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&VI~c~~31
I7 I 7 : '. I .7 7< .' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 157h KINETIC TREATMENT OF THE NUCLEATION IN SUPERSATURATED VAPORS* By R. Becker and W. During 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"nucleation in supersaturated systems" (such as the formation of fog in supersaturated water vapor, for example) was originally made amen able to quantitative treatment by Volmer and Weber (ref. 1).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 ability, proportional to e where Acrit. 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. 5), 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 bec use the ensuing functions of the molecule 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 obtained an incorrect result 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 24, 1955, pp. 719752. NACA TM 1374 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.1 The first kinetic calculation of the thermodynami cally indeterminate quantity K for crystals was made by Kaischew and Stranski (ref. 4). 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 results 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 4 and 5 a comparatively simple and clear calculation of nucleation frequency for two and three dimensional nuclei. 1Kossel's contrary opinion (Ann. d. Phys. (5), 21, p. 457, 1954) 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 under 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. 2The 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., 55, 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 1574 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 64 Lastly (section 7), 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 +kT state in the direction of growth, are given exact by Constant X e , where A is thermodynamic 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 orcrystal. 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 (pm) reads Pn dnkT In = a dF (1) Pm where dn is the increase in the number of molecules corresponding to the surface increase dF. With the radius rn of the droplet for spher ical shape n = Y rn5 P and F = 4rn2 5 m hence in n = 2om 1 (2) Pm kTp rn rn is the critical droplet radius corresponding to the pressure Pn At given pressure, 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 NACA TM 1374 _s a proportional to e k, where S is the entropy decrease associated with the formation of a droplet of radius rn from a vapor of pressure Pn 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 can be determined, according to Volmer, by the following process: 1. Removal of n molecules from the vapor space 2. Expansion of Pn to p, 3. Condensation on a flat fluid surface 4. Formation of droplet from the fluid The sum of these four operations must give the wanted quantity A; but (1) and (2) compensate one another, which leaves A = nkT In Pn + oF Pm Hence, with equation (1) borne in mind A = oF n dF (5) F dn/ Since F = Constant x n2/5, it follows that n dF 2 that is F dn 5 A = Fc (4) For the number of fog droplets produced per second, denoted hereafter by the letter J, we therefore expect OFn J = Ke 5kT (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 1574 is clearer from the methodical point of view. Aside from that, the origin of crystal nuclei is to be treated also for which this equation (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 (5) and (1), it should be noted that dF/dn is the mean growth of surface in the devel opment of a molecule. For in the thermodynamic equation (1), 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 quasistationary 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 "nucleation 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 21, 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, assume that: qdt is the probability that in time interval dt, one molecule will leave 1 cm2 of the surface of a drop of v molecules, aodt 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. NACA TM 1374 other hand is the probability that one molecule from the vapor space condenses on a surface of 1 cm2, Fy is the surface of a droplet with v molecules, and Zy' = ZyFy is the total surface of all droplets with v molecules. Applied to the constant current we get J = agZv,' qvZv' (for all v). Indicating 13 a (6) qv the initial conditions read then v' = Zv1' v J v (7) aO The factors 0 introduced by equation (6) increase monotonic with increasing v. For the critical molecule number v = n, Bn = 1. If r, denotes the radius of the droplet with v molecules, then, by equation (2) 2aM 1 SPn = ePRT rV () The factor occurring in the exponent is indicated by a. 2 (8a) pRT 4Unfortunately, the notation Zy' and Zv in the Stranski and Kaischew article are enterchanged relative to Farkas' report. We follow Farkas' notation. NACA TM 1574 7 In order to eliminate from the equations r Z+l' = Zv'Pv+1 a vl Zv+2' Zv+1 +2 0 v+2 (7a) Zs' = Zsl'ps OS the factors Zv+l', Zv+21, Zsl', the first is divided by Pv+l, the second by Bv+liv+2, etc., the last one by Bv+Piv+2 0s. When all the thus obtained equations are added up, all the Z' values lying between Zy' and Zs' cancel out leaving Zc' J 1 1 = Zy'  + + + + Pv+lPv+2 Ps ao v+l v+lPv+2 1 Pv+l v+2 Ps With it the nucleation frequency J is known as soon as one of the values of Z' is given. In view of the calculations for the crystal nucleus this method of solution is somewhat modified as follows: Through the multi plications equation (7a) takes the form Si+l = i Ji (9) with fi = and Ri = (9a) 2 P5 i aoP025 Pi The quantity Oi arises from the corresponding Z' values by divi sion by the product of all the 0 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 0203 P,.) The style of writing (equation (9)) of the equation 8 NACA TM 1374 system indicates that the current J flows from point i toward point i + 1 under the influence of the voltage difference ei Oi+l by over coming the ohmic resistance Ri. Visualizing a series connection of resistances R1, R2, etc., the entire nucleation current J can be regarded as a current driven by a given potential difference through this chain J(Rv + Rv+l + + Rs+l) = v s Now 01 is directly equal to Z1' and Os equal to zero. The whole problem therefore consists in adding the separate partial resistances. Now it is seen that the individual Ov 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 unity. Up to the value Rn the partial resistances consist, therefore, of a product of integral factors which are greater than unity; on above Rn the additive factors appearing are all less than unity. As a result the Ri 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 Ri reads S1 1 il CL + *+ +. Ri = e '( r ri Fn ao The sum of the reciprocal radii occurring here in the exponent is replaced by an integral with respect to the quantity rv ((,O Xv = rn ( (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)3, dv = 5nx2dx hence S+ + + = x xdx = (xi2 x12) r2 r3 ri rn Jv= xv rn x1 2 rn NACA TM 1574 In addition i 1 = n(xi x15) Indicating for abbreviation A' =a = aFn 2rn 5kT the term for partial resistance Ri reads Ri = e eA' (5xi22xi3)(5xl22x135) a0 Replacing the summation over the partial resistances integration, leaves J S Rydv = 5n [Xs SX R(x)x2dx = i5 eA'(3x122x3) a0 (11) also by an x s fX1 At x = 1 the integrand has a steep maximum of the order of eA'. There fore we put x = 1 + E, i.e., 5x2 2x5 = 1 52 2t3, and get the integral eA' eA' (532+263) (1 + )2dt 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 +00 co e3A'W2 d Then, the total resistance (5x12 2x15 compared to unity being disre garded in the exponent) reads R = n aR= a0 e A' V 3A' (12) eA'(3x22x3)x2dx 10 NACA TM 1574 With this the thermodynamically obtained expression for the nucleation frequency of the indeterminate constant K is defined. The final result is aZl' A' A' A' Fn (13) J = e A' (15) n \5 3kT Against this calculation the objection might be raised that the formula (1) 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 figurel, 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 Zl'. 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. 5) does dZy' not seem to be entirely acceptable. They replace Zv_1 Zv' 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 Zl1' 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 dZv' the range of small where the logarithmic term changes rather con dv dZ siderable even at minor changes in dv The formula obtained for J is now compared with the VolmerFlood measurements on fog formation at adiabatic expansion of water vapor.. The factor Z1' is, by assumption, equal to the total surface of the free molecules; aoZl' signifies thus twice the number of gas kinetic collisions per second between the Z1 vapor molecules. From the mean free path length 2 and the mean molecular velocity v the number of collisions per cm5 of vapor space follows at aoZl' = N NACA TI 1574 Since 7 is inversely proportional to the concentration N, we get aOZl, = 2 V = L /8 1 p2 VHR P0 0lQ TS = 5 x 1022 1 p2 0\/M T 4T where N number of vapor molecules per cm3 L Loschmidt number p vapor pressure mm Hg S free path length at 00 C and standard pressure For the number of molecules in the critical droplet we get n = rn P L 5 M 4 i/2opM 35 pL 5 \pRT x/ M whereby In = x. P. For water (p = 1 g/cm5; o = 75 dyn/cm) n = 240 L \T/ x3 For the attainable supersaturations (x 1.5) n amounts to about 100 molecules. NACA TM 1574 Posting A 1 OFn 4= La/2aM 1\2 A' 5 kT 5 RTopoRT x for A' the formula for J reads + In p P + 2x + 2 in x 17.7( _2 1 1p0/20c/2 in x2 pm in mm Hg This result is then water at temperatures T T= 2700, p. = 4 mm Hg, the logarithm. Hence compared of 2600 1 = 105 (15a) with the VolmerFlood measurements on and 2750. All measurements at cm, o = 75 dyn/cm are entered below in J = 52.5 + 2x + 2 in x 5.74 x 105 2 The curves obtained for Ln J are shown plotted against x in figure 2 for T1 = 275.20 and T2 = 261.00. 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 = 2.5) is plain fog. Choice of the intersection point of the curves with the straight line In J = 1, gives the following values for the critical supersaturation, which can be compared with the measurements pn x Pm T o(dyn/cm) calculated calculated measured Curve 1 275.2 75.25 1.46 4.50 4.21 Curve 2 261.0 77.28 1.64 5.14 5.05 A' amounts to 55 to 56 Worthy of note is the insensitivity of the theoretically computed factors to errors in the calculation of aoZ1'. Even a factor 10 would change the constant 52.4 only by 2, i.e., practically no change at all in the result. In J = 49 NACA TM 1574 Since all further measurements 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 (1i) 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 (1 + l)th chain of length z Is included in the growth. The growth of this new chain is analyzed. Figure 3 represents the stage in which exactly k = 5 atoms of the (I + l)th chain are condensed. The diffi culty of forming nuclei here is due to the fact that during 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, P2,  Pk, .. Then the possibility qkdt that the kth atom evaporates as a result of the thermal motion in time interval dt on a chain con .sisting of k atoms, is given by (Pk qk = F(T)e kT (14) On the other hand, the possibility adt, that a further atom settles on the chain, is independent of k and solely given by the external vapor pressure. 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 a0 (of section 2) by multiplication with the surface atom. We put a = F(T)e kT (15) NACA TM 1374 hence, where the energy is the measure of the external vapor pressure.5 The whole mechanism of growth is governed by the factors (Pk Pk = e kT (16) 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 P~ 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 5. But the new chain in the process of formation 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 nk due to deposition of an atom at a certain end of the chain k, is indi cated by nk+lI By partial current J' is meant the excess of the growth process per second which lead from the nk crystals to those of the type nk+l, through the evaporation processes, which lead from nk+l to nk. For this specific partial current J' = nka nk+lqk+1 (17) 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 buildup. 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 nO to n1 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 4 for z = 6. How the not entirely exact assumption is made that all partial cur rents leading from k toward k + 1 are equivalent. Since their sum 5At absolute zero point the heat of vaporization would have to be used. As a rule, 4 signifies a thermodynamic potential. NACA TM 1374 gives the total current J, each is equal to J= J' = 2(z k) It is readily apparent from figure 4 that rigorously correct on account of the equations partial currents. Owing to equation (17) this the assumption that all positions of the chafn frequent. this assumption cannot be of continuity between the assumption corresponds to k are identically Thus, on this premise the steady state is described by the equations nI = noo1 n2 = n1i2 nk+l = nkPk+  nz = nz1Pz  J 01 a z J 82 2a z 1 J Pk+1 2a z k J 2a These equations 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 kth equation by the produce Pk+l = 1PP2 Pk+l, they take the form ck+l = 'k JRk (19) where the individual potentials and partial resistances indicate (18) NACA TM 1574 1 Rk = 2a(z k)p but 4 = no, 3 k O =1 "o a Thus, in figure 4 the electric potential of a specific state is represented by the quotient of the number nk of crystals in state k and the product Pk = P102 Pk of 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  and Rk Rk1 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 number of crystals removed per second is called linear nucleation frequency. Hence we put %s = 0 and find 1 J= ( O, with OO = no 1 1 1  2a (1 + 1 + + 2a z (z 1)01 (z 2)piP2 1 (z s + 1)pl .1 is1 nk k = 7 0102 k (19a) NACA TM 1574 With the specialization 02 = 3 = .* = p and because s << z 2az L ls2 1 2 + B51 1 2az (p 1)pps2 is applicable also. Disregarding the 1 next to ps1 and the 2 next to 1/P1, leaves 1 P R = hence the frequency of the linear nucleation at one 2az (0 1)p1 of the nO edges, independent of s = 2az P131 (20) no13 T 1 The factor 2 01, small compared to unity, 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 RO to Rz1 must be added up J(RO + R1 + + Rz1) = O Oz or S + 1 1 1 J +f 1 + + i= no 2a z (z 1)p1 (z 2)p3l2 1 312 ". Pz1 12 P The first partial resistance 2/z in comparison with 1 (z 1)pl can always be disregarded. Putting 02 = = = = we get with the abbreviation NACA TM 1374 Sz = B + + +.z (21) 2 5 z1 nz = noP;P21 Sz (22) The sum Sz does not lend itself to elementary evaluation. The approximate value Sz(P) z (21a) Sz z In p(21a) used in the following is obtained by the following consideration: Replacing the sum (21) by an integral gives Sz f x dx 1 x hence, with the substitution x = z 1 In p y p z In p(zl) eY Sz CJ dy 1  z In 0 The approximate value is obtained by disregarding which is z ln B small compared to unity, 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 (17a), valid for the actual elementary process, is simply replaced by the relation J = noAz nzBz (23) NACA TM 1574 whereby 2ap1z1 31 Az = 2 S1 2az ln p T Sz 2a zl 3 (23a)6 Bz = S 2az In PPz Sz 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 im1 = 1 or = m (24) m is that chain length which is precisely in equilibrium with the exter nal pressure. According to (16) the definition (24) of m is equivalent to 91 + (m l)p = mr or (24a) S = (cp 91) m The mean evaporation energy of the "critical chain m" is equal to the energy characterizing the external vapor pressure. 4. TWODIMENSIONAL 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 5. The bonding energy of a single atom on the smooth base ("bond to one neighbor") is denoted with c90; 91 and c have the same meaning as in section 3. Accordingly, there are 9o^ ^lt cp9 O = e kT, 1 = e kT =e kT (25) The energy required to detach the whole plate (i, k) from the base is then p0 + (i + k 2)91 + (i l)(k 1)p OIn 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. NACA TM 1374 From the assumption that p2 = 3 = .. = are all equivalent and independent of the position of the deposited molecule on the base, inevitably follows the condition (PO + (q = 2(p hence also PBP = 012 (25a) The total bonding energy of a structure must be independent of the manner in which the growth takes place. Applied specifically to a system of 5 atoms on the base as in figure 5, the binding energy for growth in the order of 1, 2, 3, is cpo + pg + cp, but for the sequence 1, 5, 2, it is cpo + Cp + Pl. The equality yields the above relationship. The evap oration energy of the whole plate (i, k) is therefore ilqp (i + k)(cp (p1) Visulize a column of cross section i x k consisting of I whole atom layers, the deposit of the (Z + 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 ns,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 + l)(k z + 1). The total number Zsz of the columns with a plate (s,z) would then be ns, (i s + l)(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 current Js,z, 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 equation (22), nz is replaced by Jsz b, while is ns+l,z, nO by s,z and J by is)(k while Jsz is 2(1 s)(k z + 1)s NACA TM 1574 to denote the current from (s,z) to (s,z + 1). plane growth is then governed by the equations ns+l,z = n"szi l i Sz ns+l'z =szl' 4a(i s)(k z + 1) ns,z+l = ns,zIP3s On the other hand is applicable. In the steady state the s,z = 1, 2, (26) "1 a( Js, z1)( 4a(i s + l)(k z) J1 = nil = noo0o a 0o aik (26a) Now the content of these equations is described by a discussion in the sz 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 3 x 2. The current Js,z flows then horizontally from s,z toward s + 1, z, Jsz', but 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 (0.0) and branches off in partial currents Js,z and 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 Os+l,z = s,z JszRsz (27) Os,z+l = Ds,z Js,z'RS,z' 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 buildup of the plane (s + l,z), that is, Ps+I,z = PB01(s+zI)gs(z1) NACA TM 1374 The potential os,z and the partial resistances Rs,z and Rs,z' become then =sz =  n srz 0,0 = n,0,0 (27a) Psz 0 1 (s+z2)p(sl)(zl) 1 Sz Rsz = 4a(i s)(k z + 1) Ps+l,z (27b) 1 z 4a(i s + l)(k z) Ps,z+1 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 Sz gives 1 1 Rsj' s, 4a(i s)(k z + l)z In 03 pOls+zlnszsz Introduction of the critical edge length m defined by equation (24) and with the relation =  = Om following from equation (25a), the P0 01 second factor of the above Rsg becomes Pmmm(s+z)sz = gm+m2(sm)(zm) With a system of axes turned through 450 (a along the diagonal, t at right angle to it) s = a + U s a(28) becomes becomes (s m)(z m) = (a m)2 2 NACA T' 1574 hence the factor of Rs,z in question pm+m2 ( (m)2 t2 The relationship of t implies: if the partial resistances Rsz, 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, 0 Y 9, etc. Thus practically only the diagonal s a z of this lattice is conductive, while along the diagonal (change of a at = 0), the resistance has the same sharp maximum at a = m, i.e., at the critical edge length. There the partial resistances drop with the distance from this point to the 01, p4, p9th 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 Dr2, which in turn leads over a very steep "pass" at a = m. The overall resistance R of the entire network is practically concentrated on the resistance SR m m+m2 (29) 4a(i m)(k m + l)m In P at the height of the pass. Naturally, it is assumed 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 purposes 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 3. 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 outoff 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 = 1 m+2Y hence, the nucleation 4aikm in B NACA IM 1574 current for a single one of the no crystals is = 4aikm In ppmm2 (50) nO The two factors P and m still remaining are tied to one another cp1pl by the relation 3m = 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 m(cpq9l) IP9l = 4aik e kT e kT (50a) no0 kT Here, as it should be, half the freeedge energy kT is in the first e power, because ((p pi) is the freeedge energy 2 per atom, 4m2(c pl) is, therefore, the total freeedge energy. The plane energy accompanying the formation of the chain (at both of its ends!) 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 impinging upon the plate ik. The then still remaining factor kT is nearly equal to evaporation heat 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 (Pqp1)2 J = 4aik P l e (kT)2 in P (5Ob) kT This equation gives the dependence of current J on the experimen tally directly defined supersaturation 8. NACA TM 157h 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) 0,0 Oi,k = JRi,k where Ri,k is the overall 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; Ri,k, therefore, according to equation (29), may be replaced by Rm,m. If use is made of the relations 0op = p12 and = Pm in the product Ps,z' Ps,z then becomes Psz = 00l1s+z2(sl)(zl) = (sm)(zm)m2 Thus, by equations (27) and (29) i,k = n (im)(km)m2 m+(im) (k) (31) 4a(i m)(k m)m In p 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 = no,0Ai,k ni,kBi,k (52) where Ai,k = 4a(i m)(k m)m ln Bpmm2 i (32a)7 Bi,k = A, k m2(im)(km) 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 7Multiple nucleation is excluded again. At very great i and k the foregoing is therefore no longer applicable. NACA TM 1374 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,k is very slightly dependent on i and k, while Bi,k decreases rapidly with increasing size of the plane. 5. THE CRYSTAL NUCLEUS After these preparations the quantitative treatment of the nucleation frequency for spatialcrystals is easy. Again visualize in a vapor space a large number of boxlike 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 Zi,k,Z. They may, for example, change to crystals (i + 1, k, 1) by gathering of a plane (k,l), 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 ni,k by Zi+l,k,l, no,0 by Zi,k,l, i,k by k,2 and J by 1/2Ji,k,l results in Zi+lk Z (km)()m2 Ji k Om+(km)(1m) ^l k I= Zik 3) 8a(k m)(2 m)m in p where Ji,k, indicates the partial current that leads from i,k,Z to i + l,k,l. By the method previously used several times, equation (33) can again be put in the form i+l,k,i = i,k,l Ji,k,Ri,k,l (54) and identify it as the electrical current in a spatial network whose lat tice points have the potential Oi,k,Z and in which Ri,k,Z is the ohmic resistance of the piece of wire that leads from i,k,Z to i + l,k,2. The transition from (55) to (54) is accomplished again by division of (35) by the product Pi+l,k,l of all the P values which occur during the suc cessive development of the state (i + l,k,l) from (i + l)k,l single atoms. The factor for Zi,k,Z in equation (35) is precisely the product Pk,l NACA TM 1574 of the p values for that plane (k,Z) which is newly added in the par ticular reaction. The product Pi,k,l can be constructed as follows: On a single atom free columns of i 1, k 1, 2 1 atoms are depos ited along the coordinate axes, each of which gives the factor pg The spreadout rectangle sides are then filled out. It yields (i 1)(k 1) + (k 1)(1 1) + (Z )(i 1) times the factor p1 and, in addition, (i l)(k 1)(? 1) times the factor p. Hence, altogether Pi,k,l = p0i+k+i5l(il)(kl)+(kl)(21)+(2l)(i)p(il)(kl)(3l) for which the relations and  = m give l 0 Pi,k,2 pm(ik+kl+li)+ikl+3ml Pi+lk,h = Pi,k, (km)(m)m2 By division with this quantity, we obtain in equation (34) of Ji,k,I the final term for the partial resistance m (ik+kl+1 i)ikl+m22m1+l 8a(k m)(1 m)m In 0 To clarify the behavior of the line dropped from the point i,k,2 a,o,a to indicate the foot of this as factor (55) exponent visualize a perpendicular of the network on the space diagonal, perpendicular. S= o + r k = o + r 2 = o + rJ k a + r .2 (36) the construction given then rl + r2 + r3 = 0 Ri,k,I = Putting (36a) NACA TM 1574 along with rlr2 + rr + rrl = (rl2 + rl2 + r 2) r2 23 31 2 13 2 By (36) and (36a) ik + kZ + ii = 3a2 ir2 and iki = oa lor2 + rlr2r With this the exponent of equation (35) reads m(ik + kZ + li) ikI = 352m 3 + (a m)r2 rr2r Disregarding the practically nonessential term r1r2rF (the sur rounding of the diagonals being considered), the conditions for Rik, are the same as before in section 4 for the plane lattice. The factor l(am)r2 (2 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 03a2m03 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 = 1 4m3+m22m+l 8ams In p Owing to (l,l,l = Z1,l,1 the nucleation frequency is therefore defined at J = 8aZll,lm3 ln pB4m3m2+2m1 (37) Elimination of p by means of pm =e kT m In P = B1 kT NACA TM 1574 leaves 4m2 (qP p1) m(991) q1 _19 IP Cbl .Mi_ 2 S= 8aZl1, 1,m2 T e kT e kT e m (37a) The factors deciding the order of magnitude of J are aZl,,1, and the first of the three efunctions; aZl,1,1 is essentially (like the factor aoZ1' in equation (13) for droplet formation) the number of gas kinetic collisions per second. The first epower is synonymous with the Fn factor e 3kT of the thermodynamic formula (5). In fact, ( (( (1) is the surface energy per atomic the total surface energy of the cube of critical edge length 2m, therefore, is equal to (2m)2 x 6 x (p t9) = 12m2((p ); the third portion of it stands, as it should be, in the exponent. The exponent of the second efunction indicates, as shown in section edge enter 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 12 the problem involving the critical supersaturation the second and third efunctions 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 (13) and solid (37) nuclei enable a theoretical foundation and a quantitative improvement of this law to be made. Omitting the last two efunctions in (37a) and introducing the rela tion for Volmer's exponent of equation (5) A" F =_ 4m2(q ql) 3kT kT NACA TM 1374 we get by equation (37a) on the crystal nucleus Crystal: Jcrystal = 2aZl ,1A"eA" while equation (15) produced Droplet: Jdroplet = aZ eA' The factor Z1' was, according to section 2, the number of vapor molecules multiplied by its surface. Since, according to section 5, a arises from a0 by multiplication with the atom surface, aZ1,1, and aOZ are identical in order of magnitude. Thus, the factor K of equation (5) for the formation of droplets appears smaller by 1/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 m2 omitted at Jcrystal The factor A" and matter even less. crystal Ic 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 5 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 quantity aF the surface F of the 3kT 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 = Cn2/5 is the surface corresponding to the molecule number Pn n, then by equation (1), with x denoting the abbreviation of ln n Poo kx = a dF 2 aC/2 dn 5 Fl/2 NACA IT. 1574 hence the surface corresponding to x F 4 a2C3 9 (kTIx)2 The nucleus volume V for the sphere (radius r) is V = nv = r TrI thus, F = 4K7 2/5)2/5n for the cube (edge length a): V = nv = a3, hence, F = 6v2/3n2/5 Hence, for the sphere c3 = 36~v2 and for the cube c3 = 63v2 The critical area corresponding to the same greater for the cube than the droplet. To assure frequency, hence, equal values of F, it must x is = 1.91 times identical nucleation = Ji.91 = 1.58 For the critical supersaturations themselves the condition would be PO)cube p\l1h58 P/sphere (38) As an illustration for applying this relationship, the supercooling at which crystalline and fluid nuclei occur with comparable frequency is analyzed. The saturation vapor pressure of the liquid phase is denoted Xcube Ccube /2 sphere c3 sphere/ 52 NACA TM 1357 by Pl, that of the solid phase by P2. By equation (58) the condition for comparable nucleation frequency reads J2 = LP V158 P 2 1' or In p = In Pl + 2.6(ln pl In p2) (59) (The factor 2.6 is equal to 1:(1.58 1). Figure 7 shows the vapor pres sure curves In pl and In p2 plotted against T. According to equa tion (59) 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 TA at which an isothermal pressure rise would result in a simultaneous separa tion of fluid and crystalline nuclei. Below TA only solid, above TA only fluid nuclei would be observed. Naturally, it may also happen that no intersection point appears. In that event, the law of stages holds unrestrictedly.8 7. THE GENERAL RESISTANCE ANALOGY As already stated several times in the foregoing, the equations (17), (23), and (52), 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. 8Such a case seems to exist in the theoretical case treated by Stranski and Totomanow (Z. f. phy. Chem. (A), 165, p. 399, 1955). NACA TM 1574 lext, 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 nI and nII are the number of crystals of type I and II in the steady state, the equation for this specific transition process reads J = nia nIqvy+l (40) where a and qv+l are the repeatedly employed deposition and evapora tion probabilities of the atom at that particular spot. With the abbre viation gv+l = a the result is again qv+1 J(41) nII = nlPv+1 ; v+1 (41) cPv+l i kT kT, introduced, hence, Again q+1 = F(T)e kT and a = F(T)e are introduced, hence, Pv+l  Pv+l = e kT The energy of separation Pv+l 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 Pi. The individual Pi still will be dependent upon the sequence in which the atoms of the crystal are joined together. But the product P(II) of all Bi v+l !++i 9 v+l i=l 9Factor p0, which by itself corresponds to no growth process, is put equal to 1; hence cp = r. NACA TM 1574 is solely dependent upon the configuration of crystal II, because the v+ total work of growth _lpi can no longer depend upon the manner in i=1 which the growth took place. Dividing equation (4l) by PI1 gives v+l I II = JRI (42) where, for abbreviation Oi = nI I = nII (42a) Pv(I) Pv+(II) and R 1 (42b) I a P(I) 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, III, we then correlate a wire connection between the points of state I and II, to which the resistance given by equation (4lb) is allocated. The equation (42) can then be regarded as the Kirchhoff equations of this wire netting and 0 as the corresponding potential of the nodes in this net. This interpretation is possible, because Q 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 4 and 5, 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 4 and 5, a large number of wires were consistently ignored because of their high resistance and complete wire systems combined into one resistance. For the actual calculation, this general analogy is therefore of little help. However, the purely quali tative distribution of the resistances will be indicated. NACA TMl 1574 The resistance of a wire depends solely upon its initial point and has the form v R = e kT (45) a where the quantity vi pi 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 AK Rnucleus = ek (43a) with AK the work of nucleation. As is seen, the saddlelike character of the resistance distribution near the nucleus is completely independent of the model. The specific model representations merely yield information about the number 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 by Nucleus. Translated by J. Vanier National Advisory Committee for Aeronautics 36 NACA TM 1574 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. 256. 5. Kaischew, R., and Stranski, I. N.: Ztschr. f. phys. Chem. B. 26, 1954, p. 317; Stranski, I. N., and Kaischew, R.: Phys. Ztschr. 26, 1955, p. 395. 4. Kaischew, R., and Stranski, I. N.: a. a. 0., also Ztschr. f. phys. Chem. (A), 170, 1954, p. 295. 5. Kossel, W.: Ann. d. Phys. (5), 21, 1934, p. 457. NACA 'M 1374 0 Figure 1. (case 3xi2  Exponent 3xi22xi3 .5 0.5 / 1 0.5 1 1.5 x  Resistance Ri plotted against droplet radius xi A' = 10 and n = 100).  : Curve of exponent 2xi3. . y I NACA IM 1574 Figure 2. InJ plotted against x= In P at two temperatures, Po computed for water by equation (13a). NACA TM 1374 Figure 3. A specific state of crystal growth. k =0 1 2 3 zlz Figure 4. Network of partial currents for the growth of the linear chain. 40 NACA T 1574 2 Figure 5. Derivation of the relation Pe0 = PI . NACA TM 1374 k 5 4 3 2 I Figure 6. 1 2 3 4 5 6 Current network for growth of the plane. Point A: Rectangle: 3 x 2. Size of base: 8 x7. NACA IM 1574 In p In p1 Inp 2 1 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. A .. ....."."." .....*.'+**** NACA TM 1374 State 1 a. State II d. Figure 8. The most elementary process of crystal growth. Deposi tion of one atom at the emphasized spot of state I leads to state II. Evaporation of raised atom on crystal II leads to crystal I. NACALangley 9854 1000 I E a, '  cu' " Su*V .4 0J, C 2>('  Cu c cr r .2 E z U 0 C = E E i N ( < u o20 : o ui . " ^i n ^irt ja9 E di l~ ,iL 1s1 2! o a) bb ml E~ ci > u L w. C J C 0 r, r CIZ~ LOWle o U .W (U 0 "u u t W 0 1'"0 C S'6 0i a r .4. bO I)L *a~fv CQ E m m^; a a, = M c d 0 i M C Q W cu w 06 cu W 6 a.0 *,C Z WCmw LEML Ej^CS i 13,2 , oe >c ci ; fr c2i5 o llrti ;  NIC E M W 0S Crn u CU ( (. 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