Quantification of Nanoparticle Interactions in Pure Solvents and a

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Quantification of Nanoparticle Interactions in Pure Solvents and a Concentrated PDMS Solution as a Function of Solvent Quality Nupur Dutta,† Sergei Egorov,‡ and David Green*,† †

Department of Chemical Engineering and ‡Department of Chemistry, University of Virginia, Charlottesville, Virginia 22904, United States ABSTRACT: Using turbidity measurements, we quantified the interactions between PDMS-grafted silica nanoparticles (PDMS-gsilica) in pure solvents and a concentrated polymer solution with a focus on detecting the impact of solvent quality on graft layer stretching. This work is an extension of our previous work where we showed that interfacial wetting of the grafted polymer leads to depletion restabilization in semidilute and concentrated polymer solutions in good solvents (Dutta, N.; Green, D. Langmuir 2008, 24, 5260−5269). Subsequently, we showed that the criterion for depletion restabilization holds for both good and marginally poor solvents (Dutta, N.; Green, D. Langmuir 2010, 26, 16737−16744). In this work, we quantified nanoparticle interactions in terms of the second virial coefficient (B2), which captures the stretching of the brush in a good solvent in comparison to compression in a poor solvent. The transition from stretching to compression of the graft layer as a function of solvent quality was also supported by self-consistent mean-field (SCF) calculations. The PDMS-g-silica nanoparticles in a concentrated polymer solution in a good solvent within the complete wetting region behaved as though they were in a good solvent rather than in a polymer melt where on the basis of the SCF calculations the graft layers were expected to behave ideally. Overall, our results indicate that turbidity measurements can be used to determine the second virial coefficients for polymergrafted nanoparticles in solvents and concentrated polymer solutions, and the relative values of the coefficients correspond well to those from theoretical calculations.

1. INTRODUCTION Applications that involve dispersing polymer-grafted colloids in semidilute and concentrated polymer solutions often require changes in the solvent or polymer to save money, satisfy environmental regulations, or solve biocompatibility issues. Such applications include pharmaceuticals, agrochemicals, paints, inks, and personal care products. To this end, the effect of solvent and polymer on particle dispersion in semidilute and concentrated polymer solutions was recently elucidated by us for particles with a core radius, R, larger than the free polymer radius of gyration, Rg.1,2 In this environment, the size ratio R/Rg > 1 equates to the colloidal or large probe limit, where we show for particles with low interfacial curvature (R > 50−100 nm) that the free polymer penetrates the graft layer when its graft polymer molecular weight, P, is greater than that of the free polymer, N, or P/N < 1.0, resulting in brush stretching and particle stability through steric repulsion in good and marginally poor solvents. Hence, the interactions between the matrix and grafts control particle stability, which depends on the graft density, Σ, the volume fraction of free polymer, ϕp, and the Flory−Huggins interaction parameter, χ, depicting solvent quality.1−3 On this basis, we seek to use the turbidity method4,5 to quantify the interactions of polymer-grafted nanoparticles with respect to χ and ϕp in simple solvents and concentrated © XXXX American Chemical Society

polymer solutions. The method benefits experimental exploration because the primary equipment, a UV spectrophotometer, is readily available in most laboratories and can be used to quantify interactions in suspensions subject to multiple scattering over a wide range of particle sizes and concentrations. Of course, other techniques can access colloidal interactions in simple solvents and polymer solutions such as static light scattering,4−14 laser scanning confocal microscopy,15 small- and ultra-small-angle X-ray scattering,16 and neutron scattering,17,18 which have been used in studies on hard spheres, microgels, and soft spheres; however, the application of these techniques has limitations. For example, small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) are suitable techniques for studying interparticle interactions for colloids that are less than 50 nm in diameter; however, as the size increases beyond 50 nm, conventional SAXS and SANS can no longer can be applied accurately. Static laser light scattering (SLS) becomes the most suitable method beyond this length scale and can be accurately applied to colloidal systems.19 Similar to SAXS and SANS, in SLS the sample is scanned at constant wavelength over a range of scattering angles, and the Received: August 30, 2012 Revised: April 17, 2013

A

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section 2, we cover the experimental methodology and the theory behind turbidmetry and self-consistent mean-field calculations. In section 3, we present our experimental and theoretical results in which we make connections between the conformation of the graft layer and the second virial coefficient of the particles. The Article concludes with section 4, a summary of the experimental and theoretical results.

scattering intensity is analyzed to extract colloidal interactions. However, suspensions must be dilute; otherwise, multiple scattering at higher particle concentration skews the analysis. An alternative is the turbidity method that can account for multiple scattering.20 Vrij and co-workers were the first to use the turbidity method to quantify the interactions between monodisperse stearyl alcohol-coated silica particles in pure solvents.8,11 Using a UV spectrophotometer, they measured the transmittance (Tr) of a sample over a range of wavelengths and calculated the turbidity (τr) based on the Rayleigh−Debye (RD) theory of light scattering.8 Subsequently, they quantified particle attractions as a function of solvent quality through the second virial coefficient (B2) in different solvents. Several groups have also used the turbidity measurements to quantify particle interactions in pure solvents and dilute polymer solutions. For example, Ballauff and co-workers extended the method by Vrij to elucidate the effects of electrostatics on highly charged polystyrene (PS) latexes in good and theta solvents.4,5,9,12 They also studied how free polymer affects the bridging and depletion flocculation of latexes in dilute polymer solution.9,12 Zukoski and co-workers also used the method to investigate the depletion flocculation of hard spheres in dilute PS/toluene solutions.21 To tune depletion forces, they changed the polymer-to-particle size ratio (i.e., Rg/R) to investigate both the colloid limit (R > Rg) and the protein limit (R < Rg).22 From their data, they calculated the osmotic compressibility to account for particle interactions. Tan et al. employed the turbidity method to quantify interactions between pH-responsive microgel particles in aqueous solutions.23 On the basis of the developments of Ballauff and co-workers,4,5,12 Tan et al. investigated the transition of the microgel from swelling to deswelling by detecting the changes in the osmotic compressibility of the solutions as a function of pH, microgel concentration, and degree of cross-linking. Although the studies of Vrij, Balluff, Zukoski, and Tan highlight the use of the turbidity method, to our knowledge no publication exists with a focus on quantifying the interactions between polymer-grafted nanoparticles in simple solvents and concentrated polymer solutions. Hence, we seek to adapt turbidity measurements to determine the osmotic compressibility (Π) and extract the second virial coefficient (B2) for polymer-grafted nanoparticles in these environments. These measurements will provide a link to detect the stretching and collapsing of graft polymers whose conformations change with solvent quality. Conformational changes will impact the magnitudes of Π and B2, both of which are expected to increase or decrease, respectively, with the stretching or collapsing of polymer brushes. Hence, by using well-characterized PDMS-g-silica nanoparticles in pure solvents and concentrated PDMS solutions, we seek to quantify the effect of solvent quality on nanoparticle interactions using turbidity measurements. In particular, we seek to detect how solvent quality affects the stretching and compression of the graft polymer through the osmotic compressibility (Π) and second virial coefficient (B2). To this end, samples were formulated by dispersing PDMS-g-silica nanoparticles in pure solvents and a concentrated polymer solution such that the swelling ratio of the polymer on the particles was P/N < 1.0; thus, the grafts were longer than the free polymer. Moreover, the particle−polymer size ratio is R/Rg > 1; thus we operate in the colloidal, large-probe limit. In

2. EXPERIMENTAL AND THEORETICAL METHODS To quantify particle interactions in simple solvents and concentrated polymer solutions, we synthesized monodipserse silica particles, grafted PDMS to the particles, formulated well-controlled polymer/ particle suspensions, and characterized the samples using electron microscopy, elemental analysis, and light scattering. Turbidity measurements on these suspensions were analyzed to calculate the second virial coefficient (B2) for the particles. Moreover, selfconsistent mean-field calculations were used to connect the experimental results to theory. Our experimental and theoretical methods are discussed in this section. 2.1. Materials. Reagent-grade chemicals (purity ≥99%) were used as purchased from the manufacturer without further purification unless specified. Tetraethylorthosilicate (TEOS), absolute ethanol, ammonium hydroxide (30% NH3), and 1-pentanol were all obtained from Acros Organics. Cyclohexane and t-butanol were obtained from Fisher Scientific. Tetrahydrofuran was obtained from Sigma-Aldrich. All industrial-grade PDMS polymers and octamethyltrisiloxane were obtained from Gelest. The 18 MΩ deionized water was produced with a Barnstead E-pure water purification system. 2.2. Nanoparticle Synthesis and Characterization. The Stöber method24 was used to synthesize nanoparticles with diameters of 100 ± 19 nm (45 particles, 95% confidence interval) as verified by SEM (JEOL 6700 F). The particles are shown in Figure 1. Reagent

Figure 1. SEM micrograph of silica nanoparticles formulated from 0.30 M TEOS, 0.45 M NH3, and 2.96 M H2O. concentrations of [NH3] = 0.45 M, [H2O] = 2.96 M, and [TEOS] = 0.30 M in ethanol were used to make the nanoparticles as derived from an empirical relation that relates the concentrations of the initial reactants to the size of the final particles.25 The particles were then washed five times with absolute ethanol to remove any unreacted chemicals. To facilitate particle grafting in cyclohexane, the surface hydroxyl and ethoxy groups were removed by combining purified ethanol/silica solutions with an equal volume of 1-pentanol (Acros Organics) and then removing the ethanol through distillation. Subsequently, the esterification of silica with 1-pentanol was performed for 24 h at 140 °C,26,27 which was carried out to prevent the strong adsorption of PDMS onto silica.1,2 Subsequently, the B

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particles were washed five times in cyclohexane, and PDMS of N = 18 kg/mol was grafted to the surface by adapting the procedures of Edwards et al.28 and Auroy et al.29 The dispersions were then poured into centrifuge tubes and washed four times with cyclohexane to remove unreacted polymer. The graft density of PDMS was calculated from elemental analysis as described in our previous publication,1,2 and five separate measurements were performed to ensure statistical accuracy. The graft density was found to be Σ = 0.200 ± 0.002 chains/ nm2 (error calculated in 95% confidence interval for measurements on the same sample). The PDMS-g-silica nanoparticles were then washed four times in octamethyltrisiloxane and t-butanol to make purified stock solutions that were used to formulate nanoparticle suspensions as discussed in the next subsection. All of the solvents were filtered through a 0.2 μm Millipore filter to remove dust prior to making the samples. 2.3. Formulation of Nanoparticle Suspensions. The suspensions were formulated in octamethyltrisiloxane (OTS) and t-butanol (t-BuOH) as well as in concentrated PDMS solutions in the same solvents. The molecular weight of the free PDMS polymers was P = 5.84 kg/mol, and the volume fraction of PDMS in solution was held constant at ϕp = 0.70, which is above ϕp** = 0.58, the crossover volume fraction from a semidilute to concentrated polymer solution calculated using the method in our previous publication.1,2 The free polymer molecular weight was characterized using GPC as discussed previously.1,2 The polymers were weighed into clean borosilicate vials, and the requisite amount of solvent was added to the respective vials, followed by the addition of the PDMS-g-silica nanospheres, whose amount was adjusted to vary the volume fraction of silica cores, ϕn = 1.0 × 10−2−1.25 × 10−3. For data analysis, the particle volume fractions were converted into mass concentrations (c) by multiplying ϕn by the density of silica (ρp = 1.85 g/cm3) (i.e., c = ρpϕn). All solvents were prefiltered before sample preparation. 2.4. Light-Scattering Theory and Turbidity Measurements. Turbidity measurements were analyzed to determine the osmotic compressibilities and second virial coefficients of the PDMS-g-silica nanoparticles in good and marginally poor solvents as well in a concentrated PDMS solution. In this section, we highlight the essential theory to calculate the virial coefficients; a detailed theoretical treatment of colloid-based turbidity measurements was derived by Ballauff and co-workers.4,5,9,12 The turbidity measurements of colloidal samples center upon obtaining the transmittance (Tr), which is defined as the ratio of the transmitted intensity (It) to the incident intensity (Io) or Tr = (It/Io). Subsequently, the turbidity (τr) can be determined as the attenuation of the incident beam due to scattering when passing through the sample

⎛I ⎞ τr = lo−1 ln⎜⎜ o ⎟ ⎝ It ⎠

Parameter s is the sample-to-detector distance. For unpolarized light, Rr(q) is

⎛ n πR ⎞3 R r(q) = Kc ⎜ o ⎟ P(q) S(q , c)(1 + cos2 θ) ⎝ λo ⎠

(2)

where P(q) is the form factor and accounts for the scattering due to the particle shape, whereas S(q, c) accounts for scattering due to particle interactions. The optical constant K is defined by

K=

2 6no ⎛ m2 − 1 ⎞ ⎜ 2 ⎟ λρP ⎝ m + 2 ⎠

(3)

which includes the relative refractive index m = np/no. Assuming no light absorption, the turbidity is related the scattering intensity and thus to Rr(q) by

τr = 2π

∫0

π

R r(q) sin θ dθ

(4)

The benefit of using a UV spectrophotometer is that the intensity measurements are taken at small scattering angles (θ = 1−2°) for particles that are not too large, allowing the expansion of P(q) and S(q, c) as θ→ 0, corresponding to small qR. Hence, P(q) and S(q, c) for spherical particles can be approximated by the expansion for qR ≪ 1 P(q) = 1 −

1 (qR )2 + O(qR )4 3

(5)

S(q , c) = S(0, c) + α(qR )2 + O(qR )4

(6)

4

where O(qR) represents higher-order terms for the error in making the low-scattering-angle approximation. Moreover, S(0, c) is the structure factor at vanishing scattering angle, and coefficient α can be related to a correlation function that depends on the type of closure.4 Another advantage of using a UV spectrophotometer over the SLS, SAXS, and SANS setups is the ability to vary the wavelength of light during the scattering experiment. Thus, instead of scanning as a function of angle as in SLS, SAXS, and SANS, the turbidity method involves scanning as a function of wavelength, which involves modifying the turbidity by substituting eqs 5 and 6 into eq 2, and the subsequent integration of eq 4 yields

⎛ n πR ⎞3 16π ⎡ ⎞ ⎛ n π ⎞2 ⎛ 1 ⎢S(0) + 8⎜ o ⎟ ⎜α − S(0)R2⎟ τr = Kc ⎜ o ⎟ ⎝ λ ⎠⎝ ⎝ λ ⎠ 3 ⎣ ⎠ 3 ⎛ 1 ⎞4 ⎤ + O⎜ ⎟ ⎥ ⎝λ⎠ ⎦

(7)

This expression can be factored into the product of two functions Q(λ2, c) and Z(λ2, c) (i.e., the integrated form and structure factors, respectively)

(1)

where lo is the optical path length, equal to 10 mm. The turbidity was measured with a Spectra Plus Max UV−visible spectrophotometer from Molecular Devices, which casts unpolarized light on the PDMSg-silica nanoparticles that scatter light at levels much higher than that adsorbed. Accordingly, when adsorption is excluded, the conservation of energy dictates that Io = It + ΣIs, where ΣIs is the intensity of light scattered in all directions. This approximation facilitates the use of Rayleigh−Gans−Debye (RGD) light-scattering theory to extract osmotic compressibilities and second virial coefficients of the nanoparticles from turbidity measurements. To this end, the RGD theory relates the turbidity to the structure factor, S(q, c), of the particles, where S(q, c) is a convenient measure of particle interactions. More specifically, the RGD theory extends the Rayleigh scattering theory for point scatters to larger particles in the size range of R(np − n0) < λ/10, where np and n0 are the refractive indices of the particles and medium, respectively, and λ is the wavelength of light. In general, in any light-scattering experiment with colloids, the relative intensity of the scattered light per unit scattering volume is defined as the Rayleigh ratio, Rr(q) = Iss2/IoVs, where Vs is the illuminated volume “seen” by the detector in the direction θ corresponding to the scattering vector q = (4πn0/λ) sin(θ/2).

Q (λ 2 , c) = 1 −

2 ⎛ 1 ⎞4 8 ⎛⎜ noπ ⎞⎟ 2 R + O⎜ ⎟ ⎝λ⎠ 3⎝ λ ⎠

⎛n π ⎞ ⎛1⎞ Z(λ 2 , c) = S(0, c) + 8α⎜ o ⎟ + O⎜ ⎟ ⎝ λ ⎠ ⎝λ⎠ 2

(8) 4

(9)

leading to

⎛ n πR ⎞3 16π τr = Kc ⎜ o ⎟ Q (λ 2 , c) Z(λ 2 , c) ⎝ λ ⎠ 3

(10)

Our objective is to use turbidity measurements in conjunction with eq 10 to obtain Z(λ2, c) to determine S(0, c), which is related to the osmotic pressure from which the second virial coefficient B2 can be determined through a virial equation of state. Experimentally, the determination of B2 translates into making a series of suspensions that vary in particle concentration, c, and measuring the turbidity as a function of wavelength, λ, so that the dependence of the integrated form factor, Q(λ2, c), can be found in eq 10 by extrapolating the turbidity to dilute concentrations (c → 0), C

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where the integrated structure factor approaches unity, or Z(λ2, c) → 1, a characteristic of negligible particle interactions.

⎛ n πR ⎞3 16π ⎛ τr ⎞ ⎜ ⎟ = K⎜ o ⎟ Q (λ 2) ⎝ c ⎠c = 0 ⎝ λ ⎠ 3

solvent of varying quality. By employing the Derjaguin approximation, one can convert this result into PMF between spherical nanoparticles.31,32 However, it is well known that the Derjaguin approximation breaks down for particles with sufficiently high curvature.33,34 Accordingly, in the present work we account for curvature by employing a 2-D version of the lattice-based theory.35,36 Specifically, each nanoparticle core is modeled as a cylinder whose radius is equal to its height, and the SCF equations are set up on a cylindrical lattice.35−37 The lattice layers along the axis of the cylinder are numbered according to z = 1, 2,..., Nz, whereas circular arrangements of the lattice sites within each z layer are numbered as r = 1, 2, · · ·, Nr. Each grafted cylindrical nanoparticle is placed in such a way that its z axis coincides with the z axis of the cylindrical lattice, and the PMF is computed as a function of separation between the two cylindrical cores along this common axis. The grafted region (zg, rg) for the grafted end segments on the first nanoparticle is given by z0 − 1 ≤ zg ≤ z0 + 2 and rg ≤ 3, where z0 is the location of its core center on the z axis. The grafted region for the second nanoparticle is the same for its r coordinate, with the z coordinate shifted according to the value of the separation distance between the two cores. Within the lattice-based SCF approach, polymers are modeled as chains of monomers, all of which are the same size, with every monomer occupying one lattice site.36 The system under study contains three types of molecules (in addition to nanoparticle cores that are taken to be neutral in terms of their interactions with other species): graft polymer chains of length N, free polymer chains of length P, and the (monomeric) solvent. The end segments of graft polymers are restricted to the grafting region as described above. Regarding the choice of boundary conditions, for the upper and lower values of z and for the upper value of r mirrorlike reflecting boundary conditions are implemented. The solvent quality is adjusted by varying the value of the Flory−Huggins interaction parameter χ between free and graft segments on one hand and the solvent molecules on the other hand. The SCF approach operates with two key quantities: density distributions of all of the species and the corresponding self-consistentfield potentials.30 Within our 2-D formalism, the density profiles of all of the species depend on two coordinates, z and r, and the mean-field approximation is employed for the angular coordinate. In the spirit of a mean-field approach, the potentials represent average interactions of a test molecule with all of the other molecules in the system. As such, they depend on the density distributions of the molecules that, in turn, are determined by the potentials. Thus, both quantities must be computed simultaneously and self-consistently via a numerical iterative solution to obtain the Helmholtz free energy (relative to the reference state, which is denoted by an asterisk and taken as the pure, unmixed, amorphous components)30

(11)

Thus, Q(λ , c) is obtained by plotting the specific turbidity, τr/c, at a specific wavelength, λ, versus the particle concentration, c, and extrapolating τr/c to c → 0 to obtain (τr/c)c=0 at the intercept. Subsequently, Z(λ2, c) can be obtained as follows: 2

Z(λ 2 , c) =

(τr /c) (τr /c)c = 0

(12)

In the limit of infinite wavelength (e.g., the near-infrared λ ≈ 750 nm for particles of R ≤ 200 nm),5 Z(λ2, c) can be related to S(0, c) through eq 9. Thus, plotting Z(λ2, c) with respect to 1/λ2 yields S(0, c) as the intercept that is proportional to the osmotic pressure, Π, and by association the isothermal osmotic compressibility (∂c/∂Π)T through S(0) =

1 ⎛⎜ ∂c ⎞⎟ β ⎝ ∂Π ⎠T

(13)

where β = 1/kBT in which kB is the Boltzmann constant and T the temperature. In the limit of sufficiently small concentrations, the inverse of S(0, c) can be written in a virial expansion in eq 14:

S(0, c)−1 = 1 + 2B2 c + O(c 2)

(14)

The use of this relation leads to the computation of the second virial coefficient, B2, which is related to the particle interaction potential (i.e., the potential of mean force (PMF), or W(r)) through

B2 =

2πNA M

∫0

∞

(1 − e−βW (r))r 2 dr

(15)

On the basis of eq 15, the magnitude of B2 is a direct measure of interactions between polymer-grafted nanoparticles in good and marginally poor solvents as well as in polymer solutions. Hence, the turbidity measurements on these suspensions were carried out by placing the suspensions in Teflon-capped glass cuvettes that were extensively precleaned by (1) washing with a soap solution; (2) cleaning with a solution of sulfuric acid and chromium trioxide to remove adsorbed silica particles or organic impurities; (3) sonication in cyclohexane for 2 h to remove trace PDMS; and (4) drying in a convection oven at 70 °C. Finally the cuvettes were rinsed with the solvent before introducing the sample for measurement. All experiments were carried out at a constant temperature of T = 25.5 °C in the spectrophotometer, in which samples were allowed to equilibrate for 1 h prior to measurement. The suspensions in pure solvents were measured relative to the solvent blank that was pre-equilibrated for 10 min. The pre-equilibration time was increased to 30 min for the PDMS solution blanks. 2.5. Self-Consistent Mean-Field (SCF) Theory. Self-consistentfield (SCF) calculations were carried out to elucidate the effect of solvent and polymer on nanoparticle interactions from the potential of mean force, W(r), that was used to predict the relative magnitudes of the virial coefficients, B2, from eq 15. These predictions were compared to the corresponding values from the turbidity measurements. The central quantity of the SCF calculations is the mean-field free energy, which is expressed as a functional of the volume fraction profiles and SCF potentials for all components in the system. As explained below, the calculation of the volume fraction profiles and SCF potentials requires solving the SCF equations iteratively and numerically, which necessitates space discretization (i.e., the use of a lattice). In the present work, we employ the method of Scheutjens and Fleer,30 which uses the segment diameter σ as the size of the cell; throughout this work, all of the distances are reported in units of the cell size σ. In our earlier work,1,2 the 1D version of Scheutjens and Fleer theory was used to compute the potential of mean force (PMF) between polymer-grafted nanoparticles, whereby the PMF was obtained as a function of separation between two flat grafted surfaces immersed in a

β(F − F *) = β(U − U *) −

S − S* kB

(16)

where the first and the second terms give the energy and entropy contributions to the free energy, respectively. Moreover, the calculation of the energy contribution includes the Flory−Huggins parameter χ to characterize the interaction between chain segments and solvent molecules. By calculating the free energy as a function of distance z between the two nanoparticle cores, one finally obtains the potential of mean force (PMF) between the two polymer-grafted nanoparticles as follows:

W (z) = F(z) − F(z = ∞)

(17)

The PMF in eq 17 was used to simulate the effect of variable solvency on particle interactions. Hence, the Flory parameter in the calculations was varied from χ = 0.2 to 0.8 to simulate the stability of PDMS-grafted nanparticles in good through poor solvents. Hence, two sets of calculations were run: (1) graft surfaces immersed in pure solvents and (2) graft surfaces immersed in a concentrated PDMS solution. The free polymer molecular weight used for the PDMS D

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solution case was P = 5.84 kg/mol, and the volume fraction of PDMS in solution was held constant at ϕp = 0.7. These calculations were used as a basis to carry out the turbidity experiments to elucidate the impact of solvent quality on the interactions between PDMS-g-silica in simple solvents and concentrated PDMS solutions with good and marginally poor solvents.

3. RESULTS AND DISCUSSION Briefly, turbidity measurements were used to quantify nanoparticle interactions in pure solvents and a concentrated polymer solution to elucidate the effects of solvent quality on suspension behavior. To this end, SCF calculations were first carried out to gain insight on how solvent quality impacts nanoparticle interactions. The results of these calculations were used as a guide to formulate suspensions for turbidity experiments. These experiments were analyzed using Raleigh−Debye (RD) scattering theory to extract the second virial coefficient, a link to particle interactions. To relate experiment and theory, the relative magnitudes of the virial coefficients from the turbidity experiments were compared to those from the SCF calculations. 3.1. Self-Consistent-Field Calculations. In choosing the model parameters in Table 1, we followed our earlier work1,2

Figure 2. Potential of mean force, W(z), as a function of the separation distance, z, in lattice units and the Flory parameter, χ, for polymer-grafted nanoparticles in good, theta, and poor solvent as well as a concentrated polymer solution with a good solvent with a polymer volume fraction of ϕp = 0.7.

volume fraction of free chains of ϕp = 0.7 (under good solvent conditions for both free and graft chains); the corresponding result is also given in Figure 2. The results for the PMF in a theta solvent and in a concentrated polymer solution are quite similar,37 suggesting that both systems are close to ideal. On the basis of SCF calculations in Figure 2, we anticipate that the virial coefficients from the turbidity measurements will follow trends similar to the interaction potentials where the stretching of the graft polymer in good solvents will lead to virial coefficients of the PDMS-g-silica nanoparticles that will be greater than those in a marginally poor solvent and a concentrated polymer solution. This would translate to B2‑OTS > B2‑PDMS‑OTS > B2‑t‑BuOH. The second virial coefficient can also be cast in the form of eq 18, facilitating further quantification of the effect of solvent quality on particle interaction through comparison to the interaction for hard spheres, B2,HS, which is the first term on the right side of eq 18.32,38

Table 1. Experimental and Theoretical Properties of PDMS Polymers manufacturer Mw (kg/mol)

polymerization index real chainsa

polymerization index lattice chainsb

N = 18 P = 5.84

242 79

63 21

a

Mw,PDMS/Mw,DMS; Mw,DMS is the molecular weight of dimethylsiloxane monomer = 74 g/mol. bPolymerization index for lattice chains.30

where the SCF method was used to compute the interactions between two flat graft PDMS surfaces. In the current work, we set the length of the graft lattice chains to be N = 63 (corresponding to the experimental value of the polymerization index of real PDMS chains equal to 240), and the length of the free lattice chains was taken to be P = 21 (corresponding to the experimental value of the polymerization index of real PDMS chains equal to 80). After setting the grafting density equal to Σ = 0.2 chains per unit area to match the experiment, we first computed the PMF between two polymer-grafted nanoparticles in pure solvent (i.e., in the absence of free polymer) for three values of the Flory−Huggins interaction parameter χ = 0.2, 0.5, and 0.7 corresponding to good, theta, and poor solvent conditions, respectively. The corresponding results are shown in Figure 2. One sees that with decreasing solvent quality the PMF gradually becomes less repulsive and for a poor solvent an attractive minimum develops. In general, particle repulsion in a good solvent is greater than that in a marginally poor solvent (e.g., χ = 0.55, not shown in Figure 2 for clarity) as the graft polymer stretches strongly in the good solvent in comparison to in the marginally poor solvent. Strong stretching of the graft brush in a good solvent implies that particles repel each other at greater separation distances than in a marginally poor solvent where the compression of the brush leads to a closer approach before repulsion takes place. Particle attractions in the poor solvent result from unfavorable interactions between the grafts and the solvent, leading to the collapse of and attraction between the graft layers. Next, we computed the PMF between two polymer-grafted nanoparticles in a concentrated polymer solution for the

B2 = B2,HS + 2π

∫R

∞

(1 − e−βW (r))r 2 dr

(18)

In general, B2,HS = 8. Hence, based on the second term on the right of eq 18, graft polymer stretching for soft spheres in good solvents would cause B2,HS > 8, whereas depletion attraction or graft polymer collapse on soft spheres in poor solvents should product B2 < 8. 3.2. Interactions between PDMS-g-silica Nanoparticles in Solvents. Using UV spectrophotometry, we carried out wavelength scans to quantify the interactions between the PDMS-g-silica nanoparticles with graft polymers of N = 18 kg/ mol in octamethyltrisiloxane and t-butanol, the good and marginally poor solvents, respectively. The particle concentrations varied between c = 0.0023 and 0.0185 g/mL. All samples were sonicated for at least 1 h prior to measurement, and each UV scan was repeated three times to ensure reproducibility. The sample temperature was held constant at 25.5 °C in all experiments. Figure 3a shows the transmittance (Tr) with respect to wavelength (λ) for the PDMS-g-silica nanoparticles in octamethyltrisiloxane. The transmittance decreases with E

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Figure 3. UV−vis spectroscopy of PDMS-g-silica nanoparticles in octamethyltrisiloxane. (A) Transmittance (It/Io) as a function of wavelength (λ) and particle concentration (c) in g/mL. (B) Turbidity (τr) vs particle concentration (c) and wavelength (λ). (C) Specific turbidity (τr/c) vs particle concentration (c) and wavelength (λ). (D) Integrated structure factor Z(λ2, c) as a function of 1/λ2 and particle concentration (c) in g/mL. The structure factor, S(0, c), is obtained from the intercept of the dashed line that is a model fit to the data. Dashed lines are model fits to the data.

one extracts the dilute-solution-specific turbidity (τr/c)c=0 by extrapolation to the y intercept, which on the basis of eq 7 is proportional to the integrated form factor Q(λ2, c). The use of (τr/c)c=0 permits the determination of the integrated structure factor Z(λ2, c) in eq 8. On the basis of eq 9, taking the intercept of the plot of Z(λ2, c) versus 1/λ2 yields the structure factor at vanishing angle S(0, c) in Figure 3d. The second virial coefficient (B2‑OTS) can be determined from the dimensionless osmotic compressibility, or the inverse of S(0, c) in eqs 13 and 14. The value of the coefficient is B2‑OTS = (2.2 ± 0.1) × 1010 mL/mol, which is extracted from the slope of the plot of 1/S(0, c) vs c in Figure 6. Dividing the virial coefficient by M/NA makes it dimensionless and places it on a particle volume fraction basis, yielding B2‑OTS = 136 ± 6, which is more than one order of magnitude greater than the virial coefficient for hard spheres of similar core size B2‑HS = 8, indicating that brush stretching in good solvents increases the particle interaction. Similarly, wavelength scans were carried out on the PDMS-gsilica nanoparticles in t-butanol, a marginally poor solvent for PDMS. The particle concentration in these experiments was varied between c = 0.0046 and 0.0185 g/mL. Figure 4 shows

increasing particle concentration as a result of light attenuation, and the transmittance decreases with decreases in wavelength as a result of destructive interference. These behaviors are captured in the Beer−Lambert law in eq 19, characterizing the exponential dependence of Tr on the sample thickness, lo, and the sample turbidity τr, otherwise known as the absorption or extinction coefficient.39

Tr =

It = exp( −τrlo) Io

(19)

On the basis of eq 7, τr should increase with decreases in λ, leading to the drop in Tr in Figure 3a. Figure 3b shows the turbidity (τr) as a function of concentration (c) and wavelength (λ), where τr is calculated from Tr in eq 1 at different wavelengths. Figure 3b represents an intermediate step in determining the specific turbidity (τr/c) and enables a viewing of the functional dependence among τr, c, and λ, demonstrating how τr should increase linearly with c while being inversely proportional to λ. Figure 3c shows the specific turbidity (τr/c) as a function of particle concentration (c) and wavelength (λ). From Figure 3c, F

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Figure 4. UV−vis spectroscopy of PDMS-g-silica nanoparticles in t-butanol. (A) Transmittance (It/Io) as a function of wavelength (λ) and particle concentration (c) in g/mL. (B) Turbidity (τr) vs particle concentration (c) and wavelength (λ). (C) Specific turbidity (τr/c) vs particle concentration (c) and wavelength (λ). (D) Integrated structure factor Z(λ2, c) as a function of 1/λ2 and particle concentration (c) in g/mL. Structure factor S(0, c) is obtained from the intercept of the dashed line that is a model fit to the data. Dashed lines are model fits to the data.

each sample was allowed to equilibrate for 30 min at T = 25.5 °C prior to measurement. Figure 5a−d shows the results for the transmittance (Tr) versus wavelength (λ), the turbidity (τr) versus concentration (c), the specific turbidity (τr/c) as a function c, and the integrated structure factor Z(λ2, c) versus 1/λ2 to extract the change in slope of the inverse of the dimensionless osmotic compressibility with particle concentration (i.e., 1/S(0, c) versus c) to calculate the second virial coefficient in Figure 6. The second virial coefficient for this system was calculated to be B2‑PDMS‑OTS = (1.9 ± 0.4) × 1010 mL/mol, or 111 in dimensionless particle volume fraction units, indicating that brush stretching in a good solvent increases the number of particle interactions in a concentrated polymer solution, as again B2‑PDMS‑OTS > B2‑HS. 3.4. Discussion of Viral Coefficients of PDMS-g-silica in Solvents and PDMS Solutions. Figure 6 displays the inverse the structure factor, 1/S(0, c), or the dimensionless osmotic compressibility, β(∂Π/∂c)T, as a function of particle concentration, c, for the PDMS-g-silica nanoparticles in octamethyltrisiloxane, t-butanol, and concentrated PDMS/ octamethyltrisiloxane. On the basis of the relationship between 1/S(0, c) and c for the PDMS-g-silica nanoparticles, the use of

the plots of the raw data and their further analysis, which exhibit trends similar to the PDMS-g-silica nanoparticles in octamethyltrisiloxane. Figure 6 shows the results of the calculation of the dimensionless osmotic compressibility for the nanoparticles in t-butanol. The value of the second virial coefficient in the marginal solvent was B2,t‑BuOH = (3.0 ± 0.4) × 109 mL/mol or 17 ± 2 on a particle volume fraction basis, indicating that brush stretching contributes to particle interactions because the PDMS-g-silica nanoparticles behave as effective hard spheres in t-butanol as B2,t‑BuOH = 17, which is greater than the hard sphere value of B2‑HS = 8. That said, B2,t‑BuOH is roughly an order of magnitude less than that in concentrated PDMS/octatrimethylsiloxane solution as discussed in section 3.3 below. 3.3. PDMS-g-silica Interactions in PDMS/Octamethyltrisiloxane. Subsequently, wavelength scans were carried out to quantify the interactions between nanoparticles in concentrated polymer solutions. Hence, a range of samples were prepared with PDMS-g-silica nanoparticles in PDMS/ octamethyltrisiloxane solutions by varying the particle concentration between c = 0.0012 and 0.0065 g/mL while holding the volume fraction of the free polymer constant at ϕp = 0.7. All samples were sonicated for 1 h to ensure homogeneity, and G

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Figure 5. UV−vis spectroscopy of PDMS-g-silica nanoparticles in concentrated PDMS/octamethyltrisiloxane solution (polymer fraction − ϕp = 0.7). (A) Transmittance (It/Io) as a function of wavelength (λ) and particle concentration (c) in g/mL. (B) Turbidity (τr) vs particle concentration (c) and wavelength (λ). (C) Specific turbidity (τr/c) vs particle concentration (c) and wavelength (λ). (D) Integrated structure factor Z(λ2, c) as a function of 1/λ2 and particle concentration (c) in g/mL. Structure factor S(0, c) is obtained from the intercept of the dashed line that is a model fit to the data. Dashed lines are model fits to the data.

0.1) × 1010 and (3.0 ± 0.6) × 109 mL/mol, respectively. The magnitudes of the virial coefficients support the trends in the SCF calculations in Figure 2 where greater repulsive interactions between nanoparticles in good solvent octamethyltrisiloxane occur as a result of greater PDMS brush stretching. Moreover, the experimental and theoretical virial coefficient ratios in the marginally poor and good solvents in Table 2 agree well as shown by the fact that the experimental B2‑t‑BuOH/B2‑OTS = 0.13 ± 0.03 coincides with that from SCF theory at 0.11. For PDMS-g-silica nanoparticles in concentrated PDMS/ octamethyltrisiloxane, the second virial coefficient, B2‑PDMS‑OTS = (1.9 ± 0.5) × 1010 mL/mol, is equivalent to that in octamethyltrisiloxane based on the margin of error of B2‑PDMS‑OTS/B2‑OTS = 0.9 ± 0.2 in Table 2. Hence, the particles in concentrated PDMS/octamethyltrisiloxane behave as if in a good solvent rather than being in an ideal, melt-like environment as suggested by the ratio from SCF theory at 0.34, which corresponds to the theta solution value. The difference between the experimental and theoretical values in the concentrated polymer solution may suggest that the osmotic compressibility has other contributions (e.g., that

eq 14 is justified to extract second virial coefficients B2‑OTS, B2‑t‑BuOH, and B2‑PDMS‑OTS. Hence, the experimental virial coefficients were extracted using eq 14; these values, the corresponding y-axis intercepts, and the associated model fit errors are listed in Table 2. We also computed the virial coefficients for t-butanol and concentrated PDMS/octamethyltrisiloxane to octamethyltrisiloxane, or B2‑t‑BuOH/B2‑OTS and B2‑PDMS‑OTS/B2‑OTS, to facilitate comparisons to SCF theory. In Figure 6, the experiment results (symbols) and model fits (lines) exhibit trends indicative of having dilute solutions of noninteracting nanoparticles and solutions in which interactions between particle pairs (i.e., two-body interactions) govern solution properties. For example, the y-axis intercept in Figure 6, or the lower limit of the dimensionless osmotic compressibility, approaches unity with decreasing particle concentration in correspondence with eq 14, indicative of dilute solutions. Increasing the particle concentration in Figure 6 raises the dimensionless osmotic compressibility whose slope with respect to concentration yields the virial coefficients through the analysis of eq 14. As listed in Table 2, the virial coefficients in octamethyltrisiloxane and t-butanol are (2.2 ± H

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Table 2. Experimental and Theoretical Determination of the Dimensionless Compressibility Intercept, Second Virial Coefficient, and Virial Coefficient Ratio Relative to Good Solvent Octamethyltrisiloxane (OTS) experiment system octamethyl-trisiloxane t-butanol PDMS/octamethyltrisiloxane theta solvent

solvent condition good marginally poor good polymer solution ideal

theory

polymer fraction ϕp

intercept − limc→0∏

B2,exp (mL/mol)

B2,exp

B2,exp/B2,OTS

Flory parameter χ

B2,theory/ B2,OTS

0 0 0.7

1.02 ± 0.02 0.97 ± 0.02 1.00 ± 0.04

2.2 ± 0.1 × 1010 3.0 ± 0.6 × 109 1.9 ± 0.5 × 1010

136 ± 6 17 ± 2 111 ± 23

1 0.13 ± 0.03 0.9 ± 0.2

0.2 0.55 0.2

1 0.11 0.34

0.5

0.34

0

volume fraction was held constant at ϕP = 0.7. The parameters from the experiments were directly used in the SCF calculations; thus, the calculations were performed with no adjustable parameters. Using Rayleigh−Debye scattering theory and statistical mechanics, we extracted the virial coefficients from the turbidity measurements. Their values were (2.2 ± 0.1) × 1010, (3.0 ± 0.6) × 10 9 , and (1.9 ± 0.5) × 10 10 mL/mol in octamethyltrisiloxane, t-butanol, and PDMS/octamethyltrisiloxane, respectively. As shown in Table 2, the ratio of the virial coefficient in the good and marginally poor solvents agrees well with the SCF calculation. In contrast, the experimental ratio between the concentrated polymer solution and good solvent indicates that the particles behave as if they are immersed in a good solvent, which deviates from the theoretical calculation that suggests that the nanoparticles should behave ideally within the melt-like environment of the concentrated polymer solution. This deviation may indicate the need for a greater refinement of the SCF calculations to capture the interactions within concentrated polymer solutions.

Figure 6. Dimensionless osmotic compressibility (Π) of the PDMS-gsilica nanoparticles as a function of particle concentration c(g/mL) within the three suspensions as shown in the legend. The symbols are the experimental points whose relative errors per point are less than 1%. The dashed lines are model fits to the data from the parameters listed in Table 2.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected].

from the free polymer and solvent) that may not be appropriately accounted for in the Scheutjens−Fleer SCF lattice theory. Thus, future studies of polymer-grafted nanoparticles in concentrated polymers solutions are needed to enable further comparisons between experiment and theory to elucidate quantitatively the nature of nanoparticle interactions in these systems.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (NSF) in the division of Chemical, Bioengineering, Environment and Transport Systems (CBET- 0649081 - Small Grant for Exploratory Research and CBET 0644890 - NSF Career Award). We gratefully acknowledge our discussions with Matthias Ballauff.

4. CONCLUSIONS The aim of this study was to use turbidity measurements to quantify the impact of solvent quality on the interactions between polymer-grafted nanoparticles in simple solvents and a concentrated polymer solution. The ratio of the virial coefficients from these studies was compared to those from self-consistent mean-field (SCF) calculations based on the Scheutjens−Fleer lattice model using cylindrical coordinates to mimic particle curvature. To this end, monodisperse silica nanoparticles (D = 100 nm) were synthesized and grafted with PDMS chains of molecular weight N = 18 kg/mol at a graft density of 0.20 chains/nm2. The particle concentration was varied between c = 0.0012 and 0.0185 g/mL. Suspensions were formulated in octamethyltrisiloxane and t-butanol, good and marginally poor solvents, respectively, for PDMS. A Flory parameter of χ = 0.2 was used to model the interaction between PDMS and octamethyltrisiloxane, whereas χ = 0.55 was used to capture the interaction between PDMS and t-butanol. The molecular weight of the free PDMS polymer in the concentrated polymer solution was P = 5.84 kg/mol, and its

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