Chapter 1
Glasses and glass-ceramics
This chapter aims to define a glass, a glass transition temperature and a glass-ceramic. A literature review is proposed for the glass formation ability. Two structural theories have been proposed: the
"crystallite theory" of Lebedev and the "random network theory" of Zachariasen. Those two opposed theories have inspired the recent "tissue-pathway" and the "modified random network"
models. The kinetic theory, taking into account the time needed to crystallize rather than the glass structure, is however the most accepted nowadays.
1 Glasses
1.1 Definition [9, 10, 11]
Two different approaches were proposed to define a glass:
• Structural definitions: "A glass is a frozen-in undercooled liquid"(Simons, 1930) and "In the glassy state, there are solids, uncrystallized materials"(Tammann 1933);
• Operational definitions: "A glass is an inorganic product of fusion which has been cooled to a rigid condition without crystallizing"(American society for testing and materials, 1945) or
"A glass is a solid obtained by freezing a liquid without crystallizing" [11];
All those definitions are too large or too restrictive: except from Tammann, they are limited to quenching of a liquid. There are several other ways to obtain a glass than freezing the structural disorder of a liquid phase. The condensation of a gas on a cool enough surface can lead to glassy structures, favoured by the disordered character of a gas. The amorphization, by radiation or mechanical actions, leads to a disordering of the crystalline phase by breaking the crystal order.
The definition of glass proposed by Tammann was too general as silica gel should be a glass. The first operational definition was in addition limited to inorganic products.
Zarzycki [11] (1982) proposed to add the concept of vitreous transition which is a funda- mental property of glasses: "A glass is a non-crystalline solid that presents a vitreous transition phenomenon". This definition has no restriction on the way to obtain the glass, it can be organ- ic/inorganic and adds a condition on the internal stability. Indeed, non-crystalline solids that form
glasses have an excess of energy regarding the crystalline solid and are metastable.The concept of glass transition will be detailed in 1.2.
Another version, taking atomic order into account was proposed by Shelby [10](1992): "A glass is an amorphous solid completely lacking in long range, periodic atomic structure and exhibiting a region of vitreous transformation behaviour", so that any material, inorganic, organic or metallic, formed by any technique, which exhibits glass transformation behaviour is a glass.
According to this, a glass does not show any symmetry neither periodicity at long-range. Unlike crystals, the structure of glass cannot be described using atomic positions in an elementary cell and translation element to reproduce this cell in 3D space. Glasses are likely to be described using coordination distribution and inter-atomic distances.
1.2 Glass transition
As it was first introduced in Zarzycki’s definition of glass, the glass transition concept is a fundamen- tal property of glasses. It is characterized by a sharp modification of second order thermodynamic properties, i.e. heat capacity at constant pressure (cp) 1 or thermal dilatation (↵) 2 while first order thermodynamic properties, i.e. volume or enthalphy, vary continuously. The case of a glass formation from a liquid is illustrated in Figure 1.1 and Figure 1.2. Figure 1.1 illustrates the effect of temperature change on the volume and the enthalpy while Figure 1.2 illustrates the effect of temperature change on the dilation coefficient and the heat capacity.
1cp=@H@T|P 2↵=V1@V@T|P
Figure 1.1– Volume and enthalpy variation with temperature [12]
Figure 1.2 – Thermal dilatation (↵) and heat capacity cp variation - l=liquid, ls=undercooled liquid, c= crystal and v=glass [11]
On cooling, the increase in viscosity limits the motion of the atoms until inhibiting it. If cooling is fast enough, the liquid begins to freeze and no discontinuity appears at the melting temperature (Tm) neither for the enthalpy nor the volume. No crystallization occurs. The enthalpy decreases and the dilatation coefficient of the supercooled liquid is the same as for the liquid. The slope of the curves for the volume and the enthalpy changes, becoming closer to that of the crystalline solid. The vitreous/glass transition marks the change from a supercooled liquid to a glass and corresponds to a viscosity (⌘) of about 1013 poise/ 1012Pa.s.
The glass transition temperature (Tg) is defined over a range of temperatures because it depends on the cooling rate. A rapid cooling freezes the atoms more rapidly and makes Tg moving to higher temperatures. This is illustrated in Figure 1.3.
Figure 1.3– Influence of the rate of cooling (U in this Figure) on the glass transition temperature:
U1<U2<U3 [11]
2 Glass formation
Internal structure of glasses and its role on glass formation ability still gather interest. While there is a general agreement on the gross features of the interplay between glass composition and structure in oxide glasses, there are still questions on how their basic building blocks are organized to form the bulk of the vitreous material, even for one-component glasses like SiO2or B2O3. The situation is even more complicated in multi-component glasses.
Theories tend to predict that some unique features of certain melts lead to glass formation while the lack of these features prevents the glass formation. There is no single theory able to predict and specify which material is able to form a glass and under which conditions without exceptions.
There are however several theories or models that encompass most of the relevant aspects which are known to lead to a glass structure. These theories are grouped into structural theories, i.e
"wether" a material will form a glass, and thekinetic theory, i.e "how fast" a material will form a glass [13].
The structural theories can be divided into the crystallite theory introduced by Lebedev and thenetwork theory introduced by Zachariasen. For Zarzicky [11], "both models are probably the boundaries limiting a more complex reality depending on the type of glass". Nowadays, the kinetic theory is adopted in most cases [10, 11]. Even if most of structural theories are now of historical interest, some had a deep impact, especially therandom network theory proposed by Zachariasen.
Structural theories still gather attention for the comprehension and control of glass properties, for
example the comprehension of ionic diffusion mechanism in oxide glasses [14, 15].
2.1 Structural theories of glass formation [10, 11, 16]
The first model of glass structure, presented in Figure 1.4a was proposed by Frankenheim in 1835 [17] and was based on the structure of silicate crystals. The "micro-crystal" approach suggested that glasses consist of very small crystals in a disordered network. The small size of the crystal explains the lack of well defined peak in the X-Ray diffraction patterns.
Lebedev introduced in 1927 [18] the crystallite model. Crystalline are distinguished from
"micro-crystals" because they have to be considered as strongly deformed structures corresponding only approximatively to the normal lattice. A glass was defined as an accumulation of crystallites connected by highly disordered (amorphous) regions, like grain boundaries. This crystallite model is illustrated in Figure 1.4b.
Figure 1.4– Micro-crystals and crystallite representation from (a) Frankenhein [19] (b) Lebedev [20]
In 1926, Goldschmidt noted that ionic glasses (oxides, halides, chalcogenides) with formula RnOm form easily when the ionic radius ratio rc/ra ranges between 0.2 and 0.4 [11]. For oxides, a radius ratio in this range involves that cations are surrounded by four oxygen ions forming a tetrahedron. It was therefore supposed that only melts containing tetrahedrally coordinated cations could form glasses during cooling. It was the first hypothesis applicable to any system containing cations and anions. Despite its simplicity and the right description in many cases of glass-forming systems, this model encounters some exceptions and describes glass forming system as pure ionic substances. Most of bonds in glass forming system exhibit a part of covalent bonds (Si-O bonds have 50% of covalent bonding [16]), limiting the application of this rule.
Structural determinations of vitreous SiO2 and GeO2 made by Warren and Pincus [21] in the 1930s using X-ray diffraction showed the structure of glass to be typical of an amorphous solid.
Glass densities were also found to be too small to be crystalline, indicating that glasses cannot consist of crystallites.
In 1932, Zachariasen [11, 22] extended the idea of Goldschmidt that was not sufficient to describe all glass forming cases. He introduced therandom network theory of glassin which the nature of bonding in the glass is the same as in the crystal but where the basic structural units in a glass are connected in a random manner in contrast to the periodic arrangement in a crystalline material. This description of glasses is clearly not in agreement with the Lebedev’s crystallite model .
The atomic structure of a glass lacks of long range periodicity. But despite this lack of long range order, the structure of glass exhibits a high degree of ordering on short length scales due to the chemical bonding constraints in local atomic polyhedra. The atoms align themselves chemically to balance charges and to minimize bond energy by filling appropriate bond orbitals. This involves that the bonding structure of glasses, although disordered, has the same symmetry signature as for crystalline materials. The coordination number, the bond length and the bond angles are similar. Figure 1.5 illustrates a schematic planar representations of crystalline and vitreous SiO2
(RO2). Figure 1.5(a) shows a short range order with a periodic crystalline lattice structure involving tetrahedral SiO4units linked together and forming regular rings-like with six edges. The Si-O bond lengths vary between the different crystal forms: for ↵-quartz the bond length is 161 pm and for
↵-tridymite it ranges from 154 up to 171 pm. The Si-O-Si bond angle also varies from 140° in
↵-tridymite to 144°in↵-quartz and to 180°in -tridymite [23].
Figure 1.5– Schematic planar representation of (a) a crystalline form of RO2 (a fourth oxygen would be located on each cation – (b) the respective vitreous form [10]– (c) Glass structure has a broader distribution of bond angles [24]
In amorphous silica (Figure 1.5(b)), the SiO44tetrahedra form a network that does not exhibit any long-range order. However, the tetrahedra themselves represent a high degree of local ordering as every silicon atom is coordinated to 4 oxygen atoms and the nearest neighbour Si-O bond length exhibits only a narrow distribution throughout the structure. In addition, the tetrahedra also form a network of ring-like structures which lead to ordering on intermediate length scales. The lack of long range order implies also that the Si-O-Si bond angles are more variable than in the crystalline form (Figure 1.5(c)).
Zachariasen considered several structural arrangements, or rules, to describe the ability of a network to form a glass [11]:
• Rule 1: No oxygen can be linked to more than two network cations;
• Rule 2: The number of oxygen atoms surrounding the network cation must be small (three or four);
• Rule 3: The polyhedral must be connected only by the corners (no edges nor faces) so that cations are located as far as possible;
• Rule 4: At least three corners of each polyhedra must be shared with another polyhedron, forming a 3D network.
In general, all four rules should be satisfied to form a glass. RO or R2O do not satisfy the rules and cannot form glasses. R2O3, where oxygen atoms form triangle around cations, RO2
and R2O5, where oxygen atoms form tetrahedral around cations satisfy the four rules to form a glass. Zachariasen’s rules are in agreement with the first Pauling’s rule to deduce the coordination number CN3[24].
If silica is considered [24]:
From Pauling’s packing rule: r Sir O24+=0.281.4 meaning a preferential tetrahedral bonding. Zachariasen’s rule 2 is satisfied. CN(SiSi4+4+)=44=CN(OO22 )=22 meaning that the coordination number (CN) ofO2 =2.
Rule 1 is satisfied. The crystal structure is made by sharing 4 corners, satisfying both rules 3 and 4.
The four rules are respected. Glass and crystal structures have the same cation polyhedra building blocks, arranged in different patterns.
Now if rock salt MgO is considered [24]:
From Pauling’packing rule: r Mgr O22+=0.721.4 meaning a preferential octahedral bonding. Rule 2 is not
3Pauling defined five rules to explain the crystal structure of complex ionic crystals. The first rule relates the ionic radii to the coordination number and the second deduces the electrostatic valence rule. The third rule relates the share of corners and edges to stability while the fourth relates a high coordination number to the tendency of non sharing edges. The fifth rule is the rule of parcimony, i.e that the number of different constituents tends to be small [23].
satisfied. CN(MgMg2+2+)=26=CN(OO22 )=26 meaning that the coordination number (CN) of O2 =6. Rule 1 is not satisfied either. The crystal structure is an edge sharing polyhedra, that does not satisfy rule 3. MgO does not form a glass.
Finally, if alumina (Al2O3) is considered [24]:
From Pauling’packing rule: r Alr O23+=0.531.4 meaning an intermediate between tetrahedral and octahedral bondings. CN(AlAl3+3+)=36=CN(OO22 )=24. Rule 1 is not satisfied, meaning that Al2O3 does not form a glass. However, in aluminosilicate,CN(Al3+)=4 andCN(O2 )=2 so that they form a glass under particular conditions.
Zachariasen generalized the formula to RmR’nO where m and n represent the number of cation per oxygen atom [11]. For a tetrahedral network with three shared corners, n lies between 0.33 and 0.5. His theory was supported by Warren and Pincus [21]. Zachariasen’s theory has some limitations because it applies only to oxide glasses and several exceptions exist.
In a further approach, Zachariasen [22] and Warren [21] applied the concept for two and multi-components systems. In general, oxides with small cations (and small CN) form glasses.
Zachariasen named “network formers” the cations that can form oxides that vitrify alone, “network intermediates” the cations that cannot form oxides that vitrify alone but which can replace them isomorphycally and “network modifiers” the cations that are not included in the network and that break linkages within the network. The most known examples of network formers are SiO2, GeO2, B2O3. Other examples of network formers as well as network-intermediates and network modifiers are listed in [11]. Figure 1.6 and Figure 1.7 show how modifiers change the random network of a simple oxide glass by breaking bonds between tetrahedra. This forms bridging and non-bridging O; the first link forms tetrahedra, the second forms ionic bonds with the modifier cations. Each non-bridging oxygen must be associated to a certain number of cations to ensure local neutrality.
Figure 1.6– Breaking of O-Si-O by a network modifier (Bridging O to Non-bridging O) [11]
Figure 1.7– Schematic planar representation of a vitreous network with both network-formers and modifiers [11]
Zachariasen has considered the formation of more complex oxide glasses with the addition of alkaline and alkali-earth oxides and added new rules [11]:
• Rule 5: There must be a sufficient percentage of network oxides surrounded by tetrahedral or triangular oxygen atoms;
• Rule 6: The tetrahedrons and triangles must be connected only by the corners;
• Rule 7: Some oxygen atoms may be linked to only two network cations without forming any other bond with another cation.
Vitreous silica is a prototypical "Zachariasen glass" with a high Tg(1200°C), SiO4tetrahedra, bridging oxygens (BO) only and corners shared in 3 dimensions. Even with a lack of long range order involving a distribution of bond length, the tetrahedra are well defined with Si-O bonds of 1.62 Åand O-O bonds of 2.65 Å. The same are found in crystalline silica. The Si-O-Si bonds angles are about 144±10°. Adding an alcali ion modifies the structure by creating a new non-brindging oxygen (NBO), every molecule creates two, so that BO+R2O–> 2NBO or 2Q4+R2–>2Q3 (in Qx, x expresses the number of bridging oxygen). The scheme developed by Zachariasen [22] and Warren [21] allows the influence of oxides to be explained. A high modified structure implies high NBO
fraction or Q-distribution, causing a decreases in the viscosity and the Tg, but an increase in density, refractive index, coefficient of thermal expansion (CTE) and electrical conductivity [10, 24].
The alkaline-earth ions have the same effect, with two NBO created by each ion. Alumina (Al3+) forms tridimentional network of (AlO4) similar to (SiO4) so thatAl3+can replace Si4+in the network by filling vacancies with larger cation to preserve electroneutrality [25]. The Si-O-Al bond is underbonded and the alkali/alkali-earth balance the new network charge. Al2O3 increases the connectivity of an alkali modified glass by replacing NBO’s with cross-linking Al-O-Si bonds.
This involves an increase in the viscosity, an increase in the elastic modulus, a decrease in the CTE and an increase in the Na-diffusion rate (Na more weakly bonded to AlO4 than to Si-NBO ). The increase in cross-linking occurs untill [Al2O3]=[R2O], needed to neutralize the charge of the new network. Further addition of Al2O3 have different effect on glass properties with the formation of tri-coordinated oxygens.
In 1942, Dietzel [11, 26] extended the network theory by considering the size and polarizabililiy of ions. This enables to consider the influence of interaction of forces between cations and anions during solidification. He introduced the notion of field strength F=Za2c (where Zc is the cation valence and a=rc+rO if the anion is oxygen) to characterize the effect of a single cation. The correspondence with Zacchariasen’s classification indicated that cations can be classified with their field strength. This concept extended the comprehension of cooling melt which could not be explained on basis of Zacchariasen-Warren hypotheses, for example the tendency towards phase separation.
In 1951, Smekal [11, 27] proposed another theory based on the type of bonding. He postulated that melts that contain mixed bonds can form glasses. Material with pure ionic bonds involve low melting temperatures, low hardness and crystallize easily. Covalent bond are much stronger and involves high melting temperatures. Inorganic compounds that contain partially ionic and covalent bonds and organic compounds that contain covalent and Van der Waals bonds fall into the glass forming category. Smekal proposed the following classification:
• Group I: the “network formers” are cations which form bond with oxygen with an ionic character near 50% (B, Si, P, Ge, As, Sb [11, 27]);
• Group II: the “network intermediates” are cations with lower electronegativities. They can replace network former cations but they cannot form a glass (Be, Al, Ti, Zr, Sn [11, 27]);
• Group III: the “network modifiers” are cations with low electronegativities. They form highly ionic bond with oxygen and never form a glass. They are used to modify the network structure formed by the “network formers" oxides (Mg, Ca, Sr, Ba, Li, Na, K, Rb, Cs [11, 27]).
The three groups are illustrated in Figure 1.8.
Figure 1.8– Estimation of the degree of ionicity of a bond from difference on electronegativity, a=cation, b=oxygen – The smallest amount of ionic bonds provides network forming cations [11]
Stanworth [11, 28] (1946) indicated a correlation between the degree of covalence of R-O bond and the ability to vitrify. The smallest the difference in electronegativity the more covalent the bond is. Group I encompasses oxides that can form glasses (SiO2 has a 50% ionic bond4) and Group III oxides that modify the network without belonging to it. This rule has exceptions: BeF2
has 80% ionic and forms a glass. Even if Sb has the same electronegativity as Si, Sb2O3 does not form glass as easily as silica.
In 1947, Sun [11, 29] suggested that strong bonds form chains that prevent or hinder the reorganization of the melt structure into a crystalline structure during cooling, promoting glass formation. A liquid structure will persist when the liquid is cooled below the melting point and will continue until long-range movements of atoms are frozen. In order to obtain such long chains or large networks, it is necessary that the strength of a single bond is very strong. The coordination numbers of the glass-forming atoms should be as small as possible to keep the bond strength. He defined it as the energy required to dissociate an oxide into its constituent atoms in the gaseous state (Ed) divided by the number of cation-anion bond in a coordination unit. Glass formers form strong bonds with oxygen (higher than 80 kcal/Avogadro bond [29]), leading to a rigid network and high viscosity, while modifiers form weak bonds (weaker than 60-80 kcal/Avogadro bond [29] 5) leading to a disrupted network and lower viscosity. Intermediates show intermediate bonds (between 60 and 80 kcal/Avogadro bond ([29]) and cannot form glasses on their own but when associated with other oxides. There are still exceptions to this rule.
Rawson [11](1956) completed the theory of Sun. To link the ability to vitrify with the ability
4According to Pauling, the difference in electronegativity values of the two elements allows the ionic percentage to be estimated [11]
5energy required to blreak avogadro number of bonds
to break bonds at the melting temperature, the thermal energy available to break them must be considered in addition to the bond strength. As the melting temperature can be an indication of this energy, he suggested that melting temperature must be included in the model of Sun: a high melting temperature means high energy available for bond disruption. Consequently, a material with high bond strength and low melting temperature will have a small Rawson parameter and will be a good glass former (see examples in [11]).
Therandom network theory of glass established in 1931 was the most used model until the early 90’s to describe the structure of glasses. The development of Nuclear Magnetic Resonance (NMR) and X-Ray spectroscopies allowed modern models to criticize and detail the previous. The crystallite model was the basis for the recent model of tissue-pathway model of Goodman and Ingram [14] in 1991 who have completed the crystallite concept. They presented the glass as a random agglomeration of highly ordered domains (clusters) surrounded by highly disordered domains (tissues) where ionic diffusion is easier. The random network theory of glass was completed by Greaves [30] in 1985 who proposed a new model called the modified random network. The model takes the covalent nature of Si-O and the ionic interaction between alkaline and oxygen into account, but considers that network-modifiers are not randomly positioned. There are some domains enriched in network-formers and domains enriched in network-modifiers. Those domains are bonded by NBO. Network-modifiers domains are preferential ways for ionic diffusion.
Figure 1.9 illustrates the two recent models: (a) The tissue-pathway model and (b) the modified random network.
Figure 1.9– (a) Tissue-pathway model: presence of clusters and paths favoring ionic migration inside the tissue [14] (b) Modified random network in two dimensions: solid lines are covalent bonds and dotted lines ionic bonds, black atoms are modifiers, white atoms are formers and oxygen. Gray channels are domains enriched in network-modifiers were ionic diffusion occurs throughout the silicate network [30]
2.2 Kinetic theory [11, 13]
A glass forms when a liquid is cooled below its meting temperature sufficiently fast to avoid nucle- ation and growth of crystals. The structural approach aimed to explain how the internal structure inhibits nucleation and growth processes to form a vitreous structure. The more recent theories have reconsidered those structural criteria by considering the ease of a glass to form. All liquids can finally form a glass and it is rather the question "how fast" the liquid must be cooled to avoid the crystallization than "whether" a material will form a glass.
From this consideration, it comes that if a glass is forming at slow cooling rates (< 1°C/sec [13]), the compound is considered as strong glass-former. Inversely, if the compound needs very rapid cooling rates (>100°C/sec [13]), it is considered as a poor glass-former. A melt that cannot form a glass even with high rate of cooling is considered as non glass-former.
Thecritical cooling rate is controlled by the nucleation and growth processes and measures how fast a liquid must be cooled to avoid crystallization and form a glass. Nucleation and growth
processes will be detailed later in chapter 2. Poor glass-formers are liquids that form quickly a large number of nuclei close to the melting temperature that grow rapidly, and oppositely for good glass-formers. As a consequence, a visible difference can be seen in nucleation and growth rate curves. This is illustrated by comparing the cases of water and fused silica in Figures 1.10 and 1.11. An overlap is visible between the curves in the case of water, with very high rates, and both are high at the same temperature. The case of silica is very different because no overlap is visible and nucleation and growth rates are very low compared to the case of water.
Figure 1.10– Nucleation and growth rates for water [13]
Figure 1.11– Nucleation and growth rates for silica [13]
Continous Cooling Transformation curves (CCT-curves) can be used to calculate the time required at a certain temperature for a given fraction of liquid to transform into a crystal. The CCT curve is obtained by calculating the time (t) required to reach a typical crystallized fraction given by the classical theory (see later) x=⇡3(It)(Ut)3, where x is the crystallized fraction, I and U
are the nucleation and growth rates (s 1 and cm.s 1) [11]. The "noze shape" results from the equilibrium between the thermodynamic driving force and the kinetic barrier. From Ulhmann [31], the minimum detectable crystal fraction is around 10 6. Figure 1.12 shows the critical cooling rate (Rc) in the general case of a poor glass former and good-glass former. Figure 1.12 shows (a) the case of water and (b) the case of fused silica. To calculate the rate of cooling to avoid a crystallized fraction above 10 6, a tangent from the melting temperature to the CCT curve must be drawn.
Figure 1.12– CCT-curves in a general case
Figure 1.13– CCT-curves for water (a) and for silica (b) [13]- Rc is the cooling rate
In practice, heterogenous nucleation has a strong effect on the critical cooling rate. Critical cooling rates with homogeneous and heterogenous nucleations in SiO2-glass are 9.10 6and 2.10 2
°C/s, respectively [9].
3 Glass-ceramics [25, 32]
Glass-ceramics were discovered "accidentally" in 1951 by Stookey [25]. They are micro- or nano- polycrystalline materials processed by controlled crystallization of a parent glass of suitable com- position. They consist most often in a (small) amount of glassy phase and one or several crystal phases. By precipitating crystal phases in the parent glass, new properties can be obtained. As nowadays new technologies constantly required new material with special properties, glass-ceramics play an important role and are fully considered as functional materials by offering the possibility to combine in one material the special properties of sintered ceramics with the characteristics of glasses.
Only specific glass compositions are suitable because some glasses are too stable and too difficult to crystallize (e.g. window glass) whereas others crystallize too easily uncontrollably. Voids, cracks and porosity must also be avoided. This indicates the complicated development of such material and how nucleation and growth processes must be understood and mastered. Nucleation is the key factor for controlling the crystallization in glass-ceramics [33]. Thermodynamic considerations of nucleation and growth are detailed in chapter 2.
The controlled crystallization of the parent glass is obtained by a thermal process in two steps, namely a nucleation stage and a growth stage. Nucleation and growth processes are highly tem- perature dependant. Typical curves with small overlapping are shown in Figure 1.14 with their corresponding heat treatments: (1) glass melting, (2) glass shaping, (3) cooling, (4) nucleation step, (5) growth step. The nucleation step consists in producing a high density of seeds/nuclei at low temperature near Tg. A step at a higher temperature leads to the growth of those nuclei to a specific and controlled size. When the overlap is large, a single heat treatment is enough. The thermal treatment converts the parent glass into a body composed from 50 to 95%vol of crystal phase(s). One or more crystalline phases may form during the heat treatment, with a composition differing from the parent glass. In this case, the residual glass has also a different composition from the parent glass.
Figure 1.14 – Crystallization treatment to form a glass-ceramic (a) Temperature dependence of nucleation and growth rate, metastable zone of undercooling and metastable zone of high viscosity (b) heat treatment stages: (1) glass melting, (2) glass shaping, (3) cooling, (4) nucleation step, (5) growth step
Controlled crystallization can be reached by homogeneous bulk nucleation (few compositions reported [34]), heterogeneous bulk nucleation with powerful nucleating agents or by surface nu- cleation when a specific orientation is required. Figure 1.15 illustrates the typical crystallization mechanisms that are used for glass-ceramics.
In most cases, glass-ceramics are processed by a controlled internal/bulk crystallization, where nuclei arise from their own melt composition or from "nucleating agents" or "promoting agents", that enhance the nucleation and counteract the lower energy of free surface leading to uncon- trollable surface crystallization. Those species are metallic (Au, Ag, Pt or Pd) or non metallic (TiO2, P2O5 and fluorides). Bulk crystallization leads to crystals randomly oriented and uniformly dispersed throughout the glassy residual matrix, as illustrated in Figure 1.15a.
In some cases, large surface-oriented crystallization is required for the properties (e.g. for piezo- and pyroelectric properties of fresnoite, see Chapter 3). These processes are however more difficult to control and requires a crystal phase that demonstrates a clear preferred growth direction for orientation selection. For example, the c-axis of fresnoite or leucite [25] is the preferential axis for growth and crystallization from the surface to the bulk is obtained. But, except those particular cases, devitrification initiated at lower energy surface sites is undesired as it leads to uncontrollable microstructures with crystals of various sizes. As it is shown in Figure 1.15b, the result is an ice-cube like structure, where the surface-oriented crystals meet in a plane in the bulk of the sample. The flow of the uncrystallized core glass in response to change in the bulk density during crystallization forces the original shape to undergo large distortions, leading to poor mechanical
properties. By opposition, when crystallization occurs uniformly and at high viscosities, internally nucleated glasses can undergo the transformation from glass to ceramic with little or no deviation from the original shape [25].
Figure 1.15 – Glass to glass-ceramic: (a) Nuclei formation, crystal growth and glass-ceramic microstructure (b) Surface crystallization [25]
Glass-ceramics may be either opaque or transparent. Their degree of transparency is a function of the crystal size, the birefringence, the inter-particle spacing, and of the difference in refractive index between the crystals and the residual glass. When the crystals are much smaller than the wavelength of light or when the crystals have low birefringence and the indices of refraction are closely matched, excellent transparency can be achieved [25].
To consider the advantages of glass-ceramics over their parent glasses, features of crystals must be considered. When crystals meet, structural discontinuities or grain boundaries are produced.
Unlike glasses, the presence of cleavage planes and grain boundaries serves to act as an obstacles for fracture propagation, enhancing the mechanical reliability of the crystallized glasses. In addition, crystals also demonstrate specific morphologies depending on their mode of growth. The spectrum of properties available with the presence of crystals is highly broader compared with that of glasses [25].
The large variety of compositions and the possibility to develop tailored microstructures allowed
specific technological properties to be developed [25]: transparency, excellent optical properties and thermal shock resistance, machineability resulting from mica crystallization, others like sapphire, may be harder than any glass, low or even negative thermal expansion behavior with -quartz- - spodueme crystallization or biomaterials with high-strength properties. Certain crystalline families may also have unusual luminescent, dielectric, or magnetic properties. Some are semiconducting or even superconducting at liquid nitrogen temperature. If the crystals can be oriented, polar properties like piezoelectricity or optical polarization may be induced.
More than $500 million of glass-ceramic products are sold annually worldwide [32] and they are now widespread in the daily life [25]: household applications (e.g. kitchen cooktops), indus- trial application (e.g. abrasion resistant tiles), environmental applications (use of waste, waste encapsulation [35]), biomedical applications (e.g. protheses for surgical implants), architectural applications (e.g. Pyran Crystal glass (AGC), Eurokera glass (Saint-Gobain and Corning) or Pyran Platinum (Schott)), and in more advanced technological applications (telescope mirrors, warheads and composite materials).
Chapter 2
Phase transformations in glasses
In this chapter, the main phase transformations that are encountered in glasses and glass-ceramics are covered. Crystallization is firstly detailed with the classical nucleation and growth theory, and with the distinction between homogeneous and heterogeneous crystallization. Then, amorphous phase separation is treated with thermodynamic considerations, a discussion about the origin of non ideality, the mechanisms of phase separation as well as the subsequent morphologies. Finally, a review concerning the influence of a prior phase separation on crystallization is presented.
1 Introduction
This chapter is devoted to the description of the different phase transformations that are usually observed in glasses [36]:
• Crystallization: the formation of a crystal phase which may or not have the same composition than the original liquid;
• Surface crystallization: crystals nucleate at the surface and usually grow perpendicular to it.
Surface crystallization is a form of heterogeneous crystallization;
• Volume (bulk or internal) nucleation: crystals nucleate from "nucleation sites". If the initi- ating site is the same as the bulk material, it is calledhomogeneous crystallization. If it is a foreign substance (commonly, a nucleating agent), it is calledheterogeneous nucleation;
• Amorphous phase separation: the formation of non-crystalline phases which have a different composition from the original phase. As indicated by the dotted arrow, crystallization may occur after or during phase separation;
• Spinodal decomposition: within a region which separates into two liquids there will be a region where there is no energy barrier to nucleation and phase separation is only limited by diffusion.
They are illustrated in Figure 2.1
Figure 2.1 – Phase transformations in glasses [36]
Those points are detailed in the following sections.
2 Crystallization
Crystallization is the equilibrium stage of a solidifying melt and is characterized by nucleation and growth processes. When a liquid is cooled below its melting point at sufficiently slow rates, crystallization occurs by the growth of crystals from a finite number of nuclei [37].
If not controlled, crystallization is commonly not desired in glasses and constitutes the most important cause of defect in glass manufacturing. The industrial compositions were optimized through history to avoid crystallization. The control of the kinetics of crystal nucleation and growth is consequently of crucial importance in determining the glass forming abilities of melts. As detailed previously, the glass formation ability may be attributed to a low rate of crystal growth, a low rate of nuclei formation or a combination of both [37].
By opposition, if nucleation and growth of crystal phases are controlled by choosing specific glass compositions and selected thermal treatments, it leads to materials with improved and tailored properties that depend on the crystal phase formed: the glass-ceramics.
Devitrification of glasses is often used for "uncontrolled crystallization" while crystallization describes a well controlled crystallization process. Avoiding or controlling crystallization requires to master theoretically the phenomena of nucleation and growth of crystals [37]. Nucleation and
growth theories are detailed below.
The following sections detail the classical nucleation theory for nucleation and distinguish the homogeneous versus heterogeneous crystallization in glasses.
2.1 Nucleation: Classical Theory for Nucleation (CTN)
2.1.1 Homogeneous nucleation [37, 36]
Homogeneous nucleation supposes the same probability for critical nuclei formation in any given volume of the system [38]. Each element in the initial phase must be structurally, chemically and energetically identical everywhere. This case is possible if the entire volume does not present any defect.
The thermodynamic treatment of nucleation starts with the requirement that a process of phase change must be associated with a decrease in Gibbs free energy¢G of the system. The stability of a particle of the new phase will depend on three contributions: the volume contribution¢Gv, the surface contribution¢Gs and the elastic deformation energy due to a volume change during crystallization (neglected in this work due to thermal treatments conducted above Tg increasing viscosity and stress relaxation during crystallization):
• ¢Gv characterizes the difference in free energy between the liquid phase and the solid phase;
• ¢Gs characterizes the interfacial energy due to the formation of a new liquid/solid interface;
¢G= ¢Gv+¢Gs (2.1)
At the melting temperature (Tm), the free energy is the same and no crystallization occurs.
The process requires an undercooling¢T = Tm- T. For T<Tm, the volume contribution is negative due to the ordering of an amorphous region into a crystalline lattice. It corresponds to an energy which is gained.
¢Gv= Vm¢gv (2.2)
where Vm is the volume of the crystal phase and ¢gv the change of free energy during the phase transformation from the supercooled liquid to the crystal phase per unit volume.
By opposition, the surface contribution is positive because it corresponds to an energy required to form a new surface, overcoming the surface tension of the environment.
¢Gs=A (2.3) where A is the surface of the liquid-solid interface and is the surface tension, i.e, the enthalpy variation due to the interface formation at constant temperature and pressure.
Assuming a spherical shape with radius r, the change in Gibbs free energy (¢G) is:
¢G= 4
3⇡r3¢gv+4⇡r2 (2.4)
Figure 2.2 shows the free energy change with the volume and surface contributions as a function of the radius (r) for a temperature (T) below the melting temperature (Tm). For small values of r, the surface contribution dominates, but as r increases until a critical value (ddr¢G=0;r⇤= ¢2gv), the volume term will dominate and¢G becomes negative. Particles smaller thanr⇤are called embryos and are unstable, while larger particles are called nuclei and are stable since their subsequent growth is associated with a decrease in free energy. The free energy associated with a nuclei with a radius r⇤is the thermodynamic free energy barrier for nucleation: W⇤= 16⇡3¢g23
v .
The critical radius (r⇤) and the thermodynamic barrier for nucleation (W⇤) can be linked with the degree of supercooling. At T=Tm, ¢gm=¢hm-Tm¢sm=0, so that ¢hm=Tm¢sm, where ¢gm,
¢hmand¢smare the change in Gibbs energy, enthalpy and entropy per unit of volume, respectively.
Supposing a small undercooling level,¢gm=¢hm T¢Thmm=¢hm¢T
Tm. It can be deduced that W⇤ /
1
(¢T)2 and r⇤/ (¢1T) so that W⇤ and r⇤ decrease with the level of undercooling of the melt. This is illustrated in Figure 2.3.
Figure 2.2 – Free energy (¢G) of a crystalline nucleus versus its radius (r) for T<Tm, volume (¢Gv) and surface (¢Gs) contributions
Figure 2.3 – Change of the critical nuclear ra- dius with the undercooling level of the melt and the effect on the free energy (¢G) [39]
At temperatures where mobility is possible, atomic rearrangement happens continuously with thermal agitation. To evaluate the nucleation rate, a Boltzmann’s statistics is assumed: there is a distribution of embryo and critical nuclei formed by random molecular fluctuations. The probability per unit of volume that a fluctuation forms a critical nuclei is proportional to nekbTW⇤ where kB is Boltzmann’s constant, n is the total number of atoms per unit volume, i.e. n=VNm (N is the total number of atoms).
The steady-state nucleation rate is defined as the increase in the number of nuclei with time per unit volume (dndt), which is governed by diffusion. Considering the enthalpy of activation for the diffusion from a homogeneous solution to the location of nucleation (¢GD), the nucleation rate (I) can be defined as:
I= dn dt=nf e
W⇤+¢GD
kbT =Ae
W⇤+¢GD
kbT (2.5)
where f is the "jump" frequency, a statistical parameter grouping atoms vibration frequency, diffusion or interface energy. W⇤ and ¢GD are the thermodynamic and the kinetic barriers to nucleation, respectively.
Because W⇤ is proportional to (T 1
m T)2, the nucleation rate will be very sensitive to the un- dercooling level. The curve is presented in Figure 2.4. The term e
W⇤
kb·T is zero at the melting temperature, then increases with¢T until a maximum is reached. At large undercooling levels, the terme
¢GD
kb·T dominates and the nucleation rate decreases as the atomic mobility decreases.
Figure 2.4– Nucleation rate (I) as a function of temperature
The transport process at the interface is often associated with a diffusion coefficient so that [36]
I=A0Dne
W⇤
kbT (2.6)
TheDn factor is evaluated depending if the transformation isnon-reconstructiveorreconstruc- tive [36]. When the crystallization happens without change in composition (non-reconstructive), all the structural unit are already formed so that only short range diffusion is considered. Dn is evaluated by the self-diffusion coefficient of the supercooled liquid: Dn=3⇡akT0⌘ [36]:
I= A0
⌘e
W⇤
kbT (2.7)
where A’ is a constant, a0 the diffusion distance and⌘ the viscosity.
In multi-component systems, crystallization involves bond breaking (reconstructive transforma- tion) and long-distance diffusion process are considered. The activation energy is assimilated to the slowest species.
2.1.2 Heterogeneous nucleation
The term heterogeneous refers to the presence of a foreign substrate as walls or dispersed solids that facilitates the nucleation step. This effect is mainly due to the reduced thermodynamic barrier (W⇤) compared to homogeneous nucleation. This involves a decrease of the contribution of the effective surface energy to the work of critical cluster formation [38]. The thermodynamic barrier is given by [39]:
¢Ghet=¢Ghom£ (2.8)
f(✓)=£= (2+cos✓)(1 cos✓)2
4 (2.9)
Where ✓ is the wetting angle. Depending on its value, the geometrical factor £ varies from 0 to unity so that ¢Ghet is always smaller than ¢Ghom. This is illustrated in Figure 2.5 with the thermodynamic barrier versus the critical radiusr⇤ and versus¢T. Heterogeneous nucleation always occurs before homogeneous nucleation.
Figure 2.5– (a) Free energy versus radius of nuclei. The thermodynamic barrier for homogeneous nucleation is greater than for heterogeneous nucleation for the same critical radius - (b) The required undercooling for observable nuclei is smaller for heterogeneous nucleation [39] (note this case is valid without kinetic limitation)
Figure 2.6a shows a nuclei of radius r which has formed on a solid planar interface for different wetting angles✓(✓<90°,90°<✓<180°and✓=180°). Heterogeneous nucleation occurs if interatomic forces between the liquid and the substrate are larger than those within the liquid phase (✓<180°).
If✓=180°, f(✓)=1 and no heterogeneous nucleation occurs.
Figure 2.6– (a) Formation of a solid cluster on a solid substrate with different wetting angles [40]
– (b) Geometrical factor£versus contact angle✓ [39]
The steady-state nucleation rate can be adapted in the case of a heterogeneous nucleation by modifying the structural number for the calculations for the critical radius, the critical free energy and the nucleation rate. This number takes into account the new geometry of the catalysing surface. The critical radius r⇤ is the same for both homogeneous and heterogeneous nucleation (Figure 2.5a), ¢Ghet is smaller and consequently the required undercooling level for the same nucleation rate is far smaller. This is illustrated in Figure (2.5b).
Epitaxy is a controlled heterogeneous nucleation. A suitable substrate, having similar atomic spacing and arrangement than the crystal and thoroughly wetted by the liquid, is introduced, reducing the energy barrier to nucleation. The most known example is platinum (cubic) that helps nucleate lithium disilicate (orthorhombic): the (111) plane of platinum matches the (002) plane of lithium disilicate. The matching must be lower than 15 % [36]. Another example is the epitaxial growth of lithium metasilicate on Li3PO4 as illustrated in Figure 2.7.
Figure 2.7– Epitaxial growth of Li2SiO3 on Li3PO4 [25]
Two classes of nucleating agents commonly used for epitaxy can be distinguished [36]:
• Pt, Au, Ag, Cu: form crystal with low solubility in glasses;
• TiO2, ZrO2, P2O5: largely soluble in glasses with a quantity up to 20 % that can be added before nucleation occurs. They mostly precipitate as complex compound.
2.2 Growth [36, 39]
The stable nuclei will grow at a rate depending on the diffusion towards the crystal and how they are crossing the interface.
From Turnbull [11], the crystal-liquid interface can be seen as a double potential well shifted by a jump distancea0(see Figure 2.8)
Figure 2.8– Growth mechanism controlled by the interface [11] - T< Tm
If the growth of a crystal is considered at T=Tm, all molecules in the liquid arriving at the crystal-liquid interface must break their bond with the surrounding liquid molecules. This requires to overcome a barrier of potential¢G00. Due to thermal agitation, molecules are vibrating, bumping together and transferring energy. This gives to each molecule a probability to acquire an energy greater than¢G00equal to:
P(E>¢G00)=e ¢
G00
kb·T (2.10)
The net number of molecules crossing the interface per unit of surface (n), per second is given by:
dn
dt(l$c)= ⌫
6(nl nc)e ¢
G00
kb·T (2.11)
wherenlis the number of molecules per unit surface in the liquid phase,ns number of molecules per unit surface in the solid phase, and ⌫6 is the number of vibration in the right direction for the agglomeration, per second. At T=Tmand considering nl=ns, dndt=0.
Considering now the growth at a certain degree of undercooling T<Tm, once the molecule has passed though the interface to the crystal, its energy is reduced relative from that in the liquid
¢Gv=vvm¢G, where v is the volume occupied by an atom.
According to the definition of¢G00and¢Gvin Figure 2.8, the net transfer of molecules is equal to:
dn
dt|net= dn
dt(l!c)- dn dt(c!l)
= ⌫ 6nle
¢G00 kbT - ⌫
6nse
¢G00+ v vm¢G kbT
At small undercooling,nl⇡ns =n. Replacing kbv.vm by R1: dn
dt(l$c)= ⌫ 6ne ¢
G00
kbT (1 eR T¢G) (2.12)
The growth rate of the crystal (U) can be written as:
U= ⌫ 6ne ¢
G00
kbT (1 eR·T¢G) (2.13)
Considering the limiting cases:
• At T=Tm, U=0 (dynamic equilibrium);
• At small undercooling, (1 eRT¢G)⇡RT¢G so thatU= ⌫6n¢RTGe ¢G
00
kbT . U is directly proportional to the undercooling;
• At large undercooling,eRT¢G«1so thatU= ⌫6ne ¢
G00 kbT
As for the nucleation rate, there is a maximum. At the melting temperature,¢T=0. With the temperature decrease, growth rate increases with the undercooling until reaching a maximum. At large undercooling levels, the mobility, represented by the frequency parameter ⌫, decreases and the growth rate decreases, too. The maximum occurs at lower undercooling than the nucleation rate (see Figure 2.9).
Figure 2.9 – Nucleation and growth rates for crystallization as a function of undercooling - the maximum growth rate occurs at higher temperature than the maximum nucleation rate. The overlapping depends on the system considered
3 Volume nucleation versus surface nucleation in silicate glasses
Nucleation is one of the two major steps that govern the crystallization kinetics and glass forming ability of undercooled liquids [41]. In the absence of nucleating agents, only a limited number of sili- cate and borate glasses (i.e. Li2O.2B2O3, Li2O.2SiO2, CaO.SiO2, BaO.2SiO2, Na2O.2CaO.3SiO2, 2Na2O.CaO.3SiO2 and few others ) show measurable internal homogeneous nucleation rates in a laboratory time scale [34]. Most supercooled liquids crystallize heterogeneously from the external surface when heated. Surface crystallization is most often seen as a defect because it is difficult to control and nucleating agents must be introduced to obtain an internal heterogeneous nucleation, more favourable than heterogeneous surface crystallization. Surface crystallization may strongly depend on the surface quality (tips, cracks and scratches), stresses, foreign particles and surround- ing atmosphere [42]. Figure 2.10 illustrates the effect of surface state on the crystallization of a BaO.2SiO2 glass.
Figure 2.10– Number of BaO.2SiO2 nuclei on a crucible wall - scratched (s), normal (n), flame- polished surface (f) and volume crystallization (v) [43]
The ability or inability to bulk crystallize is attributed to the low nucleation and growth rates of the stoichiometric crystal, expressed by Tg, r, a local structure similarity (or similar density) or a intermediate range order. A summary of researches concerning the ability to bulk crystallize is presented in Appendix A for the interested reader. A critical review was recently made by Zanotto:
"While some observations indicate the existence of a correlation between structure similarity/dis- similarity at both short and intermediation length scales and the nucleation mechanism, this issue has not been sufficiently generalized, numerous exceptions are still not explained and further work is needed for a definite conclusion" [34].
4 Amorphous Phase Separation (APS)
A wide variety of binary or multi-component glass and melt systems exhibit amorphous phase separation (also called liquid-liquid immiscibility) and tend to phase separate into two or more liquid phases over a well defined range of compositions and temperatures called the immiscibility boundary [44, 45].Phase separation is a clear manifestation of the thermodynamic state of the glass [46]. It takes place because a system can reduce its free energy by separating into two or more phases and the phenomenon is associated with an excess free energy [46]. The exact origin of this excess free energy as well as the mechanisms that control the position and the size of the immiscibility fields has been the subjects of extensive research since the discovery of phase separation in binary borate and silicate systems by Guertler (1904) and Greig (1927) [46, 47].
Phase separation has a great technological importance. The coexistence of different phases will affect not only the optical properties of the material, but also the electrical properties and the chemical durability [46].
In the case of astable miscibility gap, phase separation occurs above the liquidus. In the case ofmetastable immiscibility, phase separation occurs below the liquidus were solid phases are more stable. The consolute temperature or critical temperature (Tc), which is the maximum temperature where phase separation occurs, is then situated below the liquidus. Figure 2.11 shows schematic binary diagrams with (a) both stable and metastable liquid-liquid immiscibility and (b) entirely metastable immiscibility. Note the S-shaped curve of the liquidus in the case of a metastable immiscibility.
Figure 2.11– Schematic binary diagrams with (a) both stable and metastable liquid-liquid immis- cibility and (b) entirely metastable immiscibility [44]
Much of the early work was concerned with stable immiscibility in which liquid separation occurs above the liquidus. Many silicate systems as CaO-SiO2, MgO-SiO2 or TiO2-SiO2 show stable immiscibility in their phase diagram. Glasses formed by cooling have either a layered structure or a strongly opalescent appearance [44] due to light scattering caused by their heterogeneous microstructures.Those regions can therefore limit the useful region of transparent glass formation [48].
Greig [49, 50] (1927) conducted a study on binary alkaline-earth-silica system and indicated for the first time a S-shaped liquidus in the BaO-SiO2 system that should probably be due to a nonmixing zone beneath the liquid phase line. More attention has been given later to metastable (or subliquidus) immiscibility, in which phase separation occurs below the liquidus [44]. In those cases, because viscosity is high and diffusion rate very slow, it is often possible to cool the glass-forming liquid through the region of metastable immiscibility without visually detectable opalescence. A very fine-scale microphase separation can thus be "frozen in" comprising droplets of one glass
distributed within another which may only be observed by electron microscopy. The required use of electron microscopy explains why metastable phase separation, which is nevertheless found in numerous silicate and borate systems, was considered later. The period from the late 50s to the early 70s was a period of intensive study of the morphological aspect of the metastable phase separation using electron microscopy and small angle X-Ray scattering. By opposition to stable phase separation, metastable phase separation is not seen as a limitation in glass forming system:
it has been stimulated by the potential effect on further crystallization and is used extensively in the Na2O-B2O3-SiO2 system to manufacture the "Vycor" type of glasses. In this process, one of the separated phases is leached out by acid treatment leaving a porous silica skeleton which may be compacted by sintering to produce a high silica glass with high thermal shock resistance. Thermal treatments of Vycor-type of glasses have a very strong influence on the phase separation process and hence the pore size distribution and morphology of the leached glass network [51].
4.1 Thermodynamic considerations
The thermodynamic quantity which governs the phase separation tendency is the free energy of mixing¢Gmwhich is defined as the free energy of the homogeneous mixture minus the free energy of the heterogeneous mixture with the same overall composition [46]. A homogeneous single phase will separate into two or more phases of different compositions if the free energy of the system with two or more distinct phases is lower than that of the system with one single homogeneous phase [36, 51].
The Gibbs free energy of mixing¢Gmat a temperature T is defined by [36]:
¢Gm=¢Hm T¢Sm (2.14)
Where ¢Hm and¢Sm are the enthalpy and entropy of mixing, respectively, both temperature and composition dependant of the mixture (see Chapter 4 - regular solution model).
Thermodynamically, phase separation occurs with a positive free enthalpy of mixing ¢Gm of the components. Figure 2.12a shows a schematic phase diagram with a stable immiscibility region (cross-hached) with a sub-liquidus extension (hached) (left) and the related free energy curves of the liquid (L) and the solids (S) at T1, T2 and T3. At T1, a mixing always involves an increase in free energy, so that a single phase is more stable over the whole range of compositions. At T2, ¢Hm becomes more positive or ¢Sm more negative, depending if the immiscibility is caused by the relation of bonding (enthalpy) or a relation of ordering (entropy). For glasses, the relation of bonding is observed in almost all systems [37], due to the importance of the relation between the network-formers and the network-modifiers. If¢Hm is large and positive, the curve will have a deflection as illustrated at T2 ("cow-boy hat" shaped). In this case, the free energy of the
system will be lowered by a separation in two liquids, over a certain range of compositions. The concentrations of the two phases are determined by the lever rule and delimit the boundary on the phase diagram. On cooling, the G curve of the liquid rises faster because @G@T= S. At T3, the G curve is still "cow-boy hat" shaped, but the solid is more stable, so that phase separation becomes metastable: the unmixing is observed only if crystallization is avoided (for example, with rapid cooling). Figure 2.12b shows the situation for an entirely sub-liquidus immiscibility, i.e. the solid phase is always more stable when a unmixing occurs.
Figure 2.12– Schematic phase diagrams – (a) Stable immiscibility– (b) Sub-liquidus immisicbility
The shape of the Gibbs free energy curve changes with temperature because as temperature rises, T¢S plays a larger role, and the magnitude of the deflection decrease. When the tem- perature is sufficiently high, the deflection disappears and the critical or consolute temperature is reached [36].
The origin of the non ideality leading to immiscibility and the tendency of a glass system towards phase separation has been a subject of debate. The structural incompatibilities were attempted to be explained with the ionic potential, the ionic field strength, the ionic radius or the coulombic repulsion. All the study that were based on the ionic nature of bond predicted correctly the tendency towards phase separation in most of cases, but exceptions were remaining. However, the introduction of the covalente nature of bond allowed a better prediction. Based on Hudon’s review [47] in 2002, a summary of the most important theories up to now is proposed in Appendix B.
4.2 Phase separation mechanisms and morphology
Phase separation occurs by two distinct mechanisms: by nucleation and growth or by spinodal decomposition. Figure 2.13a illustrates a partially convex Gibbs energy of mixing (high positive
¢Hm) in a binary system with two minima and two inflexion points, corresponding to the case where a phase separation is likely to occur over a certain composition range, i.e. between xb’ and xb”. Figure 2.13b illustrates the related phase diagram: the plain curve represents the immiscibility area corresponding to the contact points of the common tangent. In this area, a separation into two phases, i.e. an A-rich phase and a B-rich phase, is always favored. The dotted curve correspond to the inflexion points and delimits the "binodal" area and the "spinodal area". In the binodal area, a composition change due to fluctuation involves an increase in the Gibbs energy of the system. In the spinodal region, the compositional fluctuation makes the Gibbs energy to decrease, amplifying the change. Both cases are illustrated in Figures 2.13c and d, respectively.
Figure 2.13 – (a) Gibbs energy variation in a binary system with a partial miscibility against composition. A variation of xbleads to a decrease of the Gibbs energy [52] - (b) Associated phase diagram delimiting the region of unmixing as well as the binodal and spinodal regions, T<Tc where Tc is the critical temperature for phase separation [36] - (c) Decrease in Gibbs energy involving an amplification of the fluctuation leading to phase separation (grey region=spinodal region) - (d) Increase in Gibbs energy, absorbing the fluctuations (binodal region) [36]
An homogeneous system with an averaged composition x0undergoes local composition fluctu- ations due to thermal agitation. The amplitude of fluctuation depends on the second derivative of the free energy regarding concentration [36, 52, 53]. The free energy per unit volume g(c) around
