Creator:David Turnbull and Bernard Vonnegut Date Created:June 1952 Place Created:Schenectady, New York Keywords:nucleation catalysis Context:article reprinted from Industrial Enigneering Chemistry ************************************************** res. lab. reprint 1930 GENERAL0 ELECTRIC NUCLEATION CATALYSIS by DAVID TURNBULL AND BERNARD VONNEGUT SCHENECTADY, NEW YORK RES. LAB. RKPRINT 11)30 Reprinted from INDUSTRIAL AND ENGINEERING CHEMISTRY, Vol. 14, Page 12!)2, June 1052 Copyright 1952 by the American Chemical Society and reprinted by permission of the copyright owner Nucleation Catalysis DAVID TURNBULL AND BERNARD VONNEGUT GENERAL. ELECTRIC RESEARCH LABORATORY, SCHENECTADY, N. Y. It is known that the structures of crystals and the substances that catalyze their formation closely resemble each other in atomic arrangement and lattice spacing on certain low index planes. A crystallographic theory of crystal nucleation catalysis predicts that the order of catalytic potency should be identical with the order of the reciprocal of the disregistry (1/6) between the catalyst and forming crystal on low index planes of similar atomic arrangement; that for small 5 nuclei should form coherently with the catalyst—i.e., with a strain e = .aj and that for 5 very large 6 ^ e and the interface between nucleus and catalyst can be thought of as consisting of regions of good fit separated by a dislocation gridwork. The energy of this interface should be proportional to the dislocation density, hence to S — t. There is evidence that ice nuclei may form coherently on silver iodide surfaces (i = 0.0145). Experience indicates that in general nuclei form coherently with catalysts only for S < 0.005 to 0.015. ACCORDING to Volmer {28), it has been known for over a hundred years that certain solid bodies (called heterogeneities, motes, inclusions, etc.) extraneous to the system promote phase transformations, particularly condensation and crystallization. Generally, this fact has been explained on the basis that some heterogeneities catalyze the formation of nuclei of the new phase. Volmer (29) has given a formal treatment of the energetics of forming liquid nuclei on solid bodies from supersaturated vapor. The free energy difference, AF*, between a liquid nucleus of critical size and the supersaturated vapor is: A F* = 167ra3/3(A Fv)' (1) where a is the interfacial energy per area between liquid and vapor and AFy'is the free energy difference per volume between vapor and liquid phases of infinite extent. 9, the contact angle between the liquid and the surface of the solid catalytic body is given by the equation: 6 = arcos [(acv — ocl)/o] (2) where acv is the interfacial energy per area between the catalyst and vapor and ocl is the interfacial energy per area between the catalyst and liquid. The free energy of formation AFc, of a liquid nucleus of critical size on the catalyst surface is: AFC* = A F*f(d) (3) where/(») = (2 + cos 6) (1 - cos 6)'/4 (4) For 9<180<7W<1 and AF*>AFc. Becker and Doring (1) derived an expression for the frequency of formation per volume, I, of liquid nuclei in pure supersaturated vapors having the form: I = A exp.l — AF*/kT] (5) The frequency of formation per area, Ic, of liquid nuclei on the surface of the catalyst is given approximately by: Ic = 4(10-») exp.[—AF*f(6)/kT] (6) Actually the exponential factor dominates Equations 5 and 6 so completely that Ic> >/ for all 6 < 90°. The reverse process of forming bubbles in superheated liquids also is generally catalyzed by heterogeneities. Fisher (6) has published a formal treatment of this problem. Recently considerable information has accumulated on the catalysis of crystal nucleation by heterogeneities in supercooled liquids and in supersaturated solutions. These facts have come mainly from experiments on formation of snow crystals, solidi- fication of pure metals, crystallization of salt hydrates, and oriented overgrowth. The purpose of the present paper is to correlate the facts pertaining to the catalysis of crystal nucleation and to develop a theory that accounts for them. FORMAL THEORY Turnbull has developed a theory for the kinctica of crystal nucleation on the surface of heterogeneities (9, 21-24) that is analogous to Volmer's theory for the heterogeneous nucleation of liquid nuclei. The starting point of the theory is an equation derived by Turnbull and Fisher (25) for the nucleation frequency per volume of crystals in a supercooled liquid. The equation is identical in form with equation 5, but the kinetic coefficient, A has a different value. For a we must substitute ols, the interfacial energy per area between the crystal and supercooled liquid, while AFv becomes the free energy difference per volume between crystal and liquid phases of infinite extent. The numerical value of log A' is compatible with values calculated from experiments on the rate of solidification of mercury (23). It is assumed (1) that the structure of the crystal nucleus formed on the surface of the heterogeneity derives from that of the most stable macroscopic crystal phase by distortions due to the operating interfacial tensions, and (2) that solid interfacial tensions come to equilibrium in supercooled liquids with a contact angle described by an equation analogous to equation 2: 8 = arcos [(0.15 for close packed planes. Crystallization of Other Metals. Reynolds and Tottle (15) have obtained some interesting qualitative results on nucleation catalysis in the crystallization of zinc, aluminum, magnesium, lead, and copper. They coated the walls of the mold into which the metals were cast with a dispersion of small particles of various metals. The extent of grain-size refinement in the part of the casting in contact with the mold wall was assumed to be a measure of the catalytic potency of the metal particles that formed the coating. Significant grain-size refinement was observed when the disregistry between similar low index planes of the catalyst and forming metal was less than 10%. Oriented Overgrowth. Often crystals form on the surface of a foreign crystal with a definite orientation relation. This phenomenon is called oriented overgrowth or epitaxy. Generally, the orientation relation is: The planes and directions in the two crystals in which the atomic arrangement is most similar are parallel. It is generally believed that oriented overgrowth is caused by oriented nucleation on the catalyst surface (10). Therefore, we shall see how experience on oriented overgrowth relates to other nucleation catalysis phenomena. Recently Thomson (20), Van der Merwe (26), and Johnson (10) have reviewed the results on oriented overgrowth. It had been generally believed that the necessary condition for oriented overgrowth is 5 < 0.10 to 0.20. However, Johnson (10) and Schulz (17) have found instances of-oriented overgrowth for 5 as large as 0.50. This is in agreement with the fact that certain substances, such as mercurous iodide, apparently weakly catalyze the nucleation of mercury crystals, though their structures are very different from that of mercury. To test quantitative theories for oriented overgrowth, it is necessary to know how S varies with the critical supersaturation ratio, C/C0, necessary for the growth. Unfortunately, we now have very little knowledge of this variation. It seems possible that the maximum disregistry S ~0.10 to 0.20 supposedly compatible with oriented overgrowth may correspond to C/C0 values at which accidental heterogeneities on the catalyst surface or elsewhere become active nucleation catalysts. Crystallization of Salt Hydrates. Telkes (19) found that Glauber's salt (NajSOj.lOHjO) may not crystallize from an aqueous solution until the solution is undercooled on the order of 17° C. below the equilibrium temperature of 32.31° She established that the addition of 2 to 3% borax (Na2B407.10H20) reduces the undercooling necessary for recrystallization to 1° or 2°. Telkes points out that the effectiveness of borax in nucleating Glauber's salt is plausible from a crystallographic point of view, because the two crystal structures belong to the same space group (Cjj) and S = 0.015 for the basal planes. MECHANISM OF NUCLEATION CATALYSIS To summarize the experimental findings: There is good qualitative evidence that potent nucleation catalysts have low index planes in which the atomic arrangement is similar to that in certain low index planes of the forming crystal; there are tenuous indications that the order of potency of catalysts corresponds to the order of the reciprocal of the disregistries (1/5) on these low index planes. In view of these findings it is desirable to formulate a crystallo-graphic theory of nucleation catalysis. A theory may be derived based in part on concepts advanced by Frank and Van der Merwe (7) in their treatment of the energetics of formation of oriented monolayers on crystalline substrates. It may be assumed that the lattice structure in the surface of the catalyst is identical with that of similar elements in its body and that surface elements are not strained by the formation of nuclei upon them. If a nucleus is strained by the amount S in two dimensions, so that it precisely fits a surface element of the catalyst, it is said to be coherent with that element. The lattice parameter of the nucleus in a direction normal to the surface element will assume the value necessary to minimize the free energy of the system. (A nucleus formed in a notch or at a step in the catalytic surface might be coherent in three dimensions. However, for 0 very small it turns out that the free energy of nuclei is less for two- than for three-dimensional coherency.) The basic postulate of the theory is that the interfacial energy between the nucleus and catalytic surface element is a minimum when the nucleus forms coherently. In general, the condition of minimum free energy will be that the nucleus is not coherent but strained an amount e «>0 the nucleus is said to be incoherent with When the disregistry <0.20, the boundary region between the nucleus and catalyst surface can be pictured as made up of local regions of good fit bounded by line dislocations, as indicated schematically in Figure 1. [The concept that interphase boundaries may be described in terms of a dislocation model has been developed by Van der Merwe (27) and Brooks (2). ] The dislocation density per area, p, will be proportional to 8 — e. Frank and Van der Merwe (7) have shown that the energy due to dislocations in a misfitting monolayer is proportional to p. We shall assume that the interfacial energy between the nucleus and catalyst due to the dislocation gridwork is proportional to p. Therefore, we may express > t. Our considerations will apply whether the extended phase (denoted by symbol L) is a supercooled pure liquid or supersaturated solution of liquid or gas. When A Fv + ce2 < 0, A F goes through a maximum having coordinates AF* and r*. In order to survive as a nucleus for further growth, a strained embryo on a catalyst surface must attain the size r*. The critical free energy, AFc*, is given by: AFc* = 4rahs (2 + m) (1 - m)2/3(AJV + ce2)2 (15) We wish to know the relation between A Fv and S that holds when the nucleation rate is perceptible. Taking a perceptible rate to be 1 cm.-2 sec.-1 it follows from Equation 8 that corresponding to this rate Afc* ~60kT. For cS1> >|AF^r| and making the approximation 2 + m ~ 3 a linear relation between \AF'V\ and 5 obtains: |af;| = (4T/'(l3 1 and incoherent for \Afv\/eS'> 1. Assuming 0 = 0, we now find for a given S what value of e corresponds to highest probability, p, of nucleus formation. Since pa exp. [(— AFc)/kT] we require the value of e that corresponds to a minimum AF*. Setting d(AF*)/de = 0 (see Equation 15) and making the approximations 1 + m 2, (2 + m) ta 3, we find: e/S = 1 - (1 - |AfV| /c«2)'/2 (23) For/3 ^ 0, « = (S + fiaLs/a) - [(« + P c52 the root of Equation 15 is imaginary, which means that AF* = 0 and the nucleus will always form coherently withe = S as indicated in the figure. When co2 >>|AFv|,5 >>e. Elimination of efrom Equation 15 gives: A Fc* = 7ra2ffis«2(l - |AfV| /cS2)/[AFv + c52{l - (1 ~ |AfV|/c52)'/s|]2 (24) In summary, our simple crystallographic theory of nucleation catalysis for 0 =0 makes the predictions: 1. When catalysts and the forming crystals have similar low index planes, the order of the catalysts in |AF£| will be INDUSTRIAL AND ENGINEERING CHEMISTRY NUClEATION_Theory, Review. identical with their order in 5—i.e., if Si > St > S3, then | A/q , > |Af , > |AFK| 3. 2. For values of S sufficiently small so that Equation 20 is approximately fulfilled, nuclei will form coherently with the catalyst and |AF^| = c5s. [A related problem, the thermodynamics of formation of coherent precipitates from a solid solution, has been treated by Leschen and Fisher (IS).] 3. For 4 very large nuclei will form incoherently with negligible strain and |AFV | will be a linear function of 5. COMPARISON OF THEORY AND EXPERIENCE There are a few data on nucleation catalysis sufficiently quantitative to test some of the predictions of the simple theory. Values of A'exp. (Equation 8) calculated from the kinetic data on the solidification of mercury (23) and tin (14, 31) droplets coated with various films are generally in good agreement with A 'tk values calculated from nucleation theory on the assumption that the strain, e, is zero. [However, kinetic results on mercury droplets coated with mercurous iodide could not be interpreted on the basis that the mercury nucleus had the normal structure (23).] If « were not zero, the "melting point" of the nucleus would be less than Tm and A 'exp. would be larger than the apparent A'tk. Pound and LaMer's value of 10~8 A'exp. (10™) for oxide-coated tin droplets is at least a factor of 100 larger than 10 ~*Alk (10M), but the disagreement corresponds to a coherency strain of the order of only 0.01. Because the structures of the surface films in the mercury and tin experiments arc very different from the structures of the metal crystals, the crystallographic theory predicts, in good agreement with experience, that e should be negligible. Table IV. Coefficients 0.01 0.02 0.03 0.04 0.05 0.06 S Figure 4. Supercooling Corresponding to Coherent Nucleation of Ice Crystals on Catalyst Planes As function of disregistry 5 with (001) planes of ice The authors have not attempted to evaluate a and, therefore, have no predictions on the maximum value of S at which nuclei will form coherently However, it is interesting to calculate the supercooling, A?', or supersaturation ratio, a/a„ as a function of S, assuming coherent nucleation for all values of S. AT = f(S) for coherent nucleation of crystals in pure liquids is found by combining Equations 19 and 21. This function was evaluated for the nucleation of aluminum and ice crystals on their closest packed planes (111) and (001), respectively. The appropriate values of c as a function of the elastic coefficients were calculated by methods summarized by Ilearmon (8) and Zener (32) and are given in Table IV. Coefficients c' refer to a cartesian xy plane 11 (111) of the crystal and the c coefficients to xy 11 (100). As no reports of measured elastic coefficients of ice have been found, AT = f(S) was evaluated from the theoretically calculated values of Penny (13) and plotted in Figure 4. A point corre- Forming Crystal Ice Aluminum NaCl Temp., ° 273 933 300 Plane Hexagonal (001) (111) (100) In terms of elastic coefficients Cll + Cll — 2c*,/cu 2c1'»/«1» Cll + Cl) — 2cJa/ci. Numerical, dyne cm. 1.7 X 10" 7.2 X 10" 5.34 X 10" sponding to silver iodide (the only available datum, see Table IV) is shown. In view of the large uncertainty in the elastic constants, the remarkable agreement between the silver iodide point and the calculated relation (2.5 "vs. 3.1 °) may be fortuitous. However, the result gives some support to the concept that ice nuclei form coherently on a silver iodide surface. The supercooling corresponding to coherent nucleation rises very sharply with disregistry, so that at 4 ~ 0.055 AT ~ 40°. Figure 5. Supercooling Corresponding to Coherent Nucleation of Aluminum Crystals on Catalyst Planes As function of disregistry, {with (111) planes of aluminum The elastic constants for aluminum are those given by Zener, but corrected for temperature variation on the assumption that c varies with T in the same way as the rigidity modulus, M. Ke's (11) measurement of M = f(T) for an aluminum single crystal was used to make the calculation. The resulting relation AT = /(5) is shown in Figure 5. Cibula finds that titanium carbide, titanium boride, and aluminum boride for which 4 ~0.04 to 0.00 catalyze formation of nuclei for AT < 1 The authors' relation indicates that coherent nucleation for a disregistry 4 = 0.04 cannot take place until AT > 100°. It seems eit her that aluminum nuclei must form incoherently for all 4 > 0.005 or that some of the assumptions of the theory are not valid for the catalysts under consideration. Combination of Equations 18 and 21 gives a/a, = f(S) valid for the formation of coherent nuclei from a supersaturated solution. Figure 6 shows the relation a/a, = f(S), calculated from elastic coefficients tabulated by Ilearmon, for coherent nucleation of sodium chloride on (100) planes at room temperature. a/a, = 1.06 at 4 = 0.01 and 2 at 4 ~ 0.035. It is doubtful that supersaturation ratios as large as 2 are required to initiate oriented overgrowth for 4 ~ 0.035. Tt is unlikely that sodium chloride nuclei form coherently for 4 < 0.015. In view of the foregoing calculations and experimental evidence, it seems that nuclei are likely to form coherently only for 5 < 0.005 to 0.015; for 4 > 0.02 the strain, t, is probably much smaller than 4. As a has not been calculated from theory, it is interesting to 1296 INDUSTRIAL AND ENGINEERING CHEMISTRY Vol. 44, No. 6 N UCLEATI ONTheory, eview ACKNOWLEDGMENT The authors gratefully acknowledge helpful conversations with J. G. Leschen, J. C. Fisher, and R. L. Fullman concerning some of the subjects discussed in the paper. Figure 6. Supersaturation Ratio Corresponding to Coherent Nucleation of Sodium Chloride Crystals on Catalyst Planes Aa function of disregistry 5 with (100) planes of sodium chloride estimate a from the data on the crystallization of ice and aluminum. For the formation of ice nuclei the critical disregistry 5e at which nucleation becomes incoherent (see Equation 20) is estimated to be on the order of the disregistry between ice and silver iodide, «„ ~ 0.015. Taking the value of 0.02. Quantitative data are needed on the kinetics of oriented nucleation of salt crystals as a function of supersaturation, and measurements on the kinetics of heterogeneous nucleation of crystals in supercooled liquids should be extended to test the effects of additional kinds of catalysts. NOMENCLATURE activity of solute lattice parameter in low index comparison plane of structure from which nucleus derives activity of solute in saturated solution difference in lattice parameter in low index comparison plane of nucleation catalyst and structure from which the nucleus derives kinetic coefficient (1/time X volume) for nucleation of liquid in supersaturated vapor kinetic coefficient (1/time X volume) for nucleation of crystals in supercooled liquids experimental kinetic coefficient kinetic coefficient predicted by nucleation theory = coefficients of elasticity length of hexagonal axis in hexagonal unit cell concentration of solute concentration of solute in saturated solution free energy required to form a nucleus of critical size free energy required to form a nucleus of critical size on the surface of a nucleation catalyst difference in free energy per volume between two phases, of infinite volume value of A Fv corresponding to a perceptible nucleation rate nucleation frequency per volume nucleation frequency per area of surface of nucleation catalyst Boltzmann's constant cos 6 rigidity modulus probability of nucleation potency of nucleation catalyst radius of sphere, a sector of which constitutes a nucleus on catalyst surface value of r corresponding to nucleus of critical size on the catalyst surface molar gas constant entropy of phase transition per volume absolute temperature, ° K. absolute temperature at which nucleation is perceptible absolute temperature at which liquid and crystalline phases are in equilibrium Tm - Ti phase volume per mole lattice parameter of strained nucleus structural free energy per area X (S — e) of crystal- catalyst interface 1 — (acl — y )/