A Review of Classical and Nonclassical Nucleation Theories - Crystal

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A Review of Classical and Nonclassical Nucleation Theories S. Karthika, T. K. Radhakrishnan,* and P. Kalaichelvi Department of Chemical Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India ABSTRACT: Nucleation, the initial process in vapor condensation, crystal nucleation, melting, and boiling, is the localized emergence of a distinct thermodynamic phase at the nanoscale that macroscopically grows in size with the attachment of growth units. These phase changes are the result of atomistic events driven by thermal fluctuations. The occurrence of atomistic level events with the length scales on the order of 10−10 m and time scales of 10−13 S equivalent to the vibrational frequency of atoms makes the nucleation a very complicated phenomenon to study. Even though abundant literature is available about fundamental aspects of nucleation, the knowledge on these phenomena is far from complete. The classical pathway to nucleation which was once considered to have general applicability to all nucleating systems is gradually giving way to a nonclassical pathway which is now considered as a dominating mechanism in solution crystallization and other systems. In this review, an attempt is made to compare underlying physical principles involved in various nucleating systems and their theoretical treatment based on classical nucleation theory, and other important theories such as a density functional approach and diffuse interface theory. The limitations of classical theory, the gradual evolution of a nonclassical two-step pathway to nucleation, and the questions that have to be addressed in the future are discussed systematically.

1. INTRODUCTION The phenomenon of nucleation has gained attention in all aspects of science from organic, inorganic, protein, and mineral crystallization to more interesting examples which include the origin of life forms, volcanic eruptions, clouds, snow, rainfall, earthquakes, and the initiation of neurodegenerative diseases.1 Nucleation is the process of formation of a new thermodynamic phase from an old phase with high free energy to an organized structure or pattern with a low free energy. Nucleation is a challenging problem to describe, and the quantification of which is more difficult even for simple cases. It involves the description of events at the molecular level which is quite difficult since the size of the critical nucleus and its time scale of formation poses a challenge to the experimental and computational techniques. The concept of nucleation has a general applicability in a variety of phase transition systems,2−5 which include crystallization, melting, boiling, condensation, etc. To understand the mechanism of nucleation, fundamental theories explaining the nucleation pathways from the perspective of classical nucleation theory (CNT) and its applicability in various systems, density functional theory and diffuse interface approach to nucleation, the nonclassical nucleation pathway, and the experimental evidence in various systems are highlighted in this review.

of nucleation is the dominant mechanism in the formation of microcrystalline ceramics, explosions occurring when cold fluid contacts a hotter fluid, condensation on supersonic nozzles, and many other systems.8−10 Usually large supersaturations are needed to initiate this type of nucleation. Nuclei induced at the interface of vessel walls, dust particles, and impurities are considered heterogeneous, which can occur at a lower supersaturation compared to homogeneous nucleation. Various theories available to describe the homogeneous nucleation are based on phenomenological, kinetic, and microscopic approaches. The phenomenological models (which are not derived from first-principles) try to calculate the free energy formation of clusters based on macroscopic quantities. The kinetic theory of nucleation avoids the use of macroscopic surface tension; instead, it is based on molecular interactions.11−13 Molecular approaches which include Monte Carlo simulation and molecular dynamics (MD)14−16 use the first principles to calculate the free energy of cluster formation. CNT is the first theoretical treatment based on a phenomenological approach to nucleation, and the one that dominated the field of nucleation for many years is discussed in the next section. 2.1. Classical Nucleation Theory. CNT is most common theoretical model used to understand nucleation of a new thermodynamic phase such as a liquid or solid. It is an approximate theory, which gives reasonable prediction of nucleation rates. The CNT stems from the work of Volmer

2. NUCLEATION THEORIES Homogeneous nucleation6,7 is a significantly studied phenomenon that occurs when a system initially in a state of stable thermal equilibrium becomes metastable as a result of thermal fluctuations, and there is no role for foreign surfaces. This type © 2016 American Chemical Society

Received: May 26, 2016 Published: September 30, 2016 6663

DOI: 10.1021/acs.cgd.6b00794 Cryst. Growth Des. 2016, 16, 6663−6681

Crystal Growth & Design

Review

at a given supersaturation in the eq 3 gives the final expression for nucleation rate J, as

and Weber, Becker and Döring, and Frenkel.17−19 It is based on the condensation of vapor to a liquid which can be extended to other liquid−solid equilibrium systems such as crystallization from melts and solutions as well. The change in the free energy of the system during homogeneous nucleation of a spherical nucleus of radius r is given by ΔG =

−4πr 3 KT ln S + 4πr 2σ 3ν

⎡ 16πγ 3ν 2 ⎤ ⎥ J = A exp⎢ − 3 3 ⎣ 3KBT (ln S)2 ⎦

The formation free energy of the cluster, its size, and the rate are the key parameters in the nucleation research. The size of the critical nucleus has been approximated by various authors using experiments and simulations studies. The number of molecules constituting the critical nuclei usually falls in the range of 10−1000 molecules. The size of the critical nucleus can be as small as 10 molecules as reported by Garten and Head.20,21 One thousand molecules constitute a stable nucleus for phenol, naphthalene, and azobenzene as suggested by Otpushchennikov.22 In another study by Adamski, 10−15 g of barium salts forms critical nuclei which comprise a million of molecules.23 Simulation results have shown that the shape of the critical nucleus of amorphous silicon is nearly spherical and contains about 30−50 atoms.24 Yau and Vekilov25 reported that the size of critical nucleus of protein apoferritin from its aqueous solution is in the range of 40 nm. Some of the simplifying assumptions of CNT are (a) the nucleus can be described with the same macroscopic properties (density, structure, composition) of the stable phase, (b) the nucleus is spherical and the interface between the nucleus and the solution is a sharp boundary, and (c) vapor−liquid interface is approximated as planar, regardless of critical cluster size. This is known as capillary approximation and may be reasonable for large clusters, but for small clusters the surface is highly curved and the approximation leads to large discrepancies. The assumption of spherical shape is also not valid as in the case of NaCl which produces cubic-shaped nucleus. Also, if polymorphism is expected for a system, it may not necessarily nucleate in the stable form, but passes through a path in which the free energy barrier is minimum, which is not taken into account by CNT. Even though the theory is able to capture the underlying physics of the phenomena and provide good qualitative interpretation of nucleation data, its failure to provide a correct quantitative description led to the failure CNT for a variety of systems. Another shortcoming of the theory is that it is unable to explain the vanishing nucleation barrier at high supersaturations. In spite of various extensions and developments in theoretical approaches, CNT still serves as a platform to describe nucleation, since it is based on experimentally accessible information. Even though it provides reasonable estimates of critical supersaturations and nucleation rates for some systems such as water, experiments in various systems revealed the shortcomings in the predictive ability of CNT, which is discussed in the next section. 2.2. Predictions of CNT in Vapor−Liquid Systems. The homogeneous nucleation from the vapor phase is the simplest form of nucleation. Water is the widely investigated fluid for the nucleation from pure water vapor as well as its binary and ternary vapor mixtures, and its importance can be understood from the practical issues involving the water vapor condensation noticed in cloud formation, high speed wind tunnels, and power generation systems. Early studies used expansion cloud chambers, shock tubes, and diffusion cloud chambers to measure nucleation onsets and nucleation rates. The CNT does give fair approximations on the order of 1 cm−3 s−1 to the data obtained from diffusion cloud chamber experiments by various groups. Table 1 and Figure 2 indicate

(1)

where the first term represents the contribution made to ΔG by bulk free energy, 4πr3/3ν represents the number of molecules in a cluster of radius r with the volume of single molecule as ν, S = P/P* is the vapor supersaturation ratio, σ is the specific surface energy of interface between drop and the surrounding vapor. For the nucleus of larger r, the first term dominates leading to the decrease in ΔG since it represents the energy decrease upon transition from vapor to liquid. When r is less, the second term dominates which represents the creation of a new surface leading to the increase in ΔG. Thus, two terms in the eq 1 depend differently on r, and so the free energy ΔG of formation passes through a maximum = r*, where r* is the radius of critical nucleus at which the probability of formation of a nucleus goes through a minimum (Figure 1). Differentiating this equation with respect to r to find r* gives r*= 2σ/ KT ln S, and the barrier to nucleation is given by the equation ΔG* =

16πσ 3ν 2 3K 2T 2 ln S2

(2)

Figure 1. Schematic representation showing the dependence of nucleation barrier ΔG*on the radius r according to classical nucleation theory.

The critical nucleus is in metastable equilibrium with the vapor, and the addition of molecules decreases the free energy, so these nuclei are more probable. The nucleation rate, J (number of nuclei per unit volume per unit time) can be expressed in the form of the Arrhenius reaction velocity equation, ⎡ ΔG* ⎤ J = A exp⎢ − ⎥ ⎣ KT ⎦

(4)

(3)

where A is the prefactor which is determined from kinetic considerations, K is the Boltzmann constant, and T is the temperature.17 Substituting the critical nucleation barrier ΔG* 6664



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Table 1. Predictions of CNT for Water Vapor Condensation authors

equipment and temperature range

Wagner and Strey (1981)27 Miller et al. (1983)28 Viisanen et al. (1993)29 Mikheev et al. (2002)31 Brus et al. (2008)35 Brus et al. (2009)36 Manka et al. (2010)37

two piston expansion cloud chamber (276 K to 297.5 K) expansion cloud chamber (230 K to 290 K) nucleation pulse technique (217 K to 259 K) Laminar flow tube chamber (210 K to 250 K) thermal diffusion cloud chamber (290 K to 310 K) thermal diffusion cloud chamber (295 K to 320 K) Laminar flow diffusion chamber (240 K to 270 K)

nucleation rates

observation

2 × 105 to 2 × 109 rates are smaller than predicted by CNT, more deviations at high SS and nucleation rates cm−3 s−1 1 to 106 cm−3 s−1 extensive data on nucleation rates over wide range of temperatures 105 to 109 cm−3 s−1 agreement between the CNT and experiment for a temperature close to 215 K. Theoretical rates show stronger temperature dependence. 104−107 cm−3 s−1 reasonable agreement with the theory between 230 and 250 K (about 1 order of magnitude below CNT), below 220 K, CNT overestimates 3 × 10−1 to good qualitative agreement, data points are 2−3 orders of magnitude lower than CNT 3 × 102 cm−3 s−1 prediction at 290 and 320 K 3 × 10−2 to good qualitative agreement with CNT, toward lower T, deviation from isotherm 3 × 101 cm−3 s−1 increases 102 to 106 cm−3 s−1 good agreement is observed at 240 and 250 K, at high temperature CNT predicts higher rates up to 2 orders of magnitude.

orders of magnitude.41 For alcohols from methanol to nhexanol, the experimental and theoretical prediction varies significantly with their ratios ranging from 10 −10 for methanol at 273 K to 107 for n-hexanol at 257 K.42 In 1995, the research groups selected n-pentanol as the best compound to develop data over a wide range of nucleation rates. The correction factor ranges from 2 × 10−5 to 50 for the nucleation of n-pentanol from 280 to 320 K. Experiments with n-pentanol, from 260 to 290 K by another group showed that the nucleation rates are 3 orders of magnitude higher than that predicted by CNT.37,43 Figure 3 shows the comparison of results for n-pentanol by

Figure 2. Comparison of nucleation rate data of water from vapor measured by different research groups over years for different temperature ranges from 210 to 320 K [refs 37, 32, 29, 31, 28, 35, 36, 34, 30, 33 as per the order of author names mentioned in the figure]. Dashed lines in blue represent the CNT corrected for temperature dependence by Wölk and Strey. (Reproduced with permission from ref 37. Copyright 2010 AIP Publishing LLC.)

that there is a fair agreement for water vapor system measured by various groups, and it is the one system showing some agreement with CNT.26−37 In the other cases, the approximations are not quite good. The diffusion cloud chamber experiments show a fair approximation to the predicted critical supersaturations for nucleation of ethanol, methanol, and water38 and for a binary mixture of o-xylene and m-xylene, ethanol−water systems at low water concentrations.39 Although CNT could estimate the critical supersaturations in vapor− liquid systems, the nucleation rates are erroneous by many orders of magnitude. The variation of absolute nucleation rates with temperature cannot be estimated accurately. Nucleation rate data are available for n-alkanes and n-alcohols reported by different groups. All the physical properties such as vapor pressure, surface tension, and density to calculate the nucleation rates are known for the above-mentioned systems since the condensation occurs above the triple point. Early experiments on homogeneous nucleation rates for n-nonane from 233 to 315 K showed that the temperature-dependent correction factor ranging from 2 × 10−5 to 4 × 103 is required for CNT to agree with the experiments. The nucleation rates are overpredicted at high temperatures and underpredicted at lower temperatures.6,7,40 Experiments with n-alkanes from nheptane to n-nonane show a need for a correction factor of 4

Figure 3. Comparison of nucleation rates of n-pentanol reported by various research groups. Dashed lines represent predictions by CNT, and the solid line corresponds to the predictions of CNT, multiplied by 2000 [ref 44−49 and 43 as per the order of authors mentioned in figure]. (Reproduced with permission from ref 44. Copyright 2005 AIP Publishing LLC.)

various research groups over years.43−49 It is clear from the figure that the experimental rates are higher than predicted, and a factor of about 2000 is multiplied by CNT to agree with the experiment within the range. Figure 4 shows the nucleation rates for n-butanol reported for the temperature ranges between 225 K to 265 K which is fairly consistent with CNT predictions.50 For gas−liquid nucleation of nonpolar substances such as SiCl4, SnCl4, TiCl4, the predictions of CNT are moderately accurate.51,52 The critical supersaturations were in agreement with the predictions of CNT, for the nucleation of tin and titanium chloride vapors. But the CNT predictions on 6665



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concentrations, the surface is significantly enriched with ethanol (surface enrichment effect) as shown, but the classical theory predicts a significant water enrichment effect. Since ethanol has lower surface tension than water, the surface free energy of the nucleus is greatly reduced, which enhances the nucleation. This mutual enhancement of nucleation at lower ethanol activity is also observed for binary systems of water−n-alcohol (up to nhexanol), irrespective of miscibility gaps between water and higher alcohols.49 But the enhancement of nucleation is found to decrease with the increasing side chain length of alcohols. So, it is clear that the changing composition of the cluster as a function of mole fraction must be taken into account. But for an ethanol−hexanol system,55 which is nearly ideal, trends in the composition of the critical nucleus shows a qualitative agreement with the theoretical predictions. The comparison of experimental data for binary systems such as n-nonane−npropanol shows that CNT predicts the nucleation rates with reasonable accuracy or low n-nonane and high n-propanol activities. On increasing the n-nonane activity in the gas phase, the theoretical nucleation rates differ by 2 orders of magnitude from that predicted by CNT with an accurate kinetic prefactor. The total number of molecules is also approximated by CNT with reasonable accuracy.56 The value of interfacial tension was not available for all compositions in the case of partially miscible mixtures which make it difficult to apply CNT. For polar vapors such as acetonitrile, benzonitrile, nitromethane, and nitrobenzene, the predictions are not accurate. The classical theory does not properly account for the dipole− dipole interaction in calculating the free energy of the droplets.57 The vapor phase nucleation of hydrogen bonded substances such as glycols and glycerols at low temperatures is found to be much higher than that predicted by CNT.58 Strong disagreement between experiment and CNT predictions can be seen in argon nucleation from the vapor phase. Experimental measurement of nucleation of argon is very difficult because of the difficulties associated with the maintenance of cryogenic temperatures. Discrepancies of 16−26 orders of magnitude and 11−13 orders of magnitude have been reported by two different groups.59,60 Similarly for nitrogen, CNT underpredicts the rates by 9−19 orders of magnitude.61 Monte Carlo simulation studies reveal that the error in the calculation of work of cluster formation arises out of modeling the smallest clusters as liquid drops. Correction factors were needed to correct the work of cluster formation which results in the nucleation rates in good agreement with the experiments for argon and water.62 2.3. Predictions of CNT in Metallic Vapor Systems. When compared with the above-mentioned systems, CNT predictions in metallic vapors systems show larger discrepancies. Nucleation of metallic vapors is considered as a difficult phenomenon, and the surface tension of metals is about 10− 100 times higher than that of molecular fluids. CNT requires higher supersaturation to lower the energy barrier when compared with molecular fluids. The critical radius r* to form stable nucleus is very small ranging from 1 to 10. The experimental determination of the nucleation rate and the critical cluster size is very difficult. Metals vapors such as mercury, cesium, iron, silver, and sodium have been studied for homogeneous nucleation.63−67 The metal vapors and their condensed phases have different electronic structures. As the nonmetallic vapor phase condenses to a macroscopic metallic droplet, the property of the cluster varies from nonmetallic, or some intermediate state to metallic, and so the rate of

Figure 4. Experimental nucleation rate for supersaturated n-butanol vapor compared to the predictions of CNT from 225 to 265 K. It is clear from the figure that the CNT underpredicts the nucleation rates by the maximum of about only 2 orders of magnitude and predicts higher critical supersaturations. Such a quantitative agreement is quite good in nucleation theories. (Reproduced with permission from ref 50. Copyright 1994 American Chemical Society.)

the nucleation barrier did not reflect the binding energy in the critical cluster.53 The predictions for cluster properties for multicomponents mixtures are even more difficult since surface activities, nonideal mixing, and uneven distribution of components inside the clusters complicate the theoretical treatments. The first experimental observation for binary nucleation was done by Flood in 1934 for an ethanol−water system. Though fair agreement with the classical theory was noted, the discrepancies are larger than the experimental uncertainties. The major problem for theoretical descriptions in binary systems is the incorrect description of composition of critical nucleus. The free energy of formation in binary systems requires the knowledge of composition of critical nucleus to be determined accurately. For example, in ethanol−water systems,54 the surface of the cluster shows a different composition than the bulk. The variation of composition of nucleus with the gas phase activity ratio indicates a nonideal mixing behavior as shown in the Figure 5. Especially at lower ethanol

Figure 5. Composition of critical nucleus x* = neth/(nwat + neth), as a function of average composition shows sigmoid relationship as shown in red.54 This indicates that there is an enrichment effect of water on the ethanol rich side. The curve shown in black indicates the classical prediction of x*, which clearly indicates that the nucleus is enriched in water for low water partial pressures. 6666



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Figure 6. Temperature−enthalpy diagram (left) and time−temperature transformation diagram (right) showing structures that can be expected after various cooling rates (fast cooling-metallic glass forms) and also the temperatures at which such transformations takes place.

⎡ ΔG * ⎤ ⎡ ΔG* ⎤ D ⎥ exp⎢ − J ∝ exp⎢ − ⎥ ⎣ KT ⎦ ⎣ KT ⎦

nucleation and the temperature dependence of supersaturation vary accordingly. Such a transition which depends on the droplet size is sharp as it is observed in the condensation of mercury vapors, but cesium does not show such a sharp transition as that of mercury. Mercury is the first metal studied for the homogeneous nucleation in the upward diffusion cloud chamber in the temperature range of 260−400 K.63 The small clusters of mercury have a nonmetallic character which is evident from susceptibility measurements. CNT predicts a very high supersaturation when compared with molecular liquids and the size of the critical cluster containing less than 10 atoms. The measured values of critical supersaturations are about 4 orders of magnitude lower than that predicted by CNT due to the nonmetallic character of its nucleus. The study of nucleation of cesium in the cloud chamber at high temperatures and low supersaturations reveals that the capillary approximation is valid for the nucleation process in simple metals. Large clusters of metal atoms containing more than 25 atoms can be treated as liquid drops with bulk metallic properties. Below 455 K, higher supersaturations are required than predicted by the theory, due to the loss of metallic character of critical nucleus with the decreasing size.64,68 2.4. Predictions of CNT in Glass Systems. The study of kinetics nucleation is of prime importance, as it leads to crystallization which must be suppressed in the glass formation. If the crystallization proceeds in a controlled fashion, it results in the formation of ceramics. Glasses are noncrystalline amorphous solids which when heated transforms from a hard and relatively brittle “glassy” state into a molten state. Vitrification is the reverse process of cooling the molten liquid rapidly, so that it passes through the glass transition to form a vitrified solid while crystallization is suppressed. Here, there is no structural rearrangement, but the structure freezes in. Higher viscosities at the melting point and the increase of viscosities with decreasing temperature favor the formation of glass. Glass forms when the cooling is so rapid such that the nose of the TTT curve is avoided which is shown in Figure 6 (right). Glass formation can be achieved in silicate glasses at modest cooling rates since the nucleation is slow due to high viscosity and low diffusivity. But the cooling rate has to be rapid, to form metallic glasses. The steady state homogeneous nucleation rate according to CNT is given by the following equation,

(5)

The first term in the RHS of the equation arises out of the kinetic contribution, in addition to the thermodynamic barrier, which represents the barrier for an atom to cross the interface from liquid to the surface of the crystal. ⎡ ΔG * ⎤ KT D ⎥∝ exp⎢ − 3πλ 3η ⎣ KT ⎦

(6)

where η represents the viscosity of melt which is the function of temperature, and λ represents the jump distances of building units. The thermodynamic barrier to nucleation is given by 16πσ 3ν 2 ΔG* = where ΔGv represents thermodynamic driving 3ΔGv (T )

force for crystallization, which is given by the following equation for T < Tm, ΔGv Vm = ΔHm/Tm(Tm − T ) +

∫T

Tm

(ΔCp/T ) dT ′

∫T

Tm

ΔCp dT ′− (7)

The term ΔCP refers to the crystal melt difference in specific heat at constant pressure, and ΔHm represents the molar enthalpy of fusion. Most of the glasses show heterogeneous nucleation phenomenon and homogeneous nucleation are rare and exhibited only by some type of glasses, for example, lithium disilicate and sodium calcium silicate glass systems. The atomic arrangement in the nucleating systems could be similar to the structure of the final stable phase which would minimize the barrier to homogeneous nucleation. CNT often underestimates the steady state nucleation rates by several orders of magnitude even in the case of binary and ternary silicate melts. Some of the reasons for the discrepancies are (a) dependence of surface energy on temperature and the size of the nuclei, (b) elastic strain effects due to the different in specific volumes of melt and the crystal, and (c) precipitation of solid solution of varying composition. Because of the lack of direct reliable measurements of the crystal/liquid interface energy, one has to rely on the fit of experimental data to theory. When CNT is used to treat the nucleation data, the interface energy σ is commonly treated as a constant independent of size and temperature. 6667



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the size of attaching unit, showed better agreement with experimental rates.72 2.5. Predictions of CNT in Supercooled Liquid Metals and Alloy Systems. The solutions of metals and alloys can be cooled far below their equilibrium temperatures without crystallization, where a large barrier exists to the formation of a stable phase. Since the properties of the liquid and the crystal phases are quite similar, the barrier to nucleation is small, but undercooled metals and alloy solutions show unusual stability against crystallization. According to Frank, this is due to the presence of highly packed regions exhibiting slow dynamics known as icosahedral short-range order (ISRO),73 which creates a barrier to the formation of crystal phases. The first experimental study linking ISRO to nucleation barrier was performed in Ti−Zr−Ni liquid74 with decreasing temperatures. In situ X-ray diffraction studies demonstrated that the local order influences the nucleation of specific phases which show varying degrees of ISRO. In the electrostatically levitated Ti37Zr42Ni21liquid, metastable icosahedral quasicrystal called the i-phase nucleates before the stable C14 polytetrahedral crystallographic phase. The nucleation barrier for the i-phase must be lower than that of the C14 phase which indicates it is structurally more similar to liquid. The CNT often fails to consider the ordering near the interface, and the theoretical treatment of liquid/solid phase transitions in these systems should consider the fluctuations in local order parameters.75 In the natural volcanic magmatic systems undergoing decompression and devolatilization, the crystallization is driven by effective undercooling by loss of water molecules. So, there is a change in composition of melt and rheology of magma during crystallization. In the nucleation of feldspar from hydrous melts, the expression of interfacial free energy σ should incorporate the reduction in the barrier to the nucleation due to the presence of additional phases. The increase in the dissolved H2O content by a factor of 2.5 decreases σ by a factor of 5. Thus, the nucleation rate data can be modeled by the modified expression of CNT, with σ varying as a function of composition of H2O.76 The knowledge of precipitation kinetics in alloys is necessary to optimize their properties. The CNT can predict the nucleation rates with good accuracy in the binary alloy system such as Al−Zr, Al−Sc and in the ternary alloy system Al−Zr−Sc. When the short-range ordering tendency in these alloy systems are taken into account, the results are in good agreement with atomic simulation and experimental data. The extension of CNT predicts the increase of the nucleation rate with Zr addition to Al−Sc alloy.77−79

Hence, there is a large difference in the pre-exponential term computed by the theory and the experiment. So the surface energy is estimated from experimental nucleation rate data at each temperature and can be used as a fit parameter, and its temperature dependence can be obtained. But the surface energy calculated as above was observed to slightly increase with temperature as against the fact that it should decrease with increasing temperature. So, the interface energy has to be corrected for curvature and can be expressed as a function of radius of nucleus by Tolman’s equation which was originally derived for vapor/liquid equilibria which is given as σ(r) = σα/ (1 + 2δ/r), where σα is the surface energy of planar interface and δ (Tolman’s parameter) is the width of interface regions between the coexisting phases which is on the order of atomic dimensions. The value of δ was so chosen that the surface energy decreases with an increase in temperature, and it is possible to obtain the temperature dependence of σα predicted by theory.69 The composition of crystals differs from that of the parent glass, in the initial phases of phase transformation for both stoichiometric and nonstoichiometric compositions.70 The formation of solid solution is typical in silicate systems, which contain more sodium in the initial stages. If the sodium content in the glass is increased, the thermodynamic barrier is decreased due to the decrease in the interfacial free energy between crystal and melt. So, the sodium-rich solid solution having a low thermodynamic barrier than stoichiometric crystals nucleates first, and hence the composition of critical nuclei differs from that of the parent glass (Figure 7). During crystallization

Figure 7. The decrease in the sodium content of crystals in glass as a function of volume fraction at crystallized 720 °C. Dashed lines 1, 2, 3, 4 represent the corresponding composition of Na, Ca, Si, and O in the parent glass Na2O·2CaO·3SiO2 of stoichiometric composition. (Reproduced with permission from ref 70, Copyright 2003 Elsevier.)

3. FUSION OF EXTENDED MODIFIED LIQUID DROP MODEL WITH DYNAMIC NUCLEATION THEORY (EMLD-DNT) MODEL One of the successful modifications of CNT is the extended modified liquid drop (EMLD) model proposed by Reguera et al.80 which does not need any information on intermolecular potential like other molecular theories and accurately describes the properties of very small confined systems. The contribution to the free energy of formation by the translating drop in the container and the effect of fluctuation in “n” of a system of few molecules are incorporated in the EMLD model. In the subsequent work, an alternate model for the nucleation based on EMLD and dynamic nucleation theory (DNT)81 was proposed which incorporates the fluctuations relevant for small systems and is able to predict the spinodal. They proposed a new definition for a cluster which is based on the cluster

composition changes continuously, and final composition reaches stoichiometric proposition. Thus, it is clear that the surface energy of the crystals differ considerably from that of the stable phase. Deviations between the predicted and calculated data would decrease if such factors are taken into account. Some modifications of classical theory, such as diffuse interface models (given in section 5) of nucleation, decreased the problem of anomalous prefactors seen in glass forming silicate systems by considering the gradual change of thermodynamic properties from bulk crystalline phase to bulk liquid phase across the interface.71 Another modified form of classical theory for rough interfaces, which takes into account of the rough surface of nuclei over the length scale comparable to 6668



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critical density profile ρc(r). The rate of formation of critical droplets per unit volume is given by

volume that minimizes the pressure within the container. The nucleation barrier in this model is given by the following equation,

JDFT = A exp−ΔΩDFT / KBT

ΔG* = ΔΩ* = ΔF(N *, Vm) − Vm(po − P) + N *Δμo*

(9)

The rate of nucleation is calculated by taking the same prefactor from CNT, and the ΔΩDFT was evaluated through density functional methods.86 To apply this theory, knowledge of intermolecular potentials of the systems is required, which is approximated through the hard-sphere perturbation theory which is discussed in the next section. 4.1. Perturbation Approximation. Intermolecular potential ϕ(r) is divided into attractive and repulsive parts. While the repulsive potential is responsible for the structure of the fluid, the attractive potential is responsible for the specific density of the fluid. The intrinsic free energy of the system is calculated from the attractive and repulsive contribution to the spherical potential ϕ(r) = ϕrep(r) + ϕatt(r). The dependence of intrinsic free energy of the system on the density profile is given by the following equation.

(8)

where the first term of RHS is the Helmholz free energy calculated by the standard relationship, P is the sum of gas pressure and the pressure drop exerted by translating drop, N* represents size of critical nucleus, Vm (minimum of PV isotherm ∂P for a given N) is obtained by equating ∂V = 0 at Vm, and chemical potential difference Δμo* = KBT ln(po/P). The EMLD-DNT is successful in accounting the spinodal by the inclusion of translation in fluctuation. Excellent agreement with experimental data has been reported for the nucleation of npentanol.82 The nucleation rate isotherms for vapor liquid nucleation of argon which spans from 1023 < J/cm−3 s−1 < 1026 in the temperature range of 45−70 K below the triple point showed that the results deviate from the CNT by 2−7 orders of magnitude, whereas the EMLD-DNT model seems to have excellent agreement with less than 1 order of magnitude over the temperature range.83

F[ρ(r )] =

∫ dr fh [ρ(r)] + 12 ∫ ∫ dr dr′ρ(r)ρ(r′) ϕatt(|r − r′|)

(10)

where f h[ρ(r)] is the Helmholtz free energy density of the hard sphere fluid (reference system) with density ρ. The grand potential of the inhomogeneous system is related to Helmholtz free energy as follows

4. DENSITY FUNCTIONAL THEORY (DFT) Another popular approach to study the vapor−liquid equilibria is based on microscopic molecular interaction which avoids the capillary approximations of CNT. The DFT also sometimes referred to as nonclassical theory is a powerful tool to analyze various vapor−liquid, liquid−solid nucleation phenomena, and it also able to predict the spinodal. DFT which was originally developed by Oxtoby, Evans, and Zeng approached the problem based on a density functional approach which expresses the intrinsic free energy of the system as the function of molecular number density.84−87 The theory assumes that the free energy of nucleus ΔΩDFT (same as ΔG*) depends on the average spherical density profile rather than the radius. The density of a new phase at the center of nucleus is not same as that in the bulk of a new phase (Figure 8). They located the saddle point in the functional space. It is the point at which the growth of a new phase is favored when the cluster reaches the

Ω[(r )] = F[ρ(r )] − μ

∫ ρ (r ) d r

(11)

where μ is the chemical potential. The density profile of the vapor−liquid system is obtained by the minimization of grand potential (δΩ/δρ(r) = 0), at constant chemical potential, and substituting eq 8 for F[ρ(r)] results in the following expression μ h [ρ(r )] = μ +

∫ dr′ρ(r′)ϕat(|r − r′|)

(12)

where μh represents the local chemical potential of the hard sphere fluid. A major hurdle in this theory is the requirement of intermolecular potentials which are not available for complex substances. So the Lennard−Jones potential has been taken for the nucleation of noble gases like argon to validate the theory. The repulsive contribution (which constitutes a larger part of the interatomic potential) is approximated as the short-range repulsion between the hard spheres. The attractive long-range contributions (smaller part) are approximated as follows from the Weeks−Chandler−Anderson equation.88 The LJ potential has the form ϕLJ(r ) = 4 ∈

⎡⎛ ⎞12

∫ ⎢⎣⎝ σr ⎠ ⎜

⎟

−

⎛ σ ⎞6 ⎤ ⎟ ⎥ ⎝r⎠ ⎦

⎜

(13)

where ϵ is the characteristic energy and σ is the size parameter. The Weeks−Chandler−Anderson perturbation scheme divides the LJ potential into attractive and repulsive parts. ⎧ ⎪ − ϕLJ(r ) + ∈ , r < rmin ϕattWCA (r ) = ⎨ ⎪ 0, r ≥ rmin ⎩

Figure 8. Density profile ρ(r) of the nucleus of the vapor−liquid system changing from liquid-like at the center to the bulk vapor density far from it.

(14)

and a weak attractive part 6669


Crystal Growth & Design ⎧ − ∈ , r < rmin ⎪ WCA (r ) = ⎨ ϕrep ⎪ ϕ (r ), r ≥ rmin ⎩ LJ

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(15)

where rmin = 21/6σ which represents the distance at which ϕ(r) exhibits its minimum. Equation 10 is solved by iterations to derive the density profile for the liquid−vapor interface. The difference in the grand potential of supersaturated and saturated systems gives the nucleation barrier ΔΩDFT. A semiempirical approach to DFT to directly compare nucleation rates with the experimental data was also proposed.84 The same experimental properties like equilibrium vapor pressure, liquid density, and surface tension used in CNT were used to fit three parameters in the intermolecular potential. The experimental results of n-nonane studied by three different groups6,7,40 were found to agree reasonably with this version of DFT. But for polar fluids like water and n-alcohols, the results did not show much difference from CNT since approximating polar fluids by a spherical potential did not work well. Another improvement is the dynamic DFT,89 in which a pre-exponential factor emerges automatically in the model, instead of taking it from CNT. The correction for the center of mass was also analyzed, and it was found that it had almost no effect on nucleation rates. The gas− liquid transition of two-dimensional (2D) LJ fluid was studied by Zeng using DFT in which argon was used as a model system. The study showed that the CNT works well in the prediction of nucleation rates for 2D LJ systems. But the CNT fails in its predictions for 3D LJ systems which clearly show that the effect of curvature on the nucleation rate is smaller for the 2D systems.90

Figure 9. Cross interfacial enthalpy and entropy distributions under (a) equilibrium and (b) nonequilibrium conditions.91

W * = −κδ 3ΔgoΔψ (η)

where q = (1 − η)1/2, η = Δgo/Δho, Δgo = Δho − TΔS0 and ψ = 2(1 + q)η−3 − (3 + 2q)η−2 − η−1

W = −δ Δho = −δT ΔSo A

(16)

(17)

where A is the interfacial area, interfacial width δ = limR→∞ (RSRH) > 0, is the equilibrium distance of enthalpy and entropy surfaces. But δ may be smaller than the interface thickness. For the cluster larger than the interfacial thickness RS − RH > δ, the size and the free energy of formation of the cluster is given by

R S* = δ(1 + q)η−1

⎛ ζ2 ⎞ γ = γ∞⎜1 + ζ + ⎟ 3⎠ ⎝

(21)

γ∞ = −δT ΔSo = δ /RH , RN = RH

(22)

γ∞ = −δ Δho = δ /R S , RN = R S

(23)

The above equations show that interface free energy is sizedependent and the nucleation rate is calculated as J = A exp(−W*/KBT).This theory has been extensively studied and tested.92 This model was found give a significant improvement in the treatment of nucleation for both solid−liquid, vapor− liquid systems of nonpolar, weakly polar, and other metallic substances.The temperature dependence of γ can be explained well by this theory. The DIF model predicts the size-dependent interface free energy and eliminates the problem of anomalous prefactors. The DIF theory proves to be consistent with the nucleation data on a variety of substances including liquid metals, oxide glasses, and hydrocarbon liquids. The oxide glasses such as [Li2O·2SiO2(LS2), BaO·2SiO2(BS2), Na2O· 2CaO·3SiO2(NC2S3)], and the metallic glasses (Fe40Ni40P14B6) were tested, and the data were consistent with the theory.93 The problem of anomalous prefactor was not found. Nucleation data on oxide glasses crystallizing polymorphously were also found to be consistent with the theory.94 It is obvious from the discussions that all the above theories show limited success, since the evolutionary stages leading to the critical nucleus are not properly captured in the treatment. CNT is based on the assumption that the clustering and reorganization to the new phase occur simultaneously, and density is the single order parameter which differentiates old and new phases. But the growing experimental evidence of nucleation events in crystallizing solutions has shown that the formation of cluster with a higher density and its structural reorganization to form a crystal are separated in the time zone which requires at least two order parameters, density and structure, to differentiate old and new phases. This has led to the evolution of the concept of a nonclassical two-step pathway to nucleation which is discussed in the next section.

where κ = 4π/3, and RH and RS are the positions of enthalpy and entropy surfaces. The area enclosed by the curves Δh and TΔS in Figure 9 is proportional to the interfacial free energy γ, which is derived from eq 16 and is given as γ=

(20)

The interface free energy according to the new model is given as

5. DIFFUSE INTERFACE THEORY (DIT) The predictions of CNT shows an error of several orders of magnitude for the experimental vapor condensation data due to the assumption that the thickness of interface is small when compared to the size of the nuclei. The phenomenological diffuse interface theory tries to improve the droplet model by taking into account of the interface between solid−liquid, liquid−vapor systems which actually extends to several molecular layers in contrast to the assumption of a sharp interface of the classical droplet model.91 The DIT considers a strongly curvature-dependent surface tension related to a characteristic interface thickness. The model takes into account of the bulk solid and liquid values of enthalpy and entropy within the interface region. The free energy of cluster formation according to this theory is given by W = κ(RH3Δho − R S3T ΔSo)

(19)

(18) 6670



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colloidal systems.100,101 This led to the confirmation of the presence of the intermediate phase, and the nucleation is considered to proceed in two steps. Two-step mechanism102 assumes that the first step of nucleation is the formation of droplets of dense liquid called clusters, due to the critical density fluctuation97 or phase separation followed by the second step, which is the appearance of crystal nucleus due to structural ordering of molecules. A recent direct observation has provided support for the validity of a two-step mechanism in protein crystallization, tracking the evolution of clusters using dynamic light scattering and confocal depolarized spectroscopy.103 The dense droplet is considered as stable with respect to parent liquid and metastable with respect to the crystalline state. A typical protein-rich droplet size varies from several tens to several hundreds of nanometers with the volume fraction well below 10−3 of the solution. The nucleation rate increases inside the droplet because of high density, and the fluid layer acts as a buffer between the original fluid and the growing crystal. The nucleation can be selectively enhanced without increasing the rate of crystal growth which is achieved by adjusting the solvent conditions.101,104 The actual progress of two-step mechanisms can be tracked by following the typical trend in the overall kinetics of the system, which reaches a maxima when the quantity of the intermediate is maximum, after which a plateau develops due to the consumption of intermediates for growth.105 The nonclassical pathway is favored when the sum of the energy barriers in two-step nucleation is less than the single energy barrier of CNT. Some kinetic treatments are available in the literature with respect to protein nucleation, wherein the phenomenological theory was developed which takes into account the metastable intermediate.106 The formation of the metastable intermediate is considered fast due to its low energy barrier, and the formation of ordered nucleus with appreciably large activation barrier is the rate-determining step. The nucleation rate J is expressed as a function of protein concentration and temperature as follows,

6. NONCLASSICAL NUCLEATION PATHWAYS Modern experimental techniques have shown that the nucleation process in solution crystallization proceeds through intermediate stages before reaching a thermodynamically stable phase. Complex materials like proteins, colloids, minerals, and polymeric solutions show that this behavior can be interpreted in terms of the rule of stages. Oswald’s rule of stages95 states that in the course of transformation of an unstable or metastable system into a stable one, the system does not go directly to the most stable conformation, but prefers to reach intermediate stages having closest free energy to the initial state. According to Stranski and Totomonow, the phase that emerges is the one that separates from the parent phase by the smallest free energy barrier.96 The occurrence of metastable intermediates was originally proposed for protein crystallization.97,98 The phase diagrams of proteins are different because the range of interaction between the molecules is less than onequarter of the particle diameter and the range is determined by the size of solvent molecules. The phase diagrams are measured experimentally or derived from simple colloidal models which treats proteins as spheres with short-range attractions (refer to Figure 10III). When the range of interaction is very short, only

Figure 10. Phase diagram for hard sphere colloids (left) as a function of volume fraction, and the phase behavior is similar to atomic substances and they have no potential energy and the phase behavior is determined by entropy; for colloids with long-range attractions (middle); colloids with short-range attractions (right). The dashed lines in Figure III represent the spinodal and solid curve represents binodal, and the symbol ■ represents critical point.

J=

two thermodynamically stable phases exista colloidal fluid and a colloidal crystal. The metastable binodal curve is observed when the critical point for fluid−fluid phase transition lies below the fluid−solid coexistence curve. Spinodal curves lies below the binodal curve and they meet at critical point. The location of bimodal curve within the fluid−solid coexistence region varies as a function of a range of interactions. The fluid− fluid coexistence becomes metastable, and this has been observed for a variety of protein and colloidal systems. When a protein solution is quenched, two amorphous phases are separated with one phase rich in protein content and the other phase is a dilute one containing less proteins, and the nucleation is expected to occur within these phases which provides an alternate two-step nucleation mechanism for the solution to crystal phase transition. The unusual dependence of the rate of nucleation on temperature has been observed in protein systems and is found to be maximum at the critical point.99 A system which is near the critical point is subjected to large density fluctuations, and as a result the local supersaturation becomes large enough to nucleate a solid phase. This has been confirmed by theoretical studies in which employing density functional theory to the nucleation in the presence of metastable critical point showed an increase in the nucleation rate of several orders of magnitude near the critical point, which is found in most protein and

K 2C1T exp( −ΔG2*/KBT ) ⎡ ΔG* ⎤ U η(C1 , T )⎢1 + U1 exp K Tc ⎥ ⎦ ⎣ 2 B

( )

(24)

where ΔG2* is the barrier for nucleation inside the clusters, ΔGC* is the standard free energy of a protein molecule inside the clusters in excess to that in the solution, C1 is the protein concentration inside the clusters, η is the viscosity inside the clusters, the constant K2 scales the nucleation rate of crystal inside the clusters, and U1 and U0 are the effective rates of decay and formation of clusters at temperature. The kinetics of the second step is unclear because of the difficulty in experimentation. If the mononuclear mechanism is assumed, the overall nucleation rate of crystals was found to be equal to the nucleation rate of dense droplets. But the literature shows that not all the droplets nucleate crystals, and some of them gradually disappear into the solution. So the overall rate is also determined by the local nucleation rates of crystals in the droplets and the mathematical model of which is found in ref 107. The following evidence has answered the doubts about the actual role of these metastable clusters in nucleation. Proteins such as glucose isomerase, proteinase K, and hen egg white lysozyme ,to name a few, show looped macrostep formation (a mechanism of layer generation), which is triggered by the mesoscopic clusters when the crystals are exposed to the supersaturated growth solutions. There is a 100-fold reduction 6671



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possibility of a two-step mechanism. Recently, it is shown that the proximity to the critical point is not a prerequisite for enhanced nucleation; rather any point of the system below the metastable spinodal line will enhance the nucleation. The formation of a dense liquid phase is very fast and enhances crystallization both near the metastable critical point and almost everywhere below the fluid−fluid spinodal line. In contrast to the systems showing dense liquid precursor phases, the following evidence shows that relatively high local ordered structures behave as the precursors for crystal nucleation.114 This concept has been tested in colloidal systems which contribute to our understanding of the way in which polymorphs are nucleated. Theoretical studies have shown that for small supercooling, the bcc phase is favored in the nucleation of all simple fluids exhibiting weak first-order phase transitions. Simulation studies on the homogeneous nucleation of LJ systems at moderate supercooling have shown evidence for the presence of body centered cubic (bcc) like ordering at the surface critical nucleus and a face centered cubic (fcc) structure at the core of the nucleus.115 A random hexagonal close-packed (rhcp) structure is the dominant metastable solid in hard-sphere systems.116 The critical clusters observed in the homogeneous nucleation of LJ systems using MD simulation provide the evidence for the presence of two extremities of critical clusters. The smaller clusters with more fcc component are in the core and bcc component at the interface and the large clusters with a less fcc core and more bcc are at the surface. But these differences settle out as the nucleation proceeds, and all the range of paths converge to well-structured fcc clusters. The evidence of multiple metastable structures with one structure dominating the nucleation process has been recognized in colloidal systems.117 A relatively ordered liquid structure exhibits local orders close to hcp, bcc, and fcc symmetries in the colloidal model system consisting of polymethyl methacrylate. The dominance of hcp precursors was observed, and it could be due to the structural symmetry between the disordered liquid phase and the hcp precursor. So it is inferred that the definition of clusters should include the structure of clusters as well. But the CNT considers that all the critical nuclei consist of only the thermodynamically more stable phase, and the free energy landscape of CNT does not consider the strong correlation existing between cluster structure and size. Detailed reviews are available in refs 118 and 119 for more information on nucleation in colloidal systems. 6.2. Evidences of Metastable Phases in Glass Systems. The evidence of the existence of metastable states have been reported in the case of lithium disilicate and sodium calcium silicate glasses. The evidence of the existence of metastable systems dates back to 1967,120 and the other evidence121−124 listed in the Table 2 gives strong evidence in favor of metastable phases precipitated in the early stages of transformation in lithium disilicate model systems. In earlier studies in 1974, the nucleation studies of LS2 glasses of stoichiometric composition had shown only the presence of a stable phase using TEM.125 But later evidence suggested the presence of metastable states or precursor phases in LS2 glasses.122 So the nucleation in lithium disilicate glasses had remained controversial between research groups. But later direct observation of early and intermediate stages of LS2 glasses under TEM, selected area electron diffraction (SAED) showed that in the samples heat treated at 454 °C, there is a concurrent formation of distinct crystalline phase LS2 and the metastable LS, and they coexisted up to 120 h. It was also suggested that in the later

in the nucleation rate for glucose isomerase when the clusters are removed (10-fold reduction for lysozyme).108 In the crystallization of bovine beta-lacto globulin, the increased concentration of salt CdCl2 near the transition zone of the phase diagram leads to an enhanced energy barrier for nucleation of these precursors resulting in slow growth.105 Excellent reviews are available in the literature for more information102,106,109 on the two-step mechanism of crystal nucleation from solutions. In the following sections, the evidence of two-step mechanisms in various solution crystallization systems is reviewed. 6.1. Observation of Two-Step Mechanism in Colloidal Model Systems. Colloidal particles are used as model systems to study the nucleation process due to their larger size, ease of characterization due to slow diffusion in liquids, and the tunability of interactions. With high speed confocal microscopy, it is possible to observe the mechanism of nucleation in colloidal particles. Depending on the temperature and volume fraction, the behavior of colloids can the modeled as shown in Figure 10. Depending on the type of interactions, we can divide the colloidal model systems as a purely hard sphere (HS) colloidal system, hard sphere colloids with long-range interactions, and the hard sphere with short-range attractions.110 The phase diagram shows a dilute colloidal phase which is analogous to vapor, a dense colloidal phase which is analogous to liquid, and a colloidal crystal phase to solid phase. The HS colloids show the simplest type of interactions and show only fluid and crystal phases, in which below the freezing point (volume fraction φ = 0.494), the suspension is a fluid (I of Figure 10). If the attractions of long range are introduced in comparison with the diameter of colloids, the phase diagram of the colloidal suspension resembles the atomic substances like argon (II of Figure 10). If the range of attractions is very short compared to the diameter of particles, as in the case of proteins, only two stable phases occur as discussed in previous section (III of Figure 10). Theoretical and experimental evidence has shown that the nucleation is mediated by the dense liquid precursor phases (Figure 11), and this has been tested in colloidal

Figure 11. Nonclassical pathway to nucleation in the colloidal model system (polystyrene particles of diameter 0.99 μm) via amorphous dense phase (a) dilute mother phase (b) 2D amorphous dense droplets formed on glass surface (c) crystal nucleation from amorphous phase. (Reproduced with permission from ref 112. Copyright 2007 American Chemical Society.)

systems.104,111−113 Initially it was thought that the existence of the metastable fluid−fluid critical point is the prerequisite for the observed enhanced crystal nucleation due to density fluctuations which was tested for protein systems. Later, the simulation studies showed that the purely HS systems which exhibit no metastable fluid−fluid coexistence also shows the 6672


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D. L. Kinser, L. L. Hench (1968)120 sequence of phases observed is identical but small temperature changes give rise to pronounced morphology changes D. L. Kinser, L. L. Hench (1970)121 metastable phase appears during early stages of nucleation, which is totally dissimilar to lithium disilicatewhich act as J. Deubener and R. precursor to LS2 Brückner (1993)122 Two metastable states were noticed α, β α phase transforms to β phase prior to the final stable LS2 when heated at 454 Iqbal et al. (1998)124 °C for 120 to 480 h; multiple polymorphs were noticed for the samples nucleated for greater than 228 h metastable phases do not persist in lithium disilicate glass which has been heated for extended time periods in the Burgner et al. nucleation region (2000)123

stages, LS phase disappears playing no role in the nucleation of LS2.126 The large discrepancy between the theoretical nucleation rates and the experimentally observed ones is because the supercooled state is actually heterogeneous in contrast to the implicit assumption of homogeneity by CNT, and this stems from the transient medium-range bond orientation ordering in glass forming liquids which decreases the free energy of the system and promotes nucleation. Phase separation is observed to occur in many glass ceramic systems which provides an alternate pathway to nucleation.127 Phase separation and heterogeneities in the glassy structure have been noticed in some systems like aluminosilicate glasses containing ZrO2.128 The direct imaging of glass-in-glass separation with Zr/Zn-rich domains with sharp boundaries is observed, and they play a role as a template for a nucleation process which resembles two-step models proposed for protein systems. 6.3. Two-Step Nucleation in Bio- and Biomimetic Mineralization. In the bio-/biomimetic mineralization and geosciences, precursor phases/prenucleation clusters (PNCs) and small polymeric species play a major role in the early stages of nucleation.129−131 The most notable features in the biologically formed crystalline minerals is that crystallization is highly controlled, and they have a unique ability to convert the minerals into materials with a complex pattern which matches the function.132 The macromolecular matrix provides sites for nucleation and assists the formation of a critical nucleus and directs the location and orientation of crystals. Calcium carbonate (CaCO3) mineralization is studied extensively, and the prenucleation stage of amorphous calcium carbonate (ACC) before CaCO3 mineralization has been seen in the literature.133−136 In the absence of additives, the liquid precursors of calcium carbonate stabilized intrinsically by bicarbonate ions are formed before the formation of ACC. In the presence of polymer additive such as polyaspartate, the liquid condensed phase is kinetically stabilized by polymer additives in a distinct, controlled fashion suggesting the way in which biological organisms control the mineral deposition. Precursor species are noticed in the prenucleation stages where the ionic clusters are considered to be stable in contrast to the classical view of metastability of clusters. In the prenucleation stages of CaCO3, clusters in equilibrium with ionic solution are noticed which is pH dependent. There are two options after this stage. The clusters may form a critical nucleus and grow by single ion attachment, or the clusters may aggregate and precipitate different ACC phases. Studies show that different species of ACC exist in the solution which possesses some short-range order corresponding to a final crystal form. The actual role of PNCs has also been demonstrated in the template-directed crystallization of CaCO3 and CaPO4. The PNCs are formed with the size range of 0.6−1.1 nm, which are stable, and they aggregate in solution to nucleate ACC nanoparticles of 30 nm (Figure 12). Association of these particles with the template surface initiates the growth of ACC, which assembles at the template and develops randomly oriented crystallization domains at first which then later develops into a single crystal with the orientation stabilized by the interaction with the surface. Surface-induced formation of apatite from simulated body fluid seems to follow a similar mechanism observed for CaCO3. But the formation of ACP particles mainly occurs at the surface through heterogeneous nucleation, while in the previous case, the amorphous phase nucleated in the solution. PNCs with 0.87 ± 0.2 nm are reported. Aggregation of prenucleation crystals of amorphous

30 mol % Li2O-SiO2 glass analyzed with hot stage TEM

Li2O·2SiO2, Na2O-2CaO, 3SiO2 and 2Na2O, CaO·3SiO2

LS2 glass with 33.91 mol % Li2O heat treated at 454 °C for 50−551 h

lithium silicate with four different LiO2 compositions

2.

3

4

5

crystalline lithium metasilicate appeared and redissolved before final lithium disilicates

30 mol % Li2O-SiO2 glass electron microscopy and X-ray analysis 1.

system s. no.

Table 2. Various Studies Showing the Metastable Intermediates in Lithium Silicate Glasses

findings

reference


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concentration with little or no spatial order. At 7 ns local density remains constant and marks the emergence of spatial order with the number of ion increasing from 218 to 249. These regions of high density contain a sufficient quantity of water which is shown in red. As the crystal grows, the water molecules are gradually excluded, before a final anhydrous crystallite appears. This example clearly supports the two-step mechanisms proposed for proteins and biominerals. Panels c and d show the growth of crystalline region by acquiring ions from the solution. Thus, it is more evident the precrystalline nucleus is not necessarily similar to that of the final stable phase. 6.5. Two-Step Nucleation in Open Framework Materials. Unlike the protein and colloidal systems, the nucleation in open framework materials has received less attention, and only a few works are available about the nucleation pathways. These are the important classes of porous materials that occur in nature as well as are synthesized. These materials include zeolites (microporous alumina silicates), metal−organic frameworks, covalent organic frameworks (COFs), and clathrates. Evidence for the presence of amorphous precursor species has been reported in a variety of zeolite materials like MFI, LTA, and FAU frameworks.142−144 The precursors may vary from silicate monomer/oligomers to complex building units like polytetrahedra. The initial phases in the synthesis of zeolites are marked by the formation of a primary amorphous phase whose structure varies from gel like to a colloidal one. Before the formation of nuclei, a secondary amorphous phase which exhibits some short-range order has been observed in many cases. Probing the structure of the amorphous precursors formed in the early stages zeolite synthesis has shown evidence of local order that resembles the crystalline zeolite. Amorphous alumina silicate precursors used to form zeolite A reveal that the medium range order of precursors changes during crystallization. Studies show that due to a high concentration of constituents in the gel phase, more nuclei are found in the gel or gel/liquid interface rather than the solution phase which is observed in alumina silicates. They are rich in aluminates and silicates and serve as reservoirs of crystal growth. A recent study on the synthesis of zeolite L reports a self-assembly and structural evolution of precursors into wormlike particles (WLPs), which consists of silica-rich domains. Precursors grow to a maximum size until nucleation starts, and its number density decreases during the growth of zeolite.145 The presence of precursors indicates that a

Figure 12. Computer-aided visualization of tomograms recorded after reaction times of 6 min, 11 min, and 45 min in the template (stearic acid monolayer as a template deposited on supersaturated CaHCO3 solution) directed crystallization of CaCO3. The image in (a) shows the monolayer formation (red arrow) and 30 nm ACC nanoparticles in the solution, (b) clustering and growth takes place at the surface of organic matrix as shown (c) development of crystalline regions inside the amorphous particle. (Reproduced with permission from ref 134, Copyright 2009 Science.)

calcium phosphate (ACP), which acts as a precursor in the formation of apatite crystals in bones, teeth, and their role in nucleation has also been demonstrated.137 In contrast to the amorphous precursor phases, the formation of stable crystalline precursor phases has been observed in the case of gypsum precipitation from aqueous solutions. The homogeneous precipitation of nanocrystalline (10−15 nm) hemihydrate basanite (CaSO4·0.5H2O) was observed in undersaturated conditions which finally transformed into dehydrate gypsum, through oriented self-assembly of bassinite.138 6.4. Two-Step Nucleation in Salt Solutions. Efforts have also been made to find evidence of the existence of prenucleation clusters in the nucleation from salt solutions. Earlier efforts are focused on the NaCl systems which offered some insights about the early stages of nucleation. MD simulations of aqueous NaCl systems were studied for ion clustering and nucleation near saturated and supercritical conditions.139 MD simulation studies for smaller systems with the limited ions using the path sampling method show that the atomistic mechanism of NaCl nucleation is characterized by the nonhydrated sodium ions coordinated octahedrally by chloride ions, and they act as centers stability of the aggregates, which then grow into a large ion cluster.140 The MD simulations on a large scale to study the nucleation pathway in slightly supersaturated NaCl solution shows a complicated two-step pathway where the density fluctuations precedes the structural fluctuations.141 Figure 13 shows that at 3 ns, the densification starts in which the local concentration exceeds the bulk solution

Figure 13. Large scale dynamic simulations at 300 K, 1 atm pressure with 64000 particles in slightly supersaturated (3.97 m) solution of NaCl in water showing the development of density fluctuation followed by structural fluctuation in the nucleation mechanism. The sodium ions are shown in blue, chloride ions in yellow, oxygen atoms of water molecule in red, and hydrogen atoms in white (larger blue and purple refers to reference sodium and chloride ions. (Reproduced with permission from ref 141. Copyright 2013 American Chemical Society.) 6674



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nonclassical route by particle attachment is possible. As Figure 14 suggests, WLPs have a direct role in crystallization, but the

Figure 15. Transition from 5 layer square (□) lattice to 4 layer triangular (Δ) lattice structure in s−s transition is shown as an example for two-step nucleation via the formation of an intermediate liquid phase. The parent □ phase is shown in green, liquid like nucleus is shown in red, and the evolving Δ phase is shown in blue inside the liquid phase. The contact angle α of Δ lattice at the junction of □-Δliquid as shown in the inset is less than 90° and it does not wet the square lattice. (Reproduced with the permission from ref 150. Copyright 2014 Nature Publishing Group.)

Figure 14. (A−C) SEM showing the role of wormlike particles taken in the intermediate stage of crystallization of zeolite L in the presence of growth modifier 1-butanol to reduce the crystal growth rate. The wormlike particles feed the nutrients to zeolite L (disclike morphology) and the presence of layers (macrosteps with height corresponding to multiples of unit cell parameter) on the surface suggest that crystallization by particle attachment or the nonclassical route is the possible mechanism. (Reproduced with permission from ref 145, Copyright 2013 American Chemical Society.)

temperature of about −137 °C results in a different type of amorphous ice of varying densities. Ice exists in two crystalline forms which mainly consist of hexagonal crystal structure and minute traces of cubic ice. The existence of metastable states has been noted for the transition to ice at ambient pressures. Experimental evidence demonstrates that cubic ice forms as a metastable state in the Earth’s atmosphere when micron-sized water droplets freeze at a temperature below 190 K which is the temperature range found in polar stratospheric clouds.152 Tracking the trajectory of freezing using MD simulation (Figure 16) by the quenching of pure water to 230 K shows

exact trajectory of growth from the WLPs to zeolite crystal is unclear. Excellent reviews are available in the literature for more information related to the zeolite crystallization mechanism mediated by precusors.146−148 The nucleation and growth process in COFs and MOFs are least understood, although a study on COF-5 reports that the nucleation is mediated by monomers and oligomers. Study of nucleation MOF is highly difficult since they have unpredictable induction times followed by rapid growth, although some information on the occurrence of the metastable phases before the growth of MOF-5 is reported in the literature.149 6.6. Nucleation in Solid−Solid Phase Transitions. One more interesting class of phase transitions occurring in elemental crystals, alloys, and minerals and in diamond and steel production is the solid−solid transitions. Solid−solid (s− s) transitions between different crystalline structures are the most numerous of nature’s phase transitions. There has been some evidence to show that the intermediate liquid should exist in the nucleation processes of solid−solid phase transitions. The study of s−s phase transition using a colloidal model system confined between two glass walls with single particle resolution provides evidence for the presence of liquid precursors during the transformations between square (□) and triangular lattices (Δ).150 The free energy barrier for the formation of liquid nucleus is less than that of Δlattice nucleus at small values of “r”. Hence, a liquid-like nucleus is formed first which then later transforms to Δlattice nucleus at larger values of “r”. This can be understood from Figure 15. The reason for this type of transition can be attributed to the interfacial energy which is less than 200 mJ m−2 for coherent interfaces between two crystalline phases in metals and alloys, and ranges from 30 to 250 mJ m−2 for solid/liquid interfaces. Since the surface of a nucleus cannot be entirely coherent, it is inferred that a liquid nucleus should form in the early stages of s−s nucleation. This fact is also supported by using a combination of molecular dynamics and Monte Carlo simulation.151 6.7. Metastable Phases in the Freezing of Ice. In this subsection, the phase transition from water to ice and the discussion of metastable states are given. Depending on the pressure and temperature, ice molecules exist in 16 different phases. Rapid quenching of water below the glass transition

Figure 16. Molecular trajectories showing the phase transition from pure water to ice. The encircled region shows the polyhedral nucleus composed of hydrogen bonds formed spontaneously at t = 256 ns; at t = 290 ns, the nucleus expands in 3D space leading to six membered rings; images at t = 320 ns and 500 ns show the rapid growth of stacked honeycomb structure throughout the system. (Reproduced with the permission from ref 153, Copyright 2002 Nature Publishing Group.)

that the ice nucleation is found to occur when a sufficient number of relatively long-lived hydrogen bonds develop into a nucleus, which then changes size and shape slowly,153 which represent a stepwise transition. Another MD simulation study reports the existence multiple pathways in the freezing to high pressure ice VII.154 Both the direct transition as well as epitaxymediated transition is observed to occur. The nucleation of ice VII easily occurs at the grain boundaries of metastable embryos with a tetrahedral closed packed structure. Recent evidence has 6675



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shown that the ice nucleation on carbon surface proves to be in line with the concept of CNT for heterogeneous nucleation.155

7. SINGLE STEP AND/OR MULTISTEP? The literature also shows more interesting scenarios such as coexistence of single and two-step pathways, or the dominance single/two-step pathways in the same system under selected conditions. While the high concentration of glucose isomerize (100 mg/mL) shows the evidence of the nonclassical route, the lower concentration of it (0.01 mg/mL) shows one-step nucleation which is confirmed by the absence of a disordered precursor (which would be of higher energy) preceding the formation of the crystalline monolayer. As shown in Figure 17,

Figure 18. Template directed nucleation on carboxyl self assembled monolayers following a classical pathway is shown. With carboxyl terminated SAMs, free ions or ion pairs aggregate directly on the SAM to form a critical nucleus of calcite. Subsequent growth of calcite takes place by the dissolution of amorphous calcium carbonate which is formed in the solution (Reproduced from ref 157 with permission. Copyright 2012 Royal Society of Chemistry.)

studies of calcite nucleation on different functional groups and conformations (C11 and C16 chain lengths). The nucleation rates of calcite onto these substrates over different supersaturations show a linear relationship as per the predictions of CNT for all substrates. The interfacial energy in the templatedirected nucleation (which consists of three components which includes the individual contributions from crystal−liquid, crystal−substrate, substrate−liquid interfaces) relates linearly with the total binding free energy of the crystal−substrate interface which strengthens the classical descriptions. The reductions in the barrier to heterogeneous nucleation favor the formation of calcite directly without the evidence of ACC formation. The nucleation and growth of magnetite from solution also shows the classical behavior with no intermediate precursor phase.158 The primary particles of size 2 nm aggregated in branched networks shows a monodisperse size distribution which then becomes denser to form spheroidal nanoparticles of 5−15 nm. Unlike the above-discussed cases of calcium sulfate and phosphate, there is no intermediate amorphous phase noticed; instead the nanoprecursors fuse with the magnetite surface with the development of crystallinity. Another observation on CaCO3 nucleation reveals the interesting phenomenon of existence of a range of pathways simultaneously under the same conditions within a single experiment.159 Nucleation of both metastable and stable forms of CaCO3 occurred simultaneously. Nucleation of vaterite (Figure 19I and II) was observed to occur directly from solution and through the multistep pathway in which amorphous calcium carbonate appeared first and then transformed into more stable vaterite. Concurrent nucleation of multiple phases was also noticed. The growth mechanism in silicalite 1 (MFI zeolite) also shows that growth occurs through the addition of silica molecules and the attachment of nano precursor particles.160 Simultaneous occurrence of both mechanisms has also been observed in zeolite L growth.145 A recent molecular simulation study shows that simultaneous existence of direct (but only small fraction of entire pathway) and stepwise pathways exists in the nucleation of gas hydrates. These hydrates appear “ice-like” and are nonstoichiometric multicomponent systems composed of water molecules with other gases molecules such as CO2 and CH4. The majority of pathways begin with an amorphous core followed by the transformation to crystalline structure, whereas some pathways followed direct transition without the formation of amorphous nucleus.161

Figure 17. Atomic force microscopic (AFM) images showing the glucose isomerase 2D nucleation on mica which follows a classical pathway despite showing evidence for the existence of metastable states in bulk solution (a) nucleation starts at t = 0 min (b, c, d) large clusters I and II tend to add monomer units, small clusters III and IV dissolve completely. (Reproduced with the permission from ref 156.)

there exist a critical cluster beyond which it tends to add growth units (I &II), and others dissolve into the solution, which is in line with the concept of CNT.156 The same phenomenon of classical nucleation preferred at low concentration (area fraction 17% nearly) has been reported earlier when the freezing dynamics was studied in colloidal model systems with short-range attractions. But two distinct stages were observed when the area fraction was about 30%. Even in the case of CaCO3 as stated in the previous section, it is tempting to accept the nonclassical picture of nucleation as a common platform to understand CaCO3mineralization since mounting evidence confirmed the presence of prenucleation clusters and amorphous precursor phases. The calcite nucleation studies carried out at supersaturation excess of solubility limit of amorphous calcium carbonate show the evidence of prenucleation clusters. But the calcite nucleation on organothiol selfassembled monolayers (SAMs) which proceeds through direct one-step nucleation at low concentration shows no evidence of precursor phases and hence raises questions on the earlier proposed models in the nonclassical pathway.157 The direct evidence of prenucleation clusters at low concentrations at the surfaces is not available yet. As shown in the Figure 18, for carboxyl directed SAM nucleation, ACC particles do present in the solution, but they do not serve as a direct precursor to calcite nucleation. But the calcite nucleation is observed to occur independently, and the calcite nucleus grows by classical ion by ion model at the expense of dissolving ACC. This view of template-directed nucleation is also well supported by kinetic 6676


Figure 20. S*cr, S*ag, S*cl are the critical degrees of saturations at which the critical nuclei, cluster aggregates, and clusters form. The melt in the S1 state can form critical nuclei and crystallize into alumina α-Al2O3 in the melt, and the multistep nucleation process with S2, S3, and S4 can only form alumina products that are dependent on the crystallization process during cooling of the postreaction melt. (Reproduced with permission from ref 162. Copyright 2014 Nature Publishing Group).

show that there could be multiple pathways to nucleation which is mainly dictated by the free energy cost of the system under consideration. A theoretical model proposed by Baumgarner et al. based on the conceptual framework of CNT which takes into account of the energy content of single primary particle can be used to understand the range of pathways to nucleation.158 In terms of thermodynamic energy barrier to nucleation, when ΔG A *> ΔG C * (suffix A represents amorphous phase and C represents crystalline phase), direct formation of crystalline phase may be expected. When ΔGA*< ΔGC*, there is an initial formation of amorphous bulk phase, followed by the crystalline phase either through nucleation or

6677

11

9 10

8

7

5 6

4

3

1 2

relevant area of technical importance

water vapor and its binary mixtures with D2O mixtures of n-alkanes165,166

cloud formation, power generation, and turbo mechanical flows during handling and transport of natural gas, large and sudden pressure drops may induce mist formation due to the phase change of higher alkanes, separation of long chain n-alkanes and impurities from natural gas aerosol systems [acid−water systems (H2SO4, HNO3, and methanesulfonic acid), new particle formation in atmosphere which can affect climatic conditions, for example, sulfate particles (formed by binary nucleation of water-H2SO4) scatter solar radiation back to space ternary systems involving water- H2SO4 and ammonia/amines]5,167,168 nucleation from melts169−171 CSD and texture of igneous rocks and natural volcanic products is determined by nucleation event, In synthetic environments, to understand the glass forming abilities of melts, conversion of glass forming systems to ceramics by controlled heat treatment. protein crystallization172,173 structural biology, pharmaceutical preparations biomineralization (Production of inorganic oxides, carbonates, sulfides, silica and knowledge of nucleation helps to understand how the organic and inorganic phases, biominerals obtain their unique characteristics such phosphates)131,132 as shape, mechanical, optical, and magnetic properties to match the specific function. biomimetic systems (mimics the properties of biological systems)130 knowledge used in the synthesis of inorganic and hybrid material for several application such as sensors and optoelectronic devices, catalyst supports, reinforced polymers, etc. solid−solid phase transition174−176 knowledge of transformations exhibited by elemental crystals, alloys and minerals are applied in earth science, diamond and steel production, synthesis of ceramic materials, etc. freezing of water177,178 knowledge helps in understanding cloud formation, cryopreservation, formation of rain, snow, etc. open framework materials179−181 physicochemical properties, such as pore size, volume, geometry, and molecular functionality can be tuned for target applications which include gas storage, catalysis, molecular sieving, ion exchange, drug delivery, etc. gas hydrates (multi compoment crystals, composed of gas molecules like CO2, significant the areas like energy recovery, flow assurance, gas storage/transportation, global climate change, and CO2 sequestration CH4, and water molecules)182−185

163,164

8. THERMODYNAMIC TREATMENT The evidence for the existence of prenucleation clusters/ primary particles and the discussions in the previous section

systems

Figure 19. (I) In situ transmission electron microscopic (TEM) images showing the direct nucleation and growth of vaterite from solution at (a) t = 8 s (b) t = 56 s. (II) Vaterite formation through indirect pathway (a) at t = 84 s, vaterite plates forming on ACC particle (b) at t = 93 s, vaterite plates growing in size at the expense of ACC and the shrinkage may be due to expulsion of water from the amorphous phase. Note that the two phases are in constant physical contact throughout the phase transformation. (Reproduced with the permission from ref 159, Copyright 2014 Science.)

s. no.

Table 3. Nucleation in Various Systems and Their Technical Importance

Crystal Growth & Design Review



Review

cooling of the melt, and so the morphologies and structures of the alumina products in solid steel are controlled by the various species of clusters in the melt. From the examples discussed in this review, it is evident that comprehensive understanding to such a complex phenomenon requires collaborative efforts from various disciplines, given the unique advantage of its universal applicability to various vapor− liquid and liquid−solid equilibria. The physical principles of nucleation and the fundamental understanding of pathways in various systems would have direct scientific and technological significance in relevant fields as shown in Table 3.

through solid state transformation from amorphous to crystalline phase. The thermodynamic energy barrier for amorphous and crystalline phase is written as

ΔGA* =

ΔGc* =

16πσA 3 3(ΔGA )2

(25)

16πσC3 3(ΔGC)2

(26)

where ΔGA* equals ΔGC*, σA/σC = (ΔGA/ΔGB) gives the boundary which divides the direct formation of the crystalline phase and the transformation mediated by amorphous bulk phase. When the transformation is mediated by primary particles/precursors, the barrier to nucleation from precursor is given as ΔĞ * = ΔG* + 4/3πr3f P, where the second term in the RHS represents the energy content of the single primary particle and f P is the energy density averaging over both surface and bulk energies. If the primary particles/precursor are metastable ( f P < 0), we can expect a metastable amorphous bulk phase. If f P > 0, the primary particle are stable leading to a crystalline bulk phase. The boundary between these two routes is given as σA/σC = ((ΔGA − f P)/(ΔGC − f P))2/3. This can be treated as a general model to quantitatively describe various pathways to nucleation. In another study, the multistep nucleation pathways are elucidated in the steelmaking process from melt, by correlating the excess oxygen present in the systems to the various types of nucleating clusters.162 In the melting process, dissolved elements like aluminum (Al) and oxygen (O) present in Fe− O−Al melt react to form alumina products, usually α-Al2O3. However, the experimentally measured oxygen concentration in the melt exceeds the equilibrium level with respect to α-Al2O3, and the alumina products all have polycrystalline structures consisting of α-, γ-, and δ-alumina. The reaction between aluminum and oxygen is written in terms of clusters population as 2/3

2Al + 3O → (1/n)(Al 2O3)n

(27)

(1/n)(Al 2O3)n → α ‐Al 2O3

(28)

9. CONCLUSION The basic theories underlying nucleation and the evidence to support them have been reviewed. Even though vapor/liquid and liquid/solid equilibria show similarities, they also show differences at the microscopic level that must be accounted for in nucleation models. From the comparison of nucleation in different systems, it can be inferred that although CNT offers a platform to start with, more sophisticated models to capture the complexity of molecular level events accompanying nucleation are required. It appears that classical theory depicts one of the possible available pathways available to nucleation. The probability of a system to choose a direct/indirect pathway is difficult to predict with the present understanding, since the complicated kinetics and the various factors governing the formation of metastable clusters poses a major hurdle in the theoretical treatments. The future works have to focus on the governing kinetics of formation of crystals from clusters to extract more information on the controlling mechanisms. Kinetic and thermodynamic treatments would be more complicated for the system when direct and indirect pathways coexist or if it nucleates in a classical way in spite of the existence of intermediate metastable states or the dominance of single metastable component in the systems with multiple metastable components. Though the initial phase in the evolution of the nonclassical pathway has indicated complex possible routes, a much large theoretical and experimental foundation is needed to clearly ascertain the role of intermediates and to gain control over the nucleation phenomena.

■

where n is the number of Al2O3 units in a cluster. If n equals 1, the single molecule of Al2O3 is unstable and easily transforms into other alumina clusters. For n greater than 1, alumina clusters, alumina cluster aggregates, and crystal-like clusters could be present in the melt. The thermodynamic properties of clusters calculated by DFT shows that there exist a thermodynamic relationship between the clusters and the saturation levels of dissolved oxygen and aluminum in the melt. If the saturation ratios S* of Al and O are higher than that of the formation of critical nuclei (log S*cr = 3.5, ΔG = −354.169, KJ·mol−1) there is a multistep nucleation of (Al2O3)n clusters, which then coalesce to form aggregates. Then, the aggregates crystallize into the critical nucleus, followed by the formation of α-Al2O3 (ΔG = −479.688 KJ·mol−1, path S1 Figure 20). However, if S*ag< S* < S*cr the cluster aggregates cannot crystallize and equilibrium oxygen with clusters is present in the postreaction melt (path S2). If S*cl < S* < S*ag, the clusters are formed but they cannot coalesce and form aggregates (ΔG = −381.056 KJ·mol−1, path S3). If the S*< S*cl, the Al and O cannot react to form (Al2O3)n, and there are only crystal-like clusters (path S4). In other words, paths S2, S3, S4 can form products depending on crystallization pathway during the

AUTHOR INFORMATION

Corresponding Author

*Tel: +91431-2503104; fax: + 91 431 250 0133; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

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A Review of Classical and Nonclassical Nucleation Theories - Crystal

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