Ind. Eng. Chem. Res. 1988,27, 1721-1728 cyclohexane, 3178-22-1; isobutylbenzene, 538-93-2; sec-butylbenzene, 135-98-8;decane, 124-18-5;l-methyl-3-isopropylbenzene, 535-77-3; 1,2,3-trimethylbenzene, 526-73-8; 1-methyl-4-isopropylbenzene, 99-87-6;indan, 496-11-7; 1-methyl-2-isopropylbenzene, 527-84-4; sec-butylcyclohexane,7058-01-7;butylcyclohexane, 1678-93-9;1,3-diethylbenzene,141-93-5;butylbenzene, 104-51-8; 1,2-diethylbenzene, 135-01-3;1,6diethylbenzene,10505-5; trans-decaline, 493-02-7; 2-methyldecane, 6975-98-0; cisdecaline, 493-01-6;undecane, 1120-21-4; tetraline, 119-64-2;1,4diisopropylbenzene, 100-18-5; 1,3,5-trimethyl-2-ethylbenzene, 3982-67-0;dodecane, 112-40-3;naphthalene, 91-20-3; tridecane, 629-50-5;2-methylnaphthalene,91-57-6; 1-methylnaphthalene, 90-12-0; tetradecane, 629-59-4; biphenyl, 92-52-4; diphenylmethane, 101-81-5;2,3-dimethylnaphthalene,581-40-8;pentadecane, 629-62-9; acenaphthene, 83-32-9; decylcyclopentane, 1795-21-7;nonylbenzene, 1081-77-2;hexadecane, 544-76-3; decylbenzene,104-72-3;octadecane,593-45-3;phenanthrene, 85-01-8; anthracene, 120-12-7;eicosane, 112-95-8.
1721
Literature Cited Gibbons, G.; Laughton, G. J. Chem. SOC., Faraday Trans. 1984,80, 1019.
Gomez-Nieto, M.; Thodos, G. AIChE J. 1977,23,904. Merlin, J. C.; Allemand, N.; Jose, J. 2nd Codata Symposium, Paris, 1985. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Rauzy, E. Thesis, Marseille, 1982. Riedel, L. Chem.-1ng.-Tech.1954, 26, 83. Rogalski, M. Thermochim. Acta 1987, submitted to publication. Soave, G . Chem. Eng. Sci. 1972,27, 1197. Stryjek, R.; Vera, J. H. Can. J. Chem. Eng. 1986, 64, 323. Thek, R. E.; Stiel, L. I. AIChE J. 1966, 12, 599. Willman, B.; Teja, A. S. Ind. Eng. Chem. Process Des. Deu. 1985,24, 1033.
Received for review October 15, 1987 Accepted May 3, 1988
Crystallization and Agglomeration Kinetics in the Batch Precipitation of Strontium Molybdate Otakar Sohnel,?John W.Mullin,* and A l a n G. Jones Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London WClE 7JE, England Kinetic processes were studied during the batch precipitation of SrMo04(mixing equimolar solutions of SrC12and Na2Mo04at 25 "C) over a range of supersaturations and under different stirring modes. Primary heterogeneous nucleation was predominant, with homogeneous nucleation becoming significant at S > 27, followhg which diffusion-controlled growth became dominant. Secondary nucleation was not detected under the conditions studied. Soon after the induction period, the small individual crystals agglomerated orthokinetically and t h e agglomerate size depended on both t h e intensity of stirring and the initial supersaturation. Toward t h e end of t h e precipitation, the agglomerate particle size distribution stabilized and was no longer affected by prolonged agitation. 1. Introduction
The crystallization of sparingly soluble substances, generally referred to as precipitation, has been studied on a scientific basis since the time of von Weimarn.' Despite much work, however, and the vast amount of information gathered as a result, there is little agreement either on the experimental data themselves or on their interpretation, even for the same precipitating system. The majority of reported precipitation data refer to batch experiments since these are the simplest to perform on a laboratory scale, and although the time-varying data can be difficult to interpret, careful analysis can yield much useful information. The precipitation process may proceed through different nucleation and growth mechanisms, depending on the prevailing conditions; e.g., nucleation can be primary (homogeneous or heterogeneous) or secondary and crystal growth can be controlled by diffusion, by surface nucleation, or by screw-dislocation mechanisms. This enormously complicates the analysis. Moreover, the effects of secondary processes (secondary nucleation, agglomeration, Ostwald ripening, polymorphic transformation, crystal habit modification, etc.) are often ignored in determining crystallization kinetics, and their omission can lead to substantial error. Some of the widely contradictory conclusions, inferred from experimental precipitation data, which have been put forward may be illustrated for the case of BaS04,one of Permanent address: Research Institute of Inorganic Chemistry, RevoluEni 86, 400 60 Usti nad Labem, Czechoslovakia.
the most widely studied precipitated substances. For example, nucleation has been determined as primary with an insignificant proportion of secondary: and exclusively secondary;' crystal growth has been reported as being controlled by diffusion? screw-dislocation,8~9and surface nucleationlo mechanisms; the kinetic order of the crystal growth process has been evaluated as 1,11 2,899J2 3,7J3J4and 4;" agglomeration during the early stages of precipitation has been found to be virtually absent2 and substantial;15J6nuclei were observed to grow into individual discrete crystals3J7 and agglomerate soon after their formation to form pseudosingle c r y ~ t a l s . ' ~ J ~ The object of the present work was to study the batch precipitation of a model substance under carefully controlled conditions and, by comparison of the experimental data with models of nucleation, growth, and agglomeration, to deduce which processes play a decisive role during this mode of operation. SrMo04 was chosen because it (i) forms as a result of a simple ionic reaction without the formation of complexes in the solution, (ii) precipitates as compact shaped crystals of reasonable size that do not change their morphology over the range of supersaturation studied, without the formation of precursors, different crystalline modifications, or an amorphous phase, (iii) readily agglomerates, and (iv) is easily removable from the experimental equipment when deposits are formed. 2. Theoretical Section 2.1. Nucleation a n d Growth. Methods for the identification of nucleation and/or growth mechanisms predominating during the early stages of precipitation have
0888-5885/88/ 2627- 1721$01.50/0 0 1988 American Chemical Society
1722 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988
Thus, a plot of log D ( t ) against t yields a straight line, the slope of which should increase with increasing agitation rate and concentration.
250-mL beaker (diameter 63 mm) agitated either by a magnetic stirrer or by a three-bladed paddle wheel, in which case wall baffles were provided. Fifty milliliters of SrC12 solution was rapidly added to a stirred 50 mL of Na2Mo04solution, and samples for analysis were withdrawn by pipet at specified intervals. After each run, the glassware and the stirrer were thoroughly rinsed with dilute HC1 and then with water to remove all traces of SrMo04. Solutions were prepared by dissolving the required amount of the Analar-grade chemicals in deionized water and filtered through Nucleopore filters (0.8-pm pore size). The blades of the paddle-wheel stirrer were 20 X 5 mm, inclined 45O to the vertical and located 120' apart. The stirrer was powered by a Heidolph RZR-2000 unit which enabled a smooth change of speed up to 750 f 1 rpm. Induction periods of precipitation were determined visually by recording the time elapsed between mixing the reactants and the instant when turbidity was recognized when viewed against a black background. Independent determination normally varied by less than 10%. The number of particles (comprisingboth discrete single crystals and agglomerates) formed was determined by counting under a light microscope. A known volume of suspension was withdrawn from the precipitating system and quickly diluted with a known volume of water to achieve a convenient solids concentration for counting. A sample of the dilute suspension was transferred to a Burker cell counting chamber, and the number of particles present in a specific volume were counted. The number of crystals present (an agglomerate consisting of say 20 visibly distinct crystals would contribute 20 to the total crystal count) and the number of particles present (an agglomerate of say 20 crystals and a separate single crystal would both be counted as 1 each) were determined. For each case, the volume of a sample containing a t least 700 individual crystals (both separate and in agglomerates) and, when applicable,300 agglomerates was determined and the result converted into the number of particles per unit volume of original precipitated suspension. The reproducibility of particle concentration determination was usually better than 25%. The size distribution of particles in the precipitating system was determined by a Malvern laser sizer (Model 3600E) fitted with a 63-, loo-, or 300-mm focal length lens as appropriate. A sample of suspension withdrawn from the precipitating system was quickly added to filtered deionized water in the mixed sample cell of the sizer and measured immediately. Each computed result was an average of 500 individual measurements performed by the sizer in rapid succession. The whole procedure took ca. 30 s. The reproducibility of the particle size distributions was excellent. For example, median sizes from independent 10-min runs were all grouped in the range 27.3-28.6 pm. Desupersaturation curves were measured at 25 OC with a Pt conductivity cell connected to an Alpha 800 conductivity meter. Fifty milliliters of SrC12 solution was rapidly added to 50 mL of Na2Mo04solution (equimolar) in a jacketed beaker mixed by a magnetic stirrer. The specific conductivity of the system was recorded as a function of time. The shape of the crystals was observed under a light microscope in a slightly diluted original suspension. A drop of the original suspension was dried on a sample holder, suitably coated, and observed with a SCAN electron microscope.
3. Experimental Section The precipitation of SrMoO, was carried out by mixing equimolar solutions of NazMo04and SrC1, at 25 "C in a
4. Results 4.1. Induction Periods. The induction periods, in seconds, measured in a magnetically stirred system for
been described previou~ly.~J~ Briefly, measured induction periods are plotted as a function of (i) (log S ) - 2 , (ii) (log S ) - I and (iii) log (S-l). A straight line obtained from any of these plots can give an indication of the predominant mechanism viz. either (i) homogeneous nucleation or primary nucleation and diffusion-controlled growth, (ii) polynuclear growth, or (iii) screw-dislocation growth. Further support for such mechanisms, however, must be sought from the evaluation of a relevant physical parameter (e.g., interfacial tension) from the slope of such functions. 2.2. Agglomeration. Precipitated primary crystals often agglomerate to form larger secondary particles. Two basic mechanisms have been proposed, viz. perikinetic and orthokinetic agglomeration, respectively, and these can be analyzed as follows. Perikinetic agglomeration applies to monodisperse submicroscopic particles subjected to collisions during Brownian motion and is described byI9 N ( t ) = No/(l + 8rDrNOt) (1) which can be rearranged to give 2 = N o / N ( t ) = 1 + Ct
(2)
where
C
= 8aDrNo
(3) and the diffusion coefficient D = k T f 6 r r ~ .Thus, a plot of NofN(t)against t yields a straight line. The rate of orthokinetic agglomeration induced by fluid-mechanical forces acting on particles is given by19 -N = 2tGD3(t)N2(t)/3
(4)
where G is the rate of fluid shear. Recognizing that V(t) = D 3 ( t ) N ( t ) k v
(5)
eq 4 can be modified to
-N
= 2eGV(t)N(t)/3kV
(6)
If during agglomeration the total volume of solid, V(t), present in a unit volume of suspension is reasonably constant, Le., in the absence of further nucleation or growth, then V(t) = VI = constant (7) and integration of eq 6 gives N o / N ( t )= exp(2eGVlt/3Kv)
(8)
where No and N ( t )are the number of particles present in a unit volume of the initial dispersed crystal suspension and after agglomeration has proceeded for time t , respectively. combination of eq 5 and 8 leads to a corresponding expression for particle size: log D ( t ) = A + Bt (9) where A = log (N0kv/V,)1~3= log Do
(10)
B = 9.65 x 10-2eGV,/kv
(11)
and
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1723
c '5'0
I
\
I z m
600
400
200
700
rpm
Figure 2. Induction period as a function of paddle-wheel stirrer speed for an initial concentration of 20 mol m-s of SrMoOl. I
I
0.4
0.6
0.8
1 13.5 1.0
80
(log SI-2
Figure 1. Induction period and number of individual crystals as functions of initial supersaturation for a magnetically stirred system.
I
I
0
200
lr
I
I
400
600
eight different initial concentrations of SrMoO,, covering the range 10-70 mol m-3, are represented by log tind = 4.711 - 2.203 log CO (12) with a correlation coefficient of 0.98. Alternatively, eq 12 can be expressed as tind= 5.142
x
104~0-2.203
(13)
Analysis of these data in a manner similar to that described previously4J8 ruled out "growth only" as the mechanism controlling the duration of the induction period and indicated that it is governed by primary nucleation followed by diffusion-controlled growth. The log tindversus (log S function plotted in Figure 1consists of two linear parts, with a change in slope at about co = 40 mol m-3 (S = 27). From the slope of the steeper part, at = 12.9, the interfacial tension of solid SrMo04 may be calculated.20 Thus, assuming v = 2 and V, = 5.45 X m3 mo1-1,21 ys = 0 . 0 2 9 ( ~ ~ a ( / V , ~=) ~75 / ~mJ m-2
(14)
and the intercept at (log S ) - 2 = 0 is equal to -6.7. Initial supersaturations were calculated as S = co/ceq, where co is the SrMoO, concentration in the actual precipitating system, i.e., with no solid formed, and cq is the equilibrium solubility of SrMoO, in a solution with a concentration of NaCl equal to 2c0. The value of ceqwas determined from the expression ceq
= Ksp1f2/y*
(15)
where the thermodynamic solubility product of SrMoO, was taken as K, = 0.257 mol2m+,22 and the mean activity coefficient of sofute yf was calculated from the Giintelberg equation log y* = -2.036 PJ2/(1 + PI2)
(16)
where the solution ionic strength I = 0.5Ccizt. The induction periods determined for systems agitated with a paddle-wheel stirrer depended on stirrer speed, as shown in Figure 2. The induction period decreases with increasing stirrer speed for any initial supersaturation within the studied range, although quantitative measurements were only feasible for co 5 30 mol m-3. 4.2. Number of Particles. In the present work, the number of individual crystals formed in both magnetically
Time (s)
Figure 3. Degree of agglomeration, 2,as a function of the time elapsed after mixing the reacting solutions together. Initial conmol m-3 of centrations: 20 (A),30 (V),40 (01,50 (X), and 70 (0) SrMo04.
stirred and paddle-wheel agitated systems is a function of the initial supersaturation and is independent of the stirrer speed and sampling time. A plot of log No versus (log S)-2 (see Figure 1) is approximated by two straight lines with the slopes changing at S = 27 (i.e., co = 40 mol m-3). An interfacial tension of the precipitated solid SrMoO, can again be calculated from the slope of the steeper part.20 Thus, assuming v = 2 and V, = 5.45X m3 ys = 0 . 0 2 5 4 ( ~ ~ a ~ / V = , ~55 ) ~mJ / ~ m-2
(17)
and the intercept at (log S ) - 2 = 0 is equal to -17.4. The average number of individual crystals forming an agglomerate,2, (Le., the degree of agglomeration) defined as 2 = number of individual crystals per unit volume/ number of separate particles per unit volume = No/N(t) (18) was determined for five different initial concentrations as a function of time that elapsed after the reacting solutions were mixed under a constant magnetic stirring speed (see Figure 3).
1724 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988
Z was also determined as a function of the paddle-wheel stirrer speed (200-750 rpm) for five different initial concentrations (15-70 mol m-3) of &Moo4. In this case, the sampling time, the same for each initial concentration used, but different for different concentrations (the sampling time is in any case longer than tind), was chosen so that Z for a stirrer speed R = 200 rpm was close to unity. The experimental degree of agglomeration, normalized for each concentration so that Z (200 rpm) = 1,can be represented by 2, = Z(R)/Z(200) = 0.8005 + 1.178 X 10-3R (19) with a correlation coefficient of 0.90. 4.3. Desupersaturation Curves. The specific conductivity of a precipitating system can be expressed as23 ~ ( t=) A(NaCl)c(NaCl) + 2Ae(SrMo04)c(SrMo04) (20) where c(NaC1) = 2c0 in the case of precipitating equimolar solutions and c(SrMo0,) is the actual concentration of dissolved SrMo04at time t. The equivalent conductivity of SrMo04can be expressed as a function of concentration by24 he(SrMo04) = 134.3 - 489.6(c(SrM00,))”~/(1 + (c(SrM00~))l’~) (21) Numerical values in eq 21 were evaluated using the limiting ionic conductivities Ao(SrZf)= 59.8 Q-’ cm2/equiv and AO(MOO~~-) = 74.5 0-l cm2/equiv, which are valid at 25 0C,25A(NaC1) for each concentration was taken from ) converted to the ref 26. Experimental values of ~ ( twere degree of reaction, defined by (22) d t ) = [co - c(SrMo04)l/[co- ceql by using eq 20-22 and ceq calculated from eq 15 and 16. For details see ref 27. From the calculated a versus t curves, it was established that, at the end of the induction period, less than 5% of the solute is precipitated (Le., (Y = 0.05), whereas when no further growth of agglomerates is detectable more than 75% of the solute has already been precipitated. 4.4. Size Distribution. The size distribution of particles present in the precipitating system is a function of the initial concentration, the elapsed time after mixing of reacting solutions, and intensity of stirring. A typical example of the time development of the size distribution, given as a relative percentage frequency curve, for a suspension agitated by a paddle wheel and a magnetic stirrer is shown in Figures 4 and 5 , respectively. Particle distributions determined by the Malvern sizer on a volume basis28(Figures 4a and 5a) were also converted to a number basis by using the software (version M3.0) supplied with the sizer (Figures 4b and 5b). Soon after the induction period ends, the volume-based size distribution of particles moves toward higher sizes, widening at first, and after entering the 10-100-pm range thereafter remains unchanged (see Figures 4a and 5a). The corresponding number-based size distribution (see Figures 4b and 5b) shows that the “initial” suspension, in which, due to a slight unavoidable delay in analysis, agglomeration had already begun, consists mostly of particles of uniform size. Agglomeration of particles results at first in a decrease in the height of the narrow first peak corresponding to single particles and the consequent appearance of a wider peak at larger sizes corresponding to agglomerates. Later on, when all single particles are agglomerated, the first peak disappears totally and a much wider distribution, often without any distinctive peak, results at larger sizes. The time development of the size distribution (volume based characterized by the median size) for four different
I
30
a
1
p,
3kins dtes mixing
fi
bins dttr mixing
--..
B. I 38 I
I38
6Br lftes mixing
tl1
45s dter mixing k I30
9
3Br dttr mixinq /
8 30
. 15s dter mixing
&-----
I
18
Pirtlclr S i l l (u),
sa-
im
I@@
b 3hins dter mixing
oi I
I38
hins lftes
mixing
c X
38 t
E
1
~ 3 0 .
. U ‘ X 38
7
> /
-
6Br alter mixing
45s dter nixing
)
I
j I*‘\+
385
alter mixing
39
.---.
15s dter mixing
Figure 4. Relative percentage frequency curve (initial concentration 50 mol m-3 of SrMoO,.mixed by paddle-wheel stirrer a t 500 rpm) as a function of the time elapsed after mixing the reacting solutions together: (a) volume-based and (b) number-based distributions. Table I. Median of the Final Particle Size Distribution on a Volume Basis (Micrometers) of SrMoOl Precipitated from a System of Initial Concentration (co) Mixed by a Paddle-Wheel Stirrer R CO 200 350 500 650 20.3 37.1 26.4 40 51.7 27.6 22.2 38.2 50 26.7 39.5 31.3 70 62.4 32.5 28.9 90 66.0 45.1
initial concentrations and a stirring speed of 650 rpm is shown in Figure 6. The higher the initial concentration of SrMo04, the faster is the agglomerate growth and the sooner the final size is reached. This behavior is common for all the other stirring rates studied (200, 350, and 500 rpm). The final agglomerate size, for selected initial concentrations and paddle-wheel stirrer speeds, is given in Table I. Increasing the initial concentration of the system at a constant stirring rate causes the final particle size to increase, whereas increasing the stirring rate, at a constant initial concentration, decreases the final size.
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1725
50i .' I
K 36
A
I
I
I
45s after tixing
30s after Mixing
.
K 30
2k after mixing 0 Y
36
X
8( 36
*
I
-.
15s alter nixing
. 16s after
1
Mixing
1 A
38
* 1
,
5s after Mixing
+ Partiole size
0.5
I I
8
Time, t ( m i n 1
45s afteP Mixing
t
39
1
6
*
6 i
I
L
Figure 7. Median of the volume distribution as a function of the time elapsed after mixing the reacting solutions together in a system 17 (V), stirred by a magnetic stirrer for initial concentration 15 (O), 20 ( O ) ,30 (0),40 (X), 50 (A),and 70 (m) mol m-3 of SrMoOl. Arrows indicate the induction period for the respective concentrations.
2
i 38
1
2
(un)
0. /
I 0
lW
b
.
136
im
10
1
;
7-
36s after
Mixing
20s after
mixing
Table 11. Numerical Values of Constants i n Equation 9 where D ( t ) Is the Median of the Volume-Based Distribution Expressed i n Micrometers and t Is Expressed i n Seconds. Magnetically Stirred System CO A 103~ remarks"
*
I
I
/
36. . I .
I
6/ 56.
-
i ,
6 - f
'
15 17
15s ittrr nixing
20 30 40 50 70
* I
Ss alter
Mixing
-0.234 -1.218 -0.197 -1.045 0.037 -0.827 -0.422 -0.495 0.370 0.136
0.77 5.46 1.04 6.80 1.18 8.09 12.18 12.90 19.92 47.00
g S
g 8
g 8
S S S S
g and s denote values calculated from the "gradual" and "steep" parts of the dependence, respectively. Figure 5. Relative percentage frequency distribution curve (initial concentration 70 mol m-3 of SrMo04,magnetic stirrer) as a function of the time elapsed after mixing the reacting solutions together: (a) volume-based and (b) number-based distribution. I
I
I
I
I1
Time, t (min)
Figure 6. Time development of the median of the volume-based size distribution. Mixing by paddle-wheel stirrer a t 650 rpm. Initial 50 (V),70 (e),and 90 (X) mol m-s of Srconcentrations: 40 (O),
MOO^.
If the median size of the volume distribution of particles formed in a magnetically stirred system with different initial concentrations is plotted as a function of time on semilogarithmic coordinates, a linear relationship is obtained (see Figure 7). For initial concentrations, co < 20 mol m-3, two lines of distinctly different slope may be drawn through the experimental points. For co 2 30 mol m-3, only the steep second linear region can be identified with any confidence. Values of constants A and B, obtained by fitting experimental data to eq 9 by the leastsquares method, are given in Table 11. Both the shape of the size distribution curve and the median size were highly reproducible. 4.5. Particle Shape. Individual crystals of SrMoO, formed during precipitation are small (-1 pm) and of compact shape when viewed under a light microscope. Within the range of concentration and stirring rate studied, no substantial changes of individual crystal shape are observed. Agglomerates consist of readily discernible individual crystals firmly stuck together: they cannot be redispersed by any of the standard techniques such as vigorous mixing or ultrasonic irradiation in the presence of sodium citrate, sodium pyrophosphate, or surface active agents. The application of substantial mechanical force only broke the large agglomerates into several smaller, but still agglomerated, pieces.
1726 Ind. Eng. Chem. Res., Vol. 27, No. 9,1988
a
U 5 um
b
U 5 um
C
I
primary nucleation followed by diffusion-controlled growth. The mechanism of primary nucleation is heterogeneous when S < 27, but homogeneous nucleation begins to be important when S > 27, as indicated by the change of slope in both the log t b d and log N versus (log S)-2plots. These functions, however, are curved over the region of supersaturation where the predominant nucleation mechanism change~,W*~~ and therefore, the slope of the tangent drawn to the curve in that region gradually increases with increasing supersaturation,i.e., with an increasing amount of homogeneous nucleation. Values of the interfacial tension, y, calculated from the slopes of the lines in Figure 1, using eq 14 and 17 (i.e., 75 and 55 mJ m-2, respectively), are lower than that expected (100 mJ m-2),20while the absolute values of the intercept with the ordinate (i.e., -6.7 and 17.4) are smaller than those predicted (-10 and 28) for the log t b d and log N versus (log s)-2plots, respectively.2 The early stages of SrMo04precipitation appear to be governed by the "classicalnmechanism of attachment of ions to nuclei rather than by the agglomeration of either molecular aggregates present in the solution or of already formed nuclei.20 Microscopic observation of the particles and inspection of the size distribution curves reveals that at the end of the induction period crystals are mostly present as individual entities of narrow size distribution. Thus, agglomeration is negligible during the induction period during which primary nucleation is confined to a rather limited interval. Soon after the induction period ends, the size distribution of particles shifts toward higher sizes (Figure 4 and 5). Agglomeration of particles rather than primary crystal growth is responsible for most of that shift, as can be seen by comparing Figure 5 with the steepest curve in Figure 3, since they both refer to the same conditions. This is also confiimed by a mass balance. For the conditions of Figure 5,3 X 1014individual crystals are present in 1 m3 of suspension, and their number barely changes as precipitation proceeds. Assuming all solute available for further growth is deposited on those crystals, they would reach a-maximum size, r,,, given approximately by
. lo um I
r,,
= (~V,AC/~?~N~)~/~ = [(5.45 X 10-5)(70- 1.9)/4~(3X 1014)]1/3 =
1.4 X Figure 8. Particles of precipitated &Mool observed by SEM (a) a t the end of the induction period, (b) after 1min, and (c) after 30 min.
An indication of the development of particle form can be seen in the selected photomicrographs (taken from different runs) in Figure 8. At the end of the induction period, the crystals appear mainly as discrete spheres (Figure 8a). Within about 1 min, however, the crystals are elongated (Figure 8b). After about 30 min, virtually all the crystals are agglomerated (Figure 8c). 5. Discussion
The reproducibility of the results obtained on the spontaneous precipitation of SrMo04was very good; independent runs carried out under identical precipitating conditions always gave similar results within the limits of experimental error, unlike the results for BaS04 compiled in section 2 and also of other substances reported in the literat~re.~~ Analysis of the early stages of the precipitation of SrMoo4, i.e., during the induction period, over the range of supersaturation 10.5-37.6, indicates that it is governed by
lo4
m
i.e., a diameter of 2.8 pm, which is roughly 1 order of magnitude smaller than the actual final size determined by the Malvern sizer. This is accounted for by agglomeration occurring subsequent to the induction period. The onset of agglomeration is responsible for the change in slope of the time dependence of the particle median size (Figure 7). Agglomeration proceeds at a rapid rate initially, slows down in the later stages, and finally ceases. The rate of agglomeration and the final size of agglomerates is dependent both on the initial concentration and the rate of stirring. The degree of agglomeration,2, as a function of time, shown in Figure 3, indicates that 2 is an exponential rather than a linear function of time. Furthermore, estimation of the constant C in eq 2 using r = lo4 m and p = 0.9 X Pa s gives a value (6.3 X lo4 s-l) that is (depending on the initial concentration) 2-4 orders of magnitude smaller than the slope of a tangent of the steepest part of the experimental function (i.e., 0.01-2.2 s-l). Thus, the data imply qualitatively that crystals of SrMo04 in a batch precipitation agglomerate by via an orthokinetic rather than a perikinetic aggregation mechanism after the induction period.
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1727 The quantitative agreement between experiment and theory of orthokinetic agglomeration, however, is less satisfactory. Theoretically, values of B (eq 11) should increase by a factor of (70/15)'13 = 1.7, due to an increase in VIas the initial concentration increases from 15 to 70 mol m-3. The experimental B values (Table I) increases by a factor of 8.6, Le., 5 times more than expected, which could be caused by the attachment efficiency coefficient, t, also increasing with concentration (or supersaturation). Desupersaturation continues during the agglomeration period, and the aggregated particles become intergrown and develop into a stable agglomerate that cannot be redispersed into individual crystals. For this reason, the final size distribution cannot be modified by applying a mechanical force of different intensity from that used during precipitation. Secondary nucleation, Le., the formation of new particles in a precipitating system as a result of interactions either between the crystals already formed and the supersaturated solution, other crystals, or parts of the crystallizer itself, has been claimed to occur in both spontaneous' and ~ e e d e dprecipitations. ~i~~ In all the present experimental work, however, no new particles were formed after the induction period terminated, as was established both from the particle size distribution (the frequency of occurrence of small particles always decreased with time) and from the counting of individual particles present. In this work, therefore, no significant secondary nucleation occurred after the induction period. Increasing the intensity of stirring brings about a decrease in the induction period for all the initial supersaturations studied, in agreement with recent results of studies on other sparingly soluble substances.l* Neither primary nucleation nor diffusion-controlled growth of small particles can be substantially influenced by stirring. Enhancement of the supersaturation buildup after mixing the reacting solutions would result in an increase in the number of particles formed, but this did not happen. Therefore, an increase in the rate of secondary processes, mainly agglomeration (eq 9 and ll),is likely to be responsible for the decrease of the induction period. 6. Conclusions The spontaneous precipitation of SrMoOI is fairly reproducible when the reaction conditions of independent runs are repeated consistently. The induction period is governed by primary nucleation completed within a rather confined interval, followed by diffusion-controlled growth of particles formed. Neither secondary nucleation nor agglomeration is significant within its duration. Soon after the induction period, however, orthokinetic crystal agglomeration occurs, its rate being enhanced by increasing both the initial concentration and the mixing intensity. Primary crystal growth continues within the agglomerates. Agglomerates in the system reach a final size (proportional directly to the initial concentration and indirectly to the mixing intensity) that is not further influenced by prolonged mixing either of the same or different intensity. Finally, when a precipitating system that easily agglomerates is studied, results that are determined after a certain "aging" period are not suitable for the determination or comparison of nucleation and growth kinetics since the size distribution data are substantially modified by the agglomeration process. Conversely, crystallization kinetic data alone are not sufficient for estimating the characteristics of the final product formed by a precipitation process; agglomeration effects must also be taken into consideration. This work therefore emphasizes the need to account for agglomeration in addition to nucleation
and growth in analyzing precipitation processes. Acknowledgment
0. Sohnel is indebted to the Science and Engineering Research Council of Great Britain for financial support. The assistance of J. D. Murphy during the experimental work is gratefully acknowledged. Nomenclature A , B = constants co, ceq = initial and equilibrium molar concentrations Ac = supersaturation (=co - ceq) C = constant D = diffusion coefficient D ( t ) = characteristic dimension of solid particle at time t after mixing reactant solutions together D50 = median of a size distribution G = shear rate I = ionic strength of a solution k = Boltzmann constant k, = volume shape factor K = constant (=0.66tG/kV) for given conditions KBp= thermodynamic solubility product No = number of individual crystals in a unit volume N ( t ) = number of separate particles in a unit volume at time t after mixing reactant solutions together r = particle radius R = stirrer speed, rev/min S = supersaturation ratio (=co/ceq) t = time tind = induction period V , = molar volume of solid V ( t )volume of solid present in a unit volume at time t after mixing reactant solutions together zi = ionic charges 2 = degree of agglomeration: average number of crystals forming an agglomerate = N o / N ( t ) 2, = normalized value of 2 [ = Z ( R ) / 2 ( 2 0 0rpm)] Greek Symbols (YN,at= slopes of
log No and log t b d versus (log S)-2functions, respectively a ( t ) = degree of reaction = mean activity coefficient ys = interfacial tension of solid E = attachment efficiency coefficient ~ ( t=) specific conductivity A, A, = molar and equivalent conductivities, respectively 1' = limiting ionic conductivity v = number of ions in a molecular unit = viscosity Registry No. SrMoO,, 13470-04-7; SrCl,, 10476-85-4; NazMOO,, 7631-95-0.
Literature Cited (1) von Weimarn, P. P. Chem. Reu. 1926,2, 217. (2) Nielsen, A. E. Krist.Tech. 1969, 4 , 17. (3) Mealor, D.; Townshend, A. Talanta 1966, 13, 1069. (4) Sohnel, 0.; Mullin, J. W. J. Colloid Interface Sci. 1988,123,43. (5) Garten, V. A.; Head, R. B. J. Chem. Soc., Faraday Trans.1 1973, 69, 514. (6) Matz, G.; Kaufhold, G. CEPAS 78; Nielsen, A. E., Ed.; Copenhagen University: Copenhagen, 1978; p 13. (7) Gunn, D. J.; Murthy, M. S. Chem. Eng. Sci. 1972, 27, 1293. (8) Nielsen, A. E. Pure Appl. Chem. 1981,53, 2025. (9) Liu, S.-T.; Nancollas, G. H.; Gasiecki, E. A. J. Crystal Growth 1976, 33, 11. (10)Van Rosmalen, G. M.; van der Leeden, M. C.; Gouman, J. Krist. Tech. 1980, 15, 1213. (11) Nielsen, A. E. Acta Chem. Scand. 1958, 12, 951. (12) Rizkalla, E. N. J. Chem. SOC., Faraday Trans. 1 1983,79,1857. (13) Klein, D. H.; Fontal, B. Talanta 1964, 11, 1231. (14) Doremus, R. H. J. Phys. Chem. 1958, 62, 1018.
Ind. Eng. Chem. Res. 1988,27, 1728-1732
1728
(15) Melikhov, I. V.; Kelebeev, A. S. Kristallographia 1979,24,410. (16) Melikhov, I. V.; Kelebeev, A. S.; BaEiE, S. J . Colloid Interface Sci. 1986, 112, 54. (17) Serebryakov, Ju. A.; Khamskii, E. V. Kristallographia 1970,15, 1226. (18) Sohnel, 0.;Mullin, J. W. Cryst. Res. Technol. 1987, 22, 555. (19) von Smoluchowski, M. Z. Phys. Chem. 1917,92, 129. (20) Nielsen, A. E.; Sohnel, 0. J. Crystal Growth 1971, 11, 233. (21) Handbook of Chemistry and Physics; Weast, R. C., Ed.; CRC: Cleveland, OH, 1974-1975; p 8-142. (22) Linke, W. F. Solubilities of Inorganic and Metal-organic Compounds; Van Nostrand: New York, 1958. (23) Falkenhagen, H. Theor. Elektrolyte 1971, 289. (24) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959; p 145.
(25) Landolt-Bornstein Zahlenwerte und Funktionen, II. band, 7 . Teil; Springer-Verlag: Berlin, 1960; p 259. (26) International Critical Tables; McGraw-Hill: New York, 1929; Vol. 6, p 233. (27) Sohnel, 0.; HandGiovl, M. Cryst. Res. Technol. 1984,19,477. (28) Brown, D. J.; Felton, P. G. Chem. Eng. Res. Des. 1985,63, 125. (29) Nancollas, G. H.; Purdie, N. Q.Reu. 1969, 18, 1. (30) Sohnel, 0.; Mullin, J. W. J. Crystal Growth 1978, 44, 377. (31) Fiiredi-Milhofer, H.; MarkoviE, M.; Komunjer, Lj.; PugariE, B.; BabiE-IvanEiE, V. Croat. Chem. Acta 1977, 50, 139. (32) Tomazic, B.; Mohanty, R.; Tadros, M.; Estrin, J. J . Cryst. Growth 1986, 75, 339. Receiued for review November 30, 1987 Accepted May 9, 1988
Reduction Theorem for Phase Equilibrium Problems Eric M. Hendriks K o n i n k l i j k e l S h e l l Laboratorium, A m s t e r d a m ( S h e l l Research B.V.), P.O.Box 3003, 1003 A A A m s t e r d a m , T h e N e t h e r l a n d s
If the excess Gibbs free-energy function for n-component mixtures depends on composition only through a limited number, K , of linear functions (scalar products), then the set of two-phase equilibrium equations and the equations of the stability test can be reduced to a set of only K + 1 equations; the Newton-Raphson correction equations can be reduced from a system of n linear equations to one of size, K 1,and the spinodal curve can be evaluated from the criterion of positive semidefiniteness for a ( K 1)-dimensional quadratic form. Applications include examples ranging from mixtures of hydrocarbons with L non-hydrocarbons ( K = 2L 2 ) to polymer mixtures. In a number of special cases, the simplications have been exploited by other authors, among other things t o save computer time and storage. The present work shows the mathematical structure behind these examples and generalizes these to a well-defined class of models.
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1. Introduction The aim of this paper is t o present a general mathematical theorem in phase equilibrium equations. It shows, how for a well-defined class of thermodynamic models, the dimensionality of various phase equilibrium problems can be reduced. For this reason, it has been called the reduction theorem. For a number of special cases, the possibility of a reduction has been noted by other authors, who have used it among other things to save computer time and storage for multicomponent mixtures and to derive a simple criterion for the limit of stability, the spinodal curve, in polymer physics. This paper shows that the simplifications are due to the same underlying mathematical structure. So it unifies results in different fields. The conditions of the theorem are easy to verify. They are often fulfilled, at least approximately. The theorem may then serve as a basis for a perturbation expansion. The reduction in storage and computer time, such as has been achieved by other authors, is due to the algebraic structure leading to the theorem and is due to the large number of components and small number of parameters. If these conditions are fulfilled for a specific problem/ model, then we expect that computer savings are possible. After introduction of some basic concepts used in the evaluation of phase equilibria in section 2, we shall formulate the theorem in section 3, prove it in section 4, and give a number of examples in section 5. Finally, some conclusions will be given in the last section. 2. Basic Concepts
In phase equilibrium problems, the thermodynamic model is usually stated by specifying the molar Gibbs free-energy function of a homogeneous phase, directly or 0888-5885/88/2627-1728$01.50/0
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indirectly (through an equation of state, for example) as a function of composition 2,pressure p , and temperature T. Quite generally, it is of the form n
G = RT(Cxi log xi i=l
+ g($,p,T))
(1)
for a mixture of n components. When an equation of state is used, G is determined only up to a temperature-dependent but otherwise constant term, which is of no importance here. The free energy (GbJ of a multiphase system is simply the sum of the free energies of the various phases. The equilibrium situation at fixed temperature and pressure corresponds to the global minimum of G,,, in thermodynamic state space. Metastable states correspond to local minima. A necessary condition for equilibrium is that the chemical potentials of the various species be equal in all phases. For a two-phase equilibrium pi' = pi"
(i = 1, ..., n )
In addition, the conditions of material balance have to be satisfied: r$
+ n?I = niF
(i =
1,
..., n)
(3)
in which ni7 (y = I, 11, and F) are mole numbers. The superscript F denotes feed, i.e., the mixture as a whole. These equations are easily generalized to situations with more phases. After the solution has been found, it still has to be tested, whether or not it corresponds to a global minimum of Gb, (stability test). An important symmetric matrix, related to the above phase equilibrium equations, is AijT
= (api/hj)T
0 1988 American Chemical Society
(7 = I, 11)
(4)
