Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors
Various models leading to predictions of negative capacitance, C, are briefly reviewed. Their relation to the nature of electric control is discussed. We reconfirm that the calculated double layer capacitance can be negative under σ-control – an artificial construct that requires uniform distribu...
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| Zitieren: | Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors / M.B. Partenskii, P.C. Jordan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 397–412. — Бібліогр.: 32 назв. — англ. |
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irk-123456789-1196392017-06-08T03:07:38Z Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors Partenskii, M.B. Jordan, P.C. Various models leading to predictions of negative capacitance, C, are briefly reviewed. Their relation to the nature of electric control is discussed. We reconfirm that the calculated double layer capacitance can be negative under σ-control – an artificial construct that requires uniform distribution of the electrode surface charge density, σ. However, only the total charge q (or the average surface charge density σ) can be experimentally fixed in isolated cell studies (q-control). For those σ where C becomes negative under σ-control, the transition to q-control (i.e. relaxing the lateral change density distribution, fixing its mean value to σ) leads to instability of the uniform distribution and a transition to a non-uniform phase. As an illustration, a “membrane capacitor” model is discussed. This exactly solvable model, allowing for both uniform and inhomogeneous relaxation of the electrical double layer, helps to demonstrate both the onset and some important features of the instability. Possibilities for further development are discussed briefly. Представлено короткий огляд моделей, які передбачають негативну ємність C. Обговорюється роль цих моделей у явищі електричного контролю. Ми ще раз показуємо, що розрахункова ємність подвійного шару може бути негативною завдяки σ-контролю – штучній конструкції, яка вимагає однорідного розподілу густини поверхневого заряду електрода, σ. Разом з тим, тільки загальний заряд q (або усереднена густина поверхневого заряду σ) може бути експериментально зафіксованою при дослідженні ізольованої комірки (q-контроль), Для значень σ, де C стає від’ємною в умовах σ-контролю, перехід до q-контролю (тобто релаксація латеральної густини розподілу заряду шляхом фіксації її середнього значення до величини σ) веде до нестабільності однорідного розподілу і переходу до неоднорідної фази. В якості ілюстрації розглядається модель “мембранного конденсатора”. Ця точно розв’язувана модель допускає як однорідну, так і неоднорідну релаксацію і таким чином допомагає продемонструвати зародження і деякі важливі риси нестабільності. Коротко обговорюються можливості подальших досліджень. 2005 Article Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors / M.B. Partenskii, P.C. Jordan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 397–412. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 68.35.Rh, 68.35.Md, 82.45.Mp, 82.45.Rr, 82.45.Uv, 68.08.-p, 73.30.+y DOI:10.5488/CMP.8.2.397 http://dspace.nbuv.gov.ua/handle/123456789/119639 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
Various models leading to predictions of negative capacitance, C, are briefly
reviewed. Their relation to the nature of electric control is discussed. We
reconfirm that the calculated double layer capacitance can be negative under
σ-control – an artificial construct that requires uniform distribution of
the electrode surface charge density, σ. However, only the total charge q
(or the average surface charge density σ) can be experimentally fixed in
isolated cell studies (q-control). For those σ where C becomes negative
under σ-control, the transition to q-control (i.e. relaxing the lateral change
density distribution, fixing its mean value to σ) leads to instability of the uniform
distribution and a transition to a non-uniform phase. As an illustration,
a “membrane capacitor” model is discussed. This exactly solvable model,
allowing for both uniform and inhomogeneous relaxation of the electrical
double layer, helps to demonstrate both the onset and some important features
of the instability. Possibilities for further development are discussed
briefly. |
| format |
Article |
| author |
Partenskii, M.B. Jordan, P.C. |
| spellingShingle |
Partenskii, M.B. Jordan, P.C. Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors Condensed Matter Physics |
| author_facet |
Partenskii, M.B. Jordan, P.C. |
| author_sort |
Partenskii, M.B. |
| title |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors |
| title_short |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors |
| title_full |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors |
| title_fullStr |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors |
| title_full_unstemmed |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors |
| title_sort |
negative capacitance and instability at electrified interfaces: lessons from the study of membrane capacitors |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119639 |
| citation_txt |
Negative capacitance and instability at electrified interfaces: Lessons from the study of membrane capacitors / M.B. Partenskii, P.C. Jordan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 397–412. — Бібліогр.: 32 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT partenskiimb negativecapacitanceandinstabilityatelectrifiedinterfaceslessonsfromthestudyofmembranecapacitors AT jordanpc negativecapacitanceandinstabilityatelectrifiedinterfaceslessonsfromthestudyofmembranecapacitors |
| first_indexed |
2025-07-08T16:18:59Z |
| last_indexed |
2025-07-08T16:18:59Z |
| _version_ |
1837096282697498624 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 397–412
Negative capacitance and instability at
electrified interfaces: Lessons from the
study of membrane capacitors
M.B.Partenskii, P.C.Jordan
Department of Chemistry, MS–015
Brandeis University
PO Box 549110
Waltham, MA 02454–9110, USA
Received September 14, 2004
Various models leading to predictions of negative capacitance, C, are briefly
reviewed. Their relation to the nature of electric control is discussed. We
reconfirm that the calculated double layer capacitance can be negative un-
der σ-control – an artificial construct that requires uniform distribution of
the electrode surface charge density, σ. However, only the total charge q
(or the average surface charge density σ) can be experimentally fixed in
isolated cell studies (q-control). For those σ where C becomes negative
under σ-control, the transition to q-control (i.e. relaxing the lateral change
density distribution, fixing its mean value to σ) leads to instability of the uni-
form distribution and a transition to a non-uniform phase. As an illustration,
a “membrane capacitor” model is discussed. This exactly solvable model,
allowing for both uniform and inhomogeneous relaxation of the electrical
double layer, helps to demonstrate both the onset and some important fea-
tures of the instability. Possibilities for further development are discussed
briefly.
Key words: electrochemical interfaces, instabilities and phase transitions,
electric double layers, capacitance, membrane electroporation
PACS: 68.35.Rh, 68.35.Md, 82.45.Mp, 82.45.Rr, 82.45.Uv, 68.08.-p,
73.30.+y
1. Introduction
The question of the admissible sign of the differential capacitance at charged in-
terfaces and its relation to thermodynamic stability has a long and dramatic history.
It emerged in early 1970s with the so-called Cooper-Harrison catastrophe [7], an ap-
parent prediction of C < 0 for “dipolar capacitors” – lattices of reorientable dipoles
between conductive plates. Soon thereafter, similar anomalies were predicted for
some ionic double layer models [3,12]. In the mid-1980s interest in this problem was
c© M.B.Partenskii, P.C.Jordan 397
M.B.Partenskii, P.C.Jordan
renewed due to predictions of C < 0 for microscopic “relaxing gap capacitor” (RGC)
models (see [22,26,27,30] and references therein) which accounted for the metal elec-
tron contributions and the corresponding relaxation of the “electronic plate” of the
interfacial capacitor with charging. It is accepted that C must be strictly positive
for open “φ-controlled” systems, where the electric cell is connected to a source of
controllable voltage [15]. However, it was shown that C < 0 is possible as a stable
state of an isolated RGC if electric control is maintained by a macroscopically uni-
formly distributed surface charge density σ, so-called σ-control (see [10,22,26] for
review). Prior to this discovery, an attempt had been made to prove, on general
statistical-mechanical grounds, that for an equilibrium σ-controlled system C must
be strictly positive [4]. Somewhat later improved analysis [2,20,21,24,27] showed the
model [4] does not forbid negative C values. For both ionic and “relaxing gap ca-
pacitor” models, which together cover a very wide range of interfacial phenomena,
the equilibrium capacitance under σ-control can be negative. This is true for both
the “compact layer” capacitance and the total double layer capacitance, including
the “diffuse” layer contributions [22,26].
Here we show that though admissible for σ-control, which implies a uniform
charge density on the “electrode”, negative EDL capacitance is not possible for a
real isolated system, where uniformity of σ is not enforceable and only the total
electrode charge can be fixed. In other words, we show that in σ domains where C
is negative under σ-control, the system is unstable with respect to transition to an
inhomogeneous state with nonuniform lateral distributions of the electrode charge
density and of mobile electrolyte ions. To demonstrate this anomaly, we discuss a
transparent “membrane capacitor” model, which exhibits a C < 0 domain presuming
uniformity (an elastic slab with plane surfaces and uniform surface charge) while
becoming unstable in this domain if non-uniform surface charge distributions and
membrane deformations are considered.
This result addresses questions raised recently [6,11] with respect to the meaning
and physical reality of C < 0 for the isolated capacitor. Our analysis suggests that a
model demonstrating this anomaly under σ-control can be used to study transition
to an inhomogeneous interfacial state under q-control.
2. Control of electrified interfaces – theory and experiment
1. Experimental study of the electrical double layer (EDL) at electrochemical
interfaces is usually conducted under “potential” (φ-) control, where electrodes
are connected to a voltage source. Changing the applied voltage in increments
∆φ leads to corresponding changes of the electrode charge, ∆q. Similarly,
controlled modulation of the voltage, φ = φ(t), results in charge modulation
measurable by impedance techniques. Connection to a potentiostat, required
to maintain φ-control, results in an open system, which is treated by grand
canonical methods.
398
Negative capacitance and instability at electrified interfaces
The differential capacitance (per unit area) is then defined by the derivative
Cφ = ∂φσ, (1)
where the average surface charge density is σ = q/A and A is the surface area
of the electrode. The form of equation 1 is a typical response function
κF = ∂FX, (2)
where F is the external parameter (“force”) and X the conjugate intensive
variable.
2. The electrical properties of interfaces can equally well be studied by controlling
the electrode charge, q. In q-control the measured quantity is the correspond-
ing potential φ. Charge can be regulated by connecting the electrodes to a
battery for brief periods of time ∆t, measuring the current j, with the charge
increments found by ∆q = j∆t. With q fixed, the resultant φ is measured in
an isolated system, i.e. a canonical ensemble.
The corresponding response function, analogous to equations 1 and 2 is the
inverse differential capacitance
C−1
q = ∂σφ. (3)
Obviously, q-control is a synonym for σ-control: fixing the total charge q is
equivalent to fixing the average surface charge density σ. The thermodynamic
potentials of the expanded (φ-controlled ) system, Aφ, and the isolated (q-
controlled) system, Aq, are related by the Legendre transformation,
Aφ(σ, φ) = Aq(σ) − qφ. (4)
3. Most calculations of the electric double layer (EDL) assume that the electrode,
often described as a charged flat wall, has a uniform, fixed charge density. This
is effectively σ-control, which is generally not equivalent to q- (or σ-) control.
The terms are interchangeable only if the equilibrium surface charge density
is uniform on a scale exceeding atomic dimensions, i.e. if σ = σ = const. A
counter-example is the one where the optimized local charge density is non-
uniform in the electrode plane, σ = σ(rs) 6= σ (rs is the radius vector in the
electrode plane), reminiscent of “charge density wave” states in an electron
plasma. In practice there is no way to control the local charge density; in
isolated systems only the total charge can be constrained externally. σ-control
is a purely theoretical construct; its predictions must be tested to determine
if a uniform surface charge density and its corollary, a laterally-uniform ionic
distribution, correspond to a real equilibrium state.
399
M.B.Partenskii, P.C.Jordan
3. Admissible sign of the differential capacitance
3.1. φ-control: the open system
General thermodynamic [14] and statistical-mechanical [18] treatments of elec-
trified interfaces show that, under φ-control, differential capacitance must be strictly
positive. In our view attempts to circumvent this restriction [2,31,32] have somewhat
misinterpreted the concept of potential control [26,27].
Simple models [9,10,22,26,27] demonstrate that “equilibrium” states leading to
predictions of C < 0 under φ-control are inevitably unstable (and therefore ficti-
tious), corresponding to local maxima in the thermodynamic potential Aφ. Typically,
a different locally stable state (or states) for which C > 0 should exist for the same
voltage φ. Thus, predictions of C < 0 in open systems must be tested with respect
to stability and to the existence of alternative, stable states. Recent analysis [6] has
compared two theoretical approaches to the modeling of the EDL: (1) Monte-Carlo
treatment of EDL under σ-control and (2) Density-Functional Theory for the open
system (φ-control). Both approaches predict C < 0. While this prediction can be
valid in the framework of the first approach (see e. g. [26,27] and the following
discussion), it must be carefully reexamined for an open system.
Near a critical voltage φcr, defined by C−1(φcr) = 0, the system becomes unstable.
The transition to a new state is accompanied by charge flow from the potentiostat
to the electrodes, a sort of electrical “breakdown” (see [9,10,22] for more details),
which would be a unique path to phase transformation assuming lateral uniformity.
However, as discussed below, the transition can also involve formation of a later-
ally non-uniform phase accompanied by nonuniform redistribution of the electrode
charge density σ(ρ).
As the requirement that Cφ > 0 is now generally accepted, we turn to treating
isolated systems. The sense of the upcoming discussion is already implicit, once
having recognized that the first of the two aforementioned instabilities is forbidden
by the very definition of “q-control” since, once q is fixed, electric contact with the
potentiostat must be interrupted. Can C be negative under this constraint? Since
the admissible sign of C in isolated systems has almost always been analyzed in
σ-control terms, we first consider this case and postpone discussion of the more
general q-control.
3.2. σ-control in the isolated system
3.2.1. Primitive models of electrolytes
Interest in the admissible sign of C in the theory of the diffusive layer was
stimulated by work of Blum, Lebowitz and Henderson [4]. They tried to provide
a rigorous restriction on the sign of C for “primitive ionic models”: charged hard
ions in a uniform dielectric medium between two rigid, uniformly charged walls. The
corresponding Hamiltonian is quite generally [24]
H (σ, {R}) =
σ2d
2εε0
− σf({R}) + H
′
({R}), (5)
400
Negative capacitance and instability at electrified interfaces
where {R} refers to a particular configuration of the system (charge coordinates,
dipolar orientations, etc.). The first term describes a direct interaction between
the charged walls, with d the inter-wall distance. σf({R}) accounts for interaction
between the electrolyte and the electrode field (the physical significance of f will
be clear shortly) and H ′ is a σ independent interaction energy. The potential drop
between the charged plates is
φ(σ) =
σd
εε0
+ 〈f〉, (6)
where
〈(. . .)〉 =
∫
e−βH(σ,{R})(. . .)dΩ∫
e−βH(σ,{R})dΩ
is a canonical average with integration over the system’s configurational space, Ω,
and β = 1/kT. Equation 6 reveals the meaning of f : 〈f〉 is the potential drop induced
in the electrolyte by the field of the charged plates. It arises from redistribution of
free (ionic) charges shielding the applied field, and from repositioning of the bound
charges (the reorientation of molecular dipoles).
For Hamiltonians of the type of equation 5 the capacitance satisfies the general
condition [24]
C−1 =
d
εε0
−
A
kT
(
〈f 2〉 − 〈f〉2
)
, (7)
leading to a simple and self-evident result:
C−1
6
d
εε0
. (8)
Equation 8 simply means that redistribution of free charges and molecular po-
larization induced by the electric field in the electrolyte reduce the potential drop
between the electrodes and increase the capacitance C, a condition that places no
constraints on the sign of C under σ-control. One should note here that if the
distance d � λD, the characteristic Debye length in the electrolyte, then the total
inverse capacitance of the cell splits into two independent double layer contributions
belonging to two “electrodes”:
C−1 = C−1
1 + C−1
2 .
If the sign of the total cell capacitance C is unrestricted under σ-control, this is even
more true for the individual double layer contributions, C1 and C2.
Equation 7 (equation 24 of [24]) was derived in a study of a “dipolar capacitor”
(“DC”), a lattice of point dipoles embedded between the plates of a parallel-plate ca-
pacitor, a model often used for analyzing a compact layer at metal-solvent interfaces.
For the DC
f = fDC = −
1
ε0
Pz =
1
Aε0
∑
i
pi,z
is the potential drop corresponding to an arbitrary configuration of the molecular
dipoles with Pz the average surface density of the dipolar moment in the lattice
401
M.B.Partenskii, P.C.Jordan
and pi,z the projection of the individual dipole moment normal to the surface of the
lattice.
If interaction between the charged walls (the first contribution to the Hamilto-
nian, equation 5) were properly included in the Hamiltonian of the primitive ionic
model [4], it would also lead to equation 7 (see p. 68 of [24]) with
f = fion = −
1
ε0
∑
qizi ,
where qi is the charge of the i-th ion and zi is its distance from the charged wall
positioned at z = 0. Equation 7 and its analogs have been repeatedly derived and
discussed [2,21,27]; they hold for any model in which the electrodes are treated
as hard charged walls with distinctly separate regions occupied by electrode and
electrolyte. With these restrictions the interaction between the electrode and the
electrolyte can quite generally be described by a contribution ∼ σ
∑
qizi where
the summation includes both the mobile ionic and the molecular multipole charges
[27]. Such constraints clearly exclude “polarizable” models, those explicitly treating
molecular electronic polarizability, electron density penetration into regions occupied
by electrolyte, etc.
3.2.2. Relaxing gap capacitors (RGC)
Immobility of the charged “plates” in primitive models does not account for
another important phenomenon, possible displacement of the “electronic plate” of
interfacial capacitors and of the equilibrium positions of the electrolyte species in
contact with the electrode, in response to charging [22,26,27,30]. These effects are
effectively illustrated by the “relaxing gap capacitor” metaphor, which emphasizes
the dependence of the effective gap d on charging. The potential drop can be quite
generally represented as
φ =
1
εε0
σd(σ), (9)
where d(σ) is the effective separation between the “plates” of the capacitor associated
with the “centers of mass” of two microscopic charge distributions (see [10,22,30]
for review). We assume a uniform dielectric background ε between the plates, in
the volume occupied by free charges. For models using a non-uniform background,
typical of unified models used to account for both the “inner” (or Helmholtz) and
the diffuse layers [5], the expression is more complex and must also account for the
distributions of the bound (polarization) charges. However, these details are not
essential; they are omitted here.
For the electrode-electrolyte interface the effective separation is
d(σ) = zi(σ) − zi,e(σ), (10)
where
zi,e =
∫
ρσ
i,ezdz
σ
402
Negative capacitance and instability at electrified interfaces
with z the coordinate normal to the electrode surface. The inverse differential ca-
pacitance for the RGC is
(
1
εε0
C
)−1
= (σd(σ))′σ = d(σ) + σd(σ)′σ . (11)
Dependence of d on σ, a general feature of practically all double layer models, implies
that C is dependent on σ (or on the applied voltage). The “plate” displacement that
contributes to the variation of d not only reflects a shift of charge density profiles,
but is more a consequence of shape variation [26]. Quite typically, there is always a
σ range in which the charging decreases the effective gap. Elastic compression of the
lipid membrane by electric stress [8,23,25] and response of the Gouy-Chapman-Stern
(GCS) diffuse layer to the charging [5] are two representative examples.
In a range of σ where the effective gap contracts with the charging, d
′
(σ) < 0
and Cσ can be negative if
∆σ · d − σ · ∆d < 0. (12)
This inequality means that a potential increase due to a change of σ is overwhelmed
by its decrease due to gap contraction. A number of electromechanical and micro-
scopic models [10,22,26,27] show that negative capacitance (NC) under σ-control
is compatible with system stability. For instance, for every fixed σ (including the
domain where C < 0) the equilibrium gap of the elastic capacitor is defined by a
stable balance between elastic and electrostatic forces. Similarly, density functional
minimization led to a NC at metal-electrolyte interfaces due to relaxation of the
“electronic plate” of the capacitor [9,10,13].
Previous work [10,22,28] showed that accounting for “electronic plate” relaxation
in combination with traditional GCS and similar models typically leads to negative
C domains even though the GCS model itself (as does any other “local” statistical
model where ionic concentrations are local functions of the potential) does not lead
to such an anomaly [10]. Thus, while we agree that finding C < 0 under σ-control
must be both common and important for ionic models of electrolytes [11], it is not
a necessary condition for the appearance of this anomaly. If other relaxation mech-
anisms are taken in account a NC domain may arise even if the ionic contribution
is positive.
3.3. q-control in the isolated system
As discussed, σ-control is an artificial construct. Nevertheless, charges are nor-
mally distributed uniformly in the plane of the electrode and the charge q naturally
yields a uniform charge density σ, in which case σ- and q-control are identical. How-
ever, it is possible that under special conditions a non-uniform distribution of charge
in the electrode plane becomes energetically preferable. Under these conditions the
σ-control metaphor implies that artificial restraints (forces) were applied to the sur-
face charges to enforce uniformity of σ. Elimination of these artificial restraints would
result in transition to an inhomogeneous state. We will show this and its relation to
the sign of the capacitance by considering an undulating membrane capacitor, an
403
M.B.Partenskii, P.C.Jordan
exactly solvable model that contains many features common to real EDLs, which is
thus generally useful for discussing double layer behavior under critical conditions.
For a membrane capacitor under potential control the onset of instability results
from the steep increase of the electrostatic force ∼ φ2/h2 where φ is the applied
potential and h is the membrane thickness. This arises because the system is open:
as h decreases, the thinning leads to charge transfer between a battery and the plates
of the capacitor required to maintain the fixed value of the potential drop φ ∼ σh =
const. Mathematically, charge transfer is controlled by the term −qφ relating the
thermodynamic potentials of the open and isolated capacitor (equation 4). Thus,
both the charge density and the attractive force increase unrestrainedly as the plates
approach one another. At a critical voltage, no increase of the elastic repulsive force
can compensate for the catastrophic increase of the electrostatic attraction, leading
to the potential-driven instability first noted by Crowley [8].
In an isolated system this pathway to instability is forbidden. When isolated, the
total plate charge is fixed and uniform system thinning cannot increase the attrac-
tive force, which remains constant, ∼ σ2. Instability can only result from nonuniform
redistribution of the charge density in the plane of the membrane capacitor with an
associated inhomogeneity in h. Thus we are led to consider the possible instability
of an isolated flexible membrane capacitor. For φ-control we solved this problem for
electrolyte charge distributions that satisfy the Poisson-Boltzmann equation [23].
Here we consider a simplified example, a capacitor in contact with a “perfect con-
ductor,” i.e. εsolvent = ∞, so that the membrane surfaces are isopotentials, a reliable
approximation even for dilute electrolytes [23].
Consider the parallel-plate membrane capacitor with its midplane at z = 0 and
an unperturbed thickness (when q = 0) equal to h0. Charging the membrane to
charge density σ yields an electrostatic pressure which gives rise to membrane com-
pression. Introducing the thinning coefficient α = h/h0, with h=h(σ) the average
thickness of the compressed membrane, the total energy of the uniform slab is
W0 = Wd,0 + We,0 , where Wd,0 =
1
2
Ks(α − 1)2 and We,0 =
h
2εε0
σ2, (13)
the two terms are the harmonic approximation to the deformation (stretching-
compression) energy and the electrostatic energy respectively, Ks is the stretching
modulus and the index “0” refers to a uniformly deformed membrane. The equi-
librium membrane thickness h(σ) = h0α(σ) is found from the condition ∂αW = 0
leading to the thinning coefficient
α(σ) = 1 −
1
3
(
σ
σcr
)2
, (14)
where
σcr =
√
2Ksεε0
3h0
, (15)
404
Negative capacitance and instability at electrified interfaces
which is interpreted in what follows. The transmembrane potential drop and the
corresponding inverse differential capacitance are
φ =
1
εε0
σh(σ) =
1
εε0
h0σ
[
1 −
1
3
(σ/σcr)
2
]
(16)
and
C−1
σ (σ) =
dφ
dσ
= C−1
0
[
1 − (σ/σcr)
2
]
, (17)
where
C−1
0 =
1
εε0
h0
is the inverse capacitance of a capacitor with the fixed gap h0. Equation 17 reveals the
meaning of σcr. It is the charge density where the differential capacitance Cσ becomes
infinite; Cσ is negative for |σ| > σcr. The corresponding membrane thickness is
h(σcr) =
2
3
h0 ,
at σ = σcr the membrane has thinned by ∼ 33% , a value typical of “relaxing
gap” capacitor models [13,22,23,26,27,29]. The membrane is stable relative to virtual
uniform compression (thinning) under σ-control (i.e. assuming a uniform surface
charge density) for all σ including the range where C < 0. This is verified from
equation 13:
∂2
ααW |σ =
Ks
α
> 0. (18)
We now focus on the energy change, ∆W , for an isolated membrane capacitor in
response to a small charge increment, ∆σ, assuming σ-control. Using the relation
∂2
σσW (σ) = C−1(σ)
we find:
∆W = φ(σ)∆σ +
1
2
C−1(σ) (∆σ)2. (19)
This equation has important consequences. First we virtually separate the membrane
capacitor into equal patches I and II, each of area A/2, and permit the charge
∆q = A∆σ/2 to flow from I to II. For simplicity let us neglect boundary effects
and assume that each charge density is uniform (σ1 = σ−∆σ and σ2 = σ+∆σ) and
that the patches deform independently, i.e. two membrane capacitors in parallel, not
elastically coupled. Since the potential is constant in the plane of the membrane, we
find from equation 19 that the total energy change is
∆W12 =
A
2
∆W1 +
A
2
∆W2 =
A
2
C−1 (∆σ)2. (20)
Thus ∆W12 is negative if C < 0 for the chosen σ. Put differently, C < 0 provides a
driving force leading to a nonuniform charge distribution and membrane deformation
if the artificial σ-control restrictions are relaxed.
405
M.B.Partenskii, P.C.Jordan
Figure 1. Membrane slab under the effect of symmetric undulations.
The energy penalty results from the continuous transition between the properties
of the two membrane patches and can be described as a linear interfacial tension. It
is proportional to the length of the border between them and, for large A, is negli-
gible relative to ∆W12, equation 20. Thus this charge density redistribution and the
corresponding non-uniform deformation of the membrane is possible energetically
for those charge densities that lead to NC assuming σ-control. The appearance of
C < 0 in treatments that presume a uniform charge density indicates that the sys-
tem is unstable. This result is similar to the thermodynamic arguments of Nikitas
[19] who considered equilibrium conditions between two separate surface phases. We
will now show that the prediction of NC under σ-control indicates the possible for-
mation of the inhomogeneous phase under q-control. In our virtual experiment we
assumed that the charge density is uniform in each patch. Releasing this restriction
provides other pathways for transition to a non-uniform state. Consider, for exam-
ple, membrane stability relative to symmetric undulations (illustrated in figure 1),
the harmonic variation of membrane thickness:
h(x) = h + 2u cos(kx),
where u is the amplitude of the undulation of the membrane surfaces; the corre-
sponding “lower” and “upper” interfaces are described by the equations
zu,l(x) = ± z0(x), where z0(x) = h/2 + u cos(kx).
This problem has been discussed previously for φ-controlled systems (see [8,23,26]
and references therein). Now we consider a q-control environment and fix the total
charge density q = σA.
Some aspects of the solution procedure should be stressed.
1. Unlike under φ-control, under q-control the transmembrane potential drop
V is not fixed by the external source (battery, potentiostat). However, the
conductive surfaces are still equipotentials and the potential φq is constant on
the membrane plane.
406
Negative capacitance and instability at electrified interfaces
2. The value of φq depends on both the original charge density σ of the unper-
turbed membrane and the parameters u and k, characterizing the undulations.
3. φq is then determined as follows:
(a) The solution for a fixed but arbitrary φ determines the potential v(x, z)
within the membrane [1,16,23].
(b) The equation σ(x) = εε0∇nv(x, z0(x)) determines the local charge density
σ(x), where ∇n is the normal derivative taken at the interface z0(x).
(c) The total interfacial charge q̃(V ), is found by integrating σ(x) over the
interface with a weighting factor, 1/
√
1 − [∂xz0(x)]2, that accounts for
the membrane stretching associated with undulations.
(d) φq is determined from the condition q̃(V ) = σA:
φq =
σ
εmε0
h
(
1 −
k coth (kh/2)
h
u2
)
(21)
with h = h(σ) = h0α(σ). The externally fixed potential φ in the equations
for φ-control can now be replaced by φq, which completes the solution of
the problem for q-control.
The membrane’s electrostatic energy is then
W q
e = W q
e,0 + W q
e,u , (22)
where
W q
e,0=
σ2
εmε0
h(σ) (23)
is the energy of the uniform membrane slab and
W q
e,u = −W q
e,0
ku2
h(σ)
coth
[
kh(σ)
2
]
(24)
is the undulatory contribution. The onset of instability is determined by competition
between the decrease of the electrostatic energy, equation 24, and the corresponding
increase in membrane deformation energy averaged in XY-plane, Wd,u. For simplici-
ty, we consider small k (the long- wavelength limit) kh � 1. Similar to [23] where a
slightly different form of Wd,0, equation 13, was used, we can represent Wd,u as
Wd,u ∼
Ksu
2
h2
0
[
1 + O
(
(kh)2
)]
. (25)
Higher order terms in (kh)2 arise from surface tension and bending contributions
to the elastic energy and are neglected. They are analogs to the non-uniform inter-
facial contributions of the previous example. In the same limit equation 24 can be
represented as
W q
e,u ∼ −2W q
e,0u
2
[
1
h(σ)2
+ O
(
(kh)2
)]
. (26)
407
M.B.Partenskii, P.C.Jordan
The uniform distribution becomes unstable when
Wu = Wd,u + W q
e,u 6 0.
Substituting equations 24 and 14 we find the condition of instability:
| σ| > σcr ,
which is equivalent to
C−1
q 6 0.
Our thought experiment presumed that the conditions for σ-control , i.e. uniformity
of σ, could be arbitrarily relaxed anywhere within the negative C domain. In reality
there is no way to enforce uniformity when the system is unstable; thus the distribu-
tion spontaneously becomes inhomogeneous at the edge of this domain with critical
point σ = σcr, where
C−1
q (σcr) = 0, (27)
the transition actually occurs a bit earlier, at a point roughly determined by a
Maxwell construction [27].
We have analyzed a simplified model corresponding to a concentrated electrolyte
(Debye length λD → 0). Further analysis based on our previous work shows that
equation 27 also determines the onset of instability for finite λD [23]. This result
is valuable on its own. As already indicated the Poisson-Boltzmann approximation
and other local statistical models do not predict NC (see [10,27,28] and references
therein). Equations 11 and 12 show that for these models the rate of gap contraction
with the charging, d′(σ), is always less than 1/|σ| and thus Cσ is always positive.
Consequently, this class of models would not satisfy the criteria suggested in [11]
linking the model quality to the appearance of a Cσ < 0 domain. Our results indicate
that adding another relaxation mechanism immediately leads to the appearance of
a Cσ < 0 domain and instability. This further illustrates that anomalies are more
typical than expected based on purely ionic models with immobile charged plates.
In previous analysis [22] we considered two elastically coupled membrane capaci-
tors, with the extra term in the deformational energy ∼ α(u1 − u2)
2 accounting for
the non-uniformity penalty (differential thinning of the patches). Depending on the
coupling constant α, this system could exhibit NC before transition to a nonuniform
state. Our present discussion implies that such a picture is unrealistic. In terms of
the first model, the constant α must be proportional to the width of the transition
region relative to the area of the patches A and thus can become infinitesimal if A is
sufficiently large. In addition, as shown in the second example, the non-uniformity
contributions vanish at small k which provides a reasonable pathway for the onset
of instability.
Finally, we reiterate the major difference between q- and φ-control for the onset
of instability. Under φ-control (an open system connected to a potentiostat) the
stability is lost simultaneously for both uniform deformation and undulations [8,
23]. In contrast, a q-controlled (isolated) system is always stable with respect to
uniform deformation (see equation 18 and the corresponding discussion) and only
loses stability in making a transit to a nonuniform state.
408
Negative capacitance and instability at electrified interfaces
4. Perspectives for further study
It is by now well established that the capacitance can be negative for uniformly
charged surfaces under the artificial conditions of σ-control. Many statistical ionic
models have demonstrated this anomaly (see [6,11,31] and references therein). While
in the 1980s many would have viewed such predictions as model faults, the most
recent view [6,11] represents a dramatic change of mind, and it is even suggested
that the prediction of C < 0 must be considered as a criterion validating an ionic
model [11] rather than an imperfection. We do not disagree. In fact, it accords
with our observation [10,13,22,27] that such predictions are typical of models jointly
accounting for various contributions (electronic, ionic, etc.) to the charging induced
relaxation of the effective gap of the interfacial capacitor.
What is the physical significance of such predictions? Our analysis indicates
that they imply an instability with respect to a transition to an inhomogeneous
state. To analyze the consequences, the artificial assumption of a uniform surface
charge distribution must be dropped, given that in a real metallic electrodes the
electrons are free to move and thus surface charge density can become, at least in
principle, laterally non-uniform. This non-uniformity can be even more pronounced
in soft media such as lipid bilayers. The value of the models predicting this anomaly,
emphasized in [11], is even greater since they are candidates for analyzing such an
instability.
We must re-emphasize that in our usage the terms “instability” and “transition”
are not related to real interfacial critical phenomena. While the phase transition
actually occurs before Cσ becomes negative, we permit the system to enter this
domain by artificially maintaining σ-control. Relaxing the uniformity constraint at
any σ within the NC region leads immediately to a transition to an inhomogeneous
state of fixed q, q = σA. Although the initial (σ-controlled) state is artificial, the
final stable inhomogeneous phase (if it exists) is real since the equilibrium state is
unique. Thus, our approach is useful for testing and developing statistical models
that describe both uniform (C > 0) and inhomogeneous (regions with C < 0 under
σ-control) phases.
Predicting instability does not necessarily imply that the model describes the
formation of a new stable inhomogeneous phase. As the transition can be accompa-
nied by a substantial local increase of the charge density and a corresponding local
increase of the ionic density in the EDL, a model must have a stabilization mecha-
nism that interrupts the propagation of instability. This would permit the formation
of a new stable phase before the condition of “ideal polarizability” is broken and
interfacial charge transfer occurs. Ionic size and correlation effects in the electrolyte
should be of importance here.
Although the condition C < 0, obtained for a primitive ionic model of the EDL,
can be an important factor leading to instability, it is not a necessary condition.
Even with ionic models that by themselves do not lead to instability (such as GCS
model) the addition of other mechanisms of relaxation, such as a displacement of
the “electronic plate” of the interfacial capacitor, can lead to Cσ < 0 and thus
409
M.B.Partenskii, P.C.Jordan
trigger the instability [10,22,28]. In other words, this anomaly should be even more
commonplace than is implied by ionic model studies.
Another important question is a comparison of phase transitions predicted for
the open and the isolated systems. While in the first case the transition can be
accompanied by charging the electrodes, in the second case the lateral variation of
charge keeps the total charge fixed. Finally, the inherent inhomogeneity (roughness)
of an electrode surface (especially with respect to solid electrodes) must be con-
sidered. The effect of the roughness on the equilibrium properties of EDL is well
established [17], and its possible effect on the surface phase transition can also be a
promising field for further research.
Acknowledgement
Work supported by a grant from the National Institutes of Health, GM28643.
M. B. P. wishes to thank Dr. V.J.Feldman for many fruitful discussions and Professor
A.M. Brodsky for suggesting the possibility of charge density-type phases in double
layers (private communication, ∼1988).
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Негативна ємність та нестабільність біля
заряджених поверхонь: уроки, що випливають з
вивчення ємності мембран
М.Б.Партенський, П.К.Джордан
Відділення хімії університету Брендіз,
Волтхем, США
Отримано 14 вересня 2004 р.
Представлено короткий огляд моделей, які передбачають негативну
ємність C. Обговорюється роль цих моделей у явищі електричного
контролю. Ми ще раз показуємо, що розрахункова ємність подвій-
ного шару може бути негативною завдяки σ-контролю – штучній
конструкції, яка вимагає однорідного розподілу густини поверх-
невого заряду електрода, σ. Разом з тим, тільки загальний за-
ряд q (або усереднена густина поверхневого заряду σ) може бути
експериментально зафіксованою при дослідженні ізольованої ко-
мірки (q-контроль), Для значень σ, де C стає від’ємною в умовах
σ-контролю, перехід до q-контролю (тобто релаксація латеральної
густини розподілу заряду шляхом фіксації її середнього значення
до величини σ) веде до нестабільності однорідного розподілу і
переходу до неоднорідної фази. В якості ілюстрації розглядаєть-
ся модель “мембранного конденсатора”. Ця точно розв’язувана
модель допускає як однорідну, так і неоднорідну релаксацію і таким
чином допомагає продемонструвати зародження і деякі важливі
риси нестабільності. Коротко обговорюються можливості подаль-
ших досліджень.
Ключові слова: електрохімічні інтерфейси, нестабільність та
фазові переходи, електронний подвійний шар, ємність
PACS: 68.35.Rh, 68.35.Md, 82.45.Mp, 82.45.Rr, 82.45.Uv, 68.08.-p,
73.30.+y
412
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