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A crystal growth model with concentration dependent diffusion
380
Journal of Crystal Growth 67 (1984) 380--382 North-Holland, Amsterdam
LETTER TO THE EDITORS A CRYSTAL GROWTH MODEL WITH CONCENTRATION DEPENDENT DIFFUSION Michael SASKA
*
and Allan S. MYERSON School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332 -0100, (ISA Received 15 November 1983
Recent measurements of the diffusion coefficients in several solid—liquid systems indicate a rapid drop of the diffusivity in the supersaturated region. In this work the concentration profile was studied around an isolated growing surface step when the diffusivity of the growth substance is a strong (decreasing) function of concentration. The order of crystallization was depressed by up to 23~ for the parameter range studied. It appears that for systems where the diffusion coefficient decreases rapidly in the supersaturated region, the effects of variable diffusivity have to be considered when evaluating the measured growth rate — supersaturation curves.
The study of diffusion coefficients in supersaturated solutions is of fundamental importance in further understanding the mechanism of diffusion, as well as for the rational design of crystallization processes. Sorell and Myerson [1] and Chang [2] measured the diffusivity of urea, glycine and potassium chloride in supersaturated aqueous solutions employing Gouy interferometry. Results for each system show a very rapid decline in the diffusion coefficient with increasing concentration in the supersaturated region. This behavior is similar to that observed previously in liquid—liquid systems near the consolute point. Claersson and Sundelof [3] and Haase and Siry [4] reported that the binary diffusion coefficient drops rapidly to zero in the region of the critical solution temperature (consolute point) and explained this phenomenon on thermodynamic grounds. Myerson and Senol [5J applied the thermodynamic argument to supersaturated urea solutions. It is the purpose of this work to examine the effect of a rapidly declining concentration dependent diffusion coefficient
on the concentration profile and hence the growth rate of a crystal. The evaluation of the effect of a declining Concentration dependent diffusion coefficient on the crystal growth rate required the formulation of a growth model amenable to mathematical treatment. An unstirred boundary layer of thickness 6 defines the region in which the concentration profile is sought. The growing crystal face is represented by a train of equidistant identical steps of height a and separation A. The solution domain bound by the six boundary pieces (BP) is shown in fig. 1. The diffusion equation in the vicinity of a stationary step (R <
a
—
ax
(D—~~1 I+ ~x/
a
a
I a~ a~
—
=
0.
(1)
-
where x and y are the rectangular coordinates parallel and perpendicular to the crystal surface and a is the supersaturation (%). The diffusion coefficient, D, is represented for a limited range of
*
Correspondence author. Current address: Audubon Sugar
i1
Institute, Louisiana State University, Baton Rouge, Louisiana 10803, USA.
D= D
0022-0248/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
3. 0
—
~a
(2)
M. Saska, AS. Myerson
/
Crystal growth model with concentration dependent diffusion
cretized on a triangular mesh [8,91 automatically refined so that the size of the ith triangle, h,, is proportional to [o.oi + x~+ (y1 a)2] 1/3 (3)
BPZ
1
///
381
BP 3
—
(
// flBP6
8 3~i~”1\6
BP5
x
4
at the re-entrant corner (0,a). The derivatives
BP4
~a/0x
“2
$
=
0.05 rn/s.
and aa/ay are approximated by two- in-
side and one-sided differences on the boundary of
Fig. 1. The region adjacent to one surface step of a vicinal face on which the supersaturation profile given by eq. (1) is sought. The curves represent zones of constant a (%): (2) 2.2, (3) 2.5, (4) 2.9, (5) 3.2, (6) 3.6, (7) 3.9, (8) 4.3, (9) 4.7 for a run with = 5%, a = 0 and given in table 1.
where x and y1 are the coordinates of the center of the triangle. The mesh is made densest in the area
Other parameters are
where D0 is diffusivity at the saturation and a is a
constant obtained from experimental data. Eq. (2) was found to be a reasonable approximation of experimentally obtained diffusivity data in the
urea—water system. The normal derivative of the supersaturation, 0a/3n, is zero on all BP except on BP 5 (fig. 1) where [6] 3~f3 also at BP ,2 a GBuIk. The interfacial kinetics velocity, $, is estimated [7] to be of such magnitude that i0~
the domain (fig. 1). Eq. (11) is solved iteratively by employing the solution array from the previous step to calculate D(a) from eq. (2). Four or five iterations are usually sufficient to reduce the shifts
in solution to about 1% of its value. The numerical solution of eq. (1) gives supersaturation as a function of position around the growth step. The results for one run in the form of iso-supersaturation contours are shown in fig. 1. a ‘s at the three grid points on the step growth front BP 5 are then fitted with a second order polynomial and a mean supersaturation ö is then calculated as: lfaa(oy)dy (4) ö is then used to calculate the step movement velocity in the x direction R
=
~Q/3Ce/100,
(5)
Table I Numerical values of the growth parameters used in the simulations Parameter
Symbol
Concentration parameter of the dependence diffusion
a
coefficient (eq. (2)) Stepheight Interface kinetics velocity
a
Equilibrium solute concentration Boundary layer thickness Diffusion coefficient at saturation Step separation Molecular (atomic) volume Bulk supersaturation
Value used 2/s 9X ~o 13 rn 2X108m 0.5, 0.05, 0.005 rn/s
C~
3
5
1025 molecules/rn iO~ m iO~ m2/s
X £2 GB*ik
2x107 27 m3 =
[((‘
—
C*q)/C*qIXIOO
0 to 10%
m
M. Sa.vka, AS. Mverson / Crystal growth model with concentration dependent dijjuston
382
/
~
//
8
/ /
.
• .7 2
•.
. S.
4
6
8
Fig. 2. Per Cent decrease of predicted growth rate R (m/s) when including the effects of concentration dependent diffusion coefficient (a> 0) as compared to that when 13 = D 0= constant (R0) versus bulk supersaturation, GBUIS. The non.linear character causes a reduction in the order of crystallization, —) $=0.5, (...) $=0.05, (—~) $=0.005
(•)
to aR/A. If the effect of variable diffusivity on step separation is neglected, linear growth velocities can be calculated. The effect of a variable diffusion coefficient on the linear growth rate is to depress the order by a factor of I -nj. For the three cases for which data were generated, the crystallization order declined by 0.23. 0.08 and 0.01 for = 0.005. 0.05 and 0.5 rn/s. respectively. This represents up to 23% of the usual range. Con-
Balk . V.
rn/s.
(assuming an equi-distant set of identical steps) in the direction roughly perpendicular to it is equal
/3
• 2
respectiv’~1y. equal to one. The exponents when is = 0 were all The linear growth velocity of a vicinal plane
centration dependent diffusion coefficients, therefore, can depress the order of crystallization in a given crystal growth system and must he considered in studies of crystal growth mechanisms and evaluation of experimentally determined kinetic data. It appears that the predictive analytical models of crystal growth developed for a = 0 can still be
Calculated points.
where ~ is the volume of a single molecule (atom) and Ce, the equilibrium solute concentration. The step movement velocity was calculated at three different values of /3 (0.005, 0.05 and 0.5 m/s) for the constant and variable diffusion cases. Other parameters used are listed in table 1. The results given as per cent decrease of the step velocity R for the variable diffusivity case (a > 0) as compared to the case when a = 0 are given in fig. 2 for the three values of /3. Assuming the step movement velocity, R, to be proportional to ~B~Ik’ the step movement velocity data generated for the variable diffusion cases was fit to obtain the exponent m. The exponents obtained were 0.77, 0.92 and 0.99 for /3 equal to 0.005, 0.05 and 0.5 rn/s.
used employing an effective value of D, D(a~~~). For the limited rangeclose of parameters investigated in 0~fI was to ~, the average superthis study, saturation on the step growth front BP 5. References [1] L.S Sorell and A.S. Myerson, AIChE J. 28 (1982) 772.
[21 Y.C.
Chang, PhD Dissertation, Georgia Institute of Tech-
nology, Atlanta, GA, in progress. [3] S. Claersson and L.O. Sundelof. J. Chem. Phys. 54 (1957) 914. F~l R. Haase and M. Siry, Z. Physik. Chem. (NF) 57 (1968) 56. [~l AS. Myerson and D.E Senol. AIChE J. in press
[61 A.A. Chernov, Soviet Phys.-Usp. 4(1961)116. [71 P. Bennema, in: Crystal Growth, Ed. H.S.
Peiser (Per-
gamon, Oxford. 1967). [8] G. Sewell, in: Advances in Computer Methods for Partial Differential Equations III, Eds. R. Vichnevetsky and R.S. Stepleman (New Brunswick, NJ, 1979). [9] J.R. Rice ci al., ELLPACK 77/78 (Computer Science Dcpartrnent, Purdue University, 1980).
Journal of Crystal Growth 67 (1984) 380--382 North-Holland, Amsterdam
LETTER TO THE EDITORS A CRYSTAL GROWTH MODEL WITH CONCENTRATION DEPENDENT DIFFUSION Michael SASKA
*
and Allan S. MYERSON School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332 -0100, (ISA Received 15 November 1983
Recent measurements of the diffusion coefficients in several solid—liquid systems indicate a rapid drop of the diffusivity in the supersaturated region. In this work the concentration profile was studied around an isolated growing surface step when the diffusivity of the growth substance is a strong (decreasing) function of concentration. The order of crystallization was depressed by up to 23~ for the parameter range studied. It appears that for systems where the diffusion coefficient decreases rapidly in the supersaturated region, the effects of variable diffusivity have to be considered when evaluating the measured growth rate — supersaturation curves.
The study of diffusion coefficients in supersaturated solutions is of fundamental importance in further understanding the mechanism of diffusion, as well as for the rational design of crystallization processes. Sorell and Myerson [1] and Chang [2] measured the diffusivity of urea, glycine and potassium chloride in supersaturated aqueous solutions employing Gouy interferometry. Results for each system show a very rapid decline in the diffusion coefficient with increasing concentration in the supersaturated region. This behavior is similar to that observed previously in liquid—liquid systems near the consolute point. Claersson and Sundelof [3] and Haase and Siry [4] reported that the binary diffusion coefficient drops rapidly to zero in the region of the critical solution temperature (consolute point) and explained this phenomenon on thermodynamic grounds. Myerson and Senol [5J applied the thermodynamic argument to supersaturated urea solutions. It is the purpose of this work to examine the effect of a rapidly declining concentration dependent diffusion coefficient
on the concentration profile and hence the growth rate of a crystal. The evaluation of the effect of a declining Concentration dependent diffusion coefficient on the crystal growth rate required the formulation of a growth model amenable to mathematical treatment. An unstirred boundary layer of thickness 6 defines the region in which the concentration profile is sought. The growing crystal face is represented by a train of equidistant identical steps of height a and separation A. The solution domain bound by the six boundary pieces (BP) is shown in fig. 1. The diffusion equation in the vicinity of a stationary step (R <
a
—
ax
(D—~~1 I+ ~x/
a
a
I a~ a~
—
=
0.
(1)
-
where x and y are the rectangular coordinates parallel and perpendicular to the crystal surface and a is the supersaturation (%). The diffusion coefficient, D, is represented for a limited range of
*
Correspondence author. Current address: Audubon Sugar
i1
Institute, Louisiana State University, Baton Rouge, Louisiana 10803, USA.
D= D
0022-0248/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
3. 0
—
~a
(2)
M. Saska, AS. Myerson
/
Crystal growth model with concentration dependent diffusion
cretized on a triangular mesh [8,91 automatically refined so that the size of the ith triangle, h,, is proportional to [o.oi + x~+ (y1 a)2] 1/3 (3)
BPZ
1
///
381
BP 3
—
(
// flBP6
8 3~i~”1\6
BP5
x
4
at the re-entrant corner (0,a). The derivatives
BP4
~a/0x
“2
$
=
0.05 rn/s.
and aa/ay are approximated by two- in-
side and one-sided differences on the boundary of
Fig. 1. The region adjacent to one surface step of a vicinal face on which the supersaturation profile given by eq. (1) is sought. The curves represent zones of constant a (%): (2) 2.2, (3) 2.5, (4) 2.9, (5) 3.2, (6) 3.6, (7) 3.9, (8) 4.3, (9) 4.7 for a run with = 5%, a = 0 and given in table 1.
where x and y1 are the coordinates of the center of the triangle. The mesh is made densest in the area
Other parameters are
where D0 is diffusivity at the saturation and a is a
constant obtained from experimental data. Eq. (2) was found to be a reasonable approximation of experimentally obtained diffusivity data in the
urea—water system. The normal derivative of the supersaturation, 0a/3n, is zero on all BP except on BP 5 (fig. 1) where [6] 3~f3 also at BP ,2 a GBuIk. The interfacial kinetics velocity, $, is estimated [7] to be of such magnitude that i0~
the domain (fig. 1). Eq. (11) is solved iteratively by employing the solution array from the previous step to calculate D(a) from eq. (2). Four or five iterations are usually sufficient to reduce the shifts
in solution to about 1% of its value. The numerical solution of eq. (1) gives supersaturation as a function of position around the growth step. The results for one run in the form of iso-supersaturation contours are shown in fig. 1. a ‘s at the three grid points on the step growth front BP 5 are then fitted with a second order polynomial and a mean supersaturation ö is then calculated as: lfaa(oy)dy (4) ö is then used to calculate the step movement velocity in the x direction R
=
~Q/3Ce/100,
(5)
Table I Numerical values of the growth parameters used in the simulations Parameter
Symbol
Concentration parameter of the dependence diffusion
a
coefficient (eq. (2)) Stepheight Interface kinetics velocity
a
Equilibrium solute concentration Boundary layer thickness Diffusion coefficient at saturation Step separation Molecular (atomic) volume Bulk supersaturation
Value used 2/s 9X ~o 13 rn 2X108m 0.5, 0.05, 0.005 rn/s
C~
3
5
1025 molecules/rn iO~ m iO~ m2/s
X £2 GB*ik
2x107 27 m3 =
[((‘
—
C*q)/C*qIXIOO
0 to 10%
m
M. Sa.vka, AS. Mverson / Crystal growth model with concentration dependent dijjuston
382
/
~
//
8
/ /
.
• .7 2
•.
. S.
4
6
8
Fig. 2. Per Cent decrease of predicted growth rate R (m/s) when including the effects of concentration dependent diffusion coefficient (a> 0) as compared to that when 13 = D 0= constant (R0) versus bulk supersaturation, GBUIS. The non.linear character causes a reduction in the order of crystallization, —) $=0.5, (...) $=0.05, (—~) $=0.005
(•)
to aR/A. If the effect of variable diffusivity on step separation is neglected, linear growth velocities can be calculated. The effect of a variable diffusion coefficient on the linear growth rate is to depress the order by a factor of I -nj. For the three cases for which data were generated, the crystallization order declined by 0.23. 0.08 and 0.01 for = 0.005. 0.05 and 0.5 rn/s. respectively. This represents up to 23% of the usual range. Con-
Balk . V.
rn/s.
(assuming an equi-distant set of identical steps) in the direction roughly perpendicular to it is equal
/3
• 2
respectiv’~1y. equal to one. The exponents when is = 0 were all The linear growth velocity of a vicinal plane
centration dependent diffusion coefficients, therefore, can depress the order of crystallization in a given crystal growth system and must he considered in studies of crystal growth mechanisms and evaluation of experimentally determined kinetic data. It appears that the predictive analytical models of crystal growth developed for a = 0 can still be
Calculated points.
where ~ is the volume of a single molecule (atom) and Ce, the equilibrium solute concentration. The step movement velocity was calculated at three different values of /3 (0.005, 0.05 and 0.5 m/s) for the constant and variable diffusion cases. Other parameters used are listed in table 1. The results given as per cent decrease of the step velocity R for the variable diffusivity case (a > 0) as compared to the case when a = 0 are given in fig. 2 for the three values of /3. Assuming the step movement velocity, R, to be proportional to ~B~Ik’ the step movement velocity data generated for the variable diffusion cases was fit to obtain the exponent m. The exponents obtained were 0.77, 0.92 and 0.99 for /3 equal to 0.005, 0.05 and 0.5 rn/s.
used employing an effective value of D, D(a~~~). For the limited rangeclose of parameters investigated in 0~fI was to ~, the average superthis study, saturation on the step growth front BP 5. References [1] L.S Sorell and A.S. Myerson, AIChE J. 28 (1982) 772.
[21 Y.C.
Chang, PhD Dissertation, Georgia Institute of Tech-
nology, Atlanta, GA, in progress. [3] S. Claersson and L.O. Sundelof. J. Chem. Phys. 54 (1957) 914. F~l R. Haase and M. Siry, Z. Physik. Chem. (NF) 57 (1968) 56. [~l AS. Myerson and D.E Senol. AIChE J. in press
[61 A.A. Chernov, Soviet Phys.-Usp. 4(1961)116. [71 P. Bennema, in: Crystal Growth, Ed. H.S.
Peiser (Per-
gamon, Oxford. 1967). [8] G. Sewell, in: Advances in Computer Methods for Partial Differential Equations III, Eds. R. Vichnevetsky and R.S. Stepleman (New Brunswick, NJ, 1979). [9] J.R. Rice ci al., ELLPACK 77/78 (Computer Science Dcpartrnent, Purdue University, 1980).