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Particle Restabilization in Silica/PEG/Ethanol Suspensions: How Strongly do Polymers Need To Adsorb To Stabilize Against Aggregation? So Youn Kim and Charles F. Zukoski* Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana—Champaign, Urbana, Illinois 61801, United States ABSTRACT: We study the effects of increasing the concentration of a low molecular weight polyethylene glycol on the stability of 44 nm diameter silica nanoparticles suspended in ethanol. Polymer concentration, cp, is increased from zero to that characterizing the polymer melt. Particle stability is accessed through measurement of the particle second-virial coefficient, Bh2, performed by light scattering and ultrasmall angle X-ray scattering (USAXS). The results show that at low polymer concentration, cp < 3 wt %, Bh2 values are positive, indicating repulsive interactions between particles. B h2 decreases at intermediate concentrations (3 wt % < cp < 50 wt %), and particles aggregates are formed. At high concentrations (50 wt % < cp) B h2 increases and stabilizes at a value expected for hard spheres with a diameter near 44 nm, indicating the particles are thermodynamically stable. At intermediate polymer concentrations, rates of aggregation are determined by measuring time-dependent changes in the suspension turbidity, revealing that aggregation is slowed by the necessity of the particles diffusing over a repulsive barrier in the pair potential. The magnitude of the barrier passes through a minimum at cp ≈ 12 wt % where it has a value of ∼12kT. These results are understood in terms of a reduction of electrostatic repulsion and van der Waals attractions with increasing cp. Depletion attractions are found to play a minor role in particle stability. A model is presented suggesting displacement of weakly adsorbed polymer leads to slow aggregation at intermediate concentration, and we conclude that a general model of depletion restabilization may involve increased strength of polymer adsorption with increasing polymer concentration.
I. INTRODUCTION Polymers are commonly used to control interactions in colloidal suspensions. Nonadsorbing polymer produces a depletion attraction the strength of which scales on the polymer solution osmotic pressure and the range of which are controlled by the polymer radius of gyration. Increasing nonadsorbing polymer concentration leads to aggregation, gelation, liquidliquid, and crystallization phase separations. Adsorbing polymers are used as steric stabilizing aids where the approach of two particles restricts adsorbed polymer configurations resulting in an entropic repulsion. Steric stabilization is commonly used to stop aggregation at high ionic strengths. Clearly, to confer these properties, the polymer must adsorb with sufficient strength that it is not displaced by Brownian encounters. Here we address the question of the magnitude of the strength of attraction between polymer segments and the particle surface that will confer stability.13 Polymer-induced interactions, of course, occur in addition to other forces acting on the particles such as van der Waals attractions and electrostatic repulsions. In combination, these forces determine particle stability and the state of colloidal aggregation. Thus, interpreting the state of colloidal aggregation requires understanding how polymer concentration alters all interactions controlling particle stability. An example of a situation where the state of aggregation must be controlled as polymer concentration is increased comes in creating polymer nanocomposites by mixing polymer and particles in a low molecular weight solvent and subsequently removing the solvent. r 2011 American Chemical Society
These initial conditions are similar to those that result in depletion forces. The aggregation and gelation that can result from depletion attractions are often undesirable.1,4 While many studies have investigated the effects of polymers on the state of aggregation at relatively low polymer concentrations,58 fewer have investigated particle stability in concentrated polymer solutions as the melt concentration is approached. Well-understood and tested models for depletion attractions predict monotonic growth in the strength of depletion attractions with increasing polymer concentrations.3,7,9 Thus, if particles aggregate at low polymer concentration, the paths leading to stable particles following solvent removal are not obvious. Nevertheless, such paths exist. Here, we explore how stable polymer nanocomposite melts can be prepared with the solvent removal method in a system that is aggregated at low polymer concentration but stable at high polymer concentration. The system investigated involves 44 nm diameter St€ober silica particles suspended in ethanol and 400 molecular weight polyethylene glycol (PEG 400). We characterize the thermodynamic and structural properties of nanoparticles suspended in low molecular weight solvent as the polymer concentration increases toward that of the polymer melt. Our measurements suggest that particles are stabilized by electrostatic repulsion in pure ethanol.10 As polymer concentration is increased, the particle charge decreases and particles Received: January 3, 2011 Revised: March 16, 2011 Published: April 05, 2011 5211
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Langmuir aggregate due to van der Waals forces.11 At still higher polymer concentration the refractive index of the polymer and ethanol reach that of the particles, thus minimizing van der Waals interactions, and the particles restabilize. Second-virial coefficient measurements indicate that at high polymer concentration the particles are thermodynamically stable against aggregation. At intermediate polymer concentrations where there is no particle charge, there is a barrier in the pair potential which slows aggregation. These observations are interpreted in terms of models of electrostatic, van der Waals, depletion, and steric stabilization. Our models build on recent studies showing that for this system the strength of attraction of polymer segments with the particle surface increases with polymer concentration.12 Stability at high polymer concentration is thus associated with steric stabilization, while aggregation at low polymer concentration is associated with van der Waals attractions and a minimum of steric repulsions. As polymer concentration increases, the van der Waals attractions decrease and the polymer adsorbs more strongly. These conditions combine to create a condition of thermodynamic stability for the particles above a specific polymer concentration. At polymer concentrations just below this transition, the polymer adsorbs sufficiently weakly that it can be displaced during a Brownian encounter of two particles, resulting in slow aggregation into a primary van der Waals minimum. In section II we describe our experimental measurements, while in section III we discuss measures of thermodynamic stability as characterized by the particle second-virial coefficient and characterize the rate of aggregation at intermediate cp. The origin of these observations are discussed and compared with model predictions in section IV. A summary and conclusions are presented in section V.
II. EXPERIMENTAL SECTION Particle Synthesis and Sample Preparation. Colloidal silica particles used in this study were synthesized based on the method of St€ober et al.,13 which accompanies the base-catalyzed hydrolysis and condensation of tetraethyl orthosilicate (TEOS). The reaction temperature was 55 C. A 3610 mL amount of 200 proof ethanol was mixed with 96 mL of deionized water, and 156 mL of ammonium hydroxide (pH 13.8) was added as a catalyst. The reaction was allowed to run for 4 h after mixing in 156 mL of tetraethyl orthosilicate. Particle diameter (D) was determined by measuring 100 particles from SEM measurements, yielding a diameter of 43 ( 5 nm, while a fit to the form factor determined by the angle dependence of X-ray scattering from a dilute suspension of particles yielded a diameter of 42 ( 1 nm. The Kuhn length segment diameter of PEG is d = 0.7 nm (d = C¥l/cos(θp/2), where C¥ = 4.1 is the characteristic ratio of PEG in water, l = 1.5 Å is the length of a backbone bond, and θp = 68 is the angle of a backbone bond).14 The radius of gyration for PEG 400 is Rg = 0.7 nm. The small particle size was chosen to minimize the D/d value while still making the particle size large enough to work in the colloidal limit. PEG 400 was purchased from Sigma-Aldrich; 200 proof ethanol was supplied from Decon Lab. Inc. After particle synthesis, particles in ethanol were concentrated approximately 10 times by heating in a ventilation hood. During this process, the excess of ammonium hydroxide was removed. For preparation of polymer suspensions, particle dispersions were made by adding known masses of PEG to the defined mass of particles in ethanol to result in the desired concentrations of
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particle, PEG, and ethanol. PEG 400 is a liquid at room temperature, and it is completely miscible with ethanol at room temperature (the melting temperature of PEG 400 Tm,PEG400 = 8 C). In the absence of PEG, the alcosol was a transparent blue. As the silica and PEG have nearly identical refractive indices (nsilica =1.4555, nPEG = 1.4539), the polymer/particle mixture becomes increasingly transparent as cp is raised. In the absence of ethanol, the polymer nanocomposite melt is transparent. To determine particle volume fractions, the mass and density of each component were used to obtain its volume. The particle density is 1.6 g/cm3.15 Light Scattering. Dynamic light scattering (DLS- Brookhaven instruments BI-200 SM goniometer, Lexel Argon-Ion model 95 laser, λ = 514 nm) was conducted to measure the diameter of particles with varying PEG concentrations and their number distributions. Static light scattering (SLS) was carried out with the same instrument to obtain second-virial coefficients from low polymer concentrations (0 wt %) up to 60 wt % of PEG. The instrument was calibrated with the scattering of toluene at 90 to convert scattering intensity from counts per second to cm1. The samples were filtered with 0.45 μm Corning filters into glass tubes with a beam length of 1 cm. Samples were measured at 25 C. To develop the Zimm plot, 5 dilute concentrations were prepared for one set of polymer concentrations (130 mg/mL) and measurements were taken from θ = 40 to 130. As polymer concentration increases, the refractive index of the PEG/ethanol mixture begins to match that of the silica. Thus, at higher polymer concentrations, to increase the scattering intensity, higher particle concentration was used but was always less than 30 mg/mL to prevent multiple scattering. For polymer concentrations greater than 60 wt %, the scattering intensity from the particles was insufficient to determine the second-virial coefficient. SBUSAXS. At higher polymer concentrations (above 60 wt %), second-virial coefficients were measured with X-ray scattering. Side bounce ultrasmall-angle X-ray scattering experiments (SBUSAXS) were conducted at the X-ray operations and Research beamline on the 32ID-B at the Advanced Photon Source (APS), Argonne National Laboratory. The instrument employs a BonseHart camera and Si(111) optics, with extended q range, allowing measurement down to 2 103 Å1. Additional sidereflection Si(111) stages enables effective pinhole collimation, minimizing slit smearing effects. An absolute calibration converts the scattering intensity from counts per second to absolute units of cm1 based on the geometric information of the sample thickness along the path of the beam. Samples were loaded in multiposition sample holders. The sides of each cell chamber in the holder where the beam passes were sealed with two Kapton polyimide slides. All measurements were performed at 25 C. Measurements on each sample took about 30 min including loading and measurements. Background intensities were subtracted off from the sample scattering intensity, which was measured every fifth sample. The background scattering intensity at each polymer concentration was negligible compared to the scattering intensity of the particles (The background intensity from ethanol differs from 90 wt % PEG solution by a factor of 1013 cm1 when the scattering intensity of the sample having silica particles has an order of 103 cm1) This ensures our assumption that the scattering from the silica particles dominates throughout the system. Turbidity Measurement. To quantify the rate of aggregation, suspension turbidity was measured as a function of time using an HP 8453 general purpose UVvis spectrophotometer at a fixed 5212
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wavelength of 500 nm. As soon as polymer was added to the alcosol solution, the intensity of light transmitted through the sample was measured. The turbidity was measured as a function of time immediately after mixing particles and polymer. Polymer concentrations from 3 to 50 wt % were studied, while the initial particle concentration was set to ensure accurate rates of aggregation could be measured (ranging from 0.5 to 63 mg/mL). Particle concentrations were chosen to ensure that turbidity increased linearly with time, which is taken as an indication that scattering is dominated by contributions from monomers and dimers. To exclude the background scattering effects, the scattering intensity of the blank was measured each time. Turbidity measurements using the spectrophotometer are less sensitive to multiple scattering than measurement of the intensity of light scattered at a particular angle.16 Characterizing the kinetics of aggregation requires the knowledge of solvent viscosity, which was measured as a function of cp with a CFRC B50 capillary viscometer manufactured by Cannon Instrument Co. For cp = 3, 5, 7.5, 10, 17.5, 30, and 50 wt %, viscosities of 1.20, 1.26, 1.31, 1.38, 1.72, 2.68, and 5.35 mPa 3 s were measured. Conductivity Measurements. Nanoparticles suspended in ethanol are not suitable for measurement of electrophoretic mobilities with the standard electrokinetic instrumentation. As a result, we estimate particle charge by measuring the conductivity of suspensions as a function of volume fraction. All measurements were conducted using an YSI model 34 conductanceresistance meter with a model 3403 dip cell at 25 C. The cell constant, K, was 1.0 cm1. Adsorption Isotherm. The conventional solution depletion method was used to determine the adsorption isotherm. Polymer/silica/ethanol suspensions were prepared to quantify the adsorbed amount by increasing polymer concentration from 0.25 to 3.5 wt %. Each sample contained 10 wt % of silica particles. After preparation, the samples were centrifuged for 4 h at 6500 rpm. The supernatant solutions were obtained and kept in vials. To measure the polymer concentrations in the supernatants, ethanol was evaporated in a 45 C oven for 48 h. All masses were measured before and after evaporation.
III. RESULTS The stability against aggregation of 44 nm silica particles is characterized as a function of polymer concentration (cp) from pure solvent (0 wt %) up to polymer melt (100 wt %) densities. In secton A below we discuss light scattering and visual observation of aggregation as a function of cp. In section B the rate of particle aggregation at intermediate cp is characterized.
A. State of Particle Dispersions at Low and High cp. Particle Second-Virial Coefficients. The thermodynamic state of
dilute particle suspensions can be accessed with measurement of the particle second-virial coefficient. In this work, second-virial coefficients were measured by static light and X-ray scattering. The scattering intensity of light from a single component system is written as Rðq, cÞ ¼ cMKPðqÞSðq, cÞ þ B
ð1Þ
where R(q,c) is the Rayleigh ratio, c is the silica concentration (g/mL), and M is the particle molecular weight (g/mol). For light scattering, K is an optical constant, K = 2π2n2(dn/dc)2/ NAλ4, where n is the refractive index of the medium, dn/dc is the refractive index increment, NA is Avogadro’s number, and λ is the
wavelength of the incident light. q is the scattering vector, q = (4πn/λ)sin(θ/2) with the scattering angle θ. In eq 1, P(q) is the form factor containing intraparticle information, S(q,c) is the structure factor accounting for interparticle correlations, and B is background scattering of the given medium. In the limit where qD < 2, Guinier’s law yields the form factor as 5 qD 2 þ Oðq4 D4 Þ ð2Þ PðqÞ ¼ 1 9 2 where D is the z-average particle diameter.15 At small q where SLS is performed, only the leading term in qD is used in the analysis. In the q f 0 limit, S(q,c) is related Q to a virial expansion through the osmotic compressibility, ∂ /∂c lim
qf0
1 M ∂Π 2B2 NA 3B3 NA 2 2 ¼ ¼ 1þ cþ c þ ::: Sðq, cÞ NA kB T ∂c M M2 ð3Þ
where B2 and B3 are virial coefficients associated with pair and triplet interactions, respectively. In the dilute particle limit, higher order terms may be neglected lim
qf0
1 M ∂Π 2B2 NA Vc ¼ ¼ 1þ c Sðq, cÞ NA kB T ∂c M
ð4Þ
where Bh2 = B2/B2HS, the hard-sphere value is equal to 4 times the particle volume, B2HS = 4Vc, and M is the weight-average molecular weight. These two assumptions enable us to use the Rayleigh equation and rewrite it in Zimm form17 # " #" cK 5 qD 2 1 8B2 NA Vc þ ¼ 1þ c ð5Þ Rθ ðq, cÞ 9 2 M M2 Here, Rθ is the Rayleigh ratio and Vc is particle volume. An example Zimm plot is shown in Figure 1 in ref 15. In the limits where q goes to zero, the data can be extrapolated to obtain the indicated second-virial coefficients. Here, 0.0012 < q < 0.0023 Å1. In the zero concentration limit, D = 43.0 ( 0.8 nm is obtained, which is similar to that from the value of bare particles confirmed with SEM and USAXS measurements of 43 ( 5 and 42 ( 1 nm, respectively. The value of dn/dc for silica in ethanol was determined to be 0.06 cm3/g with a Milton Roy refractometer. We found the molecular weight of silica particles, M, to be 3.28 ( 0.05 107 g/mol. The weight-average molecular weight divided by the particle size gives a silica density of 1.62 g/cm3, close to what was anticipated from previous studies.15 As cp increases, the values of dn/dc varies in that the refractive index of the medium changes but particle molecular weight remains the same. As a result, with increasing cp, dn/dc was adjusted to hold M constant. As polymer concentration increases, the refractive indices of silica and the polymer solution are matched. Therefore, at polymer concentrations above 60 wt %, the second-virial measurements were made with X-ray scattering. A similar form of eq 5 can be used to extract the second-virial coefficients.15 From detailed measurements we extract second-virial coefficients, (B h2 is large and h2) as a function of cp. In pure ethanol, B positive, which supports the observed stability of the dispersion in the absence of polymer (Figure 1). Dissociated silanol groups give rise to long-range repulsions. (The electrostatic origin of this repulsion was confirmed by adding salt to the solution and 5213
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Figure 1. Calculated and experimental normalized second-virial coefficient of 44 nm silica particleparticle interaction as a function of PEG 400 concentration, cp (wt %), from light scattering measurement at 25 and 65 C. For 60, 80, 90, and 100 wt %, the coefficient values were determined with USAXS. The predicted curve is determined from the total interaction potential based on Figure 6 and eq 19.
observing Bh2 to decrease. For example, when measurements are made in ethanol solutions containing 0.02 and 0.2 M of LiCl, B h2 values drop to 2.04 and 0.32, respectively.) At intermediate polymer concentrations, 5 wt % < cp < 30 wt %, Bh2 values could not be obtained due to particle aggregation. The aggregation process is slow (see below), indicating that the process of aggregation results from diffusion over a potential energy barrier. At these concentrations, the suspensions became opaque and turbid within 0.5 h. We report here only values of Bh2 where there is no evidence of aggregation. The line in Figure 1 results from prediction of B h2 based on estimates of the particle surface potential and Hamaker coefficient as discussed below. B. State of Particle Dispersion at Intermediate cp. Aggregation Kinetics. At intermediate cp, suspension turbidity, τ, was measured as a function of time after mixing with polymer. The results are analyzed assuming that initially the suspension contains a number density of particles, F0. At time t, a number density of monomer and dimer are given as F and F2, respectively, with the initial number density of particles F0 = F þ 2F2. As monomers are lost and dimers are formed, the turbidity increases because the scattering of a particle increases with the square of its volume. At short times and in the dilute particle limit, using a pseudosteady-state approximation, we can relate the rate of monomer disappearance and the rate of dimer formation to the turbidity increment18 dτ dF kr ¼ a ð6Þ ¼ a F2 dt dt W where the coagulation rate constant kr is given by the Smoluchowski rapid aggregation limit, kr = 8kBT/3η, where η is the continuous phase viscosity and a is a constant that can be calibrated in the suspension at zero time. As mentioned above, η was measured as a function of cp to ensure accurate determination of kr. Here, W is the stability ratio that accounts for slowing of the rate of aggregation due to diffusion over a potential barrier where W is related Z ¥ expðV =kTÞ dr ð7Þ W ¼ 2R r 2 GðrÞ 2R
Figure 2. (a) Normalized turbidity as a function of time measured on the spectrophotometer at different PEG 400 concentrations (cp): 3 (blue diamonds), 5 (red rectangles), 7.5 (green traingles), 10 (blue ), 17.5 (blue asterisks), 30 (brown ovals), and 50 (blue þ) wt%. Initial particle concentrations varied depending upon cp: 63, 1, 0.5, 0.5, 1, 63, and 63 mg/mL, respectively. (b) Vmax as a function of PEG 400 concentration, cp (wt%), from turbidity experiments (red diamond) and modeling calculation (green triangles). Data for modeling caculations are taken from the height of the total interaction potential as shown in Figure 6a.
R is the particle radius, and hydrodynamic interactions of two particles diffusing along a line of centers are accounted for in G(r). This effect is small and, here, G(r) given a value of unity. The stability ratio is often approximated as W exp
Vmax kT
ð8Þ
where Vmax is the barrier height over which the particles must diffuse to aggregate.18,19 As discussed in the Experimental Section, turbidity, τ, was measured at different polymer concentrations. All data were gathered under conditions where the turbidity at t = 0, τ0, and kr/W were linear functions of particle concentration. From the slopes of the lines kr/W was calculated. The resulting values of Vmax are shown in Figure 2. We note that the values of Vmax are only as accurate as the approximations in eq 8 support. As discussed in detail below, methods to approximate the pair potential are complicated by the complexity of the ternary system 5214
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being investigated. Our goal here is to establish that there is a barrier to aggregation and that it changes with polymer concentration. Below we show that the broad trends in this data are captured by a combination of forces, suggesting the barrier results from displacement of polymer from the particle surface. For our purposes of establishing the presence of a nonelectrostatic barrier in pair potential and the limits of approximation used to describe the pair potential eq 8 yields the necessary level of accuracy. At the lowest polymer concentration where aggregation could be detected, the barrier over which the particles diffuse starts at ∼20 kT at low polymer concentrations and passes through a minimum of ∼12 kT at 10 wt % polymer. Particles are indefinitely stable at polymer concentrations below 3 wt % and above 50 wt %.
IV. DISCUSSION WITH MODELING CALCULATION The results reported above show that the particles are stably dispersed in low and high polymer concentration (cp < 3 wt % and 50 wt % < cp) and aggregate at intermediate concentration (3 wt % < cp < 50 wt %). Our goal is to understand the origin of the particle stability and the rates of aggregation at intermediate cp. This will be done by exploring the nature of the pair potential as a summation of (i) electrostatic repulsions, (ii) van der Waals attractions, (iii) depletion forces, and (iv) steric repulsions as a function of polymer concentration. We note that the number of parameters required to characterize the pair potential is large, and despite attempts to carefully estimate their numerical values, significant uncertainties exist. Nevertheless, we successfully demonstrate that the origins of particle restabilization and kinetics of aggregation result from the dependency of polymer concentration on electrostatic repulsions, variations in Hamaker coefficient, and weak polymer adsorption. 1. Electrostatic Repulsions. Following classical colloid stability theories, in the absence of polymer, we characterize particleparticle interactions with the classical assumption of a linear superposition of electrostatic repulsions, VR(r), and van der Waals attractions, VA(r).10,11 The total interaction potential, V(r) is given by ( ¥ 0 e r e ð2R þ δa Þ V ðrÞ ¼ ð9Þ VR ðrÞ þ VA ðrÞ ð2R þ δa Þ e r e ¥ where R is the particle radius and δa is the distance of the closest approach. Particles are initially negatively charged by dissociation of silanol groups. The resulting repulsive pair potential can be written 2 2 ! q0 e R VR ðrÞ ¼ 4πεr ε0 exp½ kðr 2RÞ 4πεr ε0 Rð1 þ kRÞ r ð10Þ where εr is the relative permittivity of the continuous phase, ε0 is the permittivity of vacuum, q0 is the net charge on the particle, and r is given as the particle center-to-center distance. The DebyeH€uckel parameter is expressed as κ = ((e2/(εrε0kBT))Σkzk2Fk)1/2, where zk and Fk are the valence and the bulk number concentration of the kth ion. The key parameters are the net charge on the particle and the DebyeH€uckel parameter which is experimentally estimated based on conductivity measurements.
Conductivity Measurements. We use suspension conductivity to estimate the particle surface charge as a function of polymer concentration. The zeta potential was estimated from conductivity assuming the particles and their counterions carry charge above and beyond that of the continuous phase following the analysis of Anderson et al.15 12εr ε0 2εr ε0 2 zi wi ς þ ς φc σ ¼ σ0 þ 2 ð11Þ D exp½ kD=2 3μ where σ is the suspension conductivity, σ0 is the conductivity of the solution at infinite particle dilution, D is the particle diameter, wi is the mobility of the ith ion, μ is the viscosity of the solution, zi is the valence of the counterion, and φc is the particle volume fraction. For the calculation, we assume the valence charge of all ions to be 1. The mobility of the counterion in ethanol is estimated from the standard mobility of an ion in aqueous solution (ωi,aq = 7.8 104 cm2 V1 s1), and the ratio of the continuous viscosity of ethanol to that of water is applied (ωi,EtOH = ωi,aqμaq/μEtOH, where μaq = 1.0 mPa 3 s and μEtOH = 1.2 mPa 3 s). In determining κ, the bulk ion concentration is required. Here, Fk is estimated from extrapolation of σ0 at infinite dilution, yielding an approximate value of 1/κ = 127 nm. The change of continuous phase viscosity with polymer concentration is negligible at the low polymer concentrations of interest in these measurements (cp < 5 wt %). The volume fraction dependence of the conductivity at three polymer concentrations is shown in Figure 3a. In the absence of polymer, we estimate the particle zeta potential is ς = 118 mV, indicating that electrostatic repulsions play a significant role in stabilizing the particles. The reduced conductivity with 1 and 5 wt % of polymer are also shown in Figure 3a. We find that even at 1 wt % polymer, the particles make a negligible contribution to the suspension conductivity. However, the negative slopes of the reduced conductivity at 1 and 5 wt % are much steeper than we expected based on Maxwell theory,19 implying there could be some additional mechanisms contributing to the suspension conductivity. Similar effects were observed in polymer melts of different molecular weights as discussed by Anderson.15 These results indicate that electrostatic repulsions diminish rapidly as polymer is added to an ethanolic suspension. Therefore, for cp > 5 wt %, our results suggest that particles are not charged or the charge is weak enough to be neglected in estimating pair potentials. From the conductivity measurements, when cp = 0 we estimate that ζ = 118 mV and κD = 0.35. This estimate is based on Fk determined from an extrapolated conductivity at zero volume fraction (eq 11). Trace amounts of ammonia and water can result in substantial variation in background ionic strength, and as a result, κ has large uncertainties. In the calculations discussed below, κ and surface charge q0 have been varied to test the sensitivity of our conclusions with the magnitudes of these parameters. For example, if we assume ζ = 118 mV and κD = 2.1 for the electrostatic repulsion in pure ethanol, it predicts Bh2 = 13, which is close to the experimental value of 12 while κD = 0.35 gives B h2 to κD can be understood in h2 = 307. The sensitivity of B terms of the effective increase in the volume of the particle with expansion of the electrostatic double layer. Consistent with this uncertainty, we note that measures of B h2 in pure ethanol fluctuated between different synthetic batches of particles, always remaining large and positive. 5215
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Figure 3. (a) Reduced conductivity as a function of particle volume fraction for pure ethanol (blue squares), 1 wt % (red squares), and 5 wt % (green triangles). Dashed lines show linear fits. (b) Electrostatic repulsive interaction energy at different polymer concentrations as a function of surface-to-surface distance of particles for concentrations of 0, 2, 3, and 5 wt % of cp from top to bottom.
In the presence of polymer, ζ varies systematically as dielectric permittivity decreases with increasing polymer concentration. As a result, we assume that ζ rapidly decreases from its value in pure ethanol (cp = 0 wt %) to cp = 5 wt % such that there are linear decreases with cp from 118 mV at cp = 020 mV at cp = 3 wt %, and then to 7.86 mV at cp= 5 wt %, and finally ζ = 0 for cp greater than or equal to 10 wt %. The resulting electrostatic repulsive pair potentials are selectively shown in Figure 3b. 2. van der Waals Attraction. The van der Waals attractions between a pair of particles can be approximated by
8 > < ¥, " ! ! " ## A 2R 2 2R 2 r 2 4R 2 VA ðrÞ ¼ , > : 6 r2 4R 2 þ r2 þ ln r2
r < 2R þ δa
9 > =
r g 2R þ δa > ;
ð12Þ where A is the Hamaker constant and δa is the nonzero distance of closest approach and ensures that VA does not diverge at contact. The Hamaker coefficient is approximated by20 3 ε1 ε3 2 3hνe ðn1 2 n3 2 Þ2 A ¼ kT þ pffiffiffi ð13Þ 4 ε1 þ ε3 16 2 ðn1 2 þ n3 2 Þ3=2 for the symmetric case of two identical phases 1 interacting across medium 3, where νe is the main electronic absorption frequency
Figure 4. (a) (Inset) Calculated Hamaker coefficient as a function of polymer concentration. Pair interaction potential as a function of particle surface to surface distance (rD) with selective cp (wt %) for van der Waals attraction. (b) Pair interaction potential as a function of particle surface to surface distance (rD) with selective cp (wt%) for depletion.
in the UV typically around 3 1015 s1 and ni is the refractive index of the ith component.20 Phase 1 is silica, and phase 3 is the polymerethanol mixture. As a result, ε3 and n3 vary depending on the polymer concentration while ε1 =3.8 and n1 = 1.4555 are fixed for silica. Here, we determine the dielectric permittivity and refractive index as ε3 ¼ εEtOH ð1 xp Þ þ εPEG xp n3 ¼ nEtOH ð1 xp Þ þ nPEG xp
ð14Þ
where xp is the mole fraction of PEG 400 in polymer solution. Substituting eq 14 into eq 13, the effective Hamaker coefficient can be calculated. The refractive indices of silica and PEG are matched (nPEG = nSiO2), which contributes to a reduction in the Hamaker constant as cp increases (Figure 4a). For silica in ethanol, A = AEtOH = 0.99 kT when ε3 = 24 and n3=1.333, while in a melt solution and for silica in PEG 400, A = APEG = 0.03 kT when ε3 = 6 and n3=1.4539. We note that the calculated Hamaker constants vary systematically with increases in cp as a result of changes in the dielectric properties of the suspending medium which may include some uncertainties. In Figure 4a we show how the van der Waals attractions vary with surface to surface separation at different polymer concentrations. The approximation used in eq 12 diverges when the particles come into contact. Due to the particle surface roughness, we truncate the surface to surface separation to a reasonable 5216
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value. Here, we choose a cutoff to be δa = 0.03 nm. We acknowledge the cutoff distance δa is critical to determine the depth of the primary minimum. Here, δa is chosen such that at a surface separation of δa the van der Waals attraction has a value of 101102 kT at intermediate cp. The approximation for van der Waals attractions used in eq 13 is known to overestimate the extent of attractions as the particles separate. As a result, we note eq 12 is an upper bound on the strength of attraction.19 In the absence of other forces, we expect the particles to be stable at low cp due to strong electrostatic repulsions and decrease in stability as the particle charge goes to zero. However, because A is a decreasing function of cp, the strength of attraction at contact will diminish until, at large cp, the suspension becomes stable. At a fixed volume fraction, as the contact value for the van der Waals attraction moves from many kT to near zero, we would expect to see the suspension restabilized. At cp ≈ 10 wt %, the strength of attraction will be 1020 kT and there is no barrier to aggregation. As a result, in the absence of other interactions, we would expect to see rapid aggregation. Instead, we observe slow aggregation associated with a barrier height of ∼12 kT. We attribute this barrier to particle interactions mediated by the presence of polymer above and beyond polymer-induced changes to the continuous phase dielectric constant and refractive index. To this point, we considered only indirect effects of adding polymer: how the refractive index and dielectric constant of the continuous phase are altered by addition of polymer. Below we consider the more direct effect of polymerparticle interactions. 3. Polymer-Mediated Interactions. There are two types of interactions we expect from the polymer. The first is a depletion interaction that occurs when the polymer is excluded from the gap between particles.1 The second type of interaction arises from adsorption of polymer to the particle surface. To estimate the effects of depletion forces on particle stability we apply the Polymer Reference Interaction Site model (PRISM) integral equation-based theory21 for polymer-induced depletion attractions. This implementation of PRISM is devised to provide an accurate microscopic description of polymer melts, solutions, mixtures, blends, copolymer, and star-like architectures, which enables us to see their structures, thermodynamics, and phase transitions.22,23 We start by applying this theory for nonadsorbing polymers. This theory describes polymer molecules as connected chains of spherical monomer units. The magnitude of the depletion potential is determined by the concentration of polymer and the degree of polymerization. The extent of the excluded volume interaction is characterized by the polymer density screening length, ξF.2123 We approximate the depletion attraction by Wp(r), the polymermediated contribution to the colloidcolloid total correlation function based on athermal PY closure. The particle pair potential with depletion attraction predicted by PRISM can be written Vp ðrÞ ¼ kT logð1 þ Wp ðrÞÞ πz R R ðr 2RÞ=ξF Wp ðrÞ ¼ , r > 2R e 3 r d d d πz d ¼ þ ; ¼ ξF ξc 3 ξc
rffiffiffiffiffi 12 N
ð15Þ ð16aÞ
ð16bÞ
where z = Fpd3 = (NNAcp/Mwp)d3, which is proportional to cp, N is the degree of polymerization, Fp is the polymer segment number density, d is the effective statistical segment length, (Rg = (N/6d)1/2),
Mwp is the molecular weight of the polymer, cp is the polymer concentration (in g/mL), and cp* is the semidilute polymer concentration. The density screening length, ξF, is a key parameter to determine the range of depletion interaction, defined with eq 16b. As polymer concentration increases, the excluded volume between colloids does not increase linearly with concentration. One notes from eq 16b when cp, cp*, ξF has a value on the order of Rg. When cp > cp*, ξF decreases rapidly, and in the limit of high polymer concentrations, ξF approaches the polymer segment size d. For PEG 400, cp* = (3Mwp/4πRg3NA) = 26.7 g/100 ml = 26.7 wt %, and Rg = 0.7 nm. Thus, while the strength of the depletion increases monotonically with increasing polymer concentration, the range of the attraction shrinks dramatically.9,21 The depletion potentials for selected polymer concentrations are shown in Figure 4b. These calculations show that at cp = 3 wt % a relatively long range of depletion interaction is induced but its contact value of potential is small. At cp= 10 wt %, we observe a shorter range of depletion interaction but the contact value of potential is increased. In the polymer melt (cp = 100 wt %) the depletion attraction is a very short-range interaction with a large contact value. These calculations predict that at low polymer concentration there is long-range but weak attraction. In combination with electrostatic and van der Waals interactions, pair potentials including depletion attractions result in B h2 becoming more negative with increasing polymer concentration. However, when cp is large, where van der Waals and electrostatic interactions are minimal, the depletion attraction is not sufficient to drive B2 negative, indicating that depletion interactions play a minimal role in altering the state of particle aggregation as cp is raised to melt densities. These effects result from the low polymer molecular weight where D/Rg = 63. Overall, the magnitudes of depletion attractions are small compared to the van der Waals forces. Nevertheless, these forces will only accelerate the rate of aggregation as cp is increased and will delay the onset of restabilization at higher cp. With only electrostatic, van der Waals, and depletion forces for cp > 3 wt %, the pair potential decreases monotonically as two particles approach each other. As a result, these interactions cannot explain the observed slow rate of aggregation. 4. Steric Repulsion. The model of depletion attraction discussed above does not account for the adsorption of polymer on the particle surface. Adsorption will confine polymer segments to the particle surface, and the adsorbed chains will extend into the solution. As a result, when two particles approach, the chains will be constrained lowering the chain entropy thus resulting in repulsive interaction energy. If the adsorption energy is sufficiently strong that the polymers are not displaced upon particle approach, this will result in a repulsive interaction. The range of this repulsion will be on the order of Rg.2426 On the other hand, if the adsorption energy is small, the polymers may be displaced and result in a transient barrier to aggregation. The barrier height can be estimated by the energy required to displace the polymer layer in the area of contact. To quantify the steric repulsion in this system, the amount of polymer adsorbed on the silica particles is shown in Figure 5a. Polymer concentrations were limited up to 3.5 wt % because at higher concentration particles start to aggregate and the suspensions sediment. Figure 5a shows that adsorption increases as polymer is added and approaches a plateau at near 1 wt %. The final adsorbed amount lies near 0.08 mg m2, indicating for cp > 1 wt % a monolayer of 70008000 monomers is associated with each particle. Also shown in Figure 5a is data from Trens et al.,26 5217
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Figure 5. (a) Adsorption isotherm for PEG 400 on 10 wt % of 44 nm SiO2 particles in ethanol (blue diamonds) determined experimentally and PEG 400 in water (green squares) determined from Trens et al.26 and Langmuir isotherm fitting (line). (b) Pair interaction potential as a function of particle surface to surface distance (rD) with different cp (wt %) for steric repulsion. (c) Schematic diagram of polymer segments displaced area.
who measured the adsorption isotherm of PEG 400 on silica in water. For water as a solvent, a plateau adsorption density of 0.15 mg m2 is reported. The adsorption isotherm shown in Figure 5a is fitted with a Langmuir isotherm to compare the amount of polymer adsorption in water and in ethanol. The Langmuir isotherm is given as Γ ¼ Γmax
Kl cp 1 þ Kl c p
ð16Þ
where Kl is the Langmuir equilibrium constant, cp is the molar concentration of adsorbate, Γ is the amount of adsorbed, Γmax is the maximum amount adsorbed as cp increases, and R is the gas constant. Fits of eq 17 to the data yield Kl ≈ 98 L mol1 for ethanol and Kl ≈ 5.2 103 L mol1 for water. Two distinctive features of the isotherms shown in Figure 5a indicate weak polymer adsorption onto silica particle in ethanolpolymer mixture: (i) the adsorption plateau in ethanol is approximately half of that in water, and (ii) the affinity for PEG 400 in ethanol is substantially smaller than in water. The nature of polymer adsorption onto particle surfaces differs depending on polymer molecular weight, solvent quality, and surfacepolymer segment interaction.21,26 Theories have been developed to predict the spatial distribution of polymer segments
between two approaching particles2729 from which pair potentials are estimated. The cases of end-grafted chains and irreversibly chemically adsorbed are well studied.20 When segments bind to surface through physical adsorption, however, the situation is more complicated.19,20 Weakly adsorbed polymers are dynamic with segments continually attaching and detaching from the surface, and steric forces can be weak, making it difficult to isolate the polymeric forces from other forces in solution. Scheutjens and Fleer developed a self-consistent field theory27 for polymer segment density profiles, while de Gennes developed the scaling theory for adsorbed polymer layer.30,31 Qiu et al.29 calculated steric repulsion for PEG on silica particles dividing the repulsion into “hard” and “soft” contributions. The hard contribution results from short chains which cannot be displaced, resulting in a contact value that diverges. Our experimental evidence clearly shows that PEG is weakly adsorbed to the silica particle surface under conditions where we see slow aggregation. The slowness of this aggregation indicates a barrier to the particles sampling a potential energy minimum. We attribute the fact that the barrier is low enough to allow aggregation to low polymer molecular weight and weak adsorption. To estimate the magnitude of the repulsions in the presence of weak adsorption we use the approach suggested by Mackor,32,33 which assumes repulsive interactions result from loss of the configurational entropy of the adsorbed molecules when approaching particles begin to interact. Polymers are treated as rigid rods, anchored at one end to the surface by a freely hinged joint, and the model estimates the reduction in entropy resulting from restrictions of segment movement by a similar opposing surface.32,33 The resulting pair potential can be written34 9 8 > = < Ns kTθ¥ πðδ xÞ2 D þ δ þ x , for x < δ > δ Vs ¼ > ; : 0, for x > δ > ð17Þ where Ns is the number of adsorbed sites per unit area, δ is the adsorbed layer thickness, θ¥ is the surface coverage (∼0.2 for polymer adsorption34), and x is the surface to surface separation of the two approaching particles (x = r D). From the adsorption isotherm (Figure 5a), we estimate Ns = 1.50 1018 m2, with θ¥ = 0.19. Equating the adsorbed layer thickness with the polymer radius of gyration of PEG 400 we set δ = 0.7 nm. At contact, Vs = 31 kT. If the adsorption of polymer is not altered for cp > 3 wt %, the pair potential for steric repulsion will be roughly independent of increasing polymer concentration. The calculated steric repulsion is shown in Figure 5b. The predictions made from eq 18 can be compared with the scaling theory of the polymer adsorption,30,31 which involves estimating additional parameters. Assuming Rg = 0.7 nm is sufficiently small that polymer bridging does not occur and based on the fact that the strength of attraction between polymer segments and particle surface is less than 0.55 kT,35 we neglect the bridging attraction term from the original equation36 and find the steric repulsion can be approximated when x is greater than the cut-off distance δa as ( ! ) Da RSc kT 4DSc 5=4 8Γ 9=4 1 1 9=4 Vs ¼ π Φs0 DSc 2 x1=4 δ1=4 25=4 Γ0 ξF 3
ð18Þ where Da is the diameter of a particle with the adsorbed polymer layer, ξF is the segment length of a monomer defined as the mesh 5218
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Langmuir size of the FloryHuggins lattice, and RSc is a numerical constant that is related to polymer osmotic pressure. Φs0 is the polymer concentration at the saturated surface, and DSc is the effective thickness of the adsorbed polymer layer. Here, we set Dsc = δ. Γ/Γ0 is the coverage where Γ is the total amount of polymer adsorbed on a single surface and Γ0 is the adsorbed amount at saturation. In our comparisons we choose (RSckT/ξF3) = 3 105 N/m2 and Φs0 = 0.03 to be the same as that estimated in previous studies.36,37 DSc is set as the adsorption thickness and Γ/Γ0 =1 since adsorption layers are saturated at intermediate polymer concentrations. This equation has uncertainties in several variables such as in (RSckT/ξF3) and Φs0 as Runkana pointed out36 in that not all variables can be obtained experimentally. However, we are interested in understanding the effects of weak polymer adsorption on the pair potential in the presence of van der Waals attractions and choose variables within acceptable ranges. Fritz et al. also suggested a steric repulsion model for grafted polymer on particle. The model takes account of the local increase in the osmotic pressure and loss of entropy due to the overlap of the polymer brushes. These models predict similar steric pair potentials with contact values of 2040 kT and extents of ∼δ.38 In Figure 5b, we compare the predictions of steric repulsion based on eqs 17 and 18. The deGennes scaling theory accounts for flexible chains, and it decays a little more rapidly than Mackor’s theory. On the basis of the similarity of these two predictions we focus on eq 18 to consider the total interaction potential. Figure 5b shows the contact value (at x = δa) is ∼3035 kT. Note that this contact value depends on the concentration of polymer at the particle surface. Stronger polymerparticle interactions are expected to increase the polymer concentration at the particle surface, thus increasing the contact value of the steric repulsion.34,36 To further support the predictions of this steric model, we estimate the energy required to displace a monolayer of polymer from two approaching surfaces as they come into contact. Consider two spheres in contact. We estimate the number of polymer segments that will be displaced as the shaded area in Figure 5c, where the surfaces are closer than δ. This approximation will be a lower bound on the number of segments displaced. The shaded area on both spheres sums to 2πDδ. Assuming that two segments per chain are adsorbed on average, with a monoloayer coverage of 8 105 g m2 (yielding 2.4 1017 bonds displaced/m2) with D = 44 nm and δ = 0.7 nm, ∼45 bonds will be displaced. This suggests that to achieve a contact steric repulsion of 3035 kT, the polymer segment/surface interaction energy is 0.250.5 kT per segment, in line with estimates derived by independent studies.35 5. Total Interaction Potential. The total pair potential is estimated by summing up electrostatic repulsion, van der Waals attraction, depletion attraction based on PRISM theory, and steric repulsion drawn in Figure 6a. As cp increases, there is a repulsive barrier of 1020 kT at cp ≈ 10 wt %. Despite all the approximations used in determining A and the magnitude of the steric repulsion, the barrier height predicted is surprisingly close to those measured in the aggregation kinetics experiments. One notes that for both experiments and calculations the barrier decreases up to cp ≈ 10 wt % and then climbs to 2030 kT as the Hamaker coefficient decreases. Our model explicitly assumes that the polymer can be displaced from the particle surface. However, as the strength of the van der Waals forces decreases with increasing polymer concentration, the driving force for
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Figure 6. Pair interaction potentials as a function of particle distance (r D) with different cp (wt %) for (a) total interaction potential; (b) its magnified view.
displacement decreases and the barrier height increases such that when cp > ∼50 wt % the contact value of the van der Waals interaction and the steric force are approximately equal and the system is thermodynamically stable. The transition point from aggregation to restabilization occurs at cp = 40 wt % but is sensitive to the cutoff distance (δa). For 3 wt % < cp < 50 wt %, the predicted pair potential has both primary and secondary minima. Although the magnitude of attractions at secondary minimum vary from 0.5 kT to 2 kT depending on polymer concentration, the range and depth of this attraction are not sufficient to drive phase separation. This is demonstrated by calculating the second-virial coefficient assuming hard particles from the minimum in the pair potential with the attractive tail at larger separations from Z ¥ 2π r 2 ð1 eV ðrÞ=kT Þdr δmin B2 ¼ ð19Þ 2π ðD þ δmin Þ3 3 where δmin is the surface separation at the secondary minimum. Over the range of polymer concentrations of interest B h2 is found to be 0.50.8, indicating stable particles. For cp > 50 wt %, the total interaction potential is always repulsive. The magnitude of the contact value of the pair potential remains sufficiently positive, indicating particles do not aggregate into a primary van der Waals minimum. We note that recent studies based on comparison of measured and calculated suspension microstructure, the strength of polymer segmentsurface interactions is found to decrease as ethanol 5219
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concentration increases.12 The segmentsurface contact attractive potential is found to be approximately a linear function of the polymer weight fraction starting at 0.55 kT at 100 wt % and decreasing to 0.275 kT at a polymer concentration of 50 wt %.12 The experiments are compared with predictions of PRISM theory developed for dense polymer solutions and concentrated particle suspensions. This theory becomes less accurate as cp approaches zero. However, it does predict strong depletion interactions if the segmentsurface interaction energy, εpc, is lower than ∼0.25 kT. For larger strengths of attraction between polymer segments and the particle surface, the particles are predicted to be thermodynamically stable in the absence of other interactions except volume exclusion. These recent results12 are in agreement with the estimates made here which approach the intermediate range of cp values from theories developed to characterize colloidal interactions in dilute polymer solutions. From the data presented here, the models of pair potentials based on van der Waals, electrostatic, depletion, and steric repulsions and in combination with the recent studies on the particle microstructure in dense polymer solutions, we conclude that aggregation at intermediate polymer concentrations is driven by van der Waals attractions. In this study depletion is weak compared to van der Waals attraction, although we note it can accelerate the aggregation to some extent. Aggregation is slowed by weakly adsorbed polymer segments that are displaced as a result of Brownian collisions giving rise to slow aggregation. As the polymer concentration is increased the driving force for van der Waals aggregation drops to zero while steric forces remain constant or grow giving rise to thermodynamic stability for the particles. These models suggest that polymer segments must have a contact attractive energy on the order of 0.20.3 kT to ensure that the barrier to aggregation is sufficient to limit aggregation. This study offers insights on the strength of polymer adsorption required to provide steric stability when the particles also experience van der Waals attractions. Let the dimensionless energy ratio of steric repulsion to van der Waals attraction be Rs Rs ¼
strength of steric repulsionðkTÞ strength of van der Waals attractionðkTÞ πNs θ¥ Dδ 24πNs θ¥ δδa ¼ AD A 24δa
ð20Þ
When Rs . 1 the particles will be stable, and when Rs , 1 the particles will aggregate. For 10 wt % < cp < 50 wt %, Rs is close to unity and particles aggregate. When more polymer is adsorbed, Rs will increase and the system will restabilize. For example, consider the silica/PEG/water system where van der Waals attractions are essentially the same as in the silica/PEG/ethanol system but PEG adsorbs more strongly. Under these circumstances, increases of Ns or θ¥ result in the onset of the particle restabilization at a polymer concentration lower than 50 wt %. Indeed, the strength of attraction may be sufficient to confer stability at all cp.
V. CONCLUSION In this paper, the colloidal stability of 44 nm silica particles in polyethylene glycol (PEG) 400 and ethanol mixtures has been investigated. Particles are stable at low polymer concentrations (cp < 3 wt %) due to the electrostatic repulsions. The electrostatic origin of the repulsion and its reduction with increasing polymer
concentration has been demonstrated by conductivity measurements and drops in second-virial coefficients with increasing electrolyte concentration. Since electrostatic repulsion decreases with addition of PEG, van der Waals and depletion attractions produce a driving force for aggregation. At high polymer concentration the Hamaker coefficient drops to a low value due to index matching of the solution with the particles. Under these conditions, while van der Waals forces are minimized, depletion interactions will drive aggregation in the absence of polymer adsorption. At intermediate concentration slow aggregation is observed. This aggregation is attributed to displacement of weakly adsorbed polymers as particles find the primary van der Waals minimum. Model predictions based on electrostatics, van der Waals, depletion, and steric forces show good agreement with experimental results. In this system we argue that steric repulsion controls the rate of aggregation at intermediate cp and results in permanent stability as cp approaches melt concentrations. Our conclusions are in line with many studies that demonstrate the significance of steric repulsion for depletion restabilization where particles aggregate at low cp but are stable at high cp.39,40 Here, we find that stability at high cp arises from two sources. First, due to index matching of particles and ethanolPEG solution, there is a decrease in van der Waals attractions as cp increases. Second, the strength of attraction between polymer segments and the particle surface, εpc, grows with cp, resulting in an adsorbed polymer layer that is not displaced by Brownian encounters. We suggest that the aggregation seen at intermediate polymer concentration results from weak adsorption of polymer. In the absence of other interactions, the detailed theories of Schweizer et al.35 show that depletion forces drive aggregation when there is weak attraction of polymer segments and the particle surface. At large strengths of attraction, εpc > 1 kT, these theories predict bridging aggregation. Only at intermediate strengths of attraction, 0.275 kT < εpc < 1 kT, steric stabilization is predicted. In the presence of van der Waals attractions, the window of particle stability will narrow. Nevertheless, as the strength of polymer segmentparticle surface attraction grows, there will be a greater tendency for stability until bridging aggregation occurs at large εpc. The binding energy of a polymer segment to the particle surface is a relative energy that depends on replacing polymer/ polymer and polymer/solvent interactions in the bulk with polymer/surface interactions when the segment is adsorbed. This energy will thus depend on the theta state of the polymer segment in the bulk. In a polymer melt the segments are in a theta state, marginally stable, and segments will lower their energy by adsorbing to the particle surface (large εpc).12 For the experimental system studied here, when cp is small, the polymer is in a good solvent in the bulk and polymer segments will have less tendency to adsorb to the particle surface (low εpc).12 As a result, with increases in cp, εpc will increase in magnitude. In the presence of van der Waals attractions, this would suggest that adsorbed polymer layers may be displaced in Brownian encounters at low cp but not be displaced at large cp. We suggest this tendency of polymer segments to adsorb more strongly at high polymer concentration may play a substantial role in the poorly understood phenomena of depletion restabilization. The ability of steric forces to stabilize a colloidal suspension thus depends on the ease with which polymer layers can be displaced. We suggest that the ratio of contact steric repulsive 5220
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Langmuir strength to contact van der Waals attraction, Rs, given in eq 20 can be employed as a design tool for characterizing conditions giving rise to particle stabilization. Although many of the parameters do exist in the particle/polymer/solvent system for prediction of particle stability the simplified concept of Rs provides insights into conditions that will result in depletion stabilization and where steric interactions will stabilize particles against aggregation by van der Waals forces.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We acknowledge the 32ID-I beamline at the Advanced Photon Source (APS), Argonne National Laboratory, where USAXS experiments were carried out. The APS is supported by the U.S. DOE, Basic Energy Sciences, Office of Science under Contract No. DE-AC02-06CH11357. This work was supported by the Nanoscale Science and Engineering initiative of the National Science Foundation under NSF Award Number DMR-0642573. ’ REFERENCES (1) Poon, W. C. K.; Warren, P. B. Europhys. Lett. 1994, 28 (7), 513–518. (2) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168 (2), 485–496. (3) Tuinier, R.; Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 2000, 113 (23), 10768–10775. (4) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2002, 116 (5), 2201–2212. (5) Bergenholtz, J.; Poon, W. C. K.; Fuchs, M. Langmuir 2003, 19 (10), 4493–4503. (6) Lekkerkerker, H. N. W.; Poon, W. C. K.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Europhys. Lett. 1992, 20 (6), 559–564. (7) Poon, W. C. K. J. Phys.: Condens. Matter 2002, 14, 33. (8) Shah, S. A.; Ramakrishnan, S.; Chen, Y. L.; Schweizer, K. S.; Zukoski, C. F. Langmuir 2003, 19 (12), 5128–5136. (9) Chatterjee, A. P.; Schweizer, K. S. J. Chem. Phys. 1998, 109 (23), 10464–10476. (10) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. (11) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (12) Kim, S. Y.; Hall, L. M.; Schweizer, K. S.; Zukoski, C. F. Macromolecules 2010, 43 (23), 10123–10131. (13) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (14) Flory, P. J. Statistical mechanics of Chain Molecules; Interscience: New York, 1969. (15) Anderson, B. J.; Zukoski, C. F. Macromolecules 2007, 40 (14), 5133–5140. (16) Apfel, U.; Horner, K. D.; Ballauff, M. Langmuir 1995, 11, 3401. (17) Zimm, B. H. J. Chem. Phys. 1948, 16, 62–69. (18) Evans, D. F.; Wennerstr€om, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet; VCH: Cambridge, 1994. (19) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, UK, 1989. (20) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic press: New York, 1991.
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