Viscous flow and the viscosity of melts and glasses

ISSN 1753-3562

August 2012 Volume 53 Number 4

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Physics and Chemistry of GlassesEuropean Journal of Glass Science and Technology B

Volume 53 Number 4 August 2012

Cover: Figure 1. Optical micrographs (1,2) and SEM (3-8) images of as-prepared glasses 21-02 (1,3), 21-07 (5) and 21-21 (2,7) and their surfaces after heat treatment and etching (4,6,8, respectively) of the paper: Structural features of high-Fe2O3 and high-Al2O3/Fe2O3 SRS HLW glasses by Sergey V. Stefanovsky et al (Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, August 2012, 53 (4), 158–166)

CONTENTS143 Viscous flow and the viscosity of melts and glasses

Michael I. Ojovan151 Effect of alkali cations on the compressibility of MAlSi3O8 glasses (M=Na, K,

Rb, Cs) in the pressure range up to 6·0 GPaR. G. Kuryaeva

158 Structural features of high-Fe2O3 and high-Al2O3/Fe2O3 SRS HLW glassesSergey V. Stefanovsky, Kevin M. Fox, James C. Marra, Andrey A. Shiryaev & Yan V. Zubavichus

167 Structure evolution and optical properties of Co-doped zinc aluminosilicate glass-ceramicsI. Alekseeva, O. Dymshits, V. Ermakov, V. Golubkov, A. Malyarevich, M. Tsenter, A. Zhilin & K. Yumashev

A35 Abstracts

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Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 53 Number 4 August 2012 143

Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, August 2012, 53 (4), 143–150

Email [email protected] during the International Conference on the Chemistry of Glasses and Glass-Forming Melts, Oxford, September 2011Current address: Department of Nuclear Energy, International Atomic Energy Agency, A2644 WTS, Vienna International Centre, PO Box 100, Vienna, 1400 Austria. Email [email protected]

IntroductionThe viscosity (e.g. viscosity coefficient) is one of the most important properties of a flowing material, e.g. in the glass industry viscosity determines melting conditions, working and annealing temperatures, rate of refining, maximum use temperature, and crystallization rate. On Earth the behaviour of magma and hence volcanic eruptions and lava flow rates depend directly on the viscosity of mantle. Viscosity is a property of fundamental importance both for the molten and solid states of amorphous materials. The viscosity quantifies the resistance of material to flow and indicates the ability to dissipate momentum. The momentum balance of a Newtonian fluid is described at the macroscopic level by the Navier–Stokes equa-tions.(1) At the microscopic level the viscosity arises because of a transfer of momentum between fluid layers moving at different velocities as explained in Maxwell’s kinetic theory.(1) The shear viscosity coefficient η by definition is the ratio of shear stress divided by the velocity gradient (see Figure 1):

η=(F/A)/(V/H) (1)

where F is the force applied to the surface of area A, V is the velocity and H is the separation height between two plates. In terms of imaginary layers the tighter are the bound layers the more difficult is their motion and the higher is the resulting viscosity.

The viscosity governs relaxation processes as Maxwell’s relaxation time, which gives the time required to attain stabilised parameters of a material, is directly proportional to the viscosity coefficient τM=η/G, where G is the shear modulus.(2–5) The higher the viscosity the longer the relaxation time, which

for glasses might be enormous, e.g. fused silica at room temperature has τM ~1098 y.(6) Note that such time scales are practically infinite as they are incom-mensurably longer than the lifetime of the Universe ~1·5×1010 y.

The viscosity depends on the chemical composi-tion of materials, e.g. in silicate systems viscosity attains the highest values for vitreous silica. Viscos-ity is highly influenced by temperature moreover it is commonly assumed that shear viscosity is a thermally activated process. The viscosity has been expressed in terms of an activation energy Q by the Frenkel equation

η(T)=Aexp(Q/RT) (2)

where T is the absolute temperature, R is the molar gas constant and A is approximately a constant. This relationship also follows from the Eyring equation for viscosity

η(T)=(Nh/Vη)exp(Fη/RT) (3)

where Fη is the activation free energy of the viscous flow, which is equal to the change in the free energy of the system on the isothermal transition from the equilibrium state to the activated state, Vη is the activation volume of the viscous flow, h is Planck’s constant and N is the number of kinetic units (atoms,

Viscous flow and the viscosity of melts and glassesMichael I. OjovanDepartment of Materials Science & Engineeering, University of Sheffield, Sir Robert Hadfield Building, Mappin Street, Sheffield, S1 3JD, UK

Manuscript received 21 November 2011Revised version received 5 February 2012Accepted 28 February 2012

A brief compendium of viscosity models for melts and glasses is given including recent data on viscous flow model based on network defects in which thermodynamic parameters of configurons – elementary excitations resulting from broken bonds - are found from viscosity–temperature relationships.

Figure 1. Imaginary moving layers in a flowing materialFigure 1. Imaginary moving layers in a flowing material

144 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 53 Number 4 August 2012

molecules). It was revealed that at high temperatures typical for melts the activation energy of flow is lower compared with that at low temperatures typical for glasses. Both at high (e.g. for melts) and low (e.g. for glasses) temperatures an Arrhenius-type depend-ence of viscosity is observed with low QL and high QH activation energies respectively. At intermediate temperatures, which are approximately comparable with the glass transition temperature Tg, the activa-tion energy of flow depends on temperature and the viscosity cannot be described using Equation (2). Figure 2 shows an example of such changes based on Sanditov’s results who calculated the activation free energy of viscous flow of silicate melts at intermedi-ate temperatures using the concept of a fluctuation hole within the formalism of the hole-activation model.(7)

Apparent changes in activation energies of viscos-ity with temperature remain an unresolved issue for condensed matter science.(7)

Fragility concept

The term glass transition temperature Tg is often used to refer to the temperature at which the viscosity attains a value of 1012 Pas. This definition of Tg was used by Angell to plot the logarithm of viscosity as a function of (Tg/T).(8,9) In such a plot, materials that only exhibit small changes in the activation energy of flow with temperature have a nearly linear de-pendence on (Tg/T). These materials were termed strong. In contrast, materials termed fragile have activation energies which significantly change with temperature. Note that changes are characteristic only for intermediate temperatures and the viscosity is always Arrhenian at enough high or enough low temperatures compared to Tg. Asymptotically both at

low and high temperatures the activation energies of viscosity are constant, therefore the materials can be unambiguously characterised by Doremus’ fragility ratio,(10,11) which ranges from 1·45 for silica to 4·52 for anorthite (Table 1):

RD=QH/QL (4)

Amorphous materials are either strong if RD<2 or fragile if RD≥2. The higher the value of RD the more fragile a material. The initial implication of strong–fragile classification was that strong materials are strongly and fragile ones are weakly bonded.(9) However Doremus(10,11) concluded that these widely used convenient terms are misleading, e.g. binary silicate glasses are strong although they have many nonbridging oxygens. In contrast materials such as anorthite and diopside have very high activation energies and are strongly bonded although they are classified as fragile (see Table 1).

Viscosity models

Many equations to model viscosity have been pro-posed. The first one was the Frenkel model which as-sumed that viscosity is a thermally-activated process described by a simple exponential equation (Equation (2)) with a constant activation energy of viscosity.(12) As this simple model fails to describe the behaviour of viscosity at intermediate temperatures, many other models have been developed however few have be-come widely used. For example it is well known(13,14) that the best description of viscosity is given by the two-exponential equation of Douglas.(15) However the most popular viscosity equation is that named after Vogel, Tamman & Fulcher (VTF).(3,4) Although it fails to give correct asymptotic behaviour of viscosity it gives a very good description of viscosity behaviour at intermediate temperatures which are very impor-tant for industry. The Avramov–Milchev (AM) model is another equation used to describe the viscosity in

Figure 2. Temperature dependence of the activation en-ergy of viscous flow for a silicate melt.(7) Courtesy of D. S. Sanditov

Figure 2. Temperature dependence of the activation energy of viscous flow for a silicate melt.(7)

Courtesy of D. S. Sanditov

Table 1. Asymptotic activation energies of viscosity and Doremus’ fragility parametersAmorphous material QL (kJ/mol) QH (kJ/mol) RD

Silica (SiO2) 522 759 1·45Germania (GeO2) 272 401 1·4766·7SiO2.33·3PbO 274 471 1·7280SiO2.20Na2O 207 362 1·7565SiO2.35PbO 257 488 1·959·9SiO2.40·1PbO 258 494 1·9175SiO2.25Na2O 203 436 2·1575·9SiO2 24·1PbO 234 506 2·16SLS: 70SiO2.21CaO.9Na2O 293 634 2·16Salol (HOC6H4COOC6H5) 118 263 2·2370SiO2.30Na2O 205 463 2·2665SiO2.35Na2O 186 486 2·61a-phenyl-o-cresol 103 275 2·67(2-hydroxydiphenylmethane)52SiO2.30Li2O.18B2O3 194 614 3·16B2O3 113 371 3·28Diopside (CaMgSi2O6) 240 1084 4·51Anorthite (CaAl2Si2O8) 251 1135 4·52

Proc. Int. conf. on chemIstry of Glasses and Glass-formInG melts, oxford, UK, sePtember 2011


the intermediate range of temperatures.(16) Out of the intermediate range none of these models correctly describe the behaviour of viscosity and in the limits of low and high temperatures the best description of viscosity is provided by Frenkel model with high and low activation energies. It should be noted that experiments show that at very high temperatures there is a tendency to a non-activated regime for melt viscosities.(17) Table 2 summarises temperature behaviour of viscosity and equations used to describe the viscosity coefficient.

VTF model

The VTF equation of viscosity is an empirical expres-sion which describes viscosity data at intermediate temperatures over many orders of magnitude with a high accuracy:

ln[η(T)]=AVTF+BVTF/R(T−TV) (5)

where AVTF, BVTF and TV (Vogel temperature) are constants determined by fitting the experimental viscosity data to Equation (5).(3,4)

The VTF equation can be derived from the free volume model which relates the viscosity of the melt to free (or excess) volume per molecule Vf. The excess volume is considered to be the specific volume of the liquid minus the volume of its molecules. This molecular volume is usually derived from a hard sphere model of the atoms in the molecules. Molecular transport is considered to occur when voids having a volume greater than a critical value by redistribution of the free volume.(18) The flow unit or molecule is imagined to be in a structural cage at a potential minimum. As the temperature increases there is an increasing amount of free volume that can be redistributed among the cages, leading to increased transport and this leads to an exponential relationship between viscosity and free volume:(18)

η=η0exp(BV0/Vf) (6)

where V0 is the volume of a molecule, η0 and B are constants. In terms of the specific volume V per mol-ecule it can be shown that Vf=V−V0=V0(T−T0)/T0, for some constant and low temperature T0. Clearly Equa-tion (6) is the same as Equation (5) when AVTF=lnη0, BVTF=BT0 and TV=T0. The generic problem with the free volume theory is that the specific volume of an amorphous material as a function of temperature shows a discontinuity in slope at the glass transition

temperature, whereas the viscosities of melts and glasses do not exhibit any discontinuity.(11)

Adam and Gibbs model

The Adam and Gibbs (AG) equation is obtained as-suming that, above the glass transition temperature molecules in a melt can explore many different con-figurational states over time, and that as the tempera-ture is raised, higher energy configurational states can be explored. In contrast below the glass transition temperature it is assumed that the molecules in the glass are trapped in a single configurational state. The resulting AG equation for viscosity is similar to the VTF equation:(19)

ln[η(T)]=AAG+BAG/TSconf(T) (7)

where AAG and BAG are adjustable constants and Sconf(T) is the configurational entropy. Assuming that Sconf(T)=ΔCp(T−TV)/T, where ΔCp is the relaxational part of the specific heat one can see that Equation (7) transforms into VTF Equation (5) where AAG=AVTF and BAG=ΔCpBVTF/R. The configurational entropy model of Adam and Gibbs fits a large number of viscosity data but like the free volume theory, it does not provide an accurate fit over the entire temperature range. At high and low viscosities Equation (7) does not describe the experimental temperature dependence of viscosity and increasingly large deviations from the experimental values are produced. In addition the configurational entropy model gives discontinuities in the first differential of the entropy at the glass transition, despite the fact that that there are no dis-continuities in experimentally measured viscosities, i.e. the problem with the entropy theory is the same as for the free volume theory.(11)

Avramov & Milchev model

The Avramov & Milchev (AM) viscosity model excel-lently describes the viscosity behaviour within the temperature range where the activation energy of viscosity changes with temperature. The AM model assumes that due to existing disorder, activation en-ergy barriers with different heights occur and that the distribution function for the heights of these barriers depends on the entropy. Thus viscosity is assumed to be a function of the total entropy of the system which leads to a stretched exponential temperature

Table 2. Viscous flow type and viscosity equations for various temperaturesTemperature Low (T/Tg) <<1 Intermediate (T/Tg) ~1 High (T/Tg) >>1 Extremely high (T/Tg) Æ∞ Viscous flow type Arrhenian, high Non-Arrhenian, variable Arrhenian, low Non-activated flow activation energy activation energy activation energyApproximate ACTexp(QH/RT) Aexp(B/R(T−TV)) or ATexp(QL/RT) ATviscosity equations Aexp(2·3(13·5−A)(Tg/T)Universal viscosity η(T)=A1T[1+A2exp(B/RT)][1+Cexp(D/RT)]equation



dependence of equilibrium viscosity:(16,20–22)

ln[η(T)]=AAM+2·3(13·5−AAM)(Tg/T)α (8)

where in this case Tg is defined by ln[η(Tg)/(dPas)]/2·3 =13·5, AAM is a constant and α is Avramov’s fragility parameter. The higher α the less strong is a fluid so that strong liquids have a value of α close to unity. Equation (8) fails to describe the experimental tem-perature dependence of viscosity in the limits of high and low temperatures in a similar fashion to the VFT and AG models.

Sanditov model

Developing the valence-configurational theory Nemilov has assumed that viscous flow is an acti-vated switching of valence bonds between atoms.(23) This concept was used in many other works. For example recently Sanditov(7) developed the concept of a fluctuation hole as a small scale low activation local deformation of a structural network (or bond lattice, see Ref. 24), i.e. the quasi-lattice necessary for the switching of the valence bond, which is the main elementary event of viscous flow in glasses and melts. He has shown that the free energy of activation of a flow within the range of the liquid–glass transition model has an exponential temperature dependence:

Fη/R=B+Cexp(D/T) (9)

Moreover he has shown that this dependence follows available experimental data (see Figure 2 from Ref. 6). A drastic increase in the activation free energy of viscous flow in the liquid–glass transition region was explained by a structural transformation which limits the local elastic deformation of the structural network, which in turn, originates from the excitation (critical displacement) of a bridging atom such as the oxygen atom in a Si–O–Si bridge. As at elevated temperatures there are many excited bridging atoms (locally deformed regions of the structural network), the activation free energy of viscous flow is almost in-dependent of temperature. In fact the hole-activation model (as noted by Sanditov) is closely connected with other models describing the viscous flow of in amorphous materials such as Nemilov’s model, AM or the defect (or Doremus’) model considered below.

Defect model of viscous flow

Doremus analysed data on diffusion and viscosity of silicates and suggested that diffusion of silicon and oxygen in these materials takes place by transport of SiO molecules formed on dissociation of SiO2, moreover these molecules are stable at high tem-peratures and typically result from the vaporization of SiO2.(10,11) He concluded that the extra oxygen atom resulting from dissociation of SiO2 leads to five-fold coordination of oxygen atoms around silicon. The

three-dimensional (3-D) disordered network of sili-cates is formed by [SiO4] tetrahedra interconnected via bridging oxygens ≡Si·O·Si≡, where · designates a bond between Si and O, and ·O· designates a bridging oxygen atom with two bonds. The breaking of a SiO molecule from the SiO2 network leaves behind three oxygen ions and one silicon ion with unpaired elec-trons. One of these oxygen ions can bond to the silicon ion. The two other dangling bonds result in two silicon ions that are five-fold coordinated to oxygen ions. Moreover one of the five oxygen ions around the central silicon ion has an unpaired electron, and it is not bonded strongly to the silicon ion.(10,11) Doremus suggested that this electron hole (unpaired electron) should move between the other oxygen ions in similar fashion to the resonance behaviour in aliphatic organic molecules. There is experimental evidence for five-fold coordination of silicon and oxygen at higher pressures in alkali oxide SiO2 melts from NMR, Raman and infrared spectroscopy, and evidence for five-fold coordinated silicon in a K2Si4O9 glass at atmosphere pressure.(25) Doremus showed that in silicates, the defects involved in flow are SiO molecules resulting from broken silicon–oxygen bonds and therefore the SiO molecules and five-fold coordinated silicon atoms involved in viscous flow derive from broken bonds. Although he failed to re-produce the two-exponential equation of viscosity(11) his approach was used to derive an exact equation of viscosity which can be used at all temperatures.(26,27)

Indeed assuming that viscous flow in amorphous materials is mediated by defects originating from broken bonds, which are quasi-particles termed configurons, we can find the viscosity coefficient. This one is related to the diffusion coefficient, D, of the configurons, which mediate the viscous flow, via the Stokes–Einstein equation η(T)=kT/6πrD, where k is the Boltzmann constant and r is the con-figuron radius. One can then find the equilibrium concentrations of configurons: Cd=C0f(T) where f(T)= exp(−Gd/RT)/[1+exp(−Gd/RT)] and Gd=Hd−TSd is the Gibbs free energy of formation, Hd is the enthalpy, Sd is the entropy and C0 is the total concentration of elementary bond network blocks or the concentra-tion of unbroken bonds at absolute zero. The prob-ability of a configuron having the energy required for a jump is given by the Gibbs distribution w= exp(−Gm/RT)/[1+exp(−Gm/RT)], where Gm=Hm−TSm is the Gibbs free energy of motion of a jumping configuron, Hm and Sm are the enthalpy and entropy of configuron motion. Thus as first shown in Refs 26–28 the viscosity of amorphous materials is di-rectly related to the thermodynamic parameters of configurons via:

η(T)=A1T[1+A2exp(B/RT)][1+Cexp(D/RT)] (10)

with A1=k/6πrD0, A2=exp(−Sm/R), B=Hm, C=exp(−Sd/R) and D=Hd, where D0=fgλ2zp0ν0, f is the correlation



factor, g is a geometrical factor (~1/6), λ is the aver-age jump length, ν0 is the configuron vibrational fre-quency or the frequency with which the configuron attempts to surmount the energy barrier to jump into a neighbouring site, z is the number of nearest neighbours and p0 is a configuration factor.

Experiments show that, in practice, four fit-ting parameters usually suffice(14,17) indicating that A2exp(B/RT)>>1. Equation (10) can be fitted to practi-cally all available experimental data on viscosities of amorphous materials.(17) Moreover it can be readily approximated within a narrow temperature interval by known empirical and theoretical models such as VTF, AG, AM or Sanditov models. However in contrast to those approximations it can be used at all temperatures and gives the correct Arrhenian-type asymptotes at high and low temperatures namely η(T)=ACTexp(QH/RT) and ATexp(QL/RT), respec-tively, where QH=Hd+Hm and QL=Hm.

Equation (10) also shows that at extremely high temperatures when TÆ∞ the viscosity of melts should gradually change to a non-activated, e.g. non-Arrhenius behaviour, which is characteristic of systems of almost free particles.

Note that in term of imaginary layers (see Figure 1) the configuron formation enthalpy relates to interlayer bond strength whereas the configuron motion enthalpy to interlayer friction. Obviously at high temperatures when temperature fluctuations create plenty of configurons the activation energy of viscosity reduces to a low value equal to Hm. Whereas in the glassy state in order to provide layers motion it is necessary to break some of bonds (as temperature fluctuations do not create them effectively) thus the activation energy of flow takes its full value: QH=Hd+Hm.

Configuron parameters

The five coefficients A1, A2, B, C and D in Equation (10) can be treated as fitting parameters derived from the experimentally known viscosity data. Having obtained these fitting parameters one can evaluate the thermodynamic data of the configurons (e.g. network breaking defects).(17) Hence from known viscosity–temperature relationships of amorphous materials one can characterise the thermodynamic parameters of configurons. As the number of param-eters to be found via fitting procedure is high and both equations are nonlinear a dedicated Genetic Algorithm was used to achieve the best fit between Equations (10) and experimental viscosity data.(17) An example of such evaluation is demonstrated in Figure 3, which shows the best fit for viscosity–temperature data of amorphous anorthite and diopside obtained using Equation (10).

From the numerical data of evaluated parameters A1, A2, B, C and D which provide the best fit of theo-

retical viscosity–temperature relationship (10) to ex-perimental data we can find enthalpies of formation and motion and entropies of formation and motion of configurons, e.g. parameters of bond lattice of an amorphous material.(17) Evaluated thermodynamic parameters (enthalpies and entropies of formation

Figure 3. Viscosity–temperature data for anorthite (A) and diopside (B) with experimental data fitted to Equation (10)

(B)

(A)

Table 3. Thermodynamic parameters of configurons in amorphous materialsMaterial Hd Sd/R Hm Sm/R (kJ/mol) (kJ/mol)

Strong materialsSilica (SiO2) 237 17·54 522 11·37Germania (GeO2) 129 17·77 272 2·4966·7SiO2.33·3PbO 197 25·40 274 7·3 80SiO2.20Na2O 155 17·98 207 7·79 65SiO2.35PbO 231 30·32 257 8·53 59·9SiO2.40·1PbO 236 31·12 258 6·55

Fragile materials75SiO2.25Na2O 233 30·62 203 4·22 75·9SiO2.24·1PbO 262 36·25 234 5·44 70SiO2.21CaO.9Na2O 331 44·03 293 24·40Salol (HOC6H4COOC6H5) 145 68·13 118 0·114 70SiO2.30Na2O 258 34·84 205 5·22 65SiO2.35Na2O 300 40·71 186 7·59 a-phenyl-o-cresol 172 83·84 103 0·134(2-Hydroxydiphenylmethane) 52SiO2.30Li2O.18B2O3 420 52·06 194 0·227 B2O3 258 44·2 113 9·21Diopside (CaMgSi2O6) 834 88·71 240 0·044 Anorthite (CaAl2Si2O8) 884 79·55 251 0·374



and motion) of configurons for several materials are given in Table 3.

The data in Table 3 show that the entropy of configuron formation is significantly higher than the entropy of configuron motion Sd>>Sm for most ma-terials. Accounting for values of Hm one can see that indeed most often Gm/RT>>1 and thus it is legitimate to simplify Equation (10) as four fitting parameters are usually sufficient to correctly describe the viscos-ity–temperature behaviour of a melt.(17)

Configuron clustering

The melt consists of dynamic clusters which con-tinuously change their configuration. On cooling the closer is the temperature to the glass transition temperature the larger are the atomic clusters.(29,30) Finally on further cooling the clusters agglomer-ate to a solid state to form a glass. The system of joining chemical bonds in an amorphous material also changes with temperature.(24) The bonds in a material can be either intact or broken. A broken chemical bond and associated strain-releasing local adjustment in centres of vibration form a configuron, an elementary configurational excitation in an amor-phous material.(31) The higher the temperature the more bonds are broken by temperature fluctuations hence the higher is the configuron concentration. Above and at the glass transition temperature the material is completely penetrated by a percolation cluster made of configurons. On cooling the sizes of atomic clusters increase and the sizes of configuron clusters decrease.(32,33) At temperatures below the glass transition temperature there are no percolating clus-ters made of configurons.(33) The characteristic size of both configuron and atomic clusters is described by the correlation length x(T). Below the glass transition temperature the correlation length gives character-istic sizes of clusters made of configurons, whereas above the Tg it gives characteristic sizes of clusters made of unbroken bonds, e.g. atoms. Moreover x(T) gives the linear dimension, above which the material is homogeneous: an amorphous material on a scale of size larger than x(T) has on average uniformly distributed configurons and atoms whereas on a scale of size smaller than x(T) the amorphous material is inhomogeneous and is described by fractal geometry with the fractal dimension of percolation clusters, e.g. Df=2·55±0·05.(32–34)

At temperatures much higher or smaller than Tg the correlation length becomes very small reducing to the atomic scale and the amorphous material both macro and microscopically is homogeneous. In contrast at temperatures approaching Tg x(T) diverges accordingly to x(T)=x0|f(T)−fc|ν, Here f(T)= exp(−Gd/RT)/[1+exp(−Gd/RT)] is the relative concen-trations of configurons, fc is the critical density when the percolation cluster made of broken bonds is

formed and ν=0·88 is the critical exponent.(32–34) Hence amorphous materials are dynamically inhomogene-ous at temperatures very close to Tg as measuring lengths in experiments are smaller than x(T) which diverges at Tg.(33)

Effect of radiation on viscosity

Although amorphous materials retain a disordered structure on irradiation, they can be quasi-melted or fluidised (as the temperature remains low) by radiation due to bond-breaking effects, e.g. radiation-induced formation of configurons.(35) Non-thermal quasi-melting of micro- and nano-samples during transmission electron microscope irradiation of glassy materials was recently analysed in refer-ences.(35–37) It has been shown that overheating and true melting by the electron beam is not occurring through evidence such as the ultra-sharp boundary between transformed and intact material. Moreover it has been explained that the observed fluidisation (non-thermal quasi-melting) of glasses is caused by effective bond breaking processes induced by the energetic electrons in the electron beam. Worthy of note, these experimental and theoretical findings were recently independently confirmed by the group led by Kun Zheng(38) who used the term superplastic deformability for fluidisation caused radiation in-duced decrease of viscosity.

The bond breaking processes modify the effective vis-cosity of glasses to a low activation energy regime.(35–37) Moreover the higher the electron flux density the lower the viscosity of an irradiated glass (Figure 4).

Non-thermal quasi-melting of glasses at high enough electron flux densities can result in shape modification of nano-sized particles including formation of perfect beads due to surface tension. Accompanying effects, such as bubble formation and

Figure 4. Viscosities of non-irradiated and electron beam irradiated soda–lime–silica glass 70SiO2.21CaO.9Na2O as a function of temperature calculated for two electron flux densities frad

(36,37)

Figure 4. Viscosities of non-irradiated and electron beam irradiated soda–lime–silica glass70SiO2.21CaO.9Na2O as a function of temperature calculated for two electron flux densities frad

(36,37)



foil bending were revisited in the light of the new interpretation.(37) Moreover spinodal phase separa-tion occurs in silica-rich and alkali-borate rich phases similar to that which occurs in “Vycor”-type glasses at temperatures above 700°C and which requires times of the order of several hours.(36) The behaviour of viscosity on e-beam irradiation although similar to a melting process is due to non-thermal effect of configuron generation by radiation and occurs at temperatures close to room temperature.(35–38)

Calculating the viscosity

The more-or-less randomness, the openness, and the varying degrees of connectivity allow glass structures to accommodate large variations in composition, i.e. glass acts like a solution. Moreover it was found that melts and glasses produced from them can be often considered as solutions consisting of salt-like products of interactions between the oxide compo-nents.(39) These associates are similar to the crystalline compounds which exist in the phase diagram of the initial oxide system. Calculations in this model are based on solving the set of equations for the law of mass action for the reactions possible in the system of oxides, and the equations of mass balance of the components. This approach describes well such properties as viscosity, thermal expansion, isothermal compressibility, optical parameters.(39)

For oxide glasses a small change in glass composi-tion typically causes a smooth change in glass proper-ties. The unit addition or substitution of a component can be deemed as a contribution characteristic of that component to the overall property. This notion gives rise to the additive relationships with many properties such as densities, refractive indexes obey-ing additive relationships.(4) An additive property P obeys a linear relation of the type:

P p Ci

n= ∑=

i i1

, where Ci

n

i =∑=

1001

% (11)

where pi are additivity factors for a given component i=1, 2, 3, … n, and Ci are the mass% or the mol% of that component in the glass. In oxide glasses the density follows additivity primarily because the vol-ume of an oxide glass is mostly determined by the volume occupied by the oxygen anions, the volume of cations being much smaller.(4) Additivity relations work over a narrow range of compositions and additivity coefficients of a given oxide may change from system to system. Nonlinearities appear when various constituents interact with each other. Glass properties can be calculated through statistical analysis of glass databases such as SciGlass.(4,40) Linear regression can be applied using common polynomial functions up to the second or third degrees. For viscosities of amorphous oxide mate-rials (melts and glasses) the statistical analysis of

viscosity is based on finding temperatures (isokoms) of constant viscosity log[h(Ti)]=consti, typically when viscosity is 1·5, 6·6 and 12 (the practical glass transition viscosity).(40–42) A detailed overview of the statistical analysis of viscosities and individual oxide coefficients Ci in isokom temperatures of oxide materials is given in Ref. 42. Addition of oxides to certain base compositions changes the viscosity and the impact of different oxides is different.(40–42) Figure 5 shows the effect of component addition to the base composition in mol% 73SiO2.2B2O3.2Al2O3.12Na2O.2K2O.2MgO.7CaO on the temperature where the viscosity log[h(T)/Pas]=1·5 (isokom).(42)

Conclusions

Viscous flow has a low activation energy at high tem-peratures and a high activation energy at low temper-atures with a wide intermediate temperature range where the viscosity has a non-Arrhenian behaviour. The most common models of viscosity for intermedi-ate temperatures are those of Vogel–Fulcher–Tam-man and Avramov–Milchev however the viscosity of melts and glasses is most exactly described by a

Figure 5. Effect of component addition on isokom log[η(T)/Pas]=1·5.(42) Courtesy Alexander Fluegel

Figure 5. Effect of component addition on isokom log[η(T)/Pa s]=1·5.(42) Courtesy Alexander Fluegel



two exponential equation η(T)=A1T[1+A2exp(B/RT)][1+Cexp(D/RT)], which is derived from the assump-tion that configurons resulting from bond breakages assist the flow by weakening the material bond lattice. This equation can be used over all temperatures, gives correct Arrhenian-type asymptotes at both high and low temperatures, and can also explain the effect of radiation on the viscosity of glasses.

References 1. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics, Vol. 6, Butterworth-

Heinemann, 2nd ed., Oxford, 1987. 2. Kittel, C. Introduction to solid state physics, J. Wiley and Sons Inc., New

York, 1996. 3. West, A. R. Basic solid state chemistry, J. Wiley and Sons Ltd., Chichester,

1999. 4. Varshneya, A. K. Fundamentals of inorganic glasses, Society of Glass

Technology, Sheffield, 682p., 2006. 5. Zarzycki, J. Glasses and the vitreous state, Cambridge University Press,

New York, 1982. 6. Ojovan, M. I. Viscosity and Glass Transition in Amorphous Oxides,

Advances in Condensed Matter Physics, 2008, Article ID 817829, 23 pages. 7. Sanditov, D. S. Shear viscosity of glass-forming melts in the liquid-

glass transition region, J. Exp. Theor. Phys., 2010, 110, 675–688. 8. Angel, C. A. Perspective on the glass transition. J. Phys. Chem. Solids,

1988, 49, 863-871. 9. Martinez, L. M. & Angel. C. A. A thermodynamic connection to the

fragility of glass-forming liquids, Nature, 2001, 410, 663–667. 10. Doremus, R. H. Melt viscosities of silicate glasses, Amer. Ceram. Soc.

Bull., 2003, 82, 59–63. 11. Doremus, R. H. Viscosity of silica, J. Appl. Phys., 2002, 92, 7619–7629. 12. Frenkel, Y. I. Kinetic theory of liquids, Oxford University Press, Oxford,

1946. 13. Stanworth, J. E. Physical properties of glass, Oxford University Press,

Oxford, 224p., 1950. 14. Volf, M. B. Mathematical approach to glass, Elsevier, Amsterdam, 1988. 15. Douglas, R. W. The flow of glass, J. Soc. Glass Technol., 1949, 33, 138–162. 16. Avramov, I. & Milchev, A. Effect of disorder on diffusion and viscosity

in condensed systems, J. Non-Cryst. Solids, 1988, 104, 253–260. 17. Ojovan, M. I., Travis, K. P. & Hand. R. J. Thermodynamic parameters

of bonds in glassy materials from viscosity-temperature relationships, J. Phys.: Condensed Matter, 2007, 19, 415107, 12p.

18. Turnbull, D. & Cohen, M. H. Free-Volume Model of the Amorphous Phase: Glass Transition, J. Chem. Phys., 1961, 34, 120–125.

19. Adam, G. & Gibbs. J. H. On the temperature dependence of cooperative relaxation properties in glass-forming liquids, J. Chem. Phys., 1965, 43, 139–146.

20. Avramov, I. Pressure dependence of viscosity of glass forming melts, J. Non-Cryst. Solids, 2000, 262, 258–263.

21. Avramov, I. Viscosity in disordered media, J. Non-Cryst. Solids, 2005,

351, 3163–3173. 22. Avramov, I. Interrelation between the parameters of equations of

viscous flow and chemical composition of glassforming melts, J. Non-Cryst. Solids, 2011, 357, 391–396.

23. Nemilov, S. V. Thermodynamic and Kinetic Aspects of the Vitreous State, CRC Press, Boca Raton, FL, USA, 1995.

24. Ojovan, M. I. Configurons: thermodynamic parameters and symmetry changes at glass transition, Entropy, 2008, 10, 334–364.

25. Stebbins, J. F. NMR Evidence for Five-Coordinated Silicon in a Silicate Glass at Atmospheric Pressure, Nature (London), 1991, 351, 638–639.

26. Ojovan, M. I. Viscosity of oxide melts in the Doremus model, JETP Lett., 2004, 79, 85–87.

27. Ojovan, M. I. & Lee, W. E. Viscosity of network liquids within Doremus approach, J. Appl. Phys., 2004, 95, 3803–3810.

28. Ozhovan, M. I. Topological characteristics of bonds in SiO2 and GeO2 oxide systems at glass-liquid transition, J. Exp. Theor. Phys., 2006, 103, 819–829.

29. Medvedev, N. N., Geider, A. & Brostow, W. Distinguishing liquids from amorphous solids: Percolation analysis on the Voronoi network, J. Chem. Phys., 1990, 93, 8337–8342.

30. Evteev, A. V., Kosilov, A. T. & Levchenko, E. V. Atomic mechanisms of pure iron vitrification, J. Exp. Theor. Phys., 2004, 99, 522–529.

31. Angell, C. A. & Rao, K. J. Configurational excitations in condensed matter, and “bond lattice” model for the liquid-glass transition, J. Chem. Phys., 1972, 57, 470–481.

32. Stanzione III, J. F., Strawhecker, K. E. & Wool, R. P. Observing the twinkling fractal nature of the glass transition, J. Non-Cryst. Solids, 2011, 357, 311–319.

33. Ojovan, M. I. & Lee, W. E. Connectivity and glass transition in disor-dered oxide systems, J. Non-Cryst. Solids, 2010, 356, 2534–2540.

34. Ma, D., Stoica, A. D. & Wang, X.-L. Power-law scaling and fractal nature of medium-range order in metallic glasses, Nature Mater., 2009, 8, 30–34.

35. Ojovan, M. & Möbus, G. On radiation-induced fluidization (quasi-melting) of silicate glasses, Mater. Res. Soc. Symp. Proc., 2009, 1193, 275–282.

36. Ojovan, M. I., Yang, G. & Möbus, G. On spinodal decomposition of e-beam irradiated borosilicate glasses, Proc. WM’10 Conference, March 7–11, 2010, Phoenix, Arizona, WM–10202, 8 p.

37. Möbus, G., Ojovan, M., Cook, S., Tsai, J. & Yang, G. Nano-scale quasi-melting of alkali-borosilicate glasses under electron irradiation, J. Nucl. Mater., 2010, 396, 264–271.

38. Zheng, K., Wang, C., Cheng, Y.-Q., Yue, Y., Han, X., Zhang, Z., Shan, Z., Mao, S. X., Ye, M., Yin, Y. & Ma. E. Electron-beam-assisted superplastic shaping of nanoscale amorphous silica, Nature Comm., 2011, 1:24, 1–8. DOI:10.1038/ncomms1021.

39. Shakhmatkin, B. A., Vedishcheva, N. M. & Wright, A. C. Can thermo-dynamics relate the properties of melts and glasses to their structure? J. Non-Cryst. Solids, 2001, 293–295, 220–226.

40. SciGlass 7.7 Database and Information System, 2005. www.sciglass.info (19.11.2011).

41. Martlew, D. Viscosity of Molten Glasses; In Properties of Glass-Forming Melts; Eds D. Pye, I. Joseph & A. Montenero, Taylor & Francis, Boca Raton, 485p., 2005.

42. Fluegel, A. Glass viscosity calculation based on a global statistical modelling approach, Glass Technol., 2007, 48, 13–30.


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Viscous flow and the viscosity of melts and glasses

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