Aggregation and gelation kinetics of fumed silica–ethanol suspensions

Journal of Colloid and Interface Science 304 (2006) 359–369 www.elsevier.com/locate/jcis

Aggregation and gelation kinetics of fumed silica–ethanol suspensions William E. Smith, Charles F. Zukoski ∗ Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received 3 June 2006; accepted 9 September 2006 Available online 14 September 2006

Abstract The kinetics of aggregation and gelation of fumed silica suspended in ethanol were investigated as a function of volume fraction. At low particle concentrations, gelation is well described by aggregation into a primary minimum arising from hydrogen bonding and dispersion forces. The gelation is extremely slow due to an energetic barrier (∼25kT ) in the interparticle potential associated with solvation forces. The solvation forces also contribute to the formation of a secondary minimum in the interparticle potential. The depth of this minimum (∼3kT ) is sufficient that, at a critical particle concentration, long-range diffusion is arrested due to the short-range attractions and the cooperative nature of particle interactions, as described by mode coupling theory. The presence of the secondary minimum is also observed in the microstructure of the gels studied using X-ray scattering. These observations reinforce the importance of understanding the role of solvent–particle interactions in manipulating suspension properties. © 2006 Elsevier Inc. All rights reserved. Keywords: Aggregation kinetics; Colloidal gels; Dynamic light scattering; Fumed silica; Mode coupling theory; Solvation forces

1. Introduction Fumed silica suspensions are known for their thixotropic behavior, which is associated with the build up of aggregates and gelation [1,2]. Thixotropy in general is a poorly understood phenomenon as it links mechanical properties to changes in microstructure and state of aggregation, typically in dense and strongly interacting systems [3,4]. Here we explore the gelation of fumed silica in ethanol. We present evidence that the pair interaction energy at separations on the order of the solvent diameter leads to two different gelation mechanisms. One mechanism acts at low volume fractions and is related to particles diffusing over a primary maximum in the interparticle potential. This maximum is correlated with solvent molecules that associate with the silica surface and are displaced when the particles fall into a primary attractive minimum. As particles diffuse into the primary minimum, aggregates grow that ultimately span space forming a gel. The primary minimum is sufficiently deep that aggregation is expected at all measurable volume fractions. The second gelation mechanism is * Corresponding author. Fax: +1 217 333 5052.

E-mail address: [email protected] (C.F. Zukoski). 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.09.016

associated with a shallow secondary minimum. Gelation due to this minimum is associated with cooperative phenomena and is only observed when the volume fraction exceeds a critical value. Fumed silica is composed of primary particles of silica (∼10 nm) irreversibly fused together to form an open branched particle [1], Fig. 1. Previous studies on the geometrical effects of fumed silica on suspension properties were carried out by eliminating the interparticle forces [5,6]. These “hard” fumed silica particles only experienced volume exclusion interactions and showed rheological properties similar to hard spheres when the particle concentration was scaled with the free volume in the suspension. These studies demonstrated the ability of these branched particles to occupy the same volume as an equivalent sphere but with lower mass. In the study reported here where the unaltered fumed silica particles are suspended in ethanol, we anticipate that the important forces governing the suspension properties will be dispersion and electrostatic forces as well as solvation forces due to the hydrogen bonding of the ethanol to the surface silanol groups [7]. The degree these forces influence the suspension behavior can be shown by a comparison of the results of the hard fumed silica suspensions with particles capable of hydrogen bonding to one another and to the solvent.

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radius a at a center-to-center separation r is given by   AH 2a 2 r 2 − 4a 2 2a 2 + + ln ΦA (r) = − 6 r 2 − 4a 2 r2 r2 and   εp − εc 2 3hP υe (n2p − n2c )2 3 + √ , AH = kT 4 εp + εc 16 2 (n2p + n2c )3/2

Fig. 1. Scanning electron micrograph of fumed silica particles.

These observations are significant in developing methods for manipulating particle interactions to achieve desired mechanical properties and time-dependent behavior. The strength of interaction between particles is one of the key parameters in determination of the state of aggregation and mechanical properties of colloidal suspensions. The particle–particle interaction potential can be manipulated by altering the continuous phase of the suspension, where small changes in temperature, pH, ionic strength, and concentration or molecular weight of polymer can drastically alter the properties of the suspension [8,9]. Changes in the interparticle potential not only change the state of particle aggregation, but the kinetics of transformations as well. In many applications, the time required to achieve a given state is important. This study indicates that with proper manipulation of solvent–surface interactions, one can achieve either slow gelation that continues to stiffen with time due to diffusion into the primary minimum or rapid gelation associated with a shallow secondary minimum. Below in Section 2, standard aggregation models are discussed that describe the kinetics of aggregation and gelation when particles diffuse over maxima in interparticle potentials to form essentially irreversible bonds. These models have a venerable history and the work is briefly reviewed to distinguish this approach from more recent models of gelation based on mode coupling theory (MCT), where the strength of attraction is sufficiently weak that gelation is a reversible phenomenon. In Section 3, the experimental system is described, along with the use of dynamic light scattering as a noninvasive technique for investigating the rates of aggregation and gelation. Section 4 contains the results of the study, including slow gelation at low volume fractions and a narrow volume fraction range where the gel times become shorter than can be measured with the experimental techniques. Conclusions are drawn in Section 5. 2. Theoretical background The typical starting point in investigating particle–particle interactions is through the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, which contributes interparticle attractions to van der Waals dispersion forces and repulsive forces to electrostatic interactions [9]. The DLVO theory is the sum of the two potentials between two spheres as described below. The van der Waals attraction potential ΦA between spheres of equal

(1)

(2)

where AH is the Hamaker coefficient, k is the Boltzmann constant, T is the absolute temperature, ε is the dielectric constant, n is the refractive index, the subscripts p and c correspond to the properties of the particle and continuous phase, respectively, hP is the Planck constant, and υe is the characteristic adsorption frequency. The potential energy of repulsion ΦR for spheres following the approximations for superposition of single sphere potentials at a surface separation, h = r − 2a, is  2   kT 2 ezψ0 ΦR (r) = 32πεc ε0 a (3) tanh exp(−κh) ze 4kT and



κ=

  e2 i zi2 ci0 1/2 , εc ε0 kT

(4)

where ε0 is the vacuum permittivity, z is the valence number of the ion, e is the charge of an electron, ψ0 is the potential at the interface, κ −1 is the Debye length, and ci0 is the bulk concentration of ion species i [9]. An approximation of ψ0 is the zeta potential, which is extractable from measurements of the electrophoretic mobility. The concepts of the DLVO theory can be used in studying aggregation of colloidal suspensions. In the dilute limit, multibody interactions can be ignored and the flux J of particles coming together is J=

4kT  3ηc a ∞ 2a

ρ2 exp(Φ/kT ) r 2 G(r)

,

(5)

dr

where ρ is the particle number density, ηc is the continuous phase viscosity, Φ is the interparticle potential, and G(r) is a hydrodynamic correction term approximated as being equal to one [9]. For aggregation without any repulsive forces, Φ can be approximated as being zero. Thus, 2φkT ρ , πηc a 3 where φ is the particle volume fraction

J0 =

4 φ = ρ πa 3 . 3 The stability ratio is

(6)

(7)

J0 (8) , J leading to the characteristic time for doublet formation td , where the number of particles is reduced to half of its original value

W=

td =

πηc a 3 W . φkT

(9)

W.E. Smith, C.F. Zukoski / Journal of Colloid and Interface Science 304 (2006) 359–369

An approximation relating W and the maximum in the interparticle potential Φmax is     Φmax W = W∞ + 0.25 exp (10) −1 , kT where W∞ , the stability ratio due to dispersion forces alone, incorporates the finite range of attractions and the hydrodynamic interactions [9]. In the case of an energetic barrier, the particles must diffuse over the barrier to reach the primary minimum and form an aggregate. Gels are produced by the growth of percolating aggregates. Such cases have been studied using latex particles in an aqueous suspension, where there is not only a deep minimum due to van der Waals forces but also a high electrostatic barrier. The kinetics of aggregation can be tuned with the addition of salt to these suspensions, which reduces the barrier height allowing the suspensions to aggregate and form a gel. This type of gelation phenomenon has been studied over a wide range of concentrations ranging from extremely dilute [10] to concentrated [11]. While the DLVO relationships strictly apply only in the pair limit, Bremer et al. [12] developed a model for the kinetics of bulk suspension gelation for particles interacting via a DLVO potential. This model begins by calculating gel times for particles forming fractal clusters without the presence of an energetic barrier. For systems that undergo this mechanism of gelation, the effective volume fraction φeff increases to a value of unity at the gel point. φeff can be related to the size of the aggregating cluster Rc by using a fractal dimension Df given by  3−Df Rc . φeff = φ (11) a The gelation time for fast aggregation tgel,fast for concentrated suspensions is   6Df 3Df πηc a 3 3/(Df −3) tgel,fast ≈ 1 − (12) , + φ 2Df + 3 Df + 6 kT where Df = 1.8 for diffusion limited cluster aggregation, due to the assumption that there are no energetic barriers. Bremer et al. also reported experimental data from other studies supporting the usefulness of Eq. (12) in describing slow aggregation as well, where Df = 2.2 for reaction limited cluster aggregation. This would result in tgel = tgel,fast W.

(13)

The DLVO theory was developed for a pair potential composed of a summation of van der Waals and electrostatic forces (i.e., the summation of Eqs. (1) and (3)). However, particles experience a variety of different forces, including hydrogen bonding and solvation forces [13]. If the forms of these potentials are known, Eqs. (5)–(6), (8)–(13) remain valid, allowing the use of rates of aggregation and gelation to probe the interparticle potential. Due to the form of the potential used, Eqs. (5)–(6), (8)–(13) assume that the particles diffuse into a primary minimum where the particles are irreversibly aggregated (i.e., the rate of disaggregation is negligible).

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A second gelation mechanism is now introduced that differs significantly from the primary minimum gels described by DLVO theory in that it accounts for systems where the attractive well is sufficiently shallow that for particles to diffuse together and apart. This second gelation mechanism occurs in many systems where the primary minimum is occluded by non-DLVO forces [14–17] and is characterized by a finite volume fraction being required to see gel formation. Recently the cooperative phenomena associated with this type of gelation has been formulated in mode coupling theory, MCT, where a requirement for gelation is that the particles experience an attraction that is weak and has a range much smaller than a particle radius [18,19]. MCT was originally developed to describe glass transitions where this localization phenomenon is associated with particles becoming trapped within cages of nearest neighbors [20,21]. This theory has been extended to systems experiencing attractions. While glasses form near close packing where there is limited free volume, attractions induce nonergodic states at much lower volume fractions. Thus, the volume fraction where gelation occurs becomes a function of the strength and range of the interaction. Under these conditions, the suspension is stable and shows no signs of aggregation at low volume fractions. Instead, the particle structure factor is that of a weakly attractive fluid. According to MCT, at the gel volume fraction φgel the particles are localized and unable to diffuse over large distances compared to the particle size. The result is the loss of long-range diffusion and structural arrest. In idealized MCT, the gelation transition occurs over a narrow range of volume fractions, where the suspension undergoes a change from a fluid with a small relaxation time and low viscosity to a soft solid with a yield stress and elastic modulus. If the suspension is quenched to a particular state above the gel transition, the mechanical properties would evolve to those of a gel over a time scale governed by free particle diffusion. The predictions of MCT have been tested using suspensions that undergo gelation due to depletion attractions [22–26] where the strength and range of the interparticle attraction is controlled by the nonadsorbing polymer concentration and molecular weight. In addition, MCT predictions have been tested in suspensions that undergo a gelation transition due to changes in temperature [24,27]. These thermal gel systems are produced by using silica particles with attached C18 chains suspended in a solvent such that by modifying the temperature, the solvent can be changed from a good solvent to a poor solvent for the chains, making the particles attractive. Experimental studies on both of these systems show that a relatively small well depth of ∼2–4kT can cause a suspension to gel at volume fractions of 0.1 < φgel < 0.45. Unlike particles whose state of aggregation can be described by the DLVO model, these suspensions are stable below a certain volume fraction. Above the gelation volume fraction, due to no potential barriers, there will be rapid gelation as the particles diffuse into the shallow minimum and are trapped by the attraction and cooperative phenomena. Bergenholz and Fuchs [18,19] developed an analytical expression for gelation of weakly attractive particles experiencing square well attractions from idealized MCT. The particles are assumed to interact through a square well potential ΦSQ defined

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Fig. 2. Viscosity measurements on the fumed silica–ethanol suspension (1) at a shear stress of 0.5 Pa. Measurements from hard fumed (P) are also shown [6]. The line represents Eq. (16) with φmax = 0.19.

by the strength of attraction ω and the range of the attraction Δ: ⎧ ⎫ ∞ for r < 2a ⎬ ΦSQ (r) ⎨ (14) = −ω/kT for 2a < r < 2a(1 + Δ) . ⎩ ⎭ kT 0 for r > 2a(1 + Δ) Gelation is predicted to occur when 2    ω 12 φgel exp − − 1 = 1.42. kT π 2 2a

(15)

A distinction is made here concerning φgel , which concerns the spherical volume fraction φsphere and the fumed silica volume fraction φ. Making a comparison of the volume fraction of fumed silica to the volume fraction of spheres, one must include the openness of the fumed silica particle. Previously reported values showed that hard sphere and hard fumed silica particles show the same rheological properties based on the free volume of the suspensions [6]. Therefore, a direct comparison could be made by scaling the volume fraction with the maximum packing fraction for both systems. This same scaling is carried over to this study, where maximum packing fraction of hard spheres φsphere,max = 0.64 and the maximum packing fraction of the fumed silica in ethanol suspension φmax = 0.19. φmax was determined using the low shear viscosities measurements made at short times (Fig. 2) where aggregation is assumed to have a negligible contribution and the following equation [28,29]:  −2 φ η = 1− . (16) ηc φmax

The amount of ultrasonication was determined by monitoring a diluted sample using Brookhaven Instruments fiber optics quasi-elastic light scattering (FOQELS). Additional ultrasonication was performed on each sample to ensure complete dispersion of the particles before measurements were taken. In time-dependent measurements, the completion of the ultrasonication represented the initial time. The use of the FOQELS instrument allowed standard DLS measurements to be performed on turbid suspensions without the influence of multiple scattering. The instrument consists of a 10 mW solid state laser with wavelength λ = 677 nm, BI 9000 AT digital correlator and temperature control from 5–75 ◦ C with step sizes of 0.1 ◦ C. In this study, the data was collected for two minutes for each sample and the samples were stored in a circulating bath to maintain the specified temperature for the duration of the study. The intensity autocorrelation function g(τ ) was measured as a function of the decay time τ :  2 g(τ ) = B + X1 C(τ ) , (17) where B is the background intensity, C(τ ) is the intermediate scattering function, and X1 is an optical constant. For freely diffusing particles, C(τ ) decays from a value of one at τ = 0 to a value of zero for τ → ∞. In the dilute limit, C(τ ) = exp(−Γ τ ),

where Γ is the decay rate. At short decay times, the short-time diffusion coefficient DS is related to Γ by DS =

Γ . q2

(19)

The scattering wave vector q describes the length scale investigated by the measurement and is given by     θ 4πnc sin , q= (20) λ 2 where θ is the scattering angle. With FOQELS, the scattering angle is dependent on nc and the angle between the incident beam and detector θB , which is 56.8 in the FOQELS instrument,     nair θB 180 − θ (21) sin = sin , nc 2 2 where nair is the refractive index of the air, approximately equal to one. In the dilute limit for hard spheres, the Stokes–Einstein diffusion coefficient D0 can be related to the hydrodynamic radius RH by

3. Experimental DS = D 0 = Stock solutions were formed by dispersing fumed silica particles, Cab-O-Sil M-5 manufactured by Cabot Corporation [1], in ethanol through ultrasonication with the Artek Model 300 Ultrasonic Dismembrator to render the particles into their most reduced form [30]. The concentration of the stock suspension was determined by measuring the dry weight of the particles. Using mass concentration measurements, the conversion to volume fraction of silica φ is completed by dividing the mass concentration by the density of the particle, 2.2 g/ml.

(18)

kT . 6πηc RH

(22)

For the particles in this report, RH = a = 60 ± 5 nm. Viscosity measurements were performed on a Bohlin CS10 rheometer at a temperature of 25 ± 1 ◦ C. The measurements reported here were taken utilizing a cup and bob geometry with a 14 mm bob diameter and outer cup diameter of 15.4 mm, yielding a tool gap of 0.7 mm. Prior to measuring the viscosity, the samples were ultrasonicated and the measurements represent the viscosity before aggregation was detected by FOQELS

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measurements. A solvent trap was used to prevent solvent evaporation. Instrument calibration was verified with Cannon standard oils. Electrophoresis measurements were performed using Brookhaven Instruments ZetaPALS (λ = 658 nm), utilizing phase analysis light scattering to determine the electrophoretic mobility u. The experimental measurements were made using an electric field of 25.75 V/cm and frequency of 2 Hz. At least five measurements were taken on each sample, and the reported data represent the average values. Henry’s equation provides a relationship between the mobility and the zeta potential ζ with the assumptions that the surface potential is less than 25 mV and the ion atmosphere is undistorted by the applied field. The Hückel (Cz = 2/3, for κa < 0.1) and Helmholtz– Smoluchowski (Cz = 1, for κa > 100) equations describe the relationships for the extreme cases, u=

Cz εc ε0 ζ . ηc

(23)

For κa > 10, Cz can be approximated by 3 4.5 37.5 330 (24) − 4. − + 2 κa κa 2 κa Conductivity measurements were performed using a YSI Model 34 Conductance-Resistance Meter with a Model 3403 dip cell with the temperature maintained at 25 ◦ C using a constant temperature bath. The conductivity σ can be related to the ion concentration  ezi ci0 ui , σ= (25)

Cz ≈

i

and the mobility of the ions can be calculated by using the ionic radius of the ion R ze . u= (26) 6πηc R Thus, the ion concentration contributed by the alcohol can be estimated by measuring the conductivity of the solvent and using Eqs. (25) and (26). In the manufacturing of fumed silica, there is residual HCl present with the particles. Therefore, contributions to the conductivity are expected to be from the dissociation of the alcohol (AlcOH → AlcO− + H+ ), dissociation of the HCl (HCl → Cl− + H+ ), and dissociation of the surface silanol groups on the particle (SiOH → SiO− + H+ ) causing Eq. (25) to become

363

Ultra-small-angle X-ray scattering (USAXS) experiments were performed at the UNICAT facility on the 33-ID line, Advanced Photon Source at Argonne National Laboratory [31]. The instrument utilizes a Bonse–Hart camera using Si(111) optics with additional side-reflection stages enabling effective pinhole collimation. There is an experimental q range of approximately 10−4 –0.1 Å−1 . The beam size was 0.4 mm vertical and 1.5 mm horizontal through which approximately 2 × 1013 photons/s were incident at 10 keV with λ = 0.0154 nm. The scattering beam was analyzed with a rotating Si(111) channel cut crystal and measured with a photodiode detector. The samples were loaded in 2 mm diameter glass capillary tubes with a 0.01 mm wall thickness and then flamed sealed. 4. Results and discussion Aggregation was studied using the noninvasive technique of DLS. This was essential due to the slow kinetics of aggregation in the fumed silica–ethanol suspensions. Fig. 3 shows typical data of this experimental system, where the slow increase in relaxation times is associated with the particles aggregating. At short times, C(τ ) decays to zero. At long times, C(τ ) decays to a finite value greater than zero corresponding to nonergodic suspensions. There have been many methods used to determine gel times [12,32,33]. In order to provide a systematic method of determining the gel time, suspensions are considered a gel when the C(τ ) does not decay to a value of 0.1 when τ = 0.1 s. This method agreed with the visual inspection of when the suspension did not flow. When C(τ ) does not decay to zero, the averaging technique of van Megen and Pusey [34] was used to develop the correlation functions shown. Gel times were measured as a function of volume fraction and temperature. The results are shown in Figs. 4 and 5. While there is a noticeable change in the data for these different temperatures, the FOQELS instrument and slow gelation of ethanol did not allow a wide range of temperatures to be studied. Fig. 5 shows that the data seem to collapse onto a single curve when the gel times are nondimensionalized using the Brownian time scale, tBr = a 2 /D0 , which essentially corrects for changes in

σ = e(cAlcO− zAlcO− uAlcO− + cH+ zH+ uH+ ) + e(cCl− zCl− uCl− + cH+ zH+ uH+ )   + 6πηc au2p cp + ecH+ zH+ uH+ ,

(27)

where up and cp are the mobility and concentration of the particles, respectively. To maintain neutrality, ecH+ = ezp cp = 6πηc aup cp .

(28)

The conductivity of the alcohol can be subtracted from the conductivity of the suspension to solve for the ion concentration through algebraic manipulation of Eqs. (26)–(28) and using the measured concentration and mobility of the particles.

Fig. 3. DLS measurements on a sample of fumed silica dispersed in ethanol at 25 ◦ C at a silica volume fraction of 0.15. Open circles represent gelled suspensions according to the criteria outlined above. Some data points were removed for clarity.

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Fig. 4. Concentration dependence of gel times of fumed silica suspended in ethanol at 25 (1), 40 (!), and 50 ◦ C (P). The closed symbol represents the concentration where the suspension was a gel in less time than was required to analyze the system at 25 ◦ C (∼300 s).

Fig. 6. Time-dependent diffusion coefficient measurements used to estimate the stability ratio according to Eqs. (9) and (29). The data points correspond to experimental data at φ = 0.097 (1), 0.077 (!), and 0.033 (P) corresponding to a Φmax = 25 ± 2kT . The line represents Eq. (29).

Fig. 5. Superposing of dimensionless gel times at different temperatures of fumed silica using the same symbols as in Fig. 4.

the continuous phase viscosity with temperature. This observation indicates that the mechanism of gelation is largely dependent on the Brownian motion of the particles. While Fig. 5 shows a modest change in gel times with respect to temperature even with scaling with tBr , this decrease can be explained by the hydrogen bonding interaction between ethanol and fumed silica being less stable at higher temperatures. Further investigation at a larger range of temperatures would be needed to confirm this hypothesis. To understand the slow aggregation, the aggregation kinetics at low φ were studied to determine the stability ratio. From measurements of the diffusion coefficient as a function of time t during the initial stages of aggregation, td can be determined. The procedure assumes spherical particles and includes polydispersity effects on the average short-time diffusion coefficient Dave outlined in Killman and Adolph [35]:  2 p Np (t)fp Dp Dave (t) =  2 p Np (t)fp  5/3 (t/t )(p−1) (1 + t/t )−(p+1) d d p=1 p  = D0,ave , (29) 6/3 (t/t )(p−1) (1 + t/t )−(p+1) p d d p=1 where f is the scattering amplitude, N is the number, and D is the diffusion coefficient of particle p. By fitting the experimental measurements with Eq. (29), the only fitting parameter is td . Experimentally determining Dave and using Eqs. (9) and (29),

Fig. 7. Comparison of experimental data to model calculations of gelation. Solid line is from φgel = 0.16 and the dashed line corresponds to Eq. (12).

the stability ratio of the aggregation can be estimated. Furthermore, Eq. (10) can be used to estimate the effective barrier height in the interparticle potential. For the suspensions of interest in this study, W∞ is between a value of 1–2, if the particles were treated as spheres [9]. For fractal particles, this hydrodynamic contribution is expected to be less because the liquid can permeate through the particle [12]. Both of these observations indicate that the W∞ has a negligible role in this system. While the above relationship applies only in dilute systems, this analysis could only be performed on samples that show changes in the diffusion coefficient over the experimental time frame. Therefore, it was not possible to ensure that the samples were in the dilute limit where hydrodynamic effects would be negligible. By analyzing samples at different concentrations, shown in Fig. 6, the calculated Φmax = 25 ± 2kT showed no concentration dependence within the experimental uncertainties. The potential barrier determined by studying the initial stages of aggregation can be used to estimate the gel times by using Eqs. (12) and (13). In the calculations, an effective spherical volume fraction was used due to the open geometry of the fumed silica particles determined by multiplying the silica volume fraction by φsphere,max /φmax . The dashed line in Fig. 7 shows the results of the model of Bremer et al. [12] with

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Table 1 Values from investigation concerning the role of electrostatics in fumed silica–ethanol suspensions

Ethanol Ethanol + KCl

φ

u (µm/s)/(V/cm)

ζ (mV)

κ −1 (nm)

c0 (mM)

Φmax,DLVO (kT )

0.0002 0.007

−0.28 ± 0.05 0.04 ± 0.06

−21 ± 4 2±3

770 5

0.0001 2.2

8.4 0

Φmax = 25kT . Fig. 7 demonstrates an acceptable prediction of the concentration dependence of tgel up to a critical concentration, φ ∼ 0.16, where the gel time drops very rapidly with increasing φ. Above φ = 0.16, the suspensions essentially gel instantaneously, which cannot be explained by a change in microstructure. An explanation of this observation is given below, but first the origin of the potential barrier is discussed. In media with a continuous phase of high dielectric permittivity, repulsive barriers often arise due to electrostatic repulsions. The role of electrostatic repulsions was explored by measuring the suspension conductivity and particle surface potential on fumed silica–ethanol suspensions and the same suspensions saturated with KCl. After experimentally determining the mobility, calculating the zeta potential, and using the known values of the continuous phase and particles, the only unknown in Eqs. (3) and (4) is the concentration of ions in the suspension. The ion concentrations were determined by experimental measurements made on the conductivity of fumed silica– ethanol suspension at different particle concentrations. The conductivity showed a linear behavior over the concentration range φ = 0–0.075 described by σ = σEtOH + 6.94 × 10−4 φ,

Fig. 8. DLVO interaction potential estimates of ΦA (dashed), ΦR (solid), and ΦA + ΦR (dotted) using Eqs. (1) and (3) and the values in Table 1 for the fumed silica–ethanol suspension.

(30)

where σEtOH = 7.5 × 10−6 S/m. This relationship was used to determine the conductivity of the suspension due to the addition of particles to ethanol. The addition of salt produces another term in Eq. (27) and contributes to the concentration of ions. The measured conductivity of the ethanol saturated with KCl was 1.04 × 10−2 S/m, orders of magnitude larger than the other components of the conductivity, suggesting that the only significant component to the total ion concentration was that of KCl. Measured values of the particle mobility, along with calculated values of the zeta potential, Debye length, and total ion concentration are presented in Table 1. As seen in Table 1, the addition of salt effectively screens out the electrostatic forces because the mobility and zeta potential are zero within the experimental uncertainties. Using Eq. (3), the repulsive term of the interaction potential due to electrostatics can be estimated, Fig. 8. Also shown in Fig. 8 is the combined DLVO potential ΦA + ΦR . From these estimations using the DLVO theory, the largest repulsive potential experienced by particles coming from infinite separation would be Φmax,DLVO = 8.4kT . Φmax,DLVO underestimates the Φmax determined from the initial stages of aggregation. Additionally, due to the large Debye length, the electrostatic repulsions are extremely far-reaching. As the particle spacing becomes smaller, the energy needed for the particles to experience the primary minimum decreases. For example, at a particle spacing of 5a, the particles would only need approximately 2kT of energy to overcome the barrier because the

Fig. 9. Comparison of gel times of fumed silica suspensions in ethanol (1) and ethanol + KCl (E).

potential at r = 5a is ∼6kT . An estimate of the average particle spacing rave to φ is given by   φmax 1/3 , rave = 2a (31) φ which shows that rave = 5a occurs at φ = 0.012. Thus, even at low concentrations, there would be only a small barrier for the particles to diffuse over due to the electrostatic forces. Further evidence is shown in Fig. 9, where there was no significant difference in the time scales of aggregation between the fumed silica suspensions with or without salt. While there is evidence of a potential barrier in the aggregation of these suspensions, the origin of the barrier is not electrostatics. The slow aggregation and gelation seen at low volume fractions indicates that particles must diffuse over a barrier with a height ∼25kT to sample a primary minimum. The electrophoretic mobility studies suggest that this barrier does not have an electrostatic origin. The barrier height is thus attributed to solvation forces. Previous studies have indicated that the solvation forces play a significant role in fumed silica suspensions [7,36]. The origins of these forces are thought to be

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Table 2 Calculation of the attractive strength due to dispersion forces using Eqs. (1) and (2) at different particle surface separations using the bulk properties of silica and ethanol h (nm)

ΦA (kT )

0.55 1.10 1.65

−9.2 −4.3 −2.7

through hydrogen bonds between the surface silanol groups on the particle and the hydroxyl group of ethanol. The forces associated with solvent molecules bonded to the surface of the particle are expected to be exponentially dampened, oscillatory forces due to the force required to “squeeze out” a layer of solvent [13]. The size of the solvent molecule governs the frequency of the oscillations in the solvation potential. The depth of the primary minimum can be the result of dispersion forces and hydrogen bonding between the surfaces of two particles. The presence and depth of a secondary minimum would depend on the presence of residual hydrogen bonds and dispersion forces at a separation distance of a solvent molecule diameter. Table 2 gives an estimate of this secondary minimum from particles with effective radii of RH = 60 nm interacting via dispersion forces with a minimum distance of approach of one or more solvent diameters. At a spacing of two solvent molecule diameters the dispersion forces would yield an attractive secondary minimum of ∼4kT . Atomic force microscopy (AFM) studies suggest that this spacing is not an unreasonable expectation for strongly bound ethanol on silica surfaces [37–39]. It is noted that due to the openness of the fumed silica particle, the use of the bulk properties of silica and ethanol in Eq. (2) overestimates AH . This is because part of the volume enclosed by a sphere of size RH would actually be ethanol ∼20% [6]. Even without this correction, the values reported in Table 2 provide a suitable estimate of the dispersion forces. The presence of solvation layers can provide both a barrier to fast aggregation and a secondary minimum. X-ray scattering measurements are carried out to probe the weak attractions in this system by looking at the suspension microstructure and osmotic compressibilities. Fig. 10 plots the scattered intensity I of different concentrations of the fumed silica–ethanol suspensions. Because the experimental setup utilized flame-sealed capillary tubes where some solvent could be lost during sealing, the particle concentration was estimated by comparing the results are compared to previous USAXS measurements of hard fumed silica suspensions [5]. In studying hard fumed silica suspensions, it was shown that there was a linear relationship between particle concentration and the value of q when S(q) = 0.9. This relationship is applied to the fumed silica– ethanol suspensions in this study and the results are shown in Fig. 11. It should be noted that the radius of gyration of the hard fumed silica particles Rg,hard is 53 nm and the Rg of the particles in this study is 76 nm. This is not expected to greatly influence the results as it was shown that hard fumed silica particles of different sizes followed this same line when scaled in this manner.

Fig. 10. USAXS intensity curves with the fumed silica–ethanol suspensions at φ = 0.19 (1), 0.18 (P), and 0.14 (!) superposed with the fumed silica suspended in 0.1% SDS at φ = 0.03 (F) at high q. Some data points were removed for clarity.

Fig. 11. The concentration dependence of the value of qRg when S(q) = 0.9 for hard fumed silica suspensions (!) [5] and fumed silica in ethanol (2).

Also plotted in Fig. 10 is a dilute sample of fumed silica dispersed in 0.1% sodium dodecyl sulfate (SDS). Addition of SDS is used as a stabilizing agent to keep the particles from aggregating, and it was verified by DLS that there was no change in the RH during the measurements of this study. The scattered intensity is a function of the single particle form factor P (q), structure factor S(q), scattering volume VS , scattering length distance between the particle and solvent Ω, and incoherent scattering Bi : I = φVS ( Ω)2 P (q)S(q) + Bi .

(32)

By construction, S(q) goes to one in the limit when φ goes to zero. P (q) can be determined in the dilute limit from measurement of the scattering intensity. S(q) was determined by dividing I (q) by P (q) and then normalizing them such that S(q) = 1 at high q values. The validation of this technique is seen in Fig. 10 as all the data superposes at the high q values. The results of the S(q) calculations are shown in Fig. 12. At a sufficiently high particle concentration, there is an upturn in the low q region of S(q) and a continued suppression of S(q) at intermediate q. The upturn at low q at elevated concentrations was also seen in the hard fumed silica suspensions as reproduced in Fig. 12 [5]. This upturn is poorly understood, but shear viscosity and dynamic light scattering measurements on the hard fumed silica

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367

Table 3 Tabulated values of ρb and f (Δ) as reported by Ramakrishnan and Zukoski [43] Δ/2a

f (Δ)

ρb (Δ)(2a)3

0.004 0.015

−0.5537 −0.768

2.563 2.585

Table 4 Comparison of the strength of attraction from S(q) measurements of hard fumed silica suspensions [5] and the native fumed silica–ethanol suspensions φ

Fig. 12. Structure factor calculations of ethanol-fumed silica suspensions at φ = 0.19 (1), 0.18 (P), and 0.14 (!) along with hard fumed silica suspensions at φ = 0.16 (×) and 0.14 (+) [5]; Rg,hard = 53 and Rg = 76 nm. Some data points were removed for clarity.

suspensions demonstrate that this upturn occurs in the absence of any evidence of clusters at the elevated volume fractions [5,6]. Thus, it is assumed that the low q upturn is not associated with clusters, but possibly from voids left within an increasingly uniform medium. Debye and Bueche [40] attributed this sort of scattering as occurring in uniform materials with point defects that have a correlation length between them [41]. The major difference in the scattering from hard and native fumed silica particle suspensions lies in the minimum values of S(q) achieved. The minimum value of the structure factor S(qmin ) for the fumed silica–ethanol suspensions is larger than those of the hard fumed silica suspensions, suggesting that the native particles in ethanol feel an increased attraction. For suspensions of particles undergoing an idealized MCT gel transition, the transition is not of structural but of kinetic origin. Therefore, at concentrations above φgel , the structure is essentially the frozen state of the attractive liquid structure [42]. Furthermore, even though the suspension is not in equilibrium, it is expected from idealized MCT that S(q = 0) would increase with increasing strength of attraction. Evidence of this interpretation has been seen in the case of thermal gels [24] where the microstructure of the gel shows an increase in S(q = 0) as the strength of the attraction increases. Based on this understanding, a comparison is made between the S(q) values of hard fumed silica–liquid suspensions and fumed silica–ethanol gels. As q goes to zero for suspensions of uniform particles, S(q) is a measure of the suspension compressibility. For hard spheres this can be expressed as 1 dΠ 1 + 4φ + 4φ 2 − 4φ 3 + φ 4 1 , = = S(0) kT dρ (φ − 1)4

(33)

where Π is the suspension osmotic pressure. An empirical model has been developed for spheres interacting via square well potentials [43]: 2 πρ(2a)3 ρ(2a)3 α(Δ, ω) Π + , =1+ 3 ρkT (1 − ρ/ρ0 )2 (1 − ρ/ρb (Δ))3

where

(34)

0.14 0.18 0.19

S(qmin )/Shard (qmin ) Experimental

ω = 2kT

ω = 3kT

ω = 4kT

1.3 ± 0.2 1.8 ± 0.2 1.9 ± 0.2

1.6 1.7 1.6

2.4 2.5 2.4

4.7 4.9 4.6

ρ0 = 1.605(2a)3 , (35) ω α(Δ, ω) = (36) f (Δ), kT and ρb and f (Δ) are tabulated values shown in Table 3. The attractions between the particles are expected to be modulated by the solvent. Therefore, Δ = 0.55 nm to be on the order of a solvent diameter. A relative comparison of the expected shift in the osmotic compressibility is shown in Table 4 using values of ρb and f (Δ) interpolated from the values shown in Table 3. The comparison is made by taking the ratio of experimental values of the minimum in the structure factor for the fumed silica–ethanol gels S(qmin ) to that of the hard fumed silica suspension Shard (qmin ). This experimental ratio is then compared to the ratio of S(q = 0) for spheres interacting via a square well potential at different strengths of attraction and hard spheres using Eqs. (33) and (34). In Eqs. (33) and (34), the volume fraction used was φadj = bφ. The value of b = 1.6 was chosen such that the experimental values of Shard (qmin ) matched the theoretical values of 1/kT (dΠ/dρ) at elevated volume fractions. This same shift in volume fraction is then used to calculate the equation of state for a suspension of particles interacting via a square well potential. Table 4 demonstrates, following the approximations discussed above, that an attractive strength of 2–3kT corresponds to the experimentally observed higher S(qmin ) values for the fumed silica–ethanol suspensions. For volume fractions greater than 0.16, the suspensions essentially gel as soon as mixing is terminated. Applying the MCT model suggests this dramatic transition between slow and rapid gelation can be the result of a transition from growth of aggregates produced by diffusion into the primary minimum to gelation resulting from particles that are localized by cooperative effects in the secondary minimum of the solvation forces. The data suggest that for times much shorter than td , particles experience a short-range attraction from the secondary minimum, and if conditions are correct, they will form gels by being localized within this potential minimum. This concept is tested using Eq. (15). If the range of attraction is equivalent to the size of the solvent molecule, Δ = 0.55 nm, and φgel is taken as the concentration when the fumed silica–ethanol suspension gelled at approximately the time of a DLS measurement (∼300 s),

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then Eq. (15) can estimate the strength of attraction in this system. Equation (15) predicts that an attractive well depth of 3.1kT is sufficient to result in MCT gelation at a fumed silica volume fraction of 0.16, which corresponds to an equivalent sphere volume fraction of 0.54. The X-ray scattering and MCT estimates of the presence of a secondary well of depth ∼2–3kT present a consistent story for the abrupt transition of slow gelation kinetics to essentially instantaneous gelation over a very narrow range of volume fractions. 5. Conclusions By postulating the existence of solvation forces between fumed silica particles suspended in ethanol, the aggregation and gelation properties of these suspensions can be understood over a wide range of volume fractions. The large potential energy barrier which particles must diffuse across to aggregate into a primary minimum is due to the layer of ethanol hydrogen bonded to the particle surface being “squeezed out.” The secondary minimum is developed by the combination of the oscillatory solvation forces and the attractive dispersion forces. The depth of this minimum sets a volume fraction of nearly instantaneous gelation. Further testing of this model can be accomplished by altering the surface chemistry of the particles. If surface modifications reduced the ability of the particle to hydrogen bond, then there would be a weaker interaction between ethanol and the particle surface. This would result in lower potential barriers and thus, faster gel times. Additionally, changing the solvent will modify the surface–particle interactions. The potential barrier will be altered through changes in the hydrogen bonding ability between the particle and solvent. Also the range of the forces will change due to changes in the size of the solvent molecule. Further investigation probing different solvent–particle interactions would help validate the hypothesis of solvation forces giving rise to two distinct mechanisms of gelation, as observed in the fumed silica–ethanol suspensions. Acknowledgment The authors wish to thank Cabot Corporation for the use of their Cab-O-Sil fumed silica products, along with J. Ilavsky for his assistance in gathering the scattering data. The UNICAT facility at the Advanced Photon Source (APS) is supported by the U.S. DOE under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, the Oak Ridge National Laboratory (U.S. DOE contract DE-AC05-00OR22725 with UTBattelle LLC), the National Institute of Standards and Technology (U.S. Department of Commerce) and UOP LLC. The APS is supported by the U.S. DOE, Basic Energy Sciences, Office of Science under contract No. W-31-109-ENG-38. We acknowledge the Center for Microanalysis of Materials, University of Illinois, which is partially supported by the U.S. Department of Energy under grant DEFG02-91-ER45439 where imaging of particles was made possible. Finally, the authors are grateful for financial support from the Nanoscale Science and Engineering

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