R. R., and Warrick, E. L. (to Corning Glass Works), U. S. Patents 2,389,802-7 (Nov. 27,1945). (16) Merker, R. L., Zbid.,2,456,642 (Dec. 21, 1948). (17) Murphy, C. M., and Saunders, C. E., “Investigation of Silicone
Polymer Fluids,” part IX, unpublished. (18) Murphy, C. M., and Saunders, C. E., Petroleum ReJiner, 26,111, (1947). RECEIVED December 10, 1948. The opinions or assertions contained in this article are t h e authors’ and are not to be construed as official or reflecting the views of the Navy Department.
Diffusion-Controlled Reactions in Liquid Solutions FRANK C. COLLINS
GEORGE E. KIMBALL
Polytechnic Institute of Brooklyn, Brooklyn, N. Y.
Columbia University, New York, N. Y.
I n the prevailing theories of bimolecular reaction kinetics, the frequency factor is assumed to be directly proportional to the bulk concentrations. Recent work by the authors indicates that a modified Smoluchowski behavior forms the fundamental basis of bimoIecular reactions and that in certain cases the Smoluchowski concentration gradients and transients may be extremely important. In this paper, the effect of liquid structure on the Smoluchowslii theory is discussed with the following conclusions: (1)Where the reaction radius is of the same order of magnitude as the root-mean-square “jump” length, the initial flux is not significantly greater than the limiting diffusion
rate, thus masking the transient effect. (2) The ooncentration of reactants within the Rabinowitch “liquid cage,” calculated from the a posteriori probabilities, leads to time-dependent concentrations in the cage which may be greatly different from the bulk concentrations. This effect is marked where the activation energy of diffusion is of the same order as that of the reaction itself. It is of importance in very rapid reactions, as in the quenching of fluorescence and free radical polymerizations, and may be significant in the case of certain other reactions where the activation energy is of the order of magnitude of the diffusion activation energies of the reactants.
T
The effect of liquid structure on bimolecular reaction kinetics in solution has been dealt with by Rabinowitch (2) who did not take the Smoluchowski theory into account. The present paper discusses the derivation of integrodifferential diffusion and flux equations which can be solved to give the initial rate and which, in the limit of long diffusion time, go smoothly over into solutions of Fick’s differential diffusion law. According to the Rabinowitch theory, the reactants must be adjacent to each other in a “cage” of surrounding solvent molecules in order for reaction to ensue. However, the same considerations as to a posteriori probability apply within the cage as in the gross
HE extension of the Smoluchowski (4) colloid coagulation
equations to ordinary bimolecular reaction kinetics was first proposed b y Sveshnikoff (5)in connection with the quenching of fluorescence. It has been shown by the authors (1) that certain paradoxes regarding the establishment of concentration gradients, previous to reaction taking place, are easily resolved on the basis of a posteriori probability. The knowledge that a given molecule has not reacted at a given time, t, diminishes the probability of a potential reactant being in the immediate neighborhood. It was also shown that the a posteriori probability is governed b y the ordinary laws of diffusion.
Vol. 41, No. 11
INDUSTRIAL AND ENGINEERING CHEMISTRY
2552
The general diffusion equation is then
(4) where u is the mean jumping frequency; on reversing the order of the integration this becomes
Expanding leads to
C(TQ,
t ) in a Taylor's series about r and integrating
where is the mean square jump length. If higher order derivatives are neglected and the usual substitution, D
=z
1 v 6
(7)
is made, it Ieads directly to the customary expression of Fick's law
Figure 1. Diffusion by Jump of Length s from a Volume Element, dt.0 to a Spherical Shell (r, r dr)
+
diffusion process and the concentration within the cage may become greatly depleted if the rate of reaction is faster than the diffusion of molecules into the cage. This leads to two limiting cases, neither of which is diffusion controlled, and an intermediate case where diffusion plays the major role.
Thus Fick's Iaw holds if, and only if, the higher derivatives of the density function c(r, t ) ase negligible, a condition which is ordi> 1. narily fulfilled if v t >
RANDOiM WALK DIFFUSION EQUATIONS
Diffusion of molecules in liquids may be considered to be an exaggerated Brownian motion with the diffilsing molecules making random jumps between potential minima, the frequency and length probability of the jumps being governed by the physical nature of the solvent and solute. For simplicity, attention will be limited to the spherically symmetric case. Diffusion in the absence of a sink will first be considered in order to determine the conditions under which Fick's law is valid. Referring to Figure 1, a particle originally within the volume element, dro, is assumed to have a probability, ip(s)ds, of making a jump of length betyeen s and s -tds. All directions of thc jump are assumed to be equally probable. The probability of the particle arriving in the spherical shell ( ? , 1' 4- d r ) by a jump of length s to s ds is proportional to the portion of the shell ( r , r d r ) included in the volurne, 4asds. This is an annulus of volume, 4nr2 sin ededr, where e is the angle between the vectors T O and r dran-n to the points of arrival and departure of the particle. If c(r0,t)duois the probability of the particle being originally within duo at time t, the probability of the above jump is
+
+
This, for a particle which may be anywhere in the shell (ro, ro TO) , becomes, after a trigonometric transformation,
2~r0r
-
c(t-0,
+
Figure 2. Diffusion by Jump of k x ~ g t h.q from a Spherical Shell (ri,, rc dr,,) to a Shell ( r , r f d r ) within a Sink of Radius R
+
t ) p(s) drodsdv
Integrating over all possible values of s, the following is obtained for the probability of 8 particle jumping from any shell (YO, To f dro) into the volume element dv:
The geometry involved in the flux into a spherical sink is shown in Figure 2. The probability of a particle jumping from the shell (r0,ro dro) into the shell (7, T 1- d r ) is
+
November 1949
INDUSTRIAL AND ENGINEERING CHEMISTRY
The total flux into the sink is then
2553
TABLE I. EFFECTOF RATIO OF REACTIONRADIUS,R , MEAN JUMP LENGTH, 9, ON RATIOBETWEEN INITIAL LIMITING FLUX UTES R/
P
“1
In order to avoid confusion as to the significance of c ( R , t ) , in expanding in Taylor’s series, it is expedient t o replace c with q where q(r, t) = C ( T , t ) and R < r < and where q has a form consistent with continuity between 0 and R. After replacing c(r0, t ) with the Taylor’s series about R, the integration of Equation 11 may be performed giving Q ( t ) = 7rv
1
-
(Rz
9) q(R, t )
+(-r+T R2
+ . . .I
(12)
The initial flux rate is then
Since Fick’s law, Equation 8, is applicable for 6Dt/ >> 1the above initial rate may be readily compared with the limiting steady state diffusion. The solution with the boundary conditions
r
c(r, 0 ) = cu
r
c(r, t ) = 0
The Fick’s law limiting flux is
a(-)
=ii
-d1/nDt R
4rDRc0
=
coe-kt
(19)
and is substantially independent of diffusion. These kinetics may be realised in the case where reactant A is a molecule in a photoactivated state with a very short half life. Then if the reaction with B does not immediately ensue after activation, reactant A becomes deactivated, incapable of reaction with B. The other < Y . Here the back-diffusion Bvc will be limiting case is that of k < very nearly equal to the entering flux a, Equation 17 will a p proach zero, and the concentration will remain constant at which is the common case of activation energy controlled kinetics. In the intermediate case of diffusion control as in the quenching of fluorescence where the half life of the excited state of the fluorescent molecule is of the order of 10s times the mean time b e tween diffusional jumps, the exponential terms may be neglected and Equation 18beoomes
where
+
Equations 17 and 18 have interesting implications. When a priori rate k is very large, Equation 18 assumes the form c
- e r j dc 4L RX )
= 4rDRco [I
7.38 30.0
where k is the a priori reaction rate and 0 is a fraction less than unity; this accounts for the possibility that not every diffusional jump will carry the B molecule out of the cage. Where the flux @ is approximately constant as in the case where R / < s > is not significantly larger than unity discussed earlier in this paper, Equation 17 may be integrated and leads to
R
is well known and is c(r, t ) = co (1
1.37 2.88
which molecule B must enter before reaction can occur, exists adjacent to reactant molecule A . This arrival into juxtaposition or “encounter” may be handled as an example of the modified Smoluchowski diffusional flux described earlier. The rate of change of the a posteriori concentration, c, of B molecules in the liquid cage is given by
>R Q
AND
Q?(O)/Q?(m)
1 2 5 20
Interchanging the order of the integration and assuming that the probability of jumps longer than R is negligible leads to
AND
Q
= k
(16)
When
Y
G
is small compared to k, Equation 20 further simplifies to
The ratios of the initial and final flux rates are given in Table I; in this calculation it has been assumed that which, if the observed reaction rate is taken to be R = kccA, yields the equation At comparatively small ratios (which may be taken to be the usual case in liquid state reaction kinetics), an initial transient in the flux rate will not ordinarily be observed.
R
= CAQ
(22)
which is the expected equation for diffusion control.
CONCENTRATION OF REACTANTS IN LIQUID CAGE
LITERATURE CITED
Rabinowitch and Wood (9) showed that collisions in a condensed system occur in sets. On this basis, Rabinowitoh (9)inferred that in liquid solutions molecules of solute are surrounded by a cage of neighboring solvent molecules with which the solute molecule suffers several hundred collisions before being separated by the diffusional process. In the case of bimolecular reactions, it may be considered that a reaction zone, consisting of a cage into
(1) Collins, F.C.,and Kimball, G . E., J. Colloid Sci., 4,425 (1949). (2) Rabinowitch, E.,Trans. Faraday Soc.. 33, 1225 (1937). (3) Rabinowitch, E.,and Wood, W.C.,Ihid.. 32, 1381 (1936). (4) Smoluchowski. M.v., 2.physilc. Chem.. 92,192 (1917). (5) Sveshnikoff, B.,Acta Physicochirn. (U.R.S.S.), 3, 257 (1935). RECEIVED February 24, 1949. Presented before the Division of Physical and Inorganic Chemistry at the 115th Meeting of the AMERICAN CHEMICAL Q o c m w , San Franoiaco, Calif.
