Creator:David Turnbull Date Created: Place Created:Schenectady, New York Keywords:catalysis of nucleation Context:reoprt by David Turnbull ************************************************** RES. LAB. REPRINT 1984 GENERAL 0 ELECTRIC THEORY OF CATALYSIS OF NUCLEATION BY SURFACE PATCHES BY DAVID TURNBULL SCHENECTADY/ NEW YORK THEORY OF CATALYSIS OF NUCLEATION BY SURFACE PATCHES* DAVID TURNBULLf By hypothesizing a plausible distribution of units for crystal nucleation with respect to size, and that the size of the units may be of the order of the critical size for growth of a nucleus into a supercooled liquid, the multiplicity in crystal nucleation frequency sometimes observed for the isothermal solidification of small droplets is accounted for with the use of no more than two fundamental frequencies. The theory satisfactorily explains the athermal nucleation of crystals for the solidification of small mercury droplets that have "HgX" patches on their surface. THEORIE DE L'ACTION CATALYTIQUE DES CONCENTRATIONS A LA SURFACE SUR LA GERMINATION La multiplicity dans les frequences de germination parfois observee lors de la solidification isotherme de petites gouttelettes est expliquee en employant une frequence fondamentale seulement; ceci en supposant une distribution plausible des unites pour la germination des cristaux en rapport avec les dimensions, et que les dimensions des unites sont de l'ordre des dimensions critiques necessaires pour la croissance d'un germe dans un liquide surfondu. Cette theorie explique d'une fagon satisfai-sante la germination athermique des cristaux pour la solidification de petites gouttelettes de mercure, dont la surface est couverte de concentrations de "HgX". DIE THEORIE DER KEIMBILDUNGSKATALYSE DURCH OBERFLACHENFLECKEN Es wird als Hypothese eine plausible Grossenverteilung von Bereichen der Keimbildung vorge-schlagen. Es wird weiterhin angenommen, dass die Dimensionen dieser Bereiche von der gleichen Grossenordnung wie die kritische Grosse fur das Wachstum eines Keimes in eine unterkiihlte Fliissig-keitsind. Dann lasstsich die Multiplizitat der Keimbildungsfrequenz, die manchmal bei der isother-men Erstarrung kleiner Tropfchen beobachtet wird, unter Benutzung von nur zwei elementaren Frequenzen erklaren. Die Theorie erklart zufriedenstellend die nicht-thermische Kristallkeimbildung bei der Erstarrung kleiner Quecksilbertropfchen, die "HgX" Flecken auf ihrer Oberflache haben. Introduction Recently the kinetics of solidification of small droplets of mercury [1] and tin [2] coated with various surface films were investigated. Isothermal data for certain mercury droplet dispersions coated with a given film were well described by a single nucleation frequency (7)/area or volume. However, the isothermal data on oxide-coated tin droplets and on mercury droplets having "HgX" patches (HgX, a strong catalyst for the formation of mercury crystal nuclei, is a decomposition product of certain mercury carboxvlates) seemed to require a multiplicity of I values for their description. Although various explanations for this multiplicity, all presupposing heterogeneous nucleation, were considered [1; 2], none were fully developed. In an earlier note [3] a new theory for the multiplicity in / was proposed. In this paper we shall give a more complete development of the theory and compare its predictions with experience. Theory The potency, P, of the surface of a substance in catalyzing the formation of crystal nuclei has been characterized by the contact angle 6 made by the nucleus on the flat catalyst surface in contact with supercooled liquid. The less is d, the greater is P, and we may set P cc 1 /d. The angle d is determined ^Received August 8, 1952. fGeneral Electric Research Laboratory, The Knolls, Schenectady, New York. ACTA METALLURGICA, VOL. 1, JAN. 1953 by the structure and chemistry of the catalyzing surface. Suppose that the crystal embryos have the form of spherical caps on the relatively flat catalyst surface. In order to survive as a crystal nucleus on the surface, these embryos must have attained a critical radius, r*s, given by [4]: (1) r*s= — 2cr sin 6/AFv where AFv = free energy per volume for the liquid^ crystal transition when both phases have an infinite volume-to-surface ratio, and a is the interfacial energy between liquid and crystal phases. The radius, r*, of the crystallization nucleus that forms in the body of the supercooled liquid without the aid of any catalytic surface is equal to or larger than r*s and given by: (2) r* = - 2 r*s formed on the catalyst surface do not become transformation nuclei except by fluctuations! that increase the radius of the aggregate of catalyst particle plus crystal to r*. In earlier publications [4; 1] we have developed the theory of catalysis of nucleation by catalytic JReiss [5] and Pound and La Mer [2] have discussed the theory for this process. TURNBULL: NUCLEATION BY SURFACE PATCHES units having radii greatly in excess of r*. We now consider the catalysis of nucleation by bodies whose radii are of the same order of magnitude as r*. It will be supposed, for simplicity, that these bodies (we shall hereafter call them patches) are all characterized by a single value of 9 and that their number is proportional to the surface area of the liquid.t Let nA be the number of patches per unit area with radius > R. We assume a continuous distribution of nA with respect to R, as follows: (3) nA = f(R) If dp is the contact angle made by a crystal nucleus on a patch the critical radius r*p for nucleation on the surface of the patch is given by: (4) r*„ = - 2 r*s or r*. To a good approximation: (5) AFv = A5. AT where ASv is the entropy of transition/volume and AT is the supercooling. The number of patches (nA*)/area, capable of serving as transformation nuclei is found by substituting r* or r*s in equation (3). From equations (1), (2), (3) and (5), it follows that the dependence of nA* on AT is (6) nA* = /( - 2 a/\Sc AT) or (7) nA* = /( - 2 a sin 6/ASv AT). Let Ip be the frequency of nucleation/area of patch. Then the nucleation frequency (e)/patch is (8) c = Ip/ ap where ap is the area of the patch. Now consider a collection of supercooled liquid droplets of diameter D. The average number of patches (/w)/droplet capable of serving as transformation nuclei is given by: (9) m = nA* (tt D2). c will be very strongly dependent upon temperature so that there is a very narrow temperature range in which it changes from an imperceptible to a perceptible value. For some systems m will be very much larger than unity when c becomes perceptible and the nucleation frequency/droplet for droplets of uniform size will be constant. However, there is a fThe theory to be developed is easily extended to the case where the number of patches is proportional to the liquid volume. large probability that, for some systems, c becomes perceptible when m is of the order of unity or less. Under these circumstances, the number of patches/ droplet that can serve as transformation nuclei is far from uniform and a substantial fraction of droplets will contain no supercritical patches. Let x be the fraction of a liquid sample, consisting of uniform droplets, solidified in time /. To obtain x = fit) for m of the order unity or less, a derivation due to Kimball is directly applicable. Kimball [6] assumed that all catalytic units (average m/droplet) are equally effective and that they are distributed among the droplets according to Poisson's distribution law. With these assumptions he obtained for constant AT: (10) x = 1 - expj - m[l - exp( - rf)]}. All supercritical patches are not equally effective as nucleation catalysts since their areas are different. Actually, however, the temperature range for which the rate of nucleation is measured is very small and the variation in r* or r*s is correspondingly small (e.g., 5 to 10 per cent). Hence it may be assumed, with fair accuracy, that all supercritical patches have the same area so that equation (10) satisfactorily describes the isothermal dependence of x on / for the solidification of a collection of uniform droplets containing on the average m supercritical patches droplet. As / —» oo, .t approaches a limiting value, xa < 1 given by: (11) xa = 1 — exp( — m). The fraction of droplets containing no supercritical patches is therefore 1 — xa. To obtain a more convenient relation for analysis we derive from (10): (12) dx/( 1 - x) dt =f = cm exp( - ct) and (13) In / = In {cm) — ct. Suppose that instead of a uniform droplet diameter we have a distribution of diameters such that at time t the fraction of droplets of diameter D that have not solidified is xD. When t = 0, xD = xD° and D' is defined by: t (14) h = I xD°dD. Jo Equation (12) becomes: (15) / = \cm'[F(t, D)]/( 1 - x)Z?'2jexp( - ct) where 10 ACTA METALLURGICA, VOL. 1, 1953 (16) F(t, D) = J xBD2dD, (17) = and (18) f XodD = 1 - x. Jo Xa is now given by: (19) xa = I - f xD°exp[ - m'(D/D')2]dD. Jo Although our theory and the earlier one due to Kimball give formally identical expressions for the isothermal dependence of x on t, there are important differences in the predictions of the two theories. As described by Pound and La Mer, Kimball's theory seems to require that m be indepedent of AT, while our theory requires that m increase with AT, according to equations (6) and (9). According to Kimball's theory, x will be perceived to approach an asymptote xa substantially less than unity and perceptibly greater than zero only for systems in which (approximately) 0.01 < m < 3 where m is the number of catalyst units/droplet characterized by the smallest 8 value. Our theory predicts that 0 < xa < 1 if 0 < m < 3 (approximately) when c becomes perceptible. Therefore, Kimball's theory seems to predict that the perception of 0 < xa < 1 should be a very improbable occurrence while our theory predicts that such perception should be possible in a fair number of instances. Comparison of Theory and Experience Mercury Droplets Infected with HgX Patches Mercury droplets coated with mercury laurate or mercury stearate become infected in time with patches of a decomposition product, HgX (believed to be mercury oxide formed by hydrolysis of the carboxylates), that is a very powerful catalyst for the formation of mercury crystal nuclei. Transformation nuclei formed rapidly from these patches at only 2-4° supercooling. The fraction x of droplets solidified quickly approached an asymptote xa < 1 almost as soon as the droplets reached bath temperature so that the essential features of the transformation were described by xa = f(AT), where xa increases with increasing AT. This type of transformation in which time does not enter into the description is called "athermal" and has been recognized in crystalline media [7] and designated the "athermal martensite transformation." The interpretation of these results is that c is perceptible and in fact very large, at a value of AT which corresponds to a value of m that is essentially-zero. When supercooling is increased sufficiently to make xa perceptible (m > 0.02 approximately) all the patches are already covered with crystalline mercury. Thus the potential transformation nuclei are fully formed before any transformation is perceptible. These potential nuclei become active transformation nuclei immediately when AT is increased sufficiently to make r* (or r*s) = R. Because of the athermal nature of the transformation, c cannot be determined but nA can be evaluated quite accurately from xa. The value of nA* calculated from the earlier data [1] (sample A) is approximately given by: (20) nA* = a exp(/3AT) where a and /3 are constants. Combining equations (2), (5) and (6) we have (21) nA=aexp[b/R] where b = - 2 xa. This procedure was repeated until nearly all of the infected droplets had solidified. The results are shown in Figure 1. The values of xa obtained from these data were substituted into equation (11) to obtain m. nA* was calculated! from equation (9) with D = D' = 4.0 X 10~4cm. R = r* corresponding to these nA* values was calculated from equations (2) and (5) using the value of a calculated from earlier results [1]. nA is plotted against R and AT in Figure 2. cannot serve as transformation nuclei. Therefore, it follows that the solidification rate at a given temperature, Ti, and extent of transformation x will be more rapid when the sample is quenched to T\ after having been held at a temperature a few tenths of a degree above Ti than if the sample is quenched to Ti from above the melting temperature. Pound and La Mer [2] established that two of their isotherms could be described approximately by the Kimball equation (10) with m increasing with AT. They gave no explanation for the dependence of m on AT. According to equation (13), In / should be a linear function of t. We have measured / between x = 0.3 and x = 0.7 for the four of Pound and La Mer's isotherms corresponding to the greatest supercooling and the droplet size distribution corresponding to D' = 3.75 microns (D between 2.5 and 5.0 microns). Figures 3, 4, 5, and 6 show In / Figure 2. Calculated size distribution curve for HgX patches on mercury droplets of sample B. nA — f{R) is a sigmoidal relation that is fairly symmetrical around R = Rm. The slope of nA = f(R) is satisfactorily described by: (22) dnA/dR = (7.7 X 1011) exp [ - 1.8 X 1012(i? - i?m)2]cm-3 where Rm = 3.3 X 10-°cm. and nA = 5.3 X 105 cm.-2 at R = Rm. The range of patch radii, 250 to 450A, seems plausible. Solidification of Tin Droplets Coated with Tin Oxide Pound and La Mer [2] established that the major fraction of nuclei for the solidification of oxide- coated tin droplets formed isothermally. The rate determining step is the formation of a tin nucleus on a supercritical patch (thermal nucleation). Crystalline tin may also coat subcritical patches but these fit was established from the droplet size distribution that m values calculated from equations (11) and (19) did not differ to an important extent. TIME (MIN.) Figure 3. Test of equations (18) and (20) using Pound and La Mer's data on the solidification of oxide-coated tin droplets {AT = 116.35). 50 TIME (MIN.) Figure 4. Test of equations (18) and (20) using Pound and La Mer's data on the solidification of oxide-coated tin droplets (AT = 117.25). ACTA METALLURGICA, VOL. 1, 1953 (24) ln(/ — v) = In (cot) — ct. Figure 5. Test of equations (18) and (20) using Pound and La Mer's data on the solidification of oxide coated tin droplets (A r = 117.95). as a function of t for each of the four AT values. In all cases / and hence the transformation rate dx/dt, tends to be larger at the longer times than predicted by the Kimball equation. The existing non-unifor-mity of droplet size would cause / to fall off more sharply with time (see equation (15)) than predicted by the Kimball equation. However, these results and the theory are easity reconciled by taking into account the auxiliary . TIME (MIN.) Figure 6. Test of equations (18) and (20) using Pound and La Mer's data on the solidification of oxide-coated tin droplets (AT = 118.85). steady nucleation process suggested by Pound and La Mer. Let the frequency of the auxiliary nucleation be v per droplet; then / = cm exp( — ct) + v By proper selection of v a good straight-line relation between In (J — v) and t is obtained for each of the four values of AT. These relations are shown in Figures 3, 4, 5 and 6. Thus each of Pound and La Mer's isotherms can be satisfactorily described by three parameters c, m, and v, each of which is a function of AT and droplet size, m is the number of supercritical patches per droplet, c is given by equation (8) and is the frequency of forming a tin nucleus on a patch, v is a steady nucleation frequency per droplet that may correspond to the frequency of homogeneous nucleation or possibly to the heterogeneous nucleation frequency per area of inactive surface. The values of c, v, and m that describe Pound and La Mer's data are summarized in Table I. The un- TABLE 1 Numerical Values of Constants that Describe Pound and La Mer's Data A r(°C) c (sec ') v (sec ') 107.65 110.85 113.85 116.35 117.25 117.95 118.85 0.090 0.20 0.40 0.74 0.87 1.10 1.08 2.64 X 10~4 1.67 X HT5 4.05 X 10~4 3.00 X 10"6 5.00 X 10-J 4.24 X 10"5 9.5 X 10-4 11.6 X HT5 certainty in each value is of the order of ± 10 per cent. There is an additional uncertainty due to the neglect of the non-uniformity of the droplet size. It is hoped that this uncertainty is minimized by the evaluation of/in the vicinity of x = 0.5 (x = 0.3 to x = 0.7) for all four isotherms. Also listed in the table are m values calculated from xa for the three isotherms corresponding to the smallest AT's. This calculation is approximately valid since the contribution of v to the rate is less the less is AT. Pound and La Mer's isothermal results for four different droplet size distributions indicate that m is proportional to droplet area [2].f Values of nA* = m/ivD'- were calculated from the m values listed in Table I, which correspond to D' = 3.75 X 10_4cm. As a function of AT, nA* is satisfactorily described by equation (20) with a = 1.41 X 10~6 cm-2 and /3 = 0.238°K-1. a and /? were calculated from the constants of the straight line obtained by plotting log nA* against AT (see Fig. 7). From equation (21) and the value of a estimated by Pound and La Mer jThis fact was established independently by F. C. Frank. TURNBULL: UCLEATIO we obtain the following expression for nA: (25) nA = 1.41 X l(T6exp[3.14 X lO'6/R]cm~2. R(A') AT CO Figure 7. Calculated size distribution curve for the patches that catalyzed the formation of tin crystal nuclei in Pound and La Mer's sample. Log c and log v are plotted against 1/(AFV)2T, where AF„ is expressed in ergs/c.c., in Figure 8. c is described by the equation: (26) c = 1011±6exp[ - 1.74 X 106/(AFB)2/fer], and v by: (27) v = 1017±6exp[ - 2.60 X 10*/(AF,)*kT]. Now the area of a patch of critical size is of the order of 10-13 cm2, hence I„ the frequency of nuclea-tion per cm2 of patch is (see equation (8)): (28) I„ = Ks exp[ - 1.74 X 106/(AF^kT] where Ks = 1024±6 cm-2 sec-1. This value of A"s is in fair agreement with the value 1027-5 cm-2 sec-1 predicted by the theorv of heterogeneous nucleation [1]. v might be interpreted as the frequency of homogeneous nucleation or the frequency of heterogeneous nucleation on an inactive surface. If we assume homogeneous nucleation: (29) v/vD' = I = Kv exp[ - 2.60 X / (AFtfkT] where I = nucleation frequency per volume and vD' is the volume of a droplet of diameter D' = 3.75 X 10-4cm. To describe the data on this basis Kv = 1027±6 cm-2 sec-1, a value in poor agreement with K„ = 1036 predicted by the theory of homogeneous nucleation [1]. Assuming heterogeneous nucleation on the inactive surface of the droplet we obtain: BY SURFACE PATCHES 13 (30) v/aD' = I,= Kg exp[ - 2.60 X 106/(AF„)2*71 where I, is the nucleation frequency per area of inactive surface and aD' is the area of the droplet of diameter D'. Ks is found to be 1023 4±6 cm-2 sec-1 in l/(afv)zt xio21 Figure 8. Dependence of v and c, for the nucleation of tin crystals in Pound and La Mer's sample, upon supercooling. fair agreement with the value 1027-5 predicted by the theory of heterogeneous nucleation. Crystallization of Small Water Droplets Dorsch and Hacker [8] have measured the initial freezing temperature, T(, of water droplets as a function of their diameter D and found that Tt decreased with decreasing D. In order to explain these results Levine [9] postulated that a variety of types of units that catalyze ice nucleation ordinarily exist in water, each variety characterized by a particular AT corresponding to the initiation of freezing. Let «„ be the number of units/volume that initiate freezing at a supercooling of AT or less. Levine showed that the dependence of 7\ on D is satisfactorily described by assuming «„ proportional to exp (/3A7").f He gives no theory for the increase of w„ with AT. One possible interpretation for the dependance of nv on AT in the freezing of water is that nucleation is effected by patches characterized by a single 6 and a size distribution function of the form », = a exp (b/R). However, Dorsch and Hacker made no fThe formal analogy between Levine's statistical theory for the initiation of freezing and Weibull's [10] (see also Fisher and Hollomon [11]) theory for the stress specimen-size relationship in brittle fracture is noteworthy. According to Griffith [12] brittle fracture is initiated by microcracks. The larger the radius, r, of the crack the less is the fracture stress. Weibull showed that the experimental stress-size relationship can be explained on the assumption that nA = a exp (b/r) where nA is the number of microcracks per area having a radius > r. ACTA METALLURGICA, VOL. 1, 1953 quantitative measurements on the time dependence of solidification of the droplets as a function of D and AT. It is possible, therefore, that the catalytic units responsible for solidification in their experiments were supercritical at all values of AT. The dependence of b, on AT, deduced from their results, can be accounted for on the assumption of a statistical distribution of catalyst units with respect to the contact angle, 6[n„ =/(#)] such that the number per volume, nv, characterized by a contact < 6 increases with increasing 6. The temperature coefficient of the rate of nucleation Is on a particular catalyst surface is very large so that Is changes from an imperceptible value to a magnitude too large to measure over 2-3° temperature range. Hence to a gross approximation a catalyst unit characterized by a given 6 may also be characterized by a particular value of AT. Conclusions The multiplicity in crystal nucleation frequency that has been observed for the solidification of small droplets is readily accounted for by the patch nucleation theory with the use of no more than two fundamental nucleation frequencies for a given dispersion at a given temperature. It is shown that the data are described by plausible distributions of patches with size. Also the theory satisfactorily explains the athermal formation of crystal nuclei in solidification processes. Although the existing data are accounted for on the assumption that the patches are distributed over the droplet surface, the surfaces of very small particles suspended in the droplet might under certain circumstances constitute the patches. Formally the theory is analogous for suspended particles and patches in the droplet surface. Acknowledgement The author is pleased to acknowledge valuable discusssions with E. W. Hart in developing the content of this paper. References 1. Turnbull, D. J. Chem. Phys., 20 (1952) 411. 2. Pound, G. M. and La Mer, V. K. J. Amer. Chem. Soc., 74 (1952) 2323. 3. Turnbull, D. J. Chem. Phys., 20 (1952) 1327. 4. Turnbull, D. J. Chem. Phys., 18 (1950) 198. 5. Reiss, H. J. Chem. Phys., 18 (1950) 529. 6. Kimball, G. E. Private communication to Pound and La Mer [2], 7. Fisher, J. C., Hollomon, J. H., and Turnbull, D. Trans. A.I.M.E., 185 (1949) 691. 8. Dorsch, R. G. and Hacker, P. T. Nat. Advisory Comm. Aeronaut. Repts. Tech. Mem. Notes (1950) 2142. 9. Levine, J. Nat. Advisory Comm. Aeronaut. Repts. Tech. Mem. Notes (1950) 2234. 10. Weibull, W. Roy. Swed. Inst. Eng. Research, No. 151 (1939). 11. Fisher, J. C. and Hollomon, J. H. Metals. Tech., 14 (1947), no. 5. 12. Griffith, A. A. Phil. Trans. Roy. Soc., 221A (1921) 163. SCHENECTADY, N. Y.