Kinetic treatment of the nucleation in supersaturated vapors

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Title:
Kinetic treatment of the nucleation in supersaturated vapors
Series Title:
NACA TM
Physical Description:
43 p. : ill. ; 28 cm.
Language:
English
Creator:
Becker, R
Döring, Werner, 1911-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Atmospheric nucleation   ( lcsh )
Aerodynamics -- Research   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: The equations of the individual processes of self nucleation are utilized through an electrical analogy to obtain the nucleation frequency. This process is shown to be shorter and less subject to error than that of previous investigators since the appearance of indeterminant integration constants is completely avoided. With the nucleation frequencies of crystals and spheres the Ostwald law of stages is reviewed and modified. In the final section the general resistance image is discussed and mention is made of the relation of the electrical network and Volmers formula.
Bibliography:
Includes bibliographic references (p. 36).
Statement of Responsibility:
by R. Becker and W. Döring.
General Note:
"Translation of Kinetische behandlung der Keimbildung in übersättigten Dämpfen," from Annalen der Physik, Folge 5, Band 24, 1935."
General Note:
"Report date September 1954."

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University of Florida
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sobekcm - AA00006149_00001
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Full Text
&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fog in supersaturated water vapor, for example) was originally made amen-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. 719-752.






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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.






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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 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 (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






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_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






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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 quasi-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 "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.






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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' = Zv-1' 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.






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In order to eliminate from the equations

r
Z+l' = Zv'Pv+1 a v-l


Zv+2' Zv+1 +2 0 v+2
(7a)



Zs' = Zs-l'ps OS


the factors Zv+l', Zv+21, Zs-l', 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 i-l
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






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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' (5xi2-2xi3)-(5xl2-2x135)
a0

Replacing the summation over the partial resistances
integration, leaves


J S


Rydv = 5n [Xs
SX


R(x)x2dx = i5 e-A'(3x12-2x3)
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'


e-A' (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


e-3A'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'(3x2-2x3)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 figure-l, 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 Volmer-Flood
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






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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.






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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 = 10-5


(15a)


with the Volmer-Flood 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






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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 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

(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)






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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
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 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.






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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 = nz-1Pz -


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 k-th 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)






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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


- respectively
Rk-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 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 is-1


nk
k = --7
0102 k


(19a)






NACA TM 1574


With the specialization 02 = 3 = .* = p and because s << z




2az L ls-2


1- 2 + B5-1 1
2az (p 1)pps-2

is applicable also.

Disregarding the 1 next to ps-1 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 Rz-1 must
be added up


J(RO + R1 + + Rz-1) = O Oz


or

S + 1 1 1
J +f 1 + + i= no
2a z (z 1)p1 (z 2)p3l2 1 312 ". Pz-1 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 z-1

nz = noP;P2-1 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(z-l) e-Y
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
2ap1z-1 31
Az = 2 S-1 2az ln p -T
Sz

2a zl 3 (23a)6
Bz = -S 2az In PP-z
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

im-1 = 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. 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 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-^ ^l-t cp-9
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 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 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 build-up
of the plane (s + l,z), that is,


Ps+I,z = PB01(s+z-I)gs(z-1)






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+z-2)p(s-l)(z-l)


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+z-lnsz-s-z

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-(s-m)(z-m)


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 0-1, p-4, p-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 Dr2, 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


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
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 = 1 m+2Y hence, the nucleation
4aikm in B






NACA IM 1574


current for a single one of the no crystals is

= 4aikm In pp-m-m2 (50)
nO

The two factors P and m still remaining are tied to one another
cp-1pl
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(cp-q9l) IP-9l
= 4aik e kT e kT (50a)
no0 kT

Here, as it should be,

half the free-edge energy
kT

is in the first e power, because ((p pi) is the free-edge energy
2
per atom, 4m2(c pl) is, therefore, the total free-edge 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

-(P-qp1)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 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; 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+z-2(s-l)(z-l) = (s-m)(z-m)-m2


Thus, by equations (27) and (29)


i,k = n (i-m)(k-m)-m2 m+(i-m) (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 Bp-m-m2
i (32a)7
Bi,k = A, k m2-(i-m)(k-m)


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 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 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 (k-m)()-m2 Ji k Om+(k-m)(1-m)
^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
spread-out 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+i-5l(i-l)(k-l)+(k-l)(2-1)+(2-l)(i-)p(i-l)(k-l)(3-l)


for which the relations


and


- -= m give
l 0


Pi,k,2 p-m(ik+kl+li)+ikl+3m-l



Pi+lk,h = Pi,k, (k-m)(-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+m2-2m1+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(a-m)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 03a2m-03 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+m2-2m+l
8ams In p
Owing to (l,l,l = Z1,l,1 the nucleation frequency is therefore
defined at


J = 8aZll,lm3 ln pB-4m3-m2+2m-1 (37)

Elimination of p by means of


pm -=e kT m In P =
B1 kT






NACA TM 1574


leaves
4m2 (qP p1) m(9-91) 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 e-functions; 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 e-power 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 e-function 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
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 (13) and solid (37) nuclei enable a
theoretical foundation and a quantitative improvement of this law to be
made.

Omitting the last two e-functions 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"e-A"


while equation (15) produced


Droplet: Jdroplet = a----Z e-A'


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 = 4K-7 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\l1-h58
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 V1-58
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, I--II, 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 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 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
3xi2-2xi3










-.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 z-lz

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.


NACA-Langley 9-8-54 1000




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UNIVERSITY OF FLORIDA


1262 06 54 8