9. Formation, Structure, and Properties of Glasses

- 10주차 -

▪ Crystal Growth▪ Once the nuclei are formed, they will tend to grow until they start to impinge

upon each other.

The growth of the crystals depends on the nature of the growing interface which has been related to the entropy of fusion.

It can be shown that for crystallization processes in which the entropy change is small, that is, Δ𝑆𝑆𝑓𝑓 < 2𝑅𝑅, the interface will be rough and the growth rate will be more or less isotropic. In contrast, for large entropy changes 𝑆𝑆𝑓𝑓 > 4𝑅𝑅, the most closely packed faces should be smooth and the less closely packed faces should be rough, resulting in large-growth-rate anisotropies.

Based on these notions, various models of crystal growth have been developed.

Standard growth, 𝜟𝜟𝑺𝑺𝒇𝒇 < 𝟐𝟐𝟐𝟐▪ In this model, the interface is assumed to be rough on the atomic scale, and a sizable fraction of the interfaces are available for growth to take place.

Under these circumstances, the rate of growth is solely determined by the rate of atoms jumping across the interface.

Using an analysis that is almost identical to the one carried out in Sec. 7.2.3, where the net rate of atom movement down a chemical potential gradient was shown to be

𝜈𝜈𝑛𝑛𝑛𝑛𝑛𝑛 = ν0exp(−∆𝐺𝐺𝑚𝑚∗

𝑘𝑘𝑘𝑘 ) 1 − exp(−Ξ𝑘𝑘𝑘𝑘)

where Ξ was defined as the difference between the energy barrier in the forward and backward direction, it is possible to derive an expression for the growth rate as follows.

Comparing Figs. 7.5 and 9.3, the equivalence Ξ of and Δ𝐺𝐺𝑣𝑣 is obvious. Hence, the growth rate 𝑢𝑢 of the interface is given by

where Δ𝐺𝐺𝑣𝑣 is given by Eq. (1).

It is left as an exercise to the reader to show that in terms of viscosity, this equation can be rewritten as

𝑢𝑢 ≈ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘

3𝜋𝜋𝜋𝜋𝜆𝜆21 − exp −Δ𝐻𝐻𝑓𝑓

Δ𝑘𝑘𝑘𝑘𝑚𝑚𝑘𝑘𝑅𝑅

when the growth is occurring at temperature 𝑘𝑘, with an undercooling Δ𝑘𝑘.

𝑢𝑢 = 𝜆𝜆𝜈𝜈𝑛𝑛𝑛𝑛𝑛𝑛 = 𝜆𝜆ν0 exp −∆𝐺𝐺𝑚𝑚∗

𝑘𝑘𝑘𝑘 1 − exp −Δ𝐺𝐺𝑣𝑣𝑅𝑅𝑘𝑘

This is an important result because it predicts that the growth rate, like nucleation rate, should also go through a maximum as a function of undercooling.

The reason, once more, is that with increasing undercooling, the driving force for growth Δ𝐺𝐺𝑣𝑣 increases, while the atomic mobility, expressed by 𝜋𝜋, decreases exponentially with decreasing temperature.

It is important to note that the temperature at which the maximum growth rate occurs is usually different from that at which the nucleation rate peaks.

For small Δ𝑘𝑘 values, a linear relation exists between the growth rate and undercooling (∵𝑒𝑒−𝑥𝑥 ≅ 1 − 𝑥𝑥 for small 𝑥𝑥). Conversely, for large undercooling the limiting growth rate

𝑢𝑢 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘

3𝜋𝜋𝜋𝜋𝜆𝜆2= (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)

𝐷𝐷𝑙𝑙𝑙𝑙𝑙𝑙𝜆𝜆

is predicted. Once a stable nucleus has formed, it will grow until it encounters other crystals or until the molecular mobility is sufficiently reduced that further growth is cut off.

Surface nucleation growth, 𝜟𝜟𝑺𝑺𝒇𝒇 > 𝟒𝟒𝟐𝟐▪ In the normal growth model, the assumption is all the atoms that arrive at the

growing interface can be incorporated in the growing crystal. This will occur only when the interface is rough on an atomic scale.

If, however, the interface is smooth, growth will take place only at preferred sites such as ledges or steps. In other words, growth will occur by the spreading of a monolayer across the surface.

H.W. Worked Example 9.2

▪ Kinetics of Glass Formation

▪ At this point, the fundamental question, posed at the outset of this section, namely, How fast must a melt be cooled to avoid the formation of a detectable volume fraction of the crystallized phase? can be addressed somewhat more quantitatively.

The first step entails the construction of a time-temperature-transformation (TTT) curve for a given system. Such a curve defines the time required, at any temperature, for a given volume fraction to crystallize.

If at any time 𝑐𝑐, in a total volume 𝑉𝑉, the nucleation rate is 𝐼𝐼𝑣𝑣, it follows that the number 𝑁𝑁𝑛𝑛 of new particles formed in time interval 𝑑𝑑𝜏𝜏 is

𝑁𝑁𝑛𝑛 = 𝐼𝐼𝑣𝑣𝑉𝑉𝑑𝑑𝜏𝜏

For a time-independent constant growth rate 𝑢𝑢 and assuming isotropic growth (i.e., spheres), the radius of the sphere after time 𝑐𝑐 will be

𝑟𝑟 = �𝑢𝑢 𝑐𝑐 − 𝜏𝜏 𝑓𝑓𝑐𝑐𝑟𝑟 𝑐𝑐 > 𝜏𝜏0 𝑓𝑓𝑐𝑐𝑟𝑟 𝑐𝑐 < 𝜏𝜏

and its volume will be

𝑉𝑉𝜏𝜏 =43𝜋𝜋𝑢𝑢3(𝑐𝑐 − 𝜏𝜏)3

where 𝜏𝜏 is the time at which a given nucleus appears.

Hence the total volume transformed after time 𝑐𝑐, denoted by 𝑉𝑉𝑐𝑐, is given by the number of nuclei at time t multiplied by their volume at that time, or

𝑉𝑉𝑛𝑛 = 𝑉𝑉𝜏𝜏𝑁𝑁𝑛𝑛 = �𝑉𝑉𝜏𝜏𝐼𝐼𝑣𝑣𝑉𝑉𝑑𝑑𝜏𝜏 = �𝜏𝜏=0

𝜏𝜏=𝑛𝑛𝑉𝑉𝐼𝐼𝑣𝑣(

43𝜋𝜋𝑢𝑢3)(𝑐𝑐 − 𝜏𝜏)3𝑑𝑑𝜏𝜏

Upon integration and rearranging, this gives

𝑉𝑉𝑛𝑛𝑉𝑉

=𝜋𝜋3𝐼𝐼𝑣𝑣𝑢𝑢3𝑐𝑐4

An implicit assumption made in deriving this expression is that the transformed regions do not interfere or impinge on one another. In other words, this expression is valid only for the initial stages of the transformation.

(12)

A more exact and general analysis which takes impingement into account, but which will not be derived here, yields

𝑉𝑉𝑛𝑛𝑉𝑉

= 1 − exp −𝜋𝜋3𝐼𝐼𝑣𝑣𝑢𝑢3𝑐𝑐4

This is known as the Johnson-Mehl-Avrami equation. This equation reduces to Eq. (12) at small values of time.

Given the nucleation and growth rates at any given temperature, the fraction crystallized can be calculated as a function of time from Eq. (13).Repeating the process for other temperatures and joining the loci of points having same volume fraction transformed yield the familiar TTT diagram, shown schematically in Fig. 9.5.

(13)

Once constructed, an estimate of the critical cooling rate (CCR) is given by

𝐶𝐶𝐶𝐶𝑅𝑅 ≈𝑘𝑘𝐿𝐿 − 𝑘𝑘𝑛𝑛𝑐𝑐𝑛𝑛

where 𝑘𝑘𝐿𝐿 is the temperature of the melt and 𝑘𝑘𝑛𝑛 and 𝑐𝑐𝑛𝑛 are the temperature and time corresponding to the nose of the TTT curve, respectively.

The critical cooling rates in degrees Celsius per second for a number of silicate glasses are shown in Fig. 9.6, where the salient feature is the strong functionality of the CCR on glass composition (note the log scale on y axis).

▪ Criteria for Glass Formation

▪ The question of glass formation can now be restated: why do some liquids form glasses while other do not? Based on the foregoing discussion, for a glass to form, the following conditions must exist:

1. A low nucleation rate. This can be accomplished by having either a small ∆𝑆𝑆𝑓𝑓or a large crystal/liquid interfacial energy. The lower ∆𝑆𝑆𝑓𝑓 and/or the higher 𝛾𝛾𝑠𝑠𝑙𝑙,the higher ∆𝐺𝐺𝑐𝑐 and consequently the more difficult the nucleation.

2. High viscosity 𝜋𝜋𝑚𝑚 at or near the melting point. This ensures that the growth rate will be small.

3. The absence of nucleating heterogeneities that can act as nucleating agents, the presence of which can reduce the size of the critical nucleus and greatly enhance the nucleation kinetics.

Based on items 1 and 2 before, a useful criterion for the formation of a glass is the ratio

Δ𝑆𝑆𝑓𝑓1𝜋𝜋𝑚𝑚

The smaller the product, the more likely a melt will form a glass, and vice versa.Table 9.1 shows that to be the case, indeed. Further inspection reveals that atom mobility as reflected in 𝜋𝜋𝑚𝑚 at the melting point is by far the dominant factor. It follows that a melt must have a high viscosity at its liquidustemperature or melting point if it is to form a glass.

9.3 Glass Structure▪ In principles, if the requisite data were available, the TTT diagram for any

material could be generated, and the CCR that would be required to keep it from crystallizing could be calculated. In other words, if cooled rapidly enough, any liquid will form a glass, and indeed glasses have been formed from ionic, organic, and metallic melts.

What is of interest here, however, is the so-called inorganic glasses formed from covalently bonded, and for the most part silicate-based, oxide melts.

These glass-forming oxides are characterized by having a continuous three-dimensional network of linked polyhedra and are known as network formers.

They include silica (SiO2), boron oxide (B2O3), phosphorous pentoxide(P2O5), and germania (GeO2).

Commercially, silicate-based glasses are by far the most important and the most studied and consequently are the only ones discussed here.

Since glasses possess only short-range order, they cannot be as elegantly and succinctly described as crystalline solids − e.g., there are no unit cells.

The best way to describe a glass is to describe the building block that possess the short-range order (i.e., the coordination number of each atom) and then how these blocks are put together.

The simplest of the silicates is vitreous silica (SiO2), and understanding its structure is fundamental to understanding the structure of other silicates.

Vitreous silica SiO2

▪ The basic building block for all crystalline silicates is the SiO2 tetrahedron.

In the case of quartz, every silica tetrahedron is attached to four other tetrahedra, and a three-dimensional periodic network results (see top of

Table 3.4).

Increase in the stability of the structure

▪ Silica

SiO2 (∵ O/Si ratio = 2)

each oxygen is linked to two Si (∴ no NBOs) and each Si is linked to four O, resulting in a three-dimensional network.

all allotropes of silicas which, depending on the exact arrangement of the tetrahedra, include quartz, tridymite, and cristobalite.

if long range order is lacking, the resulting solid is labeled amorphous silica or fused quartz.

The structure of vitreous silica is very similar to that of quartz, except that the network lacks symmetry or long-range periodicity.

This so-called random network model, first proposed by Zachariasen, is generally accepted as the best description of the structure of vitreous or fused silica and is shown schematically in two-dimensions in Fig. 1.1b.

Quantitatively it has been shown that the Si-O-Si bond angle in vitreous silica while centered on 144°, which is the angle for quartz, has a distribution of roughly ±10 percent. In other words, most of the Si-O-Si bond angles fall between 130° and 160°, which implies the structure of fused silica is quite uniform at a short range order, but that the order does not persist beyond several layers of tetrahedra.

In Sec. 3.6, the formation of nonbridging oxygens upon the addition of alkali or alkaline earth oxides to silicates melts was discussed in some detail.

Because, as discussed shortly, these oxides usually strongly modify the properties of a glass, they are referred to as network modifiers. The resulting structure is not unlike that of pure silica, except that now the continuous three-dimensional network is broken up due to the presence of nonbridging oxygens, as shown in Fig. 9.7.

Table 9.2 lists typical compositions of some of the more common commercial glasses and their softening points. Most of these glasses are predominantly composed of oxygen and silicon.

Multicomponent silicates

9.4 Glass Properties

▪ The noncrystalline nature of glasses endows them with certain characteristics unique to them as compared to their crystalline counterparts.

Once formed, the changes that occur in a glass upon further cooling are quite subtle and different from those that occur during other phase transitions such as solidification or crystallization.

The change is not from disorder to order, but rather from disorder to disorder with less empty space.

In this section, the implication of this statement on glass properties is discussed.

▪ The Glass Transition Temperature

▪ The temperature dependences of several properties of crystalline solids and glasses are compared schematically in Fig. 9.8.

Typical crystalline solids will normally crystallize at their melting point, with an abrupt and significant decrease in the specific volume and configuration entropy (Fig. 9.8a and b).

The changes in these properties for glasses, however, are more gradual, and there are no abrupt changes at the melting point, but rather the properties follow the liquid line up to a temperature where the slope of the specific volume or entropy versus temperature curve is markedly decreased.

The point at which the break in slope occurs is known as the glass transition temperature and denotes the temperature at which a glass-forming liquid transforms from a rubbery, soft plastic state to a rigid, brittle, glassy state.

In other words, the temperature at which a supercooled liquid becomes a glass, i.e., a rigid, amorphous body, is known as the glass transition temperature or 𝑘𝑘𝑔𝑔.

In the range between the melting and glass transition temperatures, the material is usually referred to as a supercooled liquid.

Given that (see Fig. 9.8) at the glass transition temperature, the specific volume 𝑉𝑉𝑐𝑐 and entropy 𝑆𝑆 are continuous, whereas the thermal expansivity 𝛼𝛼and heat capacity 𝑐𝑐𝑝𝑝 are discontinuous, at first glance it is not unreasonable to characterize the transformation occurring at 𝑘𝑘𝑔𝑔 as a second-order phase transformation.

After all, recall that, by definition, second-order phase transitions require that the properties that depend on the first derivative of the free energy 𝐺𝐺 such as

𝑉𝑉𝑠𝑠 = (𝜕𝜕𝐺𝐺𝜕𝜕𝑃𝑃

)𝑇𝑇 and 𝑆𝑆 = −(𝜕𝜕𝐺𝐺𝜕𝜕𝑃𝑃

)𝑃𝑃

be continuous at the transition temperature, but that the ones that depend on the second derivative of 𝐺𝐺, such as

𝛼𝛼 = 1𝑉𝑉

(𝜕𝜕𝑉𝑉𝜕𝜕𝑇𝑇

)𝑃𝑃= 1𝑉𝑉

( 𝜕𝜕2𝐺𝐺𝜕𝜕𝑃𝑃 𝜕𝜕𝑇𝑇

) and 𝑐𝑐𝑝𝑝 = (𝜕𝜕𝐻𝐻𝜕𝜕𝑇𝑇

)𝑝𝑝= −𝑘𝑘(𝜕𝜕2𝐺𝐺

𝜕𝜕𝑇𝑇2)𝑃𝑃

be discontinuous.

Thermodynamic considerations

In a very real sense, 𝑘𝑘𝑔𝑔 is a measure of the rigidity of the glass network;

In general, the addition of network modifiers tends to reduce 𝑘𝑘𝑔𝑔, while the addition of network formers increase it.

This observation is so universal that experimentally one of the techniques of determining whether an oxide goes into network of forms nonbridgingoxygen is follow the effect of its addition on 𝑘𝑘𝑔𝑔.

Effect of composition on 𝑻𝑻𝒈𝒈

What is occurring at 𝑘𝑘𝑔𝑔, however, is more complex, because it is experimentally well established that 𝑘𝑘𝑔𝑔 is a function of the cooling rate, as shown in Fig. 9.8a; the transition temperature 𝑘𝑘𝑔𝑔 shifts to lower temperatures with decreasing cooling rates.This implies that with more time for the atoms to rearrange, a denser glass will result and strongly suggests that 𝑘𝑘𝑔𝑔 is not a thermodynamic quantity, but rather a kinetic one.