MELIKHOV; KELEBEEV & BACIC 1988

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Electron Microscopic Study of Nucleation and Growth of Highly Dispersed Solid Phase IGOR V. MELIKHOV, ALEXANDER S. KELEBEEV, AND SLAVICA BACIC *'1 Faculty of Radiochemistry, Chemical Department, Moscow State University, Leninskii Gory, Moscow 11734, USSR; and *Radiation Protection Department, Boris Kidri( Institute of Nuclear Sciences--Vin(a, P.O. Box 522, 11001 Belgrade, Yugoslavia Received February 5, 1985; accepted September 27, 1985 Crystallization of barium carbonate and barium sulfate at fast mixing of reagents solutions was investigated. The method of high resolution electron microscopy showed that during crystallization ultramicrocrystals of 1- to 5-rim size are formed which have an oriented aggregation and are further transformed into polyhedrons not different in their outer appearance from the single crystals. The change of ultramicrocrystals size distribution is described by the Fokker-Planck equation. Ultramicrocrystal nucleation and growth at large supersaturation interval obey the kinetic reaction equations of the third and the second order, respectively. It is possible to consider nucleation as ion triplet associate formation with further rearrangement requiring activation energy of approximately 70 kJ mole -l. © 1986Academic Press,Inc.

INTRODUCTION

tallization from high supersaturated aqueous solutions. An extensive literature has been devoted to the crystallization of these salts but investigations were mainly focused on processes occurring in low supersaturated solutions (3-10). In high supersaturated solutions specific processes were observed (11), i.e., acceleration of nucleation and aggregation of particles with further transformation of aggregates into ordered polyhedrons not different in their outer appearance (shape) from single crystals. If we are to accept, as was done so far, that every such polyhedron was formed from only one crystallization center, what we obtain then is an incorrect picture of the whole process. This has motivated us to carry out the investigations presented in this paper.

The study of formation of highly dispersed solid phase is expected to yield information on the least investigated stages of crystallization, i.e., on nucleation and growth of crystals immediately after their formation. At formation of highly dispersed solid phase, nucleation and growth, as a rule, become complicated by fast crystal aggregation (1, 2). The study of these processes necessitates clarification of evolution of individual crystal size distribution and aggregate size distribution function. Methods of light scattering, ultrafiltration, ultracentrifugation, pulse electrometry, and electron microscopy were used. The last method plays a special role since it enables visual observation of the solid phase and the study of both individual crystals and aggregates morphology. In this study electron microscopy of high resolution was applied for investigation of barium carbonate and barium sulfate crys-

THEORY

Crystallization was induced by fast mixing of barium chloride and potassium sulfate or sodium carbonate solutions. Sample taken from the suspension was fast dried and ob-

1 TO whom all correspondence should be addressed. 54 0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All fightsof reproductionin any formreserved.

Journalof Colloidand InterfaceScience.Vol. 112,No. 1, July 1986

NUCLEATION AND GROWTH

served under transmission electron microscope. The advantage of this simple procedure was that it allowed maximum shortening of the suspension ageing time and its drawback was the presence of particle backgrounds (NaC1, KC1, etc.) complicating identification of the basic phase. A typical device used for fast mixing (10) presented in Fig. 1 consisted of a mixer (1), of a tube (2), and of a receiver (3). At crystallization reagents are injected into the mixer and flow through the tube into the receiver. The state of suspension at any given point of the system in continuum approximation may be described by the well-known balance equations Of= d i v [ D c g r a d f - v J ] Ot 0

Oc Ot

-- = div[DLgrad c - VLC]

- 3p

G~r2fdr,

[2]

~0 °°

~

5

3 ~-

7

55

at boundary conditions

f(r, O, t) = f(oo, t, x) = O, c(t, O) = c °, c(O, x) = O, J = [ G f - Gc ~r ]

[3] r=O ~

where f = ON~Or is the local crystal size (r) distribution function in the point x at the time t, N is the number of crystals per unit volume of suspension, D is the diffusion coefficient, vc is the velocity of crystal movement, G is the linear crystal growth rate, Gc is the crystal growth fluctuation coefficient, P is the rate of crystal binding into aggregates, c is the molar concentration of crystallizing matter in point x at time t, vL is the liquid flow rate, p is the crystal density, ~ is the shape factor, co is the molar concentration of crystallizing matter at the entrance into the mixer (x = 0), J is the nucleation rate; subscripts c relate to crystals and L related to liquid. When functions f(r, x, t) and P(r, x, t) are determined by selection and electron microscopic analysis of suspension sample at different VLand c °, it is possible by Eqs. [1]-[3] to establish functions G(r, c), Go(r, c), and J(c), i.e., to obtain a complete picture of crystallization. In this work the velocity VL was maintained at sufficient magnitude in order to prevent aggregation prior to arrival of suspension into the receiver, the receiver already containing an antiaggregation additive. In this way the necessity of determination of the P(r, x, t) function was avoided and nucleation and growth could be assessed on the basis of the crystal size distribution function in the receiver. MATERIALS AND METHODS

FIG. 1. Experimental arrangement for crystallization. 1--rnixer, 2--tube, 3--receiver, 4--reagent vessels, 5 - valve, 6--rnagnetic stirrer, 7--flask, 8--manometer. The tube cross section is 0.13 crn z. The length of the mixer and the tube xk = 12 cm. The volume of the mixer is 0.13 cnl 3.

The device shown in Fig. 1 was used for the experiment. Over a period of 1.3 s equimolar solutions of BaCl2 and KzSOa or Na/CO3 were injected into the mixer at equal and constant rates. Solutions were previously thermostated at 293 K. The range of initial solutions concentration was (0.1-0.4) mole dm -3 in the case of BaSO4 and (0.005-0.1) in the case of BaCO3. At BaCO3 crystallization solutions of Journal of Colloid and Interface Science. Vol.

112, No. 1, July 1986

56

M E L I K H O V , KELEBEEV, A N D BACIC

BaC12 and Na2CO3 have contained NaCI (0.1 mole dm -3) for purpose of alleviating ionic strength changes of the environment. The consumption of reagents was 38.5 cm 3 s-1, the mixture flow rate in tube was V0L= (3.0 + 0.3) m s-1. The mixture remained in the mixer for ~ 5 X 10 - 3 S and in the tube ,~5 × 10-2 s. The mixture was flowing into the receiver filled with water or ethylene diamine tetraacetic acid sodium solution (Na-EDTA) as antiaggregation additive. In this experiment Na-EDTA retarded solid phase particles aggregation without causing significant solid phase dissolution. The quantity of Na-EDTA in the receiver did not exceed 5 mole% of the Ba2÷ quantity in the system. At arrival into the receiver the mixture was diluted 2 times. When the reagents injection was completed (3060 s) the suspension sample was taken from the receiver and was diluted by water or ethyl alcohol. A drop of diluted suspension was placed on the formvar plate of the electron microscope and was dried under an infrared lamp (310 K), The time interval from the taking of sample up to complete drying of the drop was about 60 s. The dry residue of the drop was observed under an JEM-100 B electron microscope by transmission method with microdiffraction control of the particle structure and local X-ray emission composition control by means of an Kevex-5100 analyzer. During observation of particles in the electron microscope beam there occurred evaporation and melting of dry residue particles. Larger particles were melting 1-3 min after introduction of sample in the electron beam and the smaller ones after 2-4 min. In order to avoid destruction of particles microscope was focused first on one particles and then shifted to the other ones which during focusing were outside of the action of electron beam. These later ones were photographed at different degrees of magnification. Change of magnification in the range of 104-10 s has no influence on the size and shape of particle image. Under these conditions the B a S O 4 o r B a C O 3 particles were detected with certainty on the formvar plate background when their size was larger Journal of Colloid and Interface Science, Vol. 112,No. 1, July 1986

than 0.6 nm. Particles were clearly distinct from larger NaC1 or KC1 crystals formed during sample drying. The Na-EDTA impurities were not an obstruction to observation because they were adsorbed by the observation plate without formation of discernable particles. This was proved on model solutions containing NaC1, KC1, and Na-EDTA but free of BaCO3. Particle images on electron micrographs were observed under an optical microscope at 5-10 magnification. Parameters of projection of at least 1500 particles in every test were measured as well as angles between projections of neighboring particles in the manner described in Ref. (11). The main attention was focused on individual crystals and small aggregates in the interior of which it was possible to distinguish separate particles. Projections were approximated by rectangles. The error in determination of size of projection of separated particles at repeated experiments was 0.2 nm. Needlelike particles formed during crystallization were in different ways oriented toward the observation plate surface. This in no way influences measurement of width of needles but makes impossible the measurement of their length. In cases when needles are bound into aggregates, however, such configuration makes it possible to determine what orientation the needles are having. It was shown in an earlier paper (11) that the angles between needles in such BaSO4 aggregates are equal to 2~r/3 or 47r/3. Angles between needle projections in aggregates are close to these angles but only when aggregates are oriented parallel to the surface of the observation plate. In that case the projection length is identical with the length of needle. Aggregate projections with angles of 27r/3 and 47r/3 were selected for measurement of length of needles. On the basis of particle projection data their volume was calculated. Needles were considered to be rectangular parallelopipeds with square cross sections and plates the inflated cylinders. Height of cylinder was determined on the basis

NUCLEATION AND GROWTH of particles which in aggregates are oriented perpendicular to the observation plate surface. The precipitate taken out of the receiver was identified by X-ray diffraction. Five minutes after mixing of reagent, the receiver content was filtrated through the membrane filter of 5-nm pore size. Filter precipitate was washed 3 times by water and ethanol, dried at 293573 K to its constant weight and analyzed on a Dron-2 (CuK~) diffractometer. The specific surface area of the precipitate was determined by the BET method of krypton adsorption. The crystal growth rate of BaSO4 was also measured. BaS04 precipitate obtained without Na-EDTA in the receiver was stabilized for 20 h in the bidistilled water at 370 K. This precipitate was rinsed and dried as described above and was then added to a supersaturated BaSO4 solution. This supersaturated solution was prepared in the following manner: solutions of K2SO4 and BaC12 (2 X 10-4 mole dm -3) were added in drops into the bidistilled water (at 298 K) during constant stirring (Reynolds Re = 5 X 103) until the BaSO4 concentration was increased to 4 X 10 -5 mole dm -3. The obtained solution was homogeneous and did not contain solids. After introduction of BaSO4 precipitate into this solution, the system was thermostated (at 298 K) and continuously stirred while BaSO4 concentration in the solution was measured by conductometric method (8). Previously measured calibration dependency of electric conductivity of BaSO4 concentration in the KC1 solution was used. A separate experiment proved that the solid phase quantity lower than 0.1 g does not at all influence the conductometric measurements. The growth rate of BaSO4 particles was calculated according to V dc - - . - G = Spy d r '

[4]

where Vis the solution volume, Sis the specific surface area, p = 4.5 kg dm -3 solid phase density, and y = 0.03 g quantity of solid phase. RESULTS During mixing of reagents the temperature of the system practically does not change at

57

all, i.e., crystallization takes place isothermally. At BaCO3 crystallization, a colloid solution of crystallizing matter is formed in the mixer and in the tube. In the receiver this colloid solution became turbid after an induction period. The length of induction period ranges from 1 to 30 s varying inversely as the initial solution concentration increased from 5 X 10-3 to 10-~ mole dm -3. When BaSO4 was used the induction period was too small to be measured (lower than 5 X 10-3 s). In samples of colloid solution and freshly formed suspension, there were a number of individual and aggregate needlelike and platelike ultramicroparticles having a crystalline structure with parameters identical to the ASTM-Card data for BaSO4 and BaCO3. When Na-EDTA is present in the receiver, at short time intervals of sampling, individual ultramicroparticles and their small orderly aggregates and larger disorderly aggregates were observed. Without the Na-EDTA and at longer sampling (suspension ageing times) intervals the presence of individual ultramicroparticles and their aggregates was not observed. In this case the sample contained only the large orderly aggregates covered with crystalline crust no different in their outer appearance from the single crystals. Ultramicroparticles of the BaCO3 are platelike in shape (Fig. 2). The normalized function F(r) = (1/Nc)(ON/Or) of BaCO3 ultramicrocrystal size (r) distribution is shown in Fig. 3. The size of ultramicroerystal r is equal to the circle diameter having the surface equal to the surface of projection. Arc is the number of measured crystal projections. Thickness of plate was changing proportional to the change of its size and was 8.9 _+0.3 times smaller than r. The shape and size of aggregate were the same as already described in Ref. (12). Ultramicroparticles of the BaSO4 are needlelike in shape (Fig. 4a). The normalized function of BaSO4 ultramicrocrystals width distribution is presented in Fig. 5a and the length distribution in Fig. 5b. The functions F(r) are not dependent on the Na-EDTA concentration in the receiver. The width and Journal of Colloid and Interface Science, Vol. 112, No. 1, July 1986

58

MELIKHOV, KELEBEEV, AND BACI(~

FIG. 2. Electronmicrographs of BaCO3particlesat differentstages of solid phase formation: (a) ultramicrocrystalsand their disorderedaggregates;(b) disorderedaggregatesand aggregateswith formed faces.

length of needle were changing proportionally one to another and the width was 4 +_0.2 times smaller than the length. Small, orderly needle aggregates were mostly ring-shaped (Fig. 4a) and the larger orderly aggregates (pseudosingle crystals) were having the shape of hexagonal regular prisms (Fig. 4d). Some pseudosingle crystals at earlier stages of growth are also ringshaped (Fig. 4c). Stabilized precipitate consisted of mierocrystals with well-formed faces (Fig. 4e). In the volume of pseudosingle crystals during short sampling intervals oriented ultramicroparticles were observed (Fig. 4b) and later on the round inclusions were observed (Fig. 4d). The distribution function of ringlike aggregates on projection surface of their interior faces and distribution function of inclusions within pseudosingle crystals on Journal of Colloid and Interface Science, Vol. 112, No. 1, July 1986

projection surface of their interior faces are shown in Fig. 6. In the volume ofpseudosingle crystals there was 40 + 3 mole% of water which was not eliminated during drying (293-310 K). Microcrystals contained 18 +_ 2 mole% of water. The specific surface area of precipitate which consisted of pseudosingle crystals was 25 +_ 2 m 2 g-l, irrespective of the drying temperature. Stabilized precipitate had the specific surface area of 10.2 +_ 0.4 m 2 g-1. The average growth rate of microcrystals after they are introduced into the supersaturated solution of BaSO4 (5 × 10-5 mole d m -3) can be described by G ~/q

]

- 1 ,

[5]

NUCLEATION

AND GROWTH

59

be determined on the basis of their electron micrograph image irrespective of potential possibilities of change of properties of solid phase during drying and under the effects of the electron beam. This is confirmed by the following facts:

2.0-

E co I0

(1) Drying in the investigated temperature interval does not influence the value of specific surface area of solid, the quantity of occluded water in it, and the phase composition, which means that the entity of particle in the process of drying has been preserved. (2) The size and shape ofultramicrocrystal image does not depend on the electron degree r (nm) of magnification and, therefore, neither on the FIG. 3. Normalized functions of BaCO3 ultramicrocrystalsizedistribution. The points on the curverepresent strength of electron beam which indicates the experimentaldata for c°: ×--0.0025, 0---0.005, 11----0.025, sufficient stability of the particle in the beam. (3) Between the image of the ultramicroand A--0.05 mole dm-3. The continuousline denotesthe results calculated from Eqs. [1]-[3], [5], [8], and [10] for crystal and the image of the pseudosingle cryskin= 1.4× 10l~m-3s- t , m = 3 , / q = 1.1 × 10-t4ms -t, tal of BaSO4 there is a relation which proves a~ = 7.5 X 10-5 mole dm-3, b = 1 nm, a = 0.17. that in the solution ultramicrocrystals had the same habitus as in the micrographs. LI_

1.0

where the crystal growth rate coefficient k~ = (9.1 _+ 0.7) x 10 -14 m s -1 and a = (1.30 +__0.07 X 10-5 mole dm -3 are empirical coefficients, 3'_+ = 2.0461/I(1 + ~/)-1 is the mean ionic activity coefficient calculated on the basis of the Giintelberg equation (9, 13), and I is the ionic strength of solution. Equation [5] describes experimental data (Fig. 7) with a high correlation coefficient of 0.98. Calculated values are in agreement with data from Ref. (7) but disagree with data from Refs. (6, 8, 9). In (6-9), however, crystals smaller than 1 tzm were not investigated. The crystal growth rate G did not depend on the size of growing particles in the interval r = 20-100 nm. DISCUSSION

At BaSO4 and BaCO3 crystallization from high supersaturated solutions, a number of ultramicrocrystals are formed which aggregate forming initially disorderly and small orderly aggregates and later on pseudosingle crystals and their conglomerates, as shown in Fig. 8. The size and shape of ultramicrocrystals may

Individual ultramicrocrystals and the ringshaped aggregates are visible on the electron micrographs only during short aging times. At medium aging times only pseudosingle crystals and their conglomerates were observed. This difference indicates the binding of ultramicrocrystals into pseudosingle crystals during aging according to the scheme shown in Fig. 8. During such binding the interior faces of ringshaped aggregates form round inclusions in pseudosingle crystals. The size of inclusions is determined by the length of ultramicrocrystals which form the ring-shaped aggregate. Inclusions did not change during drying (at 310 K) which is proved by the unchanged quantity of occluded water. Therefore, the size of inclusions on electron micrographs was the same as in pseudosingle crystals in the suspension. Furthermore, the size of inclusions is equal to the size of faces on the electron micrographs of ring-shaped aggregates (Fig. 6). Thus, the faces as well as ultramicrocrystals on electron micrographs had the same size as in the suspension. In this way, on the basis of data on the size and shape we can assess nucleation Journal of Colloid and Interface Science, Vol. 112, No. 1, July 1986

60

MELIKHOV, KELEBEEV, AND BACIC

FIG. 4. Electron micrographs of BaSO4particles at different stages of solid phase formation. (a) needlelike ultramicrocrystals and their small ring like aggregates. (b) aggregateswithout formed faces; (c) pseudosingle crystals at earlier stages of growth; (d) pseudosingle crystals (larger orderly aggregates)in the shape of regular hexagonal prisms with round inclusions visible within them; (e) microcrystals after 2 days aged solid phase (stabilized precipitate).

and growth o f particles in supersaturated solution. Let us n o w consider nucleation and growth o f BaSO4 and BaCO3 separately. Crystallization ofBaS04. Estimating on the Journal of Colloid and Interface Science, Vol. 112, No. t, July 1986

basis o f the small induction period, BaSO4 is completely crystallized in the mixer and in the tube. Therefore, instead o f function F(r) (Fig. 5) it is m o r e appropriate to use the function f (r, Xk, t) by m e a n s o f

61

NUCLEATION AND GROWTH

'1.2~ o

1.0 0.8

e

5 E

°?o o6

'TE 0.6 o

o

04.

~0.4

0,2-

0,2

2b

r (nlmO)

r (nm) (a)

(b)

FiG. 5, Normalized functions of BaSO4 needlelike ultramicrocrystal size distribution: (a) ultramicrocrystals width distribution; (b) ultramicrocrystals length distribution. The points on the curve represent experimental data for cO: 0.05, 0.1, and 0.2 mole d m -3. The continuous lines denote the results calculated from Eqs. [3], [5], and [8]-[10], for k~ = 5 × 1014 m -3 s -l, m = 3, kn = 9 × 10 -14 m s -l, aoo = 1.04 X 10-5 mole d m -3, a = 1.3 × 10-5 mole d m -3, and for the length b = 1 nm, a = 0.04 and for the width b = 0.5 n m , a = 0.5.

NR"

Vo F(r) = - ~ R f (r, Xk, t),

[6]

where NR is the fractional concentration ofultramicrocrystals in the receiver, V0 is the total volume of solution introduced into the system, VR is the suspension volume in the receiver after the completion of process, and the xk is the length of mixer and the tube. The value NR is determined on the basis of the balance equation

where CRis the BaSO4 concentration in solution in the receiver. Taking as the characteristic value the length of ultramicrocrystals we obtain the shape factor a = 0.04. The average value of (r) calculated by means of function F(r) is presented in Table I. Table I also shows the values of NR and it is evident that NR increases with the co increase. The functionf(r, Xk, t) is described by Eqs. [1]-[3] which may be significantly simplified when applied to the realized regime of crys-

[7]

c ° V ° - CRVR = N R V R p a ( r 3 ) ,

0,7. t5~

06

°

0,5

% ~o,

'% E 0,4

--i~5 •

=~ 0,3. o

~.

6

~

1'0

s.iO ~6(m 2 )

FIG. 6. Normalized distribution function ofringlike aggregates ((3) on projection surfaces (s) of their interior faces at small aging times and normalized distribution function of inclusions within pseudosingle crystals (O) on projection surfaces o f their interior faces at longer aging times. Nc is the n u m b e r of measured interior faces.

cb 0,2 0A

0

2

4,

.6~

=8_~ 10

12

1~,

tc~F'10 rno[ dm FIG. 7. Pseudosingle crystal growth rate (G) of BaSO4 vs square of ionic activity in the solution. Journal of Colloid and Interface Science, Vol. 112, No. 1, July 1986

62

MELIKHOV, KELEBEEV, AND BACIC

lllll Ill

I I 1,,I.__~ 11 L22rl tl II 1311 I--~ II IIIII I -t Ililt llll

I1tll

LILIIIIII I I I I I I 11 1

I~'r4~I

i l Y l l Ndl I I ~'1 ~g~d N I L¢'I L/I I T'~l TN] l ~ I I INII

]~JI~N I I I I1 I I I I L~

11 1 I I ' I ~ M - I

II III I I I III [l

r~J

~r-l'~Ll~r i ~rlA~l'l

III Ill I I I II[ I I + - ~ Y ' L , ~ I ' L I ~ / " F [ ~ - ~ IILAll l I I IILIII

I I N,I I I J / q l I~

I F11

I I I I I7

L~,~qt

I

I

I i I I I I I I L I I LI

N~J~

EJJ

FIG. 8. Scheme of BaSO4 coagulation growth.

tallization. During the addition of reagents (order of magnitude--seconds) processes taking place in the mixer and the tube may be considered to be stationary (14). We can neglect the term P in the Eq. [ 1]. This term describes departure of ultramicrocrystals from the collective of growing particles during their binding into aggregates. In the mixture and the tube, however, judging by Fig. 4a, aggregates were disordered and, thus, ultramicrocrystals could extend their growth in their volume. These ultramicrocrystals were losing contact with mother solution during formation of pseudosingle crystals but that occurred only in the receiver. In Eqs. [ 1] and [2] we can neglect the terms with diffusion coefficient because in our experiment the following conditions are fulfilled: vLc >> DLgrad c and v~f >>D~gradf(14). Furthermore, we can equalize

vc and/)L because for the crystal size smaller than 100-nm sedimentation is negligible and crystals are completely withdrawn from the liquid flow. Therefore, inhomogeneity of flow in tube cross section should be neglected and flow divergency div vLfand div v~c should be considered unidimensional (14). In conclusion, we may state that in accordance with Eq. [5] crystal growth rate G is not dependent on crystal size and is determined only by solution concentration. The crystal growth fluctuation coefficient Gc is (15) b cc = ~ c,

[81

where b is the characteristic fluctuation length (b = 1 nm). Taking into consideration simplifications made, Eqs. [1] and [2] can be presented in the form

TABLEI

Salt

BaCO3

BaSO4

Concentration of background of electrolytes (mole dm-3)

co (mole dm-3)

NaC1 0.1 0.1 0.1 0.1

0.0025 0.005 0,025 0.050

4.16 3,97 4,57 4,33

KC1 0.1 0.2 0.4

0.050 0.100 0.200

1.53 _+ 0.02 1.94 _+ 0.02 2.18 + 0.02

Journal of Colloid and Interface Science, Vol. 112, No. 1, July 1986

NR X 10 -19

(rim)

+ 0.04 ___0.04 _+ 0.04 +__0.03

(drn-3)

0.404 1.05 3.37 7.79

_+ 0.005 +- 0.05 _+ 0.05 __ 0.04

3.64 + 0.03 3.57 + 0.03 5.01 + 0 . 0 3

63

NUCLEATION AND GROWTH

While transforming into crystallization centers, triplets are being rearranged with activation free energy/XF so that we obtain

VL O X = 6 ~

OC = 3(7Ofo ~ ~r2fdr. --VL -~X

[91

Equations [3], [5], and [9] applied to BaSO4 crystallization contain only one unknown value--the nucleation rate J. It is possible to evaluate it on the basis of data from the funct i o n f ( r , xk, t). We have done this on the basis of a calculation experiment. In our calculation we have taken different functions J(c) and we solve Eqs. [3], [51, and [9], by numerical methods and by means of algorithms described in (15). The calculated value f(r, xk, t) was compared with the experimental data and such J(c) dependence was selected which was in the best agreement with the experiment. Calculation has shown that the best agreement is obtained when (Figs. 4 and 5)

AF

~T~k k a®,

[121

where X is the transmission coefficient. According to Nielsen (17) for BaSO4 crystallization k" = 250 liters mole -1 and k " = 7 liters mole -~ is obtained, so that at X = 0.5, taking into evaluation our calculated value of kin, we obtain A F = 73 kJ mole -~. Crystallization of BaC03. The induction period of BaCO3 crystallization throughout the investigated co concentration interval is considerably longer than the time of mixer flow through the tube and its homogenization in the receiver (~0.1 s). Therefore, crystallization J=kmF(C3"+-)m-kk a~ / 11' [10] takes place mainly in the receiver in the homogeneous solution. Thus, in Eqs. [ 1] and [2] applied to BaCO3 crystallization it is necessary where km = (4.8 + 0.7) X 10 xa m -3 s-1 is the to neglect terms with divergences. The value nucleation rate coefficient, m = 3 is the nu- k,~ may be evaluated on the basis of(18). This cleation order, and ao~ = 1.04 X 10-5 mole work shows that the BaCO3 crystal growth with dm -3 is the solubility of BaSO4. The value 3,+ specific surface area of S = 3 m 3 g-~, at 298 was calculated according to the Giintelberg K and am/ao~ = 4.5, has a linear crystal growth equation throughout the BaSO4 concentration rate of G = 2 X 10-H m s-1, so that the /Ca interval (c o < 0.2 mole dm -3, at KC1 concen- --~ 1.1 X 10 -14 m s -1. When we take as a chartration lower than 0.4 mole dm-3). That the acteristic value the ultramicrocrystal width use of Giintelberg equation is justified was (a = 0.17) and when we solve Eqs. [1], [2], proved by the fact that in the investigated con- and [5] and compare the solution with the centration interval calculated and experimen- functionf(r, t) for t = 60 s and b = 1 nm, we tal values for Cd, Co, Mn, and Mg sulfates are obtain k m = 1.4 X 1017 m -3 s -1 and m = 3 in good agreement (16). (Fig. 3). Taking into account the closeness of The value m = 3 in Eq. [ 10] indicates the k" and k" for all 2-2 electrolytes, by Eqs. [ 11 ] possibility of considering as crystallization and [12] we obtained zXF = 75 kJ mole -1. centers activated ion triplet associates. We can Therefore, nucleation in high supersatuallow that crystallization centers are formed rated solutions may be considered as activated from simple ion triplets having thermody- rearrangement of ion triplets with energy acnamic activity (am) at any given m o m e n t of tivation of about ~ 7 0 kJ mole -1, and the crystallization equal to growth as binding reaction of the second order with crystal growth rate coefficient/% -~ 10 -14 am = kUk"(c')'+_)3, [111 m s-1 at 293 K. The description of crystallization given in where k" and k" are power constants of ion pair and ion triplet formations, respectively. this work may be deemed to be an evaluation Journal of Colloid and Interface Science, Vol. 112,No. 1, July 1986

64

MELIKHOV, KELEBEEV, A N D BACIC

II I ~ BaCO3 ""O51C% O0.5M (~025M C)

0.'5

1.()-

:~

2,

00.0514,

6/j 10

x/x k

t/t k

(a)

(b}

FIG. 9. (a) Calculation results of the function c(x) for BaSO4. Characteristic length xk = 12 cm. (b) Calculation results of the function c(t) for BaCO3. Characteristic time tk = 5 × 10-2 S.

for the following reasons: first, because of a considerable error in determination of size of very small particles, second, because the only information available is on crystal projection so that it was necessary to assume the third parameter, and third, because of limitations of continuum approximation used in calculations. This second reason is the most significant one and it could have caused differences in calculated values ofkn and km from the real ones. The third reason is important for particles having the size smaller than a nanometer. The participation of subnanometric particles in the quantity of solid phase is, however, practically negligible throughout the whole crystallization. This was proven by solution of Eqs. [1]-[3], [5], and [10] and on the basis of functions determined by them, c(x, t), and the quantity of subnanometric particles. Therefore, when analyzing crystallization it is possible to neglect discreet crystal growth even when such highly dispersed solids as barium sulfates and barium carbonates obtained from high supersaturated solutions are analyzed, Calculation results of the function c(x) for BaSO4 and c(t) for BaCO3 are shown in Fig. 9.

am ao~ b c D

F(r) ~xF

f(r, x, t) G G~ h 1 J k k" k" kn

APPENDIX: N O M E N C L A T U R E

km m

A a

Avogadro's number empirical parameter defined in Eq.

Journal of Colloid and Interface Science, Vol, 112, No. 1, July 1986

N

[5] means microcrystals solubility activity of ion triplets solubility of crystallizing matter thickness of the grown layer-characteristic fluctuation length molar concentration of crystallizing matter in point x at time t diffusion coefficient normalized crystal size distribution function activation free energy for transformation of ion triplet into active crystallization centre local crystal size distribution function in point x at time t linear crystal growth rate crystal growth fluctuation coefficient Planck's constant ionic strength nucleation rate Boltzman's constant power constant of ion pair formation power constant of ion triplet formation crystal growth rate coefficient nucleation rate coefficient nucleation order Number of crystals per unit volume of suspension

NUCLEATION AND GROWTH

N~ N~ P R Re g

S t

T V

v0

UL

x

Xk Y

Number of measured crystal proSuperscript jections Related to the distance x -- 0 Fractional concentration of ultramicrocrystals in the receiver ACKNOWLEDGMENT Rate of crystal binding into aggreS. B. was supported by a fellowship from the Internagates tional Atomic Energy Agency. Universal gas constant Reynolds number REFERENCES Crystal size 1. Heicklen, J., "Colloid Formation and Growth: A Average value of cubic crystal size Chemical Kinetic Approach," Vol. 4. Academic Specific surface area Press, New York, 1976. Time 2. Giitlich, Ph., and Leiser, K. H., Z. Phys. Chem. Neue Temperature Folge 46, 257 (1965). 3. Reddy, M. M., and Nancollas, G. H., J. Colloid InSolution volume terface Sci. 36, 166 (1971). Total volume of solutions intro4. Packter, A., and Uppaladini, S. C., Krist. Tech. 9, 983 duces into the system (1974). Suspension volume in the receiver 5. Garcia-Ruiz, J. M., and Amoros, J. L., Bull. Mindral. after the completion of process 104, 107 (1981). 6. Doremus, R. H., J. Phys. Chem. 74, 1405 (1970). Velocity of crystal movement 7. Nancollas, G. H., and Purdie, N., Trans. Faraday Soc. Liquid flow rate 59, 735 (1963). Distance from the point of injection 8. Gunn, D. J., and Murthy, M. S., Chem. Eng. Sci. 27, of the reagent into the mixer 1293 (1972). 9. Nielsen, A. E., Krist. Tech. 4, 17 (1969). Length of mixer and the tube 10. Liteanu, C., and Lingner, H., Talanta 17, 1045 (1970). Quantity of solid phase

Greek Symbols Od

2/+_ p x

Shape factor Mean ionic activity coefficient Crystal density Transmission coefficient

Subscripts 0 c

L R

65

Related to Related to Related to Related to

the time t = 0 crystal liquid the receiver

11. Melikhov, 1. V., and Kelebeev, A. S., Kristallografiya (Moscow) 24, 410 (1979). 12. Melikhov, 1. V., Vukovi6, Z., Ba6i6, S., and Lazi6, S., Chem. Eng. Sci. 39, 1707 (1984). 13. Robinson, R. A., and Stokes, R. H., "Electrolyte Solutions." Butterworth, London, 1959. 14. Caldin, E. F., "Bystrye reaktsii v rastvore,'" p. 281. Mir, Moscow, 1966. 15. Melikhov, L V., and Befliner, L. B., Chem. Eng. Sci. 36, 1021 (1981). 16. Harner, H. S., and Owen, B. B., "The Physical Chemistry of Electrolytic Solutions," p. 546. Reinhold, New York, 1958. 17. Nielsen, A. E., Acta Chem. Scand 13, 784 (1959). 18. Ba6i6, S., Lazi6, S., and Vukovi6, Z., Bull. Soc. Chim. Beograd. 48, 103 (1983).

Journal of Colloid and Interface Science, Vol. I 12, No, 1, July t986