Growth Rates of Aluminum Potassium Sulfate Crystals in Aqueous Solutions Gregory D. Botsaris and Edward 0. Denk, Jr. Department of Chemical Engineering, Tufts University, Medford, Mass. 02166
The linear growth rates of the (100), (1 lo), and (1 1 1) faces of potassium alum crystals in aqueous solutions were measured as functions of supersaturation in a flow crystallizer under conditions of constant temperaappears ture, supersaturation (0 to 18%), and liquid velocity. The region of supersaturation from 9 to 1 to be one of unstable crystal growth in which both high and low growth rates are possible. A compound growth mechanism (dislocation growth plus mononuclear two-dimensional nucleation) can correlate the growth rates of the crystal faces for the range of supersaturation studied. Where scattering was observed, the mechanism correlates the average growth rates. This model has certain advantages over previous empirical correlations.
3y0
THEORETICAL
WORK in the field of industrial crystallizations from solution has led to several mathematical models by which the behavior of a crystallizer-e.g., crystal size distribution, and where important, yield-can be predicted (Abegg et al., 1968; Han and Skinnar, 1967; Randolph and Larson, 1962). However, most of these studies have used assumed or empirical functions to represent the dependence of crystal growth rate on supersaturation, which may impose a severe limitation in their utilization. More sophisticated models for crystal growth need to be developed and the purpose of this investigation was to contribute to that development. Aluminum potassium sulfate dodecahydrate was used as the crystallizing system. Most prior studies on the growth of potassium alum crystals report only the relative growth rates of the various faces (usually expressed as the growth rates of the fastest growing faces divided by that of the slowest growing face). Since nothing is said about the absolute growth rates, the data cannot be used to develop a kinetic model. Except in special cases-for instance, when the absolute growth rate curves are linear-the relative growth rates would be functions of supersaturation. This is probably one of the reasons for the considerable disagreement among the data of the above studies (Table I).
Table 1.
Portnov (1967), as part of a study of the effect of impurities on growth, measured the growth rates of the cubic faces from pure alum solutions, but only a t three values of supersaturation. Komarova and Figurovski (1954) also measured the rate of crystallization of K alum, but did not work under conditions of constant supersaturation and consequently their results are of limited scope. Two important studies report the growth rates of potassium alum crystals as a function of supersaturation. The earliest is that of Bennema (Bennema, 1965, 1966a, b, 1967; Bennema et al., 1967). He measured the growth rates of the octahedral (111) faces only, and obtained a linear relationship between the growth rate (R) and supersaturation (8) for values of S below about 1%. Since Frank’s model of dislocation growth predicts a linear R-S relationship, Bennema concluded that a t low supersaturations growth was occurring by this mechanism. Above about 0.75% S, the data showed a considerable increase in scatter and deviated from the straight line obtained a t lower supersaturation. Bennema attributed this to the onset of two-dimensional nucleation and proposed a compound growth mechanism in which dislocation growth and polynuclear two-dimensional nucleation occurred simultaneously. Because the highest supersaturation Bennema worked a t was only 1.2%, the data supporting this
Relative Growth Rates of Potassium Alum Faces
Riw
Riio
Riii
.1.2 1.63 1.75 2.16 2.7 3.1 3.6 3.6 3.7 4.2 5.3 5.3
1.15
1 1 1 1 1 1 1 1 1 1 1 1
... ... ...
2.5 2.9 2.7 5.9 3.9 3.4 4.8 4.8
276 Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
xs
... ...
...
3-10 11 3.5 3.5 Strong 3.5 3.5 Strong Strong
f,
O C
... 30 30 32 20 20 20 19 20 20 29 29
Reference
Buckley, 1930 Keenen and France, 1927 Paine and France, 1935 Mullin and Garside, 1967a Portnov and Belyustin, 1965 Belyustin and Portnov, 1962 Belyustin and Portnov, 1962 Spangenberg, 1925 Belyustin and Portnov, 1962 Belyustin and Portnov, 1962 Spangenberg, 1925 Gunther, 1929
compound growth mechanism were not nearly as complete as those showing the linear relationship. After completion of the present work, Mullin and Garside (1967a, b) published their data on the growth of alum crystals. As in this investigation, they used seeds containing both cubic and octahedral faces. However, their data for the cubic faces are limited and rather inconclusive, and were for supersaturations ranging from 1 to 10% only. The present investigation obtained growth rate data for both the (100) and (111) faces over a range of supersaturations from 0 to By0.Some data for the (110) faces were also taken. Crystal Growth Theories
Growth of Perfect Crystals. MOKOKUCLEAR TwoDIMENSIOKAL NUCLEATION. I n the late 1920Js, Stranski (1928) and Kossel (1927) proposed the first of the so-called modern crystallization theories by considering the growth of a perfect crystal. Simply stated, their two-dimensional nucleation theory said that crystals grew by adding layers of niolecules to their surfaces. This process consisted of the formation of a two-dimensional nucleus beginning a new layer and a series of spreading steps completing the layer. The original Stranski-Kossel theory assumed that the spreading steps occurred much more rapidly than the nucleation steps. This is tantamount to assuming that every time a stable (critical) two-dimensional nucleus forms, one new crystal layer is added. The Stranski-Kossel theory is usually referred to as mononuclear two-dimensional nucleation theory, because it requires only one nucleus for every new crystal layer added. With this assumption, the following can be derived.
R where R
+
Cz exp[-CCa/ln(S l)] (1) = crystal growth rate (the rate a t which the faces of the crystal advance, distances measured normal to the faces) S = supersaturation = (C - C,)/C, C = bulk solute concentration in grams of salt per cm3 of solvent C, = saturation concentration of solute =
Cz =
4 (a frequency factor)
= (au2an)/(k2TZ) u = average interfacial free energy
a = thickness of a growth layer D = volume of a solute molecule k = Boltzmann constant T = absolute temperature
POLYNUCLEAR TWO-DIMENSIONAL NUCLEATION. Recently, two-dimensional nucleation theories have been proposed in which it is assumed that the rate a t which nuclei form is rapid relative to the rate a t which they spread. For these models, several nuclei add to the crystal surface before a complete new layer is formed; hence, this is called polynuclear two-dimensional nucleation. Various growth rate expressions based on different simplifying assumptions have been presented (Bennema et al., 1967; Brice, 1967; Nielsen, 1964). The one used by Bennema is = (C4/Sz) exp [-Cs/ln(S
+ l)]
where C4 = (~kT/h,)(u*/kT)~ a exp (--E/kT)
cg = 7r(u*/kT)2
For low supersaturation
R
(4) (5)
(6) U* = surface free energy of a side face in the critical nucleus expressed as free energy per area corresponding to one growth unit h, = Planck’s constant E = activation energy for two-dimensional nucleation
=
C1*Sz
(71
For high supersaturation
R = CIS pC,*an,d* exp(- H , / k T ) 27ru0a*/(kTz,)
(9)
C1 = pC,*nn,d* exp( - H , / k T )
(10)
where C1*
p
C8
R
If reasonable values for the physical properties of the system (such as U, a, 9, etc.) are substituted in the growth rate expressions (Equations 1 and 4), both the mononuclear and polynuclear theories predict that the growth rates will be virtually infinitesimal a t supersaturations below about 1 or 2%. Since real crystals grow a t finite rates a t these supersaturations, the growth rate behavior of real crystals at low supersaturations cannot be explained by two-dimensional nucleation theory. Other theories which note certain imperfections in the crystal structure can account for the growth rate behavior in this region. Growth of Imperfect Crystals. FRANK’S THEORYOF DISLOCATION GROWTH.Frank (1949) demonstrated t h a t a crystal possessing a particular type of imperfection, a screw dislocation, contained a self-perpetuating step and t h a t for such a crystal the need for two-dimensional nucleation mould not arise. He later developed a theory for growth from this type of dislocation in which it was assumed that the surface diffusion of crystallizing species was rate-controlling (Burton et al., 1951). As originally presented, Frank’s theory described the process of growth from the vapor, but Bennema (1965) subsequently extended the model to include growth from solution. Frank assumed (1) that once the growth units diffused to the kinks (the active growth sites), they were integrated into the crystal lattice very rapidly, and (2) ,that ,the distance between adjacent kinks was small relative to the average distance an absorbed molecule traveled on the crystal surface before being desorbed. His theory predicts a parabolic relation between the growth rate and the supersaturation a t low supersaturation, and a linear dependence a t high supersaturation:
=
(8)
= a correction factor applied when assumption 1
does not hold C,* = a correction factor applied when assumption 2 does not hold no = a constant approximately equal to the number of molecular positions per unit area of crystal surface d*= a frequency factor that may be approximated by the frequency of atomic vibrations H , = activation energy for crystal growth by a screw dislocation mechanism u0 = surface free energy of a side face in a nucleus expressed as free energy per molecule a* = distance between molecular positions on the crystal surface za = average distance an adsorbed solute molecule travels on the crystal surface before being desorbed Frank predicted that for crystallization from solution the transition from parabolic to linear growth would occur at about 0.1% supersaturation, but Bennema (1965) feels that for K alum the transition might occur a t an even lower supersaturation. Experimental Procedure
Single seed crystals of the potassium alum were grown from aqueous solutions in a flow crystallizer under conditions of Ind. Eng. Chem. Fundam., Vol. 9,
NO. 2, 1970 277
BEARING STOPPER
CY Figure 1 .
Crystallization flask
constant temperature, liquid velocity, and supersaturation. The crystals were held stationary in the crystallizer while stirring forced the solution to flow past the crystal a t velocities which depended on the stirring rate. The seed crystals were mounted in such a way that the (100) and (111) faces whose growth rates were measured were parallel to the direction of liquid flow. From the measured difference in the crystal dimensions before and after a n experiment, growth rates normal to the (100) and (111) faces were obtained over a range of supersaturations from 0 t o 18%. Some data were also taken for the (110) faces. Figure 1 shows the type of flow crystallizer used. This crystallizer was immersed in a constant temperature bath, at a nominal temperature of 35.1"C and maintained constant to within *O.0loC. The technique used to change the alum supersaturation differs from those employed in previously reported studies. The original alum solution was prepared by dissolving a certain weight (determined by the desired supersaturation) of reagent grade alum (Fisher Scientific Co. A601) in a known volume of distilled water. This solution was then poured into the crystallization flask, and when the temperature of the solution became the same as the bath temperature (35.loC), the seed crystals were inserted and the growth rates of the crystal faces measured. For the subsequent experiments, to change the alum concentration (and hence the
supersaturation), the temperature of the solution was raised to a point above the saturation temperature. A specific volume of alum solution was then pipetted from the crystallization flask and replaced by a n equal volume of distilled water a t the same temperature as the solution. Finally, the temperature of the solution was lowered to the working temperature and the growth rates were again measured. The advantage of this technique is that all growth rate experiments may be conducted at the same temperature in a solution containing the same original alum. Seed crystals were prepared in two ways. Technique A was to make up a n alum solution about 15% supersaturated, add to this enough Agi304 almost to saturate the solution with Ag, place this solution in a closed container, agitate the solution continuously, and allow the crystals to nucleate and grow. Technique B was to prepare a pure solution about 35% supersaturated and place this in an open container. No agitation was used, and the crystals were allowed to nucleate and grow as before. The seed crystals were always carefully inspected to be sure that they had no visible defects. Seeds were mounted on holders made from lengths of type 316 stainless steel rod inch in diameter. To mount the seeds, the sharpened tip of the holder was heated slightly and then plunged into the seed at the appropriate point. After the crystals resolidified, they became firmly attached to the holder. The advantages of this mounting method are that no glue is used and therefore no contaminant is introduced, and the holder does not interfere with the liquid flow or with the growth of the faces which are being measured. One final step, performed before seed crystals were used, was to insert the mounted seed crystals into a 570supersaturated solution and keep them there until a very thin (pure) crystal layer had deposited on their surfaces. The amount of alum incorporated into the seeds during a single experiment never exceeded 0.01 gram. The magnitude of the change in supersaturation caused by this incorporation was always less than about 0.02%. Even though this error was slight, it was always accounted for, so that the errors would not accumulate. The supersaturations in the experiments were calculated from the known amount of alum contained in the solution and the solubility of alum a t the crystallization temperature. The solubility data for potassium alum reported in the literature are in disagreement (Denk, 1968). I n view of this,
INDICATES AVERAGE GROWTH RATE) (110) FACES ( I l l ) FACES INDICATES ESTIMATED ERROR
A (100)FACES 0 0
I
CURVE DRAWN THROUGH THE LOWEST GROWTH RATES
LIQUID VELOCITY = 3.1 INCHES/SECOND
?e '
Figure 2. 278
Ind. Eng. Chem. Fundam., Vol. 9,
No. 2, 1970
SUPERSATURATION
Relation of growth rates to supersaturation
L
I 0
-
AVERAGE GROWTH RATE OF T H E ( 1 1 1 ) FACES LIQUID VELOCITY 3.1 INCHES/SECOND
NUCLEATION QROWTH
Yo SUPERSATURATION Figure 3.
Compound growth mechanism for (1 1 1 ) faces
the solubility of K alum was determined [using a method given by Furman (1962)l a t the same temperature, in the same crystallizer, and using the same reagent grade alum as in the growth rate experiments. Growth Rates of (loo), (1 lo), and (1 1 1 ) Faces as a Function of Supersaturation
The runs in which the growth rates were measured as functions of supersaturation can be divided into two groups, performed about 6 months apart, used different types of seeds, and were conducted with reagent grade alum from different batches (but still from the same supplier). For these or perhaps other reasons, what a t first appear t o be two different types of growth rate behavior result. Results of Last Runs. The seeds were prepared by Technique B. They contained both cubic (100) and octahedral (111) faces, and occasionally rhombic dodecahedral (110) faces. The results from one run are presented in Figure 2. Several seeds were grown a t the same time, and for each curve, every point represents a different seed. The growth rate data for the (110) and (111) faces exhibited little scatter (Figure 2). At either high or low supersaturations, the data for the (100) faces were also relatively consistent. However, a t intermediate supersaturations, the growth rates of the (100) faces of the seeds varied, despite the fact that the crystals had grown in the same solution a t the same time under the same conditions. Nevertheless, a smooth curve could be drawn through the average of the points a t each supersaturation. An expression for this curve, as well as expressions for the growth rates of the (110) and (111) faces, can be developed by assuming that crystallization occurs via a compound growth mechanism. Growth Model. The data from other runs of this study do not contradict Bennema’s conclusion that a t low supersaturations the growth rates of the (111) faces are linear with supersaturation (Denk, 1968). I n the same region, the rates of growth of the (100) faces (not measured by Bennema) are also linear with supersaturation. Beyond certain transition points, however [about 1% supersaturation for the (111) faces and about 5% supersaturation for the (100) faces], the growth rates are no longer linear with supersaturation. The compound growth mechanism (dislocation growth plus polynuclear two-dimensional nucleation) proposed by
Bennema (Bennema et al., 1967) to account for the deviation from linear growth was tested and failed to correlate our data. We then decided to try a different compound growth model. The component mechanisms chosen were Frank’s theory of dislocation growth (Burton et al., 1951) and mononuclear two-dimensional nucleation (Kossel, 1927; Stranski, 1928). I n a compound growth model such as this it is assumed that the observed growth rate is equal to the sum of the growth rates arising from the component mechanisms. For our model R R
f Rmononuolear nuoleation
(11)
+ Czexp [-Ca/ln(S + 111
(12)
&rank
or =
CB
where C1, Cg, and C3 are given by Equations 10, 2, and 3, respectively. Equation 12 was fitted to the data of this study as follows: First, the slope of the least squares straight line passing through the data of this study a t low supersaturations was calculated. The value of C1 was taken to be equal to this slope. Knowing C1, the rate of growth by mononuclear twodimensional nucleation could be found, since R,,,, = Robs CIS. Values of R,,,, were determined in this manner and the data plotted as ln(Rmono)us. l/ln(S l), because the expression for the rate of growth by mononuclear two-dimensional nucleation (Equation 1) predicts that a straight line will result. When tested in this manner, the data of this study formed a straight line. The best straight line was then drawn (by eye) through the data and from this line the values of Cz and C3 were derived. The resulting growth rate expressions are
+
Rlw, mm/hour = 5.40S Rno, mm/hour
=
1.60 S Rill, mm/hour
=
0.41S
+ 3.11 exp[-O.280/ln(S + l ) ]
(13)
+ 1.72 exp[-0.200/ln(S + l ) ]
(14)
+ 1.60 exp[-O.206/ln(S + l ) ]
(15)
The fit of the growth rate curves described by the above equations to the experimental growth rates is shown in Figure 2, where the solid curves represent the growth rates predicted by Equations 13, 14, and 15. The fit is relatively good. I n Figure 3, the contribution of each mechanism to the over-all Ind. Eng. Cham. Fundam., Vol. 9, No. 2, 1970
279
( I l l ) FACES INDICATES ESTIMATED ERROR LlQUlO VELOCITY 3.1 INCHES/SECONO
o GROWTH RATE OF THE
/
EMPIRICAL MOOEL
I R.9.OS"") R89.OS"")
COMPOUND GROWlH MODEL (R-0.419 l.SOEXPf-O.P06/LnfS+I)) WHERE Rm CRYSTAL GROWTH RATE IN mm./nr. S. SUPERSATURATION
+
/
/
Y
0.I o.2
t
00 0
1
2
4
Figure 4.
6
8
1
1
1
1
1
1
1
1
IO I 2 14 16 18 20 22 24 % SUPERSATURATION
1
26
1
28
30
Empirical and compound growth models
growth of the (111) faces is indicated. At low supersaturation, most growth is by the dislocation model, while a t higher supersaturations two-dimensional nucleation becomes increasingly important. Comparison with Previous Empirical Correlations. T o express growth rate data, many workers in the field of crystallization (Bransom, 1960; McCabe and Stevens, 1951; Mullin and Garside, 1967a) are using an empirical correlation of the form
R = ASB where A, B = empirical constants.
(16)
For example, Mullin and Garside (1967a) used such an expression to correlate their data on the growth of the (111) faces of potassium alum crystals. The growth rate curves of the present study could also be correlated by a similar equation-for instance, the data for the (111) faces in Figure 2 can be fitted by an equation in which B is about 1.65. However, there are certain advantages in using a correlation of the form of Equation 12 instead of one like Equation 16. I n Figure 4 both correlations for the (111) faces are presented. At intermediate supersaturations, both correlations fit the data of this study well. For our compound growth model, a t low supersaturation, the rate of growth by twodimensional nucleation becomes increasingly small and the model predicts that in this region the growth rates will become linear with supersaturation. Since the linearity of the K alum growth rate curve a t low supersaturation has been relatively well established by the careful measurements of Bennema (1965, 1966b), this is one advantage of our growth model. At high supersaturations the slope of the growth rate curve described by the empirical correlation continues to increase, whereas the growth rates predicted by our compound growth mechanism tail off (Figure 4). Because of this, a t high supersaturations, the compound growth mechanism, and the empirical correlation, predict significantly different growth rates. It is not yet known which model is the more accurate in this region, because (three-dimensional) nucleation occurs a t these high supersaturations. Our experimental setup could not be used under these conditions. An apparatus in which nucleation would not affect the measurement of the growth rates of seed crystals will be used in future experiments. The main advantage of the proposed compound growth mechanism over the empirical correlation is that the con280 Ind.
Eng. Cham. Pundom., Vol.
9, No. 2, 1970
stants in Equation 12 are related to physical properties of the system through Equations 2, 3, and 10. As a result, if these properties are known, the growth rate behavior of any system can be predicted beforehand. The constants in the empirical correlation (Equation 16), however, must be established experimentally for every system. The values of all of the physical properties appearing in the compound growth mechanism are not available for alum (actually this is the case for most crystallizing systems). One estimate of how reasonable the proposed compound growth mechanism is may be had by seeing how reasonable the values of the physical properties derived from the experimental constants are. Interfacial free energies of the crystal faces were derived from Equation 3, using the experimental values of Cs. Free energies of 2.44, 2.45, and 2.76 ergs per em2 were obtained for the (loo), (110), and (111) faces, respectively. The only free energy for K alum with which these may be compared is Bennema's value of 2.5 ergs per cm2 for the (111) faces (Bennema et al., 1967). Although all of these free energies seem to be low, they are in line with the findings of other investigators who, working with other systems, obtained similarly low free energies (Denk, 1968). "sing Equation 10 and the experimental values of C1, the activation energies for dislocation growth of the (loo), (110), and (111) faces were estimated to be 12.8, 13.7, and 14.0 kcal per gram mole. These values may be compared with an activation energy of 18.4 kcal per gram mole reported by Bennema (Bennema et al., 1967) for growth of the (111) faces of K alum by polynuclear two-dimensional nucleation. Although the proposed compound growth mechanism correlates the experimental results and appears to contain reasonable constants, objections to it might be raised on theoretical grounds. For instance, Cahn and Hillig (1966), on the basis of the crystal growth model proposed by Burton, Cabrera, and Frank (1951), argued that, for the case of solidification from the melt, when dislocation growth had reached the region of linear dependence on supersaturation, an increase in the number of steps on the crystal surface should have no effect on the growth rate. iiccording to this, a two-dimensional nucleation mechanism acting in conjunction with a dislocation mechanism would provide more steps but have no effect on the growth rate. However, it is an experimental fact that a mechanism in addition to dislocation growth (which operates a t all supersaturations) exists, a t least a t high supersaturations. This additional mechanism could come about for many reasons, but unless we drastically revise our thinking about crystal growth-to consider, for instance, such things as the direct attachment to a growing crystal of small crystallites formed in the bulk solution by three-dimensional nucleation-we have to examine this additional mechanism in terms of two-dimensional nucleation theory (without, of course, excluding the possibility that a yet unrecognized mechanism may be operating). What can be stated from our work, however, is that the data can be correlated by assuming that this additional mechanism is mononuclear two-dimensional nucleation. The stepped surface consisting of one type of steps resulting from dislocations and other types of steps resulting from two-dimensional nucleation is a much more complex surface than the idealized one analyzed by Burton, Cabrera, and Frank. It could be argued that instabilities arising because of such a complex surface could lead to higher growth rates than those predicted by the linear law. Controlling Resistance. The resistance controlling the
.. -
c L ' \
g
1.4
-
1.2
-
1.0 -
A 0
I
(100) FACES (111) FACES INDICATES ESTIMATED ERROR LIOUID VELOCITY
= 3.1
INCHES/SEWND
E
P
Yo SUPERSATURATION
Figure 5.
Relation of growth rates to supersaturation
rate of crystallization under the conditions of this study was estimated as follows: Dissolution data for the (100) and (111) faces were taken, and from the slopes of the dissolution rate curves, mass transfer coefficients were estimated. [It can be shown (Denk, 1968) t h a t the mass transfer coefficients derived from dissolution data must be equal to or less than the true coefficients.] Once the mass transfer coefficients were known, the bulk diffusion-controlled growth rate curves could be constructed. When this was done, the experimental growth rate curves fell below the bulk diffusion-controlled growth rate curves, indicating that, under the conditions of this study, crystallization was not bulk diffusion-controlled. Comparison of Our Results with Mullin and Garside's Data. Mullin and Garside (1967a) reported growth rate curves for the (111) faces of potassium alum crystals a t supersaturations ranging from 1.5 to 10%. As in our study, it is possible t o correlate their growth rates by both Equations 12 and 16. There are some differences between our study and theirs, however: The growth rates they report are higher than those measured in our study and they claim that, in their study, growth was bulk diffusion-controlled, whereas in our study, a t the same liquid velocities, the conclusion was that crystallization was not bulk diffusion-controlled. Impurities may have caused these differences. Mullin and Garside worked with recrystallized alum and deionized water, while in this study reagent grade alum (not recrystallieed) and distilled water were employed. This raises the possibility that Mullin and Garside's growth rates were measured in solutions that were purer than those of the present study. If so, their growth rates should be higher than those of our study. The discrepancy concerning the controlling resistance can also be accounted for. Let us assume that Mullin and Garside's growth rate curves are accurate and represent the limiting growth rates a t which crystallization first becomes surface diffusion-controlled; the growth rates reported by Mullin and Garside would then correspond to the maximum amounts of solute that can be supplied by bulk diffusion. If, as in the present study, the growth rates for some reason (such as impurities) are lower than these limiting rates, the amount of alum that can be supplied by bulk diffusion is greater than
the amount required to maintain our observed (lower) growth rates. The conclusion must then be that the limiting resistance a t our lower growth rates is something other than resistance to bulk diffusion. By this reasoning the contention of Mullin and Garside that the growth rates of their crystals were bulk diffusion-controlled is not necessarily in opposition to the conclusion of this study that growth was not bulk diffusion-controlled. A more detailed comparison between the two studies is made by Denk (1968). Hypothesis for Scattering at Medium Supersaturations
Instability of Growth. iit intermediate supersaturations there is a significant increase in the scatter of our data (Figure 2). This scatter, most pronounced for the (100) faces, is typical of the growth rate behavior in this region. A clue concerning the reasons for this may be obtained from the early runs of this study. I n these runs, the growth rate behavior was, t o say the least, unexpected. (The seeds in these early runs were prepared by Technique A.) Figure 5 shows the results of one such run. Here the growth rate curves for both the (100) and (111) faces are characterized by plateaus in the region of supersaturation from 9 to 13%. A comparison of the growth rate curves from Figures 2 and 5 (not shown) indicated that the curves were nearly the same except in this region of intermediate supersaturation and that the low growth rates from the curve of Figure 2 fell on the plateau of the curve in Figure 5. [If the growth rate curve is drawn through the lowest growth rates between 9 and 13% supersaturation, a plateau appears (Figure 2).] The growth rate behavior in the range of supersaturation from 9 to 13% is hypothesized to be the result of an instability in the growth process. At any given supersaturation in this region, two growth rates are possible and a crystal may switch from a high to a low growth rate or vice versa. Where scattering occurred, some of the crystals were growing a t the low growth rate (and gave a plateau), while the other crystals, in the same solution a t the same time, grew a t the high growth rates a t least part of the time. For the rum in which the plateau appeared, the crystals grew almost exclusively a t the low growth rate. Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
281
A 1100) FACES 0
I
0.01 0
SUPERSATURATlON = 9.52%
( I l l ) FACES INDlCAfES ESTIMATED ERROR
I
2
'
4
I
I
6
I
' , I
8
I
IO
'
I
12
LIQUID VELOCITY,
Figure 6.
Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
14
'
I
16
I
18
' 20'
I
' ' 24' '
22
6
INCHES/SeCOND
Relation of growth rates to liquid velocity
Similar jumps in the growth rates were also observed in experiments where the growth rates were measured as functions of the liquid velocity (Figure 6). Beyond the point a t which crystallization was no longer bulk diffusion-controlled, for each face both high and low growth rates were possible. This instability in the growth process may be the result of a time-dependent adsorption of impurities similar to that observed by Beck (1964). This time dependency is the result of a feedback mechanism between the rate of crystal growth and the concentration of impurities on the crystal surface. Presumably, the faster the growth rate, the shorter the time any given crystal surface is exposed to impurity, because the faster the growth rate the faster the crystal surfaces aye covered over by new crystal layers. At steady state, the rate of growth and the surface concentration of impurity would be fixed. If, however, the system was perturbed in some way so that the growth rate was suddenly decreased, this would increase the exposure time and increase the surface concentration of impurity. This would lead to a further decrease in the growth rate, which would increase the exposure time even more, the concentration of impurity a t the crystal surface would be further increased, the growth rate would be further decreased, and the effect could continue on and on. The trend might also be in the opposite direction-for example, a random increase in the growth rate would decrease the exposure time and decrease the impurity concentration a t the surface, leading t o a further increase in the growth rate, etc. This cascading effect could continue on indefinitely if there were no constraints on the system. However, constraints usually exist-for example, the growth rate would probably not exceed the growth rate in a pure solution. Therefore, in time the growth rate would proceed to some point and then stop. Besides Beck's study, similar instabilities in the crystal growth rates have been observed by Booth and Buckley (1952) for ethylenediamine d-tartrate, and by Mullin and Garside (1967a) for the (100) faces of potassium alum. Another possible explanation for the observed instabilities in the growth rates may result from the mechanism of crystal growth. If crystallization proceeds by the compound growth mechanism proposed in this study, under certain conditions the component mechanisms may no longer be independent and additive but interfere with each other. Exactly how this interference would occur is not known a t this time, however. 282
I
In any event, the observed scatter in the growth rates between 9 and 13% supersaturation may be explained by assuming a growth instability due to either (or both) of the previous phenomena-time-dependent adsorption of impurities or interference between competing mechanisms of crystal growth. While these phenomena are possible explanations for the observed scatter, neither requires a plateau to appear in the growth rate curve. At present, we have no explanation for the physical significance of the plateau, and plan further experiments. Relative Growth Rates
As shown in Table I, the relative growth rates of potassium alum reported in the literature are in serious disagreement. The relative growth rates of the (100) and (110) faces [referred to the (111) faces] calculated from our data were functions of the supersaturation (compare the curves in Figure 2). The relative growth rates are greatest a t low supersaturations-for instance, the relative growth rate of the (100) faces can be as high as 13.2-and fall off asymptotically to lower limits as the supersaturation is increased: 3.0 for the (100) faces and 1.5 for the (110) faces. As the relative growth rates are functions of supersaturation, it is not surprising that the literature values are in such disagreement. Conclusions
For potassium alum crystals, the region of supersaturation from 9 to 13% appears to be one of unstable crystal growth; a t any given supersaturation, both high and low growth rates are possible. A curve drawn through the low growth rates is characterized by a plateau in this region-Le., the low growth rates appear t o be independent of the supersaturation. On the other hand, the curve drawn through the average growth rates is smooth. This instability a t intermediate supersaturations may be the result of either (or both) time-dependent adsorption of impurities or interference between competing mechanisms of crystal growth. The growth rates of the (loo), (110), and (111) faces may be correlated by a compound growth mechanism in which it is assumed that crystallization occurs both by dislocation growth and by mononuclear two-dimensional nucleation. This compound growth mechanism has certain advantages over previously reported empirical correlations.
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Nielsen, A. E., “Kinetics of Precipitation,” p. 46, Pergamon Press, New York, 1964. Paine, P. A., France, W. G., J . Phys. Chern. 39,425 (1935). Portnov, V. N., Soviet Phys. Crystallog., 11, 774 (1967). Portnov, V. N., Belyustin, A. V., Soviet Phys. Crystallog. 10, 291 (1965).
Randolph, A. D., Larson, &I.A., A.I.Ch.E. J . 8 , 639 (1962). Spangenberg, K., 2. Krist. 61, 189 (1925). Stranski, I. N., 2.Physik. Chem. 136, 259 (1928). RECEIVED for review May 9, 1969 ACCEPTED February 27, 1970 Study supported by the National Science Foundation through Grant GK 1015. The financial assistance is gratefully acknowledged. Paper presented a t 64th National Rleeting, American Institute of Chemical Engineers, New Orleans, La., March 1969.
Rates of Evaporation and Condensation between Pure Liquids and Their Own Vapors Jer Ru Maa Distillation Research Laboratory, Rochester Institute of Technology, Rochester, -V. Y . 14623
In the process of evaporation and condensation, the random motions of the vapor molecules are distorted by their group streaming. The effect of this mass vapor movement on the rate of phase change was examined by using the jet stream tensimeter and Schrage’s theory was found satisfactory over the tested range. An approximate method i s suggested for considering the motion of the phase boundary due to the loss and gain of surface liquid in the heat transfer calculation. The agreement between the theoretical rates of evaporation and condensation so computed and those observed experimentally confirms with new certainty the unity of the evaporation and condensation coefficients of common liquids.
T H E EVAPORATIOK or condensation coefficient of ordinary liquids is unity or nearly so and the rates of these processes depend on two factors (Hickman, 1954, 1965; Alaa, 1967, 1969), the effective pressure of the vapor and the true temperature of the liquid surface. The effective pressure of the vapor is complicated by the mass movement of vapor molecules to or from the liquid surface. Direct readings of surface temperature cannot yet be made because of steep thermal gradients beneath the liquid surface. During the process of evaporation (or condensation), there is always a mass movement of vapor molecules from (or toward) the vapor-liquid interface. Schrage (1953) considered the effect of this mass vapor movement and derived a correction factor for the calculation of the net rate of phase change. This factor was adopted by many workers in this field
(Bonacci and Eagleton, 1966; Jamieson, 1965; Maa, 1967, 1969; Mills, 1965; Nabavian and Bromley, 1963). However, some of the assumptions used in his derivation were disputed by others (Standart and Cihla, 1958). Because of the importance of this method of correction to the calculated rates of evaporation and condensation, it is desirable t o examine and verify it by simple experiments. Our first objective here is to do so by using the jet stream tensimeter. Calculations for thermal gradients in the liquid have recently been offered for a liquid stream in laminar flow exposed t o vapor for -0.001 second (Maa, 1967, 1969). The yields observed experimentally for many common liquids including water agree with values calculated for a n evaporation coefficient of unity with, however, increasing divergence as the total rate of evaporation or condensation is increased-that Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
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