Mechanical relaxation in vitreous borates in the transition range

Journal of Non-Crystalline Solids 29 (1978) 119-129 © North-Holland Publishing Company

MECHANICAL RELAXATION IN VITREOUS BORATES IN THE TRANSITION RANGE I.A. JANSEN-FALKENBURG, W.J.TH. VAN GEMERT and J.M. STEVELS Laboratory of Inorganic Chemistry, Eindhoven University of Technology, Eindhoven, The Netherlands Received 6 December 1977

Dynamic mechanical relaxation measurements have been carried out in vitreous borates in the transition region at frequencies ranging from 2.5 to 10 s -1. The observed loss peaks are caused by network motions. Their temperature positions are related to the degree of cohesion of the systems. The temperature dependence of the relaxation times can be described by a free volume theory resulting in the WLF equation, as is verified in the case of B203.

1. Introduction Many mechanical relaxation measurements in vitreous systems for temperatures up to the transition range have been presented. These relaxation phenomena are sometimes called the secondary relaxations and cause, for example, the single alkali peak, the mixed alkali peak and the mixed alkali-proton peak. All these experiments show an abrupt increase of the mechanical losses when the temperature approaches the transition range. These relatively high losses occur in the softening region, and thus it is convenient to ascribe them to motions of the network and call them the primary relaxations. Dynamic mechanical relaxation measurements in the transition range have been executed by Coenen and Amrhein [ 1]. They presented bending and torsion experiments at low frequencies (between 5 and 15 s -1) on some vitreous borates, vitreous silicates and some technical glasses. These measurements reveal the presence of a relaxation peak in the softening region. Analogous measurements on vitreous borates are presented in this paper, and an attempt is made to explain the influence of alkali ions on the position of the network relaxation peak.

2. Experimental procedure and sample preparation Energy dissipation has been measured by determining the decay rate of a freely vibrating system. A major difference from the measurements of the secondary 119

120

1,4. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

relaxation phenomena is that the material under test had to be measured in the form of a surface layer on a substrate. The measurements have been carried out in a bending as well as a torsional mode. Some of the experimental features in the following are discussed below. (1) Bending experiments have been carried out using a substrate of fused silica. This material proves to be adequate since: (a) it is chemically inert in the temperature range involved; (b) the energy dissipation in fused silica is very low at temperatures up to 1000°C; and (c) the physical adhesion of borates to fused silica is good. The frequency of measurement by this method varies from 4 to 10 s -1. (2) Torsion experiments have been carried out on a P t - R h (9 : 1) substrate. Platinum is chemically inert at temperatures up to 1000°C and adhesion of the glass specimen is good. A disadvantage is that platinum has rather high losses at higher temperatures. A measure for the damping is tan 6, which can be approximately calculated from the experimental data by tan 6 -- (1/mr)" ln(tn/to) (1) where 6 is the loss angle, n the number of passages of the light beam across the two photodiodes between t = to and t = tn and to, tn are the points of time in which the light beam crosses the photodiodes in the zeroth and nth cycles, respectively. Discs have been cast after melting mixtures of B203 and alkali carbonates in a platinum crucible for 2 h at 1000°C in an electrically heated furnace. Batches containing silver were prepared using porcelain crucibles. In order to prevent reaction with water from the atmosphere the discs were stored under dry oil. Immediately before measurement the discs were ground and applied to the substrate as smoothly as possible. When the temperature was raised to a value high enough above the glass transition temperature the layer became fairly uniform. The system was then cooled and the measurements were carried out simultaneously. When a temperature was reached where the glass became rigid and cracks arose owing to the difference in volume expansion between the specimen layer and the substrate, the experiment was concluded. 3. Interpretation of the measured loss peak It is supposed that the elastic properties of the substrate and the glass specimen add up linearly. Thus, the total energy dissipated in the system is the sum of the contribution of the volume fraction of the substrate and of that of the volume fraction of the zpecimen. This leads to a relation which correlates the measured damping (tan 6total) with the damping of the specimen (tan 6spec): tan 6total = (Ospec/°total) ' tan 6spec + (Osubstr/Ototal) " tan 6substr or

tan 6spec = (Ototal " tan 6 total - Vsubstr ' tan 6substr/Ospec),

(2)

LA. Jansen-Falken burg / Mechanical relaxation in vitreous borates

T

%

8O

/

** 6o g 4O 20 0

/

./

0

121

/

Y

02

0.4

specimen weight [g]-----~ Fig. 1. Verification o f the a s s u m p t i o n o f additivity for the system 0.25 N a K O • 0.75 B 2 0 3 ; bending.

where t a n ~ s u b s t r . is the damping of the substrate, Ospec , Osubstr the volume of the specimen and that of the substrate, respectively, and vtota I is the volume of substrate and specimen toeether. When Osubstr • tan

(~substr ~

Ototal "

tan (~total,

the relation may be approximated by tan 6 spec = (Ototal/Ospec) " t a n 6 t o t a I .

(3)

For simplicity, the room-temperature values for OtotaI and Ospec will be substituted into this equation. It is assumed that neglecting the temperature dependence of the densities causes an error which is within the error of measurement. A test of the validity of the assumption that the damping of the substrate and that o f the specimen add linearly is shown in fig. 1. A linear relationship exists between the measured peak height and the weight of the specimen which has been added. This justifies the use of a formula in the form of eq. (2).

4. Results Results of bending and torsion experiments for the system x NaKO • (1 - x ) B203 are shown in figs. 2 and 3. The halfwidths of the peaks are between 40 and 50°C. This is much smaller than the halfwidths of the secondary peaks measured at temperatures below the transition range. From these figures and from figs. 4 and 5 it is apparent that Tmax as a function of x passes through a maximum. In all cases this maximum is situated between x = 0.2 and x = 0.3. Table 1 and fig. 6 show the results of measurements on the mixed alkali system

122

I.A. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

bending

xNaKO.(1 - x ) B 2 0 3

0.3

360-

/5~02s 03 /Y/ 02

% ~ 240. c

12

:0

i

"

300

400 - - T

500 {*CI

600 >

Fig. 2. T a n 8 versus t e m p e r a t u r e for t h e s y s t e m x N a K O • ( I - x ) B 2 0 3 ; b e n d i n g .

% x '.,0

c ,,

200

400

buO

- - T [*C):,Fig. 3. T a n 6 versus t e m p e r a t u r e for t h e s y s t e m x N a K O • (1 - x ) B 2 0 3 ; t o r s i o n .

1,,4. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

123

torsion

480

I

500.

U

bending

38O E 400.

1--

300

/

~ /~

280 0

0-2

0~

x Na20.(1-x)B203 xK20.(1 - x) B 2 0 3 0.(1 - x ) B 2 0 3

i

0

0"1

X

0 2 -

-

0-3 X

Fig. 4. T e m p e r a t u r e o f m a x i m u m loss as a f u n c t i o n o f the alkali c o n t e n t for the system x NaKO • (1 - x ) B 2 0 3 ; bending. Fig. 5. T e m p e r a t u r e o f m a x i m u m loss as a f u n c t i o n o f the alkali c o n t e n t for several systems; torsion.

0.25(x N a 2 0 • (1 - x ) K 2 0 ) • 0.75 B203. In fig. 7 the value of Tmax is shown as a function o f x . It is evident that a mixing effect is present, although it is a small one. Measurements for mixed silver-alkali systems are illustrated in fig. 8. At the same time it is shown that the relaxation curves in and above the glass transition region are the extension of the curves measured at approximately 1 s -1 with a tor-

Table 1 System 0.25(xNa20.(1 - x ) K 2 0 ) • 0 . 7 5 B 2 0 3 ; bending m o d e x

1.0 0.96 0.80 0.60 0.40 0.20 0.04 0.00

tan 6ma x X 103

total

specimen

60 66 60 52 56 56 53 56

474 412 376 333 356 356 331 356

fmax (s-l)

Tmax (°C)

£XWl/2 (C) (half width)

9.0 10.1 9.5 8.5 9.0 8.9 9.3 8.9

520 515 498 482 480 475 485 490

40 40 42 40 40 40 40 40

124

1.,4. Jansen-Falkenburg / Mechanical relaxation in vitreous borates bending

i

x:Oi'8.~, ~1

400

~96

0,4

.t0

fO-04

x

530

/

510

200

o

lAt///

I 400 - - T

500

x

49O

470

600 )

i*CI

/ j,/

m

\ 0

1 -

-

X

~.-

Fig. 6. Tan 6 versus temperature for the mixed alkali system 0.25[x N a 2 0 - ( 1 - x ) K20 ] • 0.75 B203; bending• Fig. 7. Temperature of maximum loss as a function of the alkali mixing ratio for the system 0.25 [x Na20 - (1 - x ) K20 ] • 0.75 B203; bending.

400

o~..o0,~o~li %

torsion,1 s-| ( - - - ) C s ~

x g3 C

200

I 0

.

200

41 I0 ---T

500

600

I "CI

Fig. 8. Tan ~ versus temperature for mixed silver-alkali borates; bending. Dashed curves refer to low-temperature torsion measurements [2].

1,4. Jansen-Falkenburg/ Mechanical relaxation in vitreous borates

125

sion pendulum below the transition region [2]. Moreover, fig. 8 shows the large difference in heights between the secondary and primary peaks.

5. Discussion 5.1. Correlation o f the experimental data with the structure of vitreous borates According to Warren [3] the boron atoms in vitreous B203 have a triangular coordination o f oxygen atoms. In the systems x M20 • (1 - x) B203 the excess o f oxygen atoms introduced by the alkali oxide can give rise to the occurrence o f fourcoordinated boron ions (N4), or can be taken up in the form of asymetrical BO a groups (with one non-bridging oxygen atom, X). Bray and O'Keefe [4] assume that the introduction of alkali oxide up to x = 0.30 causes merely the formation of BO4 tetrahedra. Beyond this composition only asymmetrical BO 3 groups will also be formed. However, Beekenkamp [5] showed that the formation of non-bridging oxygen in detectable quantities already commences nearx = 0.15. The presence of BO 4 tetrahedra increases the rigidity of the network and the presence of asymmetrical BO3 groups decreases it. Visser [6] made it plausible that the rigidity-increasing influence o f the former is nearly equal to the rigidity-decreasing influence of the latter. So he came to the conclusion that a simple correlation exists between physical properties which are related to the network rigidity and the value of (N 4 - X). For this reason Tmax in the system x M20 • (1 - x ) B203 is a maximum for the same value of x, namely 0.23, where (N 4 - X) is a maximum (figs. 4 amd 5). At this value o f x the rigidity of the network is a maximum. 5.2. The effect o f the nature o f the alkali ions present on the position o f the loss maximum The temperature positions of the network relaxations can be influenced by the nature of the network modifying ions. In the system 0.25 M20 • 0.75 B203, network motions wilt occur by preference via - O - M - O - bonds. The strengtt of these bonds is determined by the ionic character of the bonds, which is turn depends on the sizes o f the M ÷ cations. Fig. 9 shows that the rigidity of the network is smaller (lower Tmax) the larger the size of M ÷. In the system 0.25(x Na20 • (1 - x) K20 ) • 0.75 B203 a weak negative deviation occurs from a linear dependence of the temperature of the peak maximum on x (see fig. 7). This can be understood qualitatively as followings. Viscous flow will mainly occur via the weakest bonds in the network. In the case considered they are the O - K - O - bonds rather than the O - N a - O bonds. Furthermore, Na ÷ ions have a larger field strength than K ÷ ions, owing to their smaller

126

LA. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

Li

Na

025M20 0 75B203

500

a00 o

2

--Rd

t ,l 3 -

Fig. 9. Temperature of maximum loss as a function of the alkali ion volume for the system 0.25 M20.0.75 B203; torsion.

radius. Therefore, the former ions will attract the electron clouds o f the latter ions if they occur together in the network. This results in a weaker-than-normal bond for the K ÷ ions. For x - 1, the mere presence of the K ÷ ions facilitates the viscous flow, but the partial depletion of their electron clouds by the polarizing influence of the Na ÷ ions will cause a further decrease in Tma x. For x - 0, Tmax would be increased by the presence of the Na ÷ ions but, in fact, is decreased by the weakening o f all those O - K - O bonds which are in the neighb o u r h o o d o f the Na + ions.

5.3. Description o f the kinetic processes by means o f the free volume theory One may attempt to represent the thermal motions o f atoms and molecules in a liquid as a jumping o f particles from certain momentary equilibrium states into others. The time necessary for the particles to jump, i.e. the relaxation time for the process, is determined by the equation r = r 0 exp(Ea/kT)

(4)

where E a is the activation energy for the transition of the kinetic unit (atom, segment) from one equilibrium position into another. This formula holds for many liquids. However, it is not surprising that the relaxation time, the viscosity and other kinetic properties of many liquids at relatively low temperatures do not exhibit the temperature dependence given by eq. 4 , M a n y investigators found that this formula does not hold in the transition range [ 7 - 1 2 ] . In order to explain the behavior o f the material in these temperature ranges it is possible, or even necessary, to assume that the activation energy Ea is temperature dependent. Another approach is based on the assumption that the mobility o f the molecules depends on the free volume in the network. Therefore, the time necessary for the molecules to rearrange is not determined by the probability o f a j u m p o f a particle,

I.A. Jansen-Falkenburg /Mechanical relaxation in vitreous borates

127

but rather by the probability that a sufficiently large free volume exists in the neighbourhood of that particle. The kinetic units in a liquid at sufficiently high temperatures may change their positions more or less independently of each other [the activation energy in eq. (4) is a constant]. However, the change of position of each particle at temperatures lower than the transition range Tg is related to rearrangements of its direct surroundings and perhaps of regions at a somewhat greater distance. A change in the arrangement is considered as a change in the number of vacancies in the neighbourhood of a particle. The probability of a unit leaving its position depends on the number of vacancies in its neighbourhood. Appearance and disappearance of vacancies are determined by heat fluctuations. The fraction f of the free volume can be defined by

f = f g + ( T - Tg)" Aa

when T > Tg

and

(5) f = fg

when T < Tg.

Thus, f is constant for all temperatures below Tg. In these regions the volume expansion coefficient a is the result of the increase in amplitude of the molecular vibrations with temperature. Above Tg a new free volume is created. This gives rise to an increase Ac~ in the volume expansion coefficient. Williams et al. [7] proposed that the logaritm of the relaxation times of the kinetic units should vary linearly with l/f, when the temperature is not too far above Tg, according to the relation ln(T/rg) = 1If

l/fg.

(6)

From eqs. (5) and (6) follows the well-known WLF equation: ln(r/rg) = - A '

T - Tg

(7)

where A and B are empirical constants. Here, A is a virtually universal constant with a value of about 40; B depends on the nature of the material. Sanditow [13] pointed out that B can be converted into a dimensionless universal constant B o by the empirical relation B o = (1 - B/Tg) ~- 0.66 for many glass-forming liquids. Therefore, eq. (7) can be rewritten as ln(r/rg) = - A " [ ( T -

Tg)/(T-

BoTg)] ,

(S)

Tg may be defined as the temperature at which rg = 100 s. After substitution of the numerical values for A and Bo into eq. (8) one finds log r = 2 - 17.4 [(T - Tg)/(T - 0.66 Tg)].

(9)

1..4. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

128

Substitution of eq. (9) into tan 6 = c. [cot/(1 + eo2r 2)

yields the expression co- 10 (2-17.4[(T-Tg)/(T-0.66 Tg)] ) tan8 = C - 1 + 602. 102(`2--17"41(T-Tg )/(T-°'6nTg)I}

"

Introducing A = tan 6 / t a n 6max

and

tan 8max = 0 . 5 C,

we o b t a i n

60. lO (`2-17"4[(T-Tg)/(T-O'66 Tg)]} A = 2 " 1 + (,02 • 102 (`2-17.4[(T-Tg)/(T-O.66Tg)]}

•

(10)

The m a x i m u m value o f A is u n i t y , and the position o f this m a x i m u m Tm is deter-

T

,o

~

08

E

06

0"4

02

0 /-,80

5 20

560 600 640 - - T IKI > Fig. 10. Comparison of the theoretical curve of tan 6/tan 6ma x versus temperature with the experimental curve for B2 0 3.

1.4. Jansen-Falkenburg / Mechanical relaxation in vitreous borates

129

mined b y the relation m r = 1. We thus obtain T m = Tg

( 0.924 - 0.038 log 6o ) 0.885 0.057 log • "

(11)

In fig. I0, A is plotted versus T for B20 3 glass. It is seen that the ampliature positions of the curves calcultated theoretically from eq. (10) are in good agreement with the experimental curves. F r o m this it can be concluded that the WLF equation is suitable for qualitative explanation o f the observed phenomena. The hal(width o f the measured curve is larger than the calculated hal(width. This may be a result o f the fact that the vacancies have been assumed to be o f equal shape. In reality they vary in shape, causing a spectrum o f r-values.

6. Summary and conclusions The mechanical losses o f vitreous borates measured both in bending and torsion experiments at 1 - I 0 s -1 at high temperatures are characterized by the presence o f a relatively high loss peak in the transition range. This relaxation is ascribed to network motions. As expected, the temperature position o f the peak is a function of the concentration, as well as the nature o f the alkali ions present. Measuring mixed alkali glasses with varying mixing ratios reveals that there is a weak deviation from additivity in temperature o f the peak maximum. This can be explained b y assuming that a polarization in the network is introduced by the presence o f alkali ions with unequal sizes. In B203, it has been shown that the measurements are in good agreement with the free volume theory, resulting in the well-known WLF equation.

References [1] M. Coenen and E.M. Amrhein, Symp. R6sist. M6ch. du Verre, Florence (1962). [2] W.J.Th. v. Gemert, H.M.J.M.v. Ass and J.M. Stevels J. Non-crystalline Solids 14 (1974) 281. [3] B.E. Warren, J. Am. Ceram. Soc. 24 (1941) 256. [4] P.J. Bray and J.G. O'Keefe, Phys. Chem. Glasses 4 (1963) 37. [5] P. Beekenkamp, Proc. Int. Conf. Phys. Non-Crystalline Solids, Delft (1964) p. 512; Thesis, University of Technology, Eindhoven (1965). [6] T.J.M. Visser and J.M. Stevels, J. Non-Crystalline Solids 7 (1972) 376; T.J.M. Visser, Thesis, University of Technology, Eindhoven (1971). [7] M.L. Williams, R.F. Landel and J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701. [8] F.G. Fox, J. Appl. Phys. 21 (1950) 581. [9] A.K. Doolittle, J. Appl. Phys. 22 (1951) 1471. [10] F. Bueche, J. Chem. Phys. 21 (1953) 1850. [ I 1] D.S. Sanditow and I.A. Isw. Wusow, Fisika 11 (1968) 93. [12] A. Napolitano, P.B. Macedo and E.G. Hawkins, J. Am. Ceram. Soc. 48 (1965) 613. [131 D.S. Sanditow, Hochmolekulare Verbindungen (Russ.) 10B (1968) 745.