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Self-Diffusion and Collective Diffusion of Charged colloids Studied by Dynamic Light Scattering

A microemulsion of decane droplets stabilized by a non-ionic surfactant film is progressively charged by substitution of a non-ionic surfactant molecule by a cationic surfactant. We check that the microemulsion droplets remain identical within the explored range of volume fraction (0.02 to 0.18) and of the number of charge per droplets (0 to 40) . We probe the dynamics of these microemulsions by dynamic light scattering. Despite the similar structure of the uncharged and charged microemulsions the dynamics are very different . In the neutral microemulsion the fluctuations of polarization relax, as is well known, via the collective diffusion of the droplets. In the charged microemulsions, two modes of relaxation are observed. The fast one is ascribed classically to the collective diffusion of the charged droplets coupled to the diffusion of the counterions. The slow one has, to our knowledge, not been observed previously neither in similar microemulsions nor in charged spherical colloids. We show that the slow mode is also diffusive and suggest that its possible origine is the relaxation of local charge fluctuations via local exchange of droplets bearing different number of charges . The diffusion coefficient associated with this mode is then the self diffusion coefficient of the droplets.

the spectra is found to be the sum of two exponentials described theoretically 3,4,5 in terms of the collective and self diffusions respectively . These two modes have been oberved in concentrated colloidal dispersions 6,7 , in charged dispersions 8,9 and in water-in-oil droplet microemulsions 5,10 . In bicontinuous microemulsions two relaxation modes have been reported 11 which have a different origin related to the local fluctuations of the topology of the bicontinuous network. In linear polyelectrolytes two modes are again observed [12][13][14] , the origin of the slow mode is still controversial. Recently two relaxation modes have been observed in the dynamics of a globular protein and the authors claim the second mode to be due to diffusion of well-ordered clusters of the globular proteins 15 . And in viscoelastic fluids complex DLS spectra are observed with two or three relaxation modes see ref 16 and references therein. One of this mode is directly related to the transient elasticity of the system. This list is not exhaustive, it illustrates the variety of systems where DLS spectra display two or more relaxation modes the origin of which depends strongly on the type of system studied.
In the present paper, we consider the case of weakly charged colloids. Oil in water microemulsion droplets stabilized by a non-ionic surfactant film can be charged by addition of minute quantities of an ionic surfactant without modifying the size and shape of the droplets. They so provide model systems of weakly charged colloid, the charge of which is easily varied and controlled. Such charged microemulsions have been studied previously by Gradzielski and Hoffman 17 and by Evilevitch et al 18 but these authors report on only one relaxation mode in the DLS spectra of these microemulsions. Another convenient realization of a weakly charged colloid is a solution of globular proteins the charge of which is monitored by the pH. Such a system was studied by Retailleau et al 19 who also observe only one relaxation mode in the DLS spectra.
Here, we investigate the DLS pattern of weakly charged oil in water microemulsion droplets. Using small angle neutron scattering, we first check that the size and shape of the droplets remain constant upon changing their charge by addition of small amounts of an ionic surfactant . Then we present the DLS spectra which clearly exhibit two relaxation modes. We then discuss two possible mechanisms at the origin of the two modes. The mechanism first considered by Pusey et al in the early eighties involves the size polydispersity of the droplets.
To our knowledge, the second mechanism has never been considered to date: it rather relies on the charge polydispersity of the droplets. A simplified toy model based on a bidispersed distribution of charges in otherwise identical particles is presented to illustrate the probable relevance of this second mechanism.

Materials
The neutral surfactants are TX100 and TX35 from Fluka used as received. The ionic surfactant: Cetyl pyridinium chloride (CPCl) from Fluka is purified by successive recristallization in water and in acetone. Decane from Fluka is used as received.

Samples
The microemulsions 20 are thermodynamically stable dispersions in water of oil droplets surrounded by a surfactant film: O/W microemulsions. The spontaneous radius of curvature of the surfactant film is adjusted by varying the relative proportion of the surfactant and cosurfactant. All samples are prepared by weight in triply distillated water or deuterated water (for the SANS measurements). In the neutral microemulsion, the amphiphilic film is composed of TX100 (surfactant) and TX35 (co-surfactant) ; the weight ratio of TX35 to TX100 is 0.48 ; decane is added so that the sample is close to the emulsification failure limit that is the limit of oil solubilization, the weight ratio of decane to surfactant is 0.7. It is now well establish that, under these conditions, the droplets of microemulsion have a radius corresponding to the optimum curvature of the surfactant film and a narrow size distribution which remain constant over a broad range of concentration [20][21][22] . We showed previously that it was also the case for this particular microemulsion 23 . In the charged microemulsions a small part (up to 3% in weight) of ionic surfactants are added to the non-ionic surfactants, that corresponds to a number of 0 to 40 ionic polar heads among ~1600 TX neutral polar heads per droplet.
We checked, as described below, that the size of the microemulsion droplets is not modify by this addition. The samples are characterized by the volume fraction of droplets Φ and by the mean number of charges per droplet p ; p can be calculated from the composition of the samples and the mean volume of the droplets deduced from the SANS data (see below) . The parameters to calculate and Φ p are given in Table 1 .

Small Angle Neutron Scattering : SANS Measurements
The intensity scattered by a dispersion of spherical colloids with a distribution of size can be written 24 : With q (Å -1 ) the scattering vector, Φ the volume fraction, and v m < A 2 > respectively the volume and the scattered amplitude squared of the colloids averaged over the size distribution. , the measurable structure factor , reflects the interactions between the colloids, it is 1 at large q's or for very dilute samples where interactions can be neglected.
The best simulation is obtained for spherical droplets of outer radius but with three successive scattering length densities corresponding roughly to an inner sphere of decane, a corona containing the aliphatic chains of surfactant and an outer corona containing the polar heads = short PEO chains . The scattering length density profile is sketched in figure 1 . The averaged squared amplitude is calculated from the amplitude for spherical concentric shells R 3 25 and with a Schulz-Zimm distribution for the size of the droplets 26

Dynamic light scattering -DLS
The thermally activated dynamics of microemulsions can be probed by DLS 1,2 . The measurements are performed on a standard setup (AMTEC Goniometer with a BI9400 Brookhaven correlator), the light source is an argon ion laser (λ=514.5 nm). The homodyne intensity autocorrelation function is measured at different q values, ranging from 3 10 6 to 3 with n the refractive index of the solvent and θ the scattering angle ).
The normalized autocorrelation function of the scattered intensity writes: and if the scattered field obeys Gaussian statistics , this measured autocorrelation function can be expressed as a function of the first-order electric field correlation function through the , where c (0<c<1) is the experimental coherence factor. In our samples, the scattering originates from the fluctuations of the droplet concentration and the autocorrelation function of the concentration fluctuations. In a suspension of microemulsion droplets in the low concentration regime we expect these fluctuations of concentration will relax through simple brownian diffusion and , assuming a monodisperse suspension , we simply write : In this case the relaxation mode is diffusive as indicated by the q -2 -dependence of τ . This is indeed what is observed in the neutral microemulsions as illustrated in figure 3A, the small departure from a monoexponential behavior at large times is to trace back to the moderate size polydispersity of the droplets. The well-known cumulants expansion 4,28 -strictly valid for a non interacting suspension-can be used to account for the polydipersity of the microemulsion (cf fig 3A). The line in B is a fit to eq (5) with The situation is different when the microemulsion droplets are charged. As can be seen in figure 3B the autocorrelation function is no longer quasi-monoexponential and it is well fitted to the sum of two single exponentials: Two relaxation modes are present, each one characterized by its relative contribution and its decay time : . This was found to be the case for all the charged microemulsions as is further illustrated in figure 4 for samples with a constant charge p =20 and an increasing volume fraction. The lines are fits of the data to eq. (5) and the dotted lines are fits of the data to a single exponential. Each set is shifted by a factor of 10 for clarity.
The two relaxation modes are diffusif: as illustrated in figure 5 . The slopes of the straigth lines are the diffusion coefficients associated to the two relaxation modes .
Furthermore the relative amplitudes of the two modes are q-independent within experimental errors as shown in figure 6 . where the droplets interact via an overall repulsive potential (a Van der Waals attractive potential plus a hard sphere repulsive potential) 29 . From D 0 = 2.58 ×10 −11 m 2 × s −1 we can extract the apparent hydrodynamic radius 2 R app = 95 of the droplets using the Stokes-    diffusion coefficient can be written : with S(0) the structure factor in the limit q=0, ν the mobility, the osmotic pressure and N the number density of particles .
In this step the colloids readjust their distance but remain in their initial spatial arrangement furthermore the amplitude of the collective mode ( α S(0)) is suppressed by the interactions so that the two modes can be readily observed . Two modes have indeed been observed in charged colloidal systems D coll D self 8,9,30,31 .
Although this mechanism based on the size (and therefore the scattering power) polydispersity certainly contributes to our observation, we presume that a second mechanism rather based on the charge polydispersity may well bring a significant contribution.. This is much longer than the relaxation of the intensity-intensity correlation function. Anyway we do not observe, as expected in the dilute regime,, D f = D coll = D 0 . This can be understood if we recall that the electrostatic potentiel of interaction is long-ranged (unscreened coulombic potential) so that the dilute regime (in the sense that the interactions between droplets become negligible) is much lower than the lowest limit studied ( ) and extrapolation to zero volume fraction at a finite Φ = 0.017 p cannot be achieved properly .
However, extrapolates to at zero volume fraction and zero charge as it should. We find no contradiction between our description and the evolution of the two diffusion coefficients.
To illustrate our general description , let us consider the very simple model described in the next section.

A simple model for the diffusion of charged particles
Although the charge distribution of the droplets is indeed spread, we consider for simplicity two populations of particles only which differ by their number of charges per particles and . Their respective concentrations (in molar fractions) are noted and . All particles are otherwise identical (same size, shape and scattering length). We assume further that the free energy density (in molecular units ) of the dispersion comprises two terms: The first term is the enthalpic contribution arising from the repulsion between the particles due to their charges. Its functional form is , but at the scale of the inverse wave vector investigated experimentally (large scale compared to the average distance between the droplets), we can reasonably admit that it is a function of the local charge density of the dispersion only. The second term is indeed the usual contribution of the entropy of dilution of h populations 1 and 2 respectively. Expanding in terms of small concentration fluctuations f 1 X δ and δX 2 from the average values X 1 and X 2 : where A stands for the second derivative of the enthalpic repulsion with respect to the local charge density: Of course only the quadratic terms in (eq.7) are relevant for the diffusion of the particles.
For the sake of simplicity again, we further assume that the hydrodynamic interactions between the two populations can be neglected. This is a strong approximation, but we believe it not to be too dramatic at sufficient dilution. In this limit, we can therefore use the generalized Fick's law for the fluxes and write the conservation of the particles: where ν is the mobility of the particles (identical for both populations) and i μ are the chemical potentials of each population: i i X f ∂ ∂ = μ . In absence of hydrodynamic interactions, the only effect of one population onto the motion of the other arises from the term coupling the two concentrations in the free energy density (eq 7): Combining eqs (7) and (8) in the conventional way and switching to the recipocal space, we obtain the kinetic equations for the two kinds of particles: where is the wave vector. Solving these coupled kinetic equations we easily get two very simple expressions for the characteristic relaxation times (see appendix 1 for the details): where the indices and stand for slow and fast respectively. As expected, both modes are diffusive. The fast mode explicitly depends on both the temperature and the enthalpic contribution arising from the repulsions between charges. Whereas the slow mode is totally independent of the enthalpic terms. This is precisely what we expected from the discussion in the previous section: the fast mode is the collective diffusion mode whereas the slow mode only depends on the temperature and can be identified with the self diffusion mode.  (7)). The tedious complete calculations are summarized in appendix 1. Unfortunately, the general expressions are prohibitively complicated. Nevertheless (see eq-A-9), the amplitudes of the two modes are q-independent consistently with the experimental results. Moreover, the amplitude of the slow diffusive mode is proportional to (see ( p 1 − p 2 ) eq-A-9) and decreases to zero when the difference in the number of charges per droplet in the two populations vanishes.
By chance, the expressions for the amplitudes simplify considerably if we consider the case where both populations have the same concentrations: X 1 = X 2 = X / 2. This simplified case is relevant for the true situation where we expect the fluctuations of the number of charges per droplet above and below the average to be essentially symmetrical. Writing p 1 = p + dp p 2 = p − dp , δX q = δX q1 + δX q2 and A k = A k b T , we finally get for the autocorrelation function of the concentration fluctuations: Interestingly, according to eq.11 the amplitude of the slow mode remains finite when A diminishes and equals zero. However , of course, (see eqs. 10 , 11) the characteristic times of the two modes become identical in this limit so that the two modes can no longer be distinguished. This general evolution could be checked, for example, by tuning the electrostatic interaction and thus A by adding increasing amounts of salt.
In figure 8 , the relative amplitudes of the modes are found almost independent on and Φ p . This feature does not come out from eq 11 ( when X 1 = X 2 = X / 2). However we must keep in mind that we have not specified the functional dependence of the enthalpic term. From the expansion in eq 7 , A is an unknown function of p 1 X 1 , p 2 X 2 and in this context it is impossible to discuss this feature further.

Conclusion
We report here on a second relaxation mode which to our knowledge has not mentioned in the earlier study of similar weakly charged microemulsions. In the paper of Gradzielski and Hoffmann 17 which dates back to 1994 it was experimentally impracticable to ascertain such a second relaxation mode . However the correlation functions in figure 13   .Concentrations are in molar fraction. The overall concentration fluctuation, in reciprocal space, is δX q (t ) = δX 1q (t ) + δX 2q (t ) . Our objective is to calculate the autocorrelation function of the concentration fluctuations : C( q, t ) = δX q (t ) δX q (0) ( ) δX 1q (t ) and δX 2q (t ) are the solutions of the coupled equations (eqs 9 ). They are solved using matrix methods: as explained in the text. And eqs 9 write: δX q (t ) = (δX q1 (t ),δX q2 (t )) and δ ′ X q (t ) = (δ ′ X q1 (t ),δ ′ X q2 (t )), the derivative of δX q (t ) with respect to time. It can be rewritten , in tems of the diagonalized matrix , and the eigenvectors L δX dq (t ) : The two eigenvalues of are: After calculating the matrix , we can develop eq A-7 and we obtain : P , P −1 and Q δX (q,t ) = δX 1 (q,t ) + δX 2 (q,t ) = The relaxation times of the two modes are given by (eq 10):