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Why TDS fails to be a good way to measure water quality in Malta – TAPP Water (Malta) Ltd
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Why TDS fails to be a good way to measure water quality in Malta
Total Dissolved Solids (TDS) is used by Reverse Osmosis suppliers in Malta as a way of demonstrating the effectiveness of water filters, however it really does fail as a good way to measure water quality in Malta.
In this article we explain what a TDS meter measures, and more importantly what it does not test.
A TDS meter is used as a water testing method as it is cheap and easy to do. I bought a TDS tester to try out for less than €20 in Malta and it was really easy to use, just place the probe end in water and bingo you have a TDS number.
The issue is that TDS is very limited in terms of what it is measuring. Sure, it has it's uses but it is just one measure of one factor in water quality so it is really failing to give you a complete picture of water quality in Malta.
On the other hand a full detailed analysis of water quality was at least €1,000 when we had our independent test results completed for Malta water filtration with TAPP Water.
What are Total Dissolved Solids (TDS) in water?
We can start by knowing what TDS is and what it is measuring. Total Dissolved Solids are the inorganic salts and small amounts of organic matter that are dissolved in water. So really a TDS meter measures the electrical conductivity of salts and minerals in water and then calculates a TDS number.
The Total Dissolved Solids (TDS) are measured in mg per Litre unit of water (mg/L) which is also known as parts per million, ppm. That's why a TDS meter is sometimes called a PPM meter.
The following are included in measuring Total Dissolved Solids (TDS) in Malta:
So from all these that are measured the only contaminants are nitrates and chloride, the rest are all healthy minerals.
A TDS meter does not tell you if your water is healthy, for instance high levels of contaminants such as THM's, PFAS Forever Chemicals, lead and heavy metals, oil, arsenic, pesticides, herbicides and pharmaceuticals can b present in your water with a low TDS level , which is not an indicator of water quality, its an indicator of low dissolved salts and minerals.
What is a good measure of TDS in tap water?
Typically tap water in Malta has a TDS measure of up to 1,000 mg/L which is quite normal for Malta. Water Services Corporation provide test results all over Malta of the conductivity which can then be used to calculate TDS. This exactly what a TDS meter does, it measures the conductivity of water and then calculates TDS.
Areas with higher quantities of minerals in water will have higher readings.
Reverse Osmosis, ion exchange and distillation water filtration systems will reduce this measure to zero or close to zero. Activated carbon filters will not filter out TDS and with a TAPP Water Filter we do not expect conductivity, and hence TDS to change, as all healthy minerals are retained. Nitrates and Chlorine are filtered out though which reduces TDS potentially but not enough to show up as a drastically reduced reading.
Bottled mineral water will measure high in TDS, for example Evian comes in at 300 mg/L and San Pelligrino at 850mg/L; we would aspire to drink such healthy mineral waters and trust them to be doing us good, regardless of their TDS measure.
The medical importance of mineral waters depends on the contained amount of minerals and trace elements. Calcium and magnesium in mineral waters are of greatest interest with respect to their potential relevance for the health status; iodine, fluorine and lithium are the most important trace elements. The bioavailability of minerals from mineral water is good and can be compared with the values derived from milk. A positive effect of the minerals in mineral water on health status is especially apparent in the case of insufficient intake by nutrition. The apprehension that there might be an undesired hypertensinogenic effect due to the sodium content in mineral waters is not justified. Sodium is present in mineral water in small amounts only, and mainly as sodium bicarbonate, which has no effect on blood pressure. A certain advantage of a delivery of minerals and trace elements by regular drinking of mineral waters is the simultaneous intake of water, without supply of energy. [Health-related effects of natural mineral waters]
Is a high Total Disolved Solids (TDS) measure a bad thing?
A study by the World Health Organization concluded with the following measures in their Guidelines for drinking water:
50 mg/L to 300mg/L - Excellent
300 mg/L to 600 mg/L - Good
600 mg/L to 900 mg/L - Fair
900 mg/L to 1200 mg/L - Poor
1200 mg/L and above - Unacceptable
Higher TDS waters are described by testers as having a heavier feel to them with very low TDS waters being described as flat, but it is not a question of water quality, more about taste and feel.
Why do people use TDS as a measure of water quality in Malta?
The simple answer is that its cheap and easy to do. Just purchase a €20 tester, drop it in a sample before and after filtration and you have an immediate "answer". But it's not a great answer and not accurate as a measure of water quality.
This measure is particularly used by Reverse Osmosis companies to promote the use of this filtration system in Malta; but there are disadvantages too for this type of home water filtration system in Malta which we describe in our article All about Reverse Osmoisis (RO) home water filters in Malta.
Real tests of water quality focus on the levels of chlorine, pesticides, heavy metals and pharmaceuticals and these require far more analysis and complex testing equipment.
TAPP Water filters have been tested for TDS in tap water with values up to 1000 and the water taste remains good and the water is safe to drink.
Unless your tap water as a TDS value that is vastly outside of the recommended range you have no need to worry. High TDS in your Malta tap water basically means that there is mineral water coming out of your tap. Once you have filtered it to remove over 80+ common contaminants in Malta tap water with your TAPP 2 Twist you will be really happy.
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The content of these qualifications is designed to equip learners with transferable skills required in the sector. Learners can expect to develop teamwork and leadership skills, all of which are hugely important as many roles in the public services sector rely heavily on these skills.
To ensure that the qualifications are relevant and robust, NCFE work alongside experts actively employed in Public Service organisations to develop the content and ensure that the assignments are linked to realistic work-based scenarios that learners could encounter in their future careers.
Lee Mandley, Head of Sport and Public Services at Suffolk New College, commented: "These qualifications include a range of modules specific to the day-to-day roles of the uniformed Public Services which has helped prepare learners for application processes and fitness requirements through classroom taught lessons, guest speakers and practical and physical activities.
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Attempts to revive the spirit of the Tiananmen Square movement have been made before. In the quarter century since Chinese security forces killed hundreds, if not thousands of pro-democracy supporters many have tried to evoke the events of June 4, 1989, to inspire defiance anew.
In Hong Kong, for instance, democracy activists hold an annual candlelight vigil in Victoria Park. A model of the movement's Democracy Goddess is invariably present.
However, the vigils haven't garnered much momentum, and Cui's album didn't sell nearly as much as he had hoped. Essentially, attempts to reinvigorate or reinvent China's democracy movement for a newer, younger, China — often preoccupied by a struggle for a share of the country's rapid-fire economic growth — have failed.
But that hasn't stopped dissident Chinese blogger Wen Yunchao from a quixotic quest to reclaim the spirit of '89.
From the safety of his New York home, Wen launched a Twitter campaign to reignite the Tiananmen protests on June 4, the 25th anniversary of the event. He was neither in Beijing in 1989 nor espoused — at the time — the values that brought at first students and then people from all stripes of Chinese society to the monument that represents the heart of the Chinese Communist Party.
The campaign for Tiananmen 2.0 is called 重回天安门 (Chong Hui Tiananmen) — Return to Tiananmen. It has garnered support and criticism from Chinese around the world who remember the original movement, whether from TV or by having witnessed it firsthand.
"After 25 years, we find that what they were talking about — democracy — still hasn't been realized today," said Wen, a native of southern Guangdong province, who currently resides in New York. Wen, who has engaged in multiple online campaigns for human rights and free expression, refuses return to China because he fears that he will face repression, censorship and possibly retribution for his dissidence.
What resulted was a campaign designed in part to spark a protest on the ground, but to also draw the attention of international media issues of socioeconomic injustice in China.
"This event is designed to still bring media attention back to Tiananmen. It's also to return the ideas of Tiananmen to the people's hearts," said the blogger.
On a typical June 4, Beijing police will heavily restrict activity on and around Tiananmen Square. On the quarter-century anniversary of the movement, people expect extra precautions against popular movement. Police have already reportedly set up checkpoints around the square.
"A Tiananmen-inspired movement has little chance to gain traction given the full-blown, no-holds-barred efforts of the Chinese Communist Party to muzzle dissent and the flow of information," renowned U.S.-China affairs commentator Ray Kwong told Al Jazeera.
Still, Wen is confident that — although he himself will be unable to attend renewed protests at the square — others will.
"It's hard for me to estimate how many people will participate. But there must be many people planning on participating for the government to be so nervous," he said, referring to the arrests of the government's critics, like dissident journalist Gao Yu, ahead of this year's anniversary.
Qingwen Dong, a Chinese media and current affairs analyst and communications professor at University of the Pacific, expressed his opposition to the campaign.
"The past 25 years have seen lots of changes in China politically, socially and economically," Dong said, "We should not use the same thinking as we had 25 years ago to deal with today's situation." In 1989, when Dong was a student himself, he "shared lots of ideas" with the members of the movement, but was studying abroad in the United States.
Wen acknowledged that Syria's popular movement for democracy has devolved into civil war and that the military rulers who were part of the administration of ousted Egyptian President Hosni Mubarak are back in power, just three years after Mubarak was deposed. But he said the situation is reminiscent of a Chinese idiom: 不能一口吃胖子(Bu neng yikou chi pangzi), "One can't eat his corpulence in a bite" — democracy isn't gained overnight.
Wen would not reveal to Al Jazeera who had collaborated with him to launch the campaign. "It's not convenient to say," he said.
But renowned Chinese dissident Hu Jia, who has faced numerous detentions and disappearances by the Chinese government for his advocacy for democracy and AIDS rights, has been helpful in disseminating the hashtag #重回天安门 — Return to Tiananmen, not on Chinese social media Sina Weibo, where the term has been blocked well ahead of the anniversary, but on Twitter.
Twitter is only accessible in China with the help of virtual private network software, often purchased by Chinese Netizens looking to logon to foreign sites, like YouTube and Facebook, blocked by the so-called Great Firewall.
Hu sees Tiananmen-style protests as an inevitable part of the country's social and economic development.
"China is the world's second-largest GDP," Hu said, "As China reforms and develops, there will inevitably be people taking to the streets in a manner similar to the student and citizen protests of the June 4,  movement," he said.
"All that is necessary is the consciousness of the public to achieve this universal value" of democracy.
Still, Wen's campaign has ruffled some feathers in China's diaspora community. The University of the Pacific's Dong opposed a campaign orchestrated primarily on foreign social media by activists in the United States.
And Chinese youth remain detached from their recent history, it seems.
"Today's young people are post-'90s, and they have a very vague idea of the true meaning of Tiananmen," Dong said, adding that still, even as analysts continue to describe most modern Chinese youth as being disaffected from politics, they have "new ideas of democracy, equality and freedom" that are altogether independent of the those of the Tiananmen generation.
A 20-something Chinese national living in Northern California told Al Jazeera that "our generation is not really familiar with that event." The national refused to give his name, citing retribution by Chinese authorities against people who have discussed Tiananmen openly on the Internet or among peers.
As the Chinese government attempts to block information on the events of the Tiananmen Square incident, so too do many Chinese parents.
"Parents mostly fear their children would do something similar so lots of them don't really tell the whole thing about that event," he said. Several other sources based in China declined to tell Al Jazeera about their views on Tiananmen.
Just how much Chinese youth do know and how much they've withheld from the public discourse for fear of retribution remains unknown. As is the number of activists who will return to Tiananmen on June 4, either physically or by way of social media.
Even if not on Wednesday, Hu is confident the square will draw democracy activists again.


text:
\section{Introduction}

Supercapacitors are typically comprised of porous carbon-derived electrodes with an aqueous, organic, or ionic liquid electrolyte, and store charge in electrostatic double layers at all surfaces in the pore network. Models have been constructed at a varying levels of complexity to describe this system and its components, i.e.~individual pores, pore networks or electrodes. For short individual pores, it can be viable to perform molecular dynamics simulations to resolve the complicated many-body interactions in full, although this is often very computationally expensive. At the cell level, physics-based models have been developed with an averaging approach under the assumptions of porous electrode and dilute solution theories. These can incorporate electrolyte transport at multiple pore scales \citep{verbrugge2005microstructural,kroupa2016modelling}.

For concentrated electrolytes, such as ionic liquids, and systems where the pore size and electrostatic double layer widths are comparable, Poisson--Boltzmann (PB) theory for dilute solutions becomes invalid. More complicated models have been developed, incorporating ion size (steric) effects and electrostatics beyond the mean-field description. Concerned with the dynamics of concentrated electrolytes, \cite{kilic2007steric1} considered two models for an electrostatic double layer, the modified PB model of \cite{borukhov1997steric} which includes steric effects in a mean-field manner, and a composite model incorporating an inner layer of ions at the electrode surface with high packing fraction and a dilute PB outer layer. \cite{kilic2007steric1} showed that steric effects suppress the unbounded growth of the zero-frequency differential capacitance ($\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$) with voltage found in PB theory, exhibiting bell- or camel-shaped curves where either one or two maxima are present, respectively -- this finding was corroborated by \cite{doi:10.1021/jp067857o} with theoretical results for a lattice-gas model, also showing that different ion sizes would result in the asymmetry of $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$ about zero potential (relative to the potential in the bulk). At the level of equivalent circuit models, \cite{kilic2007steric1} then predict that the dynamical response time of the system (which scales linearly with the capacitance) is greatly reduced from the PB result at large voltages.

In a companion paper, \cite{kilic2007steric2} derive a modified Poisson--Nernst--Planck (PNP) system based on the free energy functional of \cite{borukhov1997steric}, and perform numerical simulations of a parallel-plate capacitor. When charging, the double layer governed by the modified PNP system initially proceeds as it would with the unmodified PNP equations, until the packing limit is reached and the double layer extends laterally with constant ion concentrations (while the unmodified PNP solution grows exponentially at the electrode surface). This is however a qualitatively incorrect description of how steric effects affect double layer structure \citep{gillespie2015review}. A phenomenological model with electrostatic correlations, improving on the usual mean-field electrostatics, was proposed by \cite{bazant2011double}. At low voltages, overscreening/charge inversion was observed, where the magnitude of charge on the electrode surface is less than that in the first layer of counter-ions.

\cite{kondrat2013charging} studied pore charging using an improved mean-field model with electrostatic correlations and the steric free energy of \cite{borukhov1997steric}, solving a spatially 2D continuity equation in long slit pores of width and length on the nanometer scale. In most of their simulations, they found that wider pores charged faster, and observed a front-like propagation of charge into empty pores, with initially filled pores charging diffusively. Additionally, \cite{kondrat2013charging} outlined an approach for optimising a porous electrode with a simplified schematic in which the pore widths and lengths are tuned to attain the best combination of energy and power density.

Another approach to model the behaviour of confined and concentrated electrolytes is classical density functional theory (DFT). DFT computes the equilibrium behaviour of inhomogeneous fluids in the presence of external potentials by minimising a free energy functional \citep{evans1979nature} (the reviews of \cite{evans1992density} and \cite{lutsko2010recent} provide an overview of the major advances in classical DFT since its inception).
In DFT, ionic liquids are typically modelled as mixtures of charged hard spheres; for these interactions, good approximate free energy functionals exist.
The hard sphere (steric) component is incorporated using the fundamental measure theory (FMT) derived by \cite{rosenfeld1989free}, or the improved White Bear versions constructed by \cite{yu2002structures,roth2002fundamental} and \cite{hansen2006density}. Theories for electrostatic correlations, based on the mean spherical approximation (MSA) for bulk systems \citep{blum1975mean}, have been derived by a number of authors \citep{kierlik1991density,rosenfeld1993free,gillespie2002coupling,gillespie2003density,roth2016shells}. Both the hard sphere and electrostatic theories used in the present work are discussed below.

Classical DFT has already been utilised extensively to analyse ionic liquid choice and pore size effects with applications in supercapacitor design. However, DFT is limited to the computation of equilibrium quantities such as the integral capacitance or $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$. \cite{jiang2011oscillation} observed that the integral capacitance can have a strongly oscillatory dependence on the pore width, indicating that the optimisation of carbonaceous electrode porosities could lead to a significant increase in the energy density of an ionic liquid supercapacitor. The influence of bulk ion densities and ion size asymmetry on $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$ was investigated by \cite{jiang2011density}, observing both camel- and bell-shape plots for the differential capacitance as a function of the pore surface potential for low and high bulk densities, respectively, as found in previous studies incorporating ion size effects \citep{kilic2007steric1,doi:10.1021/jp067857o}. Further work has focused on the influence of an added solvent \citep{jiang2012solvent} or other impurities \citep{liu2017impurity}, and the possibility of mixing ionic liquids to maximise capacitance \citep{lian2016enhancing,neal2017ion}.

Dynamic density functional theory (DDFT) extends DFT to non-equilibrium systems. \cite{evans1979nature} was the first to write down a deterministic dynamical equation which governs the evolution of the particle densities, with later derivations due to \cite{dieterich1990nonlinear} and \cite{marconi1999dynamic}. The DDFT equation has the form of a conservation law, and specifies that particle density distributions evolve according to gradients in the local chemical potentials (these are spatially varying out of equilibrium), relaxing to classical DFT equilibria. The validity of this model far from equilibrium is being investigated \citep{schmidt2013power}. \cite{te2020classical} provide an overview of the original DDFT model of \cite{evans1979nature} and its extensions, with a comprehensive survey of the many applications. The DDFT system reduces to the usual diffusion equation in the case of a non-interacting ideal gas, and classical PNP can be obtained for dilute charged particle systems with a mean-field assumption for the electrostatics.

There have been a number of DDFT studies for ionic liquids; the majority of these consider a nanoscale parallel plate capacitor set-up, such as \cite{jiang2014time,lian2016time} and \cite{qingsurface}, who studied charging dynamics. \cite{babel2018impedance} applied a sinusoidal voltage on the walls of a nanoscale parallel-plate capacitor to calculate the impedance response, performing numerical simulations both with and without mean-field electrostatics and hard-sphere effects, finding that the latter had the most significant influence on the result. Also of relevance to the present study is the work of \cite{qing2019dynamic} on ion dynamics in like-charged nano-slits. The authors fixed the density and chemical potential equal to bulk values at permeable pore walls, and allowed the slit to fill from empty, with a focus on determining the adsorption capacity and rate as a function of the pore width. All of the above works are spatially 1D, and capture the fast dynamics of ionic liquids on the nanoscale.

In the present work, we are interested in using DFT and DDFT to predict the influence of the underlying ion equilibria on the impedance response. The study of impedance response of porous electrodes was pioneered by \cite{de1963porous,de1964porous,de1967electrochemical} who modelled the system as a set of infinite and semi-infinite cylindrical pores connected in parallel. He proposed a transmission line (electrical circuit) model for the ion transport in the pores, for which the circuit parameters could be expressed in terms of physical quantities. Under a number of simplifying assumptions, including electroneutrality, infinite electrical conductivity in the electrode (valid for carbon-based supercapacitor electrodes), and dominance of ion transport by migration over diffusion, the impedance response of a single semi-infinite pore homogeneously filled with electrolyte can be reduced to the finite-space Warburg (FSW) element defined by
\begin{equation}\label{deLeviemodel} Z(\omega) = \mathcal{Z}  \frac{\coth ( \sqrt{ \textrm{i} \omega \tau} ) }{\sqrt{ \textrm{i} \omega \tau}}.\end{equation}
Here, $\textrm{i} = \sqrt{-1}$ and $\omega$ is the angular frequency of the alternating voltage/current input, $\mathcal{Z}$ is an impedance and $\tau$ is a time constant. The high and low frequency limits of \eqref{deLeviemodel} are
\begin{equation}\label{deLeviemodellimits} Z(\omega) \xrightarrow{{\omega \rightarrow 0}} \mathcal{Z} \left( \frac{1}{3} - \frac{ \textrm{i}}{ \omega \tau} \right), \qquad  Z(\omega) \xrightarrow{ {\omega \rightarrow \infty}}  \frac{\mathcal{Z}(1-\textrm{i})}{\sqrt{2 \omega \tau}},\end{equation}
respectively \citep{song2012effects}. The high frequency limit pertains to diffusive behaviour, and at low frequencies the system is capacitive.

Analytical expressions similar to \eqref{deLeviemodel} can be derived if some of the assumptions used by de Levie are relaxed, as discussed in the review of \cite{huang2020review}. However, there has been only limited success in incorporating more complex physical models into the analytical derivations of impedance response expressions. For example, in the planar electrode geometry, \cite{li2021impedance} derived the analytical expression for the impedance response using the PNP model without the electroneutrality assumption, whereas \cite{wang2012intrinsic} resorted to a numerical investigation of the same set-up with the PNP model and its modifications due to \cite{LIM2007159} and \cite{kilic2007steric2} (see also \cite{ROLING2012526} and \cite{WANG2012529}). Experimental works often attempt to fit data to \eqref{deLeviemodel}, replacing the exponent $1/2$ with an unknown (constant phase element) exponent if needed. The influence of non-trivial pore geometry was first investigated by \cite{keiser1976abschatzung}, who utilised the transmission line model of de Levie and varied the circuit parameters to account for spatial dependence of the cylindrical pore radius. This problem was recently revisited by \cite{cooper2017simulated} who performed numerical simulations of a diffusion equation in frequency space, both in open and closed pores of varying cross section and in more complicated and tortuous geometries including fractal domains. In the present work, our interest is solely in the influence of improvements to the physical model employed, keeping the geometry as simple as possible.

We study ion dynamics in idealised, spatially 2D, rectangular slit pores, for which the pore length is significantly larger than the nanoscale width. The pore is connected to an infinite particle bath, and the system is driven from equilibrium by variations of the potential on the slit walls. Due to the multi-scale nature of this problem, it would likely be intractable to solve a full 2D DDFT, let alone perform molecular dynamics simulations. We proceed with an asymptotic reduction of the DDFT system, resulting in a limit model on the timescale of supercapacitor applications for which the ion densities appear to equilibrate instantaneously in the nanoscale pore width. This model is the simplest way of incorporating both DFT, for the accurate resolution of the (out-of-plane) ion profiles, and time-dependent dynamics for the ion transport in the pore length dimension. During the preparation of this manuscript, we became aware of the work of \cite{aslyamov2020relation} who derive the same limit model to study the charging of slit pores.

We perform a frequency response analysis of the limit model, finding that it is amenable to a semi-analytical approach. We recover the FSW element \eqref{deLeviemodel} in the case where only one ion species is present in the pore, and construct effective FSW elements, valid in the low and high frequency limits, for the general case. Additionally, we carry out fully numerical simulations in which the time-dependent system is solved in response to a sinusoidal voltage input to mimic a potentiostatic electrochemical impedance spectroscopy (PEIS) experiment at the pore scale (akin to the study of \cite{wang2012intrinsic}). Exploration of the system response beyond the linear PEIS regime through full numerical simulation is left to a later study, although relevant results with discontinuous voltage inputs are presented in \cite{aslyamov2020relation}. For two parameter sets inspired by real ionic liquids, we investigate the influence of system parameters and the choice of DFT. We find that the inclusion of electrostatic correlations in the DFT modify the dynamic characteristics of the pore system for widths close to those of the ions. Without electrostatic correlations, the characteristic time of the linear response appears to be overestimated for pores that can fit a single ion. We observe peaks in the response time as a function of pore width, which shift in magnitude and location depending on the base voltage. For the optimisation of supercapacitor geometries, such dynamic results should be taken together with equilibrium results to ensure that an appropriate balance of energy and power density is achieved for the desired application.

In Section \ref{problemform}, we outline the governing equations for the simplified pore system and discuss the asymptotic model reduction. We provide the semi-analytical theory and outline the fully numerical approach for the impedance response analysis in Section \ref{ImpRes}, and present results for two sets of parameters in Section \ref{numresults}. A discussion and our concluding remarks are contained in Section \ref{Conclu}.

\section{Problem formulation and model reduction \label{problemform}}

\begin{figure}
\begin{tikzpicture}[scale=2.75]

\draw[thick]
(0,0) -- (2,0)
(0,0.5) -- (2,0.5)
(1,0) node[below] {$\psi = \mathcal{V}(t)$}
(1,0.5) node[above] {$\psi =\mathcal{V}(t)$};

 \draw[thick]
(0-0.5,0)  -- (2+0.5,0) coordinate (A4)
(0-0.5,0.5)  -- (2+0.5,0.5) coordinate (A3);

 \draw[thick]
(2+0.5,0) -- (2+0.5,0.5) ;

 \draw[thick]
(0-0.5,0) coordinate (A5) -- (0-0.5,-0.25) coordinate (A6)
(0-0.5,0.5) coordinate (A2) -- (0-0.5,0.5+0.25) coordinate (A1);

 \draw[<->]
  (-0.25,0) -- (-0.25,0.5);
  \draw[thick]
  (-0.25,0.25) node[left] {${L}$};

 \draw[<->]
 (0-0.5,-0.35) --  (2+0.5,-0.35);
  \draw[thick]
  (1,-0.35) node[below] {${H}$};

\draw[thick]
(1,0.25) node {$\rho_i(x,y,t), \quad \mu_i(x,y,t), \quad \psi(x,y,t)$}
(-1.25,0.35) node {$\rho_i^{\textrm{b}}, \quad \mu_i^{\textrm{b}}$,}
(-1.25,0.15) node {$\psi = 0$};

         \def (A1) -- (A2) -- (A3) -- (A4) -- (A5) -- (A6) --  (2+0.75,-0.25) -- (2+0.75,0.75){ (A1) -- (A2) -- (A3) -- (A4) -- (A5) -- (A6) --  (2+0.75,-0.25) -- (2+0.75,0.75)}

           \fill[gray,opacity=0.1]  (A1) -- (A2) -- (A3) -- (A4) -- (A5) -- (A6) --  (2+0.75,-0.25) -- (2+0.75,0.75);
            \fill[pattern=north east lines,opacity=0.1]  (A1) -- (A2) -- (A3) -- (A4) -- (A5) -- (A6) --  (2+0.75,-0.25) -- (2+0.75,0.75);

  \draw[->] (2+0.75+0.25+0.25,0.125) -- (2+1+0.25+0.25,0.125) node[right] {$x$};
  \draw[->] (2+0.75+0.25+0.25,0.125) -- (2+0.75+0.25+0.25,0.25+0.125) node[above] {$y$};

\end{tikzpicture}
\caption{A single slit pore connected to an infinite particle bath.}\label{schematic_main}
\end{figure}

We consider the dynamics of ionic liquids in slit pores of length ${H}$ and width ${L}$ (with an infinite third ``depth" dimension) as shown in the schematic in Figure \ref{schematic_main}. Our interest is in developing models for thin pores with ${H} \gg {L}$, where ${L}$ is on the order of nanometers (comparable to typical ion sizes). The coordinates in the pore length and width are $x$ and $y$, respectively, and in the following discussion we use $\bm{r}$ to denote a general coordinate vector. This individual slit pore represents one of many that comprise a typical carbonaceous electrode in an electrostatic double layer capacitor, specifically of nano-wall type. The filling ionic liquid is modelled as a mixture of hard spheres with point charges located at the particle centres, referred to as the primitive model; specifically, the ion--ion pair potential between particles of species $i,j$ with separation distance $r = |\bm{r}|$ is
\begin{equation}\label{interactionpot1}u_{i,j}(r) = \begin{cases}
            \infty & \text{if $r < R_i + R_j$}, \\
            \displaystyle{ \frac{ z_i z_j {\textrm{e}}^2}{4 \pi \epsilon_0 \epsilon_{\textrm{r}} r } }& \text{if $r \geq R_i + R_j$},
        \end{cases}\end{equation}
where $R_i = \sigma_i/2$ and $z_i$ denote the radius and valence of the $i^{\textrm{th}}$ particle species, respectively. Additionally, ${\textrm{e}}$ is the elementary charge, $\epsilon_0$ denotes the permittivity of free space, and $\epsilon_{\textrm{r}}$ is the dielectric constant of the medium. The electrode material is assumed to have the same dielectric constant to avoid image charges.

The time evolution of the ion density distributions in the pore, denoted $\rho_i(\bm{r},t)$, are governed by the deterministic DDFT system first written down by \cite{evans1979nature},
\begin{equation}\label{DDFT3D}
\frac{\partial \rho_i}{\partial t} = \beta D_i \bm{\nabla} \bm{\cdot} \left( \rho_i \bm{\nabla} \mu_i \right).
\end{equation}
Here, $\mu_i$ are the local chemical potentials, which are spatially varying out of equilibrium. Gradients in the local chemical potentials drive the density evolution, and without external forcing, solutions of \eqref{DDFT3D} relax to equilibrium DFT solutions with constant $\mu_i$. The product $\beta D_i$ is the mobility of the $i^{\textrm{th}}$ species, where $D_i$ denotes a single-particle self-diffusion coefficient and $\beta = 1/k_{\textrm{B}}T$ is the thermodynamic inverse temperature (both $\beta$ and the $D_i$ are assumed constant in this work). At the pore cap we impose a no-flux boundary condition, $\partial_x \mu_i = 0$ at $x=H$, as well as on the long sides of the slit pore, $\partial_y \mu_i = 0$. At $x=0$, the pore is open and connected to an infinite particle bath, with bulk densities and chemical potentials in the far field denoted by $\rho_i^{\textrm{b}}$ and $\mu_i^{\textrm{b}}$, respectively. When the system is at rest, the chemical potentials in the pore equal those in the bulk, $\mu_i = \mu_i^{\textrm{b}}$. The relationship between the densities and local chemical potentials, required to close the system, is based on that used in classical DFT for particle systems in equilibrium, as described next.

The pair interaction potential \eqref{interactionpot1} uniquely determines an intrinsic Helmholtz free energy, $F[ \{\rho_i\} ]$, a functional of the particle densities \citep{evans1979nature}. This free energy is independent of the external potentials, denoted ${{U}}_i$, which describe the influence of the pore walls on species $i$. The grand potential energy density functional for the system is given by
\begin{equation}
\label{grandpotential}
 \Omega[\{\rho_i\}] = F[\{\rho_i\}] + \sum_{i} \int  ({{U}}_i - \mu_i)\rho_i \; \mathrm{d}\bm{r}.
\end{equation}
Equilibrium DFT solutions are obtained as minimisers of $\Omega$, satisfying the Euler--Lagrange equations,
\begin{equation}\label{EL0}
0 = \frac{\delta \Omega}{\delta \rho_i} = \frac{\delta F}{\delta \rho_i} + {{U}}_i - \mu_i,
\end{equation}
which is also used to relate the constant bulk quantities $\rho_i^{\textrm{b}}$ and $\mu_i^{\textrm{b}}$ (for which $U_i = 0$). The DDFT assumption is that \eqref{EL0} remains valid close to equilibrium, and so yields the desired expression for the local chemical potentials to be used in \eqref{DDFT3D}. While there exists a unique Helmholtz free energy corresponding to the particle interactions \eqref{interactionpot1}, it cannot be written down exactly (analytical expressions for $F$ only exist for special cases of $u_{i,j}$). For an exact Helmholtz free energy, it can be shown that there is a unique correspondence between the external potential and the equilibrium density \citep{evans1979nature}, as the minimisation problem is convex, yielding a unique global minimum (unique equilibrium). The DFT approach relies critically on the ability to construct accurate approximations of $F$. However, the convexity property of the minimisation problem can be lost during this approximation process, and sometimes more than one equilibrium solution exists for a given external potential with an approximate free energy \citep{te2020classical}.

Typically, $F$ is decomposed into ideal and excess components, and for the case of pair potential \eqref{interactionpot1}, the excess free energy is split up into contributions due to hard-sphere and electrostatic interactions, with the latter broken down further into mean-field direct Coulomb interactions and electrostatic correlations (sometimes referred to as the electrostatic residual or screening contribution),
\begin{equation}\label{freeenergysplit}F = F^{\textrm{ID}} + F^{\textrm{EX}} = F^{\textrm{ID}} + F^{\textrm{HS}} + F^{\textrm{CO}} + F^{\textrm{EC}}.\end{equation}
The ideal component in \eqref{freeenergysplit} corresponds to the intrinsic Helmholtz free energy for a system of noninteracting particles, i.e.~$u_{i,j} \equiv 0$, and is given exactly by
\begin{equation}\label{idealfreeenergy}F^{\textrm{ID}}[\{\rho_i\}]  = \beta^{-1} \sum_{i} \int ( \log ( \Lambda_i^3 \rho_i(\bm{r}) ) - 1)\, \rho_i(\bm{r})\; \mathrm{d}\bm{r}, \end{equation}
where $\Lambda_i$ is an effective (de Broglie) thermal wavelength \citep{tarazona2008density}. If $F^{\textrm{EX}} = U_i = 0$, the local chemical potential is given by $\mu_i = \beta^{-1} \log \rho_i$ and returns the standard diffusion equation when substituted into \eqref{DDFT3D}. Substituting (\ref{freeenergysplit},\ref{idealfreeenergy}) into the Euler--Lagrange equations \eqref{EL0} gives
\begin{equation}\label{expform1} \rho_i(\bm{r})  = \Lambda_i^{-3} \exp\left[ - \beta \left( \frac{\delta F^{\textrm{EX}}}{\delta \rho_i} + {{U}}_i(\bm{r})  - \mu_i \right) \right].
\end{equation}
This implicit expression for the densities is commonly used as the basis of Picard iteration numerical schemes for equilibrium DFT solutions. The $\Lambda_i$ can be absorbed into the chemical potentials or, at equilibrium, can be cancelled by relating the bulk densities and chemical potentials \citep{tang2003density}.

The direct Coulomb contribution to the excess free energy is
\begin{equation}F^{\textrm{CO}} = \frac{1}{2} \frac{{\textrm{e}}^2}{4\pi \epsilon_0\epsilon_{\textrm{r}}} \sum_{i,j} \int \int  \frac{z_i  z_j \rho_i(\bm{r}) \rho_j(\bm{r'}) }{ |\bm{r} - \bm{r'}|} \; \mathrm{d}\bm{r'} \mathrm{d}\bm{r}.
\end{equation}
This is the leading order term of the expansion of the excess free energy functional corresponding to the pair interaction $u_{i,j}(r) = {\textrm{e}}^2 z_i z_j/4 \pi \epsilon_0 \epsilon_{\textrm{r}} r$, and comprises the electrostatics employed in the PB theory valid for weak electrolytes. This mean-field expression treats the particles as fully uncorrelated, only accounting for the potential created on a given particle by the mean particle distribution \citep{evans1992density,tarazona2008density}. The external potential we consider can be decomposed into the sum of hard-wall and electrostatic components, $U_i = U_i^{\textrm{HW}} + U_i^{\textrm{ES}}$, where $U_i ^{\textrm{ES}}$ can be identified as the the electrostatic potential energy of an ion of charge ${\textrm{e}}z_i$ in the presence of the electrostatic potential generated by all non-ionic charges in the system (e.g.~charged walls). The mean electrostatic potential $\psi(\bm{r})$, incorporating the electrostatic components due to ionic and non-ionic charges \citep{wang2011weighted,wu2011classical,medasani2014ionic}, is defined by
\begin{equation}\label{electrostaticincorporation}{\textrm{e}} z_i \psi(\bm{r}) = \frac{\delta F^{\textrm{CO}}}{\delta \rho_i}  + U_i ^{\textrm{ES}}(\bm{r}) = {\textrm{e}} z_i  \sum_{j} \int  \frac{ {\textrm{e}} z_j \rho_j(\bm{r'}) }{4\pi \epsilon_0\epsilon_{\textrm{r}} |\bm{r} - \bm{r'}|} \; \mathrm{d}\bm{r'} + U_i ^{\textrm{ES}}(\bm{r}).
\end{equation}
The electrostatic potential satisfies the Poisson equation in the ionic liquid phase (the non-ionic component is harmonic in this region),
\begin{equation}\label{Poisson0}\epsilon_0\epsilon_{\textrm{r}} \nabla^2\psi = - {\textrm{e}} \sum_i z_i \rho_i.\end{equation}
As indicated in Figure \ref{schematic_main}, we impose a voltage (measured relative to zero potential in the bulk) at the pore walls,
   \begin{equation}\label{PoissonBC1}\psi|_{y=0} = \psi|_{y={L}} = \mathcal{V}(t),\end{equation}
   which we use to drive the system from equilibrium.
Without contributions to the free energy from hard-sphere effects and electrostatic correlations, the local chemical potential becomes $\mu_i = \beta^{-1} \log \rho_i + {\textrm{e}}z_i \psi$. This recovers Nernst--Planck when substituted into \eqref{DDFT3D}.

For the hard sphere component, $F^{\textrm{HS}}$, we use the White Bear II functional developed by \cite{hansen2006density}, improving on the original FMT functional of \cite{rosenfeld1989free}. We also performed computations with FMT and White Bear I \citep{roth2002fundamental,yu2002structures}, finding that the influence of this choice was negligible.
Further discussion of the various hard-sphere functionals can be found in the review by \cite{roth2010fundamental}.

The earliest approximation of the electrostatic correlation contribution was derived independently by \cite{kierlik1991density} and \cite{rosenfeld1993free}. They used a functional Taylor expansion about constant bulk densities with coefficients (direct correlation functions) given by the MSA. This bulk fluid density (BFD) approach was refined by \cite{gillespie2002coupling,gillespie2003density} who used self-consistent non-homogeneous reference densities in the Taylor expansion, termed reference fluid density (RFD) theory, with the resulting equilibrium profiles often in close agreement to molecular simulations \citep{voukadinova2018assessing}. More recently, functionalised mean spherical approximation (fMSA) was developed by \cite{roth2016shells} which significantly reduces the computational complexity with only some loss of accuracy (compared to RFD) \citep{voukadinova2018assessing}. In the present work, we show results for both BFD and fMSA theories.
In their study of pore charging, \cite{aslyamov2020relation} use FMT and the weighted correlation approach (for $F^{\textrm{EC}}$) developed by \cite{wang2011weighted}, whose accuracy has not been systematically investigated.

The formulation of the grand potential energy density functional \eqref{grandpotential} is completed with the hard-wall component of the external potential, which away from the pore extremities is a function of $y$ alone,
  \begin{equation}\label{confiningpotential1} U_i^{\textrm{HW}}(y) =   \begin{cases} \displaystyle
  0  & \quad \text{if }  R_i \leq y \leq {L}-R_i, \\
    \infty  & \quad \text{otherwise.}
  \end{cases}
  \end{equation}
This enforces non-overlapping of the pore wall and the spherical ions, with the density profiles of the latter dropping to zero within one radius of the wall.

\subsection{Asymptotic model reduction}

We consider the DDFT equation \eqref{DDFT3D} in 2D from here onwards, assuming uniform densities in the infinite depth dimension. The focus of the present work is the case of thin slit pores, for which we introduce a scale separation parameter $\epsilon = {L}/{H} \ll 1$. Additionally, we are only concerned with dynamics on the timescales of supercapacitor applications (with typical frequencies ranging from 0.001 to 1000 Hz) which are significantly longer than the nanosecond diffusion times of ionic liquids (see parameter sets below). Since a full asymptotic derivation of the model valid in the limit as $\epsilon \rightarrow 0$ is presented in \cite{aslyamov2020relation}, we provide an informal derivation following the discussions in \cite{chacon2014asymptotic} and \cite{narski2014asymptotic} for the related problem of anisotropic diffusion equations.

With the auxiliary pore length variable $\xi = \epsilon x$ and slow timescale $\tau = \epsilon^2 t$, we obtain the rescaled DDFT equation
\begin{equation}\label{DDFT2Dnondimrescale}
 \frac{\partial \rho_i}{\partial \tau}  =   \beta D_i \frac{\partial}{\partial \xi} \left( \rho_i  \frac{\partial \mu_i}{\partial \xi} \right) +  \frac{1}{\epsilon^2} \beta D_i \frac{\partial}{\partial y} \left( \rho_i  \frac{\partial \mu_i}{\partial y} \right),
\end{equation}
where, in $(\xi,y)$ coordinates, the pore is a square of side ${L}$. Since we do not wish to resolve the dynamics on the fast timescale, we may formally set $\epsilon = 0$ in the above equation to obtain
\begin{equation}\label{leadingorder}
0 =  \beta D_i \frac{\partial}{\partial y} \left( \rho_i  \frac{\partial \mu_i}{\partial y} \right).
\end{equation}
This is equivalent to the statement that, to leading order, the densities satisfy a 1D equilibrium DFT system in the pore width defined by the Euler--Lagrange equations \eqref{EL0} with constant, yet undetermined, chemical potentials $\mu_i \equiv \mu_i(x,t)$. It is important to note that, while the ion densities satisfy a 1D equilibrium problem, they are not necessarily \emph{in equilibrium} with the bulk phase external to the pore.

Equation \eqref{leadingorder} is a constraint that neither fully specifies an equilibrium problem at each $x$ nor informs the time-evolution of the system; to provide the correct well-posed limit model it must be allied with the appropriate integral equation \citep{narski2014asymptotic}. The latter is obtained by integrating the DDFT equation in $y$ and applying the no-flux boundary conditions, annihilating the singular term in \eqref{DDFT2Dnondimrescale} for any $\epsilon$. Taking the limit $\epsilon \rightarrow 0$, we may use the fact that the $\mu_i$ are independent of $y$, and the integral equation becomes, in $(x,t)$ variables,
\begin{equation} \label{weakform1}\frac{\partial N_i}{\partial t}  =   \beta D_i \frac{\partial}{\partial x} \left( N_i  \frac{\partial \mu_i}{\partial x} \right),
\end{equation}
where we define the particle numbers (per unit of pore width and depth)
\begin{equation}\label{particlenumberdefn}N_i(x,t) = \int_{0}^{L} \rho_i(x,y,t) \; \mathrm{d}y.\end{equation}
Equation \eqref{weakform1} describes the evolution of the particle numbers and is the same as the limit equation derived by \cite{aslyamov2020relation}. Accordingly, we solve a canonical 1D equilibrium problem at each $x$ for the ion densities as a function of $y$, where the particle numbers \eqref{particlenumberdefn} are prescribed. This is as opposed to solving a grand canonical problem where the chemical potentials are given. The numerical approach is discussed in Appendix \ref{numericsequilibrium}. As an output of this equilibrium problem, we obtain the chemical potentials $\mu_i$ at each $x$, gradients of which are required to evolve the particle numbers in time according to \eqref{weakform1}. There exist DFTs specifically for canonical calculations with a small integer number of particles \citep{de2014full}, e.g.~hard spheres in a closed cavity. However, since our slit pore extends infinitely in the direction perpendicular to the $x$--$y$ plane, we can perform a thermodynamic limit, and use DFTs derived originally in the grand canonical framework with \eqref{particlenumberdefn} as an additional constraint.

The 3D DFT components discussed above must be restricted to a planar 1D geometry. The 1D weight functions for the hard sphere terms can be found in \cite{roth2010fundamental}, while the expressions for the correlations according to the BFD and fMSA theories can be found in \cite{gillespie2002coupling} and \cite{roth2016shells}, respectively. For the electrostatics, the left hand side of the Poisson equation \eqref{Poisson0} becomes $\epsilon_0\epsilon_{\textrm{r}} \psi_{yy}$. Using Gauss' law, we can compute the surface charge density along the long pore walls as a function of time,
 \begin{equation}\label{NeumannPoissonnonsym1} \psi_y|_{y=0} = - \frac{\mathcal{Q}_0(x,t)}{\epsilon_0\epsilon_{\textrm{r}}}, \qquad \psi_y|_{y={L}} = + \frac{\mathcal{Q}_L(x,t)}{\epsilon_0\epsilon_{\textrm{r}}}.\end{equation}
 Integrating the Poisson equation over the full pore width, $y\in [0,L]$, and applying \eqref{NeumannPoissonnonsym1} gives the instantaneous charge neutrality expression for a fixed-$x$ slice of the pore,
\begin{equation}\label{integralequationSCD1}\mathcal{Q} = \mathcal{Q}_0 + \mathcal{Q}_L = -  {\textrm{e}} \sum_i z_i N_i,\end{equation}
where $\mathcal{Q}(x,t)$ denotes the sum of the local surface charge densities on the pore walls.

When the entire system is at equilibrium, the chemical potentials in the pore equal those in the bulk and $\mathcal{Q}$ is independent of $x$ and $t$. We can use \eqref{integralequationSCD1} to compute the the zero-frequency differential capacitance, $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$ (denoting the rate of change in $\mathcal{V}$ for fixed $\mu_i$), and the integral capacitance, $\mathcal{Q}/(\mathcal{V}-\mathcal{V}_0)$, where $\mathcal{V}_0$ is the potential of zero charge (i.e.~$\mathcal{Q}(\mathcal{V}_0) = 0$).
The electrostatic component of the external potential, $U_i^{\textrm{ES}}$ (omitted above for brevity), is a function of the surface charge density, not the surface potential \citep{valisko2018systematic}; according to the uniqueness theorem of \cite{evans1979nature}, fixing $(\mathcal{Q}_0,\mathcal{Q}_L)$ uniquely determines the equilibrium densities in a planar geometry. However, the same cannot be said when prescribing surface potentials, and the surface charge--potential relation is known to be non-monotonic for certain parameters (which would provide more than one distinct equilibria for some values of the wall potential). For this reason, it is not ensured that $\mathcal{Q}_0=\mathcal{Q}_L$, or equivalently that the density profiles are symmetric about the pore centreline, $y = {L}/2$. Accordingly, all equilibrium profiles are solved for in a full pore, rather than a half pore with $\psi_y|_{y = {L}/2} =0$ (as is common in the literature), however, we did not observe any symmetry breaking or multiplicity of equilibria for the parameters considered here.

At the pore entrance and cap, the asymptotics break down and spatially 2D density profiles are expected; the full resolution of these regions is beyond the scope of the present study. At the entrance we fix the chemical potentials equal to those in the bulk,
\begin{equation} \label{entranceboundary} \mu_i = \mu_i^{\textrm{b}} \quad \textrm{at} \quad x = 0. \end{equation}
The particle numbers at $x=0$ are then computed from solving the grand canonical problem for 1D DFT equilibria (these will be time-dependent as the surface potential varies).
At the pore cap, we ignore electrostatic effects due to the charged bounding wall and simply implement a no-flux condition,
\begin{equation} \label{capboundary} \frac{\partial \mu_i}{\partial x} = 0 \quad \textrm{at} \quad x = H.\end{equation}

\section{Impedance response analysis\label{ImpRes}}

\subsection{Semi-analytical approach}

Following a similar route to that outlined in \cite{huang2020review}, we construct an expression for the frequency response of the limit model which incorporates numerically computable sensitivities. The base state for the following linear analysis is a pore at equilibrium with the bulk phase ($\mu_i = \mu_i^{\textrm{b}}$), for which the equilibrium particle numbers are $\overline{N}_i$ given a potential $\overline{\mathcal{V}}$ applied on the pore walls. While the above theory is valid for general many-component mixtures of ionic liquids, we study two-component ionic liquids exclusively in this work. Variables and constants corresponding to the cation are indicated with a subscript $+$, while anion quantities have a subscript $-$.

To mimic a potentiostatic impedance response experiment at the pore scale, we prescribe a voltage of the form $\mathcal{V}(t) = \overline{\mathcal{V}} + \Delta \mathcal{V}(t),$
where we use $\Delta$ to denote perturbation variables. Writing $\mu_i \equiv \mu_i(N_+,N_-,\mathcal{V})$, the chemical potential in the slit has the Taylor expansion
\begin{equation}
\mu_i = \mu_i^{\textrm{b}} + \frac{ \partial \mu_i}{\partial N_+} \Delta N_+ + \frac{ \partial \mu_i}{\partial N_-} \Delta N_-  + \frac{ \partial \mu_i}{\partial \mathcal{V}} \Delta \mathcal{V} + O(\Delta^2),
\end{equation}
where the partial derivatives are evaluated at the base state. We have the analytical fact that $\partial \mu_i/\partial \mathcal{V} = {\textrm{e}} z_i $, but the sensitivities with respect to the particle numbers must be computed numerically in general (these are calculated by solving the canonical equilibrium problem, as discussed in Appendix \ref{numericsequilibrium}, with perturbed particle numbers). The linearisation of \eqref{weakform1} about the equilibrium state is
\begin{equation}\label{linear1} \frac{\partial  \Delta N_i }{\partial t}  =   \beta D_{i} \overline{N}_i   \frac{\partial^2 }{\partial x^2} \left( \frac{ \partial \mu_i}{\partial N_+} \Delta N_+ + \frac{ \partial \mu_i}{\partial N_-} \Delta N_-  \right),
\end{equation}
with the boundary condition at the pore entrance \eqref{entranceboundary},
\begin{equation}\label{linear2}  \frac{ \partial \mu_i}{\partial N_+} \Delta N_+ + \frac{ \partial \mu_i}{\partial N_-} \Delta N_-  = -  {\textrm{e}} z_i \Delta \mathcal{V}   \quad \textrm{at} \quad x = 0, \end{equation}
and the no-flux condition \eqref{capboundary} at the pore cap,
\begin{equation}\label{linear3}  \frac{\partial }{\partial x} \left( \frac{ \partial \mu_i}{\partial N_+} \Delta N_+ + \frac{ \partial \mu_i}{\partial N_-} \Delta N_-  \right) = 0 \quad \textrm{at} \quad x = H. \end{equation}

With the solution of the above system, using \eqref{integralequationSCD1}, the perturbation to the local surface charge density, a function of $x$ and $t$, is calculated as
\begin{equation}\label{integralequationSCD1pert} \Delta \mathcal{Q} = - {\textrm{e}} z_+ \Delta N_+ - {\textrm{e}} z_- \Delta N_- .\end{equation}
Taking Fourier transforms in time (denoted with hats), we form the complex impedance per unit of pore depth and length (with units $\Omega$\,m$^2$),
\begin{equation} \label{Zdefn}
Z = \Delta \widehat{\mathcal{V}}\left( \frac{\textrm{i} \omega }{H} \int_0^H \Delta \widehat{\mathcal{Q}} \; \mathrm{d} x \right)^{-1},
\end{equation}
where $\omega = 2\pi f$ is the angular frequency for a temporal frequency $f$ (Hz). With this, we may obtain the frequency-dependent differential capacitance (in F\,m$^{-2}$) defined by
\begin{equation}\label{Cdiffdefn} C_{\textrm{diff}} = \frac{-1}{\omega \operatorname{Im}[Z]}. \end{equation}
The use of the average charge perturbation as opposed to the total charge perturbation in \eqref{Zdefn} is so that the differential capacitance \eqref{Cdiffdefn} is comparable with results for spatially 1D studies of slit pores, with the property that
\begin{equation}\label{Cdifflimit} C_{\textrm{diff}} \xrightarrow{{\omega \rightarrow 0}}  \frac{ \mathrm{d}\mathcal{Q}}{\mathrm{d}\mathcal{V}}.\end{equation}

\subsubsection{Single ion species}

If one of the ion species is not present the pore, the equations for the perturbations (\ref{linear1},\ref{linear2},\ref{linear3}) are significantly reduced. For example, if $L < \sigma_+$, we have $\overline{N}_+ = 0$ and it can be surmised that $\Delta N_+ = 0$. The problem that remains for $ \Delta N_-$ is
\begin{equation} \frac{\partial  \Delta N_- }{\partial t}  =   \beta D_{-} \overline{N}_-   \frac{ \partial \mu_-}{\partial N_-}  \frac{\partial^2 \Delta N_- }{\partial x^2}  ,
\end{equation}
with
\begin{equation} \frac{ \partial \mu_-}{\partial N_-} \Delta N_- = -  {\textrm{e}} z_- \Delta \mathcal{V}   \quad \textrm{at} \quad x = 0, \qquad \frac{\partial \Delta N_-}{\partial x}   = 0 \quad \textrm{at} \quad x = H. \end{equation}
In frequency space, the solution is given by
\begin{equation}\Delta \widehat{N}_- = \frac{ {\textrm{e}} z_- }{(\partial \mu_-/\partial N_-)} \left[ \tanh(\sqrt{\textrm{i} \omega \lambda/\beta} H) \sinh(\sqrt{\textrm{i} \omega \lambda/\beta} x) - \cosh(\sqrt{\textrm{i} \omega \lambda/\beta} x)  \right]\Delta \widehat{\mathcal{V}},\end{equation}
where $\lambda = 1/ D_{-} \overline{N}_- (\partial \mu_-/\partial N_-)$. With the substitution of this expression and $\Delta \widehat{\mathcal{Q}} = - {\textrm{e}} z_- \Delta \widehat{N}_-$ into \eqref{Zdefn}, we obtain precisely the FSW impedance \eqref{deLeviemodel} with parameters
\begin{equation}\label{singleionexact}\mathcal{Z} = \frac{H^2}{ {\textrm{e}}^2 z_-^2 \beta D_{-} \overline{N}_- },
\qquad  \tau = \frac{H^2}{\beta D_{-} \overline{N}_- (\partial \mu_-/\partial N_-)}.\end{equation}
Using the zero frequency limit of the FSW element from \eqref{deLeviemodellimits} with (\ref{Cdiffdefn},\ref{Cdifflimit},\ref{singleionexact}) gives
\begin{equation}\label{singleiondQdv}\frac{\mathrm{d}\mathcal{Q}}{\mathrm{d}\mathcal{V}} = \frac{\tau}{\mathcal{Z}} = \frac{{\textrm{e}}^2 z_-^2 }{ \partial \mu_-/\partial N_-}.\end{equation}
This relation can also be obtained from $\mathcal{Q} = - {\textrm{e}} z_- N_-$ and the cyclic chain rule for partial derivatives (note that $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$ denotes the derivative of $\mathcal{Q}$ respect to $\mathcal{V}$ with chemical potentials held fixed).

\subsubsection{General case}

In the situation that both ion species are present in the pore, we again take a Fourier transform in time of (\ref{linear1},\ref{linear2},\ref{linear3}) and work with the new variables
\begin{equation}\label{Ahatdefn} \widehat{A}_i =  \frac{ \partial \mu_i}{\partial N_+} \Delta \widehat{N}_+ + \frac{ \partial \mu_i}{\partial N_-} \Delta \widehat{N}_- ,
\end{equation}
with which we can write the system in the form
\begin{equation}\frac{\partial^2  }{\partial x^2}
\bm{\widehat{A}} = \frac{\textrm{i} \omega }{\beta} M \bm{\widehat{A}} , \qquad \widehat{A}_i   = -  {\textrm{e}} z_i \Delta \widehat{\mathcal{V}}   \quad \textrm{at} \quad x = 0, \qquad \frac{\partial \bm{\widehat{A}} }{\partial x}  = 0 \quad \textrm{at} \quad x = H,\end{equation}
where the matrix $M$ is
\begin{equation}
M = \frac{1 }{ |J|  }
\begin{pmatrix}
 (\partial \mu_-/\partial N_-) /D_{+} \overline{N}_+ & -  (\partial \mu_+/\partial N_-) /D_{+} \overline{N}_+  \\
-  (\partial \mu_-/\partial N_+)/D_{-} \overline{N}_- & (\partial \mu_+/\partial N_+) /D_{-} \overline{N}_-
\end{pmatrix}, \quad |J| = \frac{ \partial \mu_+}{\partial N_+}   \frac{ \partial \mu_-}{\partial N_-} -  \frac{ \partial \mu_+}{\partial N_-} \frac{ \partial \mu_-}{\partial N_+}.
\end{equation}
If $\lambda_1$ and $\lambda_2$ denote the eigenvalues of $M$, corresponding to eigenvectors $\bm{v}_1 = (v_1,1)^{T}$ and $\bm{v}_2 = (v_2,1)^{T}$, respectively, the solution to this system is
\begin{align}
\frac{\widehat{A}_+}{{\textrm{e}} \Delta \widehat{\mathcal{V}}} = & \; v_1 a_1 \left[\tanh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} H)  \sinh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} x)  -   \cosh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} x) \right] \nonumber \\
& + v_2 a_2\left[  \tanh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} H) \sinh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} x) - \cosh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} x) \right] ,\\
\frac{\widehat{A}_-}{ {\textrm{e}} \Delta \widehat{\mathcal{V}}} = & \; a_1 \left[  \tanh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} H)  \sinh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} x) - \cosh(\sqrt{\textrm{i} \omega \lambda_1 /\beta} x) \right] \nonumber \\
& + a_2 \left[ \tanh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} H) \sinh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} x)  -  \cosh(\sqrt{\textrm{i} \omega \lambda_2 /\beta} x) \right],
\end{align}
where $(a_1,a_2)^T$ satisfies
\begin{equation}\begin{pmatrix}
v_1 & v_2 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
 a_1 \\
 a_2
\end{pmatrix}
=
\begin{pmatrix}
 z_+ \\
  z_-
\end{pmatrix}.
\end{equation}
Defining time constants $\tau_i = H^2  \lambda_i /\beta$, the complex impedance \eqref{Zdefn} can be expressed as
\begin{align}
 \frac{ H^2 }{{\textrm{e}}^2 \beta} \; Z^{-1}  = & \;  z_+ D_{+} \overline{N}_+ \left[ v_1  a_1  \sqrt{\textrm{i} \omega \tau_1}   \tanh(\sqrt{\textrm{i} \omega \tau_1} ) + v_2  a_2 \sqrt{\textrm{i} \omega \tau_2}  \tanh(\sqrt{\textrm{i} \omega \tau_2}) \right]  \nonumber \\
& +  z_-  D_{-} \overline{N}_- \left[ a_1 \sqrt{\textrm{i} \omega \tau_1}  \tanh(\sqrt{\textrm{i} \omega \tau_1}) + a_2 \sqrt{\textrm{i} \omega \tau_2}  \tanh(\sqrt{\textrm{i} \omega \tau_2}) \right]. \label{fullZ}
\end{align}

For small frequencies, the complex impedance behaves according to
\begin{align}
  \frac{ H^2 }{{\textrm{e}}^2 \beta} \; Z^{-1}  \xrightarrow{{\omega \rightarrow 0}} & \;  \frac{\omega^2 }{3} \left(  z_+ D_{+} \overline{N}_+  \left[  v_1  a_1 \tau_1^2 +  v_2  a_2 \tau_2^2 \right]
 +  z_-  D_{-} \overline{N}_-  \left[  a_1 \tau_1^2 +  a_2 \tau_2^2 \right] \right)\nonumber \\
&
 +   \omega \left( z_+ D_{+} \overline{N}_+ \left[ v_1  a_1 \tau_1 + v_2  a_2 \tau_2 \right]  +  z_-  D_{-} \overline{N}_-  \left[ a_1 \tau_1  + a_2 \tau_2  \right] \right) \textrm{i} . \label{smallomegaanalytical}
\end{align}
The asymptotic behaviour \eqref{smallomegaanalytical} fully prescribes an ``effective" FSW element \eqref{deLeviemodel}, valid for small frequencies, with parameters denoted $\mathcal{Z}_0$ and $\tau_{0}$; these are found by substituting $Z^{-1} \approx \mathcal{Z}_0^{-1}(\textrm{i} \omega \tau_0 + \omega^2 \tau_0^2/3)$ into \eqref{smallomegaanalytical}. The zero frequency limit \eqref{smallomegaanalytical} can also be used to compute a relation between the chemical potential sensitivities and $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$, although more convoluted than \eqref{singleiondQdv}, and we may also obtain the asymptote of $\operatorname{Re}[Z]$,
\begin{equation}
\frac{3{\textrm{e}}^2 \beta }{  H^2} \operatorname{Re}[Z] \xrightarrow{{\omega \rightarrow 0}}  \frac{ z_+ D_{+} \overline{N}_+  \left[  v_1  a_1 \tau_1^2 +  v_2  a_2 \tau_2^2 \right]
 +  z_-  D_{-} \overline{N}_-  \left[  a_1 \tau_1^2 +  a_2 \tau_2^2 \right] }{  \left( z_+ D_{+} \overline{N}_+ \left[ v_1  a_1 \tau_1 + v_2  a_2 \tau_2 \right]  +  z_-  D_{-} \overline{N}_-  \left[ a_1 \tau_1  + a_2 \tau_2  \right] \right)^2}.
\end{equation}
In the large $\omega$ limit, we have
\begin{equation}
 \frac{ (1-\textrm{i}) H^2 }{{\textrm{e}}^2 \beta \sqrt{2\omega }  } \; Z^{-1}  \xrightarrow{{\omega \rightarrow \infty}}  z_+ D_{+} \overline{N}_+ \left[ v_1  a_1  \sqrt{ \tau_1 } + v_2  a_2 \sqrt{ \tau_2 } \right] +  z_-  D_{-} \overline{N}_- \left[ a_1 \sqrt{ \tau_1 } + a_2 \sqrt{ \tau_2 } \right], \label{largeomegaanalytical}
\end{equation}
which fixes one of the two degrees of freedom required to determine an effective FSW element with parameters $(\mathcal{Z}_{\infty},\tau_{\infty})$. We can identify that the correct time constant is $\tau_{\infty} = \min\{\tau_1,\tau_2\}$ by including the exponentially small terms in the above expansion and in the corresponding FSW limit in \eqref{deLeviemodellimits}. In the case that one ion species is negligible in comparison to the other, e.g.~$\overline{N}_- \gg \overline{N}_+ > 0 $, both $\tau_0$ and $\tau_{\infty}$ can be approximated by the FSW time constant in \eqref{singleionexact}, involving a sensitivity with respect to the non-negligible species. We observed that $\tau_0$ diverged from the single species result for much smaller co-ion particle numbers than $\tau_{\infty}$ (which can remain close to \eqref{singleionexact} for particle numbers separated by an order of magnitude only).

\subsection{Time-dependent numerical simulations\label{timedepsection}}

We additionally perform time-dependent simulations of the limit model to validate the semi-analytical impedance expressions. This step validates the numerical scheme itself, which will be used for future investigations of ionic liquids in long nano-pores. The details of the finite-volume scheme, as well as its particular implementation for frequency response simulations, are described in Appendix \ref{numericstime}. The approach is similar to that utilised by \cite{aslyamov2020relation}, but we take care to use time-stepping schemes which preserve positivity of particle numbers (a particularly desirable property for simulating pore systems for which some of the ion species can become negligibly small) and use a non-uniform spatial grid to reduce numerical expense.

For our dynamic simulations, we initialise with a system at equilibrium and apply a sinusoidal voltage signal of amplitude ${\max_t|\Delta \mathcal{V}(t)|} = 2.5$ mV. The average surface charge density in the pore is computed by integrating \eqref{integralequationSCD1} in $x$ and dividing the result by $H$ (approximated with a rectangle rule for the discretised system). After each period, we estimate the complex impedance by fitting the magnitude, $|Z|$, and phase, $\arg(Z)$, with the input voltage signal and output average surface charge density (fast Fourier transforms can be used when the data is equispaced in time).

\section{Numerical results\label{numresults}}

We present results for two parameter sets based on the monovalent ($z_+ = - z_- = 1$) ionic liquids [EMIm][TFSI] and [EMIm][BF$_4$] at 300K, summarised in Table \ref{physproptable}:
\begin{table}[H]
\begin{center}
\begin{tabular}{ |c|c|c|c| }
 \hline
Ionic liquid & I -- [EMIm][TFSI] & II -- [EMIm][BF$_4$] \\
 \hline
$\epsilon_{\textrm{r}}$ &  12  & 14.5  \\
$\sigma_+$ (nm) & 0.5 & 0.5  \\
$\sigma_-$ (nm) & 0.5 & 0.3  \\
$D_+$ ($\times 10^{-11}$ m$^2$s$^{-1}$) & 6.746 & 5.322 \\
$D_-$ ($\times 10^{-11}$ m$^2$s$^{-1}$) & 3.771 & 4.461  \\
$\sigma_+^2/D_+$ (ns) & 3.706 & 4.698 \\
 \hline
\end{tabular}
\end{center}
\caption{Physical parameters based on [EMIm][TFSI] and [EMIm][BF$_4$].}\label{physproptable}
\end{table}
The relative permittivities (dielectric constants) given in the first row are the static (zero-frequency limit) bulk values obtained by dielectric relaxation spectroscopy \citep{huang2011static,hunger2009temperature}, and are assumed fixed for the model ionic liquids. The ion diameters in the second and third rows of Table \ref{physproptable} are commonly used approximations for the asymmetric molecules under consideration as spherical particles \citep{neal2017ion}, while the (constant) self-diffusion coefficients in fourth and fifth rows of Table \ref{physproptable} are obtained from the Vogel--Tammann--Fulcher equations (based on data from NMR experiments) provided by \cite{noda2001pulsed}. The characteristic diffusion timescale is given in the final row of Table \ref{physproptable}. The parameters for case I were chosen to exemplify an ionic liquid composed of ions with similar sizes (restricted primitive model) with widely differing diffusivities, whereas case II represents ionic liquids exhibiting strong size asymmetry. Bulk densities of the cation and anion are set equal to $2.4$ nm$^{-3}$ (3.985 mol/L), providing the non-dimensional reduced bulk densities $\sigma_+^{3} \rho_+^{\textrm{b}} = \sigma_+^{3} \rho_-^{\textrm{b}} = 0.3$. This is close to the typical value for the ionic liquids which inspired the above parameter sets, and has been considered in other works \citep{qingsurface}. It follows from the limit model (and the expressions for the complex impedance) that changing the pore length results in a simple rescaling of the impedance response, and so we fix $H = 100$ $\mu$m for our numerical results. All numerical calculations were performed in a non-dimensional framework (see Appendix \ref{numericstime}). The use of dimensional variables in the figures below allows the comparison of the two ionic liquids (of differing characteristic diffusion times) with ease.

We consider four different DFTs, the simplest of these being the DCA-PB model \citep{hoffmann2013ion,gillespie2015review} which is the incorporation of steric hard-wall interactions \eqref{confiningpotential1} into the standard PB model (i.e.~$F^{\textrm{HS}} = F^{\textrm{EC}} = 0$). Without hard-wall effects, contact densities are unrealistically large and the equilibrium problem becomes difficult to resolve numerically, so the standard PB model is not included in our study. We also consider the addition of (White Bear II) hard-sphere terms to the DCA-PB model, which we refer to as the HS-PB model. The two remaining DFTs include all these elements as well as electrostatic correlations, and are identified by the model used for the latter (either BFD or fMSA). Equilibrium density profiles for the four DFTs with ionic liquid I are shown in Figure \ref{paramsetIequilibria} in Appendix \ref{numericsequilibrium}.

\begin{figure}
\centering
\begin{subfigure}{2.8in}\caption{Resistance.}
\includegraphics[width=2.55in]{RealZ_A.pdf}
\end{subfigure}
\begin{subfigure}{2.8in}\caption{Differential capacitance.}
\includegraphics[width=2.55in]{Cdiff_A.pdf}
\end{subfigure}
\caption{Impedance response for a pore of width $L = 2\sigma_+$ with the parameters of ionic liquid II and $\overline{\mathcal{V}} = 0.5$. Solid lines correspond to the full semi-analytical result \eqref{fullZ}, with the dotted and dashed lines indicating the high and low frequency effective FSW elements with parameters $(\mathcal{Z}_{\infty},\tau_{\infty})$ and $(\mathcal{Z}_{0},\tau_{0})$, respectively. The markers denote the results of time-dependent numerical simulations.}\label{BF_Cdiff_plots}
\end{figure}

In Figure \ref{BF_Cdiff_plots}, we plot the (a) resistance (real part of the complex impedance) and (b) differential capacitance \eqref{Cdiffdefn} against frequency for ionic liquid II with base voltage is $\overline{\mathcal{V}} = 0.5$ V. The solid line, computed from the full expression for the complex impedance \eqref{fullZ}, exhibits a double plateau in panel (a). This is expected for the impedance response of a system with two characteristic times ($\tau_1$, $\tau_2$). The dashed and dotted lines, denoting the results for the low and high frequency effective FSW elements, respectively, bound the solid line above and below. The BFD curves in panel (b) show a similar transition from the low to high frequency behaviour in the differential capacitance, whereas the asymptotic curves for DCA-PB are very close to the true differential capacitance across the full frequency range. The plots for HS-PB and fMSA are similar to those for DCA-PB and BFD shown in the figure, respectively, and so are not included. It is clear from Figure \ref{BF_Cdiff_plots} that the impedance response can deviate significantly from the typical FSW element obtained for a straight slit pore with a simple physical model. The markers in Figure \ref{BF_Cdiff_plots}, indicating the results of time-dependent numerical simulations, track the solid line very closely (it is too numerically expensive to simulate the system deeper into the low frequency regime). Beyond the results shown in Figure \ref{BF_Cdiff_plots}, we carried out time-dependent numerical simulations for each combination of DFT and ionic liquid, for $L/\sigma_+ = 2,4,6,8$, and a range of frequencies in the interval $0.1\; \textrm{Hz} < f < 100 \; \textrm{Hz}$. We found a similarly good agreement between the semi-analytical and fully numerical approaches in all cases (data not shown).

\begin{figure}
\centering
\begin{subfigure}{3in}\caption{High frequency response time (I).}
\includegraphics[width=2.75in]{DFTcomparisonNEW_S.pdf}
\end{subfigure}
\begin{subfigure}{3in}\caption{High frequency response time (II).}
\includegraphics[width=2.75in]{DFTcomparisonNEW_A.pdf}
\end{subfigure}
\centering
\begin{subfigure}{3in}\caption{Low frequency response time (I).}
\includegraphics[width=2.75in]{DFTcomparison_tau0_S.pdf}
\end{subfigure}
\begin{subfigure}{3in}\caption{Low frequency response time (II).}
\includegraphics[width=2.75in]{DFTcomparison_tau0_A.pdf}
\end{subfigure}
\caption{High (a,b) and low (c,d) frequency response time against pore width at $\overline{\mathcal{V}} = 0.5$ V. The results for ionic liquid I are shown in panels (a,c), with ionic liquid II in (b,d). The solid curves indicate the effective FSW time constants ($\tau_{\infty}$ and $\tau_{0}$) for each of the DFTs. The solid vertical lines correspond to the widths below which no ions can fit in the pore, with the dashed vertical lines in panels (b,d) indicating $L = \sigma_+$, below which only the smaller anion can enter the pore. For $\sigma_- < L < \sigma_+$, the curves in panels (b,d) are the exact FSW time constants for the single-ion system \eqref{singleionexact}.}\label{FSW_tau_plots}
\end{figure}

In Figure \ref{FSW_tau_plots}(a,b), we plot $\tau_{\infty}$ against pore width for each of the four DFTs, again with $\overline{\mathcal{V}} = 0.5$ V. The results for the size-symmetric ionic liquid I are shown in panel (a), with the size-asymmetric ionic liquid II in panel (b). The solid and dashed vertical lines in Figure \ref{FSW_tau_plots}(b) bound the region in which the larger cation cannot enter the slit pore; the value of $\tau_{\infty}$ here is exact and given in \eqref{singleionexact}. For the BFD and fMSA curves, the transition across the dashed line is smooth since $\overline{N}_+$ is negligible as $L \rightarrow \sigma_+$ (from above). For the DFTs without electrostatic correlations (DCA-PB and HS-PB), the response time decreases with pore width. When $L > 2 \sigma_+ $, the values of $\tau_{\infty}$ for these two theories agree quite closely (this is due to the dominance of electrostatic effects for our ionic liquid-inspired parameters, with a larger disparity found for $\epsilon_{\textrm{r}} = 40$, say). For the DFTs including screening effects, the value of $\tau_{\infty}$ when $L \rightarrow \sigma_-$ is relatively small, with a peak observed at $L = 2\sigma_+$ in panel (a), and a large $\tau_{\infty}$ regime found across $1.25 < L/\sigma_+ < 1.5$ in panel (b). There is good agreement between the results for BFD and fMSA, with a noticeable shift in $\tau_{\infty}$ for the size-asymmetric ionic liquid in panel (b) -- the shapes of the curves match well, however. In both panels (a) and (b), the high frequency response times for all DFTs converge to similar values for large pores ($L \gtrsim 6 \sigma_+$).

The corresponding plots for $\tau_0$ are shown in Figure \ref{FSW_tau_plots}(c,d). In panel (c), we see a significant deviation between the results for DCA-PB and HS-PB across all $L$, with the latter agreeing closely with BFD and fMSA for $L \gtrsim 4 \sigma_+$. For the size-asymmetric case in panel (d), we again see wide peaks in the time constant and a disagreement between the DFTs for large $L$. While the peaks in $\tau_{\infty}$ appear to be more significant for ionic liquid II, the situation is reversed for $\tau_0$. Much of the structure of $\tau_0$ for small $L$ is observed in $\mathrm{d}\mathcal{Q}/\mathrm{d}\mathcal{V}$ (as the latter determines one degree of freedom for the low frequency FSW element).

\begin{figure}
\centering
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \geq 0$ (I).}
\includegraphics[width=2.75in]{plusVscan_IL=S_NEW.pdf}
\vspace{3mm}
\end{subfigure}
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \geq 0$ (II).}
\includegraphics[width=2.75in]{plusVscan_IL=A_NEW.pdf}
\vspace{3mm}
\end{subfigure}
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \leq 0$ (I).}
\includegraphics[width=2.75in]{minVscan_IL=S_NEW.pdf}
\end{subfigure}
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \leq 0$ (II).}
\includegraphics[width=2.75in]{minVscan_IL=A_NEW.pdf}
\end{subfigure}
\caption{High frequency response time against pore width for a range of $\overline{\mathcal{V}}$ with fMSA for the DFT. Panels (a,b) show curves corresponding to $\overline{\mathcal{V}} = 0, 0.1, 0.2, 0.3, 0.4, 0.5$ V, with $-\overline{\mathcal{V}} =  0, 0.1, 0.2, 0.3, 0.4, 0.5$ V in panels (c,d). The left/right panels correspond to ionic liquid I/II, with the solid and dashed vertical lines as in Figure \ref{FSW_tau_plots}.}\label{FSW_Vscan_plots}
\end{figure}

For fMSA, we further investigated the dependence of the impedance response on the base voltage $\overline{\mathcal{V}}$, the results of which are presented in Figure \ref{FSW_Vscan_plots}. The top/bottom panel corresponds to positive/negative wall potentials, with the results for ionic liquids I and II shown in the left and right panels, respectively. The fMSA lines in Figure \ref{FSW_tau_plots} are the curves of smallest magnitude in panels (a,b) of Figure \ref{FSW_Vscan_plots}. For the size-symmetric ionic liquid in Figure \ref{FSW_Vscan_plots}(a,c), we see the large $\tau_{\infty}$ regime widen and shift from $L \approx 2 \sigma_+$ for $|\overline{\mathcal{V}}| = 0.5$ V towards $L \approx \sigma_+$ as $|\overline{\mathcal{V}}|$ is decreased to zero (potential of zero charge). A second peak of smaller magnitude shifts similarly, found at $L$ approximately double that of the first peak, and located at $L \approx 3\sigma_+$ for $\overline{\mathcal{V}} = 0$ V. The equilibrium profiles for oppositely signed wall potentials are identical, but with the cation and anion densities swapped. The differences between the corresponding pairs of curves in Figure \ref{FSW_Vscan_plots}(a,c) are solely due to the dissimilarity of the single-particle diffusivities for ionic liquid I (found in Table \ref{physproptable}). For small pores and large potentials, there is a negligible density of co-ions in the pore. In such cases, the ratio of time constants in Figure \ref{FSW_Vscan_plots}(a,c) is approximately the ratio of the diffusivities.

The plots for ionic liquid II are shown in Figure \ref{FSW_Vscan_plots}(b,d). Due to the asymmetry of the ion sizes, the equilibrium profiles for oppositely signed $\overline{\mathcal{V}}$ cannot be related as for ionic liquid I. Furthermore, the potential of zero charge is a non-zero function of $L$ (between $-0.1$ V and $0$ V for our case). To investigate the influence of the small variation in diffusivities for ionic liquid II, we computed the impedance response for $D_+$ and $D_-$ set equal to the mean of the diffusivities in Table \ref{physproptable}. When the density of one species in the pore was negligible, the response times were rescaled according to the change of the single-particle diffusivity of the dominant species. Otherwise, there was an insignificant deviation from the curves shown in Figure \ref{FSW_Vscan_plots}(b,d). In panel (b), we see smooth variation across the dashed vertical line between $\tau_{\infty}$ and the the exact single-species result, as for the BFD and fMSA results in Figure \ref{FSW_tau_plots}(b). For negative wall potentials in panel (d), the density of the smaller anions are negligible either side of $L = \sigma_+$, and so the value of $\tau_{\infty}$ is discontinuous as $\overline{N}_+$ jumps to zero. The values of $\tau_{\infty}$ to the left of the dashed line are not plotted as they correspond to a negligible quantity of ions. As observed for ionic liquid I, there are two large response time regions for the $L$-interval plotted, which shift as the base voltage varies.

\section{Discussion and conclusions\label{Conclu}}

We constructed a reduced-order limit model for the dynamics of ionic liquids in long slit pores, incorporating DFT for the equilibrium system and DDFT for the out-of-equilibrium time-evolution. We utilised this model to investigate the dependence of the impedance response on system parameters and the choice of DFT used. A semi-analytical approach was presented for the rapid evaluation of the complex impedance, with the only numerical inputs being sensitivities of the equilibrium DFT system. The derived expressions were validated with fully time-dependent numerical simulations.

The impedance response of our reduced-order model can deviate significantly from the de Levie result \eqref{deLeviemodel}, to an extent similar to that found for non-trivial pore geometries with an idealised electrolyte model \citep{cooper2017simulated}. In the high and low frequency limits, we are able to obtain effective FSW elements. We found that the inclusion of electrostatic correlations (which are vital for a physically correct description of the system \citep{voukadinova2019energetics}) drastically modified the predicted high and low frequency response times for pores with widths comparable to one or two ion diameters (see Figure \ref{FSW_tau_plots}). By varying the base voltage (at which the porescale PEIS was performed), we found that peaks in the FSW time constants shifted towards smaller pore widths as the potential difference magnitude was decreased (see Figure \ref{FSW_Vscan_plots}).

\begin{figure}
\centering
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \geq 0$ (I).}
\includegraphics[width=2.75in]{Q_plusVscan_IL=S_NEW.pdf}
\vspace{3mm}
\end{subfigure}
\begin{subfigure}{3in}\caption{$\overline{\mathcal{V}} \geq 0$ (II).}
\includegraphics[width=2.75in]{Q_plusVscan_IL=A_NEW.pdf}
\vspace{3mm}
\end{subfigure}
\caption{Charge stored at equilibrium (per unit of pore length and depth) against pore width for a range of $\overline{\mathcal{V}} \geq 0$ with fMSA for the DFT. The curves are analogous to those of $\tau_{\infty}$ in Figure \ref{FSW_Vscan_plots}(a,b) -- see the caption for details. Note the difference in vertical axes between the two panels.}\label{Qstored}
\end{figure}

As shown in Figure \ref{Qstored}, the charge stored in the pore also varies strongly for small widths as the underlying equilibrium changes. The oscillating dependence on $L$ becomes more dramatic for smaller values of $\epsilon_{\textrm{r}}$ (see the integral capacitance plots of \cite{jiang2011oscillation} with $\epsilon_{\textrm{r}} = 1$, for example). For supercapacitor geometries, a good distribution of pore sizes will provide an optimal balance of total charge storage and charge accessibility for a given application (which determines typical voltage and frequency ranges). For high frequency applications, the total charge stored by the individual electrode pores is relatively less important than their characteristic response times. In addition, the optimisation must respect the total pore fraction available, the fact that thinner pores take up a smaller proportion of the latter, and the shapes of pore size distribution that are feasible to manufacture.

For a positive electrode, Figure \ref{FSW_Vscan_plots}(a,b) and Figure \ref{Qstored} describe the dynamic and static charge storage properties of slit pores in the high frequency limit, respectively. For ionic liquid I, we find that peaks $\tau_{\infty}$ and $\mathcal{Q}$ coincide for $\overline{\mathcal{V}} = 0.5$ V, and so the two optimisation objectives are in conflict (improvements in total capacity result in reduced accessibility).  Slit pores with $L \approx \sigma_+$ have lower response times without a significant reduction in charge stored. For ionic liquid II, the optimisation objectives appear to be non-conflicting. Pores of width close to $\sigma_-$ provide large values of $\mathcal{Q}$ as well as fast response times, and for $L \approx 1.5 \sigma_+$, both the static and dynamic charge storage properties are relatively poor. It is clear that the presented theory (along with consideration of nonlinear responses \citep{aslyamov2020relation}) adds to the pool of information on which the optimisation of supercapacitor geometries can be based.

The model for the ionic liquid can be generalised with ease to incorporate non-constant permittivities and diffusivities; molecular dynamics and mean-field simulations have shown that confinement can cause significant variations in these quantities \citep{terrones2016enhanced,kondrat2014accelerating}. Relevant DDFT studies include the work of \cite{qing2020effects} incorporating density-dependent permittivities, and the theoretical investigation of packing fraction-dependent diffusivities by \cite{stopper2018bulk}. Inclusion of tortuosity effects is also kept for future work. Overall, the approach presented here is clearly applicable to more general particle systems or other models of ionic liquids; the asymptotic reduction of the DDFT system is an application of lubrication theory, and can be carried out for a large class of dynamical equations in long slit pores. The resulting limit model may then also be amenable to a semi-analytical impedance calculation.

We believe that modifications to the DDFT will be required to accurately model ion dynamics in long cylindrical pores. As discussed by \cite{te2020classical}, for the simple case of a single species hard-sphere fluid in single-file transport, particles can pass through each other when the time evolution is governed by the DDFT equation \eqref{DDFT3D}. In a slit pore, on the other hand, it can be thought that the ions slide past each other in the unbounded direction perpendicular to the $x$--$y$ plane.
Extensions of \eqref{DDFT3D} may be required to simulate pore systems further from equilibrium (e.g.~nonlinear electrochemical impedance spectroscopy). One option is provided by the recently developed power functional theory \citep{schmidt2013power} which generalises standard DDFT by including non-adiabatic effects ignored by the DDFT assumption \eqref{EL0}.

Lastly, we note that the limit model presented can be used as the basis of a simplified cell-level model, in which a set of pores representing each electrode are directly coupled to the separator region (with a modification of the entrance boundary condition \eqref{entranceboundary}), akin to the simplified model presented in \cite{kondrat2013charging} which ignores the pore-size hierarchy typical of electrode geometries; this parallels the many-particle models studied by \cite{kirk2020modelling} for Li-ion batteries. The time-dependent simulation approach detailed in Appendix \ref{numericstime} was only used for validation in the present work, but for geometries more complex than a single straight pore, for which the system is likely analytically intractable, time-dependent simulations may be the only viable approach. It would be of interest to improve the resolution of the system at pore entrances and branchings which could involve a hybridisation of the limit model with a 2D DDFT solver; we have also considered hybridisation with molecular simulations as done in \cite{borg2015hybrid} for compressible nanoscale flows.

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text:
Q: mPDF CSS class for text color of  nested in
* overridden by parent's color relevant HTML looks like:

         111095
      Foo/Bar, Some Text

... and the relevant CSS looks like ( file = icomoon's foo.css ) ...
/* this is an icomoon icon, that should be green */
.ic-p-appr:before {
  content: "\e905";
  color: #0f8040 !important;
}

... and ( file = bar.css ) ...

    .t1{color:#B92426;}
    /* .line-entry never defines a color */

I have added !important AND loaded the foo.css After the bar.css in an effort to make the icon green (#0f8040), but the icon nested within the .t1 class always inherits that parent's font color ONLY when mpfd converts the HTML to a pdf doc.  (the nested icon in HTML works as expect)
When I look at the mPDF documentation for supported CSS, I see nothing there or in SO that suggests why this occurs.  My mPDF is 7, and here is the configuration:

    $mpdf = new \Mpdf\Mpdf([
        'fontDir' => array_merge($fontDirs, [
            '../vendor/resources/fonts',
        ]),
        'fontdata' => $fontData + [
            'icomoon' => [
                'R' => 'icomoon.ttf',
            ],
        ],
        'format' => 'A4-L',
        'orientation' => 'L',
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see:
enter image description here
Help with this issue is appreciated.

A: Fixed: I discovered that because mPDF ver7 does not support CSS pseudo class :before, it was necessary to declare the font-color independently with the class name.  So for my HTML output, this worked, .ic-p-appr:before {  color: #0f8040 !important; } but I had to declare it again, minus the ':before' to register that style in the PDF document.
Summary: mPDF7 ignores CSS style attributes declared under a pseudo-class.

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Government keeps Lake Illawarra jetty repair report secret
The NSW Liberals and Nationals Government has refused to publicly release a secret report on the condition and repair of Lake Illawarra jetties, which have been closed off to the public for years.
Member for Wollongong, Paul Scully, made an application under freedom of information laws for the release of a report dealing with an assessment of Lake Illawarra infrastructure.
The Department of Planning, Industry and Environment (DPIE) notified Mr Scully this week that his application was rejected and the report would remain secret.
"The Government has closed off public access to a number of jetties on the Lake Illawarra foreshore for years now.
"The Department and the Minister have been sitting on an assessment report to repair these facilities and re-open them so the public can enjoy Lake Illawarra during this summer.
"But instead the jetties remain locked up from the public again this summer and continue to rot away on the picturesque Lake Illawarra – one of the Illawarra's most under-utilised natural assets – all because the Government wants to keep a report secret", he said.
Mr Scully said there is a growing trend in the Government refusing to publicly release information under the GIPA laws because of third-party involvement.
"Many of these reports are put together by private sector consultants and paid hundreds of thousands of dollars by the taxpayer.
"Yet these private sector consultants who have pocketed taxpayer funds refuse to allow the release of these taxpayer-funded reports", he said.
Mr Scully said he would an appeal to the NSW Information Commissioner in an effort to have the report released to the public.
"The community has a right to use the Lake Illawarra foreshore and its facilities and a right to know why they were closed in the first place.
"They have been locked up from public access for far too long", he said.
Authorised by Paul Scully MP, Shop G2, 51 Crown St, Wollongong NSW 2500

redpajama_set_name: RedPajamaCommonCrawl


text:
Desert Radiology | Premier Diagnostic Imaging Centers - Desert Radiology Supports Southern Nevada Students Going Back to School
Desert Radiology Supports Southern Nevada Students Going Back to School
LAS VEGAS (August 10, 2018) – Desert Radiology, a premier diagnostic imaging company that has served Southern Nevada for over 50 years, continues its mission to give back and help local communities. The imaging company donated nearly 300 backpacks and school supplies to students who attended the organization's annual "Operation Backpack" event, hosted at City Impact Center.
"Helping students across the Las Vegas Valley get back-to-school ready is a core component of our ongoing philanthropic initiatives," said Chief Executive Officer Richard Bodager. "This year we increased our efforts and donated 100 more backpacks filled with school supplies to low-income students at City Impact Center."
The imaging company has a continuing partnership with City Impact Center, a community outreach center that provides services to low-income individuals. More than 70 Desert Radiology staff members volunteered their time to welcome over 250 children to the "Operation Backpack" event. The students filled new backpacks in a mock store setting at Innovations Charter School within the City Impact Center campus and enjoyed breakfast sponsored by Desert Springs Hospital. This is Desert Radiology's second year supporting the backpack drive.
Desert Radiology also donated 100 backpacks to Desert View Hospital for its Back-to-School Fair on July 21 in Pahrump, Nevada. The event offered complimentary sports physicals, vision and dental screenings, backpacks and school supplies, to students in Southern Nevada.
Desert Radiology has been a key community partner in supporting Southern Nevada organizations, including City Impact Center. The imaging company assists with City Impact Center's Urban Garden and donates frozen turkeys for holiday meals every year, among other community initiatives. Desert Radiology also works with Las Vegas Diaper Bank, Volunteers in Medicine of Southern Nevada, and the Susan G. Komen Race for the Cure.
Desert Radiology has been the area's chief resource for diagnostic imaging services since 1966. The diagnostic imaging company currently serves over 30,000 patients monthly across its outpatient facilities. Desert Radiology will open its ninth imaging facility on August 13 making it the largest radiology provider in Southern Nevada.


text:
Q: Unzip objects collection into arrays of its properties Let's assume that there is some class:
public class SomeClass
{
    public string A;
    public int B;
    public double C;
}

and we have a collection of it:
var list = new List<SomeClass>()
{
    new SomeClass() { A = "A", B = 2, C = 4.0 },
    new SomeClass() { A = "B", B = 6, C = 8.0 }
};

How can we write in the most efficient way the transform of this collection into three arrays (or list of arrays) of values of A, B and C respectively? In this case it would be:
var Ar = new[] { "A", "B" };
var Br = new[] { 2, 6 };
var Cr = new[] { 4.0, 8.0 };

So far I have working LINQ query:
var lists = list.SelectMany(c => new object[] { c.A, c.B, c.C })
                .Select((v, i) => new { Index = i, Object = v})
                .GroupBy(g => g.Index % 3, g => g.Object, (k,v) => v.ToArray())
                .ToList();

but I wonder if I'm not missing something obvious and if there is possibility to do it clever, for example without the need for creating temporary subcollections within SelectMany.
Edit: my question is rather from theoretical point of view - if it can be done in single LINQ query (for example as a part of another, more complex single query).

A: Why don't you do this the simplest possible way? I think that 3 separated queries should be more efficient then SelectMany+GroupBy solution:
var Ar = list.Select(x => x.A).ToList();
var Br = list.Select(x => x.B).ToList();
var Cr = list.Select(x => x.C).ToList();

It requires only 3 iterations over list collection. With GroupBy you have only one iteration but the source collection is 3 times longer then list, so there is no difference (because of SelectMany) with additional GetHashCode calls to make the grouping.
Update
Single line version of the same approach:
var items = new [] {
    list.Select(x => x.A).ToList(),
    list.Select(x => x.B).ToList(),
    list.Select(x => x.C).ToList()
}

Update 2
If you don't want to use SelectMany, you can use Concat instead. However, it would require change in GroupBy as well:
var lists = list.Select(c => c.A)
                .Concat(list.Select(c => c.B))
                .Concat(list.Select(c => c.C))
                .Select((v, i) => new { Index = i, Object = v})
                .GroupBy(g => g.Index / list.Count, g => g.Object, (k,v) => v.ToArray())
                .ToList();