- Email: [email protected]
On the viscosity of glass composites containing rigid inclusions
March 1998
Materials Letters 34 Ž1998. 285–289
On the viscosity of glass composites containing rigid inclusions A.R. Boccaccini
)
Institute for Mechanics and Materials, UniÕersity of California, San Diego, La Jolla, CA 92093-0404, USA Received 30 June 1997; accepted 4 July 1997
Abstract A previously derived equation that treats viscosity as a field or transport property has been applied to calculate the effective viscosity of glass matrix composites containing rigid inclusions. The innovative aspect of the approach is that the shape of the particles, which is modeled by spheroids having a mean axial ratio, is explicitly considered. Thus, if information on the shape of the inclusions is available, the equation becomes predictive. Experimental data on different glass matrix composites drawn from the literature were used for comparison with the calculated values. It was shown that the equation is only applicable for volume fraction of inclusions for which the particle–particle interaction is negligible or nonexistent. The possibility of extending the application range of the approach is briefly addressed. q 1998 Elsevier Science B.V. PACS: 81.05.Pj Keywords: Viscosity; Glass matrix composites; Rigid inclusions; Creep
1. Introduction One approach to the improvement of the mechanical properties of glass, e.g. fracture strength, fracture toughness and thermal shock resistance, is to form a composite w1x. In discontinuous reinforced glass composites, the low-modulus, low-strength, brittle glass matrix are reinforced by incorporating a highmodulus, high-strength andror high-ductility second constituent in the form of particulate, chopped fibres, platelets or whiskers w1–5x. A great variety of glass matrix composite materials has been developed in the last decades. In a comprehensive recent review, the different systems investigated, the fabrication )
On leave from: TU Ilmenau, FG Werkstofftechnik, D-98684 Ilmenau, Germany. Tel.: q1-619-5348951; fax: q1-619-5348908; e-mail: [email protected].
methods employed and the properties achieved have been discussed w6x, emphasizing the significant effect of the microstructure, i.e. concentration, shape and orientation of the inclusion phase on the final effective properties. One issue that has received little attention, however, is concerning the effect of the Žrigid. inclusions on the viscous behaviour of the glass composites. Knowledge of this effect is important not only for the possible uses of the composites at moderate to high temperatures, where viscosity issues begin to be relevant, i.e. due to the direct correlation between effective viscosity and creep deformation w7–9x, but also for the optimization of the processing parameters. Thus, for example, the densification rate of the glass matrix via a viscous flow sintering mechanism may be significantly affected by the presence of the rigid, i.e. non-densifying inclusions, as demonstrated theoretically w10x and
00167-577Xr98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 5 7 7 X Ž 9 7 . 0 0 1 8 6 - 9
286
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
experimentally w11,12x. Moreover, the effective viscosity is an important parameter determining the feasibility of using hot-forming for fabricating glass matrix composite components. Of more general interest, particulate glass matrix composites may be used as model systems to study the behaviour of glass-ceramics containing a large proportion of a residual glass and for other glass bonded ceramic materials, such as porcelains. On an even broader basis, predicting the effective viscosity of particulate viscous systems is also of importance in a variety of problems involving viscous flow, including applications in biomechanics w13x. In one of the few studies dealing with the viscosity of glass containing inclusions, Demana and Drummond w14x measured the viscous deformation of a glass–crystalline alumina system using a beambending viscosimeter. The samples contained up to 50 vol% of dispersed alumina particles in the glass matrix. A significant increase of the viscosity of the glass with the volume fraction of inclusions was found. Similar results were obtained by Tewari et al. who studied the rheological behaviour of SiC Žparticulate. –borosilicate glass composites at 675 and 7008C w15x. In both studies, the comparison of the experimental results with the prediction of classical theoretical models for the viscosity of suspensions behaving as Newtonian fluids w16x did not yield successful results, especially for high concentrations of inclusions. Indeed simple approaches based only on the volume fraction of the second phase are thought to be inadequate to predict the effective viscosity of these composite systems. Other studies dealing with the viscous deformation of glass matrix composites include the work of Dutton and Rahaman on soda–lime glass containing Ni particles w17x and the early study of Dunn on silica containing tungsten inclusions w18x. In these investigations, however, no attempt was made to compare the measured values with the prediction of available models. A new equation for the viscosity of suspensions has been proposed based on the analogy between field properties of two-phase materials and viscous flow of suspensions w19x. The innovative aspect of this approach is that the shape and orientation of the suspended particles are explicitly taken into account. Besides having been applied to calculate the viscosity of several liquid suspensions, this equation has
been recently used to predict the viscosity of melted glasses containing solid phase aggregates w20x. It was found that the upper concentration limit for application of the equation was coincident with the onset of particle–particle interactions. The objective of the present study is to investigate the suitability of the equation to predict the effective viscosity of glass matrix composites containing rigid inclusions. Literature experimental data for glass composites with different inclusion morphologies are considered. The viscosities involved vary in a wide range Ž0.05–10 6 MPa s..
2. Analysis The mathematical analogy between different groups of physical properties of materials has been established in the past on the basis of these properties being governed by equations of the same form w21x. In general, field properties, such as thermal and electrical conductivity, can be defined by equations of the following type q˙ s yl= E
Ž 1.
where q˙ is a flow density, E s E Ž x, y, z . is a three-dimensional field and l is an effective field property. The viscosity of Newtonian flow, h , is defined by the following equation
tx z s h
dÕ
Ž 2.
dy
where t x z is the shear stress in the Ž x, z . plane and dÕrd y is the velocity gradient in the y direction. It was shown by Saltzer and Schulz that modifying Eq. ¨ Ž2. for the steady laminar flow of an incompressible fluid, the analogy between Eqs. Ž1. and Ž2. becomes obvious w19x. Thus, the viscosity of suspensions can be treated as a field property. On the basis of this analogy, and using earlier derived equations for the effective field property of two-phase composite materials w22x, the expression for the effective viscosity of an isotropic suspension, valid for systems containing isolated, i.e. non-interacting inclusions, has been derived as w19,20x
h s h0 Ž 1 y f .
m
Ž 3.
A.R. Boccaccinir Materials Letters 34 (1998) 285–289 Table 1 Shape factor Ž F . as a function of the axial ratio of spheroidal particles w22x Axial ratio Ž z r x .
Shape factor Ž F .
Description
0 0.002 0.02 0.1 0.2 1 2 10 20 200 2000 ™`
0 0.00157 0.01532 0.06959 0.12476 0.33333 0.41322 0.48986 0.49663 0.49994 0.49999 0.5
platelet oblate spheroid
3F y 2 3F Ž 1 y 2 F .
on melted glasses containing crystalline phases, i.e. at relatively low viscosities Ž-f 10 3 MPa s.. The application of the equation to predict the viscosity of glass matrix composite systems, which is of interest especially for higher viscosities Žup to f 10 6 MPa s., is considered in the present study. 3. Comparison with experimental results and discussion
sphere prolate spheroid
fibre
where h 0 is the viscosity of the fluid, f is the volume fraction of the rigid, i.e. non-deformable, inclusions and m is a function of the shape of the inclusions, which are modeled by spheroids having a fixed axial ratio Ž zrx . ms
287
Ž 4.
In Eq. Ž4., F is the shape factor, which depends on the axial ratio of the inclusions and is identical to the well-known deelectrification factor w19,22x. Some characteristic values of F are listed in Table 1. A complete tabulation of the values of F as a function of zrx has been given elsewhere w22x. The axial ratio of the inclusions can be determined on plane sections of the material investigated by measuring the area and perimeter of sectioned inclusions and from stereological equations w23x. Thus, Eq. Ž3. has predictive character as all variables involved can be measured independently. As mentioned before, Eq. Ž3. is applicable for composite systems containing isolated and well-distributed inclusions, i.e. when there is no contact between the inclusions and, therefore, no percolation effects are expected. It is interesting to point out that Eq. Ž3. has the same form as the equations of Brinkenman w24x and Roscoe w25x, which were derived as an extension of the original equation of Einstein, valid for dilute suspensions w26x. Eq. Ž3. has been tested successfully by comparison with experimental data on liquid suspensions and
As noted above, only limited data has been published on the viscosity of glass matrix composite materials. The earliest study found in the literature dealing with this topic is the work of Dunn on the viscous behaviour of silica with tungsten inclusions of spheroidal shape w18x. A comparison of his experimental results with the present model is difficult, however, because the temperature at which the viscosity was measured was varied along with the volume fraction of inclusions. Moreover extensive particle–particle interaction was found at concentration as low as 10 vol%. As mentioned before, Eq. Ž3. is not expected to be applicable in this case. The experimental data for the effective viscosity of a potassia borosilicate glass containing alumina particulates as inclusions, as determined by Demana and Drummod w14x using a beam-bending viscosimeter, are more suitable for comparison with the present approach. The measured values, normalized to the viscosity of the glass free of inclusions, are shown in Fig. 1 together with the prediction of Eq. Ž3.. The viscosity measurements covered a range of 10 8 –10 11 Pa s in the temperature range from 575 to 9008C. Unfortunately, no information on the morphology of the included particles was provided by the authors w14x. The theoretical curve was therefore calculated using zrx as a fitting parameter. As Fig. 1 shows, the best result was obtained for F s 0.483 which corresponds to an axial ratio zrx f 8 w22x. Micrographs shown in the original work w14x suggest some elongation of the alumina particles, however an axial ratio of f 8 seems to be too large. The micrographs of the composite microstructure at a concentration of inclusions of 40 vol% show considerable particle–particle interaction w14x. Thus, using a lower axial ratio in Eqs. Ž3. and Ž4., corresponding to the more equiaxed particles shown in the published micrographs w14x, would make the theoretical and ex-
288
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
Fig. 1. Relative viscosity of potassia borosilicate glass containing . calalumina inclusions: ŽB. measured values w14x, Ž culated curve ŽEqs. Ž3. and Ž4., F s 0.483..
perimental data to coincide only up to concentrations of f 20 vol%. This may represent a more realistic scenario for the applicability of the approach. Another set of data can be obtained from the work of Dutton and Rahaman w17x, who studied the creep deformation of soda–lime glass compacts containing spherical Ni inclusions of three different particle size ranges. A loading dilatometer was used to obtain creep strain rates at constant stresses, from which the creep viscosity was obtained. Only the composites containing the larger particles Žin the range 150–300 m m. did not show particle–particle interaction, i.e. percolation behaviour, up to concentrations higher than 10–15 vol%. The experimental data for the viscosity of glass composites containing Ni inclusions Ž150–300 m m. at 7308C are given in Fig. 2, together with the theoretical curve. This was calculated from Eqs. Ž3. and Ž4. using zrx s 1 Ž F s 0.33.. There is excellent agreement for the low volume fraction of inclusions Ž10 vol%. but beyond the percolation limit, the equation tends to underestimate the measured data, in agreement with the previous observation for the glassralumina system. The last set of available viscosity data corresponds to the data measured by Tewari et al. w15x, who studied the rheological behavior of SiC Žparticulate. –borosilicate composites at 675 and 7008C. Two size ranges of SiC inclusions Žfine and coarse grade. were considered. The samples were tested in
compression. It was found that at a high concentration of inclusions Ž40 vol%. the flow behaviour of the composites was non-Newtonian with a significantly faster increase of viscosity. The experimental data for the SiC particulate reinforced glass matrix composites of Tewari et al. w15x is shown in Fig. 3 together with the theoretical curve from Eqs. Ž3. and Ž4.. A value F s 0.2 was used in the calculation, which corresponds to an axial ratio zrx f 0.7 w20x. This value seems realistic by observation of the images of the microstructure presented in the original investigation w15x. Thus, Fig. 3 confirms the suitability of Eq. Ž3. to predict the viscosity of glassrcrystalline composite systems when the contact between particles is negligible or nonexistent. On the contrary, when particle–particle interaction is significant, the theoretical prediction underestimates the measured values. Unfortunately, the limited experimental data available is not sufficient to make a definite statement on the prediction capability of the equation. Except for the study of Dutton and Rahaman on glassrNi composites w17x, the shape of the included particles, which enters in Eq. Ž3. via the shape factor F, has not been properly characterized. Thus, new experimental investigations on the viscosity behaviour of glassrcrystal composite systems is re-
Fig. 2. Creep viscosity of soda–lime glass matrix composites containing spherical Ni inclusions Ž150–300 m m.. Ž'. Measured . calculated curve ŽEqs. Ž3. and Ž4., values w17x, Ž F s 0.333.. Note that the percolation limit was in the range 10–15 vol% for this system.
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
289
tent. A wide range of viscosities Ž0.05–10 6 MPa s. was analyzed for three glass composite systems drawn from the literature. More experimental data on composites containing particles of well-characterized shape is required, however, to make a definite statement on the prediction capability of the equation. Such measurements are the focus of current research.
References
Fig. 3. Viscosity of borosilicate glass matrix composites containing SiC particle inclusions: ŽB. measured values, coarse particles, Ž^. measured values, fine particles w15x, Ž . calculated curve ŽEqs. Ž3. and Ž4., F s 0.2..
quired. A well-characterized Al 2 O 3-platelet reinforced borosilicate glass w27x has been chosen as a model system for such on-going studies. Finally, it should be pointed out that the range of applicability of the approach could be extended for higher volume fraction of inclusions if only some of the particles become in contact, i.e. before the formation of a particle continuous network. In this case, the shape of the inclusions being modeled by spheroids would change from that of the primary particles to that of the agglomerates, clusters or chains of particles formed. These will then have a particular axial ratio, and hence a characteristic shape factor F, which should be used in Eq. Ž4..
4. Conclusions It has been shown that a previously derived equation that treats viscosity as a field or transport property can be used to predict the effective viscosity of glass matrix composites containing rigid particles. The equation includes a shape factor which can be calculated from the axial ratio of the inclusions modeled by spheroids. The approach is only applicable for volume fraction of inclusions for which the particle–particle interaction is negligible or nonexis-
w1x R.D. Rawlings, Composites 25 Ž1994. 372. w2x A.R. Hyde, G. Partridge, S. Vignesoult, Br. Ceram. Trans. 92 Ž1993. 55. w3x A.R.B. Verma, V.R.S. Murthy, G.S. Murty, J. Am. Ceram. Soc. 78 Ž1995. 2732. w4x A.R. Boccaccini, G. Ondracek, C. Syhre, Glastech. Ber. Glass Sci. Technol. 67 Ž1994. 16. w5x R.H. Moore, S.C. Kunz, Ceram. Eng. Sci. Proc. 8 Ž1987. 839. w6x A.R. Boccaccini, M. Kopf, Monatshefte ¨ Berg Huttenmann. ¨ ¨ 141 Ž1996. 110. w7x T. Yagamuchi, M. Shinagawa, J. Mater. Sci. 30 Ž1995. 504. w8x J.-H. Jean, J. Mater. Res. 11 Ž1996. 2098. w9x R. Lyall, K.H.G. Ashbee, J. Mater. Sci. 9 Ž1974. 576. w10x G.W. Scherer, J. Am. Ceram. Soc. 70 Ž1987. 719. w11x M.N. Rahaman, L.C. de Jonghe, J. Am. Ceram. Soc. 70 Ž1987. C348. w12x A.R. Boccaccini, J. Mater. Sci. 29 Ž1994. 4273. w13x P. Tong, Y.C. Fung, J. Appl. Mech. 38 Ž1971. 721. w14x B.P. Demana, C.H. Drummond III, Ceram. Eng. Sci. Proc. 16 Ž1995. 921. w15x A. Tewari, V.S.R. Murthy, G.S. Murty, J. Mater. Sci. Lett. 15 Ž1996. 227. w16x D.G. Thomas, J. Colloid. Sci. 20 Ž1965. 267. w17x R.E. Dutton, M.N. Rahaman, J. Mater. Sci. Lett. 12 Ž1993. 1453. w18x S.A. Dunn, Am. Ceram. Soc. Bull. 47 Ž1968. 554. w19x W.-D. Saltzer, B. Schulz, in: O. Brulin, R.K.T. Hsieh ŽEds.., ¨ Continuum Models of Discrete Systems 4, North-Holland, 1981, p. 423. w20x A.R. Boccaccini, K.D. Kim, G. Ondracek, Matwiss. u. Werkstofftech. 26 Ž1995. 263. w21x J.N. Shire, R.L. Weber, Similarities in Physics, Adam Hilger, Bristol, 1982. w22x G. Ondracek, Rev. Powder Metall. Phys. Ceram. 3 Ž1987. 205. w23x A.R. Boccaccini, G. Ondracek, E. Mombello, J. Mater. Sci. Lett. 15 Ž1996. 534. w24x H.C. Brinkman, J. Chem. Phys. 20 Ž1952. 571. w25x R. Roscoe, Br. J. Appl. Phys. 3 Ž1952. 267. w26x A. Einstein, Ann. Phys. 19 Ž1906. 289. w27x A.R. Boccaccini, P.A. Trusty, J. Mater. Sci. Lett. 15 Ž1996. 60.
Materials Letters 34 Ž1998. 285–289
On the viscosity of glass composites containing rigid inclusions A.R. Boccaccini
)
Institute for Mechanics and Materials, UniÕersity of California, San Diego, La Jolla, CA 92093-0404, USA Received 30 June 1997; accepted 4 July 1997
Abstract A previously derived equation that treats viscosity as a field or transport property has been applied to calculate the effective viscosity of glass matrix composites containing rigid inclusions. The innovative aspect of the approach is that the shape of the particles, which is modeled by spheroids having a mean axial ratio, is explicitly considered. Thus, if information on the shape of the inclusions is available, the equation becomes predictive. Experimental data on different glass matrix composites drawn from the literature were used for comparison with the calculated values. It was shown that the equation is only applicable for volume fraction of inclusions for which the particle–particle interaction is negligible or nonexistent. The possibility of extending the application range of the approach is briefly addressed. q 1998 Elsevier Science B.V. PACS: 81.05.Pj Keywords: Viscosity; Glass matrix composites; Rigid inclusions; Creep
1. Introduction One approach to the improvement of the mechanical properties of glass, e.g. fracture strength, fracture toughness and thermal shock resistance, is to form a composite w1x. In discontinuous reinforced glass composites, the low-modulus, low-strength, brittle glass matrix are reinforced by incorporating a highmodulus, high-strength andror high-ductility second constituent in the form of particulate, chopped fibres, platelets or whiskers w1–5x. A great variety of glass matrix composite materials has been developed in the last decades. In a comprehensive recent review, the different systems investigated, the fabrication )
On leave from: TU Ilmenau, FG Werkstofftechnik, D-98684 Ilmenau, Germany. Tel.: q1-619-5348951; fax: q1-619-5348908; e-mail: [email protected].
methods employed and the properties achieved have been discussed w6x, emphasizing the significant effect of the microstructure, i.e. concentration, shape and orientation of the inclusion phase on the final effective properties. One issue that has received little attention, however, is concerning the effect of the Žrigid. inclusions on the viscous behaviour of the glass composites. Knowledge of this effect is important not only for the possible uses of the composites at moderate to high temperatures, where viscosity issues begin to be relevant, i.e. due to the direct correlation between effective viscosity and creep deformation w7–9x, but also for the optimization of the processing parameters. Thus, for example, the densification rate of the glass matrix via a viscous flow sintering mechanism may be significantly affected by the presence of the rigid, i.e. non-densifying inclusions, as demonstrated theoretically w10x and
00167-577Xr98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 5 7 7 X Ž 9 7 . 0 0 1 8 6 - 9
286
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
experimentally w11,12x. Moreover, the effective viscosity is an important parameter determining the feasibility of using hot-forming for fabricating glass matrix composite components. Of more general interest, particulate glass matrix composites may be used as model systems to study the behaviour of glass-ceramics containing a large proportion of a residual glass and for other glass bonded ceramic materials, such as porcelains. On an even broader basis, predicting the effective viscosity of particulate viscous systems is also of importance in a variety of problems involving viscous flow, including applications in biomechanics w13x. In one of the few studies dealing with the viscosity of glass containing inclusions, Demana and Drummond w14x measured the viscous deformation of a glass–crystalline alumina system using a beambending viscosimeter. The samples contained up to 50 vol% of dispersed alumina particles in the glass matrix. A significant increase of the viscosity of the glass with the volume fraction of inclusions was found. Similar results were obtained by Tewari et al. who studied the rheological behaviour of SiC Žparticulate. –borosilicate glass composites at 675 and 7008C w15x. In both studies, the comparison of the experimental results with the prediction of classical theoretical models for the viscosity of suspensions behaving as Newtonian fluids w16x did not yield successful results, especially for high concentrations of inclusions. Indeed simple approaches based only on the volume fraction of the second phase are thought to be inadequate to predict the effective viscosity of these composite systems. Other studies dealing with the viscous deformation of glass matrix composites include the work of Dutton and Rahaman on soda–lime glass containing Ni particles w17x and the early study of Dunn on silica containing tungsten inclusions w18x. In these investigations, however, no attempt was made to compare the measured values with the prediction of available models. A new equation for the viscosity of suspensions has been proposed based on the analogy between field properties of two-phase materials and viscous flow of suspensions w19x. The innovative aspect of this approach is that the shape and orientation of the suspended particles are explicitly taken into account. Besides having been applied to calculate the viscosity of several liquid suspensions, this equation has
been recently used to predict the viscosity of melted glasses containing solid phase aggregates w20x. It was found that the upper concentration limit for application of the equation was coincident with the onset of particle–particle interactions. The objective of the present study is to investigate the suitability of the equation to predict the effective viscosity of glass matrix composites containing rigid inclusions. Literature experimental data for glass composites with different inclusion morphologies are considered. The viscosities involved vary in a wide range Ž0.05–10 6 MPa s..
2. Analysis The mathematical analogy between different groups of physical properties of materials has been established in the past on the basis of these properties being governed by equations of the same form w21x. In general, field properties, such as thermal and electrical conductivity, can be defined by equations of the following type q˙ s yl= E
Ž 1.
where q˙ is a flow density, E s E Ž x, y, z . is a three-dimensional field and l is an effective field property. The viscosity of Newtonian flow, h , is defined by the following equation
tx z s h
dÕ
Ž 2.
dy
where t x z is the shear stress in the Ž x, z . plane and dÕrd y is the velocity gradient in the y direction. It was shown by Saltzer and Schulz that modifying Eq. ¨ Ž2. for the steady laminar flow of an incompressible fluid, the analogy between Eqs. Ž1. and Ž2. becomes obvious w19x. Thus, the viscosity of suspensions can be treated as a field property. On the basis of this analogy, and using earlier derived equations for the effective field property of two-phase composite materials w22x, the expression for the effective viscosity of an isotropic suspension, valid for systems containing isolated, i.e. non-interacting inclusions, has been derived as w19,20x
h s h0 Ž 1 y f .
m
Ž 3.
A.R. Boccaccinir Materials Letters 34 (1998) 285–289 Table 1 Shape factor Ž F . as a function of the axial ratio of spheroidal particles w22x Axial ratio Ž z r x .
Shape factor Ž F .
Description
0 0.002 0.02 0.1 0.2 1 2 10 20 200 2000 ™`
0 0.00157 0.01532 0.06959 0.12476 0.33333 0.41322 0.48986 0.49663 0.49994 0.49999 0.5
platelet oblate spheroid
3F y 2 3F Ž 1 y 2 F .
on melted glasses containing crystalline phases, i.e. at relatively low viscosities Ž-f 10 3 MPa s.. The application of the equation to predict the viscosity of glass matrix composite systems, which is of interest especially for higher viscosities Žup to f 10 6 MPa s., is considered in the present study. 3. Comparison with experimental results and discussion
sphere prolate spheroid
fibre
where h 0 is the viscosity of the fluid, f is the volume fraction of the rigid, i.e. non-deformable, inclusions and m is a function of the shape of the inclusions, which are modeled by spheroids having a fixed axial ratio Ž zrx . ms
287
Ž 4.
In Eq. Ž4., F is the shape factor, which depends on the axial ratio of the inclusions and is identical to the well-known deelectrification factor w19,22x. Some characteristic values of F are listed in Table 1. A complete tabulation of the values of F as a function of zrx has been given elsewhere w22x. The axial ratio of the inclusions can be determined on plane sections of the material investigated by measuring the area and perimeter of sectioned inclusions and from stereological equations w23x. Thus, Eq. Ž3. has predictive character as all variables involved can be measured independently. As mentioned before, Eq. Ž3. is applicable for composite systems containing isolated and well-distributed inclusions, i.e. when there is no contact between the inclusions and, therefore, no percolation effects are expected. It is interesting to point out that Eq. Ž3. has the same form as the equations of Brinkenman w24x and Roscoe w25x, which were derived as an extension of the original equation of Einstein, valid for dilute suspensions w26x. Eq. Ž3. has been tested successfully by comparison with experimental data on liquid suspensions and
As noted above, only limited data has been published on the viscosity of glass matrix composite materials. The earliest study found in the literature dealing with this topic is the work of Dunn on the viscous behaviour of silica with tungsten inclusions of spheroidal shape w18x. A comparison of his experimental results with the present model is difficult, however, because the temperature at which the viscosity was measured was varied along with the volume fraction of inclusions. Moreover extensive particle–particle interaction was found at concentration as low as 10 vol%. As mentioned before, Eq. Ž3. is not expected to be applicable in this case. The experimental data for the effective viscosity of a potassia borosilicate glass containing alumina particulates as inclusions, as determined by Demana and Drummod w14x using a beam-bending viscosimeter, are more suitable for comparison with the present approach. The measured values, normalized to the viscosity of the glass free of inclusions, are shown in Fig. 1 together with the prediction of Eq. Ž3.. The viscosity measurements covered a range of 10 8 –10 11 Pa s in the temperature range from 575 to 9008C. Unfortunately, no information on the morphology of the included particles was provided by the authors w14x. The theoretical curve was therefore calculated using zrx as a fitting parameter. As Fig. 1 shows, the best result was obtained for F s 0.483 which corresponds to an axial ratio zrx f 8 w22x. Micrographs shown in the original work w14x suggest some elongation of the alumina particles, however an axial ratio of f 8 seems to be too large. The micrographs of the composite microstructure at a concentration of inclusions of 40 vol% show considerable particle–particle interaction w14x. Thus, using a lower axial ratio in Eqs. Ž3. and Ž4., corresponding to the more equiaxed particles shown in the published micrographs w14x, would make the theoretical and ex-
288
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
Fig. 1. Relative viscosity of potassia borosilicate glass containing . calalumina inclusions: ŽB. measured values w14x, Ž culated curve ŽEqs. Ž3. and Ž4., F s 0.483..
perimental data to coincide only up to concentrations of f 20 vol%. This may represent a more realistic scenario for the applicability of the approach. Another set of data can be obtained from the work of Dutton and Rahaman w17x, who studied the creep deformation of soda–lime glass compacts containing spherical Ni inclusions of three different particle size ranges. A loading dilatometer was used to obtain creep strain rates at constant stresses, from which the creep viscosity was obtained. Only the composites containing the larger particles Žin the range 150–300 m m. did not show particle–particle interaction, i.e. percolation behaviour, up to concentrations higher than 10–15 vol%. The experimental data for the viscosity of glass composites containing Ni inclusions Ž150–300 m m. at 7308C are given in Fig. 2, together with the theoretical curve. This was calculated from Eqs. Ž3. and Ž4. using zrx s 1 Ž F s 0.33.. There is excellent agreement for the low volume fraction of inclusions Ž10 vol%. but beyond the percolation limit, the equation tends to underestimate the measured data, in agreement with the previous observation for the glassralumina system. The last set of available viscosity data corresponds to the data measured by Tewari et al. w15x, who studied the rheological behavior of SiC Žparticulate. –borosilicate composites at 675 and 7008C. Two size ranges of SiC inclusions Žfine and coarse grade. were considered. The samples were tested in
compression. It was found that at a high concentration of inclusions Ž40 vol%. the flow behaviour of the composites was non-Newtonian with a significantly faster increase of viscosity. The experimental data for the SiC particulate reinforced glass matrix composites of Tewari et al. w15x is shown in Fig. 3 together with the theoretical curve from Eqs. Ž3. and Ž4.. A value F s 0.2 was used in the calculation, which corresponds to an axial ratio zrx f 0.7 w20x. This value seems realistic by observation of the images of the microstructure presented in the original investigation w15x. Thus, Fig. 3 confirms the suitability of Eq. Ž3. to predict the viscosity of glassrcrystalline composite systems when the contact between particles is negligible or nonexistent. On the contrary, when particle–particle interaction is significant, the theoretical prediction underestimates the measured values. Unfortunately, the limited experimental data available is not sufficient to make a definite statement on the prediction capability of the equation. Except for the study of Dutton and Rahaman on glassrNi composites w17x, the shape of the included particles, which enters in Eq. Ž3. via the shape factor F, has not been properly characterized. Thus, new experimental investigations on the viscosity behaviour of glassrcrystal composite systems is re-
Fig. 2. Creep viscosity of soda–lime glass matrix composites containing spherical Ni inclusions Ž150–300 m m.. Ž'. Measured . calculated curve ŽEqs. Ž3. and Ž4., values w17x, Ž F s 0.333.. Note that the percolation limit was in the range 10–15 vol% for this system.
A.R. Boccaccinir Materials Letters 34 (1998) 285–289
289
tent. A wide range of viscosities Ž0.05–10 6 MPa s. was analyzed for three glass composite systems drawn from the literature. More experimental data on composites containing particles of well-characterized shape is required, however, to make a definite statement on the prediction capability of the equation. Such measurements are the focus of current research.
References
Fig. 3. Viscosity of borosilicate glass matrix composites containing SiC particle inclusions: ŽB. measured values, coarse particles, Ž^. measured values, fine particles w15x, Ž . calculated curve ŽEqs. Ž3. and Ž4., F s 0.2..
quired. A well-characterized Al 2 O 3-platelet reinforced borosilicate glass w27x has been chosen as a model system for such on-going studies. Finally, it should be pointed out that the range of applicability of the approach could be extended for higher volume fraction of inclusions if only some of the particles become in contact, i.e. before the formation of a particle continuous network. In this case, the shape of the inclusions being modeled by spheroids would change from that of the primary particles to that of the agglomerates, clusters or chains of particles formed. These will then have a particular axial ratio, and hence a characteristic shape factor F, which should be used in Eq. Ž4..
4. Conclusions It has been shown that a previously derived equation that treats viscosity as a field or transport property can be used to predict the effective viscosity of glass matrix composites containing rigid particles. The equation includes a shape factor which can be calculated from the axial ratio of the inclusions modeled by spheroids. The approach is only applicable for volume fraction of inclusions for which the particle–particle interaction is negligible or nonexis-
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