Science Direct Materials ELSEVIER Progress in Materials Science 52(2007)1-61 cience www.elsevier.com/locate/pmatsci A review on fundamentals and applications of electrophoretic deposition (epd Laxmidhar besra a, Meilin liu b Colloids and Materials Chemistry Gr Research Laboratory ( Council of Scientific and Industrial Research) swar 751013. Orissa. India School of Materials Science and ering, Georgia Institute of Technology 'I Ferst Drive. Atlanta. GA 30332-0245. USA Received 10 January 2006: accepted 5 July 2006 Abstract This review encompasses the fundamental aspects of electrophoretic deposition technique, factors infuencing the deposition process, kinetic aspects, types of EPD, the driving forces, preparation of electrophoretic suspension, stability and control of suspension, mechanisms involved in EPD, mul ticomponent/composite deposition, drying of deposits obtained by EPD. Numerous applications including coatings, nanoscale assemb aded materials, hybrid materials, infiltration in porous and patterned thin films, near shape ceramics and glasses, woven fibre preforms for preparation of fibre reinforced ceramic matrix composites, etc have been described. The use of mathematical modeling including kinetic equations for deposit formation and volumetric particle concentration in the suspension, together with brief description of discrete ele- ment modeling of EPd process is presented c 2006 Elsevier Ltd. All rights reserved 1. Introduction 2. Electrophoretic deposition-definition 3. Factors influencing EPD m 0079-6425S see front matter 2006 Elsevier Ltd. All rights reserved doi:l0.l016 pmatsci.2006.07.001
A review on fundamentals and applications of electrophoretic deposition (EPD) Laxmidhar Besra a,*, Meilin Liu b a Colloids and Materials Chemistry Group, Regional Research Laboratory (Council of Scientific and Industrial Research), Bhubaneswar 751013, Orissa, India b School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, Atlanta, GA 30332-0245, USA Received 10 January 2006; accepted 5 July 2006 Abstract This review encompasses the fundamental aspects of electrophoretic deposition technique, factors influencing the deposition process, kinetic aspects, types of EPD, the driving forces, preparation of electrophoretic suspension, stability and control of suspension, mechanisms involved in EPD, multicomponent/composite deposition, drying of deposits obtained by EPD. Numerous applications including coatings, nanoscale assembly, micropatterned thin films, near shape ceramics and glasses, solid oxide fuel cells, laminated or graded materials, hybrid materials, infiltration in porous and woven fibre preforms for preparation of fibre reinforced ceramic matrix composites, etc. have been described. The use of mathematical modeling including kinetic equations for deposit formation and volumetric particle concentration in the suspension, together with brief description of discrete element modeling of EPD process is presented. 2006 Elsevier Ltd. All rights reserved. Contents 1. Introduction . ..................................................... 3 2. Electrophoretic deposition – definition .................................... 3 3. Factors influencing EPD. ............................................. 5 0079-6425/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2006.07.001 * Corresponding author. Tel.: +91 674 481 635; fax: +91 674 581 637. E-mail address: ldbesra@rrlbhu.res.in (L. Besra). Progress in Materials Science 52 (2007) 1–61 www.elsevier.com/locate/pmatsci
L Besra, M. Liu/ Progress in Materials Science 52(2007)1-61 3.1. Parameters related to the suspension 3.1.1. Particle size 3.1.2. Dielectric constant of liquid 667 3.1.3. Conductivity of suspension 3.1. 4. Viscosity of suspension. 3.1.5. Zeta potential 3.1.6. Stability of suspension 3. 2. Parameters related to the process 3.2.1. Effect of deposition time 0001 3.2.2. Applied voltage 3.2.3. Concentration of solid in suspension 13 3.2. 4. Conductivity of substrate 13 4. Kinetics of electrophoretic deposition 5. Role of polymer binders in EPD 6. Importance of powder washing before EPD 7. Practical considerations 17 8. Water-based EPD 9. Non-aqueous EPD 10. Charge development on powder surface in suspension 10. 1. Aqueous suspension I1. 1. The electrical double layer and electrophoretic mobility. 11. 2. DLvO theory and suspension stability. 4 12. Mechanism of EPD process 12. 1. Flocculation by particle accumulation 12. 2. Particle charge neutralization mechanism 12.3. Electrochemical particle coagulation mechanism 2. 4. Electrical double layer(EDL) distortion and thinning mechanism 13. Multi-component deposition Drying of deposits produced by EpD 15. Design of electrophoretic apparatus 16. Deposition on non-conducting substrates 17. Application of EPD 17. 1. Assembly of nanoscale particles into nanostructures and micropatterned thin films 17. 2. Near shape manufacturing of complex-shaped glasses and ceramics 17.3. Solid oxide fuel cell (SOFC) fabrication 174. Laminated materials 17.5. Functionally graded materials 48 17.6. Hybrid materials 17.7. Fibre reinforced ceramic matrix composites 18. Modeling of EPD process 19. Concluding remarks
3.1. Parameters related to the suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1. Particle size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2. Dielectric constant of liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.3. Conductivity of suspension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.4. Viscosity of suspension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.5. Zeta potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.6. Stability of suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2. Parameters related to the process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.1. Effect of deposition time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.2. Applied voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.3. Concentration of solid in suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.4. Conductivity of substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4. Kinetics of electrophoretic deposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5. Role of polymer binders in EPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6. Importance of powder washing before EPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7. Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8. Water-based EPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 9. Non-aqueous EPD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 10. Charge development on powder surface in suspension . . . . . . . . . . . . . . . . . . . . . . . . . . 19 10.1. Aqueous suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 10.2. Non-aqueous suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 11. Properties of suspension for EPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 11.1. The electrical double layer and electrophoretic mobility. . . . . . . . . . . . . . . . . . . 21 11.2. DLVO theory and suspension stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 12. Mechanism of EPD process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 12.1. Flocculation by particle accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 12.2. Particle charge neutralization mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 12.3. Electrochemical particle coagulation mechanism. . . . . . . . . . . . . . . . . . . . . . . . 28 12.4. Electrical double layer (EDL) distortion and thinning mechanism . . . . . . . . . . . 29 13. Multi-component deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 14. Drying of deposits produced by EPD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 15. Design of electrophoretic apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 16. Deposition on non-conducting substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 17. Application of EPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 17.1. Assembly of nanoscale particles into nanostructures and micropatterned thin films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 17.2. Near shape manufacturing of complex-shaped glasses and ceramics . . . . . . . . . . 41 17.3. Solid oxide fuel cell (SOFC) fabrication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 17.4. Laminated materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 17.5. Functionally graded materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 17.6. Hybrid materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 17.7. Fibre reinforced ceramic matrix composites . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 18. Modeling of EPD process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 19. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 L. Besra, M. Liu / Progress in Materials Science 52 (2007) 1–61
L Besra, M. Liu/ Progress in Materials Science 52(2007)1-61 1. Introduction The electrophoretic deposition(EPD)technique with a wide range of novel applications in the processing of advanced ceramic materials and coatings, has recently gained increas- ing interest both in academia and industrial sector not only because of the high versatility of its use with different materials and their combinations but also because of its cost -effec. tiveness requiring simple apparatus. Electrophoretic deposition(EPD) has been known since 1808 when the russian scientist ruess observed an electric field induced movement of clay particles in water. But the first practical use of the techniques occurred in 1933 when the deposition of thoria particles on a platinum cathode as an emitter for electron tube application was patented in USA. Although the basic phenomena involved in EPD are well known and have been the subject of extensive theoretical and experimental research, the EPD of ceramics was first studied by Hamaker [1] and only in the 1980s did the process receive attention in the field of advanced ceramics. There is general agree- ment in the scientific community that further r&d work needs to be done to develop a full, quantitative understanding of the fundamental mechanisms of EPD to optimise the working parameters for a broader use of EPD in materials processing. This paper presents a review of electrophoretic deposition and its application in various fields of processing 2. Electrophoretic deposition-definition Electrophoretic deposition(EPD) is one of the colloidal processes in ceramic produc tion and has advantages of short formation time, needs simple apparatus, little restriction of the shape of substrate, no requirement for binder burnout as the green coating contains few or no organics Compared to other advanced shaping techniques, the epd process is very versatile since it can be modified easily for a specific application. For example, depo- sition can be made on flat, cylindrical or any other shaped substrate with only minor change in electrode design and positioning. In particular, despite being a wet process, EPD offers easy control of the thickness and morphology of a deposited film through simple adjustment of the deposition time and applied potential. In EPD, charged powder particles, dispersed or suspended in a liquid medium are attracted and deposited onto a conductive substrate of opposite charge on application of a DC electric field. The term electrodeposition is often used somewhat ambiguously to refer to either electroplating or electrophoretic deposition, although it more usually refers to the former. Table 1 presents the distinction between the two processes [2] The basic difference between an electrophoretic deposition process(EPD)and an electro- lytic deposition process(ELD) is that the former is based on the suspension of particles in a solvent whereas the later is based on solution of salts, i.e., ionic species [3]. There can be two types of electrophoretic deposition depending on which electrode the deposition occurs When the particles are positively charged, the deposition happens on the cathode and the pro- cess is called cathodic electrophoretic deposition. The deposition of negatively charged par- ticles on positive electrode(anode)is termed as anodicelectrophoretic deposition. By suitable modification of the surface charge on the particles, any of the two mode of deposition is pos- ble. Fig. I presents a schematic illustration of the two electrophoretic deposition process. With regard to technological application the potential of electrophoretic deposition (EPD) as a materials processing technique is being increasingly recognised by scientists and technologists. In addition to its conventional applications in fabrication of wear
1. Introduction The electrophoretic deposition (EPD) technique with a wide range of novel applications in the processing of advanced ceramic materials and coatings, has recently gained increasing interest both in academia and industrial sector not only because of the high versatility of its use with different materials and their combinations but also because of its cost-effectiveness requiring simple apparatus. Electrophoretic deposition (EPD) has been known since 1808 when the Russian scientist Ruess observed an electric field induced movement of clay particles in water. But the first practical use of the techniques occurred in 1933 when the deposition of thoria particles on a platinum cathode as an emitter for electron tube application was patented in USA. Although the basic phenomena involved in EPD are well known and have been the subject of extensive theoretical and experimental research, the EPD of ceramics was first studied by Hamaker [1], and only in the 1980s did the process receive attention in the field of advanced ceramics. There is general agreement in the scientific community that further R&D work needs to be done to develop a full, quantitative understanding of the fundamental mechanisms of EPD to optimise the working parameters for a broader use of EPD in materials processing. This paper presents a review of electrophoretic deposition and its application in various fields of processing. 2. Electrophoretic deposition – definition Electrophoretic deposition (EPD) is one of the colloidal processes in ceramic production and has advantages of short formation time, needs simple apparatus, little restriction of the shape of substrate, no requirement for binder burnout as the green coating contains few or no organics. Compared to other advanced shaping techniques, the EPD process is very versatile since it can be modified easily for a specific application. For example, deposition can be made on flat, cylindrical or any other shaped substrate with only minor change in electrode design and positioning. In particular, despite being a wet process, EPD offers easy control of the thickness and morphology of a deposited film through simple adjustment of the deposition time and applied potential. In EPD, charged powder particles, dispersed or suspended in a liquid medium are attracted and deposited onto a conductive substrate of opposite charge on application of a DC electric field. The term ‘electrodeposition’ is often used somewhat ambiguously to refer to either electroplating or electrophoretic deposition, although it more usually refers to the former. Table 1 presents the distinction between the two processes [2]. The basic difference between an electrophoretic deposition process (EPD) and an electrolytic deposition process (ELD) is that the former is based on the suspension of particles in a solvent whereas the later is based on solution of salts, i.e., ionic species [3]. There can be two types of electrophoretic deposition depending on which electrode the deposition occurs. When the particles are positively charged, the deposition happens on the cathode and the process is called cathodic electrophoretic deposition. The deposition of negatively charged particles on positive electrode (anode) is termed as anodic electrophoretic deposition. By suitable modification of the surface charge on the particles, any of the two mode of deposition is possible. Fig. 1 presents a schematic illustration of the two electrophoretic deposition process. With regard to technological application the potential of electrophoretic deposition (EPD) as a materials processing technique is being increasingly recognised by scientists and technologists. In addition to its conventional applications in fabrication of wear L. Besra, M. Liu / Progress in Materials Science 52 (2007) 1–61 3
L Besra, M. Liu Progress in Materials Science 52(2007)1-61 Table I Characteristics of electrodeposition techniques [2] Electroplating trophoretic depositis oving specie Solid particles Charge transfer on deposition Ion reduction Required conductance of liquid medium Preferred liquid b + Fig. 1. Schematic illustration of electrophoretic deposition process.(a) Cathodic EPD and (b) anodic EPD resistant and anti-oxidant ceramic coatings, fabrication of functional films for advanced microelectronic devices and solid oxide fuel cells as well as in the development of novel composites or bioactive coatings for medical implants, there has been increased interest for its application in nanoscale assembly for advanced functional materials [4]. Electropho- retic deposition also offers important advantages in the deposition of complex compounds nd ceramic laminates. The degree of stoichiometry in the electrophoretic deposit is con- trolled by the degree of stoichiometry in the powder used. According to Sarkar and Nich olson [5], particle/electrode reactions are not involved in EPD, and ceramic particles do not lose their charge on being deposited which can be shown from the observation that reversal of the electric field will strip of the deposited layer [6]. Therefore, it is important to use sim- ilarly charged particles and similar solvent-binder-dispersant systems for gaining better control of layer thickness. The principal driving force for electrophoretic deposition (EPD) is the charge on the particle and the electrophoretic mobility of the particles in the solvent under the influence of an applied electric field. The EPd technique has been used successfully for thick film of silica [7]. nanosize zeolite membrane [8] hydroxyapatite coating on metal substrate for biomedical applications [9, 10] luminescent materials [11- 13), high-Tc superconducting films [14, 15), gas diffusion electrodes and sensors [16, 17]. multi-layer composites [18] glass and ceramic matrix composites by infiltration of ceramic particles onto fibre fabrics [19], oxide nanorods[20], carbon nanotube film [21], functionally graded ceramics [22, 23], layered ceramics [24], superconductors [25, 26], piezoelectric mate- rials [27], etc. Indeed, the only intrinsic disadvantages of EPD, compared with other colloi dal processes(e.g. dip and slurry coating), is that it cannot use water as the liquid med because the application of a voltage to water causes the evolution of hydrogen and oxygen gases at the electrodes which could adversely affect the quality of the deposits formed. How- ever, given the numerous non-aqueous solvents that are available, this limitation is minor
resistant and anti-oxidant ceramic coatings, fabrication of functional films for advanced microelectronic devices and solid oxide fuel cells as well as in the development of novel composites or bioactive coatings for medical implants, there has been increased interest for its application in nanoscale assembly for advanced functional materials [4]. Electrophoretic deposition also offers important advantages in the deposition of complex compounds and ceramic laminates. The degree of stoichiometry in the electrophoretic deposit is controlled by the degree of stoichiometry in the powder used. According to Sarkar and Nicholson [5], particle/electrode reactions are not involved in EPD, and ceramic particles do not lose their charge on being deposited which can be shown from the observation that reversal of the electric field will strip of the deposited layer [6]. Therefore, it is important to use similarly charged particles and similar solvent–binder–dispersant systems for gaining better control of layer thickness. The principal driving force for electrophoretic deposition (EPD) is the charge on the particle and the electrophoretic mobility of the particles in the solvent under the influence of an applied electric field. The EPD technique has been used successfully for thick film of silica [7], nanosize zeolite membrane [8], hydroxyapatite coating on metal substrate for biomedical applications [9,10], luminescent materials [11– 13], high-Tc superconducting films [14,15], gas diffusion electrodes and sensors [16,17], multi-layer composites [18], glass and ceramic matrix composites by infiltration of ceramic particles onto fibre fabrics [19], oxide nanorods [20], carbon nanotube film [21], functionally graded ceramics [22,23], layered ceramics [24], superconductors [25,26], piezoelectric materials [27], etc. Indeed, the only intrinsic disadvantages of EPD, compared with other colloidal processes (e.g. dip and slurry coating), is that it cannot use water as the liquid medium, because the application of a voltage to water causes the evolution of hydrogen and oxygen gases at the electrodes which could adversely affect the quality of the deposits formed. However, given the numerous non-aqueous solvents that are available, this limitation is minor. Table 1 Characteristics of electrodeposition techniques [2] Property Electroplating Electrophoretic deposition Moving species Ions Solid particles Charge transfer on deposition Ion reduction None Required conductance of liquid medium High Low Preferred liquid Water Organic Fig. 1. Schematic illustration of electrophoretic deposition process. (a) Cathodic EPD and (b) anodic EPD. 4 L. Besra, M. Liu / Progress in Materials Science 52 (2007) 1–61
L Besra, M. Liu/ Progress in Materials Science 52(2007)1-61 3. Factors influencing EPD The mechanism of EPD involve charged particles in a suspension being deposited onto n electrode under the influence of an applied electric field. Two groups of parameters determine the characteristics of this process; (i)those related to the suspension, and (ii) those related to the process including the physical parameters such as the electrical nature of the electrodes, the electrical conditions( voltage/intensity relationship, deposition time etc. ) For the EPd of particles, part of the current should be carried not only by the particles but by free ions co-existing in the suspension. Therefore the amount of deposited particle is not simply related to the current. However, the current carried by the free ions could be ignored when the amount of free ions is negligible. Indeed the amount of free ions is generally small in organic suspensions such as ketones. On the other hand, it is believed that the accumulation of anionic and cationic charge at the electrodes during electropho resis suppresses the subsequent deposition rate. However the effect of accumulated ions are negligible in the initial period The first attempt to correlate the amount of particles deposited during EPD with differ nt influencing parameters was described by Hamaker [l] and Avgustnik et al. [28] Hamakers law relates the deposit yield (w)to the electric field strength(E), the electropho retic m y (u), the surface area of the electrode(A), and the particle mass concentration in the sion(C) through the following equation w=/μ·E·A·C.dr Avgustinik's law is based upon cylindrical, coaxial, electrodes and the electrophoretic mobility has been expanded and is represented in terms of permittivity (a), the zeta poten tial ($), and the viscosity of the suspension(n) l·E·E·5 where /and a are the length and radius of the deposition electrode, respectively, b is the radius of the coaxial counter electrode(b>a) e Biesheuval and Verweij [29] improved upon these classical equations and developed ore complex model of the deposition process by considering the presence of three distinct phases namely (i) a solid phase(the deposit), (ii)a suspension phase, and (iii)a phase con- taining little or no solid particles. The deposit phase and the particle-free liquid phase both grow at the expense of the suspension phase. By considering the movement of the bound ary between the deposit and the suspension phase with time along with the continuity equation and expression for velocity of particles in the suspension, Biesheuval and Verweij [29]derived the following equation based on that of Avgustinik et al. [28]: 2·兀·H·l·E·Cdφ where s and d are the volumetric concentration of particles in suspension and deposit respectively, Cd is the mass concentration of particles in the deposit, u is the electropho- retic mobility (=sc/6n)
3. Factors influencing EPD The mechanism of EPD involve charged particles in a suspension being deposited onto an electrode under the influence of an applied electric field. Two groups of parameters determine the characteristics of this process; (i) those related to the suspension, and (ii) those related to the process including the physical parameters such as the electrical nature of the electrodes, the electrical conditions (voltage/intensity relationship, deposition time, etc.). For the EPD of particles, part of the current should be carried not only by the charged particles but by free ions co-existing in the suspension. Therefore the amount of deposited particle is not simply related to the current. However, the current carried by the free ions could be ignored when the amount of free ions is negligible. Indeed the amount of free ions is generally small in organic suspensions such as ketones. On the other hand, it is believed that the accumulation of anionic and cationic charge at the electrodes during electrophoresis suppresses the subsequent deposition rate. However the effect of accumulated ions are negligible in the initial period. The first attempt to correlate the amount of particles deposited during EPD with different influencing parameters was described by Hamaker [1] and Avgustnik et al. [28] Hamakers law relates the deposit yield (w) to the electric field strength (E), the electrophoretic mobility (l), the surface area of the electrode (A), and the particle mass concentration in the suspension (C) through the following equation: w ¼ Z t2 t1 l E A C dt ð1Þ Avgustinik’s law is based upon cylindrical, coaxial, electrodes and the electrophoretic mobility has been expanded and is represented in terms of permittivity (e), the zeta potential (n), and the viscosity of the suspension (g) w ¼ l E e n C t 3 lnða=bÞ g ð2Þ where l and a are the length and radius of the deposition electrode, respectively, b is the radius of the coaxial counter electrode (b > a). Biesheuval and Verweij [29] improved upon these classical equations and developed more complex model of the deposition process by considering the presence of three distinct phases namely (i) a solid phase (the deposit), (ii) a suspension phase, and (iii) a phase containing little or no solid particles. The deposit phase and the particle-free liquid phase both grow at the expense of the suspension phase. By considering the movement of the boundary between the deposit and the suspension phase with time along with the continuity equation and expression for velocity of particles in the suspension, Biesheuval and Verweij [29] derived the following equation based on that of Avgustinik et al. [28]: w ¼ 2 p l l E Cd lnða=bÞ /s /d /s t ð3Þ where /s and /d are the volumetric concentration of particles in suspension and deposit, respectively, Cd is the mass concentration of particles in the deposit, l is the electrophoretic mobility (=en/6pg). L. Besra, M. Liu / Progress in Materials Science 52 (2007) 1–61 5