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`2
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`

`

`.... ~ " ' -01~ ! . .~
`
`'.._.~ ), /~
`
`i_vJOURNAL OF APPLIED PHYSICS
`
`SC".<
`
`Vol. 92, No. 12, 15 December 2002
`CODEN: JAPJAU
`ISSN: 0021-8979
`
`LASERS, OPTICS, AND OPTOELECTRONICS (PACS 42)
`6959 Relaxation kinetics of photoinduced surface relief grating on azopolymer
`films
`
`6966 P-type macroporous silicon for two-dimensional photonic crystals
`
`6973 Modeling of laser cleaning of metallic particulate contaminants from silicon
`surfaces
`
`6983 Near-field enhanced Raman spectroscopy using side illumination optics
`
`6987 Anomalous increase of photocurrent anisotropy in a liquid crystalline
`binary mixture
`
`PLASMAS AND ELECTRICAL DISCHARGES (PACS 51-52)
`6990 Calculated characteristics of radio-frequency plasma display panel cells
`including the influence of xenon metastables
`6998 Charged species dynamics in an inductively coupled Ar/SF6 plasma
`discharge
`
`7008 Comparison of excessive Balmer a line broadening of glow discharge and
`microwave hydrogen plasmas with certain catalysts
`7022 Mechanism of C2 hydrocarbon formation from methane in a pulsed
`microwave plasma
`7032 Analytical model for ion angular distribution functions at rf biased surfaces
`with collisionless plasma sheaths
`
`Tatsunosuke Matsui,
`Shin-ichiro Yamamoto,
`Masanori Ozaki, Katsumi Yoshino,
`Francois Kajzar
`P. Bettotti, L. Dal Negro,
`Z. Gaburro, L. Pavesi, A. Lui,
`M. Galli, M. Patrini, F. Marabelli
`M. Arronte, P. Neves, R. Vilar
`
`Norihiko Hayazawa,
`Alvarado Tarun, Yasushi Inouye,
`Satoshi Kawata
`K. L. Sandhya, Geetha G. Nair,
`S. Krishna Prasad,
`Uma S. Hiremath,
`C. V. Yelamaggad
`
`L. C. Pitchford, J. Kang, C. Punset,
`J.P. Boeuf
`Shahid Rauf, Peter L. G. Ventzek,
`Ion C. Abraham,
`Gregory A. Hebner,
`Joseph R. Woodworth
`R. L. Mills, P. C. Ray,
`B. Dhandapani, R. M. Mayo, J. He
`M. Heintze, M. Magureanu,
`M. Kettlitz
`
`Laxminarayan L. Raja, Mark Linne
`
`STRUCTURAL, MECHANICAL, THERMODYNAMIC, AND OPTICAL PROPERTIES OF CONDENSED MATTER
`(PACS 61-68, 78)
`7041 Stimulation of electrical conductivity in a 1r-conjugated polymeric
`conductor with infrared light
`
`S. C. J. Meskers,
`J. K. J. van Duren,
`R. A. J. Janssen
`S. A. Basun, D. R. Evans,
`T. J. Bunning, S. Guha,
`J. 0. Barnes, G. Cook,
`R. S. Meltzer
`H. Reuther
`Shin-Pon Ju, Cheng-I Weng,
`Chi-Chuan Hwang
`Jan Genzer, Edward J. Kramer,
`Daniel A. Fischer
`H. Kim, A. J. Kellock,
`S. M. Rossnagel
`N. Panev, M.-E. Pistol,
`S. Jeppesen, V. P. Evtikhiev,
`A. A. Katznelson, E. Yu. Kotelnikov
`
`7051 Optical absorption spectroscopy of Fe2 + and Fe3 + ions in LiNb03
`
`7056 Annealing behavior of magnesium and aluminum implanted with iron ions
`7062 Damascene process simulation using molecular dynamics
`
`7070 Accounting for Auger yield energy Joss for improved determination of
`molecular orientation using soft x-ray absorption spectroscopy
`7080 Growth of cubic-TaN thin films by plasma-enhanced atomic layer
`deposition
`7086 Spectroscopic studies of random telegraph noise in lnAs quantum dots in
`GaAs
`
`TX 5·646·209
`
`1111111111111111111111111111111111111111111111111111111111111111111111
`:ti,;TX0005646?.09~
`
`(Continued)
`
`3
`
`

`

`JOURNAL OF APPLIED PHYSICS
`
`VOLUME 92, NUMBER 12
`
`15 DECEMBER 2002
`
`Viscosity of silica
`Robert H. Doremusa)
`Department of Materials Science and Engineering, Rensselaer Polytechnic Institute,
`Troy, New York 12180-3590
`
`(Received 13 May 2002, accepted 22 August 2002)
`
`Experimental measurements of the viscosity of silica (Si02) are critically examined; the best
`measurements show an activation energy of 515 kJ/mole above 1400 °C and 720 kJ/mole below this
`temperature. The diffusion of silicon and oxygen in silica have temperature dependencies close to
`that of the high temperature viscosity. Mechanisms of viscous flow and diffusion of silicon and
`oxygen in silica are proposed that involve motion of SiO molecules. Viscous flow is proposed to
`result from the motion of line defects composed of SiO molecules At temperatures below 1400 °C
`the fraction of SiO molecules in line defects changes with temperature. The relaxation of this
`fraction to an equilibrium value depends on the time. These proposed mechanisms are consistent
`with experimental measurements of silica viscosity. © 2002 American Institute of Physics.
`[DOI: 10.1063/1.1515132]
`
`I. INTRODUCTION
`
`The viscosities of liquids and melts are among their most
`important properties. In glasses, viscosities determine melt(cid:173)
`ing conditions, temperatures of working and annealing, rate
`of removal of bubbles, maximum temperature of use, and
`crystallization rate. In geology magma behavior, volcanic
`eruptions, and lava flow rate depend directly on silicate vis(cid:173)
`cosity.
`Recently, there has been much interest in mass transport
`in glass-forming
`liquids near
`the glass
`trans1t10n
`temperature. 1
`6 In these discussions the viscosity of silica
`-
`(Si02) is usually considered as an example of a "strong"
`liquid in which the activation energy for viscous flow is con(cid:173)
`stant. One purpose of this article is to provide a critical as(cid:173)
`sessment of experimental data on the viscosity of silica. The
`conclusion is that there are two separate temperature regimes
`in the viscosity of silica in which the activation energy for
`flow is quite different. Another purpose of this article is to
`examine theories for the viscosities of network liquids like
`silica, and to present some new suggestions for models of
`flow of silica in the two different regimes of different acti(cid:173)
`vation energy.
`This discussion contains the following sections: experi(cid:173)
`mental measurements of the viscosity of silica; temperature
`dependence of viscosity; theories of viscosity; viscosity of
`silica; flow and a line defect; defect concentration; diffusion
`and viscosity; and conclusions.
`
`II. EXPERIMENTAL MEASUREMENTS OF SILICA
`VISCOSITY
`
`Experimental measurements of the viscosity of silica are
`given in Table I and Fig. 1. The measurements of Urbain et
`al. 7 were made over a wide temperature range ( 1192
`- 2482 ° C) and are the values usually quoted. Their mea(cid:173)
`surements for viscosities below 106 poise (105 Pas) were
`
`a)Electronic mail: doremr@rpi.edu
`
`made with a rotating cup; for higher viscosities they used a
`penetration method (isothermal deformation). All the mea(cid:173)
`surements described in this article were on type I silica,
`which is made by melting highly pure crystalline quartz. The
`main impurity is aluminum, at about 10-50 ppm, which does
`not influence the viscosity much. Water (OH) is also present
`in small quantity ( < 10 ppm). The activation energy for vis(cid:173)
`cous flow was constant throughout the measurement range at
`515 kJ/mole in Urbain's results; the data given in Table II of
`Ref. 7 when plotted as log viscosity vs 1/T give a straight
`line with a correlation coefficient from linear regression of
`0.999 78. The data in the table were apparently selected from
`more measurements, which are plotted in Fig. 1 of Ref. 7.
`The measurements of Bowen and Taylor8 at tempera(cid:173)
`tures from 2085 to 2310 °C had about the same activation
`energy as those of Urbain et al. but were about an order of
`magnitude smaller. The measurements of Bruckner9 from
`1686 to 2006° also had about the same activation energy as
`those of Urbain et al., but were about a factor of three higher.
`The measurements of Bacon et al. 10 from 1935 to 2322 °C
`agree reasonably well with those of Urbain et al., but are
`quite scattered. Earlier measurements of Bockris et al. 11 from
`about 1920 to 2060 °C have about the same activation energy
`as those of Urbain et al. but are a factor of about ten smaller;
`the data of Solomon 12 from 1720 to 2000 °C showed a lower
`activation energy (about 373 kJ/mole) and are about a factor
`of ten smaller than those of Urbain et al. The author con(cid:173)
`cludes that from 1400 to 2500 °C the viscosities of Urbain et
`al. are the most reliable, with an activation energy of 515
`kJ/mole.
`At temperatures from 1400 to 1000 °C Hetherington et
`al. 13 measured the viscosity of silica by a fiber elongation
`technique. These authors took special care to stabilize the
`silica in an "equilibrium" condition. They found that silica
`with different fictive temperatures had quite different mea(cid:173)
`sured viscosities at temperatures below 1400 °C; from their
`measurements the glass transition temperature (the tempera(cid:173)
`ture at which the viscosity is 1013 poise) was about 1185 °C.
`
`0021-8979/2002/92( 12)/7619/11 /$19 .00
`
`7619
`
`© 2002 American Institute of Physics
`
`4
`
`

`

`- - - - - - - - - - , __________ ""'-
`
`7620
`
`J. Appl. Phys., Vol. 92, No. 12, 15 December 2002
`
`Robert H. Doremus
`
`TABLE I. Viscosity values for amorphous silica considered to be the most
`reliable.
`
`TABLE IL Values of 71DIT for organic liquids data from Ref. 22
`
`Urbain et al (Ref. 7)
`
`Hetherington et al. (Ref. 13)
`
`Benzene
`T cc
`
`D71/TX 109
`
`Pentane
`T cc D 71/TX 109
`
`Cyclopentane
`T cc
`
`D/TX 109
`
`log 7J 7J
`in poise
`
`9.81
`11.22
`12.83
`14.65
`16.82
`
`15
`25
`35
`45
`55
`65
`mean
`
`4.44
`4.44
`4.29
`4.23
`4.57
`4.50
`4.41±0.13
`
`-60
`-19.9
`0.
`20
`40
`mean
`
`4.58
`4.55
`4.27
`4.21
`3.98
`4.32±0.25
`
`0
`5
`15
`20
`25
`30
`35
`45
`mean
`
`4.19
`4.10
`4.40
`4.48
`4.51
`4.57
`4.54
`4.18
`4.37± 19
`
`et al. found an activation energy of about 712 kJ/mole for
`viscous flow of stabilized silica glass at temperatures be(cid:173)
`tween 1400 and 1100 °C, as shown in Fig. 1. At temperatures
`below about llOO °C the time to stabilize the glass (reach
`metastable equilibrium) was longer than experimental times.
`Hetherington et al. found the same absolute value of viscos(cid:173)
`ity as Urbain et al. at 1400 °C, but higher values than those
`of these authors at lower temperatures. The measurements of
`Fontana and Plummer14 at temperatures from 1236 to 1335
`°C by beam bending agree quite well with those of Hether(cid:173)
`ington et al. except below about 1280 °C, where the values
`of Fontana and Plummer become slightly higher (see Fig. 1).
`Viscosity measurements by beam bending at two tempera(cid:173)
`tures by Kimura 15 agree closely with those of Hetherington
`et al. Measurements by Volarovich and Leontieva16 from
`1332 to 1436 °C showed an activation energy of 712 kl/mole
`but were about a factor of three higher than those of Heth(cid:173)
`erington et al. Thus the measurements of four independent
`groups of investigators are consistent with an activation en(cid:173)
`ergy of about 712 kJ/mole at temperatures below 1400 °C. It
`is likely that the measurements of Urbain et al. in the tem(cid:173)
`perature range from about 1187 to 1400 °C were made on
`glass with a fictive temperature higher than the measurement
`temperature, and were therefore not the viscosities of the
`glass at metastable equilibrium. Perhaps the penetration
`method used by these authors led to a rapid measurement, so
`that the glass was not at equilibrium; the details of the mea(cid:173)
`surement methods are apparently only available in a thesis .
`I conclude that the most reliable experimental measure(cid:173)
`ments of the stable viscosity of silica are those of Urbain et
`al. 7 from 2500 to 1400 °C:
`77= 5.8( 10)- 7 exp(515 400/RT),
`(1)
`and those of Hetherington et al. 13 from 1400 to 1000 °C:
`77= 3.8( 10)- 13 exp(712 000/RT),
`
`(2)
`
`with viscosity in poise and activation energy in J/mole.
`Equation (1) extrapolates to log n= -6.24 as T approaches
`infinity, somewhat lower than the values for some other
`liquids. 5
`The reasons for concluding that the measurements of
`Urbain et al. are the most reliable are: (1). The measured
`values of viscosity fit very well to the Arrhenius equation
`from 1400 to 2500 °C. (2). The measured values have the
`least scatter of all measurements of silica viscosity over this
`wide temperature range. (3). The measurements are in be-
`
`Temperature cc
`
`log 7/ 7/
`in poise
`
`Temperature cc
`
`1400
`1300
`1200
`1100
`1000
`
`2482
`2382
`2268
`2168
`2061
`1964
`1870
`1776
`1652
`1599
`1438
`1375
`1306
`1250
`1192
`
`3.53
`3.92
`4.36
`4.80
`5.25
`5.79
`6.36
`6.90
`7.79
`8.08
`9.48
`9.97
`10.95
`11.40
`12.15
`
`The fictive temperature is the temperature at which the prop(cid:173)
`erties of glass are the same as those of the melt in metastable
`equilibrium. Thus the fictive temperature is related to the rate
`at which a glass is cooled from above the glass transition
`temperature; the more rapid the cooling rate the higher the
`fictive temperature. The measured viscosity below about
`1400 °C in silica increases or decreases with time to the
`value for metastable equilibrium at long time. Hetherington
`
`14
`
`13
`
`12
`
`(])
`-~ 11
`0..
`.23
`.ig 10
`u
`·s:
`.3 9
`
`(J)
`
`8
`
`7
`
`6
`
`5 L-.,c=.:....L--....L..-......11.---...L---.L.--....L.----l
`0.65
`0.7
`0.75
`0.5
`0.55
`0.6
`0.
`1000/T
`
`FIG. 1. Log viscosity of silica as a function of reciprocal temperature. (A)
`Urbain et al. (Ref. 7); (O) Hetherington et al. (Ref. 13); (---) Fontana and
`Plummer (Ref. 14); (X) Kimura15
`. Lines drawn through data of Urbain and
`Hetherington.
`
`5
`
`

`

`J. Appl. Phys., Vol. 92, No. 12, 15 December 2002
`
`Robert H. Doremus
`
`7621
`
`300
`
`400
`
`500
`
`700
`
`1000
`
`1500
`
`T, °C
`
`3.0
`
`2.8
`
`50
`
`IOOO/T
`2.6
`100
`
`2.4
`T, •c
`
`2.2
`
`2.0
`
`200
`
`14
`
`12
`
`10
`
`UJ 5 8
`
`0...
`
`~
`I-
`~ 6
`<.>
`U) >
`0 gf 4
`
`...J
`
`2
`
`0
`
`-2
`
`-140
`
`-120
`
`- 4 8
`
`7
`
`-100
`6
`
`5
`1000/T
`
`0 T, °C
`
`4
`
`3
`
`2
`
`salt,
`fused
`glass-forming
`a
`of
`viscosity
`Log
`3.
`FIG.
`0.4 (KN03)i0.6 Ca(N03)i (see Ref. 21) and n-butylbenzene (see Ref. 20),
`as a function of reciprocal temperature.
`
`(3)
`
`B. T0 equation
`The following three-parameter equation has enjoyed
`wide use for fitting the temperature dependence of viscosity
`data, and also is derived from theories considered in a sub(cid:173)
`sequent section:
`17= 'T/o exp[Bl(T-T0 ) J
`in which 'T/o, B , and T O are temperature-independent con(cid:173)
`stants. A number of names have been associated with this
`equation: Vogel, Fulcher, Tammann, Hesse, Doolittle, Will(cid:173)
`iams, Landel, and Ferry. Rather than try to designate devel(cid:173)
`opers of this equation, it is perhaps better to describe it with
`a distinguishing feature, such as the T O parameter; thus I will
`call it the TO equation.
`A good fit of Eq. (3) to the experimental measurements
`of viscosity of silica by Urbain et al. 7 and Hetherington et
`al. 13 [Eqs. (1) and (2)] is possible with a low value of T0 ,
`such as 600 °K. Nevertheless at the extremes of temperature
`(high and low) viscosities calculated from Eq. (3) deviate
`from the best-fit Arrhenius lines for the experimental data.
`Similar deviations from Eq. (3) occur for experimental data
`for other liquids, such as those in Figs. 2 and 3. I conclude
`that the TO equation does not fit accurate viscosity data at
`
`16
`
`14
`
`12
`
`0
`c.,
`
`8
`
`.....
`"' 10
`0
`Q.
`> t-;;;
`"' >
`S!
`"' 0
`
`6
`
`~
`
`0, .... 9---',1=----, ..... 5---'13~---L11---9..___--'1'----..J5
`
`10,000/T
`
`FIG. 2. Log viscosity of boron trioxide (see Ref. 18) and a soda-lime glass
`(70 wt% Si02 , 21 % Na20, 9% CaO) (Ref. 19) as a function of reciprocal
`temperature.
`
`tween those of four other groups. (4). The measurements of
`Urbain et al. 7 for several other silicate melts are also the
`most reliable. The measurements of Hetherington et al. 13
`were chosen because they agree well with measurements
`temperature range of Fontana and
`over a narrower
`Plummer14 and Kimura, 15 they have the least scatter, and
`their measurement at 1394 °C agrees with the data of Urbain
`et al.
`
`Ill. TEMPERATURE DEPENDENCE OF VISCOSITY
`
`A. Arrhenius dependence
`
`The pattern of two separate temperature regions of vis(cid:173)
`cosity with an Arrhenius dependence in each region is also
`found in other silicate melts, such as sodium disilicate
`(Na20 · 2 Si02) 17- 19 and albite (NaAI Si30 8) [R. H. Dore(cid:173)
`mus, unpublished analysis]. In other glass -forming liquids
`there are regions at the highest and lowest temperature (low(cid:173)
`est and highest viscosities) that follow Arrhenius depen(cid:173)
`dence, with a region of changing activation energy between
`constant activation energies at the extremes of temperature
`and viscosity. Examples are in Figs. 2 and 3 for B20 3 , a
`soda-lime-silicate glass, n-butyl benzene, and 0.4 KN03
`-0.6Ca(N03)i fused salt. A variety of other organic liquids,
`such as n-propyl alcohol, di-n-butyl phthalate, and isopropy(cid:173)
`22 Macedo and
`lbenzene (cumene) show similar curves. 20
`-
`Napolitano8 showed in detail this experimental temperature
`dependence of B20 3 , with constant activation energy at high
`and low temperature and a region of changing activation
`energy between them.
`
`6
`
`

`

`~ - - - - - - - - - ---------·~
`
`7622
`
`J. Appl. Phys., Vol. 92, No. 12, 15 December 2002
`
`Robert H. Doremus
`
`TABLE III. Values of T/ DIT for sodium chloride and sodium nitrate melts
`data from Ref. 22
`
`NaCl
`T °C
`
`825
`836
`879
`933
`936
`937
`942
`mean
`
`TJ D(Cl)IT
`
`1.26( 10)- 9
`1.28
`1.15
`1.05
`1.35
`1.12
`i.14
`1.20±0.10
`
`T 0C
`
`320.6
`328.7
`350.9
`359.4
`366.2
`377.4
`mean
`
`TJ D(N03)/T
`
`5.67(10)- 10
`5.64
`5.68
`5.64
`5.61
`5.75
`5.66±0.05
`
`TABLE IV. Sizes of flow units in different liquids, calculated from Eq. (5)
`
`Liquid
`
`Benzene
`Pentane
`Cyclopentane
`NaCl
`NaN03
`Copper
`Indium
`Silica
`
`Diameter of spherical flow unit, A
`
`0.33
`0.34
`0.34
`1.21
`2.6
`1.1
`2.2
`0.75
`
`high and low temperatures, bringing into question theories
`that result in it. Equation (3) may be useful for fitting experi(cid:173)
`mental measurements over limited temperature ranges or that
`are rather scattered, but it cannot justify theories of the tem(cid:173)
`perature dependence of viscosity.
`A better fit to experimental viscosity data is provided by
`a sum of two exponential terms, which can be written:
`7J,;,_A exp(BIRT) [l + C exp(DIRT)]
`with temperature-independent constants A, B, C, and D. This
`equation fits Arrhenius relations at high and low temperature,
`as found experimentally.
`
`(4)
`
`C. Diffusion and viscosity
`
`In fluid dynamics the force F to move a solid sphere of
`radius R through a fluid of viscosity 7J is:
`
`If this equation is valid for the motion of a spherical mol(cid:173)
`ecule through a fluid, it leads to the following relation be(cid:173)
`tween the diffusion coefficient D of the molecule and the
`viscosity:
`
`(5)
`
`(6)
`
`in which k is Boltzmann's constant [ 1.38(10)- 16 ergs/K in
`the cgs units used in this article] and T the absolute tempera(cid:173)
`ture. The units of D are cm2/s, and 1J poise (gm/cm s). It is
`quite an extrapolation to use Eq. (5), valid for macroscopic
`spheres, for molecules, although one might expect the tem(cid:173)
`perature dependencies of Eq. (6) to be valid even if the nu(cid:173)
`merical constants are not exact.
`In Table II values of 7J DIT are shown to be constant for
`some hydrocarbon liquids, as expected from Eq. (6). Values
`of 7J DIT are also constant for the fused salts NaCl and
`NaN03 , as shown in Table III, with the anion diffusion co(cid:173)
`efficients. It is interesting that if the somewhat higher cation
`diffusion coefficients are substituted in Eq. (6), the values of
`7J DIT decrease somewhat as the temperature is raised.
`Grosse23 showed that Eq. ( 6) is valid for liquids of the metals
`sodium and zinc over rather narrow temperature ranges; Eq.
`( 6) is also valid for many other liquid metals such as copper,
`lead, and indium (data from Ref. 24), although for the wider
`range of temperatures measured for lead and indium the val(cid:173)
`ues of 7J DIT change somewhat as the temperature increases,
`
`7.5(10)- 10 dynes/K
`to
`170°C
`at
`from
`decreasing
`5.6(10)- 10 dynes/K at 750 °C for indium, and increasing for
`lead. The preexponential factor for lead in Ref. 22 and earlier
`editions of Smithells' book is a factor of ten too small; it
`should be 2.3(10)- 3 cm2 /s. 25 The temperature dependence
`of the diffusion coefficient of silicon in silica also is consis(cid:173)
`tent with Eq. (6) and silica viscosity, as described in more
`detail in the section on silica viscosity.
`The diameters of flow units in these liquids as calculated
`from Eq. (6) are given in Table IV. The calculated diameters
`for the organic liquids and silica are too small for reasonable
`flow units, by at least a factor of five. The diameters for the
`fused salts are perhaps too small by about a factor of two, the
`larger value for NaN03 is consistent with the larger size of
`the nitrate ion compared to chlorine. The diameter for copper
`seems small, and that for indium about correct. Thus al(cid:173)
`though the temperature dependencies of viscosity and diffu(cid:173)
`sion coefficients are consistent with Eq. (6), there are appar(cid:173)
`ently some additional factors that influence the effective
`sizes of flow units in liquids.
`The constancy of 7JD!T with temperature for many liq(cid:173)
`uids shows that the mechanisms of viscous flow and diffu(cid:173)
`sion in these liquids are closely related. Therefore any suc(cid:173)
`cessful
`theory
`for viscosity must explain
`the close
`relationship with diffusion.
`
`IV. THEORIES OF VISCOSITY
`
`A. Introduction
`
`Many different ideas to explain the viscosity of liquids
`have been proposed. This interest has been particularly ac(cid:173)
`tive recently; in addition to Refs. 1-6, articles describing
`various calculational schemes for viscosity are appearing
`regularly.26
`27 In this section I will not try to review all this
`•
`work, but will select a few ideas that seem to be the most
`active, and a few articles for comment I have found most
`helpful.
`Douglas proposed a theory for the viscosities of silicate
`glasses. 28 Although this theory was limited to these glasses,
`it has some features of much wider applicability and has
`been completely ignored in succeeding literature. Douglas
`assumed that the flow for silicate glasses was limited by
`breaking silicon-oxygen-silicon bonds (Si-0-Si). He con(cid:173)
`sidered that the oxygen atoms between two silicon atoms
`could occupy two different positions, separated by an energy
`barrier. The distribution of the oxygen atoms between these
`two positions is then calculated from an entropy of mixing
`
`7
`
`

`

`J. Appl. Phys., Vol. 92, No. 12, 15 December 2002
`
`Robert H. Doremus
`
`7623
`
`equation. Thus, Douglas anticipated the emphasis on con(cid:173)
`figurational entropy in later theories. From these consider(cid:173)
`ations Douglas derived Eq. (4), which fits viscosity data for a
`wide variety of liquids. Although a different mechanism is
`proposed for the viscosity of silica glass from that of Dou(cid:173)
`glas, I believe his ideas are worthy of careful consideration.
`Douglas in