RESEARCH STATEMENT - Pennsylvania State University · Pei Liu Research Statement January 2019 2...

Pei Liu Research Statement January 2019

RESEARCH STATEMENTPei Liu ([email protected])

Postdoctor, Department of Mathematics, Pennsylvania State University, University Park, PA

Abstract

Broadly speaking, my research interests include the mathematical modeling, analysis and numericalsimulation on complex fluids and statistical mechanics with applications in math biology and energy de-vices. Mainly, I have been focusing on developing consistent continuum theories and correspondingnumerical fast algorithms for the non-isothermal and non-local charge transport with further biophys-ical and chemical engineering applications [24, 55, 28, 27, 25, 29, 19]. In the meantime, I also collaboratewith experimental/simulation groups by providing data analysis and algorithm design [52, 59]. In thefuture, I propose to keep working on projects related with understanding the non-equilibrium thermody-namics from different perspectives and different scales, such as the problems in the kinetic theory, thestochastic process as well as the continuum limit of particle dynamics.

1 Overview

The systems with charges are abundant in real mechanical and biological environments. They play an ex-tremely important role in a wide range of applications, for example, the Lithium-ion batteries and electrolyticcapacitors, the structure and function of bio-macromolecules, and the propagation of neuron signals. Startingfrom the pioneering work of L. G. Gouy [15] and D. L. Chapman [9], a lot of researchers from different disci-plines, including theoretical physics, chemical engineering and electrophysiology, have been deeply attractedby this complex system over the past century. The main difficulty and challenge comes from the fact thatit is a multi-scale many-body system with non-local interaction. Thanks to the development of appliedmathematics, the manner and degree of using mathematical tools have changed dramatically nowadays. Itis time to employ new mathematical techniques (asymptotic analysis, numerical simulations, etc) torevisit and improve the classical models from the standard statistical mechanics. We follow the First Principleof Energetic Variational Approach (EnVarA) [17, 53]: the conservative forces can be obtained throughthe Least Action Principle, the dissipative forces are given by the Maximum Dissipation Principle, and On-sager’s principle points out the balance between these two kinds of forces. Basically, with given free energyfunctional and entropy production rate, a unique complete PDE system can be derived. The main advantagesby doing so are, (1) the basic physical laws are automatically fulfilled; (2) the mechanisms at different scalesare consistently included. It should be emphasized that almost all biological and technological applicationsinvolve time and friction. The EnVarA extends the classical thermodynamics to deal with time and frictionconsistently. Besides analyzing the solution of the PDE system (existence, uniqueness, etc), our analysis alsofocuses on the physical interpretations, with the help of numerical simulations.

It is known that the classical Poisson–Boltzmann (PB) model (for equilibrium state) from Gouy–Chapmantheory and the classical Poisson–Nernst–Planck (PNP) model (for transport dynamics) might fail to capturethe correct physics due to their mean-field nature. For example, they cannot explain the saturation andstratification of ionic distributions near a strongly charged surface. In order to understand this phenomenon,we take into account non-uniform ionic sizes in the size-modified PB model, prove the valence-to-volume ratiois the key parameter in the density stratification and the finite volume of charges can reduce the screening[24]. The numerical results computed through an augmented Lagrange multiplier method, match with ouranalysis and molecular simulations. This publication is selected in the Journals Highlights of 2013.

Another counterintuitive phenomena is the like-charge attraction in electrolytes. We derive the freeenergy functional through the perturbation theory to go beyond the mean-field approximation, obtaininga self-energy modified PB/PNP model [25]. Using this model, we systematically investigate the occurrence oflike-charge attraction, which is in good agreements with the experimental and molecular simulation observa-tions, indicating the abnormal attraction is a consequence of the ionic correlation and dielectric depletion [28].On the other hand, we develop WKB asymptotic expansions and adopt selected-inversion algorithmto numerically solve the self-energy equation, which is a 6-dimensional PDE [55, 25]. Moreover, we performmatched asymptotic analysis to study the two-layer structure in the vicinity of interface/boundaryabout the charge transport between two electrodes [19].

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The aforementioned models as well as most models in literature are all for isothermal cases, while thesystems in batteries and ion channels are sensitive to temperature changes. In [29], we make the first attemptto propose a thermodynamically-consistent framework to describe the coupling between mechanicalfluxes and heat flux for non-isothermal electrokinetics. Through the EnVarA, we show that the variationalprinciples are still valid for non-isothermal systems along with the derived Fourier law for heat conduction.Our framework guarantees the basic laws of thermodynamics to be fulfilled automatically, extends the classicalthermodynamics to deal with time and friction. This approach can be generalized to a variety of complexfluid systems, such as liquid crystals and phase-field models.

As an application, to understand better the gating current measurements [5] in ion channels, wegeneralize the Shockley–Ramo theorem from vacuum to electrolytes [27]. By means of the charge and dipolarrenormalization, the generalized Shockley–Ramo theorem provides a simple and direct relation between themotion of a macro charged particle in electrolytes and the induced currents without tedious integration. Thistheorem can be used to determine the trajectory of a macro-particle in electrolytes through the measurementsof induced currents, e.g., the motion of the voltage sensor as well as the conformational change of an ionchannel can be inferred from the patch-clamp experiments on the electronic signal.

Another development is in the numerical methods. In [59], we develop a two-layer image chargemethod for the efficient evaluation of the Greens function in a confined domain with given boundary condi-tions. The inner-layer image charges are located near the boundary to eliminate the singularity of the inducedpolarization potential. The outer-layer image charges with fixed positions approximate the long-range tailof the potential. This method can be used for molecular simulations [58] and the numerical solution of theelliptic equations. In [52], we also design an optimization algorithm to help analyzing the experimentalmeasurements of copolymerization. Through the quantitative determination of rate constants and reactivityratios with the aid of numerical calculation in a simulated annealing scenario with bootstrap strategy, we pro-vide a widely applicable design principle for the sophisticated construction of sequence distribution-regulated,functional polyethers via a simple and efficient one-pot approach.

My future research will be focused on understanding the non-isothermal electrokinetics from differentperspectives and different scales.

• The dielectric permittivity is a coarse-grained variable to describe the atomic polarization, taking intoaccount the polarization density could help us understand the dielectric effects.

• The PDE system we obtained are highly nonlinear and nonlocal, we also need to implement efficientand accurate algorithms for solving numerical PDE, which preserves the discretized conservation laws(mass, momentum, energy, entropy).

• The temperature effects for polymers through a modified FENE model.

• Developing a fast algorithm for elliptic equations based on the two-layer image charge method.

• From the stochastic point of view, with the help of Molecular Dynamics, we would be able to find outa better free energy functional to include the correlation and temperature effects, as well as a non-localentropy production rate.

• The phase-space density distribution can be described through the Liouville’s equation, we would liketo find out the continuum limit for the non-isothermal system.

• Although there are many experimental results, the mathematical descriptions to the gating mechanismand synaptic cleft are still not understood, we plan to properly include the fluid-structure interactionwithin the electrodiffusion model and apply to the real biological system.

• The capability of the batteries and capacitors are strongly related with the nanostructure and dielectricpermittivity of the electrodes. Our multiscale modeling and analysis could help to understand anddesign better energetic storage devices.

• The integral equation theory for liquid has an intrinsic convolutional hierarchy structure. I propose tofind a higher order closure for a more reliable model, with acceptable computational cost.

In the following sections, I will explain the completed work and future projects in more detail.

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2 Completed Work

2.1 Size modified Poisson–Boltzmann model. It has been observed both in experiments andmolecular simulations [49] that the ionic density would saturate and stratify in the vicinity of a stronglycharged surface, while the PB theory always predicts unbounded monotonic ionic density profiles. The latticegas model [7] has been extended to the size modified Poisson–Boltzmann theory with nonuniform ionic radius[43, 23]. The free energy functional is written as:

F [ρ1, · · · , ρN ] =

∫ [− ϵ2|∇ϕ|2 +

N∑i=1

ziρiϕ+ kBTN∑i=0

ρi log ρivi

]dV, (1)

where ρ0 = (1 −∑N

i=1 ρivi)/v0 is the density of solvent molecules, ρi and vi are the density and volume ofith species, T is the temperature, kB is the Boltzmann constant, ϵ is the dielectric constant, ϕ is the meanelectrical potential governed by the Poisson’s equation,

−∇ · ϵ∇ϕ =

N∑i=1

ziρi. (2)

The equilibrium state distribution is given by minimizing the free energy functional (1). For the caseof uniform size vi = v, the distribution ρi = ρbie

−ziϕ/kBT /(1 + v∑N

j=1 ρbje

−zjϕ/kBT ) is like Fermi-Dirac type

distribution thus have a upper bound (ρbi is a constant representing the bulk concentration of ith species).However an explicit distribution for nonuniform sizes is generally not available, so we describe it as the implicitgeneralized Boltzmann distributions [24]:

viv0

ln(1−N∑j=1

ρjvj)− ln ρivi =ziϕ− µikBT

. (3)

Figure 1: Counterion stratifica-tion from minimizing the free en-ergy functional (1).

Here µi is the chemical potential of ith ion species. Eq. (2) and Eq. (3) to-gether form the size-modified Poisson–Boltzmann model. We then presentdetailed analysis and numerical calculations, prove the concentration ofions with the largest and smallest valence-to-volume ratios decreases andincreases monotonically respectively, with respect to ϕ. Furthermore, thetotal ionic charge density is a monotonic function of the potential.

By analyzing the asymptotic distribution when |ϕ| is large, we thenprove the value of ionic valence-to-volume ratio plays the key role in de-termining the counterion stratification phenomenon. As ϕ→ −∞ (or ∞),the concentration of ions with the largest (or smallest) valence-to-volumeratio approaches to the inverse of its corresponding volume while all otherconcentrations approach zero, which form the first layer near a stronglycharged surface. Similarly, there will be additional layers if there are moreionic species, as shown in Fig. 1.

Moreover, we report the finite sizes not only play a center role in the near field ionic distribution, but alsochange the screening in the far field where ϕ→ 0. In the original PB theory, the screened Coulomb potentiale−κr/r is characterized by the inverse Debye length κ. Through linearizing the modified PB equation, weprove the size effect reduces the inverse Debye length: 0 ≤ κ ≤ κ.

2.2 Self-energy Modified PB/PNP Model. In [25], we introduce the standard perturbation theoryin statistical mechanics, deriving the free energy functional given by the Debye charging process:

F =

∫ [− ϵ2|∇ϕ|2 +

N∑i=1

ρiziϕ+

N∑i=1

kBTρi log ρi

]dr+

∫ ∫ 1

02λ

N∑i=1

ρi(r)Ui(r;λ)dλdr. (4)

The first integral is the same with the mean-field PB and PNP theory. The second integral representsfor the contribution from ionic correlation and inhomogeneous dielectric environment, which is expressedas: Ui(r;λ) = 1

2

∑Nj=1

∫hij(r, r

′;λ)ρj(r′)vji(r

′, r)dr′ + UBorni (r). Here λ is a real number between 0 and 1,

describing the charging state. vij(r, r′) is the pairwise potential between particles of ith and jth species,

hij(r, r′;λ) represents for the total correlation function which is related with the two-body joint probability

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density distribution. The Born energy UBorni describes the solvation energy of a single ith particle [6]. By

using linear response approximation for hij , we obtain the generalized Debye–Huckel equation,−∇ · εi(r, r′)∇Gλ

i (r, r′) + 2λ2χi(r− r′)I(r)Gλ

i (r, r′) = δ(r− r′),

−∇ · ϵ0∇G0(r, r′) = δ(r− r′),

Ui(r;λ) =z2i2uλi (r) =

z2i2

limr′→r

[Gλi (r− r′)−G0(r− r′)],

(5)

where εi(r, r′) = ϵ(r) and χi(r − r′) = 1, if |r − r′| > ai. Otherwise εi(r, r

′) equals to a constant ϵ0 andχi(r − r′) = 0. The local ionic strength I(r) =

∑Ni=1 ρiz

2i /2kBT . Similar equations have also been derived

and studied in literature [4, 35, 32, 33, 47, 45, 14] using different approaches.

The equilibrium state corresponds to the minimization of the free energy functional (4). So the electrostaticpotential and the ionic concentration distribution is given by the modified PB equation:

−∇ · ϵ∇ϕ =

N∑i=1

ziρi =

N∑i=1

ziρbie

−(ziϕ+z2i2u1i )/kBT . (6)

Together, Eq. (5) and (6) form the self-energy modified PB model. The dynamic transport can be derivedthrough Fick’s law or using EnVarA [55, 25], resulting in a self-energy modified PNP model,

∂

∂tρi = ∇ ·Di

[kBT∇ρi + ρi∇(ziϕ+

z2i2u1i )

]. (7)

Simplification for the Self-energy. Equation (5) is of 6-dimension in real 3-dimensional applications thuscannot be numerically solved directly. Noticing that the ionic radius ai are usually very small compared withthe fluid system, we first investigate the self-energy of a point charge [55]. Through WKB approximation,the self-energy of a point ion between two parallel dielectric interfaces of separation D, is given by the sumof screened Coulomb potentials from reflected image charges,

uPCA(x) = −κ(x) +∑

ℓ=2,4,···

2γℓe−κ(x)ℓD

ℓD+

∑ℓ=1,3,···

γℓ

[e−κ(x)(ℓD+2x)

ℓD + 2x+e−κ(x)(ℓD−2x)

ℓD − 2x

]. (8)

x

Net

char

gede

nsity

-1 -0.9 -0.8 -0.7 -0.60

0.2

0.4

0.6

0.8

1

q = 0WKB, q = 0.05WKB, q = 0.1WKB, q = 0.2FDM, q = 0.05FDM, q = 0.1FDM, q = 0.2

( a )

T = 0.2

x-1 -0.9 -0.8 -0.7 -0.60

0.2

0.4

0.6

0.8

1

q = 0WKB, q = 0.05WKB, q = 0.1WKB, q = 0.2FDM, q = 0.05FDM, q = 0.1FDM, q = 0.2

( b )

T= 0.5

Figure 2: Comparison between WKB and FDM.

Here κ is weighted average of the inverse Debyelength. This analytical expression is in good agree-ment with direct numerical results computed fromfinite difference method(FDM). On the other hand,by noticing when the ionic radius vanishes, theGreen’s function can be understood as the inverseof the linear operator −∇·ϵ∇+2λ2I, the self-energyis then the nonsingular part of the diagonal entry.Thus we can employ the selected inversion algo-rithm [56] to evaluate the self-energy efficiently.

Since the ionic radius is usually much smaller than the length scale of the fluid, we can seek for theasymptotic expansion of the self-energy using ionic radius a as the small parameter,

uλ(r) ≈(

1

ε(r)− 1

ε0

)1

4πa+ uPλ (r) +

a

4πε(r)

[λ2κ2(r) +

∇2ε(r)

6ε(r)

]+ λ2κ2(r)a2uPλ (r). (9)

-1

-0.9

-0.8

-0.7

0 0.1 0.2 0.3

(a)

Sel

f E

ner

gy

Radius

Numer. Sol.-1/(1+x)

First orderSecond order

-1

-0.6

-0.2

0.2

0.6

1

0.1 0.2 0.3

(b)

Sel

f E

ner

gy

Radius

Numer. Sol. Zeroth order

First orderSecond order

Figure 3: Comparison between asymptotic expansionsof self-energy with exact or direct numerical solution.

We then prove the chemical potential, defined as thevariation of the Free energy with respect to the con-centration, is the same with the self-energy to theO(a) order asymptotics but different for the O(a2)term. With this relation, we prove the resulting selfenergy modified PNP equations satisfy a proper en-ergy law. Moreover, we design an IMEX scheme tosolve the self-energy modified PNP equations, whichhas second-order accuracy and preserve the discretemass conservation and energy dissipation.

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0 1 2 3 4 5 6Time (ns)

-1

-0.5

0

0.5

1

1.5

Con

cent

ratio

n D

evia

tion

(a)

( 10-15)

MF

PCA

First

Second

0 1 2 3 4 5 6Time (ns)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Dif

fusi

on C

harg

e

(b)

MFPCAFirstSecond

0.5 1 1.5 1.9x (nm)

0

10

20

30

40

Con

cent

ratio

n (m

ol/

L)

(c)

MFPCAFirstSecond

-1.9 -1 0 1 1.9x (nm)

0

1

2

3

4

5

Con

cent

ratio

n (m

ol/

L)

(d)

MFPCAFirstSecond

Figure 4: (a) Deviation of the mass conservation law. (b) Total diffused charge. (c) The cation distributionsat t = 6.4ns. (d) The anion distribution at t = 6.4ns.

Like Charge Attraction. The interaction force between two likely charged particles/surfaces is usuallyrepulsive due to the Coulomb interaction. However, the counterintuitive like-charge attraction (LCA) inthe presence of electrolytes has been observed in experiments and molecular simulations, which cannot beexplained by the PB/PNP model [12, 2]. In [28], we systematically investigate the LCA phenomenon usingthe self-energy modified PB model. The pressure between two parallel plane is computed through,

P =∂F

∂D− P∞, (10)

where F is the free energy (4), D is the separation between two planes, P∞ is the osmotic pressure at thebulk (infinite distance). The main results are shown in Fig. 5 indicating the like-charge attraction is mainlydue to the ionic depletion from the surfaces and can be regulated by the electrolyte and charged objectsproperties. As the surface charge increases or the electrolyte concentration decreases, the attraction forcebecomes weaker. Moreover, the smaller size and higher valence of the counterion could enhance the LCAphenomenon. The numerical results are consistent with the experimental results, showing the self-energymodified model is promising in describing the electrolyte system.

Deff

(nm)

0 1 2 3

P (

kB

T/n

m3 )

-0.5

0

0.5

1

1.5

(×10-2)

(a)

σ1=-0.01 , σ

2=-0.01

σ1=-0.015, σ

2=-0.01

σ1=-0.015, σ

2=-0.015

Deff

(nm)

0.5 1 1.5 2 2.5

P (

kB

T/n

m3)

-0.5

0

0.5

(× 10-2)

(b)

cb=0.05M

cb=0.07M

cb=0.1M

Deff

(nm)

0 0.5 1 1.5 2

P (

kB

T/n

m3 )

-1

0

1

2

(×10-2)

(c)

a+=0.2, a

-=0.2

a+=0.4, a

-=0.2

a+=0.2, a

-=0.4

a+=0.4, a

-=0.4

Deff

(nm)

0 1 2 3

P (

kB

T/n

m3 )

-2

0

2

4

(×10-2)

(d)

z+=1, z

-=-1

z+=2, z

-=-1

z+=1, z

-=-2

z+=2, z

-=-2

Deff

(nm)

0 1 2 3P

(k

BT

/nm

3 )

-0.5

0

0.5

1

(×10-2)

(e)

ǫ=2.5ǫ=25ǫ=80ǫ=160

Figure 5: LCA analysis. (a) Different surface charge density; (b) Different bulk concentration. (c) Asymmetricradius of ions. (d) Asymmetric valences of ions. (e) Different dielectric constant.

Boundary Layer Analysis. The singular perturbation method is a powerful tool to analyze the chargetransport process [21, 1, 42, 46]. Consider the electrolytes between two blocking electrodes, we [19] focus

on the dielectric contribution of the self-energy u(x) = γ[e−κ(x)(D+x)

D+x + e−κ(x)(D−x)

D−x

]. There are two small

parameters appearing in the model: the Debye length ℓD and the Bjerrum length ℓB. Matched asymptoticsolutions have shown that there is a two-layer structure close to electrodes when the dielectric boundaryeffect is weak. The dielectric self energy only comes into play in the first layer, while the second layerbehaves like the classical PNP solution, without any correction from the dielectric boundary effect. When thedielectric boundary effect is relatively strong, the asymptotic analysis has demonstrated that there is only one

Figure 6: Cation concentration with oscil-latory applied voltage. (a) direct numerics;(b) asymptotic solution.

layer in which the inner solution satisfies modified Poisson–Boltzmann equations with the dielectric self energy appearingin the Boltzmann factor. The charging and recharging behaveslike two capacitors with one resistor,

−C(ζL)dζL

dt=(z2−z+ − z2+z−

) V+ − V− + ζL − ζR

2,

−C(ζR)dζR

dt=(z2−z+ − z2+z−

) V− − V+ + ζR − ζL

2,

(11)

Our asymptotic analysis is robust even for time dependentvoltage applied on two electrodes, as shown in Fig. 6. Sys-tematic investigations on the ionic concentrations, electrostatic

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potential, charge density, diffuse charges, differential capacitance, and charge inversion have deepened ourunderstanding of the dielectric boundary effect on electrostatic phenomena near interfaces.

ion channel

membrane

voltage sensor

electrolyteselectrode

i1 i2

Figure 7: Schematic diagram of a voltage-gated ion channel with voltage sensor.

2.3 Generalized Shockley-Ramo theorem Ionchannel problem is one of the most important applicationsof the continuum complex fluid model. There are chargedresiduals called voltage sensor which contribute to the con-formational change under external fields in voltage-gated ionchannels. The gating mechanism is usually inferred throughthe induced currents [39, 3, 20]. With given voltage sensor motion, one can easily solve the transport equa-tions to obtain the current as a forward problem, while the computational cost for directly solving the inverseproblem is very high. The original Shockley-Ramo theorem [41, 36] provide the connection between the chargevelocity in the vicinity of an electrodes in vacuum. When applied to the electrolyte, one need to sum up allthe contribution from ions and they are nonlinearly dependent on the charged particle motion [34, 13].

Time (t)0 2000 4000 6000

Indu

ced

Cur

rent

(i)

1.8

1.9

2

2.1

2.2

(10-6)

(a)

PNP1

PNP2

GSR1

GSR2

Time (t)0 50 100

Indu

ced

Cur

rent

(i)

9

10

11

(b)

(×10-3)

(b)

(×10-3)

Figure 8: Comparison between PNP and GSR on twoelectrodes. (a): v = 0.001; (b) v = 0.05.

To overcome this difficulty, we develop a gen-eralized Shockley-Ramo theorem which relate themotion of a charged particle in electrolytes and theinduced currents in a direct and simple manner [27].Under the assumption of slow charged particle mo-tion, the electrical potential u near a grounded elec-trode which is at the bulk region can be modeledthrough the linearized Poisson-Boltzmann equation,with renormalization of charge. Introduce the aux-iliary potential Φα for the electrode labeled α:

iα =(qrEα − Φα∇qr +∇(pr · Eα)

)· v. (12)

In literature, only a renormalized charge has been discussed in colloidal applications, and the renormalizedmultipoles are usually ignored. In our problem, we should additionally introduce the renormalized dipole,which can be significantly important and we might need to include the quadrupole and even multipole expan-sion for more accurate description of a complex geometry, the extension of this theorem is straightforward.

To illustrate the idea, we performed the numerical computation of a one dimensional special case wherethe renormalized charge and dipole can be analytical computed. When the charge density of the particle isweak, the dipole can be ignored while when the charge density is high, the contribution from dipole couldbe extremely significant. And the comparisons between the generalized Shockley-Ramo (GSR) theorem andthe direct solving continuum PNP model support our derivation. Fig. 8 is the results fro different particlevelocity v. Our theorem works perfectly when the charged particle move slowly and the electrode measuringinduced current be grounded and placed into bulk region but not too far away in case the signal is too weak.

2.4 Non-isothermal Electrokinetics. The inhomogeneous and time-dependent temperature could beof great importance in the electrodiffusion processes and play a key role in many biological and chemicalapplications [8, 37]. To model the non-isothermal charge transport, we carefully couple the mechanicaldiffusion and the heat flux, through the EnVarA. Generally, the free energy in control volume V is written as,

F (V, t) =

N∑i=0

∫VΨi(ρi(r, t), T (r, t))dr+

N∑i,m=0

zizm2

∫∫Vρi(r, t)ρm(r′, t)v(r, r′)drdr′

+N∑i=0

zi

∫Vρi(r)

(ψ(r, t) +

N∑m=0

zm

∫Ω\V

ρm(r′, t)v(r, r′)dr′

)dr. (13)

The first term Ψi is a local function of density ρi(r, t) and temperature T (r, t), representing the free energydensity from the entropy contribution. It should be noted, one can include the steric effect, ionic correlationand the variable dielectrics into proper form of Ψi. The index i = 0 stands for the solvent particles, which isincompressible with constant density ρ0, and index 1, · · · , N represents the solute species. The second termrepresents for the potential energy from the Coulomb interaction and kernel v satisfies −∇·ϵ∇v(r, r′) = δ(r, r′).

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The last term is the potential energy from the external field, including the external electrical potential ψ andthe contribution from particles outside domain V . According to the Least Action Principle, the conservativeforce can be obtained through the variation of the action with respect to the flow map,

f coni (x, t) =δA

δxi(X, t)= −miρi

(∂

∂tui + ui∇ui

)−∇Pi − ρizi∇ϕ, (14)

where ϕ(r, t) = ψ(r, t) +∑N

i=1 zi∫Ω ρi(r

′, t)v(r, r′)dr′ and Pi(r, t) = ρ2i (r, t)∂

∂ρi(r,t)

(Ψi(ρi(r,t),T (r,t))

ρi(r,t)

)are the

mean electrical potential and the thermal pressure [53]. The entropy production rate density is chosen to be

∆(r, t) =N∑i=1

νiρi|ui − u0|2 + ξi|∇ · ui|2

T+

N∑i=0

λi|∇ui|2

T+

1

k| jT|2. (15)

Then the dissipative force is given by the Maximum Dissipation Principle,

fdisi (r, t) =1

2

δ∫Ω T (r, t)∆(r, t)dr

δui(r, t)=

νiρi(ui − u0)−∇(ξi∇ · ui)−∇ · λi∇ui, i = 1, · · · , N.∑N

m=1 νmρm(u0 − um)−∇ · λ0∇u0, i = 0.(16)

Combining the first law and second law of thermodynamics, we conclude,N∑i=0

ui ·(f coni − fdisi

)= j ·

(j

kT+

∇TT

). (17)

which indicates the Onsager Principle holds, f coni = fdisi , and the Fourier law: j = −k∇T . This approachpresents a general framework, indicating the Energetic Variational Principle can work perfectly for the non-isothermal system, deriving a thermodynamic consistent model.

As a special case, we present the Poisson–Nernst–Planck–Fourier (PNPF) system for the non-isothermalelectrokinetics, which is at the mean-field level and can be generalized to take into account the aforementionedcorrelation and dielectric effects,

∂

∂tci +∇ · (ciui) = 0,∇ · u0 = 0,

−∇ · ϵ∇ϕ =∑N

m=0 cmzm,

kB∇(ciTi) + cizi∇ϕ = νici(u0 − ui) +∇(ξi∇ · ui) +∇ · λi∇ui,

∇P0 + c0z0∇ϕ =N∑i=1

νici(ui − u0) +∇ · λ0∇u0,

N∑i=0

(−T ∂

2Ψi

∂T 2

)(∂T

∂t+ ui · ∇T

)+

(N∑i=1

∂Pi

∂T∇ · ui

)T = ∇ · k∇T +

N∑i=1

νici|ui − u0|2 + λ|∇u0|2.

(18)

We also present the numerical comparison between the PNPF model and the classical PNP model. We cansee the temperature increase reduces the charge transport efficiency.

0 5 10x

0.98

0.985

0.99

0.995

1

Na

/0

(a)

V=100mVV=200mVV=300mV

0 5 10x

0.98

0.985

0.99

0.995

1

Cl/

0

(b)

V=100mVV=200mVV=300mV

0 2 4 6 8 10x

24

26

28

30

32

34

36

T(° C

)

(c)

V=100mV

V=200mV

V=300mV

0 1 2 3 4 5V(Volt)

0

1

2

3

4

Cur

rent

(d)

PNPPNPF

Figure 9: When the system approaches to steady state. (a) Na+ density distribution. (b) Cl− densitydistribution. (c) Temperature distribution. (d) Voltage-Current relation of the system.

2.5 Two-layer Image Charge Method. The Image Charge Method (ICM) [54] has been widelyapplied in particle simulations for dielectric effects near interfaces, but it only exists in very limited boundarygeometries such as planar interfaces and spheres. We [59] propose a meshless fast algorithm for numericalcalculation of the Green’s function for Poisson’s equation −∇2G(r, r′) = 4πδ(r, r′) on general domain Ω withgiven boundary condition. The solution can be represented through, G(r, r′) = 1/|r− r′|+Gind(r, r

′), wherethe induced potential Gind comes from the boundary condition,

Gind(r, r′) =

∫Γ

q(p)

|r− p|dp+

∫∂Ωe

σ(p)

|r− p|dp ≈

L∑ℓ=1

qℓ|r− rℓ|

+N∑i=1

Qi

|r−Ri|. (19)

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ΩΩe

qj σ(r)

Figure 10: Diagram of thetwo-layer image charges.

Here Ωe ⊃ Ω is a larger domain with smooth boundary. Γ is in between ∂Ω and∂Ωe, representing the points where harmonic extension of G from Ω to Ωe fails.While the outer layer image charges are fixed and uniformly distributed on ∂Ωe,the position of inner layer image charges rℓ are to be numerical computed. Bynoticing the main contribution of the inner layer image charges is to removethe singularity near the boundary, we find the inner layer does not need to bevery precise and the error can be controlled by the outer layer image charges.Moreover, very small number of L is needed in practice. We then propose atwo-step algorithm: (1) Compute qℓ and rℓ by minimizing the deviation withthe boundary condition, through the steepest descent method; (2) Solve thelinear system for Qi through the singular value decomposition method.

Figure 11: Error distribution of Gind on different cross-sections.

The numerical tests are shown inFig. 11. The source charge is locatedinside a cubic box (left) or a sphere(right) with homogeneous Diricheletboundary condition. We can see theerror from this two-layer image chargemethod can be very small. Moreover,we present the detailed convergenceanalysis as well as other boundary con-ditions in [59].

Figure 12: (a) Instantaneous monomerconversion versus total conversion for thecopolymerization; (b) schematics of chaincomposition along the chain growth direc-tion for the copolymers.

2.6 Copolymerization Rate Regression. Multicom-ponent polymers can be obtained by copolymerization of twoor more different types of monomers into a sophisticated pri-mary structure. The monomer concentrations [M1] and [M2], aswell as the polymer concentrations [M∗

1 ] and [M∗2 ] are governed

by the chemical master equation,

d

dt[M1] = −k11[M∗

1 ][M1]− k21[M∗2 ][M1],

d

dt[M2] = −k12[M∗

1 ][M2]− k22[M∗2 ][M2],

d

dt[M∗

1 ] = −k12[M∗1 ][M2] + k21[M

∗2 ][M1],

d

dt[M∗

2 ] = k12[M∗1 ][M2]− k21[M

∗2 ][M1].

(20)

The time series of monomer concentrationM1,t andM2,t are measured through the real-time 1H NMR spectraexperiments. Numerical solution to Eq. (20) through the 4th-order Runge-Kutta scheme, gives the monomer

concentration series M1,t and M2,t. To find the reaction coefficients, we then combine the simulated annealingstrategy and the gradient descent method with backtracking line search to solve the optimization problem,

ϵ(k11, k12, k21, k22) =

√√√√ 1

ns

ns∑t=1

[([M1,t]−M1,t)2 + ([M1,t]−M1,t)2] + µ(k11k22 − k12k21)2. (21)

Part of the numerical results are shown below, our method is shown to be reliable in real applications.

Figure 13: Copolymerization rate regression applied for different systems.

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Current Projects

Micro-Macro Model for time dependent dielectric environment. The dielectric permittivity is acoarse-grained variable, which depends on the material concentration, the electrical field, the temperature aswell as the history. Instead of treating the ion to be a point charge and the solvent as a continuum dielectricbackground like the classical PB/PNP theory, we model each ion and each solvent molecule to be a pointcharge with a polarizable point dipole. Along with the macroscopic transport, the polarization density isevolved with the microscopic stretching and orientation. Applying the Energetic Variational Approach onthe macroscopic level, we achieve the equation describing the transport of the mass center. Applying theEnergetic Variational Approach on the microscopic level, we achieve the equation describing the microscopicdeformation. These two equations are coupled together, forming a Dipolar Poisson–Nernst–Planck model [26].

Harmonic Surface Mapping Algorithm for Solving Poisson Equation. The Harmonic Surface Map-ping Algorithm is based on the two-layer image charge method [59]. It combines the traditional image chargemethod and boundary integral method to numerically evaluate the Green’s function G of Poisson equation.More generally, the solution of −∇2u = f is given by the convolution of the Green’s function G and thesource term f , plus one surface integral to take into account the boundary condition. These integrals can becomputed through Gauss quadratures and accelerated by means of the Fast Multipole Method. The algorithmis efficient, highly-accurate, suitable for adaptive mesh and mixed boundary conditions. This work is withJiuyang Liang (Ph.D.) and Prof. Zhenli Xu from Shanghai Jiao Tong University.

Numerical Method for 3-D modified Poisson–Nernst–Planck System. In this project, we plan toinclude both the short range steric effect and the long range Coulomb correlation. For the Coulomb part,we will use the self-consistent field model as described before. For the short range steric effect, the modifiedfundamental measure theory (mFMT) [57, 38] provides the best density functionals for hard sphere mixtures.The excess free energy is approximated in terms of the weighted densities: nα(r) =

∑i

∫ρi(r− r′)ωα

i (r′)dr′,

where the weight functions ωαi (r) characterizes the geometry of the hard sphere. Through the EnVarA, these

two correlation contribution can be taken into account consistently, presenting a modified PNP model. Thenumerical solver could help us study the charge transport within complex geometry in real applications.The FMT part could be accelarated through the Fast Fourier Transform. The Poisson equation for theelectrical potential is solved through finite difference method with algebraic-multigrid preconditioner providedby HYPRE. The self-energy will be solved using the select inversion algorithms. This is a joint project withProf. Shenggao Zhou from Soochow University and Prof. Zhenli Xu from Shanghai Jiao Tong University.

Non-Isothermal Dynamics of Polymer Melts. We propose an extension of the classical model for finiteextensible nonlinear elasticity (FENE) in order to include non-isothermal effects. The non-isothermal modelis derived by a self consistent procedure through an energetic variational approach, which combines the leastaction principle and Onsager’s principle of maximum energy dissipation. The result is then supplemented bythe existence of global-in-time solutions in the whole spatial domain, for small initial data close to equilibrium.We further provide provide some numerical results. This is a joint project with Prof. Francesco De Annafrom Universitat Wurzburg, Prof. Chun Liu from Illinois Institute of Technology, and Dr. Stefano Scrobognafrom Basque Center for Applied Mathematics.

Stochastic Description of Charge Transport. The Mori–Zwanzig formulation [30, 60] is an importantmethod in developing coarse-grained models based on the dynamics of the full system, rather than the equilib-rium statistical properties. Through the time series generated through Molecular Dynamics, the free energyfunctional at the continuum level as well as the memory functional (which corresponds to the entropy pro-duction) can be obtained from the statistical properties of the charged particles. This free energy functionalwill be more robust since it includes the correlation between particles, so that we can compare it with theclassical Poisson–Nernst–Planck models and its modified versions. This work is under the supervision of Prof.Xiantao Li from the Pennsylvania State University.

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Future Interests

Integral equation theory. In general, the direct correlation function cij(r, r′) is defined through the

Orstein-Zernike(OZ) equation:

hij(r, r′) = cij(r, r

′) +N∑k=1

∫cik(r, r

′′)ρk(r′′)hkj(r

′′, r′)dr′′, (22)

and hij(r, r′) is the total correlation function related with the joint density distribution, i and j represents

different species. Once we have another closure relation between h and c, we can numerically solve for thecorrelation functions. In literature, many different closures has been proposed for homogeneous liquid, suchas the hypernetted-chain approximation (HNC), mean spherical approximation (MSA), etc. Moreover, theintegral equation for inhomogeneous liquid could be used to describe the the phase transition and the liquid-liquid/liquid-vapor interface problems. Physically speaking, the asymptotic behavior of h(r, r′) and c(r, r′)when |r − r′| → ∞ is known to be decay fast, depending on the pairwise interaction kernel. I am interestedin developing proper closure relation and corresponding fast algorithm to solve the integral equation. Thisequation has an intrinsic non-local convolutional hierarchy structure, which is known as loop-expansionin theoretical physics and diagrammatic method in liquid theory. The mean-field Poisson–Boltzmann theory isequivalent to the 0th-order loop-expansion, and the self-energy modified Poisson–Boltzmann model is actuallyequivalent to the 1st-order loop-expansion. Higher-order will give more detailed representation, but requestsa lot more computational resources. We will follow a data-driven approach from molecular simulations or usemachine learning strategy to obtain a more accurate closure relation.

Kinetic Description of Temperature. From microscopic point of view, the motion and distribution ofensemble of particles can be described through the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierar-chy. Under the molecular chaos hypothesis, it reduces to the Boltzmann’s equation, and the continuum limitcould be Euler or Navier–Stokes equation (for ideal gas), Poisson–Nernst–Planck equation (for electrolytes).One important concept is the Maxwellian for velocity, which is a Gaussian distribution of variance proportionalto kBT [22]. However, if we are interested in microcanonical ensemble (NVE) instead of canonical ensemble(NVT), then the temperature might vary in both space and time. Our goal is to obtain the continuum limitof the temperature equation and the fluid equation [51].

Energy Devices. In batteries, the energy is restored in the form of chemical energy, which can be convertedinto electrical energy via chemical reactions. In supercapacitors, the energy is restored through the electricaldouble layer in the form of electrical energy without chemical reaction. Under the same condition, thesupercapacitor could provide higher power density with exceptional cycling stability, but lower energy density,compared with batteries. To better understand and design for the energy devices, we need to understandmore about the charge transport, dielectric properties as well as the temperature effects [50, 18].

Biological Applications. (1) Ion Channel. Ion Channels are the transistors of life. Although there are lotsof experiments to understand their functions and structures, the proper mathematical model for the gatingmechanism is still missing. On one hand, the intra- and extracellular ions could diffuse and transport; on theother hand, the ion channel itself could deform. This problem is multi-scale in both time and length, andinvolves with the fluid-structure interaction [48, 10, 16].(2) Tripartite Synapse. Two adjacent neuron cells are linked by gap junctions, which is called synapticcleft. These two cells are asymmetric in structure and function, one is presynaptic neuron while the other ispostsynaptic. Moreover, the astrocyte, which is a star-shaped glial cells in the brain and spinal cord, has closeassociation with the synaptic cleft and can modulate synaptic activity. Together, they are named TripartiteSynapse. Between two adjacent neurons, the neural signal is transported in the form of chemical signal, whichis accomplished by neurotransmitters. We are interested in the signal transport in the synaptic cleft and therole of astrocytes, which is related with mental diseases [40, 44].(3) Temperature effects. Biological system is very sensitive to temperature and temperature is related withmany diseases such as fever, cold and stroke. We are interested in understanding how human body controlthe temperature distribution in vivo and take advantage of the temperature for curing diseases [31, 11].

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