Electric field in a 2D wedge

Capillary flow in V-grooves

6.3 Electric field distribution in conducting groove with thin fluid

6.3.2 Electric field in a 2D wedge

Note that the analysis in this section will be carried out in dimensional variables, with a fixed fluid midline height ofh =d. Consider a wedge with opening half angle α and an outer radial limitb, as in Figure 6.4. Assume the walls and fluid interface are perfect conductors (and hence have ψ = 0), and the outer boundary has some potential distribution ψ|_r=b = ψ^outer(β). In polar coordinates(r, β), we must then solve

∇²ψ=∂rrψ+ 1

r∂rψ+ 1

r²∂ββψ= 0 (6.21)

d b

β=0

β=α

β=-α β

boundary of tractable annular wedge domain

fluid interface conduc

ting groove w

all

ψ|

β=-α=0 ψ| ^β==^α0

ψ|_r=b=ψ^outer(β)

r

Vacuum domain

Figure 6.4: Cross-sectional schematic of the system depicted in Figure 6.1 (b), with quantities relevant to the electric field solution in a 2D wedge emphasized. The system is described in polar coordinates{r, β}, with β= 0 at the center of the groove and β =±αbeing the groove walls. The groove walls are grounded, so thatψ|_β=±α = 0, and the electric potential at the top of the groove, a distance bfrom the corner, isψ|_r=b =ψ^outer(β). The boundary of the annular wedge domain, tracing outr =d, is shown, as is the fluid interface, shown with interface midline thicknessd. Note that the variables shown in this diagram are dimensional.

with boundary conditions

ψβ=±α = 0 (6.22a)

ψ_r=b =ψ^outer(β) (6.22b)

ψ_r=σ(β)= 0, (6.22c)

wherer =σ(β) =d×Σ(β) is the fluid interface.¹

Solving Equation (6.21) with only the wall condition Equation (6.22a) by separation of variables yields

ψ=

∞

X

j=−∞

c_jr(j+1/2)(π/α)cos

j+1 2

π αβ

, σ(β)≤r≤b. (6.23)

Solving for c_j such that the field vanishes at the fluid surface is analytically intractable, even when the surface has a simple shape such as a semicircle, and thus a numerical approach will be necessary. However, we can analytically determine the scaling of the electric field gradient in the limit of a distant electrode, so long as the fluid interface is sufficiently well-behaved. To do

1Note that in reality the interface depends on z andtas well, having the formσ(β, z, t) =d×Σ(β, Z, T).

However, since we are solving in 2D “slices” due to the assumptions that the electric field is time-independent and slow-varying inZ, we simplify notation by omittingz andtfromσin this section.

so, we will first solve for the field in the tractable case of a homogeneous Dirichlet condition on a circular boundary with negative-curvature (instead of the realistic fluid interface, which has positive curvature and is not circular), forming an annular wedge domain. The following section will show that when δ = (d/b) ≪ 1, the Maxwell pressure scales as (d/b)^(π/α)−2, where b is the distance to the outside electric field (defined as the groove wall length or the distance to the nearest charge or counter-electrode, whichever is shorter) and d is the inner radius of the annulus. In the subsequent section, the annular wedge analysis will be extended to the case of a more general fluid interface, where it will be shown that the dominant contribution to the Maxwell pressure still scales as (d/b)^(π/α)−2. This scaling result will be verified numerically in Section 6.7.

Electric field in a 2D annular wedge

To find the annular wedge solution, we solve forc^a.w._j using the boundary conditionsψa.w.|_r=b= ψ^outer(β) andψ_a.w.|_r=d= 0. Defining

ψ_k^outer = 1 α

Z α

−α

cos

k+1 2

π αβ

ψ^outer(β)dβ, (6.24)

the outer and inner boundary constraints become

ψ_k^outer =c^a.w._k b(k+1/2)(π/α)+c^a.w._−k−1b−(k+1/2)(π/α), (6.25a) 0 =c^a.w._k d(k+1/2)(π/α)+c^a.w._−k−1d−(k+1/2)(π/α), (6.25b) yielding

c^a.w._k = b−(k+1/2)(π/α)

1−δ2(k+1/2)(π/α)ψ_k^outer (6.26)

and ψa.w.=

∞

X

j=0

"

1

1−δ2(j+1/2)(π/α)

r b

(j+1/2)(π/α)

+ 1

1−δ−2(j+1/2)(π/α)

r b

−(j+1/2)(π/α)#

×ψ_j^outercos

j+ 1 2

π αβ

=

∞

X

j=0

δ(j+1/2)(π/α)

1−δ2(j+1/2)(π/α)

"

r d

(j+1/2)(π/α)

− r

d

−(j+1/2)(π/α)#

ψ_j^outercos

j+1 2

π αβ

(6.27) We are interested in the behavior of the field near the fluid surface, i.e., at r = d. By the assumption that d < b, the lowest powers of δ = (d/b) will dominate the result. The ratio in magnitude of the kth term in the series to the 0th term then has order δ^k(π/α)|ψ_k^outer/ψ₀^outer|, which is ≤δ^2k|ψ_k^outer/ψ^outer₀ |, where we have used the fact thatα < π/2 (α must be less than π/2 for the substrate to form a groove instead of a plane or an external corner). Therefore, so long as |ψ^outer_k /ψ₀^outer| < O(δ^−2(k−1)) (the smoothness condition which we assumed a priori),

then the 0th order term will dominate the series and all higher order terms will be at most O(δ²). The electric field near the inner annulus surface is then well-approximated by

ψa.w.=

"r b

(1/2)(π/α)

−δ^(π/α) r

b

−(1/2)(π/α)#

ψ₀^outercos 1

2 π

αβ h

1 +O(δ²)ⁱ. (6.28) The Maxwell pressure of the annular wedge solution atr =d,β = 0is then given by

pa.w.|_r=d,β=0=− 1 2ϵ0

(∂rψa.w.)²+ 1

r² (∂βψa.w.)²

_r=d

=−ϵ0(ψ^outer₀ )² b²

π² 8α²

"

r b

(π/α)−2

+ r

b

−(π/α)−2

δ^2(π/α)+ r

b −2

δ^(π/α)cos πβ

α #

r=d,β=0

=−ϵ0(ψ^outer₀ )² b²

π²

4α²δ^(π/α)−2

1 + cos πβ

α

β=0

h

1 +O(δ²)ⁱ

=−ϵ₀(ψ^outer₀ )² b²

π² 2α²

d b

(π/α)−2

h1 +O(δ²)ⁱ. (6.29)

Note in particular thatp_a.w.|_r=d scales as δ^(π/α)−2= (d/b)^(π/α)−2. Note also that the pressure of the annular wedge solution differs from the pressure of the pure wedge solution evaluated at r=d(the latter has an extra multiplicative factor of 1/4).

Electric field in a 2D wedge with a nontrivial fluid interface

Having solved the annular wedge solution, we now consider the general solution to the electric potential ψ satisfying Equation (6.21) and boundary equations Equations (6.22a) to (6.22c).

But because it is difficult to deal with a boundary condition onr =σ(β), we instead construct an equivalent boundary conditionψ_r=d=ψ^inner(β). That is,ψ^inner(β)is the boundary condition on the annular wedge domain which gives a solution with ψ|_r=σ(β)= 0.

In the same manner asψ^outer_k , we define ψ^inner_k = 1

α Z α

−α

cos

k+1 2

π αβ

ψ^inner(β)dβ, (6.30)

and the outer and inner boundary constraints become

ψ^outer_k =c_kb(k+1/2)(π/α)+c_−k−1b−(k+1/2)(π/α), (6.31a) ψ_k^inner=ckd(k+1/2)(π/α)+c−k−1d−(k+1/2)(π/α)

, (6.31b)

yielding

c_k = b−(k+1/2)(π/α)

1−δ2(k+1/2)(π/α)

hψ_k^outer−δ(k+1/2)(π/α)ψ^inner_k ⁱ. (6.32) Comparing the magnitudes of a term withk̸= 0 to the 0term:

c_kd(k+1/2)(π/α)

c₀d(0+1/2)(π/α)

= δ(k+1/2)(π/α)[1−δ2(0+1/2)(π/α)] δ(0+1/2)(π/α)[1−δ2(k+1/2)(π/α)]

ψ_k^outer−δ(k+1/2)(π/α)ψ_k^inner ψ^outer₀ −δ(0+1/2)(π/α)ψ₀^inner

=O(δ²)×

1−δ(k+1/2)(π/α)ψ_k^inner/ψ^outer_k 1−δ(0+1/2)(π/α)ψ₀^inner/ψ₀^outer

. (6.33)

The annular boundaryr=dand the true interface boundary r=σ(β) =dΣ(β) are a distance O(d)apart. IfΣ(β) is sufficiently smooth, then it may be the case that lim_δ→0ψ^inner_k /ψ^outer_k <

O(δ^−k(π/α)), and hence that thekth series term isO(δ²) smaller than the0th series term. This would be nontrivial to prove, and indeed would not be true ifΣ(β)were to have a sharp corner or cusp. The contact line, where the interface meets the wall, is certainly not smooth ifθ >0;

however, being an internal rather than external corner, it is not expected to lead to a singularity inducing an arbitrarily large electric field (Jackson, 2012). We will take it as an assumption here that the interface is smooth and that the contact line does not significantly affect the electric field, and later confirm the results numerically. Therefore, the field near the fluid surface can be approximated using only thek= 0 andk=−1terms:

ψ= ("

r b

(1/2)(π/α)

−δ^(π/α) r

b

−(1/2)(π/α)#

ψ^outer₀ +δ^(1/2)(π/α) r

b

−(1/2)(π/α)

ψ₀^inner )

×cos 1

2 π

αβ h

1 +O(δ²)ⁱ

=δ^π/(2α) ("

r d

(1/2)(π/α)

− r

d

−(1/2)(π/α)#

ψ₀^outer+δ−(1/2)(π/α)r d

−(1/2)(π/α)

ψ₀^inner )

×cos 1

2 π

αβ h

1 +O(δ²)ⁱ, σ(β)≤r≪b. (6.34)

Evaluatingψ at the fluid interfacer =dΣ(β),

0 =ψ|_r=dΣ(β)=ⁿO(δ^(1/2)(π/α))ψ^outer₀ +O(δ⁰)ψ^inner₀ ^ocos 1

2 π

αβ h

1 +O(δ²)ⁱ. (6.35) In order forψto vanish at the fluid surface, it must then be the case thatψ^inner₀ =O(δ^(1/2)(π/α)), yielding the same scaling as the annular wedge solution. In other words, despite all the ugly calculations, we conclude that we can use the Maxwell pressure from an annular wedge as a good approximation to the Maxwell pressure for a different fluid interface shape in a wedge, so long as the fluid is sufficiently shallow compared to the wedge depth and the assumptions of sufficiently smooth fluid interface and outside electric field distribution hold. In the remainder of this work, we will approximate Maxwell pressure as

p_Maxwell|_r=h,β=0 =Cpb _a.w.|_r=h,β=0 =−ϵ₀(ψ₀^outer)² b²

π² 2α²

h b

(π/α)−2

Cb^h1 +O(δ²)ⁱ, (6.36) whereC^bis anO(1)correction to the annular wedge pressure. Note that by adopting the annular wedge solution atr=h, we are explicitly taking into account the way the electric field changes as the fluid thickness changes.