Interactions between polyelectrolytes mediated by ordering and orientation of multivalent non-spherical ions in salt solutions Hossein Vahid1 2 3Alberto Scacchi1 2 3Maria Sammalkorpi2 3 4and Tapio Ala-Nissila1 5 6

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Interactions between polyelectrolytes mediated by ordering and orientation of

multivalent non-spherical ions in salt solutions

Hossein Vahid,1, 2, 3 Alberto Scacchi,1, 2, 3 Maria Sammalkorpi,2, 3, 4 and Tapio Ala-Nissila1, 5, 6, ∗

1Department of Applied Physics, Aalto University, P.O. Box 11000, FI-00076 Aalto, Finland

2Department of Chemistry and Materials Science,

Aalto University, P.O. Box 16100, FI-00076 Aalto, Finland

3Academy of Finland Center of Excellence in Life-Inspired Hybrid Materials (LIBER),

Aalto University, P.O. Box 16100, FI-00076 Aalto, Finland

4Department of Bioproducts and Biosystems, Aalto University, P.O. Box 16100, FI-00076 Aalto, Finland

5Quantum Technology Finland Center of Excellence, Department of Applied Physics,

Aalto University, P.O. Box 11000, FI-00076 Aalto, Finland

6Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences,

Loughborough University, Loughborough, Leicestershire LE11 3TU, UK

(Dated: October 11, 2022)

Multivalent ions in solutions with polyelectrolytes (PE) induce electrostatic correlations that can

drastically change ion distributions around the PEs and their mutual interactions. Using coarse-

grained molecular dynamics simulations, we show how in addition to valency, ion shape and con-

centration can be harnessed as tools to control like-charged PE-PE interactions. We demonstrate a

correlation between the orientational ordering of aspherical ions and how they mediate the eﬀective

PE-PE attraction induced by multivalency. The interaction type, strength and range can thus be

externally controlled in ionic solutions. Our results can be used as generic guidelines to tune the

self-assembly of like-charged polyelectrolytes by variation of the characteristics of the ions.

Electrostatic interactions between charged molecules

and ions in solution are ubiquitous in colloidal, soft, and

biological systems [1]. Systems such as some polyelec-

trolytes (PEs), synthetic and biopolymers, DNA [2, 3],

nanotubes in phospholipids [4], actin ﬁlaments [5, 6], mi-

crotubules [7], viruses [8, 9], and even bacteria [10], can

often be approximated by charged cylinders immersed in

an electrolyte solution consisting of a solvent and mobile

ions [11]. Understanding the ion distribution in such sys-

tems is paramount since solution-mediated interactions

are greatly aﬀected by the ionic environment, especially

due to electrostatic screening eﬀects [12] and ion redistri-

bution [13, 14]. Both the nature and concentration of ions

play a signiﬁcant role, whence ion valency is an important

handle for tuning the properties of macroions [15–19].

Many chemically speciﬁc ions, such as, e.g., diamine,

spermine, and spermidine, exhibit elongated, cylindrical

shapes and are multivalent [7, 20]. Some anions in bat-

tery electrolytes are non-spherical, which inﬂuences ion

transport and conductivity in solution [21, 22]. Ionic

liquids are typically composed of highly non-spherical

ions, which inﬂuences their cohesion energy and main-

tains their liquid character, but also inﬂuences ionic

transport [22, 23]. Consequently, ion speciﬁcity is

paramount in controlling interactions between charged

macromolecules.

For modeling purposes, traditional mean-ﬁeld ap-

proaches such as Poisson-Boltzmann (PB) theory treat

mobile ions as point charges in the weak-coupling regime.

The standard PB theory cannot describe general chem-

ically speciﬁc ions [24, 25]. However, successful models

∗tapio.ala-nissila@aalto.ﬁ

incorporating ion size properties exist, such as those in

Refs. [26–30]. In the case of like-charged PE-PE in-

teractions, the PB theory always predicts repulsion. To

this end, the soft-potential-enhanced-PB (SPB) theory

has been shown to accurately predict ion distributions

around PEs [31] for ion sizes up to the PE radius, and

like-charged PE-PE repulsion for small monovalent ions

such as Na+and Cl−[32]. In Ref. [33], however, like-

charged PE-PE attraction was reported for large mono-

valent ions and high salt concentration, and attributed

to short-range charge correlations beyond the PB theory.

For multivalent ions, charge-charge correlations natu-

rally appear and cause charge reversal (see, e.g., Refs. 34–

36) in the strong-coupling regime even with point-like

ions [15, 37–45]. Both valency and ion size have been

considered in the context of classical density functional

theory [46]; see, e.g., the recent advances on electric dou-

ble layers [47–49]. There also exist Monte Carlo stud-

ies on dumbbell-like (two separated point charges), yet

volumeless, ions focused on counterion-mediated interac-

tions between charged plates [50–55] or cylinders [56].

References 54, 56–59 have suggested that a bridging

mechanism is responsible for the attraction between like-

charged surfaces. Nevertheless, to our knowledge none

of these approaches have simultaneously considered both

correlations and steric eﬀects of aspherical multivalent

ions.

In this Letter, we extend the ion-mediated interaction

scenario in the case of spheroidal multivalent ions. Us-

ing coarse-grained (CG) molecular dynamics (MD) sim-

ulations, we focus on systems composed of single and

double rod-like PEs. We ﬁrst investigate the condensa-

tion and orientation response for diﬀerent ion speciﬁcities

around a single PE. We then address the order-mediated

arXiv:2210.03492v2 [cond-mat.soft] 10 Oct 2022

interactions between two like-charged PEs and underpin

the eﬀect of ion orientation, valency, and shape. Tun-

ing the balance between attractive and repulsive forces

allows for controlling the thermodynamic properties of

assembled systems, their stability, and their response to

external conditions [7, 8, 20, 60–62]. We show how this

control can be achieved by adjusting ion valency, shape,

and salt.

Model and theory. The setup consists of a periodic cu-

bic box ﬁlled with charged spheroidal mobile ions and

one (or two) ﬁxed charged rod(s). The simulations are

carried out in the NV T ensemble where we employ the

Nosé-Hoover thermostat [63, 64] with a coupling con-

stant of 0.2ps and reference temperature T= 300 K.

The equations of motion are integrated using a velocity

Verlet algorithm with a time step of 2fs to satisfy en-

ergy conservation. The production run lasts 20 ns, out of

which the ﬁrst 5ns are omitted in the data analysis (equi-

libration). The initial conﬁgurations are prepared using

Moltemplate [65] and Packmol [66]. Figure 1 shows the

schematic representation of our model.

The LAMMPS software was used for the simula-

tions [67, 68]. The interactions between components i

and j(also between PE and ions) at a distance rare

modeled via a soft repulsive version of the orientation-

dependent Gay-Berne potential [69, 70], which is ob-

tained by shifting and truncating the potential as

Uij (ˆ

ui,ˆ

uj,rij ) = ij (ˆ

ui,ˆ

uj,ˆ

rij )4(Σ12

ij −Σ6

ij )+1(1)

at rij < rij

c(ˆ

ui,ˆ

uj,ˆ

rij ), where

Σij =σij

rij −σij (ˆ

ui,ˆ

uj,ˆ

rij ) + σij

.(2)

Here ˆ

uiand ˆ

ujare the unit vectors along the molec-

ular axes, σij

0the minimum contact distance for the ij

pair, σij the orientation-dependent separation distance

2 nm

0.4 nm

(a)

0.317 nm

0.635 nm

0.159 nm

0.207 nm

0.277 nm

0.83 nm

(c)

0.25 nm

1 nm

(d)

(e)

(b)

FIG. 1. Ions with (a) Ac= 1, (b) Ac= 2, (c) Ac= 3, and (d)

Ac= 4. The charges are separated from the center by σmaj/4

(green dots). (e) Snapshot of the simulation box of size (20

nm)3with periodic boundary conditions containing one PE

(grey), cations (red), and anions (cyan), where Ac=Aa= 3.

at which attractive and repulsive contributions cancel,

ij =ij

0[ij (ˆ

ui,ˆ

uj)]ν[˜ij (ˆ

ui,ˆ

uj,ˆ

rij )]µthe orientation-

dependent well depth, and rij

c(ˆ

ui,ˆ

uj,ˆ

rij )the position

of the potential minimum (see Supplementary Material

(SM)). Following Ref. 71 we set ν= 1 and µ= 2.

We set σi

0= 0.4nm (common hydrated diameter of

ions) and i

0= 0.1kcal mol−1. The major (σmaj) and

minor (σmin) axes deﬁne the aspect ratio A=σmaj/σmin.

We set σmaj =σi

0A2/3and σmin =σi

0A−1/3, which, re-

gardless of A, provides a volume equivalent to the one

of a sphere (A= 1) with diameter equal to σi

0. Here we

consider A= 1,2,3,and 4. The choice of i

maj/min follows

Ref. 72.

The rigid PE is built of charged spherical beads

(force centers) interacting via the Weeks-Chandlers-

Andersen [73] potential UPE(r)=4PE[(σPE/r)12 −

(σPE/r)6] + PE for r≤21/6σPE. Here, the bead di-

ameter is σPE = 1.2nm, and the depth of the potential

well PE is equal to i

0. The beads are ﬁxed at a distance

b= 0.27 nm apart so that a smooth equipotential sur-

face is experienced by the ions. The PE dimensions are

in line with common synthetic and biopolymers, such as

poly(styrene sulfonate) (PSS). Lorentz-Berthelot mixing

rules ij

0=qi

0j

0and σij

0= (σi

0+σj

0)/2are used. We

use 74 beads, each with charge ZBe=−e, providing a

line charge density λ=ZBe/b ≈ −4e/nm, close to that

of PSS (−3.7e/nm). The surface charge density λ/πσPE

is close to that of DNA molecules (−1e/nm2).

The electrostatic interactions are modelled via

Coulombic potentials, which, for two charges Zieand

Zje, read βeV ij (r) = ZiZj`B/r, where β= 1/kBT, the

Bjerrum length `B=βe2/(4πε) = 0.7nm measures the

coupling strength by specifying the distance at which two

unit charges have interaction energy of ≈kBT. In this,

ε=εrε0is the eﬀective dielectric constant, εrbeing the

solvent dielectric constant (for water, εr= 78 at 300 K

FIG. 2. (a) Illustration of ˆ

er,ˆ

u.ˆ

eris normal to the cylinder

surface, whereas ˆ

upoints along the ion major axis. ˆ

udeﬁnes

the orientation of the ion and characterizes the relative ori-

entation with respect to the PE via ˆ

er·ˆ

u. Conﬁgurations

relative to the PE surface and their orientation order param-

eters Cand S. (b) Order parameter χis deﬁned between two

PEs. χ= 1 if the ion is parallel to the xaxis and χ=−1if

perpendicular.

and 1atm [74]) and ε0the vacuum dielectric constant.

These contributions are obtained in reciprocal space,

after a real space cutoﬀ of 1.2nm, using the Particle-

Particle Particle-Mesh summation method [75] with rel-

ative force accuracy of 10−5. Monovalent, divalent, and

trivalent charges are equally split into two points at dis-

tances of σmaj/4from the center of the ions along the

major axis, as sketched in Fig. 1. Finally, for valency

(Z) and aspect ratio (A) of cations (c) and anions (a) we

use the notation Zc:Zaand (Ac, Aa), respectively.

The PE is neutralized with counterions from multiva-

lent added salt. The case of monovalent counterions with

multivalent added salt is discussed in the SM. An extra

monovalent anion is added for systems containing triva-

lent salt counterions. Finite size eﬀects are checked by

repeating the simulations for boxes with sides of 4,6,10,

20,40, and 60 nm. All density proﬁles converge for boxes

with sides of 20 nm.

The single-charge number density distribution of

species iis obtained via

ni(r) = h

k=1

2δ(|r−ri

k|)it/Vk(r),(3)

where r= (x, y)is the distance vector on the xy plane

from the center of the backbone of the PE, Nithe number

of charges of type i,h...itthe time average, ri

k= (xi

k, y i

are planar vectors pointing on single charges and Vk(r)

the volume of a cylindrical shell located at r. Hereafter

cation and anion charge densities are denoted as n+and

n−, respectively.

To characterize the orientation along the PE zaxis, we

deﬁne the order parameter S(r)≡2h| ˆ

ez·ˆ

uk|it,r −1,

where h...it,r is both time average and average over par-

ticles at r,ˆ

ukthe unit vector along the major-axis of

the kth ion and ˆ

ezthe unit vector along the zaxis. For

ions oriented perpendicular to the PE S=−1, parallel

S= 1, and randomly oriented S= 0. As an additional

measure of the tendency of the ions to be tangential to

the PE surface, we deﬁne C(r)≡1−2h| ˆ

er·ˆ

uk|it,r,

where ˆ

eris the unit vector normal to the PE surface. If

all ions are tangent to the PE surface C= 1, if perpen-

dicular C=−1, if randomly oriented C= 0. Finally,

to quantify the tendency to orient along the xaxis we

use χ(η)=2h|sin θ(η) cos ϕ(η)|it,η −1,where θand ϕare

deﬁned in Fig. 2(b) and η= (x, y). Speciﬁcally, χ= 1 for

parallel, χ=−1for perpendicular and χ= 0 for random

orientations. Figure 2 shows diﬀerent ion orientations

and the respective values of S,Cand χ.

Results for a single PE. We ﬁrst focus on the case of ion

condensation around a single PE. We have recently shown

that, for monovalent salt, a soft-potential-modiﬁed PB

theory gives accurate results in the case of a cylindrical

PE for a wide range of salt and ion sizes [31]. When mul-

tivalent ions are introduced into the system, such mean-

ﬁeld approximation breaks down. To this end, we have

considered three diﬀerent cases at ionic strengths of 0.5M

in detail. (i) Trivalent cations with spherical monovalent

FIG. 3. (a) n+and (b) |n−|(enm−3) in 3:1salt with

ionic strength of 0.5M, Aa= 1 and varying Ac. (c) Order

parameters Sc(r)and (d) Cc(r)for the systems shown in (a).

The gray bar indicates the PE with a radius of 0.6nm. Error

bars in this and the other ﬁgures are comparable to the symbol

sizes or smaller.

anions, i.e. case 3:1and Aa= 1 with Ac= 1 −4. (ii)

Trivalent cations and anions, i.e. case 3 : 3 with Aa= 3

and Ac= 2,3,4. (iii) Trivalent cations and anions (3 : 3)

with Ac= 3 and Aa= 2,3,4. Additional data for the

eﬀect of the ionic strength I,Zc, and λare shown in

Figs. S2, S3, and S4 of SM, respectively.

The strong electrostatic attraction between the mul-

tivalent cations and the PE results in overcharging, as

shown in Refs. 15, 76, and 77 and a large peak in n+.

The excess charge attracts anions, resulting in the for-

mation of a second layer. This can be clearly seen in

Figs. 3(a)-(b) and 4(a)-(b), where we show results for

the cases (i) and (ii), respectively. Results for (iii) are

very similar to (ii) and are shown in Fig. S6.

Interestingly, the n+data of the case (i) shows the

aspherical cations have a much lower density near the

PE surface than the spherical ones. This is in line with

Monte Carlo simulations for dumbbell-like ions [50, 54].

Furthermore, increasing the spacing between charges in

the cations decreases the charge density close to PE (cf.

Fig. S7).

The position of the spheroidal cation peaks indicates

orientational ordering near the PE. To quantify this, the

two order parameters are shown in Figs. 3(c)-(d) and

4(c)-(d). They indicate that the spheroids have a ten-

dency to align along the backbone of the PE, as expected

from electrostatics. Such ordering is enhanced with in-

creasing electrostatic interactions, as shown in Fig. S4.

Curiously, the order parameters show small negative min-

ima indicating a tendency to align perpendicular to the

PE. For case (ii), the closest spheroidal anions show a

tendency to align with the cations whilst exhibiting no

minimum. The positions of the order-parameter minima

for cations correspond to r≈(σPE +σmaj)/2, as shown

in Fig. 4(f).

Results for PE-PE interactions. We now turn to the

interesting question of how multivalent spheroidal ions

inﬂuence PE-PE interactions. In Ref. 32, we investigated

the interactions between two negatively charged rods in

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Interactionsbetweenpolyelectrolytesmediatedbyorderingandorientationofmultivalentnon-sphericalionsinsaltsolutionsHosseinVahid,1,2,3AlbertoScacchi,1,2,3MariaSammalkorpi,2,3,4andTapioAla-Nissila1,5,6,1DepartmentofAppliedPhysics,AaltoUniversity,P.O.Box11000,FI-00076Aalto,Finland2DepartmentofChemistryan...

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Interactions between polyelectrolytes mediated by ordering and orientation of multivalent non-spherical ions in salt solutions Hossein Vahid1 2 3Alberto Scacchi1 2 3Maria Sammalkorpi2 3 4and Tapio Ala-Nissila1 5 6

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