Structure and Yielding of Colloidal Silica Gels Varying the Range of

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Structure and Yielding of Colloidal Silica Gels Varying the Range of Interparticle Interactions Fabrice Brunel, Isabelle Pochard, Sandrine Gauffinet, Martin Turesson, and Christophe Labbez J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b04047 • Publication Date (Web): 10 Jun 2016 Downloaded from http://pubs.acs.org on June 12, 2016

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The Journal of Physical Chemistry

Structure and Yielding of Colloidal Silica Gels varying the Range of Interparticle Interactions Fabrice Brunel,∗,† Isabelle Pochard,‡ Sandrine Gauffinet,¶ Martin Turesson,§ and Christophe Labbez¶ †C2P2, UMR 5265 CNRS - CPE, villeurbanne, France ‡UTINAM, UMR 6213 CNRS, Université de Bourgogne-Franche-Comté, Besançon, France ¶ICB, UMR 6303 CNRS, Université de Bourgogne, Dijon, France §Lund University, Sweden E-mail: [email protected] Phone: +0033 (0)4 72431767. Fax: +0033 (0)4 724317957

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Abstract The relationship between interaction range, structure, fluid-gel transition and viscoelastic properties of silica dispersions at intermediate volume fraction, Φv ≈ 0.1 and in alkaline conditions, pH = 9 was investigated. For this purpose, rheological, physicochemical and structural (synchrotron-SAXS) analyses were combined. The range of interaction and the aggregation state of the dispersions was tuned by adding either divalent counterions (Ca2+ ) or polycounterions (PDDA). With increasing calcium chloride concentration, a progressive aggregation was observed which precludes a fluid-gel transition at above 75 mM of calcium chloride. In this case, the aggregation mechanism is driven by short range ion-ion correlations. Upon addition of PDDA, a fluid-gel transition, at a much lower concentration, followed by a reentrant gel-fluid transition was observed. The gel formation with PDDA was induced by charge neutralization and longer range polymer bridging interactions. The re-fluidification at high PDDA concentrations was explained by the over-compensation of the charge of the silica particles and by the steric repulsions induced by the polycation chains. Rheological measurements on the so-obtained gels reveal broad yielding transition with two steps when the size of the silica particle clusters exceeds ≈ 0.5 µm.

Introduction Colloidal gels, formed by a percolated network of chain and cluster of particles, present many applications principally because they behave as a solid at rather low particle volume fractions. Silica colloidal gels are probably among the most used due to the prevalence of silicon on the earth crust. They can also serve as a model system for yielding and flow of hydrated cement paste, due to their chemical stability and homogeneity relative to cementitious systems, 1,2 the particularity we are the most interested in. Indeed, both silica particles and the main cement hydrate, i.e. calcium silicate hydrate nanoparticles (C-S-H), bare a negative surface charge in alkaline conditions originating from the titration of the silanol 2


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groups which cover their surface, and thus present similar surface and interfacial properties. As an example, multivalent counterions such as calcium ions are known to overcharge C-S-H and silica particles in alkaline conditions 3 due to the strong ion-ion correlations prevailing in these systems. Ion-ion correlations were also shown to lead to short range attractive interactions between C-S-H particles, one of the key of cement setting. 4 Despite the importance of the process of yielding and flow in colloidal gels, for their casting and applications, their mechanisms are still under debate, especially for low volume fraction, while high volume fraction gels have been fairly well-studied and understood. However, it has received a recent interest in colloidal gels at intermediate volume fraction for which the transition from solid-like behaviour to flow is marked by a two step transition. Pham et al. first reported a two-step yielding for colloidal glasses with hard-sphere interaction and an adjustable short-range depletion attraction and suggested that the first step corresponds to the breaking of bonds between particles while the second at larger strain corresponds the breakup of nearest-neighbour cages. 5,6 Koumakis et al. later reported a two step yielding process for colloidal gels (Φv < 0.6) and noted that the second yield strain increased as volume fraction decreased due to enhancement of structural inhomogeneities. 7 The second yield step could not be explained as a consequence of cage breaking and was interpreted as the breakup of larger structural units. Shao et al. attributed the first step in the strain response to the rupture of the attractive gel network while the second step was attributed to the breakup of dense clusters. 8 Hsiao et al. suggested a totally different mechanism to connect the microstructure of soft materials to their non linear rheology by visualizing colloidal gels using confocal microscopy. 9 They observed that non linear yielding behaviours could be due to erosion of rigid clusters that persist far beyond the first yield strain. However, in this case, one would see a monotonous slope change in the G′ versus γ curves rather than two well-defined yielding steps. Mohraz and co-workers proposed that the first yield point stems from bond rotations and structural rearrangements while the second yield point corresponds to the rupture of interparticle bonds. 10,11 Finally, using a combination of oscillatory rheology

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with simultaneous USANS measurements Kim et al. identified large-scale microstructure to be critical in the yielding process of heterogeneous colloidal gels. 12 They suggested that after the first yielding-step the compression of the fractal domains led to the formation of large voids between dense agglomerates which were later destroy at high strain amplitude. These results call for expanded studies in order to more directly connect the microstructure of attractive colloidal gel with non-linear yielding behaviour.

In this paper we propose to investigate the importance of the range (and magnitude) of interparticle interactions on the two step yielding process of colloidal gels which has been previously overlooked. The study of the yielding process is further combined with a detailed analysis of the electrokinetic and structural properties, obtained via Synchrotron-SAXS experiments, of the silica suspensions. Short range interactions were achieved by addition of calcium divalent counterions known to induce attractive ion correlation forces while long range bridging forces were obtained by replacing Ca2+ with PDDA polycations. PDDA exhibits permanent positive charge (quaternary ammonium) and is therefore widely used for adsorption and flocculation of negatively charged particles, see e.g. 13,14 The magnitude, on the other hand, was tuned by a progressive addition of counterions, either calcium or polycation. Finally, the mechanisms of the two step yielding process are discussed in light of these new experimental data.

Experimental Section Polyelectrolytes Poly(diallyl-dimethyl-ammonium chloride) linear polycations (PDDA), see Figure 1, with an average molecular weights of M w ≈ 100,000 g/mol (35 wt. % in water, R pH = 4.5) was purchased from Sigma-Aldrich . The molar mass was verified from the

value of the intrinsic viscosity, in 0.5 M NaCl, according to the MHKS coefficient given by Dautzenberg et al.. 15 The result was 42,000 g/mol.

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Figure 1: Chemical structure of the linear poly(diallyldimethylammonium chloride)

Sample preparation The colloidal silica suspension, supplied by Rhone Poulenc as RP1, had a weight concentration of 10.25 %wt., corresponding to a volume fraction Φv = 0.041. The particle size was measured by dynamic light scattering (see ζ-potential section for details) and the radius found was of 43 ± 14 nm, which corresponds to a polydispersity index of 0.32. 10 %wt. Silica suspensions were prepared by mixing suitable amounts of silica suspension, NaOH (1 mol/L, Prolabo) solution and polymer solution. The added amounts of NaOH and polymer solutions are sufficiently low not to vary significantly the silica mass concentration which remains at about 10 %wt. The purpose of the NaOH addition was to raise the pH suspensions in the alkaline region (pH = 9) so that the silica particles bear a negative surface charge density (≈ 0.4e/nm2 ). In these conditions, the PDDA adsorption and ion-ion correlations are promoted. Suspensions were also prepared replacing PDDA by calcium chloride (Prolabo) in exactly the same way as the one used for the silica/PDDA gel making. After addition of calcium chloride or PDDA, the samples were left overnight under mild shaking conditions. Sweep strain as well as SAXS experiments were performed on the pellets obtained after centrifugation (9000 rpm, 15 min) of these equilibrated samples, see below. The volume fractions of the pellets were determined by weighing the initial suspension amount to be centrifuged and the withdrawn solvent amount after centrifugation. A volume fraction of Φv = 0.10 ± 0.02 was obtained for all the studied samples, see Figure 2. 5



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Figure 2: Volume fraction of the silica dispersions Φv after centrifugation at 9000 rpm for 15 min as a function of the amount of added PDDA (g/L) or Ca2+ (mM = mmol/L).

ζ-potential The size distribution and the ζ-potential of the silica particles in all the prepared suspensions were determined at 173◦ with a Zetasizer NanoZS (Malvern instrument Ltd, Worcestershire, UK) equipped with a 4 mW He/Ne laser beam operating at λ = 633 nm. All measurements were performed at 25.0 ± 0.2 ◦ C . The self-correlation function was expanded in a power series (Cumulants methods). The value of the ζ-potential was derived using Smoluchowski’s equation from electrophoretic mobility measurements. 16 For sample close to neutralization, partial flocculation might occurs and measurement were carried out on the upper part of the dispersion. Rheological measurements Rheological measurements were made with a controlled strain rheometer (ARES Rheometric Scientific, USA). In this case, the samples were submitted to a sinusoidal strain γ, and the resulting stress τ was recorded. The complex modulus of the studied material (G∗ =

τ ) γ

can be written as G∗ = G′ + iG′′ , with G′ the storage

or elastic modulus and G′′ the loss or viscous modulus. In this work, we used a classical parallel plate geometry, modified in order to keep the sample in water-saturated air. The 6


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study sample consisted of a disk of paste, with a diameter of 50 mm and a 2 mm gap. With these dimensions and the characteristics of the rheometer, the lowest strain possible was 0.01 % (10−4 ). In addition, serrated plates were also used to avoid wall slip. Prior to the measurement, silica suspensions were spun down by centrifugation at 9000 rpm for 15 min. The volume fraction of each silica paste was assayed after centrifugation (see Figure 2). A sharp increase in the volume fraction was observed below 1 g/L of added PDDA or 25 mM of added calcium chloride. This hindered the rheological investigations in these ranges of PDDA or Ca2+ concentrations as the rheological response strongly depends on the solid volume fraction. Therefore, the rheological measurements were only reported for samples with added PDDA equal or greater to 1.6 g/L and added CaCl2 equal or greater to 50 mM for which the volume fraction is fairly constant at 0.10. Samples were pre-sheared (ω = 1 rad.s−1 for 10 s) before each measurement to remove the effect of shear history. After 1 min waiting time, increasing strain sweep measurements (from 0.01% to 1000%) were performed at 20 ◦ C, ω = 1 rad.s−1 .

SAXS measurements The structural aspect of the silica nanoparticle dispersions was analysed by SAXS measurements at the SWING beamline of the French synchrotron source SOLEIL (Soleil, Saint Aubin, France). Experiments were carried out at an energy of 15 keV with a sample to detector distance of 6.5 m, using the 2D AVIEX CCD detector. Data were normalized to take into account the beam decay, detector sensitivity and sample transmission. The parasitic scattering intensity (cell windows and air) was subtracted from the total scattering intensity. The radial integration of the 2D SAXS at SWING was performed using the FOXTROT software developed at Soleil. Intensities were subsequently converted to absolute units using a sample of water as reference. SASfit 0.93.3 software was used for analysing and plotting small angle scattering curves.

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Figure 3: Example of a SAXS curve (black points) and corresponding fitting curves (blue and red curves) Silica/PDDA dispersions (Φv = 0.10)

At high q-range, all samples exhibits a q −4 slope characteristic of spherical particles (see Figure 3). These feature are identical in all samples. However, at very low q-range a slope change is observed depending on the sample, revealing some differences in the large lengthscale organization of the particles. Therefore, for large q values, ranging from 0.02 to 0.2 Å−1 , scattering intensities are fitted using the form factor of polydisperse spheres with a log-normal radii distribution (r = 50 nm and σ = 0.5) which remains constant for all samples. Then, at low q-range, the structure factor of a mass fractal is added to account for the different structural organization of each gels. The size of the fractal initiator is set equal to the radius of the sphere determined above. With these prescriptions, the structure factor can be written as:

D S(q) = 1 + D r0

Z

∞

rD−3 h(r, ξ) 0

sin qr 2 r dr qr

(1)

With ξ size of the fractal aggregate, r0 the size of the fractal initiator (i.e. equal to the radius of the particle r), D the fractal dimension and h(r, ξ) = exp(r/ξ) the cut-off functions. The fitting parameters of structure factor are of course not very accurate due to the limited 8


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low q-range. However, the structural information given by both the fractal dimension and the size of the fractal aggregates can be summarized within the calculation of the packing density (dpacking ) defined as the volume fraction of particles in a fractal aggregate of radius ξ: 17

dpacking =

ξ r0

(D−3)

(2)

Results ζ-potential measurements Figure 4 displays the ζ-potential of silica suspensions (Φv = 0.041, pH = 9) upon addition of calcium chloride or PDDA. The initial ζ-potential of the particles is negative (-35 mV) due to the presence of deprotonated silanol groups on the silica particle surface at pH 9. A charge reversal is observed at about 4 g/L of added polycation. Above this value, the adsorbed polymer layer gives rise to a positive effective charge on the particles. A macroscopic gelation is observed around 4 g/L of added polycation, indicating strong attractive interactions. This macroscopic gel will be refer as strongly attractive gels. This macroscopic flocculation vanishes up for higher PDDA concentrations for which the ζ-potential becomes positive. On either side of the charge reversal (i.e. lower and higher concentrations) the resulting gels are obtained only after settling of the clusters via centrifugation and will be refer as weakly attractive gels.

The addition of Ca2+ as counterions of the silanol groups leads to a partial neutralization of the effective surface charge from -35 mV initially to -10 mV at [Ca2+ ] = 100 mM. ζ-potential measurements for greater calcium chloride concentrations were not reliable due to high ionic strength and values close to zero. Under these conditions, no charge reversal 9



0

[Ca2+] (mM) 40 60

20

80

100

120

40

ζ-potential (mV)

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PDDA Ca2+

20

0

−20

−40 −2

0

2

4 6 [PDDA] (g/L)

8

10

Figure 4: ζ-potential of silica particles at pH 9 during addition of (a) PDDA and (b) calcium chloride. Lines are only guide to eyes

was observed. However, the suspension is destabilized i.e. an aggregation is observed for [Ca2+ ] > 25 mM through the strong increase of the average diameter measured by dynamic light scattering (data not show).

SAXS experiments Figure 5 shows the absolute intensity of the scattered light as a function of the scattering vector (0.002 < q < 0.5 Å−1 ) for silica dispersions at various PDDA and calcium chloride concentrations. The fitting parameters obtained with the mass fractal structure factor as set forth in the experimental part are reported in Table 1.

First of all, with calcium chloride addition, the size of the fractal aggregates ξ continuously increases from 110 to more than 10 µm and the fractal dimension D decreases from 1.85 to ≈ 1.4. Second, with PDDA addition, the evolution of the fractal structure is not linear over the concentration range. From 0.8 to 3.2 g/L polymer, ξ continuously increases and large flocs (ξ ≥ 10 µm ) having a loose or ramified structure (D = 1.3) are observed

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Figure 5: SAXS curves of silica dispersions (Φv = 0.10) obtained with increasing (a) PDDA or (b) calcium chloride concentrations.

around 3 g/L. Above 3.2 g/L, smaller and denser clusters are formed. As an example, at 8.1 g/L, ξ decreases to reach 130 nm and D increases up to 1.72. The information given by both the fractal dimension and the size of the fractal aggregate were summarized through the calculation of the packing density (dpacking ) given on Figure 6. dpacking decreases upon calcium addition from 0.3 down to approximately 10−4 . With PDDA, dpacking reaches a minimum (≈ 10−4 ) for intermediate concentrations ([PDDA] ≈ 3 g/L) which corresponds to large flocs (ξ > 10 µm) having a loose and ramified structure (D < 1.3). On each side of this minimum, dpacking increases up to 0.06 for the lowest PDDA concentration and 0.2 for the highest.

Rheological measurements Typical non linear oscillatory response of the colloidal silica gels with 4.8 g/L of PDDA is presented in Figure 7 at ω = 1 rad.s−1 . At low strain, G′ shows a typical plateau with G0N , the plateau modulus, characteristic of the linear viscoelastic domain. The size of this region provides information on how far the material can be deformed before the onset of the structural breakdown at a given strain. The linear viscoelastic domain is followed by a drop in G′ values at a given γ value named γc1 , accompanied by a maximum in G′′ versus γ. For most of colloidal dispersions, G′ typically exhibits continuous decrease with γ for

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Table 1: Parameters obtained by fitting the SAXS curves of the silica particles gels at various PDDA or calcium concentrations. [PDDA] (g/L) 0.8 1.6 2.4 3.2 4.9 6.4 8.1 [Ca2+ ] (mM) 10 25 50 75 100 150 200

Size of the fractal aggregate ξ (nm) 260 710 > 10,000 > 10,000 1800 280 130 Size of the fractal aggregate ξ (nm) 110 260 390 3700 7200 > 10,000 > 10,000

Fractal Dimension D 1.38 1.32 1.31 1.33 1,34 1.52 1.72 Fractal Dimension D 1.85 1.57 1.50 1.42 1.41 1.39 1.41

γ > γc1 . However, in most of the cases reported here, this drop is not continuous. Actually, upon increasing the deformation, the rheological curves exhibit a second critical strain γc2 (i.e. G′ reaches a plateau and G′′ a second maximum). Above γc2 , G′ becomes smaller than G′′ , indicating that the material exhibits liquid-like behaviour after yielding. The values of γc1 and γc2 provide information on the dispersion yielding features, like the length of interactions and the energies required to rearrange or disrupt the percolation network of the solid.

All the G′ and G′′ versus γ curves are shown in Figure 8a and b respectively at various added amount of PDDA, and in Figure 9 for CaCl2 . Table 2 reports the rheological features of all curves i.e., G0N , γc1 and γc2 (when present) values. The values of γc1 and γc2 have been determined, examining both the G′ and G′′ versus strain curves (i.e. the elastic modulus decreases dramatically and the viscous modulus reaches a maximum). For the calcium chloride concentration of 25 and 50 mM, the dispersion was too fluid to get accurate measurements

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Figure 6: Packing density dpacking of silica dispersions as a function of PDDA or Ca2+ concentration.

Figure 7: Storage G′ and loss G′′ modulus versus strain γ obtained from a typical strain sweep experiment on a gel with [PDDA] = 4.8 g/L.

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of the critical strain.

Figure 8: Strain sweep measurements on silica dispersions (Φv = 0.10) obtained with increasing PDDA concentration.

Figure 9: Strain sweep measurements on silica dispersions (Φv = 0.10) obtained with increasing calcium chloride concentration.

A one-step yielding behaviour is observed for [Ca2+ ] < 150 mM while a two-step yielding is noticed for 150 and 200 mM of added CaCl2 . With polymer addition, two-step yielding is observed between 2.4 and 4.8 g/L and one-step yielding is observed above and below this range of concentrations i.e., at 1.6, 6.2 and 8.1 g/L of PDDA. Values of the first yield strain γc1 are below 1 % for all cases except for the two highest PDDA concentrations, for which γc1 is highly enhanced with 6 % and 10 % respectively. γc2 presents strikingly different values between calcium (10 %) and PDDA (40 %). 14


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For [Ca2+ ] > 50 mM, G0N is fairly constant between 70 and 91 kPa. In the case of PDDA, the evolution of G0N is qualitatively different. A non monotonous variation as a function of the polycation concentration is observed. G0N increases up to 136 kPa for 3.2 g/L and then decreases with further polycation addition to finally reach its smallest values at the highest PDDA addition, i.e. G0N = 2 103 Pa for 8.1 g/L.

Table 2: Estimated values of the plateau modulus and critical strains defined as in Figure 7 for silica dispersions at various PDDA or calcium chloride concentrations.

G0N (kPa) from G′ γc1 (%) from G′′ 2 γc (%)

1.6 42 0.3 0.6 -

2.4 125 0.1 0.3 40

[PDDA] 3.2 136 0.1 0.3 40

(g/L) 4.8 80 0.1 0.3 40

6.2 5 6 10 -

8.1 2 10 26 -

25 1.1 0.1 -

50 22 0.1 -

[Ca2+ ] 75 70 0.4 0.8 -

(mM) 100 91 0.4 0.8 -

150 75 0.4 0.5 10

200 75 0.4 0.5 10

Discussion Sol-gel transitions Without any CaCl2 or PDDA addition, the suspension of silica nanoparticles, with a radius of 42 ± 14 nm, remains stable due to a strong electrical double layer repulsion (ζ-potential is largely negative). The increase in ζ-potential upon calcium chloride addition attests for the presence of Ca2+ at the negatively-charged silica surface and is in full line with previous studies, see e.g. Labbez et al. 18 Multivalent counterions like Ca2+ are known to give rise to an attractive interaction between equally charged surfaces due to ion-ion correlations, 4,18 leading to the formation of small clusters which eventually flocculate into a larger percolated network as described by A. Zaccone. 19 This phenomenon is well-observed here on the CaCl2 /silica system through the macroscopic aggregation observed for [Ca2+ ] > 25 mM and the increase in fractal aggregates and decrease in fractal dimension (Table 1) corroborated 15



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by the decrease in the packing density (Figure 6).

With PDDA addition, the close relationship between the size of fractal aggregates ξ and the effective charge neutralisation of the silica particles shows that the cluster-cluster aggregation is mostly governed by electrical double-layer features as already observed elsewhere. 20 For the low added amount of PDDA (typically below 2 g/L), flocculation is limited (size of fractal aggregates below 1 µm) due to still significant electrostatic repulsions (ζ-potential ≪ 0). Between 2 and 4 g/L of polymer, the charge of the particles is further screened up to neutralization, as seen from ζ-potential curve which increases from -30 mV to 0 mV. As a result the clusters grow until they eventually percolate to form a gel (ξ > 10 µm). The large second critical strain observed further (γc2 = 40 %) demonstrates that polymer bridging is taking place (see section below). In other words, the aggregation which eventually leads to gelation is both the results of charge screening and polymer bridging. Upon further addition of polycation, above 4 g/L, polycations overcompensate the charge of the particles as characterized by the sign reversal of the ζ-potential curve. This reentrant fluid transition, schematized in Figure 10, is rather generic, see e.g. the experimental work of Bauer 14 and Pickrahn et al., 21 and is well explained by electrostatic interactions, see e.g. the simulation work of Skepo and Linse 22 and theoretical work of Shklovskii et al. 23 at the level of the primitive level. Recently, the group of Borkovec has provided similar results by combining electrophoretic measurements, dynamic light scattering and interparticle force measurements on diluted latex particle dispersions which clearly demonstrates the electrostatic restabilization at high polyelectrolyte dosage. 24,25

The significant changes in the gel structure revealed by small angles X-ray scattering intensities are in good agreement with the two limiting regimes of colloid aggregation. 26 That is, a fast aggregation (i.e. strong attractive interactions between particles) leads to large aggregate (ξ > 10 µm) having a loose and ramified structure (D ≪ 2). This is the

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Figure 10: Flocculation/redispersion of silica particles upon PDDA addition. From left to right: stable colloidal suspension (electric double layer repulsion), flocculated particles (attractive polymer bridging and charge neutralization), redispersed colloidal suspension (electrostatic and steric repulsion).

case for the intermediate PDDA concentration and for the high end calcium concentration. This could also be seen in Figure 6 which shows the variation of the packing density. On the contrary, a slow aggregation hamper by electrostatic repulsions results in denser structures with D ≈ 2 and ξ < 1 µm, as it is the case at low calcium concentrations and of high PDDA concentrations. One further remarks, that in the range of concentration studied where ([Ca2+ ] ≥ 75 mM or [PDDA] ≥ 1.6 g/L), the particle volume fraction, Φv ≈ 10%, is well below the critical value for adhesive hard sphere, Φc = 26%, 27 as well as below the lowest critical volume fraction for square well hard spheres, Φc = 14 % 28 (i.e. for an interaction range comparable to the diameter of the particle whatever the range of inter-particle interactions). This result seems to indicate that all our gels are formed by homogeneous percolation.

Rheological behaviours The rheological properties and gel structure are nicely consistent as illustrated in Figure 11. Indeed, the plateau modulus G0N , the fractal dimension D and the volume fraction are ruled by a power-law relationship. Shih et al. demonstrated the existence of a scaling relationship to explain the elastic properties of colloidal gels by considering the structure of the gel network as a collection of close-packed fractal flocs of colloidal particles. 29 G0N depends on the number of contact points in the percolated network of particles, which is in turn related to

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D and Φv .

Figure 11: Power-law relationship between the plateau modulus G0N , the fractal dimension D and the volume fraction Φv .

In the literature, it is clearly shown that repulsive systems give rise to a one-step yielding behaviour while attractive colloids may lead to a two-step yielding behaviour. 5,7–9,11,30 These features are well-observed here for silica nanoparticles in presence of calcium chloride or PDDA counterions. More particularly, the G′ versus γ curves exhibit a one-step yielding for weakly attractive gel ([CaCl2 ] < 150 mM, [PDDA] < 2.4 g/L and [PDDA] > 4.8 g/L) while a two-step yielding is observed for which a strongly attractive gel is formed ([CaCl2 ] ≥ 150 mM, 2.4 ≤ [PDDA] ≤ 4.8 g/L). This is to our knowledge the first time that two-step yielding behaviour is reported for colloidal particles interacting via ionic correlations. Two yielding-steps in the rheological curves mean two different yielding mechanisms, i.e. two kinds of interactions (length and energy) in the gel network. A two-step yielding behaviour is well established for colloidal systems at high volume fractions (i.e. colloidal glass), and was attributed to, first, bond breaking between particles and, second, cage breaking. 5,30 Koumakis 7 et al and Shao et al 8 extended this interpretation to low volume fraction attractive gels. Other interpretation of complex yielding in low volume fraction attractive gels were given. 9,10 While, Chan et al 11 proposed bond rotation and breakage to explain the observed

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two-step yielding process. Hsiao et al 9 claimed for bond breakage by erosion of rigid clusters. However, complex yielding behaviour in low volume fraction gels are still under debate.

The first yield strain γc1 observed here is below 1 % in the case of CaCl2 addition and for unsaturated-PDDA samples (< 6.2 g/L), attesting for short-length structural changes in the system, followed by the yielding of the system. The significantly greater γc1 value for [PDDA] > 4.8 g/L is a clear signature of the longer-range interaction between particles induced by polymer bridging. Moreover, this first critical strain is always present, whatever the structural behavior of the gels (weak or strong attractive gels as defined previously). Therefore, this first critical strain is attributed to bond breaking between particles. These interactions are of short length and weak energy in the case of calcium (ionic correlations) and become of longer range when the particles are polymer-saturated revealing polymer bridging.

For strongly attractive gels ([CaCl2 ] ≥ 150 mM and [PDDA] = 2.4 and 4.8 g/L), γc1 is followed by a second yielding step which appears to be correlated to the long range fractal organization observed by SAXS. The interpretation of the second yield strain, observed here, is far less straightforward. In this case, γc1 probably corresponds to the breakage of the junctions between large flocs having a ramified structure. 12,31,32 At higher strain amplitude (γc1 < γ < γc2 ) the reorganization of the large flocs could lead to the densification of the structure. The gel would then undergo a micro-structural change, from a fractal gel to a more glassy state, resulting in the strain-hardening. This mechanism would rapidly reach a maximum and at increasing strain amplitude (γ > γc2 ) would cause the breakage of intercluster bonds and the depercolation of the whole network into a suspension of disconnected clusters (G′ < G′′ ) as suggested by Kim et al. 12 This structural breakdown mechanism provides a reasonable explanation to the current experimental results for the two-step yielding process and can be associated to the presence of large-scale structures.

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Finally, the value of γc2 is four times greater with PDDA (40 %) than those with Ca2+ (10 %). Moreover, the second step yielding appears as a plateau of the elastic modulus in the case of calcium where the elastic modulus rises up with PDDA. These differences can not be explained by structural differences since the fractal dimension and the size of the fractal aggregates are very much the same in both cases. One possible explanation relies on the range of the interactions. Indeed, ion correlation forces, in the case of Ca2+ , are much shorter range than bridging forces induced by the PDDA chains. An increased interaction range via polymer bridging may reasonably explain a greater viscoelastic reorganization under shear and, thus, higher γc2 .

Conclusions The scope of this paper was to rationalize the yielding properties of low volume fraction silica gels in relationship with the microstructural organization and the range of interparticle interactions. We observed fluid-gel transition with Ca2+ concentration and a fluid-gel transition followed by a reentrant fluid transition upon increasing PDDA addition. In the case of Ca2+ addition, the gel formation (particle aggregation) is due to short range attractive interactions induced by ionic correlations, while in the case of PDDA , gels form due to longer range attractive interactions mediated by bridging forces. The re-fluidification in the latter case is due to the overcharging of the silica particles by PDDA and steric repulsions. When the interaction is strongly attractive (i.e. [PDDA] ≈ 3 g/L or [Ca2+ ] > 100 mM), cluster aggregation occurs rapidly leading to loosely packed clusters network having large dimensions, as observed by SAXS. In this case, a two-step yielding process is observed. This result confirms the link between the two-step yielding process and the presence of largescale heterogeneities recently evidenced by Helgeson et al.. 12,31 The first yielding step can be attributed to the breakup of junctions between large flocs followed by the compression

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and elongation of dense fractal domains resulting in strain-hardening up to the destruction of intercluster bonds above the second critical strain. The two-step yielding have been observed in a wide variety of colloidal gel formed using depletion attraction, 5–7 screening of electrostatic interaction 33 and polymer bridging. 21,31 This is, to our knowledge, the first time that a two-step yielding behaviour is reported in a low volume fraction gel as a result of short-range ionic correlation attraction. Furthermore, using either short-range ionic correlation interaction or longer range interaction mediated by polymer bridging we were able to investigate the impact of the interaction range on the two step yielding process, which has been previously overlooked. The range of the interparticle interactions significantly change the magnitude of the G′ overshoot as well as the strain amplitude where the second yielding is observed. We hypothesized that this behaviour can be explained by the longer range interaction induced by polymer bridging, as compared to interactions mediated by ion-ion correlations, which would allow for greater viscoplastic reorganization under shear, see e.g. 11 Such reorganization of colloidal gels has been reported from various attractive particle interactions, 5–9,11,12 as well as, induced dipoles from applied electric and magnetic fields. 34–39 Indeed, when a magnetic field is applied, ferromagnetic particles align themselves to form well-defined columnar structures with tougher backbone networks compare with the non-magnetic suspension. 40,41 The tunable and reversible rheological properties of magnetic fluids are of great commercial interest for many engineering applications such as shock absorbers and dampers in aerospace. However, high-quality magnetorheological fluids are expensive and therefore limited in commercial feasibility for large scale use as commodity materials. On the contrary, tuning rheological properties using polymer bridging interactions could be used for the development of ductile cementitious material, as described in a previous study. 13 Further research is needed to clarify the impact of the interaction range on the structural reorganization in colloidal gels under shear by using polycations having different molar masses or topologies.

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Acknowledgement The author thanks Aurélien Thureau for his help on the SOLEIL Synchrotron : SWING beamline. This work was supported by the "Agence Nationale de la Recherche" grant number ANR-BRIDGE-10-JCJC-080901

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Structure and Yielding of Colloidal Silica Gels Varying the Range of

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