Printing System Models - Control of Color Imaging Systems: Analysis and Design - page 578

the potential is increased, the particle layer ruptures and splits when the sum of the
electrostatic forces and mechanical stress exceeds the cohesive strength. If z s is the
spatial distance at which the layer splits, the fraction F of the material attached to
layer 1 is de ned as the transfer ef ciency, which is

z d 2

F ¼ 1 ( 10 : 35 )

The transfer ef ciency depends on (1) the space charge density of particle layer,

(2) the electric field between electrodes, and (3) the cohesive forces between

particles. The stress caused by space charge and electric field can be modeled

using Euler is equation:

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ð

z
0 r z ðÞ E 2 ( z 0 )d z 0 þ P m e 3 E 3

[N=m 2 ] P ( z ) ¼ ( 10 : 36 )

where
r m 3 ] (z) is the space charge density [C =

E 2 (z) is the electric field in the particle layer [V = m]

The last two terms represent the constant of integration resulting from the electro-
static attraction between the two electrodes. At the boundary between layers 2 and
3(z ¼ m 2 ) 0), the total stress must equal the sum of the mechanical stress P m (in N =

e 3 E 3 due to electrostatic attraction between the two metal
electrodes, where E 3 is the electric

and the compression

field in layer 3. Cohesive forces between particles
are represented by the quantity C(z) (in N

m 2 ) resulting from the particle

= - particle

interaction and particle - electrode adhesion due to electrostatic, dispersion, or chem-

ical forces.
When the mechanical stress at the boundary, P m , is increased, the layer separates
where P(z), which is the sum of the electrostatic forces and mechanical stress,
exceeds the cohesive force, C(z). At the particle layer separation (i.e., at z ¼ z s ), the
value of P m can be found by determining the value of z that corresponds to a
minimum in total stress P(z) - C(z). Therefore,

d
d z [ P ( z ) C ( z )] ¼

0 and P ( z ) C ( z ) ¼ 0 ( 10 : 37 )

The solution that gives the smallest value of P m corresponds to the mechanical stress
for which P(z) rst exceeds C(z).

When the particle layer does not have a spatial variation, then Equations 10.36
and 10.37 reduce to the condition

r( z ) E 2 ( z ) ¼ 0 ( 10 : 38 )

Equation 10.38 has the trivial solution r (z s ) ¼ 0orE 2 (z s ) ¼ 0. The field within the
particle layer, E 2 (z), is derived by using Poisson is equation:

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