Notes on 22.76/3.S75 Ionics and Its Applications Ju Li, MIT, December 21, 2021 1 Electrostatics and Electrochemical Potential 3 1.1 Coulomb Explosion argument: bulk electroneutrality principle ........ 6 1.2 Parallel Capacitor ................................. 7 1.3 Equalization of Price: electronic versus ion/atom pair ............. 9 1.4 Electrocapillarity and Point-of-zero-charge (PZC) ............... 16 2 Electrons and Valence 20 3 Solvation Model and Debye–Huckel Equation 27 4 Electrode Kinetics 33 4.1 Butler-Volmer and Exchange Current Density ................. 34 4.2 Tafel Approximation ............................... 38 4.3 Exchange current density and limiting curent density ............. 38 5 Long-range Mass Transport 39 1
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Pt and Ru does have different PZCs (generally there is almost linear relationship between
18
the workfunction W of a noncrystalline metal with its PZC). And so we will have
〈φ〉Pt − 〈φ〉aq 6= 〈φ〉Ru − 〈φ〉aq (1.71)
and qM(Pt) 6= qM(Ru), due to, say different preference for Cl− adsorption on the surface.
While the treatment above was specific to electrons, it can be generalized to the sweepage
of any thermodynamic potential for ions or neutrals. The PZC is also similar to the zeta-
potential in colloidal science. What people find is that as one tunes the pH of the solution,
the colloidal particles could turn from positively charged at low pH, to negatively charged at
high pH. One could think of the H+ in solution as “electrons” that’s strongly attached to the
colloid, and the rest of the ions as the compensation. In this case, pH is like the potential
U here, and there will be a charge-neutral pH where colloidal particles are the easiest to
aggregate and lead to flocculation.
If we think of EDL capacitors as energy-storage device, then it might be curious why the
free-energy γ is turning downward as −Cd(U−UPZC)2
2instead of turning upward, since we
usually think of capacitor as accumulating positive potential energy. This have to do with
the particular way Gibbs free energy is defined
G = F − work (1.72)
For example, in a mechanical system, if the external pressure is P ext,
G = F + P extV = F0 +B(V − V0)2
2V0
+ P ext(V − V0) (1.73)
and so while at equilibriumB(V − V0)
V0
+ P ext = 0 (1.74)
we have F − F0 = B(V−V0)2
2V0= (P ext)2V0
2B, the Gibbs free energy is actually going down by the
same amount:
G = F0 −(P ext)2V0
2B. (1.75)
So in fact, the energy stored in the supercapacitor is really +Cd(U−UPZC)2
2.
19
Chapter 2
Electrons and Valence
The electronic structure theory makes chemistry a much deeper science. For example, the
notion of acid and base has three famous, gradually more inclusive but mutually consistent
definitions: by Arrhenius in 1884 (won Arrhenius the Nobel Prize in Chemistry in 1903)
with the help of Ostwald, by Brønsted and Lowry, and by Lewis (1923). The Arrhenius
scale is obsessed with liquid water as the solvent, the Brønsted-Lowry scale is obsessed with
proton transfer and both Arrhenius and Brønsted-Lowry are “proton cults”, but the Lewis
acidity/basicity is much more general. The Lewis octet rule for sp-bonded main-group ele-
ments (s-block: groups 1 and 2, p-block: groups 13 to 18), and the Lewis acid/base concept
have direct electronic frontier orbital correspondences, which can be readily generalized to
group 3 to 12 d-block transition metals as well. Frontier moleular orbitals (MO) are those
that control a molecular fragment (could be anion/cation or charge neutral)’s reactivty. The
most famous frontier orbitals are the highest occupied molecular orbital (HOMO) and lowest
unoccupied molecular orbital (LUMO). An electron lone pair (LP) is quite often HOMO,
doubly occupied (if singly occupied, it’s called a radical, and can’t be called lone pair), and
sterically unhindered. Reactivity derived from this is called Lewis base / nucleophile, be-
cause it attracts other molecular fragments with LUMO of similar electronic energy (Lewis
acid / electrophiles) to overlap and bond with it. This kind of quantum-mechanical res-
onance and overlap between HOMO(Lewis base) and LUMO(Lewis acid) can be so strong,
that pieces of the original Lewis acid or the original Lewis base could be torn off after
the reaction, causing ionization (although it does not necessarily have to, to be a Lewis
acid/Lewis base pair). And this auto-ionization often happens inside liquid solutions, im-
parting mobile ions (ionic strength), and controlling solubilities and electrolyte transference
20
numbers, by affecting the solvation atomic structures. For all these reasons, it is important
to talk about the meaning of acids and bases, radicals, and redox processes, which are all
related but distinct concepts.
Figure 2.1: AlCl3 and BBr3 are Lewis acids but not Brønsted–Lowry acid, i.e. no proton istransferred. Taken from Web.
While the Lewis acid-base reactivity can be more general, the following equations covers
most of the discussions:
A + B = A− B adduct or complex = cb(A) + ca(B) (2.1)
where A stands for acid of the reaction, B stands for base of the reaction, cb stands for con-
jugate base - what becomes of A, and ca stands for conjugate acid - what becomes of B,
after the reaction. A, B, cb(A), ca(B) can all be charged (cation or anion) or charge-neutral.
The acid-base equilibria is about very localized electrons and orbitals on molecules/ions.
Later, with delocalized electronic states (e.g. metals and semiconductors), one can further
develop solid-state chemistry as a limit of really large clusters.
The most rudimentary acid-base language is the Arrhenius definition, which is just for aque-
ous (liquid-phase H2O) solvent. Any substance that increases the hydronium concentration
[H+] by reacting with H2O is called Arrhenius Acid (AA), any substance that increases the
hydroxide concentration [OH−] by reacting with H2O is called Arrhenius Base (AB). We
often use molarity (1 molar≡ 1 M ≡ mol(solute)/Liter(solution)) to denote concentra-
tion. But occasionally for ease of recipe-making we also use molality (1 molal≡ 1 m ≡mol(solute)/kg(pure solvent)) which can have a nonlinear relationship with molarity at high
21
concentrations. Since 1 kg(pure H2O solvent) corresponds to 1000/(2 × 1.008 + 15.999) =
55.5093 mol(H2O), assuming the addition of AA/AB and perhaps even some metal salts
does not change the volume (this is indeed a big if), then 1 M of [H+] (pH=0) roughly
corresponds to 1 H+ per 55 unsplit H2O molecules. Even if the proton grabs one H2O to
form H3O+ (the actual hydronium - this is also a Lewis adduct), and also has three water
molecules surrounding the H3O+ to form the solvation shell, there are still 51 free water
molecules left! So 1 M is not very high concentration yet, and one can certainly go to negative
pH≡ − log10([H+(aq)]/M) if one wants! But on the other hand, something like 10M should
be quite high concentration aqueous solution, since one may start to run out of relatively
free H2O solvent molecules. Also, the equilibrium constant Keq of the auto-ionization:
H2O(aq) = H+(aq) + OH−(aq) (2.2)
is a temperature dependent quantity
a[H+(aq)]a[OH−(aq)]
a[H2O(aq)]= Keq(T ) (2.3)
where a stands for activity - a proxy function of the chemical potential of the species in the
bracket - and in the dilute limit a for the solutes can be substituted for by the mole fraction.
So [H+(aq)] and [OH−(aq)] are constrained. AB would increase [OH−(aq)] and pH, while
AA would increase [H+(aq)] and decrease pH. However, [H+(aq)][OH−(aq)] ≈ 10−14M2 is
only approximately true near room temperature (indeed the concept of “room temperature”
varies from 300K=26.85 Celsius to 25 Celsius to 20 Celsius, and generally means “not very
well controlled experimental temperature” condition).
Brønsted-Lowry acid (BLA) and Brønsted-Lowry base (BLB) also talk about the proton H+
and is part of the “proton cult”, but not necessarily in liquid H2O solvent:
BLA + BLB = cb(BLA) + ca(BLB) (2.4)
where BLA donates a H+ to BLB. For example, acetic acid CH3COOH is a polar solvent,
and can dissolve oils. Methylamine CH3NH2 is a gas. The two can react like
the three-LPs hydroxide anion (OH−). The charge-neutral ·OH is the second most
oxidative species in nature after fluorine, and can be generated by electrochemical
means to remove organic wastes in water [8]. So radicals can be associated with
EA/IP operations, but not always, for example the breakup of Cl2 molecule (with 6
LPs) into Cl2 = Cl· + ·Cl, each with 3 LPs and 1 radical, does not involve redox.
In the interaction of radiation with matter, plasma chemistry, etc., there is a lot of
radicalization also.
3. Promotion, hybridization: an isolated atom has spherical symmetry, and s-orbital
φl=0(x) = s(x) has lower energy than p-orbitals φl=1(x) = px(x), py(x), pz(x). Hy-
bridization means some linear combination of atomic orbitals (LCAO) into
h1(x) = s(x) + px(x) + py(x) + pz(x) (2.6)
h2(x) = s(x) + px(x)− py(x)− pz(x) (2.7)
h3(x) = s(x)− px(x) + py(x)− pz(x) (2.8)
h4(x) = s(x)− px(x)− py(x) + pz(x) (2.9)
These sp3 hybridized orbtials are higher-energy mixed states, and are thus called pro-
moted. For example, a carbon atom = 2.4 (1s22s22p2), when promoting its own 4
electrons to h1(x), h2(x), h3(x), h4(x) would have four radicals and quite high energy
(the promotion energy), but when forming CH4 would satisfy the octet rule for carbon
(and doublet rule for hydrogen), in a tetrahedral geometry. Resonance
4. Resonance, delocalization/localization:
5. Lone pair to bond transition, which is the essence of the Lewis acid/base concept.
In the Lewis perspective, we should focus on what happens to the electronic LP, in-
stead of on the proton, because the story of the electronic LP (who provided it, who
accepts it) is a main storyline of chemistry. A Lewis Acid (LA) is a cluster that will
increase/accept #LPs (note, not increasing #radicals, as most chemical reactions we
are familiar with do not involve radicals), because LA loves electronic LP (electrophile
before the reaction). A Lewis Base (LB) is a cluster that will decrease/donate #LPs,
because LB loves electron-barren proton or other LP-barren entities (at least in the
spatial direction of bonding), and thus is called nucleophile before the reaction. Lewis
base is most often also a BLB, but a Lewis acid (such as AlCl3, BBr3) does not have
to be a Brønsted–Lowry acid as it does not necessarily eject protons.
25
Figure 2.2: Acetic Acid is a Lewis Acid. Methylamine is a Lewis Base. Lewis Acid ejects aLP-poor subcluster (H+), leaving behind a LP-rich residual (conjugate base). Lewis Base,which originally is LP-rich, accepts a LP-poor subcluster, forms a Bonding Pair (BP) usingits orginal LP electrons, and becomes LP-poorer (conjugate acid).
The mnemonics for Lewis acid/base is “A|A b|d” where | is like a mirror, where Acid is
electron-lone-pair Acceptor (electrophile), and base is electron-lone-pair donor (nucleophile).
Crudely speaking, electrophile acid often ejects a nucleus, like an angry wife ejecting the
husband, keeping the children to herself. While nucleophile base often takes in such ejected
nucleus, because she has excess children (LP) that can be shared. The exchanged nucleus can
also be subcluster (or even the entire LA cluster itself), and certainly does not have to be a
proton. A deal can be struck between LA and LB parties to exhange the nucleus/subcluster,
with associated electronic-configuration changes (mainly LPleftrightarrow bonding pair BP,
and with associated collective relaxations), in the thermodynamically favorable direction,
because well, chemistry is chemistry, and the same nucleus/subcluster may find a better fit
after the exchange.
26
Chapter 3
Solvation Model and Debye–Huckel
Equation
At equilibrium, all the free agents are seeking maximum marginal happiness (or minimum
price). Given
pi = zieφ+ µi + kBT ln ai (3.1)
and in dilute solution, we have
pi ≈ zieφ+ µi + kBT ln ci/1M (3.2)
we see that if there is meso-scale φ(x) variation, we need to have
ci = c∞i e−zieφ/kBT . (3.3)
which is the same as distribution of O2 molecules on Earth surface despite mO2gh gravita-
tional potential energy.
But this would surely violate electroneutrality! (cation concentrations increase, while anion
concentrations decrease)
The total charge density therefore would vary as
ρfree(x) =∑i
ziec∞i e−zieφ/kBT , (3.4)
27
and the equation
∇2φ(x) = −∑i ziec
∞i e−zieφ(x)/kBT
ε(3.5)
where c∞i is the far-field concentration at φ = 0 (electrostatic potential reference) that must
also satisfy bulk charge neutrality.
(3.5) is called the Poisson-Boltzmann equation. It is a nonlinear PDE and generally requires
a numerical solver. But assuming
eφ(x) kBT (3.6)
we can linearize the equation, and get
∇2φ(x) =e2∑
i z2i c∞i
kBTεφ(x) (3.7)
The ionic strength is defined as
I ≡ 1
2
∑i
z2i c∞i (3.8)
Thus 0.1M MgCl2 would have
I =1
2(0.1× 4 + 0.2× 1) = 0.3M (3.9)
and we can define Debye length
λ ≡√kBTε
2Ie2=
3.04A√I
(3.10)
So, for 0.1M MgCl2 in water, we have λ = 5.55A.
Truly, 5.55A is very short and it is not clear that the continuum approximation would work
well. However, there are more extreme examples than this, which is a metal! A metal can be
thought as a high density soup of electrons and holes. The carrier concentration in copper is
1e/11.81E-30 = 141 M, and the equivalent I = 70.3M (partly thanks to the atomic density
of copper). Thus, λ = 0.36A. Because λ a0 the lattice constant, this is just saying that
electronic screening is so strong that it probably does not even make sense to talk
about ions in good metals like Cu. Now actually since pe include quantum effects also,
one should really use the Thomas–Fermi screening model instead of Debye–Huckel screening
that uses Boltzmann statistics, but the basic conclusion holds, which is that a metal is
too good an “electrolyte” (for electrons, not ions) to even talk about ionization. Now in a
28
semiconductor, where the free carrier density is much lower, it still makes sense to talk about
ionization, e.g. a Boron dopant capturing a valence band edge electron and becomes an
anion B−Si (with paired electron) embedded in Silicon lattice, while leaving a free hole [h] in
Bloch state inside the silicon.
So
∇2φ(x) =φ(x)
λ2, x 6= 0 (3.11)
can be solved for the parallel-plate and point-charge situations also.
First imagine a flat electrode at x = 0, and electrolyte in x > 0:
d2φ
dx2=
φ(x)
λ2, x = (0,+∞) (3.12)
the stable solution is obviously
φ(x) = φ0e−x/λ, E(x) =
φ0
λφ0e
−x/λ, (3.13)
with areal charge densityρ
A= ε∇ · E =
εφ0
λ(3.14)
at x = 0, and volumetric screening charge density
ρfree = −ε∇2φ = −εφ0
λ2e−x/λ (3.15)
It is clear that this is effectively a double-layer capacitor, with equal and opposite charge
− ρA
at the mean distance
d = λ. (3.16)
Thus, the effective capacitance would be
C
A=
ε
d=
ε
λ=
√2Iεe2
kBT(3.17)
that fully screens out whatever surface charge density we put into the electrode.
In the point-charge case, we have
∇2φ(x) = −zeδ(x)
ε+φ(x)
λ2, x 6= 0 (3.18)
29
φ(x) =zee−r/λ
4πεr(3.19)
since
∇2φ = r−2∂r(r2∂rφ) =
ze
4πεr−2∂r(r
2(−r−2e−r/λ − r−1e−r/λ/λ))
= − ze
4πεr−2∂r(e
−r/λ + re−r/λ/λ)
= − ze
4πεr−2(−e−r/λ/λ+ e−r/λ/λ− re−r/λ/λ2)
=ze
4πεr−1e−r/λ/λ2
=φ
λ2. (3.20)
However, just like in the solvation-shell theory, in reality there is excluded volume that
counter-ions cannot get into. Imagine we have a cation z+e at r = 0, and the closest that
the anion can get to is rcut = r+ + r−, where r+, r− are the cation and anion ionic radius, so
rcut is a hard core. In this case, the solution needs to be modified, as,
φ(r) =
A−z+e4πεrcut
+ z+e4πεr
, r < rcut
A exp((rcut−r)/λ)4πεr
, r > rcut
(3.21)
where the total screening charge must still be −z+e:
−z+e =∫ ∞rcut
4πr2dr(−ε)A exp((rcut − r)/λ)
4πεrλ2= −
∫ ∞rcut/λ
Ardr exp(rcut/λ− r)
= −A(rcut/λ+ 1), (3.22)
and so
A =z+e
rcut/λ+ 1(3.23)
so we get the excess stablization electrostatic potential due to screening charge is
A− z+e
4πεrcut
= − z+e
4πεrcut
(1− 1
rcut/λ+ 1) = − z+e
4πεrcut
rcut
rcut + λ= −z+e
4πε
1
rcut + λ, (3.24)
which is an eminently sensible result after such laborious derivations.
But remember this stabilization energy is equally shared between the central cation and
surrounding anions, so there is a factor of 1/2, and the excess stablization energy can be
30
identified as
kBT ln γ+ = −z2
+e2
8πε(rcut + λ)(3.25)
We also have, complimentarily for the anion,
kBT ln γ− = −z2−e
2
8πε(rcut + λ)(3.26)
Generally speaking suppose the binary salt formula looks like Z+ν+Z−ν− , we have
ν+z+ + ν−z− = 0 (3.27)
(note that ν+, ν− > 0, while z− < 0) and we can define
ν ≡ ν+ + ν− (3.28)
with mean activity a± defined as
νkBT ln a± ≡ ν+kBT ln a+ + ν−kBT ln a− (3.29)
aν± ≡ aν++ a
ν−− . (3.30)
Given c M of Z+ν+Z−ν− , we will have
c+ = ν+c, c− = ν−c (3.31)
and so
a+ = γ+ν+c, a− = γ−ν−c (3.32)
so
aν± = (γ+ν+c)ν+(γ−ν−c)
ν− = (γ+ν+)ν+(γ−ν−)ν−Xν (3.33)
so we can also define mean activity coefficient
γν± ≡ γν++ γ
ν−− , (3.34)
to get
aν± = (γ±c)νν
ν++ ν
ν−− . (3.35)
31
Finally, given the Debye–Huckel result, we will have
ln γ± =ν+ ln γ+ + ν− ln γ−
ν= −
ν+z2+e
2 + ν−z2−e
2
8πεkBT (rcut + λ)ν= −ν−z−z+e
2 + ν+z+z−e2
8πεkBT (rcut + λ)ν
= − z+z−e2
8πεkBT (rcut + λ)(3.36)
where λ ∝ I−1/2.
The Debye–Huckel model works quite well for binary electrolyte in the dilute limit.
32
Chapter 4
Electrode Kinetics
In Chap. 1 we have shown the basic thermodynamic relation between a free electron in
the metal (or semiconductor) electrode of µe ≡ EFermi ≡ −eU ,
f(ε) =1
1 + eε−µekBT
=1
1 + eε+eUkBT
(4.1)
and ion/atom pair ∆G(T, P ) + kBT lnQ. The gist is: the electron (“child”) abandoned
behind in the metal is the outcome of an ionization reaction of the atom (living in some
phase) and jumping into perhaps another phase as an ion, resulting often in the ion in
phase and the electron child in another phase (unless auto-ionization like dopant in silicon).
This involves a change in the chemical identity of the atom. Such change in chemical
identity, specifically the chemical valence of some elements, causes a Faradaic current.
More generally, we can replace the word atom by any molecular fragment.
The definition of U thus must be covariant with the definition of ∆G(T, P ) for the
ion/atom pair. If U = 0 is taken to be the vaccuum (4.44 eV above SHE with aqueous
electrolyte), then the definition of ∆G(T, P ) ≡ price(ion)−price(atom), when talking about
price(ion), need to talk about the true electrochemical price of the ion living in the liquid, and
thus would have to include the electrostatic potential average of the liquid. (price(atom) does
not care, but price(electron) does care about what its environmental electrostatic potential
average is).
If U = 0 is taken to be the average electrostatic potential of the liquid phase, then ∆G(T, P )
just contains the chemical part of the ion’s price in standard condition. This is a nice sim-
33
plification conceptually, because then the entire ∆G(T, P ) + kBT lnQ is chemical (local)
and free of electrostatics. Then, under this definition, the environmental electrostatic po-
tential average of the metal electrode with respect to the liquid it is in contact with need to
be added to the price(electron) (but the quantum and the thermal are also there inside the
price(electron), which are also local). This is the most “physical” definition, because then
the equilibriating process (when the price does not match) by a small change in the EDL
can shift this environmental electrostatic potential difference, without significant change in
the bulk composition. This occurs because 1ML of Mg dissolving can cause 36.19×2 =72.38
Volt shift, if all the electrons left behind stays on the metal surface (which they are likely to
do). 0.01ML of anything dissolving is not observable at bulk scale, and thus the equilibra-
tion process can establish the equilibrium (and the open-circuit potential) with “barely any
visible” change.
If U = 0 is taken to be the SHE, then the electron energy ε is in reference to the Fermi
level of the bubbling Pt (but could be any other metal, because the ion/atom pair we are
actually talking about is H+/H2
2, not Pt - Pt is just a host, although a kinetically facile
one) in contact with the same liquid electrolyte, and ∆G(T, P ) would actually have the
µH+ −µH subtracted off, for every electron left behind that we are counting. So you can say
the ∆G(T, P ) in this metric to be not “pure”. But measurement wise, it is the most easy
thing to do.
4.1 Butler-Volmer and Exchange Current Density
Once the thermodynamic picture is clear, the kinetics is easy. Some atoms strips the electrons
and jump into ocean (anodic), and some ions climb out of the electrolyte onto the metal and
recombine with the electron (cathodic). The total curent density on electrode surface can
be net anodic (by convention, this give positive i [Amp/m2] – in other words, if the current
vector (which is oppostite to the electron flow vector) is pointing into the electrode), or
net cathodic (by convention, this gives negative i – in other words, if the the current vector
(which is oppostite to the electron flow vector) is pointing out of the electrode). Consider
the following paradigm:
R = O + e− (4.2)
where in this example, both R and O are living inside the liquid electrolyte. (this does not
have to be case, but we are using this as pedagogical example).
34
The actual microscopic ionization process could be very complicated. To get close enough
to the metal electrode, R may need to break up its solvation shell, to get onto the inner
Helmholtz plane (IHP) (see Fig. 3.2 of [4]) and be directly adsorbed. Such ad-ion configura-
tion could be labelled Rs where superscript s is a surface site. The same for Os, so generally
the process could be R → Rs → Os → O. It turns out that anions, because of weaker
interaction with its solvation shell (larger), are preferentially found in IHP than cations.
Or, perhaps electron transfer could happen when R is in the outer Helmholtz plane (OHP),
which is the distance of closest approach for a solvated ion that keeps its solvation number
intact.
For simplicity, let us take dilute liquid solution with known concentrations cR, cO near
the surface, and assuming the redox is happening for OHP situation, we have
ia = kacRe−Qa/kBT , ic = kccOe
−Qc/kBT (4.3)
where Qa ≡ E∗ − ER and Qc ≡ E∗ − EO+e are the microscopic energy barriers. Note
that EO+e contains the energy of both O and e−, separated by the EDL, and thus EO+e
will depend on U . If we take the liquid environmental electrostatic background to be zero,
then ER, being living fully in the liquid phase, does not depend on U . The saddle point
configuration in between, as the nuclide crosses the EDL, will have energy E∗ that depends
partially on U , but not as much as EO+e. So we can say that
∂ER
∂U= 0,
∂E∗
∂U= −e(1− β),
∂EO+e
∂U= −e, (4.4)
where 0 < β < 1 is called the symmetry factor.
At equilibrium potential U eq (for the known and fixed cR, cO), we have zero net current, so
kacRe−Qeq
a /kBT = kccOe−Qeq
c /kBT (4.5)
from our micro-kinetics model, or
Qeqa = Qeq
c − kBT lnkccO
kacR
. (4.6)
and so
E∗ − ER = E∗ − EO+e − kBT lnkccO
kacR
. (4.7)
The activated state cancel from both sides, and we see that at electronic equilibrium, we
35
have
0 = ER − EO+e − kBT lnkccO
kacR
, (4.8)
This is equivalent to our stated thermodynamic principle for dilute solution
0 = ∆G = ER − EO+e − kBT lnkccO
kacR
, (4.9)
if we recognize that the Gibbs free energy driving force ∆G contains configurational entropy
contributions kBT ln cO − kBT ln cR, similar to our stated thermodynamic driving force of
∆G = pR − (pO − eU) (4.10)
for net-anodic reaction. This is called detailed balance principle, or thermodynamics-
kinetics equivalence. The kinetic constants kBT ln kc, kBT ln ka, can be absorbed into ER,
EO+e as some kind of “vibrational free energy”, by normalizing with the constant prefactor
k0cref , in other words if we identify
pR ≡ ER + kBT ln kacR/k0cref , pO ≡ EO + kBT ln kccO/k0cref , Ee ≡ −eU (4.11)
so the prices include vibrational and configuratinal entropy terms, and by definition of U eq,
we have
0 = ∆G = pR − (pO − eU eq), (4.12)
that we have asserted in Chapter 1 without detailing.
We can therefore define the exchange current as
i0 = kacRe−Qeq
a /kBT = k0crefe−(E∗−pR)/kBT , (4.13)
i0 = kccOe−Qeq
c /kBT = k0crefe−(E∗−pO+eUeq)/kBT (4.14)
Note that E∗ above is right at U = U eq for the given cR, cO.
For U 6= U eq for the given cR, cO, there will be
η ≡ U − U eq, (4.15)
which is defined as surface polarization overpotential, and so
ia = i0ee(1−β)η/kBT , ic = i0e
e(−β)η/kBT (4.16)
36
so we derived the famous Butler–Volmer equation
i = i0[ee(1−β)η/kBT − e−eβη/kBT ] (4.17)
for electronic (voltage) out-of-equilibrium situations.
It is important to recognize that i0 is defined at U = U eq(cR, cO) and thus depends on both
cR and cO. For example, even though (4.13) appears to depend on cR only on first look,
through the
∆E∗ = −e(1− β)∆U eq = (1− β)∆kBT lncR
cO
(4.18)
dependence, one ends up having
i0 ∝ c1−βO cβR (4.19)
This turns out to be important in discussing polarization voltage partition between long-
range transport and short-range charge-transfer reaction in mass-transport limited corrosion
(page 386 in [4]).
For multi-electron transfer reactions, such as
R = O + ne− (4.20)
we will generally have
i = i0[eeαaη/kBT − e−eαcη/kBT ] (4.21)
with the anodic transfer coefficient αa and cathodic transfer coefficient αc:
αa + αc = n, αa ≡ n(1− β), αc ≡ nβ (4.22)
and the exchange current density would scale as
i0 ∝ cαaO c
αcR . (4.23)
Note that the above is not cαaR c
αcO that one might be confused about!
37
4.2 Tafel Approximation
There are three regimes in (4.21): exponential, linear, and somewhere in between. The
exponential regime occurs when eαη kBT , where one can ignore one of the two terms,
and get
log10 |i| ≈ log10 i0 +eα|η|
2.3026kBT(4.24)
(4.24) is called the Tafel Approximation. Since 2.3026kBT = 59.16 meV, this means if
α = 0.5, every 118 mV increase in surface overpotential would cause 10× the current density.
This approximation is generally only valid when |η| 100mV. This calls for semi-log plot,
where the voltage is linear, but the current is plotted in log10-scale.
When |η| 10mV, we can also approximate (4.21) as
i = i0neη
kBT, (4.25)
this calls for linear-linear plot, but in linear-log plot would appear as a deep cusp at η = 0
or U = U eq, since log is unbounded in the negative value side.
4.3 Exchange current density and limiting curent den-
sity
Butler–Volmer is very “near-sighted” and describes electron transfer across nm distance,
and associated chemical indentiy change. The fact that η is defined based on local chemistry
(at so-called x=0)
38
Chapter 5
Long-range Mass Transport
Electrode kinetics, aka, charge-transfer (CT) reaction at phase boundaries, is very short
ranged, as electron can only tunnel a short distance. After some reactions have happened,
the products need to be transported away, and the reactants are exhausted and need to be
resupplied, otherwise U eq would shift and the net reaction would stop. How the reactants
are resupplied and products transported away is the topic of this chapter.
In a fluid electrolyte, ion/molecule would generally have flux
Ji = ci(vi + vCM) (5.1)
where vCM is center-of-mass translation of the fluid, and vi is the average velocity of species
i with respect to the center-of-mass frame. Thus, by definition
∑i
micivi = 0. (5.2)
Note that ci has unit #/m3, Ji has unit #/m2/s.
Solving for vCM (such as by solving Navier-Stokes equation with knowledge of viscosity) is
the business of fluid mechanics,
ρ
(∂tvCM
∂t+ vCM · ∇vCM
)= ρg −∇p+ η∇2vCM (5.3)
39
while modeling vi requires chemical solution thermodynamics. We generally model
vi = Mi(−∇pi) (5.4)
note that Mi has the unit m/s/N, and
pi = zieφ+ µi + kBT ln ai (5.5)
is the electrochemical potential, and so in the dilute solution limit, there is
pi = zieφ+ µi + kBT ln ci/1M (5.6)
and
−∇pi = −zie∇φ− kBT∇cici
(5.7)
and so
Ji = −Micizie∇φ−MikBT∇ci + civCM. (5.8)
The first term is called migration (or drift), which is driven by the electrostatic part of the
price. The second term is called diffusion, where we identify
Di = MikBT (5.9)
as the Einstein relation, and the third term is called convective term.
and then the cation is also then freed from the anion equation. In this scenario, one can
think of the electrolyte as a superconductor, where everywere inside the liquid has the same
electrostatic background potential. This reactive cation is consumed/injected at different
electrode interfaces, so there is a concentration variation, and the equilibrium potential
would vary as a function of this liquid concentration right at the interface, that affects the
electrodic Faradaic process still. The mathematical treatment would certainly be simplier
with such excess supporting electrolyte.
52
5.5 Convective Mass Transfer
In reality, it is quite difficult to get rid of convection completely, even in 102µm spaces. The
ion flux at interface is often expressed as:
J · n′ ≡ Ji = k(c∞i − cinterfacei ) (5.81)
where k is the mass-transfer coefficient, with unit of m/s (i.e. velocity unit), and n′ is
the interfacial normal pointing from inside-the-electrolyte toward the electrode. Note that
cinterfacei is still ion concentration inside the liquid, but limiting toward the interface, and c∞i
is the ion concentration inside the liquid far away from the interface.
When there is no convection, the mass-transfer coefficient is of the form
k =D
L(5.82)
where L is the distance from c∞i to cinterfacei . So we see ∞ is not really infinity, just L!
When convection is turned on, there will be an enhancement of mass transfer,
k = ShD
L(5.83)
where Sh is the dimensionless Sherwood number, which is the enhancement over diffusive
transport.
This enhancement factor is generally a function of the Reynolds number (Re) and Schmidt
number (Sc):
Sh = f(Re, Sc) (5.84)
where
Re =ρvCML
η(5.85)
where ρ is fluid mass density [kg/m3], η is fluid viscosity [Pa · s], and v∞CM is the characteristic
fluid velocity away from the interface. Re is the ratio of the nonlinear inertial term in the
Navier–Stokes equation to the linear viscous term in fluid velocity. The larger Re is, the
more turbulent the flow is, and the more dominant the convective transport becomes.
53
The Schmidt number is
Sc =η/ρ
D(5.86)
It is the ratio of momentum diffusivity (by viscosity) to mass diffusivity. If the Stokes–Einstein
relation is correct, then
Di = MikBT =1
6πriηkBT (5.87)
where ri is the hydrodynamic (Stokes) radius of ion (this would include some tightly bound
solvent molecules in the solvation shell), then we can see that
Sc =6πrη2
ρkBT(5.88)
where for binary salt electrolyte:
r−1 =(z+ − z−)r−1
+ r−1−
z+r−1+ − z−r−1
−. (5.89)
and in the case of ν+ = ν−, just
r =r+ + r−
2. (5.90)
With a large Schmidt number, the momentum would diffuse faster than the mass, and this
also will boost the convective mixing. So generally there is scaling relation
Sh ∝ ReaScb (5.91)
where a, b typically ranges between 0.2 and 0.6. (5.91) is for forced convection, for example
in a flow battery where there is external pumping.
There is also natural convection driven by buoyancy due to temperature change or other
reasons. This happens in weather patterns and ocean streams. In electrochemical systems
with liquid electrolytes, electrodic processes could also alter the liquid mass density directly
(for instance if some heavy ion like Pb2+ is electroplated, and also the electrodic process can
also release or absorb heat, causing a net liquid mass density change (∆ρ). This leads to the
Grashof number (Gr):
Gr =gL3∆ρ
ρ(η/ρ)2(5.92)
which is the ratio of buoyancy inertia to viscosity. In natural convection, we will have
Sh ∝ GraScb (5.93)
54
So in the end we would get
Ji = ShDi
L(c∞i − cinterface
i ) (5.94)
when convection is dominant (more significant than migration and diffusion). The limiting
current is defined as
J limiti = Sh
Di
Lc∞i (5.95)
Consider the electrorefining of Cu. An impure copper alloy may have Ni, Ag, etc. inside.
Ag is more noble than Cu, while Ni is less noble. If we control the anodic voltage to be
just slightly higher than 0.337 V, the equilibrium voltage of pure Cu, then only Cu and less
noble metals may be dissolved into the solution. But if we make the cathodic voltage to
be only slightly lower than 0.337 V, then only Cu and metals more noble than that can be
deposited. Then, we can refine Cu from 99% to 99.99% pure.
Liquid water has kinematic viscosity η/ρ = (1mPa · s/1e3kg/m3) = 1e − 6m2/s, which is
faster than mass diffusivities. After adding 0.25 M CuSO4 and supporting electrolyte, the
density changed to ρ = 1094kg/m3 (Illustration 4.7 of [4]), and η/ρ = 1.27e−6m2/s. For the
electrodic current we are considering, ∆ρ = 32kg/m3, L = 0.96m, so the Grashof number
Gr =9.8× 0.963 × 32
1094× 1.27e− 62= 1.5724e+ 11 (5.96)
so the buoyancy inertia is truly quite large compared to viscous term in Navier-Stokes equa-
tion. The effective mass diffusivity is
D =kBT
6πrη= 5.33e− 10m2/s (5.97)
if we take r = 4.1A. So the Schmidt number is
Sc =1.27e− 6
5.33e− 10= 2383 (5.98)
The following correlation is recommended:
Sh = 0.31(GrSc)0.28 = 3732 (5.99)
This means we have 3732× amplification over the diffusive-migrative limiting current!
55
So we have
J limiti = Sh
5.33e− 10
0.96
0.25× AV O1e− 3
= 3.1198e+ 20/m2/s (5.100)
and because it is Cu2+ that is transported, the limiting current is
J limitq = 99.97A/m2. (5.101)
In order to produce 1000 tons of Cu per day, we would still require 3.5158e+ 05m2 of area.
This sounds like a lot, and indeed copper electrorefinning plant is usually a huge-footprint
operation.
The above is about solid-electrode/liquid electrolyte interface. For gas-evolving electrode, the
gas bubbles that detach from the solid surface can assist convective mass transfer. Stephak
and Vogt developed a correlation for gas-evolving electrode,
Sh = 0.93Re0.5Sc0.487 (5.102)
where Re is computed using the break-off diameter of the gas bubble (usually ∼ 50µm) and
steady-state velocity of these gas bubbles.
5.6 Concentration Overpotential
Convective mixing flattens the concentration profile in the bulk, and accelerates mass transfer
by Sh× amplification of the flux in the bulk, almost like a “superconductor” (this is an
advantage of liquid electrolyte versus solid electrolyte). However, near the electrode surface
there is a “boundary layer” of thickness δ µm, within which convective transport is not
dominant, and significant concentration variation can build up. Within this boundary layer,
the price
pi = zieφ+ kBT ln ci (5.103)
gradient −∇pi still drives the transport.
In
The treatment of φ and ci will follow our previous “purely diffusive” treatment in Sec. 5.3.
In particular, we note that certain ion species n are consumed/ejected by the electrode
56
(“non-blocking” or “Faradaic”), while others i 6= n are inert (“blocking” or “non-Faradaic”
or “polarizable”). As the electrode is consuming n, it still needs to be transported across
δ, at the expense of pn. Indeed, if the electrodic process is very efficient (i0 is large), it
will deplete cn(x = 0) more severely, and cause bigger driving force pn(x = δ) − pn(x = 0)
to increase the across-boundary-layer flux. The situation is quite like on-land fishermen
depleting a particular fish stock (say “tuna”) near the shore.
The other fishes are not consumed by the fishermen on the shore, but to maintain elec-
troneutrality they have to make some adjustments in concetration to match with that of
cn(x).
If certain ion species are not
57
Appendix A
Review of Bulk Thermodynamics
Equilibrium: given the constraints, the condition of the system that will eventually be
approached if one waits long enough.
Example: gas-in-box. Box is the constraint (volume, heat: isothermal/adiabatic, permeable/non-
permeable). One initialize the atoms any way one likes, for example all to the left half side,
and suddenly remove the partition: BANG! one gets a non-equilibrium state. But after a
while, everything settles down.
Atoms in solids, liquids or gases at equilibrium satisfy Maxwellian velocity distribution:
dP ∝ exp
(−m(vx − vx)2
2kBT
)dvx, 〈v2
x〉 =kBT
m. (A.1)
kB = 1.38× 10−23 J/K is the Boltzmann constant, it is the gas constant divided by 6.022×1023. If I give you a material at equilibrium without telling you the temperature, you could
use the above relation to measure the temperature.
But in high-energy Tokamak plasma, or dilute interstellar gas, the velocity distribution could
be non-Gaussian, bimodal for example. Then T is ill-defined. Since entropy is conjugate
variable to T , entropy is also ill-defined for such far-from-equilibrium states.
Equilibrium is however yet a bit more subtle: it is possible to reach equilibrium among a
subset of the degrees of freedom (all atoms in a shot) or subsystem, while this subsystem is
not in equilibrium with the rest of the system.
58
This is why engineering and material thermodynamics is useful for cars and airplanes. Imag-
ine a car going 80 mph on highway: the car is not in equilibrium with the road, the axel
is not in equilibrium with the body, the piston is not in equilibrium with the engine block.
Yet, most often, we can define temperature (local temperature) for rubber in the tire, steel
in the piston, hydrogen in the fuel tank, and apply equilibrium materials thermodynamics
to analyze these components individually.
This is because of separation of timescales. The atoms in condensed phases collide
much more frequently (1012/second) than car components collide with each other. Thus,
it is possible for atoms to reach equilibrium with adjacent atoms, before components reach
equilibrium with each other.
Define “Type A non-equilibrium”, or “local equilibrium”: atoms reach equilibrium with
each other within each representative volume element (RVE); the RVE may not be in
equilibrium with other RVEs.
For “Type A non-equilibrium”, we can define local temperature: T (x), and local entropy.
In this course, we will be mainly investigating “Type A non-equilibrium”, and study how the
RVEs reach equilibrium with each other across large distances compared to RVE size. Type
B non-equilibrium, such as in Tokamak plasma, or radiation knockout in radiation damage,
can be of interest, but is not the main focus of this course.
Consider a binary solid solution composed of two types of atoms, N1, N2 in absolute numbers
(we prefer to use absolute number of atoms instead of moles in this class). Helmholtz free
energy F ≡ E − TS = F (T, V,N1, N2): dF = dE − TdS − SdT is a complete differential.
For closed system dN1 = dN2 = 0, the first law says dE = δQ − PdV , where PdV is work
(coherent energy transfer) and δQ is heat (incoherent energy transfer via random noise).
For open system, dE = δQ− PdV needs to be modified as
dE = δQ− PdV + µ1dN1 + µ2dN2 (A.2)
µ1, µ2 are the chemical potentials of type-1 and type-2 atoms, respectively. To motivate
the additional terms µ1dN1 +µ2dN2 for open systems, consider a process of atom attachment
at P = 0, T = 0. And for simplicity assume for a moment N2 = 0 (just type-1 atoms).
In this case, before and after attaching an additional atom, kinetic energies K are zero.
E = U + K = U(x1,x2, ...,x3N1). U(x1,x2, ...,x3N1) is called the interatomic potential
59
function, a function of 3N1 arguments. For some materials, such as rare-gas solids, it is
a good approximation to expand U(x1,x2, ...,x3N1) ≈∑i<j uij(|xj − xi|), where i, j label
the atoms and run from 1..N1, and uij(r) is called the pair potential (energy=0 reference
state is an isolated atom infinitely far away). Clearly then, E will change, since there is
one more atom in the sum, within interaction range from the previous set of atoms. Since
P = 0, PdV = 0. In order to maintain T = 0, δQ = 0. To do this there must be an
“intelligent magic hand” to drag on the atom to have a “soft landing”. The energy input by
the “intelligent magic hand” is coherent energy transfer, δQ = 0 (if not convinced, consider
a layer of atoms adding on top of solid by a “forklift” - the added layer will move like a
piston - no heat is needed). Also, the “intelligent magic hand” or “forklift” accomplishes
so-called “mass action” (addition or removal of atoms), and is different from traditional PdV
work, which describes a process of changing volume without changing the number of atoms.
And thus µ1 is motivated. In fact, from this microscopic idea experiment we have derived
µ1(T = 0, P = 0) =∑j uij(|xj − xi|)/2 when xj runs over lattice sites.
A well-known pair potential is the Lennard-Jones potential:
uij(r) = 4εij
[(σijr
)12
−(σijr
)6], (A.3)
which achieves minimum potential energy −εij when r = 21/6σij = 1.122σij. For an atom
inside a perfect crystal lattice, its number of nearest neighbors (aka coordination number) is
denoted by Z. For instance, in BCC lattice Z = 8, in FCC lattice Z = 12. To further simplify
the discussion, we can assume the pair interaction occurs only between nearest-neighbor
atoms, and the Lennard-Jones potential is approximated by expansion uij(r) = −εij +
kij(r− 21/6σij)2/2 (perform a Taylor expansion on Lennard-Jones potential and truncate at
u = 0).
The simplest model for a crystal is a simple cubic crystal with nearest neighbor springs
uij(r) = −εij + kij(r− a0)2/2 (Kossel crystal), where a0 is the lattice constant of this simple
cubic crystal. With Z nearest neighbors (Z = 4 in 2D and 6 in 3D), µ(T = 0, P = 0) =
−Zε/2.
From dimensional argument, we see µ is some kind of energy per atom, thus on the order of
minus a few eV (eV=1.602 × 10−19J), in reference to isolated atom. To compare, at room
temperature, thermal fluctuation on average gives kBTroom = 4.14 × 10−21J ≈ 0.0259 eV =
eV/40 per degree of freedom.
60
Second law says TdS = δQ when comparing two adjacent equilibrium states (integral form
(T, V,N1, N2) describes the outer characteristics of (or outer constraints on) the system, and
(A.4) describes how F would change when these outer constraints are changed, and could
go up or down. But there are also inner degrees of freedom inside the system (for example,
precipitate/matrix microstructure, which you cannot see or fix from the outside, and can
only observe when you open up the material and take to a TEM). When the inner degrees
of freedom change under fixed (T, V,N1, N2), the 2nd law states that F must decrease with
time.
From theory of statistical mechanics it is convenient to start from F , since there is a direct
microscopic expression for F , F = −kBT lnZ, where Z is so-called partition function [10,
11]. Plugging into (A.5), one then obtains direct microscopic expressions for P , the so-called
internal pressure (or its generalization in 6-dimensional strain space, the stress tensor σ, in
so-called Virial formula), as well as S, µ1, µ2. This then allows atomistic simulation people to
calculate so-called equation-of-state P (T, V,N1, N2) and thermochemistry µi(T, V,N1, N2), if
only the correct interatomic potential U(x3(N1+N2)) is provided. The so-called first-principles
CALPHAD (CALculation of PHAse Diagrams) [12] is based on this approach, and is now a
major source of phase diagram and thermochemistry information for alloy designers (metal
hydrides for hydrogen storage, battery electrodes where you need to put in and pull out
lithium ions, and catalysts). Since atomistic simulation can access metastable states and
even saddle-points, there is also first-principles calculations of mobilities, such as diffusivities,
interfacial mobilities, chemical reaction activation energies, etc. So F is important quantity
computationally.
For experimentalist, however, most experiments are done under constant external pressure
instead of constant volume (imagine melting of ice cube on the table, there is a natu-
ral tendency for volume change, illustrating the concept of transformation volume). For
discussing phase change under constant external pressure, we define Gibbs free energy
61
G ≡ F + PV = E − TS + PV . The full differential of G is
dG = V dP − SdT + µ1dN1 + µ2dN2 (A.6)
so
V =∂G
∂P
∣∣∣∣∣T,N1,N2
, S = −∂G∂T
∣∣∣∣∣P,N1,N2
, µ1 =∂G
∂N1
∣∣∣∣∣T,P,N2
, µ2 =∂G
∂N2
∣∣∣∣∣T,P,N1
. (A.7)
The above describes how a homogeneous material’s G would change when its T, P,N1, N2
are changed, which could go up or down. If the system has internal inhomogeneities that
are evolving under constant T, P,N1, N2, however, then G must decrease with time. Internal
microstructural changes under constant T, P,N1, N2 that increase G are forbidden.
Also,
d(E + PV ) = δQ+ V dP + µ1dN1 + µ2dN1 (A.8)
so if a closed system is under constant pressure, the heat it absorbs is the change in the
enthalpy H ≡ E + PV = G + TS. H is also related to G through the so-called Gibbs-
Helmholtz relation:
H =∂(G/T )
∂(1/T )
∣∣∣∣∣N1,N2,P
. (A.9)
Putting ∆ before both sides of (A.9), the heat of transformation ∆H is related to the free-
energy driving force of transformation as
∆H =∂(∆G/T )
∂(1/T )
∣∣∣∣∣N1,N2,P
. (A.10)
Now we formally introduce the concept of thermodynamic driving force for phase transfor-
mation. Consider two possible phases φ = α, β that the system could be in. Both phases
have the same numbers of atoms N1, N2, the same T and P . Consider pressure-driven phase
transformation, dGα = V αdP , dGβ = V βdP . Suppose V α > V β, when we plot Gα and Gβ
graphically on the same plot, we see that at low pressure, the high-volume phase α may win;
but at high pressure, the low-volume (denser phase) β will win. As a general rule, when P
is increased keeping T fixed, the denser phase will win. So liquid phase will win over gas,
and typically solid phase will win over liquid. Consider for example Fig. A.1(a). Density
ranking: ε > γ > α. For fixed T,N1, N2, there exists an equilibrium pressure Peq where the
62
8
P
T0.0098°C
0.00603atm
SOLID
VAPOR
LIQUID
1 atm
220 atm
374°C
Phase Diagram of H2O
sublimationdeposition
vaporizationcondensation
meltingfreezing
Figure A.1: (a) Figure 1.5 of Porter & Easterling [13]. (b) Phase diagram of pure H2O:the solid-liquid boundary has negative dP/dT , which is an anomaly, because ice has largervolume than liquid water.
Gibbs free energy curves cross, at which
Gα(Peq, T,N1, N2) = Gβ(Peq, T,N1, N2). (A.11)
At P > Peq, the driving force for α → β is ∆G ≈ (V α − V β)(P − Peq). Vice versa, at
P < Peq, the driving force for β → α is ∆G ≈ (V α− V β)(Peq−P ) (by convention, we make
the driving force positive). P−Peq (Peq−P ) may be called the overpressure (underpressure),
respectively.
We could also have temperature-driven transformation, keeping pressure fixed: dGα =
−SαdT , dGβ = −SβdT . So G vs T is a downward curve. The question is which phase
is going down faster, Gα or Gβ. The answer is that the state that is more disordered (larger
S) will go down faster with T ↑. So at some high enough T there will be a crossing. Liquid
is going down faster than solid, gas is going down faster than liquid, with T ↑ holding P
constant. For a fixed pressure, there exists an equilibrium temperature Teq where the Gibbs
free energy curves cross, at which
Gα(P, Teq, N1, N2) = Gβ(P, Teq, N1, N2). (A.12)
Consider for example solid↔liquid transformation. In this case, Teq = TM(P ), the equilib-
rium bulk melting point. α=liquid, β=solid, Sα > Sβ. At T > Teq, the more disordered
63
phase is favored, and the driving force for β → α transformation, which is melting, is
∆G ≈ (Sα − Sβ)(T − TM). Vice versa at T < Teq, the more ordered phase is favored, which
is solidification, and the driving force for α → β is ∆G ≈ (Sα − Sβ)(TM − T ). Because we
are doing first-order expansion, it is OK to take Sα−Sβ to be the value at TM. However, at
TM we have Eα + PV α − TMSα = Hα − TMS
α = Hβ − TMSβ = Eβ + PV β − TMS
β, we have
Sα − Sβ = (Hα −Hβ)/TM. Hα −Hβ is in fact the heat released during phase change under
constant pressure, and is called the latent heat L. So we have
∆G ≈ L
TM
|TM − T |. (A.13)
|TM − T | is called undercooling / superheating for solidification / melting. We see that the
thermodynamic driving force for phase change is proportional to the amount of undercooling
/ superheating (in Kelvin), with proportionality factor LTM
= ∆S. Later we will see later
why a finite thermodynamic driving force is needed, in order to observe phase change within
a finite amount of time. (If you are extremely leisurely and have infinite amount of time,
you can observe phase change right at Teq).
solid/liquid: melting, freezing or solidification. liquid/vapor: vaporization, condensation.
solid/vapor: sublimation, deposition. At low enough pressure, the gas phase is going to
come down in free energy significantly, that the solid goes directly to gas, without going
through the liquid phase.
Thus, typically, high pressure / low temperature stabilizes solid phase, low pressure / high
temperature stabilizes gas phase. The tradeoff relation can be described by the Clausius-
Clapeyron relation for polymorphic phase transformation (single-component) in T − P
plane. The question we ask is that suppose you are already sitting on a particular (T, P )
point that reaches perfect equilibrium between α, β,
Gα(N1, N2, T, P ) = Gβ(N1, N2, T, P ) (A.14)
in which direction on the (T, P ) plane should one go, (T, P )→ (T+dT, P+dP ), to maintain
that equilibrium, i.e.:
Gα(N1, N2, T + dT, P + dP ) = Gβ(N1, N2, T + dT, P + dP ) (A.15)
Gα(N1, N2, T, P )− SαdT + V αdP = Gβ(N1, N2, T, P )− SβdT + V βdP. (A.16)
64
So:
−SαdT + V αdP = −SβdT + V βdP. (A.17)
and the direction is given by
dP
dT=
Sα − Sβ
V α − V β=
L
T (V α − V β). (A.18)
The above equation keeps one “on track” on the T − P phase diagram. It’s like in pitch
darkness, if you happen to stumble upon a rail, you can follow the rail to map out the whole
US railroad system. The Clausius-Clapeyron relation tells you how to follow that rail. L is
called “latent heat”. V α − V β is the volume of melting/vaporization/sublimation, you may
call it the “latent volume”.
In above we have only considered the scenario of so-called congruent transformation α↔ β,
where α and β are single phases with the same composition. We have not considered the
possibility of for example α ↔ β + γ, where γ has different composition or even structure
from β. To understand the driving force for such transformations which are indeed possible
in binary solutions, we need to further develop the language of chemical potential.
The total number of particles is N ≡ N1 + N2. Define mole fractions X1 ≡ N1/N , X2 ≡N2/N . Since there is always X1 + X2 = 1, we cannot regard X1 and X2 as independent
variables. Usually by convention one takes X2 to be the independent variable, so-called
composition. Composition is dimensionless, but it could be a multi-dimensional vector if
the number of species C > 2. For instance, in a ternary solution, C = 3, and composition
is a 2-dimensional vector X ≡ [X2, X3]. Composition can spatially vary in inhomogeneous
systems, for instance in an inhomogeneous binary solution, X2 = X2(x, t). In order for
α↔ β+γ to happen kinetically, for instance changing from X2(x) = 0.3 uniformly (initially
α phase) to some region with X2(x) = 0.5 (in β phase, “solute sink”) and some region with
X2(x) = 0.1 (in γ phase, ‘solute source”). This requires would require long-range diffusion
of type-2 solutes over distances on the order of the sizescale of the inhomogeneities, which
is called solute partitioning.
We can define the particle average Gibbs free energy to be g ≡ G/N = G(T, P,N1, N2)/(N1 +
N2). Like the chemical potentials, g will be minus a few eV in reference to isolated atoms
ensemble. It can be rigorously proven, but is indeed quite intuitively obvious, that g =
g(X2, T, P ), which is to say the particle average Gibbs free energy depends on chemistry
but not quantity (think of (N1, N2) ↔ (N,X2) as a variable transform that decomposes
65
dependent variables into quantity and chemistry). It is customary to plot g versus X2 at
constant T, P . It can be mathematically proven that µ1, µ2 are the tangent extrapolations
of g(X2) to X2 = 0 and X2 = 1, respectively. Algebraically this means
µ1(X2, T, P ) = g(X2, T, P ) +∂g
∂X2
∣∣∣∣∣T,P
(0−X2)
µ2(X2, T, P ) = g(X2, T, P ) +∂g
∂X2
∣∣∣∣∣T,P
(1−X2). (A.19)
It is also clear from the above that g(X2, T, P ) = X1µ1 +X2µ2, so
G(T, P,N1, N2) = N1µ1 +N1µ2 = N1∂G
∂N1
∣∣∣∣∣T,P,N2
+N2∂G
∂N2
∣∣∣∣∣T,P,N1
(A.20)
On first look, the above seems to imply that particle 1 and particle 2 do not interact. But
this is very far from true! In fact, µ1 = µ1(X2, T, P ), µ2 = µ2(X2, T, P ).
For pure systems: X2 = 0, g(X2 = 0, T, P ) = µ1(X2 = 0, T, P ) ≡ µ1(T, P ); or X2 = 1,
g(X2 = 1, T, P ) = µ2(X2 = 1, T, P ) ≡ µ2(T, P ). µ1(T, P ), µ2(T, P ) are called Raoultian
reference-state chemical potentials (they are not the isolated-atoms-in-vaccuum reference
states, but already as interacting-atoms). In this class we take the µ1, µ2 reference states to
the same structure as the solution, but in pure compositions (so-called Raoultian reference
states).
When plotted graphically, it is seen that g(X2) is typically convex up with µ1(X2, T, P ) <
µ1(T, P ) and µ2(X2, T, P ) < µ2(T, P ) (if not, what would happen?) This negative difference
is defined as the mixing chemical potential
µmixi ≡ µi(X2, T, P )− µi(T, P ), i = 1, 2 (A.21)
and mixing free energy
gmix ≡ X1µmix1 +X2µ
mix2 = g −X1µ1(T, P )−X2µ2(T, P ), Gmix = Ngmix (A.22)
respectively. Clearly, by definition, Gmix = 0 at pure competitions. gmix(X2, T, P ) can be
interpreted as the driving force to react pure 1 and pure 2 of the same structure as the
solution to obtain a solution of non-pure composition, per particle in the mixed solution.
∆G = −Ngmix(X2, T, P ) is in fact the chemical driving force to make a solution by mixing
66
pure constituents.
It turns out there exists “partial” version of the full differential (A.6):
dg(X2, T, P ) = vdP − sdT +∂g
∂X2
∣∣∣∣∣T,P
dX2 (A.23)
dµi(X2, T, P ) = vidP − sidT +∂µi∂X2
∣∣∣∣∣T,P
dX2 (A.24)
where
v1 ≡∂V
∂N1
∣∣∣∣∣T,P,N2
, v2 ≡∂V
∂N2
∣∣∣∣∣T,P,N1
, s1 ≡∂S
∂N1
∣∣∣∣∣T,P,N2
, s2 ≡∂S
∂N2
∣∣∣∣∣T,P,N1
,
e1 ≡∂E
∂N1
∣∣∣∣∣T,P,N2
, e2 ≡∂E
∂N2
∣∣∣∣∣T,P,N1
, h1 ≡∂H
∂N1
∣∣∣∣∣T,P,N2
, h2 ≡∂H
∂N2
∣∣∣∣∣T,P,N1
, ... (A.25)
Generally speaking, for arbitrary extensive quantity A (volume, energy, entropy, enthalpy,
Helmholtz free energy, Gibbs free energy), “particle partial A” is defined as:
ai ≡∂A
∂Ni
∣∣∣∣∣Nj 6=i,T,P
. (A.26)
The meaning of ai is the increase in energy, enthalpy, volume, entropy, etc. when an ad-
ditional type-i atom is added into the system, keeping the temperature and pressure fixed.
The particle-average a is simply
a ≡ A
N=
C∑i=1
Xiai. (A.27)
For instance, the particle average volume and particle average entropy
v ≡ V
N= X1v1 +X2v2, s ≡ S
N= X1s1 +X2s2, (A.28)
is simply the composition-weighted sum of particle partial volumes and partial entropies
of different-species atoms, respectively. While (A.27) relates all ai(X2, ..., XC , T, P )s to
a(X2, ..., XC , T, P ), it is also possible to obtain individual ai(X2, ..., XC , T, P ) from a(X2, ..., XC , T, P )
67
by the tangent extrapolation formula:
ai(X2, ..., XC , T, P ) = a(X2, ..., XC , T, P ) +C∑k=2
(δik −Xk)∂a(X2, ..., XC , T, P )
∂Xk
, (A.29)
where δik is the Kronecker delta: δik = 1 if i = k, and δik = 0 if i 6= k. Note in (A.29),
although the k-sum runs from 2 to C, i can take values 1 to C. (A.19) is a special case of
(A.29): for historical reason the particle partial Gibbs free energy is denoted by µi instead
of gi.
The so-called Gibbs-Duhem relation imposes constraint on the partial quantities when com-
So by taking derivative against l on both sides, and then setting l = 1, there is
hi−1∂ai∂hi−1
+ hi∂ai∂hi
+ hi+1∂ai∂hi+1
= ai. (B.84)
In 3D, there is ai(lhi−1, lhi, lhi+1) = l2ai(hi−1, hi, hi+1) and hi−1∂ai∂hi−1
+hi∂ai∂hi
+hi+1∂ai∂hi+1
= 2ai.
Comparing the two equations, we see that
... =γi−1
hi−1
=γihi
=γi+1
hi+1
= ... = β (B.85)
for all i, will be a variational extremum. In fact, dFsurface = dFbulk = (Pint − Pext)dV
is the original Young-Laplace pressure argument (Fig. ??(a)), and the facet-independent
Lagrange multiplier β can be identified to be simply the Young-Laplace pressure difference
∆P = Pint − Pext. So in 2D, we have ∆P = γihi
.
The above means that the inner envelope formed by all Wulff planes (a Wulff plane lies
perpendicular to γ(n)n at γ(n)n) gives the equilibrium shape of a free-standing nanocrystal.
This is called Wulff construction, which minimizes the total surface energy of a free-
standing nanoparticle. Note that the Wulff construction serves a different purpose from the
tangent circle theorem. The tangent circle theorem deals with the stability of one surface
constrained to have overall inclination n′ because it must conform to the substrate, whereas
the Wulff construction needs to optimize all facets of the nanocrystal simultaneously.
In 3D, there is an extra factor of 12
on RHS, and we get
... =γi−1
hi−1
=γihi
=γi+1
hi+1
= ... =β
2=
∆P
2(B.86)
or ∆P = 2γihi
to be the pressure increase inside the solid particle. We see that for isotropic
surface energy and spherical particle, this reduces to the familiar expression ∆P = 2γR
.
100
B.3 Gradient Thermodynamics Description of the In-
terface
First-order phase transition is characterized by finite jump in the order parameter ηα → ηβ
as soon as T = T±e (the nucleation rate may be very small, but theoretically suppose one
waits long enough one can witness this finite jump at T±e ). For example, melting of ice
at P = 1atm is a first-order transition because as soon as T rises up to 0.0001C and
melting can occur, there is a finite density change from ice to liquid water, and there is an
obvious change in the viscosity as well. Also spatially, the transition from η(x) = ηα to
η(x′) = ηβ typically occurs over a very narrow region: the shortest distance between x and
x′ (interfacial thickness w) is typically less than 1nm. Previously, we assigned a capillary
energy γ to this interfacial region without discussing this region’s detailed structure. Such
“sharp interface” view, where one ignores the detailed interfacial structure and represent it
as a geometric dividing surface, is sufficient for most first-order phase transition problems.
If one is really interested in the physical thickness of this interfacial region however, one
must use so-called gradient thermodynamics formulation [17] to be introduced below, where
the capillary energy∫γdA in the sharp-interface representation is replaced by a 3D integral
involving a gradient squared term∫K|∇η(x)|2d3x with K > 0. The above replacement
makes sense intuitively, since the interfacial region is characterized by large gradients in
η(x), absent in the homogeneous bulk regions of α or β. Nucleation and growth is a must
for all first-order phase transitions, where large change (ηα → ηβ) occurs in a narrow region
(the interface) even during nucleation.
In contrast, second-order phase transition is characterized by initially infinitesimal changes
over a wide region. These initially infinitesimal changes appear spontaneously in the system
and grow with time, without going through a nucleation (large change in a small region)
stage. For example, in the paramagnetic (α)→ferromagnetic (α1,α2) transition of pure iron
as T is cooled below Tc = 1043K (the Curie temperature, also called the critical point), both
the spin-down α1 and the spin-up α2 phase have very small magnetic moments: ηα1 = −m,
ηα2 = m, with m ∝ (Tc − T )1/2. Microscopically, going from α1 to α2 near Tc would
involve the flipping of a very small number of spins. So the high-temperature paramagnetic
phase, and the two low-temperature ferromagnetic phases are very similar to each other
near Tc: |ηα − ηα1|, |ηα − ηα2| ∝ (Tc − T )1/2, where η is the magnetic moment. The breakup
of a uniform paramagnetic domain into multiple ferromagnetic domains upon a drop in
temperature below Tc is spontaneous and instantaneous and does not require a nucleation
101
stage: it is growth, off the bat. In other words, no under-cooling is required for observing
the start of second-order phase transition within a given observation period. The growth
happens essentially instantaneously at T = T±c . Although, to see the growth and coarsening
to a certain amplitude would require time.
The way a system can accomplish second-order transition vis-a-vis first-order transition is
best illustrated using the binary solution example: gsoln(X2, T ) ≡ Gsoln(N1, N2, T )/(N1+N2).
Suppose Ω1 = Ω2 = Ω, we may define specific volume free energy as
gv(c2) ≡ Ω−1gsoln(X2 = c2Ω) (B.87)
so the bulk solution free energy for a homogeneous system is just
Gsoln =(∫
d3x)gv(c2). (B.88)
gv(c2) is the same function as gsoln(X2) after horizontal and vertical scaling. So the tangent
extrapolation of gv(c2) to c2 = 0 (corresponding to X = p1) would give Ω−1µ1, and tangent
extrapolation of gv(c2) to c2 = Ω−1 (corresponding to X = p2) would give Ω−1µ2. c2(x) is
our order parameter field η(x) here. For an inhomogeneous system, the solution free energy
should intuitively be written as
Gsoln =∫d3xgv(c2(x)). (B.89)
Using the above as reference, the total free energy then looks like:
G =∫d3x(gv(c2(x)) +K|∇c2(x)|2) +Gelastic (B.90)
where the gradient squared term replaces the capillary energy∫γdA. Gelastic = 0 if Ω1 =
Ω2 = Ω. (B.90) is a unified model that can be used to investigate both finite interfacial
thickness in first-order transitions [17], as well as second-order transitions [23]. Since K > 0,
the model (B.90) punishes sharp spatial gradients, the origin of interfacial energy. On the
other hand if all changes occur smoothly over a large wavelength with small spatial gradients,
then G approaches Gsoln. Since Gsoln is the driver of phase transformation (gradient/capillary
and elastic energies are typically positive), let us consider what Gsoln wants to do first.
For a closed system, c2 is conserved:∫d3xc2(x) = const (B.91)
102
which means it is possible to partition the solutes, but it is not possible to change the
total amount of solutes in the entire system. For instance, if one starts out with a uniform
concentration c2(x) = cα2 , a partition may roughly speaking occur as:
cα2 = fα1cα12 + fα2cα2
2 , (B.92)
where volume fraction
fα1 =cα2
2 − cα2cα2
2 − cα12
, fα2 = 1− fα1 =cα2 − cα1
2
cα22 − cα1
2
(B.93)
of the region has c2(x) = cα12 and c2(x) = cα2
2 , respectively, separated by sharp interfaces.
The solution free energy of the partitioned system is then
Gsoln =(∫
d3x)
(fα1gv(cα12 ) + fα2gv(c
α22 )) (B.94)
compared to the unpartitioned and uniform original system (∫d3x) gv(c
α2 ).
Local stability means Gsoln is stable against small perturbations in c2(x). The necessary and
sufficient condition for local stability is that
∂2gv∂c2
2
> 0. (B.95)
If ∂2gv∂c22
< 0, a small partition with cα12 ≈ cα2 ≈ cα2
2 would be able to decrease Gsoln. For
example, with cα22 = cα2 + ∆c, cα1
2 = cα2 −∆c, fα1 = fα2 = 1/2, one has
Gsoln∫d3x
=1
2gv(c
α2 −∆c) +
1
2gv(c
α2 + ∆c) = gv(c
α2 ) +
1
2
∂2gv∂c2
2
(cα2 )(∆c)2 + ... (B.96)
which would be lower than uniform gv(cα2 ) if ∂2gv
∂c22< 0. A sinusoidal perturbation
c2(x) = cα2 + a(t) sin(k · x) (B.97)
would also have equal amount of “ups and downs”, and would thus also reduce Gsoln. The
reason sinusoidal perturbation is preferred (at least initially) compared to the step function
between cα2 − ∆c and cα2 + ∆c is that it minimizes the gradient energy by spreading the
gradients around. Therefore if ∂2gv∂c22
< 0, its amplitude a(t) will increase with time. This is
the trick behind spinodal decomposition, or more generally second-order phase transitions,
103
which can reduce the system free energy without nucleation. Nucleation is not needed here
because the system’s initial state does not have local stability. The loss of local stability is
induced by temperature, i.e.
∂2gv∂c2
2
(cα2 , T+C ) > 0,
∂2gv∂c2
2
(cα2 , T−C ) < 0 (B.98)
thus∂2gv∂c2
2
(cα2 , TC) = 0. (B.99)
During initial growth of the sinusoidal profile in the unstable composition range, the solutes
appears to diffuse up the concentration gradient (Fig. 5.39 of [13]). According to the
phenomenological Fick’s 1st law J2 = −D∇c2, this would mean a negative interdiffusivity
D(c2) < 0. This is in fact not surprising, because D (from D1, D2) contains thermodynamic
factor 1 + d ln γ2d ln c2
, which can be shown to be X2(1−X2)kBT
∂2g∂X2
2and thus have the same sign as
∂2g∂X2
2. When ∂2g
∂X22
is negative, D is negative. This means that at the most fundamental level,
diffusion is driven by the desire to reduce free energy or chemical potential, and not by the
desire to smear out the concentration gradient.
Mathematically, while a positive diffusivity tends to smear out the profile (the shorter the
wavelength, the faster the decay of the Fourier component amplitude), a negative diffusivity
would tend to increase the roughness of the profile. The growth of very-small wavelength
fluctuations in spinodal decomposition will be punished by the gradient energy, though.
Thus an optimal wavelength will be selected initially, which can be tens of nms. Later,
after the compositions have deviated largely from cα2 , the microstructural lengthscale may
further coarsen, although the interfacial lengthscale will sharpen. Because α1 and α2 do not
come out of a nucleation and growth process, but amplification of sinusoidal waves of certain
optimal wavelength, they lead to unique-looking interpenetrating microstructures.
In contrast to spinodal instability, in a first-order phase transition the system’s initial state
has never lost its local stability. At T = T+e , one is in a globally stable uniform composition,
which means
gv(cα2 , T
+e ) < fα1gv(c
α12 , T+
e ) + fα2gv(cα22 , T+
e ) (B.100)
for small and large deviations |cα22 − cα2 | alike (thus a globally stable system must be locally
stable, but not vice versa). Then at T = T−e , c2(x) = cα2 becomes locally stable only, which
means small deviations would still induce the system energy to go up, but large deviations
104
may induce the system energy to go down. Thus, small perturbations like (B.97) would decay
and die, but large enough perturbations may survive. The chance survival of large enough
perturbations/fluctuations in the order-parameter field is just nucleation.
(B.90) can be used to estimate interfacial thickness w in the following manner. Since ∇c2 ∝(cβ2 − cα2 )/w inside the interface, the gradient energy integral scales as K(cβ2 − cα2 )2/w, so the
wider the interface the better for the gradient energy. On the other hand, right at T = Te,
gv(c2) of the first term connects two energy-degenerate states gv(c2 = cβ2 ) = gv(c2 = cα2 ),
with a bump g∗v − gv(cα2 ) in between. The solution free energy first term thus gives an excess
∝ (g∗v − gv(cα2 ))w, that punishes wide interfaces. The best compromised is thus reached at
w ∝ K1/2|cβ2 − cα2 |(g∗v − gv(cα2 ))−1/2, with interfacial energy γ ∝ K1/2|cβ2 − cα2 |(g∗v − gv(cα2 ))1/2.
It turns out that for Te near Tc, |cβ2 − cα2 | ∝ (∆T )1/2, where ∆T = Tc−Te, and g∗v − gv(cα2 ) ∝(∆T )2, so the interfacial width near the critical temperature would diverge as (∆T )−1/2, and
the interfacial energy would vanish as (∆T )3/2 [17].
Science advances greatly when two seemingly different concepts are connected, for instance
the Einstein relation M = D/kBT . Cahn and Hilliard made a similar contribution when they
connected interfacial energy to critical temperature and second-order phase transformation.
Based on the insight that gradient term should be added to thermodynamic field theories
(fundamentally this is because of atomic discreteness), they developed gradient thermody-
namics formalism for chemical solution systems that predict finite interfacial width, interfa-
cial energy, as well as wavelength selection in spinodal decomposition [23], under one unified
framework. The development can in fact be traced back to the work of van der Waals for
single-component systems, using density as order parameter[24]. Another offshoot of this
approach was provided by Ginzburg and Landau in the theory of superconductivity.
Finally, if Ω1 6= Ω2 the 1-rich α1 phase and 2-rich α2 will have different stress-free volumes,
and to accommodate this mismatch coherently would involve finite elastic energy Gelastic >
0. Growth of the sinusoidal concentration wave would require growth of the associated
transformation strain wave. This would delay the onset of the spinodal instability.
105
Appendix C
Neuromorphic equivalent circuit -
Mantao model
Proton-based electrochemical synapses [1, 25] can operate at very short timescale ∼ ns. Its
structure is similar to that of a battery. We may call the hydrogen reservoir (PdHx) region-0,
the electrolyte region-2, and the HyWO3 region-4. The 0/2 interface will be called 1, and
2/4 interface called 3, where charge-transfer reactions happen.
There is a change of chemical identity at interface 1, where
H(0) ↔ H+(2) + e−(0) (C.1)
and at interface 3,
H+(2) + e−(4) ↔ H(4) (C.2)
where the “(4)” means free electron or H (a bound electron+proton pair) inside the HyWO3
phase, and “(0)” means free electron or H inside the metallic PdHx phase.
Equation C.1, C.2, if they are consummated, are called Faradaic current. Note that Faradaic
current always involves a free electron or a free ion crossing the interface. In C.1 reaction
from left to right, a charge-neutral H(0) auto-ionizes, the proton crosses the interface 1,
leaving an orphaned electron e−(0) behind. In C.1 reaction from right to left, the process
reverses and the proton crosses the interface backwards.
It is also possible for the interface 1 to store excess H+(2), and excess e−(0) across the
106
interface of physical width d1, without consummating the reaction C.1 (that is, no H(0) is
added/subtracted in 0). After all, PdHx is metallic and can store excess electrons on its
surface, and the electrolyte can store excess proton in its surface as well, without these two
reacting. This is called non-Faradaic or capacitive current. The non-Faradaic current leads
to equal and opposite charge ±Q1, stored across the interface ravine d1 that are typically a
few Angstroms. The interfacial area is A1, and typically there is relation
dQ1
dV1
≡ C1 ≈A1ε
d1
(C.3)
where V1 is the electrostatic potential drop across the interface 1, and ε = εrε0 is the dielectric
constant, ε0 = 8.8541878128E − 12 Farad/meter is the vacuum permittivity.
When Equation C.1 reaches thermodynamic balance, there is
U eq(0) = U(0) +e
kBTlnaH+(2)
aH(0)
(C.4)
where −eU(0) = EF(0) is the electronic Fermi energy in PdHx, taking the average electro-
static potential inside 2 to be zero. Similarly, when Equation C.2 reaches thermodynamic
balance, there is
U eq(4) = U(4) +e
kBTlnaH+(2)
aH(4)
, (C.5)
where a’s are the thermodynamic activities, that may be taken to be concentration in ideal-
solution approximation. For simplicity, if we assume (a) H+(2) is the only mobile ion in solid
electrolyte 2, and (b) rigorous electroneutrality inside 2, then aH+(2) is identical everywhere
inside 2, and then
V eq ≡ U eq(0)− U eq(4) = V +e
kBTlnaH(4)
aH(0)
≈ V +e
kBTlncH(4)
cH(0)
(C.6)
where cH(4), cH(0) are the concentrations right at the interface.
It is customary to model these interfaces as R1,C1 and R3,C3, where
V1 ≡ V eq1 +R1I
Faradaic1 , V3 ≡ V eq
3 +R3IFaradaic3 (C.7)
IFaradaic1 can follow the Butler–Volmer equation
IFaradaic1 = A1i
01[ee(1−β1)η1/kBT − e−eβ1η1/kBT ] (C.8)
107
and
IFaradaic3 = A3i
03[ee(1−β3)η3/kBT − e−eβ3η1/kBT ] (C.9)
for electronic (voltage) out-of-equilibrium situations, where i01 and i03 are exchange current
densities. Thus,
η1 ≡ V1 − V eq1 = R1A1i
01[ee(1−β1)η1/kBT − e−eβ1η1/kBT ] (C.10)
η3 ≡ V3 − V eq3 = R3A3i
03[ee(1−β3)η3/kBT − e−eβ3η3/kBT ] (C.11)
We see that
R1 →kBT
eA1i01, R3 →
kBT
eA3i03(C.12)
for small η1, η3 25 mV, but they gets exponentially smaller for larger η1, η3 25 mV:
R1 →η1e
e(β1−1)η1/kBT
A1i01, R3 →
η3ee(β3−1)η3/kBT
A3i03. (C.13)
So the systems of equations are:
V ≡ V eq + η = V +e
kBTlncH(4)
cH(0)
+ η (C.14)
η ≡ η1 + η2 + η3, (C.15)
I = C1dη1
dt+ A1i
01[ee(1−β1)η1/kBT − e−eβ1η1/kBT ]
=A2
L2
κ02
sinh(eη2h/2kBTL2)
eh/2kBTL2
= C3dη3
dt+ A3i
03[ee(1−β3)η3/kBT − e−eβ3η3/kBT ] (C.16)
where the middle equality assumes only ionic Ohmic polarization and no no ionic con-
centration polarization for simplicity, κ02 [unit S/m] is the proton conductivity of the solid
electrolyte at zero field, h is the H+(2) one-hop distance in the thickness direction along 2,
and L2 is the thickness of 2. In this we have also assumed the electronic conductivity in 0
(gate) and 4 (source-drain) is always much better than the ionic conductivity in 2, otherwise
the electronic Ohmic loss in 0 and 4 would also need to be added.
108
There are also two PDEs, one in region-0 and one in region-4, with
∂c
∂t=
∂
∂z
(D0
H
∂c
∂z
), z > z0 (C.17)
∂c
∂t=
∂
∂z
(D4
H
∂c
∂z
), z < z4 (C.18)
for the charge-neutral H (a bound electron+proton pair) concentration. There are the flux
boundary conditions:
A1eD0H
∂c
∂z(z0) = A1i
01[ee(1−β1)η1/kBT − e−eβ1η1/kBT ] (C.19)
A3eD4H
∂c
∂z(z4) = A3i
03[ee(1−β3)η3/kBT − e−eβ3η3/kBT ] (C.20)
In Electrochemical impedance spectroscopy (EIS), equation (C.17), (C.18) correspond to the
Warburg impedance element.
The condition that we work in is huge constant voltage pulse η(t = 0+) (on the order of
4Volt), but very very short time duration. Let us take A1 = A2 = A3 = A. At the beginning,
all the η is to drive ionic Ohmic current inside 2:
I(t = 0+) =A
L2
κ02
sinh(eηh/2kBTL2)
eh/2kBTL2
(C.21)
and so
η3(t) ≈ At
C3L2
κ02
sinh(eηh/2kBTL2)
eh/2kBTL2
(C.22)
and so the Faradaic current (when it is still smaller than the non-Faradaic current) goes as
Ai03[ee(1−β3)η3/kBT − e−eβ3η3/kBT ] (C.23)
it is likely that η3(t) will exceed 25mV, and the Faradaic current gets into the exponential
regime:
Ai03ee(1−β3)η3/kBT (C.24)
but in this regime, the time-dependence should be exponential in time, not powerlaw in time.
The powerlaw may come later, but before there is matching
A
L2
κ02
sinh(eη2h/2kBTL2)
eh/2kBTL2
= Ai03ee(1−β3)η3/kBT (C.25)
109
C.1 Variable resistor solution
Standard RC circuit solves the following equation
CdV
dt= −V
R(C.26)
and this gives the well-known τ = RC time constant.
But what happens when R depends on V ? Suppose
R = R0 exp(− VV0
) (C.27)
then instead of a single relaxation time, the system has a spectrum of relaxation times, which
is very very fast when V is large, but slows down appreciably when V is decaying down.
We havedV
dt= −
V exp( VV0
)
R0C(C.28)
we can define dimensionless voltage and time:
v ≡ V
V0
, s ≡ t
R0C(C.29)
thendv
ds= −vev (C.30)
and we are in the v 1 regime.
Suppose at s = 0, we have v = v0 1. Then the relaxation kinetics is
−e−vdv
v= ds (C.31)
or ∫ v
v0−e−vdv
v= s = Ei(−v0)− Ei(−v) (C.32)
where Ei(·) is a special function called exponential integral.
For positive v, there is also
Ei(−v) = −E1(v) (C.33)
110
so alternatively
s = E1(v)− E1(v0) (C.34)
111
Bibliography
[1] XH Yao, K Klyukin, WJ Lu, M Onen, S Ryu, D Kim, N Emond, I Waluyo, A Hunt,
JA del Alamo, J Li, and B Yildiz. Protonic solid-state electrochemical synapse for