1. Spatial heterogeneity of glass-forming liquids and crystal nucleation
V.M. Fokin, A.S. Abyzov, J.W.P. Schmelzer
Vavilov State Optical Institute, St. Petersburg, Russia
NSC Kharkov Institute of Physics and Technology, Kharkov, Ukraine
Institut für Physik der Universität Rostock, Rostock, Germany
I hope, that in our presentation we convince you that crystal nucleation in glass forming liquids correlated with its spatial heterogeneity.

2. 𝐼 st (𝑇) for Li2O 2SiO2-glass
Problem:
The classical nucleation theory (CNT) does not describe quantitatively the steady state nucleation rate of crystals 𝐼 st in glasses at temperatures T< Tmax lower than the temperature of its maximum.
𝐼 st (𝑇) for Li2O 2SiO2-glass
The problem is that the classical nucleation theory (CNT) does not describe quantitatively the steady state nucleation rate of crystals in glasses at temperatures lower than the temperature of its maximum. This plot shows the steady state nucleation rate for Lithium disilicate glass, solid line is calculation, dots present measured data. We can fit very well the high-temperature slope of the dependence of nucleation rate on temperature, but the fit of low-temperature slope is very bad – some order of magnitude.

3. Thermodynamic barrier for nucleation versus temperature
The thermodynamic barrier for nucleation (estimated from the experimental values of steady-state nucleation rates and the time-lag for nucleation) increases at temperatures below the nucleation rate maximum
This slide shows the thermodynamic barrier for nucleation, estimated from the experimental values of steady-state nucleation rates. We see, that at low temperature the linear temperature dependence of thermodynamic barrier on temperature breaks down, and it seems unnatural.
Thermodynamic barrier for nucleation versus temperature

4. The same behavior of thermodynamic barrier we see for different glasses
Reduced thermodynamic barrier for nucleation, 𝑊 c 𝑇 / 𝑘 B 𝑇, versus reduced temperature, 𝑇/ 𝑇 m , for different glasses 𝑇 m is the liquidus temperature.

5. The steady-state nucleation rate and time-lag for nucleation
BASIC EQUATIONS
The steady-state nucleation rate and time-lag for nucleation
𝐼 𝑠𝑡 =𝑐 𝜎 𝑘 𝐵 𝑇 𝐷 𝑑 0 exp − 𝑊 𝑐 𝑘 𝐵 𝑇 ,
𝜏= 𝑘 B 𝑇𝜎 ∆ 𝐺 𝑉 2 𝑑 0 2 𝐷 , 𝑊 𝑐 = 16𝜋 3 𝜎 3 ∆ 𝐺 𝑉 2 ,
𝜎 is the specific interfacial energy, D is the effective diffusion coefficient governing the processes of aggregation of ambient phase particles to crystal clusters, 𝑑 0 is its characteristic size, 𝛥 𝐺 𝑉 is the thermodynamic driving force for nucleation, 𝑊 𝑐 is
the work of critical cluster formation, c is the number density of “structural units” in the melt
𝑐≈1/ 𝑑 0 3
These equations of CNT for one component (or pseudo one component) systems commonly employed for analysis of nucleation rate and time lag in glass forming liquids. 𝜎 is the specific interfacial energy, D is the effective diffusion coefficient governing the processes of aggregation of ambient phase particles to crystal clusters, 𝒅 𝟎 is its characteristic size, 𝚫 𝑮 𝑽 is the thermodynamic driving force for nucleation, c is the number density of “structural units” in the melt, 𝑊 𝑐 is the work of critical cluster formation

6. BASIC EQUATIONS
Since the same effective diffusion coefficient, D, should control both 𝐼 st and 𝜏, 𝐼 st may be rewritten as:
𝐼 st = 𝑘 B 𝑇 1/2 𝜎 3/2 ∆ 𝐺 𝑉 2 𝑑 0 6 𝜏 exp − 𝑊 c 𝑘 B 𝑇
Since the same effective diffusion coefficient, D, should control both nucleation and time-lag, in this way we express the diffusion coefficient D through the experimentally determined time-lag and obtain this formula.

7. 𝐼 st = 16 3 𝑘 B 𝑇 1/2 𝜎 3/2 ∆ 𝐺 𝑉 2 𝑑 0 6 𝜏 exp − 𝑊 c 𝑘 B 𝑇
𝑊 𝑐 = 16𝜋 3 𝜎 3 ∆ 𝐺 𝑉 2
𝜎= 𝜎 𝛿 𝑅 c
Tolman’s equation
The experimental and calculated by CNT nucleation rate in B2S glass versus temperature
𝝈 𝟎 and 𝜹 were used as adjustable parameters to achieve the best agreement with experiment at 𝑇> 𝑇 𝑚𝑎𝑥 and then we employed the same parameters for 𝑇< 𝑇 𝑚𝑎𝑥
𝝈 𝟎 is the specific surface energy of a planar interface,
𝜹 is the Tolman length, which is of the same order of magnitude as 𝑑 0 .
We employed the Tolman’s equation. 𝜎 0 and 𝛿 were used as adjustable parameters for a description of the nucleation rate data at temperatures higher than that of the nucleation rate maximum, 𝑇 max - blue line in the figure.
We see, that in this way it is impossible to describe the nucleation rates measured below Tmax without additional hypotheses of its explanation

8. 𝐼 st = 16 3 𝑘 B 𝑇 1/2 𝜎 3/2 ∆ 𝐺 𝑉 2 𝑑 0 6 𝜏 exp − 𝑊 c 𝑘 B 𝑇
As was already noted such decrease of the experimental nucleation rate could be explained at least formally by increase of the thermodynamic barrier for nucleation at T<Tmax. Such result contradicts CNT. Thus in the framework of CNT It is not possible to describe the nucleation rates measured at T<Tmax without additional assumption.
One of possible assumption is the reduction of the thermodynamic driving force due to the elastic stresses caused by disagreement of liquid and crystalline nucleus densities.

9. 1. The effect of elastic stresses on the thermodynamic barrier for crystal nucleation
Work of critical cluster formation and elastic stresses
Taking into account the elastic stresses, we have to decrease the thermodynamic driving force by the value of the elastic stress energy ∆𝑮 𝒔𝒕𝒓
To test such possibility we calculated elastic stress to valuate its influence on the thermodynamic barrier for nucleation and hence on nucleation rate.

10. 1. The effect of elastic stresses on the thermodynamic barrier for crystal nucleation
Work of critical cluster formation and elastic stresses
Taking into account the elastic stresses, we have to decrease the thermodynamic driving force by the value of the elastic stress energy ∆𝑮 𝒔𝒕𝒓
The thermodynamic driving force decreases by the value of the elastic stress energy

11. Elastic stress energy in the case of a purely elastic solid
Elastic stress energy in a viscoelastic glass-forming liquid
Δ𝐺 str = Δ𝐺 str 0 𝑓(𝜃
𝜉= 𝜌 c − 𝜌 m ) 𝜌 m
The elastic stress energy must be computed with account of its relaxation
𝜏 rel = 𝜂 1+ 𝛾 m 𝐸 m
𝜃= 𝜏 𝜏 rel ,
𝑬 is the Young modulus, 𝜸 is the Poisson number, ρ is density. The subscripts “c” and “m” indicate the crystal and melt, respectively.
𝑓(𝜃) takes into account both the development and the relaxation of stress via the Kohlrausch relaxation

12. Nucleation rates vs temrpeature
The dotted lines show nucleation rates calculated without account of stresses. The solid lines show the rates taking into account the elastic stresses and their relaxation.
The results are presented in this slide for two glasses. The dotted lines show nucleation rates calculated without account of stresses. The solid lines show the nucleation rates taking into account the elastic stresses and their relaxation. We see, that the effect of elastic stresses is little or negligible.

13. In the framework of CNT, the effect of elastic stresses cannot explain the rapid decrease of the experimental nucleation rates at deep undercoolings corresponding to the low temperature side of the nucleation rate maximum
The results were published in this paper
A.S. Abyzov, V.M. Fokin, A.M. Rodrigues, E.D. Zanotto, J.W.P. Schmelzer, The effect of elastic stresses on the thermodynamic barrier for crystal nucleation. J. Non-Cryst. Solids 432 (2015)

14. Proposed solution:
Incorporating into CNT the concept of spatial heterogeneity of glass- forming liquid near Tg. Hereby it is assumed that:
crystals nucleation mainly occurs in “liquid-like” regions, and is suppressed in “solid-like” regions
volume fraction of “liquid-like” regions decreases with decreasing temperature.

15. Concept of spatial heterogeneity
𝐼 st = 𝑽 𝒏 𝑽 𝑐 𝑘 B 𝑇 𝜎 ∆ 𝐺 𝑉 2 𝑑 0 3 𝜏 exp − 𝑊 c 𝑘 B 𝑇
𝑐= 1 𝑑 𝑜 3 ⟾𝐶=𝑐 𝑉 𝑛 𝑉
This formula for the nucleation rate differs from the usual one by the ratio Vn/V, because the nucleation occurs in the liquid-like regions only, but we see crystals in the total volume
Vn – volume of liquid-like regions V – total volume of the system

16. Concept of spatial heterogeneity
𝐼 st = 𝑉 𝑛 𝑉 𝑐 𝑘 B 𝑇 𝜎 ∆ 𝐺 𝑉 2 𝑑 0 3 𝜏 exp − 𝑊 c 𝑘 B 𝑇
𝑉 𝑛 𝑉 = 1 at 𝑇≥ 𝑇 𝑠𝑤 exp − 𝐾 𝑉 𝑇 sw −𝑇 𝛽
To take into account volume of disposed for nucleation liquid-like regions we use the fitting function for the ratio Vn/V
Vn – volume of liquid-like regions V – total volume of the system
𝐾 𝑉 , 𝛽, and 𝑇 sw , are fit parameters

17. a) Nucleation rates for L2S glass versus temperature
a) Nucleation rates for L2S glass versus temperature. The circles denote experimental data, the red line represents a fit of the effective nucleation rate, taking into account the change of the liquid-like volume 𝑉 𝑛 /𝑉 and blue line shows nucleation rate in liquid-like region only. 𝑇 g and 𝑇 sw are denoted by vertical dashed lines. b) 𝑉 𝑛 /𝑉 versus temperature is estimated for L2S glass.

18. Crystal nucleation rates and the volume fractions of liquid-like regions for several silicate glasses versus temperature
Li2O2SiO2
Li2O2B2O3, BaO2SiO2, 1Na2O2CaO3SiO2, 2Na2O1CaO3SiO2, 1Na2O1CaO2SiO2, 44Na2O56SiO2
Similarly to lithium disilicate glass, a theoretical treatment of nucleation rate data was also performed for the different glasses of stoichiometric and non-stoichiometric compositions.

19. Some consequences of the assumed heterogeneity of glass-forming liquids in the glass transition interval
Let us consider some consequences of the assumed heterogeneity of glass-forming liquids in the glass transition interval

20. 1. Decoupling of diffusion and viscosity
𝜏= 𝑘 B 𝑇𝜎 ∆ 𝐺 𝑉 2 𝑑 2 𝐷 τ
𝐷 𝜂 =𝛾 𝑘 𝐵 𝑇 𝑑 0 𝜂
This slide shows effective diffusion coefficients for L2S glass calculated from time-lag for nucleation (Dτ) and from viscosity by the Stocks-Einstein equation (Dη) versus temperature. We see, that diffusivity proceeds more quickly, than viscosity, because Dη refers to the whole system, while Dτ refers only to the liquid-like regions.
Dη refers to the whole system, while Dτ refers only to
the liquid-like regions !
Effective diffusion coefficients for L2S glass calculated from time-lag for nucleation (Dτ) and from viscosity by the SEE equation (Dη) versus temperature.

21. 𝑇 𝑔 𝑉 is defined from 1− 𝑉 𝑛 𝑉 dependence like Tg from DSC curve.
2. The temperature change of the volume fraction of solid-like regions and the apparent activation energy of viscous flow.
Volume fraction of solid-like regions
𝑉 𝑠𝑙 𝑉 = 1− 𝑉 𝑛 𝑉
This plot presents the dependence of solid-like volume fraction on temperature. We can determine TgV like Tg from DSC curve
𝑇 𝑔 𝑉 is defined from 1− 𝑉 𝑛 𝑉 dependence like Tg from DSC curve.

22. 𝑇 𝑔 𝑉 is defined from 1− 𝑉 𝑛 𝑉 dependence like Tg from DSC curve.
2. The temperature change of the volume fraction of solid-like regions and the apparent activation energy of viscous flow.
𝛥𝑇= 𝑇 𝑠𝑤 − 𝑇 𝑔 𝑉
𝛥𝑇 reflects the rate of the temperature evolution of the solid-like region volume fraction
The difference Tswitch minus Tgv reflects the rate of the temperature evolution of the solid-like region volume fraction
𝑇 𝑔 𝑉 is defined from 1− 𝑉 𝑛 𝑉 dependence like Tg from DSC curve.

23. 2. The temperature change of the volume fraction of solid-like regions and the apparent activation energy of viscous flow.
𝐸 𝑒𝑓𝑓 ≡ 𝑑 𝑑 1 𝑇 log(η) 𝑇= 𝑇 𝑠𝑤
This plot presents dependence of Delta T the effective activation energy of viscous flow. We see that the faster is an increase of the volume fraction of solid-like regions with decreasing temperature (narrower ΔT interval) the higher is the effective activation energy of viscous flow
The faster is the increase of the volume fraction of solid-like regions (narrower ΔT interval) with decreasing temperature the higher is the effective activation energy of viscous flow.

24. 3. The glass transition and the temperature evolution of volume fraction of solid-like regions
Tg versus q
𝑇 𝑔 𝑞 = 𝑇 𝑔 𝑉
This figure shows, that the temperature evolution of volume fraction of solid-like regions strongly correlates with a DSC-curves, this slide is for the lithium disilicate.
Black line – Vr /V
Color lines – DSC curves
with heating rates 1, 2, 5, 10 K/min

25. 3. The glass transition and the temperature evolution of volume fraction of solid-like regions
The volume fraction of solid- like regions, where according to the proposed model the nucleation process is suppressed, is well correlated with the change of the DSC signal that reflects the transition of the glassy state into metastable liquid.
Similar correlations for several other silicate glasses are shown in this slide. So, the volume fraction of solid-like regions, where according to the proposed model the nucleation process is suppressed, is well correlated with the change of the DSC signal that reflects the transition of the glassy state into metastable liquid.
Volume fractions of solid-like regions in several silicate glasses as a function of temperature and the corresponding shifted DSC curves

26. Scheme of proposed model
This slide shows the scheme of proposed concept of spatial heterogeneity of glass-forming liquid. At high temperature melt is homogeneous, whole volume is disposed for nucleation.

27. Scheme of proposed model
When temperature decreased, the solid-like regions arise, but it fraction is small and not affects significantly on the nucleation rate

28. Nevertheless, when temperature decreased more, the solid-like regions occupied significant part of the volume and the nucleation rate decreases. Let us note, that number of clusters we calculate in the total visible volume, shown as a circle in this figure, but induction time of nucleation does not dependent on the volume fraction of liquid-like regions, it dependent on the temperature only

29. Scheme of proposed model
At the temperatures significantly below the glass transition temperature the volume fraction of liquid-like regions is very small, and nucleation rate decreases more and more

30. Scheme of proposed model
In the hypothetical case, when we could see nucleation in the liquid-like regions, nucleation rate will be significantly higher – blue line in this figure

31. Conclusions
Proposed model incorporating into CNT the concept of spatial heterogeneity of glass-forming liquid near Tg. The evolution of such heterogeneous structure with decreasing temperature, is the origin of the decrease of the nucleation rate.

32. Conclusions
Proposed model incorporating into CNT the concept of spatial heterogeneity of glass-forming liquid near Tg. The evolution of such heterogeneous structure with decreasing temperature, is the origin of the decrease of the nucleation rate.
The correlations found between the temperature evolution of the volume fraction of the solid-like regions and typical features of the glass transition corroborate the proposed model.

33. Conclusions
Proposed model incorporating into CNT the concept of spatial heterogeneity of glass-forming liquid near Tg. The evolution of such heterogeneous structure with decreasing temperature, is the origin of the decrease of the nucleation rate.
The correlations found between the temperature evolution of the volume fraction of the solid-like regions and typical features of the glass transition corroborate the proposed model.
Any explanation of nucleation process at 𝑇<𝑇 𝑚𝑎𝑥 must be correlated, consequently, with an appropriate glass transition model.

34. Thank you for your attention !