2. Kauzmann Paradox

Another topic of intensive current debate connected with entropy variations in the course of the cooling of glass-forming melts and the glass transition is the behavior of the specific entropy difference, Δs, between liquids and crystals denoted commonly as Kauzmann paradox [24,52,121,122,123]. As mentioned by Cahn [121], “It is rare indeed for a scientific paper to remain central to current concerns several decades after its publication …” and an “…increasing number of physicists …keep coming back to Kauzmann and his eponymous paradox …”. The Kauzmann paradox and its possible consequences have been reanalyzed in detail in a preceding paper [64]; here we would like to review briefly some of its main results, adding some more details.

As suggested first by Tammann [27,28] and about two decades later by Kauzmann [124], specific entropy differences between liquids and crystals decrease with decreasing temperature and may become equal to zero at a temperature denoted today as the Kauzmann temperature, TK, or even less than zero below TK (see Figure 6). Tammann did not consider such a type of behavior as being in conflict with basic laws of physics, supposing that below the Kauzmann temperature, the crystallites should transform into a glass. Additionally, Kauzmann did not see any principal problem in this respect, noting in a footnote in his paper, “Certainly, it is unthinkable that the entropy of the liquid can ever be very much less than that of the solid. It could conceivably become slightly less at finite temperatures because of a ‘tighter’ binding of the molecule in the highly strained liquid structure…”. Perhaps this may be the main reason that he was not so happy with the notation “Kauzmann paradox” introduced by Angell (for details, see [125,126]). In line with such considerations, Dyre posed in [44] the question of how seriously the Kauzmann paradox has to be taken and gave some examples in which a crossing of the entropy curves is realized in nature.

Figure 6. Specific entropy difference between metastable liquids and crystals given in dependence on temperature. (a,b) is adopted from the papers by Tammann (Figure 4 in [27]) and Kauzmann (Figure 4 in [124]). Δsm is the heat of melting or fusion.

Despite his comment in the footnote of his paper [124], Kauzmann in detail discussed possibilities to avoid the crossing of the entropy curves and negative values of Δs below the Kauzmann temperature. This discussion has been continued intensively until now (see [64] for details). As a possible mechanism, he supposed the existence of intensive crystallization near to the Kauzmann temperature connected with the existence of a “pseudo-spinodal curve”. He wrote the following: “Suppose that when the temperature is lowered a point is eventually reached at which the free energy barrier to crystal nucleation becomes reduced to the same height as the barriers to the simpler motions…At such temperatures the liquid would be expected to crystallize just as rapidly as it changed its typically liquid structure to conform to a temperature or pressure change in its surroundings …There are good theoretical reasons for believing in the existence of such a ‘pseudo-critical temperature’”. As evident from this statement, Kauzmann supposed the pseudo-spinodal curve to be reached if the Maxwellian relaxation time, τR, becomes equal to the average time of formation of the first supercritical nucleus, ⟨τ⟩, that is, if the relation

τR≅⟨τ⟩

(11)

is satisfied. He supposed this condition, resulting in intensive crystallization, to be fulfilled near to the Kauzmann temperature.

As also can be traced from the quotation given above, Kauzmann connected the rate of crystallization with the average time of formation of the first supercritical nucleus at steady-state conditions with a steady-state nucleation rate, Jss. Only in this case is the average time, ⟨τ⟩, of formation of the first supercritical crystallite correlated with the work of critical cluster formation (the energy barrier to crystal nucleation), Wc, via [127,128,129]:

⟨τ⟩=⟨τ⟩ss≅1JssV,Jss=J0exp(−WckBT)

(12)

Here V is the volume of the melt, J0 is a factor determined by the kinetics of crystal nucleation, kB is the Boltzmann constant, and T is the absolute temperature.

In order to illustrate the consequences of this preposition, we adopt here the Adam–Gibbs model for viscosity, identifying Δs with the specific entropy difference of liquid and crystal phases. In this case, the temperature of the divergence of viscosity—as frequently supposed to be the case—is identical to the Kauzmann temperature. As outlined in detail in [64], the general consequences are independent of this particular assumption. Accounting for the Maxwell relation [24]:

τR≅ηd30kBT

(13)

we obtain

τR≅τ0exp(ATΔs)

(14)

Here d0 is a measure of the size of the ambient phase particles in the liquid, η is the viscosity of the liquid, and A is a parameter specific for the system under consideration. A combination of Equations (11) and (12) yields the following in the limit Δs→0 or T→TK:

JssV≅1⟨τ⟩ss≅1τR≅1τ0exp(−ATΔs)∣∣∣Δs→0⟹Jss(T→TK)=0

(15)

At the Kauzmann temperature, the steady-state nucleation rate tends to zero; intensive crystal nucleation as suggested by Kauzmann, consequently, does not occur at his supposed pseudo-spinodal curve.

In a similar analysis, instead of ⟨τ⟩ss, Angell et al. [130] identified the characteristic time of crystallization with the time-lag in nucleation, τn. As already demonstrated in [64,129], this estimate is, in the range of temperatures Kauzmann was interested in, a much better approximation for the average time of formation of the first supercritical nucleus as compared to ⟨τ⟩ss. The time-lag can be determined via Equation (16) [24,64,129]:

τn≅ωηd0σn2/3c≅CτRn2/3c,C=ωkBTσd20

(16)

In Equation (16), ω may vary in the range from 1 to 4 in dependence on the theory employed for the determination of the time-lag, and σ is the specific interfacial energy. With estimates of the parameter C as given in [130] in its discussion (C≅102−103; similar estimates are obtained also in [24]), the mentioned authors arrived at the conclusion that Kauzmann’s condition, Equation (11), cannot be fulfilled, and for this reason, a pseudo-spinodal is absent in melt-crystallization. This result was taken as the starting point for the search of alternative mechanisms to prevent the Kauzmann paradox, such as the concept of an ideal glass transition. However, as is demonstrated below (see also [64]), such additional mechanisms are not required for the resolution of the Kauzmann paradox.

In the estimates of the parameter C by Angell et al. [130] and also in [24], the capillarity approximation was employed. More correct estimates involving a size dependence of the specific interfacial energy result, however, in different values of the parameter C, allowing the fulfilment of Kauzmann’s condition, Equation (13). In addition, as shown in [129], for isothermal conditions, the average time of formation of the first supercritical nucleus is, in a good approximation, equal to the sum of the induction time, τind, widely equal to the time-lag, τn, and average time of formation of a critical nucleus at steady-state conditions ⟨τ⟩ss:

⟨τ⟩≅⟨τ⟩ss+τind

(17)

For the low-temperature range Kauzmann had in mind, it is nearly equal to the time-lag; for higher temperatures, it is determined by the time of formation of the first supercritical nucleus at steady-state conditions. These results are illustrated in Figure 7.

Figure 7. Structural relaxation time, τR; the induction time, τind, required to establish steady-state nucleation; and the average time of formation of a supercritical cluster at steady-state nucleation conditions, ⟨τ⟩ss≅1/(JssV), are shown in dependence on temperature (a,c). The average time of formation of the first supercritical nucleus, ⟨τ⟩, can be expressed generally in a good approximation as the sum ⟨τ⟩≅⟨τ⟩ss+τind with τind≅τn [129]. The dependence of ⟨τ⟩ on temperature is illustrated in the lower part of (b,d). For temperatures near to the melting temperature, ⟨τ⟩ is always determined by ⟨τ⟩ss. At the intersection of ⟨τ(T)⟩ss with τind(T), ⟨τ(T)⟩ becomes dominated by the values of τind. Values of the parameters employed in the computation of the nucleation rates and related quantities are taken for 2Na2O⋅1CaO⋅3SiO2 [131,132]. In (b,d), the Vogel temperature is replaced by the Kauzmann temperature (see also text).

In the upper curves of Figure 7a,c, the structural relaxation time, τR; the induction time, τind, required to establish steady-state nucleation; and the average time of formation of a supercritical cluster at steady-state nucleation conditions, ⟨τ⟩ss≅1/(JssV), are shown in dependence on temperature. The induction time, τind, is, except for in the immediate vicinity of the melting temperature, Tm, practically identical in the relevant time scales to the structural relaxation time. It does not depend on the volume of the system in which crystallization may take place.

In contrast to τind, the average time of formation of a supercritical crystal cluster at steady-state nucleation conditions, ⟨τ⟩ss≅1/(JssV), does depend on the volume of the system. For small volumes, there may be no intersection of the temperature dependencies of τR(T) and ⟨τ⟩ss(T), as is evident from the figure. However, there always exists a certain value of the volume of the system at which such crossing occurs. Following the interpretation of Kauzmann, one has then to suppose that the existence and, if it exists, the location of the pseudo-spinodal depend on the volume of the system. Moreover, these would be located not at temperatures near to the Kauzmann temperature but at temperatures higher than the temperature of the maximum of the steady-state nucleation rate. Provided the average time of formation of the first supercritical nucleus is estimated via Equation (12) assuming steady-state nucleation, then there exists always a value of the volume of the system at which this condition is fulfilled. It follows as an additional conclusion that the fulfillment of Equation (11) at steady-state nucleation conditions also does not necessarily result in intensive nucleation. For example, if this condition is fulfilled only in a system with a very large volume, then the fulfillment of Equation (11) does not imply intensive nucleation.

The dependence of ⟨τ⟩ on temperature is illustrated in the lower part of Figure 7b,d. For temperatures near to the melting temperature, ⟨τ⟩ is always determined by ⟨τ⟩ss. At the intersection of ⟨τ(T)⟩ss with τind(T), ⟨τ(T)⟩ becomes dominated by the values of τind. Again, this switch in the quantity dominating the value of ⟨τ(T)⟩ does not take place at temperatures near to the Kauzmann temperature but at temperatures higher than the temperature of the maximum of the steady-state nucleation rate, Tmax.

For the curves shown in Figure 7a,b, the values of the parameters employed in the computation of the nucleation rates and related quantities are taken for 2Na2O⋅1CaO⋅3SiO2 [131,132]. These parameters are briefly summarized also in [64], where they are employed for the description of similar dependencies in a different form (Figure 6 there). In addition, here in the computations leading to Figure 7c,d, the Vogel temperature is replaced in the equation for the description of the viscosity by the Kauzmann temperature, TK. As mentioned earlier, the Kauzmann temperature can be determined from the condition of the maximum of the thermodynamic driving force [65]. As is evident, this replacement leads only to minor changes in the behavior.

The above discussion is performed under the assumption that the liquid is brought very fast to the respective temperature, T. However, in reality, processes of clustering may take place already in the course of the cooling process. The account of such prehistory effects may further reduce the time of formation of the first supercritical nucleus. This reduction is particularly significant in the range of temperatures near to the glass transition temperature, respectively, near to the maximum of the steady-state nucleation rate (Figure 7 in [64]). Consequently, the crossing of the ⟨τ(T)⟩ and τR(T) curves, if taking places, is favored to occur in this range of temperatures and not near to the Kauzmann temperature. However, in any of these cases, the average time of formation of the first supercritical nucleus is either proportional to or widely determined by the relaxation time. This relaxation time diverges in the approach of the Kauzmann temperature, preventing any crystal nucleation (for details, see [64]). Consequently, in the range of temperatures in which Kauzmann expected the pseudo-spinodal to be located, independently of whether or not the condition of Equation (11) is fulfilled, intensive crystal nucleation does not take place.

This conclusion refers not only to Kauzmann’s original suggestion concerning the location of the pseudo-spinodal curve. Additionally, beyond the near vicinity of the Kauzmann temperature, his assumption does not hold. It is not the pseudo-spinodal defined via Equation (11) that governs the maximum of crystal nucleation. In agreement with experimental investigations performed first by Tammann [63] and confirmed by all subsequent experimental and theoretical analysis of crystal nucleation [24,61], there exists one and only one maximum of the steady-state nucleation rate located near to the traditionally defined glass transition temperature (identifying the glass transition temperature with values of the viscosity in the order of η≅1012 Pa·s [40]). It is determined by the interplay of the decrease in the work of critical cluster formation and the increase in viscosity with decreasing temperature [60,64,133,134,135]. The location and magnitude of this maximum nucleation rate or the maximum of the overall crystallization rate are not determined by Equation (11), but by other relations both for crystallization caused by variations of temperature or pressure (see [60,133]).

A schematic representation (in discussing Kauzmann’s suggestion of the existence of a pseudo-spinodal curve) of the average structural relaxation time of a supercooled liquid and the average nucleation time is given in Figure 2 in [52], assuming steady-state nucleation to hold. Is is stated that another “vital concept related to supercooled liquids, which is not well-known within the glass research community, is the liquid stability limit or kinetic spinodal temperature…” Here, “the supercooled liquid becomes unstable against crystallization and crystal growth immediately proceeds”. The shape of the curves for the average time of formation of the first supercritical nucleus presented in their Figure 2 is quite different from our computational results shown in Figure 7, and the dependence of ⟨τ(T)⟩ss on the volume of the system is also not accounted for. It is correctly noted in the further discussion that sufficiently above Tg, steady-state nucleation dominates. Consequently, employing this condition of steady-state nucleation in their Figure 2, they implicitly expect a location of the pseudo-spinodal at temperatures above Tg. In such an interpretation, Kauzmann’s suggestion underlies a direct experimental proof. Existing experimental data do not exhibit any peculiarities indicating the existence of a pseudo-spinodal curve above Tg. We note also that under such conditions, its location should depend on the volume of the system.

In the further discussion of these topics in [52], our result ⟨τ⟩≅⟨τ⟩ss+τind with τind≅τn, derived in [129], is adopted partly for the analysis of Kauzmann’s suggestions at low temperatures. However, as already noted, even if critical nuclei would be formed intensively, they would not grow, as the maximum of the growth rate is located at much higher temperatures compared to the maximum of the nucleation rates [60]. This is the origin for why Tammann’s two-stage development method is so widely employed in the analysis of crystallization processes [24,61,136]. However, they also do not form with the intensity supposed by Kauzmann at the low temperatures al supposed by him because of the very low values of the kinetic coefficients, which may also become equal to zero, as discussed in connection with flow processes in glasses.

Summarizing our point of view, we come to the following conclusions: Independent of the results of the estimates of ⟨τ⟩ and whether this condition can be fulfilled or not, Kauzmann’s suggestion concerning the possibility of the existence of a pseudo-spinodal causing intensive crystal nucleation does not hold by the following reasons derived above: (i) The values of the kinetic parameters are too low and prevent any crystallization near to the Kauzmann temperature. (ii) The maximum nucleation, growth, and overall crystallization rates are defined by other relations. Consequently, the pseudo-spinodal and the properties assigned to it by Kauzmann are not a “vital concept related to supercooled liquids” but are irrelevant with respect to the crystallization behavior of glass-forming melts, at least, in the sense that Kauzmann assigned to them.

By the above argumentation, it also follows that Kauzmann’s suggestion concerning the existence of a pseudo-spinodal in melt-crystallization is not sufficient–as he believed—to avoid the realization of the Kauzmann paradox. However, there exists another mechanism preventing this and, much more importantly, preventing also possible contradictions to the third law of thermodynamics. Indeed, according to the general kinetic criterion of glass formation, Equation (3), the rate of cooling determines the value of the glass transition temperature, Tg, where the metastable liquid is frozen-in into a glass. Assuming a behavior of the relaxation time as given by Equation (14), for any finite value of the rate of change of temperature, the glass-forming melt transforms to a glass at temperatures Tg>TK. The Kauzmann temperature can be reached only in the limit of zero cooling rates. Such a process cannot be realized in an experiment. The same conclusions can be drawn also by employing other physically reasonable assumptions concerning the temperature dependence of the viscosity and the relaxation time, as discussed in detail in [64].

These conclusions are in full agreement with statements by Simon, who noted already in 1931 that, in cooling, the glass-forming liquids either crystallize or save themselves from crystallization by going over into the vitreous state [32]. This is the mechanism for the prevention of contradictions with the third law of thermodynamics, the only consequence of Tammann’s and Kauzmann’s observations on the behavior of specific entropy differences, which could lead to conflicts with basic laws of nature. Consequently, Simon in fact resolved the Kauzmann paradox about 20 years prior to its formulation. We note also that Simon considered the transition of the melt into a glass as a final process not supplemented by any subsequent crystallization. Consequently, also according to Simon, crystallization is not the ultimate fate of a glass (cf. [52]).