Adiabatic Nucleation

Journal of Non-Crystalline Solids 274 (2000) 162±168

www.elsevier.com/locate/jnoncrysol

Section 12. Nucleation and crystallization II

Adiabatic nucleation in supersaturated liquids Elon M. de S a, M aximo F. da Silveira, Erich Meyer, Vitorvani Soares * Instituto de Fõsica da Universidade Federal do Rio de Janeiro, C.T., Bl. A Cidade Universit aria, 21945-970 Rio de Janeiro, Brazil

Abstract Adiabatic nucleation theory (ANT) is shown to be in good agreement with experimental data of superheated liquids (boiling). Pure superheated liquids nucleate just a few K before reaching the spinodal, calculated by the Peng±Robinson equation of state. In spite of the correlation between the nucleation curve and the spinodal, nucleation is usually explained by the (isothermal) classical nucleation theory (CNT), which does not take into account the proximity of the spinodal. An alternative explanation is given by ANT. When the liquid is close to the spinodal, the spinodal is reached and overpassed by volume ¯uctuations and the nucleus appears as the result of spinodal phase separation. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Adiabatic nucleation theory (ANT) has been successfully applied to liquid metals [1,2], oxide glasses [3±5], polymers [4,5], gel-derived glasses [6,7], and metallic and chalcogenide glasses [8,9]. All these cases are related to the liquid±solid phase transition, which is a special case in the sense that the coexistence curve does not terminate at a critical point and that there are no spinodals. In contrast, the liquid±vapor (boiling/cavitation) phase transition is related to a coexistence curve, which terminates at a critical point and there are spinodals, which terminate at the same point. In the present work, ANT is adapted to this type of transition. In the enthalpy±entropy±pressure (Mollier) diagram, how the system looses stability near the spinodal [10,11] is shown. The spinodal

curves can be determined by Peng±Robinson equation of state [12] (an improved van der WaalsÕ equation) and are de®ned by 

op ov

 ˆ 0;

where p, v, and T are the pressure, molar volume, and absolute temperature, respectively. With the relation [13,14] ÿ op
2


v ; cp ÿ cv ˆ ÿT ÿ oT op

…2†

ov T

where cp and cv are the molar speci®c heats at constant pressure and volume, respectively, one sees that at the spinodal cp ˆ 1

* Corresponding author. Tel.: +55-21 560 0191; fax: +55-21 560 0191. E-mail address: [email protected] (V. Soares).

…1†

T

…3†

because cp > cv and because …op=oT †v is a function which does not diverge. The speci®c heat cv is in a

0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 1 1 - 8

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

163

®rst approximation, a function of the temperature only 1 and also because of this does not diverge at the spinodal. Because of   os ; …4† cp ˆ T oT p where s is the molar entropy, one obtains at the spinodal,    2  oh oT T ˆ ˆ ˆ 0; …5† os2 p os p cp where h is the molar enthalpy. We see that the spinodals are represented by in¯ection points, where the curvature changes from positive to negative. In areas of stability we have    2  oh oT T ˆ ˆ >0 …6† os2 p os p cp and in areas of instability  2    oh oT T ˆ ˆ < 0: 2 os p os p cp

…7†

It follows that the h…s†…p ˆ const:† curve must qualitatively be as shown in Fig. 1, where point 1 and 2 represent the in¯ection points. Point A and B are points of the coexistence curve and DhL and DsL are the molar latent heat and latent entropy, respectively. In contrast to the liquid±solid phase transition, where it is easy to calculate and design the h(s) curves quantitatively [1,2,15], in the present case it is more complicated, because only with the help of Eq. (2), together with the Peng±Robinson equation of state, do we know in which way cp diverges, approaching the spinodal. However, the qualitative Fig. 1 helps to understand the nucleation phenomenon and the exact position of the spinodal can directly be calculated by the Peng± Robinson equation, as a function of p and T. When the system approaches the spinodal, volume ¯uctuations have larger e€ects and to get an idea of what may happen, we consider the mean square ¯uctuations of temperature and volume. Interest1

See [14], p. 229.

Fig. 1. A qualitative isobaric curve is shown in the enthalpy (h)±entropy (s) (Mollier) diagram. Point A and B are points of the coexistence curve of the liquid and vapor and DhL and DsL are the latent heat and latent entropy, respectively. Point 1 and 2 are spinodal (in¯ection) points. Point 3 is a point in the unstable range, supposed to be attained by volume ¯uctuations. Arrow 3-C shows the resultant entropy increase of the adiabatic nucleation process by ÔspinodalÕ phase separation. Note that this ®gure can be understood with the enthalpy and entropy per mol, per molecule or by any number of molecules. The curves remain qualitatively the same. One also has to consider that volume ¯uctuations may not occur isobarically, as suggested for simplicity in this ®gure. What is important, is that they may overpass the spinodal when the sample variables are suciently close to it.

ingly, the volume, which undergoes a discontinuity in the ®rst order phase transition, has also mean square ¯uctuations, which go to in®nity at the spinodal [14]. The temperature, on the other hand, shows mean square ¯uctuations, which do not diverge (in Refs. [10] and [11] we used only temperature ¯uctuations), 2

h…DT † i k ˆ ; 2 T CV 2

h…DV † i kT ˆÿ 2 V2 V

…8† 

oV op

 ;

…9†

T

where CV is the heat capacity per number of ¯uctuating molecules and at constant volume and k is the Boltzmann constant. CV is a function of the temperature only 2 and because of this does not diverge at the spinodal. 2

See [14], p. 229.

164

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

The arrow (3-C) in Fig. 1 schematically shows the adiabatic nucleation process. When the system is between point A and point 1, close to point 1, a volume ¯uctuation may transport a minimum quantity of molecules to any position of the curve between point 1 and 2, as e.g., point 3. This ¯uctuation process is not necessarily isobaric, as shown for simplicity in Fig. 1. What is important, is that the ¯uctuation overpasses the spinodal. This group of molecules may separate into two parts, one part evolving in the direction of point B and the other backwards in the direction of point A. The arrow shows the resultant process. Of course the two parts exchange enthalpy (heat) between themselves. Both parts together, however, may be considered as an adiabatically evolving subsystem (at least at the ®rst moment), which in this way increases the entropy, as postulated by the second law of thermodynamics, for systems which are not in equilibrium, but are moving irreversibly to it. Note that Fig. 1 can be understood with entropy and enthalpy per mol or per molecule or per any number of molecules. The curves remain qualitatively the same. Interestingly, no interfacial tension problem is involved at the very ®rst moment of this adiabatic nucleation process. At the ®rst moment of phase separation (at point 3 in Fig. 1), the phases are almost identical, so that the interfacial tension is negligible. The interfacial tension develops then more and more with increasing phase separation. Fig. 2 shows two isotherms, calculated by the Peng±Robinson equation of state (for H2 O), in a pr ±vr graph. There, pr ˆ p=pc ; vr ˆ v=vc and Tr ˆ T =Tc , where pc , vc and Tc are the critical pressure, volume and temperature, respectively. The spinodal curve, which surrounds the (hatched) instable range, is also shown. It can be seen that at the left (liquid) side …ov=op†T is much smaller, except very close to the spinodal (where …ov=op†T ˆ 1). That is why, in the liquid state, only very close to the spinodal, do volume ¯uctuations occur which can overpass the spinodal and create in this way the necessary conditions for adiabatic nucleation. On the other (vapor) side, however, the situation is opposite, …ov=op†T is much larger everywhere, even near the coexistence curve (not shown in Fig. 2). That is why the nu-

Fig. 2. Two isotherms, calculated by the Peng-Robinson equation of state (for H2 O), are shown in a pr ±vr (reduced pressure±reduced volume) graph. The spinodal curve, which surrounds the (hatched) instable range, is also shown. It can easily be seen that at the left (liquid) side (ov/op)T , which is proportional to the mean square of volume ¯uctuations, is very small, except very close to the spinodal (where …@v=@p†T ˆ 1). That is why, in the liquid state, only very close to the spinodal, volume ¯uctuations occur which can overpass the spinodal and create in this way the necessary conditions for adiabatic nucleation. On the other (vapor) side, however, the situation is opposite, (ov/op)T is large everywhere, even near the coexistence curve (not shown in Fig. 2). That is why the nucleation of the liquid in the vapor can occur already far from the spinodal. This prevision is in complete agreement with experimental observations, as shown for supersaturated liquids in Figs. 3 and 4.

cleation of the liquid can occur already far from the spinodal. Eq. (9) results in 1/n for the limit of the ideal gas, where n is the number of molecules participating in the ¯uctuation. This analysis is in qualitative agreement with experimental observations, as shown for supersaturated liquids below. 2. Determination of the spinodal The Peng±Robinson equation of state [12,13] pˆ

RT a…x; Tr † ÿ v ÿ b v…v ‡ b† ‡ b…v ÿ b†

…10†

is used because it can be (quantitatively) applied to the vapor as well as to the liquid range, while van der WaalsÕ equation is only qualitatively correct, especially for the liquid. R is the gas constant and

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

x is the acentric factor. The parameters a, b and x are related as shown below a…x; Tr † ˆ a…Tc †a…x; Tr †; a…Tc † ˆ 0:45724

2

where a0 ˆ 0:45723553;

…17†

b0 ˆ 0:07779607;

…18†

…11†

Tc2

R ; pc

…12†

and zc ˆ 0:30740131:

2

a…x; Tr † ˆ ‰1 ‡ j…1 ÿ Tr1=2 †Š ;

…13†

j ˆ 0:37464 ‡ 1:54226x ÿ 0:26992x2 ;

…14†

…19†

The spinodals were calculated using Eqs. (1), (16)± (19) and the data of Table 1 [16]. 3. Results

RTc : b ˆ 0:07780 pc

…15†

For our purpose it is more convenient to write Eq. (10) in reduced variables [16] 2

pr ˆ

165

Tr a0 ‰1 ‡ j…1 ÿ Tr1=2 †Š ÿ 2 2 ; z c v r ÿ b0 zc vr ‡ 2zc b0 vr ÿ b20

…16†

The results are shown in Fig. 3, where the experimental points are data of maximum supersaturations in liquids. The right line is the spinodal, calculated by Peng±Robinson equation of state and using the acentric factors from Table 1. The interrupted line at the left is the calculated (Peng± Robinson) coexistence curve and the dotted line in

Table 1 Acentric factor of liquids Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

H2 O He4 H2 N2 O2 CH4 Ar Kr Xe C5 H12 C6 H14 C7 H16 C8 H18 C6 H12 C6 H6 CH4 O C2 H6 O C3 H6 O C3 H8 C4 H10 O ClCHF2 C5 F12 C6 F14 C7 F16

Water Helium Hydrogen Nitrogen Oxygen Methane Argon Krypton Xenon n-Pentane n-Hexane n-Heptane n-Octane Cyclohexane Benzene Methanol Ethanol Acetone n-Propane n-Butanol Chlorodi¯uoromethane Per¯uoropentane Per¯uorohexane Per¯uoroheptane

Acentric factor x

Refs. of experimental values collected in [17]

0.344 )0.365 )0.218 0.039 0.025 0.011 0.001 0.005 0.008 0.251 0.299 0.349 0.398 0.212 0.212 0.556 0.644 0.304 0.153 0.593 0.221 0.432 0.514 0.556

[18±21] [22,23] [24] [25±27] [25,28] [25] [27,29,30] [29] [29] [19,31±35,44] [18,19,32,34±41] [19,36,42,43] [43,45,46] [31,39,41,43,47,48] [18,19,21,32,34,35,39,49,50] [19,31,35,51±53] [19,31,35,37,49±52] [19,31,49,50] [46,54,55] [19,30,49,50,52] [54] [56] [56] [56]

166

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

Fig. 3. The experimental points for H2 O, Ar, N2 , O2 , H2 , He4 , CH4 O, CH4 , Kr, Xe, C2 H6 O, C3 H8 , C5 F12 , C5 H12 , C6 F14 , C6 H12 , C6 H6 , C6 H14 , C7 F16 , C7 H16 , C8 H18 , and ClCHF2 are data of maximum supersaturations in the liquid phase. The right line is the spinodal, calculated by Peng±RobinsonÕs equation of state and using the acentric factors from Table 1. The interrupted line at left is the calculated (Peng±Robinson) coexistence curve and the dotted line is a qualitative extrapolation. Literature data for measurements of di€erent estimated nucleation frequencies are given with di€erent symbols (in cmÿ3 sÿ1 ), as indicated for each substance. The general trend is that liquids can be supersaturated up to very close to the spinodal, in excellent agreement with ANT.

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

Fig. 3 (H2 O) is a qualitative extrapolation. Measurements of di€erent estimated nucleation frequencies are given with di€erent symbols (in cmÿ3 sÿ1 ), as indicated in each ®gure. The general trend is that liquids can be supersaturated close to the spinodal, in agreement with ANT. A few exceptions, where experimental points appear on the right side of the spinodal, may be due to experimental errors and/or inexactness of the calculated spinodals. Fig. 4 shows all experimental points of Fig. 3 in a single graph. The line at right is a mean spinodal …x ˆ 0:201† obtained by using weighted acentric factors. (For each substance, in Fig. 3, its related acentric factor was multiplied by the number of corresponding experimental points and all these numbers were summed and the result divided by the total number of experimental points.) The line in the middle is the spinodal calculated by van der WaalsÕ equation. Almost all experimental points (except those of He and H2 , which have negative acentric factors) are at the right side of this spinodal. This shows the limitation of van der WaalsÕ equation (except for He and H2 , van der Waals spinodal can rather be considered as an Ôempirical

167

curve of critical supersaturationÕ below which homogeneous nucleation frequencies are very low) and explains, why the present ANT for liquids could only be proposed after the publication of the Peng±Robinson equation in 1976. The interrupted line at the left is the coexistence line as calculated by van der WaalsÕ equation and the dotted line is a qualitative extrapolation of it.

4. Discussion Nucleation in boiling/cavitation phenomena is usually explained by the (isothermal) classical nucleation theory (CNT), which does not take into account the strong correlation between nucleation and spinodal curves and which uses for the vapor the equation of the ideal gas. An alternative way, which explains this correlation, is given by ANT. Consider a sample at a temperature between point A and point 1, close to point 1, in Fig. 1. A group of molecules can then reach and overpass the stability limit, de®ned by the spinodal curve, by means of statistical volume ¯uctuations, giving rise to nuclei, adiabatically and with increasing entropy. ANT is completely di€erent from CNT. However, Mokross [57] showed that there are similarities between ANT for the liquid to vapor transition (in the enthalpy±entropy diagram) and the Cahn±Hilliard [58,59] theory (in the free energy±concentration diagram).

5. Conclusions

Fig. 4. All experimental points of Fig. 3 are shown in a single graph. The line at the right is a mean spinodal …x ˆ 0:201† obtained by using weighted acentric factors. The line in the middle is the spinodal calculated by van der Waals equation. Almost all experimental points (except those of He and H2 ) are at the right side of this spinodal. This shows the limitations of van der Waals equation. The interrupted line at the left is the coexistence line as calculated by van der Waals equation and the dotted line is a qualitative extrapolation of it.

The results of this work con®rm that ANT can be applied to the liquid±vapor phase transition as it has been successfully used in the case of liquid± solid phase transitions. However, in both cases it is not yet possible to calculate nucleation frequencies. At present, it is only possible to indicate at what temperatures and pressures strong homogeneous nucleation is expected in the liquid to vapor and in the liquid to solid phase transitions. It is also explained why a similar general indiction cannot be made for the vapor to liquid transition.

168

E.M. de S a et al. / Journal of Non-Crystalline Solids 274 (2000) 162±168

Acknowledgements The institutions CEPG/UFRJ, CNPq and FINEP are acknowledged for ®nancial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

E. Meyer, J. Cryst. Growth 74 (1986) 425. E. Meyer, J. Cryst. Growth 76 (1986) 525. E.D. Zanotto, J. Non-Cryst. Solids 89 (1987) 361. E. Meyer, J. Cryst. Growth 84 (1987) 533. E. Meyer, J. Non-Cryst. Solids 102 (1988) 148. E. Meyer, E.D. Zanotto, M.A. Aegerter, J. Non-Cryst. Solids 121 (1990) 279. E.D. Zanotto, J. Non-Cryst. Solids 147&148 (1992) 820. V. de B. Brasil, E. Meyer, J. Non-Cryst. Solids 219 (1997) 75. E. Meyer, J. Non-Cryst. Solids 219 (1997) 30. M.F. da Silveira, E. Meyer, Mater. Sci. Forum 179±181 (1995) 753. E. Meyer, in: J. Menon (Ed.), Current Topics in Crystal Growth Research, vol. 2, Research Trends, Trivandrum, India, 1995, p. 123. D.-Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59. M. Modell, R.C. Reid, Thermodynamics and its Applications, Prentice-Hall, Englewood Cli€s, NJ, USA, 1983. L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon, London, 1958. M.F. da Silveira, E. Meyer, in: A. Ferraz, F. Oliveira, R. Os orio (Eds.), Non-Linear Physical Phenomena, World Sci. Publ., 1990, p. 173. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. C.T. Avedisian, J. Phys. Chem. Ref. Data 14 (1985) 695. P.A. Pavlov, V.P. Skripov, High Temp. (USSR) 8 (1970) 540. V.P. Skripov, P.A. Pavlov, High Temp. (USSR) 8 (1970) 782. L.J. Briggs, J. Appl. Phys. 21 (1950) 721. L.J. Briggs, J. Chem. Phys. 19 (1951) 970. D.H. Sinha, J.S. Semura, L.C. Brodie, Phys. Rev. A 26 (1982) 1048. D.N. Sinha, PhD thesis, Portland State University, Portland, OR, 1982. J. Hord, R.B. Jacobs, C.C. Robinson, L.L. Sparks, J. Eng. Power 86 (1964) 485. V.G. Baidakov, V.P. Skripov, Russ. J. Phys. Chem. 56 (1982) 499. K. Nishigaki, Y. Saji, Jpn. J. Appl. Phys. 20 (1981) 849. J.W. Beams, Phys. Fluids 1 (1959) 1. A.Ye. Dianov, V.G. Mal'tsev, V.G. Baydakov, V. Skripov, Heat Transfer Sov. Res. 12 (1980) 75. V.P. Skripov, V.G. Baidakov, A.M. Kaverin, Phys. Sci. 95A (1979) 169.

[30] V.G. Baidakov, V.P. Skripov, A.M. Kaverin, Sov. Phys. JETP 38 (1974) 557. [31] Ye.N. Sinitsyn, N.N. Danilov, Heat Transfer Sov. Res. 12 (1980) 66. [32] Ye.N. Sinitsyn, V.P. Skripov, Russ. J. Phys. Chem. 42 (1968) 440. [33] Ye.N. Sinitsyn, V.P. Skripov, Prib. Tekh. Eksp. 4 (1966) 178. [34] V.P. Skripov, Ye.N. Sinitsyn, Russ. J. Phys. Chem. 42 (1968) 167. [35] V.P. Skripov, Metastable Liquids, Wiley, NY, 1974. [36] V.P. Skripov, G.V. Ermakov, Russ. J. Phys. Chem. 37 (1963) 1047. [37] V.P. Skripov, P.A. Pavlov, E.N. Sinitsyn, High Temp. (USSR) 3 (1965) 670. [38] Ye.N. Sinitsyn, V.P. Skripov, Ukr. Phys. J. 12 (1967) 100. [39] B.S. Holden, J.L. Katz, AIChE J. 24 (1978) 260. [40] V.P. Skripov, V.L. Kukushkin, Russ. J. Phys. Chem. 35 (1961) 139. [41] H. Wakeshima, K. Takata, J. Phys. Soc. Jpn. 13 (1958) 1398. [42] C.T. Avedisian, I. Glassman, J. Heat Transfer 103 (1981) 272. [43] J.G. Eberhart, W. Kremsner, M. Blander, J. Colloid Interface Sci. 50 (1975) 369. [44] V.P. Skripov, G.V. Ermakov, Russ. J. Phys. Chem. 38 (1964) 208. [45] C.T. Avedisian, J. Heat Transfer 105 (1982) 750. [46] T.A. Renner, G.H. Kucera, M. Blander, J. Colloid Interface Sci. 52 (1975) 319. [47] G.V. Ermakov, V.P. Skripov, Russ. J. Phys. Chem. 43 (1969) 1242. [48] E.N. Sinitsyn, N.N. Danilov, V.P. Skripov, Teplo®zika 425 (1971) 22. [49] N.N. Danilov, Ye.N. Sinitsyn, Heat Transfer Sov. Res. 12 (1980) 71. [50] N.N. Danilov, Ye.N. Sinitsyn, V.P. Skripov, in: Gidrodinamika i Teploobmen v Energeticheskih Ustanovkakh, Academy of Sciences of the USSR, 1975, p. 3. [51] J.G. Eberhart, E.J. Hathaway, M. Blander, J. Colloid Interface Sci. 44 (1973) 389. [52] C.T. Avedisian, J.R. Sullivan, Chem. Eng. Sci. 39 (1984) 1033. [53] J.R. Patrick-Yeboah, PhD thesis, MIT, Cambridge, MA, 1979. [54] Y. Mori, K. Hijikata, T. Nagatani, Int. J. Heat Mass Transfer 19 (1976) 1153. [55] W. Porteous, M. Blander, AIChE J. 21 (1975) 560. [56] G.V. Ermakov, V.P. Skripov, Russ. J. Phys. Chem. 41 (1962) 39. [57] B.J. Mokross, in: J.W.P. Schmelzer, G. R opke, V.B. Priezzhev (Eds.), Nucleation Theory and Applications, Joint Institute of Nuclear Research, Dubna, Russia, 1999, p. 445. [58] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [59] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 31 (1971) 688.