DOI:
10.1039/C7SM01825H
(Paper)
Soft Matter, 2018,
14, 331343
Charge regulation of nonpolar colloids†‡
Received
11th September 2017
, Accepted 10th November 2017
First published on 10th November 2017
Individual colloids often carry a charge as a result of the dissociation (or adsorption) of weaklyionized surface groups. The magnitude depends on the precise chemical environment surrounding a particle, which in a concentrated dispersion is a function of the colloid packing fraction η. Theoretical studies have suggested that the effective charge Z_{eff} in regulated systems could, in general, decrease with increasing η. We test this hypothesis for nonpolar dispersions by determining Z_{eff}(η) over a wide range of packing fractions (10^{−5} ≤ η ≤ 0.3) using a combination of smallangle Xray scattering and electrophoretic mobility measurements. All dispersions remain entirely in the fluid phase regime. We find a complex dependence of the particle charge as a function of the packing fraction, with Z_{eff} initially decreasing at low concentrations before finally increasing at high η. We attribute the nonmonotonic density dependence to a crossover from concentrationindependent screening at low η, to a high packing fraction regime in which counterions outnumber salt ions and electrostatic screening becomes ηdependent. The efficiency of charge stabilization at high concentrations may explain the unusually high stability of concentrated nanoparticle dispersions which has been reported.
I. Introduction
Virtually all colloids carry a charge when immersed in an electrolyte. The subsequent Coulombic interactions are crucial to a wide variety of processes including understanding the programmed selfassembly of nanoparticles,^{1} the phase stability of suspensions,^{2,3} and the hierarchical architecture of virus structures.^{4} A practical challenge is that the charge is not fixed a priori but typically free to adjust through a chemical equilibrium.^{5} Strongly acidic or basic groups tend to be fully dissociated, regardless of system parameters such as salt concentration or pH, while the dissociation of weak acid or base surface groups depends on the electrochemical potential.^{5} As a result, colloids carrying weak ionic groups are often referred to as charge regulated, in the sense that the effective surfacecharge density is not fixed but adapts to minimize the free energy of the system with ions migrating on and off surface sites. To date, the concepts of charge regulation (CR) have been applied almost exclusively to aqueous systems. Examples include analysis of the electrostatic doublelayer interactions between surfaces covered with protonated groups,^{5–8} explanation of the extremely longrange attractive forces that operate between proteins with dissociable amino acid groups close to their point of zero charge,^{9–11} or the role of charge regulation effects in determining not only the magnitude but the sign of the force between asymmetricallycharged particles.^{12,13} In contrast, little is known regarding the role of CR in nonpolar systems, where the degree of dissociation of surface ionic groups is small yet finite and complex charging processes operate.^{14} Indeed the strong longrange repulsive interactions arising from Coulombic charges in low dielectric solvents make nonpolar systems a fascinating testing ground for CR concepts.
The doublelayer interactions between chargeregulating particles are different to those between fully dissociated particles.^{15,16} For example, from Poisson–Boltzmann theory the repulsive disjoining pressure Π(h) between identicallycharged surfaces spaced by h, in the limit of small separations (h → 0), is fixed directly by the osmotic pressure of the ions which are trapped in the gap between the two surfaces by electroneutrality constraints.^{17} In a CR system the degree of dissociation of surface groups and hence the concentration of released counterions is controlled by the total electrostatic potential Φ(h), which is itself a function of the separation h between the charged surfaces. As a result, the effective surface charge density and Π differ substantially from the predictions of a constantcharge (CC) model as the two chargeregulated surfaces are brought together. In general, it has been found that CR significantly weakens the repulsive interactions between surfaces,^{5,6,18} as the increase in counterion osmotic pressure is relieved by a shift in the dissociation equilibria towards the uncharged state.
The effect of charge regulation on the repulsive interactions between particles in concentrated dispersions has been examined in a number of studies.^{15,16} To simplify the manybody electrostatic problem, a Poisson–Boltzmann (PB) cell approximation has been frequently employed in which the dispersion is divided up into identical spherical cells, each containing just one colloid in osmotic equilibrium with a salt reservoir of Debye length κ_{res}^{−1}. Cell calculations using different CR schemes^{15,16} show that, independent of the specific surface chemistry, the colloidal particle discharges monotonically with increasing packing fraction η – in the sense that the colloid charge Z_{eff} reduces with increasing η. This is in line with the asymptotic dependence observed for chargeregulated plates as h → 0. The charge reduction is predicted to be most severe in dispersions of small particles and solutions of low ionic strength where κ_{res}R → 0.
The experimental situation is however less clear, which may be accounted in part by the focus so far on systems with relative large κ_{res}R. Royall et al.,^{19} using charges estimated from radial distribution functions measured at κ_{res}R ≈ 1, have argued that the sequence of reentrant transitions observed upon increasing the colloid density in some charged nonpolar suspensions (fluid BCC fluid FCC), is a consequence of a steady reduction in Z_{eff} with increasing η. In recent work, Kanai et al.^{20} have explored the crystallization of large (R > 0.46 μm) colloids, charged by the addition of the surfactant AerosolOT (sodium bis(2ethyl1hexyl)sulfosuccinate or AOT) in a nonpolar solvent mixture. They observed close agreement between the reentrant phase boundaries measured and numerical calculations of the electrostatic charging effects produced by the reverse micelles. In an alternative approach, Vissers et al.^{21} measured the electrophoretic mobility of concentrated nonpolar dispersions at κ_{res}R ≈ 0.5. Using an approximate cell model to take into account doublelayer overlap, they showed that the particle charge reduced with increasing concentration.
If the strength of the electrostatic repulsions between chargeregulated particles decays rapidly with increasing colloid packing fraction, it is feasible that concentrated dispersions of charged nanoparticles could become colloidally unstable at very low ionic strengths, due to the absence of a strong enough osmotic repulsion to counter attractive van der Waals forces.^{22} Indeed, the notion that a high colloid surfacepotential (and consequently a large particle charge) at low colloid concentrations does not necessarily guarantee stability has been proposed for some time. Over fifty years ago,^{23,24} noted the absence of any correlation between the electrokinetic ζpotential of waterinbenzene emulsions and their stability against coalescence. Later work by Mishchuk, Sanfeld, and Steinchen^{25} suggested that a chargestabilized waterinoil emulsion should be unstable above a critical volume fraction, which reduced as the ion concentration fell. Indeed there have been repeated experimental reports^{26–29} of rapid coagulation at high colloid concentrations and low electrolyte concentrations in dispersions with high surface potentials, which would be sufficient to stabilize a more dilute system.
In this article, we present a study of the packing fraction dependence of the effective charge and the bulk correlations in nonpolar colloids in the weak screening regime (κ_{res}R ≪ 1). Experiments were carried out over an extended range of concentrations (10^{−5} ≤ η ≤ 0.3) using small nanoparticles in solutions of very low ionic strengths, so the dimensionless screening parameters κ_{res}R studied are considerably smaller than in any previous work. This has the advantage that the charge reduction due to regulation is evident at significantly lower packing fractions. By using two different methods to prepare charged nonpolar dispersions, we systematically varied the background ion concentration within the range 0.0 ≤ κ_{res}R ≤ 0.24. We determined the effective charge Z_{eff}(η) as a function of η by an analysis of (i) the particle structure factor obtained from Xray scattering measurements, and (ii) the density dependence of the electrophoretic mobility at η < 10^{−2}. At low packing fractions, we observe the charge reduction predicted by theory for a chargeregulated system at low κ_{res}R. However at high particle concentrations (η > 10^{−2}) we demonstrate a surprising density dependence of the effective charge, with Z_{eff} increasing with η. Extensive measurements reveal that this is a general feature of concentrated charged dispersions at low ionic strengths. Our dispersions always remain entirely in the fluid regime so the changes identified are not a consequence of phase changes, such as crystallization. Our paper is organized as follows: in Section II existing CR models are summarized and we show that, although details differ, all models predict a monotonic reduction in the effective charge with increasing packing fraction. Details of the nanoparticle systems and the analysis methods used in our experiments are detailed in Section III. In Section IV(A) we confirm that, without added charge, our colloids display pure hardsphere interactions. The electrophoretic mobility of dilute charged dispersions is analysed in Section IV(B) and shown to be consistent with a simple CR model. The charge at high packing fractions is extracted from measured structure factors in Section IV(C). Interestingly, we find that the measured charge increases, rather than decreasing with particle concentration as would be expected naively. This observation suggests that charge stabilization is much more effective in highly concentrated suspensions than is generally believed. We discuss the origins of the surprising increase in the particle charge with concentration in Section IV(D), before concluding in Section V.
II. Charge regulation models
We consider dispersions of N charged nanoparticles of radius R and charge Ze suspended in a solvent of volume V, and relative permittivity ε_{r} at a temperature T. The colloid number density is ρ_{C} = N/V and the corresponding packing fraction is η = (4π/3)ρ_{C}R^{3}. The doublelayer repulsions between particles are controlled by two length scales: the solventspecific Bjerrum length _{B} = βe^{2}/(4πε_{0}ε_{r}) where β = 1/k_{B}T, k_{B} is the Boltzmann constant, e is the fundamental charge, and ε_{0} is the permittivity of a vacuum, and the (reservoir) Debye screening length , with 2ρ_{s} the number density of univalent salt ions in the reservoir. In dodecane at T = 293 K, the Bjerrum length is 28.3 nm.
Although at high packing fractions or for thick double layers manybody interactions between charged nanoparticles can become significant,^{30,31}ab initio computer simulations^{32,33} show that the colloidal structure of a concentrated dispersion can often be approximated remarkably well by representing the effective pair potential U_{eff} by the hardsphere Yukawa (HSY) function,

 (1) 
where
x =
r/
R is the dimensionless pair separation,
r is the centretocentre distance, and the range and strength of the interparticle repulsions are evaluated from the effective charge
Z_{eff} and the screening length
κ_{eff}^{−1}, according to
and

 (3) 
While the HSY potential has been widely used, there is no rigorous route to construct such a potential at a finite nanoparticle density. In the low density limit
η → 0,
eqn (1) does reduce, under Debye–Huckel conditions (
Z_{eff}
_{B}/
R < 1), to the classical expressions derived by Derjaguin and Landau
^{2} and Verwey and Overbeek
^{3} with
Z_{eff} =
Z and
κ_{eff} =
κ_{res}. In general, however the effective parameters
Z_{eff} and
κ_{eff} will be density dependent. Denton, for instance, has derived
^{34} an expression similar to
eqn (1) from a rigorous statistical mechanical treatment but with a modified screening parameter that incorporates both the counterions released from the particle and corrects for excluded volume effects. For a colloid of charge
Z_{eff}, and added 1
:
1 electrolyte of density
ρ_{s} the screening parameter is obtained as

 (4) 
where
ρ_{C} is the colloid number density. In this work, we follow this approach and fix the effective screening parameter
κ_{eff} using
eqn (4).
To illustrate the consequences of charge regulation, we determine the effective charge Z_{eff} using a PBcell model of the dispersion, in which each particle is placed at the centre of a Wigner–Seitz spherical cell of radius R_{WS} = Rη^{−1/3}. The electrostatic potential ϕ(r) is then a sphericallysymmetric function of the radial variable r, measured from the centre of cell. To facilitate a direct comparison between theoretical predictions and experiment we compute the scaled electrostatic potential Φ = βeϕ and the reduced effective charge Z_{eff}_{B}/R as a function of η by solving numerically the nonlinear Poisson–Boltzmann equation on the interval 1 ≤ x ≤ η^{−1/3},

2Φ′(x) + xΦ′′(x) = (κ_{res}R)^{2}xsinhΦ(x)  (5) 
where
x =
r/
R.
Eqn (5) is solved subject to the appropriate charge regulation boundary conditions
^{15,16} together with the constraint,
Φ′(
η^{−1/3}) = 0, which follows from the overall charge neutrality of the Wigner–Seitz cell. The effective charge
Z_{eff} is obtained from the numericallyobtained electrostatic potential by matching
Φ(
x) at the edge of the cell to a solution of the linearized PB equation.
^{35}
The simplest theoretical model of a charged colloid is as a spherical insulator with a constant charge (CC model). The CC model implies no exchange between surface binding sites and free ions in solution so that the bare charge remains frozen, independent of the electrolyte concentration and the particle packing fraction η. Recently, there have been a number of attempts^{15,16,36} to develop more realistic statistical models of charge regulation in a low dielectric environment. Fig. 1 shows numerical predictions for the dimensionless charge Z_{eff}_{B}/R as a function of the packing fraction η, for a number of different surface chemistries. Crucially, we see that while the details of each model differ, they share a common qualitative behavior in that the particles are predicted to discharge continuously with increased colloid density.

 Fig. 1 Packing fraction dependence of the reduced charge Z_{eff}_{B}/R predicted by charge regulation models for κ_{res}R = 0.25. (A) Constant potential (CP), and (B) single site association/dissociation model (CR_{1}). Model parameters are chosen such that, in both cases, the reduced charge is fixed in the dilute limit.  
Everts, Boon, and van Roij,^{15} for instance, have proposed that the net colloid charge is determined by a balance between two competing surface ionization reactions,
S_{1} + P^{+} ⇄ S_{1}P^{+}, 

S_{2} + N^{−} ⇄ S_{2}N^{−},  (6) 
where S
_{1,2} denote different sites on the colloidal surface which bind either positive (P
^{+}) or negative ions (N
^{−}). We label the generic twosite charge regulation scheme, outlined in
eqn (6), as an example of a CR
_{2} model. Numerical solution of this model reveals that the charge
Z_{eff} carried by the colloidal particle reduces monotonically with increasing colloid density, in contrast to the CC model. Effectively, the particle discharges continuously as a function of
η, with the charge asymptotically tending to
Z_{eff} ≈ 0 as
η → 1.
In the limit, where there is a significant adsorption of both positive and negative ions onto the particle, the CR_{2} boundary condition reduces to the simpler constant potential (CP) model. The ionization of the positive and negative surface groups adjusts so that as particles approach each other counterions migrate back onto surface sites to maintain a fixed surface potential, and the surface charge density decreases monotonically with increasing colloid density. Theoretical arguments for the validity of a CP model in nonpolar dispersions have been made by a number of authors.^{15,16,36,37} Roberts et al.^{36} have, for instance, analysed a model of charge regulation in which charged micelles adsorb onto the surface of a colloid and shown that it is equivalent to assuming constant potential boundary conditions provided that (a) both positive and negative micelles are able to adsorb and (b) the surface coverage of micelles is below the saturation limit.
Finally, Smallenburg et al.^{16} has compared numerical predictions for Z_{eff} as a function of η from a singlesite association/dissociation equilibrium (CR_{1} model)
with CP calculations and found that although the predicted surface charges are indistinguishable at low densities, there are significant variations at higher
η. These observations are consistent with more general predictions
^{18} that the repulsions between surfaces with both positive and negative sites more closely resemble the constant potential limit than a surface with only a single ionizable site. The CR
_{1} scheme however still predicts a qualitatively similar charge dependence to the CR
_{2} model, in that
Z_{eff} decreases monotonically from a finite low
η value to essentially 0 at
η ≈ 1.
III. Materials and methods
A. Colloids
The systems studied consisted of three batches of nanoparticles (NP1–NP3), approximately 50 nm in radii, dispersed in dry dodecane (dielectric constant ε_{r} = 2.01) at a packing fraction η. Each system consisted of a core of poly(methyl methacrylate) [PMMA] surrounded by a chemicallygrafted shell of poly(12hydroxystearic acid) [PHSA]. The particles were prepared inhouse^{38,39} by a freeradical dispersion polymerization of methyl methacrylate (MMA) and methacrylic acid (MAA) in a mass ratio of 98:2 using 2,2′azobis(2methylpropionitrile) [AIBN] as initiator and a preformed graft copolymer poly(12hydroxystearic acid)gpoly(methyl methacrylate) as dispersant. The synthesis was carried out at 80 °C in a mixed solvent of dodecane and hexane (2:1 by wt) for 2 h, before the temperature was raised to 120 °C for a further 12 h, in the presence of a catalyst, to covalently lock the stabilizer to the surface of the particle. All particles were undyed. Colloids were purified by repeated cycles of centrifugation and redispersion in freshlydried dodecane to remove excess electrolyte and stabilizer. Once cleaned, the average hydrodynamic radius R_{h} was determined by dynamic light scattering (DLS) using a Malvern Zetasizer nano S90 (Malvern instruments, UK). The sizes are given in Table 1. Nanoparticles were stored under nitrogen to prevent water uptake and ion generation. The conductivity of the purified dispersions was checked prior to use with a Scientifica (UK) model 627 conductivity meter.
Table 1 Nanoparticles studied^{a}
Nanoparticle 
R (nm) 
R
_{h} (nm) 
R
_{c} (nm) 
s

R is the effective hardsphere radius, R_{h} the hydrodynamic radius, R_{c} the core radius, and s the normalised polydispersity.
SAXS data not recorded, so assume R_{c} = R_{h} − δ with δ = 10 nm.

NP1 
43.3 ± 1.3 
49.0 ± 2.9 
38.3 ± 0.3 
0.10 
NP2 
58.0 ± 3.0 
63.0 ± 3.0 
53.0^{b} 
0.10 
NP3 
33.7 ± 1.2 
36.0 ± 2.5 
28.7 ± 0.2 
0.13 
Table 2 Constantpotential charging parameters: obtained from cellmodel fits to experimental μ_{red}(η)^{a}
Experimental system 
Fitted parameters 
Nanoparticle 
C
_{AOT} [mmol dm^{−3}] 
κ
_{res}
R

Φ
_{s} = eϕ_{s}/k_{B}T 
κR

Φ
_{s} is the scaled surface potential and κR is the effective screening parameter, obtained from a nonlinear fit to eqn (8).

NP1 
5.0 
0.033 
−2.0 ± 0.1 
0.060 
NP1 
25 
0.078 
−2.1 ± 0.1 
0.088 
NP1 
50 
0.11 
−2.4 ± 0.1 
0.090 
NP1 
250 
0.24 
−2.6 ± 0.1 
0.54 

NP2 
5.0 
0.049 
−4.0 ± 0.1 
0.076 
NP3 
0.0 
0.0 
+2.6 ± 0.1 
0.037 
We employed two routes to generate charge: batches NP1 and NP2 were differentlysized PMMA particles, with the same surface chemistry, which were both charged negative^{36,37} by the addition of the surfactant AOT while nanoparticles NP3 had a different surface chemistry, and were charged positive by the dissociation of lipophilic ionic groups^{39,40} introduced into the particle during synthesis. AOT (98%, Aldrich, UK) was purified by dissolution in dry methanol and centrifuged prior to use to remove residual salts. It was used at molar concentrations C_{AOT} above the critical micellar concentration^{41}C_{CMC} ≈ 0.13 mmol dm^{−3}, where although the majority of the reverse micelles are neutral conductivity measurements^{36,42} show that a small fraction of the micelles (≈1 in 10^{5}) are ionized by thermal fluctuations. Measurement of the conductivity and viscosity of AOT in dodecane confirmed that the total molar concentration of reverse micellar ions C_{ion} = C_{+} + C_{−} increased linearly^{36} with C_{AOT}, C_{ion} = χC_{AOT} with χ = 5.5 × 10^{−7}. A range of AOT concentrations, from C_{AOT} = 5–225 mmol dm^{−3} was employed, which correspond to total micellar ion concentrations of 2.8–125 nmol dm^{−3}. Suspensions were prepared by dilution of the same concentrated particle stock for all AOT concentrations to ensure accurate relative concentrations. The dominant ionic species are singlycharged positive and negative AOT micelles. The colloid charge is not fixed but regulated by a competitive adsorption of cationic and anionic micellar ions onto the particle surface.^{36} Batch NP3, in contrast, contains noadded salt and is a counteriononly system. Particles were charged by copolymerization into the core of the particle of approximately 4 wt% of the ionic monomer ntridodecylpropyl3methacryloyloxy ammonium tetrakis [3,5bis(trifluoromethyl)phenyl]borate ([ILM–(C_{12})]^{+}[TFPB]^{−}). The molecular structure of the polymeric NP3 nanoparticles is described, in greater detail, in the supplementary information. The ionic monomer and nanoparticles were prepared following the procedures outlined in previous work.^{39,40} Ionic dissociation of surfacebound [ILM–(C_{12})]^{+}[TFPB]^{−} groups generated a positive colloid charge of +Z together with Z negative [TFPB]^{−} counterions in solution.^{40}
B. Electrophoretic mobility
The electrophoretic mobility μ as a function of packing fraction η was measured at 25 °C using phaseanalysis light scattering (Zetasizer Nano, Malvern, UK). The mobility μ = v/E, where v is the electrophoretic velocity induced by an applied electric field of strength E, was determined from the modulation in the phase of scattered light produced by a periodic triangular Efield. Equilibrated samples were transferred into clean 10 mm square quartzglass cuvettes and a nonaqueous dip cell (PCS1115, Malvern) with a 2 mm electrode gap placed in the cell. In a typical measurement, a series of runs were performed at different driving voltages between 10 V and 50 V and the scattering from a 633 nm laser was collected at a scattering angle of 173°. No systematic dependence of μ on E was observed. Any measurement where the phase plot significantly deviated from the expected triangular form was discarded, and the measurement repeated. Since PMMA nanoparticles in dodecane are weakly scatterers of light, reliable electrophoretic mobilities were recorded over a relatively wide range of packing fractions, 10^{−5} ≤ η ≤ 10^{−2}.
C. Modelling of electrophoretic mobility
The electrophoretic mobility μ of a colloidal particle is determined by a balance between electrostatic and hydrodynamic forces. In the limits of low concentration and κ_{res}R → 0 the reduced mobility, defined by the expression μ_{red} = 6πν_{B}μ/e (where ν is the solvent viscosity), is equal to the scaled particle charge Z_{eff}_{B}/R. At finite concentrations however, the mobility decreases approximately logarithmically^{43} with increasing η, as frictional forces grow because of strengthening particle–particle interactions. To model the effects of interactions, we follow the theoretical analysis of Levine and Neale^{44} who proposed a cell model for the mobility in a concentrated dispersion, valid in the linear PB limit. Ohshima^{45} derived an equivalent expression, 
 (8) 
where Φ_{D} = Φ(η^{−1/3}) is the potential at the cell boundary and, the integrand H(x) is a function of the scaled potential within the cell, 
 (9) 
with x = r/R.
D. Smallangle Xray scattering (SAXS)
SAXS measurements were performed at a temperature of 20 °C on the Diamond Light Source (Didcot, UK) using the I22 beamline at a wavelength of λ_{0} = 0.124 nm and a sample to detector distance of 10 m leading to a useful qrange of approximately 0.015–0.7 nm^{−1}. The scattering wave vector q is defined as 
 (10) 
where θ is the scattering angle and λ_{0} is the incident Xray wavelength. Dispersions were loaded into reusable flowthrough quartz capillary cells that were filled alternately with samples of the background solvent and the dispersion to allow accurate subtractions of the background. At least ten 2D images of 10 s each were collected, azimuthally averaged, transmission and background corrected according to established procedures to yield the scattered intensity I(q), as a function of q. An additional series of higher resolution measurements were made at the ESRF (Grenoble, France) on the ID02 beamline with a qrange of approximately 0.01–0.8 nm^{−1}.
The intensity scattered by a dispersion of spherical particles with a narrow size distribution f(R) can be factored as

I(q) = AηP_{M}(q)S_{M}(q),  (11) 
where
A is an instrumental factor,
η is the packing fraction,
is the polydisperse particle form factor, and
S_{M}(
q) is the measured structure factor.
^{46} This was determined experimentally from the intensity ratio,

 (12) 
where the dilute scattered intensity
I_{dil}(
q) =
Aη_{dil}P_{M}(
q) was measured at a sufficiently low packing fraction (
η_{dil} ≈ 10
^{−3}) to ensure that all interparticle interactions were suppressed and
S_{M}(
q) = 1.
E. Charge screening
The effective screening length in the micellecontaining systems can not be calculated directly from eqn (4) because the micellar ions are in equilibrium with neutral micelles, through an autoionization reaction of the form M^{+} + M^{−} ⇌ 2 M. The mixture of charged and neutral micelles acts effectively as a charge buffer. To model the buffering process, we characterize the selfionization equilibrium constant as K = ρ_{+}ρ_{−}, with ρ_{±} the number density of the ±micelles. If the colloid has a surface charge of −Z_{eff}, then, from charge neutrality, the number density of positive ions in solution is ρ_{+} = ρ_{−} + ρ_{C}Z_{eff}. Adopting the particle radius R as a natural length scale and introducing the scaled ion densities _{±} = ρ_{±}R^{3} then we may express the scaled positive ion density simply as _{+} = _{−} + 2Δ, where Δ = 3ηZ_{eff}/(8π).
Substituting this expression into the law of mass action gives the equilibrium ion concentrations, in the presence of charged nanoparticles, as

 (13) 
The total ion concentration is therefore
, which is less than the value of 2(
Δ +
KR^{3}) obtained by naively adding the salt and counterion densities together, demonstrating the charge buffering effect. From
eqn (13) the corresponding screening parameter is

 (14) 
which in the limit of no background ions (
K = 0) reduces to the counteriononly limit

 (15) 
IV. Results and discussion
A. Nanoparticles
We used nanoparticles with a core of poly(methyl methacrylate) [PMMA] of radius R_{c} sterically stabilized by a chemicallygrafted shell of poly12hydroxystearic acid [PHSA] of thickness δ suspended in dodecane. To determine the core radius R_{c}, the excess (nanoparticle dispersion minus solvent) SAXS scattering profiles I(q) were measured from dilute dispersions (η ≈ 10^{−3}). The Xray scattering length density of the core was calculated to be 10.8 × 10^{−6} Å^{−2} and the stabiliser layer as (8 ± 1) × 10^{−6} Å^{−2}, where the uncertainty is due to the variation in mass density reported in the literature. The calculated Xray scattering length density of dodecane is 7.34 × 10^{−6} Å^{−2}, so the shell contrast is weak and the scattering arises predominately from the PMMA core. To model the dilute particle data, we used a polydisperse core–shell model with a Schulz size distribution^{47} adjusting the (mean) core radius R_{c} and the polydispersity s to best describe the measured I(q). The shell thickness was fixed at δ = 10 nm on the basis of previous measurements.^{47,48} Agreement between the model calculations and the lowη SAXS data is very good, with the fitted values of R_{c} and s listed in Table 1.
A thick polymeric shell is a highly efficient way to stabilize a nanoparticle but interpenetration of polymers in the shell can, particularly at high concentrations, result in a softness in the mutual interactions between grafted particles. To test if the core–shell structure of the synthesised nanoparticles was altered in concentrated dispersions we conducted a series of SAXS measurements, using the procedures outlined in Section III(D), to determine the structure factor of uncharged particles as a function of packing fraction. The experimental results for S_{M}(q) are depicted by the symbols shown in Fig. 2(a). The solid lines show fits to the measured S_{M}(q) using a polydisperse hardsphere (HS) model, using as adjustable parameters the effective HS radius R, and the ratio η/η_{c}, where η_{c} is the experimentallydetermined core packing fraction. On the basis of the dilute form factor results, the size polydispersity was fixed at s = 0.10. The HS calculations can be seen to describe the experimental structure factors extremely well over a wide range of wavevectors and nanoparticle concentrations. The hardsphere character of the nanoparticles was confirmed further by comparing the lowq limit of the measured inverse colloid–colloid structure factor lim_{q→0}1/S_{M}(q) with the Carnahan and Starling prediction for the isothermal compressibility of a hardsphere fluid. The agreement evident in Fig. 2(b) is very good. Overall, we found an effective HS radius of R = (43.3 ± 1.3) nm, which is 5 nm smaller than the magnitude of the core–shell radius R_{c} + δ = 48.3 nm estimated from the form factor analysis. We attribute this discrepancy to interpenetration of interlocking polymer shells at high packing fractions. We therefore fix the effective HS radius for our systems as R = R_{c} + 5 nm. The resulting values for the effective HS radius (R), the hydrodynamic radius determined by dynamic light scattering (R_{h}), the core radius (R_{c}) from SAXS analysis, and the normalized polydispersity for the systems studied are collected together in Table 1.

 Fig. 2 (a) The evolution with packing fraction of the measured structure factors S_{M}(q) for uncharged dispersions in dodecane (particles NP1, no added AOT). The qaxis in the plot is logarithmic. Packing fractions are defined in the legend. The symbols denote the experimental data, while the lines are calculated polydisperse Percus–Yevick hard sphere structure factors. (b) Comparison between the lowq limit of the measured inverse structure factor (filled circles) and the reduced isothermal compressibility β/(ρ_{C}χ_{T}) calculated from the quasiexact Carnahan–Starling equation of state for hard spheres (dashed line).  
B. Charge regulation
To generate a particle charge we have used two approaches: (a) surface modification, and (b) adsorption of charged reverse micelles. Although the molecular mechanism of particle charging in low polarity solvents is not well understood, different hypotheses have been proposed which emphasise either ‘charge created’ on the particle by the dissociation of surface groups or ‘charge acquired’ by the adsorption of charged surfactant species. Our experiments use examples from both categories. System NP3 was charged by the addition of a lipophilic ionic comonomer to the dispersion synthesis (for details, see Section III(A)). Dissociation of an anion from the surface of the nanoparticle generated a positive particle charge. The charge equilibrium can be represented by the singlesite CR_{1} dissociation process,where S^{+} denotes a positivelycharged surfacebound group, and N^{−} a negative species. Nanoparticle NP1 and NP2 were, in contrast, charged negative by addition of the oilsoluble ionic surfactant AerosolOT at molar concentrations C_{AOT} above the critical micellar concentration so that spherical reverse micelles form in solution. It has been proposed^{14,36} that charge regulation in systems containing AOT is a multisite CR_{2} process with two independent association reactions,
S_{1} + M^{+} ⇄ S_{1}M^{+}, 

S_{2} + M^{−} ⇄ S_{2}M^{−},  (17) 
where M^{±} refer to charged reverse micelles and the balance between the two competing adsorption processes (and the net charge) depends on the hydrophilicity of the particle surface.
The effect of charge regulation on the surface charge was demonstrated by measurement of the reduced electrophoretic mobility μ_{red} as a function of packing fraction η, with the data shown in Fig. 3. For a comparison between different nanoparticle batches, we consider all mobilities in reduced units μ_{red} = 6πν_{B}μ/e where ν is the viscosity of the solvent, and _{B} is the solventspecific Bjerrum length. For isolated particles, the reduced mobility assumes the value μ^{0}_{red} = Z_{eff}_{B}/R = Φ_{s} in the Hückel limit (κ_{res}R → 0), where μ^{0}_{red} is the reduced mobility at infinite dilution and Φ_{s} = βeϕ_{s} is the dimensionless surface potential.^{43} However as dispersions become more concentrated the electrophoretic mobility drops. When the electrostatic interactions are strongly screened (κ_{res}R ≫ 1) the electrophoretic mobility shows only a relatively weak concentration dependence, since the electricfielddriven dynamics originates only from within a thin interfacial region at the particle's surface.^{49} The mobility drop by a factor of ≈10 as η is increased from 10^{−4} to 10^{−2} seen in Fig. 3 is therefore quite surprising. A similar dependence of μ_{red}(η) was seen for all samples studied, with the data plotted in Fig. 3(A) suggesting that concentrations of η ≈ 10^{−5} are still not sufficiently low enough to reach the infinitedilution limit μ^{0}_{red}.

 Fig. 3 The reduced electrophoretic mobility μ_{red} measured as a function of volume fraction in dilute dispersions of (A) positivelycharged NP3 particles (filled circles) and (B) negativelycharged NP2 particles (5 mM AOT, open squares). The solid lines are mobilities calculated using a Kuwabara cell model (eqn (8)) assuming a constant potential (CP) boundary condition at the surface of the particle. Fitted charge parameters are listed in Table 2. The insets show the density dependence of the reduced particle charge predicted from the CP fit. The dashed vertical lines indicate the predictions for the concentrations where the screening clouds of neighbouring particles start to overlap.  
Qualitatively, the sharp drop in the electrophoretic mobility μ_{red} occurs because of the strong mutual interactions between charged nanoparticles in the weak screening limit. The reduction in μ_{red} will be significant at concentrations η* where the electrical doublelayers of neighbouring particles first begin to overlap. If we approximate a charged nanoparticle and its ionic atmosphere as a new effective particle of radius R + κ_{res}^{−1} then mutual overlap will occur when the effective volume fraction η_{eff} = η[1 + (κ_{res}R)^{−1}]^{3} is of order unity, or equivalently η* = [1 + (κ_{res}R)^{−1}]^{−3}. In the NP3 batch, where κ_{res}R ≈ 0.04, concentration effects will be important at concentrations as low as η ≈ 10^{−4}. The vertical dotted lines in Fig. 3 depict the concentration η* where double layer overlap is significant and, as evident from the plot, these lines also pretty effectively delineate the regime where μ_{red} begins to decrease.
To quantify the dramatic change in the electrophoretic mobility we use a Kuwabara cell model, first proposed by Levine and Neale,^{44} to predict μ_{red} as a function of η. We work in a spherical Wigner–Seitz cell of radius R_{WS} = Rη^{−1/3} containing a single particle together with neutralizing coions and counterions. In the treatment detailed by Ohshima,^{45} which is accurate for low surface potentials and for all κ_{res}R values, the reduced electrophoretic mobility μ_{red} is a function of the equilibrium electric potential Φ(r) (and its derivative) inside the cell. At low packing fractions, simulations have shown^{15,16} that the particle charge predicted by either of the CR_{1} and CR_{2} models can be accurately mimicked by assuming a constant potential boundary condition. Fixing the packing fraction η, we solve the nonlinear Poisson–Boltzmann equation varying the surface potential Φ_{s} and the ionic strength of the reservoir and calculate μ_{red}(η) (for details see Section III(C)). The results of these calculations are plotted as the solid lines in Fig. 3. The calculations are seen to be in excellent agreement with the experimental data and confirm that our dispersions are indeed charge regulated. The consequences of regulation are revealed in the inset plots of Fig. 3, where the dependence of Z_{eff}_{B}/R on the volume fraction η is plotted. We found a similar level of agreement between measured and calculated electrophoretic mobilities as the screening parameter κR was changed. The comparison between the experimental and numerical mobilities μ_{red}(η) as κR was varied over almost an order of magnitude is presented in the ESI.‡
C. Charge at high concentrations
The low density mobility data of Fig. 3 suggest that the particles should continuously discharge with increasing particle concentration. Testing this prediction in concentrated dispersions is however tricky. Light scattering measurements of μ_{red} are limited to low concentrations by multiple scattering effects. At high η, less direct methods must be used. To this end, we have recorded the positional correlations between nanoparticles, which arise as a result of both charge and excluded volume interactions between particles, using smallangle Xray scattering (SAXS) techniques. As we demonstrate below, careful modelling of the measured structure factor S_{M}(q) yields a robust measure of the effective charge Z_{eff}.
The microstructure of chargeregulating dispersions has been investigated as a function of packing fraction. The symbols in Fig. 4(a)–(d) summarizes the measured structure factor S_{M}(q) for four different charged systems. With increasing η, interparticle interactions start to dominate and the dispersions become more highly structured. This results in the emergence of a broad nearest neighbour peak at q_{max} ∼ 0.07 nm^{−1} while at the same time, the structure factor at low q decreases, behaviour that is characteristic of a purely repulsive fluid system. The lack of sharp correlation peaks in the structure factors indicates a fluid state of nanoparticles. We see no evidence for particle crystallization. Indeed since the strength of Coulombic repulsion scales linearly with size (at the same scaled charge Z_{eff}_{B}/R, see eqn (3)), charge correlations are expected to be weak in a dispersion of nanoparticles with R ∼ _{B}. To confirm charge interactions are not strong enough to drive crystallization at the densities studied here we refer to the phase transition data obtained recently on much larger charged colloids.^{20} In the case where R ≫ _{B} the crystallization boundary was accurately modelled by the onecomponent plasma condition Γ ≥ 106,^{20} where Γ is defined as Z_{eff}^{2}_{B}/d, with d = ρ_{C}^{−1/3} the typical spacing between particles. In scaled units, the coupling constant is . Using values of Z_{eff}_{B}/R ≈ 2 and R/_{B} ≈ 2 we estimate Γ ≈ 3 for our nanoparticle system at η ≈ 0.3, confirming the conclusion from the S_{M}(q) data that all of the suspensions studied here are disordered fluids.

 Fig. 4 Evolution with packing fraction (η) of measured structure factors S_{M}(q) from four different charged PMMA systems in dodecane: (a) particles NP1 with 5 mM AOT (κ_{res}R = 0.033), (b) NP1 with 25 mM AOT (κ_{res}R = 0.078), (c) NP1 with 225 mM AOT (κ_{res}R = 0.22), and (d) NP3 (κ_{res}R = 0.0). Packing fractions are defined in the legends. The experimental data is denoted by symbols, while the solid lines represent the best fits to S_{M}(q), calculated from the MPBRMSA approximation assuming a hardsphere Yukawa fluid. The main peak of the structure factor at q_{max} ≈ 0.07 nm^{−1} corresponds to particle separations of 2π/q_{max} ≈ 90 nm.  
To begin our analysis of the scattering data, we determine the concentration dependence of the inverse isothermal osmotic compressibility . Here Π is the osmotic pressure of the suspension measured at a colloid number density of ρ_{C} and the derivative is evaluated at a constant chemical potential of salt.^{33} From the Kirkwood–Buff relation^{50} the infinitewavelength limit of the colloid–colloid structure factor, S(0) = lim_{q→0}S(q), is related to χ_{T} by the identity,

 (18) 
which is exact for a monodisperse suspension. Experimentally, we determined the low
q limit by extrapolating a linear plot of
S_{M}(
q)
versus q^{2} to
q = 0. The resulting estimates of the reduced inverse osmotic compressibility
β/(
ρ_{C}χ_{T}) = 1/
S_{M}(0) as a function of the packing fraction
η for dispersion NP1 with AOT concentrations of
C_{AOT} = 5, 25 and 50 mmol dm
^{−3} are plotted in
Fig. 5. At each volume fraction and AOT concentration, the effective particle charge
Z_{eff}_{B}/
R was determined using a mean spherical approximation (MSA) expression for the structure factor of a HSY fluid, recently derived by VazquezRodriguez and RuizEstrada.
^{51} The only free parameter in the computation is
Z_{eff}, since the screening length
κ_{eff}^{−1} is fixed by
Z_{eff} (
eqn (14)) and the selfionization equilibrium constant
K is known from previous work.
^{36}

 Fig. 5 Lowq limit of the inverse structure factor, lim_{q→0}1/S_{M}(q), for charged dispersions as a function of colloid packing fraction η. Squares (black) 5 mM AOT, up triangles (red) 25 mM AOT, down triangles (blue) 50 mM AOT. Solid line indicate reduced isothermal compressibility β/(ρ_{C}χ_{T}) calculated from Carnahan–Starling equation of state for hard spheres. Inset scaled effective charges Z_{eff}_{B}/R calculated from experimental lowq data using MSA (see text for details).  
The inset plot in Fig. 5 shows the particle charges computed from the longwavelength limit of the structure factor. Surprisingly, we see substantial disagreements at intermediate packing fractions from the CP predictions, as is evident from a quick comparison between Fig. 1 and 5. The effective charge does not decrease smoothly with increasing η but instead displays a minimum at η ≈ 0.06 before finally increasing in magnitude with particle density.
To confirm the validity of the trends identified above, we have determined the particle charge Z_{eff}(η) using a second independent technique. The fullq dependence of the measured structure factor S_{M}(q) was compared to a S(q) calculated using the quasiexact MPBRMSA integral equation scheme,^{52} assuming a HSY effective pair potential. The effect of size polydispersity was neglected since polydispersity indices are small (see Table 1) and S_{M}(q) contains no sharp peaks. In all, two adjustable parameters were used in our theoretical modelling of S_{M}(q): the scaled particle charge Z_{eff}_{B}/R and the ratio η/η_{c}, where η_{c} is the experimentally assigned core volume fraction. The screening parameter κ_{eff}R was derived selfconsistently from the fitted values for Z_{eff}_{B}/R and η using eqn (14). To check the reliability of this analysis, we first fitted the structure factors measured in the uncharged system (without any added AOT). The fitted charge was, as expected, close to zero with an average of 〈Z_{eff}_{B}/R〉 = 0.05 ± 0.05, validating our approach.
Reassured by the accuracy of our fitting strategy, Z_{eff} was determined for the charged systems. The resulting particle charges as a function of η, are reproduced in Fig. 6. The increase in the effective charge at high densities, evident in the compressibility data, is particularly clear for the counteriononly system (NP3, main body of Fig. 6). The inset plot confirms that the same qualitative trend persists when small amounts of salt are present (system NP1). Remarkably, we find that all of the experimental data is consistent with an asymptotic scaling of the particle charge, , at high packing fractions. The evidence for this expression is strongest from the data collected on the nosalt system (NP3) although a dependence is also clearly visible in the NP1 dataset, particularly at low C_{AOT}. Finally, we note that the onset of the scaling regime moves to higher packing fractions with increasing background salt concentrations. So, for instance, in the nosalt system the charge starts to increase at η ≈ 0.02, while at κ_{res}R = 0.033 the scaling regime is delayed until η ≈ 0.07, η ≈ 0.10 for κ_{res}R = 0.078, and η ≈ 0.20 for κ_{res}R = 0.11. Consistent with this picture, at the highest salt concentration (C_{AOT} = 225 mmol dm^{−3}) where κ_{res}R = 0.22, the charge decreases monotonically with increasing η (data not shown) and there is no sign of any charge minimum.

 Fig. 6 Measured charge versus volume fraction, for different salt conditions. The scale in the plot is logarithmic. Main figure contains data for nosalt system (NP3, circles). Inset show experimental data for NP1 dispersion, with salt concentration increasing from bottom to top. Symbols same as Fig. 5.  
D. Origins of charge increase
We emphasise that the pronounced minimum in the effective charge Z_{eff}(η) seen in Fig. 6 is not caused by a change in the microstructural order of the suspension, such as crystallization. First, the height of the main peak in the structure factor S_{M}(q_{max}) is significantly smaller than the Hansen–Verlet criterion of S_{M}(q_{max}) = 2.85 for freezing^{53} even after allowing for polydispersity,^{54} second the structure factor contains no sharp peaks indicative of crystal formation, and finally comparison with the phase boundaries measured in suspensions of much largersized charged particles^{20} suggest our samples lie deep within the fluid phase of charged particles. To rationalize the increase in Z_{eff} at high densities we go back and reconsider the consequences of constant potential boundary conditions, recognizing that the CP model fairly accurately mimics charge regulation. The reduced particle charge Z_{B}/R is a dimensionless quantity, so it can depend only on combinations of dimensionless parameters. In the usual description^{16} of a dispersion of constantpotential spheres the set of dimensionless variables is chosen as the packing fraction η, the screening parameter κ_{res}R, and the scaled surface potential Φ_{s} ≡ eϕ_{s}/k_{B}T. However, a more intuitive picture emerges if we use the average separation d between particles in units of the Debye length κ_{res}^{−1}, in place of η. The dimensionless ratio λ ≡ κ_{res}d details the extent of the overlap between the electrical double layers surrounding neighbouring particles, with large λ corresponding to widely separated particles. λ dictates the curvature of the electrostatic potential between particles. From Gauss's law, the charge on the surface of the nanoparticle is 
 (19) 
where the prime denotes a derivative with respect to the radial variable. So a smaller value of λ will correlate naturally with a smaller particle charge. Numerical solutions of the CP model for Z_{B}/R as a function of λ, plotted in Fig. 7, confirm this picture. When particles are widely spaced the colloidal charge plateaus at an asymptotic value, which in linearscreening theory is 
 (20) 
Moving the particles closer together results in progressive overlap of the counterion atmospheres around each particle and subsequent colloid discharge, with Z_{B}/R → 0 as λ → 0. Furthermore, Fig. 7 reveals that the charge dependence Z(λ) is only weakly dependent on κ_{res}R, under the weakscreening conditions (κ_{res}R ≪ 1) appropriate to our experiments.

 Fig. 7 Charges predicted by constant potential model as a function of the particletoparticle separation d, in units of the Debye length κ_{res}^{−1}, for several screening parameters κ_{res}R. The surface potential Φ_{s} is chosen so that the charge in the widelyseparated limit is Z_{eff}_{B}/R = 3.  
Before we compare directly the experimental results and cell model predictions we emphasise a key distinction between the two approaches. The cell model calculations, in common with most theory and simulation studies, utilize a grandcanonical treatment of the electrolyte in which the charged dispersion is assumed to be osmoticallycoupled to an external ion reservoir with a constant chemical potential, so that calculations are performed at a fixed value of κ_{res}R. By contrast, in experiments there is no ion reservoir. Measurements are conducted in a canonical ensemble, in which the equilibrium concentration of mobile ions and hence κ_{eff}R varies with both the packing fraction η and charge Z_{eff}. To estimate the densitydependent screening in experiments we assume all ions are univalent and use eqn (4), or its equivalent in scaled units,

 (21) 
The dimensionless range
λ_{eff} =
κ_{eff}d is accordingly,

 (22) 
Inspection of
eqn (22) reveals a subtle dependence of
λ_{eff} on particle concentration. In the conventional saltdominated regime, where the added electrolyte exceeds the number of counterions released from the surface of the particles, the overlap of the double layers between particles grows (
λ_{eff} ∼
η^{−1/3} reduces) as the dispersion is concentrated. This behaviour is however reversed in the nosalt limit, where the righthand term of
eqn (22) dominates. In this regime,
λ_{eff} ∼
η^{1/6} so that, rather counterintuitively, colloids become less strongly interacting at high packing fractions. The crossover from a saltdominated to a counteriondominated screening regime occurs at a particle volume fraction
η_{c} which may be identified with the location of the turning point, where ∂
λ_{eff}/∂
η = 0. Differentiation of
eqn (22) yields the estimate for the crossover packing fraction
η_{c},

 (23) 
above which we enter the counteriondominated regime. Since typically 
Z_{eff}_{B}/
R ≈
(1)
eqn (23) reveals that the counteriondominated regime will be accessible only if
κ_{res}R ≪ 1.
To quantify in which screening regime our experiments lie, we determine the dimensionless range λ_{eff} as a function of η using the particle charge Z_{eff} measured by SAXS (Fig. 6), and the in situ screening parameter κ_{eff}R calculated from Section III(E). Fig. 8 shows the resulting packing fraction dependence for the nosalt system (NP3, circles), and for the NP1 system at AOT concentrations of C_{AOT} = 5 (squares), 25 (uptriangles) and 50 mmol dm^{−3} (downtriangles) in the inset plot. We identify three interesting features. First, Fig. 8 reveals that λ_{eff} increases with increasing η, so our SAXS measurements all lie within the counteriondominated screening regime. This conclusion agrees with the crossover volume fractions estimated from eqn (23). So, for instance, the minimum charge measured in the C_{AOT} = 50 mmol dm^{−3} (κ_{res}R ≈ 0.11) system is Z_{eff}_{B}/R ≈ 1.3 and eqn (23) predicts η_{c} ≈ 0.006, while SAXS experiments were performed at concentrations of η > 0.02. Second, the lowη data is consistent with the λ ∼ η^{1/6} dependence expected for the counteriononly limit. This is particularly evident in the experimental data for the NP3 system, where the background ion concentration is close to zero. Finally, we note that this asymptotic dependence only holds for low η: at intermediate and high volume fractions a new powerlaw regime with λ ∼ η^{0.42±0.01} appears. This is most clearly seen in the counteriononly data. We note from eqn (23) that a powerlaw of λ ∼ η^{0.42±0.01} suggests that the particle charge must scale as Z_{eff} ∼ η^{0.51±0.02} for a nosalt system, consistent with Fig. 6.

 Fig. 8 Measured particletoparticle separation (circles) in nosalt (NP3) dispersions, as a function of packing fraction η. Inset shows NP1 data in presence of added salt. Symbols same as Fig. 5.  
The observation that the SAXS data lie within the counteriondominated screening regime rationalizes qualitatively why the measured particle charge increases with increasing particle density. Numerical solutions of the nonlinear Poisson–Boltzmann equation show that CPparticles should chargeup (see Fig. 7) as the range of the electrostatic interactions becomes significantly smaller than a typical particle spacing. In the nosalt limit, the thickness 1/κ_{eff} of the double layer shrinks faster with increasing colloid concentration (1/κ_{eff} ∼ η^{−1/2}) than the mean spacing d between particles (d ∼ η^{−1/3}) so that as the packing fraction is increased, charged particles become less highly correlated, λ_{eff} grows, and the particle charge should accordingly be boosted. To facilitate a direct comparison, we plot in Fig. 9 experimental values for the dimensionless particle charge Z_{eff}_{B}/R as a function of the range λ_{eff} of the interactions. The data are represented by the symbols and should be compared to the CPmodel predictions plotted in Fig. 7. Fig. 9 confirms that Z_{eff}(λ_{eff}) is an increasing function of λ_{eff}, at least at large λ_{eff}, in broad agreement with CP predictions. However, a pronounced minimum appears in the experimental data at λ_{eff} ≈ 1.5, which is not predicted by the cell model. This may be a consequence of the neglect of threebody and higher order correlations^{30,31} in a spherical approximation. As λ_{eff} is reduced, each charged particle interacts not only with its immediate shell of neighbours but increasingly with the next nearest neighbour shell. Threebody interactions between charged particles are attractive within Poisson–Boltzmann theory,^{31} a fact which has been interpreted in terms of electrostatic screening by macroions.^{30} We speculate that this manybody mechanism of charge screening, which will be strongest at small λ_{eff}, could enhance the dissociation of surface charge groups and so increase the particle charge above the predictions of a simple spherical approximation.

 Fig. 9 Experimental charges replotted as a function of the particletoparticle separation d, in units of the Debye length κ_{res}^{−1}. Main figure contains data for counteriononly system (NP3, circles). Inset show experimental data for NP1 dispersion, with salt concentration increasing from bottom to top. Symbols same as Fig. 5.  
V. Conclusions
We have determined the density dependence of the structure factor S_{M}(q) in weaklycharged nonpolar colloidal dispersions using smallangle scattering techniques. An extended range of particle number densities was employed, with colloid packing fractions varying by a factor of approximately 10^{3}. By utilizing small radii nanoparticles and low ionic strengths, typical of nonpolar systems, we ensure measurements remain in the weak screening regime (κ_{res}R ≪ 1) even up to packing fractions approaching 30%. The effective charge eZ_{eff} of the particles was determined from a comparison between S_{M}(q) and structure factors calculated from the highlyaccurate modified penetratingbackground corrected rescaled mean spherical approximation (MPBRMSA) introduced by Heinen et al.^{52} The Debye screening parameter κ_{eff} was calculated selfconsistently from the effective charges and the measured background ion concentrations. The resulting Z_{eff}(η) values are in near quantitative agreement with effective charges measured by electrophoresis, at least for low packing fractions where mobility measurements are feasible. Our results for Z_{eff}(η) cover a broader range of packing fractions than previous studies and allow us to scrutinize in great detail predictions for the densitydependence of electrostatic interactions in concentrated charged dispersions.
This work suggests that the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory widely used to describe the interaction between two uniformlycharged particles, cannot describe nonpolar colloids. We test this conclusion by examining two different experimental systems, in which charge was generated by, either the dissociation of surfacebound groups or, alternatively by the adsorption of charged surfactant micelles. DLVO theory assumes a fixed sphericallyuniform charge distribution. The results presented here clearly demonstrate that such an assumption does not hold in the case of nonpolar dispersions where colloids are chargeregulated and surface charge density varies as a function of packing fraction. We find that a charge regulating system is less repulsive than an equivalent dispersion with a fixed charge distribution. This is due to a reduction in the dissociation of weaklyionizing groups as their separation decrease, driven by the need to avoid a local increase in the counterion density in the region between neighbouring colloids.
Theoretical models of charge regulation^{15,16} predict a strong decrease in the effective charge upon increasing particle concentration. While we find that Z_{eff} does indeed decrease at low packing fractions, our data reveals unexpectedly a pronounced minimum in the effective charge at η ≈ η_{c}. For η > η_{c}, the effective charge increases with particle concentration in apparent disagreement to existing models. We observe in all samples a near squareroot scaling of the measured effective charge, Z_{eff} ∼ η^{1/2}, at high densities. The packing fraction at the charge minimum is η_{c} ≈ 10^{−2} in the case of a nosalt system and shifts to progressively higher volume fractions (reaching η_{c} ≈ 10^{−1} at κ_{res}R = 0.22) as the salt concentration in the system is increased.
Our findings can be explained in terms of a crossover from backgroundion to counteriondominated screening as the colloid concentration is increased. The ideas are illustrated in Fig. 10. At low η, background salt ions dominate the screening (since ions from the dissociation of surface groups on the particles are negligible – see Fig. 10(a)) so κ_{eff}R is independent of the colloid concentration. The predictions of charge regulation models (which assume constant κ_{eff}R) apply directly, and the effective charge Z_{eff} reduces with increasing η. By contrast, at high packing fractions, ions from the particles outnumber the bulk salt ions (see Fig. 10(b)). As the number of ions released from the surface of the particles depends on the number of colloids the screening parameter κ_{eff}R becomes ηdependent. Electrostatic screening then increases as the particle volume fraction grows. The increased screening at high η reduces the coupling between chargeregulated particles, which allows the particle charge to grow. The nonmonotonic density dependence of Z_{eff} seen in experiments is therefore a consequence of a crossover from a saltdominated to a counteriondominated screening regime at high η. Finally, we note that the enhanced charge repulsion at high particle concentration evident in our data may account for the unusual high colloidal stability of nanoparticle dispersions which has been reported.^{1}

 Fig. 10 Illustration of backgroundion and counteriondominated screening regimes. (a) At low nanoparticle packing fractions, the majority of the ions in solution are contributed by the bulk electrolyte (shown in yellow). (b) At high η, most of the ions in solution arise from the dissociation of surface groups on the nanoparticles (shown in blue).  
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
JEH was supported by EPSRC CDT grant EP/G036780/1 and DAJG by a studentship from Merck Chemicals. We thank Marco Heinen for his MPBRMSA code. Finally, the authors would like to thank the ESRF and the Diamond Light Source for Xray beam time (experiments SC3655 and SM8982).
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Footnotes 
† PACS numbers: 82.70.y, 82.70.Dd, 42.50.Wk. 
‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm01825h 

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