Problems
Application 3.2 The Ionization Potential of a Metal Cluster
4.3 Electric Dipole Layers
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98 ELECTRIC MULTIPOLES: APPROXIMATE ELECTROSTATICS FOR LOCALIZED CHARGE
Figure 4.6: Fluorescence microscopic image (40µm×160µm) looking down onto a liquid mixture of cholesterol and a phospholipid trapped at an air-water interface parallel to the image. The black regions are long, thin “drops” of the liquid, with a thickness of exactly one molecule. Figure from Seul and Chen (1993).
4.2.6 The Dipole-Dipole Interaction
We can apply (4.26) to compute the total electrostatic energyUEof a collection ofNpre-existing point dipoles located at positionsr1,r2, . . . ,rN. As in the corresponding point charge problem (Section 3.6), it costs no energy to bring the first dipolep1into position. The work done byusto bringp2into position is exactly the interaction energy (4.25), with the electric field (4.10) ofp1playing the role ofEext:
W12= −p2·E1(r2)= 1 4π 0
p1·p2
|r2−r1|3 −3p1·(r2−r1)p2·(r2−r1)
|r2−r1|5
+
. (4.30)
Repeating the logic of the point charge example leads to the total energy:
UE =W= 1 4π 0
1 2
N i=1
N j=i
pi·pj
|ri−rj|3 −3pi·(ri−rj)pj·(ri−rj)
|ri−rj|5
+
. (4.31)
A more compact form of (4.31) makes use of the electric fieldE(ri) at the position of theithdipole produced by all theotherdipoles:
UE = −1 2
N i=1
pi·E(ri). (4.32)
4.3 Electric Dipole Layers 99
b b
Figure 4.7: Side view of a dipole layer: two oppositely charged surfaces with areal charge densities±σ separated by a distanceb. In the macroscopic limit,b→0 andσ→ ∞with the dipole moment densityτ=σb held constant.
microscopic distanceb(see Figure 4.7). This abstraction makes no attempt to model the actual charge density of each molecule. Rather, we pass to the macroscopic limit (b→0 andσ → ∞withτ=σb held constant) where the size of the molecules (and the separation between them) is unresolved as in Figure 4.6. This makes the surface electric dipole densityτ a macroscopic property exactly like the surface charge densityσ.
The construction illustrated in Figure 4.7 is called adipole layer.5In our example, the macroscopic surface dipole density of the dipole layer derives from the intrinsic dipole moment of the constituent molecules. More commonly, dipole layers arise (in the absence of intrinsic dipoles) when mobile charges exist near surfaces or boundary layers. This occurs frequently in plasma physics, condensed matter physics, biophysics, and chemical physics. An example is the surface of a metal (see the graph ofρ0(r) in Figure 2.7) where the electrons “spill out” beyond the last layer of positive ions. After Lorentz averaging (Section 2.3.1), the electrostatics of this situation is well described by a surface distribution of electric dipole moments.
4.3.1 The Potential of a Dipole Layer
The most general dipole layer is a charge-neutral macroscopic surfaceS(not necessarily flat) endowed with a dipole moment per unit area that is not necessarily uniform or oriented perpendicular to the surface. This leads us to generalize (4.33) to
τ= dp
dS. (4.34)
The electrostatic potential produced by such an object is computed by superposing point dipole potentials as given by (4.13). Therefore,
ϕ(r)= − 1 4π 0
S
dp(rS)· ∇ 1
|r−rS| = − 1 4π 0
S
dSτ(rS)· ∇ 1
|r−rS|. (4.35) A judicious rewriting of (4.35) reveals the singular charge densityρL(r) associated with a dipole layer.
The key is to convert the surface integral (4.35) to a volume integral using a delta function. For a dipole layer which coincides withz=0, we writerS =xˆx+yˆyandr=rS+zˆzto get
ϕ(r)= − 1 4π 0
d3rδ(z)τ(rS)· ∇ 1
|r−r|. (4.36)
Now change the argument of the gradient operator fromrtor and integrate (4.36) by parts. Two minus signs later, we get the desired result,
ϕ(r)= − 1 4π 0
d3r∇·
τ(rS)δ(z)
|r−r| . (4.37)
Comparing (4.37) with (4.1) shows that a surface dipole layer atz=0 is equivalent to the volume charge density6
ρL(r)= −∇ · {τ(rS)δ(z)}. (4.38)
5 Some authors reserve the term “double layer” for a dipole layer where every dipole points normal to the surface.
6 Comparing (4.38) to (2.51) shows that the polarization of this dipole layer isP(r)=τ(x, y)δ(z).
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Figure 4.8: The charge density (solid curve) and electrostatic potential (dashed curve) associated with a microscopic dipole layer of finite thickness. In the macroscopic limit,ϕis discontinuous by an amountϕand ρLis singular atz=0.
To interpret (4.38), we define a “surface gradient”,
∇S≡ ˆx ∂
∂x+ˆy ∂
∂y, (4.39)
and an effective surface charge density,
σ(rS)= −∇S·τ(rS). (4.40)
In this language, the singular charge density (4.38) associated with an arbitrary dipole layer is ρL(r)= −τz(rS)δ(z)+σ(rS)δ(z). (4.41) The first term in (4.41) is the charge density associated with the component of the layer dipole moment perpendicular to the surface. To see this, compare the capacitor-like charge density in Fig- ure 4.7 with the solid curve in Figure 4.8 labeledρL(z).
The latter is a smooth microscopic charge density which Lorentz averages to the macroscopic delta function derivative in (4.41). The associated electrostatic potential (dashed curve in Figure 4.8) is similarly reminiscent of the potential of a parallel-plate capacitor. In detail, the Poisson equation relates the curvature of the potential directly to the charge density:
0
d2ϕ(z)
dz2 = −ρL(z). (4.42)
The surface charge densityσ(rS) in (4.41) arises exclusively from the components ofτ(rS) parallel to the surface plane. Qualitatively, the negative end of each in-plane point dipole exactly cancels the positive end of the immediately adjacent in-plane dipoleunlessthere is a variation in the magnitude or direction of the dipoles along the surface. The resulting incomplete cancellation is the origin of (4.40).
4.3.2 Matching Conditions at a Dipole Layer
The matching condition,0nˆ2·[E1−E2]=σ(rS), tells us that the second term in (4.41) generates a jump in the normal component of the electric field atz=0. To discover the effect of the first term, we write out the Poisson equation (4.42) using only the first term on the right-hand side of (4.41) as the source charge:
0 d2
dz2ϕ(rS, z)=τz(rS)δ(z). (4.43)
4.3 Electric Dipole Layers 101
Up to an additive constant, integration of (4.43) overzimplies that 0
d
dzϕ(rS, z)=τz(rS)δ(z). (4.44)
The constant disappears when we integrate (4.44) fromz=0−toz=0+. What remains is a matching condition for the potential:
ϕ(rS, z=0+)−ϕ(rS, z=0−)=τz(rS)/0. (4.45) The condition (4.45) shows that the electrostatic potential suffers a jump discontinuity at a dipole-layer surface. The variation inϕ(z) sketched as the dashed curve in Figure 4.8 makes this entirely plausible.
The potential rises from left to right across this microscopic dipole layer because the electric field inside the layer points from the positive charge on the right to the negative charge on the left.
The discontinuity (4.45) implies that the tangential component of the electric field is not continuous if the dipole-layer density varies along the surface. In the language of Section 2.3.3, the reader can show that
ˆ
n2×[E1−E2]= ∇ × {ˆn2( ˆn2·τ)}/0. (4.46) Under certain conditions, (4.46) generates corrections to the Fresnel formulae (Section 17.3.2) which describe the reflection and transmission of electromagnetic waves from interfaces.7