Nudeation of Quiet Supersaturated Potassium Chloride Solutions Effect of Some lonic Crystals GEORGE W. P R E C K S H Q T ‘
AND
GEORGE GRANGER BROWN
UNIVERSITY O F M I C H I G A N , ANN ARBOR, M I C H .
T h e purpose of this work was to determine t h e effect of crystallographically similar, insoluble, ionic crystals i n nucleating quiet supersaturated solutions of potassium chloride referred t o t h e spontaneous nucleation of these same solutions which were saturated a t 49.453’ C. T h e t i m e required for t h e formation of nuclei, ‘Lwaitingtime,” on these foreign crystals and i n t h e bulk of t h e solution was measured by a conductometric instrument and saturation concentrations were determined from temperature readings. Nucleation was found t o depend on t h e insoluble foreign crystal present. T h e lattice constants of all foreign crystals used were within 10% of t h a t of potassium chloride and no correlation appeared possible on t h e basis of lattice constants. Activation energies, surface free energies, and sizes of crystal nuclei were determined for nuclei of potassium chloride over t h e range of temperatures 32’ t o 42’ C .
T
HE formation of crystal nuclei froin supersaturated solutions when none previously existed has been the object of many investigations. Spontaneous formation of crystal nuclei was reported by Dehlinger and Wertz (3jand Amsler ( 1 ) to take place a t temperatures below the saturation temperature, if sufficient “waiting time” was allou ed. The presence of insoluble foreign crystals reduced the supersaturation required to form crystals from their solutions as reported by Volmer and Weber (24), Stranski and coa.orkers (19, 20), and Vonnegut ( 2 5 ) . The purpose of the work reported here is the determination of the effect of crystallographically similar insoluble ionic crystals such as lead selenide, lead telluride, stannous telluride, and naturally occurring lead sulfide on the nucleation of quiet aqueous potassium chloride solutions a t various amounts of supersaturation. The term, “nucleation” used in this paper is defined as the process for the formation of the crystal nucleus when none of this nucleus previously existed. I t may include the deposition of crystalline solids on dusts, metallic surfaces, other crystalline surfaces, or spontaneously in the bulk of the solution, but excludes the deposition on surfaces of the same material.
Concentration changes a t the moment of the nucleus formation are measured conductometrically and are used to determine the waiting time, which is defined as the time required for a nucleus to form on the other insoluble crystal surfaces or in the bulk of the solution. This waiting time is a measure of t,he frequency of nucleus formation. The purpose of the folloning derivation is to show how this measured waiting time may be used t o evaluate the effectiveness of crystals serving as nuclei and to calculate the surface free energies of the crystals relative to the solution and the insoluble crystalline solids, the activation energy, and the side length of the crystal in equilibrium with its solution. This derivation ( I S ) is not new but is modified to meet the conditions of the present work. The basic relation of the equilibrium between a solid and its solution, which links the solubility expressed as the thermodynamic potential to the particle size, is the Thompson equation ( 2 3 ) : Pr
1 Present
1314
- II.,
=
m
address, University of Minnesota, Minneapolis, Minn.
(2:
The frequency of the formation of these crystal nuclei from their respective saturated solutions has been given (29) as:
(1)
K X
J
X e-A/RT
e-C‘i/RT
(3)
Usually J is not large, so that equilibrium conditions previously imposed still prevail. For a given volume of solution the frequency of nucleus formation is inversely proportional to the waiting time, e, the time requir,ed for the formation of the firqt nucleus. After taking logarithms of both sides, log
TH EOR ET1C A I
(0.239 X 10-7) 2uN2
Only when the crystal has a radius r will it be in equilibiiunl with the solution, and under these conditions the thermodynamic potentials of the two coexisting phases are equal: When Equation 1 is written for the equilibrium between macroscopic particles and a particle of radius T 71-ith their respective solutions, then the work of formation of the nucleus of the n e x phase of radius 1’ in equilibrium with its solutions is expressed on a molar basis as:
e
=
log K-’
u, + 2.303RT ~
A
T-
For an isotherm, the term Lr,/2.303R7‘is a constant and Equation 4 takes the form of log6 = B f
A 2.303RT ~
In order to evaluate the activation energy, A , simplification in the form of the first approximation ( I S ) is taken for the quantity ( P ~ - Pas: ~
pi
-
io = pr
- ,urn = 2.303RT
log(z/zo)
(61
If Equations 2 and 6 are substituted in Equation 5, the result is: log e = B
4uu3V2,N + (0.239 3(X2.303)3R3T3
1
1 for iaolog2(~lxo) The slope of this line is:
Equation 7 gives a straight line of log e veraus thermal measurements.
(71
4w03V;A-
m = (0.239 X 10-7)3-__
INDUSTRIAL AND ENGINEERING CHEMISTRY
~
q2.303j
w
~
3
VOl. 44, No. 6
NUCLEATION-From Liquids All quantities in this slope a r e k n o w n e x c e p t u, t h e specific surface energy of the solid relative to its solution. The initial saturation conditions in the present work were maintained constant, and the supersaturation was varied by varying the amount of subcooling over a range of 10" C. Equation 7, however, applies strictly t o isothermal measurements, but i t may apply over a limited temperature range, provided the term B does not vary a great deal. Approximate values of B were calculated for 30" and 40" C. by using calculated heats of crystallization (1.4) and an estimation for R ( 2 1 )in the nucleation rate equation 3. This variation in B was found to be only 25%. If log 8 plotted versus I/ Iog*(s/x,) is linear when the temperature is varied and when B is essentially constant, then over the range of temperatures the quantity u / T is essentially a constant. Thus, just as in the isothermal case, the specific surface energy of the solid relative to its solution may be calculated from the slope of the line in Equation 7 as 2.303RT 3m ]lis u = (9) 0.239 X lo-' 4wViN There will be no effect of particle size of the insoluble solid on the temperature of nucleus formation at a constant waiting time, if its size is large compared t o r, the radius of the sphere inscribed in the solute crystal nucleus in equilibrium with the solution. It is expected that different solids would cause nuclei t o form at different (18, 22, 26, 26) temperatures for constant waiting time and certainly at temperatures different from those when there are no solids present. This may be looked upon as being dependent upon the activation energies for nucleation-the smaller the activation energy, the greater is the tendency for nucleation to occur on the solid. This activation energy may be calculated from the results of the experiment as:
[
A =
2.303R Trn log2(x/xo~
(10)
Finally, r, the radius of the sphere inscribed in the nucleus in equilibrium with its solution, may be computed from Equations I, 2, and 6 as:
The side length of a cubic crystal in equilibrium with its solution is u = 2r and should be of the order of l o + em. EXPERIMENTAL PROCEDURE
Apparatus. The photograph in Figure 1 shows the apparatus for measuring conductometrically the time a t which nucleation begins on the various insoluble crystals singly exposed t o supersaturated solutions of potassium chloride. A sensitivity of 0.16 p.p.m. per mm. movement of the bridge slide-wire is achieved with this equipment. ~
Unit A is the control panel board; B, C, and D are constant temperature baths and contain controls, saturator, and conductivity cells, respectively; E is the alternating current Wheat-
June 1952
Figure 1.
Nucleation Apparatus
stone bridge; F is an audio-oscillator; G is a preamplifier; H i s the electronic filter; I is an oscilloscope; J is the conductivity water steaming unit; and K is a unit for producing and storing dust-free conductivity water. Various controls on the panel board allow the regulation of the thermostatic baths, B and D, to a temperature of lfO.002" C. Bath B contains water and houses the saturator vessel; bath C contains transformer oil and houses the conductivity cells during filling and preheating and is manually controlled to within 0.1' C. of the desired temperature. Bath D contains transformer oil and houses the conductivity cells a t the temperature of measurement. The close temperature control for baths B and D is achieved by use of the standard mercury-in-glass thermal regulator, protected against moisture and oil contaminations, in conjunction with a low current electronic relay (6). Temperatures were measured with calibrated Beckmann thermometers. The saturator vessel housed in bath B is a 2-liter glass bottle containing solid analytical grade potassium chloride and its solution made with double-distilled water. The vessel contains a lass stirrer, a n ultrafine biological filter, 40 mm. in diameter $Corning No. 36060 UF, with a maximum pore size of 1.2 microns), and means for filling and closingit off fromthwtmosphere. A pair of specially designed conductivity cells shown in Figure 2 are housed in baths C and D. Each contains a mercury valve t o allow for expansion and contraction of the liquid during the course of the experiment, and a stopcock t o close the cell off from the atmosphere. The platinum electrodes A , have bee8 deposited directly on the glass from an ethyl aicohol-ether solution of chloroplatinic acid. The L cell, which is required to reduce the effect of temperature variations on the bridge performance, contains 41 inches of borosilicate glass tubing 5 mm. in inside diameter between the electrodes. The R cell is of similar construction and contains 52 inches of 5-mm. borosilicate glass tubing between the electrodes. I n addition, R has a large groundglass joint ahead of the mercury valve for the introduction of the selected insoluble crystals. Silicone stopcock grease held at 0.001 mm. of mercury and 350" C. is used as a lubricant for all joints and stopcocks. The standard Wheatstone bridge, E, oscillator F , preamplifier G , filter H , and oscilloscope I are designed t o detect changes in the null signal of the order of 10-7 volt at 1000 cycles. The sensitivitv of the detector unit is found to be 2.5 X 10-6 volt per inch signal reading on the screen when the oscilloscope is operated a t 50% of the maximum gain or amplification.
A small but gradual change in the slide-wire reading amounting to about 10 mm. of slide-wire movement per hour was noted when pure equivalent resistances were introduced in the arms of the
INDUSTRIAL AND ENGINEE'RING CHEMISTRY
1315
NUCLEATION-From
Liquids
bridge. This was due to temperature changes in the circuit elements and these small changes did not affect materially the progress of the nucleation experiment. !Then the balance cell, L, and the test cell, R, were introduced in their respective arms of the bridges, temperature variations of 0.003 C. caused a shift in the null setting of the slide wireof 1 mm. Thussmall temperature variations could be tolerated, provided both L and R experienced these temperature changes simultaneously.
Figure 2.
Conductivity Cells
Left.
B a l a n c e c o n d u c t i v i t y cell, L R i g h t . T e s t c o n d u c t i v i t y cell, R
Conductivity water steam is used to steam glassware t o clirninate dust contamination and to fill conductivity cells with dust-free conductivity Lvater. This water \vas stored in clean, dust-free containers and was used as needed to purge conductivity cells after each experiment. Materials. Double-distilled water and analytical grade potassium chloride without further purification are used in the freparation of solutions. The insoluble crystals, lead selenide, ead telluride, and stannous telluride, which are used in these nucleation experiments, were obtained from A. D. McKay, 192 Broadway, Kew York 7 , N. Y. The elements from which they were made were a t least 99.570 purity. No attempt was made to analyze these compounds for their major constituents or for traces of their elements. No attempt was made t o purify these compounds further. Inclusion-free crystals of naturally occurring lead sulfide were selected for the preparation of lead sulfide specimen used in the experiments. All are used in the form of - 100- to fl50-mesh hand ground particles. These particular crystalline compounds are chosen for this xork essentially because of their ionic sodium chloride structure, In this respect they are similar to potassuim chloride, which has four molacules per cell and has a coordination number of 6. In addition, they are insoluble in potassium chloride solutions and they do not react with water or the solute. T h e various crystals, however, differ in their chemical nature and in lattice dimension. These dimensions are given in Table I.
Table I . Crystal PbS PbSe KC1 SnTe PbTe
Lattice, Dimensions of Crystals yo Deviation from
Unit Cell Dimensions ( 9 7 ) 5,920
6 . I40 6.278 6.285
6.340
KC1 Unit Cell -5.7
-2.2
0.0
0.1 1.0
Procedures. A11 glassware which comes in contact with conductivity water or potassium chloride solutions is cleaned with a solution containing a strong washing owder (Calgonite) and a detergent (FAB). After washing, al?glassware is rinsed with t a p water, distilled water, and finally conductivity water and
1316
steamed with conductivity water before being filled with either dust-free conductivity water condensed in place or potassium chloride solution. The specially treated silicone lubricant used in the conductivity cells is removed with Decalin (decahydrona hthalene, C10H18)before the cell is washed with the detergent soition. After the prescribed n-ashing and rinsing, all joints are again lubricated. The cells are then steamed with conductivity water for 1 hour before steam is condensed in t'hem. Freshly distilled mercury is then placed in the mercury valve of the balance cell, L, to seal the water from the atmosphere. En the case of the test cell, R , a specially prepared sample of insoluble crystals is introduced before mercury is placed in the expansion valve. Adsorbed gases are removed from the respective crystalline samples by evacuating them at' mm. of mercury and a t 350" t o 400" C. prior to sealing them off in special tubes containing two capillaries. A slight amount of evaporation of the crystal is noted before the tube is sealed off. The crystal sample is then brought in contact Kith dust-free conductivity water by breaking off one capillary under rvater. The broken end is then immersed in the conductivity water in R through the large ground-glass joint, which is opened only briefly for the introduction of the sample. The other capillary is theB broken off, allowing the crystals to enter R. The cell is then quickly closed m d freshly distilled mercury is placed in the expansion valve. Mercury is added to the elect'rical contact tubes and the cell then is ready for connection in the bridge circuit. The saturator vessel housed in bath B is filled initially with potassium chloride solution saturated a t 100" C. in order to assure the presence of crystals at 50' C. The temperature of this bath is controlled a t 49.453' i 0.002° C. for 2 hours while the solution is stirred vigorously. After this time the solution is considered saturated at t,his temperature and transfer to the conductivity cell is begun. At this time cell R is placed in bath C maintained a t 65' C. The saturator vessel is closed o f f from the atmosphere and is connccted to the test conductivity cell, R, by a removable section of the heated transfer line. During transfer, B is maintained a t the control temDerature. 49.453' i 0.002" C. and C a t about 65" C. Filtered a i r forces the saturated solution through the ultrafine biological filter into the transfer line and finally into the conductivity cell, displacing the conductivity water. About 400 nil., or about seven times the cell volume, of saturated solution are pumped through R before the saturator is disconnected. Relative bridge measurements showed that the solution concentration is essentially constant after the passage of about 300 nil. of saturated solution. After R is filled and closed off, the removable section of the t.ransfer line is disconnected and -washed. For about 10 to 15 minutes dust-free conductivity water is flowed into t.he fixed portion of the transfer line to prevent crystallization in it and on the downstream side of the filter. The balance conductivity cell, L, is filled in the same manner, except that the potassium chloride solution is saturated a t about 30" C. L is always unsaturated at the temperatures a t ivhich measurements are made and is filled only once. Experimental determinations begin after these cells have been held a t 65" C. for 1.33 hours or longer. The residence time a t elevated temperatures was found not to affect the spontaneou, nucleation temperature for quiet potassium chloride solution (6). Holding the solutions a t this temperature assures the a,bsence of very small potassium chloride cryst'als which may havp passed through the filter. Just rior t o the run C is cooled to and controlled a t 50" 2 0.05" In addition, D, whose temperature was held a t 1 i0.002° C., is cooled to a temperature below this cont'rol temperature so that when the two conductivity cells and their metallic support from C a t 50" C. are placed in D, the temperature of the latter rises just to the control temperature, t. This amount of undercooling amounts to 0.3" to 0.6" C. and depends essentially on t.he supersaturation desired in cell R and on the room temperature. After several runs the undercooling is eatimated readily for the particular experiment. The experiment officially starts when the cells are immersed in bath D simultaneously with the recording of time for the determination of waiting time. Bridge readings as well as temperature readings are taken as coon a.3 possible, usually within 3 to 5 minutes after the &art of the experiment. The reading of the slide-wire of the bridge is related to the concentration of the solutions in L and R and follows the following pattern. When the temperature of the cells decreases by 0.003' C. the slide wire reading, 82, increases by about 1 mm. and indicates the net increase in resistance of R over L. When the temperature of the cells approaches t h a t of the bath, within 3 t o 5 minutes, the slide-wire reading remains relatively constant and indicates the constancy of the resistances of the cells. When
8.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 44, No. 6
NUCLEATION-From Liquids Figure 3. Graphical Determination of Waiting T i m e for Representative Runs
IO&
so00
-
mau
V I -
. 0
+
E=4 W*-
a
70 WAITING
TIdE,
8 , NIN.
Table 11. Tabulated Results of Experiments i n the Nucleation of Supersaturated Potassium Chloride Solutions w i t h Various Insoluble Crystals SubExpt. NO.
Sat.
Temp., ta,
c.
cooling Meas. Temp., t = t a t ,
t,
c.
O
c.
Waiting Time, 8, Min.
Log
e
1 Log* (a/zo)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
49.452 49.453 49.453 49.452 49.450 49.452 49.452 49.453 49.453 49.450 49.452 49.454 49.455 49,453 49.455
41.412 41.396 41.394 39.556 39.537 39,539 38,013 37.966 37.952 35.655 35.652 35.643 32.180 32.178 32.178
Lead Selenide 53 8.040 107 8.057 42 8.059 19 9,896 18.5 9.913 9.913 20 7.5 11.439 18.5 11.487 6.5 11,501 13.795 120 91 >317
>2.323 >2.210 2.079 1.960 >2.501 1.846 1.654 1.176 1.218 1.322 1.176 1.146
1512 1239 1112
0.721 20.700 0.846 S0.602
1112 1103 930 929 852 749 749 748 747 746 528 463 463 462
Lead Sulfide 11.494 160 13.818 26 13.819 24 16.241 6.5 17.276 4.5
2.204 1.415 1.381 0.814 0.654
1104 746 746 528 463
No Crystals 13.798 26.5 14.178 80.5 14.286 69.5 14.291 39 15.656 60 15.772 39 17,263 9 17.281 14.5 17.285 32.5 17.293 25
1.424 1.906 1.842 1.592 1.679 1.592 0.955 1.162 1.512 1.398
749 708 696 695 573 564 463 462 462 461
70
45 15 16.5 21 15 14 10 5.25 55 7 24
1.000
nuclei form, the bridge slidewire reading increases sharply and indicates an increase in resistance of cell R over cell L. Bridge readings are taken until the slidewire reading, SZ, reads 100 cm. or more. After this time in each experiment involving insoluble crystals, potassium chloride crystals were visible only on t h e foreign crystals and not anywhere else in the cell. The waiting time, e, is determined by plotting s2 against time and is taken a t the break of the curve, examples of which are shown in Figure 3. After visual observations confirm t h e presence of nuclei, cell R is transferred to bath C at 65’ C. in preparation for t h e I 1 1 next experiment. About 200 ml. of dust60 free conductivity water purges the cell of saturated potassium chloride solution and potassium chloride crystals. The cell is held a t 65’ C. for a t least 1 hour before being refilled with new solution saturated a t about 49.453O C. This procedure of purging and filling is used only when the samples of insoluble crystals are not removed from the cell. Insoluble crystals from one series of runs are always removed chemically before proceeding with a new series of experiments. The major portion of the crystals is removed by flushing with water and the remainder is removed with the appropriate reagent (11). The chemically cleaned cell is then subjected to t h e cleaning and filling routine described previously. RESULTS
The summarized data on waiting times, 0, and other calculated results are grouped in Table I1 for each of the insoluble crystals. The concentrations are obtained from the solubility-temperature data (10,17) and the saturation temperatures recorded in Table 11. The densities of potassium chloride (8) and its aqueous solutions (9) are used in the calculations. The same amount of crystalline surface is exposed for each experiment related arbitrarily to that of lead selenide. T h e weights of the various samples are listed in Table I11 for each of the experiments. Table Ill. Crystal PbSe PbSe PbSe PbTe SnTe SnTe SnTe PbS
Weights of Crystals Used i n Experiments Weight, Gram 0.1172 0.1173 0.1169 0.1245 0.0942 0.0966 0.0940 0.1088
Used in Experiments 1,4,5,7,10,11 2, 6 , 8 , 12, 13, 14 3.9, 15 16, 17, 18, 1 9 , 2 0 , 2 1 22,24, 25, 30,31,32,33, 36,37, 38 26,28,34,35 23, 27, 29 39,40, 41, 42, 43
The data presented in Table I1 are again shown in Figures 4 to 8 and show t h a t log e is linear with l/log*(z/z,), just as predicted by Equation 7 as applied to nonisothermal measurements. This relationship is strikingly shown for crystals of lead selenide and lead sulfide, especially in Figures 4 and 6. The data for lead telluride, stannous telluride, and the blank (in the absence of all foreign bodies) given in Figures 5, 7, and 8, respectively, are somewhat less striking, yet straight lines were drawn by using the lines for lead selenide and lead sulfide as guides. The results and the composite curves of Figures 9 and 10 olearly show that nucleation of aqueous potassium chloride solutions is dependent on the crystalline solid present. Potential energy “wells” in crystals are located a t regular intervals corresponding to the lattice spacing, and the deposit atoms orient themselves a t these minimums. It might be expected that crystals such as stannous telluride with a lattice spac in@:essentially equal t o that of potassium chloride (see Table I) would be most effective in nucleating potassium chloride solu-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1311
NUCLEATION-From
Liquids
tions. This is not borne out by the measurements, which instead show that stannous telluride is less effective than either lead selenide or lead telluride. I n addition, experimental results show t h a t lead telluride is less effective than lead selenide, contrary t o considerations based on lattice constant difference. Several factors may be responsible for these deviations from the expected behavior. The first is that the chemical nature might be allx
0
400 600
200
1000 1200 1400 1600 I800
800
I/
periments are concerned. These pure elements will collect in the grain boundaries and should have little or no influence on the behavior of the intermetallic crystal. Thus it is believed by the degassing of the samples, surfaces of the intermetallic compounds r e r e exposed to the supersaturated solutions. There appears t o be no simple correlation of this nucleation effect on the basis of the standard lattice constant or the difference in lattice constant between the insoluble and potassium chloride crystals. Work of other investigators (16, 22, 8 6 ) points t o such a relationship, but no other rule can be applied, except that an insoluble solid will act as a nucleating agent if its lattice dimension does not differ by more than 15% from that of the solute crystal. The nucleation data for a particular insoluble nucleating agent are coneistent within themselves, even though the crystallographic plane exposed is different from insoluble crystal to insoluble crystal. However, there can be no correlation of the nucleation data of the several insoluble nucleating crystals unless their orientations are known. A knowledge of the orientation is necessary for a fuller understanding of this problem. Further work along this path is already under way.
2000 2200 2400 2 6 0 0
LOZf
Figure 4. Lead Selenide Crystals Present i n Supersaturated Potassium Chloride Solutions
controlling because force fields differ from the different chemical ion pairs. The second is that no precautions were taken to expose the same crgstallographic planes for each of the crystals. Force fields from different crystallographic planes for the same crystal are different. These two factors may be experienced in the form of different activation energies. The third is that there may have been a variable change in particle size or different sized cracks may have been produced for the several crystals during or after the standardization procedure. These changes in particle size could have occurred by splitting due to heating or by condensation of vaporized crystals. No measurements were made to test this. This effect of particle size is particularly critical with small particles of the order of 10-7 to 1 0 - 8 em. in diameter. 2.0
-
XO
Values of the specific surface energy,
U,
are presented in Table
IV, from the experimental data and Equation 9. The parenthe-
1.0 1.0-
/ 8
ZOO
'
400
,
8
600
Calculated Values of u for Potassium Chloride
Crystalline Sample PbSe PbTe SnTe PbS None
s -
I I ,' 8 8 0 0 1000 1200 I/ LOG'
+
,I
*I
LL
1400 1600 1800
Figure 5. Lead Telluride Crystals Present i n Supersaturated Potassium Chloride Solutions
Phase diagrams (12) give an invariant composition for each of these insoluble crystalline compounds and information on their manufacture indicates that these materials are very stable at temperatures around 400' C. Calculations of decomposition pressures from thermodynamic data in the literature point to the extreme stability of these compounds a t 400' C. The presence of parent pure elements in the intermetallic compound is believed to be of little or no significance as far as nucleation ex-
1318
LO82
Figure 6. Lead Sulfide Crystals Present i n Supersaturated Potassium Chloride Solutions
Table IV.
(I
I
I/
ses indicate extrapolated values which lie outside the range of the experimental data.
+ -
0
/
(318.2) (1.70) (2.16) (2.60) (2.80) (2.94)
Temperature, K. 313.2 308.2 303.2 1.66 1.67 1.62 2.09 2.12 2.06 2.66 2.52 2.48 2.72 2.76 2.67 2.85 2.89 2.80
(297.2) (1.59) (2.02) (2.43) (2.62) (2.74)
u is the surface energy of the crystal relative to its solution and the insoluble crystalline sample present. The surface tension of the solid crystal can be computed from these values if the surface tension of the supersaturated solution is known. It can be seen from these values of u that the surface tension of the solid crystal would not differ very greatly from the surface tension of the supersaturated solution. The activation energies in gram calories per gram mole which follow from Equation 10 and the experimental data are recorded in Table V. The calculated activation energies are lower for substances which are more effective in forming nuclei of the solute from its supersaturated solution. They are higher for substances which are not as effective in forming nuclei of the solute from its supersaturated solution. Side lengths of cubic nuclei which are in
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 44, No. 6
NUCLEATION-From Liquids Table V.
Activation Energy for Nucleation of Potassium Chloride Crystals Temperature, K. (318.2) 313.2 308.2 303.2 (297.2) Activation Energy, A , Gram Cal./Gram Mole
Crystalline Sample PbSe PbTe SnTe PbS None
Table VI.
560
1400 2900
(9 200) (14:OOO) (24 400) (30:600) (35,300)
320 610 1100 1300
1200 2000 2500 2900
5090
6400
7400
(220) (330)
(570) (720) (830)
1500
Side Length a of Cubic Nucleus
Cr stalline
Temperature, ' K. 313,2 308.2 303.2 Side Length, a, A.
(318.2)
Ample
(297.2)
O
equilibrium with the solutions stem from Equation 11 for the same temperature range. These data are presented in Table VI. The data of Table VI show that the size of the nuclei of potassium chloride in equilibrium with the supersaturated solution varies with the temperature as well as with the insoluble substance present. The effect of the insoluble substance is the same as a reduction in temperature. For isothermal measurements the size of the nucleus is smaller for the more effective nucleating insoluble crystal. The crossing of the curves in the composite graphs in Figures 9 and 10 cannot be fully explained, but may be due in part to the assumption that log e is linear with l/log2(z/z,) over such a great temperature range which includes the region of high subcooling where measurements are least reliable. Another contributing factor may be the lack of constant crystallographic orientation. It is believed that the character of the curve and log e us. l/logz (z/z,) changes sharply for low values of e. In this region the measurements are least reliable. /
I-
1ic /2.0
n*1/4To
110
+ 300
*oo
*
500
,
,
600
700
,
.
,
,
,
,
,
L
810' 900 1000 1100 le00 1300 1400 1500 1 " I/LOG'f
B
Figure 7. Stannous Telluride Crystals Present Supersaturated Potassium Chloride Solutions
in
A comparison of the results of the work on spontaneous nucleation with that of previous investigators is somewhat difficult because of the differences in experimental conditions. Studies in quiet supersaturated potassium chloride solutions over a wide range of compositions (3-6)where cooling rates were of the order of 1 'C. per 5 to 10 minutes gave spontaneous nucleation temperatures which were on the average 19.6" C. below the saturation temperature. This is in contrast to curve 5 of Figure 10, which shows that nucleation may take place essentially a t any temperaJune 1952
ture, provided the waiting time is long enough. It is doubtful if a single value of this spontaneous nucleation temperature would have been obtained in the cited works (3-6) if the cooling rate had been varied over a wide range. Comments may be made on the relation of the present work in quiet solutions with the work in stirred potassium chloride solutions ( 1 ) . I n the work with stirred solutions the investigator maintained that diffusion controlled the formation of the nucleus during spontaneous nucleation and indicated that rapid stirring would eliminate this diffusional effect, provided that stirring was started after the solution was cooled to the temperature at which measurements were to be made. Waiting times were obtained for subcoolings of 7 to 9 C. below the saturation temperature of 29.9 O C. The waiting time-subcooling curve had the same shape as curve 5 of Figure 10 in the present work, but it was displaced toward smaller subcoolings. It is doubtful that this great difference is due entirely to the differences in the saturation temperatures-that is, 29.90' C. as compared to 49.45' C. Some of t h e reported difference, therefore, must be due to stirring.
2.0
lo;,
c
,
/
100 2 0 0 300 4 0 0
,
BOO
,
,
600 7 0 0
Figure 8. No Crystals Present i n Supersaturated Potassium Chloride Solutions
,
800
,
There are two compensating effects which may produce the above difference due to stirring. Stirring breaks up molecular aggregate. If this were the only effect, a greater subcooling would be required t o produce nucleation in stirred than in quiet solutions. When shock was imparted to mildly agitated solutions (88-30),a smaller subcooling was required for nucleation. This latter effect, therefore, on the basis of comparison of the data in quiet and stirred solutions, is larger than that due to the destruction of molecular aggregates, thus giving the net result of a smaller subcooling for stirred solutions. No comparison of the present work for nucleation by solids is possible with the literature, for no data are reported for the nucleation of potassium chloride solutions by insoluble crystals. The results of the present work are in contrast, in one respect, with the results of the work on nucleation of other salt solutions by other insoluble crystals ( 7 , 15, 19, 20, 24). These investigators reported a unique nucleation temperature for a particular supersaturated solution. Their experiments undoubtedly were carried out under a constant cooling rate. It is doubtful that a unique nucleation temperature would have been obtained had the cooling rate been varied over a wide range. The present results show that potassium chloride solutions can be nucleated by insoluble crystals over a range of temperatures below saturati&, provided the waiting time is long enough. CONCLUSIONS
A conductometric instrument was devised to detect nycleus formation in quiet potassium chloride solutions and techniques were developed t o present reproducible surfaces for determining
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NUCLEATION-From
Liquids
2.0 r
/
tM ,I , ,
0
200
400
600
800
1000 1200 1400 Id00 I800 2000 2200 2400 2600 I/
LO'",
f:
Figure 9. Comparison of Crystals Present i n Supersaturated Potassium Chloride Solutions
the effect of crystalline solids in nucleating quiet supersaturated potassium chloride solutions. The nucleation of quiet aqueous potassium chloride solutions saturated a t 49.453 a C. depends on the presence of the insoluble crystalline solid. For a constant waiting time, lead selenide was most effective in nucleating potassium chloride solutions and required the smallest subcooling. Lead telluride was less effective and required a larger subcooling. Stannous telluride was still less effective, requiring a still larger subcooling. Of the solids investigated, lead sulfide was least effective and required the largest subcooling. Nucleation spontaneously in the bulk of the solution required the largest subcooling. Nuclei formed on the solid crystals a t different amounts of subcooling, if the waiting time was long enough. Nuclei also formed spontaneously in the bulk of the solution, in the absence
of foreign nuclei a t different but larger amounts of subcooling' if the waiting time was long enough. A linear relationship between log e and l/log2(s/x,) held for a small range of temperature of about 10' C. Activation energies for the nucleation of potassium chloride on various crystals were calculated for temperatures of 32' to 42" C. from the linear relationship of log e us. 1/log2(x/x,). For each of the crystals the activation energy increased rapidly as the supersaturation decreased. The isothermal activation energies for these same crystals increased in the same sequence as the decrease in their effectiveness in nucleating potassium chloride solutions-that is, in the order of lead selenide, lead telluride, stannous telluride, lead sulfide, and spontaneous nucleation. For example, for subcooling of 10" C. (2' = 313.2" K.) the activation energies were 1400, 2900, 5000, 6400, and 7400 gram calories per gram mole, respectively. Surface energies of potassium chloride crystals relative t o their supersaturated solutions and the foreign crystals Tere calculated to be 1.5 to 3 ergs per square centimeter. The side lengths of the nuclei of potassium chloride in cquilibrium with their solutions were calculated and range from 8 to 30, depending upon the temperature and the insoluble crystalline solid present. ACKNOWLEDGMENT
The advice and suggestions of George E. Uhlenbeck and Lars Thomassen have been of great value in the present work. Initial financial assistance given by the University of Michigan Engineering Research Institute and the Swenson Evaporator Co. is acknowledged. The assistance given by the School of Chemistry and Department of Chemical Engineering of the University of Minnesota was helpful in the completion of the final phases of this work. Valuable assistance of many others has been given in the construction of the instruments and equipment. NOMENCLATURE
160'
140-
130100 110
-
p 100+
90-
4 ea700
z
so-
I
so -
w90
-
D E G R E E S SUBCOOLING-AT.
"c
Figure I O . Effect of Crystals on Waiting T i m e for Nucleation of Supersaturated Potassium Chloride Sol u t ion s
9320
I
-
u1
w
c
= enerm of activation for formation of nucleus. gram cal. pergram mole a = side length of crystal in equilibrium with solution, A. B = a constant e = natural logarithm base J = frequency of formation of nuclei, (min. -I)( ml. -1) K = a constant (approximately) M - 2 = molecular weight of solid phase m = slope of line plot of e us. l/log2 (z/zo) A' = Avogadro's number R = gas constant r = radius of sphere inscribed in the crystal nucleus in equilibrium with solution, em. s2 = bridge slide-wire reading, em. 7' = temperature, K. t = temperature, O C. t, = saturation temperature, O C. = energy of activation for the molar transition from phase 1 (solution) t o phase 2 (crystal), gram cal. per gram mole v 2 = molar volume of crystal?cc. per gram mole x = mole fraction of solute in the supersaturated salt solution at temperature t xo = mole fraction of solute in the salt solution saturated a t temperature t PI = thermodynamic potential of supersaturated solution, gram Eal. per gr&mmole PI0 = thermodynamic potential of saturated solution, gram cal. per gram mole PLY2 = thermodynamic potential of nucleus of radius T , gram cal. per gram mole Prn = thermodynamic potential of nucleus of infinite radius, gram cal. per gram mole P2 = density of phase 2 (crystal), grams per ml. u = specific surface energy of crystal compared to its solution, and foreign crystals, ergs per sq. em. 6 = "waiting time," minutes nucleus surface w = dimensionless factor, r2 A
153-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 6
NUCLEATI ON-From L I T E R A T U R E CITED
Amsler, J., H e h . Phys. Acta, 15, 699-732 (1942). Dehlinger and Wertz, Ann. Physik, 39, 226-40 (1941). Gopal, R., J . Indian Chem. SOC.,20, 183 (1943). Ibid., 21, 103-8, 145-7 (1944). Ibid., 24,279-84 (1947). Hersh, R. E., Fry, E. M., and Fenske, M. R., IND.ENG.CHEM., 30, 363 (1938). Heverly, J. R., Trans. Am. Geophys. Union, 30, 205-10 (1949). International Critical Tables, Vol. 3, p. 43, New York, McGrawHill Book Co., 1928. Ibid., p. 106. Ibid., Vol. 4, p. 239. Mellor, J. W., “Comprehensive Treatise on Inorganic and Theoretical Chemistry,” Vols. 7, 10, 11, London, Longmans, Green and Co., 1922. Ibid., Vol. 10, p. 786, Vol. 11, pp. 2-40 ff., 55. Neumann, K., and Miess, A., Ann. Physik, 41, 319-23 (1942). Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., pp. 246, 1052, New York, McGraw-Hill Book Co., 1940. Rau, Schriften deut. Akad. Luftfahrt, 8, 65 (1944).
Liquids
(16) Rhodin, T. N., Discussions Faraday Soc., 5, 215 (1949). (17) Shearman, R. W., and Menzies, A. W. C., J . Am. Chem. SOC., 59,185 (1937). (18) Smith-Johannsen, R., Science, 108, 652-4 (1948). (19) Stranski, I. N., and Kuleliew, K., 2. physik. Chem., 142, 467 (1929). (20) Stranski, I. N., and Mutaftschiew, Z. C., Ibid., 150, 135 (1930). (21) Turnbull, D., and Fisher, J. C., J . Chem. Phys., 17,71-3 (1949). (22) Van der Merwe, J. H., Discussions Faraday Soc., 5 , 2 0 1 (1949). (23) Volmer, M., “Thermodynamik der Phasenbildung,” Dresden and Leipzig, Theodor Steinkopff, 1939. (24) Volmer, M., and Weber, A., 2. physik. Chem., 119,295 (1926). (25) Vonnegut, B., Chem. Revs.,44,277-89 (1949); J . Applied Phys., 18,593-5 (1947). (26) West, D. C., The Frontier, 8, No. 2, 12 (1945). (27) Wyckoff, R. W. G., “Crystal Structures.” New York, Interscience Publishers, 1948.(28) Young, S.W., J . Am. Chem. Soc., 33, 148 (1911). (29) Young, S.W., and Cross, R. G., Ibid., 33, 1375 (1911). (30) Young, S. W., and Van Sicklen, W. J., Ibid., 35, 1067 (1913). RECEIVED for review January 14, 1951.
ACCEPTED4pril 1, 1952.
NUCLEATION FROM SOLIDS
Theory of Nucleation in Solids ROMAN S M O L U C H O W S K I CARNEGIE INSTITUTE OF TECHNOLOGY, PITTSBURGH, P A ,
T h e basic problem of a theory of nucleation in solids is outlined and t h e complications caused by a departure from t h e conditions for applicability of Volmer’s theory are described. T h e various theories and their comparisons w i t h experimental data are summarized. In particular t h e nucleation theory of recrystallization and i t s successful interpretation of observations is discussed. It is pointed o u t t h a t t h e qualitative agreement w i t h experim e n t confirms general ideas about nucleation, while t h e absence of quantitative agreem e n t does n o t permit a critical appraisal of t h e specific assumptions.
A
T PRESENT there is no satisfactory theory of nucleation in
solids. This embarrassing situation is due t o the difficulty of experimental investigations of that phenomenon and also to the lack of theoretical estimates of the numerous factors which influence formation of nuclei in solids. Observations are made at relatively late stages of reactions when nucleation has been followed by appreciable growth and thus other unknown factors are added to the already complex picture. I n a few instances, as in recrystallization which is discussed here, it has been possible to extrapolate back to the process of nucleation without making too drastic assumptions about the rate of growth. The theories give at best a qualitative dependence of nucleation on time, temperature, composition, and other factors. A comparison with experiment sometimes leads to reasonable values for the adjustable constants. T H E BASIC PROBLEM
A comparison of nucleation in a solid, as for instance during precipitation in a supersaturated solid solution, with nucleation in a vapor phase during condensation illustrates well the various additional factors which have to be considered. The fair agreement between the various variants of Volmer’s theory ($00)of condensation and experiments indicates that the basic notions about the roles of free energy, surface energy, fluctuations of density, critical radius of the nucleus, etc., are in that particular case sound and convenient. How do these relatively simple concepts apply t o precipitation from a supersaturated solid solution? Considering first the free energy per mole of the precipitating compound, the first difJune 1952
ficulty is encountered: Information about its composition or its crystal structure is not certain, even if this information about the final precipitate obtained a t equilibrium were available. In fact there are many instances where it is definitely known that there are several intermediate stages in the development of a precipitate (8), and thus an unambiguous extrapolation to zero time is not possible. Here the inherent limitations of the finest x-ray techniques become apparent: Usually only the later stages of the formation of a nucleus corresponding t o the development of its crystalline structure can be observed. Only in a few particularly favorable cases the presence and shape of clusters of (ultimately precipitating) atoms within the crystal lattice of the original matrix has been determined (9). Thus the notion of free energy per mole of the early nuclei becomes vague and it would be perhaps better to avoid that concept altogether and consider the phenomenon on a purely atomic basis. Clearly this will not be a n easy task. Next there is the surface energy. Does a nucleus in a solid have definite surface? There is little doubt that embryos-i.e., unstable nuclei corresponding, in the case of condensation, t o nuclei smaller than the critical size-are coherent with the matrix, and both their composition and atomic configuration gradually merge with the surrounding lattice. It could be assumed that a certain layer of atoms in this transition zone is the surface of the nucleus, but the ambiguity of t h a t concept is obvious. Thus the surface energy cannot be as uniquely defined as for condensation nuclei. For the same reason the critical radius or size of the nucleus in a solid is not rigorous concept. This is particularly true since solid nuclei are seldom spherical.
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