Creator:David Turnbull Date Created:July 18, 1949 Place Created:Schenectady, New York Keywords:heterogenous nucleation Context:article reprinted from Journal of Chemical Physics ************************************************** Reprinted from Thk Joirkal of Chkmical Physics, Vol. 18. No. 1, 198-203, February. 1 <>50 Printed in U. S. A. Kinetics of Heterogeneous Nucleation I). TYrnbi'LI. Research Laboratories, (tmeral Electric Company, Schenectady, New York (Received July 18, l')49) Consider a phase transition ai^ti. Stable 0 embryos tan be retained in the temperature range of or-stability in cavities of suitable extraneous solids present in the system. Retained 0-embrvos may nucleate the «—►/?-transformation in the range of /S-stability I'sing these concepts a theory for I he effect of thermal history upon the kinetics of phase transformations is proposed The kinetics of heterogeneous nuclealion in the absence of retained embryos is discussed. INTRODUCTION MOST of the theories for the kinetics of nucleation that have been advanced apply to "homogeneous nucleation" which is initiated in the interior of a single phase. The nucleation theory of Becker and Doring' was successful semiquantitatively in accounting for the experimental results of Yolmer and Flood5 on the rate of nucleation of liquids in supersaturated vapors. Although theories of homogeneous nucleation have had some qualitative success in explaining certain features of the kinetics of nucleation in condensed systems, many investigations have indicated that nucleation in these systems is "heterogeneous," that is nuclei form preferentially at some interface in the system. Particularly for liquid-solid transformations in relatively large volumes of liquid the evidence seems decisive that at least the first few nuclei that appear originate by a heterogeneous mechanism at the surfaces of the containing vessel or of colloidal inclusions in the system. If the growth rate is rapid these few nuclei may transform the entire sample without the formation of homogeneous nuclei. An important characteristic of liquid-solid transformations in relatively large samples is the very marked effect of thermal his ton- upon the rate of the process first observed by Othmer.3 Richards and coworkers^ have made comprehensive studies of the phenomenon and obtained convincing evidence that it must be explained in terms of extraneous structures present in the system. Webster® has shown that the effect also is important in the solidification of certain bulk metals. Similarly, the author has observed that the solidification of gallium is very thermal history dejiendent. Gallium is advantageous for study l>ecause of the very rapid growth rate (at least 1 cm/'sec. at 0°C) of the crystal so that the kinetics of the transformation are generally controlled by the nucleation rate alone. The effect of thermal history upon the kinetics of many liquid-solid bulk transformations is illustrated 1 R. Becker and W. Doring, Ann. d. Phvsik (3) 24, 719 (1935). »M. Volmer and H. Flood, Zeits. f. physik. Chemie 170A, 273 (1934). • P. Othmer, Zeits. f. anorg. allgem. Chemie 91, 235 (1915). 1 W. T. Richards, J. Am. Chem. Soc. 54, 479 (19.32). 'Richards, Kirkpatrick, and Hutz, J. Am. Chem. Soc. 5S, 2243 (1936). • W. L. \Vclster, Proc. Roy. Soc. 140A, 653 (1933). schematically in Fig. 1. A7'+ is the number of degrees above the melting point, Tm, to which the liquid is heated after melting. AT- is the number of degrees that the liquid must be subsequently sub-cooled in order for nucleation to occur during some specified holding time at the lower temperature. In the region of the curve ab which usually extends only a few degrees above the melting temperature, A7'_ is strongly dej>endent upon AT+ while in the region br AT. is virtually independent of A7\. For example, in gallium samples the region ab extended to about 20° above the melting point but for a given sample A7'_ was found to be independent of A7',, within experimental error, for values of the latter ranging between 50 and 1050°. The following are a numl>er of other important features of the thermal history effect. 1. The length of time in the sub-cooled range does not alter the relation between AT- and AT+ significantly. In exjxriments with gallium the whole curve was shifted only 1-2° toward higher temperatures when the holding time was shifted from the order of a few minutes to one day. V 2. Increasing the time at temperature above the melting point, after a certain minimal period—usually of the order of a few minutes, does not significantly affect the value of AT subsequently observed. i. AT- is generally characteristic of the highest temperature attained in the Al\ region. 4. When a sample is solidified after having been heated in the Ar-region the AT vs. AT+ relation origi nally obtained is usually not changed significantly. Certainly this is true for gallium. However, there are exceptions to this behavior. Hinshelwood and Hartley7 observed that after sulfur samples are heated into the -region and then solidified the A7'_ :s. AT: relation is altered so that the AT+ coordinate of b is only an insignificant fraction of its former value. 5. Richards demonstrated that the phenomenon was highly sensitive to added inclusions such as activated charcoal or powdered quartz. Thfte inclusions greatly increased the temperature (by as much as 100-150° in some cases) in the AT+ region that it was necessary to reach before A T- became independent of A TV 6. According to Richards the maximum sub-cooling ' C. N. Hinshelwood and H. Hartley, Phil Mag. 43, 78 (1922). 199 HETEROGENEOUS NUCLEATION obtainable for bulk specimens is independent of added inclusions. However, the author observed that this value was specific for particular gallium specimens and ranged from 35 to 55°. The specificity was not significantly changed by heating to A7+ values of 1050°. Richards proposed a qualitative theory for the effect of thermal history upon the transformation kinetics based on the concept that extraneous structures retain a "crystalline adsorbate" above the melting temperature. This adsorbate is stabilized in the regjon of liquid stability by a negative heat of adsorption on the extraneous structures. As the temperature of the liquid is increased the crystalline adsorbate becomes less stable and finally disappears altogether at some elevated temperature. He supjxises that its rate of reformation is very slow so that the process of heating into the for-region is essentially irreversible, as observed. Other investigations that demonstrated the role of extraneous surfaces in promoting the nucleation of solid phases in* liquids are those of Goetz,* Hammer3 and Meyer and Pfaff."' Although it has been recognized (see Johnson and Mehl") that a large number of solid-solid transformations are nucleated preferentially at grain boundaries, there seems to be little systematic information regarding the role of extraneous particles of other phases located either at the grain lx>undary or within the grain in promoting nucleation in solid-solid transformations. In view of the evidence on liquid-solid transformations it is expected that such influences may prove to be very imjx>rtant in many reactions involving only solid phases. Avrami1- has developed a theory for the kinetics of such transformations in which it is supposed that all nuclei originate from pre-existing embryos, some of which presumably might be stabilized by extraneous particles of other phases. It is the purpose of this pa[>er to develop further the theory of the kinetics of heterogeneous nucleation. An «A. Goetz, Phys. Rev. 35, 193 (1930). ' « C. Hammer, Ann. d. Physik 33, 445 (1938i. 10 J. Meyer and W. Pfaff, Zeits. f. anorg. ailgem. Chemie 217, 257 (1934)'. "W. A. Johnson and R. F. Mehl, Trans. A.I M E 135, 416 (1939). u M. Avrami, J. Chem. Phys. 7, 1103 (1939). explanation for the effect of thermal history upon transformation kinetics will be offered along the lines of the basic concepts suggested by Richards, but in more explicit terms containing the interfacial energies involved rather than heats of adsorption. A more a -- Kiu. 2. ti = embryo on a flat surface. 0 - equilibrium contact angle. mechanistic explanation will be offered for the irreversibility of the phenomena with respect to temperature. THEORY Volmer13 has calculated the work of nucleating a phase 6 in the transformation a—>ft, where £ is considered to be the low temperature modification, on a tlat solid substrate denoted by Let the contact angle between the tf-embryo and S be 6 as shown in Fig. 2. Then for 18O°>0>O Oa-s — er- atom, v= volume per atom, and r is the radius of the sphere. In the region of a-stability there is no possibility of retaining /8-embrvos on tlat surfaces for which the condition OX) holds since the term involving surface energy in (2) is always |x«itive for this condition. Volmer" shows that the work of forming a nucleus of critical size is, [16T^aV/3(/fl-/a)!I(2+co^)(l-cos0jV4] (3) which differs from the work of forming a nucleus in the interior of a by the factor [(24-cos6)(1 — cos0)*/4]. When cra-s> [2 sin7/cos '("y+0)3 X[1 — sin(7+0)]. Figure 4 shows the free energy relations involved in forming /3-embryos in cavities for T> Ta for this situation. At each temperature above To there is a stable position of the a—^-interface specified by r0. As the tem-)>erature increases r0 becomes less and the embryos will disappear altogether when r0 becomes of the order of the dimensions of the bottom of the cavity. After the embryos have been destroyed by the high temperature treatment a positive free energy at a lower temperature must be expended (.see Fig. 4b) to reform them because this process must begin on surfaces which are approximately tlat (i.e., on the rounded bottom of the cavity). In cylindrical cavities (see Fig. 5) the work of embryo formation is given by: AF= Tr-h{{s—ja) Vj)+2irrtja.^ X [r( 1 - cos0)/sin20 - h cos0]. (5) For values of h sufficiently large the surface energy term in Eq. (5) will be negative. However, the factor that determines whether or not jtf-embryos will be retained in the pores above the transformation tem- Fig. 4a. Free energy oi embryo formation in conical cavity al-ove transformation temperature. ])erature is the coefficient p, of h in (5) p— Trr-(Je~*fa) vA-2vrr. (6) The work of embryo formation as a function of h and r is shown schematically in Fig. 6. Again it is noted that after retained /3-embryos are destroyed by heating to high temperatures their reformation at a lower temperature involves an increase in free energy. For each temperature there is a critical value of r=m such that no irf-embrvos are retained in pores exceeding this size. m decreases sharply as the temjjerature increases. The existence of retained embryos in cavities on the container wall or in extraneous particles explains satisfactorily the effect of thermal history on the rate of nucleation. Suppose tnat the 0-phase is heated above Ta by varying amounts = AT^. The maximum values of 7 and r for conical and cylindrical cavities in which embryos are retained are shown as functions of ST+ in Fig. 7. When a is now sub-cooled into the /9-region, the embryos retained in the conical or cylindrical cavities will immediately grow to the mouth of the cavity. However, they are not nuclei for Lhe a —/^-transformation unless the radius at the cavity opening equals or exceeds the critical radius for nucleation of a spherical 2(11 HI I'KRO G K N IOCS \ 1' C L K A T I O X sector of fi on the flat surface 2ecimens quenched to 2 will have retained 0-embryos which are. supercritical, and these will nucleate immediately. If the growth rate is very slow, some embryos which were only slightly sub-critical may nucleate and grow. In any event, the nucleation rate will be a maximum at zero time and will fall off sharply thereafter. When the sample is quenched to a A 7' value of .v, all retained -embryos are sub-critical with respect to nucleation at this temperature. The additional free energy necessary to nucleate them may l)e so large that no nuclei will form in a reasonable period of time. At a ST indicated by v there are a number of retained embryos which are very close to the critical size for nucleation. These retained embryos may nucleate in Fig. 5. Nucleation in cylindrical cavity. Fig. 6. Free energy of embryo formation in a cylindrical cavity as a function of h. time, hut still later the nucleation rate will decrease as the retained embryos are used up. In general, the rate of nucleation, I, for this case- will go through a maximum with time as shown in Fig. 8. If y is very close to the curve A7\_ vs. A7'+ or if the rate of transport across the interface is very rapid, the initial increase, O.r, may not be observed at all. For example, Hammer" has published curves showing the variation in / with time for some liquid-solid transitions which resemble the curve in Fig. 8 excepting that the initial increase is absent. The way in which the total number of nuclei appearing isothermally in the range yz changes with temperature will be governed by the size distribution curve for cavities in which embryos are retained. Since the number of cavities generally will increase with decreasing cavity size, the nucleation number will increase as the temperature is lowered. From Fqs. (6) and (7) and setting fa—fa ~ A (7'0 — 7')/ To, where X = heat of transformation per atom, a simple equation relating A7' and AT"f can be derived for cylindrical cavities, Ar_M7V=tan0. (8) This equation holds only for one tv(>e of catalyst and \ D T I R N B r I. L 202 FlO. 8. Nucleation rate as function of time for a limited number of nucleation sites. where the rate of production of nuclei on flat surfaces or in the interior of a by thermal fluctuations is practically nil. This will be so for relatively small values of A7"_. For larger values of AT- the number of nuclei produced by thermal fluctuations becomes very large in comparison with the number originating from embryos retained in cavities; under these conditions, AT- is constant with respect to AT+ as shown by the be portion of Fig. 1. In order to find the order of magnitude of cavity sizes necessary to explain the observed effects in terms of the theory developed, some calculations were made for the gallium liquid-solid transformation catalyzed by a substrate such that 9 between the solid gallium (in equilibrium with the liquid) and the substrate was 60°; thus, Ar_«1.7A7V. It was assumed that the interfacial energy between liquid and solid gallium18 is 55 ergs/cm2. For AT+ = 5 embryos of solid gallium are retained in cylindrical cavities having diameters ~140A. These embryos are capable of growing at AT_~8.5, and the number of gallium atoms contained in the spherical sector at the mouth of the cavity is of the order of 6000. It is evident that nucleation by thermal fluctuations at this value of AT- in the absence of retained embryos is impossible. For AT+=20C the maximum diameter of cavities in which embryos are retained is =35A and AjT_k=34°. However, the number of atoms contained in the spherical sector at the cavity mouth is only about 100, so that nucleation by thermal fluctuations must become predominant at this temperature or slightly below it for the conditions assumed in the calculation. When a specimen is cooled into the ^-region from a temperature in the a-range where retained embryos have l>een destroyed the nucleation problem becomes one of forming nuclei on catalytic surfaces which are essentially flat. At steady state the number, n, of embryos of critical size per unit area of a given surface is n««a exp(—AF*/kT), (9) * where n$ is the number of a-surface atoms per unit area and AT* is given by Fq. (4). The steady state rate of nucleation per unit surface, Is, is Is-K exp[— (AF*+(])/kTlft where q is the activa- tion energy required to move an atom across the a — j3-interface. AF* differs from the free energy required for interior nucleation, AF*, only by a factor involving the contact angle. Therefore, AF* will vary with temperature in approximately the same way as AF* while q re-' mains essentially constant. Consequently Is will generally pass through a maximum as the temperature is lowered from T0 just as does I for interior nucleation. Also before the establishment of a steady slate of surface nucleation a transient in which Is accelerates with time should be observed as pointed out earlier.14 These features of heterogeneous nucleation in the absence of retained embryos are very similar qualitatively to those predicted by the theories of homogeneous nucleation. However, since AF* is much smaller, the rate of heterogeneous nucleation will generally be much greater at a given temperature than the rate of homogeneous nucleation. In some reactions there may be only a limited number of sites that promote nucleation. In this case the observed rate of nucleation may fall off with time after the rising transient in a way similar to that shown in Fig. 8. In the absence of a transient the rate of nucleation will change with time according to the equation given by Avramiu providing that all of the preferred nucleation sites are equivalent. In order to test the theories of homogeneous nucleation rigorously it is important to devise experiments in which nucleation catalysts are eliminated or localized. Experiments are usually performed on bulk samples where one or more nuclei originating from catalytic sites are able to transform the entire sample under observation. In order to prevent this from occurring, it is desirable that the sample*be broken into a large number of small isolated parts. This is accomplished in the well-known experiments of V. Schaefer16 on the nucleation of snow crystals in a fog of water droplets. B. Vonnegut17 has used a somewhat similar technique in his study of the liquid-solid transition in tin particles. Also, the author" has applied this procedure to mercury and gallium. In these investigations a much larger degree of sub-cooling was obtained than can generally be obtained when bulk samples are sub-cooled. Observations of the kinetics of transformations on fine dispersions have the following advantages to the study of nucleation: 1. Sincc the linear growth rate in transformations is approximately constant and the rate of nucleation is proportional to the volume, it is highly probaifle that in very' small droplets one nucleus transforms the whole mass of the droplet. Therefore, observations on the rate of transformation constitute a direct measure of the nucleation rate. 2. Only a negligible number of drbplets contact the container walls at any point, so that any points orr these walk which promote nlicleation are no longer effective. 18 D. Turnbull, Metals Tech. 15, T.P. 2365 (1948). V. J. Schaefer, Bull. Am. Meteorolog. Soc. 29, 175 (1948). » B. Vonnegut, J. Colloid Science 3, 563 (1948). r>. Turnbull, J. App. Phvs 20, 817 (19491 203 II E T E R O G ENEOrs NT r I. K ATION J. Extraneous particles promoting nucleation. if their number is not large, may Ik- localized on a negligible proportion of the droplets. 4. If the interfaces are properly chosen, nucleation takes place at the interior of the droplet rather than at the surface. This requires that the material at the interface "wet" the parent phase better than the phase which is forming. Thus, the technique of observing transformation kinetics on finely disjiersed droplets of Lhe mother phase covered by a good wetting agent appears to offer the most promise in measuring rates of homogeneous nucleation in condensed systems. For this purpose it is highly desirable to have dispersions in which the droplet size is very uniform, and accurately known. Under these conditions a simple kinetic expression may be written for the rate at which volume is transformed per unit time as follows: dV/dt=Iv(VB- V), (10) where 1= total volume transformed in time, t, F0= total volume at 1=0, 7 — rate of nucleation per unit volume, and r= volume of the droplet. This expression assumes that the transient in the nucleation rate is over and a steady state obtains. Equation (10) may be integrated to give: V= F<0-exp(-/t'0] for the volume transformed in time 1.