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Polymer-graft-mediated interactions between colloidal spheres Jeanette Ulama, Malin Zackrisson-Oskolkova, and Johan Bergenholtz Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b04739 • Publication Date (Web): 07 Mar 2016 Downloaded from http://pubs.acs.org on March 9, 2016

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Polymer-graft-mediated interactions between colloidal spheres Jeanette Ulama,† Malin Zackrisson Oskolkova,‡ and Johan Bergenholtz∗,†,‡ Department of Chemistry and Molecular Biology, University of Gothenburg, SE-41296 Göteborg, Sweden, and Division of Physical Chemistry, Center of Chemistry and Chemical Engineering, Lund University, SE-22100 Lund, Sweden E-mail: [email protected]

Abstract

INTRODUCTION

Aqueous dispersions of fluorinated colloidal spheres bearing grafted poly(ethylene glycol) (PEG) are studied as a function of salt and particle concentration with the aim of improving the understanding of interactions among polymer-grafted particles. These dispersions can sustain large concentrations of salt, but crystals nucleate in dilute dispersions when a sufficient Na2 CO3 concentration is reached, which is attributed to the presence of attractions between particles. On further increasing the Na2 CO3 concentration the solvent is rapidly cleared of particles. Small-angle X-ray scattering and cryogenic transmission electron microscopy are employed in order to quantify the attractions. The former is used to extract a second virial coefficient and the latter shows that the PEG-graft contracts as a function of increasing salt concentration. The contraction not only leads to a reduction in excluded volume, but it is also accompanied by attractions of moderate magnitude. In contrast, dispersion of the particles in ethanol, in which bulk PEG solutions crystallize, lead to fractal structures caused by strong attractions.

The stability of colloidal particles can usually be drastically enhanced by adsorbing or grafting polymer on particle surfaces relative to particles stabilized only through electrostatic effects. The stability of such sterically stabilized systems is governed largely by the solvency of the dispersion medium for the stabilizing polymer. Thus, for changes in temperature and solvent composition the stability may be compromised. There is a large body of literature on the stability of sterically stabilized colloidal systems, but there seems to be no consensus as to the origin of the attractions that underlie the instability that is encountered when the solvent quality is decreased sufficiently. In some cases a corecore van der Waals attraction, just as in the DLVO theory of electrostatically stabilized systems, is thought to be the dominant cause. 1–3 In this view the polymer corona simply contracts in marginal and poor solvents to expose more of this attraction. Other studies focus on polymer-polymer interactions and describe them either semi-macroscopically through variations of Hamaker-Lifshitz theory 4–9 or through polymer-mixing models. 10 The extent to which core-core van der Waals attractions contribute to the net attraction presumably depends on the thickness and grafting density of the polymer corona and on the material make-up of the particles and on the solvent so the two mecha-

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To whom correspondence should be addressed University of Gothenburg ‡ Lund University †

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nisms need not be mutually exclusive. 11 The question of the origin of attractive interactions between particles with tethered chains has resurfaced within the field of metallic and semiconductor nanocrystals, which under the right conditions form a variety of crystalline arrays of interest for a wide range of applications. 12 The structural diversity observed has been attributed to attractions among the nanocrystals variously thought to be dominated by core-core van der Waals attractions due to large Hamaker constants 13–16 or ligandmediated attractions, 17–19 as well as more complex, directional interactions. 15 In addition, the possibility that the polymer corona undergoes a structural transition to a crystalline state resulting in strong attractions 20 has been revisited in experiments on non-aqueous dispersions of sterically stabilized silica in organic solvents. 21,22 It has been suggested that the freezing of the stabilizing chains cause these systems to gel. This mechanism has also been called upon to rationalize the appearance of amorphous structures of gold nanocrystals. 23,24 A suitably chosen model system might shed light on the mechanisms behind attractions between polymer-grafted particles and we propose that fluorinated particles with grafted poly(ethylene glycol) (PEG) can be used to challenge the mechanistic viewpoints mentioned. Fluorinated particles can be synthesized with short- and longer-chain grafts with quite narrow size distributions. 25 Like polymer particles in organic solvents, they can be refractive index matched but in predominantly aqueous solvents. As such, the core-core van der Waals interaction can be minimized. Furthermore, in aqueous solvents addition of salt screens any remaining electrostatic effects, which may be an issue in non-aqueous systems. 26–28 In addition, the particles can be transferred into ethanol, a solvent in which bulk solutions of free PEG crystallize at room temperature, 29 while remaining nearly refractive index matched. Thus, the generality of polymer-graft ordering as a possible mechanism behind attractions between particles can be investigated.

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In this study PEG-grafted spheres are studied as marginal and poor solvent conditions are approached through addition of Na2 CO3 , which is known to cause phase separation in bulk PEG solutions 30 and aggregation in dispersions of PEG-grafted particles. 31–33 The study is restricted to salt-induced destabilization under rather dilute particle concentrations. The PEG corona is investigated using cryogenic transmission electron microscopy (cryo-TEM) and the resulting interactions are studied using ultrasmall-angle X-ray scattering (USAXS). Combination of the results allows for separating the reduction in the excluded volume caused by the contraction of the polymer corona and the appearance of attractions.

EXPERIMENTAL TION

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Materials Similar particles used in this study have been carefully characterized previously. 33 The particles are of core-shell type where the core is fluorinated and the shell consists of PEG with a molecular weight of 2000 g/mol. The synthesis procedure is known to produce monodisperse particles with a high PEG grafting density. Due to the fluorinated core, a low refractive index is obtained. Two new batches (L5) following the synthesis procedure given in 33 were synthesized, referred to as L5 and LE5 throughout the text. L5 is used to study the destabilization process as Na2 CO3 is added and also to elucidate the collective behavior as particle concentration is increased. LE5 was transferred to ethanol in order to investigate the collective behavior in a less polar solvent. The radius of L5 is 98 ± 2 nm and 91 ± 1 nm as measured by DLS and SAXS, respectively, corresponding to the hydrodynamic radius and the radius of the fluorinated core. The corresponding radii for LE5 are 82.5 ± 2 and 77 ± 1 nm. The core polydispersities of L5 and LE5 have been determined from analysis of SAXS data and resulted in approximately 4% for both. The PEG-layer thickness can be roughly estimated by the dif-


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ference in radii from DLS and SAXS, 33,34 leading to grafted polymer layers extending some 7 ± 2 and 5.5 ± 2 nm from the particle surface for L5 and LE5, respectively. Note that the radii quoted for LE5 apply to aqueous dispersions of the particles before transferring them into ethanol. The ethanol (99.7%) was supplied by Solveco and was used as received.

Small-angle X-ray scattering (SAXS) measurements were conducted at the ID2 beamline at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. All scattering measurements employed a wavelength λ of 0.995˚ A and a sample-to-detector distance of 20 m leading to a scattering vector (q) range of 0.0035 ≤ q ≤ 0.38 nm−1 . A flow-through quartz capillary with an inner diameter of 1.6 mm was used for all dispersions and solvent backgrounds. Exposure times varied between 0.1 and 1 s depending on the particle concentration. Typically, after a preliminary check for sample beam damage, 10 consecutive spectra were recorded, which were radially averaged and examined for beam damage, after which they were averaged. Solvent scattering, measured in the same capillary, was subtracted as background and a measurement of water was used to bring the scattering curves onto an absolute scale. More concentrated suspensions were obtained by centrifugation using Jumbosep filter devices (Pall Inc. 30kD cutoff), and later diluted with either Na2 CO3 or stabilizing media. As stabilizing medium we use a 10 mM aqueous electrolyte (7 mM NaCl and 3 mM NaN3 ). Sodium azide was added because it is known to prevent bacterial growth and sodium chloride was added to screen any residual charges from the initiator. To replace water with ethanol as a solvent, aqueous samples were placed in a Jumbosep filter device and centrifuged against ethanol until the filtrate had roughly the same density as pure ethanol. The ethanol content in the final filtrate was 99.5% as determined from an ethanol/water density standard curve. The density was measured at 25◦ C using a density meter (DMA 5000, Anton Paar).

Methods Dynamic light scattering (DLS) measurements were performed using an ALV CGS3 instrument, equipped with a 632.8 nm HeNe laser (22 mW) and an avalanche photodiode detector positioned at a scattering angle of 90◦ . All measurements were recorded at 25◦ C and results are reported as an average of at least 3 runs per sample. The hydrodynamic radius was extracted from a second-order cumulant analysis. The samples were typically prepared a few days in advance except for measurements close to the phase boundary which were left to equilibrate for over a week. The samples were prepared by mixing a stock solution of Na2 CO3 with water and then adding a portion of a particle stock dispersion to yield the desired concentrations of both the electrolyte and particles. Samples for cryo-TEM were prepared in a climate chamber kept at a temperature of 25-28◦C and a relative humidity close to 100% to prevent evaporation from samples during preparation. A 5µL sample drop was placed on a lacey carbon-coated film supported by a copper grid. Excess sample was removed by blotting with filter paper, leaving a thin (20-400 nm) liquid film in the holes of the carbon film. The grid was subsequently plunged rapidly into liquid ethane at −180◦ C and transferred into liquid nitrogen at −196◦ C. The vitrified samples were stored in liquid nitrogen and moved into a Philips CM120 BioTWIN TEM, equipped with a postcolumn energy filter (Gatan GIF 100) using an Oxford CT 3500 cryo-holder and its workstation. The acceleration voltage was 120 kV and the working temperature was kept below −182◦ C. The images were recorded digitally with a CCD camera (794IF) under low-dose conditions with an underfocus of less than 1 µm.

RESULTS AND DISCUSSION Phase behavior In Fig. 1 the phase behavior of dilute dispersions is shown as a function of added Na2 CO3 and particle concentration. In terms of volume


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Here, D0 is the single-particle diffusion constant and η is the solvent viscosity. For salt concentrations corresponding to unstable conditions in Fig. 1, either the redispersion process does not result in single particles or some form of growth process ensues. This can be seen in Fig. 2 where the particle concentration is held constant at 0.1 mg/mL and the Na2 CO3 concentration is increased. Beyond 0.65 M Na2 CO3 the intensity correlation functions no longer superimpose on the result for fully dispersed particles. For some of the dilute samples in Fig. 1 the growth process was very slow and could not be established until after three days.

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fractions the particle concentration in this diagram varies from roughly 3.6 · 10−5 at the low end to 6.4·10−3 for the least dilute samples. Below ∼ 0.7 M Na2 CO3 the dispersions are stable in this range of particle concentration except for the least dilute sample. This was ascertained through monitoring samples after mixing by dynamic light scattering. Samples in the vicinity of the border between stable and unstable regions aggregate rapidly upon mixing dispersions with salt solution. However, for states labeled as stable in Fig. 1 these aggregates slowly redisperse over time, resulting eventually in fully dispersed, single particles. This is shown by the intensity correlation functions in Fig. 2 for the stable samples at 0.65 M Na2 CO3 , which agree well with the result for the same particles with no added Na2 CO3 once variations in solvent viscosity and refractive index are accounted for by scaling the delay time t by the relaxation time (q 2 D0 )−1 ∝ (q 2 /η)−1 .

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Figure 2: Intensity correlation function as a function of the delay time, scaled to account for variations in solvent viscosity and refractive index, as a function of (top panel) particle concentration at a Na2 CO3 concentration of 0.65 M, and as a function of (bottom panel) Na2 CO3 concentration at a constant particle concentration of 0.1 mg/mL, as labeled. Shown also is a dilute sample of the same particles dispersed in 10 mM monovalent electrolyte. For high enough Na2 CO3 concentrations the dispersions contain structures that are larger than single particles. At higher particle concentrations these can be seen by the naked eye and appear to be crystallites suspended in the solvent. These eventually settle under gravity,


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absent for the depletion systems. Strong attractions between particles might push such an upturn to very low particle concentrations as has been argued by Napper and co-workers 38 in the context of the so-called critical flocculation boundary for similar dispersions of PEGgrafted particles. They observed a similar lack of curvature of this boundary with respect to temperature, which was used to worsen the solvent quality in their experiments. However, Vincent et al., 39,40 also for PEG-grafted particles, determined a critical flocculation temperature that decreased with increasing particle concentration. The discrepancy was attributed to differences in the PEG-graft thickness, which might cause the attraction to become very strong over a small temperature interval. 40 The determination of the stability limit in Fig. 1 differs somewhat from the critical flocculation boundary examined in these studies. Aside from the solvent conditions being controlled entirely by added salt rather than temperature, the results in Fig. 1 are made after allowing systems to relax, for days for the most dilute systems. Thus, we believe these results reflect the equilibrium behavior, which may or may not govern the critical flocculation boundary. In addition, given that, at least at somewhat higher particle concentrations, the systems crystallize, it is unlikely that the flatness of the stability limit comes from very strong attractions between particles. A more likely explanation is that the dilute-most samples in Fig. 1 belong to the fluid-crystal coexistence region. In that case it is the fluid-crystal binodal that is quite flat with respect to salt concentration, which may be caused by moderate overall attractions if they become very short ranged. Such a flattening of the fluid-crystal binodal on shortening the range of the attraction is observed with the model of Charbonneau and Frenkel. 41

leaving behind a very dilute dispersion. These states have been labeled as ‘fluid-crystal coexistence’ in Fig. 1. This behavior can be expected because the particles are of very low polydispersity. Such crystals are not visible to the eye in more dilute samples (≤ 0.1 mg/mL) but the samples nonetheless contain larger structures as seen by the slower decay of the intensity correlation function, shown in Fig. 2, when the salt concentration reaches 0.7 M. These dilute samples have been labeled as ‘slowly restructuring’ because only after a few days could a growth process be firmly established. Whether this process corresponds to nucleation and growth of crystals or formation of disordered aggregates is unknown. The remaining states in Fig. 1, which are labeled ‘demixed’, behave differently in that the particles show a strong tendency to avoid the solvent. Part of them collect at the bottom of the sample as a sediment but some of them collect at the sample surface, presumably captured there by surface tension due to the solvent not wetting the PEG-covered particles. The solvent inbetween the particle-laden surface and the bottom with the sediment is essentially free of dispersed particles. Part of the phase behavior reported in Fig. 1 is similar to observations made for dilute colloidal systems subject to the depletion effect whereby added polymer controls the attraction between particles. 36,37 In particular, RomeroCano and Puertas 37 note in their study of the phase diagram below volume fractions of 0.01 a qualitatively similar sequence of structures as a function of increasing polymer concentration, i.e. increasing attraction strength. At the lower polymer concentrations they observe fluid-crystal coexistence, followed by aggregation and sedimentation of particle clusters, leaving a dilute dispersion or particle-free solvent at the highest polymer concentrations. The latter case is reminiscent of the demixing between solvent and particles that occurs here. The absence of an upturn of the instability at low particle concentrations is a feature of the phase diagram that was not noted in the depletion case. 37 This could be due to the long lag times before detecting growth, as observed in our measurements, or that such behavior is

Structure and interactions Small-angle X-ray scattering (SAXS) has been employed to study the interactions that lead to crystallization and ultimately to an expulsion of the particles from the solvent. However, to


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amplify the effect of interactions these measurements have been carried out at higher particle concentrations than those in Fig. 1. Also, recognizing that quantification of the interactions from analysis of scattering data at higher concentrations requires quantitative model fitting, which is done most reliably by minimizing the number of model parameters, we begin by examining the behavior under good solvent conditions before turning to worsening solvent conditions. In this way we can use the good solvent scattering data to calibrate the particle concentration scale because the model requires particle number densities whereas experimentally only mass fractions are known.

ing. As described in the Supporting Information, the small polydispersity of the present system allows for modeling the form factor out to the first three minima using a mixture formalism in which the size distribution is approximated by three discrete components. This simplification makes for an easier incorporation of structure factor effects that need to be added to describe interacting systems at higher concentrations. As seen in Fig. 3, the form factor can be quantitatively described by a slightly polydisperse homogeneous sphere model. In other words, the scattering from the PEG graft is negligible in comparison to the scattering from the fluorinated particle core. The core radius corresponding to the main peak in the size distribution (see Supporting Information) is found as 91 nm with a standard deviation of 3.8 nm. As the concentration is increased the intensity in the low-q region is found to increase only very weakly, far less than what is expected in the dilute limit where the intensity varies linearly with concentration as I(q) = nP (q). Here, n is the number density of particles and P (q) is the so-called form factor. In addition, the first form factor minimum becomes increasingly more shallow with increasing particle concentration. We attribute this to multiple scattering caused by the high electron density of the fluorinated core material. In the Appendix, we describe how this effect is quantitatively modeled in an a priori fashion. The suppression of the intensity at lower q relative to the dilute-limiting behavior can be accounted for by incorporating in this case a hard-sphere interaction with diameters taken as the core radii with an added constant shell thickness δ. Setting the electron densities of the particle cores and the solvent as 1.36 · 10−3 and 9.4 · 10−4 nm−2 , respectively, the only fitting parameters are the overall number densities and the shell thickness that is to account for the PEG graft. In Fig. 3, δ was kept constant at 6 nm and a linear relation between mass fraction m and fitted number densities n was found as n = (2.13 · 10−7 nm−3 ) m. Adding Na2 CO3 to the dispersions worsens the solvent conditions. In Fig. 4 scattering curves for varying amounts of added Na2 CO3 are shown at a constant particle concentration.

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Figure 3: Scattering intensity as a function of q for aqueous dispersions in the absence of Na2 CO3 with varying particle concentration, as labeled. Shown also is a very dilute sample of the same particles labeled as form factor. The solid lines are fits using hard-sphere mixture model. To model the scattering data quantitatively we employ the multicomponent version 42 of the adhesive sphere model, 43 which has been used in the past for modeling scattering data from more polydisperse systems. 44–50 In Fig. 3 the scattering from a series of dispersions under good solvent conditions, i.e. without Na2 CO3 , is shown. Due to the low polydispersity and the finely collimated X-ray beam, several form factor minima can be observed in the scatter-


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Figure 4: Scattering intensity as a function of q for 13 wt.% aqueous dispersions with various amounts of Na2 CO3 , as labeled. The solid lines are fits using the adhesive sphere mixture model. The model fits and data have been shifted along the intensity axis for clarity using the multiplicative factors shown to the right. The inset shows the intensity recorded for [Na2 CO3 ]=0.54 M along with the fit from the main panel and the intensity from the same sample recorded 30 min later.

Figure 5: Normalized second virial coefficient (top panel) as a function of added Na2 CO3 concentration as obtained from the scattering analysis in Fig. 4. Baxter parameter (bottom panel) as a function of added Na2 CO3 concentration and PEG layer thickness δ that in combination yield the above B values. The green symbols have been obtained by using δ values from the analysis of cryo-TEM measurements (cf. Fig. 6). Lines are shown in both panels as guides to the eye.

In the presence of Na2 CO3 the scattering at lower q is greater than what would be expected for hard-sphere interactions that capture the behavior under good solvent conditions. In addition, at the highest salt concentration in Fig. 4 an additional upturn at low q is observed, suggesting the presence of some larger aggregates. Since the dispersions for this part of the study were prepared on site at the synchrotron beamline, and aggregates that eventually dispersed were noted in more dilute dispersions in determining the phase behavior, care was taken to examine the time dependence of the scattering. As shown in the inset, the enhanced low-q scattering decreased with time, indicating that aggregates were indeed in the process of dispersing. The time scale for this dispersion process appears to be hours and the scattering from a fully relaxed structure was not recorded. In order to fit these data with the particle concentration kept constant at 13 wt.%, corre-

sponding to n = 2.8 · 10−8 nm−3 , it is necessary to introduce attractions. In the adhesive sphere model these are quantified by so-called stickiness parameters. In the modeling a single stickiness parameter, independent of particle size, was used. With this model there remain only two adjustable parameters, the stickiness parameter τ and the shell thickness δ that goes into the interaction diameters, e.g. for component i with core radius ri , σi = 2(ri + δ). The stickiness parameter subsumes the effect of the range and depth of short-range attractions 51 into a single parameter that is moreover directly related to the second virial coefficient. When salt is added to these systems not only may the attraction range and depth change, but the grafted layer may contract. This effect has been observed in recent computer simulations of the potential of mean force between particles covered with polymer brushes. 52 This


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tion and appearance of attraction from one another under worsened solvent conditions, cryoTEM imaging has been carried out with varying concentration of Na2 CO3 in the solvent. In these measurements particles are collected in a monolayer before being frozen along with the solvent. The sample images in Fig. 6 show that the particles tend to order. In addition, many particles can be found as part of close-lying pairs, where the separation appears to be governed by the extent of the PEG-grafts. Image analysis was used in this way to determine the thickness of the PEG layers as a function of the Na2 CO3 concentration, the result of which is shown in Fig. 6. There is a modest contraction of the PEG-graft for salt concentrations of 0.50.6 M. Such a contraction can be expected for preferential hydration of ions compared to the ethylene oxide groups of the PEG chains, 55 provided the ions avoid the PEG chains. There is indeed evidence for salt-deficient regions around PEG in bulk solution 56 and around surfactant head groups of ethylene oxide type. 57 The data in Fig. 6 are far from complete. However, if we assume that the contraction of the PEG layer occurs in a gradual manner, we can use a linear correlation for graft thickness and salt concentration to extract δ at the intermediate salt concentrations the scattering curves in Fig. 4 were recorded for. In this way the τ values can be determined that together with these δ values fit the scattering curves. The result is shown in the lower panel of Fig. 5. It shows that attractions are introduced gradually as the PEG-graft contracts, which is in qualitative agreement with the computer simulation results for densely grafted systems. 52 In addition, at the higher salt concentrations in Fig. 5, it is clear that the scattering model becomes independent of the shell thickness δ and the scattering is determined in this limit only by the attraction between particles. Grafted chains may undergo a structural transition to an ordered state when the solvent conditions change. 20–22 We assume such a transition to be correlated with the behavior of the corresponding bulk polymer solution. With this in mind we note that sufficient addition of Na2 CO3 to aqueous PEG solutions leads to seg-

effect is also evident experimentally in singleparticle measurements involving particles with adsorbed PEG. 53 In the modeling of the scattering data in Fig. 4 it was discovered that the several combinations of τ and δ yielded model fits that were indistinguishable from one another. In other words, based on an analysis of scattering data alone one cannot separately determine τ and δ. Instead, the scattering is controlled by the second virial coefficient for the adhesive sphere mixture model, which is given as 54 3 N 2π X 1 σi + σj B= xi xj 1− (1) 3 i,j=1 2 4τij where xi is the mole fraction of component i with hard-core diameter σi = 2(ri + δ) and core radius ri . The functions τij are the stickiness parameters that govern the attraction, which have been assumed to be given by a single parameter τ for interactions among all components. It is important that B in Eq. 1 not be normalized by its hard-sphere value, PN σi +σj 3 x x B HS = 2π , because the rei j i,j=1 3 2 sulting quantity just reflects changes in τ and misses the fact that the scattering is generally also affected by changes in δ. Therefore, we report results in terms of B/B0HS , where B0HS is the hard-sphere second virial coefficient with δ = 8 nm. In Fig. 5 the reduced second virial coefficient B/B0HS , extracted from single-parameter fits to the scattering curves in Fig. 4, is shown as a function of the concentration of added Na2 CO3 . The reduced second virial coefficient decreases as the Na2 CO3 concentration increases. This trend can be brought about through an increased attraction or a reduction in excluded volume or a combination of both. Figure 5 illustrates also that combinations of τ and δ values yield the same B value by showing two curves that depict τ values that give the B values necessary to fit the scattering data given a shell thickness of 8 nm as well as a vanishing shell thickness. Combinations of τ and δ that generate appropriate values of B can be found in the shaded area in the lower panel of Fig. 5. To disentangle the effects of polymer contrac-


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[Na2CO3] (M) Figure 6: Cryo-TEM images of particles dispersed in, from left to right, aqueous solutions of [Na2 CO3 ]=0, 0.5, and 0.6 M. A PEG-layer thickness was extracted from image analysis of 12, 55, and 65 close-lying particle pairs, respectively, for [Na2 CO3 ]=0, 0.5, and 0.6 M, which is shown in the graph along with a line to guide the eye. The scale bars correspond to, from left to right, 100, 100, and 200 nm in the cryo-TEM images. The left-most image is reproduced here from a previous study. 33 regation into a two-phase system comprising a dilute polymer solution with most of the salt and a semi-dilute polymer solution with little salt. 58 Further addition of salt causes precipitation of salt crystals. 59 The situation is drastically altered when PEG is dissolved in ethanol. At high temperatures PEG and ethanol are miscible but on lowering the temperature toward room temperature the PEG crystallizes. 29 It follows that we can expect to observe differences in interactions between PEG-grafted particles when they are dispersed in ethanol. Similar PEG-grafted pHFBMA particles as those

studied in aqueous solvents have been transferred into ethanol and the scattering intensity has been determined, which is shown in Fig. 7. The form factor for the same particles dispersed in aqueous solvent is also shown, which demonstrates that these non-crosslinked particles do not swell when introduced into ethanol. The scattering in ethanol is dramatically different from the scattering curves in Fig. 4. It decays at low q as q −d with the exponent d close to 2, which indicates an aggregated fractal structure. We note that the core-core van der Waals interaction is expected to be weaker in this case


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at the low particle concentrations investigated. At high salt concentrations particles become expelled from the solvent completely and collect at the sample surface or bottom. Results from analysis of cryo-TEM images show that the thickness of the PEG-graft decreases in what appears to be a gradual fashion as a function of added Na2 CO3 . Incorporating this trend in the analysis of SAXS measurements leads to a scenario in which attractions between particles appear as PEG layers contract. These attractions are subtle in that they combine with reductions in excluded volume to affect properties, such as the low-q behavior of the scattering intensity. The situation is dramatically altered when particles are dispersed in ethanol. Particles aggregate then into fractal structures.

ethanol 10 mM

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0.1 -1

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Figure 7: Scattering intensity pHFBMA particles dispersed in ethanol as a function of q, shown along with the scattering from dilute dispersions of the same particles dispersed in an aqueous solvent with 10 mM monovalent electrolyte. The form factor has been fit with a slightly polydisperse homogeneous sphere model shown by the solid line. The scattering data recorded in ethanol have been shifted vertically by a factor of 2 to better show the correspondence between the form factor minima.

ASSOCIATED CONTENT Supporting Information Treatment of size polydispersity in modeling scattering from interacting systems (Figure S1) and a table of parameter values obtained from the SAXS modeling in Figs. 3 and 4.

because ethanol matches the refractive index of the fluorinated particle cores even better than does water. It follows that the PEG-grafts induce a strong enough attraction between particles in ethanol to trigger non-equilibrium aggregation. In contrast, in aqueous solvents, when the concentration of Na2 CO3 is increased, PEGlayers are found to contract gradually, which is accompanied by increasing attractions of moderate magnitude between particles.

Appendix: Multiple scattering The replacement of hydrogen for fluorine has the advantage of greatly increasing the electron density and the scattering contrast of the particles. However, it was noted that the scattering contrast is increased so much that multiple scattering occurs to an appreciable extent. In Fig. 8 an enhancement is shown about the first few minima in the scattering from the concentration series with no added Na2 CO3 , shown in Fig. 3. As seen, the first intensity minimum becomes more shallow with increasing particle concentration. Increasing particle concentration also leads to an overall stronger scattering, which implies that the probability for multiple scattering increases. This effect is seen more clearly if comparisons are made with the scat-

CONCLUSIONS Well-defined colloidal dispersions of nearly monodisperse spheres with grafted PEG have been shown to crystallize or possibly aggregate into clusters under dilute conditions when a sufficient amount of Na2 CO3 is added. This occurs at a boundary in the phase diagram that shows little dependence on the particle concentration


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tensity is Hankel-transformed as Z ∞ I˜sngl (r) = 2π J0 (qr)Isngl (q)qdq

tering model, which for this range of q values is given essentially by the form factor. While the form factor fits the data well at low concentrations there is an increased mismatch around the first minimum as the particle concentration is increased.

After application of Eq. 2 the multiply scattered intensity then follows from an inverse Hankel transform as Z ∞ 1 Imult (q) = J0 (kr)I˜mult (r)rdr (4) 2π 0

4

20.33 wt% 15.536 wt% 10.511 wt% 5.636 wt% form factor

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10

The above prescription has been applied in modeling the scattering data in Fig. 8. The theoretical model intensity was used as the singly scattered intensity using for d the thickness of the sample-containing capillary (1.6 mm). This procedure avoids any problems with transforming experimental data extending over a finite q range and instead accounts for multiple scattering by smearing the model intensity. In addition, it is a prediction of the multiply scattered intensity, free of any adjustable parameters aside from those entering the model intensity. As shown in Fig. 8, taking multiple scattering into account yields near-perfect agreement with scattering data for all particle concentrations studied.

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0.12

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q (nm )

Figure 8: Enhancement about the first few minima in the scattering from samples in the absence of added Na2 CO3 as a function of particle concentration as labeled. The thin solid lines are fits to the theoretical model assuming single scattering only, whereas the thick solid lines have been obtained by accounting for multiple scattering. Some data have been shifted for clarity along the intensity axis by a multiplicative factor as labeled to the right.

Acknowledgement Financial support from the Swedish Research Council Röntgen ˚ Angström program is gratefully acknowledged. The authors thank Gunnel Karlsson (Biomikroskopienheten, Materialkemi, Kemicentrum, Lund Uiversity) for her expert help with the cryo-TEM measurements. The ESRF is acknowledged for providing the synchrotron beam time at ID2 and we thank Dr. Gudrun Lotze and Dr. Theyencheri Narayanan for assistance during the measurements.

Schelten and Schmatz 60 considered the effect of radiation multiply scattered in the forward direction, well suited to the situation commonly encountered in SAXS and SANS measurements. For radiation elastically scattered in the forward direction with a characteristic cross section that decays rapidly with increasing q, the intensity Imult (q) is related to the intensity from a singly scattering but otherwise identical system Isngl (q) as

References

k2 ˜ 2 2 ˜ I˜mult (r) = e−Isngl (0)/(k0 /d) 0 eIsngl (r)/(k0 /d) − 1 (2) d

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