## SciPost Submission Page

# Multiscale response of ionic systems to a spatially varying electric field

### by Jesper Schmidt Hansen

#### This is not the current version.

### Submission summary

As Contributors: | Jesper Schmidt Hansen |

Arxiv Link: | http://arxiv.org/abs/1611.02131v2 (pdf) |

Date submitted: | 2016-11-10 01:00 |

Submitted by: | Hansen, Jesper Schmidt |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approaches: | Theoretical, Computational |

### Abstract

In this paper the response of ionic systems subjected to a spatially varying electric field is studied. Following the Nernst-Planck equation, two forces driving the mass flux are present, namely, the concentration gradient and the electric potential gradient. The mass flux due to the concentration gradient is modelled through Fick's law, and a new constitutive relation for the mass flux due to the potential gradient is proposed. In the regime of low screening the response function due to the potential gradient is closely related to the ionic conductivity. In the large screening regime, on the other hand, the response function is governed by the charge-charge structure. Molecular dynamics simulations are conducted and the two wave vector dependent response functions are evaluated for models of a molten salt and an ionic liquid. In the low screening regime the response functions show same wave vector dependency, indicating that it is the same underlying physical processes that govern the response. In the screening regime the wave vector dependency is very different and, thus, the overall response is determined by different processes. This is in agreement with the observed failure of the Nernst-Einstein relation.

### Ontology / Topics

See full Ontology or Topics database.###### Current status:

### Submission & Refereeing History

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## Reports on this Submission

### Anonymous Report 2 on 2016-12-22 Invited Report

- Cite as: Anonymous, Report on arXiv:1611.02131v2, delivered 2016-12-22, doi: 10.21468/SciPost.Report.57

### Strengths

1. The development of simple relations between dynamic properties in the presence of an external field and equilibrium properties of ionic systems is an interesting and important problem.

2. The authors proceeds to compute inputs to the theory from simulation.

### Weaknesses

1. Many issues that need to be clarified regarding the physical interpretation of the results (see report).

2. There are issues with the simulation procedure. In particular, is the use of cutoffs reasonable for all systems studied.

3. The ionic liquid model needs to be described in much greater detail.

4. The author never actually uses the theory to describe the response of fluids to external fields. Even though this is the purpose of the theory.

### Report

\newcommand{\kT}{\ensuremath{k_{\rm B}T}}

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In this paper, Hansen investigates the response of ionic systems to a spatially varying electric field through

a combination of theory and molecular dynamics simulations.

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\textbf{This work may be suitable for publication eventually, but only after significant revision.}

%

There are some shortcomings that need to be addressed, as detailed below.

%

However, addressing these issues should significantly improve the paper.

%

\begin{itemize}

\item After Eq. (5), the author assumes that the response function $\chi_N'$ can be separated into distinct spatial

and temporal components. Can the author explain why this assumption is valid and when it is expected to break down?

\item The sentence ``The symbol $\chi_i$ is a matter of choice." is not necessary, because all notation is a matter of choice.

\item The author refers to the Debye-H\"{u}ckel regime as the ``zero screening'' limit throughout.

The meaning behind this statement would be better appreciated by the reader if the author mentioned the dielectric constant, $\varepsilon$.

For example, Eq. (17) implies

\begin{equation}

\frac{1}{\varepsilon(k)} = 1+\frac{2\pi n \tilde{D}(k)}{\kT k^2},

\end{equation}

such that $\varepsilon(k)\rightarrow 0$ as $k\rightarrow 0$,

assuming $\tilde{D}(k)$ does not impact the small $k$-dependence of $\varepsilon$.

Note that $\varepsilon(0)=\infty$ in the complete or perfect screening limit.

See Chapter 10 of Reference 2 for further details.

\item Similarly, in the ``screening regime,'' the dielectric constant is (using Eq. 18)

\begin{equation}

\frac{1}{\varepsilon(k)} = 1+\frac{2\pi n D_0}{\kT k^2} S_{zz}(k).

\end{equation}

Using $\lim_{k\rightarrow 0} S_{zz}(k) = q^2 k^2 / [k_{\rm D}^2 (1+k^2/k_s^2)]$, where $k_{\rm D}$ is the Debye wavelength,

$k_s=A k_{\rm D}^2$, and $A$ is related to the isothermal compressibility (See Chapter 10 of Ref. 2),

one can readily obtain $\lim_{k\rightarrow 0} \varepsilon^{-1}(k)=1+D_0/2$.

Thus, in the perfect screening regime, $\varepsilon(k)=\infty$ implies $D_0=-2$.

Therefore, when complete screening is obtained, $D_0$ is negative! Is there a sign error somewhere in the text?

Or is $\tilde{D}(k)<0$?

It is strange that Equation (10) of the manuscript (and those that follow from it) show that the nonuniform

density $n$ is proportional to the negative of $\chi$; typically $n\propto \chi$ and $n\propto - S_{zz}$.

Similarly, Eq. 18 shows that $\chi\propto S_{zz}$, while the linear response function is typically

proportional to the negative of $S_{\rm zz}$. Can the author provide some insight here?

\item On page 6, ``$\tilde{\rho}_q=n \tilde{\phi}^{\rm ext} / \kT$'' is missing a negative sign, and should read

``$\tilde{\rho}_q=-n \tilde{\phi}^{\rm ext} / \kT$''.

\item I have serious concerns with the simulation procedure.

The author uses the same fixed cutoff radius to evaluate the Coulomb interactions in the various systems.

However, unless this cutoff is significantly larger than the screening length in the systems studied,

artifacts will be introduced. Such artifacts may substantially alter the expected behavior in the system,

especially with respect to subtleties like electroneutrality that are important for examining scaling relations.

What are the screening lengths of the various systems, and how do the results change with different cutoffs?

Cutoff procedures may be a good approximation for some aspects of uniform systems, but typically break

down in nonuniform systems due to the importance of long-ranged interactions (like systems in the presence of an external field).

Why not use Ewald summation?

\item The author ``simulates'' the Debye-H\"{u}ckel regime by simply removing the charges on the ions.

Why not just simulate at a high temperature?

In this case, the author is simply studying a WCA fluid at the prescribed temperature and pressure.

Such a system has no electrostatic correlations, and the resulting charge correlations are trivial

(they are just the WCA number density correlations).

\item ``Yurukawa potential'' should be ``\textbf{Yukawa} potential'' throughout.

\item The ionic liquid model cited by the author (Ref. 15) is not what is being used in this work.

That model consists of hard sphere and square well potentials, in addition to Yukawa potentials.

Ref. 15 discretizes the Yukawa potential to simulate this model with discrete molecular dynamics simulations.

In contrast, Hansen has modified the model to consist of WCA potentials and a continuous Yukawa potential.

The author needs to describe the model in much greater detail.

What are the WCA parameters for each site? How were they chosen? What are the charges, units, etc.?

\item With regard to the fits using Eq. 23, what are the values of $D_0$, $\alpha$, and $\beta$ that are obtained?

\item At the end of page 8, Hansen states ``$\tilde{\chi}(k)\propto S_{\rm ZZ}(k)$.''

The typical charge-charge linear response function for the response of an ionic fluid to a general external field

is equal to $-\frac{n}{\kT} S_{\rm ZZ}(k)$, where $n$ is the bulk density of the fluid.

How does $\tilde{\chi}(k)$ relate to the usual charge-charge linear response function,

and in particular, how does its physical interpretation differ?

\item Do the charge-charge structure factors satisfy the zeroth and second moment conditions of Stillinger and Lovett?

\item How are the values of $l$ obtained on page 10?

\item The discussion of the ``$T_\infty$'' model on page 10 is trivial (it is an uncharged, WCA fluid).

I would argue that the results for the $T=1.0177$

and $T=0.0177$ systems should be more interesting. This data should be included and discussed.

\item Page 10, last paragraph, ``exited'' should be ``excited''

\item Fig. 4, what about larger $k$-values for the $T=0.0177$ curve? What does this data look like?

\item For all the simulations in the presence of an external field, the author should compare the results

of the theory in Section II with the simulation results.

Without this comparison, there is little reason to develop the theory at all.

The author should also do this for the ionic liquid section.

\end{itemize}

### Requested changes

1. Clarify assumptions in the theory.

2. Clarify language regarding "zero screening" and "screening" regimes.

3. Correct typographical errors.

4. Verify the correctness of the simulation procedure.

5. Give more details regarding the simulations.

6. Discuss in more detail the physical meaning of the response function $\chi$ and how it relates to the typical charge-charge linear response function.

7. Compare the theory developed to the simulation results in the presence of an external field.

### Anonymous Report 1 on 2016-12-15 Invited Report

- Cite as: Anonymous, Report on arXiv:1611.02131v2, delivered 2016-12-15, doi: 10.21468/SciPost.Report.50

### Strengths

1. The calculations seem to be correct in theoretical part.

### Weaknesses

1. The situation is unrealistic.

2. The molecular simulation has no validation, and then the agreements between the theoretical results and the numerical results are not acceptable (at least for me).

### Report

1. The authour calculates the response of ions under spatially varying (but stationary) electric field, as shown in Eq.(3). However, as he admits,

''The experimental realization of this field is not straightforward''. This point greatly decrease the significance of the present manuscript. The authour should propose a possible experimental realisation of this ''unrealistic'' electric field.

2. In numerical simulation, the authour uses the shifted Coulomb force, introducing some references. However, I can never believe the validity of this approximation, because the long-range nature of Coulomb interaction is disregarded in this method. The authour should compare his results with other methods treating the Coulomb interaction sincerely, e.g. the Ewald method.

### Requested changes

Minor comment:

1. In molecular simulations, the numerical unit of the charge is given by $\sqrt{\epsilon\sigma}$. How the authour relate this and $q$?

2. In page 7, ''Yurukawa'' should be corrected as ''Yukawa''. In the caption of Fig.1, text overlap of ''lines'' in the third line.

I have attached my reply to the referee comments.

(in reply to Report 2 on 2016-12-22)

I have attached my reply to the referee.

## Attachment:

reply-ref1.pdf