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Materials Science and Engineering A 442 (2006) 263–267

Temperature dependence of the viscosity through the glasstransition in metaphosphate glasses and polystyrene

Hiroshi Kobayashi a,∗, Haruyuki Takahashi b, Yosio Hiki c

a National Metrology Institute of Japan, AIST, Retired, 8-1-309, 2-Chome, Tomooka, Nagaokakyo 617-0843, Japanb Faculty of Engineering, Ibaraki University, Nakanarusawa, Hitachi 316-0033, Japan

c Faculty of Science, Tokyo Institute of Technology, Emeritus, 39-303, Motoyoyogi, Shibuya-ku, Tokyo 151-0062, Japan

Received 5 July 2005; received in revised form 24 February 2006; accepted 6 March 2006

bstract

We measured the temperature dependence of the viscosity of metaphosphate glasses and polystyrene in the liquid and glass states using a methodeveloped by authors. The results show the Vogel–Tammann–Fulcher equation on the viscosity–temperature relation above the glass transitionemperature Tg and the Arrhenius one below Tg. In order to explain this viscosity change, the intermediate range order (IRO) is introduced. The

ffect of composition on the viscosity of metaphosphate glass and that of thermal and mechanical treatments on the viscosity of polystyrene wereeasured. The kinetic (viscous) transition was discovered at a temperature lower by 20 K than Tg determined by differential scanning calorimetry.e propose that the glass transition is a self-organization of the IRO of nanometer orders in size.2006 Published by Elsevier B.V.

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eywords: Glass; Viscosity; Glass transition; Complex system; Intermediate ra

. Introduction

The glass transition is a fundamental phenomenon expectedo be clarified not only on the physical viewpoint but also on thendustrial one for developments of inorganic, organic, polymericnd metallic glasses. It should be noted that the glass transitions a typical example of the complex system, which is very inter-sting in the recent physics. As temperature decreases, a glassorming liquid tends to freeze through a supercooled liquid intosolid through cooperative rearrangements of groups of parti-

les. In this temperature region, the properties of the liquid andhe glass show little changes, except that the viscosity variesy about 10 orders of magnitude. An explanation of the viscos-ty change through the glass transition is the key to solve the

echanism of the glass transition. This paper presents a phe-omenological model aiming at explaining the transition.

As a glass is in a non-equilibrium state, it is supposed that

here are intermediate range orders (IROs) in the glass state,hich are called cooperatively rearranging regions (CRRs) [1],

patial heterogeneities [2] or clusters [3], which are studied the-

∗ Corresponding author. Tel.: +81 75 952 6786; fax: +81 75 952 6786.E-mail address: hkoba@mx2.canvas.ne.jp (H. Kobayashi).

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921-5093/$ – see front matter © 2006 Published by Elsevier B.V.oi:10.1016/j.msea.2006.03.122

rder; Self-organization

retically and experimentally. The origin of heterogeneity haseen discussed from the standpoint of density fluctuation byhe small angle X-ray scattering [4] and micro-crystallites byhe high-resolution electron microscopy [5]. These experimentalesults show that small heterogeneous regions of a few nanome-ers in size are present in silica glass.

We suppose that the IRO is strongly related to the staticnd dynamic properties of glasses including the glass transition.here are many theories aiming at explaining the glass transition;

he Vogel–Tammann–Fulcher (VTF) [6,7], free-volume [8],dam–Gibbs [1], mode-coupling [9] and molecular dynamic

10] theories. The concept of fragility was reviewed on relax-tion liquids and glasses by Angell [11]. Debenedetti and Still-nger [12] reviewed the phenomenology of vitrification andupercooling by a model of the energy landscape, which haseen actively studied by computer calculations.

All theories introduce the relation η ∼ 1/(T−T0), where η ishe viscosity, T the temperature, T0 a constant, which meanshe divergence of viscosity at T0. However, Plazek and Mag-ll [13] measured the viscosity of aromatic hydrocarbon in a

ide temperature range and reported the tendency of the non-ivergence of viscosity, which follows the VTF equation aboveg but exhibits the Arrhenius behavior below Tg. The relaxation

ime τ is connected with the viscosity η by the Maxwell rela-

2 e and Engineering A 442 (2006) 263–267

tτ

[ttτ

iToo

dgciaigtait

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((hTee

η

Fig. 1. Temperature dependence of the viscosity η of metaphosphate glass(AgI)0.5(AgPO3)0.5. The data indicated by (�) were obtained by an oscillat-ing disk viscometer and those indicated by (©) were obtained by the sandwichmethod.

64 H. Kobayashi et al. / Materials Scienc

ion τ = η/G, where G is the shear modulus. It is expected thatshows the same temperature dependence as η. Alegria et al.

14] measured the relaxation time τ–temperature T relation byhe dielectric method for a polar glass polymer. They calculatedhe temperature dependence of the free volume vf deduced forfrom the free volume theory, in which vf ≈ vt − vc, where vt

s the total volume and vc is the volume occupied by the CRRs.hey reported that, in the region T > Tg, the VTF behavior isbserved in the τ−T relation, while the Arrhenius behavior isbtained in the region T < Tg.

Yamamuro et al. [2] have calculated the temperature depen-ence of the number of molecules in the CRR for severallass forming liquids following the Adam–Gibbs theory. Theyoncluded that, for all the molecules examined, the CRR sizencreases with decreasing temperature and becomes constant atbout four to eight molecules bellow Tg. From their assertion, its assumed that the size of the IRO is frozen-in bellow Tg and thelass state is established, which would explain Arrhenius-typeemperature-dependence of viscosity in the glass state. Martin etl. [15] reported an experiment in which the IRO can be producedn amorphous materials. This work is helpful for applications ofhe IRO, such as the developments of new glasses and devices.

In our recent papers [16–19], we have shown the temper-ture dependence of the viscosity of inorganic and polymericlasses from high (liquid) and low (glass) temperature, whichoes not tend to diverge at the critical temperature T0. In thisaper, we assert that the IRO should be strongly related to thishenomenon.

. Experimental methods

The sandwich method for measuring high viscosity waseveloped by the authors [16], which can be applied to deter-ine the viscosity in the range of 108–1014 Pa s and from room

emperature up to 150 ◦C. The principle of the shear deforma-ion of solids is realized in this method. The deformation of apecimen of 10−2 m × 10−2 m × 10−2 m under constant sheartress is measured by an optical interferometer with a sensitiv-ty of 10 nm in displacement, which can detect high viscosityelow Tg. The results were analyzed by the visco-elastic theory.or measuring low viscosity, a commercial oscillating disk vis-ometer was used. This viscometer constructed of two parallellates between which the specimen is putted and an upper plates rotated or oscillated.

. Results

The viscosityηof metaphosphate glasses; (AgI)x(AgPO3)1−x

x = 0.0, 0.5) is shown in Figs. 1 and 2. In Fig. 1, the low values�) were obtained by an oscillating disk viscometer and theigh values (©) were obtained by the sandwich viscometer.he former data in the high temperature region follow the VTFquation, Eq. (1) well and the latter in the low temperature region

xhibit the Arrhenius behavior, Eq. (2).

= η0 exp

{DT0

T − T0

}, (1)

Fig. 2. Temperature dependence of the viscosity η of metaphosphate glasses.The data indicated by (©) are of (AgI)0.5(AgPO3)0.5 and those indicated by (�)are of AgPO3.

H. Kobayashi et al. / Materials Science and Engineering A 442 (2006) 263–267 265

Fii

wc

η

w

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Fig. 4. Temperature dependence of the viscosity η of polystyrene. The dataindicated by (©) were obtained for the specimen, which is as-received, the dataindicated by (�) were obtained for the specimen which is annealed for 24 h at7k

Doacivrodoes not exceed the increase by the modification of the heat-ing rate. This result means that the origins of the two transitiontemperatures are different. It is assumed that, below the kinetictransition temperature, the glass is truly in a non-equilibrium

ig. 3. Temperature dependence of the viscosity η of polystyrene. The datandicated by (�) were obtained by an oscillating disk viscometer and thosendicated by (©) were obtained by the sandwich method.

here η0 is the pre-exponential constant, D a constant, T0 aonstant.

= η0 exp

(E

RT

), (2)

here R is the gas constant, E the activation energy.In Fig. 2, the temperature dependence of viscosity below Tg

ith a different composition in metaphosphate glass is shown,hich is more sensitive to the composition in the low temper-

ture region than in the high temperature region, because thelass is in a non-equilibrium state. Figs. 3 and 4 show the vis-osity of polystyrene, also showing the same tendency of theon-divergence of viscosity as metaphosphate glasses. In Fig. 3,he data indicated by (�) were obtained by an oscillating diskiscometer, and those indicated by (©) were obtained by theandwich viscometer. These data also obey the VTF equationn the high temperature region but follow the Arrhenius rela-ion in the low temperature region. In Fig. 4, the temperatureependence of viscosity below Tg with different specimen treat-ents in polystyrene is shown. The viscosity in the glass region

s shown sensitive to the specimen treatments; it is similar to theffect of composition on the viscosity of metaphosphate glasseshown in Fig. 2.

The glass transition temperatures Tg shown in Figs. 1 and 3

ere determined thermally by the differential scanning

alorimetry (DSC) at a heating rate of 10 K/min. The η − 1/Turves in these figures show another kinetic (viscous) transition,hich appears lower by about 20 K than Tg determined by the

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0 ◦C and the data indicated by (�) were obtained for the specimen which isept for one week under the weight of 9.8 N.

SC. The existence of the two transition temperatures has beenbserved in aromatic hydrocarbons by Plazek and Magill [13]nd in several polymeric glasses by the authors [20]. In order toonfirm that the appearance of the two transition temperaturess not due to the difference in the heating rate, we measured thealues of Tg on polystyrene by changing the heating rate q; theesults are shown in Fig. 5. The viscosity was measured at a ratef about 1 K/min. The difference in Tg of the two transitions

ig. 5. Dependence on the heating rate q of the glass transition temperature Tg

f polystyrene. The q in the DSC measurement was 10 K/min and the q in theiscosity measurement was 1 K/min.

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66 H. Kobayashi et al. / Materials Scienc

tate, because the viscosity–temperature relation below this tem-erature is sensitive to the specimen composition and treatments mentioned above. The attempt to connect such thermody-amic properties with kinetic behavior is very interesting andmportant [21]. However, we do not have any idea about thehysical origin of this critical temperature.

. Discussions

The Adam–Gibbs theory [1] gives the following equationetween the relaxation time τ and the temperature T,

= τ0 exp

(C

TSc

), (3)

here τ0 is the pre-exponential constant, C a constant related tohe free energy barrier, Sc the configurational component of theotal entropy. When Sc changes with temperature as (T−T0)/T0,he Eq. (3) becomes the VTF equation, Eq. (1). When Sc isonstant, Eq. (3) equivalent to the Arrhenius equation, Eq. (2).ccording to Eq. (1), the viscosity η tends to diverge at the

emperature T0 by the vanishing of Sc.The temperature dependence of the viscosity shown in

igs. 1–4 means that their activation process (VTF) in the super-ooled liquid changes to another (Arrhenius) in the glass acrosshe glass transition temperature. This change of the process isxplained by the following idea. While Sc of Eq. (3) in the super-ooled liquid state is dependent on temperature and follows theTF equation, it is constant in the glass state and follows therrhenius equation. We believe that a quantitative study of Sc iskey to clarify the glass transition mechanism.

Later, we present another model to explain this transitiony application of the IRO. When the activation process fol-ows the Arrhenius equation in the whole temperature rangen Figs. 1 and 3, Eq. (2) applies to our results, and the valuef E is related to the height of the potential barrier separatingouble-well potential minima, which is constructed by the IROsoth in liquid and glass states. It is assumed that the barriereights in the liquid state increase with decreasing temperaturend increase abruptly at a temperature near T0, and become con-tant below Tg in the glass state. This assumption means that theize and number of the IRO increase with decreasing tempera-ure above Tg and become constant below Tg as the number of

olecules in the CRR mentioned above. We can obtain the acti-ation energy E1 in the limited temperature range slightly aboveg in Fig. 2, where the viscosity follows the VTF equation, and2 in the range below Tg, where the viscosity follows the Arrhe-ius equation. The activation energies for (AgI)0.5(AgPO3)0.5re E1 = 465 kJ/mol and E2 = 60 kJ/mol [18]. The energy mapssociated with the structural configurations, which the liquidnd glass may adopt would show a field pattern in which thearrier heights are lower in the glass state than those in the liq-id state.

It should be emphasized that the glass transition depends on

he cooperative rearrangement of groups of molecules but notf a single molecule. We reported [22] that the viscosity changes a complex relaxation, that is, a cooperative phenomenon byomparing the relaxation times obtained by measurements of

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Engineering A 442 (2006) 263–267

he dielectric constant and viscosity of metaphosphate glass.t is known that materials composed of complex molecules orith a composition of different sizes tend to become glasses.etallic glasses developed recently are composed of the multi-

omponents more than three elements, and include the interme-iate range ordered atomic configurations.

Because it is assumed that the size of the IROs is very small,few nanometers, their boundary and size are important to esti-ate the properties of glass. From this viewpoint, many studies

f the effect of heat treatments including aging at different tem-eratures below Tg have been carried out. In this paper, weescribed the effect of composition on the viscosity of metaphos-hate glass in Fig. 2 and of thermal and mechanical treatmentsn the viscosity of polystyrene in Fig. 4. Their viscosity is moreensitive to the composition and treatment in the glass state thann the supercooled state. From these considerations, we sup-ose that the IRO, which increases in size and number with theemperature decreasing down to Tg, suppresses the crystalliza-ion of the glass forming liquids and generates their glass stateelow Tg. The glass transition process is considered to be a self-rganization of the IRO in glass forming materials.

. Conclusions

The temperature dependence of the viscosity of glasses doesot show the tendency of divergence at any temperature. Theiscosity of glasses is sensitive to the composition, and thermalnd mechanical treatments in the glass state. A new transitionn the viscosity–temperature change was discovered at a tem-erature lower than Tg determined thermally. The idea that thelass transition is a self-organization of the IRO is proposed.

cknowledgements

The authors are very grateful to Professor S. Kojima ofsukuba University and Dr. H. Yamashita of AIST for manyelpful discussions.

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