On the equilibrium isothermal compressibility of soda-lime silicate glass

Journal of Non-Crystalline Solids 128 (1991) 101-108 North-Holland

101

On the equilibrium isothermal compressibility of soda-lime silicate glass Ren~ Gy Saint-Gobain Recherche, 39 Quai L.-Lefranc, BP 135, 93303 Aubervilliers, France Received 28 August 1990

An estimate of the equilibrium isothermal compressibility in the transformation range is obtained for soda-lime silicate glass. This work is based on previously published uniaxial and pure shear static experimental data and on specific theoretical derivations. It is shown that a more accurate determination of this quantity could be performed by measuring the total elastic recovery in two experiments of pure shear and uniaxial stress on glass samples in the transformation range.

1. Introduction The mechanical behaviour of a linear isotropic viscoelastic material (such as silicate glasses in the transformation range but undergoing only stresses of limited magnitude) under any type of mechanical loading at a given temperature can be derived theoretically from the knowledge of two time-dependent viscoelastic moduli: Gl(t ) and GE(t ). The shear viscoelastic modulus, Gl(t), that relates the shear time-dependent stress-response to the amplitude of a step-wise shear-strain deformation m a y be expressed as Ga(t ) = 2G~/'l(t ),

(1)

where G is the (instantaneous) shear modulus and ~/'t the shear relaxation function. ~/'t(0)= 1 and ~/'1(oo) = 0, since there remains no residual stress once the shear relaxation is over. The bulk viscoelastic modulus, G2(t), that relates the hydrostatic stress (three times the interhal pressure in the material) to the amplitude of a step-wise dilatational strain may be expressed as G2(t ) = 3K e - (3Kc -

3Kg)'/'2 (t).

(2)

In this expression, K s is the instantaneous (or glassy) bulk elastic modulus, Ke the equilibrium

bulk elastic modulus and '/P2 the hydrostatic stress relaxation function. (xl'2(0) = 1 and 92(00 ) = 0.) 1 / K c is the equilibrium (liquid) isothermal compressibility of the glass-forming liquid. Among the quantities involved in these definitions, the instantaneous elastic moduli G and Kg are the most readily obtainable: they are related to the other well-known instantaneous elastic constants, the Young's modulus, E, and Poisson's ratio, v. Conversely, E can be written in terms of G and Kg: 1/E = 1/(9Kg) + 1/(3G).

(3)

Since the instantaneous elastic constants are mainly related to short-range interactions in the glass, they are not expected to be strongly affected by the temperature change (only a few percent of related variation from ambient temperature up to a temperature about 100 K above the glass transition temperature for those glasses such as silicate glass whose short range order is not markedly affected by the glass transition). Such a poor temperature sensitivity of the instantaneous elastic constants has been experimentally confirmed in the case of pure vitreous silica [1]. Therefore, quite accurate determinations of G and K~ in the transition range can be made with

0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

102

R. Gy / Compressibility of soda-lime silicate glass

the low-temperature value of the Young's modulus and the Poisson's ratio conveniently obtained in conventional static or ultrasonic experiments. The shear relaxation function, xol, is known from pure shear experiments in the transformation temperature range. Up to 100 K over the transition temperature, the soda-lime silicate glass is shown to exhibit a simple thermorheological behaviour. This allows a unique, temperature-independent, shear relaxation function to be given on a dimensionless time scale. The reduced time, u(t/¢l), is the real time, t, divided by a temperature-dependent reference time, Zr The knowledge of the two other quantities, Kc and g'2, involved in the bulk viscoelastic modulus requires more work. The measure of the limiting low-frequency velocity of the longitudinal acoustic waves is theoretically related to the equilibrium bulk modulus. However, such a dynamic method is generally performed at a temperature far above the transition range and gives the adiabatic modulus, not the isothermal one. The determination of Ke and ko2 requires a specific experimental device. Corsaro [2] made use of the 'acoustic dilatometer' to determine the volumic viscoelastic properties. The only published data obtained with this experimental device are those for vitreous B203 at about 300 o C. The bulk modulus of the liquid was found to be 0.3 times that of the glass and thermorheological simplicity was found to apply within the investigated temperature range. Thus, the isothermal equilibrium bulk modulus, K,, of soda-lime silicate glass in its transformation range is not known. For industrial glasses, the knowledge of K~ would be of practical interest to enable more accurate viscoelastic computations, especially in the case of, triaxial stress [3]. An example of such a situation might be the simulation of the .residual stress field in a thick glassware after tempering. It also has a theoretical interest, since it allows the determination of the configurational contribution to the isothermal compressibility. This quantity is required to estimate the Prigogine-Defay ratio of the glass, the value of which has theoretical implications on the nature of the glass transition [4]. The purpose of this paper is to show that K,

can be computed from uniaxial and pure shear experimental relaxation data.

2. Theoretical background Most of the usual mechanical loads cause neither pure voluminal changes nor pure shear, but mixed. In such cases, the two viscoelastic moduli, G 1 and G2, are conveniently replaced by other specific viscoelastic moduli. In the case of the simple uniaxial loading, the uniaxial viscoelastic modulus, E(t), may be defined as

E(t) = Eff'u(t),

(4)

where E is the (instantaneous) Young's modulus and ~/'u the uniaxial relaxation function. The relaxation functions ~ (i = 1, 2 or u) all have a corresponding relaxation spectrum H i defined in the following way: +o0

• ,(t) = f

exp(-t/¢)Hi(ln ¢) d In ¢.

(5)

--OO

It is convenient to make use of the Laplace transform and of the viscoelastic analogy [5]: let p be the Laplace variable and f * ( p ) the Laplace transform of any time function, f ( t ) . The Laplace transformations on both sides of eqs. (1, 2, 4) lead to the following equations, respectively:

G~(p) = 2GXOl*(p);

(6)

G : ( p ) = 3Ke/p - (3K c - 3Ks) ~/'2"( p ) ;

(7)

E*(p) = EJ'* (p), (8) Also, since z/(1 + .cp) is the Laplace transform of exp( - t/¢): ~*(P) =

f+oo oo

~ Hi(ln ¢) d In ¢ 1+¢p

+oo

.

+oo

= E (-1)JPJf_ "rJ+lHi(In ~') d In ¢. j=0 o0

(9) By identification with the Taylor series of ~ * ( p):

dJ * (p = 0) dp j .

= (-1)gflf

+OO

,rJ+lHi(ln "r) d In "r.

(10)

103

R. Gy / Compressibility of soda-lime silicate glass

Retaining only only the two first terms:

(0) = f

+oo -oo

~'H,(ln ~') d In • = (+i),

dR* dp (p=0)=

-

f+o~~-2Hi(ln r)

(11)

d In ~-

-oo (12)

= --Oi 2 -- ( T i ) 2,

where (~'i) and oi (i = 1, 2 or u) are the mean and the standard deviation, respectively, of the distribution of relaxation times (Hi) associated to the relation function ~ . The constitutive equations of linear viscoelasticity have the very simple form of the equations of linear elasticity, provided that the elastic constant 2G, 3K and E are replaced by their viscoelastic analogous: pG~'(p), pG~(p) and pE*(p), respectively. Moreover, using this analogy, the relation between the viscoelastic moduli Gi(t ) (or E(t)) and the viscoelastic compliances J/(t) [resp. the uniaxial compliance, D(t)] reduces to

G i * ( p ) 4 * ( p ) = 1/p 2 (resp. E * ( p ) D * ( p )

= 1/p2).

(13)

From eqs. (6, 7, 8, 11-13), the following expressions of the vioscoelastic compliances can be derived (cf. appendix):

instantaneous elastic compliance, the second term is the delayed elastic compliance and the third term is the contribution of the viscous flow to the compliance. From eqs. (15) and (16), the full magnitude of the delayed elastic deformation can be shown to be inversely proportional to the instantaneous elastic modulus, and proportional to the square of the normalized standard deviation ( o,.2/(~'i)2 ). This quantity is a measure of the broadness of the relaxation spectrum. It is consistent with the wellknown fact that no delayed elasticity can be accounted for by using a single Maxwell element as model of the viscoelasticity. In the temperature range where the glass is a simple thermorheological material, the normalized standard deviation of the relaxation spectrum can be shown to be temperature-independent. As a consequence, the ratio of delayed/instantaneous elastic deformation is also temperature-independent. The relationship between the magnitude of the delayed elasticity and the spread of the relaxation spectrum can be obtained in a other way [6]. It is also known from rheology [7,8]. From eqs. (15) and 16 the total elastic compliances, Jl,tot and Dtot, that account for both the instantaneous and delayed elasticity, and the corresponding total elastic moduli, Gtot and Etot, may be written as

J 2 ( / ) = ( 3 K g ) -1

+ [(3K~)-' - (3Kg)-'] (1 - ~ 2 ( ' ) ) ,

(14) J , ( t ) - ( 2 G ) - ' + (2G)-'(o?/('r,)2)(1 -- ~ l ( t ) ) + (t/2n),

D(t) = (e)-'

+ (e)-'(,,~/0-u)~)(1

+ (t/3n),

(15)

- ~o(t))

J,,tot = ( 2 G ) - ' ( 1 + o 2 / 0 " 2 ) ) = 1/(2Gtot),

(15a)

Oto t = ( E ) - I ( 1

(16a)

q- o.J/(,.r2)) = 1/Eto ,.

The relation between the uniaxial viscoelastic modulus, E(t), and the viscoelastic moduli, Gl(t ) and G2(t), is the viscoelastic analogous of eq. (3):

1/(3pG~ ) = 1 / ( pE* ( p ) ) - 1/(3pG~' ( p ) / 2 )

(16)

where the ~i (i = 1, 2 or u) are the corresponding retardation functions (~i(0) = 1 and ~i(oo) = 0). On the right side of eq. (14), the first term is the instantaneous elastic compliance and the second term is the delayed elastic compliance. On the right side of eqs. (15) and (16), the first term is the

(17) or, involving the relaxation functions ~ * , with the help of eqs. (7-9), 1 Eq'u*

1

a

3G~P~" + 9K¢ - 9 p ( K ~ -

. (18) Kg)q, ff

R. Gy / Compressibilityof soda-lime silicate glass

104

According to eq. (11), in the limiting case of p = 0, it reduces to 3G ('gu)

= -'g-- ( T 1 ) =

3 2(1 + v) (rl)"

(19)

Finally, the following value for the K e / K g ratio in terms of the Poisson ratio can be derived from eq. (20): K,

In the same way, the first derivative of eq. (18), with respect to p, in the limiting case of p = 0 and according to eq. (11) and eq. (12), reduces to

1( 1 + (%)2 t] = ~-~

1 +-~]

1 + 9Ke1

or, in terms of the total elastic moduli Etot and Gtot, defined in eqs. (15a) and (16a): 1/Etot = 1/(3Gtot) + 1/(9K~).

[+°°t~(t) ao

dt = (*i), dt = o? + (ri) 2.

3 1+--

-2(1+v)

1+-(24)

This equation shows that in the temperature range of simple thermorheologlcal behaviour, the equilibrium compressibility of the glass should depend only slightly on the temperature (as do the instantaneous elastic constants).

(21)

This equation is the same as eq. (3), the shear and uniaxial instantaneous moduli being replaced by the corresponding total moduli and the instantaneous bulk elastic modulus being replaced by the equilibrium bulk elastic modulus. It allows K~ to be evaluated, provided the magnitude of the total elasticity (instantaneous + delayed) is measured both in pure shear and pure uniaxial stress. The total elasticity is probably measured more accurately than either of its two components, because distinguishing the strictly instantaneous elastic deformation from the beglning of the delayed elastic deformation may be rather difficult (it might even be hopeless because the existence of truly instantaneous deformation is questionable). Since the 'instantaneous' moduli are expected to be very close to the well-known conventional low temperature elastic moduli, eq. (20) also allows Ke to be computed, provided that the uniaxial and pure shear relaxation spectra are known. The spectra of relaxation times are not in fact needed, only their mean values and variances. These quantities are conveniently computed directly from the relaxation functions since the following equations can be derived from eq. (11) and eq. (12): fo+~/(t)

1 - 2v

3. Application with experimental data for soda-lime

glass Uniaxial data have been published by DeBast and Gilard [9] and pure shear data by Kurkjian [10] and Rekhson and co-workers [11,12]. All these authors investigated soda-lime silicate glasses that were not rigorously of the same chemical composition but have only small differences as can be appreciated in table 1. It is assumed in the following that their viscoelastic behaviour is not significantly affected by this variation in composition.

Table 1 Chemical composition of glasses (wt%) DeBast Gilard [9] SiO2 Na20 K20 CaO MgO MnO Fe20 ~

72.22 13.71 0.23 8.01 4.04 0.04 0.115

A12%

1.09

(22)

As203 TiO2

0.02 0.03

s%

0.40

(23)

Fire loss

0.18

Kurkjian [10]

Rekhson and Ginzburg [11]

SiO2 Na20 K20 CaO MgO A1203 B203 Others

SiO2 Na20 K20 CaO MgO AI203 Others

67.7 15.1 1.2 5.6 4.0 2.8 1.5 0.1

72.08 13.11 0.59 6.64 4.88 2.00 0.89

R. Gy / Compressibility of soda-lime silicate glass

DeBast and Gilard proposed the following equation for the best fit of their experimental uniaxial relaxation function:

t

( t ) = exp( - at b),

(25)

with b = 0.5435 ( + 0.01), constant in the investigated temperature range. They also gave the theoretical relation between the magnitude of the delayed elasticity and the exponent b. From their calculations, the quantity (1 + Ou2/2) can be related to b, involving the gamma (Eulerian) function:

1+

o~ 0,,> 2

= bF(2/b) [F(1/b)] 2"

(26)

With the above value of b, 1 + o2/(%> 2 = 2.493. They found experimentally a mean ratio of delayed/instantaneous deformation of 1.39. The agreement with the theoretical value 1.493 can be considered satisfactory since a wide spread is reported for the measures of the ratio of delayed/ instantaneous deformation [9]. Apparently, Kurkjian did not study the torsional delayed elasticity, nor did Rekhson and co-workers report the delayed/instantaneous deformation ratio in their pure shear recovery experiments. Therefore, in the following, the computed variance of the spectrum derived from the experimental relaxation data will be used. These data from refs. [10] and [12] are listed again in table 2. In fig. 1, the experimental values of Uq'l(U ) versus u are plotted in both cases (u is reduced time: t/(rl> ). It is clear on this chart that the best accuracy on the numerically computed integral of ug'l(u ) (i.e., on 1 + o 2 / ( r 1 > 2) is obtained with Kurkjian's data that give more information on the long-time relaxation. In fact, the temperature scanning technique [12] used by Rekhson et al. was specifically devised to measure with an improved accuracy the short-time relaxation. For our purpose however, the accuracy is not required on the short-time relaxation, but on the long-time relaxation. It is also clear that even with Kurkjian's data, the entire integral over an infinite time range

105

Table 2 Tabulated experimental pure shear relaxation data R e k h s o n et al., [12]

Kurkjian [10]

log(t)

q'1(t)

- 5.540

0.982 0.974 0.966 0.955 0.945 0.933 0.920 0.904 0.878 0.841 0.795 0.731 0.639 0.507 0.339 0.167 0.047

-5.140 -4.74 -4.34 - 3.94 -3.54 -3.14 - 2.74 - 2.34 - 1.94 - 1.54 - 1.14 -0.740 -0.340 0.060 0.460 0.860

t(s)

XOl(t) 0 10 20 30 50 80 100

150 200 400 600 800 1000 2000 4000 6000 8000 10000

20000 40000 60000 80000 100000

150000 170000 200000 300000 500000 600000

1 0.955 0.941 0.933 0.923 0.913 0.9O8 0.897 0.888 0.860 0.840 0.825 0.810 0.762 0.697 0.650 0.610 0.578 0.462 0.335 0.265 0.215 0.180 0.120 0.102 0.082 0.047 0.013 0.007

0,4 u ~' (u) ]

0,3

0,2

~

=

Ke uk rh ks jo in ad nds 'as R ' a ttaa

0,i

0,0 2

4 6 8 (dimensionless

i0 12 14 r e d u c e d time) u

Fig. 1. A determination of the instantaneous/total elasticshear moduli ratio (1 + O2/<'rl> 2) can be m a d e from experimental pure shear relaxation data b y c o m p u t i n g the integral o f over an infinite reduced time range.

u~I'l(u)

R. Gy / Compressibility of soda-lime silicate glass

106

remains underestimated because of the cut-off at the last experimental point. Since the numerical value of the truncated integral (computed with the trapezium method) is found to be 1.947, the following inequality holds: 1 + o'?/~T1) 2 > 1.947. An attempt can be made to reach a closer value of 1 + o ] / ( T 1 ) 2 by extrapolation. Rekhson [131 fitted Kurkjian's experimental data with a six-term Prony series: i=6

qZi(t ) = y~ wa,i exp(--t/'rl.i), i=1

where i=6

i=6

~'1"1) = E Wl,i'rl,i

and

0 2 + ~'l'a) 2 =

i=l

E Wl,iT12,i• i=1

With the numerical values of wI i and r~ i, listed in table 3 from ref. [13], the result'is 1 + a~/(~l) 2 = 2.23. The Poisson ratio of the soda-lime silicate glass being 0.22, eq. (24) can now be used to estimate the Ke/Kg ratio: with 1 "t-Or?//~'/'l)2--~-2.23 and 1 + o2/(~u) 2 = 2.39, K e / K ~ = 0.32; with 1 + o]/(~'a) 2 = 2.23 and 1 + o2/(~-u) 2 = 2.493, K J K g = 0.27. These estimations of the Ko/Kg ratio for the soda-lime silicate glass are close to the analogous ratio measured with vitreous boron oxyde in its transformation range [2]. With K J K g = 0.3, E = 7.1 x 10 l° Pa and ~ = 0.22, the isothermal compressibility of the soda-lime silicate liquid,

1/K~ --- 3(1 - 2~,)/(0.3E) = 7.9 x 10 -11 Pa -1, and the configurational contribution to the compressibility,

4. Conclusion The method used in this paper to derive the isothermal compressibility from pure shear and pure uniaxial stress relaxation data would have given a more accurate result, if the experimental relaxation data had been given with accuracy for the longer times as is required for the precise computation of the integral of u ~ ( u ) . This requirement raises the following experimental difficulties: (i) accurate recording the lower force levels; (ii) recording the force over a long period of time without drift. A alternative method would be to measure the total (instantaneous + delayed) elastic recovery both in pure shear and pure uniaxial creep experiments. This method could well give a better result since it should be less sensitive to the experimental errors, since the distinction between the instantaneous and delayed recovery is not required. The author is grateful to Drs J. Barton, C. Guillemet and S.M. Rekhson for assistance with the manuscript.

Appendix: Derivation of the viscoelastic compli-" antes

The following classical properties of the Laplace transform are used in this appendix: l i m f ( t ) = lim p f * ( p ) ,

t "-'~0

p---, oc

ml0= moPi*(p),

A ( 1 / K ) = 1 / K e - 1/Kg -- 5.5 X 10 - n Pa -~

f'*(p) =pf*(p) -f(0+),

can be estimated.

f ' being the first derivative of f with respect to t.

Table 3 Coefficients for the best fit of Kurkjian's data with a six-terms Prony series after Rekhson [13]. wa,i: ra,i:

0.0427 1.900 X 101

0.0596 2.919 )< 102

0.0877 1.843 X 103

0.2454 1.180 × 104

wld , weight of ~'1,i in the Prony series. ~a,i, discrete (finite) distribution of relaxation times in a Prony series expression for gq(t).

0.2901 4.949 x 104

0.2498 1.717 × 105

R. Gy /Compressibility of soda-lime silicate glass The initial values of the compliances can be derived: lim Ji ( t ) = lim ( p J / * ( p ) ) = p ---, ~:~

lim

(1)

[1(1

= p~O lim -~ G i . ( p ) _ (d(1/Gl*)

pGi,(p )

t ---* 0

dp 1 lim G i ( t ) '

107

1}]

Gi.(O)

/ = - (dG,*/dp)p=O j~=o (G,*(O)) 2

Also, according to eq. (12), in the case of pure shear:

t ---~ 0

1(

hence

°12t

.11(0) = 1 / 2 G , J2(0) = 1 / 3 K g and D ( 0 ) = 1 / E .

t l i m ( J a ( t ) - t/2*/) = ~-~ 1 + <,1.1>2 ],

Similarly, the final values of the compliances can be derived:

and in the case of pure uniaxial stress:

lim

t~oo

J/(t)=

lim ( P J i * ( P ) ) =

p~O

lim (

1

p~o k PGi* ( p )

)

1 lim Gi(t ) '

lim (Ju(t)-t/3rl)=T

1(

t

1+ (%)2j-

As conclusion, the compliances can be expressed as

t - - ~ O0

hence

J2(t) = (3Kg) - '

Jl(oo) = + oo, Ju(OO)= + oo and 1

+ [(3Ke) - 1 -

J2(~) = ~

= 3K¢"

Jx(t) = ( 2 G ) -1 + ( 2 G ) - ' ( o 2 / ( ~ 1 > 2 ) ( 1

In the case of pure shear or pure uniaxial stress, the limit is not finite, but the limit of the first derivative is finite: lim l ~

t--, ~

( 3 K g ) - l ] (1 - ~ 2 ( t ) ) ;

1

=

lim

p-*O +

(( p

'

pGi* ( p )

J~(0)

))

1

Gi* (0) "

- q/ia(t))

+ (t/2*/);

D(t) = ( E ) - ' + (E)-a(O2u/<'r.>2)(l -

qbu(t))

+(1/3./). where the ~i (i = 1, 2 or u) are normalized decreasing functions of the time ( q ' i ( 0 ) = 1 and ~ i ( o o ) = 0).

For pure shear: G,* (0) = 2 G ~ * (0) = 2 G ( ¢ a ) ( a c c o r d i n g to eq. (11)) = 2,/ (definition of the viscosity ,1). F o r uniaxial stress: Gi* (0) = E g ' * (0) = E ( % ) = 3G(~'t>(cf. eq. (19)). = 3*/

T h e linear c o m p o n e n t of the compliance can be substracted from the total compliance and the limiting behaviour of the remainder is lim ( J ~ ( t ) - t/Gi*(O)) t~oO

= lim[p(Ji*(p)-l/(Gi*(O)p2))] p--*O

References

[1] J.A. Bucaro and H.D. Dardy, J. Appl. Phys. 45 (1974) 5324. [2] R.D. Corsaro, Phys. Chem. Glasses 17 (1976) 13. [3] G.W. Scherer, Relaxation in Glass and Composites (Wiley, New York, 1986). [4] C.A. Angell and W. Sichina, Ann. NY Acad. Sci. 279 (1976) 53. [5] See, for example, ch. 7 in ref. [3]. [6] N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behaviour. An Introduction (Springer, Paris, 1989) (see ~4.6 and problem 4.6.10). [7] A.G. Frederickson, Principles and Applications of Rheology (Prentice-Hall, Englewood Cliff, NJ, 1964) ch. 6. [8] H.A. Barnes, J.F. Hutton and K. Waiters, An Introduction to Rheology (Elsevier, Amsterdam, 1989) sections 4.2 & 4.5. (The quantity O = G[o~ + (¢~>] is called the 'elastic

108

R. Gy / Compressibilityof soda-lime silicate glass

normal stress difference coefficient'. It is shown to be related to the amount of 'recoverable shear' and is a "measure of how elastic a liquid is".) [9] J. Debast and P. Gilard, Comptes-Rendus de Recherches IRSIA, Travaux du Centre Technique et Scientifique de l'Industrie Belge du Verre 1 (32) (1965) 192 pp. [10] C.R. Kurkjian, Phys. Chem. Glasses 4 (1963) 128.

[11] S.M. Rekhson and V.A. Ginzburg, Sov. J. Glass of Phys. Chem. 2 (1976) 422. [12] S.M. Rekhson, N.O. Gonchukova and M.A. Chcmousov in: Proc. 11th Int. Cong. on Glass, Prague, 1977, Vol. 1. [13] S.M. Rekhson, Glass Science & Technology, Vol. 3, Viscosity and Relaxation, exls. D.R. Uhlmann and N.J. Kreidl (Academic Press, New York, 1986) p. 20.