Structural order parameters, the Prigogine–Defay ratio and the behavior of the entropy in vitrification

Journal of Non-Crystalline Solids 355 (2009) 653–662

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Structural order parameters, the Prigogine–Defay ratio and the behavior of the entropy in vitrification Jürn W.P. Schmelzer a,*, Ivan Gutzow b a b

Institut für Physik der Universität Rostock, Universitätsplatz, 18051 Rostock, Germany Rostislaw Kaischew Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

a r t i c l e

i n f o

Article history: Available online 9 April 2009 PACS: 64.70Q 64.70.kj Keyword: Glass transition

a b s t r a c t The glass transition is theoretically described in terms of a generic approach employing de Donder’s structural order parameter method, appropriate expressions for the relaxation behavior of glass-forming systems and an order parameter related to the free volume of the system. Employing this approach the behavior of a variety of thermodynamic quantities describing glass-forming systems in vitrification and devitrification processes is interpreted theoretically. Particular attention is devoted here to the estimation of the value of the Prigogine–Defay ratio P and the analysis of the behavior of the entropy in vitrification. In contrast to previous findings it is shown that treating vitrification correctly as proceeding in some interval of temperature and/or pressure the use of only one structural order parameter is sufficient to explain the generally obtained experimentally results P > 1. In addition, an analysis of some alternative approaches of introduction of different order parameters is performed and relationships between different structural order parameters are analyzed. It is shown how the concept of an internal (fictive) pressure can be introduced straightforwardly in terms of the generic approach described. Finally, within the framework of the generic approach utilized here, a general model-independent definition of internal (fictive) pressure and fictive temperature is given for the case that an arbitrary number of structural order parameters is required for the description and some further consequences are discussed briefly. Ó 2009 Elsevier B.V. All rights reserved.

‘Nobody knows what entropy really is, and if you use the word ‘entropy’ in an argument, you will win every time’. John von Neumann

1. Introduction In the course of the International Workshop on ‘Glass and Entropy’ in Trencin in June 2008, different approaches to the theoretical interpretation of the glass transition and the description of properties of the systems undergoing such transition have been discussed intensively. It is the aim of the present contribution to give a brief overview on the method (Sections 2 and 3) and the results of treating vitrification and devitrification in terms of de Donder’s structural order parameter approach (see also [1–4]) concentrating the attention to the problems which have been central in the meeting in Trencin. In particular, it is shown that a theoretical interpretation of the experimentally determined values of the Prigogine–Defay ratio P > 1 can be given employing only one structural order parameter. Moreover, the behavior of the entropy * Corresponding author. Tel.: +49 381 4986943; fax: +49 381 4986942. E-mail address: [email protected] (J.W.P. Schmelzer). 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.12.014

in vitrification and devitrification is discussed in detail (Section 4). In addition, an analysis of different alternative possibilities of introduction of structural order parameters is performed and a general model-independent definition of internal (fictive) pressure and fictive temperature is given for the general case that f independent structural order parameters are required for an appropriate description of the system under consideration (Section 5). A summary and discussion of the results and further possible developments completes the paper (Section 6). 2. Basic thermodynamic equations In treating vitrification processes, we have to consider systems in thermodynamic non-equilibrium states [5,6]. In such states, the change of the entropy, S, in any process is determined, in general, by two terms [7,8],

dS ¼ de S þ di S;

de S ¼

dQ ; T

di S P 0:

ð1Þ

Here diS describes changes of the entropy connected with irreversible processes taking place in the system under consideration, while the term deS describes changes of the entropy connected with an exchange of energy in the form of heat, dQ, with surrounding systems (denoted further as heat bath). With T here originally the

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temperature of the heat bath is denoted (assumed to be in an internal thermal equilibrium) exchanging energy in form of heat with the system, where the processes under consideration take place. In our analysis, we assume that the heat bath and the system analyzed are always sustained at states near to internal thermal equilibrium and that, with respect to temperature, a state near to equilibrium between heat bath and system is established at any time. In such cases, the temperature of the system under consideration and the temperature of the heat bath are equal. In other words, we do not consider here changes of the entropy due to the transfer of energy in form of heat between systems sustained at different temperatures. Following de Donder [7,8], we can describe systems in nonequilibrium states and processes of relaxation of glass-forming melts to the respective metastable equilibrium state by the introduction of additional appropriately chosen structural order parameters, ni. Here we assume that the system can be described with a sufficient degree of accuracy by only one such additional structural order parameter, n. This restriction is introduced here due to the following considerations: (i) the method can be generalized easily to any required number of structural order parameters by replacing one term in the respective basic equations (a detailed discussion of the problem how many structural order parameters may be required in order to appropriately describe a glass-forming system is performed in [1–3] (see also [9–14]; by similar considerations as outlined in [14] – not to confuse the issue with the theory of critical phenomena and second-order phase transitions – we employ generally instead of the term ‘order parameter’ the notation ‘structural order parameter’).); (ii) as will be demonstrated, for the description of the basic nature of the glass transition including the theoretical interpretation of the experimentally observed values of the Prigogine–Defay ratio (P > 1) and finite values of the zero-point entropy of glasses (S(T ? 0) > 0), the restriction to one structural order parameter is sufficient. Similarly, the model employed for the description of glass-forming systems (the generalized equation of state as discussed in Section 3) is chosen as simple as possible allowing one to describe the basic features of the glass transition in a relatively straightforward way. It can be easily generalized or replaced by other similar models being eventually more adequate in treating specific features of particular systems under consideration. Assuming that a single structural order parameter is sufficient for the description of the degree of deviation of the glass-forming melt from the respective stable or metastable equilibrium state, the combined first and second laws of thermodynamics can be written in the form [7,8]

dU ¼ TdS  pdV  Adn;

ð2Þ

here U is the internal energy and V the volume of the system, p the external pressure, and A the affinity of the process of structural relaxation considered. Since the number of particles of the different components remains constant (i.e., since we consider closed systems), terms describing their possible changes do not occur in Eq. (2) in the applications analyzed. Since, for any irreversible processes taking place at constant entropy and volume, the internal energy can only decrease [15,16], the relation Adn P 0 has to be fulfilled. Note that the generalization of Eq. (3) to the case of several structural order P parameters can be performed easily by replacing Adn via Ai dni [1]. A combination of Eqs. (1) and (2) yields

dU ¼ Tde S þ Tdi S  pdV  Adn:

ð3Þ

Alternatively, for any (reversible or irreversible) process of a thermodynamic system, we have according to the first law of thermodynamics [15,16]

dU ¼ dQ  pdV ¼ Tde S  pdV: Consequently, the relations

ð4Þ

Tdi S ¼ Adn or di S ¼

  A dn T

ð5Þ

have to be fulfilled, in general. It follows from latter equation that equilibrium states are characterized by A = 0. Only for such states, the production term of the entropy, diS, connected with possible irreversible processes in the system, is equal to zero. Moreover, since the changes of entropy and structural order parameter take place in the same time interval, dt, we can rewrite Eq. (5) as

T

di S dn ¼A dt dt

or

  di S A dn ¼ : dt T dt

ð6Þ

In this way, we rederived the basic relationship connecting variations of entropy and structural order parameter [8]. However, in order to employ this relation, we have to express also the affinity, A, via the basic thermodynamic functions. Employing pressure and temperature as the thermodynamic state parameters and assuming, in addition to thermal equilibrium, that the conditions of mechanical equilibrium are fulfilled (equality of pressure of the heat bath and of the system under consideration), the characteristic thermodynamic potential is the Gibbs free energy, G, defined generally via

G ¼ U  TS þ pV:

ð7Þ

Eq. (2) yields

dG ¼ SdT þ Vdp  Adn;

ð8Þ

and the relation for the affinity can be obtained in the form

A¼

  oG : on p;T

ð9Þ

At constant pressure and temperature, the thermodynamic equilibrium state is characterized by a minimum of Gibbs’ free energy, G. Let us denote further by ne the value of the structural order parameter, n, in this equilibrium state. Since in thermodynamic equilibrium, the state of the considered closed system is determined by two independent variables, the equilibrium value of the structural order parameter can be represented as an unambiguous function of pressure and temperature, i.e.,

ne ¼ ne ðp; TÞ:

ð10Þ

A truncated Taylor expansion of the partial derivative of Gibbs’ free energy with respect to the structural order parameter, n, in the vicinity of the respective equilibrium state, n = ne, results in

  oG ffi Gð2Þ e ðp; T; ne Þðn  ne Þ; on p;T

Gð2Þ e ðp; T; ne Þ

¼

! o2 G   on2 

: p;T;n¼ne

ð11Þ The first term in the expansion, ðoG=onÞp;T;n¼ne , vanishes since Gibbs’ free energy has a minimum at equilibrium (n = ne). For the affinity, A, we obtain from Eqs. (9) and (11)

A¼

Gð2Þ e ðn

 ne Þ;

Gð2Þ e

¼

Gð2Þ e ðp; T; ne Þ

¼

! o2 G   on2 

> 0:

ð12Þ

p;T;n¼ne

The parameter Gð2Þ e has to be positive, since the equilibrium state is characterized by a minimum of Gibbs’ free energy. Due to the basic nature of the glass transition, in heating or cooling (or varying any other control parameters like pressure), in general, the structural order parameter will differ from the respective equilibrium value. The rate of relaxation we describe by the relation [17–19]

dn 1 ðn  ne Þ: ¼ dt sðp; T; nÞ

ð13Þ

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In this equation, time t can be replaced by temperature (or pressure) assuming a certain rate of change of temperature,

q¼

dT ; dt

dt ¼

1 dT; q

MgF2

300

MnF2

ð14Þ CaCl 2

The equations outlined in the previous section hold generally. In order to apply them to glass-forming melts, the essential properties of glass-forming melts have to be introduced based on some model considerations, i.e., we have to connect the conventional thermodynamic parameters of the glass-forming system with the structural order parameter. The respective relationship we denote as generalized equation of state. For the specification of the thermodynamic properties of glassforming melts, we employ here relations derived from a simple lattice-hole model of liquids discussed in detail in [1–3]. The structural order parameter is connected in the framework of this model with the free volume of the liquid and defined via the number of unoccupied sites (or holes), N0, per mole of the liquid each of them having a volume, v0(p,T), identical to the volume of a structural unit of the liquid at the same values of pressure and temperature. According to this model, the molar volume of the liquid is determined as [1–3]

Vðp; T; nÞ ffi NA v 0 ðp; TÞð1 þ nÞ;

N0 N0 n¼ ffi ; NA þ N0 NA

ð15Þ

here NA is the Avogadro number. The thermodynamic functions of the system are described in the framework of this model by the sum of contributions resulting from the thermal motion of the molecules of the liquid and, in addition, from the configurational contributions described by the structural order parameter, n. The configurational contribution to the volume (or the excess volume) is given, consequently, by

V conf ffi NA v 0 ðp; TÞn:

ð16Þ

The configurational contribution to the enthalpy, Hconf, of one mole of the liquid is described in the framework of this lattice-hole model via the molar heat of evaporation, DHev(Tm), of the liquid at the melting temperature as

Hconf ¼ v1 DHev ðT m Þn:

ð17Þ

Experimental data show that the molar heat of evaporation can be expressed for a wide class of liquids in accordance with Trouton’s rule (c.f. Fig. 1) as

DHev ðT m Þ ffi v2 RT m

with

v2 ¼ 20;

ð18Þ

giving a good approximation for the group of substances included in Fig. 1. The parameter v1 will be determined later. The configurational part of the entropy per mole is described in this model via the conventional mixing term

 Sconf ¼ R lnð1  nÞ þ

 n ln n : 1n

ð19Þ

With H = U + pV, we obtain for the configurational contribution to the internal energy the relation

U conf ¼





v1 DHev ðT m Þ  pv 0 ðp; TÞ n:

ð20Þ

Finally, employing the definition of Gibbs’ free energy, Eq. (7), we arrive at

 Gconf ¼ v1 DHev ðT m Þn þ RT lnð1  nÞ þ

 n ln n : 1n

ð21Þ

Hev, kJ/mol

(or pressure, etc.). Here we will restrict the analysis to processes of glass formation and devitrification if the temperature is changed. 3. Generalized equation of state: Results of a lattice-hole model

CaF2

LiF NaF

BeF2 AgCl

200

BaCl 2

NdI2 CuCl NaCl PbBr2 LiCl MgCl 2 ZnCl 2 CdCl2 ZnBr2 CdBr2 100 PbI2 TiCl H 2O GeCl 4 BiCl 2 ZnI2 VF5 n-But WCl 6 AgBr

AlCl 3 SF6 CCl 4 0

300

600

900

1200

1500

Tm , K Fig. 1. Molar heat of evaporation, DHev, in dependence on melting temperature, Tm, for different halide and oxide substances with very different compositions (circles refer to typical glass-formers, full dots to substances not vitrified as yet). Black squares represent data for simple non-halide glass formers [20]. The straight line is given as a confirmation of Eq. (18) employed in the analysis.

The equilibrium value of the structural order parameter, n = ne, is determined via the relation (oGconf/on)p,T = 0. With Eqs. (18) and (21), we obtain the following result:

  ð1  ne Þ2 1 T ¼ where ln ne v Tm

v ¼ v1 v2 :

ð22Þ

Knowing the value of v2 (c.f. Eq. (18)), we determine the value of the parameter v1 demanding that at T = Tm the value of ne should be approximately equal to 0.05 (corresponding to experimentally observed density differences between liquid and crystal at the melting temperature, Tm [1]). In the computations performed here we set v2 = 20 and v1 = 0.166 resulting in v = 3.32. As it should be the case, in the vicinity of the state of configurational equilibrium, we obtain from Eq. (21) after performing a truncated Taylor expansion the result

1 o2 Gconf Gconf ðp; T; nÞ ffi Gconf ðp; T; ne Þ þ 2 on2

!    

ðn  ne Þ2 :

ð23Þ

p;T;n¼ne

2 2 The value of Gð2Þ e ¼ ðo Gconf =on Þjn¼ne at equilibrium and, consequently, the affinity of the process of structural relaxation (c.f. Eq. (12)) can now be easily calculated based on Eqs. (21) and (22). For small values of n, we get as an estimate Gð2Þ e ffi ðRT=ne Þ. Moreover, knowing the dependence of the structural order parameter on temperature (c.f. Eq. (22)), one can establish the deviations of all thermodynamic functions discussed above from the respective equilibrium values. For this reason, the determination of the function n = n(T) is of basic importance for the understanding of the behavior of the thermodynamic properties of vitrifying melts. An example for the change of the structural order parameter, n, in dependence on temperature, obtained via a numerical integration, is shown in Fig. 2 [2]. We start the process at Tl = Tm and cool the system down to Ts = T0 = Tm/2. Then we heat the system again with the same absolute value of the heating rate up to Tm. For convenience, we introduce the reduced temperature, h = (T/Tm). The results allow one a direct interpretation of the change of the volume (or density) of glass-forming systems in vitrification (c.f. Eq. (15) and Fig. 3) and of density maxima in devitrification at heating [22].

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0.25

0.0085

0.20

Cp,conf /R

0.0080

coo

g lin

0.15 0.10

h

0.0075

ng

O r d e r- p a r a m e t e r,

heatin g

0.30

0.05

ng eati

co

ol

i

0

0.0070 0.64

0.65

0.66

0.67

0.68

-0.05

0.69

0.64

Reduced temperature, T/Tm Fig. 2. Dependence of the structural order parameter, n = n (p, T, q) (or n = n(h) with h = T/Tm), in a cyclic cooling-heating process (full curve). It is assumed here that both cooling and heating proceeds with the same constant rate of temperature change. Arrows indicate the direction of change of temperature. By a dashed curve, the equilibrium curve ne = ne(h) is shown. In undercooling a liquid, the structural order parameter cannot follow, in general the change of the external parameter and deviates from the equilibrium value, ne, resulting in n > ne. Such kind of behavior is found in the vicinity of the glass transition temperature when the time scales of change of the external parameter become comparable with the characteristic relaxation times of the system. In heating, n decreases first until, after intersection with the equilibrium curve, it rapidly approaches it from below due to the exponential decrease of the relaxation times with increasing temperature. For the computations, we used here Tm = 750 K, T0 = Tm/2, Tg = (2/3)Tm, v1 = 0.166, v2 = 20, v = 3.32 and the relations Ua(Tg)/RTg = 30 and U a =RT g ¼ 7:5 (for the details, see [2]).

Having at ones disposal the relation for the temperature dependence of the structural order parameter in heating, respectively, cooling processes, one can also immediately compute the heat capacity. Results are shown in Fig. 4. Here the configurational contributions Cp,conf to the heat capacity are shown. They are defined, in accordance with the general definition,

C p ðp; T; nÞ ¼

0.66

0.68

0.70

Reduced temperature, T/Tm

      dH oH oH dn ¼ þ ; dT p oT p;n on p;T dT

ð24Þ

Fig. 4. Configurational contribution, Cp,conf(p,T,n), to the specific heat, Cp = Cp(p,T,n), as obtained via Simon’s model approach (dashed curve) and the model of a continuous transition as employed here (full curve). In Simon’s model (see [1,5]), it is assumed that the system remains in a metastable state until it goes over suddenly at some given temperature into a fully frozen-in non-equilibrium state, the glass. Note that, in describing vitrification more appropriately in terms of a continuous transition, the specific heats turn out to be different for cooling and heating runs (c.f. experimental data shown in Fig. 5).

parameter leads to the hysteresis of the configurational contribution to the heat capacity. Of course, the curves will be different for different heating and cooling rates, but qualitatively the shape of the figure will remain the same. A comparison with experimental data shown in Fig. 5 proves that the theoretical results are in full qualitative agreement with experimental data reproducing the same shape of the Cp(T)-curves. A variety of other results can be explained straightforwardly by the same concepts. In particular, we will show in the next section that it can be understood easily why the Prigogine–Defay ratio (a quantity with similar meaning as the Ehrenfest relation in second-order equilibrium phase transitions [1,16]) has to differ from unity [3] and has to be, in agreement with experimental data, as a rule larger than one.

via

C p;conf ðp; T; nÞ ¼

  oH dn : on p;T dT

ð25Þ

It reflects the contributions of structural reorganization to heat capacity. Consequently, the hysteresis in the structural order

2.50

4. The Prigogine–Defay ratio and the behavior of the configurational entropy in vitrification While first-order equilibrium phase transitions are characterized by a change of the structure of the system in response to variations of an external control parameter, second-order equilibrium phase transitions are connected with a qualitative change of the response. This change of the response of the system is described by jumps of the thermodynamic coefficients, the heat capacity Cp, Eq. (24), the thermal expansion coefficient, a, and the compressibility, j (Eq. (26))

2.51

, g /c m3

a¼ 3 2.52

Tg1

1

Tg2

4

2.53

aconf ¼ 400

500

j¼

  1 oV : V op T

ð26Þ

The situation in vitrification is different but similar. Here the system is transferred upon cooling from a metastable equilibrium state, characterized by non-zero values of the configurational contributions to the thermodynamic coefficients, Cp,conf, aconf, and jconf,

Tg3

2

  1 oV ; V oT p

600

T, oC Fig. 3. Temperature dependence of the density of a borosilicate glass measured for three different cooling rates: (1) 1 K/min; (2) 2 K/min; (3) 10 K/min; (4) annealing curve [21].

   1 oV dn  ; V on p;T dT p

jconf ¼ 

   1 oV dn ; V on p;T dpT

ð27Þ

i.e., (dn/dT) – 0, (dn/dp) – 0) to a frozen-in non-equilibrium state ((dn/dT) = 0, (dn/dp) = 0). Following Prigogine and Defay [8], the jumps in the thermodynamic coefficients in vitrification are governed by the

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0.5

1

0.4

Cp , cal/K.g

Consequently, employing de Donder’s approach, assuming that one structural order parameter is sufficient for the description of the glass-forming melt and treating in addition vitrification in terms of Simon’s model, one comes to the conclusion that P has to be equal to one (c.f. [9–13]). In addition, since – in terms of Simon’s model – cooling and heating results in an evolution of the properties of the systems under consideration proceeding via the same states but in opposite directions, hysteresis effects cannot be explained in principle employing Simon’s model assumption. Simon’s simplified model is of sufficient accuracy in a certain range of applications, only (c.f. e.g. [1] and the subsequent analysis). However, such picture neglects essential features of vitrification like relaxation and the accompanying entropy production [2].

Tg

2 0.3

3

0.2

373

473

573

T,K 0.8

3

0.6

2

2

(T )

Cp , cal/K.g

0.7

0.5

1 3

0.4

1 0.3 10

20

30

40

50

60

70

T, oC 0

T0

Fig. 5. Top: Experimental heat capacity curves of B2 O3 -melts upon vitrification after Thomas and Parks ([23], see also [24]): (1) cooling run curve; (2) heating curve after slow cooling run; (3) heating curve after fast cooling run. Bottom: Cp(T)-curves obtained in the process of heating of a glass-forming polymer melt for different heating rates according to the measurements of Zhurkov and Levin [25,26]. Curve (1) corresponds to a heating rate of 0.1 K/min, Curve (2) to 0.4 K/min, and Curve (3) to 1.5 K/min. Note that in this figures the full specific heat is shown and not only the configurational contribution presented in Fig. 4.

1 VT

( ) DC p Dj   ðDaÞ2 

T¼T g

¼

 hp;T  A þ hp;T T¼T g

;

ð28Þ

hp;T

T¼T g

1

  oS ¼ A þ hp;T on p;T

  oS ¼ A þ hp;T > 0 on p;T

0

4

tin hea

T0

Tg

Tm

Temperature, T

ð30Þ

hold. Based on this general result, now different model assumptions concerning the glass transition need to be employed to arrive at an estimate for the Prigogine–Defay ratio, P. Following Simon [5], the glass transition is frequently assumed to occur suddenly at the glass transition temperature, Tg (n = ne for T P Tg; n = ne(Tg) for T < Tg; c.f. Fig. 6). Employing this additional assumption, we get from Eq. (28) utilizing Eqs. (9) and/or (12):



AðT g Þ ¼ Geð2Þ ðT g Þ nðT g Þ  ne ðT g Þ ¼ 0 ) P ¼ 1:

g

ð29Þ

and the following inequalities:

hp;T > 0;

3

2

where

  oH ¼ ; on p;T

n oli

g

P¼

  oH  on p;T  ¼     T oS  on p;T  

Tm

co

Order-parameter,

configurational contributions to the thermodynamic coefficients. Based on this assumption and employing exclusively de Donder’s method without any additional assumptions the following general relation can be obtained [1,8]:

Tg

Temperature, T

ð31Þ

Fig. 6. Qualitative illustration of possible dependencies of the order parameter, n, on temperature in cooling of glass-forming melts. Curve (1) corresponds here schematically to the dependence of the equilibrium value of the structural order parameter on temperature, i.e., ne = ne(p,T). In cooling a liquid, the order parameter cannot follow, in general the change of the external parameter and deviates from the equilibrium value, ne (Curve (2)), resulting in n > ne. Such kind of behavior is found in the vicinity of the glass transition temperature when the time scales of change of the external parameter become comparable with the characteristic relaxation times of the system. According to Simon’s classical model of vitrification, the order parameter is identical to ne down to T = Tg and at this temperature, the structure becomes suddenly frozen-in (Curve (3)). The upper curve shows in the simplest possible way the behavior of the order parameter for cooling while the lower curve includes the behavior in heating runs (Curve (4)).

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Gustav Tammann nearly at the same time [6] as Simon advanced his concept mentioned already: Es ist . . . nicht richtig von einem Transformationspunkt der Gläser zu sprechen, richtiger wäre es . . . Transformationsintervall zu sagen . . . (It is incorrect to talk about a transformation point, instead one has to use transformation interval . . .) Employing this concept, the generic model of vitrification and relaxation leads to an essentially different result with respect to estimates of the Prigogine–Defay ratio (c.f. Fig. 6) as compared with the case when Simon’s model is utilized: In the vitrification range, the affinity A is not equal to zero and the Prigogine–Defay ratio is not equal to one even if only one structural order parameter is employed for the description of the vitrifying system. This ratio, computed straightforwardly via Eq. (28), is larger than one for cooling (n > ne, A < 0, P > 1) and less than one for heating (n < ne, A < 0, P < 1). However, as discussed in detail in [3], in order to give a theoretical interpretation of the experimentally determined values of the Prigogine–Defay ratio P(exp), the values for the parameters entering Eq. (28) at cooling runs have to be chosen for a theoretical interpretation leading to the general consequence ðexpÞ

P

1 ¼ VT

( ðexpÞ ) DC p DjðexpÞ    ðDaðexpÞ Þ2

¼ T¼T g

Cooling hp;T  > 1: A þ hp;T T¼T g

ð32Þ

Employing the lattice-hole model discussed, we get, in particular,

PðexpÞ

Cooling   1    ffi : Tg nne  1  vT n m

e

ð33Þ

T¼T g

This equation shows that the Prigogine–Defay ratio depends both on the thermodynamic properties of the system under consideration (reflected in the model by the ratio (Tg/Tm) and the parameter v) and the cooling rate (resulting in different values of the structural order parameter n and the difference (n  ne)). Employing the same method of analysis, the jumps of the thermodynamic coefficients can be determined as well separately. This analysis has been performed in [3]. Here we would like to add a comment, central to the meeting in Trencin, concerning the behavior of entropy in vitrification. In the framework of the model analyzed, the configurational entropy is, according to Eq. (19), uniquely determined by the order parameter n. Since the relation

oSconf ln n > 0; ¼ R on ð1  nÞ2

ð34Þ

holds generally, we get ðeÞ

Sconf P Sconf ðeÞ

for n P ne ;

ðeÞ

Sconf 6 Sconf

for n 6 ne ;

ð35Þ

here Sconf denotes the configurational contribution to the entropy ðeÞ referring to n = ne. For T ? 0, the relations ne ? 0 and Sconf ! 0 are fulfilled for the (hypothetical) extension of the metastable liquid to zero temperatures, while for the frozen-in liquid, the glass, a configurational contribution Sconf(T ? 0) > 0 is obtained (c.f. Fig. 5 in [2]). The configurational contribution Sconf(0) is identical to the respective value frozen-in in the vitrification interval, it depends on the rate of cooling, i.e., on the value of the structural order parameter n frozen-in in the vitrification interval. Based on the model employed we arrive – in contrast to [27–29] – at the ‘conventional’ conclusion [1,13,30] that glasses and not only glasses but also disordered crystals and similar systems [1,30] do have a nonzero configurational entropy at absolute zero. This point of view has been reconfirmed also in a recent analysis of these problems by Goldstein [31]. An analysis of experimental data confirming the results of our analysis (the ‘conventional’ point of view) is given in the accompanying paper [32].

Note as well that only in terms of Simon’s model the relations

DHðT g Þ ¼ 0; DGðT g Þ ¼ 0;

DSðT g Þ ¼ 0; DUðT g Þ ¼ 0

DVðT g Þ ¼ 0; ð36Þ

(D denotes the difference between glass and liquid) are fulfilled. Treating vitrification and devitrification in terms of the generic approach, the respective differences are, in general, not equal to zero since, in general, n – ne holds at Tg or, more generally, in the vitrification interval (c.f. [28,29]). Note as well the following aspect: A linear extrapolation of the temperature dependence of the structural order parameter – as indicated by a dashed curve in Fig. 2 – to temperatures below the vitrification range would lead to an intersection with the abscissa at some finite temperature T*. In contrast, the correct (for the considered model) dependence ne(T), given by Eq. (22), does not cross the abscissa but approaches it at T ? 0. In this way, employing not linear extrapolations but the correct (for the considered model) result, Kauzmann type paradoxes do not occur here. 5. Fictive (internal) pressure and fictive temperature as structural order parameters For the first time, the concept of a structural order parameter was introduced into glass science by Tool [33–35] in terms of a fictive temperature. Davies and Jones [9] noted that instead of fictive temperature one can employ equivalently the concept of fictive pressure as the determining structural order parameter. Gupta [12] developed a theoretical approach treating both fictive temperature and fictive pressure as independent parameters. There are a variety of further attempts to employ these concepts in the treatment of glass-forming systems, some of them are discussed below. All the consequences discussed above in the present analysis can be derived without relying on the concept of an internal (fictive) pressure or fictive temperature as a structural order parameter. Anyway, it is of interest to see in which way such concepts may be introduced in the framework of de Donder’s approach avoiding as far as possible any simplifying assumptions, and what results from this procedure. This task will be performed in the present section. 5.1. Fictive (internal) pressure as a structural order parameter In the already cited paper, Gupta [12] introduced two structural order parameters ni, i = 1, 2 (in our notations) assumed to be chosen in such a way that for stable or metastable equilibrium (o2Ge/ oninj) = 0 holds for i – j. Similarly to Eq. (10), in equilibrium these parameters are considered to be functions of pressure and temperature, i.e., nie = fi(p, T) for i = 1, 2. Having at one’s disposal the functions fi(p,T) connecting the equilibrium values of n with pressure and temperature, fictive pressure, pf, and fictive temperature, Tf, are determined generally via

ni ¼ fi ðT f ; pf Þ;

i ¼ 1; 2:

ð37Þ

Employing certain approximations (e.g., a linear dependence of fi on pressure and temperature, linear relaxation laws with kinetic coefficients being independent of the structural order parameters) the relaxation behavior in such systems is studied. Landa et al. developed an interpretation of the glass transition and accompanying effects employing the concept of negative internal pressure as the structural order parameter [36–38]. In [36], it was stated by Landa et al. that their approach is based on de Donder’s principle. This statement is expressed by mentioned authors in the form ‘Non-equilibrium systems can be described by equalities that include the notion of affinity and internal parameter’ continuing then in the following way: ‘In case of the glass-forming

J.W.P. Schmelzer, I. Gutzow / Journal of Non-Crystalline Solids 355 (2009) 653–662

systems, their excess volume is affinity to glass transition and relaxation, and negative pressure – internal parameter’ ([36, p. 8]). However, a proof of this statement is missing or, at least, not given in a convincing form. As will be shown here the concept of internal pressure can be introduced straightforwardly, employing the results outlined above, as follows. According to Eq. (16), the excess volume Vconf (or the configurational contributions to the volume) is (in a good approximation) linearly connected with the structural order parameter, n. Considering v0(p,T) in an also good approximation as nearly constant, Eq. (2) can be rewritten as

dU ¼ TdS  pdV 

A dV conf : NA v 0 ðp; TÞ

ð38Þ

By introducing or, more precisely, defining the internal pressure via

 ¼

A ; NA v 0 ðp; TÞ

ð39Þ

we obtain with A ¼ Gð2Þ e ðn  ne Þ (Eq. (12)) and the estimate Gð2Þ e ffi ðRT=ne Þ [2] the result

 ¼ Gð2Þ e

ðn  ne Þ ðn  ne Þ RT ffi : NA v 0 ðp; TÞ ne NA v 0 ðp; TÞ

ð40Þ

The sign of the internal pressure is thus directly determined by the sign of the difference (n  ne) or of the ratio (n  ne)/ne, its magnitude by the ratio (RT/(NAv0(p,T))). So, indeed, a parameter having the dimension of pressure can be introduced, having the essential properties as assigned to them by Landa et al. [36], for example, it is equal to zero in stable or metastable equilibrium (here n = n(e) holds) and has negative values in the glass transition range and below at cooling. However, in the subsequent heating it can and even has to assume a positive value until in the further heating stable or metastable equilibrium states are reached again. Moreover, as it turns out, in contrast to the mentioned at the beginning statement of Landa et al. [36], the internal pressure, defined in such a way, is proportional to the affinity while the structural order parameter is connected with the excess volume. Some further points of agreement and disagreement with the statements of Landa et al. formulated in [36–38] are discussed in detail in [4].

5.2. On some recent attempt of definition of fictive temperatures In a series of papers (see e.g. [39–41]), Nieuwenhuizen attempted to develop a thermodynamic description of the glassy state introducing the concept of an effective temperature similar to the concept of fictive temperatures developed originally by Tool [33–35]. His analysis he motivates by the statement that ‘the general consensus reached after more than half a century of research was: Thermodynamics does not work for glasses, because there is no equilibrium’ [39,40]. This statement is, however, not true. As already discussed in detail earlier [1–3], vitrification in cooling of glass-forming melts (or similar processes in response of the system to changes of some other external control parameters) is treated extensively for decades by thermodynamic methods, however, so far this treatment is performed commonly in terms of Simon’s model [1,5,13], only. According to this concept, the glassforming melt remains in a state of (metastable) thermodynamic equilibrium until in the process of cooling of the glass-forming melt the glass transition temperature, Tg, is reached (see also Fig. 6). Consequently, for T P Tg classical equilibrium thermodynamics is fully applicable. At Tg, the systems become – according to Simon’s model – suddenly frozen-in. This statement implies that configurational or structural changes of the melt become impossible and the configurational contributions to entropy, enthalpy and

659

volume become fixed and do not change any more in the further cooling to temperatures T < Tg (c.f. [1,5]). Taking into account this result and assuming – following Simon’s concept – that relaxation processes do not occur below Tg in the considered time scales of the experiment, further cooling and subsequent heating proceeds also in a reversible way. For this reason, equilibrium thermodynamics (in form of the combined first and second laws and their consequences) is fully applicable and the thermodynamic potentials and all other relevant thermodynamic properties of glasses can be determined. Such procedure is performed in detail in [1]. In this way, a complete description of the thermodynamics of glass-forming melts and glasses can be given employing classical thermodynamics with the following modifications: (i) The glass transition temperature (and, as the result, the configurational contributions to the thermodynamic potentials) depend on cooling rate (or, equivalently, on some additional structural order parameter or a set of order parameters which become frozen-in at the respective values of Tg). (ii) The frozen-in configurational contributions of the thermodynamic potentials lead to specific thermodynamic properties of glasses as compared with equilibrium systems with a broad spectrum of possible applications. However, these effects can be treated fully in terms of equilibrium thermodynamics (c.f. for details [1]). (iii) The third law of thermodynamics in its conventional formulation assigned to Planck [15,16] is not fulfilled, since glasses are frozen-in non-equilibrium systems characterized by excess values of the thermodynamic potentials and, in particular, the entropy. For this reason, the entropy of glasses does not follow the third law of thermodynamics in the conventional form ([1], a more detailed discussion of the limits of applicability of the third law of thermodynamics to glasses is given in [17–19] and in [32] of the present volume). According to his papers and the cited at the beginning of this section statement, Nieuwenhuizen is not aware of these results. In order to develop the desired thermodynamic description of both glass-forming melts and glasses, he instead postulates a generalization of the second law of thermodynamics in form of the relation [39,40]

dQ ¼ TdSep þ T ef dSef :

ð41Þ

According to Nieuwenhuizen, Sep is the entropy of equilibrium processes (ep) with characteristic time-scales less than the observation time (denoted as b-relaxation) and Sef is the entropy of the slow processes or configurational changes (denoted as a-relaxation), while dQ is called the ‘change in heat’ or the ‘heat of the combined system’ (note that these definitions are thermodynamically questionable since the entropy is a state function and heat is a characteristic of the process and not of the state of the system, but these weaknesses are not the most important gap). Tef is denoted as the effective temperature. Moreover, it is supposed that the total entropy is given by the sum

S ¼ Sep þ Sef :

ð42Þ

Both Eqs. (41) and (42) are postulates. The question is, however, whether these postulates describe correctly the phenomena under consideration or not. Some of the problematical issues are mentioned by the author himself. In particular, one of the problems the author discusses is the separability of time scales and/or phase space allowing one to identify uniquely the different entropy contributions. The realization of this task will be even more complicated if several structural order parameters (or here effective temperatures) have to be introduced. Another question is whether the respective effective entropies and temperatures can be always connected with energy transfers in form of heat via the form as given by Eq. (41). Already by the mentioned considerations alone, the introduction of different entropy terms into the basic equations of thermodynamics cannot be considered, to our opinion, as an advantage of

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the theoretical approach. However, there are some even more serious shortcomings of the approach analyzed which will be discussed below. Nieuwenhuizen discusses also some other assumptions which could be proposed instead of Eq. (42) [40] proceeding finally with this one. However, above equations are supposed by the author to be valid in the absence of currents (or fluxes, flows) only (c.f. e.g. Eq. (2.1) in [40] and the discussion by the author of this equation and, in comparison, Eq. (6.1) in the same reference). But for reversible processes, i.e. in the absence of currents, the well-known thermodynamic relation

dQ ¼ TdS;

ð43Þ

has to be fulfilled. Eqs. (41)–(43) yield then

ðT  T ef ÞdSef ¼ 0:

ð44Þ

Consequently, the thermodynamics of glass-forming systems, as attempted to be developed by Nieuwenhuizen, reproduces – if correct – merely existing theoretical descriptions. It allows one to describe metastable glass-forming melts (for T = Tef) and frozen-in non-equilibrium states, i.e., glasses (for dSef = 0 resulting in Tef = constant). It cannot describe correctly the thermodynamics of glass-forming melts in the vicinity of the glass transition temperature, where both entropy variations due to the exchange of energy in form of heat and entropy production effects have to be accounted for. Based on Eq. (41), the combined first and second law of thermodynamics is written by Nieuwenhuizen then as [39,40]

dU ¼ TdSep þ T ef dSef  pdV:

as generally valid. This is, as it follows from cited papers, however, not the original intention of mentioned author. 5.3. Model-independent definition of fictive (internal) pressure and fictive temperature As outlined above, the concept of fictive or internal pressure can be introduced employing appropriate models of glass-forming systems. However, starting with Eq. (2) a model-independent definition of fictive pressure and fictive temperature can be given. For this purpose it is required that the variation of the external parameters is performed in such a way that the structural order parameter can be represented as a differentiable function of the two independent thermodynamic state parameters (we consider closed systems). As such state parameters, we chose total entropy, S, and the total volume, V, of the system. Then we may write

n ¼ nðS; VÞ

dn ¼

    on on dS þ dV: oS V oV S

ð46Þ

In order to avoid confusion, we would like to stress once more the assumption underlying Eq. (46). In general, the structural order parameter is an independent parameter required for the description of the deviations of the system from equilibrium. However, if some well-defined heating or cooling rules (and/or changes of pressure) are assumed leading to a dependence like n = n(S,V) (c.f. Figs. 2 and 6) then for the given trajectory of vitrification or devitrification we have the right to formulate Eq. (46). With Eq. (2) we obtain

ð45Þ

Employing this relation, so-called modified Maxwell equations are derived by him assuming, in addition, that the effective temperature is some well-defined function of pressure and temperature (depending on the details of the cooling and heating processes), i.e., Tef = Tef(p,T;q). Latter assumption is true under certain conditions and evidently fulfilled for the model employed in the computations leading to Figs. 2 and 6 of the present analysis (where the respective dependence is shown for the alternative structural order parameter chosen by us). However, the respective course is determined by the interplay between relaxation and cooling respectively heating (and eventually change of pressure), i.e., it cannot be reproduced without an incorporation of irreversible flow processes (currents) into the theory (independent on the choice of the structural order parameter). However, as a consequence of Eq. (44) (resulting from the neglect of currents), Eq. (45) and the resulting modified Maxwell relations cannot give a correct description of the variations of the thermodynamic state parameters along the actual path of evolution of the system in the vitrification interval. Eq. (45) is restricted in its applicability to processes proceeding either at T = Tef or Sef = constant. The interplay of cooling and relaxation determines not only the value of the structural order parameter at the glass transition temperature but, as a consequence, also the value of the Prigogine–Defay ratio in vitrification. Since such effect is not incorporated into the thermodynamic description of Nieuwenhuizen, his attempts [40,41] to explain theoretically this ratio also cannot be considered as correct (c.f. in contrast our approach given in terms of de Donder’s treatment as described here and in more detail in [3]). By this reason, it is also no wonder that the (only) estimate of a Prigogine– Defay ratio for a model system made by Nieuwenhuizen is in conflict with the original estimates of the authors [42] (and also with the predictions of our approach [3]). In a recent analysis of the concept of fictive temperatures, Garden et al. [43] came to the conclusion that the approach by Niewenhuizen is equivalent to de Donder’s approach. This could be the case (neglecting for a while the other problems discussed above) if Eq. (2.1) in [41] and not Eq. (6.1) would be considered

)

dU ¼

      on on dS  p þ A dV: T A oS V oV S

ð47Þ

As a consequence, the fictive pressure and the fictive temperature can be determined similarly to the definition of the respective parameters in classical equilibrium thermodynamics via

    oU on ¼T A ; oS V oS    V oU on pfictiv e ¼  ¼ pþA : oV S oV S

T fictiv e ¼

ð48Þ ð49Þ

For stable and metastable thermodynamic equilibrium states, the so-defined parameters are equal to normal pressure and temperature (since for these states A = 0 holds) while in general these parameters depend on the path the system is transferred into the respective state (c.f. Fig. 7). Finally, employing again results of the model of glass-forming melts discussed above, i.e.,

A ffi RT

  n  ne ; ne

V ffi NA v 0 ðp; TÞð1 þ nÞ;

ð50Þ Sconf

    n ¼ R lnð1  nÞ þ ln n ; 1n ð51Þ

we obtain

    oV on 1 ¼ NA v 0 ðp; TÞ ; ¼ ; on S oV S NA v 0 ðp; TÞ       oSconf oS ln n on ð1  nÞ2 ¼ ¼ R ; ¼ : 2 on V oS V on V R ln n ð1  nÞ

ð52Þ ð53Þ

A substitution into Eqs. (48) and (49) yields finally

(

)     n  ne ð1  nÞ2 n ffiT ; T fictiv e ¼ T 1  ne ne ln n   RT n  ne pfictiv e ¼ p  : NA v 0 ðp; TÞ ne

ð54Þ ð55Þ

J.W.P. Schmelzer, I. Gutzow / Journal of Non-Crystalline Solids 355 (2009) 653–662

thermodynamic equilibrium states, the so-defined parameters are again equal to normal pressure and temperature (since for these states Ai = 0,i = 1, 2, . . . , f holds). In general, these parameters depend on the path the system is transferred into the respective state reflecting in this way the non-equilibrium nature of the system. (ii) Having at one’s disposal the generalized caloric equation of state U = U(S,V,n1,n2, . . . nf) (i.e., the dependence of the internal energy on the external control parameters S and V and the chosen set of structural order parameters {ni} to be obtained from statistical mechanical model computations), fictive temperature and fictive pressure can be determined uniquely via Eqs. (59) and (60) independent of the number of structural order parameters required for the description of the system under consideration. (iii) Employing the definition of fictive pressure and fictive temperature as developed by us (Eqs. (59) and (60)), the fundamental law of thermodynamics for closed systems and arbitrary numbers of structural order parameters can be written as

S

T Tg

Tfict Sg

0

U

Ug

Fig. 7. Illustration of the definition of fictive temperature and fictive (or internal) pressure via Eqs. (46)–(49) (here shown for the case of fictive temperature).

Note that fictive pressure and temperature are thus uniquely determined via the knowledge of the value of n. Moreover, in such general definition (fulfilling the usually assumed limiting condition that in stable and metastable equilibrium fictive pressure and temperature are equal to external pressure and temperature), the fictive pressure (in contrast to the result following from the definition given via Eqs. (39) and (40)) may be not less than zero in the glass transformation range and below even in cooling a melt. In the opposite limiting case, for temperatures tending to zero (since for T ? 0, we have ne ? 0 and (T/ne) ? 1), the relations:

A ! 1;

T fictiv e !1; T

pfictiv e ! 1 for T ! 0 p

ð56Þ

hold. The definition of fictive temperature and fictive pressure can be generalized easily to the case of an arbitrary number of structural order parameters. Indeed, in this more general situation, when e.g. f structural order parameters have to be introduced into the description, we have to replace Eq. (2) by

dU ¼ TdS  pdV 

f X

Ai dni :

ð57Þ

i¼1

Along a given path of evolution of the system, the different order parameters are functions of the external control parameters chosen here to be entropy and volume. Instead of Eqs. (47)–(49), we get then

( dU ¼

T

f X i¼1

661

) ( )     f X oni oni dS  p þ dV: Ai Ai oS V;fnj ;j–ig oV S;fnj ;j–ig i¼1 ð58Þ

dU ¼ T f dS  pf dV:

ð61Þ

In such definition, fictive pressure and fictive pressure have a much wider meaning as to represent structural order parameters, they represent the generalizations of the external control parameters (external pressure, p, and external temperature, T) governing the thermodynamic behavior of closed systems for equilibrium states and reversible processes proceeding in between them. The property listed above as (ii) – general validity of the definition of fictive pressure and temperature Eqs. (59) and (60) independent of the number of structural order parameters – is not fulfilled, for example, for the definition of these parameters (Eq. (37)) given by Gupta [12]. In particular, latter relation does not allow one uniquely to determine pf and Tf for the case that only one structural order parameter is required. The extension of Eq. (37) to f independent structural order parameters is as a rule also impossible since f sets of equations of the form as given by Eq. (37) will not have, in general, a solution. Note as well that even in the case that only one structural order parameter is required for the description of the system, both Tf and pf are different, in general, from T and p (c.f. Eqs. (54) and (55)). Finally, we would like to note that above given definitions of fictive temperature and fictive pressure can be generalized – if required – also to the case of open systems in a similar form as done here.

6. Discussion The generic model of vitrification gives, in addition to the topics discussed in detail, a sound basis for the analysis of a variety of additional features of vitrification and devitrification processes. In particular: (i) It gives the possibility to account for entropy production terms in the kinetics of vitrification [2]. (ii) It allows one to define the glass transition temperature via [2]

As a consequence, the fictive pressure and the fictive temperature can be determined similarly to the definition of the respective parameters in classical equilibrium thermodynamics via

   d T  2   dt T s ffi 10 m

    f X oU oni ¼T Ai ; oS V;fnj ;j–ig oS V;fnj ;j–ig i¼1     f X oU oni ¼ pþ Ai : pfictiv e ¼  oV S;fnj ;j–ig oV S;fnj ;j–ig i¼1

As a direct conclusion it follows that for systems characterized by a relaxation time like:

T fictiv e ¼

ð59Þ ð60Þ

Eqs. (59) and (60) represent – as we believe – the most general and appropriate definition of these internal parameters. (i) It is a straightforward generalization of the basic thermodynamic definitions of temperature and pressure as known from classical equilibrium thermodynamics (e.g. [15,16]). For stable and metastable

s ¼ s0 exp

at T ¼ T g :

 U0 ; ðT  T 0 Þ

ð62Þ



ð63Þ

a glass transition near T0 leading eventually to a hypothetical ‘ideal glassy state’ cannot be realized (c.f. also [45]). Note further that in our approach it is the interplay between the relaxation time and the characteristic time of change of the external parameters (and not of an observation time (c.f. [28,29])) which determines the glass transition. This distinction is by no means

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irrelevant. One of the basic problems in equilibrium statistical physics is how averaging over time and averaging over ensembles can be reconciled. In order to give some foundation of the equivalence of both methods, the ideas of ergodicity have been introduced without a final resolution of the problem (recurrence times, etc.). Considering systems at time-dependent conditions, one has in addition to the observation time also to account for the characteristic times of change of the thermodynamic control parameters. Consequently, observation times and time of change of the control parameters are by no means equivalent quantities. The introduction of the observation time as an essential time scale of vitrification is employed in [28,29] as the starting point for the derivation of the conclusion of zero values of the configurational entropy. This conclusion is in contrast with the ‘conventional view’ we arrived at in the present analysis and which is expressed also in the already mentioned recent analysis by Goldstein [31]. Consequently, to stress that it is the characteristic time of change of external control parameters (as we interpret it) and not of the observation time (c.f. [28,29]) gives an additional support to the conventional point of view of a non-zero value of the configurational entropy. (iii) The generic method of treating vitrification allows one to understand also the peculiarities in the behavior of kinetic parameters of glass-forming systems (equilibrium versus non-equilibrium viscosities) considering the viscosity g, for example, as a function of pressure, temperature and structural order parameter. The change of the viscosity can be described then similarly as the respective variation of the thermodynamic coefficients via (for more details, see [2])

        dg og og dn ¼ þ : dT p oT p;n on p;T dT p

ð64Þ

(iv) It allows one to understand an increased intensity of fluctuations in glasses (the spectrum of fluctuations has to be different as compared with fluctuations in a stable or metastable equilibrium system, the intensity of fluctuations will be as a rule higher since there is no restoring force [44]). (v) Finally, we would like to mention that the approach is compatible with amorphous polymorphism. Such effects could be described developing models exhibiting more than one metastable equilibrium states as attractors the order parameters can tend to in dependence on the initial conditions. 7. Conclusions Simon’s model of glass transition, employed so far in most analyses of glass formation, allows one to perform a sufficiently accurate determination of a certain spectrum of properties of glasses. In particular, it allows one to compute the thermodynamic functions of glass-forming melts in cases where entropy production terms are of minor significance. It does not cover, however, adequately a variety of other features (in particular, relaxation, hysteresis effects and the Prigogine–Defay ratio). The generic approach to vitrification, based upon de Donder’s structural order parameter approach, is able to give a comprehensive description both of thermodynamic and kinetic properties of glasses and glass-forming melts. In particular, it is shown that treating vitrification as a process proceeding in some given temperature (or pressure) interval – the experimentally observed val-

ues of the Prigogine–Defay ratio P > 1 can be understood theoretically employing only one structural order parameter and that glasses have non-zero values of the entropy at absolute zero its magnitude depending on the path/rate the system was frozenin to a glass.

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