Modeling of Electrophoretic Deposition (EPD)

Problem statement

Due to the wide range of specific Electrophoretic deposition (EPD) parameters (voltage, electrode surface, electrode distance, solids loading), as well as suspension specific parameters such as effective powder charge, electrophoretic mobility, specific resistivity of the liquid, specific resistivity of the deposit, it is very useful to simulate the deposition process allowing to calculated the yield of deposition as function of deposition time taking into account the growing deposit on the electrode, the changing suspension concentration and the changing electric field strength.

Especially in the case of functionally graded materials, it is essential to be able to predict the proper experimental EPD parameters, taking into account the powder specific EPD parameters for a specified suspension, needed to establish the pre-defined concentration gradient in the deposit.

Simulation of the EPD process

A simulation of the electrophoretic deposition process has been developed, which allows to predict the yield of deposition as a function of time for different values of the electrical properties of the liquid and the powders used. It is shown that with special precautions, conditions can be defined where the resistance of the deposit will limit its thickness. Under most conditions however, particularly if a liquid with reasonable electrical conductivity is used, the resistance of the deposit should not limit its growth.

The empirical equation of Hamaker gives the relation between the instantaneous yield rate and the electric field.

(1)

with the yield Y (kg), time t (s), the electrophoretic mobility µ (cm²/V.s), the electric field strength E (V/m), the solids loading c (kg/m³) of the powder in suspension and the surface area A (m²) of the electrode; f is a factor that takes into account that not all powder brought to the electrode may be incorporated in the deposit. In this work f is assumed 1. The electric field at the deposit can be calculated from a simplified equivalent electric circuit as shown in Fig. 1. For the suspension the resistances are:

(2)

(3)

Where d is the electrode distance (cm); d_l the deposit thickness (cm); Q_eff the effective powder charge (C/g); r_lsus is the specific resistivity of the liquid in suspension (W.cm). In the suspension, current is carried by individual ions dissolved in the liquid and by the powder with its associated ion cloud.

The deposit is assumed to be made-up of a powder bed with resistance R_pdep with a packing fraction of 0.5 as a typical value and an interparticle liquid phase with total resistance R_ldep. The corresponding resistances are :

(4)

(5)

where r_pdep is the specific resistivity of the (dry) powder in the deposit (W.cm) and with r_ldep = k.r_lsus with k £ 1. Thus the resistance in the interparticle liquid in the deposit is related to that in the suspension, but it is likely to be smaller due to the fact that the particles release their cloud of counter-ions at the electrode.

Fig. 1: The currents and voltages during the EPD process are calculated from the equivalent electric circuit shown.

Using this model, one can derive the yield as function of time for different values of the different resistances involved. Voltage drops due to polarization processes at the electrodes are neglected in our simulations.

It is commonly believed that the electrical resistance of the deposit may reduce the yield and eventually stop the EPD process (see Link). Fig. 2 shows the total yield for a value of the electrical conductivity of the suspension which we use commonly in our work on free standing objects. The conclusion is that thick deposits can be created by EPD of the order of several cm even for powders which in dry condition are excellent insulators. We do see from this graph that the thickness of a deposit can be controlled by the total supply of powder available.

Fig 2 : Yield as a percentage of the total available powder in the suspension. The specific resistivity of the suspension liquid r_lsus is assumed to be constant at 200 kW.m. Different curves show the effect of a changing specific resistivity for the dry powder r_pdep from a good conductor to a good insulator.

It is observed in practice that the resistivity of the suspension increases as deposition proceeds. This can be understood if one considers that the powder concentration is reduced by the deposition process itself and that the ionic concentration in the liquid can also be reduced by electrochemical reactions at one or both electrodes. If this change in resistivity of the liquid is also incorporated in the model, it is possible to find conditions where the thickness of a deposit can be controlled by the electric field going to zero at the suspension-deposit interface. An example is shown in Fig. 3 for a dry powder which is a good insulator and the resistivity of the suspension increasing exponentially with deposition time. It should be useful to take advantage of this effect if one is interested in laying down coatings of an insulating substance with a very uniform thickness. This will require work under clean conditions avoiding contamination of the EPD process (e.g. by the powders themselves) and the use of solvents with low ionic concentration.

Fig. 3a : Variation of the resistivity of the liquid during EPD used in the calculations of the results in Figure 3b. ρ_lsus¹andρ_lsus²are constant at respectively 1500 and 10⁶ Ωm. ρ_lsus^{3
,} ρ_lsus⁴and ρ_lsus⁵start at 1500 Ωm and then increase respectively linearly, parabolically and exponentially.

Fig. 3b : Yield as a percentage of the total available powder in the suspension. The specific resistivity of the dry powder r_pdep is assumed to be constant at 10¹²W.m. Different curves show the effect of a specific resistivity for the liquid in suspension, changing during EPD as shown in Fig. 3a.

EPD model for functionally graded materials

Electrophoretic deposition (EPD) allows the formation of plate-shaped binary functionally graded materials (FGM) by depositing from a powder suspension to which a second suspension is continuously added during the process. The deposition yield is described as a function of time and starting composition of both suspensions, resulting in a model of the EPD process that allows to predict the composition gradient in the green deposit as well as in the sintered material. The model enables to calculate the composition gradient in the FGM material from the starting composition of the suspensions, the EPD operating parameters and the powder-specific EPD characteristics. The powder-specific parameters, i.e. the effective charge, the electrophoretic mobility, and the specific conductivity of the intermicellar liquid, incorporated in the model were experimentally determined from EPD of the individual homogeneous powder suspensions. The model was verified in the ZrO₂-Al₂O₃ and TiCN-WC-Co system by the actual preparation and analysis of different FGM materials. Because of the excellent correlation between the predicted and the measured concentration profiles, illustrated below, the described model allows to precisely engineer and design the composition profile in single gradient as well as symmetrically graded FGM materials produced by EPD.

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