J. Phys. Chem. 1995,99, 14182-14187
14182
Diffuse Interface Analysis of Ice Nucleation in Undercooled Water Laszlo Granasy? Research Institute for Solid State Physics, H-1525 Budapest, POB 49, Hungary Received: April 20, 1995; In Final Form: July 10, 1995@
Experiments on ice nucleation in undercooled water are analyzed in terms of the classical and the diffuse interface theory (CNT and DIT, respectively) of crystal nucleation. It is shown that neither a constant interfacial energy in the CNT nor a constant interface thickness in the DIT can satisfactorily describe the experiments. It has been found that the temperature-dependent interfacial parameters, evaluated by Ewing's model (Ewing, R. H. Philos. Mug. 1972, 25, 779) from the structural information available on undercooled water, yield an improved description both within the framework of the CNT and the DIT. In contrast with the CNT, the DIT is fully consistent with the experiments. Both analyses (CNT and DIT) indicate a bulk (volumetric) heterogeneous nucleation mechanism.
Introduction The freezing of pure undercooled water starts with nucleation,' Le. with stochastic formation of icelike particles, which are able to grow above a critical size determined by the undercooling and the free energy y of the interface region (homogeneous nucleation). Container walls, surfaces, or floating foreign particles may help this process (heterogeneous nucleation). A quantitative model of ice nucleation would be essential from both scientific and practical viewpoints (atmospheric sciences, meteorology, cryobiology, etc.). As a consequence of this interest and also because of the optical transparency of the system, this is probably the most frequently studied crystal nucleation process. The nucleation rate vs temperature relation is known from several works,2 while in a few cases2a-deven the presence of a volumetric nucleation process was demonstrated allowing a critical test of theoretical predictions. Another unique feature is that the hexagonal (Ih) ice-water interface has been studied at the melting point (Tf) by various experimental methods3 from which the magnitude and anisotropy of the interfacial free energy could be assessed. Furthermore, the relevant physical properties (specific heat,4 diffusion ~oefficient,~ structure,6 etc.) of undercooled water are known better than for other substances. Thus, despite the difficulties originating from the existence of an easily nucleating metastable phase2gs7 (IC) and the anomalous properties of undercooled water,* ice nucleation is an important test ground for theories of crystal nucleation. The experiments have so far been interpreted exclusively in terms of the classical nucleation theory' (CNT), an approach not without problems. In the CNT the nuclei are treated as particles which show bulk physical properties although containing only 30 to a few hundred molecules, assuming thus an extremely sharp interface. Computer simulations9 and more advanced theoriesi0indicate, however, that the crystal-liquid interface is several molecular layers thick and that the nucleation rates predicted by the CNT may seriously be in error.Ii Unfortunately, a direct test of CNT for crystal nucleation is not possible since (i) an input parameter of the CNT, the free energy of the undercooled crystal-melt interface, cannot be determined independently of nucleation, and (ii) it is difficult to avoid volumetric heterogeneities that catalyze nucleation. An indirect test is still possible knowing the nucleation rate-temperature
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[email protected]. Abstract published in Aduunce ACS Absfrucrs, September 1, 1995
0022-365419512099-14182$09.00/0
relation, I(T), and having a guess of the temperature dependence of the interfacial free energy, y(T): The classical expression for the rate of bulk heterogeneous nucleation is I TC XNIO,hom exp{ -WhOm*fTV)/kr), where XN is the fraction of molecules active from the viewpoint of nucleation ( X N = 1 for a homogeneous process and XN < 1 in a heterogeneous case), IO,hom and Whom* = 4 ~ y ~ ( A g o + are ) - ~the prefactor and the work of formation of nuclei for a homogeneous process, K is a geometrical factor,I2 Ago+ is the volumetric Gibbs free energy difference between the bulk crystal and melt, f ( V ) accounts for the reduction of W by the heterog&eities, and V is the contact angle between the crystal-melt and crystal-heterogeneity interfaces. Then, plotting log(I/IO,hom) vs XCNT = x ~ ~ ( A ~ ~ + ) - ~ T (consistency plot), whereXy = y(T)ly(Tf), one should obtain a straight line intersecting the vertical axis at log(xN) with a slope proportional to Y ( T ~ )While ~. the temperature dependence of the nucleation rate is known for various substances, there are only a few theoretical attemptsI3to determine the interfacial free energy in the undercooled state. It is now well established for crystal nucleation in metallic meltsi4 and oxide glassesI5 that the assumption xy = 1 leads to unphysical XN values (lo6known as "anomalous nucleation prefactors".' Similar results were reported for ice nucleation by Wood and Walton2a using Hoffmann's expression' for the Gibbs free energy difference. (Other analyses2b-dbased on Tumbull's approximation' (which assumes that the specific heat difference AC,, = 0) claim a fair agreement between the experiments and the combination of the homogeneous CNT with = 1. However, Turnbull's approximation is unrealistic for undercooled water which shows an anomalously strong temperature dependence of the specific heat.4a No analysis has been performed using the experimental Gibbs free energy data.) Although the "anomalous prefactors" can be eliminated assuming a suitable temperature dependence of the interfacial free energy (a linear i n c r e a ~ e ' . ~with ~ . ' ~the temperature), it seems possible that the unphysical XN values indicate a general failure of the CNT. This view is strongly supported by the fact that in case of vapor condensation, where the interfacial free energy is known with a high accuracy, the predicted and measured nucleation rates deviate by several orders of magnitude.I6 The more advanced models of crystal nucleation'' cannot be easily applied since the Helmholtz free energy should be known as a function of a suitable order parameter for the intermediate states between the equilibrium ones, a relation which is experimentally inaccessible and can be calculated only after
-'
xy
0 1995 American Chemical Society
Ice Nucleation in Undercooled Water numerous approximations and with a substantial error. Furthermore, the applied square gradient approximation is valid for slow order parameter changes in the interface, which may be unrealistic for the crystal-liquid interface as happened, e.g., for vapor condensation far from the critical point.” In summary, there is not a well-proven quantitative description of nucleation; thus a different approach that accounts for interface diffuseness may be of some interest. Such a theory,I8 relating the work of formation of nuclei to a characteristic thickness 6 of the interface expressible in terms of bulk properties, has been proposed recently (diffuse interface theory, DIT) for both homogeneous’8a-e and heterogeneousIgf nucleation. It has been shown that near equilibrium the DIT and the CNT are equivalent, while far from equilibrium the DIT gives an improved description of experiments.’8b.dFor example, without adjustable parameters, the DIT describes the condensation of nonpolar vapors remarkably better than the CNT.I8”gbIf it is assumed that the interface diffuseness is essentially independent of undercooling (6 = constant), the DIT predicts a size-dependent interfacial free energy for the crystal-melt interface yielding an approximately linear y(T) function that eliminates the problem of anomalous prefactors in case of metallic melts and oxide glasses18c-f and offers a method to distinguish bulk heterogeneous and homogeneous nucleation. Note that the structural changes6 of undercooled water, which give rise to its anomalous physical properties, are expected to influence the structure of the ice-water interface as well, a phenomenon to be considered in the DIT model of ice nucleation. In this work the experimental data available on ice nucleation are analyzed in terms of the CNT and DIT. First it is demonstrated that using recent experimental data the combination of the CNT with = 1 leads to anomalous prefactors as high as XN = 105-1025. In contrast, the DIT analysis without correction for the structural changes of water (6 = constant) leads to slightly too low XN values and an inconsistency with the measured y(Tf) data. Then, Ewing’s model’3b is used to infer and x d = 6(T)/6(Tf) from structural data.6 These corrections improve the agreement between experiments and both theories, although for some data sets of Wood and Walton the CNT analysis still yields unphysical XN values (=lo4). It is concluded that, unlike the CNT, the DIT model is consistent with the experiments on ice nucleation. The present analysis indicates a bulk (volumetric) heterogeneous nucleation mechanism for all cases studied.
xv
xv
Diffuse Interface Analysis of Ice Nucleation Since the interfacial free energy for the ice-water interface is known with a poor accuracy, the diffuse interface analogue of the consistency test described in details in ref 18f will be applied. The physical basis of this approach can be summarized as follows. Characterizing the local physical state in the interface region by number density (N), specific intemal energy (u), and entropy (s) distributions, the work of formation of nuclei at constant extemal pressure po and temperature T can be expressed as
+
where Ah+ = N { u - uo po(v - V O ) } and As+ = N { s - SO}, u is the molecular volume, and V is the volume of the system after nucleation, while subscript 0 denotes properties of the parent phase. Equation 1 is equivalent with the respective equation of the field theoretic approach;”-’9 however, instead of using the square gradient approximation and solving the Euler
-
J. Phys. Chem., Vol. 99, No. 38, 1995 14183 Hg0: S=l (planar) T=273 K
m
c
,
1
S=10 (s herical) T=273 I!
I
I
,
I
Figure 1. Typical interfacial distributions and the definition of the characteristic thickness in (a) stable and (b) unstable equilibrium. (The distributions were calculated for the vupor/Ziquid interface using the van der WaalsKahn-Hilliard theory.I9 Here S = pdp, is the supersaturation, where PO andp, are the extemal and the equilibrium pressure, respectively.)
equation for the order parameter profile, we relate W to a Characteristic thickness expressible in terms of bulk physical properties. The procedure is illustrated on Ah+ and As+ profiles calculated for the vapor-liquid interface by the van der Waals/ Cahn-Hilliard theoryI9 (Figure 1). If we introduce step functions of the same integral and amplitude as the profiles, a straightforward choice of the characteristic thickness is 6 = RS - RH, where RH and Rs are the positions of the step functions (“enthalpy” and “entropy” surfaces henceforth). In stable equilibrium (planar interface) the area enclosed by either the Ah+ and As+ profiles or the step functions is equal to the interfacial free energy. Then, 6= yvmAHw-l,where um and AHt, are the molar volume of the new phase and the molar heat of transformation. In case of crystal nucleation, the anisotropy of the interfacial free energy should also be considered. The nuclei are expected to minimize their free energy, thus it is reasonable to assume7athat in case of a homogeneous process they have the equilibrium crystal shape defined by the Wulff rule,*’ Yhkl/ Rhkl = Constant (k.,6hkl/Rhkl = Constant), where Yhk1 iS the interfacial free energy of the crystal plane hkl and Rhkl is its distance from the center of the nucleus. Then, the enthalpy and entropy surfaces are similar polyhedra, and eq 1 transforms to
where K = ’/3ChklahklBhk?, a h k l = Ahki/Rhk?, B h k l = Rhkl/Rref, Ahki is the area of the face hkl, Rref and dref are the distance of the entropy surface of an arbitrary reference face from the center and the respective characteristic thickness, while A h + and Asof are the values of Ah+ and As+ at the center. It can be shown (see Appendix) that using the radius of the inscribed sphere as reference, K = 4.3‘” and 4n(l ~ ) ~ / { 3 (1 for the equilibrium shape of ice IC (octahedron) and Ih (ellipsoid of revolution with E = (rmax - rmin)/(rmax rmin) = 0.33g). Then, assuming (i) that bulk physical properties prevail at least at the center of the fluctuation (Aha+ .and ASO+are related to bulk thermodynamic properties), and (ii) that 8 h k l are independent of undercooling, a simple nucleation theory can be obtained for diffuse interfaces. Maximizing Whom with respect to Rref the size and work of formation on nuclei are Rref* = dreY(1 q}v-’ and Whom* = -Kdre$AgO+v, where $J = 2(1 q ) ~ -- ~ (3 2 q ) ~ - ~v-’, = (1 - v)”*, v = Ago+lAho+ and Ago+ = A h + - TAso+. Following the classical route, the nucleation rate for a homogeneous process reads as Ihom = IO,hom exp{Whom*/kT), where, as in the case of the C m , IO.hom = N o T z , except that the new formulas for Whom* and the critical size are used (0 = N A * , Ns is the surface density of molecules, A* =
+
+
+
+
+
+
Grhnby
14184 J. Phys. Chem., Vol. 99, No. 38, 1995
TABLE 1: Physical Properties Used in the Analysis 3~(R,,f*)~ is the surface area of nuclei, r = 6D/12,D is the diffusion coefficient, A is the jump distance of molecules, while 2.16 the factor Z = {(2nkr)-' ld2Whom*/di21p}1'2 accounts for the 0.917 23.7 f 5.4 dissolution of nuclei, i is the number of molecules in the 0.774 f 0.176 fluctuation, while * denotes quantities refemng to the nucleus). 6.019 Although in the DIT the relation between w* for the hetero273.15 geneous and homogeneous processes is more complex than in exp{79.33/(T - 215.4 K)) 3.45 x the CNT, an analysis of bulk (volume) heterogeneous nucleation [205400.1 - 3124.6797+ 17.85156F 4.53811 x 10-2p + 4.33084 x l O - T ] implies that in analogy to its classical counterpart, the plot log(Z/ -3.928 3.220 x 10-'T - 5.190 x 10-'11 XY IO.hom) vs XDIT= -AgofWT can be used to assess X N . ' ~Here ~ 7.757 - 4.879 x lO-*T+ 8.812 x 1O-'P Xd the slope is proportional to the cube of an apparent characteristic thickness 6,ff which is equal to 6 in case of a homogeneous negligible (as in our case), the rate of homogeneous nucleation process and smaller otherwise. Since the preexponential factor should be the smallest; Le., the plots yielding XN 1 should lie IO,hom depends on 6 weakly, a self-consistent plot can be found above the one calculated for a homogeneous process with (1 by a simple iteration scheme.Igd The assessed XN can be used A)yexp,where yexpis the experimental value. to test the applicability of nucleation theories. Besides XN 5 1, a lower limit can also be inferred from the fact that in the liquid Experimental Data Used in the Analysis dispersion method usually applied, a single nucleation event is enough to crystallize the sample. Then, XN = nNh,,/N should The planned analysis is known to be extremely sensitive to satisfy XN 2 1 ~ x 1 ,where n is the average number of active the thermal data.'Jse The relevant properties of water are known molecules on the surface of a heterogeneity, Nhetis the number accurately down to -37 "C and can be extrapolated to -40 "C density of heterogeneities, XI = l/NV, while NV is the number with a satisfactory accuracy. Therefore, nucleation rate data of molecules in the sample. Unfortunately, experimental from refs 2a-e are used which refer to this temperature range. information on n is nonexistent. The most potent nucleation Between -40 "C and the glass transition temperature (-136 areas (facilitated by local geometrical or chemical conditions) K) the properties of liquid water are very uncertain and are the probably cover only a fraction of the surface of a heterogeneity; subject of recent discussions.22 Thus, data on the nucleation Le., the lower limit of n is determined by the contact area of ice crystals in amorphous ice2f or in highly undercooled between the heterogeneity and the nucleus, which close to either water2g (T -73 "C) are excluded from the present analysis. the ideal or the nonwetting limits may be as low as 5-10 The five data sets read off the figures of ref 2a are molecules. The 10x1 is the absolute lower limit of XN (single representative for the more than 20 sets described in that work. heterogeneous nucleation site/sample), valid for surface-induced The results of Bertolini et alS2,were reanalyzed to infer the and bulk heterogeneous processes alike. Thus 10x1 -= XN 5 1 nucleation rate. These experiments usually indicate a faster and is a necessary criterion of consistency with the experiments. a slower nucleation process. Only data for the slower one were To apply for ice nucleation this approach should be modified considered since the accuracy of the nucleation rate is then much in two respects: higher. Special attention is devoted to the data by Wood and (a) In the light of the structural changes responsible for the WaltonZaand Taborek2dwhich refer to bulk nucleation; Le., the anomalous properties of undercooled water, the condition 6 = possibility of a homogeneous mechanism cannot a priori be constant probably cannot be retained. In the lack of experiments excluded. The other experiments may refer to surface induced a theoretical estimate is to be made. To the author's knowledge heterogeneous or volume nucleation alike. the only theory of the crystal-liquid interfacial that relates the interfacial free energy to the liquid structure is by E ~ i n g . ' ~ ~ The physical properties used in the present analysis are listed in Table 1. The amplitudes of the step functions were calculated Combining a nearest-neighbor broken-bond energetic contribufrom the experimental heat of fusion (A&), melting point and tion with an entropic term, obtained for noninteracting hard specific heat data as Aho+ = -{AHf lf AC, dT)v,-' and spheres from the assumption that the distribution of molecules Asof= -{AHf/Tf [ACdr] dT)v,-{. AC, was deternormal to the crystal surface is given by the pair correlation mined by fitting polynomials to the measured specific heat function g(r),the interfacial free energy can be expressed as y values of w a t e e and ice Ih.4b Since the respective data for = YE where modifications Ih (stable) and IC (metastable) cannot be distinguished experimentally,sthe same thermodynamic driving force is attributed to them. The interfacial free energy data from different sources3 derived with the assumption of isotropy deviate by more than while for the Ih (wurtzite) basal and the IC (diamond) 111 planes a factor of 2. A recent measurement of the equilibrium shape of water inclusion in ice Ih indicates almost complete isotropy a = 0.289,'3d AH(r) is the molar heat of fusion in the undercooled state, NAis the Avogadro number, R is the universal in the basal plane, but a strong anisotropy in planes including gas constant, and z is a spatial coordinate normal to the crystalthe c axis (6 = 0.3, which, according to the Wulff rule, corresponds to ye = 1.857yb, where subscripts b and e stand liquid interface with z = 0 at the dividing surface. Applying our definitions for the planar undercooled (non-equilibrium) for the basal and edge planes3g) far exceeding the anisotropy of free energy of the ice Ih-vapor interface.23 This fact crystal-liquid interface, and adopting the dividing surface used by Ewing, one finds that 6 = 6~ 6s, where 6~ = yEvmAH(T)-' emphasizes the importance of the liquid structure in determining the anisotropy of the ice-water interface and can account for and 6s = ysum{TAS(T)}-l. To take this information into account in the assessment of XN, log(Z/Zo,hom)should be plotted the contradictory experimental results, which are differently weighted averages for all orientations. Thus, only the ye = 44 vs XDIT= - ~ a ~ A g o + v- TI , where ~a = d(T)/d(Tf). (b) Although Y h k l is known with a poor accuracy ( A = f 10 mJ/m2 derived by Jones3b from the shape of the grain f22.7%), it offers still another criterion to be satisfied by a boundary groove in the basal plane (where the interfacial free sound nucleation theory. When the transient effects2' are energy is indeed isotropic) is to be considered, yielding Yb =
+
-'
+
+ JF,
+
+
+
J. Phys. Chem., Vol. 99, No. 38, 1995 14185
Ice Nucleation in Undercooled Water
7"
4 ' 235
I
Temperature (K)
0
241
Figure 2. Nucleation rates from various experiments: (0)Wood and ~ ~ Bertolini ,~ et Walton;2a (W) Taborek;2d (0)Butorin and S k r i p ~ v ;(0) aLZe)
XCNT
150 DIT; X 6 = l
1
23.7 f 5.4 mJ/m2. Since some of the average values are comparable to this ye, the real value is probably closer to the upper limit given by Jones. The 111 plane of IC is isostructural to the basal plane of Ih; thus it is reasonable to assume that the respective interfacial free energies are equal (7111= yb). 0 350 Unfortunately, there are no experiments on the equilibrium shape of ice IC in water. Therefore, following previous author^'^,^ Figure 3. Consistency analysis for the CNT (a) and DIT (b) assuming xy = 1 and x 6 = 1. (Xcm = x?/[(Ago+) 2 T ]is given in units of (A&/ the octahedral shape of atmospheric IC crystals7e is adopted, vJ2Tf-', while X D = ~ -xd3Ago+qT - I is presented in units of which can be derived also from the broken-bond theory. This AfZfu,-ITf-l. Notations: short dashed line, theoretical curve for shape is expected to hold, especially at low temperatures where homogeneous nucleation of IC with y l l l = 29.1 mJ/m2 (maximum the broken-bond equilibrium shape becomes a fair approximaallowed by the measurements of Jones); solid line, linear functions leastt i ~ n . *Then, ~ since the driving forces are about the same, the squares fitted to data for volume nucleation; long dashed line, same for experiments which may refer to both surface induced or volume ratio Wl,*/Wrh* x K I J K ~ < ~ 1; i.e., the nucleation of ice IC is preferred, as found in previous ~ ~ r k ~ .Accordingly, ~ g ~ ~ ~the. ~ , ~nucleation; xl. For other symbols see Figure 2. nucleation of only ice IC is considered in the rest of this work. Analysis for xu = 1 and xa = 1. The consistency plots for The temperature dependence of the diffusion coefficient is the CNT and DIT are presented in Figure 3. In accordance described by a Vogel-Fulcher expression fitted to lowwith the results of Wood and Walton, the combination of CNT temperature data.5 with xv = 1 yields anomalous prefactors (XN x lo5 + for Structural data6 from X-ray diffraction on D20 were used in almost all data studied here. The only exceptions are two sets Ewing's model. The pair correlation function at T < TOwas of experiments by Bertolini et al., where a heterogeneous calculated from the measured g(r,To) and Ag(r,51.O "C) by linear mechanism is indicated. In contrast, the DIT analysis results extrapolation: g(r,T) = g(r,To) (T - To)Ag(r,51.0 "C)/51, in XN < 1 in all cases, implying a heterogeneous mechanism as where TO = 11.2 "C is the temperature of maximum density expected from the slopes and relation of the Z(r) data. Note, for D20, and Ag(r,51.0 "C) is the isochoric differential pair however, that the XN values for the T, BS, and one of the B correlation function for temperature difference of 51 "C. The data sets are too low to satisfy the criterion 10x1 < XN (see interfacial free energy calculated for the Ih basal plane, yb(Tf) representing XI in Figure 3). To clarify the significance of these = 19.5 mJ/m2, falls within error of the experimental value, results, both the statistical error of XN and the uncertainty however, close to the lower limit. After being corrected for originating from the experimental error of the input parameters the difference of TOfor H20 and D20, second-order polynomials were investigated. It has been found that the main contributor were fitted to the computed xY(nand xa(T) relations. to the uncertainty of XN is the scattering of the nucleation rate data (0.6 + 5.6 0.m. for CNT and 0.2 - 2.2 0.m. for the DIT; 0.m. = orders of magnitude). Considering even these figures, Results and Discussion the majority of the CNT results are incompatible with the criterion XN < 1. In contrast, the DIT does not meet the Before reviewing the results of the consistency analysis it is condition 10x1 < XN, although this failure is much less worth comparing the experimental data on ice nucleation from pronounced than that of the CNT. It is, however, alarming that various sources. The nucleation rate vs temperature relations the homogeneous nucleation rates from the DIT (short dashed are shown in Figure 2. The filled symbols and solid lines line in Figure 3b), computed using even the highest 6 allowed indicate data for volume nucleation. It is remarkable that, by the experiments of Jones, are considerably larger than the although the data by Wood and Walton2" (WW, of which only measured values, a situation clearly unphysical. Were the two representative sets are shown) are about 2-4 orders of nucleation rate higher for the homogeneous process than for a magnitude higher than those by TaborekZd(T), their slope is heterogeneous one (Ihom > Ihet) the heterogeneous process could steeper (even considering the scattering of the points), indicating not be observed at all. Thus, despite its success in case of other a higher work of formation of nuclei. This is only possible if nucleation processes,18 the combination of DIT with d = the experiments refer to heterogeneous processes, of which one constant fails to describe ice nucleation in undercooled water. is of higher heterogeneity concentration and larger contact A possible reason for this failure is the structural change of angle B than the other. The data by Butorin and SkripovZb(BS) water with increasing undercooling. Therefore, in the next and Bertolini et aLZe(B) are rather similar to the former ones, paragraph a modified analysis is presented which utilizes the raising the possibility that bulk heterogeneous nucleation took structural information available on undercooled water. place in all cases examined here. 1"
*,
+
*
Grininhy
14186 J. Phys. Chem., Vol. 99, No. 38, 1995
0 230
-40' 0
Temperature (K)
XCNT I
-40' 0
270
I
120 I
I XDlT
400
Figure 4. The temperature dependence of y and 6 computed from experimental pair correlation functions using Ewing's model (a); and the modified consistency analysis for the CNT (b) and DIT (c) based on the predicted x y ( T ) and xa(T) functions. (For notations and units see Figure 3.)
Analysis for ~~(2')and ~s(2'). The temperature dependence of the interfacial free energy and the characteristic thickness calculated from structural data using Ewing's model and the results of the modified consistency analysis are presented in Figure 4. In accordance with other theoretical predictions'2a-c.d the interfacial free energy decreases with increasing undercooling (AT). In Ewing's model this dependence originates from (i) the explicit temperature dependence of the interfacial free energy (y s = -TS,), and from the variation of (ii) the enthalpy difference and (iii) the surface entropy S, = -Rv,,,-'G'g In g dz.. In contrast, the change of 6 is entirely of entropic origin. Note, that S,is the entropy reductiodinterface area (relative to the bulk liquid) due to the ordering of the liquid in contact with the crystal, an effect also borne out by computer simulationsg and the density functional theory.l0.ll a The temperaturedependent measurements of g ( r ) show6that the nearest-neighbor coordination is essentially constant, while the most pronounced change with increasing AT is the reduction of the number of next nearest neighbors, indicating the development of a more open structure25 with stronger structural correlation. While owing to the stronger structural correlation in the liquid the entropy difference between the bulk undercooled water and ice reduces, the thickness of the transient layer is expected to increase.26 The resultant of these opposing effects determines the change of the interfacial entropy reduction. Ewing's model predicts that the second effect dominates, yielding a surface entropy and a characteristic thickness which increase with undercooling (see Figure 4a).
The results of the modified CNT analysis are summarized in Figure 4b. The most pronounced change compared to the xr = 1 case is that now the assessed XN values are much lower, and the majority of them satisfy the consistency criterion 10x1 < XN 5 1. Still the plots for two of the five WW data sets examined here yield anomalous prefactors although much less pronounced ones than in the previous section, log XN = 4.0 f 2.3 and log XN = 2.9 f 1.7, where the error is again dominantly from the statistical scattering of the nucleation rate data (2.0 and 1.4 o.m., respectively). The examination of the work of Wood and Walton shows that four further data sets would lead to comparably unphysical results. It is worth noticing that the XN values for the rest of the experiments indicate a heterogeneous nucleation mechanism. The modified consistency plots for the DIT are shown in Figure 4c. The correction for the temperature dependence of 6 increased XN; thus all the respective values satisfy now the criterion 10x1 < XN 1, indicating a heterogeneous nucleation mechanism for the experiments investigated. It should be noted that the same WW data sets which remained problematic in the modified CNT analysis lead to log XN = -1.5 f 2.6 and log XN = -2.3 f 2.1, which may seem a little too high. Taking, however, the lowest values allowed by the error, one obtains log XN = -4.1 + -4.4, which implies that about 50 + 100 ppm. of the water molecules are in contact with heterogeneities, a figure not unrealistic for the applied experimental technique. It is also reassuring that the nucleation rates calculated for a homogeneous process now fall below the experimental ones, as they should. The relative positions of the respective plots imply that slightly higher undercoolings in the experiments of Taborek or Butorin and Skripov would have led into the domain of homogeneous nucleation. In summary, we may conclude that, in contrast with the classical approach, the diffuse interface theory is fully consistent with the experiments on ice nucleation in undercooled water provided that the structural change of water is taken into account. Finally, a more elegant test could be possible were a more accurate value for the ice-water interfacial free energy available. The author hopes that this work raises enough interest to repeat Hardy's accurate measurements on grain boundary grooves,3chowever, this time between crystals with their c-axes perpendicular to the plane of observation as done by Jones.3b
Concluding Remarks Experiments on ice nucleation in undercooled water have been analyzed in terms of the classical and the diffuse interface theory of nucleation. It has been shown that none of them describes the experiments satisfactorily until the structural changes of undercooled water are taken into account: the CNT yields anomalous prefactors, while the DIT results are inconsistent with the interfacial free energy data. A modified analysis based on the temperature-dependent interfacial parameters evaluated from structural information using Ewing's model yields an improved agreement between both theories and the experiments. While the CNT still leads to anomalous prefactors (although much less pronounced than before), the DIT is fully consistent with the experimental data. In the cases studied both analyses (CNT and DIT) indicate a bulk heterogeneous nucleation mechanism.
Acknowledgment. A part of this work was performed during the tenure of an Alexander von Humboldt Research Fellowship of the author spent in the Institut fiir Raumsimulation, DLR, Cologne, Germany. The author expresses his thanks to Prof. B. Feuerbacher and D. M. Herlach for their hospitality, and to Dr. L. Ratke, Prof. I. Egry and Dr. D. M. Herlach for the valuable
Ice Nucleation in Undercooled Water
J. Phys. Chem., Vol. 99, No. 38, 1995 14187
discussions. This work has been supported by the Hungarian Academy of Sciences under contracts OTKA-T-017485 and OTKA-T-017456.
Appendix: Geometrical Factors for IC and Ih nuclei IC: The equilibrium shape is an octahedron7abound by 111 Taking Rref = R I I I , a1I I = 33’2/2and PII I = 1, thus K I ~ = (8/3)alllP1113= 4x3Il2. Ih: The equilibrium shape is an ellipsoid of revolution3g of E = 0.3. Taking Rref = rmin,where rminis the radius of the inscribed sphere, and utilizing that Vn = (4n/3)(1 ~ ) ~ -( 1 c)-*rmjn3,one obtains KIh = (4n/3)(1 €)*(I - c)-2.
+
+
References and Notes (1) A recent review on crystal nucleation: Kelton, K. F. Solid State Phys. 1991, 25, 75. (2) (a) Wood, G. R.; Walton, A. G. J. Appl. Phys. 1970,41, 3027. (b) Butorin, G. T.; Skripov, V. P. Kristallografiya 1972, 17, 379. (c) Skripov, V. P. In 1976 Crystal Growth and Materials: Kaldis., E., Scheel., H. J., Eds.; North-Holland: Amsterdam, 1977; p 327. (d) Taborek, P. Phys. Rev. 1985, 832, 5902. (e) Bertolini, D.; Cassettari, M.; Salvetti, G. Phys. Scr. 1988, 38, 404. (0Koverda, V. P.: Skripov, V. P.: Bogdanov, N. M. Kristallografiya 1974, 19, 613. (g) Huang. J.; Bartell, L. S. J. Phys. Chem. 1995, 99, 3924. (3) (a) Skapski, A.: Billups, R.; Rooney, A. J. Chem. Phys. 1957, 26, 1350. (b) Jones, D. R. H. J. Mater. Sci. 1974, 9, 1; Philos. Mag. 1973,27, 569. (c) Hardy, S. C. Philos. Mag. 1977, 35, 471. (d) Femandez, R.; Barduhn, A. J. Desalination 1967, 3, 330. (e) Kotler, G. R.; Tharshis, L. A. J. Cryst. Growth 1968,3-4,603. (f) Coriell, S. R.; Hardy, S. C.; Sekerka, R. F. J. Cryst. Growth 1971, 11, 53. (g) Koo, K. K.: Ananth, R.; Gill, W. N. Phys. Rev. 1991, A44, 3782. (4) (a) Angell, C. A.; Oguni, M.: Sichina, W. J. J. Phys. Chem. 1982, 86, 998. (b) in CRC Handbook of Chemistry and Physics; 51st ed.; Weast, R. C., Ed.; The Chemical Rubber Co.: Boca Raton, FL, 1970-1971; p D-129. ( 5 ) Gillen, K. T.; Douglas, D. C.: Hoch. M. J. R. J. Chem. Phys. 1972, 57, 51 17. (6) Bosio, L.; Chen, S. H: Teixeira, J. Phys. Rev. 1983, A27, 1468. (7) (a) Takahashi, T. J. Cryst. Growth 1982, 59, 441. (b) Knight, C. A. J. Cryst. Growth 1983, 62, 633. (c) Kiefte, H.; Clouter, M. J.; Whalley, E. J. Chem. Phys. 1984, 81, 1419. (d) Vigier, G.;Thollet, G.; Vassoille, R. J. Cryst. Growth 1987, 84, 309. (e) E. Whalley J. Phys. Chem., 1983, 87, 4174. (8) For example: Water-A Comprehensive Treatise; Franks,F., Ed.; Plenum: New York, 1982; Vol. 7. (9) For example: Broughton, J. Q.: Bonissent, A,; Abraham, F. F. J. Chem. Phys. 1981, 74, 4029. Broughton, J. Q.: Gilmer, G. H. J. Chem. Phys. 1983, 79, 5095, 5105, 5119: 1986, 84, 5741, 5749, 5759. Laird, B. B.: Haymet, A. D. J. J. Chem. Phys. 1989,91,3638; Chem. Rev. 1992.92, 1819. (10) For example: (a) Oxtoby, D. W.: Haymet, A. D. J. J. Chem. Phys. 1982, 76, 6262. (b) McMullen, E. D.; Oxtoby, D. W. J. Chem. Phys. 1988,
88, 1967. (c) Curtin, W. A. Phys. Rev. 1989, 839, 6775. (d) Man, D. W.: Gast, A. P. J. Chem. Phys. 1993, 99, 2024. (11) (a) Hanowell, P.: Oxtoby, D. W. J. Chem. Phys. 1984, 80, 1639. (b) Bagdassarian, C. K.; Oxtoby, D. W. J. Chem. Phys. 1994, 100, 2139. (12) It can be shown that for equilibrium crystal shapes given by the Wulff rule (see ref 20) K = V J P , where V, is the volume of the nucleus, while r is the radius of the inscribed sphere. (13) (a) Skapski, A. S. Acta Metall. 1956, 4, 576. (b) Ewing, R. H. J. Cryst. Growth 1971, 11, 221. Waseda, Y.; Miller, W. A. Trans. Jpn. Inst. Metals 1979,19,546. (c) Spaepen, F. Acta Metall. 1975,23,729. Spaepen, F.;Meyer, R. B. Scr. Metall. 1976, 10, 257. (d) GrBnBsy, L.; Tegze, M. Mater. Sci. Forum 1991, 77, 243. (14) Tumbull, D. J. Chem. Phys. 1952,20,411. Miyazawa, Y.: Pound, G. M. J. Cryst. Growth 1974, 23, 45. (15) James, P. F. J. Non-Cryst. Sol. 1985, 73, 517. (16) For example: Wagner, P. E.: Strey, R. J. Chem. Phys. 1983, 80, 5266. Hung, C. H.; Krasnopoler, M. J.; Katz, J. L. J. Chem. Phys. 1989, 90, 1856; 1990,92,7722. Wright, D.; El-Shall, M. S. J. Chem. Phys. 1993, 98, 3369. (17) Abraham, F. F. J. Chem. Phys. 1975, 63, 157. (18) (a) GrhAsy, L. Europhys. Lett. 1993, 24, 121. (b) Granby, L.; Egry, I.; Ratke, L.; Herlach, D. M. Scr. Metall. Mater. 1994, 30, 621. (c) GrBnisy, L. J. Non-Cryst. Sol. 1993, 162, 301. (d) Grhasy, L. Mater. Sci. Eng. 1994, A178, 121. (e) GrAnlsy, L. Scr. Metall. Mater. 1995, 32, 1611. (0GrBnisy, L.: Egry, I.; Ratke, L.; Herlach, D. M. Scr. Metall. Mater. 1994, 31, 601. (19) Van der Waals, J. D. Verhand. Kon. Akad. u. Werensch. (le Sect.) 1893, I , 1. Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28, 258; 1959, 31, 688. For the crystal-liquid interface the Aht and As+ profiles could be obtained from laborious calculations using the density functional theory.” (20) Wulff, G. Z. Kristallogr. 1901,34, 449. (21) Nucleation transient: A period of reduced nucleation rate until the steady-state cluster distribution is established. In the temperature range of experiments Kashchiev’s expression for the transient time (Kashchiev, D. Surf Sci. 1969, 14, 209) predicts about 1 ns for ice nucleation, implying that the transient effects can safely be neglected on the experimental time scale. (22) See e.g.: Speedy, R. J. J. Phys. Chem. 1992, 96,2322. Angell, C. A. J. Phys. Chem. 1993,97,6339. Johari, G. P.; Fleissner, G.: Hallbruckner, A.; Mayer, E. J. Phys. Chem. 1994, 98, 4719 and references therein. (23) Furukawa, Y.; Yamamoto, M.; Kuroda, T. J. Cryst. Growth 1987, 82, 665. Gonda, T.; Yamazaki, T. J. Cryst. Growth 1978, 45, 66. (24) Moore, K. I.; Zhang, D. L.; Cantor, B. Acta Metall. Mater. 1990, 38, 1327. (25) The density, viscosity, pair correlation function, and SAXS data are not inconsistent with the presence of an increasing number of “clathratelike” cages (of approximately 5-6 Adiameter) at lower temperatures. See e.g.: Stillinger, F. H. Science 1980,209,451. Halfpap, B. L.; Sorensen, C. M. J. Chem. Phys. 1982, 77, 466. Sorensen, C. M. J. Chem. Phys. 1983, 79, 1455. Xie, Y.; et al. Phys. Rev. Lett. 1993, 71, 2050. (26) That the interface thickness increases with increasing similarity of the bulk phases is not unusual. In systems that show a critical point, the thickness of the interface between the coexisting phases (coherence length) diverges at the critical point. JP951117Q
