ceramic slurries with bimodal particle size distributions: rheology ...
CERAMIC SLURRIES WITH BIMODAL PARTICLE SIZE DISTRIBUTIONS: RHEOLOGY, SUSPENSION STRUCTURE AND BEHAVIOUR DURING PRESSURE FILTRATION R. Oberacker*, J. Reinshagen, H. von Both, M. J. Hoffmann Institut für Keramik im Maschinenbau, Universität Karlsruhe (TH) Haid-und-Neu-Straße 7 D-76131 Karlsruhe (Germany)
ABSTRACT Experiments were carried out with bimodal alumina suspensions under variation of the volume fraction of fine and coarse particles and of the interparticuluar repulsive forces. The suspensions were characterized by rheometry and direct visualisation of the suspension structure by CRYOSEM. The pressure filtration behaviour was characterised by the filtration kinetics and the density and microstructure of the filter cakes. The rheological behaviour, as well as the permeability of the filter cakes depend on both, the particle size distribution and the interparticular forces. INTRODUCTION It is common practice, to use powders with bimodal particle size distributions in ceramic suspension processing. These powders are known to favour high densities of the consolidated green bodies, due to their enhanced packing characteristics. The packing of bimodal particle systems is in general simply explained by existing theories and experimental results1 derived for coarse particles, where particle interaction forces are of minor importance. This is by far not sufficient to understand the mechanisms, which control the processing behaviour of ceramic suspensions and the particle arrangement in green bodies fabricated by the consolidation of such suspensions. Due to their colloidal character, interaction forces2 are to be expected to contribute significantly to the structure of the suspensions and the consolidated green bodies. In recent investigations3 we could demonstrate, that suspension and green body structures can be visualised by CRYO-SEM, a method which is common in bio and food science, but has been not applied in material science, so far. In the present work, we use this method to visualise the structure of suspensions from bimodal ceramic particle systems and to correlate them to suspension rheology and green body density. EXPERIMENTAL The Experiments were carried out with two alumina based powders (ALCOA CT 3000SG and CT19FG) with a mean particle size of 0.8 and 6 µm respectively. The isoelectric point (IEP) of both powders is close to pH 9. They were suspended in deionised water, varying the suspension stability in the absence of commercial deflocculants by simply changing the pH. Suspensions with bimodal particle size distributions were created by blending the two master suspensions with different coarse/fine ratios. The suspension rheology was characterised with a rotating rheometer (Rheolab LC 10, Physica, Stuttgart, Germany) with coaxial cylinders at a constant temperature of 20°C. The viscosity was determined at constant shear rate steps from 10 to 4000 s-1.
Consolidation of the suspensions was carried out by pressure filtration under constant pressure conditions of 1 MPa. A laboratory pressure casting equipment was used4, which allows for continuous measurement of the filtrate flow rate. The permeability of the filter cake, which is an indicator of the cake pore structure5, can be derived from the measured flow rate6. The liquid saturated filter cakes (denoted as filter cakes in the following text) were dried under controlled conditions in a climatic cabinet. Dimensional shrinkage and weight loss were measured during drying. The green body porosity was measured by Archimedes method after infiltrating the pores with paraffin. The porosity of the filter cakes was calculated from the green body porosities taking into account the drying shrinkage. The structure of the suspensions as well as of the filter cakes were analysed with a scanning electron microscope (SEM Hitachi S-3200N) equipped with cryogenic analysis system (CT1500, Oxford Instruments, Wiesbaden, Germany). RESULTS Suspension Structure and Rheology The rheology of the unimodal master suspensions depends on their pH and solids volume fraction cv. Newtonian behaviour was observed at low solids loading (cv < 20 %). Suspensions with medium cv exhibit pseudoplastic behaviour. At cv > 50% the suspensions become dilatant at higher shear rates. A constant solids loading of 35 vol.% was chosen for the bimodal suspensions. This solids loading ensures suitable handling of all suspensions over the investigated range of pH. The influence of the pH on the viscosity η of the master suspensions is shown in fig. 1. As expected, a viscosity maximum occurs at the IEP. The lowest viscosities are observed in the acidic region between pH 2 and 4, where the positive surface charges stabilise the suspensions. The viscosity is higher and the viscosity maximum is more pronounced for suspensions with fine particles. 10
viscosityeta [Pa s] Viskosität η [Pas]
dispersed
1
agglomerated
CT3000SG (fine)
fein grob
0,1
CT19FG (coarse) cV=35 Vol.%
IEP
D=100 [1/s]
0,01 2
3
4
5
6
7 pH
8
9
10
11
12
Fig. 1 Viscosity of the master suspensions (coarse and fine) with 35vol.% solids loading at a −1
constant shear rate of 100 s . Fig. 2 shows CRYO-SEM images taken from the interior of suspension droplets at a pH close to the IEP (pH ≅ 9) and at pH = 3.5. The coarse particles are well dispersed at pH 3.5 and form an agglomerate network at pH 9.4. The fine particles form a heavily textured skeleton of large agglomerates close to the IEP and a network of small and equiaxed agglomerates at pH 3.5. The
electrostatic repulsive interparticle forces are obviously not high enough to prevent agglomeration, even though the Zeta Potential at pH 3.5 was estimated to > 60 mV for the CT3000 suspension. This is significantly below the Zeta Potential of the CT19 suspensions at the same pH. This finding correlates with the rheology of the suspensions. Close to the IEP, both suspensions show pseudoplastic behaviour due to agglomerate disintegration with increasing shear rates. At pH 3.5, only the CT3000- suspension with the fine particles exhibits pseudoplastic characteristics, while the CT19suspension is purely Newtonian.
coarse pH 3.5
coarse pH 9.2
30µm
30µm fine pH 9.4
fine pH 3.5
30µm
30µm
Fig. 2 CRYO-SEM images of the coarse and fine fraction taken from the interior of suspension droplets at pH 3.5 and pH 9.2 (9.4), respectively. Bimodal suspensions were investigated at pH 3 and pH 6, which represent the state of maximum repulsive interparticle forces and a state, where agglomeration can be expected to have a significant influence on the structure and properties of the suspensions. The influence of the coarse/fine ratio on the suspension viscosity is shown in fig. 3. At a shear rate of 100 s-1, the viscosity increases with decreasing volume fraction of the coarse particles for both pH. At high shear rates (not included in fig. 3), the tendency is quite opposite. This is explained by the fact, that the viscosity of the fine unimodal suspension at high shear rates falls below
that of the coarse suspension, due to its pronounced pseudoplasticity. Thus, the rheological properties seem to follow a simple rule of mixture.
viscosity η [Pa Viskosität eta [Pas]s]
1
0,1
pH6 c V=35 Vol.% D=100 [1/s]
pH3 0,01
100 0
90 10
80 20
70 30
60 40
50 50
40 60
30 70
20 80
10 90
0 100
fraction ofFeinanteil coarse particles [%] [%]
Fig. 3 Viscosity versus fraction of coarse particles in mixtures at pH 3 and pH 6. ( Total solids loading is 35vol.%, shear rate 100 s − 1 ). The suspension structure visualised by CRYO-SEM is shown in fig. 4 for a pH of 3.5 and a volume fraction of 70% of coarse. A particle network is formed, where the coarse particles are in a more ore less homogeneous loose arrangement, interconnected by chains consisting of fine particles. This is again in good correlation with the slight pseudoplastic behaviour of the bimodal suspension, which can be explained by a network disintegration with increasing shear rates.
coarse/fine 70/30 pH 3.5
coarse/fine 70/30 pH 3.5
10µm
3µm
Fig. 4 CRYO-SEM images of a bimodal suspension at pH 3.5 with a solids loading of 35vol.%. Filter Cake and Green Body Characteristics The permeability and the drying shrinkage of the filter cakes is plotted in fig. 5. The permeability increases from pH 3 to pH 6. Up to 50 vol.% of coarse particles, the permeability decreases in the pH 3 suspension while it increases slightly at pH 6. Beyond 50 vol.% coarse, the permeability increases strongly in both cases.
3
1000
2,5
drying shrinkage [%]
1200
Trockenschwindung [%]
permeability [E-18 m^2] Permeabilität [E-18 m^2]
Essentially no drying shrinkage is observed beyond 70 vol.% of coarse particles, independent of the pH. At pH 3, shrinkage decreases continuously with increasing coarse/fine ratio. At pH 6, shrinkage has a pronounced maximum at about 50 vol.% of coarse particles.
800
pH6 600 400 200
pH3
2
pH6 1,5 1 0,5
pH3 0
100 0
90 10
80 20
70 30
60 40
50 50
40 60
30 70
20 80
0
10 100 0 90
100 80 30 70 40 60 50 50 40 90 20 0 10 60 30 70 20 80 fraction ofFeinanteil coarse particles [%] [%]
fraction of coarse particles [%] Feinanteil [%]
10 90
0 100
Fig. 5 Pemeability and drying shrinkage as a function of the fraction of coarse particles in mixtures at pH 3 and pH 6, respectively. The relative density of the green bodies after drying is given in fig. 6. A density maximum of about 75% of TD is obtained at a volume fraction of 60% of coarse particles at pH 3. At pH 6, the density optimum amounts only 60% of TD and shifts to 50 vol.% of coarse particles in the mixture. 100
relative density relative Dichte[%TD] [%]
90 80 70
pH3
60 50
pH6
40 30 20 10
max. 0
70 40 100 80 30 60 50 40 90 20 0 10 60 30 70 20 80 [%] fraction ofFeinanteil coarse particles [%] Grobanteil [%]
10 90
0 100
Fig. 6 Relative density of the green bodies versus the fraction of coarse particles in mixtures at pH 3 and pH 6, respectively. DISCUSSION Common approaches in suspension rheology, which consider bimodal size distributions are hard and soft sphere repulsion models7, e.g. the well known model of Krieger and Dougherty8. Such models relate the relative viscosity ηr to the solids volume fraction cv of the suspension and the maximum packing fraction cvm of the particle system. cvm is controlled by the size distribution and could be estimated from particle packing theories. According to such models, the viscosity of bimodal suspensions should always be lower than that of the unimodal base suspensions. Regard-
ing fig. 3, there is no experimental evidence from our results. However, it turns out that the suspension viscosity of bimodal suspensions follows more or less a simple rule of mixture. The final density of the green bodies is in good agreement with particle packing theories. The well dispersed suspensions (pH 3) result in an optimum coarse/fine ratio, which is not far away from the 73/27 ratio predicted by bimodal sphere models1. It becomes obvious, however, that agglomerated suspensions (pH 6) result in lower densities and a shift in the optimum coarse/fine ratio. The permeability and the drying shrinkage are strongly influenced by the state of agglomeration. Shrinkage ceases, when the coarse particles come into contact and form a skeleton. The permeability depends mainly on the diameter of the pore channels of the filter cake. For homogeneous particle arrangements, this diameter should be minimum at the coarse/fine ratio of the density maximum, which is true for pH 3. The filter cakes from the agglomerated suspensions (pH 6) thus obviously cannot consist of homogeneous packings. Despite of the relatively high consolidation pressure, the suspension structure is propagated into the filter cake. CONCLUSIONS The results confirm, that the processing behaviour of suspensions from bimodal ceramic particles is rather complicated and cannot be understood exclusively on the basis of existing packing theories. This is especially true, if the particles are not ideally dispersed, which is the case for many slip and pressure casting suspensions. More information is needed on the structure of such suspensions and consolidated filter cakes. The CRYO-SEM method has a high potential to visualise such structural details. REFERENCES 1 D.J. Cumberland and R.J. Crawford, "The Packing of Particles". Handbook of Powder Technology, Vol. 6, Elsevier Science, 1987 2 R.G. Horn, "Surface Forces and their Action in Ceramic Materials". Journal of the American Ceramic Society, 73 [5] 1117-1135 (1990) 3 H.v. Both, R. Oberacker, M.J. Hoffmann, "Characterisation and Consolidation of Aqueous SiC Suspensions". Zeitschrift für Metallkunde 90 [12] 996-1001 (1999) 4 R. Oberacker, A. Trendler, A. Geyer, "Optimierter Schlickerdruckguß als Formgebungsverfahren für additivarme Siliziumnitrid- Hochtemperaturwerkstoffe", pp. 551-556 in Werkstoff- und Verfahrenstechnik. Edited by G. Ziegler et al. DGM Informationsgesellschaft, Frankfurt 1997 5 F.A.L. Dullien, "Porous Media -Fluid Transport and Pore Structure", 2nd ed. Academic Press, San Diego, New York, 1992 6 M. Dröschel, R. Oberacker, M. J. Hoffmann, W. Schaller, Y. Y. Yang, D. Munz, "Silicon Carbide Evaporator Tubes with Porosity Gradient Designed by Finite Element Calculations", pp. 814-819 in Functionally Graded Materials 1998. Edited by W.A. Kaysser. Volumes 308-311 of Materials Science Forum, Trans Tech Publications, Zürich 1999 7 R. Fries and B. Rand, "Slip-Casting and Filter-Pressing", pp.155-185 in Processing of Ceramics, Part I. Edited by R.J. Brook. Volume 17A of Materials Science and Technology, VCH Weinheim, 1996 8 I.M Krieger and T.J. Dougherty, Trans. Soc. Rheol. 3 137-152 (1959)
ABSTRACT Experiments were carried out with bimodal alumina suspensions under variation of the volume fraction of fine and coarse particles and of the interparticuluar repulsive forces. The suspensions were characterized by rheometry and direct visualisation of the suspension structure by CRYOSEM. The pressure filtration behaviour was characterised by the filtration kinetics and the density and microstructure of the filter cakes. The rheological behaviour, as well as the permeability of the filter cakes depend on both, the particle size distribution and the interparticular forces. INTRODUCTION It is common practice, to use powders with bimodal particle size distributions in ceramic suspension processing. These powders are known to favour high densities of the consolidated green bodies, due to their enhanced packing characteristics. The packing of bimodal particle systems is in general simply explained by existing theories and experimental results1 derived for coarse particles, where particle interaction forces are of minor importance. This is by far not sufficient to understand the mechanisms, which control the processing behaviour of ceramic suspensions and the particle arrangement in green bodies fabricated by the consolidation of such suspensions. Due to their colloidal character, interaction forces2 are to be expected to contribute significantly to the structure of the suspensions and the consolidated green bodies. In recent investigations3 we could demonstrate, that suspension and green body structures can be visualised by CRYO-SEM, a method which is common in bio and food science, but has been not applied in material science, so far. In the present work, we use this method to visualise the structure of suspensions from bimodal ceramic particle systems and to correlate them to suspension rheology and green body density. EXPERIMENTAL The Experiments were carried out with two alumina based powders (ALCOA CT 3000SG and CT19FG) with a mean particle size of 0.8 and 6 µm respectively. The isoelectric point (IEP) of both powders is close to pH 9. They were suspended in deionised water, varying the suspension stability in the absence of commercial deflocculants by simply changing the pH. Suspensions with bimodal particle size distributions were created by blending the two master suspensions with different coarse/fine ratios. The suspension rheology was characterised with a rotating rheometer (Rheolab LC 10, Physica, Stuttgart, Germany) with coaxial cylinders at a constant temperature of 20°C. The viscosity was determined at constant shear rate steps from 10 to 4000 s-1.
Consolidation of the suspensions was carried out by pressure filtration under constant pressure conditions of 1 MPa. A laboratory pressure casting equipment was used4, which allows for continuous measurement of the filtrate flow rate. The permeability of the filter cake, which is an indicator of the cake pore structure5, can be derived from the measured flow rate6. The liquid saturated filter cakes (denoted as filter cakes in the following text) were dried under controlled conditions in a climatic cabinet. Dimensional shrinkage and weight loss were measured during drying. The green body porosity was measured by Archimedes method after infiltrating the pores with paraffin. The porosity of the filter cakes was calculated from the green body porosities taking into account the drying shrinkage. The structure of the suspensions as well as of the filter cakes were analysed with a scanning electron microscope (SEM Hitachi S-3200N) equipped with cryogenic analysis system (CT1500, Oxford Instruments, Wiesbaden, Germany). RESULTS Suspension Structure and Rheology The rheology of the unimodal master suspensions depends on their pH and solids volume fraction cv. Newtonian behaviour was observed at low solids loading (cv < 20 %). Suspensions with medium cv exhibit pseudoplastic behaviour. At cv > 50% the suspensions become dilatant at higher shear rates. A constant solids loading of 35 vol.% was chosen for the bimodal suspensions. This solids loading ensures suitable handling of all suspensions over the investigated range of pH. The influence of the pH on the viscosity η of the master suspensions is shown in fig. 1. As expected, a viscosity maximum occurs at the IEP. The lowest viscosities are observed in the acidic region between pH 2 and 4, where the positive surface charges stabilise the suspensions. The viscosity is higher and the viscosity maximum is more pronounced for suspensions with fine particles. 10
viscosityeta [Pa s] Viskosität η [Pas]
dispersed
1
agglomerated
CT3000SG (fine)
fein grob
0,1
CT19FG (coarse) cV=35 Vol.%
IEP
D=100 [1/s]
0,01 2
3
4
5
6
7 pH
8
9
10
11
12
Fig. 1 Viscosity of the master suspensions (coarse and fine) with 35vol.% solids loading at a −1
constant shear rate of 100 s . Fig. 2 shows CRYO-SEM images taken from the interior of suspension droplets at a pH close to the IEP (pH ≅ 9) and at pH = 3.5. The coarse particles are well dispersed at pH 3.5 and form an agglomerate network at pH 9.4. The fine particles form a heavily textured skeleton of large agglomerates close to the IEP and a network of small and equiaxed agglomerates at pH 3.5. The
electrostatic repulsive interparticle forces are obviously not high enough to prevent agglomeration, even though the Zeta Potential at pH 3.5 was estimated to > 60 mV for the CT3000 suspension. This is significantly below the Zeta Potential of the CT19 suspensions at the same pH. This finding correlates with the rheology of the suspensions. Close to the IEP, both suspensions show pseudoplastic behaviour due to agglomerate disintegration with increasing shear rates. At pH 3.5, only the CT3000- suspension with the fine particles exhibits pseudoplastic characteristics, while the CT19suspension is purely Newtonian.
coarse pH 3.5
coarse pH 9.2
30µm
30µm fine pH 9.4
fine pH 3.5
30µm
30µm
Fig. 2 CRYO-SEM images of the coarse and fine fraction taken from the interior of suspension droplets at pH 3.5 and pH 9.2 (9.4), respectively. Bimodal suspensions were investigated at pH 3 and pH 6, which represent the state of maximum repulsive interparticle forces and a state, where agglomeration can be expected to have a significant influence on the structure and properties of the suspensions. The influence of the coarse/fine ratio on the suspension viscosity is shown in fig. 3. At a shear rate of 100 s-1, the viscosity increases with decreasing volume fraction of the coarse particles for both pH. At high shear rates (not included in fig. 3), the tendency is quite opposite. This is explained by the fact, that the viscosity of the fine unimodal suspension at high shear rates falls below
that of the coarse suspension, due to its pronounced pseudoplasticity. Thus, the rheological properties seem to follow a simple rule of mixture.
viscosity η [Pa Viskosität eta [Pas]s]
1
0,1
pH6 c V=35 Vol.% D=100 [1/s]
pH3 0,01
100 0
90 10
80 20
70 30
60 40
50 50
40 60
30 70
20 80
10 90
0 100
fraction ofFeinanteil coarse particles [%] [%]
Fig. 3 Viscosity versus fraction of coarse particles in mixtures at pH 3 and pH 6. ( Total solids loading is 35vol.%, shear rate 100 s − 1 ). The suspension structure visualised by CRYO-SEM is shown in fig. 4 for a pH of 3.5 and a volume fraction of 70% of coarse. A particle network is formed, where the coarse particles are in a more ore less homogeneous loose arrangement, interconnected by chains consisting of fine particles. This is again in good correlation with the slight pseudoplastic behaviour of the bimodal suspension, which can be explained by a network disintegration with increasing shear rates.
coarse/fine 70/30 pH 3.5
coarse/fine 70/30 pH 3.5
10µm
3µm
Fig. 4 CRYO-SEM images of a bimodal suspension at pH 3.5 with a solids loading of 35vol.%. Filter Cake and Green Body Characteristics The permeability and the drying shrinkage of the filter cakes is plotted in fig. 5. The permeability increases from pH 3 to pH 6. Up to 50 vol.% of coarse particles, the permeability decreases in the pH 3 suspension while it increases slightly at pH 6. Beyond 50 vol.% coarse, the permeability increases strongly in both cases.
3
1000
2,5
drying shrinkage [%]
1200
Trockenschwindung [%]
permeability [E-18 m^2] Permeabilität [E-18 m^2]
Essentially no drying shrinkage is observed beyond 70 vol.% of coarse particles, independent of the pH. At pH 3, shrinkage decreases continuously with increasing coarse/fine ratio. At pH 6, shrinkage has a pronounced maximum at about 50 vol.% of coarse particles.
800
pH6 600 400 200
pH3
2
pH6 1,5 1 0,5
pH3 0
100 0
90 10
80 20
70 30
60 40
50 50
40 60
30 70
20 80
0
10 100 0 90
100 80 30 70 40 60 50 50 40 90 20 0 10 60 30 70 20 80 fraction ofFeinanteil coarse particles [%] [%]
fraction of coarse particles [%] Feinanteil [%]
10 90
0 100
Fig. 5 Pemeability and drying shrinkage as a function of the fraction of coarse particles in mixtures at pH 3 and pH 6, respectively. The relative density of the green bodies after drying is given in fig. 6. A density maximum of about 75% of TD is obtained at a volume fraction of 60% of coarse particles at pH 3. At pH 6, the density optimum amounts only 60% of TD and shifts to 50 vol.% of coarse particles in the mixture. 100
relative density relative Dichte[%TD] [%]
90 80 70
pH3
60 50
pH6
40 30 20 10
max. 0
70 40 100 80 30 60 50 40 90 20 0 10 60 30 70 20 80 [%] fraction ofFeinanteil coarse particles [%] Grobanteil [%]
10 90
0 100
Fig. 6 Relative density of the green bodies versus the fraction of coarse particles in mixtures at pH 3 and pH 6, respectively. DISCUSSION Common approaches in suspension rheology, which consider bimodal size distributions are hard and soft sphere repulsion models7, e.g. the well known model of Krieger and Dougherty8. Such models relate the relative viscosity ηr to the solids volume fraction cv of the suspension and the maximum packing fraction cvm of the particle system. cvm is controlled by the size distribution and could be estimated from particle packing theories. According to such models, the viscosity of bimodal suspensions should always be lower than that of the unimodal base suspensions. Regard-
ing fig. 3, there is no experimental evidence from our results. However, it turns out that the suspension viscosity of bimodal suspensions follows more or less a simple rule of mixture. The final density of the green bodies is in good agreement with particle packing theories. The well dispersed suspensions (pH 3) result in an optimum coarse/fine ratio, which is not far away from the 73/27 ratio predicted by bimodal sphere models1. It becomes obvious, however, that agglomerated suspensions (pH 6) result in lower densities and a shift in the optimum coarse/fine ratio. The permeability and the drying shrinkage are strongly influenced by the state of agglomeration. Shrinkage ceases, when the coarse particles come into contact and form a skeleton. The permeability depends mainly on the diameter of the pore channels of the filter cake. For homogeneous particle arrangements, this diameter should be minimum at the coarse/fine ratio of the density maximum, which is true for pH 3. The filter cakes from the agglomerated suspensions (pH 6) thus obviously cannot consist of homogeneous packings. Despite of the relatively high consolidation pressure, the suspension structure is propagated into the filter cake. CONCLUSIONS The results confirm, that the processing behaviour of suspensions from bimodal ceramic particles is rather complicated and cannot be understood exclusively on the basis of existing packing theories. This is especially true, if the particles are not ideally dispersed, which is the case for many slip and pressure casting suspensions. More information is needed on the structure of such suspensions and consolidated filter cakes. The CRYO-SEM method has a high potential to visualise such structural details. REFERENCES 1 D.J. Cumberland and R.J. Crawford, "The Packing of Particles". Handbook of Powder Technology, Vol. 6, Elsevier Science, 1987 2 R.G. Horn, "Surface Forces and their Action in Ceramic Materials". Journal of the American Ceramic Society, 73 [5] 1117-1135 (1990) 3 H.v. Both, R. Oberacker, M.J. Hoffmann, "Characterisation and Consolidation of Aqueous SiC Suspensions". Zeitschrift für Metallkunde 90 [12] 996-1001 (1999) 4 R. Oberacker, A. Trendler, A. Geyer, "Optimierter Schlickerdruckguß als Formgebungsverfahren für additivarme Siliziumnitrid- Hochtemperaturwerkstoffe", pp. 551-556 in Werkstoff- und Verfahrenstechnik. Edited by G. Ziegler et al. DGM Informationsgesellschaft, Frankfurt 1997 5 F.A.L. Dullien, "Porous Media -Fluid Transport and Pore Structure", 2nd ed. Academic Press, San Diego, New York, 1992 6 M. Dröschel, R. Oberacker, M. J. Hoffmann, W. Schaller, Y. Y. Yang, D. Munz, "Silicon Carbide Evaporator Tubes with Porosity Gradient Designed by Finite Element Calculations", pp. 814-819 in Functionally Graded Materials 1998. Edited by W.A. Kaysser. Volumes 308-311 of Materials Science Forum, Trans Tech Publications, Zürich 1999 7 R. Fries and B. Rand, "Slip-Casting and Filter-Pressing", pp.155-185 in Processing of Ceramics, Part I. Edited by R.J. Brook. Volume 17A of Materials Science and Technology, VCH Weinheim, 1996 8 I.M Krieger and T.J. Dougherty, Trans. Soc. Rheol. 3 137-152 (1959)
