An approximate analytical technique for extracting the concentration dependence of the interdiffusion coefficient was derived from diffusion measurements carried out with the diaphragm cell. The systematic error incurred in using the technique was estimated and found to be no greater than the random error ordinarily encountered in these experiments

Baird, J. K.

English

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THE UNIVERSITY OF ALABAMA IN HUNTSVILLE
Huntsville, Alabama 35899
OSTWALD RIPENING THEORY
QUARTERLY REPORT
April 1 - June 30, 1985
NAS8-35986
BY
Professor James K. Baird
Department of Chemistry
(NASA-CB-178862) CSTWALD EIPENING THEORY N86-27433
Quarterly Report, 1 Apr. - 30 Jun.,1985
{Alabama Uuiv., Huntsville.) 10 p
HC A02/MF A01 CSC1 07D Unclas
G3/25 43166
Prepared for
George C. Marshall Space Flight Center
National Aeronautics and Space Administration
Marshall Space Flight Center, Alabama 35812
https://ntrs.nasa.gov/search.jsp?R=19860017961 2020-03-20T14:10:57+00:00Z
(1) Personnel
Dr. Li Kao Lee continues to be supported under this contract.
(2) Expenditures
The cumulative amount, $31,392.00, or 46% of the contract budget
has been expended by the end of this the third quarter.
(3) Technical Accomplishments
3.1 Concentration Dependence of the Interdiffusion Coefficient.
It now seems certain that the proper coefficient of diffusion to be
used in the theory of Ostwald ripening of a two component system is the
interdiffusion coefficient. This is the diffusion coefficient measured
with respect to the center of volume of the system. This diffusion
coefficient is conveniently determined with the diaphragm cell.
In our first quarterly report, we showed that the concentration
dependence of the interdiffusion coefficient can be easily measured using
this cell if the volumes V, and Vp of the top and bottom compartments are
eq^al. In that report, we showed that the interdiffusion coefficient,
Dfc), evaluated at c, the arithmetic mean of the top and bottom concentra-
tions satisfied the equation
'c'(t) - c"(t)~| _ 2AD(c)t ,, ,
 M"
J' (0) - c'TOj
where £ and A are the thickness and cross-sectional area, respectively,
of the cell; V=V,=V2; and c'(t) and c"(t) are the bottom and top concen-
trations, respectively, at time t, while c'(0) and c"(0) are the analagous
concentrations at t=0. All other quantities being known, if c'(t) and
c"(t) are followed as a function of t, Dfc) can be determined directly
from Eq (1).
We have found a new and more direct derivation of Eq (1), which exploits
a Taylor series expansion of D(c) about c = c rather than c = 0 as done
previously. The new derivation is much shorter and more direct. It leads
to a new formula for estimating the systematic error, e, associated with
Eq (1) (which is approximate). We find
\6 D(c)
—
where AC = c'(0) -_ c"(0) and Dv '(c) is the second derivative of D(c)
evaluated at c = c. Using reasonable values for the quantities in Eq (2),
we find
e - 1.6%- (3.1.3)
which is well within the random error ordinarily encountered when using
the diaphragm cell.
A research paper on this subject was delivered at the European Materials
Research Society Meeting at Strasbourg, France May 13-15, 1985 and w i l l be
published in Journal de Physicue. A copy is attached.
3.2 Grain Size Distribution in Ostwald Ripening as a Function of
Volume Fraction of Grains
We consider the precipitation of a single solute from a two component
solution. Define the following quantities
c(o°) - Equilibrium solubility in molecules/Cm of the solute in
the solution.
v - molecular volume of the solute in a precipitate grain in
Cm /molecule.
D - Interdiffusion coefficient of solute and solvent in the
parent phase in Cm /sec.
7 - Surface tension between a precipitate grain and the parent
solution in erg/Cm .
kT - Product of Boltzmann's constant times the absolute temperature,
T, in ergs.
R - Radius in Cm of a given precipitate grain.
It is helpful to define some intermediate quantities. Let
a = 2-rv/kT (3.2.1)
Define a dimensionless time, T,
T = tDvc(«) /a2 (3.2.2)
and a dimensionless radius, a,
a = R/a (3.2.3)
We find that the grain distribution, F(a,i) (where F(a,-r)da is the number
of grains with radii between a and a + da) .is given by
F(a,t) = t'4/3 F0(z) (3.2.4)
Here
z = a/T1/3 (3.2.5)
and
FQ(z) = CQz2 (zo-zr8(z+3/z2r6 exp(-6/(z0-z)), Z<ZQ (3.2.6)
where , -,
3zo(2o + 5)
e = 2 + ° ° (3.2.7)
B = 1 + > 27 ,, (3.2.8)
6 = 3zo (3.2.9)
(z3
 + 3)
The cut-off radius, z , satisfies the quartic equation
- 3 = 0 (3.2.10)
where
e = (4.55<p)1/2 (3.2.11)
and q> is the precipitate grain volume fraction which occurs once the
steady state of the Ostwald ripening has been established. The normaliza-
tion constant, C , depends upon <p.
Figure 1 shows the function F (z) for various values of 9. These curves
were computed as follows: A value of <p was chosen and e was calculated
using Eq (3.2.11). Eq (3.2.10) was solved for z using this value of e.
Next, the exponents e, e, and 6 were computed using Eqs (3.2.7) - (3.2.9).
The value of the normalization constant was determined from the integral
°r\dz z- FQ(z) = 1 (3.2.12)
which guarantees the conservation of solute mass.
Figure 1 demonstrates that competition for solute between solute
grains has a substantial effect on FD(Z), even for small values of <p.
The grain size distribution becomes broader and the maximunuradius
(cutoff radius) increases with <p. Because the area under z F (z) is
conserved (See Eq (3.2.12)), the brodened distributions for 9 > 0 are
also shorter. Detailed derivations of Eqs (3.2.1) - (3.2.12) will be
presented in the final report.
1
FoCz) us z in dl case for d i f f e r e n t
e q u i l i b r i u m volume fractions
^ -e
ON THE CONCENTRATION DEPENDENCE OF THE
1NTERD1FFUS10N COEFFICIENT
James K. Baird
Department of Chemistry
University of Alabama in Huntsville
Huntsville, Alabama 35899
USA
ORIGINAL PAGE IS
OF POOR QUALITY
Abstract - We derive an approximate analytical technique for extracting the
concentration dependence of the interdiffusion coefficient from diffusion
measurements carried out with the diaphragm cell. The systematic error
incurred in using the technique is estimated and found to be no greater
than the random error ordinarily encounted in these experiments.
1 - INTRODUCTION
In microgravity, Ostwald ripening in phase separating immiscible liquids can be
observed without interference from convection and sedimentation. Two component
systems such as succinonitrile/water or succinonitrile/ethanol are easily studied,
because they are optically transparent. In order to compare results obtained with
these'systems to the theory of Ostwald ripening /1,2,3/, the interdiffusion coef-
ficient, and the interfacial energy and densities of the separating phases must be
measured.
If the two components form an ideal solution, the interdiffusion coefficient is
independent of composition and can be measured directly in earth's gravity using
the diaphragm cell /4/. In the cese'of the two monotectic systems, NC(Ch' )?CN/H?0
and NC(CH~)2CN'/C2H5°"' mentioned above, however, the interdiffusion coefficient is
a function1 of composition. In as much as the diaphragm cell yields the integral of
the interdiffusion coefficient over a range of concentration, the integral data must
be unfolded to obtain the desired quantity /5/.
Below, we show that under certain conditions, the concentration dependence of the
interdiffusion coefficient can be obtained directly from the data without unfolding.
There is a systematic error incurred in this procedure, but it is of second order
and no larger than the random error ordinarily incurred with the cell.
11 - THEORY OF THE MEASUREMENT
The diaphragm cell consists of twD volumes, V and V~, separated by a sintered glass
diaphragm disk of cross sectional 6re=, A, and nor.inol thickness, s. /4/. The sen-
sitivity of cell depends upon the ratio, A/£. Because of the torturous nature of
the diffusion paths among the glass particles in the diaphragm. £ cannot be directly
observed. Rather, the cell constant, A/f. is determined bv observina the diffusion
coefficient of a solution (often aqueous KC1) whose interdiffusion coefficient is
absolutely known.
This being done,we assume that the cell is loaded at t=0 with a solution of the two
components of concentration c'(0) occupying V. and concentration c"(0) occupying
V~. (To prevent convection, the cell is ordinarily oriented so that * is parallel
to the gravity vector, and the less dense solution is above the more dense.) Thus
take c'(0) > c"(0). If the volumes V. and V-, are well stirred, a concentration
gradient, dc/dx, exists only across the diaphragm. The gradient is related to the
flux, J, through Pick's first law
J = -D(c) dc (1)
dx"
where the interdiffusion coefficient /6/ is allowed to be a function of the concen-
tration, c. Eq. (1) can also be written, Jdx =-D(c)dc, and integrated on the left
side from 0 to £ and on the right side from c1 to c" to obtain
c"
j£ = - j D(c)dc . (2)
c 1
'By conservation of mass, the rate of loss of solute from V., -V.dc'/dt, must exactly
equal the rate of gain of solute, V-dc'Vdt by V~. Hence,
c 1
d(c'-c") _ A /I 1 \ f
 n, ., (4)
Eqs. (2) and (3) can be combined to obtain
^EM . -± (;*;) i D(odcai f yv1 v2y c,,
As written, Eq. (4) cannot be integrated directly without knowledge of the func-
tional form of D(c), which is ordinarily not known.
In what follows, we shall show that Eq. (4) can be integrated accurately after-a
Taylor series development of D(c), provided that V1 = V^.V. In th_i_s circumstance,
conservation of solute mass dictate_s that the mecri concentration c = (c' 4 c")/2,
is independent of time and if c' = c 4 AC, then c" = c - AC. Define the variable
y = c - c" and transform the integral, 1, on the right hand side of Eq. (4) to
c1 AC
I = J D(c )dc = ( D(c + y) dv ORIGINAL PAGE-IS (5)
c1 -AC DC POOR QUALFTY
Expand Dfc 4 y) about D(c~) in c Tay lo r ser ies in y,
D("c 4 y) = f VC) y (6)
n=0 n!
where DnCc) = (dD(c)/dc) k = c". Substitute Eq. (6) into Eq. (5) anc integrate
term by 'term. The result is
n=0 (n+1)!
When written to third order in AC, Eq. (7) reads
, ORIGINAL PAGE-IS
1 = 2D 0 (c)Ac + ( l /3)D2(c)(Acr OF POOR QUALTTY (8)
2
Note that the term of order (AC) cancels out. Factoring Eq. (8) and using
AC = (l/2)(c'-c"), we have
= D(c)(c'-c") (9)
where the "zerotrT derivative D0(c) = D(c). The integral, 1, thus can be accurately
represented by D(c)(c'-c") assuming that the systematic error, e,
DJc~)(Ac)2
. -^ - (10)
D(c)
is small.
We replace 1 by D (c~ ) ( c ' - c " )and integrate Eq. (4) to obtain
c ' ( t ) - c - ( t ) \ . 2AD(c) t ,
Eq. (11) is identical with the usual result /4,7/, except V = V. = V~ emd the
interdiffusion coefficient is a function of the mean concentration, c. If the
concentrations, c'(t) and c"(t), are monitored as a function of time, then a plot
of the left_hand side of Eq. (U) as a function of t, yields a straight line with
slope -2AD(c)/iV from which D(c) can be extracted if V and A/£ are known.
III - DISCUSSION
o
If_Ac is not too large, we can safely assume that D~fc)(Ac) is a small fraction of
D(c), say 1/10. In this circumstance, using Eq.(lof,.we estimate e = 1/60, which
is the same order of magnitude as the random error ordinarily incurred with the
diaphragrr, cell. To test Eqs. (10) and (11), a range of solutions of increasing AC
should be prepared and tried with the cell. So long as for each solution D(c)
determined using Eq. (11) is independent of AC, it can be concluded that the sys-
tematic error is small. In as much as the diffusion w i l l be slow when AC is smell,
specie! care should be taken to keep the cell isothermal over the course of the
experi merit.
IV - ACKNOWLEDGEMENT
This work was supported by Grant NAS8-35936 provided by the National Aeronautics
and Space Admini strati on, Marshall Space Flight Center, Alabama, 35812, USA.
REFERENCES
IM I. K. Lifshitz and V. V. Slyozov, Phys. Cheir.. Solids j_9 (1961) 35.
121 C. Wagner, Z. Electroche-;. 65 (1961) 581.
/3/ J. A. Marqusee and J. Ross, J. Che*. Phys. 79 (1953) 373.
/4/ R. K i l l s end L. A. Woolf, The Diaphragrr" CelTT Report DRu-RRi, Diffusion
Research Unit, Research School of Physical Sciences, Australian National
University, Canberra, A.C.T., Australia, 1965.
/5/ R. H. Stokes, J. AT. Che;:.. Soc. 72 (1950) 2243.
Ill G. S. Hartley and J. Crank, Trans. Far. Soc. 45 (1949) 801.
lit R. A. Robinson and R. H. Stokes, Electrolyte Solutions. 2nd Ed, Butterworths,
London, 1959.
FINANCIAL REPORT
NAS8-35986
Cumulative costs incurred as of report date $33,647.48
Estimate of cost to complete contract as of June 30,1985 36,863.27
Estimated percentage of physical completion of contract 60%


Ostwald ripening theory

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