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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Poisson-Boltzmann Electrostatics and Ionic Partition Equilibration of Charged Nanoparticles in Aqueous Media Jerome F.L. Duval, Raewyn M. Town, and Herman P. Van Leeuwen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05168 • Publication Date (Web): 11 Jul 2018 Downloaded from http://pubs.acs.org on July 14, 2018

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Poisson-Boltzmann Electrostatics and Ionic Partition Equilibration of Charged Nanoparticles in Aqueous Media

*

Jérôme F. L. Duval,1,2 Raewyn M. Town,3,4 Herman P. van Leeuwen4

1

CNRS, Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), UMR 7360 CNRS,

15 avenue du Charmois, 54500 Vandoeuvre-les-Nancy, France. 2

Université de Lorraine, LIEC, UMR 7360 CNRS, 15 avenue du Charmois, 54500 Vandoeuvre-les-

Nancy, France. 3

Systemic Physiological and Ecotoxicological Research (SPHERE), Department of Biology,

University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium. 4

Physical Chemistry and Soft Matter, Wageningen University & Research, Stippeneng 4, 6708 WE

Wageningen, The Netherlands. *

Corresponding author, [email protected] / Tel: 00 33 3 72 74 47 20.

Abstract Most nanoparticles (NPs) dispersed in aqueous media carry a net charge. The ensuing electric field plays a fundamental role in determining the thermodynamic and chemodynamic features of the interactions between NPs and dissolved metal species, and their lability and bioavailability in environmental and biological matrices. While increasing attention is being paid to the analysis of metal ion speciation in dispersions of charged complexing NPs, so far the electrostatic features of NPs have only been described on the sole basis of their structural charge properties, i.e. the number of (potentially) charged groups they carry. This approach intrinsically ignores the impact of counterion accumulation at/within the particle body/surface during equilibration of the system, which effectively lowers the magnitude of the net NP charge density. Herein we present the first analysis of the potential profile of NPs after their physicochemical equilibration with the aqueous medium, and we discuss the implications thereof in terms of counterion accumulation within and/or in the vicinity of hard, soft and core-shell NPs. The focus is on soft or core-soft shell NPs in the thick double layer limit

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for which bulky Donnan features are not applicable. The new conceptual framework identifies the spatial zone over which divalent counterion accumulation is significant as a function of the size, charge density, and type of NP, as well as the ionic strength and electrolyte composition (1-1 and 2-1) of the aqueous medium for the most common case of negatively charged NPs.

1. INTRODUCTION Many natural and engineered aqueous nanoparticles (NPs) contain structural charges due to the presence of certain functional groups in their polymer backbone or crystal structure.1-6 The spatial distribution of charges may range from homogeneous to highly heterogeneous, depending on the NPs considered.7 Upon dispersal in aqueous media, counterions of opposite charge will be electrostatically attracted by, and accumulated within, the electric field of the particle. Depending on the type of particles, the charge may be a surface charge (hard particles),1-3,6 a volume charge (soft or porous particles),1,2,5 or a combination of the two e.g. in core-soft shell particles.1,2,8 As a consequence, the particles have their own physicochemical micro-environment at an electric potential where the ion concentrations and ionic strength generally differ from those in the bulk medium. At equilibrium, in chemically inert non-complexing electrolyte media, the dispersed NPs carry a net electric charge that is typically lower in magnitude than the structural charge due to accumulation of counterions at the NP surface and/or within the NP body.5 In addition to that, the particle/medium interface generally features an electric double layer whose extension and composition mediates the reactivity of NPs e.g. stability against aggregation,9-11 interactions with neighboring colloids,12,13 or electrokinetic features.14-18 In the case of a soft spherical nanoparticle (i.e. nanoparticle characterized by a 3D distribution of structural charges and a given permeability to electrolyte ions and flow), the electrostatic conditions prevailing within the particle body depend on the relative magnitudes of the particle radius, rp, the distance of separation between structural charges carried by the NP,  C , and the Debye screening length,   1 .1 Ohshima further evidenced that the typical decay of the electrostatic potential from the center of a soft NP to its surface may span over a spatial region that is thinner than the


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typical Debye layer thickness   1 provided that the ionic strength in solution is sufficiently low and/or the density of NP charges is sufficiently large.19 The high charge density regime corresponds to

  C  1 and represents the situation in which the electric potential due to the structural charge predominantly results from the cooperative forces of many adjacent charged sites.1 At the other extreme, in the low charge density regime   C  1 , the potential is dominated by the local coulombic fields around each individual charged site. The potential profile arising from the structural charges depends on both the charge density and the radius of the particle, rp. For sufficiently large NPs with high charge density, i.e. rp >>   1 >  C , a Donnan potential difference,  D , is established between the bulk of the soft NP phase and the bulk aqueous medium. The magnitude of  D influences the equilibrium partitioning of any type of ion between the soft particle phase and the bulk electrolyte solution. At equilibrium, a true Donnan phase carries no net charge and the electric field therein is zero. In such particles, counterion condensation,20 as governed by cooperative electrostatics, may occur in the electric double layer zone at the particle/medium interface where the potential gradient is steepest.21 A different type of potential profile is found for small and highly charged NPs whose radius is comparable to the Debye layer thickness (rp    1 ).21,22 In such case, the particle body is too small for a separate Donnan phase to be established. The potential profile is then typically bellshaped, and as the ionic strength in the medium decreases, the electrostatic field distribution becomes increasingly broadened and extends to greater distance beyond the physical size of the NP body.17 For sufficiently small NPs and/or low ionic strength, the thickness of the extraparticulate counterionic atmosphere can significantly exceed the particle radius. Accordingly, counterion accumulation in the extraparticulate charged double layer zone can be significant relative to that within the particle body.23 Recent studies evidenced the paramount role played by electrostatics in defining the reactivity of NPs toward charged analytes or ions.2 For example, the chemodynamic behavior of metal ion (M)nanoparticle associates in aqueous media, which includes their rates of formation/dissociation24-26 and their overall stability,24,27,28 is intimately dictated by the distribution of the electrostatic potential at the NP/medium interphase. In particular, NP electrostatics impacts on both the kinetics of chemical binding of M to reactive sites distributed within the NP volume, and the dynamics of conductive




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diffusion of M to/from the charged NP surface.2 Such basic knowledge is relevant for assessing e.g. the speciation, fate and toxicity of metallic contaminants in natural aqueous media,1,2,29 for a proper evaluation of their labilities and bioavailability,1,2,29 and for in-depth analysis of mechanisms of metal biouptake by microorganisms.30-33 To date, descriptions of the electrostatic features of NPs within the context of metal ion binding by NPs have been based on their structural charge characteristics, i.e. the number of (potentially) charged groups they carry (e.g. carboxylic and phenolic groups for particulate fulvics17). In the light of the key role played by electrostatics in governing NP reactivity toward ionic species, it is timely to provide a suitable description of the equilibrated potential profile that explicitly integrates counterion accumulation by soft NPs for situations where Donnan features are not in effect. In passing it is noted that Donnan representations are arbitrarily adopted in popular ready-to-use equilibrium metal-binding computational codes employed to ‘predict’ the metal-binding strength and capacity of nanoparticles, regardless of their size.34,35 As a consequence, any apparent “goodness of fit” between modelled and measured parameters is purely empirical. For example, such strategies are not appropriate for environmentally ubiquitous small NPs, such as fulvics, with dimensions comparable to the Debye screening length over a large range of practical salinity conditions.17 Herein we thus elaborate the equilibrium potential profile of a small and highly charged NP (in line with the condition rp    1 ) in 1-1 and 2-1 inert electrolytes and mixtures thereof, with account of the fraction of the NP charges neutralized via Boltzmann accumulation of counterions from the background electrolyte. The conditions are typical of those encountered in environmental and biological media. The results are fundamental for e.g. discriminating between electrostatic versus chemical contributions to the stability of divalent ion-NP complexes and their defining chemodynamic features.27,28 The elaboration involves an iterative procedure for solving the non-linear Poisson Boltzmann equation. This is shown to allow for detailed evaluation of (i) the net, equilibrated NP charge density that differs from the NP structural charge density due to counterion accumulation, and underlying formation of electrostaticallystabilized counterion-NP charge pairs, as well as (ii) Boltzmann partitioning of all types of ions in the medium.


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2. THEORY Unless otherwise specified, we shall consider a spherical soft NP of radius rp where (negative) structural charges are randomly distributed over the NP body. These charges may electrostatically associate with and covalently bind transition type metal ions M2+ (M in short) present in solution at bulk concentration c M . The density of the structural charges in the NP body is denoted as  o(s) . The NP is dispersed in a solution containing a 1-1 electrolyte (e.g. KNO3) and/or a 2-1 electrolyte (e.g. Ca(NO3)2) in large excess over a target complexing metal ion M. The purely electrostatic binding features of divalent metal ions M2+ (e.g. Cd2+, Cu2+) are considered to be similar to those of the indifferent electrolyte cations, e.g. Ca2+. These conditions correspond to those experimentally adopted for measuring the stability constant of complexes formed between NPs and M.27,28 We seek to determine the equilibrated NP potential profile, and thereby the Boltzmann partitioning coefficient, operative for the divalent trace metal ions M in the given electrolyte media. For the sake of simplicity and demonstration, the water content in the NP volume is considered sufficiently high to circumvent the necessity to introduce differentiated dielectric permittivity across the NP/medium interface.17 Introduction of a spatial profile for this quantity from the NP center to the outer electrolyte solution would however be possible as detailed elsewhere.31 For a mixture of 2-1 and 1-1 electrolytes, the Poisson-Boltzmann equation to be solved for evaluation of the potential distribution  from the NP center (positioned at r  0 with r the radial coordinate system adopted) to the bulk solution r   is written after some algebra (see Supporting Information):

  f  r  exp   2 y  r    exp  y  r    x sinh  y  r    2 y  r    2  o     2 FI  3 x  

(1)

, where 2 is the Laplacian operator in spherical geometry and  o a dummy NP charge density variable. 

1

is the Debye layer thickness defined by 

1

 2I  F 2    R T  o r g

  

1/2

with F the Faraday number,

Rg the gas constant, T the temperature, and  o r the dielectric permittivity of the aqueous medium, y




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is the dimensionless potential defined by y  F / RgT . I  is the solution ionic strength (including co- and counter-ions contributions) given here by I    3  x  cX2+ where cX2+ is the bulk concentration of divalent cations X2+ from the background electrolyte and the dimensionless x refers to the ratio between bulk concentrations of monovalent cations (denoted as cX+ ) and cX2+ . The 1-1 and 2-1 electrolyte situations thus correspond to the limits x  1 and x  1 . The function f  r  involved in eq 1 is defined by the step-function like distribution of NP charges across the whole radial

  r  rp   space ( r  0 to r   ) according to f  r   1  tanh    / 2 taken here in the asymptotic limit    

 / rp  0 where  is the characteristic decay length of f  r  .36 In this work, we adopted the ratio value  / rp  103 . For the situation of core-shell NPs with core radius a,  is given by  / d  103 with d the shell thickness. Introduction of this function avoids the necessity to define boundaries other than those operational at the NP center ( r  0 ) and far from the NP surface ( r   ). The reader is referred to previous reports on that issue.36,37 The boundary conditions associated with eq 1 thus reflect the symmetry of the potential profile, i.e. dy  r  / dr r 0  0 and the electroneutrality condition in the bulk electrolyte, y  r     0 , which holds for sufficiently dilute NP suspensions such that electric double layer overlap between neighbouring NPs is insignificant (i.e. the average separation between neighbouring NPs is much larger than  1 ). The accumulation of counterions (cations for cases where  o  0 ) and the exclusion of coions within the NP body and in the extraparticulate electric double layer will lower the charge density of the NP to a net value we term

 oeq  . The magnitude of  oeq  is the relevant quantity to be considered for a proper interpretation of e.g. electrostatic contribution to metal-NP binding.27,28 In this study, the density of NP structural charges is set independent of pH for the sake of simplicity and demonstration. Without loss of generality, the current theory can be easily extended to include such pH-dependence within the Poisson-Boltzmann framework along the lines specified in previous reports.16,17,36


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2.1. Evaluation of the NP Equilibrium Charge. The potential profile y s  r  corresponding to the structural charge density  o(s) is taken as the starting point for an iterative process in order to obtain the electrostatic potential profile y  eq   r  and the corresponding net NP charge density,  o eq  , at ion partition equilibrium between NPs and the aqueous medium. For any given value  o from  o(s) to  o eq  , the solution y  r  of eq 1 was obtained using the numerical procedure COLSYS,38 based on the approximation of the solution by a piecewise polynomial with adaptive collocation at Gauss points. This algorithm was successfully tested in previous reports e.g. on electrohydrodynamics of soft surface layers.39 For a given  o and potential distribution y  r  , the amount of charges Q per particle stemming from counterions accumulated over the spatial region spanning from r  0 to

r  R (where R is the chosen distance from the NP center) may be written in the form: r2  x  exp  2 y  r   exp   y  r  dr  3 2 x   0

R

Q  8 FI  

(2)

, where the second exponential term in the integral stems from the contribution of the monovalent counterions in the NP charge neutralisation process. Starting from the situation  o   o(s) , we seek to evaluate the potential distribution y  eq   r  such that the overall amount of accumulated counterion charges Q is in equilibrium with the amount of net structural charges  o eq  carried by the NP body. For that purpose, for a prescribed expansion of the counterion accumulation region (i.e. at given R), the value of the charge density  o and the corresponding potential distribution y  r  (eq 1) were iteratively updated/solved until the following condition was satisfied: Q    o eq Vp

(3)

, where Vp is the volume of the soft nanoparticle. Following the logic of Manning for high charge density linear polyelectrolytes,40,41 the coions are considered to be completely excluded from the zone of counterion accumulation around a high charge density NP. Numerically, the iteration is stopped as soon as the criterion



o

Vp  Q  /  o Vp    is verified, where  is an arbitrary positive scalar

much lower than unity (taken equal to 10-4 in this work). Within the framework of the current study,




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the situation where  o(s) and  o eq  differ in sign (NP charge reversal) is not considered (i.e.  is strictly positive), which is a reasonable assumption for electrolytes where cations and anions have similar hydrated radius.42,43 In addition, NP charge reversal would come to metal-binding sites coverage values above unity, which generally is not relevant from the point of view of trace metal ions complexation capacity of NPs. It is straightforward to verify that the percentage of NP structural charges effectively neutralized by counterions accumulated in the region from r  0 to r  R is simply provided by



(s) o

  oeq   /  o(s) . The averaged equilibrium concentration cX eq2  of divalent

counterions in the volume 4 R 3 / 3 is then given by: R

cXeq2 

r2 eq 0 3  x exp 2 y    r dr

3Vp o(s)   oeq 

r2  x  4 R3 eq eq 0 3  x exp 2 y    r   2 exp  y    r  dr R

(4)

2F

, where the ratio of integrals corresponds to the fraction of NP charges that are compensated by divalent ions. Recalling that the electrostatic binding features of the divalent cations from the background electrolyte are similar to those of the trace metal ions M, accumulation of the latter by soft NPs proceeds according to the equilibrium Boltzmann partitioning coefficient f B

eq 

pertaining to

the divalent counterions of the background electrolyte. In turn, the Boltzmann factor f B

eq 

for the

accumulation of M from r  0 to r  R is simply provided by: f B eq   cX eq2  / cX2  .

(5)

For the sake of comparison, the Boltzmann factor f B  derived from the potential distribution y s  r  s

evaluated on the basis of the structural NP charge density  o(s) is defined by: R

f Bs   3R 3  r 2 exp  2 y s   r  dr .

(6)

0

Figure 1 summarizes the key steps of the iterative procedure detailed above to derive the situation

corresponding to ion partition equilibration between NPs and the aqueous medium.


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Figure 1. Schematics of the iterative procedure adopted for evaluation of ion partition equilibration between NPs and aqueous medium. ‘PB’ stands for ‘Poisson-Boltzmann’.

It is stressed that the classical solution of Poisson-Boltzmann equation refers itself to an equilibrium situation. However, this equilibrium simply stems from the presence of the structural charges (i.e. the charged groups) carried by the NPs and it does not integrate the decrease (neutralization) of the net charge carried by the NP upon accumulation of counter-ions and the implications thereof for the subsequent accumulation of incoming counter-ions that feel a lower NP electric field as a result of this structural charge compensation. Given this element, the iterated equilibration process detailed above kind of mimics the situation in which electrolyte is added to an aqueous dispersion of "pristine" NPs where ion partition between medium and NP body is modelled. Inclusion of simple ion-pair formation could possibly serve the same purpose but the procedure we follow has the merit to use the well-known mean field Poisson-Boltzmann framework without the need to introduce often ill-defined molecular parameters describing the formation of ion pairs in complex nano-environments. Unless otherwise stated, we illustrate below the approach for the case of a soft NP with rp of 1 nm and a randomly distributed structural charge of -2000 mol m-3 (expressed in equivalent concentration of anionic monovalent charges), which are typical size- and charge-features measured for environmentally relevant soft nanoparticles, e.g. fulvic acids.17,44 We further emphasize that the approach elaborated in this study is generic and applicable to all types of NPs, i.e. from hard to soft




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NPs (see section 3.4). Evidently, the average magnitude of counterion accumulation will depend on the spatial zone R considered in eq 2. For orientational purposes, we note that for line charges Manning identified a counterion condensation layer thickness of approximately 2 nm,41 and the same thickness was found for an intraparticulate condensation zone in large soft NPs.27,28 For the small NPs considered herein, the accumulation zone should be expected to extend into the extraparticulate region. In line with this, we firstly consider a counterion accumulation region that comprises the particle body plus an extraparticulate shell of thickness 1 nm, i.e. R  2 nm. The potential profiles corresponding to (i) the density of NP structural charges ( y s  r  ) and to (ii) the effective charge of the NP particle after attainment of Boltzmann equilibrium ( y  eq   r  ) are shown in Figure 2. As expected, the potential evaluated with adopting the structural charge density (  o   o(s) in eq 1) is greatest (in magnitude) at lower ionic strength, and at a given ionic strength is greater in 1-1 as compared to 2-1 electrolyte. The opposite trend is seen in the potential profiles for the equilibrated case, reflecting the extent to which counterions accumulate and thereby reduce the effective NP structural charge. Under the conditions examined in Figure 2, the potential profiles y  eq   r  are achieved after compensation of 36.8% and 32.3% of the NP structural charges in 100 mM (ionic strength) 2-1 and 11 electrolytes, respectively, which corresponds to f B

eq 

 1.40 and f B eq   1.70 (eq 5), respectively.

This extent of charge compensation dramatically increases to 97.1% ( f B

eq 

 36.50 ) and 95.7% (

f B eq   25.80 ) in 10 mM 2-1 and 1-1 electrolytes, respectively. The f B eq  value reflects the extent to which ions will partition from the bulk medium into the given spatial zone of the NP. That is, at Boltzmann equilibrium, the local concentration of species X2+, cX2+ , will be equal to f B concentration of X2+ in the bulk medium, cX2+ . Accordingly, the so-derived f B

eq 

eq 

times the

values will increase

with decreasing ionic strength. For the sake of comparison, the f B  estimated from the density of NP s

structural charges increases from 3.50 (7.40) to 17.20 (84.40) with decreasing the ionic strength of a 2-1 (1-1) electrolyte from 100 mM to 10 mM. f B  and f B s

eq 

thus refer to two inherently different


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electrostatic configurations: unlike f B  , the magnitude of f B s

eq 

correctly accounts for the decrease of

the net NP structural charge upon achievement of equilibrium as a result of counterion accumulation. The forthcoming sections report on computational illustrations highlighting how f B

eq 

depends on the

nominal NP charge density  o(s) , the solution ionic strength I  , the particle size rp and the considered extension R of the counterion accumulation zone. Some of the results below are further reported as a function of the parameter x  cX+ / cX2+ , thus covering the 2-1 electrolyte situation ( x  1 ), the 1-1 case ( x  1 ) and their mixtures.

Figure 2. Potential profiles computed on the basis of structural charge ( y s  r  ) and of the net effective charge after attainment of Boltzmann equilibrium ( y  eq   r  ) for a soft NP with radius rp of 1 nm and structural charge density  o(s) of -2000 mol m-3 (expressed in equivalent concentration of anionic monovalent charges). The profiles are computed for 1-1 and 2-1 electrolytes at ionic strengths I   10 mM and 100 mM (indicated). On the x-axis, r = 0 denotes the center of the particle body; and R = 2 nm indicates the radius of the considered counterion accumulation volume (see text for details).

3. RESULTS AND DISCUSSION 3.1. Dependence of f B

eq 

on Counterion Accumulation Layer Thickness and Electrolyte

Composition. The computation of f Beq  is coupled to the choice of the spatial zone over which the integration in eqs 2-5 is made. Figure 3A shows the dependence of f B

eq 

on x  cX+ / cX2+ for

different extensions R of the accumulation region under fixed ionic strength condition (10 mM). Figure 3B further reports the degree of NP charge neutralisation in the form of the ratio  o(eq) /  o(s) .




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Results indicate that for a given value of R, the extent of NP charge neutralisation decreases with increasing x, which is reflected by increasing values of  o(eq) /  o(s) (Figure 3B). This feature indicates the better efficiency of divalent cations - as compared to that of monovalent cations - for compensating the negative NP charges as a result of their larger (attractive) electrostatic interactions with the NP in the Boltzmann distribution process. In line with this, the Boltzmann partitioning coefficient f B

eq 

(Figure 3A) increases with decreasing the bulk concentration of monovalent

counterion (decreasing values of x): divalent counterions then tend to accumulate within the NP/medium interfacial region to a larger extent than their monovalent counterparts. For a fixed ratio

x  cX+ / cX2+ , Figure 3A further illustrates the key impact of the ion accumulation layer thickness R on the Boltzmann partitioning coefficient f B

eq 

. Namely, f B

eq 

decreases with increasing R for two

reasons: there is an increased dilution of the overall number of counterion charges over the volume of the accumulation layer, and the number of accumulated counterions decreases with R as then the accumulation zone extends over parts of the electric double layer where electrostatic potential tends to zero. In the limit  R  1 , the ratio  o(eq) /  o(s) necessarily goes to 1. Indeed, the electroneutrality of the ensemble consisting of the charged NP and the peripheral electric double layer is written Q   o Vp  Q  0 where Q  identifies to Q defined by eq 2 taken in the extreme  R  1 , and Q  (  0 ) is the equivalent of Q  for the co-ions. In turn, at sufficiently large  R , more specifically

for   R  rp   1 (i.e. the thickness of the extraparticulate ion accumulation shell is comparable to 1 /  ), the criterion



o

Vp  Q  / o Vp used for defining the convergence of the iteration procedure

(see §2) becomes negative. This reflects the impossibility to accommodate more counterions within the electric double layer to compensate the NP structural charges.


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 eq 

Figure 3. (A) Evolution of the equilibrium Boltzmann factor f B

versus x  cX+ / cX2+ for different values of

R (indicated) and (B) corresponding ratio  o(eq) /  o(s) . The radius rp of the considered soft NP is 1 nm and structural charge density is -2000 mol m-3 (expressed in equivalent concentration of anionic monovalent charges). The data are computed at the ionic strength I   10 mM. R indicates the radius of the considered counterion accumulation volume (see text for details). eq 3.2. Dependence of f B  on Counterion Accumulation Shell Thickness and  rp . We report

in Figure 4A the dependence of f B

eq 

on the radius R of the counterion accumulation shell for

different particle sizes rp at fixed 10 mM ionic strength of a 2-1 electrolyte solution (i.e. x  0 ). For the sake of demonstration, results are plotted versus   R  rp  in order to distinguish the situations where M accumulation operates in the intraparticulate volume (   R  rp   0 ) from cases where it extends in the extraparticulate electric double layer beyond the physical NP body (   R  rp   0 ). It is stressed here that the situation   R  rp   0 comes to consider an intra-particulate volume fraction where ions may accumulate and neutralize NP structural charges. In accordance with the conclusions derived from Figure 3, f B

eq 

increases with decreasing R at fixed particle size (Figure 4A) in

agreement with the corresponding decrease of the volume where an increased amount of counterion charges are accumulated (leading to decreasing  o(eq) /  o(s) ratios). The new important features here are (i) the decrease (increase) of f B

eq 

with increasing particle size at fixed   R  rp   0 (




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  R  rp   0 ), respectively, and the quasi-independence of f Beq  on rp for particle sizes such that

 rp is significantly above unity, and (ii) the evidence that accumulation of M in the extraparticulate double layer zone is slightly favoured upon increase of the particle size. This feature (ii) is reflected by the increase with rp of the threshold value of   R  rp  where  o(eq) /  o(s)  1 (red dashed lines in Figure 4A). The evolution of f B

eq 

with particle size is intimately connected to the peculiarities of the

potential distribution y s  r  at the soft NP/medium interface with changing  rp .22 Namely, upon increase of rp at fixed 1 /  , the potential gradient within the bulk NP body is gradually suppressed, in line with the establishment of a Donnan potential y D and the concomitant setting of electroneutrality over the intraparticulate medium (zero net charge density therein). The magnitude of y D is then determined by the solution y of the (transcendental) eq 1 taken in the extreme

2 y  r   0 .21 In turn, the possibility to accumulate M in the intraparticulate NP volume (

  R  rp   0 ) necessarily decreases (and so does f Beq  ) with increasing rp (Figure 4A) because, otherwise, such accumulation would act against the development of a Donnan potential and its accompanying intraparticulate electroneutrality. The accumulation of M in the NP body then becomes limited to the thin intraparticulate region that extends within the NP volume over a distance called the intraparticulate double layer thickness 1/  p , the latter being smaller than 1 /  upon increasing  o(s) and/or decreasing solution ionic strength.19 This latter argument stems from e.g. the expression

 p1   1  cosh  y D  

1/2

valid for a 1-1 electrolyte with the dimensionless Donnan potential defined

by y D  sinh1  o(s) /  2FI   .21,45,46 The occurrence of M accumulation within this thin layer of thickness 1/  p is in good agreement with the two-state Counterion Condensation Donnan (CCD) model, as recently proposed by Town and van Leeuwen for humic acid nanoparticles satisfying Donnan electrostatic representation.27,28 Increasing the particle size rp at fixed 1 /  is further accompanied by a reduction of the intraparticulate double layer extension 1/  p that follows the


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gradual establishment of the (maximum achievable) intraparticulate Donnan potential y D .22 This also leads to an increase in magnitude of the potential distribution operational in the extraparticulate double layer. In turn, at a given 1 /  , the extent of M accumulation in the extraparticulate double layer region increases with rp , and so does f B

eq 

(Figure 4A). Obviously, this increase levels off

upon full establishment of y D over the whole NP body, i.e. at sufficiently large  rp compared to unity, which is in agreement with the results displayed in Figure 4A. The finding that accumulation of M in the extraparticulate double layer zone is somewhat favoured with increasing rp , relative to the slight shift of the counterion accumulation region frontier R  rp away from the NP surface with increasing rp , is a consequence of the underlying increase of the electrostatic potential in the extraparticulate double layer region. Following the above identification of the quantity  rp in governing the respective magnitudes of extraparticulate and intraparticulate M accumulation processes, we report in Figure 4B the pendant of Figure 4A for a fixed particle size and different ionic strength values I  of a 2-1 electrolyte solution. Results indicate that the thicker is the extraparticulate electric double layer following a decrease in I  , the more the accumulation of M extends into the extraparticulate double layer region, as materialized by the shift of the red dashed lines in Figure 4B to larger R  rp values. For extreme values of I  leading to  rp >> 1 (not shown) (i.e. thin extraparticulate electric double layer compared to particle size), the accumulation of M becomes possible predominantly via the intraparticulate double layer component (this is the Donnan limit27,28) whereas at sufficiently low I  (i.e. low  rp ), M accumulation operates most significantly within the extraparticulate double layer region. At fixed value of the radius R of the counterion accumulation volume, f B

eq 

further decreases

with increasing I  as a result of NP charge screening and ensuing reduction in y s  r  at any position r . Again, it is verified that the limit  o(eq) /  o(s)  1 corresponds to R  rp  1 /  where 1 /  increases from ca. 1 nm to 4.3 nm and 9.6 nm for I   100 mM, 5 mM and 1 mM, respectively.




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 eq 

Figure 4. (A) Evolution of the equilibrium Boltzmann factor f B

versus the dimensionless quantity

  R  rp  , where R is the radius of the counterion accumulation volume, at different soft NP particle sizes

(indicated) for I   10 mM ( 1 /   3 nm) and a 2-1 electrolyte. The nominal NP structural charge density  eq 

is -2000 mol m-3 (expressed in equivalent concentration of anionic monovalent charges). (B) Evolution of f B

versus R  rp at different 2-1 solution ionic strengths (indicated) and at fixed particle radius rp  1 nm. Nominal NP structural charge density is -2000 mol m-3. In panels (A) and (B), the lowest value of R  rp considered corresponds to -90% of the particle radius rp and values between brackets correspond to the extremes of

 rp over the ranges of sizes and ionic strengths tested. See text for details.

3.3. Dependence of f B  on Counterion Accumulation Shell Thickness and Nominal NP eq

Charge Density. In Figure 5 we provide the impact of the NP structural charge density  o(s) on f B eq  at fixed ionic strength and particle size (  rp  0.33 with rp  1 nm, x  0 ). For a given radius R of the counterion accumulation shell thickness, results indicate that f B

eq 

increases with increasing

 o(s) as a result of the increasing amount of accumulated counterions required to reach electrostatic equilibration with the aqueous medium. This increase of f B

eq 

is particularly marked under conditions

where extraparticulate M accumulation takes place (   R  rp   0 ) whereas the amount of ions





accumulated within the intraparticulate region (  R  rp  0 ), if applicable, remains basically unaffected with changing  o(s) . As a result of this, the frontier corresponding to the condition


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 o(eq) /  o(s)  1 is shifted to positions further away from the NP surface: the larger is  o(s) (in absolute value), the more pronounced is the extension of the counterion accumulation region toward the peripheral side of the electric double layer following increase in the magnitude of the potential distribution. The underlying reason is similar to that invoked for explaining the dependence of f B

eq 

on solution ionic strength (Figure 4B, dotted red lines therein). Figure 5. Evolution of the equilibrium Boltzmann

factor

f B eq 



versus



the

dimensionless quantity  R  rp , where R is the radius of the counterion accumulation shell radius, at different values of the soft NP structural charge density  o(s) (indicated) for

I   10 mM ( 1 /   3 nm), x  0 (2-1 electrolyte) and rp  1 nm (  rp  0.33 ).

3.4. Dependence of f B  on Counterion Accumulation Shell Thickness and Type of NP eq

Considered. As a final illustration, we provide in Figure 6 computations that highlight how f Beq  depends on the nature of the NP considered, i.e. a soft NP devoid of an ion-impermeable core component (the case treated in the Figures 1-5), a core-shell particle and a hard particle with no peripheral shell layer. For that purpose, the boundary associated to the Poisson-Boltzmann equation (eq 1) at r  0 now applies at r  a with a the radius of the NP core whose surface is taken as uncharged for simplicity (the reader is referred to ref. 31 for situations where this condition is relaxed). In addition, the iterative procedure described in §2 remains valid pending replacement of the lower integration limit in eqs 2, 4-6 by a, replacement of the term R3 in eqs 4-6 by R 3  a 3 and Vp by

4  rp3  a 3  / 3 with rp  a  d the overall NP radius and d the thickness of the soft shell part of the




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NP body. The procedure was applied for different ratios a / rp . The computations given in Figure 6 further refer to conditions where the ionic strength of the considered 2-1 electrolyte ( x  0 ) is 10 mM and the total particle radius rp  a  d is fixed at 2 nm with  o(s)  2000 mM . Figure 6 shows that over the range   R  rp   0 , f B

eq 

remains quasi-invariant with increasing the ratio a / rp from 0

(soft NP case) to 0.75 (core-shell NP with a / d  3 ). Under the ionic strength condition examined and over this range of a / rp values, the extraparticulate potential distribution remains indeed practically unaffected by the presence or not of a core component. The only difference stems from the replacement of part of the soft NP component by a core, which for increasing a / rp at fixed rp is obviously accompanied by a gradual suppression of intraparticulate accumulation of M. This feature is marked by the presence of an asymptote for f B

eq 

at   R  rp   0 with increasing a / rp . For

values of a / rp in the range 0.75 to ca. unity, i.e. switching from a core-shell NP to a hard NP, f B

eq 

decreases with increasing a / rp at fixed   R  rp   0 , and the frontier of the extraparticulate accumulation region is further shifted to lower   R  rp  . These findings originate from the decrease (in magnitude) of the potential distribution y s  r  for any position r between the core surface and the bulk solution upon increase of a / rp . In turn, a lesser amount of M accumulates in the extraparticulate double layer region (leading to reduction in f B

eq 

) and the extension of the counterion accumulation

zone is gradually reduced with increasing a / rp . This latter finding is analogous to that detailed in Figure 5 when decreasing  o(s) at fixed a / rp .


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Figure

6.

Evolution  eq 

Boltzmann factor f B





of

the

equilibrium

versus the dimensionless

quantity  R  rp , where R is the radius of the counterion accumulation shell thickness, at different values of the ratio a / rp for

 o(s) / F  2000 mM , I   10 mM ( 1 /   3

(2-1 electrolyte) and x0 rp  a  d  2 nm (  rp  0.66 ). The lowest

nm),

value of R  rp considered corresponds to -90% of the shell thickness d.

4. CONCLUSIONS AND OUTLOOK A theoretical framework is established for computation of the electrostatic potential profile of small, highly charged NPs equilibrated in 1-1 and 2-1 aqueous electrolyte media, and for evaluation of the ensuing Boltzmann ion partitioning in/at the NP body/surface. Within the context of the analysis of electrostatic metal ion binding by negatively charged NPs, this study represents the first elaboration of the electrostatic features of NPs that accounts for ion partitioning equilibration between the dispersed NPs and the aqueous medium. The extent to which the equilibrated potential differs from that derived on the basis of structural charge density reflects the extent to which counterions accumulate within the applicable spatial zone. The Boltzmann factor corresponding to the equilibrium electrostatic profile,

f B eq  , is derived for a range of conditions. The magnitude of f B eq  is shown to depend on the thickness of the counterion accumulation layer, the electrolyte composition, the size of the NP, the type of NP considered and the NP charge density. The f B

eq 

values derived herein are applicable to

e.g. prediction of the extent to which trace cations such as divalent transition metal ions, will accumulate in and/or close to a charged NP. This application will be the subject of a forthcoming




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analysis where confrontation with experimental data is undertaken to support the basis of the theoretical development detailed in the present work. The theoretical framework is an essential physicochemical tool for understanding and predicting the reactivity of NPs in aqueous media. In particular, for small highly charged NPs (

 rp  1 ) our approach highlights the importance of both intraparticulate and extraparticulate counterion accumulation at equilibrium. For the smallest NPs with radius of order 1 nm, characteristic of ubiquitous environmental particles, the spatial zone over which counterion accumulation is significant, extends at least 1 nm from the NP/medium interface, i.e. over a volume 8 times greater than that of the particle body. We highlight that the spatial zone over which counterion accumulation is significant in the equilibrated NP-ion system depends on the particle properties (size, charge density) as well as the ionic strength and electrolyte composition of the medium, e.g. 1-1 versus 2-1 indifferent electrolyte. These findings pave the way for incorporating mechanistically sound descriptors for nanoparticulate complexants into classical equilibrium codes that are widely employed to compute speciation of ionic species. For example, the assumption that Donnan partitioning is operative over an arbitrary/fitted volume regardless of particle size, as invoked in the NICA-Donnan model,35 is bound to grossly overestimate the electrostatic contribution to ion association with small NPs. On the other hand, the assumption of the WHAM model34 that ion association with charged particles, regardless of size, occurs via extraparticulate Donnan partitioning is physically meaningless/impossible. Metal ion-NP complexes are ubiquitous in environmental and biological media. The chemodynamic features of M-NPs govern their rates of dissociation and thus e.g. the magnitude of signals obtained by dynamic analytical techniques (voltammetries,47 diffusive gradients in thin film,48,49 etc.) in dispersions thereof, and their bioavailability.2 The chemodynamic features of metal ions that are electrostatically associated with NPs may differ significantly from those that are covalently bound. The theoretical framework presented herein enables robust prediction of the electrostatic contributions to the stability of metal ion-NP complexes.50 In particular, the determination of the counterion accumulation region operational within and/or in the vicinity of the NP may be performed from analysis of the electrostatic contributions to metal ion binding by NPs and


Page 21 of 28


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comparison to appropriate electrochemical data collected under various 1-1 and 2-1 electrolyte concentration conditions.50 The capability to discriminate between electrostatic versus chemical contributions to the stability of metal ion-NP complexes will establish a fundamental basis for elaboration of their defining biochemodynamic features.

Main symbols and abbreviations a

radius of impermeable core of an NP (m)

CCD

counterion condensation - Donnan

PB

Poisson-Boltzmann

d

thickness of the soft shell of a core-shell NP (m)

f B eq 

Boltzmann partitioning coefficient corresponding to the net NP charge density at equilibrium

f B s

Boltzmann partitioning coefficient corresponding to the structural NP charge density

F

Faraday number (C mol-1)

I

solution ionic strength

C

average intraparticulate distance between structural charges (m)

M

metal ion

NP

nanoparticle

Q

amount of counterion charges per particle (C)

r

distance from particle center (m)

rp

particle radius (m)

R

outer limit of the counterion accumulation region from the particle center, i.e. thickness of this accumulation shell layer (m)

Rg

gas constant (J K-1 mol-1)

T

temperature (K)

Vp

particle volume (m3)

x

ratio of concentrations of mono- and divalent cations in the bulk medium, cX+ / cX2+




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y (eq ) ( r )

dimensionless potential profile corresponding to the net charge density at equilibrium

y (s ) ( r )

dimensionless potential profile corresponding to the structural charge density

Greek

 1

screening length in the bulk electrolyte medium (m)

 p 1

intraparticulate screening length for a soft (and core-shell) NP

o(eq)

net NP charge density in an NP body at equilibrium (C m-3)

o(s)

volume charge density due to structural charges in an NP body (C m-3)

D

Donnan potential (V)

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: xxx Detailed demonstration of eq 1 (PDF).

ACKNOWLEDGEMENTS RMT conducted this work within the framework of the EnviroStress Center of Excellence at the University of Antwerp. The authors declare no competing financial interest.

REFERENCES (1)

van Leeuwen, H. P.; Buffle, J. Chemodynamics of aquatic metal complexes: from small ligands to colloids. Environ. Sci. Technol. 2009, 43, 7175-7183.

(2)

van Leeuwen, H. P.; Duval, J. F. L.; Pinheiro, J. P.; Blust, R.; Town, R. M. Chemodynamics and bioavailability of metal ion complexes with nanoparticles in aqueous media. Environ. Sci.: Nano 2017, 4, 2108-2133.


Page 23 of 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(3)

Madden, A. S.; Hochella, M. F.; Luxton, T. P. Insights for size-dependent reactivity of hematite nanomineral surfaces through Cu2+ sorption. Geochim. Cosmochim. Acta 2006, 70, 4095-4104.

(4)

Li, Z.; van Dyk, A. K.; Fitzwater, S. J.; Fichthorn, K. A.; Milner, S. T. Atomistic molecular dynamics simulations of charged latex particle surfaces in aqueous solution. Langmuir 2016, 32, 428-441.

(5)

Garcia-Fernandez, E.; Paulo, P. M. R.; Costa, S. M. B. Evaluation of electrostatic binding of PAMAM dendrimers and charged phthalocyanines by fluorescence correlation spectroscopy. Phys. Chem. Chem. Phys. 2015, 17, 4319-4327.

(6)

Barisik, M.; Atalay, S.; Beskok, A.; Qian, S. Size dependent surface charge properties of silica nanoparticles. J. Phys. Chem. C 2014, 118, 1836-1842.

(7)

Polyakov, P. D.; Duval, J. F. L. Speciation dynamics of metals in dispersion of nanoparticles with discrete distribution of charged binding sites. Phys. Chem. Chem. Phys. 2014, 16, 19992010.

(8)

del Pino, P.; Yang, F.; Pelaz, B.; Zhang, Q.; Kantner, K.; Hartmann, R.; Martinez de Baroja, N.; Gallego, M.; Möller, M.; Manshian, B. B. et al. Basic physicochemical properties of polyethylene glycol coated gold nanoparticles that determine their interaction with cells. Angew. Chem. Int. Ed. 2016, 55, 5483-5487.

(9)

Xu, C. -Y.; Deng, K. -Y.; Li, J. -Y.; Xu, R. -K. Impact of environmental conditions on aggregation kinetics of hematite and goethite nanoparticles. J. Nanopart. Res. 2015, 17:394.

(10) Graf, C.; Gao, Q.; Schütz, I.; Noufele, C. N.; Ruan, W.; Posselt, U.; Korotianskiy, E.; Nordmeyer, D.; Rancan, R.; Hadam, S. et al. Surface functionalization of silica nanoparticles supports colloidal stability in physiological media and facilitates internalization in cells. Langmuir 2012, 28, 7598-7613. (11) Huang, R.; Carney, R. P.; Stellacci, F.; Lau, B. L. T. Colloidal stability of self-assembled monolayer-coated gold nanoparticles: the effects of surface compositional and structural heterogeneity. Langmuir 2013, 29, 11560-11566.




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(12) Praetorius, A.; Labille, J.; Scheringer, M.; Thill, A.; Hungerbühler, K.; Botter, J. -Y.

Heteroaggregation of titanium dioxide nanoparticles with model natural colloids under environmentally relevant conditions. Environ. Sci. Technol. 2014, 48, 10690-10698. (13) Wang, H.; Adeleye, A. S.; Huang, Y.; Li, F.; Keller, A. A. Heteroaggregation of nanoparticles with biocolloids and geocolloids. Adv. Colloid Interf. Sci. 2015, 226, 24-36. (14) Jalil, A. H.; Pyell, U. Quantification of zeta-potential and electrokinetic surface charge density for colloidal silica nanoparticles dependent on type and concentration of the counterion: probing the outer Helmholtz plane. J. Phys. Chem. C 2018, 122, 4437-4453. (15) Mei, L.; Chou, T. -H.; Cheng, Y. -S.; Huang, M. -J.; Yeh, L. -H.; Qian, S. Electrophoresis of pH-regulated nanoparticles: impact of the Stern layer. Phys. Chem. Chem. Phys. 2016, 18, 9927-9934. (16) Moussa, M.; Caillet, C.; Town, R. M.; Duval, J. F. L. Remarkable electrokintic features of charge-stratified soft nanoparticles: mobility reversal in monovalent aqueous electrolyte. Langmuir 2015, 31, 5656-5666. (17) Duval, J. F. L.; Wilkinson, K. J.; van Leeuwen, H. P.; Buffle, J. Humic substances are soft and permeable: evidence from their electrophoretic mobilities. Environ. Sci. Technol. 2005, 39, 6435-6445. (18) Ohshima, H. Electrokinetics of soft particles. Colloid Polym. Sci. 2007, 285, 1411-1421. (19) Ohshima, H. Electrophoresis of soft particles. Adv. Coll. Interf. Sci. 1995, 62, 189-235. (20) Manning, G. S. The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides. Q. Rev. Biophys. 1978, 2, 179-246. (21) Ohshima, H.; Kondo, T. Relationship among the surface potential, Donnan potential and charge density of ion-penetrable membranes. Biophys. Chem. 1990, 38, 117-122. (22) Ohshima, H. Donnan potential and surface potential of a spherical soft particle in an electrolyte solution. J. Colloid Interf. Sci. 2008, 323, 92-97.


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(23) Town, R. M.; van Leeuwen, H. P.; Buffle, J. Chemodynamics of soft nanoparticulate

complexes: Cu(II) and Ni(II) complexes with fulvic acids and aquatic humic acids. Environ. Sci. Technol. 2012, 46, 10487-10498. (24) Wonders, J. H. A. M.; van Leeuwen, H. P. Voltammetric metal titration of particle dispersions. J. Electroanal. Chem. 1996, 401, 103-112. (25) van Leeuwen, H. P.; Buffle, J.; Duval, J. F. L; Town, R. M. Understanding the extraordinary ionic reactivity of aqueous nanoparticles. Langmuir, 2013, 29, 10297-10302. (26) Town, R. M.; Buffle, J.; Duval, J. F. L.; van Leeuwen, H. P. Chemodynamics of soft charged nanoparticles in aquatic media: fundamental concepts. J. Phys. Chem. A 2013, 117, 7643-7654. (27) Town, R. M.; van Leeuwen, H. P. Intraparticulate speciation analysis of soft nanoparticulate metal complexes. The impact of electric condensation on the binding of Cd2+/Pb2+/Cu2+ by humic acids. Phys. Chem. Chem. Phys. 2016, 18, 10049-10058 (28) Town, R. M.; van Leeuwen, H. P. Metal ion – humic acid nanoparticle interactions: role of both complexation and condensation mechanisms. Phys. Chem. Chem. Phys. 2016, 18, 1802418032. (29) van Leeuwen, H. P.; Town, R. M.; Buffle, J.; Cleven, R. F. M. J.; Davison, W.; J. Puy, J.; van Riemsdijk, W. H.; Sigg, L. Dynamic speciation analysis and bioavailability of metals in aquatic systems. Environ. Sci. Technol. 2005, 39, 8545-8556. (30) Jansen, S.; Blust, R.; van Leeuwen, H. P. Metal speciation dynamics and bioavailability: Zn(II) and Cd(II) uptake by mussel (Mytilus edulis) and carp (Cyprinus carpio). Environ. Sci. Technol. 2002, 36, 2164-2170 (31) Duval, J. F. L. Dynamics of metal uptake by charged biointerphases: bioavailability and bulk depletion. Phys. Chem. Chem. Phys. 2013, 15, 7873-7888. (32) Duval, J. F. L.; Présent, R. M.; Rotureau, E. Kinetic and thermodynamic determinants of trace metal partitioning at biointerphases: the role of intracellular speciation dynamics. Phys. Chem. Chem. Phys. 2016, 18, 30415-30435.




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(33) Duval, J. F. L. Coupled metal partitioning dynamics and toxicodynamics at biointerfaces: a

theory beyond the biotic ligand model framework. Phys. Chem. Chem. Phys. 2016, 18, 94539469. (34) Tipping, E. Humic ion-binding model VI: an improved description of the interactions of protons and metal ions with humic substances. Aquat. Geochem. 1998, 4, 3-48. (35) Koopal, L. K.; Saito, T.; Pinheiro, J. P.; van Riemsdijk, W. H. Ion binding to natural organic matter: general considerations and the NICA-Donnan model. Colloids Surf. A 2005, 265, 40-54 (36) Duval, J. F. L.; Ohshima, H. Electrophoresis of diffuse soft particles. Langmuir 2006, 22, 35333546. (37) Duval, J. F. L.; Gaboriaud, F. Progress in electrohydrodynamics of soft microbial particle interphases. Curr. Opin. Coll. Interf. Sci. 2010, 15, 184-195. (38) Ascher, U.; Christiansen, J.; Russel, R. D. Collocation software for boundary-value ODEs. ACM Trans. Math. Software 1981, 7, 209-222. (39) Zimmermann, R.; Kuckling, D.; Kaufmann, M.; Werner, C.; Duval, J.F.L. Electrokinetics of a poly(N-isopropylacrylamid-co-carboxyacrylamid) soft thin film: evidence of diffuse segment distribution in the swollen state. Langmuir 2010, 26, 18169-18181. (40) Manning, G. S. Comments on “A comparison of Manning’s polyelectrolyte theory with the cylindrical Gouy Model” by D. Stigter. J. Phys. Chem. 1978, 82, 2349-2351. (41) Manning, G. S. Limiting laws and counterion condensation in polyelectrolyte solutions. IV. The approach to the limit and the extraordinary stability of the charge fraction. Biophys. Chem. 1977, 7, 95-102.

(42) Walker, D. A.; Kowalczyk, B.; de la Cruz, M. O.; Grzybowski, B. A. Electrostatics at nanoscale. Nanoscale 2011, 3, 1316-1344. (43) Guerrero-Garcia, G. I.; González-Tovar, E.; de la Cruz, M. O. Effects of the ionic sizeasymmetry around a charged nanoparticle: unequal charge neutralization and electrostatic screening. Soft Matt. 2010, 6, 2056-2065.


Page 27 of 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(44) Lead, J. R.; Wilkinson, K. J.; Balnois, E.; Cutak, B. J.; Larive, C. K.; Assemi, S.; Beckett, R.

Diffusion coefficients and polydispersities of the Suwannee river fulvic acid: comparision of fluorescence correlation spectroscopy, pulsed-field gradient nuclear magnetic resonance, and flow field-flow fractionation. Environ. Sci. Technol. 2000, 34, 3508-3513. (45) Duval, J. F. L.; van Leeuwen, H. P. Electrokinetics of diffuse soft interfaces. 1. Limit of low Donnan potentials. Langmuir 2004, 20, 10324-10336. (46) Duval, J. F. L. Electrokinetics of diffuse soft interfaces. 2. Analysis based on the nonlinear Poisson-Boltzmann equation. Langmuir 2005, 21, 3247-3258. (47)

Buffle, J.; Tercier-Waeber, M. -L. In situ voltammetry: concepts and practice for trace analysis and speciation. In: J. Buffle and G. Horvai (eds), In Situ Monitoring of Aquatic Systems: Chemical Analysis and Speciation, John Wiley & Sons: Chichester, 2000, pp. 279-405.

(48) Davison, W.; Zhang, H. Progress in understanding the use of diffusive gradients in thin films (DGT) – back to basics. Environ. Chem. 2012, 9, 1-13. (49) Galceran, J.; Puy, J. Interpretation of diffusion gradients in thin films (DGT) measurements: a systematic approach. Environ. Chem. 2015, 12, 112-122. (50) Town, R. M.; Duval, J. F. L.; van Leeuwen, H. P. The intrinsic stability of metal ion complexes with nanoparticulate fulvic acids. Environ. Sci. Technol. Submitted.




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TOC Graphic Poisson-Boltzmann Electrostatics and Ionic Partition Equilibration of Charged Nanoparticles in Aqueous Media Jérôme F. L. Duval,1,2* Raewyn M. Town,3,4 Herman P. van Leeuwen4 1

CNRS, Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), UMR 7360 CNRS, 15 avenue du Charmois, 54500 Vandoeuvre-les-Nancy, France. 2 Université de Lorraine, LIEC, UMR 7360 CNRS, 15 avenue du Charmois, 54500 Vandoeuvre-lesNancy, France. 3 Systemic Physiological and Ecotoxicological Research (SPHERE), Department of Biology, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium. 4 Physical Chemistry and Soft Matter, Wageningen University & Research, Stippeneng 4, 6708 WE Wageningen, The Netherlands.


Poisson-Boltzmann Electrostatics and Ionic ... - ACS Publications

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