Ions in concentrated electrolytes: from the screening length to energy storage

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1 Ions in concentrated electrolytes: from the screening length to energy storage Alpha Lee Department of Physics, University of Cambridge

2 The standard model DLVO theory U(r) Z 2 l B r e r/ D r l B = e2 k B T D =1/ p 4 l B c

Concentration a 12 λ S (nm) 10 8 6 DebyeHückel screening length!

3 Concentration a 12 λ S (nm) DebyeHückel screening length! [C 4 C 1 Pyrr] [NTf 2 ] _ Propylene carbonate 4 2 Na Cl H 2 O b c 1/2 (M 1/2 )

4 Concentration a 12 λ S (nm) DebyeHückel screening length! [C 4 C 1 Pyrr] [NTf 2 ] _ Propylene carbonate 4 2 Na Cl H 2 O b c 1/2 (M 1/2 )

5 Concentration a 12 λ S (nm) DebyeHückel screening length! [C 4 C 1 Pyrr] [NTf 2 ] _ Propylene carbonate 4 2 Na Cl H 2 O b c 1/2 (M 1/2 )

6 Concentration a 12 λ S (nm) DebyeHückel screening length! [C 4 C 1 Pyrr] [NTf 2 ] _ Propylene carbonate 4 Na Cl H 2 O 2 b c 1/2 (M 1/2 ) AAL et al., Faraday Discussions, 199, 239 (2017)

7 Dimensional analysis Salient length scales: Bjerrum length l B = e2 k B T Debye length D =1/ p 4 l B c Ion diameter a

8 Scaling analysis of the screening length Pure aprotic ILs Pure protic ILs IL in PC NaCl in water LiCl in water KCl in water CsCl in water 2 λ S / λ D S l B ca a / λ D AAL et al., Phys. Rev. Lett., 119, (2017)

9 Activity coefficient Compute the field around an ion S Debye charging process µ ex = 1 2 l B a a S a

10 Comparison with experimental data NaCl in water µ ex Direct measurement Prediction using measured screening length Concentration / M W. J. Hamer and YC Wu, J Phys. Chem. Ref. Data, 1, 1047 (1972)

11 Ionic crystal analogy Complete charge ordering Behaves as a dielectric

12 Schottky defects Defects behave as charges The defects are mobile Debye screening

13 Concentrated ionic solutions as a defectladen ionic crystal A perturbative theory around a disordered ionic crystal, not a dilute gas S = 4 k BT q 2 d e2 c d 1/2 The role of defects played by neutral solvent molecules Incompressibility: c tot = c ion c d

14 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

15 Scaling theory of the fluctuation energies The buffet of length scales: D, l B, a Energy density of a dilute electrolyte e DH 3 D Energy density of a concentrated electrolyte e CE l 3 B Fluctuation energy of a single defect e d q 2 d e CE a 3 e d =) q 2 d (a/l B ) 3

16 Scaling theory of the screening length S D (4 (c tot c ion )a 3 /l 2 B ) 1/2 (4 c ion l B ) 1/2 C a D 3 λ S / λ D Pure aprotic ILs Pure protic ILs IL in PC NaCl in water LiCl in water KCl in water CsCl in water a / λ D AAL et al., Phys. Rev. Lett., 119, (2017)

17 When is the scaling regime reached? r Fluctuation energy E fluct k B T l B r Gaussian statistics Q 2 N ion r 3 E fluct k B Tl B r 2 Q 2 Minimal blob k B Tl B a 2 k B T =) a/ D 1

18 From the screening length to energy storage Differential capacitance at zero voltage 1 C = 1 1 C S C d C 0 = 4 1 s a

19 The good, the bad, and the ugly C d /µf cm Expt. [C 2 Im][NTf 2 ] Prediction using measured screening length of [C 4 C 1 Pyrr][NTf 2 ] (Eq 23) Prediction using measured screening length of pure [C 2 Im][NTf 2 ] Concentration/M 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) AAL et al., Faraday Discussions, 199, 239 (2017)

20 The interface matters V Monolayer (Stern) layer of ions adsorbed onto the electrode Quasi2D confinement Ionion interactions U ion ion 1/r 3 Ionophilicity Ionsurface interaction energy relative to solventsurface interaction energy AAL, S. Perkin, J. Phys. Chem. Lett, 7, 2753 (2016)

21 Capacitance maximum as a function of dilution C D (0)/ µ F cm δ µ Ionophilicity D. J. Bozym et al., J. Phys. Chem. Lett., 6, 2644 (2015)

22 AAL, S. Perkin, J. Phys. Chem. Lett, 7, 2753 (2016) Mean field theory predicts voltageinduced phase transition µ = 2 Charge density ca µ =2 µ = u Applied potential

23 MD simulations of phase transition in the electrical double layer C. Merlet et al., J. Phys. Chem.: C, 118, (2014) B. Rotenberg, M. Salanne, J. Phys. Chem. Lett., 6, 4978 (2015)

24 Increasing the energy density by increasing electrode surface area capacitance hysteresis power V V

25 Smaller pores are better at storing ions Ion size Largeot et al., JACS, 130 (9), 2730 (2008)

26 Using ionophilicity to resolve the capacitancepower dilemma capacitance (µf /cm 2 ) d 2 ρtot pore width strongly ionophilic pore width, L/d (a) weakly ionophobic capacitance ( /cm 2 ) wide pores 6= low capacitance! AAL et al., Phys. Rev. X., 6, (2016)

27 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! I will be presenting a poster on liquid state theory and deep learning. Liquid State Theory Meets Deep Learning Alpha A.

The challenge lies in separating direct variablevariable interactions (e.g. cause and eﬀect) and transitive correlations.

Maximum entropy model OrnsteinZernike theory 0 X 1 p( ) = exp @ Jij Z i<j J matrix i j X i 1 hi i A Intermolecular potential: u(rij ) Radial distribution function: g(rij ) Covariance matrix u(r) 5 =

28 Physics of long range correlations also features in machine learning! I will be presenting a poster on liquid state theory and deep learning. Liquid State Theory Meets Deep Learning Alpha A. Lee Department of Physics, University of Cambridge (aal44@cam.ac.uk) Unsupervised machine learning A large class of problems in machine learning pertains to making sense of unlabelled data. The challenge lies in separating direct variablevariable interactions (e.g. cause and eﬀect) and transitive correlations. We develop an OrnsteinZernike approach for data analysis with a closure parameterised by deep learning to disentangle correlations in datasets. Maximum entropy model OrnsteinZernike theory 0 X 1 p( ) = Jij Z i<j J matrix i j X i 1 hi i A Intermolecular potential: u(rij ) Radial distribution function: g(rij ) Covariance matrix u(r) 5 = 0.47 g(r) rij r/σ r/σ Friend of a friend 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. The exact inference algorithm (Boltzmann learning): hn1 = hni h i idata h i ihn,jn i n1 n Jij = Jij h i j idata h i j ihn,jn Computationally intractable in the big data limit! The OrnsteinZernike model defines the direct correlation function h(rij ) = c(rij ) = c(rij ) Z Z drk c(rik )c(rkj ) Z drk Z drl c(rik )c(rkl )c(rlj ) drk c(rik )c(rkj ) A closure is needed to solve the OrnsteinZernike equation u(r) = f [h(r), c(r); ] The commonly used closures (e.g. MSA, HNC, PY) are local functions (i.e. not functional) Come find me during the poster session!

Lecture 3 Charged interfaces

Lecture 3 Charged interfaces rigin of Surface Charge Immersion of some materials in an electrolyte solution. Two mechanisms can operate. (1) Dissociation of surface sites. H H H H H M M M +H () Adsorption