Capillary flow in V-grooves
6.4 Equation of motion for perfectly conducting thin film in a V-groove
this is a good approximation (see Section 6.7 for quantification of the error induced by this approximation).
It is convenient now to define a rescaled pressure
P = Ca(α, θ)R(α, θ)Pb =Pcapillary+PMaxwell=−1
H −χ(Z)Hm, (6.42) where, similarly, Pcapillary= CaRPb capillary andPMaxwell = CaRPb Maxwell.
In terms of the rescaled pressure P defined by Equations (6.41) and (6.42), the governing equation of motion becomes
∂H2
∂T =− ∂
∂Z
"
H4 −∂P
∂Z
!#
=− ∂
∂Z
H4 ∂
∂Z 1
H +χ(Z)Hm
, (6.43)
wherem= (π/α)−2∈(0,∞) andχ(Z) = (WeR/2)(ψb oouter(z)/ψc)2C, and we have used theb fact thatCa = Φ(α, θ) = Γ/(RbA).b
Equation (6.43) is the reduced-order evolution equation of fluid interface midline thickness we have sought. Keep in mind that it applies only to a perfectly conducting fluid in a groove with perfectly conducting walls, and requires that the fluid thicknessdbe less than the characteristic length L and the groove height b, specifically, (d/L)2 ≪ 1 and (d/b)2 ≪ 1. It was further assumed that the internal groove angle, α, and contact angle, θ, were constant, that inertial and gravitational terms could be ignored, and that the applied electric field is constant in time and slow-varying in z. Furthermore, it was assumed that the electric field is sufficiently weak that the thickness sensitivity of the Maxwell pressure (∂PMaxwell/∂H) is less than that of the capillary pressure (∂Pcapillary/∂H); this requirement will be expressed as χ < χthresh. and explained shortly. For more details on the assumptions required for the above equation to be in the regime of validity, see Section 6.2.1.
For large H, PMaxwell may dominate the capillary pressure, making the overall pressure P = PMaxwell +Pcapillary satisfy ∂P/∂H < 0. In this case, the equation of motion ∂T(H2) =
∂Z H4∂ZP will become anti-diffusive and ill-posed, and breakup or instability is expected.
But for small H, PMaxwell → 0 in a groove, and so the capillary terms will dominate and the equation of motion should be diffusive. Therefore, we expect that there will be a threshold thickness (which is dependent on the strength of the applied field), above which the fluid in the groove is unstable. Rather than define a local threshold thickness, it is convenient to define a worst-case threshold thickness based on the location with the greatest electric field χmax = maxZχ(Z). Then, the threshold thickness is computed by
0 = ∂
∂H 1
H +χmaxHm
H=Hthresh.
=−Hthresh.−2 +mχmaxHthresh.m−1
=⇒Hthresh. = (mχmax)−1/(m+1). (6.44)
The threshold thickness also defines a threshold pressure Pthresh. =− 1
Hthresh. −χmaxHthresh.m
=−(mχmax)1/(m+1)−χmax(mχmax)−m/(m+1)
=−m1/(m+1)+m−m/(m+1)χ1/(m+1)max
=−(1 +m)m−m/(m+1)χ1/(m+1)max . (6.45) Alternatively, given a maximum fluid thickness Hmax = maxZH(Z) in the groove, one can define a threshold electric field strength above which the system is ill-posed:
Hmax = (mχthresh.)−1/(m+1)
=⇒ χthresh. =Hmax−(m+1)m−1. (6.46)
Recalling that the equation of motion is well-posed only when χ < χthresh. = Hmax−(m+1)m−1. Assuming d is the maximum fluid thickness, Hmax = 1, and χthresh. = m−1. The constraint on the characteristic electric potential at the groove top to ensure the fluid remains in the well-posed regime is then
ψ0outer< α π
s2b2 ϵ0
γ dRb
1 mCb
b d
m
. (6.47)
In the special case of a constant electric field, χ(Z) =χ0 [note that χ0 is so named simply to indicate that it is a constant; not to refer toχ(Z = 0)], the equation of motion Equation (6.43) can be rewritten as
∂H2
∂T = ∂2
∂Z2 1
3H3−χ0 m
m+ 4Hm+4
. (6.48)
The Cb factor
The exact value of (∇Ψ|Y=H)2 depends on the geometric factorsα and θ, as well as the form of the electric field at the top of the groove; these details lead to an O(1) deviation, C, fromb the annular wedge solution.
The case of a groove covered by a flat electrode of constant potential ψclocated aty=bwas tested numerically (see Section 6.7), with internal groove half angles of α ∈ {15◦, 30◦, 45◦, 60◦,75◦}and contact angles θ∈ {10◦,15◦,30◦,45◦,60◦}. Cb increased with θand decreased with α, ranging from a minimum of Cb = 0.45 for (α = 75◦, θ = 10◦) to a maximum of 1.15 for (α= 15◦, θ= 60◦).
6.4.1 Estimate of MEP threshold values
We can make some order-of-magnitude estimates of the critical fluid interface midline thickness in grooves on the MEP emitter. Suppose the counter-electrode sits a heightℓ above the array
substrate floor and has voltage gapV0, and the grooves on the substrate floor have depthb. The electric field at the floor is then approximately V0/ℓ. Since the electric potential in the groove goes as ψ ≈ ψouter0 (r/b)π/(2α), ψouter0 can be approximated by matching the field strength at the groove top with the field at the substrate floor.
V0
ℓ ≈ π 2α
ψouter0 b
=⇒ ψ0outer ≈ 2α π
V0
ℓ b. (6.49)
The Maxwell pressure at the fluid surface is then pMaxwell≈ −ϵ0(ψ0outer)2
b2
π2 2α2
h b
(π/α)−2
Cb=−2ϵ0 V0
ℓ 2h
b
(π/α)−2
C,b (6.50) where we used the annular wedge solution, Equation (6.29), and the overall pressure at the surface is
p≈ − γ
R(α, θ)hb −2ϵ0 V0
ℓ 2
b−mhm, (6.51)
whereR(α, θ) = sinb α/(cosθ−sinα) is the surface curvature factor, m = (π/α)−2, and we have takenCb to be 1 (sinceCbis O(1) and we are constructing an order of magnitude estimate).
The threshold thickness is then given by
0 = γ
R(α, θ)hb 2thresh. −2ϵ0 V0
ℓ 2
b−mmhm−1thresh.
=⇒ hthresh.=
"
bm γ mR(α, θ)b
1 2ϵ0−1
ℓ V0
2#1/(m+1)
. (6.52)
Solving forV0,
Vthresh. = s ℓ2
2ϵ0d b
d
m γ
mR(α, θ)b , (6.53)
where we have substituted in d, the maximum fluid midline thickness, for hthresh. to then yield the threshold voltageVthresh. above which the equation of motion is ill-posed and instability is expected.
Let us substitute in ϵ0 ≈8.85×10−12 F/m, the permittivity of free space, andγ ≈0.57 N/m, the surface tension of liquid indium (Chentsov et al., 2011; Tiesinga et al., 2019).
Let the counter-electrode be250microns from the substrate surface. Supposing typical grooves are b= 10microns deep, the maximum film thickness in the thin film regime is perhaps d= 3 microns. Using these values, a set ofVthresh.is shown in Table 6.2 for a variety of internal groove half angles α and contact anglesθ. Note that 1/(mR)b is monotonically decreasing in both α andθ, and so largerαor θwill always lead to smaller threshold voltage. Results are reported to
θ= 0◦ θ= 15◦ θ= 50◦ α= 15◦ 5700kV 5570kV 4110kV α= 30◦ 144kV 139kV 77kV α= 45◦ 39 kV 37 kV
α= 60◦ 19 kV 16 kV α= 75◦ 10 kV
Table 6.2: Approximate values of threshold counter-electrode voltageVthresh. [Equation (6.53)]
for MEP values: permittivity of free spaceϵ0 = 8.85×10−12F/m (Tiesinga et al., 2019), liquid indium surface tensionγ ≈0.57N/m (Chentsov et al., 2011; Tiesinga et al., 2019), grooves of depthb= 10microns and film thickness d= 3microns, and a flat counter-electrode a distance 250microns above the top of the groove. These values are order of magnitude estimates, having omitted the O(1)multiplicative correction factorC. Results are reported to the nearest kV notb to imply accuracy, but merely in order to make the results distinguishable from one another.
Results are shown for α∈ {15◦, 30◦,45◦,60◦,75◦} andθ∈ {0◦,15◦,50◦} (note that results can be reported only within the valid regime in which the Concus-Finn condition,α+θ < π/2, is satisfied).
the nearest kV not because that is the known accuracy of the MEP system (it is not; we have omitted theO(1)factor C), but in order to distinguish the results for different geometries fromb each other.
Grooves of different internal angles show stark differences, ranging from Vthresh. ≈5700kV for α = 15◦ to Vthresh. ≈10kV for α = 75◦. This is due both to the power m = (π/α)−2 being larger for narrow grooves (for α = 15◦, m = 10; for α = 75◦, m = 0.4) and due to Rb being smaller for narrow grooves (for α= 15◦, θ= 0◦,Rb≈0.35; forα= 75◦, θ= 0◦,Rb ≈28).
While the results of Table 6.2 are only order of magnitude approximationes, they are all larger than the typical MEP running values of 4-5kV (Marrese-Reading, 2016). This rough analysis therefore suggests that the film in the groove will likely exit the regime in which is it very thin compared to groove depth before it exceeds the threshold thickness at which one would need to worry about electrocapillary instability. However, a more detailed numerical study would be a worthwhile future research effort. In particular, the exact values of C,b α, and θ should be applied.
6.5 Analysis of equation of motion