Reappearance of structure in colloidal suspensions

– Static structure factors S(q) of deionized aqueous suspensions of charged polystyrene particles with similar radii but strongly differing bare charges have been measured for volume fractions 3.5 × 10−4 ≤ φ ≤ 1.55 × 10−2, using a cross-correlation light scattering technique which allows for the extraction of single scattered light from highly turbid samples. Measurement of absolute intensities allowed to determine unambiguously normalized values of S(q). With decreasing volume fraction, the amplitude of the first peak of S(q) reflecting the order of the suspension does not monotonically decrease, but rather shows a distinct minimum at φ ∼ (5− 8)× 10−3. This reappearance of structure is compared with theoretical predictions on the basis of a jellium model for the effective interparticle potential Ueff(r). The forces governing the structure and phase behavior of suspensions of charged colloidal particles are of prime importance for many technological applications, such as ceramic processing, drug delivery or the stability control of paints. In the standard picture of Derjaguin, Landau, Verwey, and Overbeek (DLVO) [1], the electrostatic interaction between two charged colloidal particles in an electrolyte arises from the repulsion of the electric double layers that form due to the competition between configurational entropy of the small ions and their potential energy in the Coulomb field of the macroions with negative charge −Ze0, e0 being the elementary charge. When the bare charge Z is large, a part of the surrounding counterions is strongly bound to the particle and thus reduces the bare charge to an effective charge Z̃ < Z which then governs the ion distribution at distances comparable to the Debye screening length κ−1 [2]. In contrast to the situation in simple liquids, however, the strength of the effective interparticle potential Ueff(r) determined by the effective charge Z̃ and its range κ−1 does not only depend on the bare charge Z but also on the concentrations of coand counterions and the particle volume fraction φ [3]. This density dependence of the effective potential is reflected by the fact that even at volume fractions as low as 10−4 strongly charged particles show considerable order, as reflected by the strong first peak in the colloid-colloid structure factor S(q) measured in light scattering experiments from deionized suspensions [4,5]. When, at higher volume fractions, the number of counterions balancing the particle charge becomes larger than the number of excess salt ions, the screening will become dominated by the former, L. F. Rojas et al.: Reappearance of structure in colloidal suspensions 803 and increasing the macroion number density np will result in enhanced screening. This has a twofold effect: it will reduce κ−1 and thus the range of the pair potential, but it will also affect a prefactor in Ueff(r). Both effects together can cause an initial increase followed by a reduction of order as φ increases. At still higher volume fractions, though, the packing of the macroions will finally be expected to dominate the structure of the suspension. The intrinsic density dependence of the interaction potential between charged colloidal particles can thus show a rather complex behavior, as the present investigation confirms. Starting from a dense suspension and measuring S(q), we find that with decreasing φ structure is first reduced, but reappears at lower volume fractions, in striking contrast to the situation in simple liquids where a reduction of the packing fraction always leads to a monotonic decrease of inter-particle structure. Although S(q) has been measured for deionized bulk suspensions of well-characterized, highly charged particles using light, X-ray and neutron scattering [6], an investigation of the range of volume fractions where such a reappearance of structure might occur has to date been hampered, in particular for light scattering experiments, by the large scattering contrasts leading to strong multiple scattering when the experiments are performed with aqueous solvents. In this paper we present an investigation of the structure of deionized suspensions of highly charged polystyrene particles. Static structure factors S(q) have been measured over a wide range of volume fractions 3.5 × 10−4 ≤ φ ≤ 1.55 × 10−2 by light scattering combined with a cross-correlation scheme for the detection of single scattered light from the highly turbid samples. Absolute measurements of the single scattering intensities allow an unambiguous normalization of S(q), even for the highest volume fractions, where S(q) shows strong oscillations at the largest values of q accessible with the light scattering experiment. Quantifying the order in the suspensions by the height of the first peak S(qmax) of the structure factor, we find comparable, non-monotonic φ-dependencies of S(qmax) for particles differing only in their bare charge Z. For our experiments we used sulfate polystyrene latex particles (Interfacial Dynamics) with radii a = 54.9 nm and a = 58.7 nm, respectively, whose charge is independent of pH and volume fraction. Accurate values of the particle size and of the size polydispersity were obtained from small-angle neutron scattering experiments; they agreed well with the values measured with static and dynamic light scattering from dilute samples and transmission electron microscopy. The samples were prepared by diluting the aqueous stock suspension into a mixture of ethanol and water containing a volume fraction of 70–75% pure ethanol. This reduces the solvent dielectric constant to values between 37.9 and 40.6 [7], resulting in a reduction of the screening length which is sufficient to prevent crystallization. Large aggregates and dust particles were removed by filtering the samples through 0.8μm cellulose filters into quartz glass cuvettes containing mixed bed ion exchange resin (BioRad). The cell caps were then sealed with Teflon tape and silicone putty; the degree of deionization was repeatedly checked by measuring the static light scattering at angles where a peak in the static structure factor was expected. Deionization was typically completed within 2 weeks. The resulting samples were highly turbid, with turbidities between 0.5 cm−1 and 15.5 cm−1. In order to isolate single scattered light containing the direct information on S(q) from the background arising from multiply scattered photons we employed a 3D cross-correlation scheme [8–11]; light scattered at angles 10◦ ≤ θ ≤ 150◦ was collected by two single-mode fibers and detected by two silicon avalanche photodiodes (EG&G) whose TTL output signals were cross-correlated by a multi-tau correlator (ALV). Details of the experimental setup are given in [12]. Single-scattering count rates I(q) were determined from the reduction of the measured amplitude of the cross-correlation function of the raw count rates. A turbidity correction has been applied in order to account for the angle-dependent optical path length in the square sample cells. 804 EUROPHYSICS LETTERS Fig. 1 – Static structure factors S(q) of deionized suspensions of polystyrene latex particles measured with light scattering as a function of the magnitude of the scattering wave vector q at 25 ◦C. (a) Average particle radius a = 54.9 nm, size polydispersity p = 11.8%, Bjerrum length lB = 1.383 nm, and solvent refractive index n = 1.3645. (b) a = 58.7 nm, p = 15.7%, lB = 1.482 nm, n = 1.3656. Particle volume fractions increase from left to right. Laser wavelengths were λ0 = 632.8 nm in (a) and λ0 = 680.4 nm in (b). Solid lines: best-fit polydisperse Rogers-Young calculations using a Yukawa interaction potential with adjustable volume fraction and effective charge. With incident laser powers of 30–50mW it typically took about 200–600 s to get I(q) to within 10% at a particular wave vector q. By normalization of the single-scattering count rate with the measured form factor P (q) and the count rate of a dilute reference sample containing small polystyrene particles (a = 10.5 nm) in the identical solvent, the quantity cS(q) was computed; here, c is the particle weight fraction. By fitting the peak position of a polydisperse Rogers-Young (RY) S(q) [13] to the observed peak position of cS(q) and using a particle density ρ = 1.05 g/cm, we determined the particle volume fraction φ = c/ρ, which in turn allowed to accurately determine S(q); the differences between the resulting values of φ and the values calculated from the volume fraction of the stock and the dilution factors are consistent with adsorption of a particle monolayer onto the ion exchange resin. Measured structure factors are shown in fig. 1. For both particle sizes and bare charges the measured S(q) show distinct liquid-like structure, with a strong first peak reflecting the well-defined interparticle separation; at the lowest volume fractions investigated, our data also show well-developed secondand third-order peaks with reduced heights. Interestingly, at low q the structure factor decreases to values that compare very well to the RY calculations. This is remarkable insofar that at these low angles the contribution of single scattered photons to the total signal detected is as small as 20%. Over a comparable range of volume fractions 1.3×10−3 ≤ φ ≤ 1.55×10−2 the structure factors of the L. F. Rojas et al.: Reappearance of structure in colloidal suspensions 805 Fig. 2 – (a) Peak heights S(qmax) of the measured structure factors as a function of volume fraction φ for particles with diameter a = 54.9 nm (squares) and a = 58.7 nm (triangles). Error bars represent standard deviations of S(qmax) due to photon statistics. (b) The height of the first peak of the colloid structure factor, S(qmax), of a charge-stabilized colloidal suspension as a function of the volume fraction φ calculated solving the Ornstein-Zernike equation with the RMSA closure relation with effective pair potentials based on the jellium approximation. The curves are labelled with the respective bare colloidal charges Z. larger particles with a = 58.7 nm show a similar behavior to that of the smaller ones, albeit with a less pronounced structure in S(q); the larger values of S(0) between 0.2 and 0.3 reflect not only the weaker coupling due to a smaller effective charge, but also the larger incoherent scattering due to the larger-size polydispersity. The rise of the peak height S(qmax) as φ increases is in good agreement with earlier light scattering results by Wagner et al. [14] on smaller particles. At higher volume fractions between about 0.3% and 0.5%, however, the structure factor peak drops significantly to what appears to be a minimum (see fig. 2a) at φ ∼ 5× 10−3 for the a = 54.9 nm particles and φ ∼ 8×10−3 for the a = 58.7 nm particles; at higher volume fractions, S(qmax) is observed to increase again. Although both the Bjerrum lengths lB and the particle sizes a for the two data sets differ only by about 7%, the amplitude of the S(qmax) data for the a = 58.7 nm particles is significantly smaller than the one of the slightly smaller particles; this indicates that their respective bare charges Z do indeed grossly differ. Such a non-monotonic dependence of S(qmax) on volume fraction has been predicted by the jellium model (JM) of Beresford-Smith et al. [3]. The density dependence of the structure factor is due to a density-dependent effective colloid-colloid interaction potential Ueff(r). For an isolated pair of colloidal particles and at large distances r, this pair potential is known to be a Yukawa potential∼ ȳ2e−κr/r with a screening constant κ and a prefactor ȳ which is the square of the amplitude of the electrostatic mean-field potential around a single particle. For two colloidal particles in a suspension of finite concentration, the JM assumes the effective pair potential to be still Yukawa-like; however, both ȳ and κ now depend on the colloid density, leading ultimately to the φ-dependence of S(qmax). In the following, we briefly recall how to calculate ȳ and κ in the JM; for a complete derivation of the theory and a thorough discussion of the underlying assumptions we refer the reader to the original work of Beresford-Smith et al. [3]. The dependence of ȳ and κ on the colloid density np = 3φ/4πa originates from the fact that the total number of counterions in the system is proportional to the number of colloids. To 806 EUROPHYSICS LETTERS determine ȳ, one first has to calculate the electrostatic potential Ψ(r) (normalized by kBT/e0) around one single colloid. Let ns be the concentration of the monovalent salt. Znp+ns then is the concentration of counterions, ns that of the co-ions. Within the Poisson-Boltzmann (PB) theory, the charge distribution of the positively and negatively charged microions is given by ρ+(r) = (Znp + ns)e−Ψ(r) and ρ−(r) = nse, respectively. In the JM the requirement of electroneutrality is ensured by assuming that all colloidal particles surrounding the central one contribute equally to a homogeneously distributed background charge density ρbg = Znp. From Poisson’s equation, linking Ψ(r) to the total charge density ρ+ − ρ− − ρbg, one then obtains