Theory of glass transition: chemical equilibria approach

Journal of Non-Crystalline Solids 129 (1991) 259-265 North-Holland

259

Theory of glass transition: chemical equilibria approach O.V. M a z u r i n Institute of Silicate Chemistry of the Academy of Science of the USSR, ul. Odoevskogo 24/2, 199155 Leningrad, USSR

The main principles of a chemical approach to the analysis of glass-forming melt structure and its change with temperature are formulated. It is shown that such an approach leads to the concept of the existence of chemical groupings in melts. The use of the basic equations of chemical thermodynamics provides conclusions about the temperature dependences of structural changes in glass-forming melts. Taking into account the highly probable dependence of the activation energies of structural relaxation on temperature and the distribution of relaxation times values which is inevitable for any amorphous substance, one can develop an algorithm to obtain a good fit between the calculated and experimental dependences of properties. In principle, this algorithm nearly coincides with that for the phenomenological Tool-Narayanaswamy model. Thus the approach described can be considered as a theoretical foundation for this model.

1. Introduction The p h e n o m e n o n of the glass transition is p r o b a b l y the key factor for understanding the nature of the glassy state. The correct model of the glass transition is also essential for the interpretation and prediction of all complex changes of glass properties inside the glass transition region. Hence, it is not surprising that the p h e n o m e n o n of the glass transition has been studied by m a n y scientists (see, for example, reviews in refs. [1-5]). In addition to a great n u m b e r of experimental results, m a n y authors have proposed various hypotheses and theories of the glass transition phen o m e n o n . At present their n u m b e r is large and steadily increasing. Such a situation can have an adverse effect on the further development of the glass science. Sometimes it is difficult for specialists with different views on the glass transition even to understand each other. Thus it seems reasonable to try to select a few of the most dependable and the most developed theories and concentrate discussions on these particular approaches. One can suggest two main conditions that have to be satisfied by such theories. One is the high level of the universality and precision of the theory; the other is the absence of any basic ideas in the theory that

do not follow from the generally accepted knowledge on the physics and chemistry of condensed matter. In other words, the theory that is the logical consequence of a b r o a d knowledge base seems preferable to the theory that is based on some specific postulates that at present c a n n o t be either proved or disproved. There is no d o u b t that the phenomenotogical a p p r o a c h to glass transition by Tool [6] and N a r a y a n a s w a m y [7] (it m a y be called the T o o l N a r a y a n a s w a m y model) can be considered at present to be the most b r o a d and precise. The use of this theory yields a good fit between experimental and calculated data for the t i m e - t e m p e r a t u r e dependences of properties in the glass transition regions for oxide [8,9], salt [10], polymeric [11] and even metal [12] glass-forming substances. Therefore, the theory meets the first of the conditions formulated above. The present paper shows the possibility of deriving this theory on the basis of a chemical a p p r o a c h to the liquid state.

2. Chemical equilibria in glass-forming melts It is perhaps not an exaggeration to state that nearly all the basic ideas of moclern glass science have originated from the famous paper by

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260

O. V. Mazurin / Theory of glass transition

Zachariasen [13]; the idea of random distributions of atoms in liquids is among them. Perhaps this particular fact was the main reason why at the end of the 1950s most glass scientists did not pay proper attention to the spectacular findings of the two-phase structure of some fluoride glasses made by Vogel and his co-authors [14]. Only after m a n y additional experimental findings was the possibility of metastable phase separation in glass-forming liquids universally accepted. The phenomenon of phase separation in liquids shows the importance of the chemical interaction between atoms of liquids. In one-phase liquids such interaction leads inevitably to deviations from the random distributions of atoms. This is the starting point of the chemical approach to the structure of glass-forming melts and glasses [1519]. The basic idea of such an approach is as follows. In any melt with two or more components, there are practically inevitably differences between the energies of various chemical bonds. It is obvious that the higher the bond energy, the higher the probability of formation of these particular bonds in the melt. This factor is superimposed on the statistical probability of atomic distributions. As a result, some particular groups of atoms must exist in a melt in much greater concentrations than follows from the law of random distribution of atoms. To denote such groups, it is not reasonable to use the term chemical compound because it is connected with the idea of molecules which is not consistent with the notion of a polymeric glassforming network. Hence the terms complexes, coordinating complexes, or chemical groupings seem to be more appropriate. In the following, the last term will be used. The idea of the existence of specific chemical groupings in melts leads inevitably to conclusions about the chemical equilibria involving such groupings. It is well known how difficult it is to study the exact structure of glass-forming melts. However, in some cases we can state the existence of chemical equilibria in melts with certainty. The best known example of it is the equilibrium between two coordination states of boron in alkali borate [20] and alkali-borosilicate melts. F r o m the chemical point of view, we can consider it as an

equilibrium metaborate:

between

two

forms

of

sodium

[BO,/2]Na ~ BO2/2NaO 2.

(I)

The fractional indexes used in eq. (1) mean [15] that the corresponding oxygens belong not only to the described polyhedron but to certain adjacent polyhedra as well. There is no doubt about the existence of a certain equilibrium in B203 and B203-SiO 2 melts [21]. Many scientists suggest that in these cases the equilibrium between boroxyol rings and single b o r o n - o x y g e n triangles takes place:

B30303/2 ~ 3BO3/2.

(2)

These single triangles also form rings but with a greater number of members, i.e. they are considerably less compact. There are m a n y other probable or possible examples of equilibria in glass-forming melts [22].

3. Specific features of chemical equilibria in melts with a polymeric network It is essential to answer the following question. Can we use the formulas of chemical thermodynamics to describe the temperature shifts of chemical equilibria between atom groupings incorporated in an essentially covalent network? To answer this question, one needs to discuss in some detail processes involving changes in atomic configurations inside such a network. Let us return to equilibrium (1). Figure 1 shows a portion of a b o r o n - o x y g e n network incorporat-

.~ ¢\ i Q BI o

I

~"B-

.'Q

O- B- O- B

I

0 Na b B \

/ \B

~x ~

"0

/oNa

\B \

B-O-B I B

~/ \~

Fig. 1. Schematic representation of the restructuring of a glass-forming network connected with changes in the coordination number of a boron atom.

O. II. Mazurm / Theory of glass transition

ing two different configurations of sodium metaborate which are expected to be in equilibrium. Certainly such a representation of the two structures is quite approximate. However, it gives some idea about the extreme complexity of the atomic regroupings that are needed for the transformation of one structural form to another. Presumably this particular reasoning led some scientists to the idea about the inevitability of the cooperative character of structural transformations in glass-forming melts, both inorganic and organic (see for example, ref. [23]). Let us consider these ideas in more detail. The cooperative process of the change in the mutual distribution of atoms is characterized by the shifts of several atoms. One can divide all cooperative processes into two types. One type covers processes which involve simultaneous activated j u m p s of two or more atoms. Proposing such a process, one must assume that the directions of all these j u m p s should be well coordinated to ensure the formation, after these cooperative jumps, of a new thermodynamically stable configuration. Using fig. I it is easy to see that to ensure a change of configuration A to configuration B or back at least nine atoms must change their environment significantly. It is clear that the probability of simultaneous j u m p s of nine atoms in the directions given is practically equal to zero. The term cooperativity can also be used in a quite different sense. In the case of the condensed state, for any change of its surroundings an atom or ion must not only break chemical bonds with surrounding particles but also force its way through the particles of the corresponding coordinating sphere. This is connected with the temporary elastic deformation of the corresponding parts of the substance. The more compact the substance, i.e. the less free volume it possesses, the greater the volume around the jumping atom that will be affected by such elastic deformation. One can say that during the jump, an atom has to 'squeeze out' a certain free volume from the surrounding space. The energy necessary for this squeezing out must be added to the energy needed for breaking bonds with surrounding atoms. The sum of these two energies is the so-called activation energy which determines the probability of a

261

(a)

(b)

G

................

A....

i--]"

Path of regroupings Fig. 2. Schematic representation (a) of a potential profile for a transition from one comparatively stable arrangement of atoms in a chemical grouping to another and (b) of a simplified version of the same profile.

single atom displacement. Due to the involvement of movements of a considerable number of atoms in the act of the displacement of one atom, the term cooperativity here also seems appropriate. A specific feature of this type of cooperativity is the return of all atoms surrounding the jumping one to their initial places (at least at the first approximation) after the completion of the jump. The atom grouping shown in fig. 1 is unable to transform itself by one collective j u m p from one comparatively stable configuration to another one. Therefore, one must expect that for the shift from one stable configuration to another the grouping must go through quite a few unstable configurations. One assumes that in a certain configuration space for the atomic groupings there are numerous configurations with shallow potential energy wells but only two configurations with deep potential wells: these correspond to two structural forms described by eq. (1). This is shown schematically in fig. 2(a). The probability for an i th configuration can be described by the equation [24]

Pi =

n

exp( -

GiRT )

,

(3)

~, e x p ( - G k / g T ) k=l

where G i is the free energy of the ith configuration, pi is the probability of the existence of this configuration, T is the absolute temperature, R is

262

O.V. Mazurin / Theory of glass transition

the gas constant, and n is the number of potential wells. It follows from eq. (3) that with a decrease of G i the probability of the ith configuration increases very rapidly. Although each chemical grouping explores all possible configurations during its lifetime, we can neglect the existence of all configurations except the two most stable ones and use the equations describing the chemical equilibria between these two configurations. The same reasoning may be used in connection with any possible equilibria in glass-forming melts.

4. Temperature shifts in first order chemical equilibria In the following a series of equations for describing the equilibrium A~ B

(4)

is given. A is assumed here to be the more stable, 'low-temperature' configuration.

K = x / ( 1 - x),

(5)

x = K / ( 1 + K ) = (1 + K - a ) -1,

(6)

K = exp( - A G ° / R T ) ,

(7)

x = [1 + exp( A G ° / R T ) ] - 1 ,

(8)

x = (1 + e x p [ ( A n ° - T A S ° ) / R T ] )

-t.

(9)

It is assumed that the sum of concentrations of A and B equals unity (the concentrations of B and A are equal correspondingly to x and 1 - x). K is the equilibrium constant; AG °, A H °, and AS ° are the free energy, enthalpy and entropy of chemical equilibrium (4). R is the gas constant and T is the absolute temperature. Using eq. (9) one can derive the general dependence of x on temperature which is presented in fig. 3 [19]. It follows from the figure that in the low-temperature region there is a temperature interval where the value of x is practically equal to zero. This dependence yields several conclusions about the nature of supercooled hquids and glasses [3]. Here only one result will be considered, namely the so-called Kauzmann paradox. More than 40 years ago Kauzmann [25] called attention to the following fact. If one plots the

1

2

0.4

3 0.2

1

2

3

RT/AH° Fig. 3. Temperature dependences of x for different values of S °. S °, c a l / ( m o l K): curve 1, + 1 ; curve 2, 0; curve 3, ( - 1 ) .

temperature dependence of the entropy of a glass-forming substance (see fig. 4) and extrapolates the dependence for a supercooled liquid below the glass-transition region, one finds that at temperatures not very far below the glass-transition region the extrapolated line crosses the line of temperature dependence of the entropy for the crystalline state. Below this temperature, and presumably down to absolute zero, the entropy of the supercooled liquid will be lower than the entropy of corresponding crystals. It is well known that such a conclusion contradicts the third law of thermodynamics. The problem has been discussed intensively for many years and also recently [26,27]. The approach described allows the temperature dependence of configurational entropy S k [28] to be calculated, i.e. the part of the entropy of the melt which is connected with structural changes in the melt, or, according to the ideas considered S

1

/ T Fig. 4. Schematic illustration of K a u z m a n n ' s paradox. 1, crystalline state; 2, liquid state; 3, glassy state; 4, extrapolation according to K a u z m a n n .

O.V. Mazurin / Theory of glass transition 1 1.6

vE 0.8

0.5

1.0

1.5

RT/d Ho Fig. 5. Temperature dependence of the configurational entropy Sk for the liquid state according to the chemical theory of the glass transition [28]. S °, cal/(mol K): curve 1, + 1; curve 2, 0; curve 3, ( - 1).

in the present paper, with shifts in chemical equilibrium. Figure 5 shows that in this case Kauzm a n n ' s p a r a d o x is solved in a quite natural way. Changes in the configurations of corresponding a t o m groupings must affect the values of all the properties of melts and glasses.

263

expected to be m a n y times greater than the difference in free energy AG o between two configurations of the same a t o m grouping (see fig. 2). Usually AG * are equal to several tens of k c a l / m o l and AG o have values of several k c a l / m o l . Then, according to eq. (11), the relaxation times will increase rapidly with a decrease in temperature. It is reasonable to expect that the velocity of both types of a t o m movements, namely those p r o d u c ing changes in the configuration of chemical groupings and leading to viscous flow, is limited by the same kind of potential barriers. Therefore the temperature dependence of relaxation times and viscosity coefficients must be quite similar. Experimental data in the literature support this conclusion [3]. Equations (8), (10) and (11) can be used for the qualitatively correct description of quite a few property changes inside glass transition regions.

01f 0.8

5. Kinetics of shifts of chemical equilibria To calculate the temperature dependence of the rate of shifts of chemical equilibria, one needs information about the difference between the therm o d y n a m i c characteristics of the potential wells for two equilibria configurations, A and B. One must also k n o w the height of the potential barrier between the two wells, AG* (cf. fig. 2(b)). The simplest calculations leads to the following equation [28]: dx/dt

= - ( 1 / ~ )[ x ( t ) - x ( o o ) ] ,

0.8

0

E

v

0.4

,5_ (.9

0

0.4

0.2

(10) I

I

0.5

1.0

:

where exp( AG*/RT) + = k0[1 + e x p ( A G O / R T ) ]

~"

.

(11)

Here k 0 is a constant and ~- is the relaxation time. Potential barrier heights AG* (which, as was stated above, consist of the energy needed for breaking chemical b o n d s and squeezing out the free volume needed for a successful j u m p ) are

I 1.5

RT/L~H ° RTg/L~ H o

Fig. 6. Temperature dependence of x, configurational heat capacity Cp, and configurational entropy Sk in the case of cooling of a glass-forming substance in which the temperature shift of equilibrium between two atom groupings takes place. AS° = 0; dashed lines describe the equilibrium dependences; arrows show the direction of temperature changes.

264

O. V. Mazurin / Theory of glass transition

Examples of such calculations are presented in fig. 6. However, to achieve a quantitative fit of calculations to practically all the known experimental results, one needs to introduce two additional considerations. First, it can be expected that any structural change (i.e. any shift in any chemical equilibria) must lead in principle to a change in AG*. Evidently the low-temperature configuration must be expected, in most cases, to be more compact and more dense. Accordingly, potential barriers for the jumping atoms must be higher in the case of low-temperature configurations. This leads to the conclusion that a decrease of the melt temperature results in an increase of values of AG* which must be taken into account when using eqs. (10) and (11). Second, it is necessary to keep in mind one of the main distinctions of any amorphous substance, namely the absence of regularity of its structure. Thus all characteristics of an amorphous substance connected directly or indirectly with features of its microstructure reveal a distribution of values around the most probable one. A m o n g other characteristics of glass-forming substances, such distributions must also be consistent with relaxation times.

6. Conclusions

F r o m the above sections, using a chemical approach to the melt structure, we came to the same algorithm used successfully by m a n y scientists during the past twenty years and based on the model by T o o l - N a r a y a n a s w a m y [3,7,8]. There are two main conclusions resulting from the present work. First, we can state that the T o o l N a r a y a n a s w a m y model can no longer be considered as phenomenological, but rather as derived from sound theoretical foundations. Second, if we accept the present knowledge of the main features of the structure and behaviour of liquids, we come to the inevitable conclusion about the kinetic, relaxation nature of the glass-transition phenomenon. The coincidence of the equations describing this phenomenon with those used in the T o o l - N a rayanaswamy model, the only model that thus far leads to precise quantitative descriptions of all

property changes in the glass-transition region, gives additional support to the validity of the approach presented here. Therefore, the chemical theory of glass transition described here should be seriously considered by everybody who is interested in this important phenomenon.

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O. V. Mazurin / Theory of glass transition [24] B. Wunderlich and H. Baur, Fortschr. Hochpolim. Forschung 7 (1970) 151. [25] W. Kauzmann, Chem. Rev. 43 (1948) 219. [26] E.A. DiMarzio, Ann. NY Acad. Sci. 371 (1981) 1.

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[27] B.J. Zelinski, H. Yinnon and D.R. Uhlmann, Glastechn. Ber. 56K (1983) 822. [28] O.V. Mazurin and V.K. Leko, Sov. J. Glass Phys. Chem. 9 (1983) 113.