1068 Journal of the American Ceramic Society--Bao and Nicholson Vol. 90. No. 4 on the other, and then current is passed, particles will infiltrate the medium and deposit on the back electrode. For flat sub- strates, Hamaker's lawdescribes the deposit yield as Y mobility, E is the electric field strength, S the electrode surface rea,cs the colloidal particle concentration in the suspension and t the tim For electrophoretic infiltration deposition of a flat board with parallel capillaries directed along the electric field to an electrode of opposite sign, this equation can be modified NVncs dsdt I/N(VEP+),ds dr ⅹ(m) x104 where Ip is the particle velocity, VEP the particle electrophoretic elocity, Vi the liquid velocity, S the capillary cross-section Fig 14. Electric field(E) along the center line of the fiber bundle area, and n the number of capillaries. When the suspension volume is large, cs is constant, i.e.: Y= Nc,// VEP dS+Ncs/VidS the debye length Following the procedure developed by Wang k can be estimated from the suspension conductivity. It is found that K Now, Is VL ds=0 due to net zero liquid flow across the ension with a conductivity of 2.5 cross section in a closed capillary. Thus, assuming that the field S/cm. Assuming that the fiber has the same zeta potential as alumina particle, then otal inter- action potential energy (VT=VA+VR between the fiber fila- Y =(NCSHEPEXS)r (11) ment and particle is estimated as per Fig. 15 It is clear that the electrical double-layer repulsive interaction where k= NCSHEPExS between the particles and fiber is significant and opposes particle F deposition as they pass. The maximum energy barrier is a 60kT at 10 nm and the equilibrium ation distance >100 nm k=CsHEPEx PA (12 Thus, particles can be considered" lubricated, as they diffuse the fiber preform and deposit on the back electrode where P is the porosity and d is the surface area According to Eq (11), the deposit yield depends on the elec- tric field in the suspension. The electric field, E, between two (3) Modeling EPID of Matrix Particles into a Non- parallel plates is Conductive, Porous Substrate A non-conductive porous medium(e.g. a fiber preform) can be modeled as a collection of parallel cylindrical micro-capillaries in the electric field direction. If an electrode is placed on one side of the porous medium and a suspension with the other electrode where /is the current, R is the suspension resistance, inter-electrode distance. Now.r is 70 S is the electrode a,os is the suspension con- ivity, andf is the t depending on the suspen sion volume electrode ition and the cell design. If allel electrode plates, then f= 1. If the suspension fills all around the electrodes, /<l. sc I fA where A is the current density During EPID, the suspension conductivity is assumed to re- main unchanged (<+3%); thus, os is constant at 2.5 HS/cm, the Separation Distance(nm) value measured before epid. the electric field is related exclu sively to the current density and cell coefficient, f. Fig 15. Total interaction energy as a function of the separation dis- Dividing the epid cell into two sub-cells. i.e. inside tance between a fiber filament and a particle capillaries and outside, the sub-cell coefficient is I insidewhere e is the dielectric permittivity of ethanol, xp and xf are the zeta potential of the particle and fiber, respectively, and k1 is the Debye length. Following the procedure developed by Wang30 k1 can be estimated from the suspension conductivity. It is found that k1 is 61 nm for an alumina suspension with a conductivity of 2.5 mS/cm. Assuming that the fiber has the same zeta potential as an alumina particle, then zp 5 zf 5 80 mV. Thus, the total inter￾action potential energy (VT 5 VA1VR) between the fiber fila￾ment and particle is estimated as per Fig. 15. It is clear that the electrical double-layer repulsive interaction between the particles and fiber is significant and opposes particle deposition as they pass. The maximum energy barrier is 60 kT at 10 nm and the equilibrium separation distance 4100 nm. Thus, particles can be considered ‘‘lubricated,’’ as they diffuse the fiber preform and deposit on the back electrode. (3) Modeling EPID of Matrix Particles into a Non￾Conductive, Porous Substrate A non-conductive porous medium (e.g. a fiber preform) can be modeled as a collection of parallel cylindrical micro-capillaries in the electric field direction. If an electrode is placed on one side of the porous medium and a suspension with the other electrode on the other, and then current is passed, particles will infiltrate the medium and deposit on the back electrode. For flat sub￾strates, Hamaker’s law33 describes the deposit yield as; Y ¼ Z t 0 mEPEScs dt (8) where Y is the deposit mass, mEP the particle electrophoretic mobility, E is the electric field strength, S the electrode surface area, cs the colloidal particle concentration in the suspension, and t the time. For electrophoretic infiltration deposition of a flat board with parallel capillaries directed along the electric field to an electrode of opposite sign, this equation can be modified: Y ¼ Z t 0 Z S NVpcs dS dt ¼ Z t 0 Z S NðVEP þ VLÞcs dS dt (9) where Vp is the particle velocity, VEP the particle electrophoretic velocity, VL the liquid velocity, S the capillary cross-section area, and N the number of capillaries. When the suspension volume is large, cs is constant, i.e.: Y ¼ Ncst Z S VEP dS þ Ncs Z S VL dS (10) Now, R S VL dS ¼ 0 due to net zero liquid flow across the cross section in a closed capillary. Thus, assuming that the field, Ex, is uniform inside the capillary, Eq. (10) reduces to Y ¼ ðNcsmEPExSÞt ¼ kt (11) where k ¼ NcsmEPEX S: For a porous medium, k can be written as k ¼ csmEPEX PA (12) where P is the porosity and A is the surface area. According to Eq. (11), the deposit yield depends on the elec￾tric field in the suspension. The electric field, E, between two parallel plates is E ¼ IR L (13) where I is the current, R is the suspension resistance, and L is the inter-electrode distance. Now, R is R ¼ f L ssS (14) where S is the electrode surface area, ss is the suspension con￾ductivity, and f is the cell coefficient depending on the suspen￾sion volume, electrode, size and position, and the cell design. If the suspension just fills the rectangular space between two par￾allel electrode plates, then f 5 1. If the suspension fills all around the electrodes, fo1. So, E ¼ f I ssS ¼ f L ss (15) where L is the current density. During EPID, the suspension conductivity is assumed to re￾main unchanged (o73%); thus, ss is constant at 2.5 mS/cm, the value measured before EPID. The electric field is related exclu￾sively to the current density and cell coefficient, f. Dividing the EPID cell into two sub-cells, i.e., inside the capillaries and outside, the sub-cell coefficient is 1 inside the Fig. 14. Electric field (Ex) along the center line of the fiber bundle shown in Fig. 13. −40 −30 −20 −10 0 10 20 30 40 50 60 70 0 50 100 150 200 Separation Distance (nm) VT/kT Fig. 15. Total interaction energy as a function of the separation dis￾tance between a fiber filament and a particle. 1068 Journal of the American Ceramic Society—Bao and Nicholson Vol. 90, No. 4