)IVcAt^'I3'\1 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1374 
 
 KINETIC TREATMENT OF THE NUCLEATION 
 
 IN SUPERSATURATED VAPORS 
 
 By R. Becker and W. Doring 
 
 Translation of *Kiiietische Behandlung der Keimbildung in iibersattigten 
 Dampfen,* Annalen der Physik, Folge 5, Band 24, 1935. 
 
 Washington 
 
 UNIVERSITY OF FLORIDA 
 September 1954 dc jjs DEPARTT/ENT 
 
 J SCIENCE LIBRARY 
 
l^-jcff j^tf(^ 3roiyi^^ 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM IJT^J- 
 
 KINETIC TREATMENT OF THE NUCLEATION 
 
 IN SUPERSATURATED VAPORS* 
 
 By R. Becker and W. Doring 
 
 INTRODUCTION AND SYNOPSIS 
 
 The "nucleation in supersaturated systems" (such as the formation of 
 fog in supersaturated water vapor, for example) was originally made amen- 
 able to quantitative treatment by Volmer and Weber (ref. 1). To every 
 saturation there corresponds a certain critical droplet size of the new 
 phase of such a type, that the vapor is supersaturated only with respect 
 to those droplets which are bigger than the critical droplet, but not to 
 those which are smaller. The formation of fog is therefore contingent 
 upon the origin of "kernels" or nuclei, i.e., droplets of precisely that 
 critical size by a typical phenomenon of fluctuation. The frequency of 
 such processes is, according to the relationship between entropy and prob- 
 
 -^crlt. 
 
 - — ^Om 
 
 ability, proportional to e , where Acj-^-j^^ is the energy required 
 for the reversible creation of such droplet. Volmer 's treatment is 
 briefly reviewed in section 1. The proportionality factor K, as yet 
 indeterminate (in our equation (5))> was calculated by Farkas (ref. 2) for 
 the case of droplet formation by a kinetic treatment, the results of which 
 are fully confirmed (in section 2) by a more lucid method of calculation. 
 The drawback of Farkas' calculations, as well as the arguments advanced 
 by Stranski and Kaischew in connection with it (ref. 3), is that these 
 writers' first convert the elementary equations of the kinetic theorem, 
 each of which refers to the evaporation and condensation of a single mole- 
 cule, in a differential equation which, when integrated, produce new and 
 not always lucid constants. The change to the differential equation is 
 risky because the ensuing functions of the moleciile number n are at 
 first def. led only for integral values of n and at the transition from 
 n to n + 1 change frequently so much that the differential quotient 
 loses its significance. By disregarding this risk Kaischew and Stranski 
 oD-cained an incorrect resuj-t which differs from that of Farkas. On the 
 other hand, the change into differential equation is entirely unnecessary 
 (as will be shown in section 2). The algebraic equations for the indi- 
 vidual processes give the wanted result by a simple, purely algebraic 
 process of elimination. This method is shorter and less subject to errors 
 than that of Farkas. Furthermore, the appearance of indeterminate 
 
 "Kinetische Behandlung der Keimbildung in ubersattigten Dampfen, " 
 Annalen der Physik, Folge 5, Band 2h , 1935, pp. 7I9-752. 
 
NACA TM 1371+ 
 
 integration constants is completely avoided. Thus Farkas ' final for- 
 mula, for example, still contains a constant which he himself designates 
 as indeterminate, while in reality, an accurately estimable value cor- 
 responds to it, which is in optimum agreement with the Volmer and Flood 
 measurements , 
 
 The next three sections deal with the origin of critical nuclei, to 
 which the general thermodynamic analysis of section 1 is applicable as 
 for the droplets. The first kinetic calculation of the thermodynami- 
 cally indeterminate quantity K for crystals was made by Kaischew and 
 Stranski (ref. k) . This important investigation prompted the present 
 study. With regard to the highly idealized crystal model, use is made 
 of the simple cubic lattice, utilized by Kossel as well as by Stranski, 
 which consists of nothing but cubic basic elements, which are in ener- 
 getic reciprocal action only with its six nearest neighbors. However, 
 our resiilts are largely independent of this special model conception. 
 The kinetic analysis, like that of Stranski and Kaischew, results in a 
 confirmation of Volmer ' s formula. On top of that, we succeeded in 
 defining the absolute value of K for this case too. 
 
 Our algebraic method of eliminating the intermediate states not of 
 direct interest affords an instructive representation suitable for 
 the discussion of the particular nucleation process on the passage of 
 an electric current through a network of wires of specific electric poten- 
 tial differences at the ends of the network and given ohmic resistances 
 of the individual wires forming the network. 2 
 
 The whole discussion of the system of algebraic equations is then 
 equivalent to an investigation of the conductivity properties of this 
 network. This method produces in sections k and 5 a comparatively simple 
 and clear calculation of nucleation frequency for two- and three- 
 dimensional nuclei. 
 
 -'-Kossel 's contrary opinion (Ann. d. Phys . (5), 21, p. i+57, 193^) 
 stems from a misconstrued conception of the nature of thermodynamic 
 considerations, which never refer to individual molecules but to those 
 average values which in technically feasible experiments, come \mder 
 observation. For example: the work of separation of the single molecules 
 in a lattice plane may jump back and forth arbitrarily; but in the evap- 
 oration of the total lattice plane, only the mean separation work enters 
 
 the balance of the thermodynamic process as heat of evaporation, 
 p 
 The possibility of such a representation was originally voiced by 
 
 R. Landshoff in a conversation. Another, even more instructive repre- 
 sentation is that of a diffusion process. (Cf. Volmer, Z. f. E., 35, 
 p. 555, 1929- ) But for the purposes of a quantitative treatment, our 
 electrical pattern should be superior to the diffusion pattern, especially 
 when a change from droplet to crystal is involved. 
 
NACA TM 137^ 
 
 As an example for the application of the obtained results, the expla- 
 nation and limits of validity of Ostwald's step rule are discussed in sec- 
 tion 6^ Lastly (section j) , mention is made of the unusual and rather 
 general fact that in our electrical representation of the process of 
 
 growth the resistances of all separate wires, which start from a specific 
 
 A 
 
 "^ Vrn 
 
 state in the direction of growth, are given exact by Constant X e •^, 
 where A is theiTnodynamic potential of this state with respect to the 
 initial state (vapor, for instance). The kinetic interpretation of 
 Volmer ' s formula (5) amounts then to indicating that the total resistance 
 of the network is dependent solely on those pieces of wire which lie in 
 the region of the point related to the critical droplet or crystal. 
 
 1. THERMODYNAMICS OF NUCLEATION 
 
 If n denotes the number of molecules contained in a droplet, F 
 its surface and a its surface tension, the relationship between its 
 vapor pressure pn and that of a flat fluid surface (poo) reads 
 
 dnkT In — = a dF (l) 
 
 Poo 
 
 where dn is the increase in the number of molecules corresponding to 
 the surface increase dF. With the radius r^ of the droplet for spher- 
 ical shape 
 
 n = — rn5 ° and F = ^itrn^ 
 
 hence 
 
 jj,hk^2aml_ (2) 
 
 Poo kTp r^ 
 
 r^ is the critical droplet radius corresponding to the pressure Pn- 
 
 At given pressirre, droplets with smaller radius evaporate, those with 
 
 larger radius grow. A droplet which is exactly in equilibrium with a 
 given pressure, according to equation (2), is hereinafter also designated 
 as critical droplet or as nucleus corresponding to the particular pres- 
 sure. A condensation of the supersaturated vapor can therefore take 
 place only when a nucleus originates as a result of a fluctuation phe- 
 nomenon associated with entropy decrease. 
 
 According to the Boltzmann relationship between entropy and proba- 
 bility, the probability for the appearance of such a droplet is 
 
k NACA TM 157^4- 
 
 _ £l 
 proportional to e ^, where S is the entropy decrease associated with 
 the formation of a droplet of radius rn from a vapor of pressure p^ 
 at constant volume and constant energy. If the number of molecules con- 
 tained in the vapor space is excessively great with respect to n^ this 
 entropy decrease is equal to 1/T times the work A that must be per- 
 formed in order to produce such a droplet in the vapor space isothermally 
 and reversibly. This work cem be determined, according to Volmer, by the 
 following process: 
 
 1. Removal of n molecules from the vapor space 
 
 2. Expansion of j)-^ to Poo 
 
 3. Condensation on a flat fluid surface 
 
 h. Formation of droplet from the fluid 
 
 The svun of these four operations must give the wanted quantity A; 
 but (l) and (2) compensate one another, which leaves 
 
 A = -nkT In ^ + oF 
 
 Poo 
 
 Hence, with equation (l) borne in mind 
 
 (l -^^] . (3) 
 
 V F dn/ 
 
 A = aF 
 
 Since F = Constant X n^'^ , it follows that S. ^ = i., that is 
 
 ' F dn 3 
 
 A = i Fa (k) 
 
 3 
 
 For the number of fog droplets produced per second, denoted hereafter 
 by the letter J, we therefore expect 
 
 oFn 
 J = Ke~ 3kT (5) 
 
 where Fn is the surface of the critical droplet corresponding to the 
 given pressure p. The factor K still remains indeterminate in the 
 thermodynamic study, and must be defined by kinetic analysis as origi- 
 nally made by Farkas. The subsequently chosen method of computing K 
 
NACA TM 137^4- 
 
 is clearer from the methodical point of view. Aside from that, the origin 
 of crystal nuclei is to be treated also for which this eqijation (5) must, 
 naturally, be applicable too. It will be seen that the factor K for 
 fluid nuclei and crystal nuclei of equal order of magnitude is given by 
 the gas kinetic collision factor. 
 
 Regarding the differential quotient dF/dn in equations (3) and (l), 
 it should be noted that dF/dn is the mean growth of siorface in the devel- 
 opment of a molecule. For in the thermodynamic equation (l), dn 
 still must always contain a multiplicity of molecules, although the equa- 
 tion is inapplicable as yet to single molecules. If this averaging of 
 surface growth per molecule is not carried out over a greater number of 
 molecules, the surprising result is that the concept of vapor pressure 
 loses its simple meaning for crystals, as shown by Kossel (ref. 5)^ 
 because the increase of crystals in the growth of a molecule is mostly 
 zero, but now and again very great too. 
 
 2. FLUID NUCLEI 
 
 Consider the following qimsi-stationary condensation process. The 
 vapor pressure p in a very large tank is kept constant by addition of 
 single molecules. Droplets are then produced continuously which would 
 increase infinitely without outside interference. To prevent this, each 
 droplet, as soon as a certain number s of molecules is reached, shall 
 be removed from the tank and counted. 
 
 With regard to s it is simply stipulated that it shall be greater 
 than the critical number n. The number of droplets per second counted 
 under these conditions is termed "nucleatlon frequency. "5 In this pro- 
 cedure, a steady distribution of droplets of various sizes will occur 
 within the tank, which must be examined a little closer . Suppose that 
 Zy is the number of droplets containing exactly v molecules. The num- 
 ber of free vapor molecules kept constant in our tank by addition is 
 then Z]_, while Zs is held to zero. If J is the number of droplets 
 counted per second, J may be regarded as a constant current that passes 
 through all Z. 
 
 Next, assvune that: 
 
 qydt is the probability that in time interval dt, one molecule will 
 leave 1 cm2 of the surface of a drop of v molecules, agdt on the 
 - 
 
 This term coined in the literature is somewhat misleading insofar 
 as the actual number of nuclei formed per second is exactly twice as great 
 because there is precisely a 50 percent probability for each nucleus to 
 continue to grow or to evaporate. 
 
6 NACA 1M I37I1 
 
 other hand is the probability that one molecule from the vapor space 
 condenses on a surface of 1 cm2, F^ is the surface of a droplet with 
 
 V molecules, and Z^ ' = ZyFy is the total surface of all droplets with 
 
 V molecules. 
 
 Applied to the constant current we get 
 
 J = aoZ^.l' - qyZ^> (for all v). 
 
 Indicating 
 
 Pv = ? (6) 
 
 the initial conditions read then 
 
 Zv* = Zv_i' 3v - — Pv (7) 
 
 aQ 
 
 The factors p introduced by equation (6) increase monotonic with 
 increasing v. For the critical molecule number v = n, ^^ = 1. If 
 r^ denotes the radius of the droplet with v molecules, then, by 
 equation (2) 
 
 20M/L _ l\ 
 Pv=i^=ePRTlj-n r,] ^q^ 
 
 The factor occurring in the exponent is indicated by 
 
 a = ^ (8a) 
 
 pRT 
 
 Unfortunately, the notation Zy ' and Zy in the Stranski and 
 Kaischew article are enterchanged relative to Farkas' report. We follow 
 Farkas' notation. 
 
NACA TM 137^ 
 
 In order to eliminate from the equations 
 
 Zv+l' = Zv'Pv+1 - ^ Pv+1 
 
 2v+2' = ^v+l'Pv+2 
 
 
 
 _ J_ 
 
 Pv+2 
 
 = Zs-l'p. 
 
 (7a) 
 
 the factors Zy+i', Zv+2% 
 
 Zs_]_', the first is divided by Pv+1, the 
 
 second by Pv+lPv+2^ etc., the last one by Pv+lPv+2 
 the thus obtained equations are added up, all the Z' 
 Zy ' and Zs ' cancel out leaving 
 
 . . Pg. When all 
 values lying between 
 
 Pv+lPv+2 
 
 = Zv' 
 
 1 + 
 
 ^0' 
 
 Pv+1 Pv+lPv+2 
 
 Pv+lPv+2 
 
 Ps-1. 
 
 With it the nucleation frequency J is known as soon as one of the values 
 of Z' is given. In view of the calciilations for the crystal nucleus this 
 method of solution is somewhat modified as follows: Through the multi- 
 plications equation (7a) takes the form 
 
 ^i+1 
 
 = ^-. 
 
 JR-i 
 
 (9) 
 
 with 
 
 Oi = 
 
 Zi' 
 
 P2P3 
 
 Pi 
 
 and R-i = 
 
 ^0P2P5 
 
 Pi 
 
 (9a) 
 
 The quantity ^^ . arises from the corresponding Z' values by divi- 
 sion by the product of all the p values which occur during the succes- 
 sive growth of the droplet characterized by subscript i from single 
 molecules. (By this method the equations are divided by the common fac- 
 tor 
 
 P2P3 
 
 P^.) The style of writing (equation (9)) of the equation 
 
8 NACA TM I37I+ 
 
 system indicates that the current J flows from point i tovard point 
 i + 1 imder the influence of the voltage difference <^± - Oi+i by over- 
 coming the ohmic resistance K^. Visualizing a series connection of 
 resistances R]_, ^.2, etc., the entire nucleation current J can be 
 regarded as a current driven by a given potential difference through this 
 cliain 
 
 J(Rv + Hv+1 + . . . + Rs+l) = 3>v 
 
 Now $2. is directly equal to Z,2_' and Og equal to zero. The whole 
 problem therefore consists in adding the separate partial resistances. 
 Now it is seen that the individual Pv values increase in such a way 
 that Pn is exactly equal to unity, while the preceding ones are all 
 smaller and those that follow all greater than vmity. Up to the value 
 Rn the partial resistances consist, therefore, of a product of integral 
 factors which are greater than unity; on above R^ the additive factors 
 appearing are all less than unity. As a result the Rj[ values plotted 
 against i have a distinct maximum at i = n. Owing to the importance 
 (8) of the quantities p the exact term for a partial resistance Rj_ 
 reads 
 
 . . ^ - i^ 
 
 The sitm of the reciprocal radii occurring here in the exponent is 
 replaced by an integral with respect to the quantity 
 
 = £v ^ Ml/5 
 
 I'n V"/ 
 
 XV = ^ = rz)"^ (10) 
 
 The integration variable x indicates, therefore, the ratio of a 
 particular droplet radius to the critical radius. By solution of equa- 
 tion (10) with respect to v 
 
 V = n(xv)5, dv = Jnx^dx 
 hence 
 
 xi 
 
 ^ xdx = ^ — (xi^ - xi^) 
 2 r^' ^ ^ ' 
 
NACA TM 137^+ 
 
 In addition 
 
 i - 1 = n(xj^5 _ X2_3) 
 
 Indicating for abbreviation 
 
 2Ei 
 
 2rn 3kT 
 
 fli _ cm _ "-^n 
 
 the term for partial resistance R-^ reads 
 
 Ri = J. eA' {(3xi2-2xi5)-(3xi2_2xi3)} (H) 
 
 aQ 
 
 Replacing the summation over the partial resistances also by an 
 integration, leaves 
 
 r R^dv = 3n f"^ R(x)x2dx = ^ e'^' i^^l^-^^1^) f ^ e^' 0^^-2x^)x^dx 
 Jl ^xi ^0 ^xi 
 
 At x = 1 the integrand has a steep maximum of the order of e^ . There- 
 fore we put X = 1 + I, i.e., 3x2 _ 2x3 _ ]_ _ j^2 _ 2|3^ ^^^^^ gg-f- -j-j^g 
 integral 
 
 ,A' r g-A'(3|2+2|5) (1^ ^)2, 
 
 di 
 
 The variation of the integrand is represented in figure 1. The 
 factor A' is fairly high, say about equal to 20 to 50, in practical 
 cases, as will be shown later. So, without appreciable error the above 
 integral can be replaced by 
 
 p" e-3A'^2 ^^ 
 
 Then, the total resistance (3x]_2 - 2xj^ compared to unity being disre- 
 garded in the exponent) reads 
 
 R=2n /^eA' (12) 
 
 ao \/5A' 
 
10 • NACA 'm 137U 
 
 With this the thermodynamically obtained expression for the nucleation 
 frequency of the indeterminate constant K is defined. 
 
 The final result is 
 
 5kT 
 
 Against this calculation the objection might be raised that the 
 formula (l) had been applied to droplets of as low as two or three mole- 
 cules, for which the concept of surface tension is certainly perfectly 
 meaningless. But, when considering the curve of the partial resistances 
 in figure 1, it is clear that the resultant total resistance is definitely 
 defined by the partial resistance in the neighborhood of v = n. There- 
 fore it is practically immaterial whether the partial resistances at the 
 start of the chain had been chosen by a factor 100 too great or too small. 
 Equation (13) is exactly identical with Farkas ' formula (ref. 2), when 
 bearing in mind that his constant C on the basis of its introduction 
 (p. 259) has "the significance Z-j_'. Since Farkas did not notice that 
 the extrapolation of his formula to droplets of only two or three mole- 
 cules is positively unobjectionable, he failed to recognize the signifi- 
 cance of this constant. 
 
 In comparison, the calculation of Kaischew and Stranski (ref. 3) does 
 
 cLZv ' 
 
 not seem to be entirely acceptable. They replace Zy_i - Z^ by , 
 
 dv 
 
 which serves no useful purpose in the subsequent calculation, since no 
 
 integration along this differential quotient is ever made. It merely 
 
 obscures the significance of their constant C which simply is -Z]_'. 
 
 But, contrary to Farkas, they use the calculating method of logarithms 
 and subsequent substitution of the differential quotient for the differ- 
 ence quotient for great v also. This certainly is inadmissible in 
 
 Ay. t 
 
 the range of small , where the logarithmic term changes rather con- 
 
 dv 
 
 siderable even at minor changes in 
 
 dZ, 
 
 dv 
 
 The formula obtained for J is now compared with the Volmer -Flood 
 measurements on fog formation at adiabatic expansion of water vapor.. 
 The factor Z]_' is, by assumption, equal to the total surface of the 
 free molecules; ^n^l' signifies thus twice the number of gas kinetic 
 collisions per second between the Z-|_ vapor molecules. From the mean 
 free path length l_ and the mean molecular velocity v the number of 
 collisions per cm^ of vapor space follows at 
 
 aoZi' = N f 
 
NACA TM 137^ 11 
 
 Since Z is inversely proportional to the concentration N, we get 
 
 = L /8~ ^0 1 P^ 
 yrtR PqZq\/M TV^ 
 
 = 5 X 1022 ^^ ^ 
 
 Zq\/M tn/t" 
 
 where 
 
 N number of vapor molecules per cm5 
 
 L Loschmidt number 
 
 p vapor pressure ram Hg 
 
 Iq free path length at 0° C and standard pressure 
 
 For the number of molecules in the critical droplet we get 
 
 n = ^ r 3 E L 
 3 M 
 
 _ l^-rt/2aM l\3 p 
 5 \pRT xj M 
 
 whereby In — = x. 
 
 For water (p = 1 g/cm^; a = 75 dyn/cm) 
 
 For the attainable super saturations (x « 1.5) n amounts to about 
 100 molecules . 
 
12 
 
 NACA TM 137*+ 
 
 Posting 
 
 = 1 f^n ^ ^ La/goM iV 
 3 kT 3 RTVpRT x/ 
 
 for A' the formula for J reads 
 
 In J = 1+9 + in 
 
 Poo P 
 
 Zj3M5/2a3/2 
 
 + 2x + 2 IB X - 17. 7W^^^ ^ (13a) 
 
 Poo in mm Eg 
 
 This result is then compared with the Volmer-Flood measurements on 
 water at temperatures T of 260° and 275°. All measurements at 
 T = 270°, Poo = ^ mn Hg, l_ = 10~5 cm, a = 75 dyn/cm are entered below 
 the logarithm. Hence 
 
 In J = 52.5 + 2x + 2 In X - 5-7^ X lo3 /^V -i- 
 
 The curves obtained for In J are shown plotted against x in 
 figiore 2 for T]_ = 275.2° and T2 = 26l.0°. But there is a certain 
 
 uncertainty as to which value of J is to be designated as condensation. 
 According to the graph the curves intersect the x axis at such a slope 
 that it is practically immaterial, when defining the critical supersatu- 
 ration, whether J = l(ln J = 0) or J = 10(ln J = 2.3) is plain fog. 
 Choice of the intersection point of the curves with the straiglrit line 
 In J = 1, gives the following values for the critical supersaturation, 
 which can be compared with the measurements 
 
 
 T 
 
 ct( dyn/cm) 
 
 x 
 
 calculated 
 
 Pn 
 
 Poo 
 
 calculated 
 
 measured 
 
 Curve 1 
 Curve 2 
 
 275.2 
 261.0 
 
 75.23 
 77-28 
 
 1.1+6 
 1.61+ 
 
 I+.30 
 5.11+ 
 
 1+.21 
 5.03 
 
 am' 
 
 ounts to 55 to 56 
 
 Worthy of note is the insensitivity of the theoretically computed factors 
 to errors in the calculation of aQZ-|_'. Even a factor 10 would change 
 
 the constant 52.1+ only by 2, 
 result. 
 
 I.e. 
 
 practically no change at all in the 
 
MCA TM 137*+ ■ 13 
 
 Since all further meas'orements on other substances in the Volmer- 
 Flood report have been compared with those measurements on water and 
 gave a good confirmation of Farkas ' formula, it is concluded that the 
 present formula (l3) reproduces the whole available test material very 
 satisfactorily , 
 
 3. THE LINEAR CHAIN 
 
 Preparatory to the problems of the actual crystal growth, the fol- 
 lowing process is analyzed: It is assumed that the rectangular area ABCD 
 is the base of a simple cubic crystal, on which as the beginning of a 
 new surface, a layer of edge lengths z and I is available and on which 
 the (l + l)th chain of length z is included in the growth. The gi'owth 
 of this new chain is analyzed. Figure 3 represents the stage in which 
 exactly k = 5 atoms of the (Z + l)th chain are condensed. The diffi- 
 culty of forming nuclei here is due to the fact that diiring the start of 
 a new chain the first and possibly also the second and third atom are 
 less solidly bound than those following, which are all bound with the 
 same energy (repeatable steps, according to Kossel, bond at "half crystal" 
 according to Stranski). So, unless there is too much supersaturation 
 after a complete chain has formed, there is a considerable lapse of time 
 before - as the start of a new chain - a linear nucleus capable of growing 
 has formed. The energies, with which the single atoms are bound in the 
 successive formation of the chain, are indicated with cpi, cp2j • • • 
 cpk^ .... Then the possibility qk<it that the k-th atom evaporates 
 as a result of the thermal motion in time interval dt on a chain con- 
 sisting of k atoms, is given by 
 
 9k 
 qjj. = F(T)e' kT (l^) 
 
 On the other hand, the possibility adt, that a fiirther atom settles on 
 the chain, is independent of k and solely given by the external vapor 
 pressiire. It is assumed that there is no slip of atoms at the crystal 
 surface. In that event a is essentially equal to the number of vapor 
 atoms per second arriving at the surface of a single crystal atom. The 
 quantity a introduced here follows from the aQ (of section 2) by 
 multiplication with the surface atom. 
 
 We put 
 
 a = F(T)e ^'^ (15) 
 
Ik NACA TM 137h 
 
 hence, where the energy \lf is the measure of the external vapor 
 pressure. 5 The whole mechanism of growth is governed by the factors 
 
 which for normal growth at the beginning of the chain (k = 1) are sub- 
 stantially smaller and for greater k a little above unity. In conjunc- 
 tion with Stranski and Kossel, this behavior is then schematized so that 
 Pi is regarded as very small compared to unity and all other p values 
 as equivalent and greater than unity. For the investigation of the growth 
 of a chain the following steady process is analyzed: A space under con- 
 stant vapor pressure contains a very large number of crystals which are 
 in the stage of growth represented in figure 3. But the new chain in 
 the process of fonnation may be of any possible length and assume any 
 possible position on the raised side of the rectangle. The number of 
 crystals on which the new chain has exactly the length k and is at a 
 specific position at the growing edge is indicated with nk; correspond- 
 ingly, the number of crystals arising from the crystals of the type nj^ 
 due to deposition of an atom at a certain end of the chain k, is indi- 
 cated by njj+i. 
 
 By partial current J' is meant the excess of the growth process 
 per second which lead from the nj^ crystals to those of the type nk+i, 
 through the evaporation processes, which lead from nk+1 to nk- For 
 this specific partial current 
 
 J' = n^a - 111^+1^15.^.1 (^'^) 
 
 Each chain has then two possibilities of adding an atom corresponding 
 to its two free ends. In the two positions of chain k in which one end 
 coincides with one end of the base, there is only one possibility of 
 build-up. Since, for the chain k, there are (z - k + 1) various posi- 
 tions possible, there are altogether 2(z - k + 1) - 2 = 2(z - k) partial 
 currents J', which collectively lead from all the crystals with chains 
 of length k to those with chains of length k + 1. But in the case 
 of transition from ng to ni there are only z partial currents 
 corresponding to the z deposition possibilities of the first atom of 
 the new chain. This branching of the current is represented in figure k 
 for z = 6. 
 
 Now the not entirely exact assumption is made that all partial cur- 
 rents leading from k toward k + 1 are equivalent. Since their sum 
 
 At absolute zero point the heat of vaporization would have to be 
 used. As a rule, ^ signifies a thermodynamic potential. 
 
NACA TM 157^ 
 
 15 
 
 gives the total current J, each is eqioal to 
 
 J' = 
 
 2(z - k) 
 
 It is readily apparent from figure k that this assumption cannot be 
 rigorously correct on account of the equations of continuity between the 
 partial currents. Owing to equation (I7) this assimiption corresponds to 
 the assumption that all positions of the chafn k are identically 
 frequent . 
 
 Thus, on this premise the steady state is described by the equations 
 
 ni 
 
 noPi 
 
 J £1 
 a z 
 
 = ^lP2 - 37 
 
 P2 
 
 2a z 
 
 ^k+1 - ^kPk+1 ~ •^ 
 
 J Pk+1 
 
 ^ 
 
 2a z 
 
 n. 
 
 = "z-lPz - ^ Pz 
 
 (18) 
 
 These eqixations are treated the same way as those of the droplet forma- 
 tion (in section 2), by regarding them as equations for the passage of 
 current through a series of specified partial resistances. By division 
 of the k-th equation by the produce Pk+1 = 3lP2 • • • Pk+1^ they take 
 the form 
 
 Ok+1 = 'I'k - J^k 
 
 (19) 
 
 where the individual potentials and partial resistances indicate 
 
16 NACA TW 137^+ 
 
 \ = , Rk = , 
 
 h^2 • • • Pk 2a(z - k)Pi • • . 3k 
 
 but *o = ^0> Ro = i 
 
 (19a) 
 
 Thus, in figure k the electric potential of a specific state is 
 represented by the quotient of the nxjmber nk of crystals in state k 
 and the product Pk = PiPp • • • Pk °f "the p values of all atoms bound 
 
 in this state. Specific experimental interogatory forms are synonymous 
 with the corresponding statements regarding the electric potential dif- 
 ference placed at the ends. However, it is to be noted that, in contrast 
 to the electrical picture, the absolute value of the potential in the 
 growth process itself has a well defined meaning 
 
 *k ^ *k ^. -, 
 — and respectively 
 
 ^k ^k-1 
 
 namely are the number of individual processes taking place in unit time 
 from k to k + 1 and k to k - 1. 
 
 As application of (19) the actual linear nucleation as well as the 
 growth of the rectangle layer about a whole chain is now analyzed. 
 
 A. Linear Nucleation 
 
 The procedure for defining the nucleation frequency is the same as 
 for the droplet formation. All the crystals for which the chain has 
 reached a certain arbitrarily chosen length s are removed; s is to 
 be very small compared to the length z of the edge. The n-umber of 
 crystals removed per second is called linear nucleation frequency. 
 Hence we put <I'g = and find 
 
 J = - Oq, with $Q = no 
 and 
 R = R3_ + Rg + . . . + Rs_i 
 
 - i/i . 
 
 + 
 
 2a\z (z - l)Pi (z - 2)3i|32 (z - s + l)pi . . . Pg.i 
 
 I 
 
NACA TM 131^ 17 
 
 With the specialization P2 = P3 = . . • = P and because s « z 
 
 R = -^l2 + \ ^ (P^~^ + P^-5 + . . . + 1) 
 PlP" 
 
 2az 1 - "S-2 
 
 ^f2 ^ P^-^ - I 
 2az\^ (p _ l)p^p 
 
 s-2 
 
 is applicable also. 
 
 Disregarding the 1 next to p^-l and the 2 next to l/Pi, leaves 
 
 1 P 
 R = ; , hence the frequency of the linear nucleation at one 
 
 2az (p - l)p3_ 
 of the nQ edges, independent of s 
 
 J- = 2az ^-^ Pn (20) 
 
 The factor 2 ^— ^ — p-]_, small compared to lonity, is regarded as a 
 
 probability that one of the atoms striking the edge (their number per 
 second amounting to az) grows up to a new chain. 
 
 B. Deposition of a Whole Chain of Length z 
 
 In this event all the partial resistances from Rq to Rz-1 must 
 be added up 
 
 j(Rq + Ri + . . . + R^.i) = $0 - "^2 
 
 or 
 
 no 
 
 -(- ^ . ^ . + 7 ^ • • • i 1-^^.- 
 
 2a\^z (z - 1)P3_ (z - 2)P3_P2 p^Pg . • . Pz_i/ P1P2 
 
 The first partial resistance 2/z in comparison with 
 
 ^z 
 
 (z - l)Pi 
 
 can always be disregarded. Putting Pg = . • . = Pz = P> ve get with 
 the abbreviation 
 
18 NACA ™ 157^4- 
 
 (21) 
 z - 1/ 
 
 n^ = noPiP^-l - ^ Sz (22) 
 
 The siom S2 does not lend Itself to elementary evaluation. The 
 approximate value 
 
 Sz(p) = P^ (21a) 
 
 z In 3 
 
 used in the following is obtained by the following consideration: 
 Replacing the sum (21) by an integral gives 
 
 hence, with the substitution x = z - — ^ 
 ' In p 
 
 pz Pin p(z-l) y 
 
 s^ = —7—7 / ^y 
 
 z In p Jq 1 _ y 
 
 ~ z In p 
 
 y 
 
 The approximate value is obtained by disregarding whxch is 
 
 z Ln p 
 
 small compared to lonity, which, however, presents only a rough approxi- 
 mation near the upper limit of the integral. Equation (22) enables the 
 deposition of a whole chain z to be treated as an elementary process. 
 The equation (l7a), valid for the actual elementary process, is simply 
 replaced by the relation 
 
 J = noAz - n^B^ (25) 
 
NACA TM 157^ 19 
 
 whereby 
 
 2a|3ipz-l 31 
 
 Az = -^ = 2az In (3 -f 
 
 Bz = 1^ = 2az In pp-^ 
 
 (23a)6 
 
 Az is slightly dependent on z, while Bz decreases exponentially with 
 z. Both quantities become equivalent at a critical value of z, which 
 is denoted by m, and is defined by 
 
 PlP"i-l = 1 or -^ = P°^ ' (2^) 
 
 m is that chain length which is precisely in equilibrium with the exter- 
 nal pressure. According to (16) the definition {2k) of m is equivalent 
 to 
 
 cpi + (m - l)cp = mif 
 
 or (2ila) 
 
 9 - Hf = - (cp - cpi) 
 m 
 
 The mean evaporation energy of the "critical chain m" is equal to 
 the energy \|/ characterizing the external vapor pressure. 
 
 k. TWO-DIMENSIONAL NUCLEUS 
 
 Equation (22) makes it possible to analyze a chain of length z as 
 an element, through whose deposition or evaporation the growth of plane 
 nuclei or of whole rectangular plates is controlled. In this instance 
 the growth of a plane nucleus on a given base of edge lengths i and 
 k is involved. A specific stage of this growth is represented in fig- 
 ure 3. The bonding energy of a single atom on the smooth base ("bond to 
 one neighbor") is denoted with cpoj 9l ^^^ ^ have the same meaning 
 as in section 5- Accordingly, there are 
 
 Pq = e kT ^ 31 = e kT ^ p = e kT (25) 
 
 The energy required to detach the whole plate (i, k) from the base 
 is then 
 
 cpo + (i + k - 2)cpi + (i - l)(k - l)(p 
 
 In this calculation it is assumed that at no time two nuclei are 
 simultaneously existent on the same chain and then grow together to one 
 chain. When z is not extremely great, this assumption is well justified. 
 
20 NACA TM 157^+ 
 
 From the assumption that P2 = P3 = • • • = 3 a-^e all equivalent 
 
 and independent of the position of the deposited molecule on the base, 
 inevitably follovs the condition 
 
 90 + T = atpi 
 hence also 
 
 PqP = Pi^ (25a) 
 
 The total bonding energy of a structure must be independent of the 
 manner in which the gro-vrth takes place. Applied specifically to a system 
 of 3 atoms on the base as in figure 5, the binding energy for growth in 
 the order of 1, 2, 5, is cpQ + cpQ + cp, but for the sequence 1, 5, 2, it 
 is CpQ + cp-]_ + cp-j^. The equality yields the above relationship. The evap- 
 oration energy of the whole plate (i, k) is therefore 
 
 ikp - (i + k)(cp - cpi) 
 
 Visulize a column of cross section i x k consisting of I whole 
 atom layers, the deposit of the {l + l)th layer being located on a 
 rectangle s x z. In analyzing the full growth of this deposit into a 
 whole layer the procedure is the same as in section 5. A multiplicity 
 of columns and rectangles of every possible size and position is assumed, 
 with ng z denoting the number of those at which the nucleus (s,z) has 
 a specific position on the base. The number of the possible positions 
 is, obviously (i - s + 1) (k - z + 1) . The total nLimber Zg ^ of "the 
 col;;mns with a plate (s,z) would then be ng ^(i - s + 1) (k - z + 1) , if 
 it is assumed, as in section 3, that each position of the rectangle (s,z) 
 on the base occurs with the same frequency. For the cirrrent Jg 2, 
 leading from s,z to s + l,z, there are altogether 2(i - s)(k - z + 1) 
 possibilities, namely, two each for each specific position of the rec- 
 tangle (s,z), with exception of those positions at which it lies to 
 the left or right at the edge. In these cases there is only one possi- 
 bility for depositing a new chain. It is assumed again that these 
 2(i - s)(k - z + 1) partial currents, all of which lead from s,z to 
 s + l,z, are equivalent. 
 
 Now, in order to describe the growth of the plane crystal after the 
 foregoing arguments with the above eqiiation (22), n^ is replaced by 
 
 %+l,z^ "0 ^y ^s,z ^^^ J ^y 7- ^ —' while J32' is 
 
 ' ' 2(1 - s) (k - z + 1) 
 
NACA TM 13lh 
 
 21 
 
 to denote the current from (s,z) to (s,z +1). In the steady state the 
 plane growth is then governed by the equations 
 
 ns+l,z = ns^zPipz-l 
 
 J.c 
 
 ns,z+l = ^s^zPiP 
 
 s-1 
 
 On the other hand 
 
 is applicable. 
 
 ka.{± - s)(k - z + 1) 
 
 W(i - s + l)(k - z) 
 
 nil = nooPo - ^JJ PO 
 
 s,z = 1, 2, 
 
 (26) 
 
 ^ 
 
 (26a) 
 
 Now the content of these equations is described by a discussion in 
 the s-z plane (fig. 6) : 
 
 Suppose that a certain lattice point s,z represents the crystallite 
 defined by the edges s and z. Thus, in figure 6, for example, the 
 point A corresponds to the crystallites 3x2. The current Jg ^ 
 
 flows then horizontally from s,z toward s + 1, z, Jg 2% bi^t vertically 
 upward from s,z toward s,z + 1. The whole lattice extends to s = i 
 and z = k. The problem then consists in computing a total current J 
 that enters at (O.O) and branches off in partial currents Jg 2 ^■nd 
 Js,z', according to equation (26). The obvious method is to regard the 
 entire figure 6 as the image of a material network through which passes 
 a current J under the effect of a certain electrical direct current 
 voltage. To complete the picture, the several equations of (26) must be 
 expressed in the form of Ohm's law 
 
 ^s+l,z = $s,z - Js,z^s,z 
 
 $s,z+l = 'l's,z - Js,z'Ks,z' 
 
 (27) 
 
 which describes the current in the separate pieces of wire of the network 
 in figure 6. It can be accomplished by dividing the first equation of 
 (26) by the product P of all the p values occurring in the build-up 
 of the plane (s + l,z),, that is, 
 
 Ps+l,z = ^Q^-^(s+^-^)fis{z-l) 
 
22 
 
 NACA TM 137h 
 
 The potential (t>s z and the partial resistances Rg 2 ^^nd Rg 2' 
 become then 
 
 
 s,z p R (s+z-2)p(s-l)(z-l) 
 
 >• <I'0,0 = nO,0 (2Ta) 
 
 ^s.z 
 
 Ua(l - s)(k - z + 1) P, 
 
 s+1, z 
 
 R 
 
 s,z 
 
 4a(i - s + l)(k - z) P, 
 
 s,z+l 
 
 (27b) 
 
 The whole problem now consists in the discussion of the electrical 
 properties of the network built up of partial resistances (27b). 
 
 Introduction of the approximate value (21a) for S™ gives 
 
 R 
 
 s, z 
 
 ' Ua(i - s)(k - z + l)z In p g o s+z-losz-s-z 
 
 Introduction of the critical edge length m defined by eqiiation (2^4-) 
 
 and with the relation — = — = pm following from equation (25a), the 
 
 Po Pi 
 second factor of the above Rg 2 becomes 
 
 gmpm(s+z)-sz = pm+m2-(s-m) (z-m) 
 
 With a system of axes turned through l+5° (a along the diagonal, ^ at 
 right angle to it) 
 
 (28) 
 
 becomes 
 
 (s - m)(z - m) = (a - m)^ - ^' 
 
NACA W 137^4- 23 
 
 hence the factor of Rg 2 in question 
 
 pm+m2p-(a-m)2p^2 
 
 The relationship of ^ implies: if the partial resistances Rg 2,, 
 which are met by a normal to the diagonal, are examined, they are seen 
 to have a sharp minimum on the diagonal itself (at ^ = 0) . One, two, 
 three lattice points away from the diagonal, the resistances increase by 
 
 the factors p, p, , (3-^, etc. Thus practically only the diagonal s ^ z 
 of this lattice is conductive, while along the diagonal (change of a 
 at ^ = 0), the resistance has the same sharp maximum at ct = m, i.e., 
 at the critical edge length. There the partial resistances drop with 
 the distance from this point to the p-l, p"^, 3-9-th fraction, when 
 the distance from a = m amounts to one, two, three lattice points. 
 Therefore, the entire current must flow along the diagonal in a narrow 
 gorge, characterized by the factor pC2^ which in turn leads over a very 
 steep "pass" at a = m. 
 
 The over-all resistance R of the entire network is practically 
 concentrated on the resistance 
 
 R - Km m = P"^^"^^ (29) 
 
 ' Ua(i - m)(k - m + l)m In p 
 
 at the height of the pass. Naturally, it is assimied then that the spot 
 s = z = m still lies substantially within the lattice. A further discus- 
 sion of the branching conditions in the immediate vicinity of the spot 
 m,m could contribute to the expression R no more than a factor close 
 to unity, which, however, would be useless for the piirposes involved here. 
 Now the two questions concerning the frequency of plane nucleations, as 
 well as the growth of the rectangular column i X k around a whole layer, 
 can be answered exactly as for the chain in section 5. 
 
 A. The Formation of Plane Nuclei on a Rectangle Base 
 
 To determine the nucleation frequency, all the columns for which 
 the sum of the edges s + z has reached an arbitrarily specified value 
 s + z = n are withdrawn from the vapor space and counted, that is, the 
 lattice points lying on the straight line s + z = n are grounded. The 
 saddle point s = z = m is found to be located still within the thus 
 out-off triangle and also that n and hence m itself should be very 
 small with respect to the edges i and k of the base. The result is 
 
 the satisfactory approximation: R = •; 3"^"^^^ hence, the nucleation 
 
 iJ-aikm In P 
 
2k NACA m 137^4- 
 
 current for a single one of the nQ crystals is 
 
 — = i^-aikm In P3-m-m2 (50) 
 
 The tvo factors p and m still remaining are tied to one another 
 
 by the relation |3°i = e kT ^ according to equation (24), with the aid of 
 which one of the two can then be eliminated from equation (50) . 
 
 Elimination of p leaves 
 
 "i(<P-9l) 9-9i 
 
 / = l,aik '^-^^ e" M e" ^T (50a) 
 
 "0 kT 
 
 Here, as it should be, 
 
 half the free -edge energy 
 
 icT 
 
 is in the first e power, because — (cp - q)-|_) is the free-edge energy 
 
 per atom, Um— (cp - cp-i ) is, therefore, the total free-edge energy. The 
 2 
 
 plane energy accompanying the formation of the chain (at both of its 
 ends I) is in the second e power. The factor before the already thermo- 
 dynamically required e function is extremely simple: its order of 
 magnitude is defined by the number aik of the vapor atoms per second 
 
 l+(q) - cp]_) 
 
 impinging upon the plate ik. The then still remaining factor 
 
 kT 
 
 -, -, , evaporation heat -rj_ , . .„. ^ ^.-u 
 is nearly equal to . It has no significance for the 
 
 RT 
 only interesting order of magnitude of J. 
 
 On the other hand, when m substitutes for m + 1, the elimination 
 of m from equation (30) gives 
 
 J = Uaik ?_L^ e (kT)2 in p (30b) 
 kT 
 
 This equation gives the dependence of current J on the experimen- 
 tally directly defined supersaturation p. 
 
NACA m 137^ 
 
 25 
 
 B. At the formation of a whole plane i X k it is necessary to ana- 
 lyze the entire rectangular network on which the current J is introduced 
 at point (0,0) and channeled off at (i,k) 
 
 Oo,o - *i,k = JRi,k 
 
 where R^^ j^- is the over -all resistance of the entire rectangular network. 
 
 Here, also, the only case of interest is that where the point (m,m) 
 lies still within the net; Rj^ j^., therefore, according to eqijation (29), 
 may be replaced by Rm m- If use is made of the relations PqP = P]_2 
 
 and -^ = p™ in the product Pg ^.t ^s z then becomes 
 
 Ps,z = Po3l^'^^~2p^^'^)^^-^) = p(s-m)(z-m)- 
 
 •m 
 
 Thus, by equations (27) and (29) 
 
 /. N /, \ p Rm+(i-m) (k-m) 
 m 1, = no pp(i-m)(k-m)-m^ _ j P_^ '^ '. 
 
 ka.{± - m) (k - m)m In p 
 
 (31) 
 
 On the basis of this equation (31), the accumulation of a whole 
 plane can henceforth be treated as elementary process. The equation 
 applicable to it reads 
 
 J = '^0,oAi,k - '^i,k%,k 
 
 (32) 
 
 where 
 
 -m2 
 
 ^i k = ^^^^ - m)(k - m)m In pp-m-m' 
 Bi,k = A.^j^prn2_(i_^)(i,_^) 
 
 (32a )'^ 
 
 Both quantities are equivalent at m2 - (i - m)(k - m) =0 or 
 ik = (i + k)m. For square plates, which are practically the only ones 
 
 7 
 Multiple nucleation is excluded again. At very great i and k 
 
 the foregoing is therefore no longer applicable . 
 
26 NACA TM 157^4- 
 
 occurring at the subsequent growth of the spatial crystal, it results in 
 i = k = 2m. The probability for evaporation and condensation will not 
 be identically great until the edge length of the plane is twice as great 
 as the critical chain length m. Naturally, this result was to be fore- 
 seen by reason of the fact that the equilibrium vapor pressure is defined 
 by the mean evaporation energy, as already predicted by Stranski and 
 Kaischew. It is to be noted that Ai ^ is very slightly dependent on 
 i and k, while Bj^ -^ decreases rapidly with increasing size of the 
 plane. 
 
 5, THE CRYSTAL NUCLEUS 
 
 After these preparations the quantitative treatment of the nucleation 
 frequency for spatial crystals is easy. Again visualize in a vapor space 
 a large number of box-like crystals of all possible edge lengths i, k, 
 I in steady distribution so that the vapor pressure remains constant 
 and that all crystals, as soon as they have reached a certain size, are 
 removed from the space and counted. The number of crystals with the edge 
 lengths i, k, I is assumed at Z^ j^ ^ • They niay, for example, change 
 to crystals (i + 1, k, Z ) by gathering of a plane (k, 1), that is, this 
 plane can be deposited on two different sides of the little box. The law 
 for this process was defined in equation (51) . Replacing n-j_ -^ by 
 
 Zi+l,k,l. no,0 ^y ^i,k,^' ^'^ ^^' ^'^ ^^^ "^ ^y V2Ji,k,Z results 
 in 
 
 ^v Ui ^ 2 Ji k 7 3m+(k-m)(Z-m) 
 
 Zi+1 k I = ^i k M^-^^)(^-^)-^'' V ' . (53) 
 
 1+1, k,i i,k, i 8a(k - m)(Z - m)m In p 
 
 where Ji k 2 indicates the partial current that leads from i^k, 1 to 
 
 i + l,k, Z. By the method previously used several times, equation (33) 
 can again be put in the form 
 
 *i+i,k,z = «'i,k,z - Ji,k,zRi,k,z (3^) 
 
 and identify it as the electrical current in a spatial network whose lat- 
 tice points have the potential ^i k Z ^^^ ^ which ^i k Z ^^ *^^ ohmic 
 resistance of the piece of wire that leads from i,k, Z to i + l,k, Z. 
 The transition from (33) to (34) is accomplished again by division of (35) 
 by the product Pi+i k Z °^ ^-^ "the p values which occur during the suc- 
 cessive development of the state (i + l,k, Z) from (i + l)k, Z single atoms. 
 The factor for Z^^-^ i in equation (55) is precisely the product Pk i 
 
NACA m 131^ 27 
 
 of the p values for that plane (k,Z) which is newly added in the par- 
 ticular reaction. The product ^i ^ Z ^^'^ ^^ constructed as follows: 
 
 On a single atom free colimms of i - 1, k - 1, Z - 1 atoms are depos- 
 ited along the coordinate axes, each of which gives the factor (5q. The 
 
 spread-out rectangle sides are then filled out. It yields (i - 1) (k - 1) + 
 (k - l)(Z - 1) + (Z - l)(i - 1) times the factor p-]_ and, in addition, 
 (i - 1) (k - l)(Z - 1) times the factor p. Hence, altogether 
 
 Pi,k,Z = Poi+k+Z-33-L(i-l)(k-l) + (k-l)(Z-l)+(Z-l)(i-l)p(i-l)(k-l)(Z-l) 
 
 for which the relations — = — = p"^ give 
 
 Pi P 
 
 p ^ D-m(ik+kZ+Zi)+ikZ+5m-l 
 i,k, Z 
 
 and 
 
 Pi+i,k,z =v^^^^^^(^-^)ii-^)-^^ 
 
 By division with this quantity, we obtain in equation (5^) as factor 
 °^ ^i k Z '^^^ final term for the partial resistance 
 
 R, , , = 1 nm(ik+kZ+Zi)-ikZ+m2-2m+l (55) 
 
 ' ' 8a(k - m)(Z - m)m In p 
 
 To clarify the behavior of the exponent visualize a perpendicular 
 line dropped from the point ijk, Z of the network on the space diagonal, 
 0,0,0 to indicate the foot of this perpendicular. 
 
 Putting 
 
 1 = a + r]_ 
 k = a + rp 
 Z = a + r 
 
 > 
 
 (56) 
 
 5 
 J 
 
 the construction given then 
 
 T-^ + T2 + r^ = (56a) 
 
28 NACA TM 1374 
 
 along with 
 
 By (36) and (56a) 
 
 ik + kZ + Zi = 3a2 - ir2 
 
 2 
 
 and 
 
 ikZ = a^ - — or^ + r-ir^T:! 
 
 2 
 With this the exponent of equation (35) reads 
 
 m(ik + kZ + Zi) - ikZ = 3cr m _ cr3 + —(a _ m)r2 - rnTpTz 
 
 Disregarding the practically nonessential term r-^rpTv (the sur- 
 roiznding of the diagonals being considered), the conditions for Ri k Z 
 are the same as before in section k for the plane lattice. The factor 
 
 1/' ^ 2 
 p2 in the region a > m solely considered here, effects such a 
 
 rapid rise of the resistance on leaving the diagonal, that the current 
 
 can flow practically only on the diagonal r = 0. On the diagonal itself 
 
 the factors ^^^ m-a-^ has such an enormously steep maximum at a = 2m, 
 
 that the entire voltage drop along the diagonal is practically defined 
 
 solely by the partial resistance 
 
 R2m,2m = ^ pW4-m2-2m4-l 
 
 8am-^ In p 
 
 Owing to "^1 1 1 ~ ^1 1 1 't^® nucleation frequency is therefore 
 defined at 
 
 J = 8aZi^i^im3 In pp-W5-m2+2m-l (37) 
 
 Elimination of 3 by means of 
 
 9-91 
 
NACA m 137^ 29 
 
 leaves 
 
 a, 1,1' 
 
 li e kT e kT g ^i^r " ^ / 
 
 J = 8aZ-, T .m^ i e kT e" kT g" "kT^r " ^y (57a) 
 
 kT 
 
 The factors deciding the order of magnitude of J are aZ]_ 2. l and 
 
 the first of the three e-functions; aZ]_ ]_ ]_ is essentially (like the 
 
 factor a.QZ-|_' in equation (l3) for droplet formation) the number of gas 
 kinetic collisions per second. The first e-power is synonymous with the 
 
 _ crFn 
 factor e 5kT of the thermodynamic formula (5). In fact, — (cp - cpQ_) 
 
 is the surface energy per atomj the total surface energy of the cube of 
 critical edge length 2m, therefore, is equal to (2m) 2 x 6 x — (q) - q)i) = 
 
 12m2((p - cp-|); the third portion of it stands, as it should be, in the 
 
 exponent. The exponent of the second e-f unction indicates, as shown in 
 
 1 edge energy 
 
 section 4, ^^ for a critical plane nucleus. This factor 
 
 2 kT 
 
 occurs in similar manner in the report by Stranski and Kaischew too. 
 Admittedly, its appearance hinges on the exact knowledge of the factor 
 m, as is apparent from the fact that in equation (57) the term with m2 
 
 can be made to disappear completely, if m is replaced by m - — . For 
 
 the problem involving the critical supersaturation the second and third 
 e-functions are ignored. 
 
 6. THE OSTWALD LAW OF STAGES 
 
 This law states that in the formation of nuclei from supersaturated 
 vapor the liquid phase is separated first, as a rule, even when the tem- 
 perature of the vapor is considerably below the freezing point. Our 
 results on the nucleation of liquid (l5) and solid (57) nuclei enable a 
 theoretical foimdation and a quantitative improvement of this law to be 
 made. 
 
 Omitting the last two e-functions in (57a) and introducing the rela- 
 tion for Volmer's exponent of equation (5) 
 
 A" = ^L = ^^^^^ - ^1^ 
 
 5kT kT 
 
30 NACA TM I57U 
 
 ve get by equation (57a) on the crystal nucleus 
 
 Crystal: Jcrystal = 2aZi^i^iA"e-A" 
 while equation (l3) produced 
 
 Droplet: Jdroplet 
 
 The factor Z-]_' was, according to section 2, the number of vapor 
 
 molecules multiplied by its surface. Since, according to section 3, a 
 arises from a^ by multiplication with the atom siorface, aZ]_ 1 1 ^^^ 
 
 a„Z-, ' are identical in order of magnitude. Thus, the factor K of 
 
 equation (5) for the formation of droplets appears smaller by l/n than 
 for the formation of the crystal, where n denotes the molecule number 
 of the nucleus. Although n is the order of magnitude of 100, this 
 factor is not decisive in the problem involving the critical supersatura- 
 tion. Moreover, it would be considerably overbalanced by the factor 
 
 P~"^ omitted at J + 1 • The factor A" and 1— matter even less. 
 
 crystal ^3j^ 
 
 As long as no direct measurements of J are planned, but merely the order 
 of magnitude of the critical 'supersaturation, the simple result is : The 
 factor K in Volmer's nucleation formula is simply equal to the number 
 of gas kinetic collisions, for the droplet as for the crystal. This 
 statement applies, as seen in section 3 and section 4, to linear and plane 
 nuclei; naturally, involved here is solely the number of collisions per 
 second at the base. 
 
 The decisive reason for the validity of the law of stages remains 
 then solely the fact that in the qtiantity ■— — the surface F of the 
 
 nucleus corresponding to a certain supersaturation is greater on the cube 
 than on the sphere. The difference in shape is the deciding factor, not 
 the crystalline structure. Its effect is computed on the assumption that 
 the molecular volume v and the surface tension a for fluid and crystal 
 are equivalent . 
 
 If F = Cn /5 is the sixrface corresponding to the molecule number 
 
 Pn 
 n, then by equation (l), with x denoting the abbreviation of In — ^ 
 
 Poo 
 
 kTx = a ^ = ^ ^^/^ 
 
 dn 
 
 3 f1/2 
 
NACA IM 157^ 31 
 
 hence the siirface corresponding to x 
 
 F=i -'C3 
 
 9 (kTx)2 
 The nucleus volume V for the sphere (radius r) is 
 
 V = nv = ^r5, thus, F = hJ^^'^ ^^l^n^h 
 
 for the cube (edge length a) : 
 
 V = nv = a5, hence, F = 6v2/5n2/3 
 
 Hence, for the sphere 
 
 C5 = 56itv2 
 
 and for the cube 
 
 c3 = 63v2 
 
 The critical area corresponding to the same x is — = I.9I times 
 
 greater for the cube than the droplet. To assure identical nucleation 
 frequency, hence, equal values of F, it must 
 
 ^cube / '-' cube 
 
 3.„.. \l/2 
 
 = /l79l = 1-58 
 
 ^sphere VC^sphere/ 
 For the critical supersaturations themselves the condition would be 
 
 p\ /p\l-38 
 
 t-J = [¥) (38) 
 
 ""^cube V^'^/sphere 
 
 As an illustration for applying this relationship, the supercooling 
 at which crystalline and fluid nuclei occior with comparable frequency 
 is analyzed. The saturation vapor pressixre of the liquid phase is denoted 
 
32 MCA ™ 157^ 
 
 by Pi> that of the solid phase by P2. By equation (58) the condition 
 for comparable nucleation frequency reads 
 
 or 
 
 Inp =lnpi + 2.6(lnp]_- lnp2) 
 
 (59) 
 
 (The factor 2.6 is equal to 1: (I.58 - 1). Figure 7 shows the vapor pres- 
 sure curves In p-|_ and In P2 plotted against T. According to equa- 
 tion (39) the curve for In p would then have about the shape of that 
 indicated by the broken curve. Below this limiting curve, more crystal- 
 line nuclei, above it, more fluid nuclei are to be expected. However, 
 this theoretically interesting solution is meaningless in practice as 
 long as the nucleation frequency lies below a limit amenable to observa- 
 tion. For that reason it is necessary to determine, in the same manner 
 as in section 2, the curve of that pressure at which a formation of fluid 
 nuclei occurs at all in observable amounts (dotted curve). The intersec- 
 tion point A of the two curves characterizes the temperature T^. at 
 which an isothermal pressure rise would result in a simultaneous separa- 
 tion of fluid and crystalline nuclei. Below T^ only solid, above Tj^ 
 only fluid nuclei would be observed. 
 
 Natiorally, it may also happen that no intersection point appears. 
 In that event, the law of stages holds unrestrictedly. 8 
 
 7. TEE GENERAL RESISTANCE ANALOGY 
 
 As already stated several times in the foregoing, the equations (I7), 
 (23), and (32), applicable to the elementary process, can be so trans- 
 formed by extension with a suitable factor that they could be interpreted 
 as the Kirchhoff equations of a suitably chosen network of wires. It can 
 be proved that this electrotechnical analogy is possible in complete 
 generality for the condensation and dissolution process of any structure 
 consisting of atoms. Again it is assumed that, besides the vapor phase 
 of a substance, some fractions of another phase are present in a con- 
 tainer. These fractions are hereinafter called crystals, without in any 
 way infringing upon the general character. An uninterrupted input or 
 transport of vapor and removal or addition of crystals of random specific 
 size assures the steady distribution of the crystals of various sizes and 
 shapes . 
 
 _ 
 Such a case seems to exist in the theoretical case treated by 
 Stranski and Totomanow (Z. f. phy. Chem. (a), I63, p. 399, 1933). 
 
NACA TM 157'+ 33 
 
 Next, we consider any random specified type of crystals, say, of 
 the shape represented in figure 8, for example. For a full description 
 of such a crystal, a greater number of parameters are usually advantageous, 
 a single one of which is, say, the number v of atoms in this crystal. 
 By deposition of an atom at a well defined spot of this crystal, a crys- 
 tal of type II with v + 1 atoms is produced. J is the excess per sec- 
 ond of the growth processes which lead from I to II, through the evapora- 
 tion processes which lead from II to I. Then, if nj and nji are 
 
 the number of crystals of type I and II in the steady state, the equation 
 for this specific transition process reads 
 
 nja - njiq^+i (1+0) 
 
 where a and q.v+1 are the repeatedly employed deposition and evapora- 
 tion probabilities of the atom at that particular spot. With the abbre- 
 viation Py+i = — — the result is again 
 ^v+1 
 
 nil = niPv+l - J Pv+l (^1) 
 
 _ 9v+l ijf 
 
 kT ~ W 
 
 Again qv+-l = -^(t)^ °-n<^ ^ ~ -^(T)® > ^^^ introduced, hence, 
 
 'Py+l - "^ 
 3v+-l = e l^T , 
 
 The energy of separation <Pv+i depends, as a rule, on all the param- 
 eters of state of the states I and II, rather than on v alone. 
 
 Now, visualize the crystal II built up successively from single 
 atoms. To each one of the v + 1 single processes, there corresponds 
 a specific Pj_. The individual ^^ still will be dependent upon the 
 sequence in which the atoms of the crystal are joined together. But the 
 product P(ii) of all p^ 
 v+1 
 
 X±^ 9 
 
 2_.cpi-(v+l)i 
 
 v+1 i^i 
 
 p(ii) = riPi = e kT 
 
 v+1 i=l 
 
 ■^Factor p-j,? which by itself corresponds to no growth process, is 
 put equal to Ij hence cp]^ = \if. 
 
3h 
 
 NACA Wi I57U 
 
 is solely dependent upon the configuration of crystal II, because the 
 
 v+1 
 total work of growth ^ cpj can no longer depend upon the manner in 
 
 i=l 
 which the growth took place. 
 
 Dividing equation (4l) by Pjj gives 
 
 v+1 
 
 Oj - Ojj = JRj 
 
 (U2) 
 
 where, for abbreviation 
 
 Ot = — ^ 
 
 ^11 
 
 n 
 
 II 
 
 Pv(l) ^^ Vl(ll) 
 
 (i^2a) 
 
 and 
 
 R - ^ ^ 
 
 (l|2b) 
 
 Every possible form of crystal can now be characterized by a point 
 in the space of the parameters, which define this form. To every possible 
 transition, I — w-II, we then correlate a wire connection between the 
 points of state I and II, to which the resistance given by equation (Ulb) 
 is allocated. The equation (42) can then be regarded as the Kirchhoff 
 equations of this wire netting and as the corresponding potential of 
 the nodes in this net. This interpretation is posgible, because $ is 
 merely dependent upon the state, but not the manner in which a crystal 
 is built up. 
 
 Obviously, this network of wires does not have to be multidimensional. 
 Since the number of possible forms is finite, theoretically one parameter 
 that counts the possible forms, may be sufficient. But for the represen- 
 tation the use of two or three parameters, as in sections h and 5j is more 
 convenient so that the net becomes two or three dimensional. 
 
 This network itself is rather complicated even in simple cases. In 
 the networks treated in sections k and 5^ a large number of wires were 
 consistently ignored because of their high resistance and complete wire 
 systems combined into one resistajice. For the acti:ial calculation, this 
 general analogy is therefore of little help. However, the purely quali- 
 tative distribution of the resistances will be indicated. 
 
NACA TM 137^ 35 
 
 The resistance of a wire depends solely upon its initial point and 
 has the form 
 
 V 
 
 1 ^=^^ 
 
 R = - e kT (1+5) 
 
 a 
 
 V 
 
 where the quantity v^ - y (fi in the exponent is the work to be per- 
 
 i=l 
 formed to produce the system corresponding to the initial point of the 
 wire by reversible process from the vapor. Of all the wire joints which 
 lead from crystals with v atoms to those with v + 1 atoms, the wires 
 proceeding from the crystals with the least work of growth are therefore 
 the wires of the smallest resistance. Whether this minimum is always as 
 sharp in more general cases as on the model used above, requires further 
 study. During the advance along the road of minimum resistance from 
 smaller to larger crystals, the work of growth must, at some time, reach 
 a maximum value, because, while for very small systems it certainly 
 increases with v, it must, at very great v become proportional with 
 V negatively arbitrarily great, so far as the vapor is supersaturated 
 at all with respect to the very large crystals. The crystal on which 
 the resistance reaches its (absolute) maximum with regard to advancing 
 with V and a minimum in comparison to the other wires with the same v, 
 is called the Volmer nucleus. The resistance at this point is 
 
 % 
 ^nucleus = | ^^'^ (^5a) 
 
 with Aj,;; the work of nucleation. As is seen, the saddle-like character 
 of the resistance distribution near the nucleus is completely independent 
 of the model. The specific model representations merely yield information 
 about the nimiber of parallel wires of equal resistance in the saddle 
 point, the distinct character of the saddle, and the extent to which any 
 secondary maximums in the otherwise very jagged curve of the resistance 
 become evident. The order of magnitude of the total resistance between 
 two points with very small and very great v is, however, solely defined 
 
 ^y ^nucleus. 
 
 Translated by J. Vanier 
 National Advisory Committee 
 for Aeronautics 
 
1 
 
 36 NACA TM 137^ 
 
 I 
 
 REFERENCES 
 
 1. Volmer, M., and Weber, A.: Ztschr. f. phys . Chem. 119, 1926, p. 277; 
 
 VoLmer, M. : Ztschr. f. Elektrochem. 35, 1929, p. 555. 
 
 2. Farkas, L. : Ztschr. f. phys. Chem. 125, 1927, p. 236. 
 
 3. Kaischew, R., and Stranski, I. N. : Ztschr. f. phys. Chem. B. 26, 
 
 193^, p. 317; Stranski, I. N., and Kaischew, R. : Phys. Ztschr. 26, 
 1935, p. 393. 
 
 k . Kaischew, R., and Stranski, I. N. : a. a. 0., also Ztschr. f. phys. 
 Chem. (A), I70, 193^, p. 295- 
 
 5. Kossel, W. : Ann. d. Phys. (5), 21, 193^, p. ^57- 
 
NACA m 137^ 
 
 57 
 
 pi Exponent 
 3Xi -2x •* 
 
 Figure 1.- Resistance Rj^ plotted against droplet radiiis x^^ 
 (case A' = 10 and n = 100). : Curve of exponent 
 
58 
 
 NACA IM 137^4- 
 
 A In J 
 
 
 
 
 
 
 
 
 T=275.2» 
 
 T=261.0° 
 # 
 
 
 6 
 
 
 
 \ 
 
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 5 
 4 
 
 — 
 
 
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 3 
 2 
 
 1 
 
 
 
 
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 1 
 
 
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 -2 
 
 1 
 
 1 
 
 1 1 1 / 1 1/ 
 
 1 1 
 
 
 - 1.0 
 
 1.1 
 
 1.2 13 1.4/ 1.5 1.6/ 
 
 1.7 1.8 
 
 In P/P^ 
 
 -3 
 
 — 
 
 
 / / 
 
 
 
 -4 
 
 — 
 
 
 / / 
 
 
 
 -5 
 
 
 
 / / 
 
 
 
 Figures.- InJ plotted against x = In — at two temperatures. 
 
 Poo 
 
 computed for water by equation (13a). 
 
NACA TM 137^ 
 
 39 
 
 Figure 3.- A specific state of crystal growth. 
 
 k = 12 3 
 
 z-1z 
 
 Figure 4.- Network of partial currents for the growth of the linear chain. 
 
1*0 
 
 NACA TO I57U 
 
 Figures.- Derivation of the relation Pq|3 = P]_' 
 
NACA 1M 157'+ 
 
 in 
 
 Figure 6.- Current network for growth of the plane. Point A: 
 Rectangle: 3x2. Size of base: 8x7. 
 
k2 
 
 NACA IM 157^ 
 
 Ainp 
 
 Figure 7.- Ostwald's law of stages, : Curve of equal frequency 
 
 of critical crystalline and fluid nuclei. Above OA: Excess of fluid 
 
 nuclei. Below OA: Excess of crystalline nuclei Curve of 
 
 nucleation frequency 1. 
 
NACA TO 157^ 
 
 1^5 
 
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 Figure 8.- The most elementary process of crystal growth. Deposi- 
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