Coulombic surface-ion interactions induce non-linear and chemistry-speciﬁc charging kinetics W.Q. Boon1M. Dijkstra2and R. van Roij1 1Institute for Theoretical Physics Utrecht University Princetonplein 5 3584 CC Utrecht The Netherlands

2025-05-06 1 0 391.93KB 10 页 10玖币

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Coulombic surface-ion interactions induce non-linear and chemistry-speciﬁc charging kinetics

W.Q. Boon,1M. Dijkstra,2and R. van Roij1

1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands

2Soft Condensed Matter, Debye Institute for Nanomaterials Science,

Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands

While important for many industrial applications, chemical reactions responsible for charging of solids in wa-

ter are often poorly understood. We theoretically investigate the charging kinetics of solid-liquid interfaces, and

ﬁnd that the time-dependent equilibration of surface charge contains key information not only on the reaction

mechanism, but also on the valency of the reacting ions. We construct a non-linear differential equation describ-

ing surface charging by combining chemical Langmuir kinetics and electrostatic Poisson-Boltzmann theory.

Our results reveal a clear distinction between late-time (near-equilibrium) and short-time (far-from-equilibrium)

relaxation rates, the ratio of which contains information on the charge valency and ad- or desorption mechanism

of the charging process. Similarly, we ﬁnd that single-ion reactions can be distinguished from two-ion reactions

as the latter show an inﬂection point during equilibration. Interestingly, such inﬂection points are character-

istic of autocatalytic reactions, and we conclude that the Coulombic ion-surface interaction is an autocatalytic

feedback mechanism.

Charged solid-liquid interfaces play a central role in a wide

variety of industries such as food and coating production [1–

3], mining [4–6], medicine [7–9], soil remediation [10–12]

and even carbon capture [13]. With the advent of nanoscale

ﬂuidics one expects that charged surfaces become ever more

important [14, 15]. In water and other polar solvents chem-

ical reactions are a common mechanism by which surfaces

obtain their charge. For ionic solids the de- or adsorption

of a dissolved ionic compound is often preferred over the

sorption of its own counterion [16–18]; for covalent solids

such as polymers and metal oxides the acidic nature of sur-

face groups ensures that the surface (de)protonates in polar

solvents and hence becomes charged [17–21]. However, for

many processes of industrial and environmental importance

relatively little is known about the surface chemistry [17–19]

as the electrolytes in realistic applications contain a large vari-

ety of ions that can all undergo multiple reactions [17, 18, 22].

Due to experimental limitations the majority of studies inves-

tigating surface charging are performed at (quasi)-equilibrium

conditions [17, 18], with the notable exception of pressure-

jump experiments [23, 24]. Only recently, however, it has

been shown that the kinetics of chemical surface reactions

can strongly couple to electrokinetic ﬂuid ﬂows, thereby af-

fecting the physical surface properties on macroscopic scales

[18, 25–29]. Furthermore, with the recent advent of fast and

surface speciﬁc non-linear spectroscopy the dynamic mea-

surement of surface charge has become feasible [29–34]. In

this context it has been explicitly stated that there is an ur-

gent need for theoretical models to describe such experiments

[35]. Traditionally, sorption kinetics is typically described by

(pseudo)-ﬁrst-order reactions [12, 36, 37] that exhibit single-

exponential relaxation towards equilibrium; the inﬂuence of

a time-dependent surface charge is usually neglected entirely

[35, 38, 39]. We are aware of one theoretical work [40] and

associated review [41] that considers a surface charge that af-

fects the rate constants of ion-association, which, however,

does not consider the (chemistry-speciﬁc) non-linear dynam-

ics induced by the electrostatic feedback as we do here.

In this Letter we present a theory for the charging dynamics

of solid surfaces. We include the Coulombic ion-surface

interactions and reveal an intricate dependence on the

reaction mechanism and the valency of the reactive ions

already present in a mean-ﬁeld description. The Coulomb

interactions not only affect the time constant of the late-time

exponential decay of the surface charge towards equilibrium

after an ion concentration (or pH) shock, but they also

induce strongly nonlinear dynamics at early times far from

equilibrium. Combined with the present-day capability to

experimentally measure the time-dependent surface charge

density, our theory forms a ﬁrst step to unveil the surface

chemistry of technologically important but ill-understood

materials [18, 35], such as silica [22, 42] and graphene [43],

and of processes such as the clean-up of radioactive and

heavy metals [10, 16, 44, 45].

Surfaces, for instance silica, in water commonly charge

either by desorption of ionic species from neutral surface

groups or by adsorption of ionic species onto neutral surfaces.

While the exact charging mechanism of the silica-water inter-

face is complex, there is support for charging by desorption

of protons at high pH and adsorption of protons at low pH

[22, 42, 46, 47],

SiOH(s)

−−*

)−−

kaρ

SiO−

(s)+H+

(aq),(1a)

SiOH(s)+H+

(aq)

kaρ

−−*

)−−

SiOH2+

(s),(1b)

where SiOH(s)is a neutral silanol group that is covalently

bound to the (solid) glass and where SiO−

(s)and SiOH2+

(s)de-

note a silanol group with a proton desorbed or adsorbed in

Eqs. (1a) and (1b), respectively. Here ρdenotes the proton

density at the solid surface, and the dissociation and associ-

ation rate kdand kawill be discussed below for the charging

kinetics of a single desorptive and a single adsorptive reac-

tion, not only for monovalent reactive ions as in Eqs. (1a) and

(1b) but for general valency z. While adsorption isotherms of

real materials can rarely be described by just a single charging

reaction [16, 47], we show in Supplemental Material I (SM I

[48]) that charging by multiple reactions can actually be well-

arXiv:2210.15426v3 [cond-mat.soft] 7 Feb 2023

approximated by the single-reaction kinetics presented in this

Letter for a wide range of experimental conditions.

We consider a macroscopic surface with a density Γof iden-

tical surface groups. A group can only be in either a neutral or

a charged state. The charging is assumed to take place either

by desorption (labeled by −) of a cation of charge ze, or by

adsorption (labeled by +) of a cation of charge ze, with z≥0

and ethe proton charge. The surface densities of charged and

neutral groups are denoted by σ±>0 and Γ−σ±>0 respec-

tively, and the surface charge density is given by ±zeσ±. Note

that the charging dynamics is invariant under the sign of the

reacting ions, and without loss of generality we can restrict at-

tention to reactive cations of (strictly positive) valency z. As-

suming the chargeable surface sites to be independent, we can

describe the reaction kinetics in terms of the time-dependent

surface density σ±(t)>0 which satisﬁes Langmuir kinetics

described by [49, 50]

∂tσ−=kd(Γ−σ−)−kaσ−ρ(σ−)(2a)

for desorptive charging reaction (1a), and

∂tσ+=ka(Γ−σ+)ρ(σ+)−kdσ+(2b)

for adsorptive charging reaction (1b). Here kdand kaare the

rate constants of the dissociation and association of the re-

active ion and ρ(σ±)is the volumetric concentration of re-

active ions at the surface, which is deﬁned at the position

where the rate-limiting step for the reaction occurs [50, 51].

We consider this surface to be impermeable to non-reacting

ions and therefore do not account for any Stern layer other

than the charged surface groups [27]. The equilibrium sur-

face charge follows from ∂tσ±=0 and is given by σ±,eq =

Γ(1+ (kaρeq/kd)∓1)−1, which reduces to an explicit “Lang-

muir isotherm” in the case that the equilibrium concentration

of the reactive ions ρeq ≡ρ(σ±,eq)is a constant independent

of σ±,eq. In general, however, this Langmuir isotherm is a

self-consistency equation for σ±,eq that requires an additional

“closure” relation ρ(σ±)for an explicit equilibrium solution

σ±,eq. Without (Coulombic) interactions between surface and

ions, the local concentration ρ(σ±)of reactive species in the

vicinity of the surface would be equal to the bulk concentra-

tion ρbof the reactive ions far from the surface (which is in-

dependent of σ±and hence also independent from the reac-

tion mechanism), such that Eqs. (2a)-(2b) would be linear dif-

ferential equations whose solution can be written as s±(t) =

1+ (s±(0)−1)exp[−(kd+kaρb)t]with the dimensionless

charge s±=σ±/σ±,eq such that s±,eq =1; here s±(0)−1

is the integration constant and denotes the relative deviation

from equilibrium at the initial time t=0. Note that the con-

dition that 0 ≤σ±(t)≤Γimplies that 0 <s±(0)<Γ/σ±,eq,

where the lower bound corresponds to an initially neutral sur-

face whereas the upper bound can be as large as O(10 −100),

since typical equilibrium conditions have a charge occupancy

of only a few percent of the total number of chargeable groups

[52]. Thus from measurements of σ±(t)at various concentra-

tions of reactive (dissolved) species both kdand kacould in

this non-interacting case be determined.

However, as the charged surface attracts or repels reactive

ions, Eqs. (2a) and (2b) are complicated by a nontrivial rela-

+++++++

++++++++

+++++++++

+++++++

+++++

++++

++++++++++

+ϕeq=4

○ ϕeq=2

(a)

z=0

z=1

z=2

z=3

012345

0.5

-0.5

-1

(2z+1)kaρeqt

s-(t)-1

++++++++

+++++

++++

++++++++

+++++++++

012345

10-1

10-2

+++++++

++++++

+++++

+++++++

+++++++++

++++++++++

(b)

012345

0.5

-0.5

-1

(

2z+1

)

kdt

s+(t)-1

++++++++

++++++++++

++++++++

++++++

+++++

++++

012345

10-1

10-2

++++++

+++++

+++

(c)

012345

-0.5

-1

kaρeqt

s+(t)-1

012345

10-1

10-2

FIG. 1. Time-dependent relative deviations s±(t)−1 from the equi-

librium charge density as follows from the kinetic Langmuir-Gouy-

Chapman equations (2a)-(3) in (a) and Eqs. (2b)-(3) in (b) and (c),

for equilibrium zeta potentials (kBT/e)|φeq|equal to 50 mV (cross)

and 100 mV (circle) for valencies z=0,1,2,3 (colors), in (a) s−(t)

for desorptive reactions when σ−,eq Γ, in (b) s+(t)for adsorptive

reactions when σ+,eq Γ, and in (c) s+(t)for adsorptive reactions

when σ+,eq 'Γ. Insets denote semi-logarithmic representations of

|s±(t)−1|. The case σ−,eq 'Γ(not shown) is trivial with single-

exponential decay for all z.

tion ρ(σ±), which causes a charge-dependent decay rate and

introduces deviations from purely single-exponential relax-

ation of σ±(t). In fact, an explicit function ρ(σ±)is needed

to investigate and solve the dynamics, which we will develop

here. We consider the planar and homogeneous chargeable

solid surface discussed above in contact with a bulk solvent

with permittivity εand temperature Twith a three-component

1 : 1 : zelectrolyte of bulk concentrations ρs:(ρs−zρb):ρb.

For convenience we assume trace amounts of reactive ions and

therefore set ρbρs, where ρsis the bulk salt concentra-

tion. We also assume the electrolyte volume to be macroscop-

ically large such that ρband ρsdo not change due to surface

charging. Furthermore, we assume the charging timescale τ±,

which remains to be derived, to be the slowest timescale of

the system. Given that the typical timescale for electric double

layer (EDL) equilibration is around 10−9−10−6s and that the

(geometry and ﬂow dependent) transport timescale for ions

in stirred reactors can be as short as 10−4s [53], we ﬁnd a

large window τ±10−4s for reactions to be well-described

by our (reaction-limited) theory [54]: for example phosphate

desorption shows characteristic reaction timescales of hours

[55] and adsorption of transition metals can occur on mil-

lisecond timescales [23, 24]. The slow-reaction assumption

allows us to describe the EDL within an equilibrium theory,

for which we take the Gouy-Chapman solution of Poisson-

Boltzmann (PB) theory for simplicity [20, 56]. Although PB

theory is based on a mean-ﬁeld assumption for a system of

point ions, it is known that for all but the highest salt concen-

trations this theory is quite accurate for 1 : 1 and even 1 : 2

aqueous electrolytes [57], and we expect a similar accuracy

for 1 : 1 : zelectrolytes in the limit ρbρsof our inter-

est. Within these assumptions the concentration of reactive

ions at the surface is determined by a Boltzmann distribution

ρ(σ±) = ρbexp[−zφ(σ±)], where kBTφ(σ±)/eis the elec-

tric potential at the surface with a surface charge ±zeσ±, with

kBthe Boltzmann constant. For desorptive charging the sur-

face and ions have opposite charge and hence zφ(σ−)<0,

while for adsorptive charging ions and surface have the same

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Coulombicsurface-ioninteractionsinducenon-linearandchemistry-specicchargingkineticsW.Q.Boon,1M.Dijkstra,2andR.vanRoij11InstituteforTheoreticalPhysics,UtrechtUniversity,Princetonplein5,3584CCUtrecht,TheNetherlands2SoftCondensedMatter,DebyeInstituteforNanomaterialsScience,UtrechtUniversity,Princetonp...

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Coulombic surface-ion interactions induce non-linear and chemistry-speciﬁc charging kinetics W.Q. Boon1M. Dijkstra2and R. van Roij1 1Institute for Theoretical Physics Utrecht University Princetonplein 5 3584 CC Utrecht The Netherlands.pdf

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Coulombic surface-ion interactions induce non-linear and chemistry-speciﬁc charging kinetics W.Q. Boon1M. Dijkstra2and R. van Roij1 1Institute for Theoretical Physics Utrecht University Princetonplein 5 3584 CC Utrecht The Netherlands

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