3316 Journal of the American Ceramic Society--Zok duces a gradual increase in ph and a corresponding reduc- t mechanism (n=3 for vapo tion in the s potential between par ching the iso- for lattice diffusion. and n=7 for surface dif- electric point, the slurry coagulates. The process is designed to fusion). F a/R<l, Youngs modulus of the bonded affect coagulation after completion of vacuum bagging such that aggregate scales linearly with junction radius in accordance integrity for subsequent handling Properties of Porous Matrices Ep=(2x八(R Significant progress has been made in the understanding of the where Ep is Youngs modulus of the particles; z is the particle mechanical properties of porous matrices and their dependence oordination number (approximately six for random packing) n the topology of the constituent phases as well as their evo- D is the relative packing density; and 40. 76(calculated by lution with time. The key results from analytical models, nu- the discrete element method (DEm), described in Sidebar A) merical simulations, and experimental measurements are Combining Eqs. (4)and (5)yields the time dependence of the presented below (1) Monophase Particle Networks E 0.76 work follows a power law of the form*4s nophase particle net- Junction growth due to sintering in a m by surface diffusion(n=7), the time lus increase of only This has important consequences on the long-term durability of mullite-based CFCCs under typical where a is the junction radius, R is the particle radius, I is the The relationship between toughness and junction radius for a sintering time IR is a reference time, and n is a constant; both Ir bonded particle aggregate has been obtained from numerical Sidebar A. Numerical Simulation of Bonded Particle Aggregates Youngs modulus of a bonded particle aggregate is simulated numerically using the discrete element method (DEM). " The junction response that defines the element properties is derived from finite element analysis( FEA)of a single particle in a iodic array. The interaction between particle junctions. characterized by the displacement of one junction due to the force acting on another, is also derived from FEA. An isotropic 0.8 ensure equilibrium, each particle is required to touch at least u o. three neighbors upon placement onto the aggregate. The final g particle packing density is 55% and the average coordination 3 o4 number is 6: consistent with measured values for random loose measuremt packing of spherical particles For monophase systems, junction growth is simulated by Discrete element niformly expanding the particles and re-distributing the overlapping material uniformly over the free surface of the particles. In contrast, for systems containing a precursor derived binder, the material is modeled as an aggregate of Relative density touching monophase particles, each coated with a uniform layer of the second phase. The elastic response of the junctions 50r is calculated by FEA of a periodic array with the two phases xplicitly discretized For both mono-and two-phase systems the particle network is then subjected to a prescribed macroscopically uniform strain field and the effective ela sponse is determined using DEM. Typical numerical k Its =1.0 and comparisons with experimental measurements"" are shown in Fig. Al(a). rr;=12(aR2 The toughness of the aggregate is also computed by DEM (Fig. Al(b). In this case, a crack is defined by a plane eparating particles that have had the junctions between them broken(inset of Fig. Al(b). The simulation proceeds by incrementally increasing the remote displacement(for tension or the remote rotation(for bending), while allowing the method simulatior junctions at the crack tip to fail at a critical junction stress, Oe given by 020.3 Junction size, a/R Inction toughness. The results of the simula- lus and (b)toughness of monophase-bonded particle aggregates. E tions(Fig. Al(b)are well described by Eq (7)in the text. perimental measurements in(a)are for alumina(from Green et al.produces a gradual increase in pH and a corresponding reduc￾tion in the z potential between particles. Upon reaching the iso￾electric point, the slurry coagulates. The process is designed to affect coagulation after completion of vacuum bagging such that the green preform exhibits substantially increased mechanical integrity for subsequent handling. V. Properties of Porous Matrices Significant progress has been made in the understanding of the mechanical properties of porous matrices and their dependence on the topology of the constituent phases as well as their evo￾lution with time. The key results from analytical models, nu￾merical simulations, and experimental measurements are presented below. (1) Monophase Particle Networks Junction growth due to sintering in a monophase particle net￾work follows a power law of the form44,45: a R ¼ t tR
1=n (4) where a is the junction radius, R is the particle radius, t is the sintering time, tR is a reference time, and n is a constant; both tR and n depend on the transport mechanism (n 5 3 for vapor transport, n 5 5 for lattice diffusion, and n 5 7 for surface dif￾fusion). Provided a/R  1, Young’s modulus of the bonded aggregate scales linearly with junction radius in accordance with46,47 E Ep ¼ x zD 2p
a R  (5) where Ep is Young’s modulus of the particles; z is the particle coordination number (approximately six for random packing); D is the relative packing density; and x0.76 (calculated by the discrete element method (DEM), described in Sidebar A). Combining Eqs. (4) and (5) yields the time dependence of the modulus E Ep ¼ 0:76 zD 2p
t tR
1=n (6) When sintering occurs by surface diffusion (n 5 7), the time dependence is weak: a 10-fold increase in time leads to a modu￾lus increase of only 10%. This has important consequences on the long-term durability of mullite-based CFCCs under typical service conditions. The relationship between toughness and junction radius for a bonded particle aggregate has been obtained from numerical Sidebar A. Numerical Simulation of Bonded Particle Aggregates Young’s modulus of a bonded particle aggregate is simulated numerically using the discrete element method (DEM).47 The junction response that defines the element properties is derived from finite element analysis (FEA) of a single particle in a periodic array. The interaction between particle junctions, characterized by the displacement of one junction due to the force acting on another, is also derived from FEA. An isotropic random aggregate of touching spherical particles is then generated using a computer algorithm (inset of Fig. A1(a)). To ensure equilibrium, each particle is required to touch at least three neighbors upon placement onto the aggregate. The final particle packing density is 55% and the average coordination number is 6: consistent with measured values for random loose packing of spherical particles. For monophase systems, junction growth is simulated by uniformly expanding the particles and re-distributing the overlapping material uniformly over the free surface of the particles. In contrast, for systems containing a precursor￾derived binder, the material is modeled as an aggregate of touching monophase particles, each coated with a uniform layer of the second phase. The elastic response of the junctions is calculated by FEA of a periodic array with the two phases explicitly discretized. For both mono- and two-phase systems, the particle network is then subjected to a prescribed macroscopically uniform strain field and the effective elastic response is determined using DEM. Typical numerical results and comparisons with experimental measurements48 are shown in Fig. A1(a). The toughness of the aggregate is also computed by DEM (Fig. A1(b)).42 In this case, a crack is defined by a plane separating particles that have had the junctions between them broken (inset of Fig. A1(b)). The simulation proceeds by incrementally increasing the remote displacement (for tension) or the remote rotation (for bending), while allowing the junctions at the crack tip to fail at a critical junction stress, sc, given by: sc ¼ 2 ffiffiffiffiffiffiffiffiffiffi EpGj pa r where Gj is the junction toughness. The results of the simula￾tions (Fig. A1(b)) are well described by Eq. (7) in the text. Fig. A1. Discrete element method simulations of (a) Young’s modu￾lus and (b) toughness of monophase-bonded particle aggregates. Ex￾perimental measurements in (a) are for alumina (from Green et al. 48). 3316 Journal of the American Ceramic Society—Zok Vol. 89, No. 11