F. C . GOODRICH
3660
OH radicals was not observed even on prolonged irradiation. One must conclude, therefore, that the reactions leading to the formation of NHz radicals are excitation of NH3 molecules and dissociation of the excited molecules according to reaction 9, additional to a photoionization according to NH3
+ hv +ISH3++ e-
(19)
with trapping of the electron in the matrix as a negative polaron followed by the formation of NH2 radicals according to reaction 11. Thus here again the formation of the greater part of
the NHz radicals in the frozen ammonia-water systems is due to the interaction of the species from ammonia and from water. One will therefore have a situation similar to the case discussed above for the y irradiation and one should expect again the maximum NH2 yield at a mole fraction of 0.5, which is in agreement with the observations
Acknowledgments. P. N. XI. wishes to express his thanks to the Atomic Energy Establishment Trombay, Bombay, for deputation under the Colombo Plan during the tenure of which the present work has been carried out.
On Diffusion-Controlled Particle Growth: the Moving Boundary Problem
by F. C. Goodrich Department of Chemistry, Clarkson College of Technology and Institute of Colloid and Surface Science, Potsdam, New York (Received J u l y 16, 1966)
A condensed phase growing in the presence of its vapor or by solution deposition creates a diffusion field in its neighborhood which is influenced by the fact that the phase boundary must necessarily advance into the vapor or solution. This problem is analyzed by a new method, and it is shown for most experimental conditions likely to be of interest that the rate of growth of the new phase is influenced only negligibly by the interaction of its moving boundary with the diffusion field.
I. Introduction Rates of particle growth via a diffusion-controlled mechanism are complicated by the fact that the flux of condensing substance into the particle takes place in the presence of the outward-moving boundary of the growing but otherwise stationary particle. That this motion must perturb the diffusion field has long been recognized, and various attempts’-’ have been made to estimate the magnitude of the effect. Of these the most profound is that of E ’ r i ~ c h and , ~ ~ in ~ this paper many of his results are confirmed by an independent method. The author also hopes to clarify the physical meaning of the mathenlatical approximations which must necessarily be made in the solution of this probT h e Journal o,f Physical Chemistry
lem and to discuss the errors inherent in the use of the parabolic growth law formula in current use by experimentalists. *
II. The Conventional Approach Let us consider a single sphere of colloidal dimensions (1) H. G. Houghton, Physics, 2, 467 (1932). (2) H. Reiss and V. K. La Mer, J . Chem. Phys., 18, 1 (1950). (3) F. C. Frank, Proc. Roy. SOC.(London), A201, 586 (1950). (4) H. L. Frisch, Z . Elektrochem., 56, 324 (1952). (5) H. L. Frisch and F. C. Collins, J . Chem. Phys., 21, 2158 (1953). (6) P. L. Chambre, Quart. J . Mech. A p p l . Math., 9 (II), 224 (1956). (7) J. S. Kirkaldy, Can. J . Phys., 36, 446 (1958). (8) N. A. Fuchs, “Evaporation and Droplet Growth in Gaseous Media,” Pergamon Press Ltd., London, 1959.
DIFFUSION-CONTROLLED PARTICLE GROWTH
3661
immersed in an infinite bath in which are dispersed molecules of a substance which diffuse to the surface of the sphere and are there deposited. Let the sphere at m y time t have a radius R(t) and let distance from its center be measured by a radial coordinate r. Then the concentration #(I.) of diffusing species in the region r > R external to the sphere satisfies the diffusion equation
ments with single fluid droplets bathed in an atmosphere of their own vapor.
111. A Deeper Analysis The unsatisfact,ory nature of this derivation is selfevident from the number of simplifying assumptions which are made for mathematical convenience. Rather than ignore the effect produced on the diffusion field by the moving boundary, we may proceed as follows: in (1) and its boundary conditions, substitute U ( T ) =
MY). in which D is the diffusion constant. At the boundary of the particle r = R, perfect absorption of incident molecules requires #(R) = 0, and a t t = 0 we demand that external to the sphere, # = C = the uniform initial concentration of the diffusing species. At this point in the argument, it is conventional to solve the diffusion equation with the stated boundary and initial conditions ignoring completely the fact that the radius R of the particle must be an increasing function of the time. Instead, the known solution for the problem with R = constant, is used
dU
- =
dt
d2U
Ddr
(at r = R
u = C(Ro u=0
in which erfc(z) = ( 2 / ~ ~ ) ~ m e x p ( -dz. z 2 )The total
’
4aR2D 1 d7’
=
4sRDC[!
+ R/(aDt)”2]
r=R
+4sRDC as t
-+=
(4)
= RO+ p(t))
+ x)
X
> Ro)
The quantity p(t) has the dimensions of distance and is evidently defined by p ( t ) = R - Ro, so that p ( 0 ) = 0. Introduce a new space coordinate x = r - Ro p ( t ) . Equations 4 become
R erfc[(r - R)/(4Dt)”’j}
flux of diffusing substance into the spherical surface is then readily calculated to be
> R)
(at t = 0 and r
u = Cr u=0
(for r
(at t
(at x
=
=
(5)
0)
0)
where p’ dp/dt is the velocity of the moving boundary. The reader may readily confirm for himself that for arbitrary p an exact solution to the first two members of eq 5 is
u = I / ~ C((R O
+ +
+
+
+
5 P) [I erf(x P ) / ( G ~ ) ” ~ I (4Dt/a)’/’ exp[-(x ~ ) ~ / 4 D t]j
large
+
To obtain the growth law, the rate of increase of volume V = (4a/3)R3 of the particle is equated to the asymptotic flux over the houndary dV/dt
=
(6)
4aR2(dR/dt) = 4aRDuC
in which u is the volume per molecule in the condensed phase. Integration leads immediately to R 2 - Ro2 = 2vCDt
(2)
with Ro being the initial radius of the growing particle. Because this law was derived only under the assumption of t large, it is safe to assume that whatever validity it has can be obtained only in the asymptotic limit, so that with increasing time R + (2vDCt)”’
(3)
This dependence of the particle radius on the square root of time has been confirmed by Fuchss in experi-
in which 4(t) is an arbitrary function of the time. The origin of eq 6 is to be found in an application of the Wiener-Hopf techniquee to the first member of ( 5 ) , but it is more expedient to confirm the correctness of (6) by direct substitution into the partial differential equation. The function erf(x) is the standard error integral erf(x) = ( 2 / 6 ) j exp(-z2) dz 0
To satisfy the third member of ( 5 ) ) choose 4 so that u = Oatx = 0 (9) I. N. Sneddon, “Fourier Transforms,” McGraw-Hill Book Co., Inc., New York, N. Y., 1951,p 262.
Volume 70,Number 11 AvovembeT 1966
F. C. GOODRICH
3662
{
L + ( r ) [ ~ a ~-( 7t ) 1 - ~ / 2 exp - [P@> -
+
li2C((Ro
P)
[1
P(T)J}
4D(t -
dr
=
7)
+ erfp/(4Dt)"'I +
(4Dt/T)'" exP(-P2/4Dt))
(7)
which is an integral equation for the determination of 4. Finally, to make contact with the material balance condition at the particle boundary, the rate of increase of volume of the particle must be equated to the flux of material depositing on it, and this may be made equivalent to the condition
A completely self-consistent solution to the problem of diffusional growth in the presence of a moving boundary would then consist of choosing an initial form for p, solving (7) for +, calculating (bu/bx),,o from (6), and recalculating p from (8). The procedure would then be iterated until self-consistency was achieved.
IV. Approximate Methods It need hardly be emphasized that a completely selfconsistent calculation of the type suggested above would tax the capacities of both the programmer and the computer. To avoid such difficulties, it is necessary to turn to approximate methods, and fortunately good approximations to the exact solution can be madk for most experimental conditions which are likely to be of interest. Frisch4 expanded the exact solution in a series whose successive terms satisfied an infinite set of differential equations. In the present case, an approximation will be made based less upon mathematical elegance than upon physical intuition, but which correspondingly has the advantage of yielding physical insight into its limitations. By examination of (6), we observe that the dimensionless quantity p/(4D2)1/*and quantities related to it occur in several places. From the theory of Brownian motion, it is known that (2Dt)'l' is the root-meansquare distitnce traveled by a diffusing molecule in a time t, and by definition p is the distance by which the radius of a growing spherical particle is increased in the same period. Elecause the particle is growing from the deposition of molecules diffusing to its surface, it seems intuitively evident that the radius cannot increase a t a rate greater than the mean displacement velocity of a molecule executing Brownian motion. Indeed, the rate of increase of the particle radius can be diminished at will by the experimenter simply by reducing the concentration C of the condensing species, whereas the root-mean-square displacement (2Dt)"' of a diffusing The Journal
o,f Physical Chemistry
molecule is independent of C. In the limit of low concentration, it should therefore be possible to make the ratio p / ( 4 D t ) ' / ' arbitrarily small for all values of the time. These considerations lead to the replacement of eq 6 and by
+ +
'/zC{(Ro z
u =
p)
[l
+ erf z/(4Dt)"'] +
( 4 ~ t / n ) ' exp(-x2/4Dt)} /' X + ( r ) [ 4 7 r D ( t - r ) ] - ' / ' exp[-zz/4D(t
- r ) ]d r
(9)
+ + (4Dt/n)'/']
(10)
and
'/zC[(Ro
P)
in which all quantities p/(4Dt)'/' have been neglected. The validity of this approximation can be established a posteriori. For the moment, its justification rests upon the heuristic argument posed above. Introducing the Laplace transform C(f) = /mf(t)e-ptdt 0
transformation of eq 10 leads to C(4) = C[Dp-l
+ R&'/'p-'/' + D"'p'~zC(p)] (11)
Additional transformation of (9) in the limit x also yields
+
0
lim c(bu/bx) = Z-0
l/zC(p-l
+ R&-'/zpp-'/a) + ( 1 / 2 D ) C ( 4 )
(12)
It follows that the integral L(bu/bx),,dt possesses a Laplace transform which is p-' times the right-hand side of (12). Substituting from ( l l ) ,we have
L
6:
(bu/bz),-odt =
C[p-2 + R&-'/a P
+ l/2D-1/~p-1/z~(p)]
whence upon inversion of the transform and substitution into (8) there results an integral equation for p
(Ro
+ p)'
- Ro2 = 2vDC t
+ 2 R o ( t / ~ D ) ' +/ ~
DIFFUSION-CONTROLLED PARTICLE GROWTH
3663
Y. Solution of the Integral Equation
and only the positive root
Because it is nonlinear in the unknown function p , it has not been possib!e to obtain a general analytic solution to the integral eq 13. Only in the limit of t very small and t very large is an analytic formula conveniently accessible. The computations are simpler if carried out in terms of the dimensionless variables
Y
e
=
P/RO
= Dt/Ro2
X =
vc
so that y represents the increase in particle radius measured in units of the initial radius Ro, 0 is time measured in units of the time necessary for a diffusing molecule to diffuse a mean-square distance 2R02, and X is the ratio of the number density of the condensing substance when dispersed in the solution phase to its number density in the condensed phase. The parameter X should thus for most cases of interest be very small, and indeed the success of the approximation which replaced eq 6 and 7 with eq 9 and 10 depends upon being small, a point to which we shall return in due course. I n terms of the dimensionless variables, eq 13 is
- a)-'/'y(a)
(1/21/ir)h['(e
+ y2x)e + . . .
(15)
or in terms of dimensioned variables p = ~C[2(Dt/7r)"'
+ (1 + '/zvC)(Dt/Ro) + . . . ]
For large times when the particle has grown to such
a size that its original dimensions are negligible, then in (14) neglect y with respect to y 2 and drop the term in in comparison with e y* = 2x8
+ (1/2.\/a)~J
e
(e - . ) - ' / ~ y ( ~ )da
0
By direct substitution, this last equation is shown to be satisfied by y = be"' provided that the constant b is chosen to be a root of the quadratic equation
b 2 - (X&/2)b
- 2X
=
0
(16)
has any physical meaning in the present context. Everything we have done so far has been predicated upon the assumption that p/(4Dt)'/' = ~ / ( 4 e ) ~is/ ' a very small number. It follows that our procedures are valid only if b is also very small, and from a study of (16) we may in turn imply that X must be small. However, this is precisely the conclusion to which we were led by heuristic reasoning when we remarked that our results could be valid only if the ratio X of the number density of monomer when dispersed in the solvent to its number density in the condensed phase is small, and thus we establish the validity of the approximation a posteriori. These remarks lend physical substance to the nature of the approximate method and define its experimental limitations. It may be noted that Frisch's4 series expansion is rapidly convergent only under identical conditions. To continue, expand the right-hand side of (15) in a power series in
b = (2X)"'
+ (dx/4)X + (s/321/5)Xa'/' + . . .
(17)
Using the leading term of this series and reverting to dimensioned variables, the large time asymptotic solution to the moving boundary problem is
(14)
When t (and e) are very small, p (and y) are also small, and the term in y 2 may be neglected. The resulting linear integral equation may be solved by standard Laplace transform techniques, but the exact result is of less interest than its series expansion for t (and e) small
+
+ '/2[8X + ( T X ~ / ~ ) ] ' / ~
p = (2vCDt)l"
da
0
y = a ~ ( e / , ) ~ / ~ x(i
b = Xd?r/4
in exact agreement with the conventional solution (3). While the use of additional terms from (17) might be said to constitute a correction to (3), it is unlikely that experimental conditions could be found in which the higher terms play a significant role. S o t only would the use of higher terms imply that X is no longer sniall, thus invalidating our assumption that p/(4Dt)'"
