Ions in concentrated electrolytes: from the screening

length to energy storageAlpha Lee

Department of Physics, University of Cambridge aal44@cam.ac.uk

The standard model

DLVO theory

lB =e2

✏kBT

�D = 1/p

4⇡lBc

�U(r) ⇠ Z2 lBre�r/�D

r

12

10

8

6

4

2

0

λ S (n

m)

2.01.61.20.80.40.0

c1/2

(M1/2

)

Propylene carbonate

[C4C1Pyrr]+

[NTf2] _

Na+ Cl – H2O

Debye-Hückel screening length!

Concentration

b

a

12

10

8

6

4

2

0

λ S (n

m)

2.01.61.20.80.40.0

c1/2

(M1/2

)

Propylene carbonate

[C4C1Pyrr]+

[NTf2] _

Na+ Cl – H2O

Debye-Hückel screening length!

Concentration

b

a

12

10

8

6

4

2

0

λ S (n

m)

2.01.61.20.80.40.0

c1/2

(M1/2

)

Propylene carbonate

[C4C1Pyrr]+

[NTf2] _

Na+ Cl – H2O

Debye-Hückel screening length!

Concentration

b

a

12

10

8

6

4

2

0

λ S (n

m)

2.01.61.20.80.40.0

c1/2

(M1/2

)

Propylene carbonate

[C4C1Pyrr]+

[NTf2] _

Na+ Cl – H2O

Debye-Hückel screening length!

Concentration

b

a

AAL et al., Faraday Discussions, 199, 239 (2017)

Dimensional analysisSalient length scales:

lB =e2

✏kBTBjerrum length

Debye length �D = 1/p

4⇡lBc

Ion diameter a

Scaling analysis of the screening length

81

2

4

6

810

2

4

68

100λ S

/ λ D

6 80.1

2 4 6 81

2 4 6 810

a / λD

1

3

Pure aprotic ILs Pure protic ILs IL in PC NaCl in water LiCl in water KCl in water CsCl in water

AAL et al., Phys. Rev. Lett., 119, 26002 (2017)

�S ⇠ lBca3

Activity coefficientCompute the field around an ion

Debye charging process

�µex

= �1

2

lBa

a

�S + a

�S

Comparison with experimental data

NaCl in water

W. J. Hamer and Y-C Wu, J Phys. Chem. Ref. Data, 1, 1047 (1972)

0 2 4 6Concentration / M

-0.8

-0.6

-0.4

-0.2

0µ

ex

Direct measurementPrediction using measuredscreening length

Ionic crystal analogy

+

+

+

+

--

-

-

-

-

-+-

+

+

+

+

--

-

-

-

-

+-+-+

-+-+-+-+

+

+

+

+

-

-

-

-

-+-+-+-+

+

+

+

+

-

-

-

-

-+-+-+-+

• Complete charge ordering

• Behaves as a dielectric

Schottky defects

• Defects behave as charges

• The defects are mobile - Debye screening

+

+

+

+

--

-

-

-

-

-+-

+

+

+

--

-

-

-

-

+-+-+

-+

+-+

+

+

+

+

+

-

-

-

-

-+-+-+-+

+

+

+

-

-

-

-

-+-+-+-+

Concentrated ionic solutions as a defect-laden ionic crystal

• A perturbative theory around a disordered ionic crystal, not a dilute gas

• The role of “defects” played by neutral solvent molecules

• Incompressibility: ctot

= cion

+ cd

�S =

✓4⇡

✏kBT

q2de2cd

◆1/2

What is the effective charge of a defect?

Defect charge determined by a balance between: • Self energy of an ion • Self energy of a defect

+-

+- +-+

-+

-

+-

+

-+

-+-

+

-+-

+-+- +

-+ -

+

-

+-

+- +-+

-+

-

+-

+

-+

-+-

+

-+-

+-+- +

-+ -

+

-

Scaling theory of the fluctuation energies

Energy density of a dilute electrolyte eDH ⇠ ��3D

Energy density of a concentrated electrolyte

eCE ⇠ l�3B

The buffet of length scales: �D, lB , a

Fluctuation energy of a single defect

ed ⇠ q2d

eCEa3 ⇠ ed =) q2d ⇠ (a/lB)

3

Scaling theory of the screening length

�S

�D⇠ (4⇡(c

tot

� cion

)a3/l2B)�1/2

(4⇡cion

lB)�1/2

⇠ C

✓a

�D

◆3

81

2

4

6

810

2

4

68

100

λ S / λ D

6 80.1

2 4 6 81

2 4 6 810

a / λD

1

3

Pure aprotic ILs Pure protic ILs IL in PC NaCl in water LiCl in water KCl in water CsCl in water

AAL et al., Phys. Rev. Lett., 119, 26002 (2017)

When is the scaling regime reached?

+

- +

-

--

+--+-+

+ --⌦�Q2

↵⇠ N

ion

⇠ ⇢r3

Gaussian statistics

Efluct ⇠ kBTlBr

⌦�Q2

↵Fluctuation energy

Efluct ⇠ kBT lB⇢r2

Minimal blobkBT lB⇢a

2 ⇠ kBT =) a/�D ⇠ 1

r

From the screening length to energy storage

Differential capacitance at zero voltage

C0 =✏

4⇡

1

�s + a

C =@�

@V

1

C=

1

CS+

1

Cd

The good, the bad, and the ugly

• Predicts the maximum in capacitance at pzc as a function of concentration

• Does not capture the capacitance of pure/concentrated ILs

• Why do ILs have large capacitance but long screening length?

Bozym et al., J. Phys. Chem. Lett., 6, 2644 (2015)

0 0.5 1 1.5 2 2.5 3 3.5Concentration/M

0

5

10

15

20

25

Cd/µ

F c

m-2

Expt. [C2Im][NTf

2]

Prediction using measured screening length of [C

4C

1Pyrr][NTf

2] (Eq 23)

Prediction using measuredscreening length of pure[C

2Im][NTf

2]

AAL et al., Faraday Discussions, 199, 239 (2017)

The interface matters+V-

----

-

+

+

• Monolayer (Stern) layer of ions adsorbed onto the electrode

• Quasi-2D confinement

• Ion-ion interactions

Uion�ion

⇠ 1/r3

• Ionophilicity Ion-surface interaction energy relative to solvent-surface interaction energy

AAL, S. Perkin, J. Phys. Chem. Lett, 7, 2753 (2016)

Capacitance maximum as a function of dilution

D. J. Bozym et al., J. Phys. Chem. Lett., 6, 2644 (2015)

-5 0 5δ µ

25

30

35

40

45

50

55

CD

(0)/ µ

F c

m-2

“Ionophilicity”

Mean field theory predicts voltage-induced phase transition

0 10 20 30 40 50u

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-ca2

Cha

rge

dens

ity

Applied potential

�µ = �2

�µ = 2

�µ = 0

AAL, S. Perkin, J. Phys. Chem. Lett, 7, 2753 (2016)

MD simulations of phase transition in the electrical double layer

C. Merlet et al., J. Phys. Chem.: C, 118, 18291 (2014) B. Rotenberg, M. Salanne, J. Phys. Chem. Lett., 6, 4978 (2015)

Increasing the energy density by increasing electrode surface area

-+- +

++ --

-

power

capacitance

hysteresis

- +

+V

+V

--- --

--

Smaller pores are better at storing ions

Largeot et al., JACS, 130 (9), 2730 (2008)

Ion size

Using ionophilicity to resolve the capacitance-power dilemma

wide pores 6= low capacitance!

0

30

60

90

1 1.5capa

cita

nce

(µF

/cm

2)

pore width, L/d

strongly ionophilic

ionophobicweakly

(a)

0

1

1 1.5

d2ρto

t

pore width

5

10

15

20

1.5 2capa

cita

nce

(µF

/cm

2)

pore width, L/d

d = 0.7nm

(b)

0

0.2

0.4

1.5 2

η 3D

pore width

10

20

30

1.5 2capa

cita

nce

(µF

/cm

2)

pore width, L/d

(c)

0.5nm

0.6nm

0

0.2

0.4

1.5 2

η 3D

pore width

strongly ionophilic weakly ionophobic

AAL et al., Phys. Rev. X., 6, 21034 (2016)

The upshot• Scaling theory derived by considering an ideal crystal

with defects

• The inconsistency between the long screening length and large interfacial capacitance in ILs highlights the role of the interfacial layer

• Ionophilicility controls the structure of the interfacial layer

• Engineering ionophilicity is key to designing nanoporous supercapacitors

Physics of long range correlations also features in machine learning!

Liquid State Theory Meets Deep LearningAlpha A. Lee

Department of Physics, University of Cambridge (aal44@cam.ac.uk)

A large class of problems in machine learning pertains to making sense of unlabelled data. The challenge lies in separating direct variable-variable interactions (e.g. cause and effect) and transitive correlations. We develop an Ornstein-Zernike approach for data analysis with a closure parameterised by deep learning to disentangle correlations in datasets.

Unsupervised machine learning

Maximum entropy model

Q: What is the probability distribution that generates this dataset?

A: The Ising model is the maximum entropy model that captures the first and second order statistics.

p(�) =1

Zexp

0

@X

i<j

Jij�i�j +

X

i

hi�i

1

A

J matrix Covariance matrix

“Friend of a friend”

The “exact” inference algorithm (Boltzmann learning):

Computationally intractable in the “big data” limit!

hn+1i = hn

i + ⌘⇣h�iidata � h�iihn,Jn

⌘

Jn+1ij = Jn

ij + ⌘⇣h�i�jidata � h�i�jihn,Jn

⌘

Ornstein-Zernike theory

E. T. Jaynes, Physical Review, 106, 620 (1957)D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, Cognitive Science, 9, 147 (1985)

rij

Intermolecular potential:

1 2 3 4r/σ

1

2

3

4

5

g(r)

0

1 r/σ

u(r) ∞

u(rij)

g(rij)Radial distribution function:

The Ornstein-Zernike model defines the direct correlation function

h(rij) = c(rij) +

Zdrk c(rik)c(rkj) +

Zdrk

Zdrl c(rik)c(rkl)c(rlj) + · · ·

= c(rij) +

Zdrk c(rik)c(rkj)

A closure is needed to solve the Ornstein-Zernike equation

u(r) = f [h(r), c(r); ⇢]

The commonly used closures (e.g. MSA, HNC, PY) are local functions (i.e. not functional)

u(r) ⇡ g(h(r), c(r); ⇢)

Deep learning meets Ornstein-Zernike

Covariance matrix Cij h(rij)

Direct correlation function

“Direct coupling” matrix Dij c(rij)

Total correlation function

Cij = Dij +X

k

DikDkj +X

kl

DikDklDlj + · · ·

D = (I� C)�1

Ornstein-Zernike theory for data analysis

Closure parameterised by deep learning on simulations of Ising models using the locality approximation

Jij = F (Cij , [C�1]ij , h�ii , h�ji)

⌘ = 0.47

hi = G

0

@tanh�1 h�ii ,⇥C�1

⇤ii,X

j 6=i

Jij h�ji ,X

j 6=i

Cij h�ji

1

A

Σ

Generative model in biochemical spaceThe deep learning closure is generalisable

1D ferromagnetic Ising model

�J ⇠ uniform(0.5, 2.5)

�h ⇠ uniform(0.5, 2.5)

Jij = Jji ⇠ N (0,�J/pp)

hi ⇠ N (0, 0.3�h)

Training data: 20 independent simulations with

Sherrington-Kirkpatrick model

Virtual screening for drug discovery

HO

HO

OHHN

(0 1 0 1 0 0 1 0 1 0 1 0 …)

C-C C-O C-N C-C-NO-C-O

Chemical structure to binary vector using the Morgan algorithm

Predicting protein function from evolutionary data

Binding to endrogen receptor

Samples from patients

Sequence of HIV Gag protein

A A Lee, arXiv:1706.08466 (2017)

0 0.2 0.4 0.6 0.8 1False positive

0

0.2

0.4

0.6

0.8

1

True

pos

itive

In-sample classification(AUC = 0.88)Out-of-sample classification(AUC = 0.75)

0.5 1 1.5 2 2.5 3 3.5β

00.10.20.30.40.50.60.70.8

RM

S(J ij -

J ijpre d

)

Neural NetworkThouless-Anderson-Palmerapproximation

1 1.5 2 2.5J

0

1

2

3

4

5

6

RM

S(J

- Jpr

ed)

0 0.5 1RC/RCWT

-15

-10

-5

0

EW

T -E

r = 0.86

I will be presenting a poster on liquid state theory and deep learning.

Come find me during the poster session!