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Conditions for metastable crystallization from undercooled melts
Journal of Non-Crystalline Solids 337 (2004) 220–225 www.elsevier.com/locate/jnoncrysol
Conditions for metastable crystallization from undercooled melts I. Avramov a, C. R€ ussel b
b,* ,
K. Avramova
a
a Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Otto-Schott-Institut, Friedrich-Schiller-Universit€at Jena, Fraunhoferstrasse 6, Jena D-07743, Germany
Received 10 November 2003; received in revised form 29 March 2004
Abstract The limits of validity of Ostwald’s rule of stages are investigated theoretically in the case of crystallization of undercooled melts. The treatment is within the limits of capillary theory. Two basic models are compared: (1) According to the first one (model A), the phase with lower energy of formation of critical nucleus is predominantly formed. In an enantiotropic-type phase diagram there is no region of homogeneous preferential formation of the low temperature phase. If the phase diagram is monotropic-type there is a certain temperature below which the metastable crystalline phase is preferentially formed. (2) The second assumption takes into consideration that the nature of extremely small phases is somewhat undefined. One certainly cannot determine whether, say 3particle complex, is of phase 1 or of phase 2. Moreover, it is known that properties of extremely small clusters could be different from the corresponding volume phase. The main assumption is that there is a certain crucial size (n-particle complex) at which the nature of the two phases can be distinguished. Complex of the phase, which has lower chemical potential at the crucial size, will be formed first. According to the model, in the case of enantiotropic-type transition there is a critical temperature. 2004 Elsevier B.V. All rights reserved.
1. Introduction
2. The model
Empirically, Ostwald formulated [1] a rule according to which the thermodynamically less stable phase is always formed first, Ostwald’s rule of stages. Later on, Stranski and Totomanov [2] suggested that this is a consequence of preferential formation of a critical nucleus of the metastable phase. Therefore, the activation energy, not the affinity, controls the nature of the process. As discussed in Refs. [3,4], conditions for the predominant formation of the metastable phase 2 can be formulated by comparing the nucleation rates of the stable phase J1 and of the metastable one J2 . As in Refs. [3,4] the treatment was limited to deposition from vapor phase, in the present article the crystallization from undercooled melts is investigated.
2.1. Basic properties of crystalline phases
* Corresponding author. Tel.: +49-3641 948 501/636 105; fax: +493641 948 502. E-mail addresses: [email protected] (I. Avramov), [email protected] (C. R€ ussel).
0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.012
The problem is which of the two crystalline phases (denoted hereafter with indexes 1 resp. 2) will be formed first when a melt (denoted with index m) is cooled to a temperature T below the melting points of the two phases. Hereafter, T12 stands for the equilibrium point between phase 1 and 2, the melting points of the corresponding phases are denoted as T1 and T2 . The possible phase transitions between two crystalline phases are illustrated in Figs. 1 and 2 in coordinates chemical potential l against temperature: Fig. 1 shows a reversible transition (enantiotropic transition) at a certain temperature and pressure (the inversion point T12 ). The thick solid lines represent the phase that is stable at the given temperature. Dotted lines are the extrapolations of chemical potentials of the corresponding phases (liquid, crystal 1, and crystal 2) in the metastable region. The transition points appear in the following order: T12 < T2 < T1 :
ð1Þ
I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
221
ð2Þ
T2 < T1 < T12 :
In both figures the dashed lines (parallel to the corresponding solid lines) give the chemical potential of small (n-molecular) cluster of the corresponding phases. For the present investigation, important properties are: the crystallization driving force Dli (i ¼ 1, 2), accounting for the difference in chemical potentials of the melt and of the ith crystalline phase, the corresponding enthalpy difference, Hi , and the corresponding surface energy ri . We use the following assumptions:
Fig. 1. Phase diagrams with coexisting two crystalline phases in coordinates chemical potential l against temperature. T12 is the inversion point of a reversible transition (enantiotropic) transition. The thick solid lines represent the phase that is stable at the corresponding temperature. Dotted lines are the extrapolations of chemical potentials of the corresponding phases (liquid, crystal 1, and crystal 2) in the metastable region.
• The driving force Dli depends on the melting enthalpy Hi and temperature T T Dli ¼ Hi f : ð3Þ Ti The explicit form of the dependence of supersaturation on temperature is given usually by the most widespread Thomson equation T T f ð4Þ ¼1 : Ti Ti This approximation is valid at low undercoolings or when the specific heats of the glass and of the isochemical crystal are similar. • The crystal/melt interface energy is expressed using the Skapski–Turnbull equation (see [6,7]): ri ai
Fig. 2. An irreversible transition (monotropic transition) from the metastable to the stable polymorphic form. The thick solid lines represent the stable phases. The thin solid line represent the metastable phase 2. Dotted lines are the extrapolations of the corresponding phases (liquid, crystal 1, crystal 2). The transition temperature T12 is above the melting point and cannot be realized.
Example [5] could be the transition of SiO2 (quartztype) to tridymite-type at 1143 K and to cristobalitetype at 1743 K. Fig. 2 shows an irreversible transition (monotropic transition) from the metastable to the stable polymorphic form. An example [5] is the transition of metastable CaCO3 (aragonite-type) to the stable calcite-type. Again the thick solid lines represent the stable phases. The thin solid line represent the metastable phase 2. Dotted lines are the extrapolations of the corresponding phases (liquid, crystal 1, crystal 2). The transition temperature T12 is above the melting point and cannot be realized. The transition points appear in the following order:
Hi 1=3 2=3 Na Vi
;
ð5Þ
where Na is Avogadro’s number, Vi is molecular volume and ai is a dimensionless constant [9], which should vary between 0:30 < ai < 0:55. In fitting the experimental temperature dependence of the nucleation rate J for several silicate glasses, Zanotto and coworkers [10] found that a varies within 0:40 < ai < 0:5 (under the assumption that r is temperature and size independent). The melting enthalpy of the phase 2 can be determined from the equilibrium condition (temperature T12 ) of coexistence of the two phases as follows: T12 T12 H1 f1 H2 f2 ¼ 0: ð6Þ T1 T2 It follows that f1 TT121 1 TT121 ¼ H1 h: H2 ¼ H1 H1 f2 TT122 1 TT122
ð7Þ
With Eqs. (5) and (7) the surface energy of the second phase can be expressed as
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I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
r2 ¼ r1
a2 a1
V1 V2
T 1 12 where the relative temperature parameter h ¼ TT121 is
2=3 h:
ð8Þ
If r1 is the radius of a small cluster, consisting of n molecules, formed on a foreign substrate, the radius of the same cluster on the same substrate in state 2 will be 1=3 V2 U 1 r2 ¼ r1 ; ð9Þ V1 U 2 where the dimensionless parameters Ui account for the influence of active substrates. The wetting function 0 6 Ui 6 1 account for the lowering of the energy of formation of a critical nucleus in heterogeneous nucleation and at the same time they account for the volume of a small complex on the substrate as compared to the volume of a sphere with the same radius. 2.1.1. Model A (nucleation kinetics model) We assume that at any temperature the phase with higher nucleation rate will be formed first. According to the capillary nucleation theory [6–8] the steady state nucleation rate is given as Ai Ji ¼ const: exp ; ð10Þ kT where the work of formation of a critical nucleus A depends on the crystal/melt interface energy r, on the molar volume Vi and driving force Dli as follows: Ai ¼
16pr3i Vi 2 Ui : 3Dl2i
ð11Þ
With the above assumptions work of formation of critical nucleus is given by the formula Ai
a3i Hi Ui ð1 TTi Þ
2
:
ð12Þ
Phase 2 will predominantly be formed under the condition ln J1 ln J2 < 0:
ð13Þ
Since both crystals are formed from the same melt, it is natural to assume that the preexponential constants in the nucleation rate equations (Eq. (10)) are equal, so that Eq. (13) is satisfied when the work of formation of a critical nucleus of first phase A1 is larger then the work of formation of a critical nucleus of the second phase A2 . A1 > 1: A2
ð14Þ
Combining Eq. (14) with Eq. (12), two times (for A1 and for A2 ), and taking into account Eq. (13) we come to the following condition for predominant formation of phase 2: " #2 1 TT2 W > h; ð15Þ 1 TT1
1 T
2
determined by Eq. (7) while the relative structure factor W stands for 3 a1 U 1 W¼ : ð16Þ a2 U 2 The dimensionless parameter W accounts for the possible heterogeneous nucleation. In the homogeneous case U1 =U2 ¼ 1; therefore W varies within the limits given by the ða1 =a2 Þ3 ratio. The reasonable range in which W could vary is 0:4 6 W 6 2:6 (for 0:40 < ai < 0:55 and U1 =U2 ¼ 1). Here we compare the two possible nucleation rates, completely neglecting non-steady state effects, i.e. a possible time delay of nucleation is not taken into account. It should be noted that taking into account the time delay does not lead to considerable differences in final result. Moreover, above the glass transition temperature, in the temperature range of practical interest, the time delay is of minor importance. 2.1.2. Model B (small complexes equilibrium model) In our previous paper [4] we already discussed the case of possible formation of two condensed phases from vapours. Usually, the size of the critical nucleus is so small that it does not have the full characteristics and symmetry of the macrophase. Note, that the increase of the chemical potential is proportional to the corresponding surface energy ri (i ¼ 1, 2). Therefore, below certain size the stable and metastable phases could change their places; i.e. it could happen that l1 ðr1 Þ < l2 ðr2 Þ, although below T12 for infinitely large phases the situation is opposite l1 ð1Þ > l2 ð1Þ. In model A the nucleation rate, respectively the state of clusters having the size of critical nucleus were considered. However, it is important to note that at low supersaturations the nucleation rate is also low so that phase transition is impeded. On the other hand, at high supersaturations, the critical nucleus consists of a few molecules only. It is impossible to state that, say, 3molecular cluster is of phase 1 or of phase 2. We assume that the smallest cluster for which the two phases are distinguishable consists of n molecules, i.e. in state 1 it will have radius r1n . According to the model, the phase for which the n-molecular cluster has lower chemical potential will be formed first. Therefore the condition for predominant formation of phase 2 is l2 ðr2n ; T Þ l1 ðr1n ; T Þ < 0:
ð17Þ
At any temperature T , the chemical potential of infinitely large phase li ð1; T Þ can be related to the corresponding supersaturation in respect to the undercooled melt Dli . The chemical potential of small cluster depends on the surface energy according to the Gibbs equation
I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
l1 ðr1n ; T Þ ¼ l1 ð1; T Þ þ
2r1 V1 : r1n
ð18Þ
Eq. (18) is derived in isotropic approximation, see the Appendix A. When Eqs. (3), (4), (18) are introduced into Eq. (17), and Eqs. (7) and (8) are taken into account (see the Appendix A), the condition for predominant formation of phase 2 becomes T T 2r1 V1 h 1 h 1 þ 1 < 0: ð19Þ T1 T2 H1 r1n W1=3 It is useful to express the radius r1n in dimensionless form as qi ¼ rdini , where di is the mean intermolecular distance of the ith phase. It is clear that a reasonable estimation of qi is in the limits 1 < qi < 3 because the critical cluster cannot be smaller than one repeatable unit cell and in the same time is not much larger than it. Introducing qi and taking into account Eq. (5), the last term of Eq. (19) becomes 2r1 V1 2a1 1 ¼ : q1 H1 r1n q1
ð20Þ
Thus, (in enantiotropic case, i.e. h > T2 =T1 , see Eq. (7)) the critical temperature Tc beyond which the inequality (19) is satisfied is becoming i h h 1 h 1 2a 1 1=3 q1 W Tc < T2 : ð21Þ h TT21 On the other hand, in monotropic case, it follows from Eq. (7) that 1 TT121 T ðT T12 Þ T2 ¼ 2 1 < ð22Þ h¼ T12 T1 ðT2 T12 Þ T1 1 T2 so that instead of Eq. (21) the critical condition for predominant formation of phase 2 is defined as h i 1 1 h 2a 1 Wh1=3 q1 Tc > T2 : ð23Þ T2 h T1 As soon as the phase 2 could be formed only at temperatures lower T2 , an additional condition for the formation of the second phase appears in the form h q1 T2 6 1 1 : ð24Þ 2a1 T1 W1=3
3. Discussion 3.1. Heterogeneous crystallization The idea to compare nucleation rates was used in Ref [11]; however, the authors, instead of Eq. (7), assume that melting enthalpy is proportional to melting temperature. The latter is certainly wrong. Thus, it is readily
223
seen that in the enantiotropic case the lower melting phase has higher melting enthalpy. Two extreme cases are easily seen: The first case is the presence of crystals of phase 1 (or seeds extremely active for the formation of phase 1). This case corresponds to W ! 0. It is readily seen that neither Eq. (15) nor one of Eqs. (21) and (23) is satisfied, i.e. instead of phase 2 always phase 1 is preferentially formed. In the opposite case, when seeds of phase 2 (or active in respect to formation of phase 2 substrates) are introduced W ! 1. In this case the conditions given by Eqs. (15), (21), (23) are always satisfied so that phase 2 is preferentially formed. 3.2. Homogeneous crystallization Most important, however, is the case W 1 of substrates equally active with respect to both phases or of homogeneous crystallization, which will be discussed hereafter. An important discrepancy between the predictions of the two models could appear. It is clear (see Eqs. (15) and (21)) that the results depend crucially on the h value or, in other words, on the nature of the phase transition. In the enantiotropic case h is always larger than one. Note that for W=h 6 1, Eq. (15) is never satisfied, therefore, according to model A, always phase 1 will be formed preferentially. On the other hand, according to model B phase 2 will predominantly be formed in the region formulated by Eq. (21). Moreover, for W1=3 1 Eq. (21) simplifies to 2a1 ð25Þ Tc < 1 T12 : q1 Recently [13] a crystallization behaviour of poly(trimethylene 2,6-naphtalene) was reported. Two crystalline phases could appear. Phase 2 is formed at temperature below 413 K and has melting point T2 ¼ 470 K. Both phase 2 and phase 1 are formed simultaneously in the interval between 413 and 433 K. Above 433 K phase 1 is formed with T1 ¼ 496 K; the transition point is T12 ¼ 425 K. The data correspond to predictions of Eq. (21), i.e. model B is fulfilled while model A failed to describe the results. With the above data one can determine h ¼ 1:495. Taking into account Eq. (25), for q1 W1=3 ¼ 1 one obtains 2a ¼ 1:97, a value close to the 1 expected one. In the case of a monotropic phase diagram, the dimensionless parameter h is always less than one. Therefore, in frameworks of model A, it is possible to satisfy Eq. (15). To determine the temperature below which Eq. (15) is satisfied, it is useful to take a square root of both sides of this equation, keeping in mind that the solution has a physical meaning only for W=h > 1. It follows that Eq. (15) is satisfied for temperatures T 6 TJ , where TJ stands for
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I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
0
qffiffiffi 1 h 1 W C B qffiffiffi AT2 : TJ @ 1 TT21 Wh
ð26Þ
According to the model B, the phase 2 will predominantly be formed in the temperature range between T2 and a certain critical temperature defined by Eqs. (23) and (24). For W1=3 1 Eq. (23) simplifies to 2a1 Tc > ð27Þ 1 T12 : q1 An interesting situation could appear. There could appear a temperature range above T2 in which the condition determined by Eqs. (23) and (26) is still satisfied. This means that the smallest clusters will prefer to be in state 2, although they are smaller than critical nucleus. However, if a fluctuation starts to develop as a phase 2 cluster it will never reach size of critical nucleus; it could become a critical nucleus only if it undergoes phase transition to state 1 (see Ref. [12]). The trace is more complicated so that it is less likely that the system will crystallize easily. The models A and B are apparently similar: in both cases the small clusters have higher chemical potentials due to the surface energy. Than, it is logical to ask why results are quite different. In model A the phase with larger nucleation rate appears first. At high supersaturations, however, the critical nuclei are extremely small and application of capillary theory is doubtful. Therefore the model A fails to predict properly the nature of the first phase formed. On the other hand, the model B deals with relatively larger cluster for which the two virtual phases are becoming different for the first time. Model B claims that at sufficiently high supersaturations of practical importance the critical nucleus is too small to distinguish the nature of the phase that will grow in the future. The nature of the growing phase is becoming definite at a given critical size. As soon as it has nothing to do with the nucleation rate, the nature of the new phase is controlled solely by thermodynamic causes.
becoming distinguishable. This model is able to describe existing experimental evidence. Additional advantage of the approach is that the size of the cluster under consideration is relatively large as compared to that of the critical nucleus. Although the size of the critical nucleus increases hyperbolically when temperature approaches the melting point, it is well known that nucleation process takes place at reasonable rate for temperatures at which the size of critical nucleus is very small.
Acknowledgements Thanks are due to the Deutsche Akademische Auslandsdienst (DAAD) for financial support. Appendix A In isotropic approximation the free energy F of a complex of radius r is given by 4 r3 F ¼ p l1 þ 4pr2 r; 3 V1
ðA:1Þ
where V1 is the volume of the building unit of the system. The number n of building units in the cluster is 4 r3 4pr2 n¼ p respectively dn ¼ dr: 3 V1 V1
ðA:2Þ
Therefore, the chemical potential of small cluster is given by lðrÞ ¼
oF V1 oF 2rV1 ¼ ¼ l1 þ : on 4pr2 or r
ðA:3Þ
A.1. Derivation of Eq. (19) Taking into account Eqs. (3), (4), (7) the difference Dl1 ð1; T Þ Dl2 ð1; T Þ can be presented as T T 1 h 1 H1 : Dl1 ð1; T Þ Dl2 ð1; T Þ ¼ T1 T2 ðA:4Þ On the other hand
4. Conclusions Two models predicting which is the phase predominantly formed from undercooled melts are compared. It is shown that the model comparing the nucleation rates of the two crystalline phases fails at least in the case of enantiotropic transition. The other model takes into account that the nature of the phase in extremely small complexes cannot be determined. Therefore, first the phase will be formed, with the lower chemical potential at the smallest size at which the two phases are
2r1 V1 2r2 V2 2r1 V1 r2 V2 r1n ¼ 1 : r1n r2n r1n r1 V1 r2n
ðA:5Þ
The last expression is easily simplified by taking into account Eq. (9) 2r1 V1 r2 V2 r1n 2r1 V1 1 ¼ r1n r1 V1 r2n r1n
2=3
1
r2 V2
2=3
r1 V1
U1 U2
1=3 ! : ðA:6Þ
I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
The next step is to introduce here Eq. (8) so that 1=3 ! 2=3 2r1 V1 r2 V2 U1 1 2=3 r1n U2 r1 V1 1=3 ! 2r1 V1 a2 U1 1 h ¼ r1n a1 U2 2r1 V1 h 1 1=3 : ¼ ðA:7Þ r1n W The Eq. (19) is obtained straightforward combining Eqs. (A.4) and (A.7) References [1] W. Ostwald, Z. Phys. Chem. 22 (1897) 282.
225
[2] I. Stranski, D. Totomanov, Z. Phys. Chem. A163 (1933) 399. [3] I. Gutzow, I. Avramov, J. Non-Cryst. Solids 16 (1974) 128. [4] I. Avramov, I. Gutzow, Mater. Chem. 5 (1980) 315. [5] IUPAC Compendium of Chemical Terminology, in: A. McNaught, A. Wilkinson (Eds.), [ISBN 0-8-654-26848], 2nd Ed., 1997. [6] Volmer, Kinetik der Phasenbildung, Th. Steinkopf, Dresden, 1939. [7] D. Turnbul, M. Cohen, Crystallization Kinetics and Glass Formation, in: Mackenzie (Ed.), Modern Aspects of Vitreous State, 1960, p. 38. [8] I. Gutzow, J. Schmelzer, The Vitreous State, Springer, 1995. [9] M.C. Weinberg, J. Non-Cryst. Solids 167 (1994) 81. [10] M.C. Weinberg, S. Manrich, E.D. Zanotto, Phys. Chem. Glasses 33 (1992) 99. [11] Z. Zhou, W. Wang, L. Sun, Appl. Phys. A 71 (2000) 261. [12] I. Avramov, J. Bartels, in: F. Schweitzer, H. Ulbricht, Rostock (Eds.), Nucleation-Clusters-Fractals, 1991. [13] Y. Jeonga, W. Jo, Polymer 44 (2003) 3259.
Conditions for metastable crystallization from undercooled melts I. Avramov a, C. R€ ussel b
b,* ,
K. Avramova
a
a Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Otto-Schott-Institut, Friedrich-Schiller-Universit€at Jena, Fraunhoferstrasse 6, Jena D-07743, Germany
Received 10 November 2003; received in revised form 29 March 2004
Abstract The limits of validity of Ostwald’s rule of stages are investigated theoretically in the case of crystallization of undercooled melts. The treatment is within the limits of capillary theory. Two basic models are compared: (1) According to the first one (model A), the phase with lower energy of formation of critical nucleus is predominantly formed. In an enantiotropic-type phase diagram there is no region of homogeneous preferential formation of the low temperature phase. If the phase diagram is monotropic-type there is a certain temperature below which the metastable crystalline phase is preferentially formed. (2) The second assumption takes into consideration that the nature of extremely small phases is somewhat undefined. One certainly cannot determine whether, say 3particle complex, is of phase 1 or of phase 2. Moreover, it is known that properties of extremely small clusters could be different from the corresponding volume phase. The main assumption is that there is a certain crucial size (n-particle complex) at which the nature of the two phases can be distinguished. Complex of the phase, which has lower chemical potential at the crucial size, will be formed first. According to the model, in the case of enantiotropic-type transition there is a critical temperature. 2004 Elsevier B.V. All rights reserved.
1. Introduction
2. The model
Empirically, Ostwald formulated [1] a rule according to which the thermodynamically less stable phase is always formed first, Ostwald’s rule of stages. Later on, Stranski and Totomanov [2] suggested that this is a consequence of preferential formation of a critical nucleus of the metastable phase. Therefore, the activation energy, not the affinity, controls the nature of the process. As discussed in Refs. [3,4], conditions for the predominant formation of the metastable phase 2 can be formulated by comparing the nucleation rates of the stable phase J1 and of the metastable one J2 . As in Refs. [3,4] the treatment was limited to deposition from vapor phase, in the present article the crystallization from undercooled melts is investigated.
2.1. Basic properties of crystalline phases
* Corresponding author. Tel.: +49-3641 948 501/636 105; fax: +493641 948 502. E-mail addresses: [email protected] (I. Avramov), [email protected] (C. R€ ussel).
0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.012
The problem is which of the two crystalline phases (denoted hereafter with indexes 1 resp. 2) will be formed first when a melt (denoted with index m) is cooled to a temperature T below the melting points of the two phases. Hereafter, T12 stands for the equilibrium point between phase 1 and 2, the melting points of the corresponding phases are denoted as T1 and T2 . The possible phase transitions between two crystalline phases are illustrated in Figs. 1 and 2 in coordinates chemical potential l against temperature: Fig. 1 shows a reversible transition (enantiotropic transition) at a certain temperature and pressure (the inversion point T12 ). The thick solid lines represent the phase that is stable at the given temperature. Dotted lines are the extrapolations of chemical potentials of the corresponding phases (liquid, crystal 1, and crystal 2) in the metastable region. The transition points appear in the following order: T12 < T2 < T1 :
ð1Þ
I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
221
ð2Þ
T2 < T1 < T12 :
In both figures the dashed lines (parallel to the corresponding solid lines) give the chemical potential of small (n-molecular) cluster of the corresponding phases. For the present investigation, important properties are: the crystallization driving force Dli (i ¼ 1, 2), accounting for the difference in chemical potentials of the melt and of the ith crystalline phase, the corresponding enthalpy difference, Hi , and the corresponding surface energy ri . We use the following assumptions:
Fig. 1. Phase diagrams with coexisting two crystalline phases in coordinates chemical potential l against temperature. T12 is the inversion point of a reversible transition (enantiotropic) transition. The thick solid lines represent the phase that is stable at the corresponding temperature. Dotted lines are the extrapolations of chemical potentials of the corresponding phases (liquid, crystal 1, and crystal 2) in the metastable region.
• The driving force Dli depends on the melting enthalpy Hi and temperature T T Dli ¼ Hi f : ð3Þ Ti The explicit form of the dependence of supersaturation on temperature is given usually by the most widespread Thomson equation T T f ð4Þ ¼1 : Ti Ti This approximation is valid at low undercoolings or when the specific heats of the glass and of the isochemical crystal are similar. • The crystal/melt interface energy is expressed using the Skapski–Turnbull equation (see [6,7]): ri ai
Fig. 2. An irreversible transition (monotropic transition) from the metastable to the stable polymorphic form. The thick solid lines represent the stable phases. The thin solid line represent the metastable phase 2. Dotted lines are the extrapolations of the corresponding phases (liquid, crystal 1, crystal 2). The transition temperature T12 is above the melting point and cannot be realized.
Example [5] could be the transition of SiO2 (quartztype) to tridymite-type at 1143 K and to cristobalitetype at 1743 K. Fig. 2 shows an irreversible transition (monotropic transition) from the metastable to the stable polymorphic form. An example [5] is the transition of metastable CaCO3 (aragonite-type) to the stable calcite-type. Again the thick solid lines represent the stable phases. The thin solid line represent the metastable phase 2. Dotted lines are the extrapolations of the corresponding phases (liquid, crystal 1, crystal 2). The transition temperature T12 is above the melting point and cannot be realized. The transition points appear in the following order:
Hi 1=3 2=3 Na Vi
;
ð5Þ
where Na is Avogadro’s number, Vi is molecular volume and ai is a dimensionless constant [9], which should vary between 0:30 < ai < 0:55. In fitting the experimental temperature dependence of the nucleation rate J for several silicate glasses, Zanotto and coworkers [10] found that a varies within 0:40 < ai < 0:5 (under the assumption that r is temperature and size independent). The melting enthalpy of the phase 2 can be determined from the equilibrium condition (temperature T12 ) of coexistence of the two phases as follows: T12 T12 H1 f1 H2 f2 ¼ 0: ð6Þ T1 T2 It follows that f1 TT121 1 TT121 ¼ H1 h: H2 ¼ H1 H1 f2 TT122 1 TT122
ð7Þ
With Eqs. (5) and (7) the surface energy of the second phase can be expressed as
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I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
r2 ¼ r1
a2 a1
V1 V2
T 1 12 where the relative temperature parameter h ¼ TT121 is
2=3 h:
ð8Þ
If r1 is the radius of a small cluster, consisting of n molecules, formed on a foreign substrate, the radius of the same cluster on the same substrate in state 2 will be 1=3 V2 U 1 r2 ¼ r1 ; ð9Þ V1 U 2 where the dimensionless parameters Ui account for the influence of active substrates. The wetting function 0 6 Ui 6 1 account for the lowering of the energy of formation of a critical nucleus in heterogeneous nucleation and at the same time they account for the volume of a small complex on the substrate as compared to the volume of a sphere with the same radius. 2.1.1. Model A (nucleation kinetics model) We assume that at any temperature the phase with higher nucleation rate will be formed first. According to the capillary nucleation theory [6–8] the steady state nucleation rate is given as Ai Ji ¼ const: exp ; ð10Þ kT where the work of formation of a critical nucleus A depends on the crystal/melt interface energy r, on the molar volume Vi and driving force Dli as follows: Ai ¼
16pr3i Vi 2 Ui : 3Dl2i
ð11Þ
With the above assumptions work of formation of critical nucleus is given by the formula Ai
a3i Hi Ui ð1 TTi Þ
2
:
ð12Þ
Phase 2 will predominantly be formed under the condition ln J1 ln J2 < 0:
ð13Þ
Since both crystals are formed from the same melt, it is natural to assume that the preexponential constants in the nucleation rate equations (Eq. (10)) are equal, so that Eq. (13) is satisfied when the work of formation of a critical nucleus of first phase A1 is larger then the work of formation of a critical nucleus of the second phase A2 . A1 > 1: A2
ð14Þ
Combining Eq. (14) with Eq. (12), two times (for A1 and for A2 ), and taking into account Eq. (13) we come to the following condition for predominant formation of phase 2: " #2 1 TT2 W > h; ð15Þ 1 TT1
1 T
2
determined by Eq. (7) while the relative structure factor W stands for 3 a1 U 1 W¼ : ð16Þ a2 U 2 The dimensionless parameter W accounts for the possible heterogeneous nucleation. In the homogeneous case U1 =U2 ¼ 1; therefore W varies within the limits given by the ða1 =a2 Þ3 ratio. The reasonable range in which W could vary is 0:4 6 W 6 2:6 (for 0:40 < ai < 0:55 and U1 =U2 ¼ 1). Here we compare the two possible nucleation rates, completely neglecting non-steady state effects, i.e. a possible time delay of nucleation is not taken into account. It should be noted that taking into account the time delay does not lead to considerable differences in final result. Moreover, above the glass transition temperature, in the temperature range of practical interest, the time delay is of minor importance. 2.1.2. Model B (small complexes equilibrium model) In our previous paper [4] we already discussed the case of possible formation of two condensed phases from vapours. Usually, the size of the critical nucleus is so small that it does not have the full characteristics and symmetry of the macrophase. Note, that the increase of the chemical potential is proportional to the corresponding surface energy ri (i ¼ 1, 2). Therefore, below certain size the stable and metastable phases could change their places; i.e. it could happen that l1 ðr1 Þ < l2 ðr2 Þ, although below T12 for infinitely large phases the situation is opposite l1 ð1Þ > l2 ð1Þ. In model A the nucleation rate, respectively the state of clusters having the size of critical nucleus were considered. However, it is important to note that at low supersaturations the nucleation rate is also low so that phase transition is impeded. On the other hand, at high supersaturations, the critical nucleus consists of a few molecules only. It is impossible to state that, say, 3molecular cluster is of phase 1 or of phase 2. We assume that the smallest cluster for which the two phases are distinguishable consists of n molecules, i.e. in state 1 it will have radius r1n . According to the model, the phase for which the n-molecular cluster has lower chemical potential will be formed first. Therefore the condition for predominant formation of phase 2 is l2 ðr2n ; T Þ l1 ðr1n ; T Þ < 0:
ð17Þ
At any temperature T , the chemical potential of infinitely large phase li ð1; T Þ can be related to the corresponding supersaturation in respect to the undercooled melt Dli . The chemical potential of small cluster depends on the surface energy according to the Gibbs equation
I. Avramov et al. / Journal of Non-Crystalline Solids 337 (2004) 220–225
l1 ðr1n ; T Þ ¼ l1 ð1; T Þ þ
2r1 V1 : r1n
ð18Þ
Eq. (18) is derived in isotropic approximation, see the Appendix A. When Eqs. (3), (4), (18) are introduced into Eq. (17), and Eqs. (7) and (8) are taken into account (see the Appendix A), the condition for predominant formation of phase 2 becomes T T 2r1 V1 h 1 h 1 þ 1 < 0: ð19Þ T1 T2 H1 r1n W1=3 It is useful to express the radius r1n in dimensionless form as qi ¼ rdini , where di is the mean intermolecular distance of the ith phase. It is clear that a reasonable estimation of qi is in the limits 1 < qi < 3 because the critical cluster cannot be smaller than one repeatable unit cell and in the same time is not much larger than it. Introducing qi and taking into account Eq. (5), the last term of Eq. (19) becomes 2r1 V1 2a1 1 ¼ : q1 H1 r1n q1
ð20Þ
Thus, (in enantiotropic case, i.e. h > T2 =T1 , see Eq. (7)) the critical temperature Tc beyond which the inequality (19) is satisfied is becoming i h h 1 h 1 2a 1 1=3 q1 W Tc < T2 : ð21Þ h TT21 On the other hand, in monotropic case, it follows from Eq. (7) that 1 TT121 T ðT T12 Þ T2 ¼ 2 1 < ð22Þ h¼ T12 T1 ðT2 T12 Þ T1 1 T2 so that instead of Eq. (21) the critical condition for predominant formation of phase 2 is defined as h i 1 1 h 2a 1 Wh1=3 q1 Tc > T2 : ð23Þ T2 h T1 As soon as the phase 2 could be formed only at temperatures lower T2 , an additional condition for the formation of the second phase appears in the form h q1 T2 6 1 1 : ð24Þ 2a1 T1 W1=3
3. Discussion 3.1. Heterogeneous crystallization The idea to compare nucleation rates was used in Ref [11]; however, the authors, instead of Eq. (7), assume that melting enthalpy is proportional to melting temperature. The latter is certainly wrong. Thus, it is readily
223
seen that in the enantiotropic case the lower melting phase has higher melting enthalpy. Two extreme cases are easily seen: The first case is the presence of crystals of phase 1 (or seeds extremely active for the formation of phase 1). This case corresponds to W ! 0. It is readily seen that neither Eq. (15) nor one of Eqs. (21) and (23) is satisfied, i.e. instead of phase 2 always phase 1 is preferentially formed. In the opposite case, when seeds of phase 2 (or active in respect to formation of phase 2 substrates) are introduced W ! 1. In this case the conditions given by Eqs. (15), (21), (23) are always satisfied so that phase 2 is preferentially formed. 3.2. Homogeneous crystallization Most important, however, is the case W 1 of substrates equally active with respect to both phases or of homogeneous crystallization, which will be discussed hereafter. An important discrepancy between the predictions of the two models could appear. It is clear (see Eqs. (15) and (21)) that the results depend crucially on the h value or, in other words, on the nature of the phase transition. In the enantiotropic case h is always larger than one. Note that for W=h 6 1, Eq. (15) is never satisfied, therefore, according to model A, always phase 1 will be formed preferentially. On the other hand, according to model B phase 2 will predominantly be formed in the region formulated by Eq. (21). Moreover, for W1=3 1 Eq. (21) simplifies to 2a1 ð25Þ Tc < 1 T12 : q1 Recently [13] a crystallization behaviour of poly(trimethylene 2,6-naphtalene) was reported. Two crystalline phases could appear. Phase 2 is formed at temperature below 413 K and has melting point T2 ¼ 470 K. Both phase 2 and phase 1 are formed simultaneously in the interval between 413 and 433 K. Above 433 K phase 1 is formed with T1 ¼ 496 K; the transition point is T12 ¼ 425 K. The data correspond to predictions of Eq. (21), i.e. model B is fulfilled while model A failed to describe the results. With the above data one can determine h ¼ 1:495. Taking into account Eq. (25), for q1 W1=3 ¼ 1 one obtains 2a ¼ 1:97, a value close to the 1 expected one. In the case of a monotropic phase diagram, the dimensionless parameter h is always less than one. Therefore, in frameworks of model A, it is possible to satisfy Eq. (15). To determine the temperature below which Eq. (15) is satisfied, it is useful to take a square root of both sides of this equation, keeping in mind that the solution has a physical meaning only for W=h > 1. It follows that Eq. (15) is satisfied for temperatures T 6 TJ , where TJ stands for
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0
qffiffiffi 1 h 1 W C B qffiffiffi AT2 : TJ @ 1 TT21 Wh
ð26Þ
According to the model B, the phase 2 will predominantly be formed in the temperature range between T2 and a certain critical temperature defined by Eqs. (23) and (24). For W1=3 1 Eq. (23) simplifies to 2a1 Tc > ð27Þ 1 T12 : q1 An interesting situation could appear. There could appear a temperature range above T2 in which the condition determined by Eqs. (23) and (26) is still satisfied. This means that the smallest clusters will prefer to be in state 2, although they are smaller than critical nucleus. However, if a fluctuation starts to develop as a phase 2 cluster it will never reach size of critical nucleus; it could become a critical nucleus only if it undergoes phase transition to state 1 (see Ref. [12]). The trace is more complicated so that it is less likely that the system will crystallize easily. The models A and B are apparently similar: in both cases the small clusters have higher chemical potentials due to the surface energy. Than, it is logical to ask why results are quite different. In model A the phase with larger nucleation rate appears first. At high supersaturations, however, the critical nuclei are extremely small and application of capillary theory is doubtful. Therefore the model A fails to predict properly the nature of the first phase formed. On the other hand, the model B deals with relatively larger cluster for which the two virtual phases are becoming different for the first time. Model B claims that at sufficiently high supersaturations of practical importance the critical nucleus is too small to distinguish the nature of the phase that will grow in the future. The nature of the growing phase is becoming definite at a given critical size. As soon as it has nothing to do with the nucleation rate, the nature of the new phase is controlled solely by thermodynamic causes.
becoming distinguishable. This model is able to describe existing experimental evidence. Additional advantage of the approach is that the size of the cluster under consideration is relatively large as compared to that of the critical nucleus. Although the size of the critical nucleus increases hyperbolically when temperature approaches the melting point, it is well known that nucleation process takes place at reasonable rate for temperatures at which the size of critical nucleus is very small.
Acknowledgements Thanks are due to the Deutsche Akademische Auslandsdienst (DAAD) for financial support. Appendix A In isotropic approximation the free energy F of a complex of radius r is given by 4 r3 F ¼ p l1 þ 4pr2 r; 3 V1
ðA:1Þ
where V1 is the volume of the building unit of the system. The number n of building units in the cluster is 4 r3 4pr2 n¼ p respectively dn ¼ dr: 3 V1 V1
ðA:2Þ
Therefore, the chemical potential of small cluster is given by lðrÞ ¼
oF V1 oF 2rV1 ¼ ¼ l1 þ : on 4pr2 or r
ðA:3Þ
A.1. Derivation of Eq. (19) Taking into account Eqs. (3), (4), (7) the difference Dl1 ð1; T Þ Dl2 ð1; T Þ can be presented as T T 1 h 1 H1 : Dl1 ð1; T Þ Dl2 ð1; T Þ ¼ T1 T2 ðA:4Þ On the other hand
4. Conclusions Two models predicting which is the phase predominantly formed from undercooled melts are compared. It is shown that the model comparing the nucleation rates of the two crystalline phases fails at least in the case of enantiotropic transition. The other model takes into account that the nature of the phase in extremely small complexes cannot be determined. Therefore, first the phase will be formed, with the lower chemical potential at the smallest size at which the two phases are
2r1 V1 2r2 V2 2r1 V1 r2 V2 r1n ¼ 1 : r1n r2n r1n r1 V1 r2n
ðA:5Þ
The last expression is easily simplified by taking into account Eq. (9) 2r1 V1 r2 V2 r1n 2r1 V1 1 ¼ r1n r1 V1 r2n r1n
2=3
1
r2 V2
2=3
r1 V1
U1 U2
1=3 ! : ðA:6Þ
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The next step is to introduce here Eq. (8) so that 1=3 ! 2=3 2r1 V1 r2 V2 U1 1 2=3 r1n U2 r1 V1 1=3 ! 2r1 V1 a2 U1 1 h ¼ r1n a1 U2 2r1 V1 h 1 1=3 : ¼ ðA:7Þ r1n W The Eq. (19) is obtained straightforward combining Eqs. (A.4) and (A.7) References [1] W. Ostwald, Z. Phys. Chem. 22 (1897) 282.
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[2] I. Stranski, D. Totomanov, Z. Phys. Chem. A163 (1933) 399. [3] I. Gutzow, I. Avramov, J. Non-Cryst. Solids 16 (1974) 128. [4] I. Avramov, I. Gutzow, Mater. Chem. 5 (1980) 315. [5] IUPAC Compendium of Chemical Terminology, in: A. McNaught, A. Wilkinson (Eds.), [ISBN 0-8-654-26848], 2nd Ed., 1997. [6] Volmer, Kinetik der Phasenbildung, Th. Steinkopf, Dresden, 1939. [7] D. Turnbul, M. Cohen, Crystallization Kinetics and Glass Formation, in: Mackenzie (Ed.), Modern Aspects of Vitreous State, 1960, p. 38. [8] I. Gutzow, J. Schmelzer, The Vitreous State, Springer, 1995. [9] M.C. Weinberg, J. Non-Cryst. Solids 167 (1994) 81. [10] M.C. Weinberg, S. Manrich, E.D. Zanotto, Phys. Chem. Glasses 33 (1992) 99. [11] Z. Zhou, W. Wang, L. Sun, Appl. Phys. A 71 (2000) 261. [12] I. Avramov, J. Bartels, in: F. Schweitzer, H. Ulbricht, Rostock (Eds.), Nucleation-Clusters-Fractals, 1991. [13] Y. Jeonga, W. Jo, Polymer 44 (2003) 3259.