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Generic Progressive Heterogeneous Processes in Nucleation X. Y. Liu Department of Physics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received October 12, 1999. In Final Form: May 23, 2000 The generic heterogeneous effect of foreign particles on 3D nucleation was examined in terms of the relative size and the interfacial interaction and structural match with the crystallizing phase in terms of a so-called interfacial correlation function f(m,x). At low supersaturation, nucleation is controlled by foreign bodies with a relatively small curvature and/or an optimal interfacial interaction and structural match with the crystallizing phase. This is characterized by a small interfacial correlation function f(m,x) (f(m,x) f 0). At high supersaturation, nucleation on foreign particles having a weak interaction and poor structural match with the nucleating phase (f(m,x) f 1) will govern the kinetics. It follows that if a wide range of supersaturations are applied in a crystallization system, a sequence of progressive heterogeneous nucleation processes, characterized by different interfacial correlation function f(m,x)s, can be identified, where genuine homogeneous nucleation can be regarded as the up-limit of heterogeneous nucleation. To check the above results, experiments on the nucleation of organic crystals and CaCO3 from aqueous solutions were carried out under gravity and microgravity. The results are in excellent agreement with our prediction.
I. Introduction Nucleation plays an important role in almost all aspects of materials science.1-8 The relevancy of the prediction and control of nucleation behavior goes across many areas of modern surface and materials science and technologies. Depending on the role of foreign bodies, nucleation can occur via either homogeneous nucleation or heterogeneous nucleation. The first corresponds to the process where the nucleation probability is homogeneous thorough out the system, while the latter corresponds to the process where the probability is much higher around some foreign bodies than in other parts of the system. In the past 30 years, although many models have been published to describe the kinetics of nucleation,4-7 much confusion remains. The major issue associated with this subject is the effect of foreign bodies on the general kinetics of heterogeneous nucleation. In most cases of crystallization, it is almost impossible to remove completely from crystallizing systems foreign bodies. These foreign bodies range from solid or liquid particles, gas bubbles, macromolecules, the wall of crystallization vessels, and even the clusters of metaphase of nucleating materials. In fact, the effect of these foreign bodies on nucleation is far beyond the range which current published models can cover. Concerning the impact of foreign bodies on nucleation, the attention was focused solely on the effect on (1) Allen, T. M. Curr. Opin. Colloid Interface Sci. 1996, 1, 645. (2) Hascal, V. C.; Lowther, D. A. In Biological Mineralisation and Demineralization; Nancollas, G. H., Ed.; Springer-Verlag: Berlin, 1982; p 179-198. (3) Arends, J. In Biological Mineralisation and Demineralization; Nancollas, G. H., Ed.; Springer-Verlag: Berlin, 1982; pp 303-324. (4) Chernov, A. A. Modern Crystallography III-Crystal Growth; Springer-Verlag: Berlin, 1984. (5) Mutaftschiev, B. In Handbook on Crystal Growth; Hurle, D. T. J., Ed.; North-Holland: Amsterdam, 1993; p 187. (6) Lacmann, R.; Schmidt, P. In Current Topics in Materials Science; Kaldis, E., Scheel, H. J., Eds.; North-Holland: Amsterdam, 1977; pp 301-325. (7) Skripov, V. P. In Current Topics in Materials Science; Kaldis, E., Scheel, H. J., Eds.; North-Holland: Amsterdam, 1977; pp 327-378. (8) Chayen, N. E. J. Appl. Crystallogr. 1997, 30, 108-203.
the nucleation barrier.9-18 The nucleation rate is given, according to classical heterogeneous nucleation theories,4,5,9-18 as
{
}
Ω2γcf3 16π J ) B exp -f 3 (kT)3[ln(1 + σ)]2
(1)
with
0 1), then Z ∼ n1, thus
The first boundary condition applies because whenever a (gc - 1)-mer becomes a gc-mer, it disappears from our population to begin a new stagesgrowth. But this event does not disturb the distribution of other embryos (mainly monomers). The second boundary condition applies because both ng and fg are large numbers when g is small. It follows that the perturbed concentration of monomers is almost equal to the equilibrium concentration. Proceeding, rearranging eq (14)
ng = n1 exp(-∆Gg/kT)
J′ ) Rgng[(fg/ng) - (Rg + 1/Rg)(fg + 1/ng)]
(for all g, g ) 2, 3, 4, ...) with the effective total number of “molecules” per unit volume
Z ) n1 +
∑ ng
(10)
g)2
(11)
Equation (11) is the expression usually quoted for the distribution of embryo sizes in supersaturated media. We notice that embryos are not static entities in the microsense. Each particular one is growing larger or smaller as molecules are added or leave the embryo, respectively. To obtain the rate of formation of critical size embryos, we need to derive a general expression of the growth rate of a single embryo. For this purpose, we will apply the approach of the three-dimensional homogeneous nucleation to a heterogeneous nucleation case.19 Let R be the rate of molecule addition. That is
R ) βst′ ν*
(12)
where ν* is the collision rate of monomers with an embryo and βst′ is the sticking probability at the surface of embryo, which is defined as
βst′ ) ν
a exp(-∆G* step/kT) λo
(13)
(ν denotes the vibration frequency of structural units in the neighborhood of the surface, a is the dimension of structural unit in the direction parallel to the crystal surface, λo is the average distance between two kinks at the surface, and ∆G* step is the activation free energy for step integration). Also let R′ be the rate at which embryos lose molecules. Rates R and R′ should be equal if the system were in equilibrium since no embryo would then experience a net growth or disintegration. For an embryo of radius r, the value of R is determined by the surface concentration of monomers, n1. Thus, at the detailed balance between the growth and disintegration of embryos, one has
Rgng - R′g+1ng+1 ) 0
and then using eq 18 we obtain
J′ ) Rgng[(fg/ng) - (fg + 1/ng + 1)]
(19)
Substituting g ) 1 ∼ gc - 1 into eq (19), a set of equations can be obtained. Summing up these equations, and replacing approximately the sum by the integral, one will obtain
J′ ) [
∫1g (agng)-1 dg]-1 c
(20)
The evaluation of the integral in eq (20) is somewhat difficult. But it can be simplified by determining the dominant terms within the range from g ) 1 to gc, on which we can focus our attention. Certainly ng decreases exponentially with g so that for large values of g the inverse ng term is large. Therefore, the key issue is to find the variation of Rg with g. According to eq (11), to find Rg, we need to derive the expression of ν*. ν* can be defined as the frequency of growth units colliding into the surface of an embryo with a radius of r. Obviously, this frequency is proportional to the surface area of the embryo. For an embryo created on a spherical substrate as shown in Figure 1, this area is given by
Scf ) 4πr2f ′′(m,x)
(21)
with
f ′′(m,x) )
1 + (1 - xm)/w 2
(22)
It follows then that
(14)
On the other hand, if there is growth, then the equilibrium distribution is perturbed. Let fg be this perturbed surface concentration of g-mers (fg e ng). Assuming a steady-state growth process, J, the formation rate of critical nuclei per unit volume-time around a (19) McDonald, J. E. Am. J. Phys. 1962, 30, 870.
(18)
ν* = n14πr2f ′′(m,x)
(23)
Obviously, ν* (or Rg) increases as the cubic root of g (r ∼ g1/3). For large values of g, Rg is then only a weak function of g. Returning to eq (20), if regions of high g contribute most and, in these regions, Rg is not a strong function of g, one may remove Rg from under the integral sign and, to
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approximate it as a constant equal to its value when r ) rc, i.e.
Rg ∼ Rgc ) 4πaβst′n1(rc)2f ′′(m,x)
(24)
As long as we have concluded that the important region for g in the integral of eq (20) is where g ∼ gc, we can simplify the evaluation by rewriting it, as
∫1g (ng)-1 dg
(J′)-1 ) Rgc-1
c
(25)
and with eq (10)
∫1g
(J′)-1 ) (Rgcn1))1
c
exp(∆G/kT) dg
(26)
Expanding ∆G about gc in a Taylor series, we find
∆G ) ∆Gc + (∂∆G/∂g)c(g - gc) + (1/2)(∂2∆G/∂g2)c(g - gc)2 + ‚‚‚ (27) Define
y ≡ g - gc
(28)
then 0 exp[(∆G′)c(y)/kT + ∫1-g
J-1 ) (Rgcn1)-1[exp(∆Gc/kT)]
2
(∆G′′)c(y) /2kT + ‚‚‚] dy (29) But ∆G′ at g ) gc is zero, and truncating after the second term. Approximating the lower limit as -∞, then
J′ ) 2Rgc[Q/(2πkT)]1/2
(30)
Q ) -(∂2∆G/∂g2)g)gc
(31)
where Q is positive so that the integral is an error function. The last remaining step is to evaluate Q. For the heterogeneous nucleation, the second derivative of ∆G is very complex. Note that the derivatives are evaluated at g ) gc and so are not a function of g. In most cases (where x is not small or large), f(m,x) turns out to be constant. We can approximate the value of (∂2∆G/∂g2)g)gc by f(m,x)(∂∆Ghomo/∂g2)g)gc; therefore Q can be expressed as
( ) 25πΩ2 34
Q ≈ γcf
1/3
g-4/3 f(m,x)
(32)
or, with eq 7
Q ≈ γcfΩ2f(m,x)/(2πrc4)
(33)
By combining eqs 23-25 and 30, an expression for J′ is obtained, as
[
J′ ) 4aβst′f ′′(m,x)Ω
] [
γcf f(m,x) kT
1/2
(n1)2 ×
exp -
]
∆Ghomo c f(m,x) (34) kT
The average nucleation rate in the solution depends on the density and size of foreign particles occurring in the system and is given, according to J ) 4πa(Rs)2NoJ′, by
Figure 2. Dependence of factor f(m,x) on the relative particle size x ) Rs/rc and the interaction parameter m between the nucleating phase and the foreign particle.
J ) 4πa(Rs)2 No f ′′(m,x)1/2B × 16πγcf3Ω2 f(m,x) (35) exp 3kT[kT ln(1 + σ)]2
[
]
with
( )
B ) (n1)24aβst′Ω
γcf kT
1/2
(36)
Introducing the term 4πa(Rs)2No is based on the fact that the heterogeneous nucleation takes place only in the liquid layers adjacent to the nucleating particles. Evidently, only for this part of solutions, the nucleation rate will be effectively influenced by the foreign particles. Therefore, the relative effective volume fraction for heterogeneous nucleation is proportional to the density and surface area of nucleating particles occurring in the system, namely, 4πa(Rs)2No. In the case of homogeneous nucleation, one has f ′′(m,x) ) f(m,x) ) 1, and 4πa(Rs)2No f 1. In this case, eq (35) is converted to (1) (f ) 1). This implies that eq (35) is capable to describing both homogeneous and heterogeneous nucleation. III. Generalized Role of Foreign Bodies As indicated in eqs (1) and (35), f(m,x) and f ′′(m,x) characterize the major difference between homogeneous and heterogeneous nucleation kinetics. As mentioned before, the occurrence of foreign particles will lower the nucleation barrier, resulting in an increase in nucleation rate. This effect, characterized by f(m,x), will be escalated by maximizing the interaction between the crystal phase and the substrate and/or increasing the relative size of foreign particles x (Rs/rc). The dependence of f(m,x) on m and x is shown in Figure 2. A similar plot can also be obtained between f ′′(m,x) and m and x. On the other hand, foreign particles exert also a negative impact on the nucleation kinetics. As shown in Figure 1, nucleation on a foreign particles will cause a reduction in the “effective surface of embryos”, where the growth units are incorporated into the embryos. This tends to slow the nucleation kinetics, which cancels the effect of lowering
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ln τ ∼ ln(1/J) )
16πγcf3Ω2
f(m,x) 3kT[kT ln(1 + σ)]2 ln{4πa(R2)No f ′′(m,x)[f(m,x)]1/2B} (37)
Figure 3. Plot of ln τ vs 1/[ln(1 + σ)]2 at different m. The plot illustrates the effect of interfacial parameter m on the nucleation rate at different supersaturations. Foreign particles with a high m (strong interaction and better structural match with the crystallizing phase) will control the kinetics at low supersaturation while those with a low m will control the kinetics at high supersaturation. Rs ) 1000a, γcfΩ2/3/kT ) 1.5 (a is the demension of growth units).
the nucleation barrier. This negative effect is mainly described by f ′′(m,x) and f(m,x) in the pre-exponential term of J given by eq (35). In this concern, the increase of m and/or x will slow the nucleation kinetics (see eq (35)). In different regimes, these factors will control the nucleation kinetics in very different ways. At low supersaturation, the nucleation barrier is very high (cf. eq (2)). The top priority to accelerate the kinetics is to lower the nucleation barrier. Therefore heterogeneous nucleation will occur preferentially on the particles with a low f(m,x). Conversely, at higher supersaturations, the exponential term becomes unimportant. Due to the pre-exponential factors f(m,x) and f ′′(m,x), nucleation on the substrates having a larger f(m,x) and f ′′(m,x) becomes kinetically favored. In the following, we will examine the effect of the size and surface structural properties of foreign particles on the nucleation kinetics under different supersaturations. To compare with experimental results, we will look at the relation between the incubation time of nucleation τ (∝1/ J) and supersaturation. Given that τ is proportional to 1/J, we should obtain a linear relation between ln τ and 1/[ln(1 + σ)]2 for a given f(m,x)
A. The Interaction and Structural Match between the Crystalline Phase and the Substrate. In Figure 3, ln τ (ln(1/J)) is plotted via 1/[ln(1 + σ)]2 for different values of m. As expected, for a nucleating system, foreign bodies having different surface structures and properties will control nucleation at different supersaturations. Indeed, nucleation on the foreign particles with a strong interaction and good structural match between the crystalline phase and the substrate (m f 1, small f(m,x) and f ′′(m,x)) will be dominant at low supersaturation, whereas nucleation with a weak interaction and/or poor structural match between the crystalline phase and the substrate (m f 0, large f(m,x) and f ′′(m,x)) becomes kinetically favorable at high supersaturation. We notice that epitaxial growth and crystallization templating is based on the principle of heterogeous nucleation.4-7 To facilitate the nucleation and growth of the desired crystal phase or the specific crystallographic orietation, people always choose the substrates with m f 1 (f(m,x) f 0). This implies that a good structural match is required, a part from strong atom-atom interactions (see Figure 4.) Nevertheless, if we wish to tranfer the structural information from the substrate to the crystallizing phase, or in other words, if we wish obtain a better structural match, the supersaturation should not be too high, otherwise the kinetics lead to a significant deviation of growing crystals from the desired orientation of the substrates (m f 0, -1, f(m,x) f 1; cf. Figure 2). B. Nonlinear Size Effect of Foreign Particles. f(m,x) and f ′′(m,x) are determined not only by the interaction parameter m but also by the relative size of foreign particle x. According to (3), the relative size of foreign particle x is defined as the ratio between the radius of the particle Rs and the critical radius of nuclei. Therefore, f(m,x) and f ′′(m,x) are functions of m, Rs, and σ. The change of particles size Rs will alter x and consequently f(m,x) and f ′′(m,x). This will correspondingly change J and τ. Figure 5a shows that change of J with supersatuartion at different R s. As shown in the figure, the increase of particle size will to an extent lower f(m,x) and f ′′(m,x) and promote the nucleation rate at low supersaturation. In contrast, the reduction in particle size will enhance f(m,x) and f ′′(m,x) (see Figure 2). Similar to m, it follows that particles or substrates with a large curvature will play a dominant role at high supersaturation. (See Figure 5a.)
Figure 4. Illustration of the relationship between structural match (δ) and interfacial paramenter m and the interfacial correlation function f(m,x). A small δ corresponds to a good structure match between the substrate and nucleus, which results in a large m and a very small f(m,x). In this case, the structural information is transferred to a high degree from the substrate to the nucleating phase.
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Figure 5. Effect of particle size on the nucleation rate at different supersaturations. (a) Foreign particles with a large radius of curvature will control the kinetics at low supersaturation while those with a small radius of curvature will control the kinetics at high supersaturation. m ) 0.8, γcf/kT ) 1.5. (b) The effect of particle size will result in the nonliberality of ln τ ∼ 1/[ln(1 + σ)]2.
The second conclusion is pratically very important. In many cases, such as biomineralization, nucleation often occurs at organic matrices instead of at relative flat biomembranes.2 On the basis of the principle of our model, it can be understood that this is attributed to the high supersaturation which causes the occurrence of nucleation at the substrate with a large curvature. It should be noted that for the system with foreign particles of a given size and surface properties (m and Rs are constatnt), f(m,x) and f ′′(m,x) will still change with supersaturation due to the variation of rc. Unlike the effect of interfacial property m, the plot of ln τ vs 1/[ln(1 + σ)]2 will not give rise to a linear relation, since f(m,x) and f ′′(m,x) are fuctions of σ. This nolinearity can be clearly identified in Figure 5b. However, the curve corresponding to Rs ∼ 100a in Figure 3 turns out to be most linear. This can be attributed to the following two reasons. (1) Looking at Figure 2, f(m,x) becomes almost constant for a given m if the radius of particles Rs is much larger than the critical radius of nuclei rc (Rs g 50rc). If the variation of rc takes place in this regime, a linear relation between ln τ and 1/[ln(1 + σ)]2 should be obtained. On the other hand, if the foreign particles are very small and the supersaturation is low, Rs can become comparable to the
critical radius of nuclei rc (Rs e 30rc). In this case, f(m,x) will change drastically with supersaturation, and ln τ ∼ 1/[ln(1 + σ)]2 appears as a nonlinear curve rather than a straight line (Figure 5b). (2) The parabolic relation between rc and σ (see eq (7)) will have the following implications: rc varies drastically with σ at low supersaturation and becomes approximately constant at high supersaturation. Therefore, we can expect the nonlinear size effect will be predominant at low supersaturation. From the above analysis, we can conclude that if the size of foreign particles has a significant effect on nucleation, a nonlinear relationship between ln τ and 1/[ln(1 + σ)]2 should be expected. C. Homogeneous Nucleation: The Upper Limit of Heterogeneous Nucleation. From the discussions given above, we can conclude that in a wide range of supersaturations, instead of a single process, heterogeneous nucleation will appear as a sequence of progressive processes which reveal a wide spectrum of heterogeneous characteristics described by f(m,x) and f ′′(m,x). If we regard growth as the lower limit of heterogeneous nucleation (f(m,x) ) 0), and homogeneous nucleation as the upper limit of heterogeneous nucleation (f(m,x) ) 1),
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Chart 1
heterogeneous nucleation can then be regarded as one of the most important processes in crystallization in the whole range of supersaturations. This can be summarized in Chart 1. In this sense, we can consider heterogeneous nucleation as one of the most general processes occurring in crystallization. IV. Experiemets To check above results, experiments on the nucleation of N-lauroyl-L-glutamic acid di-n-butylamide in isostearyl alcohol solutions and CaCO3 were carried out, under both gravity and microgravity. (1) Nucleation of N-Lauroyl-L-glutamic Acid Di-n-butylamide. N-Lauroyl-L-glutamic acid di-n-butylamide (>95%) and isostearyl alchohol (>99.0%) are obtained from Aijnomoto and Unichema, respectively. The induction time for N-lauroylL-glutamic acid di-n-butylamide nucleation was observed by the change of the gelation property of the solution. This is because N-lauroyl-L-glutamic acid di-n-butylamide crystals form fiber networks. Once the nuclei occur in the solution, the growth will take place rapidly. It was observed that a tiny amount of the crystals occurring in the system will gel liquid locally. From the change of the rheology property of the solutions, we can then measure the incubation time of nucleation. (2) Nucleation of CaCO3 under Gravity and Microgravity. The nucleation kinetics of CaCO3 from aqueous solutions was examined by Tsukamoto and co-workers using an advanced fast dynamic light scattering method (FDLS).20,21 This method allows to measure the size and size distribution of nucleus in the range from a few nanometers in radius to a few tens of micrometers with the successive time interval of a few seconds. Therefore, an in situ measurement of nucleation and size increase of nuclei can be made. To obtain a comprehensive and generic understanding, the experiments were carried out under both gravity and microgravity. In the experiments, the ionic strength is fixed at 0.11 M. The nucleation experiments were carried out in using the FDLS and a stopped-flow system. 20,21 To start with the experiments, the solutions were mixed and introduced to a rectangular glass cell (10 × 10 × 44 mm3) for light scattering experiments. The measurement of incubation time τ started immediately soon after two solutions were mixed together. The stopped-flow system allows mixing two solutions homogeneously in about 20 ms, which leaves sufficient time for rapid measurements. A good reproducibility is obtained by using this experimental setup.20,21 For more details about the experiments, see refs 20 and 21.
V. Results and Discussion A. Progressive Heterogeneous Nucleation. We have in the previous section shown that foreign bodies (20) Tsukamoto, K. Extended Abstract of AIST Workshop; Oct. 1998, Sapporo, pp 29-33. (21) Tsukamoto, K.; Maruyama, S.; Shimizu, K.; Kawasaki, H.; Morita, T. S. Reports of Microgravity Experiments by Aircraft 1998, 7, 51-57. Liu, X. Y.; Tsukamoto, K.; Sorai, M. Langmuir 2000, 16, 54995502.
Figure 6. Dependence of ln τ on 1/[ln(1 + σ)]2 for N-lauroylL-glutamic acid di-n-butylamide nucleating from isostearyl alcohol solutions. Within the range of supersaturations where the experiments were carried out, three straight lines with different slops intercept with each other, dividing the space into three regimes. This indicates that three different f(m,x)s control nucleation in three different regimes in a similar way as predicted in Figure 3.
Figure 7. ln τ vs 1/[ln(1 + σ)]2 is plotted for the nucleation of CaCO3 in gravity and microgravity. In gravity, three straight lines of different slops intercept with each other, dividing the space into three regimes. Under gravity, the slope is four times larger than the similar experiment obtained from the gravity experiments. In regime III the slope for gravity is 12.5 and for microgravity is 42.
with different f(m,x) and f ′′(m,x) will control nucleation in different regimes. According to eq (37) and Figure 3, the slope of ln τ ∼ 1/[ln(1 + σ)]2 is proportional to f(m,x) for a given system. A large slope corresponds to a large f(m,x). Therefore, the relative f(m,x) at different supersaturations can be obtained by measuring the change of the slope. The nucleation induction time of N-lauroyl-L-glutamic acid di-n-butylamide in isostearyl alcohol solutions τ was measured as a function of supersaturation σ in Figure 4. The results are in plotted a form of ln τ vs 1/[ln(1 + σ)]2 in Figure 6. As shown by the graph, three straight lines with different slops intercept with each other, which is similar as we predicted in section III. This indicates that foreign bodies with three different f(m,x) values control nucleation in three different regimes, namely, regime I,
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Figure 8. Microgravity will suppress heterogeneous nucleation and promote homogeneous nucleation: In gravity, convection, due to temperature or concentration gradient caused by nucleation, will help to transport growth units to the surface of substrates. This will compensate for the concentration (or supersaturation) depression during nucleation so that the effective supersaturation at the surface of foreign bodies is approximately the same as the bulk supersaturation. In this case, homogeneous supersaturation occurs at very high supersaturations if σ e σhomo (in gravity, σhomo ) [σhomo]gravity). Under microgravity, convection due to the temperature or concentration gradient is suppressed. This leads then to a drop of the effective surface supersaturation. To reach a nucleation rate similar to that obtained under gravity, much high bulk supersaturation is required. Therefore, heterogeneous nucleation is suppressed to a large extent by gravity. On the other hand, gravity has almost no effect on homogeneous nucleation. Therefore, σhomo ) [σhomo]microgravity < [σhomo]gravity. Homogeneous nucleation may be achieved more easily. Table 1. f(m,x) for Nucleation in the Three Different Regimes under Gravity f(m,x)
regime I
regime II
regime III
0.162
0.179
0.298
II, and III. If we denote f(m,x) and f ′′(m,x) occurring in these three regions as [f(m,x)]I, [f ′′(m,x)]I; [f(m,x)]II, [f ′′(m,x)]II; and [f(m,x)]III, [f ′′(m,x)]III, respectively, according to the stopes of the straight lines, we have then [fJ(m,x)]I < [fJ(m,x)]II < [fJ(m,x)]III (and [f ′′(m,x)]I < [f ′′(m,x)]II < [f ′′(m,x)]III). This agrees completely with our predictions given in section III. (See Figure 3). Similar results were also achieved for CaCO3 nucleation. Looking at the linearity of ln τ vs 1/[ln(1 + σ)]2 obtained in Figure 6, we can conclude that the change of f(m,x) at different supersaturations is attributed to the change in m rather than the size of foreign bodies. B. Microgravity Effect and Solid-Fluid Interfacial Free Energy. For the sake of the solid-fluid interfacial free energy measurement, and the control of nucleation, whether and how genuine homogeneous nucleation can be obtained is very crucial (cf. section I). In the presence of foreign bodies, to achieve genuine homogeneous nucleation may require substantially higher supersaturations, which may be difficult to achieve. To remove these particles by filtering solutions may be one of the key steps in achieving homogeneous nucleation. On the other hand, the effect of the interfaces between crystallization systems and their surroundings (such as the wall of crystallization vessels) on nucleation is impossible to be eliminated under gravity. The nucleation experiments of CaCO3 show that microgravity has a promoting effect for homogeneous nucleation. In Figure 7, ln τ ∼ 1/[ln(1 + σ)]2 was plotted for the nucleation of CaCO3 under both gravity and microgravity with the similar supersaturation regime. So¨hnel and Mullin argued that under gravity the nucleation of CaCO3 within such a regime was virtually homogeneous nucleation.7 They applied then eq (1) (f ) 1) to estimate the interfacial free energy of CaCO3 crystals from the data in regime III (under gravity). As shown by Figure 6, under microgravity, the slope of line ln τ vs 1/[ln(1 + σ)]2 is four times larger than that obtained under gravity. This implies
that under gravity, we still have f(m,x) ∼ 1/4 in the supersaturation range, meaning that the nucleation is still heterogeneous nucleation in nature. On the other hand, the nucleation under microgravity is very much close to genuine homogeneous nucleation. It is indicated that the interfacial free energy obtained from the microgravity experiments is very close to that obtained from the theoretically estimated interfacial tension.23 On the basis of this result, f(m,x) for the nucleation in the three different regimes under gravity can be obtained from the slopes of the straight lines in Figure 7. The results are given in Table 1. The microgravity effect on nucleation can be understood in terms of convection. Under gravity, convection due to temperature or concentration gradient caused by nucleation and growth helps to transport growth units to the surface of substrates. This will compensate for the concentration (or supersaturation) depression during nucleation. It will also help to bring new seed crystals from substrate surface to the bulk, which virtually enhances the “number” of effective foreign bodies for heterogeneous. In general, convection will substantially promote the effect of foreign bodies, such as tiny particles and organic bodies or the wall of crystallization vessels, which inevitably occur in all crystallization systems. Once heterogeneous nucleation occurs at relatively low supersaturation, nucleating materials are consumed quickly. This leads then to the drop of bulk supersaturation. Therefore, the supersaturation required for genuine homogeneous nucleation becomes extremely difficult to reach. Under microgravity, the convection is suppressed. This was observed in our microgravity experiments. The image obtained in microgravity is much more homogeneous compared with the image obtained in normal gravity. The inhomogeneity in normal gravity is caused mainly by an upstream convection plum. This indicates that the convection due to concentration and temperature gradient is eliminated in microgravity. Associated with this, the gravitational sink disappears in microgravity, which is very prominent in gravity when the diameter is 600 nm (22) Tsukamoto, K.; Liu, X. Y. To be submitted for publication. (23) Michimnomae, M.; Mochizuki, M.; Ataka, M. J. Cryst. Growth 1999, 197, 257-262.
Heterogeneous Processes in Nucleation
for CaCO3. Microgravity, therefore, will significantly slow the transport of growth units toward the substrate surface and cause the concentration (or supersaturation) depression during nucleation on the surface of substrates. This implies that in order to achieve a similar nucleation rate for heterogeneous nucleation, much larger supersaturation is needed. Therefore, in microgravity, heterogeneous nucleation is suppressed to a larger extent so that homogeneous nucleation may be achieved more easily. (See Figure 8.) We notice that the results of this work have also significant implications for the understanding of protein crystallization. It was reported24,25 that for lysozyme
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crystallization, some clusters (so-called “rise balls”) are formed prior to the crystallization within a certain regime. Once these clusters form, the nucleation and growth of lysozyme become much easier. Obviously, the clusters serve as the substrate (or precursor) for the lysozyme nucleation. LA991333G
(24) Tanaka, S.; Yamamoto, M.; Kawashina, K.; Ito, K.; Hayakawa, R.; Ataka, M. J. Cryst. Growth 1996, 168, 44-49. (25) Tanaka, S.; Yamamoto, M.; Ito, K.; Hayakawa, R.; Ataka, M. Phys. Rev. E 1997, 56, R67-R69.
