Charged systems in, out of, and driven to equilibrium : from nanocapacitors to cement

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from nanocapacitors to cement

Ivan Palaia

To cite this version:

Ivan Palaia. Charged systems in, out of, and driven to equilibrium : from nanocapacitors to cement. Soft Condensed Matter [cond-mat.soft]. Université Paris Saclay (COmUE), 2019. English. �NNT : 2019SACLS398�. �tel-02926717�

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and driven to equilibrium:

from nanocapacitors to cement

Th`ese de doctorat de l’Universit´e Paris-Saclay pr´epar´ee `a l’Universit´e Paris-Sud Ecole doctorale n◦_{564 Physique en ˆIle-de-France (EDPIF)}

Sp´ecialit´e de doctorat : Physique

Th`ese pr´esent´ee et soutenue `a Orsay, le 15 novembre 2019, par

I

P

Benjamin Rotenberg

Directeur de recherche, Physicochimie des ´Electrolytes et

Nanosyst`emes Interfaciaux, Sorbonne Universit´e et CNRS Pr´esident

Rudolf Podgornik

Professeur, Institute of Physics, Chinese Academy of Sciences Rapporteur

Ren´e van Roij

Professeur, Institute for Theoretical Physics, Utrecht University Rapporteur Emanuela Del Gado

Professeur, Department of Physics, Georgetown University Examinatrice

Patrick Guenoun

Directeur de recherche, Laboratoire sur l’Organisation

Nanom´etrique et Supramol´eculaire, CEA Saclay et CNRS Examinateur

Manoel Manghi

Maˆıtre de conf´erences, Laboratoire de Physique Th´eorique,

Universit´e de Toulouse et CNRS Examinateur

Emmanuel Trizac

Professeur, Laboratoire de Physique Th´eorique et Mod`eles

This thesis is the result of my work and of the patient and bright supervision of my PhD advisor and scientific father Emmanuel Trizac. When I became his student, I thought that understanding a physical phenomenon amounted to solve a few equations: the more difficult the equations, the more honorable the accomplishment. In the last three years, he taught me that mathematical rigor serves physical intuition, as interesting science is not in proving the obvious, but in seeing the non-obvious. Emmanuel has been at my side when I needed and has let me go on my own when I was ready to. He has been firmly demanding when he felt I could give more and honestly shared my excitement when I made important progress. He is the researcher I aspire to become.

I would like to thank Emanuela Del Gado for warmly welcoming me at Georgetown University, where she enthusiastically introduced me to the fascinating unanswered questions in the physics of cement. Working with her and Abhay Goyal, her student, was hard but exciting: the quality of the results we obtained was a reward beyond any expectation. I would also like to thank Benjamin Rotenberg and Adelchi J. Asta, his student, for the ongoing collaboration, for teaching me Lattice-Boltzmann and for the numerous enlightening discussions we had. Thanks to Patrick B. Warren, who hosted me at Unilever and taught me a lot about numerical calculus and diffusiophoresis, to Alexandre Pereira dos Santos and Roland R. Netz for their hospitality in Berlin, and to Ladislav Šamaj, who was my academic uncle during my first year.

I would like to acknowledge the European Union’s Horizon 2020 research and in-novation program for giving me the privilege to work as a Marie Skłodowska Curie fellow. I was honored to be part of the big NANOTRANS family, guided by Daan Frenkel, Jure Dobnikar, and Benjamin Rotenberg, who accompanied me during these three years. I met in there young hard-working and resourceful scientists, who are now my friends. I wish them all the best and cannot wait for our next late-hour chat on the origin of happiness and love.

The Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS) has been a great work environment, thanks to all of its components. I would especially like to mention Mathieu Isoard and Nicolas Pavloff for their useful comments on parts of this manuscript, Martin Lenz and Alberto Rosso for helping me planning my future in research, and Claudine Le Vaou and Karolina Kolodziej for their tireless work and the gourmand encouragements of the last months. It has been a delight to share my PhD life with all the students and postdocs who passed through LPTMS: our lab weekends were awesome. I will especially acknowledge my academical brother Lucas for creating since the beginning a great connection and so boosting productive interference between our complementary knowledge of electrostatics and colloid science. And I cannot not mention my historical office mates Mathieu, Samuel and Thibault, our unconventional tennis matches, our improvised dinners and our peculiar taste in office decoration.

I would like to thank the many people that have always supported me, filling my life with their enthusiasm: all my family, in particular my cousin Celeste, my long-date friends from Viterbo, and all the Italian-French-Tunisian friends who orbited around our home spreading joy and laughter over our evenings, nights and weekends. I hate to admit that my housemates Bartolo, Luca and Orazio have been amazing: they do deserve a full-fledged acknowledgement here, as I would have probably starved to death without them, during the writing of this thesis. They have been and will always be my strong-opinioned, fake bobo, noisy and trouble-seeking Parisian family.

Last but not least, my mum and dad, to whom I owe my passion for Physics. Without their patient education, that made me curious and interested in the most different topics since my childhood, I would have not gone so far. No matter all the difficulties that they, we, have gone through, they have always backed me up when it came to following my inclinations and dreams. Despite all, they coped with distance in the best way possible and never questioned my life choices. They would think they are lucky to have me, but I am actually the lucky one.

Acknowledgements iii

General introduction 1

1 Dynamics of the electric double layer in mean-field 5

1.1 Introduction to mean-field . . . 5

1.1.1 The Poisson-Boltzmann equation . . . 5

1.1.2 Relevant lengths . . . 8

1.1.3 The Poisson-Nernst-Planck equation . . . 10

1.2 The Electric Double Layer Capacitor . . . 11

1.3 Numerical methods . . . 13

1.4 Relaxation dynamics in planar geometry . . . 15

1.4.1 Linear regime (v → 0) . . . . 16

1.4.2 Depletion . . . 24

1.4.3 Purely nonlinear regime . . . 27

1.4.4 Fully depleted nonlinear regime . . . 30

1.5 Relaxation dynamics in cylindrical geometry, linear regime . . . 33

1.6 Shortcut to adiabaticity . . . 36

1.6.1 Introduction to STA . . . 36

1.6.2 Accelerating the electric double layer relaxation in an EDLC . . 38

1.6.3 Accelerating electroosmotic flow relaxation in an EDLC . . . 44

1.7 Conclusions and perspectives . . . 48

2 From weak to strong coupling: a correlation-hole approach 51 2.1 Introduction to strong coupling . . . 51

2.1.1 Ionic correlations, like-charge attraction and overcharging . . . . 51

2.1.2 The coupling parameter . . . 52

2.1.3 The Wigner crystal . . . 54

2.1.4 The two-wall system . . . 56

2.1.5 Open questions . . . 62

2.2 A correlation-hole approach . . . 64

2.2.1 Model and notation . . . 64

2.2.2 Inclusion of the correlation hole . . . 65

2.2.3 Analytical results . . . 67

2.2.4 Numerical results . . . 68

2.2.5 Extension to the two-wall problem . . . 75

2.3 Conclusions . . . 77

3 Strong coupling: the case of cement 79

3.1 Cement . . . 79

3.1.1 Generalities . . . 79

3.1.2 A long history . . . 80

3.1.3 A multiscale Physics . . . 84

3.2 The model and the effect of solvent . . . 93

3.2.1 The model . . . 93

3.2.2 Ion density . . . 94

3.2.3 Pressure . . . 94

3.2.4 Correlations . . . 96

3.3 Ion hydration under confinement . . . 98

3.3.1 n-mers and the limited resources argument . . . . 98

3.3.2 The ‘dry water’ picture . . . 100

3.4 Zoology of n-mers . . . 102

3.4.1 Energy and ground state . . . 102

3.4.2 Free energy and water adsorption . . . 104

3.4.3 Effect of soft walls and electric field . . . 106

3.5 Analytical equation of state . . . 108

3.5.1 Effective electric field and correlation-hole approach . . . 108

3.5.2 Ion density . . . 109

3.5.3 Generalized contact theorem . . . 110

3.5.4 Pressure . . . 112

3.6 Conclusions and perspectives . . . 113

General conclusion 115

Summary in French 119

Nature is charged. The electrostatic force is perhaps the one humans experience and underestimate the most in their everyday life: from friction to electronics, from scotch tape to building materials.

Soft matter is no exception. Indeed, soft systems are more often than not in contact with water. Ionic compounds dissociate into ions at contact with such polar solvent. Likewise, covalent compounds tend to release ions to the solution, often H+ or OH–, thus acquiring themselves a charge. This happens for small molecules, but also for macromolecules and for bigger and structurally more complicated colloids, in contact with water through their surfaces. The presence of nanoscopic mobile ions and of solvent intimately affects interactions between bigger charged objects. A key feature of soft matter systems appears here: the contrast between the size of the solute and that of solvent molecules and mobile dissolved ions. This is at the origin of the remarkable properties of such systems, and at the basis of their theoretical treatment.

Proofs of the importance of electrostatics at small scale are countless and come from inorganic matter as much as from the living world, from natural to synthetic compounds. Practically all consituents of living cells (and viruses) are charged: pro-teins, membranes, nucleic acids. The presence of strong electrolytes (dissolved salts) is even crucial for specific phenomena to take place and for the correct functioning of organisms. Particularly fruitful, in recent and less recent times, has been the study of the interaction between DNA elasticity and electrostatics: we mention DNA condensa-tion, [1–5], close-packing in viruses [6], and single-filament mechanics [7]. The folding and the functioning process of proteins, too, have made the object of numerous stud-ies, with the involvement of several disciplines, from biochemistry to machine learning: both these processes are profoundly influenced by electrostatic interactions among dif-ferent parts of a protein or between a protein and its surroundings [8, 9]. Neurons are thought to communicate via spikes of membrane electric potential, whereas ion exchange, often related to osmotic equilibrium, is essential in many processes, such as message transporting and energy storage at cellular level.

Aggregation and self-assembly of charged colloidal particles has also been a major field of research [10]. For that, the DLVO theory, managing to account both for the presence of ions in solution and for van der Waals forces between colloids, has been a successful tool [11]. To name but a few, a problem that has kept puzzling physicists, geophysicists and engineers for a while is that of clay swelling in water, and particular effort has been devoted to grasping the roles of geometry, confinement and correlation [12,13]. The recent advent of microgels with tunable surface charge density has allowed for new experimental evidence on the role of ion-ion correlation, which rekindles interest in this field [14,15].

Much motivation to pursue studies on the statics and dynamics of charged systems at smaller and smaller scales, comes from technology. Recent advances in microfluidics made the nanometric and even sub-nanometric scale accessible for controlled experi-ments on charge transport [16–18], with possible applications ranging from drinkable water filtering to renal replacement therapy [19]. Progress has been made in the field of ionic liquids and of nanocapacitors, porous materials with metallic behavior that can be used as fast rechargeable batteries [20]. In particular, the possibility to employ sim-ilar materials to extract energy, by constructing thermodynamic cycles alternatively flushing two electrolyte solutions at different concentrations, has given birth to the promising field of blue and lukewarm energy [21, 22]. All this, in turn, called for a more comprehensive study of phoretic phenomena, in particular of those concerning ions subjected to gradients of electric potential, pressure, temperature and concentra-tion [23, 24]. The applications of these studies reach areas as diverse as drug delivery and fabric cleaning [25].

From the theoretical point of view, charged systems are complex and elusive: ev-ery ‘particle’ interacts with evev-ery other ‘particle’. They do so in a long range fashion, through the Coulomb potential: this makes it impossible to use standard tools of sta-tistical mechanics, for instance the virial expansion. To overcome this difficulty, several routes have been followed. In this work, we will be interested in two of them, that are complementary as for their range of applicability. First, we will employ mean-field theory, which represents a sufficient level of description for many systems and is today standard textbook material [26,27]: in particular, mean-field may become exact in the high-temperature low-density regime, and even describe accurately the thermodynamic behavior of suspensions with colloids bearing a large charge, as shown by experiments on osmotic pressure [28], for instance. On the other hand, mean-field proves unfit in situations where energy largely dominates over entropy and correlations become impor-tant. For these systems, which are better described by a close-to-ground-state physics, we will need to account for the discrete nature of ions. We will also address the question of closing the gap between mean-field theory and strong correlation theories, which has attracted sizeable effort [29].

The thesis is divided into three parts, each starting with an introduction to the problem addressed, with specific reference to the theoretical tools employed. In Chapter 1, we use mean-field theory to study the relaxation dynamics of an electric double layer capacitor (an elementary model of a nanocapacitor) when it is driven out of equilibrium. We then tackle the problem of how to find strategies to drive it from an equilibrium state to another in a shorter time than its natural relaxation time, by means of a smart time-dependent protocol.

In Chapter2, after explaining the strong coupling regime and presenting some new ideas about like-charge attraction, we propose a bridging theory between the mean-field picture and the highly correlated regime, that makes use of the correlation-hole concept.

Lastly, in Chapter 3, we explore the blurred boundary between statistical physics and material science and lay down a theoretical model for set cement: by analyzing the unexpectedly strong electrostatic correlations due to nontrivial solvent effects, we explain the nanoscopic origin of cohesion in this fundamental material, which is the most produced artificial substance in the world [30].

From this thesis were extracted Ref. [31], related to Chapter2, and Ref. [32], par-tially related to Chapter 1. Two more publications are being extracted from Chapters 1and3. We thank our collaborators, that are mostly acknowledged in the text and in-clude Adelchi J. Asta, Emanuela Del Gado, Abhay Goyal, Katerina Ioannidou, Roland J.-M. Pellenq, Benjamin Rotenberg, Ladislav Šamaj, Martin Trulsson, Patrick B. War-ren.

Dynamics of the electric double

layer in mean-field

In this Chapter, we introduce the mean-field description of electrolyte systems, through the Poisson-Boltzmann formalism. We discuss the Poisson-Nernst-Planck equation and use it to analyze the charging dynamics of an electric double layer capacitor, whose electrodes are connected to a power generator. A systematic study of the relaxation time required to reach equilibrium upon switching on the voltage source is presented. This is carried out in planar and cylindrical geometry, for the linear and nonlinear regimes, for binary electrolytes with symmetric ionic valences and diffusivities. Finally, the question of identifying a switch-on protocol specifically designed to fasten relaxation is addressed.

1.1

Introduction to mean-field

1.1.1 The Poisson-Boltzmann equation

A widely used and fruitful approach for dealing with electrolyte solutions has been since the beginning of the last century the Poisson-Boltzmann approximation. It owes its name to the fact that it results from the combination of a Poisson equation, describing the dependence of the electric potential on the charge density, with the equilibrium distribution of particles subject to a potential at finite temperature. Such equilibrium distribution is reminiscent of the Boltzmann weight, used, for instance, in partition functions. For reasons that will become more clear later in this Section and in Chapter 2, the Poisson-Boltzmann theory is a mean-field approximation, and often goes under the very name of mean-field theory. With a slight semantic transfer, it is sometimes also referred to as DLVO theory, where the acronym stands for the last names of B. Derjaguin, L. Landau, E. Verwey and J. Overbeek. DLVO theory includes van der Waals forces and is more properly aimed at explaining the interaction between two colloids in an electrolyte solution, but certainly makes use of the (linearized) Poisson-Boltzmann formalism.

Obviously not due to a direct collaboration between Poisson and Boltzmann, which would have violated a few laws of Physics, the Poisson-Boltzmann approximation dates back to the works of Gouy [33] and Chapman [34]. We will present it here in two equivalent ways: a very simple one, where a key assumption is made directly on the charge density, and a second one where a free energy functional is defined, whose

minimization yields the desired charge density [26,35,36].

For the time being, we analyze a system made of N particles, labeled by index i, that can be classified in different ionic species, labeled by index α or α_i, of identical particles. Each particle of species α has charge ¯qαe, where e is the elementary charge

and |¯qα| = qα. These particles generate, possibly together with external charges or

fields, an electrostatic potential. We call φ the average electrostatic potential and

nα the average density of species α, where the average is intended over the canonical distribution of the ions’ positions. Both φ and nαare functions of position r = (x, y, z)

in space, or, when dealing with systems with planar symmetry, as we often will, only of coordinate z. The space dependence will be usually made explicit. Particles (ions) move in a solvent, which is a structureless uniform dielectric of permittivity ε₀εr (with

ε0 the permittivity of vacuum).

In SI units, the ones that will be used through all this thesis, the Poisson equation reads −∇2_{φ(r) =} 1 ε0εr X α ¯ qαenα(r) , (1.1)

where the sum represents the average electric charge density ρ(r). This equation is exact: indeed, the Poisson equation for the microscopic electric potential and micro-scopic charge density takes the same form as (1.1) and taking a canonical average on both sides gives precisely (1.1). Exploiting translational invariance, one can assume the origin of our spatial reference frame to be the position of a particle of type α0: in doing so, on the right-hand side the sum becomes

ρ(r) =X

α

¯

qαen∞_α gα0_α(r) , (1.2)

where n∞_α is the average density of particles of kind α at infinite distance and gα0_α(r)

is the pair correlation function related to the distribution of particles α at distance

r from a particle of kind α0. It can be shown [37] that the pair correlation function

gα0_α(r) is linked to the pair potential of mean force w_α0_α(r). This object is a potential

energy, whose negative gradient equals the average force between two particles of kind

α0 and α placed at distance r one from the other (the canonical average is taken over the positions of the remaining N − 2 particles). The exact relation between gα0_α(r) and wα0_α(r) is

gα0_α(r) = e−βwα0α(r)_{, (1.3)}

where β = (k_BT )−1, k_Bis the Boltzmann constant and T is temperature. If substituted into Eqs. (1.2) and then (1.1), this yields

−∇2φ = 1 ε0εr X α ¯ qαen∞_αe−βwα0α(r)_{. (1.4)}

The Poisson-Boltzmann approximation consists precisely in replacing the poten-tial of mean force w_α0_α by the potential energy ¯qαeφ(r) of a particle α immersed in

the average potential created around the reference particle α0. In doing so, Eq. (1.4) becomes −d 2_φ dz2(z) = e ε0εr X α ¯ qαn∞α e−β ¯qαeφ(z), (1.5)

where, in passing, we replaced r by z: we can indeed suppose our reference particle

spatial variable is the distance z from its (practically planar) surface. wα0_α and ¯qαeφ

are not equal: while the former keeps into account how the average distributions of the surrounding ions (and therefore the electric potential) is modified by the presence of particle α approaching the reference particle α0, the latter completely neglects the discrete nature of ions and supposes that the approaching particle interacts with an average potential. The approximation is justified when moving an ion has little influence on the average potential created around any other particle. This usually happens in situations where ions are sufficiently small for steric effects to be negligible, and where the importance of the discrete detail of ion-ion electrostatic interactions is overcome by the smearing effect of temperature. A more quantitative assessment of this energy– entropy balance will be provided in Chapter 2.

Neglecting correlations

The mean-field nature of Eq. (1.5) becomes more evident when a density functional theory is written down for the charged liquid at hand. The average potential energy for a Coulombic system reads:

U = 1

2 Z

hρ_micro(r)φ_micro(r)i dr . (1.6)

In Eq. (1.6), the microscopic charge density is defined by ρ_micro(r) = ρ_ext(r) + P

αqαen¯ microα (r), where nmicroα =

P

i : αi=αδ(r−ri) is the microscopic density of particles

of kind α and ρext is some constant charge density imposed by the specific problem of interest (for instance the charge of the macroion of type α0 from the previous para-graph, that we replaced by a charged plane in writing Eq. (1.5)); φmicro is the electric potential generated by the microscopic charge density ρmicro. According to the super-position principle, φ_micro can always be expressed in term of some Green function G (which for Coulombic potentials is easily obtainable), so that

φmicro(r) =

Z

G(|r − r0|)ρ_micro(r0) dr0. (1.7)

In order to perform a minimization on a free energy functional to extract ex-tremal average densities nα, it is necessary to make a mean-field-like assumption on

Eq. (1.6): this consists in considering hρ_micro(r)φ_micro(r)i ' ρ(r)φ(r), or, equivalently, hρmicro(r)ρmicro(r0)i ' hρmicro(r)i hρmicro(r0)i = ρ(r)ρ(r0). Mathematically this amounts to setting identically to 0 the covariance of ρmicro(r) with ρmicro(r0), or, equivalently, with φ_micro(r). This is the definition of mean-field.

Under such assumption, Eq. (1.6) can be incorporated in a functional of the aver-age densities nα representing the grand-potential, that is the relevant thermodynamic

potential in the grand-canonical ensemble. Such functional is written as follows:

Ω = 1 2 Z ρ(r) z }| { ρext(r) + X α ¯ qαenα(r) ! G(|r − r0|) ρ(r0₎ z }| { ρext(r0) + X α ¯ qαenα(r0) ! dr dr0 + k_BTX α Z nα(r)ln(Λ3_αnα(r)) − 1dr − X α µα Z nα(r) dr . (1.8)

Here, Λα is the De Broglie length of species α, µα = kBT ln(Λ3αn0α) is the chemical

potential of species α and n0_αis the density of particles of species α in the reservoir. The last term enforces chemical equilibrium with an infinite reservoir (where we will take the potential to vanish, for convenience). The second term represents configurational entropy, as in an ideal gas. The first term is the average potential energy, derived from Eqs. (1.6) and (1.7) with the assumption that averages of products of microscopic densities can be substituted with products of average densities. Minimizing with respect to nα and recalling the definition of chemical potentials in terms of n0α’s, one gets

nα(r) = n0_αe−β ¯qαeφ(r). (1.9) Together with the Poisson equation (1.1) and taking into account the presence of ex-ternal charges, this gives

−∇2φ(r) = ρext(r) ε0εr + e ε0εr X α ¯ qαn0_αe−β ¯qαeφ(r). (1.10)

Eq. (1.10) is completely analogous to Eq. (1.5): in Eq.(1.5), external charges, which are usually singular, are to be imposed through boundary conditions, and the bulk far away from the plate plays the role of a reservoir (n∞_α = n0_α). It is useful to think of n0_α as the density of species α at φ = 0, irrespectively of the reference potential chosen for

φ.

It is important to remark that the reservoir densities can be retrieved by exploiting the electroneutrality condition of the reservoirP

αqαn¯ 0α = 0. Taking as example a 1:1

electrolyte (meaning that ¯q− = −1 and ¯q+ = +1, with α ∈ {+, −}), electroneutrality gives n0₊ = n0₋= n0 and the following holds

n+(r) n−(r) = n0−e+βeφ(r)n0+e −βeφ(r)

= (n0)2. (1.11) This is true for any r, provided that the system is at equilibrium (otherwise the whole Poisson-Boltzmann picture fails). A direct application of such property is of interest in transient dynamical situations: the fact that the product n₊n−in Eq. (1.11) should be independent of r can be used as a necessary condition for identifying the attainment of equilibrium.

We conclude this Section remarking that even though it was presented in its grand-canonical version, the Poisson-Boltzmann approximation is valid in the same form (1.10) in the canonical ensemble as well. In this situation, the total number of particles is fixed and the constants n0_α, representing the densities that a hypothetical reservoir would have to have to be at equilibrium with the system, are determined by normal-ization. While the grand-canonical formalism is the appropriate one to describe open systems with salt, the canonical formalism describes either closed systems, or salt-free systems, where the number of counterions is imposed by the fact that their charge has to balance exactly the fixed external charge.

1.1.2 Relevant lengths

We define here a few noticeable lengths that will appear from the beginning until the end of this thesis.

The Bjerrum length

lB=

βe2 4πε0εr

is the natural length unit corresponding to the electrostatic potential. It is the distance at which two elementary charges e have an interaction energy equal to k_BT . It appears

naturally from the Poisson equation, for instance, if one considers e as unit for charges and kBT /e as unit for potentials. Intuitively, lB measures how close to each other free monovalent ions can be.

In problems with salt, such as the one we deal with in this Chapter, several kinds of ions are present and the relevant length is the Debye screening length λD. The Debye length arises if one linearizes the Poisson-Boltzmann equation and is the scale on which the electric potential varies. If we assume to be inside a solution, where ρ_ext = 0, and potential to be sufficiently small, Eq. (1.10) can be linearized as follows:

∇2φ(r) ' βe 2 ε0εr X α q_α2n0_αφ(r) = 4πlB X α q2_αn0_αφ(r) . (1.13)

This equation, besides being simpler to solve than the exact one, brings up the following length scale λD= 1 p 4πl_BP αqα2n0α . (1.14)

In the 1:1 electrolyte case, using again n0₊= n0₋ = n0, the Debye length (1.14) reduces to λD= 1 √ 8πlBn0 . (1.15)

In the simple 1D case of a semi-infinite system, the general solution to Eq. (1.13) is a decaying exponential with characteristic length λ_D; related short-range decays are found in other geometries. The coefficient in front of the exponential is proportional to the external charge placed at the boundary plane. This exponentially decaying behavior signals an efficient screening phenomenon in proximity of a fixed charge: assuming, for instance, this charge to be positive, positive ions (coions) will tend to stay away from it and negative ions (counterions) will tend to stay closer to it. At equilibrium, ions are arranged around the fixed charge in such a way that at a distance of a few λ_D’s from it, the object appears to be almost neutral.

The last length we examine is related to salt-free systems, where only counterions are present. We will deal with them in Chapters 2 and 3, and partially also in this Chapter. In this counterion-only case, screening is not efficient: in 1D, for instance, the density that solves the Poisson-Boltzmann equation decays only algebraically and the corresponding potential is logarithmic and unbounded. It remains possible and in fact convenient to define a characteristic length, giving information on the distance from a fixed charge, within which most counterions are placed (these counterions are sometimes said to be ‘condensed’). Such distance µ, called the Gouy-Chapman length, appears naturally from the solution of the Poisson-Boltzmann equation, but can also be retrieved through the following single-particle argument, thus proving the robustness of the information µ contains, reaching beyond mean-field (see Chapter2for a non-mean-field theory). Let us consider a flat wall of surface charge density eσ; this generates a constant electric field _2εeσ_0εr in front of it. An ion of charge qe interacts with this electric field through a linear potential. The distance from the wall at which the ion interacts with the wall with energy k_BT , in absolute value, is

µ = kBT

qe2 σ

2ε0εr

= 1

It is interesting to notice that µ depends on the charge of the fixed charged object: the higher σ, the closest ions tend to stay to the wall. This is not true in a salty system, as

λD does not depend on the charge of the object to be screened (too large charges can entail nonlinear effects that affect the shape of the potential at short distance, but not its intrinsic length scale, at least within the Poisson-Boltzmann formalism).

Both in systems with and without salt, around an external charge (often represent-ing the surface of a colloid or of a microscopic device), an oppositely charged region of space is formed. This is why ions within a distance λ_D (or µ if coions are absent) from the charged object are said to form an electric double layer.

It is important to underline at this point that mean-field theory is a valid approxi-mation only when l_B  λ_D or l_B  µ. In the opposite case, when l_B is large, a large energy penalty must be payed to move ions within the electric double layer, compared to the available thermal energy: hρmicro(r)ρmicro(r0)i in Eqs. (1.6) and (1.7) is dominated by a strong correlation among the positions of different ions and cannot be reduced to a product of average values. In other words, when l_B is large, the discrete nature of ions becomes relevant and, when computing averages, the Boltzmannn weight is dominated by steep minima of the configurational energy; conversely, when l_B is small compared to other lengths in the system, ionic populations can be treated as charged gases of in-homogenous densities nα, regardless of the precise microscopic structure of such gases.

A more detailed explanation of the regime where mean-field breaks down is presented in the first half of Chapter2.

1.1.3 The Poisson-Nernst-Planck equation

In a dynamical situation, all the observables presented so far will depend not only on space, but also on time t. There exists a dynamical variant of the Poisson-Boltzmann equation, governing the time evolution of average charge densities nα(r, t): the

so-called Poisson-Nernst-Planck equation, which consists of a drift-diffusion (or advection-diffusion) model for ionic currents, endowed with a continuity equation.

Let j_α(r, t) be the average current of ions of type α. If ions did not interact and the electric potential were due to an external force, average currents could be written as follows:

j_α(r, t) = −βDαnα(r, t) ¯qαe∇φ(r, t) − Dα∇nα(r, t) . (1.17)

The first term represents a drift current, obtained as the product of a density nα, of a

force expressed as the gradient of a potential energy, and of a mobility βD_α expressed in terms of the diffusivity Dα in agreement with the Einstein relation. The second term

has the form of a diffusion current, as defined by Fick’s law. The diffusivity tensor is here diagonal, meaning that the diffusion current of a species is not influenced by other species (cross-diffusion is neglected). Within the Poisson-Nernst-Plank approximation, Eq. (1.17) is assumed to hold for interacting ions, even though φ is the average electric potential that ions themselves contribute to generate, through the Poisson equation (1.1). Such an assumption is intuitively of the same kind as the one used in Sec.1.1.1 to introduce the Poisson-Boltzmann formalism. It is indeed straightforward to show that, in the static case, when currents vanish, Eq. (1.17) reduces exactly to Eq. (1.9).

For currents j_α, the following exact continuity equation must hold, ensuring mass and charge conservation in absence of generation or recombination phenomena:

∂nα

Substituting (1.17) in Eq. (1.18) yields

∂nα

∂t (r, t) = Dα∇ · ( nα(r, t) ∇β ¯qαeφ(r, t) + ∇nα(r, t)) , (1.19)

which, completed with the Poisson equation (1.1), constitute the Poisson-Nernst-Planck theory. Considered its mean-field nature, the limitations and range of application of Eq. (1.19) include the ones discussed in the previous Sections for the Poisson-Boltzmann equation.

It is interesting to note that Eqs. (1.19) can be easily coupled with the other rele-vant equation describing the dynamics of a solution, the Navier-Stokes equation. The simplest way to do so is to imagine an incompressible fluid (with density and viscosity equal to those of the solvent, to a first approximation), which at every point r and time t is subjected to a volume force of electrostatic origin given by the coupling be-tween ρ(r, t) and φ(r, t). Possibly, an external field can be applied: we will explore this possibility in Sec. 1.6.3, when we will deal with the electroosmotic flow induced by an external electric field on an electric double layer at equilibrium.

1.2

The Electric Double Layer Capacitor

The system we deal with in this Chapter is the so-called Electric Double Layer Capacitor (EDLC). This consists, like a normal capacitor, of two conductive surfaces connected to a power generator, which provides, in general, a time dependent potential difference. Inside the capacitor, there is a globally electroneutral ionic solution. The conductive surfaces are also called blocking electrodes: they are ideal metals at which no Faradaic reaction occurs (no electrons can pass from the electrode to the solution, or viceversa, so no oxidation or reduction of any chemical species is possible).

The study of the statics and, most importantly, the dynamics of electric double lay-ers in presence of constant potentials is of paramount importance in electrochemistry. Recent interest in the field has been particularly motivated by the study of nanocapaci-tors based on a porous conductive material, such as carbon, immersed in an electrolyte solution or in an ionic liquid. These nanocapacitors develop huge capacitances per unit mass, of the order of tens or hundreds of F/g: this is partly due to the large surface per unit mass of the porous materials they are composed of and partly to the micro-scopic charging mechanism, influenced by the pores microstructure and polarizability [38]. Carbon nanocapacitors are therefore a good and technologically viable candidate to build supercapacitors: these are objects that not only can store a lot of energy per unit mass, like batteries, but can also store it and release it fairly quickly, like usual capacitors, also because of the absence of any chemical reactions in their functioning mechanism.

Recent advances in nanofluidics also motivate interest in electrodynamics in pres-ence of charged surfaces at controlled potentials. For instance, the community studying ionic circuits has been trying to influence ionic currents by shaping channels appropri-ately and by inducing different charges on their surfaces [39,40]. An example is that of ionic diodes, devices that present properties of current rectification, analogous to those of electronic p-n junction diodes.

Another interesting field related to ion dynamics and to studying optimal protocols with respect to energy storage and energy loss, is that of so-called blue energy. The idea is to extract energy from a thermodynamic cycle, where an EDLC is alternatively

Figure 1.1: A cartoon representing a charged EDLC, as described in Sec.1.2, in planar geometry, for a 1:1 electrolyte. Red cations and blue anions are treated at mean-field level, as suggested by the color of the solution, representing charge density and shading continuously from red to blue. If V is constant from a certain time on, the charge density eventually reaches equilibrium, and the thickness of the electric double layer is

λD (assuming L large enough), as described in Sec. 1.1.2.

at contact with an electrolyte solution at high salt concentration and with one at low salt concentration [21]. An improvement of this concept has been lukewarm energy, i.e. the introduction of a temperature difference between the two solutions, which can result in improved work extraction [22].

In the following, we take inspirations from these fields of application and analyze a simple model of an EDLC that neglects any nonidealities, possibly due to non-infinite dielectric constant of the metal in the electrodes, inhomogeneous or irregular surface, violation of the mean-field assumption, or steric effects. This model can shed light on the physics of the system and can be used as a building block for more elaborate theories.

The model

Our ideal EDLC, depicted in Fig. 1.1, is treated at Poisson-Nernst-Planck level. For the time being, we neglect hydrodynamics, and suppose purely Coulombic interactions between ions and Coulombic and hard-core interactions between ions and walls. No absorption phenomena are considered, so the compact part of the electric double layer, usually called Stern layer, is assumed infinitely thin and neglected. The two plates, distant 2L from each other, are connected to a time-dependent ideal voltage source, imposing a potential difference 2V (t) between its two terminals.

The salt is a strong binary electrolyte (α ∈ {−, +}) and is therefore completely dis-sociated in the solvent: average densities are denoted n− and n+, integer ionic valences ¯

q− and ¯q+, and diffusion coefficients D− and D+. We will mainly deal with valence-symmetric (q− = q+= q) and diffusivity-symmetric (D− = D+= D) electrolytes, but for the sake of completeness we try to keep the notation general as far as possible. While the valence symmetry situation is verified by many ionic compounds, the assumption of equal diffusivities might need further justification: beside an argument of pure math-ematical convenience (having just one diffusion coefficient significantly simplifies the problem), it is possible to find anions and cations with similar diffusion coefficients, especially when they have similar effective sizes in the solvent (Stoke’s law).

In the planar geometry, the one that we analyze first, electrodes are parallel plates modeled as infinite charged planes. If the z axis is the one perpendicular to the planes, densities n± and potential φ are functions of the sole spatial coordinate z and of time

t, due to translational symmetry along the x and y directions, parallel to the planes.

Fig. 1.1 shows a sketch of the system, with size 2L and z = 0 in the center of the capacitor. We defer to Sec. 1.5 the description of the coaxial capacitor in cylindrical geometry.

Finally, it must be stressed that we work in the canonical ensemble, i.e. with a fixed number of ions per unit surface 2n0L. This defines n0, that from here on denotes the uniform density of salt when the power source is off and the capacitor discharged and at equilibrium. Since we will deal with applied voltages of the kind V (t) = V (t)Θ(t), with Θ the Heaviside function, n0also corresponds to the initial salt density, where the term ‘initial’ refers to time t = 0. In most regimes, more precisely when densities in the bulk solution (at the center of the capacitor) remain constant during the dynamics, the canonical ensemble is not quantitatively different from the grand-canonical ensemble. At the same time, choosing to work in the canonical ensemble allows to avoid the question of where exactly ions are injected into the system from the reservoir during the dynamics, which is very system-dependent.

1.3

Numerical methods

To solve the Poisson-Nernst-Planck equations (1.19) and (1.1), we developed a code in C that uses a flux-conservative finite-differences method, in collaboration with Patrick B. Warren (UNILEVER, Port Sunlight, UK). The code efficiently solves the planar case and its extention to the cylindrical geometry is straightforward, with no consequence, in principle, for its efficiency. Its main features are described in this Section.

Space is discretized into nodes, positioned at z_k+1 2

for k = 0, ..., N − 1, and edges, located at zk for k = 0, ..., N . If the spacing between nodes and between edges is

constant, we have z_k+1 2

= −L + (k +1₂)∆z and z_k= −L + k∆z, with ∆z = 2L/N . The extension to a nonlinear spacing is straightforward and useful. For simplicity, we will describe the algorithm assuming constant spacing.

Ion densities n± and potential φ are defined on nodes z_k+1 2

, while electric field and ionic currents are defined on edges zk; this entails a reduction of the error associated

with numerical derivation or integration, that is consistent and systematic throughout the whole algorithm. n± is initially set to its t = 0 value n0, for all nodes. Then the density and potential profiles are evolved in time in steps of ∆t. More precisely, at every step i:

– The ionic contribution to the electric field is computed via Gauss’ theorem, by numerically integrating the charge density defined as

ρ(z_k+1 2, i ∆t) = ¯q−n− (z_k+1 2, i ∆t) + ¯q+n+ (z_k+1 2, i ∆t) .

– The ionic contribution to the potential is computed by numerically integrating the ionic contribution of the electric field.

– The electrodes’ (constant) contribution to the electric field is computed by impos-ing that the overall potential difference across the capacitor (ions’ and electrodes’ contributions) be 2V (i ∆t). This corresponds to computing the surface charge density σ of the electrodes at time i ∆t, which is proportional to the electrodes’ electric field through the dielectric permittivity.

– The overall electric field (ions’ and electrodes’ contributions) determines the ionic currents through the discrete analogous of Eq. (1.17).

– Ion densities at time (i + 1)∆t are computed through the discrete analogous of Eq. (1.18), starting from the just determined currents at time i ∆t.

Iteration stops when the prescribed final time is reached.

The algorithm is flux-conservative [41] because it conserves the numerical integral of the density, for both ionic species, i.e. the total number of ions, up to machine precision. This is a consequence of the fact that density updates are computed by the finite-difference equivalent of Eq. (1.18). At any time t = i ∆t, the total number of ions is indeed X k n±(z_k+1 2, t + ∆t) ∆z = X k  n±(z_k+1 2, t) + ∆t j±(zk, t) − j±(zk+1, t) ∆z  ∆z = X k n±(z_k+1 2, t)∆z + ∆t (j±(z0, t) − j±(zN, t)) = X k n±(z_k+1 2, t)∆z , (1.20)

where the last equality follows from the zero-current condition on the electrodes, placed at z0 = −L and zN = +L. This property is exact for constant spacing ∆z, as shown,

but also for irregular spacings, provided that ∆z in Eq. (1.20) is replaced everywhere by (∆z)_k+1

2 = zk+1 − zk. The usage of irregular spacings, with nodes and edges

more dense close to the walls and less dense in the bulk, is essential to speed up the calculation when L is orders of magnitude larger than λ_D and it also favors stability in the nonlinear regime, as briefly discussed below. The irregular spacing often used in the following presents a linear distribution of nodes in the bulk region, transitioning to an exponentially denser distribution closer to the electrodes.

A necessary condition for the stability of the algorithm is that the time step ∆t ver-ify the Courant-Friedrichs-Lewy condition [41], requiring that ∆t < ∆z2/ maxα{Dα}.

Otherwise said, assuming a purely diffusive regime, the distance traveled by the fastest ‘particles’ in a time ∆t must be smaller than the lattice spacing. In the simulations reported in the following, the Courant factor ∆t/(∆z2/ maxα{Dα}) is between 0.2 and

0.9.

Lastly, it is necessary to have a sufficient number of nodes inside the double layer, for the results to be accurate and stable. This is particularly important when nonlinear

effects set in (densities and potential curves become much steeper in the double layer) and may require an irregular spacing. In addition, nonlinear effects favor advection with respect to diffusion and can elicit a rather violent response to the perturbation. A detailed study of the stability of the algorithm, including how the Courant condition is modified in the nonlinear regime is outside of the scope of this work.

The results of our numerical scheme were successfully compared to those of constant-potential Lattice-Boltzmann Electrokinetics [32], where Lattice-Boltzmann is coupled with an iterative resolution of the Poisson equation. For this and for our discussions concerning the next Section, we thank Adelchi J. Asta and Benjamin Rotenberg (Sor-bonne Université, Paris).

1.4

Relaxation dynamics in planar geometry

The question we address in this Section, with the help of the numerical tool described in Sec. 1.3, concerns the relaxation dynamics of the electric double layer in a planar EDLC, when the applied voltage passes abruptly from 0 to a certain value V₀ (i.e.

V (t) = V0Θ(t)).

The time-evolution of the system is governed by the 1D versions of the Poisson-Nernst-Planck equation (1.19), ∂n± ∂t (z, t) = D± ∂ ∂z  n±(z, t) ∂ ∂zβ ¯q±eφ(z, t) + ∂n± ∂z (z, t)  , (1.21)

and of the Poisson equation (1.1),

−∂ 2_φ ∂z2(z, t) = ρ(z, t) ε0εr , (1.22) with ρ(z, t) = q+en+(z, t) − q−en−(z, t).

The fact that ions are confined within the two slabs of the capacitor imposes the boundary condition of vanishing currents at +L and −L. From the definition (1.17):

−∂n± ∂z (−L, t) ∓ βeq±n±(−L, t) ∂φ ∂z(−L, t) = 0 −∂n± ∂z (+L, t) ∓ βeq±n±(+L, t) ∂φ ∂z(+L, t) = 0 . (1.23)

Finally, the potential must be continuous between slab and solution, so that

φ(±L, t) = ±V (t) , (1.24)

where the zero of the potential is arbitrary.

We focus on the case of a symmetric electrolyte (we take q− = q+ = q = 1, to simplify notation), with equal diffusivities D₊ = D− = D. The problem of studying relaxation in this case has been addressed previously, namely in [42] and lately in [43]. It will appear that the known solutions can be significantly simplified; besides, the question of characterizing clearly the nonlinear regime is still open. From a simple dimensional analysis, a few characteristic times can be constructed combining the two lengths L and λD with the diffusion coefficient D. In partial contrast with common intuition, the relaxation time in the linear regime has been shown in [42] to be of order

LλD/D. However, [42] assumes λD  L and makes use of a Padé approximant in the Laplace-domain, that only gives the correct relaxation time in the limit λ_D/L → 0. This

is not always the case in physical situations, where the distance between nanoporous parts of the electrodes is not so large compared to the Debye length. In addition, the functional form of relaxation is not evident in general. In principle, it is not even necessarily exponential, as demonstrated by the textbook case of a single plate of charge density σ0Θ(t) [27] (in this case the system is semi-infinite, i.e. L → ∞).

We show here that in the linear regime relaxation is multi-exponential and we com-pute the infinitely many relaxation times involved, using two different methods. We then perform a weakly nonlinear analysis to identify the onset of depletion phenomena. We finally analyze the fully nonlinear regime, that, depending on the importance of de-pletion, we subdivide into a purely nonlinear regime, a partially screened fully depleted nonlinear regime and an unscreened fully depleted nonlinear regime.

1.4.1 Linear regime (v → 0)

Since in Eqs. (1.21) the product n±φ appears, and n±and φ are linearly related through the Poisson equation (1.22), Eqs. (1.21) are in fact nonlinear. However, they can be linearized around the equilibrium values that φ and n±assume when the voltage source is off: namely, φ(z, 0−) = 0 and n±(z, 0) = n0. The linearization will give accurate results as long as |φ(z, t)|  k_BT /e and |n±(z, t) − n0|  n0during the evolution of the system; these conditions logically translate into |V (t)|  k_BT /e and |ρ(z, t)|  2n0e, respectively. Subtracting the linearized equation for n− from the linearized equation for n₊ yields the so-called Debye-Falkenhagen equation for the charge density ρ:

∂ρ ∂t = D ∂2ρ ∂z2 + 2βe 2_n 0 ∂2φ ∂z2 ! . (1.25) Using (1.22), we get ∂ρ ∂t = D ∂2ρ ∂z2 − κ 2_ρ ! , (1.26) where κ2 = 2n0βe 2

εrε0 = 8πlBn0 and corresponds to the inverse squared Debye screening

length λ_D (1.15), at salt concentration n₀.

We proceed by making the equations nondimensional: we measure lengths in units of L, times in units of L/(κD), potentials in units of kBT /e, electric charge density in units of 2en₀. From now on, x and t will denote the nondimensional coordinates and

ρ(z, t) and φ(z, t) will be nondimensional functions of nondimensional coordinates. The

system is completely described by the two dimensionless parameters  = (κL)−1 and

v = βeV0. Note that  can but need not be a small quantity, while, as long as we deal

with the linear regime, v has to be much smaller than unity. This change of units is summarized in Tables1.1and 1.2. Whenever it is useful to come back to physical units to discuss results, it will be stated explicitly that such units are used, unless clear form the context.

In dimensionless notation, Eq. (1.26) is readily rewritten as

∂ρ

∂t = 

2∂2ρ

Observable Symbol Unit Time t, τn L κD Inverse time s, sn κD L Distance z L

Volumic ion density n± n0

Volumic charge density ρ 2en₀

Electric potential φ, V 1

βe

Electric field E 1

βeL

Integrated charge density Q 2n0eL

Surface density σ ε0εr

βe2_L =

1 4πlBL

Table 1.1: List of units of the nondimensional quantities used in this Chapter. Note that they do not form a coherent system of units: for instance, the unit of distance L and the unit of volumic density n₀ are unrelated; also, Q is not measured in the same units as surface density (modulo a factor e), even though it is indeed a charge per unit surface.

Dimensionless Debye length  = 1

κL= 1 L r _ε 0εr 2n₀βe2  = λD L if n±(0, t) ' n0 

Dimensionless potential step v = βeV0

Table 1.2: The two dimensionless parameters governing the statics and dynamics of the system.

and the Poisson equation (1.22) reads

−2∂ 2_φ

∂z2 = ρ . (1.28)

The boundary condition (1.23) can be written in terms of the charge density upon linearization; its dimensionless version reads

−∂ρ

∂x(±1, t) − ∂φ

∂x(±1, t) = 0 . (1.29)

Finally, boundary condition (1.24) reads

Now that we have laid down a convenient system of units and have equations de-pending only on ρ and φ, we turn our attention to the relaxation dynamics. We present two equivalent derivations of the exact relaxation times in the linear regime: in the first one we make use of the Laplace transform to solve the Poisson-Nernst-Planck equation, as in [42], whereas in the second one we start from an ansatz about its solution and work in the time domain.

Laplace domain

Eq. (1.27) can be easily solved by using Laplace transforms, that we denote by hats and define as follows:

b

ρ(z, s) = L{ρ(z, t)} =

Z ∞

0

ρ(z, t) e−stdt . (1.31)

In the Laplace domain, knowing that ρ(x, 0) = 0 since we start from a discharged capacitor, Eq. (1.27) reads

1 + s

2 ρ =b

∂2ρb

∂z2. (1.32)

Any z-antisymmetric solution will be of the type

b

ρ(z, s) = A(s) sinh(kz) , (1.33)

where k(s) is in principle any of the two solutions of the equation

k2 = 1 + s

2 . (1.34)

In the following, we will always assume k to be the solution with positive imaginary part (thus operating a branch cut of the square-root function along the positive real axis) and we will see in a while that the complementary choice gives the same results. Using the the Laplace-domain version of Eq. (1.28) with Eq. (1.33), one gets the general form ofφ. Fixing the null-potential plane at z = 0, and imposing boundary conditionsb (1.29) and (1.30), again, in their Laplace-domain version, the following results can be obtained: b φ(z, s) = sinh(kz) + zk( 2_k2_{− 1) cosh(k)} sinh(k) + k(2_k2_{− 1) cosh(k)} V (s) ,b (1.35) b ρ(z, s) = −  2_k2_sinh(kz) sinh(k) + k(2_k2_{− 1) cosh(k)}V (s) .b (1.36) Other interesting (dimensionless) quantities are the surface charge density σ(t) on the left plate, that we express in units of ε0εr_βeL, the integrated charge density Q(t) = R0

−1ρ(z, t) dz, in units of 2n0eL, and the electric field E(0, t) at z = 0, in units of

(βeL)−1 (see Table1.1). Their Laplace transforms read

b σ(s) = −∂φb ∂z(−1, s) = −k coth(k) 2k2 1 + k(2_k2_{− 1) coth(k)}V (s) ,b (1.37) b Q(s) = Z 0 −1b ρ(z, s) dz = tanh( k 2) k 2_k2 1 + k(2_k2_{− 1) coth(k)}V (s) ,b (1.38) b E(0, s) = −∂φb ∂z(0, s) = " tanh(k₂) 2_k − k coth(k) # 2k2 1 + k(2_k2_{− 1) coth(k)}V (s) .b (1.39)

Note that these three quantities satisfy exactly Gauss’ theorem, in the form

b

σ(s) + 1

2Q(s) =b E(0, s) .b (1.40)

All these expressions for Laplace transforms of real observables are invariant under the transformation k → −k, meaning that considering the second solution of equation (1.34) gives the exact same results.

To study the time relaxation of the system to equilibrium, we recall that, by the Bromwich-Mellin inversion formula, we have

ρ(z, t) = L−1{ρ(z, s)} =_b 1

2πi

Z S+i∞

S−i∞ b

ρ(z, s) estds , (1.41)

for some real S such thatρ(z, s) has no pole lying on the right of S. We observe that,_b

if the integrand function is such that its integral over the semi-circumference of infinite radius in a left half-plane goes to zero, the integral amounts to the sum of the residues of the integrand. Hence, each non-zero pole of the function ρ(z, s) will contribute to_b

the inverse Laplace transform with a term esntRes b

ρ(z,s)(sn). The non-zero poles of ρ,b b

σ, Q orb E are thus the relaxation rates of our system. If a pole at 0 is present, itsb

residue will determine the equilibrium state at t → ∞.

For a step potential V (t) = vΘ(t), one has V (s) = v/s, so that the non-zero polesb of functions (1.35) to (1.39) all coincide. Indeed, such quantities can all be expressed as products of entire functions (except in 0) times the following transfer function H(s):

H(s) =  2_k2 1 + k(2_k2_{− 1) coth(k)} = 1 + s 1 + s√1 + s coth √ 1+s   (1.42)

The same branch cut introduced before for the square root is used.

We denote the poles of H(s) by s_n, with n ∈ {0, 1, 2...}, and we order them such that the higher their index n, the more distant they are from the origin. They all lie on the negative real axis and are all single poles. To identify them, we analyze first the case s < −1_, and then s > −1_; at s = −1_, H(s) has a removable singularity, which is immaterial.

In the case s < −1_, we set √1 + s = iY with Y real and positive. Finding the poles of (1.42) amounts therefore to finding the roots of the transcendental equation

tan _Y   = Y   Y2+ 1, (1.43)

taken from equating to 0 the denominator of Eq. (1.42). They can be easily found graphically and numerically; they are included in the interval πn < Y_n < π(n +1₂), with n ≥ 1 if  ≤ √1

3, or n ≥ 0 if  > 1 √

3. By definition of Y , this corresponds to poles

sn such that −1  − π 2_{n +}1 2 2 < sn< −1 − π 2_n2 _(1.44) with    n ≥ 1, if  ≤ √1 3 n ≥ 0, if  > √1 3 (1.45)

0 0 0.1 0.2 0.3 ϵ 0.7 0.8 0.9 1 1 τ0 (a) 10-6 10-3 ₁ 103 ϵ 10-4 10-2 1 τ0 (b)

Figure 1.2: (a) In blue, the longest relaxation time τ0 as a function of , obtained numerically from Eq. (1.46); in orange and green, estimates of the relaxation times of

ρ(−1, t) and Q(t), respectively, according to [42]: the analysis of Ref. [42] is valid only

for small  and the fact that the green and the orange curve differ and that they differ from the blue curve is an artifact of the analytical procedure followed therein. (b) The same plot in logarithmic scale. The gray line indicates  = √1

3, so that the points on its left were obtained from Eq. (1.46), while the points on its right from Eq. (1.43). At small , τ0 is constant (∼ Lλ_DD in physical units), while at large  it decays as −1 (∼ L

2 D

in physical units).

As n → ∞, s_n approaches the left extremum of the interval and scales as n2.

In the case s > −1_, we set √1 + s = Z with Z real and positive. Now, find-ing the zeroes of the denominator of H(s) amounts to findfind-ing the roots of another transcendental equation: tanh _Z   = −Z   Z2− 1 (1.46) If  ≤ √1

3, this equation has only one root − 1

 < s0 < 0. If  >

1 √

3, it has no roots. Summarizing, the infinitely many poles sn of observables (1.35) to (1.39),

corre-sponding to their relaxation rates in the time domain, are to be found as follows. For

n ≥ 1, the poles sn are the roots of Eq. (1.43) over the intervals (1.44); for n = 0, the

pole s₀ corresponds to the root of Eq. (1.43) over the interval −1_ −π2

4 < s < − 1

 if  ≥ √1

3, and to the root of (1.46) if  < 1 √

3. Out of all these infinitely many relaxation rates sn, the most interesting is probably s0, because it corresponds to the slowest relaxation time τ₀ = −s−1₀ . The behavior of τ₀ as a function of  is illustrated in Fig. 1.2, where it is compared with existing estimates from [42]; Fig. 1.3 represents instead poles sn and times τn up to n = 10, for  = 0.1 (Fig. 1.3a) and for  = 10

(Fig.1.3b).

Studying numerically the value of these relaxation rates, one can identify two limits. When   1, s₀ ∼ −1, so the slowest time scale τ₀ = −s−1₀ tends to 1 (in dimensioned time LλD

D ), as shown in Fig. 1.2; all the other sn lie on the left of −

1

, corresponding

to infinitely many shorter relaxation times τn, smaller than  (in dimensioned time λ2

D

D ) and accumulating around 0 (see Fig. 1.3a). When   1, s0 ∼ − π2

4 , so the slowest time scale τ₀ is close to _π42_ (in dimensioned time 4L

2

π2_D, see again Fig. 1.2b); 0

is still an accumulation point for the other time scales τn (see Fig. 1.3b). To sum up,

● ● ● ● ● ● ● ● ● ● ● 5 10 n -50 -100 sn 0.0 0.2 0.4 0.6 0.8 1.0 τ (a) ● _● ● ● ● ● ● ● ● ● ● 5 10 n -5000 -10 000 sn 0.00 0.01 0.02 0.03 0.04 τ (b)

Figure 1.3: (a) In blue, at the top, poles s_n for  = 0.1; at the bottom, a number line plot of the corresponding relaxation times τ_n. The first pole is s₀ = −1.057 and the longest relaxation time is τ0 = −s−10 = 0.946 (see Fig. 1.2a). (b) The same thing for  = 10. In this case the first pole is s₀ = −24.693 = −10.008π₄2 and the longest relaxation time is τ₀ = −s−1₀ = 0.04050. In orange, both in (a) and in (b), the lower bound for poles sn from Eq. (1.44).

many time scales that are all of diffusive nature (∝ L_D2). Conversely, when   1, λD is the shortest length and relaxation goes through an infinity of very fast processes, happening on timescales ∝ _DnL22 for n large enough, and approaching

λ2_D

D as n becomes

smaller; these processes eventually give way to a slower process, happening on a time scale τ0 ∝ Lλ_DD.

The derivation of the exact time scales τn, although with a heavier notation, can

be found in Ref. [43], whose authors arrived independently and at the same time as us to the same results. Compared to Ref. [42] we found discrepancies at non-vanishing  (see Fig.1.2a).

Computing explicitly the residue

Res_H(s)(sn) =

2sn(1 + sn)

3 − s2

n

(1.47)

allows to retrieve immediately expressions such as

ρ(z, t) = −vsinh z  sinh1_ − v ∞ X n=0 2(1 + sn) 3 − s2 n sinhz √ 1+sn  sinh √ 1+sn  esnt, (1.48)

that a cumbersome calculation can show to be equivalent to Eq. (40) in [43].

What happens from a physical point of view in the EDLC must also be clear after this discussion on time scales. The electrodes, that are discharged until t = 0− (i.e.

σ(0−) = 0), acquire a charge σ(0+) = −v on the left and +v on the right, as soon as the power source is switched on (the electric potential is indeed φ(z, 0+) = vz). Implicit is the assumption that the migration of charges within the electrodes, necessary for establishing the desired voltage ±v, takes place on much shorter time scales than those