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Dependence of the linear growth rate of individual ammonium dihydrogen phosphate microcrystals on temperature
Journal of Crystal Growth 52 (1981) 820—823 © North-Holland Publishing Company
DEPENDENCE OF THE LINEAR GROWTH RATE OF INDIVIDUAL AMMONIUM DIHYDROGEN PHOSPHATE MICROCRYSTALS ON TEMPERATURE S.E. BOZIN Faculty of Natural Sciences, 11001 Belgrade, Yugoslavia
and B. ZIZIC Faculty of Natural Sciences and Institute of Physics, 1101 Belgrade. Yugoslavia
The relationship between growth rate of ammonium dihydrogen phosphate microcrystals and temperature is deduced from measurements made on one and the same crystals grown in aqueous solutions of constant relative supersaturation (0.03) at temperatures in the range from 18 to 38°C.It is found that growth rates increase with temperature in an exponential manner. Derived activation energies of the overall growth process have not a unique value for all observed crystals. An explanation is given in the framework of BCF theory.
1. Introduction
2. Experimental arrangement and procedure
According to the theory of Burton, Cabrera and Frank [1], as modified by Bennema [2] for crystal growth from solution, the growth rate depends on external conditions (supersaturation, temperature) and on individual characteristics (activity of the dominating dislocation group) of the growing crystal. The experimentally determined relation between the growth rate of ammonium dihydrogen phosphate (ADP) crystals and supersaturation is in agreement with the theory [3—5]. The predicted influence of dislocation group activity on the growth kinetics has recently been confirmed for ADP crystals [5]. However, there are no reports on an experimentally determined relationship between the growth rate and temperature for ADP crystals. In the present study the investigation of growth rate versus temperature dependence for ADP crystals is described. Linear growth rate in the direction [010] was measured for microcrystals grown in aqueous solutions of constant relative supersaturation at various temperatures. In each experimental run the measurements were carried out on one and the same microcrystals.
Crystals were nucleated and grown in a cylindrical observation cell similar to that described by Davey and Mullin [6]. In order to follow the growth of one and the same crystals in several solutions without dislodging them from the cell, a particular technique was used. The cell was connected by a plastic tube to a small glass pipe branching out towards five containers situated in a thermostatic bath. The flow of solution from each container was controlled by means of screw clamps fixed on plastic tubes leading from the branching pipe to the containers. Until the start of any experimental run the thermostatic bath was kept at sufficiently high temperature to prevent undesirable seeding. The whole system was filled with solution at this elevated ternperature by producing continuous flow from all containers simultaneously. This was accomplished in a short time and the clamps were closed subsequently. In this manner the flow from each container could be started or stopped at any moment. The thermostat’s thermometer was then adjusted to the value of highest working temperature and the corresponding solution allowed to flow. Crystal seeds were created in 820
SE. Bolin
/
Linear growth rate of ADP and temperature
the cell by lifting its lid for a few seconds. As soon as the temperature had settled the measurements started. After obtaining the first set of data, the flow was stopped and the next solution let to stream, gradually replacing the previous one in the cell. Measurements were continued when growth conditions became steady. The linearity of obtained interfacial distance—tirne diagrams was taken as a proof that stability was achieved. By performing the replacements always with solutions of lower temperature, spontaneous crystallisation in the flow system was avoided. The aqueous solutions of pure ADP (pH = 3.5) were prepared on the basis of solubility data in such a way as to be of the same relative supersaturation at appropriate working temperatures. Throughout all experiments the relative supersaturation was 0.03 (concentration being measured in g/l). The flow rate of 3 mI/mm was also kept constant. Interfacial distances were measured by means of a microscope, using transmitted light. In each experimental run the data were taken for several microcrystals grown simultaneously. 3. Results and discussion Growth rate values were determined from plots of interfacial distances versus time for temperatures 18.0, 23.0, 28.0, 33.0 and 38.0°C. The obtained diagrams consisted of straight line segments of different slopes, corresponding to various temperatures. Measurements were performed on 14 microcrystals, whose initial inter-
Table I Growth rates of ADP microcrystals at various temperatures for a- = 0.03 8m/s) Growth rate (10~ Temperature (°C) Mm. Max. 18.0 23.0 33.0 38.0
0.6 0.9 1.7 2.0
1.8 2.7 4.1 4.8
semilogarithmic plots of R against l/T for three microcrystals. The diagrams F and G belong to crystals grown simultaneously, while K is from another run. The values of the activation energy (E) of the overall growth process were calculated from such diagrams and two separate groups of values for E were obtained. In the first, 9.3 ~ E ~ 12.4 kcal/mol (average value 11.2), and in the second 5.2 ~ E ~ 7.6 kcal/mol (average value 6.7). The fact that activation energies for crystals grown simultaneously in the same solutions have values belonging to either group is taken as evidence that such grouping is not due to some failure in the experiments. However, the scatter of activation energy values in each group can be attributed to experimental errors. The interpretation of obtained results in the framework of modified BCF theory [2], may be as follows. The general expression for the growth rate dependence on relative supersaturation R
=
2 tanh(o-
(C/cr 1)cr
facial distances were from 0.05 to 0.15 mm. Linear growth rates of individual microcrystals obtained at each working temperature are spread over a variations wide rangewere of values, as shown in table 1. Such observed earlier [5] for ADP crystals grown at 23.0°C. The growth rates (R) for almost all observed microcrystals fit well an exponential relation of the form (see section 5 for list of symbols)
821
1/cr),
(2)
reduces to a parabolic law 2
(3)
R = (C/cri)ofor 0 <01, and to a linear one R
=
Ccr
(4)
when r > o~.Since R
=
R 0 exp(—B/T),
(1)
where R0 and B are parameters. In fig. 1 are
kT / L~Gdeh\ C=—f3coA(~Noexp— h \ kT /I
(5)
822
SE. Boz~in/ Linear growth rate of ADP and temperature
al. [5]. Besides, its value decreases with ternperature, according to eq. (6). Therefore, at various temperatures, but for a constant value of relative supersaturation o-, individual crystals may be classified into one of the following three groups. (I) Activation energy E equals ~Gdeh if a- > o~in the whole temperature range, as indicated by eq. (8). This is the case with crystals for which the
5
4
3
observed energyestimated has a value close to L~Gdeh=activation 12.8 kcal/mole. forvery ADP by Davey et al. [5], and similarly by Mullin et al. [3].
F I
2
(ii) If for the whole temperature range a~> 0-, then the experimentally determined activation energy E should be, for such crystals, less than ~Gdeh, according to eq. (7). This is the case with the observed crystals having smaller values of activation energy. (iii) It should be possible that some individual crystals have a~ a- for lower temperatures. In such cases the In R versus lIT plots would consist of two straight line segments with unequal slopes.
0 I
K
3~2
327
332
338
+10
344
K
j
Fig. 1. Semilogarithmic plots of the growth rate versus inverse temperature for three microcrystals.
~4exp(
=
~‘~T
(6)
deads),
the growth rate dependence on temperature is R
=
k2T2 ~ —h— V x (
f3coAflNoa-2 2~Gdeh+ ~GSdIn
exp\,
—
L~Gdeads\
I
2kT
(7\ “ ‘
4. Conclusions The linear growth rate of ADP microcrystals in the direction [010] at constant relative supersaturation depends exponentially on temperature. Activation energies for the overall growth process are grouped around two values, one of which is very close to the previously determined activation free energy for dehydration for ADP, and the other is significantly smaller. These results may be explained in terms of relative magnitudes of the parameter a1 and the value of constant relative supersaturation.
when R
=
for
a- <
kT -~—
cry, and I i~G~\
f3coAS?Noo- exp~— kT
)
(8)
a- > o~.
The parameter a-~ at constant supersaturation and temperature may have various values, depending on r, as shown for ADP by Davey et
5. Symbols E h k N0
Parameter in the BCF theory Activation energy of the overall growth process Planck’s constant Boltzmann’s constant Equilibrium volume solute density
S.E. Bo~in/ Linear growth rate of ADP and temperature
R T /3 y A
Linear growth rate Absolute temperature Retardation factor for entering the step Specific edge energy Mean free path of a growth unit in solution ~Gdeh Activation free energy for dehydration ~ Activation free energy for surface diffusion L~Gdeads Activation free energy for deadsorption [1 Molecular volume in the crystal Activity of the dominating dislocation group
823
References [1] W.K. Burton, N. Cabrera and F.C. Frank. Phil. Trans. Roy. Soc. London A 243 (1951) 299. [2] P. Bennema, J. Crystal Growth I (1967) 278. [31J.W. Mullin and A. Amatavivadhana, J. AppI. Chem. 17 (1967) 151. [4] 1-tV. Alexandru, J. Crystal Growth 10 (1971) 151. [51R.J. Davey. RI. Ristlé and B. ~ J. Crystal Growth 47 (1979) 1. [6] R.J. Davey and J.W. Mullin, J. Crystal Growth 26 (1974) 45.
DEPENDENCE OF THE LINEAR GROWTH RATE OF INDIVIDUAL AMMONIUM DIHYDROGEN PHOSPHATE MICROCRYSTALS ON TEMPERATURE S.E. BOZIN Faculty of Natural Sciences, 11001 Belgrade, Yugoslavia
and B. ZIZIC Faculty of Natural Sciences and Institute of Physics, 1101 Belgrade. Yugoslavia
The relationship between growth rate of ammonium dihydrogen phosphate microcrystals and temperature is deduced from measurements made on one and the same crystals grown in aqueous solutions of constant relative supersaturation (0.03) at temperatures in the range from 18 to 38°C.It is found that growth rates increase with temperature in an exponential manner. Derived activation energies of the overall growth process have not a unique value for all observed crystals. An explanation is given in the framework of BCF theory.
1. Introduction
2. Experimental arrangement and procedure
According to the theory of Burton, Cabrera and Frank [1], as modified by Bennema [2] for crystal growth from solution, the growth rate depends on external conditions (supersaturation, temperature) and on individual characteristics (activity of the dominating dislocation group) of the growing crystal. The experimentally determined relation between the growth rate of ammonium dihydrogen phosphate (ADP) crystals and supersaturation is in agreement with the theory [3—5]. The predicted influence of dislocation group activity on the growth kinetics has recently been confirmed for ADP crystals [5]. However, there are no reports on an experimentally determined relationship between the growth rate and temperature for ADP crystals. In the present study the investigation of growth rate versus temperature dependence for ADP crystals is described. Linear growth rate in the direction [010] was measured for microcrystals grown in aqueous solutions of constant relative supersaturation at various temperatures. In each experimental run the measurements were carried out on one and the same microcrystals.
Crystals were nucleated and grown in a cylindrical observation cell similar to that described by Davey and Mullin [6]. In order to follow the growth of one and the same crystals in several solutions without dislodging them from the cell, a particular technique was used. The cell was connected by a plastic tube to a small glass pipe branching out towards five containers situated in a thermostatic bath. The flow of solution from each container was controlled by means of screw clamps fixed on plastic tubes leading from the branching pipe to the containers. Until the start of any experimental run the thermostatic bath was kept at sufficiently high temperature to prevent undesirable seeding. The whole system was filled with solution at this elevated ternperature by producing continuous flow from all containers simultaneously. This was accomplished in a short time and the clamps were closed subsequently. In this manner the flow from each container could be started or stopped at any moment. The thermostat’s thermometer was then adjusted to the value of highest working temperature and the corresponding solution allowed to flow. Crystal seeds were created in 820
SE. Bolin
/
Linear growth rate of ADP and temperature
the cell by lifting its lid for a few seconds. As soon as the temperature had settled the measurements started. After obtaining the first set of data, the flow was stopped and the next solution let to stream, gradually replacing the previous one in the cell. Measurements were continued when growth conditions became steady. The linearity of obtained interfacial distance—tirne diagrams was taken as a proof that stability was achieved. By performing the replacements always with solutions of lower temperature, spontaneous crystallisation in the flow system was avoided. The aqueous solutions of pure ADP (pH = 3.5) were prepared on the basis of solubility data in such a way as to be of the same relative supersaturation at appropriate working temperatures. Throughout all experiments the relative supersaturation was 0.03 (concentration being measured in g/l). The flow rate of 3 mI/mm was also kept constant. Interfacial distances were measured by means of a microscope, using transmitted light. In each experimental run the data were taken for several microcrystals grown simultaneously. 3. Results and discussion Growth rate values were determined from plots of interfacial distances versus time for temperatures 18.0, 23.0, 28.0, 33.0 and 38.0°C. The obtained diagrams consisted of straight line segments of different slopes, corresponding to various temperatures. Measurements were performed on 14 microcrystals, whose initial inter-
Table I Growth rates of ADP microcrystals at various temperatures for a- = 0.03 8m/s) Growth rate (10~ Temperature (°C) Mm. Max. 18.0 23.0 33.0 38.0
0.6 0.9 1.7 2.0
1.8 2.7 4.1 4.8
semilogarithmic plots of R against l/T for three microcrystals. The diagrams F and G belong to crystals grown simultaneously, while K is from another run. The values of the activation energy (E) of the overall growth process were calculated from such diagrams and two separate groups of values for E were obtained. In the first, 9.3 ~ E ~ 12.4 kcal/mol (average value 11.2), and in the second 5.2 ~ E ~ 7.6 kcal/mol (average value 6.7). The fact that activation energies for crystals grown simultaneously in the same solutions have values belonging to either group is taken as evidence that such grouping is not due to some failure in the experiments. However, the scatter of activation energy values in each group can be attributed to experimental errors. The interpretation of obtained results in the framework of modified BCF theory [2], may be as follows. The general expression for the growth rate dependence on relative supersaturation R
=
2 tanh(o-
(C/cr 1)cr
facial distances were from 0.05 to 0.15 mm. Linear growth rates of individual microcrystals obtained at each working temperature are spread over a variations wide rangewere of values, as shown in table 1. Such observed earlier [5] for ADP crystals grown at 23.0°C. The growth rates (R) for almost all observed microcrystals fit well an exponential relation of the form (see section 5 for list of symbols)
821
1/cr),
(2)
reduces to a parabolic law 2
(3)
R = (C/cri)ofor 0 <01, and to a linear one R
=
Ccr
(4)
when r > o~.Since R
=
R 0 exp(—B/T),
(1)
where R0 and B are parameters. In fig. 1 are
kT / L~Gdeh\ C=—f3coA(~Noexp— h \ kT /I
(5)
822
SE. Boz~in/ Linear growth rate of ADP and temperature
al. [5]. Besides, its value decreases with ternperature, according to eq. (6). Therefore, at various temperatures, but for a constant value of relative supersaturation o-, individual crystals may be classified into one of the following three groups. (I) Activation energy E equals ~Gdeh if a- > o~in the whole temperature range, as indicated by eq. (8). This is the case with crystals for which the
5
4
3
observed energyestimated has a value close to L~Gdeh=activation 12.8 kcal/mole. forvery ADP by Davey et al. [5], and similarly by Mullin et al. [3].
F I
2
(ii) If for the whole temperature range a~> 0-, then the experimentally determined activation energy E should be, for such crystals, less than ~Gdeh, according to eq. (7). This is the case with the observed crystals having smaller values of activation energy. (iii) It should be possible that some individual crystals have a~
0 I
K
3~2
327
332
338
+10
344
K
j
Fig. 1. Semilogarithmic plots of the growth rate versus inverse temperature for three microcrystals.
~4exp(
=
~‘~T
(6)
deads),
the growth rate dependence on temperature is R
=
k2T2 ~ —h— V x (
f3coAflNoa-2 2~Gdeh+ ~GSdIn
exp\,
—
L~Gdeads\
I
2kT
(7\ “ ‘
4. Conclusions The linear growth rate of ADP microcrystals in the direction [010] at constant relative supersaturation depends exponentially on temperature. Activation energies for the overall growth process are grouped around two values, one of which is very close to the previously determined activation free energy for dehydration for ADP, and the other is significantly smaller. These results may be explained in terms of relative magnitudes of the parameter a1 and the value of constant relative supersaturation.
when R
=
for
a- <
kT -~—
cry, and I i~G~\
f3coAS?Noo- exp~— kT
)
(8)
a- > o~.
The parameter a-~ at constant supersaturation and temperature may have various values, depending on r, as shown for ADP by Davey et
5. Symbols E h k N0
Parameter in the BCF theory Activation energy of the overall growth process Planck’s constant Boltzmann’s constant Equilibrium volume solute density
S.E. Bo~in/ Linear growth rate of ADP and temperature
R T /3 y A
Linear growth rate Absolute temperature Retardation factor for entering the step Specific edge energy Mean free path of a growth unit in solution ~Gdeh Activation free energy for dehydration ~ Activation free energy for surface diffusion L~Gdeads Activation free energy for deadsorption [1 Molecular volume in the crystal Activity of the dominating dislocation group
823
References [1] W.K. Burton, N. Cabrera and F.C. Frank. Phil. Trans. Roy. Soc. London A 243 (1951) 299. [2] P. Bennema, J. Crystal Growth I (1967) 278. [31J.W. Mullin and A. Amatavivadhana, J. AppI. Chem. 17 (1967) 151. [4] 1-tV. Alexandru, J. Crystal Growth 10 (1971) 151. [51R.J. Davey. RI. Ristlé and B. ~ J. Crystal Growth 47 (1979) 1. [6] R.J. Davey and J.W. Mullin, J. Crystal Growth 26 (1974) 45.