Electrostatic potential distribution of a soft spherical...

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Electrostatic potential distribution of a soft spherical particle with a charged

core and pH dependent charge density

Kyle McDaniel, Fedra Valcius, Joseph Andrews, and Siddhartha Das*

Department of Mechanical Engineering,

University of Maryland, College Park,

Maryland 20742, United States

*E-mail: [email protected]

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Abstract

In this paper we provide numerical results for the electrostatic potential profile of a soft charged

particle with a charged core and pH dependent charge density. The present study can be

conceived as a useful platform for better quantification of the electrostatic potential profile of the

MS2 bacteriophage virus. In this context the present study improves the analysis of a similar

recent study [Phan et al., J. Chem. Phys. 139 (2013) 244908] in two aspects. First, as compared

to this previous study, we provide numerical rather than analytical results for the electrostatic

potential profile. This is significant given the fact, as we demonstrate here, that analytical

treatment forbids the analysis of cases with intermediate to large charge density (as compared to

the bulk charge density of the electrolyte) of the viral surface. Second, consideration of the effect

of pH-dependent charge density in electrostatic potential profile is a more credible representation

of the electrostatics of the virus, which is known to demonstrate pH-dependent variation in the

electrophoretic mobility. In fact, we establish that the different predictions on the electrostatic

potential profile made in the previous study gets substantially modified when the effect of pH is

considered, and the result must now be described in terms of the difference pH−pKa, where Ka

refers to the ionization of the acid that produces the PEL ions.

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1. Introduction

Estimating the electrostatic potential profile of a soft charged spherical particle [1-6] in an

electrolyte solution is essential to describe a plethora of biological and non-biological

phenomena, such as quantifying electrophoretic mobility of biological cells [7-10], pinpointing

the interactions intrinsic to bacterial adhesion to solid surfaces [11-14], establishing the role of

surface “softness” and EPS (extracellular polymeric substance) in electrophoretic mobility of

bacterial strains [15-19], controlling agglutination and shape transition of red blood cells [20,21],

predicting the charge distribution and electrophoretic mobility of the bacteriophage MS2 virus

[22,23], controlling transport and stabilization of polyelectrolyte-coated nanoparticles [24-29],

quantifying electrokinetic properties of pNIPAM microgels [30], etc. A soft charged sphere has

been approximated as a rigid spherical particle covered by a layer of charged polyelectrolyte [1-

6] (see Fig. 1). This polyelectrolyte layer (or PEL) harbors a particular kind of ion, denoted as

the PEL ion, which remains excluded from the surrounding electrolyte, although the

corresponding electrolyte ions can be present both inside and outside the PEL. In addition to

these geometric and structural specifications, there are two more essential issues related to the

PEL structure. First is the charge characteristic of the rigid core. Quite surprisingly, in most of

the analyses on electrostatics and electrokinetics of soft charged particles, this rigid core has

been considered as uncharged and without a finite dielectric constant, represented by a zero

potential gradient boundary condition at the interface between the PEL and the rigid core

[2,4,7,31,32]. Such a condition is appropriate if the rigid core is a perfectly conducting material,

such as a metal; therefore the corresponding calculations may not be appropriate to account for

the behavior of biological entities (e.g., cells, bacteria, viruses), whose inner core may possess a

finite dielectric constant or a finite charge [33-37]. The recent study by Phan et al. [22] marks an

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important breakthrough in this regard; they considered a charged soft particle with a charged

core and explained that the corresponding electrostatic potential profile can represent the

electrostatics of the MS2 bacteriophage virus [36]. A similar study by Dingari and Bui [37]

considered a soft particle electrokinetics model with a charge-less inner core having a finite

dielectric constant, and used it to interpret the electrophoretic transport of fibrillated and

unfibrillated S. salivarius bacteria. The second important aspect regarding the modeling of EDL

electrostatics of soft charged moieties is the correct representation of the mechanism that

produces the PEL ions. These ions are typically produced by the acid-like (when the negative

PEL ions and H+ ions are simultaneously released) or base-like (when the positive PEL ions and

OH− ions are simultaneously released) dissociation of the PEL [3,38-43]. Thus the extent of this

dissociation and the resulting PEL charge density and the number density of the PEL ions

strongly depend on the pH of the electrolyte solution.

In this paper, we provide numerical solution for the electrostatic potential profile of a

charged soft particle with a charged core and pH-dependent charge density. Our study provides a

mathematical framework that ensures a more rigorous analysis of the electrostatics of the MS2

bacteriophage virus (known to demonstrate pH-dependent electrophoretic mobility [36]) beyond

the analysis of Phan et al. [22]. There are two issues with respect to which we improve the model

of Phan et al. [22]. First and foremost, a numerical solution technique allows us to probe large

ranges of electrostatic potential values (or alternatively large ranges of different parameters),

which was not possible in the analytical model provided by Phan et al. [22] [analytical treatment

necessitates employment of Debye-Hückel approximation, which is valid only for small enough

electrostatic potential (< 25 mV)]. For example, the analytical treatment forbids the analysis of

the case where the PEL charge density is comparable or greater than the bulk electrolyte charge

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density. Secondly, consideration of pH-dependent charge density of the PEL in the description of

the electrostatics allows us to redress the inferences made by Phan et al. [22]. For example, Phan

et al. noted that the properties of the charged core primarily affected the electrostatics within the

particle; however, we demonstrate that depending on the operating value of the pH, variation of

the properties of the charged core may severely affect the electrostatic potential profile outside

the particle core as well. Our model also identifies the necessary dimensionless parameters that

dictate the problem and establishes that the influence of each parameter in dictating the potential

profile both inside and outside the particle strongly depends on the value of the operating pH.

2. Theory

We consider a soft spherical particle of exactly the same structure as considered in Phan et al.

[22] (see Fig. 1). Thus the spherical particle has a rigid positively charged core of radius of a and

a polyelectrolyte layer (PEL) of thickness b−a. The spherical particle is in contact with an

electrolyte solution. The PEL harbors (or consists of) a particular kind of ion (henceforth denoted

as the PEL ion). The PEL ion is present only within the PEL and remains excluded from the

electrolyte solution. On the contrary, the electrolyte ions are present both inside and outside the

PEL. The number density of the PEL ions is a function of pH. Under these conditions, the

equations governing electrostatic potential ψ are (considering positively charged monovalent

PEL ions and monovalent electrolyte ions):

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d 2ψdr2 + 2

rdψdr

= − ρel

ε rε0

= −e n+ − n−( )

ε rε0

[b ≤ r ≤ ∞],

d 2ψdr2 + 2

rdψdr

= − ρel + Neε rε0

= −e n+ − n−( ) + Ne

ε rε0

[a ≤ r ≤ b],

d 2ψdr2 + 2

rdψdr

= − ρcore

εcoreε0

[0 ≤ r ≤ a].

(1)

In the above equations, r is the radial coordinate, ρel is the mobile charge density within the

liquid (both inside and outside the PEL), e is the electronic charge, n± are the number densities of

the cations and anions of the electrolyte, ρcore is the charge density within the solid core, N is the

number density of the PEL ions, ε0 is the permittivity of free space, εr is the relative permittivity

of the liquid medium (assumed identical for both inside and outside the PEL) and εcore is the

relative permittivity of the solid core. Also both the PEL ions and the electrolyte ions are

assumed to be monovalent. Further the number density of the electrolyte ions are dictated by the

Boltzmann distribution, expressed as:

n± = n∞ exp eψkBT

⎛⎝⎜

⎞⎠⎟, (2)

where n∞ is the bulk electrolyte number density (i.e., the number density far away from the

particle surface) and kBT is the thermal energy.

In fact, the same Boltzmann distribution can be invoked to describe the number density (in units

of 1/m3) of hydrogen and hydroxyl ions, i.e.,

nH + =103NA10− pH∞ exp − eψ

kBT⎛⎝⎜

⎞⎠⎟

, nOH − =103NA10− pOH∞ exp eψ

kBT⎛⎝⎜

⎞⎠⎟

, (3)

where NA is the Avogadro number and pH∞ and pOH∞ are the bulk pH and pOH values of the

electrolyte. Using eq.(3), it is possible to express the PEL ion number density as (considering a

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monoacidic base is producing the positively charged monovalent PEL ion; for details, kindly

refer to [43]):

N = γ

1+10 pKb−pOH∞ exp eψkBT

⎛⎝⎜

⎞⎠⎟

, (4)

where Kb is the ionization constant of the base whose dissociation produces the positively

charged PEL ion and γ is the maximum surface site density (in units of 1/m3) of the PEL ions

within the PEL. Also considering the fact that at T=298 K, pH∞ + pOH∞ = pKb + pKa (where Ka

is the ionization constant of the conjugate acid of the base that produces the positively charged

PEL ion) we can re-write eq.(4) as [43]:

N = γ

1+10 pH∞−pKa exp eψkBT

⎛⎝⎜

⎞⎠⎟

. (5)

Using eqs.(2,5), we can re-write eq.(1) in dimensionless form as:

d 2ψdr 2 + 2

rdψdr

= 1λ

⎛⎝⎜

⎞⎠⎟

2

sinh ψ( ) [1≤ r ≤ ∞],

d 2ψdr 2 + 2

rdψdr

= 1λ

⎛⎝⎜

⎞⎠⎟

2

sinh ψ( )− 1Kλ

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1+10 pH∞−pKa exp ψ( )⎡

⎣⎢

⎤

⎦⎥ [a /b ≤ r ≤1],

d 2ψdr 2 + 2

rdψdr

= − 1λ

⎛⎝⎜

⎞⎠⎟

2

ρcoreε rεcore

⎛⎝⎜

⎞⎠⎟

[0 ≤ r ≤ a /b],

(6)

where r = rb

, ψ = eψkBT

, λ= λb

λ = ε0εrkBT2n∞e

2

⎛

⎝⎜⎞

⎠⎟, Kλ =

λPEL

λ λPEL =

ε0εrkBTγ e2

⎛⎝⎜

⎞⎠⎟

, ρcore =ρcore2en∞

.

Here λ is the EDL thickness, λPEL is an equivalent double layer thickness within the PEL (see

[32] for a discussion on the significance of λPEL and Kλ) and 2en∞ is the bulk electrolyte charge

density. Eq. (6) is solved numerically in presence of the following boundary conditions:

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ψ r →∞( ) = 0, ψ r = 0( ) ≠ ∞,

ψ( )r = 1( )− = ψ( )r = 1( )+ , dψdr

⎛⎝⎜

⎞⎠⎟ r = 1( )−

= dψdr

⎛⎝⎜

⎞⎠⎟ r = 1( )+

,

ψ( )r = ab

⎛⎝⎜

⎞⎠⎟− = ψ( )r = a

b⎛⎝⎜

⎞⎠⎟+ , dψ

dr⎛⎝⎜

⎞⎠⎟ r = a

b⎛⎝⎜

⎞⎠⎟−= ε rεcore

dψdr

⎛⎝⎜

⎞⎠⎟ r = a

b⎛⎝⎜

⎞⎠⎟+.

(7)

The last condition in eq.(7) refers to a condition of continuity of electric displacement at the

interface of the PEL and the rigid core; such a condition accounts for the scenarios where the

solid possesses a finite dielectric constant.

3. Results and Discussions

Quantifying the limitation of the analytical approach of Phan et al. [22]

Phan et al. [22] provided the variation of the dimensionless (made dimensionless in terms of the

Donnan potential ψDON) electrostatic potential, and hypothesized that this potential distribution is

an apt representation of the electrostatics at the MS2-bacteriphage-virus-electrolyte interface. In

the model provided by the Phan et al. [22] ψDON expressed in linearized form is (considering

uniform pH-independent number density N of the PEL ions)

ψ DON = Neλ 2

ε0εr. (8)

Experiments report all the parameters, except N; hence it is not possible to determine the

corresponding value of ψDON directly from eq.(8). Analysis of Phan et al. [22], however, provide

us the limits of values of N (i.e., the values of PEL charge density of the MS2 bacteriophage

virus) corresponding to which analytical treatment is valid. Analytical results are shown for

potential values that can be as large as four times the value of ψDON. Therefore, for the validity of

the analytical solution, we must have [using eq.(8)]:

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4Neλ 2

ε0ε r< kBT

e⇒ N

n∞< 12. (9)

Eq. (9) also implies that the applicability of the analytical solution also requires

eN2en∞

= PEL Charge DensityBulk Electrolyte Charge Density

< 14

. This severely restricts the applicability of the

analytical model, making the numerical treatment mandatory.

Numerical solution of the potential profile with pH-dependent PEL charge density

The electrostatic potential profile depends on six dimensionless parameters:

a , λ , ρcore , ε r / εcore , pH∞ − pKa and Kλ . Figs. 2(a-i) show the effect of variation of these

parameters on the numerical solution of the electrostatic potential profile for different values of

the difference pH∞ − pKa . Fig. 2(a) demonstrates the case where all other parameters are held at

unity and only pH∞ − pKa is varied. For pH∞ − pKa < 0 , the difference between the cases with

pH-dependent and pH-independent charge densities gets substantially diminished [see Fig. 2(a)].

This can easily be noted from the second part of eq.(6), and has already been identified by Das

[43]. We can also provide a physical justification to this behavior. Here we are concerned with

positively charged PEL ions; hence as pH∞ − pKa < 0 suggests pOH∞ − pKb > 0 , we are

contemplating the case where the concentration of OH− ions are substantially reduced. Such a

reduction signals a depletion of OH− ions from the system, thereby favoring the forward reaction

that produces the OH− ions. This enhanced forward reaction will also imply enhanced production

of the positively charged PEL ions, thereby ensuring substantially less difference between the

pH-independent case (where there is a maximum possible generation of the PEL ions) and the

pH-dependent case. On the other hand, as the difference pH∞ − pKa increases, there will be much

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lesser generation of the PEL ions; consequently there is a much more enhanced difference

between the pH-dependent and the pH-independent cases, reflected in a much reduced value of

the electrostatic potential profile (for the pH-dependent case) at a given radial distance. Such a

decrease can also be hypothesized from eq.(6), as well as by invoking the physical argument

discussed above. Figs. 2(b-i) explicate the effect of variation of other parameters; for most of

these figures the above-discussed effect of the difference pH∞ − pKa becomes clearly evident.

Figs. 2(b,c) show the impact of the variation of the parameter λ=λ /b . Small λ corresponds to

a small screening length; hence the potential profile quickly decays outside the PEL (i.e., r >1)

[see Fig. 2(b)]. On the contrary, for larger λ , the decay occurs over much larger r [see Fig.

2(c)]. Further as Kλ = 1 in both Figs. 2(b,c), smaller bulk electrolyte ion number density

(corresponding to larger λ ) also signifies smaller number density of the PEL ions (i.e., smaller

γ). This is reflected in large difference in the magnitude of the electrostatic potential between the

cases of Figs. 2(b) and 2(c). Also it is useful to note how such variation in λ regulates the

influence of pH-dependence in the variation of the electrostatic potential profile within the

particle core. Thin EDLs, requiring a quick decay of the electrostatic potential, leads to much

lesser relative difference between different pH-dependent cases (both inside and outside the

particle); however the difference becomes substantially larger when EDLs are thick and the

decay occurs over greater spatial distances. These findings are important additions to the

understanding proposed in Phan et al. [22] about the relative variation of the electrostatic

potential inside and outside the particle. Figs. 2(d,e) show the effect of variation of the parameter

Kλ . Smaller Kλ implies large number density of PEL ions for a given EDL thickness (or a given

bulk concentration of the electrolyte ions); consequently there is a substantially large difference

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in the magnitude of the electrostatic potential between the cases of Kλ = 0.1 and Kλ = 10. There

are two more intriguing differences between the cases corresponding Kλ = 0.1 and Kλ = 10.

Firstly, for Kλ =10 , there is virtually no difference between the cases with varying pH∞ − pKa ;

on the other hand there is a substantial impact of the variation of pH∞ − pKa in the electrostatic

potential profiles for the case of Kλ = 0.1 . This can be easily explained from eq.(6); large Kλ

makes the pH-dependent contribution [dictated by the term 1Kλ2

11+10 pH∞−pKa exp ψ( ) ] negligible,

causing imperceptible difference between pH-independent and pH-dependent cases. Secondly,

for small Kλ the spatial variation of the potential profile outside the core is much steeper as

compared to that inside the core. This, too, can be associated with the large values of the

potential, which makes the term 1Kλ2

11+10 pH∞−pKa exp ψ( )

less significant as compared to

sinh ψ( ) [see eq.(6)]. Here too, we can present our results in the context of those forwarded by

Phan et al. [22]. We find that the variation in the properties of the PEL (reflected through a

variation in Kλ ) substantially affects the potential profile both inside and outside the particle. In

fact for a given value of the difference pH∞ − pKa the spatial variation is much weaker inside the

particle as compared to outside the particle for small Kλ , although the variation in pH∞ − pKa

has greater impact within the particle as compared to outside the particle. Figs. 2(f,g) show the

effect of variation of the dimensionless charge density of the core ρcore . Increase in the value of

ρcore increases the magnitude of the potential profile and also ensures a much steeper variation of

the potential profile (within the core) and a much-reduced effect of the pH dependence of the

PEL charge density. All these observations can be trivially inferred from eq.(6). Please note that

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Phan et al. [22] hypothesized that the impact of variation of the properties of the core influenced

the potential profile inside the particle more strongly as compared to that outside the particle.

Figs. 2(f,g) establish that this inference needs to be redressed in the light of consideration of pH-

dependent charge density – depending on the operating value of pH∞ − pKa , changes in the

properties of the core may profoundly affect the potential profile outside the particle as well.

Figs. 2(h,i) quantify the effect of variation of the permittivity ratio ( ε r / εcore ). Very much

intuitively, larger ε r / εcore reflects a much steeper variation in the electrostatic potential profile,

as predicted by Phan et al. [22] as well. Here too we find that impact of variation of ε r / εcore on

the relative variation of the potential profile inside and outside the core strongly depends on the

operating value of the pH∞ − pKa ; in this light the inference of Phan et al. [22] gets redressed

again.

4. Conclusions

In this paper, through the numerical results of the electrostatic potential profile of a soft charged

spherical particle with charged core and pH-dependent charge density we shed new light

necessary for a more complete analysis of electrostatics of possible bio-particles like viruses and

bacteria. This electrostatic information considers hitherto missing information about the pH-

dependence, and substantially improves the inferences made by the recent study of Phan et al.

[22]. This knowledge will form the founding basis for analyzing the pH-dependent

electrokinetics (reflected in the pH-dependent electrophoretic mobility) of the MS2

bacteriophage virus [36], to be considered by the authors in a future publication.

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Figure 1: Schematic of the soft charged particle with a rigid positively charged core. Here the

PEL ions are positively charged and shown in green, whereas the electrolyte ions are shown in

blue (cations) and red (anions).

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Figure 2: Dimensionless electrostatic potential profile as a function of dimensionless radial

distance. For all the plots, we provide four different cases: 1. pH independent PEL charge density

(represented in black bold line), 2. pH dependent charge density, with pH∞−pKa=−1 (shown in

dashed blue line), 3. pH dependent charge density, with pH∞−pKa=0 (shown in green line with

circular markers), and 4. pH dependent charge density, with pH∞−pKa=1 (shown in red line with

diamond markers). For all the cases, we consider a = 0.5 and all other parameters (i.e.,

λ , Kλ , ρcore , ε r / εcore ) equal to unity except for (b) λ = 0.3 , (c) λ =10 , (d) Kλ = 0.1 , (e) Kλ =10

(for this case there is imperceptible difference between the pH dependent and pH independent

cases), (f) ρcore = 0.1, (g) ρcore = 10 , (h) ε r / εcore = 0.1 , and (i) ε r / εcore = 10 .

0 5 100

0.1

0.2

0.3

r̄

!̄

0 5 100

0.5

1

1.5

r̄!̄

0 5 100

2

4

x 10−3

r̄

!̄

0 5 100

2

4

6

r̄

!̄

0 5 100

0.05

0.1

r̄

!̄

0 5 100

0.1

0.2

r̄

!̄

0 5 100

0.5

1

r̄

!̄

0 5 100

0.1

0.2

r̄

!̄

0 5 100

0.2

0.4

0.6

r̄

!̄

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Electrostatic potential distribution of a soft spherical...

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