944
P. L. DIETZ, JR., J. S. BRUKNER AND C. A. HOLLINGSWORTH
Vol. 61
LINEAR CRYSTALLIZATION VELOCITIES OF SODIUM ACETATE IN SUPERSATURATED SOLUTIONS1 BY PAULL. DIETZ,JR., JOHNS. BRUKNER AND C. A. HOLLINGSWORTH Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania, Received February $9, 1067
The linear crystallization velocity (herein abbreviated to C.V.) of sodium acetate trihydrate in supersaturated solutions was measured for several bath temperatures and concentrations of sodium acetate. The effects on the C.V. of the presence of varying amounts of acetic acid and sodium hydroxide were investigated. The temperature of the mixture of crystals and solution immediately after crystallization also was measured for several bath temperatures and concentrations of the supersaturated solution. Explanations are proposed for some of the results obtained.
Rates of precipitation from supersaturated solutions have been investigated by many workers. Most of this work has been concerned with the rate of nucleation and growth of single crystals. Investigations of the linear crystallization velocity (C.V.) usually have dealt with the crystallization of supercooled melts2J although in some cases supersaturated solutions have been used. The theory of the C.V. in a supercooled melt is complicated by the rate of escape of the heat of fusion, while in supersaturated solutions diffusion of the solute, as well as heat flow, is involved. The work reported herein deals with certain aspects of the C.V. of sodium acetate trihydrate from its supersaturated solutions. Experimental *sb
The C.V. was measured in tubes one to two cm. inner diameter, 10 to 60 cm. in length and one mm. wall thickness. The C.V. was found to be independent of the tube parameters over these ranges and to be independent of the time, that is, constant along the tube, after the crystals had grown a cm. or two beyond the oint of seeding. Crystallization was initiated by dusting t t e supersaturated solution a t one end of the tube with trihydrate crystals. In most cases the precipitation formed a distinct boundary that moved down the tube, which, in such caaes, was oriented vertically and seeded a t the top. However, for bath temperatures within about 8’ of the saturation temperature, TaRt,a distinct boundary was not present. Instead, the crystallization took place by a multitude of individual needle-shaped crystals growing in apparently almost random directions as though new grains of random orientation were continuously being nucleated. The values of the C.V. in these cases were obtained from the time interval between the times when the first crystal crossed two given marks on the tube. Since the crystals were not usually growing parallel to the axis of the tube, the crystal which was first to cross one mark was not the crystal which was first to cross the second mark. For this type of growth, and in all cases in which the crystallization boundary was not stable enough t o prevent settling of the cryAtals during growth‘, the tubes were oriented horizontally and seeded a t one end. The final temperatures, i.e., the temperatures immediately after crystallization had taken lace, were determined in separate experiments by means ao! small thermocouple placed in the center of a 250-ml. flask containing the supersaturated solution. Except for bath temperatures near Teat, the thermocouple reached the final temperature within a (1) Abstracted from a portion of the Ph.D. thesis of P . L. Dietz, University of Pittsburgh, 1955, and the M.S. thesis of J. S. Brukner, University of Pittsburgh. 1953. (2) Gustav Tammann, “States of Aggregation,” D. Van Nostrand C o . , New York, N. Y.,1925. (3) M. Volnier, “Kinetik der Phasenbildung,” Leipzig, Verlag yon Theodor Steinkopff, 1939. (4) See J. R . Partington, “An Advanced Treatise on Physical Cheniistry,” Vol. 111, Longmans, Green and Co., New York, N . Y., 1952, for a review and a large list of references. (5) Some recent work with supersaturated solutions is reported by A. C. Chatterji and R. P . Rastogi, J . Indian Chem. Soc., 28, 599 (1951): THISJOURNAL, 59, 1 (1955).
few seconds after the thermocouple was surrounded by precipitate, and this temperature then remained unchanged for many minutes. When the bath temperature was near Tsat, the crystals, instead of forming a network, settled as the precipitation took place. In order to obtain reproducible results it was necessary in such cases to shake the flask during the precipitation. The sodium acetate used was certified, reagent grade NaCzH30~.3HzO,Fisher. It was found, however, that a technical grade anhydrous sodium acetate gave the same results. The sodium hydroxide was reagent pellets, Baker Analyzed. U. S. P. Glacial, 99.5% acetic acid, Baker and Adamson, was used. The water was distilled, and had a specific conductance of about 1.5 X 10” ohm+ cm.-l, before degassing by boiling. The concentrations of the sodium acetate were determined by evaporating samples of the solutions to dryness, with final heating at 130°,and weighing the residues, which were anhydrous sodium acetate. The mole fractions of the acetic acid and sodium hydroxide were calculated from the known weights of the substances which were added to known weights of sodium acetate solutions of known compositions.
The Variation of C.V. with Bath Temperature.In Fig. 1 are shown curves of C.V. as a function of the bath temperature for several concentrations of sodium acetate. The data published by Chatterji and Rastogi6 are included. These curves are of the same general shape as those which are obtained from the crystallization of supercooled melts when conditions are such that the heat of fusion cannot escape readily while the solidification is taking place. This is characteristic of rapid crystallization in wide tubes. This typical behavior of the C.V. in supercooled melts was described by Tammann2 and is illustrated in Fig. 2. This curve can be divided into three regions ab, be and cd as indicated in the figure. The region be of almost constant C.V. has been difficult to explain. Perhaps the most nearly satisfactory theory is that proposed by Forster.’ He pointed out that when the heat of fusion is not removed sufficiently rapidly from the crystallizing front, there should be a radial variation of temperature across this front. Thus if the measured values for the C.V. refer to the portion of the front that is moving most rapidly, there should be a range of bath temperatures over which the C.V. is constant. A curve of the C.V. as a function of temperature of the growing surface would have the form of the broken curve in Fig. 2. Experimental support for the contention that the broad temperature range over which the C.V. is nearly equal to its maximum value ( = V,,,) is caused by the retention of heat of fusion during crystallization was obtained by Neu(6) A. C. Chatterji and R . P . Rastogi, J . Indian Chem. Sac., 28, 599 (1951). (7) T. Forster, 2. physik. Chem., 8176, 177 (1936).
c
July, 1957
LINEARCRYSTALLIZATION VELOCITIESOF SODIUM ACETATE
945
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mann and Micus.* These 0.70investigators obtained a much sharper maximum in the C.V., by using supercooled salol, which has a comparatively low Vmax, in tubes of small diameter and thin walls. It was knowng that in wider tubes salol gave the usual type of curve. Forster's explanation leads to the prediction that the crystallization surface would be curved radially in a way that agrees only partially with what has been observed. The theory preFig. 1.-C.V. m a function of the bath temperature for several concentrations of diets a surface concave to the melt in region ab and sodium acetate (expressed as molality of the anhydrous salt). Points and solid curves experimental. Broken curves are those given by eq. 5 with A' = eSa.", (E' - E ) / R a surface toward the are = 7809, EIR = 2820, and B / R = 23.5. melt in region cd. and this is the behuavior described by Tammann.2 However, the flat surface which, according to Tammann, occurs in the region bc is not predicted by this theory. I n the present work with the sodium acetate solutions the following observations were made. I n region ab the observed behavior depends upon the sodium acetate concentration and can be divided into three types as follows. (i) When the concentration of sodium acetate was high (12 molal or greater) so that a lot of precipitation took place, Imox. there was a spiral growth next to the wall of the TEMPERATURE __+ tube, and crystallization was much slower in the Fig. 2.-Typical behavior of the C.V. in supercooled melts. center of the tube. (ii) At moderate sodium acecurve represents the observed C.V. as a function of the tate concentrations (9 to 12 molal) the crystalli- Solid bath temperature. Broken curve re resents the theoretical zation boundary was ragged but flat, and the in- rate of crystal growth as afunction ofthe temperature of the dividual crystals fanned out with branches as they crystal surface. grew. (iii) At low concentrations (less than 9 crystals slightly ahead of the others. It is clear molal) the growth of the crystals no longer formed that if the rate of escape of the heat of fusion is not a network, and there was nothing that could be called a liquid-solid boundary. The precipitation sufficiently great, the crystals out in front will be coolest, while those farthest behind will be warmtook place by the growth of needle-shaped crystals est, since the latter will have not only their own as described in Experimental. It perhaps should heat, also some of that given off by the crystals be pointed out that the concentrations a t which one whichbut have advanced farther on. Suppose that type of behavior changed to another were by no the bath temperature is slightly below Tmsx,where means definite, and that a t temperatures suffi- T m a x is the surface temperature that will produce ciently near t o the saturation temperatures even the maximum velocity of longitudinal growth of a the most concentrated solutions exhibited behavior single crystal. It should be expected that the type (iii). crystals out in front will grow a t a slower rate than I n region bc the boundary was always flat. The the crystals just behind them and these slower than region cd was not obtained in this work, presumably the ones just behind them, etc., until a region is because experiments were not carried out a t suffi- reached where the temperature is equal to T m a x . ciently low temperatures. The tendency should then be for the advancing The flat boundary which was observed in the bc front of crystals to grow in such a way that the region in both supercooled melts and in the super- crystals which are out in front will tend to be saturated solution indicates that the radial varia- equally far advanced and to have just enough space tion in temperature (and concentration) is not the between them so that the temperature in this adonly factor involved in producing the range bc of vancing front will be equal to !!,.' If a crystal nearly constant C.V. It seems possible, however, somehow manages to get ahead of the others i t will to explain the observed facts in this manner. move into a region of temperahre lower than T,,, Imagine a mass of crystals growing parallel to and will be slowed. A crystal which falls behind each other in a supercooled melt with some of the the others will be slowed, also, because it will be in a region of temperature greater than T,,,. If, (8) K. Neumann and G. Micus, 2. physik. Chem., 2 (New Series). however, any large number of crystals should fall 25 (1954). behind, then the temperature at the advancing (9) H.Pollatschek, ibid., i M A , 289 (1929).
,
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P. L. DIETZ,JR.,J. S. BRUKNER AND C. A. HOLLINGSWORTH
Vol. 61
1
front will drop, and the region where the temperature is equal to T m a x will be located farther back. Crystals in that region will grow faster than those in front and eventually catch up and increase the density of crystals in the advancing front, and, thereby, cause the temperature to be raised again to Tma,. The precipitation is not imagined to occur all a t the advancing front, but continues farther back, probably in part by side growth from larger crystals. According to this theory, the rate of advance of the crystal front need not necessarily be very different near the wall of the tube from what it is a t the center. If heat is being taken up by .04 .03 .02 .01 0 .01 .02 .03 .04 the wall, this means that the crystals in MOLE FRACTION M~LERACTION the advancing front must be closer toOF ACETIC ACID OF SODIUM HYDROXIDE gether in the region next to the wall Fig. 3.-C.V. as a function of the amount of additives for a solution than in the region a t the center. This which is 9.25 molal in anhydrous sodium acetate. possibility - would cause the crystal front to tend to be flat. If the bath temperature is greater than T,,,, then the crystal front can progress most rapidly if the crystals are far apart so that the temperature a t the crystal surfaces will be as close as possible to that of the bath temperature. Growth near the wall would be easier and one would expect a greater density of crystals there. If the bath temperature is too far below m!' a, the heat of fusion is insufficient to raise the temperature to T,,,, even if the crystals grow as close together as they possibly can. This would result in a phase boundary convex toward the melt. The preceding argument applies to the freezing of supercooled melts. I n the case of supersaturated solutions the problem is further complicated by diffusion. The general conclusions, however, I 1 I I remain the same. lo .22
-
0
.2i
.02
.04
.03
.05
.06
.07
MOLE FRACTION OF SODIUM HYDROXIDE, L
Fig. 4.-C.V. as a function of the amount of additives for solution which is 10.45 molal in anhydrous sodium acetate.
TABLE I THEVALUESOF VmaxA N D OTHERQUANTITIES FOR VARIOUS CONCENTRATIONS OF SODIUM ACETATE Concn. (ni of anhydrous NaCeHaOz)
5
4
3
2
Satn. temp.," T.,t (OK.)
Temp.b a t
Max. which Vmax value of C.V. was attained VmBx(cm./sec.) (OK.)
7.28 308.5 0.055 286 8.85 318.5 .155 294 ,180 295 9.25 319.5 10.44 324.0 .28 297 .31 299 10.68 325.0 11.93 327.5 .45 300 13.75 330.0 .55 302 14.53 330.5 .60 303 16.96 331.0 .68 . .E These values were obtained froin a plot of the solubilities given in the International Critical Tables. Solubilities determined in this Laboratory a t three tem eratures also lie on this curve (Brukner, M.S. thesis). b5he temperature above which the C.V. decreases with increasing temperature. this value was not determined. The precipitation of anhydrous crystals caused difficulty as described in .the text. Q
I
05
--
03
01
MOLE FRACTION OF ACETIC ACID
0
01
03
05
MOLE FRACTION OF SODIUM HYDROXIDE
Fig. 5.-C.V. as a fuiiction of the amount of additivcs for a solution which is 12.25 molal in anhydrous sodium acetate.
The Effect of Sodium Acetate Concentration and Bath Temperature.-Table I shows values of (10) For a more lengthy discussion of the effect of diffusion see Paul L. Diets, Jr., Doctoral Thesis, University of Pittsburgh, 1955.
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LINEARCRYSTALLIZATION VELOCITIESOF SODIUM ACETATE
July, 1957
Tsat and Vm,,, where V m a x is the maximum value of the C.V. for a given concentration of sodium acetate. A plot of In Vma,as a function of Ts,t-'is a straight line. A simple model based upon the assumption that the C.V. is the difference between a forward (crystallization) process and a reverse (dissolving) process gives this type of behavior. Thus, we assume that V
= Ae-E/RT
- A'e-E'/RT
(1)
where V is the value of C.V., E and E' are the activation energies of the crystallization and dissolution processes, respectively. A' is a constant and A is a function of the concentration of the supersaturated solution. Use of (dV/ d7')vmax = 0 and V(Tsat) = 0 in eq. 1 gives E E/(E'-E) = A' (jp)
6
,4
.3
3
5
.2
.I
0
.w4 .02
= Ae-E/RTe-AG/RT
(3)
(4)
where B is a constant. While Volmer was primarily concerned with crystallization from melts, Rastogi and ChatterjiI2 have discussed the general properties of an equation of the same form in connection with their work on the linear crystallization in supersaturated solutions. Equation 3 is supposed to apply for a given concentration of the supersaturated solution. To take into account the effect of concentration it must be modified. One way to obtain such a modification is to assume that the dependence of A upon the concentration can be found from the fact that V = 0 at T,t and by using eq. 1. This gives = A'e-(E'-E)/RTsat
e - E / R T e-BTsnt/RT(Tsat-T)
02
.04
.06
.08
MOLE FRACTION OF SODIUM HYDROXIDE
Fig. 6.-C.V. as a function of the amount of additives for a solution which is 13.8 molal in anhydrous sodium acetate.
where A and E have the same meanings that they have in eq. 1, and AG is the free energy required to form a surface nucleus of critical size. Volmer obtained
V
0
MOLE FRACTION OF ACETIC ACID
which gives the observed linearity between In V,,, and Tsat-', The simple theory underlying eq. 1 does not, however, reproduce in det8ailthe regions in Fig. 1 that correspond to region ab in Fig. 2, whether T is taken to be either the bath temperatures or the final temperatures given in Fig. 7. I n particular this theory leads to curvesll that approach the points (Tsat, V = 0) with a negative value of d2V/dT2, which is in disagreement with the experimental curves, Fig. 1. A modification of the preceding theory to take into account the possibility that the rate-determining 'step involves surface nucleation gives results that are more nearly in agreement with experiment. Volmer3 obtained an equation of the form V
947
(5)
Equation 5 leads to the observed linear relation(11) P. L. Dietz, Jr., ref. 10, pp. 53-54. (12) R. P. Rastogi and A. C. Chatterji, THIS JOURNAL, 69, 1 (1955).
I'
0'
IO
20 30 40 INITIAL TEMPERATURE
50
60
("C.).
Fig. 7.-Final temperature as a function of the bath temperature for various initial concentrations of sodium acetate (expressed as molality of anhydrous salt.)
ship between In V,, and TSat-l. Figure 1 shows an attempt to fit eq. 5 to the sodium acetate data. It will be observed that eq. 5 gives qualitatively the proper behavior for the regions between V,,, and V = 0. It may be that a part of the discrepancy between theory and experiment is the result of the neglect of the effect of screw dislocations, which according to modern the01-y'~is important in many systems for crystallization near Tat. However, it seenis more likely that more serious factors, such as diffusion, have not been taken into account properly. This theory cannot be expected to reproduce the region bc of Fig. 2 for the reasons discussed in the previous section. Also, it is possible that the type of crystallization that takes place a t undercoolings less than 8" and that which takes place a t greater undercoolings should be treated separately. The Growth of the Trihydrate in the Presence (13) F. C. Frank,
Phil. M u g . , 1, 91 (1952).
948
OLE LAMM
of Anhydrous Crystals.-At concentrations above 16 molal of the anhydrous salt, crystals of the anhydrous sodium acetate usually precipitated spontaneously while the solution was being cooled to the bath temperature. These anhydrous crystals formed a network, which, a t room temperature, caused the solution t o hold its shape as a solid. When this network was dusted with the trihydrate, crystallization took place with a C.V. which was, within experimental error, independent of the concentration that existed in the solution before the anhydrous crystals had precipitated. This is what one would predict, since the concentration of the solution in contact with the anhydrous crystals would, assuming equilibrium existed, be equal to the solubility of the anhydrous crystals a t the bath temperature. The values of the C.V. obtained in the presence of the anhydrous network a t 22.8" were 0.599, 0.606 and 0.602 (cm./sec.) for the molalities (prior to the precipitation of the anhydrous) of 16.77, 16.82 and 17.50, respectively. A solution saturated with respect to the anhydrous crystals a t 22.8" is 15.2 molal. The value of Vm,, for such a solution is about 0.63 cm./sec. The small discrepancy here indicated might mean that the network of anhydrous crystals in some way slightly obstructs the growth of the trihydrate. The anhydrous crystals grew slowly during their spontaneous precipitation. For example, a t a bath temperature of 22.4" in a 16.8 molal solution the C.V. for the anhydrous crystals was found to be 0.0027 cm./sec. The Effects of Acetic Acid and Sodium Hydroxide.-The values obtained for the C.V. in solutions to which various amounts of acetic acid or sodium hydroxide had been added are shown in Figs. 3, 4, 5 and 6. Although these effects are complex, attention may be called to the following generalizations : (A) acetic acid always decreases the C.V.; (B) small amounts of sodium hydroxide
VOl. 61
increase the C.V. in solutions of lower sodium acetate concentration; (C) small amounts of sodium hydroxide decrease the C.V. in solutions of higher sodium acetate concentration; ID) for an intermediate concentration of sodium acetate small amounts of sodium hydroxide cause an increase in the C.V. a t low bath temperatures but a decrease in the C.V. a t higher temperatures; (E) large amounts of sodium hydroxide always decrease the C.V. The Final Temperatures after Crystallization.I n Fig. 7 are shown the final temperatures after crystallization as a function of the bath temperature for several concentrations of sodium acetate. It will be noted that for the solution of molality 16.83 the final temperature is in all cases 58.4". Presumably, if measurements could have been made a t a sufficiently low bath temperature, lower final temperatures would have been obtained. At this high a concentration there is a great tendency for anhydrous crystals to precipitate, and this caused serious difficulty at lower temperatuies. The temperature 58.4" can be only very slightly lower than the incongruent melting point of sodium acetate trihydrate,I4 where.the trihydrate decomposes into solution and anhydrous crystals. Since the solubility of sodium acetate trihydrate is known as a function of the temperature, the data in Fig. 7 allow one to calculate the change in state which takes place when a supersaturated solution of a given concentration is seeded at a given temperature, and precipitation takes place adiabatically. However, it is not known that the final temperature given in Fig. 7 is related in any simple manner to the steady state thermal condition a t the advancing crystallization front. Nothing that seems significant is obtained by the use of these final temperatures instead of the bath temperatures in equations such as eq. 1 and 5. (14) W.F. Green, THISJOURNAL, l a , 655 (1908).
AN ANALYSIS OF THE DYNAMICAL EQUATIONS OF THREE COMPONENT DIFFUSION FOR THE DETERMINATION OF FRICTION COEFFICIENTS. I. BY OLE LAMM Division of Physical Chemistry, Royal Institute of Technology, Stockholm, Sweden Received February 96, 1967
The theory has been systematically developed with regard to the measurement of the frictional coefficients in threecomponent systems from diffusion experiments. For this purpose, equations have been developed which connect the frictions with diffusion coefficients, thermodynamic factors and concentrations. The use of activity factors based on volume fraction, rather than mole fraction, is shown to result in a special simplification. This leads to exact and rather simple equations for the frictional magnitudes, without restrictions regarding the composition of the diffusing mixture: The thermodynamic factors have been given a form which shows the analogy between this theory and the simple equation for two components. The result is a generalization to three non-dilute components of the well-known relation D = RT/@.
The dynamical theory of the diffusion of more than two components' was left in a mathematically undeveloped form. It has, however, been used in connection with problems of self d i f f u s i ~ n . ~ -I~n (1) 0. Lamm, Arkiu. Kemi. Mineral. Ceol., ISA, No. 2 (1944). (2) 0. Lamm, ibid., 18B,No. 5 (1944). (3) 0. Lamm, Acta Chem. Scand., 6, 1331 (1952). ( 4 ) 0. Lamm. ibid., 8, 1120 (1954).
view of the present possibilities of performing experimental work in this Institute in the field of three-component diffusion, it has become of increasing interest t o analyze the diffusion equations and to develop an adequate treatment of the coefficients. This will be tried here in a more general sense and returned t o later in order to find the solution in some special and simpler cases of interest for
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