Nucleation Catalysis DAVID T U R N B U L L
AND
BERNARD VONNEGUT
GENERAL ELECTRIC RESEARCH LABORATORY, S C H E N E C T A D Y , N . Y.
It i s known t h a t t h e structures of crystals and t h e substances t h a t catalyze their formation closely resemble each other i n atomic arrangement and lattice spacing on certain low index planes. A crystallographic theory of crystal nucleation catalysis predicts t h a t t h e order of catalytic potency should be identical w i t h t h e order of t h e reciprocal of t h e disregistry (1 /6) between t h e catalyst and forming crystal on low index planes of similar atomic arrangew i t h a strain E m e n t ; t h a t for small 6 nuclei should form coherently with t h e catalyst-i.e., = 6 ; and t h a t for 6 very large 6 E and t h e interface between nucleus and catalyst can be thought of as consisting of regions of good fit separated by a dislocation gridwork. T h e energy of this interface should be proportional t o t h e dislocation density, hence t o B - E. There is evidence t h a t ice nuclei m a y form coherently on silver iodide surfaces ( 6 = 0.0145). Experience indicates t h a t i n general nuclei form coherently with catalysts only for 6 2 0.005 t o 0.015.
>>
A
CCORDING to Volmer (R8), 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, A F * , between a liquid nucleus of critical size and the supersaturated vapor is: 4F* = 1 6 s ~ ~ / 3 ( A F v ) ~
(1)
where u is the interfacial energy per area between liquid and vapor and AFv is the free energy difference per volume between vapor and liquid phases of infinite extent. e, the contact angle between the liquid and the surface of the solid catalytic body is given by the equation :
e
= UTCOS [(uCv -
UCL)/U]
(2)
n-here ucv is the interfacial energy per area betneen the catalyst and vapor and UCL is the interfacial energy per area between the catalyst and liquid. The free energy of formation APc, of a liquid nucleus of critical size on the catalyst suiface is: AFc" = AF*f(0)
+
(3 1
nheref(0) = (2 cos e) (1 - COS 0)*/4 (4) For e < l 8 0 " f ( e ) < l and AF*>AFc". Becker and Doring ( 1 ) derived an evpression for the frequency of formation per volume, I , of liquid nuclei in pure supersaturated vapors having the form:
I = A exp.[-AF*/kT]
(5)
The frequency of formation per area, IC, of liquid nuclei on the surface of the catalyst is given approximately by:
IC
= A(10-8) exp.[-4F*j(O)/kT]
(6)
Actually the exponential factor dominates Equations 5 and 6 so completely that IC> >I for all e 7 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-
1292
fication of pure metals, crystallization of salt hydrates, and onented overgro7Tth. The purpose of the present paper is to COIrelate 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 kinetics of crjstal nucleation on the surface of heterogeneities (9, 21 -24) that is analogous t o Volmer's theory for the heterogeneous nucleation of liquid nuclei. The starting point of the theory is an equation derived by Turnbull and Fisher ( 2 5 ) 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, '4'. has a different value. For u we must substitute U L S , the inteifacial energy per area betmeen the crystal and supercooled liquid vihile AFv becomes the free energy difference per volume betneen 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 (WS). 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:
e
=
wcos
[(uCL- U ~ S ) / ~ L , , I
(7)
where C denotes catalyst, L liquid, and S the forming crystal. Equation 7 also implies that U L S is independent of crystal oiientation. The validity of assumption ( 2 ) is supported mainly hy evidence of the kind presented by Smith (18). With these assumptions the following expression is derived for the frequency of heterogeneous formation of crystal nuclei.
IC
N
10-8A4' exp.[-4F*f(B)/kT]
(8)
Equation 8 makes the important prediction that the kinetics of heterogeneous nucleation of crystals can be described as a function of temperature in terms of the parameters that describe homogeneous nucleation and a single additional parameter, 0 This prediction is fulfilled by recent results (RS)on the frequency of crystal nucleation in supercooled mercury droplets coated with mercury acetate. The nucleation frequencl per droplet is proportional to droplet area and the measured kinetic coefficient, 10-*A'Cxp, is in excellent agreement with the value 10-8A' predicted by Equation 8. Kinetic coefficients calculated from isothermal experiments on mercury stearate-coated droplets ( 2 3 ) and on oxide-coated tin droplets ( f 4 , S l )are in fair agreement with
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 6
NUCLEATION-Theory, the values calculated from the theory. In these experiments a given isotherm could not be described by a single value of IC, so the results are not so conclusive as those on acetate-coated mercury droplets. This agreement between theory and experiment indicates t h a t assumption (1) is approximately valid insomeheterogeneous nucleation processes. The formal theory implies little about the molecular mechanism of nucleation catalysis. I n order t o obtain a better understanding of this mechanism, we shall examine how the catalytic potency of a heterogeneity is affected by its structural relationship with the forming crystal. STRUCTURAL RELATIONS
IN
N U C L E A T I O N CATALYSIS
Ice Formation. I n most heterogeneous nucleation processes, the temperature dependence of the nucleation rate is so great that there seems to be a narrow temperature range in which the nucleation catalyst becomes active. The crystal nucleation rate becomes perceptible at the threshold temperature, T,, of catalytic activity. Often Ti is shifted 1' or less by changing the cooling rate (from the equilibrium temperature into the supercooled range) a factor of 10. Vonnegut (SO) and Schaefer (16) have'measured Ti for the formation of ice crystals in a supercooled fog containing suspended particles of various substances. Some of the results are summarized in Table I. a. and co are the dimensions of a hexagonal unit cell. Aa and A(c,/a,) are the amounts that the dimensions of the catalyst unit cell exceed the ice unit cell. T , is the melting temperature of the forming crystal, and A T , = T , Ti is the supercooling a t which the catalyst becomes active. 8 is the disregistry on a low index plane; for the (001) plane of ice 8 = ~ A a / u o ~The . most potent catalyst known for the formation of ice crystals is silver iodide. Vonnegut (SO) has proposed that the effectiveness of silver iodide derives from the remarkably close fit between its lattice structure and that of ice.
-
Table I. Supercooling (T, - T, = AT,) Corresponding t o M a x i m u m Temperature a t Which Various Substances Catalyze Formation of Ice Nuclei Substance AT' C. 6 l A ~ / a o I A(co/ao)/(co/~o) Silver iodide (SO) 2 5 f0 1 0 0145 + O 0043 Lake 4lbany clay (16) 11 1 Volcanic ash (Crater Lake) ( 16 ) 16 f 1 Cryolite (16) 20 1 Topaz (16) 23 & 1
* *
Schaefer (16) measured the catalytic activity of several naturally occurring inorganic substances, and part of his data are given in Table I. Most of these substances are chemically complex and it is not certain what structure caused the formation of ice in a given instance. Crystallization of Mercury. A T < has been measured for various surface films on small mercury droplets ( 2 3 ) . The results for droplets about 10 microns in diameter are given in Table 11. A cooling rate -1' per minute was used in these measurements. Table II.
AT, for Nucleation of Mercury Crystals Coated by Various Films (23) Film
A Ti
Mercury sulfide (black), HgS Mercury acetate, HgAc Mercury iodide Hg& Mercury stearake, Hg(St) Mercury laurate, Hg (Lau)
3 . 0 f0.5 12 1 45 f 1 48 f 1 58 & 1 77 f 1
HEX
*
0.19
Table I1 gives the major component of the film; it is possible t h a t catalysis was effected, in some instances, by minor components of unknown character. Therefore, AT4 for the major component must be equal to or greater than the value given in June 1952
Review
the table. Kinetic evidence indicates t h a t mercury nuclei form in the body of mercury laurate-coated droplets without the aid of the film (homogeneous nucleation). These results demonstrate again that the potency of different nucleation catalysts varies widely. We may think of the existence of a spectrum of catalysts with bands corresponding t o different catalytic structures ranging all the way from the equilibrium temperature t o the temperature of homogeneous nucleation. Data adequate for comparing the structure of the major film component with t h a t of solid mercury exist only for mercurous iodide and mercuric sulfide. The arrangements of mercury atoms in solid mercury and mercurous iodide on low index planes are not similar. Mercury-mercury separations in the two crystals differ by 10 to 30%. The arrangements of mercury atoms on a (111) plane in solid mercury and on the closest packed plane in mercuric sulfide are similar with atomic separations differing by 19%. Therefore, i t seems that the structure of solid mercury resembles more closely that of mercuric sulfide than of mercurous iodide, and the experiments are compatible with the concept t h a t the order of catalytic potency is the same as the order of structural similarity. HgX, the most effective catalyst for the nucleation of mercury crystals, formed from the mercury laurate or mercury stearate films. The chemical evidence suggests t h a t HgX may be a mercury oxide resulting from hydrolysis of the carboxylates. Mercuric oxide has no low index plane, populated by mercury atoms, structurally similar t o t h e low index planes of pure mercury; the mercury-mercury separations in the two crystals differ by 5 t o 10%. It is possible t h a t H g X is mercurous oxide, but the authors found no information on the structure of mercurous oxide. Crystallization of Aluminum, It has long been known that certain addition elements, such as titanium and zirconium, markedly refine the as-cast grain size of aluminum. For example, 0.1% titanium additions to a pure aluminum melt result in a fine-grained equiaxed grain structure as compared with the very coarse columnar structure t h a t often forms in castings to which no additions have been made (6). It is important in practice to promote the formation of metals of fine-grained structure, as their mechanical properties are often markedly superior to those of metals having a coarse grain structure. The mechanism of grain size refinement by additions to the melt has long been disputed. However, recent results of Eborall (6) and Cibula (3, 4)on the casting of aluminum and its alloys strongly support the idea t h a t grain refinement is largely effected by nucleation catalysts. Aluminum melts with or without elements t h a t did not cause grain-size refinement supercooled about 1' before solidifying, but when grain-size refining elements were present solidification began without perceptible supercooling. Cibula found the average grain diameter of aluminum with titanium additions, when centrifuged immediately before solidification, is about ten times larger than in a casting that had not been subjected to a centrifugal field when in the molten state. Therefore, he concluded t h a t the grain-size refinement is largely effected by insoluble impurities and t h a t the mechanism is nucleation catalysis. From indirect evidence (5) i t was inferred that the nucleation catalysts were carbides of the effective addition elements, with the exception t h a t the potent grain-size refinement effect of titanium and boron in combination is ascribed to titanium boride ( 4 ) . Titanium carbide was identified in a casting to which titanium was added and the arrangement of metal atoms in the closest packed planes of the carbides of the effective addition elements (and of titanium boride) is similar to t h a t in the closest packed plane of aluminum (111). Table I11 compares the nucleating effect of various metal additions (ascribed to carbides and borides) in the crystallization of aluminum. The carbides of elements not effectivein refining grain size contain no low index
INDUSTRIAL AND ENGINEERING CHEMISTRY
1293
NUCLEATION-Theory,
Review
Table 1 1 1 . Nucleation Catalysis by M e t a l Carbides and Borides i n Casting of A l u m i n u m Face-Centered Cubic Struct u re (After Cibula, 3) Metal Compound Vanadium carbide Titanium carbide Titanium boride Aluminum boride Zirconium carbide Niobium carbide Tungsten carbide Chromium carbide Manganese carbide Iron carbide
Formula of Compound
VC
TiC TiBa hlB2 ZrC IibC W2C
CraCn, CrrC
MnsC. N n r C FesC
.
Crystal Structure Cubic Cubic Hexagonal Hexagonal Cubic Cubic Hexagonal Complex ComDlex Complex
6 For Close
Packed Planes 0.014 0.060 0.048 0.038
0.148 0.086
0.035
...
... ., .
Xucleating Effect Strong Strong Strong Strong Strong Strong Strong Weak or nil Weak or nil Weak or nil
plane having an atomic arrangement similar to the atomic arrangement in a low index aluminum plane. For the effective 0.15 for close packed planes. catalysts 6 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 t h a t formed the coating. Significant grain-size refinement was observed when the disregistry bet\? een 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 ovex growth or epitaxy. Generally, the orientation relation is: The planes and directions in the two crystals in n-hich the atomic arrangement is most similar are parallel. It is generally believed t h a t oriented overgrowth is caused by oriented nucleation on the catalyst surface (10). Therefore, n e shall see how experience on oriented overgrowth relates to other nucleation catalysis phenomena. Recently Thomson ( d o ) , Van der hlerwe ( d B ) , and Johnson (10) have reviened the results on oriented overgrowth. It had been generally believed that the necessary condition for oriented overgrowth is S 0.10 to 0.20. However, Johnson ( I O ) and Schulz ( 1 7 ) have found instances of oriented overgrowth for 6 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 t h a t of mercury. To test quantitative theories for oriented overgrowth, it is necessary t o know how 6 varies with the critical supersaturation ratio, C/C,, necessary for the growth. Unfortunately, we n o x have very little knowledge of this variation. It seems possible that the maximum disregistry 6 -0.10 to 0 20 supposedly compatible with oriented overgrowth may correspond to C/C, values at Fhich accidental heterogeneities on the catalyst surface or elsewhere become active nucleation catalysts. Crystallization of Salt Hydrates. Telkes (19) found that Glauber's salt (Sa2S04.10HzO) may not crystallize from an aqueoua solution until the solution is undercooled on the order of 17" C. below the equilibrium temperature of 32.34'. She estab~O) lished that the addition of 2 t o 370 borax ( K a ~ B ~ O ~ . l O Hreduces the undercooling necessary for recrystallization t o 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 (C:,) and 6 = 0.015 for the basal planes. M E C H A N I S M OF N U C L E A T I O N CATALYSIS
To summarize the experimental findings: There is good qualitative evidence that potent nucleation catalysts have low Endex
1294
planes in which the atomic arrangement is similar to that in certain low index planes of the forming crystal; there are tenuous indications that t.he order of potency of catalysts corresponds to the order of the reciprocal of the disregistries (1,'s) on these low index planes. In view of these findings it is desirable to formulate a crystallographic theory of nucleation catalysis. A theory may be derived based in part on concepts advanced by Frank and Van der Merwe (7) in their trcatrnent 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 t h a t surface elements are not strained by the formation of nuclei upon t,hem. If a nucleus is strained by the amount 6 in two dimensions, so that' it precisely fits a surface element of the catalyst, it is said to be coherent Tvith that element. The lattice parameter of the nucleus in a direction normal to t'he surface element will assume the value necessary to minimize the free energy of the syst,em. ( A nucleus formed in a notch or a t a step in the catalytic surface might be coherent in three dimensions. However, for e very small it burns out that the free energy of nuclei is less for h o - 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 coherent'ly. In general, the condition of minimum free energy will be that the nucleus is not coherent but strained an amount E < 6, where E =
j (z
- adiae I
(9)
and a, and 2 are the lattice parameters of the nucleus in the strain-free and strained condit,ion, respectively. Therefore, the actual disregistry between a nucleus and the catalytic element is 6 - E. When the nucleus is said to be incoherent with the catalytic element. W h e n t h e d isr egi st r y < 0.20, the boundary region ? ? 7 9 ? 7 ? between the nucleus and + + i i + + i i : y y : catalyst surface can be pictured as made up of local regions of good fit bounded by line dislocations, as indicated s c h e m a t i c a l l y i n Figure 1. [The concept that interphase boundaries may Figure 1. Cross Section I be described in terms of a dist o (100) Plane of T w o Mish a s been fitting Simplecubical Latlocation tices developed by Van der Showing t h a t interface between Merwe ( 2 7 ) and Brooks ( d ) . ] t h e m can be described by small The dislocation density per regions of relatively good fit area, P , will be proportional to bounded by dislocation network 6 - E . Frank and Van der Z'lerwe (7) have shown that the energy due to dielocations in a misfitting monolayer is proportional to p. We shall assume that the interfacial energy between the nucleus and catalyst due t o the dislocation gridwork is proportional to p. Therefore, we may express gSc as the sum of two terms : U8C = y a(6 - E ) (10)
~>EYO
7
+
where y is an interaction term due to bond type, chemistry, etc., and a(6 - E ) a p is a structural term. It follows from Equation 10 that: m = C O @~ =
[UCL
and 1
-m
=
p
1
6
-
y
-
a( 6-
+ a(6 -
E)]/ULS
E)/ULS
(11) (12)
JThere
- (UCL -
(13) Now suppose that a nucleus having the shape of a sector of a sphere of radius, T , forms on a catalyet surface (see Figure 2, 2 d ) ; =
INDUSTRIAL AND ENGINEERING CHEMISTRY
Y)/ULS
Vol. 44, No. 6
NUCLEATION-Theory, then the free energy increase due t o formation of the nucleus is (6): A F = "(AFv
+ ce2) + 2.rrr2(l -.rrr2(1 - +m2) (usc ~ ) U L , S
ULC) (14) where Ti = the volume of the sector; ce2 = the strain energy per volume of the nucleus; c will be expressed in terms of the appropriate elastic coefficients as the need arises. 2~732(1- m ) = area of the nucleus in contact with extended phase d ( 1-m2) = area of nucleus in contact with the catalyst
/
Review
lAF:l = ( 4 a u ~ s / B O k T ) ' / ~ a 6 (18) When lAF;I given by Equation 18 becomes larger than cP, the nucleus forms coherently and we shall have instead of Equation 18 a parabolic relation between /AF;/ and 6 : 1AF:I = cS2 (19) The approximate condition for coherent nucleation is:
7
6 (4~rur,s/60kZ')'/~ (a/c) (20) Figure 2 shows schematically the variation of lAF;l with 6 predicted by our theory. In the solidification of pure liquids lAF;I can be closely approximated by: 1AF:I = IASvl A T (21) where A S v is the entropy of fusion per volume Since ASv is assumed indepen,dent of temperature, A T will vary with 6 in the same way as AF In precipitation of crystals from a supersaturated solution:
1
1.
IAF:I
= / ( R T / V )In (ala,)
(22)
where V is the molar volume of the precipitate, asis the activity of the solute in equilibrium with the precipitate, and a is the solute activity in the supersaturated solution. Therefore, In (ala,) varies with Sin the same way as 1AF;I.
Figure 2.
I
I
I
8 jAF'I as a Function of Disregistry, 6
Parabolic p a r t of curve indicates coherent nucleation. Straight llne portion indicates Incoherent nucleation w i t h 6 > C.
>
Our (solisideratioils will apply whether the extended phase (denoted by symbol L ) is a supercooled purc liquid or supersaturarcd solution of liquid or gas. Khen U v - ce2 < 0, AF goes through a masiinum having coordinates AFa and T * . In order to survivc as a nucleus for iurther growth, a strained embryo on a catalyst surface must attain the size r * . The critical tree energy, AFC*,is given by: A F ~ *= IXU'LS ( 2 T m ) (1 - ~ 1 : ' , ' 3 ( 1 F vT ~ € 2 ) ' (15) lye wish to know the relation between A F v and 6 that holds when the nucleation rate is perceptible. Taking a perceptible rate to be 1 cim-2 see.-' it follows from Equation 8 that corresponding to this rate AFc* 6OX.T. For cS2> > AF;, and making /n 3 a linear relation between ,AJ';,, the approximation 2 and 6 obtains:
+
--
lAF;I = ( 4 r u ~ s ; 6 0 k T ) '(/ f~u r s f as) (16) where /AF;.l is the value Gf ;IF;,' corrcsponding to IC = 1 cni. --? sec -1. The potency, P, oi a nucleation catalyst for a given trailsformation is: a l , ' ( B u ~ s nS) I n general, P tlcpends upon tlie disregktry, 6, and an iiiteract?on term, 6, that may not he related t o 6. Froni this relation, it is predicted that, the order of catalytic potency will not always be identical with the order of 1 , 6 . The condition that P be R function of 1'8 only is that 3 = 0. From Equation 13, 8 = 0 when
+
UCL
- 7' =
ULS
(17.
.ktually Equation 17 may be approximately valid for many nuclcation processes and it is interesting to find what the theory predicrs when the equation holds. (Equation IT follows approsiniately for the iiuclcation of crystals in supercooled liquids if the interaction energy due t o bond type, chemistry, etc., between the catalyst and the crystal is the same as between the catalyst and liquid. Such a relation is plausible.) For 3, =0, Equation 16 becomes: 'June 1952
Figure 3.
e/6 as Function of
I/1Fv j/c62,
Nucleatlon Is coherent for AFvj/cSQ> 1 and incoherent for
IAFvl/cP> 1.
Assuming B = 0, we now find for a given 6 what value of E corresponds t o highest probability, p , of nucleus formation. Since pa exp. [( - A F z ) / k T ] we require the value of E t h a t corresponds to a minimum AFZ. Setting d(AF,*)/de = 0 (see Equation 15)and making the approximations 1 m ts 2, ( 2 m ) ts 3, w e find:
+
e/6 = 1
+
- (1 -
lAFvl /cS2)'/'
+
+
(23)
For p # 0, E = (6 p u ~ s / a) [(a P U L S / ~) ~/ A F v ] / c I ~ / ~ Figure 3 shows e / 6 as afunction of lAFvl/cP. For 1AFvl > cS2 the root of Equation 15 is imaginary, which means that AF; = 0 and the nucleus will always form coherently withe = 6asindicatedinthefigui-e. When& > > [ A F v l , 6 > > E . Elimination of e from Equation 15 gives: AFc*
I :
-
nCu2@Lsli2(1
-+
IAFvl / C S ~ ) / [ A F V cS2( 1 (1 IAFvl /C6z)1/2~]z(24)
-
I n summary, our simple crystallographic theory of nucleation catalysis for p = 0 makes the predictions:
1. When catalysts and the forming crystals have similar low index planes, the order of the catalysts in lAF:I will be
INDUSTRIAL AND ENGINEERING CHEMISTRY
1295
Review
NUCLEATION-Theory,
identical with their order in 6-i.e., if 61 > 62 > a3, then I AF;1 > lAF{i > lAF{/ 3. 2 . For values of 6 sufficiently small so that Equation 20 is approximately fulfilled, nuclei will form coherently with the catalyst and lAF;I = cS2. [A related problem, the thermodynamics of formation of coherent precipitates from a solid solution, has been treated by Leschen and Fisher (12).] 3. For 6 very large nuclei will form incoherently with negligible strain and lAFvl will be a linear function of 6. COMPARISON OF T H E O R Y A N D 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 (83) and tin ( I C , 32) droplets coated with various fdms are generally in good agreement with A!', values calculated from nucleation theory on the assumption that the Ftrain, 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 (831.1 If E were not zero, the "melting point" of the nucleus would be less than T , and A'exp. would be larger than the apparent Ai;. Pound and LaMer's value of 10-8 A'exp. (1030) for oxide-coated tin droplets is at least a factor of 100 larger than 10-8At; ( 1 0 * 8 ) , 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 are 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. Forming Crystal Ice
In terms of
Temp., K. 273
I
I
I
I
I
I
Plane Hexagonal (001)
Aluminum
933
(111)
KaC1
300
(100)
elastic coefficients CII cn 2c?,/csa
liumerical, dyne em.-%
+ -
,
1 . 7 X 10"
I
c l ~ c $ ~ ~ ~ 7l .- 2 X 10" Ell c12 2cf,/c1, 5 . 3 4 x 10"
+ -
sponding t o 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 t o coherent nucleation rises very sharply with disregistry, so that a t 6 0.055 A T 40 ',
-
I I6O 120
*-r *O
I
Coefficients
I
tc
-
I
1
i
-
-
I
/-I
I
t
4/
I
-I
0.01
0.02
0.03 0.04 0.05 8
Figure 5. Supercooling Corresponding t o Coherent Nucleation of A l u m i n u m Crystals on Catalyst Planes
30
AT
As function of disregistry, 8 w i t h (111) planes of aluminum
201/$/lAg;1
10 Ag 1
0.01
0.02
0.018 0.02
0.03 0.04 0.05 0.06
Figure 4. Supercooling Corresponding t o Coherent Nucleation of Ice Crystals on Catalyst Planes As f u n c t i o n of disregistry 6 w i t h (001) planes of ice
The authors have not attempted to evaluate a: and, therefore, have no predictions on the maximum value of 6 at which nuclei will form coherently. However, it is interesting to calculate the supercooling, AT, or supersaturation ratio, ala,, aa a function of 6, assuming coherent nucleation for all values of 6. AT = f ( 6) 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 (OOl), respectively. The appropriate values of c as a function of the elastic coefficients were calculated by methods summarized by Hearmon (8) and Zener ( 3 2 ) and are given in Table IV. Coefficients c' refer t o a Cartesian zy plane [ / (111)ofthecrystalandtheccoefficientstozy [ / (100). As no reports of measured elastic coefficients of ice have been found, AT = f( 6) was evaluated from the theoretically calculated values of Penny ( I S ) and plotted in Figure 4. A point corre-
1296
The elastic constants for aluminum are those given by Zener, but corrected for temperature variation on the assumption that c varies with 2' in the same way as the rigidity modulus, M. Kb's (11) measurement of hf = f( 2') for an aluminum single crystal was used to make the calculation. The resulting relation A T = f ( 6 ) is shown in Figure 5 . Cibula finds t h a t titanium carbide, titanium boride, and aluminum boride for which 6 0.04 to 0.06 catalyze formation of nuclei for AT < 1', The authors' relation indicates that coherent nucleation for a disregistry 6 = 0.04 cannot take place until A T > 100". It seems either that aluminum nuclei must form incoherently for all 6 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 ala, = f( 6 ) valid for the formation of coherent nuclei from a supersaturated solution. Figure 6 shows the relation a/a, = f ( a), calculated from elastic coefficients tabulated by Hearmon, for coherent nucleation of sodium chloride on (100) planes at room temperature. ala, = 1.06 a t 6 = 0.01 and 2 at 6 0.035. It is doubtful that supersaturation ratios as large as 2 are required t o initiate oriented overgrowth for 6 0.035. It is unlikely t h a t sodium chloride nucleiform coherently for 6 ? 0.015. In view of the foregoing calculations and experimental evidence, it seems that nuclei are likely to form coherently only for 6 7 0.005 to 0.015; for 6 > 0.02 the strain, E, is probably much smaller than 6. ,4s 01 has not been calculated from theory, it is interesting to
-
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
Vol. 44, No. 6
NUCLEATION-Theory,
m
2.60
2.20
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.
a/o, 1.80
a
1.00 0.01
0.02 0.03 004 0.05
8 Figure 6. Supersaturation Ratio Corresponding t o Coherent Nucleation of Sodium Chloride Crystals on Catalyst Planes As function of disregistry 6 w l t h (100) planes of sodium chloride
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NOMENCLATURE
= activity of solute
a,
1.40
Review-
a, Aa
= =
A
=
A‘
=
A *p. = Ath
=
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 (I/time X volume) for nucleation of liquid in supersaturated vapor kinetic coefficient (l/time X volume) for nucleation of crystals in supercooled liquids experimental kinetic coefficient kinetic coefficient predicted by nucleation theory
c11 C12
‘13 C3a
= coefficients of elasticity
C’
estimate a from the data on the crystallization of ice and aluminum. For the formation of ice nuclei the critical disregistry 6, at which nucleation becomes incoherent (see Equation 20) is estimated to be on the order of the disregistry between ice and silver iodide, 6, 0.015. Taking the value of ULS = 32 ergs per sq. cm. we calculate from Equation (20) a M 180 ergs per sq. om. For 6 - E = 0.10 we have a structural interfacial energy a( 6 - e) = 18 ergs per bq. cm. For aluminum 6,
