Shear-Thickening in Mixed Suspensions of Silica Colloid and

Sep 2, 2014 - Over a narrow composition range, around the gel to liquid transition region, shear-thickening flow is observed. It is found that, for gi...
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Shear-Thickening in Mixed Suspensions of Silica Colloid and Oppositely Charged Polyethyleneimine Huan Zhang,†,‡ Guangcui Yuan,*,† Junhua Luo,†,‡ and Charles C. Han*,† †

State Key Laboratory of Polymer Physics and Chemistry, Joint Laboratory of Polymer Science and Materials, Beijing National Laboratory for Molecular Sciences, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China ‡ University of Chinese Academy of Sciences, Beijing 100049, China S Supporting Information *

ABSTRACT: The liquid−gel−liquid transition tuned by increasing concentration of linear and hyperbranched polyethyleneimine in suspension of silica colloids, and the accompanying shear-thickening phenomena, were investigated by rheological measurements. The influence from linear and hyperbranched polymer conformation and from different sizeratio between particle and polymer on the rheological properties of suspensions flocculated by absorbing polyelectrolyte were considered. Charge neutralization and bridging mechanism are the main reasons for the flocculation of silica colloid in this study. Because of charge reversal, the irreversible bridges are turned into flexible reversible bridges with increasing adsorption amount of oppositely charged polymer, which leads to an abrupt transition from gel to liquid. Over a narrow composition range, around the gel to liquid transition region, shear-thickening flow is observed. It is found that, for given particle volume fraction, the composition region exhibiting shear-thickening for mixed suspension with linear polyethyleneimine is broader than that for mixed suspension with hyperbranched polyethyleneimine, and the onset of shear-thickening depends only on size-ratio, regardless of the actual size of particle and polymer in the range of this study. The relationship between the gel to liquid transition and shear-thickening was discussed.

I. INTRODUCTION

mixed suspension, among which the most important one is the rheological behavior.8 A great number of studies have been made in the rheological behavior of mixed suspensions of colloid and adsorbing polymer.9 Generally, flocculation occurs when the surface coverage is low, and steric stabilization occurs when the particle surface is saturated with polymers. However, for some systems, such as aqueous colloidal dispersions consisting of Laponite and weak adsorbing poly(ethylene oxide), over a narrow composition range, close to the threshold for complete surface saturation, some samples exhibited abrupt shear-thickening behavior.10,11 It should be noted that the concept of surface coverage is rather imprecise since it depends greatly on the conformation of the adsorbed species, which in turn depends on the adsorbed amount.1,5 The shear-thickening behavior in colloidal suspension flocculated by adsorbing polymer is phenomenally attributed to the network formation through shear-induced bridging. The chance for bridge formation may increase during shear when the surfaces are not completely saturated, which will increase the size of aggregates and ultimately leading to the formation of gel.10,11 A similar shear

It is well-known that addition of small amounts of adsorbing polymer into stable colloidal dispersions can lead to flocculation. The absorbing polymers may be anionic, cationic, or nonionic in nature.1 Flocculation induced by absorbing polymers typically proceeds through either charge neutralization2 or particle bridging mechanism.3 In general, charge neutralization results from the adsorption of polyelectrolyte with relatively low molecular weight on oppositely charged particle surfaces, while bridging flocculation results from the adsorption of high molecular weight polymer simultaneously on more than one particle.4 No matter whether flocculation occurs by charge neutralization or by bridging mechanism, several processes are initiated when an polymer solution is added to a stable colloid suspension: mixing of polymers among the particles, adsorption of polymer chains on the particles, rearrangement of the adsorbed chains from their initial state to an eventually equilibrium conformation, collisions between particles with adsorbed polymer to form flocs, and breakup of flocs.5 From the kinetic point of view, the rates of each process are affected by a number of factors such as concentration and adsorption affinity, which will determine the final arrangement of adsorbed polymers.6,7 Any particular conformation of adsorbed polymers is related to the macroscopic properties of © 2014 American Chemical Society

Received: April 22, 2014 Revised: September 2, 2014 Published: September 2, 2014 11011

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In this paper, we will elucidate the inherent conformation effect arising from different structure of linear and hyperbranched PEI and the size-ratio effect by mixing particles and polymers with different size on the shear-thickening behavior of mixed suspension of silica and PEI. We will answer the question that for systems showing similar liquid−gel−liquid transitions why shear-thickening exhibits only in some of them (such as mixed suspension of PEI and silica) but not in others (such as mixed suspensions of poly(N-isopropylacrylamide) (PNIPAM) microgel and polystyrene (PS) microsphere). We will also answer the questions that why shear-thickening exhibits just over a narrow composition range and what the relationship is between this particular composition range and liquid−gel− liquid transition. The general explanation of liquid−gel−liquid transition in flocculated system is based on the surface coverage, but shear-thickening flow is resulted from the nonlinear elasticity combining with the entropic effect of extended bridges and forced desorption due to hydrodynamic effect. When the entropic effect is enhanced by the bridging with enough flexibility as in the cases of small particle mixed with long polymer chain10−17 and/or desorption is exacerbated by the weak affinity between polymer chains and particle surface as in the case of charge reversal,18 shear-thickening can be observed. The onset shear rate and the strength of the shearthickening flow that we can observe are determined by the characteristic relaxation time and the flexibility of bridges.

effect is also observed in nanosized silica suspensions in the presence of adsorbing poly(ethylene oxide) long chain.12−17 In all these reported systems, there is one thing in common: the systems are constituted with small particles and very large macromolecules in which the sizes of particles are comparable to or smaller than the characteristic sizes of polymer coils.10−17 Therefore, flexible polymers that are as large as possible are considered to be necessary for shear-thickening in a flocculating system, with weak affinity originated from the large curvatures of nanoparticle surface.15−17 In our previous work,18 the rheological properties of mixed suspensions of silica colloid (with hydrodynamic radius Rh of about 500 nm) and hyperbranched polyethyleneimine (hPEI, with Rh of about 50 nm) were studied as functions of composition ratio and pH value. In a very narrow range of composition ratio, shear-thickening was observed under a steady shear flow. Polyethyleneimine (PEI) is a weak cationic polyelectrolyte. Driven by the electrostatic force and other nonelectrostatic affinity, PEI is absorbable to the surface of negatively charged silica.19,20 Depending on the amount of added hPEI, i.e., whether the concentration of hPEI (Cp) is enough or not enough to fully cover the surface of a given volume fraction of particles, the addition of hPEI will induce stabilizing or flocculating effect between silica particles.18 The liquid−gel−liquid transition with increasing Cp is observed and shear-thickening behavior exhibits over a narrow Cp range following the occurrence of an abrupt gel to liquid transition. The shear-thickening is explained by the nonlinear elastic model of suspensions flocculated by reversible bridging proposed by Otsubo,21 in which the concept of reversible bridge is the key. That is, under certain conditions, the adsorption−desorption of polymer bridges on the particles can reversibly take place, and shear-thickening results from the rapid extension of reversible bridges under a steady shear flow. Because of the self-suppressed ionization of polyelectrolyte and the charge reversal of colloid, the affinity and conformation of the polymer bridge can be changed by the concentration of added polyelectrolyte. Thus, we suggest that the self-arrangement of polyelectrolyte bridge conformation with solution environments is crucial in understanding shear-thickening in mixed suspension of silica colloid and PEI, and the Cp* (characterizing the abrupt gel to liquid transition) is taken as an important concentration ratio of polymer to particle denoting the transition from irreversible bridging to reversible bridging.18 In our further study, on one hand, we find that, even when very small hPEI with Rh of 2 nm is mixed with silica colloid with Rh of 500 nm, similar shear-thickening effect is also observed. That is, the shear-thickening behavior in colloidal suspension flocculated by absorbing polymer can be observed not only in systems with small particles and very large macromolecules where the size of particles are comparable to or smaller than the characteristic size of polymer coil but also in systems with large colloids and very small polymer where the polymer−particle interaction can be approximated as adsorption on a flat surface. On the other hand, according to different synthetic methods, PEI possesses linear or hyperbranched structures. We investigated the rheological properties of mixed suspension of silica with linear PEI (lPEI) and found that shear-thickening occurs over a wide composition range even when the mixed suspension is in gel state. A Newtonian region at low shear rate (characterizing a reversible adsorption−desorption process) is not necessary for the appearance of shear-thickening.

II. EXPERIMENTAL SECTION Materials. The values of averaged weight molar mass (Mw), hydrodynamic radius (Rh), and radius of gyration (Rg) used in the present study are determined by light scattering measurement (ALV/ DLS/SLS-5022F) unless noted otherwise. Three batches of silica colloid with different sizes were synthesized according to Stöber’s method.22 The Rh of silica colloids used in the present study are 505 ± 10, 214 ± 10, and 130 ± 8 nm. Correspondingly, the zeta potentials of silica colloids measured in deionized water at 0.2 mg/mL are −51.4 ± 2, −45.9 ± 1 and −35.8 ± 2 mV, denoting the negative charge on the surface. One linear polyethyleneimine (lPEI) with Mw of 250K g/mol (Mw was provided by the supplier which caculated from precursor of lPEI) and Rh of 136 ± 8 nm was purchased from Polysciences Inc. The conformation parameter f (with f = Rg/Rh) of lPEI is about 1.5, indicating that the lPEI shows coil-like structure in solution.23 One hyperbranched PEI (hPEI) with Mw of 830K g/mol was also purchased from Polysciences Inc. This hPEI sample is the same as we used in our previous paper.18 But the Mw value indicated in previous work18 (Mw = 750K g/mol) is provided by the supplier which is determined from the viscosimeter method. The viscosimeter method cannot provide absolute Mw of a hyperbranched polymer since there is no standard reference of hyperbranched polymer. In this work, we remeasure the Mw value of hPEI by using static light scattering with differential refractive index (dn/dc) of 0.210 mL/g. The other two hPEIs with Mw 540K and 8K g/mol were purchased from Alfa Aesar. The Rh of three hPEI are 58 ± 2, 39 ± 3, and 2 ± 0.5 nm, respectively. The hPEI contains the primary, secondary, and tertiary amine groups in a 1:2:1 ratio. The f parameter for hPEI is about 1.0, indicating its star-like conformation in solution.23 Mixed solvent of dimethyl sulfoxide (DMSO) and H2O (87/13 v/v) was used to match the refractive index of the silica colloid in order to minimize the van der Waals force. Powders of silica colloid were carefully dispersed in the mixed solvent through 20 min of ultrasonication. An appropriate amount of PEI solution was added into silica suspension and homogenized by ultrasonication for another 20 min. Mixed suspensions were used for measurements immediately after preparation. For particle suspensions used in this study, the volume fraction of silica colloid (ΦSiO2) was kept as 0.3. The effect of ΦSiO2 (in the range of 0.1−0.4) on the critical 11012

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shear rate of the onset of shear-thickening (γ̇crit) has been studied in our previous paper.18 With the decrease of the average interparticle distance by increasing ΦSiO2, the onset of shear-thickening shifts to lower shear rate value, and the viscosity enhancement in shearthickening region increases. Based on this consideration, ΦSiO2 is kept as 0.3 to ensure that the onsets of shear-thickening under different situations in this study (different silica size and polymer size) occur in a reasonable and measurable shear rate region. The ΦSiO2 was calculated by the bulk density of silica colloids, ρ = 1.96 g/cm3. The rheological properties are closely related to the surface coverage ratio of polymer on silica surface. The amount of polymer mixed with the silica particle is quantified by the mass of polymer per unit surface area of silica (Cp) in units of mg/m2 (equivalent to adsorption density under unsaturated adsorption state). Since silica particles with different sizes are used in this study, the advantage of using Cp in units of mg/ m2 (relative polymer concentration to particle surface area instead of total volume) is that the Cp value can be compared regardless of the actual particle size. The Cp (mg/m2) was obtained through the following equation: Cp = (4/3)πR3ρmPEI/(4πR2mSiO2), where R (in units of nm) is the hydrodynamic radius of silica colloid, ρ is the density of silica colloid (ρ = 1.96 g/cm3), mPEI (with unit of mg) is the mass of PEI, and mSiO2 (in units of mg) is the mass of silica colloid. Rheological Measurement. The rheological properties of the suspensions were investigated by a stress-controlled rheometer (Haake MARS). Experimental details for rheological measurement are the same as our previous study.18 All the rheological tests were performed in a 35 mm cone−plate geometry. The gap angle between the cone and plate was 1°. DMSO/H2O mixed solvent was poured into a trap surrounding the cone−plate to avoid solvent evaporation during measurements. Samples were presheared for 100 at 1 s−1 to ensure being homogenized. The initial conditioning of suspensions before the measurements did not affect the rheological data. We wait for 5 min before each rheological measurement. Viscosity was measured by gradually changing the steady shear rate. The storage and loss moduli, G′ and G″, were measured by dynamic frequency sweep with a maximum stress of 1 Pa.

Figure 1. Liquid−gel−liquid transition of mixed suspensions with ΦSiO2 = 0.3 and various concentrations of polyelectrolyte: (a) hPEI (Mw = 830K g/mol, the same data from our previous study18) and (b) lPEI (Mw = 250K g/mol). The storage modulus G′ and loss modulus G″ are the corresponding value at ω = 1 rad/s from oscillatory frequency sweep measurements. The shadow area indicates the concentration region with shear-thickening occurrence.

further increasing concentration of polyelectrolyte, both kinds of mixtures show an abrupt transition from gel to liquid. The Cp which leads to an abrupt transition of mixed suspension from gel to liquid on the modulus−concentration profile is marked as Cp*. The differences and similarities of these two kinds of mixtures are related to the flocculation mechanism in this specific system. As we will discuss later, both charge neutralization and bridging play a part in flocculation of silica particles by PEI polyelectrolyte. Conformation Effect on Shear-Thickening Behavior. The gel to liquid transition is accompanied by a concentration region with shear-thickening behavior, which is marked as the shadow area in Figure 1. The shadow area for mixed suspension with lPEI is broader than that for mixed suspension with hPEI. For mixed suspension with hPEI, the shear-thickening phenomenon shows up only after the occurrence of gel to liquid transition, but for mixed suspension with lPEI, the shearthickening phenomenon shows up even in gel state. The corresponding steady rheological properties of samples without PEI and with PEI concentration around the shadow area are shown in Figure 2. To compare the relative extent of shearthickening, the suspension viscosity (with particles, η) is normalized by the solution viscosity (without particles, ηp). Therefore, the coordinates of vertical axis in Figure 2a,b for particle suspensions is relative viscosity ηr with ηr = η/ηp. The ηp determined by a Ubbelohde viscometer is also provided in Figure 2c as a function of PEI concentration. The abscissa C in units of mg/mL is used to distinct with the polymer concentration per unit surface area Cp (mg/m2) which is used in other figures because Figure 2c shows the viscosity of PEI solution without silica particle. But there is one−one correspondence between the actual PEI concentration for data

III. RESULTS Liquid−Gel−Liquid Transition. The liquid−gel−liquid transition of mixed suspensions tuned by increasing the concentration of hPEI has been shown in our previous work.18 Without hPEI, the suspension of silica (ΦSiO2 = 0.3) is a mobile liquid. But when a small amount of hPEI is mixed with the suspension of silica, the mixture becomes a gel which cannot flow by gravity. With further increasing concentration of hPEI, the mixed suspension becomes fluid again. Similar reentrance transition triggered by increasing concentration of lPEI has also been found. Figure 1 compares the liquid−gel− liquid transitions of mixed suspensions with ΦSiO2 = 0.3 and various concentration of hyperbranched and linear PEI. The Rh of silica colloid is 505 nm. The storage modulus G′ and loss modulus G″ are obtained from the corresponding value at ω = 1 rad/s from oscillatory frequency sweep measurements. Data for mixed suspensions with hPEI have been shown in Figure 4 of our previous study.18 The liquid−gel−liquid transition can be told from the comparison of the modulus values: The G′ is smaller than G″ at small Cp values corresponding to liquid state, G′ becomes larger than G″ at medium Cp values corresponding to gel state, and then G′ becomes smaller than G″ at large Cp values corresponding to liquid state again. The transition from liquid to gel with the addition of a small amount of polyelectrolyte is abrupt in mixtures with hPEI but is gradual in mixtures with lPEI. It means that hPEI is a more effective flocculant in triggering gel formation. While with 11013

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connection of colloids. Therefore, the flow becomes Newtonian with enough PEI. With PEI concentration in an intermediate range, the rheological response between mixtures with hPEI and lPEI is different. For mixed colloid suspensions with hPEI (Figures 1a and 2a), shear-thickening behavior is observed only when the mixture across the gel−liquid transition line with increasing hPEI concentration. Shear-thickening flow observed in liquid state is always accompanied by a Newtonian flow at low shear rate. In gel state, the progressively breaking down of the network under shear leads to shear-thinning over a wide range of shear rate. For mixed colloid suspensions with lPEI (Figures 1b and 2b), shear-thickening behavior is observed in a wider range of lPEI concentration, spanning over the gel−liquid transition zone. In liquid state, there is a Newtonian region at low shear rate followed by a shear-thickening region at high shear rate, which is similar to the situation in mixed suspensions with hPEI. However, for shear-thickening observed in gel state with existence of a persistent network, there is no Newtonian region at low shear rate. As we will discuss later, a reversible adsorption−desorption process (condition) characterized by a Newtonian flow at low shear rate is a sufficient, but not necessary, condition for occurrence of shear-thickening. Size-Ratio Effect on Shear-Thickening Behavior. On one hand, the size-ratio effect could be investigated by changing the size of silica particles. Figure 3 shows the shear-thickening

Figure 2. Relative viscosity as a function of shear rate for mixed suspensions with ΦSiO2 = 0.3 and various concentrations of polyelectrolyte: (a) hPEI (Mw = 830K g/mol, the same data from our previous study18) and (b) lPEI (Mw = 250K g/mol). (c) Viscosity of PEI solutions without particles (ηp) as a function of PEI concentration.

Figure 3. Relative viscosity as a function of shear rate for various silica colloids at fixed volume fraction ΦSiO2 = 0.3 mixing with the same hPEI. The hydrodynamic radius of silica colloids are 130, 214, and 505 nm, and the corresponding Cp* is 1.14 (brown ■), 1.05 (red ●), and 0.98 mg/m2 (blue ▲).

points in Figure 2c and the PEI concentration used in Figure 2a,b in units of mg/m2. It can be seen that there is no significant change in the viscosity of PEI solution in the experimental PEI concentration range. Significant change of viscosity in mixed suspension arises from different structure formed at different mixing ratio. Without PEI, the silica suspension is electrostatically stabilized and the flow is Newtonian. For suspensions with enough PEI, either hyperbranched or linear, the flows are also Newtonian. The reason, which has been proposed in our previous work18 with mixed suspension of silica and hPEI being investigated, is that the surfaces of silica particles are totally covered by the adsorbed PEI. The saturated adsorption PEI layer will reverse the charge of colloid and in turn supply repulsive electrostatic force and steric force to restabilize silica colloids. Furthermore, the excess free moving PEI molecules will inhibit the bridging formation when the adsorption of free polymers to the surface is much faster than the diffusion and

behavior in mixed suspensions of silica colloids with different sizes at their Cp*. The volume fraction of silica colloid is ΦSiO2 = 0.3. The hPEI with Mw = 830K g/mol is used. The suspension viscosity η is normalized by the solution viscosity ηp (ηr = η/ ηp). It can be found that the onset of shear-thickening shifts to lower shear rate value for smaller particles. On the other hand, the size-ratio effect could be investigated by changing the molecular weight of PEI. Three hPEI macromolecule with different Mw were mixed with silica colloid with Rh = 505 nm. The shear-thickening behavior for mixtures with three PEIs at their Cp* are showed in Figure 4. It is interesting to found that, even when the polymer is very small (2 nm for Mw = 8K g/mol) as compared to the size of silica, shear-thickening, although not very strong, is observed in a narrow composition range when the shear rate reaches a certain value. The Cp* for hPEI with Mw = 830K g/mol is higher than 11014

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repulsion between particles with similar charges. Bridges form only when the tails or loops of an absorbed polymer can span the gap between particles. The lPEI has a coil-like structure. A small amount of linear cationic polyelectrolyte tends to rearrange with a rather flat conformation on the vast surface of negative charged particles. Therefore, the addition of lPEI initially do not greatly enhance flocculation because the flatly coated polymer have no tails or loops extending into the solution phase far enough to catch another stable particles.2 With increasing amount of lPEI, the bridging mechanism begins to play a more important part. Bridges form as the average surface area of particles that could be occupied by each lPEI macromolecules decreases; meanwhile, the adsorbed lPEIs remain flexible because the affinity becomes weak as a result of self-suppressed ionization and charge neutralization. However, for hPEI with a star-like structure, the expanding of absorbed hyperbranched cationic polyelectrolyte on the particle surface is restricted, which allows the standing arms of hPEI to span the gap between particles and form a bridge. Therefore, even when the concentration of polyelectrolyte is low, hPEI is effective in forming a bridging network, and less amount of hPEI is needed for gel formation. The hPEI bridges connecting the silica particles into a standing network has been well identified in our previous work18 particularly by scanning electron microscopy. Although there are differences in the initial growth period, the modulus of bridging gels in mixed suspensions with lPEI and hPEI reach almost the same maximum strength. The abrupt transitions from gel to liquid in mixtures with hPEI and that with lPEI indicate charge reversal of silica particles. The situation is quite clear when comparing with other neutral mixed suspensions of colloid and uncharged polymers, such as our previous study on the mixed suspensions of PNIPAM microgel and PS microsphere.25,26 The mixed suspensions of PNIPAM microgel and PS microsphere show similar liquid−gel−liquid transition with increasing concentration of microgel, but the transition from gel to liquid is gradual.25,26 We believe the difference is related to the driving force of adsorption. In mixed suspensions of PNIPAM microgel and PS microsphere, hydrophobic interaction is the driving force for adsorption; thus, the adsorption free energy per segment (or attaching point) is independent of dosage. At high surface coverage, the gel to liquid transition directly results from the gradual decreasing of the number of bridges, which is proportional to θ(1 − θ) (here, θ is the ratio of surface area covered by polymer to the total surface area of the colloidal particles). However, in mixed suspensions of PEI and silica colloid, the adsorption affinity of weak cationic PEI to the negatively charged silica surface is dosage dependent because the adsorption is greatly affected by the solution environment (i.e., electrostatic force).18−20,27−30 The adsorption features and electrokinetic properties of PEI on silica surfaces have been studied in detail by other research groups, such as Mészáros’s20 and Ong’s.29 On one hand, the charge density of PEI is a function of pH and its degree of protonation or dissociation increases gradually from zero at pH 12.0 to 1.0 at pH 2.0.19 Therefore, with increasing concentration of alkaline PEI, the ionization is self-suppressed due to the increase of solution pH.19 On the other hand, the adsorption of PEI onto the negatively charged surface will cause charge neutralization and subsequent charge reversal, which will suppress further adsorption.20 Overall, due to the self-suppressed ionization of polyelectrolyte and the charge reversal of colloid, the affinity

Figure 4. Relative viscosity as a function of shear rate for various hPEIs mixing with the same silica at ΦSiO2 = 0.3. The Rh of silica is 505 nm. The molecular weight of hPEI used and the corresponding Cp* for mixed suspension are 0.98 mg/m2 (brown ■) for Mw = 830K g/mol, 0.89 mg/m2 (red ●) for Mw = 540K g/mol, and 0.88 mg/m2 (blue ▲) for Mw = 8K g/mol.

that for 540K and 8K g/mol because the adsorbed layer is thicker. It can be found that γ̇crit shifts to high shear rate for smaller polymers, and the bridges formed by larger polymers tend to have stronger elasticity which enhances the shearthickening effect. A question arising naturally is that if the shear-thickening profile disappears when the molecular weight of polymer is decreased to a certain level. The essential of this question is related to the relative sizes of the interparticle distance and polymer chains. As revealed by Shibayama et al.,24 from a percolation-based viewpoint, the size of the polymer chains is necessary to be comparable with the interparticle distance for a shear-induced sol−gel transition. It is shown in our previous paper that,18 by using silica particle with Rh = 505 nm and hPEI with Mw = 830K, the increase of the average distance between particles (decrease of particle volume fraction) leads to a shift of γ̇crit to high shear rate value and leads to a decrease of viscosity enhancement in shearthickening region. Similar results, enhancement and absence of shear-thickening at higher and lower ΦSiO2, respectively, are observed in mixed suspension with hPEI of Mw 8K. The data of particle volume fraction effect on shear-thickening are provided in the Supporting Information (see Figure S1) as a confirmation of the occurrence of shear-thickening in suspensions with low molecular weight polymer.

IV. DISCUSSION Flocculation Mechanism. Since PEI has positive charges while the surface of bare silica has negative charges, both charge neutralization and bridging mechanism may account for the flocculation of silica particles. The gradual increase of modulus in lPEI mixtures and the abrupt increase of modulus in hPEI mixtures during the liquid−gel transition indicate that charge neutralization may play a significant part, at least in dilute concentration of polyelectrolyte where the polymer chains have a strong affinity to the particle surface. Adsorption occurs on condition that the adsorption free energy of the whole polymer chain is able to compensate for the entropy loss which occurs on transfer of polymer from solution to an interface. The initial absorbed polymer chains which have a flexible and extended conformation tend to rearrange to an eventually flat configuration with long trains attaching to the surface, if space allows. There exists a range of effective electrical 11015

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and conformation of the polymer bridges can be changed by the concentration of PEI.18−20 Charge reversal and decreasing number of bridges with increasing surface coverage are the reasons for the abrupt transition from gel to liquid in mixtures with hPEI or lPEI. The concentration of PEI in which an abrupt transition of mixed suspension from gel to liquid occurs is Cp*. Cp* in units of mg/m2 should be a constant assuming that the affinity is independent of solution environment. It should be noted that, in Figure 3, for silica colloid with a given volume fraction, Cp* in units of mg/m2 is slightly increased for smaller particles. The total surface area increases with decreasing radii of the particle. Theoretically, to neutralize the surface charge of silica with the same surface density, the mass of PEI needed for small particle with Rh = 130 nm is 3.9 times (505/130 = 3.9) of that for large particle with Rh = 505 nm. However, experimentally, due to the self-suppressed ionization of hPEI at high concentration, the mass of PEI needed to cause gel−liquid transition in suspension of small particles with Rh = 130 nm is 4.5 times that in suspension of large particles with Rh = 505 nm, which results in the increase of Cp* (in units of mg/m2) for small particles. Shear-Thickening Mechanism. Various theories have been proposed to account for the molecular origin of shearthickening, such as shear-induced non-Gaussian chain stretching,31 network reorganization,32 and/or shear-induced crosslinking.33 Unfortunately, rheological experiments do not give direct evidence of chain stretching and structural evolution. Actually, when shear is applied to suspensions flocculated by bridging, several processes including alignment of particles, stretching of bridges, desorption and recapture of polymer, etc., may take place simultaneously, which has been observed by Cabane et al.14 and Shibayama et al.24 through small-angle neutron scattering equipped with in-situ shear field. Rheological response reflects the combination of all these process. Otsubo21 proposed a nonlinear elastic model of suspensions flocculated by bridging based on the competition between hydrodynamic effect of forced desorption and entropic effect of extended bridges. Correspondingly, two primarily factors used to explain the appearance of shear-thickening are total adsorption affinity and flexibility of bridge, both of which cannot be presented in quantitative terms. It is worth noting that the total adsorption affinity refers to a whole change (or a bridge) and the adsorption affinity refers to a segment. Usually, for an adsorbed polymer, a portion of chain segments is in direct contact with particle surface. The conformation of an absorbed polymer can be explained by the proportion of chain segments attaching on the surface (trains) to chain segments extending into the solvent (loops and tails). Considering one polymer adsorbed on the surface, when the fraction of loops which extend into the solution increases, the anchoring points which attach to the surface decrease. Total adsorption affinity and flexibility of bridge are interdependent and are both affected by a number of factors such as composition mixing ratio. Therefore, bridging conformation, the embodiment of all influencing factors, is the available variable in qualitatively or semiquantitatively discussing the flow behavior. Here, we use Otsubo’s model21 concerning the bridging conformation to explain the shear-thickening behavior in mixed suspension of silica and oppositely charged polyelectrolyte. According to Otsubo,34 there are two kinds of bridges between particles, irreversible and reversible, depending on the adsorption affinity. The critical adsorption free energy needed for polymer adsorption is of the order of 0.3kT per segment,35

and there are various interactions that can easily satisfy this requirement.1 So it is possible that the adsorption−desorption of an segment reversibly occurs when the adsorption energy on the order of thermal energy. When the adsorption affinity is strong, desorption of polymer chains hardly takes place because of the multipoint attachments to the surface. All anchoring sites of an adsorbed polymer are not able to desorb simultaneously; thus, the bridging flocculation is essentially irreversible. When the affinity of polymer chains to the surface is not strong enough, the adsorbed polymer chains remain a loose and flexible conformation. Weak affinity and the resulting smaller fractions of anchoring points cause the adsorption−desorption process to occur reversibly by Brownian motion. The bridges under such conditions are called reversible bridges. As discussed above, the abrupt transition from gel to liquid at Cp* denotes the transition of affinity from strong to weak, hence the transition of bridge from irreversible to reversible. When the bridges become reversible at Cp > Cp*, with increasing shear rate, a Newtonian−shear-thickening−shearthinning flow transition is observed in mixed suspensions with hPEI and lPEI (Figure 2a,b). At low shear rate, the adsorption− desorption process reversibly occurs, so the flow is Newtonian; with increasing shear rate, the flexible bridges are rapidly extended before desorption, and shear-thickening flow appears as the relaxation time scale of extension is shorter than that of desorption. With further increasing the shear rate, the very high extension causes desorption of polymer from the particles and results in shear-thinning. The intrinsic mechanism of shear-thickening is the restoring forces of extended bridges in shear fields. Inherent flexibility of adsorbed polymers is a necessary condition for shear-thickening in flocculating system, and weak affinity is a sufficient condition for the expression of the inherent flexibility. The viewpoint above is achieved from several aspects. First, a reversible adsorption−desorption process characterized by a Newtonian flow in experimental shear rate range before the occurrence of shear-thickening is a manifestation of weak affinity, but in average sense (since rheological measurement provide averaged information), the extending bridges which contribute to the shear-thickening effect is not necessarily reversible. The hPEI has a star-like structure while lPEI has a coil-like structure. Shear-thickening was observed in gel state of mixed suspension with lPEI but not in mixed suspension with hPEI (Figure 2a,b). As mentioned above, an initial absorbed polymer chain which has a flexible and extended conformation will tend to rearrange to an eventually flat conformation with long trains attaching to the surface. But if the chance for the adsorbed polymer chain to reach equilibrium state is reduced, flexible bridge will exist. For example, when adsorption and bridging are faster than rearrangement of initial absorbed polymer chains and/or there is no more space for initial absorbed polymer chains to rearrange at some dosage, the initial flexible conformation will be kept.36 Shear-thickening can be observed even when the average affinity is not very weak and the suspension is still in gel state. Second, the bridges formed by larger polymers tend to have stronger elasticity under shear. At Cp ≈ Cp*, the extent of shearthickening is enhanced by hPEI with higher molecular weight (Figure 4). Third, the self-suppressed ionization of polyelectrolyte and the charge reversal of colloid with increasing adsorption amount of oppositely charged polymers can weaken affinity, leading to the formation of flexible bridges in certain 11016

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composition range and thus providing the opportunity for shear-thickening occurrence in silica and PEI mixed system. In another system we have studied for the mixed suspension of PS microsphere and PNIPAM microgel,25,26 which shows very similar liquid−gel−liquid transition, shear-thickening has not been observed in the whole experimental composition range. The main differences of these two systems include the driving force for adsorption and the flexibility of bridges. PNIPAM microgel is absorbed on the surface of PS microsphere with a pancake-like structure due to the strong and dosage independent affinity. Still more, flexibility is absent in PNIPAM microgel due to the internal cross-linked structure. These two reasons inhibit the occurrence of shear-thickening in mixed suspension of PS microsphere and PNIAPM microgel. Relaxation Time of the Bridge. The hydrodynamic effect forces desorption which leads to shear-thinning. The entropic effect leads to the restoring of extended bridges which provides additional resistance to flow. Therefore, shear-thickening flow is supposed to appear when the relaxation time scale of extension is shorter than that of desorption. The onset of shearthickening could be explained by considering the Weissenberg number (γ̇τ), which represents the relative strength of the shear field. Shear-thickening flow occurs at γ̇critτ ≈ 1.37,38 γ̇crit is the critical shear rate of the onset of shear-thickening, and τ is characteristic relaxation time of bridges. The shear rate dependence of viscosity data under various conditions (composition ratio, size-ratio, etc.) can be superimposed by aligning the viscosity curve along the x-axis with different shift factor τs, so that the onset of shear-thickening under various conditions appear at the same reduced shear rate γ̇τs = 1. The resulting shift factor τs is then proportional to the relaxation time of the bridge. Comparing the shift factor τs will allow us to discuss various effects on the relaxation time of the bridge. Figure 5a shows the Cp dependence of τs obtained through superimposing the data in Figure 2b. For given concentration of silica particle, τs drastically decreases with the increasing PEI concentration. Thus, a direct conclusion is that the relaxation time of the bridge decreases with the increasing PEI concentration, which is consistent with our discussion that the interaction becomes weaker due to increase in pH and the “self-suppressed ionization”. The data of Figure 3 (particle-size effect) and Figure 4 (polymer-size effect) are superimposed to give a generally description of the size-ratio effect on shearthickening. The obtained the size-ratio effect on τs is shown in Figure 5b. Rp and Rc are the hydrodynamic radius of polymer and colloid, respectively. It should be noted that threshold of shear-thickening depends on the averaged interparticle distance which could be tuned by particle volume fraction and/or particle size. But for a given particle volume fraction, it is found that the dependence of τs on the size-ratio is uniform and τs increases with larger Rp/Rc. It seems that the onset of shearthickening depends only on the size-ratio, regardless of the actual size of particles and polymers (in the range of the current study). We can imagine that if the particles used are the same, the shortest chain will first reach the limits of its extensibility during stretching. So the shear-thickening in mixed suspension with smallest hPEI appear at largest shear rate. And reversely, flexible polymers as large as possible cause significant shearthickening effect at small shear rate. That is the reason for why most previous reported systems10−17 with shear-thickening effect are generally constituted with small particles and very large macromolecules.

Figure 5. (a) Cp dependence of τs obtained through superimposing the data in Figure 2b. (b) Rp/Rc dependence of τs. In (b), date points symbolized by blue stars were originally from Figure 3 and date points symbolized by red triangles were originally from Figure 4.

V. CONCLUSION The rheological properties of mixed suspension of silica colloid with linear and hyperbranched PEI were studied. For concentrated suspensions with ΦSiO2 = 0.3, liquid−gel−liquid transition induced by increasing concentration of PEI was observed, and it was compared with the liquid−gel−liquid transition observed in mixed suspension of polystyrene microsphere flocculated by poly(N-isopropylacrylamide) microgel bridge. Charge neutralization and bridging formation account for the flocculation of mixed suspension of silica colloid and polyethyleneimine. Charge reversal with increasing adsorption amount of oppositely charged polymer leads to an abrupt transition from gel to liquid, which denotes the transition of affinity from strong to weak, hence the transition of bridge from irreversible to reversible. The formation of flexible bridges due to the weakening of affinity provides the chance to observe the shear-thickening behavior in colloidal suspension flocculated by polymer bridges, even when the size of polymer is very small. The shear-thickening arises from the rapid extension of the bridges when the relaxation time scale of extension is shorter than that of desorption. The characteristic relaxation time and the flexibility of the bridge will determine at which shear rate can we observe the shear-thickening flow and how strong is the observed shear-thickening. Longer polymer bridges contribute to stronger shear-thickening behavior in a wider composition range.

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ASSOCIATED CONTENT

S Supporting Information *

Particle volume fraction effect on the shear-thickening for mixed suspension of silica particle with Rh 505 nm and hPEI with Mw 8K g/mol. This material is available free of charge via the Internet at http://pubs.acs.org. 11017

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (G.Y.). *E-mail: [email protected] (C.C.H.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Basic Research Program of China (973 Program, 2012CB821503).

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