Novel Approach to Analyze Metastable Zone Width ... - ACS Publications

Novel Approach to Analyze Metastable Zone Width Determined by the Polythermal Method: Physical Interpretation of Various Parameters K. Sangwal

CRYSTAL GROWTH & DESIGN 2009 VOL. 9, NO. 2 942–950

Department of Applied Physics, Lublin UniVersity of Technology, ul. Nadbystrzycka 38, 20-618 Lublin, Poland ReceiVed July 2, 2008; ReVised Manuscript ReceiVed October 14, 2008

ABSTRACT: Experimental data on the maximum supercooling ∆Tmax, a measure of metastable zone width, for solutions saturated at a temperature T0, as a function of cooling rate R are analyzed, for some solute-solvent systems chosen as examples, using Ny´vlt’s semiempirical approach and a new approach based on the classical theory of three-dimensional nucleation combined with the formation of n-sized embryos from monomers according to the law of mass action. Instead of a linear relation between ln(∆Tmax) and lnR of the Ny´vlt’s approach, the new approach predicts a linear dependence of (T0/∆Tmax)2 on lnR with slope F1 and intercept F. The quantity F1/F is independent of saturation temperature T0, characteristic of a solute-solvent and is associated with the growth of the stable three-dimensional nuclei to visible entities. The value of F1 is determined by thermodynamic and solvation processes, while that of F is governed by thermodynamic and kinetic parameters as well as processes associated with solvation of solute ions/molecules and their transport in the solution. Limitations of Ny´vlt’s approach and advantages of the new approach in terms of its physical basis are exposed. It is pointed out that the new approach can also be extended to explain the value of metastable zone width by the isothermal method and to explain the effect of saturation temperature and impurities on metastable zone width. Introduction Metastability in supersaturated solutions of different types of solutes is a characteristic property of their crystallization. A measure of metastability of solutions is their metastable zone width, which is associated with the nucleation of solutes from solutions. Metastable zone width of solute-solvent systems is experimentally determined by either isothermal or polythermal method. In the isothermal method, induction period for nucleation from supersaturated solutions is usually measured. In the method, a saturated solution at temperature T0 contained in some vessel is rapidly cooled down to a predefined temperature, and the time elapsed from cooling to the first appearance of change in the state of the solution is measured. In this method, metastable zone can also be measured directly from the initially preset temperature difference when spontaneous nucleation occurs immediately after attaining the preset temperature. However, in the polythermal method, the solution is cooled at a constant cooling rate from saturation temperature T0 to a temperature when visible crystals appear in the solution,1-7 or there is sudden discontinuity at a temperature Tlim in the temperature dependence of some property such as electrical conductivity,8,9 ultrasound velocity,10,11 density11 or turbidity of the solution.8,11,12 The temperature difference (T0 - Tlim) is taken as the metastable zone width. Metastable zone width for a substance depends on a variety of factors such as saturation temperature, solvent used for preparation of supersaturated solutions, presence of impurities dissolved in the solution, presence of crystalline seeds in the solution, solution stirring and cooling rate of solution from saturation temperature. In fact, during the last four decades voluminous literature has emerged on the subject of determination and prediction of metastable zone width of a variety of systems. In the case of the polythermal method, Ny´vlt’s relation between temperature difference ∆Tmax ) (T0 - Tlim), called maximum supercooling, and cooling rate R is widely used for the study of kinetics of metastable zone width.1,2,4,5,9,10,13 The main drawback of this approach is that the nucleation rate is

given on mass basis. Kubota13 derived an equation, similar to Ny´vlt’s, relating supercooling ∆Tmax with cooling rate R, considering the number density of accumulated primary nuclei approaching a fixed value. Ny´vlt’s and Kubota’s approaches indeed describe the experimental data satisfactorily. However, as pointed by Kim and Mersmann,14 it is difficult to predict the metastable zone width using these approaches. Using different nucleation processes in industrial crystallizers, Mersmann and Bartosch,15 and Kim and Mersmann14 made attempts to predict metastable zone width for both seeded and unseeded systems at various cooling rates and compositions. In the case of isothermal method, using the classical theory of threedimensional (3D) nucleation, metastable zone width for substances from solutions without and with impurities can also be explained,16,17 and it is possible to predict its value for primary nucleation.16 The aim of this paper is 2-fold: (1) to present briefly the limitations of the Ny´vlt’s and Kubota’s approaches of metastable zone width based on power-law description of nucleation rate, and (2) to propose a new approach for the interpretation of metastable zone width measured by the polythermal method, using the classical 3D nucleation theory, and confront the experimental data for some selected solute-solvent systems. The physics of different processes involved in 3D nucleation is explained. Previous Theoretical Interpretations of Metastable Zone Width The polythermal method of metastable zone width is based on the determination of the maximum supercooling ∆Tmax, as illustrated schematically in Figure 1. A solution of known concentration c2 at the saturation temperature T2 (point B) is cooled at a constant cooling rate R, from a temperature 5 to 7 K above T2 (point A) to a temperature T1 at which first crystals are detected in the solution (point C in Figure 1). When the temperature T1, at which crystallization sets in, is denoted by Tlim, the saturation concentration c1 corresponding to Tlim by

10.1021/cg800704y CCC: $40.75  2009 American Chemical Society Published on Web 01/15/2009

Metastable Zone Width

Crystal Growth & Design, Vol. 9, No. 2, 2009 943

cooling rate R. However, in reality, nucleation occurs continuously even before ∆cmax and ∆Tmax are reached. Kubota’s Approach. The metastable zone width of a system depends on the method of detection of the first nucleation events. Consequently, even for the same system the value of metastable zone width determined by different techniques is different. The above model does not consider this feature of metastable zone width. To explain this feature, Kubota13 proposed another model, which takes into account the number of detectable nuclei Ndet in volume V after some time t:

Ndet/V )

∫0t J(t) dt

(5)

Assuming that nucleation rate is given by (cf. eq 2a)

J(t) ) k1(∆T)q Figure 1. Schematic illustration of the basis of determination of the maximum supercooling ∆Tmax in the polythermal method of metastable zone width.

c0, and the saturation temperature T2 by T0, then the maximum supercooling ∆Tmax ) (T0 - Tlim) and the maximum concentration difference ∆cmax ) (cmax - c0). Ny´vlt’s Approach. Following Ny´vlt et al.,1 we assume that the supersaturation ∆c is related with supercooling ∆T by

∆c )

( )

dc0 ∆T dT T

(1)

and that, in the vicinity of metastability, the nucleation rate J is related with maximum solution supersaturation ∆cmax by the power-law relation

J ) K(∆cmax)

(2a)

m

and with the cooling rate R ) ∆T/∆t by

J)

( )

dc0 R dT T

(2b)

where m is the apparent nucleation order, (dc0/dT)T is the temperature coefficient of solubility at temperature T and K is the nucleation constant, the value of which depends on the processes of formation and growth of stable nuclei into visible entities and the experimental method used for the measurement of metastable zone width. Substituting the value of ∆cmax from eq 1 in eq 2a and equating the nucleation rate J given by eq 2a and eq 2b, one obtains

∆Tmax )

( ) dc0 dT

(1-m)/m

( KR )

T

1/m

(3)

Taking logarithms on both sides of Eq. (3), upon rearrangement one obtains

ln ∆Tmax )

( )

dc0 1-m ln m dT

T

1 1 ln K + ln R m m

(4)

Equation 4 predicts a linear dependence of ln ∆Tmax on ln R. This linear dependence enables to calculate the values of m and K because (dc0/dT)T can be determined from solubility data. It should be mentioned that eq 2a describes the dependence of nucleation rate J on the maximum concentration difference ∆cmax at constant temperature Tlim whereas eq 2b describes the relationship between J and linear cooling rate R from T0 to Tlim, which results in a linear increase in the concentration difference ∆c with time. Both of these equations refer to the situation when ∆cmax and ∆Tmax are attained during cooling at a constant

(6)

where k1 ) K[(dc0/dT)T]q with q as a constant, and the cooling rate R ) d(∆T)/dt (i.e., dt ) d(∆T)/R), from Eq. (5) one obtains

Ndet k1 ) V R

∫0∆T

max

(∆T)q d(∆T) )

k1 (q + 1)R

(∆Tmax)q+1 (7)

Taking the logarithm on both sides, upon rearrangement, one gets

ln(∆Tmax) )

( )

( )

Ndet k1 1 1 1 + ln ln ln R q+1 V q+1 q+1 q+1 (8)

One finds again that the plot of ln(∆Tmax) against lnR is a straight line with a slope 1/(q + 1). Since the concentration of detectable nuclei Ndet/V depends on the method used for their detection, eq 8 predicts a constant slope 1/(q + 1) irrespective of the method of measurement of metastable zone width. There is a close similarity in the forms of relations (4) and (8), which are derived on the assumption of the validity of power law relations (2a) and (6) between nucleation rate J and concentration difference ∆c or temperature difference ∆T from the equilibrium state. In reality, there is poor justification of this assumption, especially when the so-called embryos reach the size of critically sized nuclei, as is envisioned in the classical theory of three-dimensional (3D) nucleation. Therefore, below we discuss the main limitations of these approaches. In the discussion, we consider Ny´vlt’s approach, but the same limitations are also inherent in Kubota’s approach. Limitations of Ny´vlt’s Approach. The main drawbacks of Ny´vlt’s approach are as follows: (1) the physical significance of the nucleation constant K and the nucleation order m of the empirical power-law relation (2a) is not known, and (2) the empirical power-law relation (2) does not reproduce satisfactorily the data on the dependence of nucleation rate J on supersaturation ln S predicted by the classical theory of 3D nucleation. Here the supersaturation ratio S ) c/c0, where c and c0 are the actual and the saturation concentration of solute at a given temperature T. With reference to Figure 1, the concentrations c and c0 are c2 and c1, respectively, at the temperature T1. According to the classical 3D nucleation theory, the dependence of rate of homogeneous nucleation J on supersaturation ln S is given by1,18

J ) A exp[-B/(ln S)2]

(9) 15

where A is the pre-exponential factor lying between 10 to 1042 m-3 · s-1, and the parameter B for spherical nuclei is given by

944 Crystal Growth & Design, Vol. 9, No. 2, 2009

B)

( )

16π γΩ2/3 3 kBT

Sangwal

3

(10)

where γ is the solid-liquid interfacial energy, Ω is the volume of a solute molecule (i.e., molecular volume), kB is the Boltzmann constant equal to RG/NA (NA is the Avogadro number), and T is the solution temperature. In Figure 1 this temperature refers to T1, while in the polythermal method it refers to Tlim. In Eq. (10) the factor 16π/3 is a result of geometry of a spherical nucleus. Since the precise value of A is not known, in our analysis we consider the ratio J/A in eq 9. The data generated by eq 9 were subsequently analyzed according to the power-law relation (cf. eq 2a)

J ) K1(ln S)b

(11)

where, following the convention of eq 9, we have used ln S for supersaturation instead of the usually defined supersaturation σ ) ∆c/c0 because ln S ≈ σ when σ , 1, and the new constant K1 ) Kc0b with b as a constant. Equation 11 may be rewritten in the form

ln(J/A) ) ln(K1/A) + b ln(ln S)

(12)

which predicts a linear dependence of ln(J/A) on ln(ln S) with intercept ln(K1/A) and slope b. The intercept ln(K1/A) ) ln(J/ A) when ln(ln S) ) 0 (i.e., when ln S ) 1). Figure 2a shows data of J/A as a function of ln S predicted by eq 9 of the classical 3D nucleation theory for various values of constant B in the form of plots of ln(J/A) against ln(lnS) of eq 12 of power-law approach in the ln S range of practical interest (i.e., ln S < 0.4) during crystal growth of various compounds soluble in water. It may be seen from the plots of Figure 2a that, for a given B, the linear relation (12) is not followed in the entire investigated range of supersaturation lnS. In fact for a given B, the values of slope b and intercept ln(K1/ A), determined from tangents to the plot of ln(J/A) against ln(lnS) at different lnS, continuously decrease with an increase in lnS. Assuming that the linear relation (12) is followed for the ln(J/ A) versus ln(ln S) data in the intervals of three successive lnS values considered in Figure 2a, the values of slope b and intercept ln(K1/A) at different ln S were calculated. It was found that, for a particular value of constant B, the values of slope b and intercept ln(K1/A) decrease with increasing value of ln S. Figure 2b shows, as an example, the dependence of exponent b on ln S for B < 4. It is observed1 that for fairly soluble compounds crystallizing from aqueous solutions, the value of the exponent b < 25 and ln S < 0.3. From Figure 2b it may be inferred that the value of

B < 1 for these compounds. The value of B < 1 corresponds to γΩ2/3/kBT < 0.39 (cf. eq 10), which is a reasonable value for compounds fairly soluble in water. Since the solubility of different compounds decreases with their increasing interfacial energy γ,1,16 the constant B and the maximum supersaturation lnS for metastable zone width increase with an increase in γ for a compound. It is frequently observed that maximum supercooling ∆Tmax increases linearly with increasing cooling rate R. Some examples of the linear dependence of maximum supercooling ∆Tmax on cooling rate R are illustrated in Figure 3. Figure 3a shows the ∆Tmax(R) data for fluoranthene-trichloroethylene system obtained by using density, ultrasound velocity and solution turbidity measurements, Figure 3b presents the ∆Tmax(R) data for KSF-water system obtained by using conductivity measurements, while Figure 3c shows the ∆Tmax(R) data for KTB-water system determined visually by observing the appearance of the first crystals. The abbreviations KSF and KTB denote potassium sulfate and potassium tetraborate, respectively. The data of Figure 3, parts a and c, were recovered from the published figures in the papers by Marciniak11 and Sahin et al.,4 respectively, while those of Figure 3b were taken from Lyczko et al.9 It may be seen from Figure 3 that, in all cases, maximum supercooling ∆Tmax increases linearly with increasing cooling rate R, but the value of the intercept ∆Tmax corresponding to R f 0 and that of the slope d(∆Tmax)/dR in a system depend on the method of measurement of ∆Tmax (Figure 3a), effects of external factors such as applied ultrasound power P (Figure 3b), and saturation temperature T0 of solutions (Figure 3c). Figure 4a presents a typical example of the dependence of maximum supercooling ∆Tmax/T0 on cooling rate R for NTO-water system for four different temperatures, where NTO denotes nitroazolone (i.e., 3-nitro-1,2,4-triazol-5-one). The data of Figure 4a do not follow a linear relationship between ∆Tmax/ T0 and R, but follow the relation

∆Tmax/T0 ) a(R - Rc)p

(13)

Here the constants a, p, and Rc are given in Table 1. It seems that the constant a decreases with an increase in saturation temperature while the constants Rc and p are temperatureindependent constants equal to about 0.20 K · h-1 and 0.091, respectively. Physically, the constant Rc may be interpreted as a threshold cooling rate above which ∆Tmax increases with R. It should be noted that use of ratio ∆Tmax/T0 for analysis of experimental data on ∆Tmax(R) is physically meaningful when

Figure 2. (a) Data of J/A as a function of ln S predicted by eq 9 of the classical 3D nucleation theory for various values of constant B shown as plots of ln(J/A) against ln(lnS) of eq 12 of power-law approach in the ln S range of practical interest. (b) Dependence of exponent b of eq 11 on ln S for different B. See text for details.



Figure 4. (a) Dependence of maximum supercooling ∆Tmax/T0 on cooling rate R for NTO-water system for four different temperatures. (b) Data of (a) represented as plots of ln(∆Tmax/T0) against lnR. In (a) best-fit curves are drawn according to eq 13 with constants of Table 1, while in (b) the straight lines are best-fit plots according to eq 14 with constants of Table 2. Original data from Kim and Mersmann.14 Table 1. Best-Fit Constants of a, p, and Rc of Equation 13

Figure 3. Examples of the linear dependence of maximum supercooling ∆Tmax with increasing cooling rate R for saturated solutions of different systems: (a) fluoranthene-trichloroethylene system for saturated solution temperature between 304.41 and 309.03 K, (b) KSF-water system without and with applied ultrasound power for solutions saturated at 303.15 K, and (c) KTB-water system for solutions saturated at four different temperatures. See text for details.

values of a, p, and Rc are analyzed as a function of saturation temperature T0. This aspect is considered later. It is interesting to note that the plots of ∆Tmax/T0 against R are approximately parallel straight lines for cooling rate R exceeding about 30 K · h-1. However, it should be remembered that these data cover a much wider range of R in comparison with that of Figure 3. Therefore, in a relatively narrow R range, all data may be approximated by straight lines, and, depending on the range of applied cooling rate R used for measurements of ∆Tmax, the slope of ∆Tmax(R) plots can also differ, as observed in Figure 3c. The experimental data of Figure 4a are shown in Figure 4b as plots of ln(∆Tmax/T0) against lnR, as predicted by Eq. (4), in the form

ln(∆Tmax ⁄ T0) ) a1 + b1 ln R

(14)

where a1 and b1 are empirical constants. The best-fit values of the constants a1 and b1, and regression coefficient RC, and the ratio b1/a1 are given in Table 2. Here instead of ∆Tmax in eq 4, we have considered the dimensionless quantity ∆Tmax/T0 for

T0 (K)

a (10-2)

Rc (K · h-1)

p (10-2)

338.15 348.15 358.15 367.15

1.98 1.77 1.57 1.43

0.19 0.27 0.14 0.22

8.83 8.66 9.84 9.22

Table 2. Best-Fit Constants of a1 and b1 of Equation 14, Regression Coefficient RC, and Ratio b1/a1 T0 (K)

-a1

b1 (10-2)

-b1/a1 (10-2)

RC

338.15 348.15 358.15 367.15

3.94 4.07 4.17 4.28

9.49 9.67 10.32 10.03

2.41 2.38 2.47 2.35

0.997 0.997 0.999 0.998

analysis because it is more informative than ∆Tmax due to the fact that it excludes the effect of saturation temperature T0 on the values of constants a1 and b1. The constant a1 corresponds, but is not equal, to the first two terms on the right-hand side of Eq. (4), while b1 ) 1/m. It may be noted from Table 2 that the quantity a1 decreases while b1 increases with an increase in saturation temperature T0. A decrease in a1 suggests that, with an increase in T0, the nucleation constant K increases more pronouncedly than the term containing the temperature coefficient of solubility (dc0/dT)T. Moreover, a change in the nucleation order m ) 1/b1 with T0 has no physical sense. It is also interesting to note that the ratio b1/a1 is a temperature-independent constant equal to -(2.40 ( 0.05) × 10-2. However, it is difficult to assign any physical significance to this constancy of b1/a1 satisfactorily from the standpoint of Ny´vlt’s approach because: (a) determination of the temperature coefficient of solubility (dc0/dT)T at a given temperature T from solubility data is usually erratic due to


Sangwal

different values of (dc0/dT) in different temperature intervals, and (b) constants like K and m are empirical. Apart from the above general trends of the dependence of a1 and b1 on saturation temperature T0, one may discern a small positive curvature in the ln(∆Tmax/T0) data with an increase in lnR. This positive curvature is best seen in the data for 367.15 K. By the term “positive curvature” we mean the bulging out of the curve in the central part above an expected linear dependence. If this bulging out of the curve is below the expected linear dependence, the curvature is negative. No explanation is known so far for this positive curvature. Another Approach for the Interpretation of Metastable Zone Width. We propose here a completely different approach for the calculation of metastable zone width using the classical theory of three-dimensional nucleation rather than empirical relations (2a) and (6). The main advantage of this approach is that the two parameters, i.e. the parameter A associated with the kinetics of formation of nuclei in growth medium and the solid-liquid interfacial energy γ, of the classical nucleation theory (see eq 9) account for both homogeneous nucleation and heterogeneous nucleation caused by impurities or by existing seeds. In the case of heterogeneous nucleation, the value of the interfacial energy γ is lower than that in homogeneous nucleation, but in both cases the value of interfacial energy γ has a well-defined physical background. Using the theory of regular solutions, one may write the relationship between the ratio of solution concentrations c1 and c2 corresponding to temperatures T1 and T2, respectively, and supercooling ∆T in the form

(

c2 ∆Hs ∆T ) exp c1 RGT1 T2

)

(15)

where ∆T ) (T2 - T1) such that T2 > T1 and c2 > c1, ∆Hs is the heat of dissolution and RG is the gas constant. For various compounds fairly soluble in water, ∆Hs/RGT1 lies between 2 and 10. Since the typical value of ∆Hs/RGT1 is about 4 and ∆T/ T2 ≈ 0.04, eq 15 may be rewritten in the form

(

∆Hs ∆T ∆c ) c1 RGT1 T2

)

(16)

where ∆c ) (c2 - c1). As in the case of Ny´vlt’s approach described above, corresponding to temperature difference ∆T ) (T2 - T1), we assume that the nucleation rate J is related with rate of change of solution supersaturation ∆c/c1 [i.e., J ∝ (∆c/c1)/∆t]. Then using eq 16, one may write

J)f

( )( )

∆Hs R ∆c ∆c ∆T ) f )f c1∆t c1∆T ∆t RGT1 T2

(17)

where the cooling rate R ) ∆T/∆t and the proportionality constant f is defined as the number of entities (i.e., particles, ions or clusters) per unit volume. The value of f is governed by aggregation and diffusion processes in the solution. When the classical theory of three-dimensional nucleation applies, the rate of formation of stable three-dimensional spherical nuclei may be given by18

[ ( ) ( ) ( )]

J ) A exp -

T2 16π γΩ2 ⁄ 3 3 RGT1 2 × × 3 kBT1 ∆Hs ∆Tmax

2

(18)

where the constant A is associated with the kinetics of formation of nuclei in the growth medium. Here we have used eq 9, where we substituted the value of ln S from eq 15. Equation 18

describes the rate J of homogeneous nucleation or heterogeneous nucleation caused by adsorption of impurity molecules on the nucleus surface such that they decrease the value of γ. However, the adsorption of impurities does not change the kinetic factor A. The kinetic effect of impurities on nucleation rate J will be discussed in another paper. It should be mentioned that the physical basis of eq 17 is associated with the probable concentration C(n) of n-sized cluster (embryo) in mutual equilibrium between n-mers of all possible sizes n ) 1, 2,..., and may be given by18

C(n) ) C0(C1 ⁄ C0)n exp{-[W(n) - nW1]/kBT

(19)

where C1 is the concentration of monomers, C0 is the concentration of sites in the supersaturated system on which the clusters (embryos) of the new phase can form, W(n) is the work of formation of n-sized cluster, and W1 is the value of W(n) at n ) 1. Equation 19 describes the development of n-sized clusters from monomers during progressive cooling of a solution saturated at temperature T0 to temperature Tlim. It also shows that the equilibrium cluster size distribution satisfies the law of mass action. It is these embryos that grow into stable critically sized 3D nuclei and, subsequently, into crystals of detectable size. The solid-liquid interfacial energy during heterogeneous nucleation caused by impurities and during secondary nucleation occurring in the presence of seed crystals is usually lower than the true interfacial energy γ. Therefore, denoting the solid-liquid interfacial energy in these situations by an effective interfacial energy γeff and using the notations T0 and Tlim for T2 and T1, respectively, from eqs 17 and 18, one may write

[ ( ) ( ) ( )]

exp -

2⁄3 3 RGTlim 2 T0 2 16π γeffΩ × × 3 kBTlim ∆Hs ∆Tmax ∆Hs R )f RGTlim AT0

( )

(20)

which upon taking logarithm on both sides and rearrangement gives

(T0/∆Tmax)2 ) F1(X + ln T0 - ln R) ) F - F1 ln R (21) with the constant F ) F1(X + ln T0), where

F1 )

[ ( ) ( )] { } ∆Hs 3 kBTlim 3 × 2/3 16π γ Ω RGTlim eff X ) ln

A RGTlim F ∆Hs

2

(22)

(23)

According to eq 21, at a given constant saturation temperature T0, the quantity ∆Tmax/T0 increases with an increase in cooling rate R. This behavior has been observed for all of the investigated systems.1,2,4,5,9,10,13 Equation 21 also predicts that the quantity ∆Tmax/T0 decreases with an increase in saturation temperature T0 and a decrease in cooling rate R. This general trend has been observed in numerous publications.3,6,7,12,19 Equation 21 predicts that, at a given saturation temperature T0, the quantity (T0/∆Tmax)2 decreases linearly with an increase in ln R, with slope F1 and intercept F. It may be noted that the value of the slope F1 depends on the effective interfacial energy γeff and the heat of dissolution ∆Hs, while the intercept F depends on γeff, ∆Hs, the kinetic factor A associated with the integration of growth units to the growing nuclei, and on the factor f determined by aggregation and diffusion processes in



the solution. Obviously, F1 is determined by thermodynamic and solvation processes, while F is governed by thermodynamic and kinetic parameters as well as processes associated with solvation of solute ions/molecules and their transport in the solution. The values of F1 and F may be estimated independently. For substances fairly soluble in water the typical value of (∆Hs/ RGTlim) lies between 2 and 5, that of (γeffΩ2/3/kBTlim) between 0.2 and 0.7, and A is about 1026 m-3 · s-1 (cf. Kashchiev18). The upper limit of the factor f may be estimated from solute concentration in the saturated solution or from the molecular volume Ω of solute. For fairly soluble substances, the typical solution concentration at 303 K lies between 0.2 and 0.5 mol · dm-3, which gives f equal to about 1029 m-3. If one takes (γeffΩ2/3/kBTlim) ) 0.2, (∆Hs/RGTlim) ) 5 and f ) 1027 m-3, one finds F1 ≈ 200, X ) 1.8 and F ) 360. Equation 21 can equally be adapted for the determination of metastable zone width by isothermal method, which is based on cooling down of a solution saturated at a particular temperature T0 to a predefined temperature T1 as fast as possible to obtain a desired supersaturation. The method usually involves measurement of induction period for various supersaturations defined as ln S, but one can also determine metastable zone width of a supersaturated solution simply from observation of spontaneous nucleation as soon as the predefined supersaturation is attained. Then the metastable zone width is taken as lnSmax, and is defined by eq 15. In the isothermal method, the cooling rate R in an experiment usually does not exceed 500 K · h-1 (i.e., about 0.14 K · s-1), and the limiting cooling rate Rlim is probably its lower value (see next section). Thus, the term lnR is a constant equal to 6.2 and -2.0 when Rlim is taken in K · h-1 and K · s-1 units, respectively, and is insensitive to the chosen value of Rlim. Then from eq 21, one obtains

Figure 5. Plots of (T0/∆Tmax)2 against lnR for: (a) KSF-water system without and with applied ultrasound power for solutions saturated at 303.15 K, and (b) KTB-water system for solutions saturated at four different temperatures. Data of parts a and b are from Lyczko et al.9 and Sahin et al.,4 respectively, and are presented in Figure 3, parts b and c, above.

(T0/∆Tmax)2 ) F1(X - ln Rlim + ln T0) ) F2 + F1 ln T0 (24) with the constant F2 ) F1(X - ln Rlim), where the constants F1 and X are given by eqs 22 and 23, respectively. When supersaturation is defined as ln S ) ln(c2/c1) such that ln Smax ) ln(cmax/c0), upon substituting the value of T0/∆Tmax in eq 24 from eq 15, one obtains

{(

Smax ) exp

B X1 + ln T0

)} 1⁄2

(25)

where X1 ) (X - lnRlim) and B is given by eq 10. The approach proposed here can be extended to explain the effect of impurities on metastable zone width by considering changes caused by them in the effective interfacial energy γeff, the heat of dissolution ∆Hs, the kinetic factor A associated with the integration of growth units to the growing nuclei, and the factor f determined by aggregation and diffusion processes in the solution. However, examination of the effect of saturation temperature and impurities on metastable zone width is beyond the scope of this paper. Confrontation of New Approach with Experimental Data Some General Trends. In order to confront the new approach of determination of metastable zone width in terms of maximum undercooling ∆Tmax as a function of cooling rate R, we use the data for KSF-water system without and with applied ultrasound power for solutions saturated at 303.15 K, KTB-water system for solutions saturated at four different temperatures (Figures

Figure 6. Dependence of (T0/∆Tmax)2 on lnR for NTO-water system for four different temperatures. Best-fit linear plots are drawn according to eq 21 with constants of Table 1. Original data are from Kim and Mersmann14 and are presented in Figure 4. See text for details.

3b and 3c), and NTO-water system for four different temperatures (Figure 4a). The data of Figure 3a for fluoranthenetrichloroethylene system are not considered for the analysis because of a relatively narrow R range used for the measurement of ∆Tmax. All the above data were obtained by primary nucleation (i.e., for unseeded systems). Parts a and b of Figure 5 show the plots of (T0/∆Tmax)2 against lnR, as expected from eq 21, for the KSF-water system without and with applied ultrasound power for solutions and for the KTB-water system for solutions saturated at four different temperatures, respectively. Figure 6 presents another example of the dependence of (T0/∆Tmax)2 on lnR for the NTO-water system for four different temperatures. In this figure the linear plots shown by the dashed lines refer to the as-applied cooling rate R. The values of the intercept F and the slope F1 for the three systems are collected in Table 3.


Sangwal

Table 3. Best-Fit Constants F and F1 of Equation 21, Regression Coefficient RC, and Ratio F1/F system a

KSF

KTBb

NTOc

T0 (K)

F (103)

F1 (102)

F1/F

RC

303.15 303.15USPd 306.15 313.15 323.15 333.15 338.15

1.30 2.65 0.686 1.217 3.527 20.54 2.656 (2.353)e 3.397 (3.007) 4.199 (3.693) 5.176 (4.563)

2.37 6.05 0.244 0.414 1.186 6.994 0.371 (0.276) 0.483 (0.362) 0.621 (0.462) 0.753 (0.562)

0.182 0.229 0.356 0.340 0.336 0.341 0.140 (0.117) 0.142 (0.120) 0.148 (0.125) 0.146 (0.123)

0.926 0.887 0.997 0.976 0.985 0.991 0.984 (0.998) 0.977 (0.999) 0.985 (0.999) 0.982 (0.998)

348.15 358.15 367.15

a KSF: potassium-sulfate-water. b KTB: potassium tetraborate. NTO: nitroazolone-water. d USP - ultrasound power P ) 0.05 W · g-1. e Values in parentheses refer to best-fit for corrected cooling rate R* ) (R - 0.5).

c

The experimental data of Figures 5 and 6 reveal negative curvatures and the regression coefficient RC is not very high (see Table 3). This feature is clearly seen for the experimental data at T0 ) 333.15 K (i.e., 60 °C) of KTB-water system and for all the four investigated T0 of the NTO-water system. However, when the cooling rate R for a system is corrected such that the corrected cooling rate R* ) (R - Rc), where the threshold Rc ) 0.5 K · h-1, the data exhibit a better fit with a higher regression coefficient RC. In Figure 6 of the NTO-water system, the data referring to corrected cooling rates R* are shown by solid lines. The value of Rc was determined by analyzing the original data with changing the value of Rc in steps of 0.1 K · h-1. The value of Rc for the KTB-water system is about 0.2 K/h. The following features may be noted from Figures 5 and 6: (1) The values of F and F1 increase with an increase in saturation temperature T0 and with applied ultrasound power P. In the latter case, it appears that F and F1 somewhat increase with the ultrasound power, but this feature is masked due to large scatter in the data. (2) The linear plots of (T0/∆Tmax)2 against ln R converge to a particular maximum value of lnR, denoted hereafter as lnRmax, which is characteristic of the investigated system. The values of ln Rmax for KTB-water and NTO-water systems are, respectively, about 2.9 and 6.0 for uncorrected cooling rate R and about 3.2 and 8.2 for corrected cooling rate R*. (3) In the case of the KSF-water system where the data exhibit large scatter, the values of ln Rmax are about 4.6 and 4.1 without and with applied ultrasound power, respectively. These values of ln Rmax are lower than that for the NTO-water system but higher than that for the KTB-water system. (4) The ratio F1/F is a constant characteristic of the compound and is the lowest in the case of the NTO-water system, highest for KTB-water system and intermediate for the KSF-water system. This trend is the same as in the case of ln Rmax. However, as judged by the study of the effect of ultrasound power, the value of the ratio F1/F also depends on factors such as presence of seeds, presence of impurities and stirring, which inhibit or promote nucleation in a system. The above features are discussed below. Physical Interpretation of Threshold and Limiting Cooling Rates and Factor F. First we discuss the physical significance of threshold cooling rate Rc, maximum cooling rate Rmax and reference cooling rate Rref corresponding to lnR ) 0. It may be seen that the threshold cooling rate Rc and the

maximum cooling rate Rmax correspond to the lower and the upper limits of the cooling rate R when relation (21) applies. Physically, Rc and Rmax correspond to situations when a system begins and ceases to respond to the cooling procedure, respectively. The former cooling rate is associated with the setting up of a thermal equilibrium between the solution and the environment, while the latter is connected with the induction period, as discussed below. Following Kim and Mersmann,14 from the maximum supercooling ∆Tmax attained in the system at a constant cooling rate R, we define the induction period tind by the relation:

tind ) ∆Tmax /R

(26)

where the time when the first crystal is detected in the system is taken as the induction period tind. Taking a typical value of ∆Tmax equal to 10 K, from eq 26, one finds tind equal to 90 and 10 s, respectively, with uncorrected Rmax ) 400 K · h-1 and corrected Rmax ) 3460 K · h-1 for NTO-water system. This value of tind is probably associated with the growth of critically sized nuclei to visible entities. The value of lnRmax is characteristic of a solute-solvent system. Therefore, the differences in its value for different systems may be attributed to the above value of induction period tind required for the growth of critically sized nuclei to visible entities. Since the trend of the value of F1/F is similar to that of ln Rmax for difference systems, the difference in the value of F1/F for different systems may be attributed to the differences in the growth rate of their critically sized 3D stable nuclei into visible entities. The higher the growth rate of the stable nuclei, the higher is the value of the constant F1/F and ln Rmax. It should be noted that the plots of ln(∆Tmax/T0) against ln R, as applied to Ny´vlt’s approach, also give a T0-independent constant ratio b1/a1 for a solute-solvent system (see Table 2). This ratio b1/a1 for a solute-solvent system has the same physical significance as the ratio F1/F in our new approach. However, this feature does not follow from the original Ny´vlt’s approach when applied in the form of eq 4. The factor F, equal to (T0/∆Tmax)2 at ln R ) 0, in eq 21 may be interpreted in terms of a dimensionless quantity connected with the diffusion of solute molecules in the solution with reference to the saturation solution. If τ is the mean jump time for an ion/molecule in solution to cover the mean diffusion distance l, its jump frequency ν (i.e., the mean number of jumps made by the ion/molecule per second; ν ) 1/τ) may be given by (for example, see Chapter 4, ref 20)

ν ) ν0exp(-E/RGT)

(27)

where E is the activation energy for the process, and ν0 is the vibration frequency of the ion/molecule at its saddle point before diffusion. From eq 27, the ratio of jump frequencies νsat and νmet of ions in solutions corresponding to saturation and metastable states, respectively, may be given by

νmet ) νsat exp[(Esat - Emet)/RGT0)

(28)

from which one obtains

exp(-Esat/RGT0) νsat ) ∆νmax exp[(Esat - Emet)/RGT0] - 1

(29)

where the jump frequency difference ∆νmax ) (νmet - νsat). Taking (T0/∆Tmax) ) F1/2 ) νsat/∆νmax, from eq 29, one obtains

F1 ⁄ 2 ) β exp(-Esat/RGT0) where the constant

(30)



β ) [exp{(Esat - Emet)/RGT0} - 1]-1

(31)

Taking logarithm on both sides of eq 30, one gets

ln(F1 ⁄ 2) ) ln β - Esat/RGT0

(32)

1/2

Equation 32 predicts that plots of ln(F ) against 1/T0 are linear with slope Esat/RG and intercept ln β. This linear dependence is indeed observed, as illustrated in Figure 7 from the data of F1/2 as a function of saturation temperature T0 for KTB-water and NTO-water systems. The values of constant lnβ and activation energy Esat for the two systems are listed in Table 4. In the case of NTO-water system the values of β and Esat, calculated from the linear plots of ln(F1/2) against 1/T from F1/2(T0) data for uncorrected R and corrected R*, are also included, but these data are not shown in Figure 7. It should be mentioned that eq 32 predicts a linear dependence when the constant β is independent of saturation temperature T0. Deviations from the linear fit, as observed from a somewhat poorer fit of the data for KTB-water system, may be attributed to the dependence of β on T0. From Table 4 the following features may be noted: (1) The values of β and Esat, calculated from F1/2(T0) data for uncorrected R and corrected R* are practically the same. This means that even the uncorrected F1/2(T0) data may be used to obtain β and Esat. (2) The values of β and Esat differ enormously from each other for the two systems. This difference is expected from the physicochemical properties of the two solute because of the constitution of their ions/molecules, i.e. K2B4O7 and NTO molecules. For example, potassium tetraborate is highly soluble in water and crystallized from aqueous solutions as K2B4O7 · 4H2O, whereas NTO is sparingly soluble. It is known (section 6.4.2, ref 20) that the activation energy ED for self-diffusion in pure liquid electrolytes (e.g., Na in NaCl) is usually a constant, independent of temperature, and follows the relation

ED/RG ) 3.7Tm

(33)

where Tm is the melting point of the electrolyte. Since the liquid in which diffusion of solute ions/molecules occurs is water,

Figure 7. Plots of ln(F1/2) against 1/T0 for KTB-water and NTO-water systems (cf. eq 32). Table 4. Values of ln β and Esat of Equation 32 for Two Systems system

ln β

β

Esat/RG (103 K-1)

Esat (kJ · mol-1)

RC

KTBc NTOa,d

23.9 ( 2.7 8.03 ( 0.09 8.13 ( 0.07

2.4 × 1010 3.1 × 103 3.4 × 103

6.35 ( 0.86 1.40 ( 0.03 1.42 ( 0.03

52.7 ( 7.2 11.7 ( 0.3 11.8 ( 0.3

0.982 0.999 0.999

b a

b Corrected data, Uncorrected data. d tetraborate-water. NTO: nitroazolone-water.

c

KTB:

potassium

taking Tm ) 373 K for water, from eq 33, one obtains ED ) 13.8 kJ · mol-1. This value of ED is in good agreement with Esat ) 11.8 kJ · mol-1 for NTO-water system (deviation 14% only!). In fact, this agreement is not surprising because eq 32 was derived on the assumption of diffusion of solute ions/molecules in the solution. Equation 33 has been interpreted in terms of the hole theory of liquids, and holds good for simple liquids in which the size of the holes is similar to that of ions/molecules which jump into them (section 6.5.6, ref 20). This ideal behavior of diffusion occurs at very low solute concentrations when solute ions/ molecules do not associate with solvent molecules forming large-sized solvated entities and with themselves forming largesized complexes. This situation occurs in the case of NTO-water system. As indicated by low solubility of NTO in water, solute-solvent interactions are poor in this system. In contrast to this NTO-water system, solute-solvent interactions are strong in KTB-water system and this is the cause of high solubility of KTB in water system. It is well-known (section 3.8.7, ref 20) that, with increasing concentration of fairly soluble solutes like potassium tertraborate, electrostatic interactions between individual oppositely charged ions result in clusters of two, three or more ions. A departure from the ideal behavior of diffusion in the KTB-water system may be attributed to the participation of such clusters in diffusion, requiring higher active energy Esat (section 6.6.2, ref 20). Summary and Conclusions Ny´vlt’s approach of interpretation of metastable zone width is based on the assumptions that: (1) when cooled at a constant cooling rate R, a solution saturated at temperature T0 develops a solution concentration ∆c which is proportional to supercooling ∆T, and (2) in the vicinity of metastability, the nucleation rate J is related with maximum solution supersaturation ∆cmax by a power-law relation and cooling rate R (see eq 2a and eq 2b). The main drawbacks of this approach are that: (1) the physical significance of the nucleation constant K and the nucleation order m of the empirical power-law relation (2a) is not known, and (2) the power-law relation (2a) between nucleation rate J and solution supersaturation ∆c do not reproduce satisfactorily the data on the dependence of nucleation rate J on supersaturation lnS predicted by the classical theory of 3D nucleation. In the present paper, a new approach is proposed to predict metastable zone width using: (1) the theory of regular solutions, where the nucleation rate is presented as a function of cooling rate R, saturation temperature T0 and heat of dissolution ∆Hs, and (2) the supersaturation dependence of nucleation rate J is given by the classical theory of three-dimensional nucleation. The final expression predicts that the quantity (T0/∆Tmax)2 decreases linearly with an increase in lnR, with slope F1 and intercept F (see eq 21). The value of the slope F1 depends on the effective interfacial energy γeff and the heat of dissolution ∆Hs, while the intercept F depends on γeff, ∆Hs, the kinetic factor A associated with the integration of growth units to the growing nuclei, and on the factor f determined by aggregation and diffusion processes in the solution. Obviously, F1 is determined by thermodynamic and solvation processes, while F is governed by thermodynamic and kinetic parameters as well as processes associated with solvation of solute ions/molecules and their transport in the solution. The ratio F1/F is a constant characteristic of the compound, and is independent of saturation temperature T0. However, as seen from the effect of ultrasound power on metastable zone


width in KSF-water system, the value of the ratio F1/F also depends on factors such as presence of seeds, presence of impurities and stirring, which inhibit or promote nucleation in a system. Analysis of the published ∆Tmax(R) data, according to eq 21, revealed the presence of two specific cooling rates: (1) threshold cooling rate Rc, and (2) maximum cooling rate Rmax. The threshold cooling rate Rc and the maximum cooling rate Rmax correspond to the lower and the upper limits of the cooling rate R when relation (21) between (T0/∆Tmax)2 and ln R applies. Physically, Rc and Rmax correspond to situations when a system begins and ceases to respond to the cooling procedure, respectively. The former cooling rate is associated with the setting up of a thermal equilibrium between the solution and the environment, while the latter is connected with the induction period. The value of Rmax is characteristic of a solute-solvent system. The new approach can be used to predict the effect of saturation temperature T0 on the metastable zone width as well as to predict metastable zone width by isothermal method (see eq 23). Acknowledgment. The author is indebted to the anonymous referees for their valuable advice and suggestions for the improvement of the manuscript. He also expresses his gratitude to Dr K. Wo´jcik for his constant support and technical assistance.

Sangwal

References (1) Ny´vlt, J.; So¨hnel, O.; Matuchova, M.; Broul, M. The Kinetics of Industrial Crystallization, Academia: Prague, 1985. (2) Sayan, P.; Ulrich, J. Cryst. Res. Technol. 2001, 36, 411–417. (3) Rajesh, N. P.; Lakshmana Perumal, C. K.; Santhana Raghavan, P.; Ramasamy, P. Cryst. Res. Technol. 2001, 36, 55–61. (4) Sahin, O.; Dolas, H.; Demir, H. Cryst. Res. Technol. 2007, 42, 766– 772. (5) Omar, W.; Ulrich, J. Cryst. Res. Technol. 2007, 42, 432–439. (6) Ananda Babu, G.; Ramasamy, P. Cryst. Res. Technol. 2008, 43, 626– 633. (7) Li, G.; Xue, L.; Su, G.; Li, Z.; Zhuang, X.; He, Y. Cryst. Res. Technol. 2005, 40, 867–870. (8) Graber, T. A.; Taboada, M. E.; Alvarez, M. N.; Schmidt, E. H. Cryst. Res. Technol. 1999, 34, 1269–1277. (9) Lyczko, N.; Espitalier, F.; Louisnard, O.; Schwartzentruber, J. Chem. Eng. J. 2002, 86, 233–241. ¨ zdemir, B. J. Cryst. Growth 2003, 252, 343–349. (10) Gu¨rbu¨z, H.; O (11) Marciniak, B. J. Cryst. Growth 2002, 236, 347–356. (12) Taboada, M. E.; Palma, P. A.; Graber, T. A. Cryst. Res. Technol. 2003, 38, 21–29. (13) Kubota, N. J. Cryst. Growth 2008, 310, 629–634. (14) Kim, K.-J.; Mersmann, A. Chem. Eng. Sci. 2001, 56, 2315–2324. (15) Mersmann, A.; Bartosch, K. J. Cryst. Growth 1998, 183, 240–250. (16) Sangwal, K. J. Cryst. Growth 1989, 97, 393–405. (17) Sangwal, K.; Mielniczek-Brzo´ska, E. J. Cryst. Growth 2004, 267, 662– 675. (18) Kashchiev, D. Nucleation: Basic Theory with Applications, Butterworth-Heinemann: Oxford, U.K., 2000. (19) Omar, W.; Ulrich, J. Cryst. Res. Technol. 2003, 38, 34–41. (20) Bockris, J.O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum: New York, 1970; Vol. 1.

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