Energy Variation and Stability of Charged Conductive Liquids

Chapter V: Shape Analysis and Energy Stability of Conductive Liquids

5.5 Energy Variation and Stability of Charged Conductive Liquids

through the formula (5.126). It’s now evident that, integral (5.128) would always be non-negative without the help of spatial acceleration v˙ and hence violate volume con- servation.

on the potential energy functional of any spatial configuration ω_liq where V[ωliq] = 3 ^Z

ω_liqdω− |Ω_liq|, |Ω_liq|=^Z

Ω_liqdΩ. (5.133) Here|Ω_liq|is the dimensionless total volume of the undeformed configuration. In other words, only energies of different shapes that share identical liquid volume should be compared. The factor 3 in V[ω_liq] arises from the earlier observation (5.127) that volume integralV[ωliq]can be reformed into a shape functionalV[γ]through divergence theorem,

V[ω_liq] =^Z

γ

x·ndγ−3|Ω_liq|=V[γ]. (5.134) In the rest of this chapter, we will adopt one unique global definition of the surface normal vector n(orN) pointing from the liquid to the vacuum.

Now let’s consider a configuration mapχ(X, ) which transforms the material domain Ω_vacand boundaryΓ to their spatial counterpartsω_vacandγ. Although the derivatives we developed in the last section hold for arbitrary function χ and value of as long as the map χ(·, ) is well-defined, in energy stability analysis we are mostly interested in the effect of small perturbations to the undeformed geometry of the conductor. We choose to align the material frame with the spatial frame at = 0, i.e. χ(X,0) =X is an identity map. Assuming sufficient continuities in parameter, then in the spirit of Taylor expansion we can represent any near-identity mapχ(·, )in the general form

χ(X, ) =X+V +1

2²A+O(³) (5.135)

for some smallwhereV andAare the material velocityv and accelerationv˙ evaluated at = 0, respectively. We reemphasize that the terminologies “velocity” and “acceler- ation” do not refer to physical quantities in time but rather parametric dependence of the map χ(·, ) on 0 ≤ 1. We also do not specify behaviors of χ(X ∈Ω_liq, ) in the liquid volume other than mass conservation and continuity of χ(X ∈ Γ, ) on the liquid/vacuum interface.

Constrained first shape variation of potential energy

The boundary Γ is said to be a critical shape of the energy functional F[Γ] if its first shape variation vanishes for all admissible infinitesimal deformations under certain constraint, in our case the conservation of total liquid volume,

dF[γ]

d

₌₀ = 0 for all χ(·, ) such that V[γ] = 0. (5.136) Equation (5.136) is a constrained derivative. If the potential energy and volume con- straint in (5.136) were multivariate functions instead of functionals, one could enforce the constraint through the multi-variable chain rule. In an abstract variational problem

of functionals, it’s not clear how to apply “chain rule" directly. To overcome this diffi- culty we instead formulate the Lagrangian functional (not to confuse with Lagrangian coordinates)

L[γ, λ] =S[γ] +E[γ]−λV[γ], (5.137) where constantλis the Lagrange multiplier. Then an unconstrained critical point(Γ, Λ) of the Lagrangian (5.137) is also a critical shape of the original constrained potential.

The critical condition reads dL[γ, λ]

d

_=0,λ=Λ= dF[γ]

d −ΛdV[γ] d

₌₀ =δ⁽¹⁾F[Γ;V]−Λδ⁽¹⁾V[Γ;V] = 0, (5.138) dL[γ, λ]

dλ

_=0,λ=Λ=−dV[γ]

d

₌₀ =−δ⁽¹⁾V[Γ;V] = 0, (5.139) where Λ is the correct constant of multiplier λ to be determined. Using the results (5.51) and (5.104) derived earlier, we arrive at the necessary conditions for Γ to be a critical shape of the constrained potential energy F[γ],

0 =^Z

Γ

− Ec

C[Γ]² |DΨ|²−2H−ΛN·V dΓ +^Z

ΓDivΓVΓdΓ, 0 =^Z

Ω_liq dΩ− |Ω_liq|,











(5.140)

where^R_ΓDivΓV_ΓdΓ = 0on a boundaryless manifold Γ. Note the definition of N in (5.140) is opposite to the normal vector in (5.49). For Γ to be an equilibrium shape, the critical condition (5.140) must hold for all admissible velocity fieldV which is only possible if

− Ec

C[Γ]² |DΨ|²−2H=Λ on Γ, (5.141) where the constantΛ is chosen such that the volume of the liquid equals|Ω_liq|. In dimensional form, the equilibrium condition (5.141) represents a point-wise balance between the electrostatic and capillary pressure everywhere on the liquid surface only up to a residue pressure that’s globally constant over the entire surface. This constant pressure level is implicitly determined by the total volume of the liquid. The potential energy of shape Γ satisfying equilibrium condition (5.141) would have a vanishing first energy variation for all volume-conserving shape deformations. Recall the stability near an equilibrium point of a mechanical system largely attributes to the local convexity of the same point in the energy landscape. We cannot tell if a critical shapeΓ (if exist) is a local minimum, maximum or saddle point of the potential energyF[γ]from the first order conditions of the Lagrangian L[γ, λ]alone.

Note volume constraintV[γ] = 0must hold for every spatial configuration which means its derivatives at all orders of must vanish,

dV[γ]

d = d²V[γ]

d² =...= d^kV[γ]

d^k =...= 0 for ≥0. (5.142)

Applying volume variation (5.120) todV[γ]/dat= 0 yields a first order constraint Z

ΓN·V dΓ = 0 (5.143)

on the velocity fieldV.

Constrained second shape variation of potential energy

The energy stability of a critical shape Γ is encoded in the functional structure of its second shape variation, which essentially maps out the local energy landscape near the critical shape and thereby allows us to determine the type of local extreme (i.e.

maximum, minimum or saddle). The second variation of the total potential energy with respect to at = 0 has two contributions from material velocity V and acceleration A, d²F[γ]

d²

₌₀ =δ⁽²⁾F[Γ;V,A] =δ⁽²⁾F[Γ;V] +δ⁽¹⁾F[Γ;A]. (5.144) Note the material acceleration field A cannot be an arbitrary vector field since χ(·, ) must conserve total liquid volume. We can eliminate this indirect constraint on A by invoking the first order critical condition (5.138) on Γ,

δ⁽¹⁾F[Γ;V] =Λδ⁽¹⁾V[Γ;V], (5.145) which is expected to hold for any admissible vector field V. Now if we replace V in (5.145) byA and then substitute (5.145) back to equation (5.144), we arrive at

d²F[γ]

d²

₌₀=δ⁽²⁾F[Γ;V] +Λδ⁽¹⁾V[Γ;A]. (5.146) We next evaluate the second order condition in the volume constraint (5.142),

0 = d²V[γ]

d²

₌₀=δ⁽²⁾V[Γ;V,A] =δ⁽²⁾V[Γ;V] +δ⁽¹⁾V[Γ;A], (5.147) which again has two contributions fromV andA. It’s now evident that the variational contributions ind²F[γ]/d² are functionals of material velocityV alone after substitut- ing identity (5.147) into (5.146). In the end, we arrive at the constrained (i.e. volume conserving) second shape variation of the potential energy F[γ] evaluated at a critical shape Γ provided the material velocity field V of a configuration map χ(·, ) satisfies the constraint (5.143),

d²F[γ]

d²

₌₀=δ⁽²⁾F[Γ;V]−Λδ⁽²⁾V[Γ;V]

=− Ec C[Γ]²

(

δ⁽²⁾C[Γ;V]−2(δ⁽¹⁾C[Γ;V])² C[Γ]

)

+δ⁽²⁾S[Γ;V]−Λδ⁽²⁾V[Γ;V], (5.148)

where the Lagrange multiplier constant Λ is determined by the first variation. We make a comment here regarding the elimination of material accelerationA: it does not implyA is dispensable. It’s merely a consequence of volume conservation under which the contribution of Ato the second variation is equivalent to some integral expression involving V only. It is computationally more convenient if we keep V as the only independent variable.

We again employ the surface-adapted curvilinear coordinates(ξ¹, ξ², ξ³)defined in (5.86) and project material velocity V into a surface vector field U(ξ¹, ξ²) tangent toΓ and a normal componentΘ(ξ¹, ξ²)N,

V =U +ΘN, U =U^αGα(ξ¹, ξ²,0). (5.149) Through the use of the second derivative of area differential (5.116), total volume (5.128) and self-capacitance (5.94) (recall normal vector is flipped in C[γ]), we can explicitly write down the expressions for the three functionals in the second shape variations (5.148),

δ⁽²⁾C[Γ;V] =^Z

Γ2(N·DΨ)(N·DΨ⁰)Θ+ 2HΘ²(N·DΨ)²dΓ

− Z

Γ[U ·D_ΓΘ−(DivΓU)Θ+UIIU] (N·DΨ)²dΓ, (5.150) δ⁽²⁾S[Γ;V] =^Z

Γ

D_ΓΘ·D_ΓΘ+ 2KΘ²−KU·UdΓ +^Z

Γ(DivΓU)²−( ˆ∇βU^α)( ˆ∇αU^β) + 2II_β^α∇ˆα(ΘU^β) dΓ

− Z

Γ4H(DivΓU)Θ−2HUIIUdΓ, (5.151) δ⁽²⁾V[Γ;V] =^Z

Γ−2HΘ²+ 2(DivΓU)Θ−UIIUdΓ. (5.152) Here K is the Gaussian curvature of surface Γ. At first glance, the integrals (5.150), (5.151) and (5.152) appear to be formidable. Fortunately many terms would cancel out each out thanks toΓ being a critical shape. To see this, we first expand the constraint Λδ⁽²⁾V[Γ;V]. NoteΛ is a constant determined by the critical condition (5.140) which therefore can be moved inside the integral ofδ⁽²⁾V[Γ;V],

Λδ⁽²⁾V[Γ;V] =^Z

Γ

− Ec

C[Γ]²|DΨ|²−2H ^h−2HΘ²+ 2(DivΓU)Θ−II_αβU^αU^βi dΓ

=− Ec C[Γ]²

Z

Γ|DΨ|^2h−2HΘ²+ 2(DivΓU)Θ−II_αβU^αU^βi dΓ +^Z

Γ4H²Θ²+ 2H^hII_αβU^αU^β−2(DivΓU)Θⁱ dΓ. (5.153) Substituting equations (5.150), (5.151) and (5.153) into the second shape variation

(5.148) yields a relatively clean expression, d²F[γ]

d²

₌₀=− Ec C[Γ]²

Z

Γ2(N·DΨ)(N·DΨ⁰)Θ+ 4HΘ²(N·DΨ)²dΓ +^Z

ΓD_ΓΘ·D_ΓΘ+ (2K−4H²)Θ²dΓ + Ec

C[Γ]² Z

ΓDivΓ(ΘU)(N·DΨ)²dΓ +^Z

Γ2II_β^α∇ˆα(ΘU^β) dΓ +^Z

Γ(DivΓU)²−( ˆ∇βU^α)( ˆ∇αU^β)−KU·UdΓ + 2Ec

C[Γ]³(δ⁽¹⁾C[Γ;V])²,

(5.154)

where material velocity V must fulfil ^R_ΓN·VdΓ = 0. The second variation (5.154) contains four parts: a purely normal and extrinsic contribution (line 1 and 2), a cross term (line 3) which couples the tangential vector fieldU with the normal scalar fieldΘ of material velocity V, a purely intrinsic integral (line 4) plus a contribution from the first order variation (line 5).

We next show that, for an equilibrium shapeΓ, the integrals on line 3 and 4 of (5.154) can be significantly simplified. We begin with the second integral on line 3 of (5.154),

Z

Γ2II_β^α∇ˆα(ΘU^β) dΓ =^Z

Γ−2( ˆ∇αII_β^α)ΘU^βdΓ =^Z

Γ−2( ˆ∇βII_α^α)ΘU^βdΓ Codazzi

=^Z

Γ−2( ˆ∇β2H)ΘU^βdΓ =^Z

Γ4HDivΓ(ΘU) dΓ. (5.155) We then substitute identity (5.155) into the integrals on line 3 of (5.154) and compute,

Ec C[Γ]²

Z

ΓDivΓ(ΘU)(N·DΨ)²dΓ+^Z

Γ2II_β^α∇ˆα(ΘU^β) dΓ

=^Z

Γ

(N·DΨ)² Ec

C[Γ]² + 4H

DivΓ(ΘU) dΓ

=^Z

Γ(−Λ+ 2H)DivΓ(ΘU) dΓ =^Z

Γ−2(U ·∇ˆH)ΘdΓ, (5.156) where in the last line of (5.156) the equilibrium condition (5.140) on Λ is invoked for the critical shapeΓ. The integral from line 4 of (5.154) is actually zero,

Z

Γ(DivΓU)²−( ˆ∇αU^β)( ˆ∇βU^α)−KU ·UdΓ

=^Z

Γ(DivΓU)²+U^β∇ˆα∇ˆβU^α−KU ·UdΓ

=^Z

Γ(DivΓU)²+U^β∇ˆβ∇ˆαU^α+U^βR^α_ναβU^ν−KU^αG_αβU^βdΓ

=^Z

Γ(DivΓU)²+U^β∇ˆβ(DivΓU) +U^α(Ric_αβ −KG_αβ)U^βdΓ

=^Z

Γ(DivΓU)²−( ˆ∇βU^β)(DivΓU) dΓ = 0, (5.157)

where in the last line of (5.157) we have used the fact that the Ricci tensor of a two- dimensional manifold is intrinsic (Kreyszig,1991), i.e. Ric_αβ =KG_αβ. With identities (5.156) and (5.157) substituted into (5.154) we obtain the final form of the second shape variation of the total potential energyF[γ]about an equilibrium shapeΓ,

d²F[γ]

d²

₌₀=− Ec C[Γ]²

Z

Γ2(N·DΨ)(N·DΨ⁰)Θ+ 4HΘ²(N·DΨ)²dΓ +^Z

Γ

D_ΓΘ·D_ΓΘ+ (2K−4H²)Θ²dΓ

− Z

Γ(U ·D_Γ2H)ΘdΓ + 2Ec

C[Γ]³(δ⁽¹⁾C[Γ;V])² for ^Z

Γ

ΘdΓ = 0, (5.158) where the Eulerian derivativeΨ⁰is the harmonic solution to the boundary value problem,

DivDΨ⁰ = 0 in Ω,

Ψ⁰ =−(N·DΨ)Θ on Γ.



 (5.159)

Later we will show that the integrals in the first two lines of the second shape variation (5.158) directly lead to the famous Rayleigh charge limit Q_Ra. However, to author’s knowledge, the terms in the last line of (5.158) have not been derived before in literature.

This is because when a liquid body in equilibrium has both constant mean curvature and constant charge density over its surface, e.g., the shape of a perfect sphere or cylinder considered by most literature, the integrals in the last line of (5.158) are identically zero. However, when the equilibrium shape has a heterogeneous distribution of surface charges, the first variation of capacitance δ⁽¹⁾C[Γ;V] from equation (5.52) may not vanish for an arbitrary volume-conserving material velocity V. Similarly, if boundary Γ is not a surface of constant curvature then the surface-advection termU·D_Γ2His not necessarily zero for every tangential velocity field U.

Energy stability of an isolated charged spherical drop

LetΓ be a dimensionless unit sphere (so that the dimensional radius of a physical drop becomes the characteristics length scaleRc). It’s natural to use the spherical coordinates X = (R, ϑ, ϕ) centered at origin of the drop for the material frame Ω. It’s easy to verify that a uniformly charged spherical drop is a critical shape of the total potential energy F[γ], the analytic solution of which is given by

Ψ(X) = 1

R, N·DΨ|Γ =−1, C[Γ] = 4π, H=−1, K= 1. (5.160) For a general equilibrium shape, the term (δ⁽¹⁾C[Γ;V])² in the second shape variation (5.158) is nonzero. However in the special case (5.160) of a spherical drop the dimen- sionless surface chargeN·DΨ is constant on the entire spherical surface which would completely eliminate this term since

δ⁽¹⁾C[Γ;V] =^Z

ΓΘ(N·DΨ)²dΓ = (N·DΨ)^2Z

ΓV ·NdΓ = 0 (5.161)

where the first order condition of volume constraint is used. The contribution from tangential variationV_Γ also drops out due to the constant mean curvature of a spherical surface. Substituting critical solution (5.160) into second shape variation (5.158) yields a surface integral involving normal variation Θ only,

d²F[γ] d²

₌₀ = Ec 8π²

Z

Γ(N·DΨ⁰)Θ+ 2Θ²dΓ − Z

ΓΘ∆ΓΘ+ 2Θ²dΓ, (5.162) where we have replaced the termD_ΓΘ·D_ΓΘ by the Laplace-Beltrami operator∆Γ =

∇ˆ^α∇ˆα via integration by parts on a boundary-less manifold (surface of a sphere). The quadratic formsΘ² andΘ∆ΓΘare local in the sense that only multiplication and differ- ential operator are involved. On the other hand, the termN·DΨ⁰ is truly nonlocal: the solution to the boundary value problem (5.159) of the Eulerian derivativeΨ⁰requires the knowledge of normal variation Θ everywhere on Γ and therefore is considered nonlocal with respect to the intrinsic coordinates of surface Γ. Analytic solution to (5.159) for a general shape Γ is often not possible. However, for the geometry of a unit sphere, it can be treated by the means of spherical harmonics expansion. In what follows, we refer to the elementary treatise by Byerly (1895) for various properties of spherical harmonics functions.

Let Y<sub>`</sub><sup>m</sup>(ϑ, ϕ) be the standard notation for spherical harmonics of integer degree`≥0 and integer order −`≤m ≤`. Normal field Θ on the unit sphere can be expanded in terms of the spherical harmonics Y_`^m,

Θ(ϑ, ϕ) =^X∞

`=0

X` m=−`

a^m_{`</sub> Y<sub>`}^m(ϑ, ϕ), (5.163) where each complex coefficient

a^m_{`</sub> =hY<sub>`}^m, Θi (5.164)

is given by the orthonormal projection under inner product integration h·,·i over the surface of a unit sphere (or equivalently the solid angle) defined as

hf, gi=^Z

Γ

f(ϑ, ϕ)^†g(ϑ, ϕ) dΓ. (5.165) Here (·)^† is the complex conjugate operator. We will suppress the (ϑ, ϕ)-dependence from now on and only refer to the projected coefficients instead. Note the zeroth degree coefficient must vanish due to the volume constraint,

a⁰₀ =hY₀⁰, Θi ∝ Z

Γ

ΘdΓ =^Z

Γ

V ·NdΓ = 0. (5.166)

Any linear combination of spherical harmonics above `= 0 would have a zero spherical mean which implies the expansion (5.163) of Θ is an admissible variation as long as a⁰₀ = 0. We then replace boundary condition for the harmonic potential Ψ⁰ in the

auxiliary boundary value problem (5.159) by the expansion (5.163). The solution to the harmonic problem for Eulerian derivativeΨ⁰ exterior to the spherical drop is found to be

Ψ⁰ =^X∞

`=0

X` m=−`

1

R<sup>`+1</sup>a<sup>m</sup><sub>`</sub> Y_`^m in Ω_vac. (5.167) Normal vectorN on the surface of a unit sphere is simply the unit vector in the radial direction which yields

N·DΨ⁰ =^X∞

`=0

X` m=−`

−(`+ 1)a<sup>m</sup><sub>`</sub> Y_{`</sub><sup>m</sup> on Γ. (5.168) On the other hand, the Laplace-Beltrami operator ∆Γ defined for the surface of a unit sphere is diagonalized by the spherical harmonicsY<sub>`}^m with eigenvalue−`(`+ 1),

∆ΓΘ =^X∞

`=0

X` m=−`

−`(`+ 1)Y<sub>`</sub><sup>m</sup>. (5.169) Substituting expansions (5.169), (5.168) and (5.163) into the second shape variation (5.162) results in a double summation over two sets of integers{`, m} and{`⁰, m⁰},

d²F[γ]

d²

₌₀=^X

`,m

X

`⁰,m⁰

Ec

8π²(1−`) +`(`+ 1)−2a<sup>m</sup><sub>`</sub> a^m_`0⁰

Z

Γ

Y_{`</sub><sup>m</sup>Y<sub>`}^m0 ⁰dΓ. (5.170) Orthonormality of spherical harmonics doesn’t directly apply yet since Y_`^m under the integral (5.170) is missing the complex conjugate defined in the inner product (5.165).

We note that Y_{`</sub><sup>m†</sup> can be recovered through manipulation of the spherical harmonics identityY<sub>`}^m = (−1)^mY_`^−m†. We first consider the partial summation in integerm⁰,

`⁰

X

m⁰=−`⁰

a^m_{`</sub> a<sup>m</sup><sub>`}0⁰

Z

ΓY_{`</sub><sup>m</sup>Y<sub>`}^m0 ⁰dΓ

= ^`

X0

m⁰=−`⁰

a^m_{`</sub> a<sup>m</sup><sub>`}0⁰hY_{`</sub><sup>m†</sup>, Y<sub>`}^m0 ⁰i= ^` X0

m⁰=−`⁰

a^m_{`</sub> a<sup>m</sup><sub>`}0⁰(−1)^mhY_{`</sub><sup>−m</sup>, Y<sub>`}^m0 ⁰i

= ^`

X0

m⁰=−`⁰

hY_{`</sub><sup>m</sup>, Θia<sup>m</sup><sub>`}0⁰ 1

(−1)^mhY_{`</sub><sup>−m</sup>, Y<sub>`}^m0 ⁰i= ^` X0

m⁰=−`⁰

hY_{`</sub><sup>−m†</sup>, Θia<sup>m</sup><sub>`}0⁰hY_{`</sub><sup>−m</sup>, Y<sub>`}^m0 ⁰i

= ^`

X0

m⁰=−`⁰

hY_{`</sub><sup>−m</sup>, Θi<sup>†</sup>a<sup>m</sup><sub>`}0⁰hY_{`</sub><sup>−m</sup>, Y<sub>`}^m0 ⁰i NoteΘ is a real function

= ^`

X0

m⁰=−`⁰

a^−m†_{`</sub> a<sup>m</sup><sub>`}0⁰δ_(−m)m0δ_{``</sub>0 =a<sup>−m</sup><sub>`</sub> a<sup>−m†</sup><sub>`0</sub> δ<sub>``}0. (5.171) Identity (5.171) reduces the double summation to an infinite sum of real quadratic forms,

d²F[γ] d²

₌₀ =^X∞

`=1

F<sub>`</sub> X` m=−`

a^m_{`</sub> a<sup>m</sup><sub>`} ^†, F_`= Ec

8π²(1−`) +`(`+ 1)−2, (5.172)

where −m in identity (5.171) is relabeled as m provided symmetry in the summation of m =−`, . . . , `. Since a^m_{`</sub> a<sup>m</sup><sub>`} ^† ≥0 is always non-negative, the coefficients F_{`</sub> in the second shape variation (5.172) play a key role in determining the contribution of each spherical harmonics deformationY<sub>`}^m to the total potential energy.

The `= 1 coefficient F₁ is always zero regardless of the value of the electric-capillary number Ec and hence doesn’t contribute to any change in the potential energy F[γ].

This is due to the translational symmetry in the electrostatic energy and in the surface energy. The three harmonic modes Y₁⁻¹, Y₁⁰ and Y₁¹ correspond to infinitesimal shifts in the direction of three axes of the three-dimensional Euclidean space R³ which must leave each component of the total potential energy individually invariant.

Rayleigh’s charge limit Q_Ra controls the sign of F₂ in the`= 2modes. These are the modes of prolate and oblate deformations, the combination of which leads to elongation or depression of the spherical drop about an arbitrarily oriented symmetry axis. With some nonzero electric-capillary numberEc =Q<sup>2</sup>/2ε0σR<sup>3</sup><sub>c</sub>, it is possible to set the sign of every `= 2 quadratic form a^m₂ a^m₂^† to be strictly negative if the following inequality is satisfied,

F₂ =−Ec

8π² + 4<0 if Ec>32π² ⇔ Q >8π^qσε₀R³_c=Q_Ra. (5.173) We can make a more general statement regarding the sign of coefficientF₂: given some integer`⁰ >1,

if Q_{`</sub>0 < Q < Q<sub>`}0+1 then





F<sub>`</sub><0 for 2≤`≤`⁰,

F<sub>`</sub>>0 for `≥`⁰+ 1, (5.174) where

Q_{`</sub>0 = 4π<sup>q</sup>(`⁰+ 2)σε0R³_c (5.175) is the`<sup>0</sup>-th Rayleigh charge. If the total charge of a spherical drop falls betweenQ<sub>`} and Q_{`+1</sub>, then the modes of harmonic deformation V ∝Y<sub>`}^mN equal to or below degree

`<sup>0</sup> contribute negatively to its potential energy F[γ] while the modes above degree ` act to increase the energy. The Rayleigh charge limit Q_Ra = Q₂ is the special case where all deformation modes are only allowed to increase the total potential energy.

In other words, an isolated perfectly conductive spherical drop, if charged below Q_Ra, is forbidden to release any of its potential energy into other forms such as kinematic energy or viscous dissipation.