Distribution in the Grafting Density of End-Functionalized Polymer

Article pubs.acs.org/Macromolecules

Distribution in the Grafting Density of End-Functionalized Polymer Chains Adsorbed onto Nanoparticle Surfaces Folusho T. Oyerokun* and Richard A. Vaia Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433-7132, United States ABSTRACT: We have developed a simple model to quantify the role of curvature on the surface composition of polymer−nanoparticle hybrids formed as a result of the adsorption of end-functionalized homopolymer chains onto spherical nanoparticles in dilute polymer solutions. The analytic model incorporates relevant contributions from changes in the translational and conformational entropies of the polymer chains, and the enthalpy gained during adsorption, to derive an expression for the distribution of the grafting density as a function of nanoparticle curvature and solution properties of the end-adsorbing chains. Our model predicts that the grafting density distribution is Gaussian. The mean grafting density is found to increase with increasing nanoparticle curvature and polymer concentration. However, the degree of polydispersity in the grafting density is also found to increase with curvature, suggesting the near impossibility of synthesizing monodispersed polymertethered nanoparticle hybrids via the grafting-to technique as the radius of the nanoparticles approaches the unperturbed radius of gyration of the polymer chains in the solution.

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INTRODUCTION Polymer-tethered nanoparticle hybrids (PTNH) consisting of high molecular weight polymer chains covalently attached onto inorganic nanoparticle surface are important in a variety of scientific and medical applications.1−3 The attached chains serve as steric stabilizers4,5 of the nanoparticles by preventing their agglomeration due to attractive van der Waals and depletion forces as well as controlling the structure and thermomechanical properties of the assembly. The strength of the repulsive interactions between two polymer-grafted nanoparticles depends on the corona architecture, which in turn depends on the grafting density and size of the polymer chains, i.e., its degree of polymerization. These hybrid nanoparticles have been proposed as a means of creating ordered nanoparticle assemblies with well-controlled structures at high (>50%) inorganic nanoparticle loadings (e.g., suitable for ultrahigh-density capacitors and high refractive index coatings) without the necessity of a polymeric matrix, in situ templating of the inorganic, or blending with agglomeration-prone nanoparticles.6 They have also been touted in other engineering applications such as in the design of efficient water purification membranes,7 conductive lubricants for RF MEMS switches,8 organic photovoltaics with improved efficiency,9 sensors and diagnostic applications,10,11 and polymer-based drug delivery systems.12,13 One common protocol employed in the synthesis of PTNHs involves the preferential adsorption of end-functionalized polymeric ligands from solution onto highly curved nanoparticle surfaces, the so-called grafting-to reaction.14,15 In addition to solvent selectivity, the resulting composition of the bound polymer corona depends on the binding strength of the functional groups and the size of the nanoparticles. While the grafting-to approach is known to afford good control on the © 2012 American Chemical Society

size of the attached chains, it usually suffers from low grafting density. Furthermore, the adsorption process often leads to a distribution in the grafting density of the polymer hybrid nanoparticles. Several pioneering theoretical16−20 and experimental studies21−23 have examined the role of nanoparticle size on the adsorption of end-functionalized homopolymers and diblock copolymers onto curved nanoparticles in very dilute solution. They found that the average surface coverage increases with polymer concentration and particle curvature (due to improved packing of the chain molecules on highly curved surfaces). While these previous studies have provided crucial insight into the variation of the average grafting density with the properties of the polymer and nanoparticles, no studies, to our knowledge, have adequately treated the role of the surface curvature on the resulting distribution in the grafting density. Understanding the relationship between the breadth of this distribution is crucial, since it is well-known that the structure formed during selfassembly and the resultant dynamical properties are intimately linked to the structure of the corona.24−29 Insight gained from such study will prove beneficial in determining the limitations and level of control possible in large-scale synthesis of these materials via the grafting-to approach. The focus of this study is to examine the role of curvature on the equilibrium adsorption of end-functionalized homopolymer chains onto spherical nanoparticles. Specifically, we are interested in how the radius of curvature of the nanoparticles and the size of the polymer chains affect the equilibrium distribution of the grafting density for an ensemble of Received: June 15, 2012 Revised: August 14, 2012 Published: September 4, 2012 7649

dx.doi.org/10.1021/ma301218d | Macromolecules 2012, 45, 7649−7659

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Exploiting the close analogy with micellization, we write the expression for the fraction of nanoparticles with n polymer chains attached to their curved interfaces as

nanoparticles. In the next section, we shall derive a model that addresses the role of curvature on the statistical properties of the grafted layer when end-functionalized chains are adsorbed from solution onto surfaces of spherical particles. The Results section presents model calculations for variation of the distribution of the grafting density (and its first and second moments) with nanoparticle radius. Also included in our analysis are the effects of the nanoparticle and polymer concentrations on the average surface coverage. The section concludes with a comparison of our model predictions with limited experiments on the adsorption of polyisobutelyne succinimide onto calcium carbonate nanoparticles. We conclude by summarizing our results and comparing our model predictions with other theoretical and experimental studies in the literature.

⎛ Δf ⎞ xn = x0xpn exp⎜ −n ⎟ ⎝ kBT ⎠

where x0 and xp are the number fractions of “bare” nanoparticles (i.e., nanoparticles with no polymer chains attached) and singly dispersed polymer chains, respectively. The parameter Δf in the exponent of eq 3 is the standard free energy change during adsorption of a single polymer chain from solution onto the nanoparticle surface. Although derived heuristically, eq 3 does have a physical meaning: It simply states that the probability of forming an n-chain polymer−nanoparticle complex is proportional to the probability of finding a nanoparticle surrounded by n polymer chains (given by the product of x0 and xnp) weighted by a Boltzmann factor which accounts for the free energy difference when n chains adsorb from solution onto the nanoparticle surface.22 Following Ligoure and Leibler, we write the change in the free energy per chain of the grafted layer on the highly curved nanoparticle surface as a sum of three contributions:33

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THEORY Consider the adsorption of np end-functionalized homopolymer chains onto the surfaces of nNP spherical nanoparticles of radius R in a dilute solution. Each polymer chain is characterized by its molecular weight or the number of statistical (Kühn) segments, N, each of length, a. Hence, the total number of Kühn monomer units nm = npN. For the sake of simplicity, we shall consider only the case where both the polymer chains and the nanoparticles are monodisperse. (Our model can be generalized to account for polydispersity in both species at the expense of additional complexity in the formulation.) In addition to the polymer chains and the nanoparticles, the solution is assumed to be comprising ns solvent molecules, each of volume vs. If we further assume that the volume of each solvent molecule is identical to that of a Kühn segment, i.e., vs = vm ∼ a3, then the total solution volume V ≈ a3(ns + npN + nNPρ3). The dimensionless quantity ρ = R/a is the ratio of the nanoparticle radius to the statistical segment length. It has long been recognized that there is a close similarity between equilibrium adsorption of end-functionalized polymer chains onto nanoparticle surfaces and the well-studied thermodynamics of micelle formation by polymeric surfactants.22 In micellization, aggregate formation occurs in order to reduce the unfavorable contacts between the hydrophobic portion of the chains with the solvent.30−32 However, for nanoparticles, chain adsorption results in order to decrease the unfavorable particle−solvent surface energy and to establish the favorable particle−chain interactions. At equilibrium, the size of the micellar aggregates is determined by equating the chemical potential of the micelles to those of the singly dispersed chains in solution: 1 μs0 + kBT log xs = (μn0 + kT log xn) (1) n Here, μ0n (μ0s ) is the standard chemical potential of a surfactant aggregate consisting of n chains (single chain), and xn (xs) is the mole fraction of these aggregates (single chains) in the solution. The distribution of the aggregate size follows directly from the rearrangement of eq 1: ⎛ Δμ0 ⎞ xn = xsn exp⎜⎜ −n n ⎟⎟ kT ⎠ ⎝

Δμ0n

μ0n/n

(3)

Δf = −ε + fconf + log σ

(4)

In eq 4, the first term accounts for the enthalpy gained (per unit kBT) as a result of the attachment of the terminal group onto the curved interface. The second term is the conformational free energy (per chain) of the end-attached homopolymer chains in the grafted layer. The last term is the contribution from the translational entropy of the physiosorbed chains on the curved interface. At equilibrium, and for sufficient enough grafting density, the adsorbed chains will form a brush on each nanoparticle interface.34 The conformational properties of the attached chains in the spherical brush depend on the size and geometry of the nanoparticle. Analysis based on a simple blob-like picture (the Daoud−Cotton model) has shown that, for sufficiently high tethering density, the grafted chains extend radially from the curved nanoparticle interface in a manner quite similar to those of star-shaped polymers.16,35,36 As a result, each radial branch of the grafted chains can be treated as a succession of nonoverlapping spherical “blobs” whose diameter varies with the distance from the grafting surface, i.e., ξ(r) ≈ aσ−1/2(r/R).35 Here, the dimensionless grafting density σ = na2/(4πR2) quantifies the surface coverage of the end-functionalized chains on the nanoparticles. Within this description, the thickness of the grafted is given as h = (1 + γn1/3)3/5 − 1 R −1/3

(5)

where γ = 5/3(4π) (RF/R) . The parameter RF = N w a is the Flory radius and is a measure of the unperturbed meansquared end-to-end radius of a single polymer chain in a good solvent. The conformational free energy per chain of the Daoud−Cotton spherical brush is 5/3

fconf = βn1/2 log(1 + γn1/3)

(2)

3/5 1/5

(6) −1/2

μ0s

In eq 2, ≡ − is the difference in the standard chemical potential between a surfactant chain present in the nchain aggregate and that of a singly dispersed chain in the solution.

where in the above equation the constant β = 3/5(4π) . (Without loss of generality, we have ignored all the unknown scaling prefactors, typically constants of order unity, from eqs 5 and 6.) 7650


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Figure 1. Distribution of the number of end-functionalized polymer chains adsorbed per nanoparticle at three particle sizes: (a) R/RG = 0.5, (b) 1, and (c) 5. xn was determined from eqs 7 and 8 using the following parameters of the polymer chains and nanoparticles: N = 1250, w = 0.0182, ε = 10, ϕNP = 10−3, and ϕp = 0.1. (d) illustrates the prediction of the Carvalho model at the same nanoparticle and polymer concentrations and the binding strength of the end-adsorbing group. Note that the prediction of the Carvalho model is independent of the nanoparticle radius.

accessible volume fractions of the two species: ϕp ≈ ΨpN and ϕNP ≈ ΨNPρ3. Once the initial concentrations Ψp and ΨNP are specified, eqs 7 and 8 are solved numerically to obtain xn as a function of the geometrical and physical parameters of the endadsorbing chains and nanoparticles in the solution. It is worthwhile to comment on the similarity between our model and the one derived by Carvalho and co-workers to study the adsorption of end-functionalized polymer chains onto colloidal nanospheres. Both models utilized the aforementioned analogy between polymer adsorption onto curved interfaces and surfactant aggregation into spherical micelles to derive identical expressions for the distribution of polymer−nanoparticle complex (eq 3). However, the two models differ in their treatment of the conformational properties of the adsorption layer. First of all, Carvalho and co-workers ignored the mobility of the adsorbed polymer chains (i.e., contribution from the translational entropy) on the nanoparticle surface, i.e., Δf Carvalho = −ε + TΔSconf. Contributions from chain mobility are particularly relevant for brushes formed on spherical nanoparticles that is of relatively weak or intermediate strength, such as physioadsorption due to differences in hydrophobicity, electrostatic, or weak specific interactions (H-bonding). Second, the authors assumed that, in the limit of low surface coverage on the NP surface, the change in the conformational entropy of the attached chains is independent of both the nanoparticle curvature and the number of adsorbed polymer chains. Using scaling analysis, the authors determined ΔSconf from the number of self-avoiding walks (SAW) for a single polymer chain attached to a planar interface: ΔSconf ≈ 0.472kB log N. The predictions and validity of the model of Carvalho and co-workers will be discussed later. Suffice now to say that the assumption that the conformational entropy of the grafted chains is independent of the nanoparticle size and grafting density likely limits its applicability to a class of real system where significant overlap occurs between the adsorbed chains

Substitution of eq 6 into eq 3 leads to an expression for the number fraction of nanoparticles with n end-functionalized polymer chains adsorbed on their interface: ⎧ ⎡ ⎛ a 2n ⎞ ⎢ xn = x0xpn exp⎨ n ε − log − βn1/2 ⎜ 2⎟ ⎪ ⎠ ⎝ 4 π R ⎣ ⎩ ⎪

⎤⎫ ⎬ log(1 + γn1/3)⎥⎪ ⎦⎭ ⎪

(7)

Since the total number of nanoparticles and polymer molecules are conserved, we expect nmax

x NP +

∑ xn

=ΨNP

n=1 nmax

xp +

∑ nxn

=Ψp (8)

n=1

Here, nmax is defined as the maximum number of spheres of radius a (the footprint of the end-adsorbing group is taken to be approximately a2) that can be packed on a given nanoparticle of radius, R. This quantity has been determined by Miracle and co-workers:37 ⎛R⎞ nmax ⎜ ⎟ = ⎝a⎠

4π ⎡ 10 arccos⎢sin ⎢⎣

( π5 )

1−

1 2

( Ra + 1)

⎤ ⎥ − 3π ⎥⎦

(9)

The quantities Ψp and ΨNP on the right-hand side of eq 8 are the total number fractions of the polymer chains and nanoparticles, respectively. By definition, the number fraction of a solute i (where i can be a polymer chain or nanoparticle) in the solution is Ψi = ni/∑ini ≈ ni/ns. In the dilute solution limit, the number fractions are related to the more experimentally 7651


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decrease exponentially with the number of adsorbed polymer chains per NP from its peak value at n = 1. Because the model of Carvalho and co-workers assumes that the conformational entropy of the attached chains is independent of nanoparticle curvature, it gives identical result for the grafting density distribution at all radii for fixed concentrations and binding strength. Once the distribution of the number of adsorbed chains per nanoparticle is known, it is straightforward to determine the mean or average number of chains per nanoparticle:

on the curved interface, i.e., at grafting densities beyond the mushroom−brush transition.

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RESULTS Distribution of End-Adsorbed Polymer Chains on the Nanoparticle Surface. We begin our parametric studies by investigating how curvature affects the distribution of the number of end-functionalized polymer chains adsorbed per nanoparticle in a dilute solution containing both species. By dilute solution, we mean that the total volume fraction of the polymer chains is much lower than the overlap volume fraction in a good solvent, i.e., ϕp < ϕ*p ∼ N−4/5w−3/5, and that the average distance between two nanoparticles is so large that particle−particle interactions can be conveniently ignored. For purposes of illustration, we shall first consider the adsorption of polystyrene of fixed molecular weight, N = 1250 (arbitrarily chosen so that the unperturbed radius of gyration of a single PS chain in the solution, RG = N3/5w1/5a/√6 ≈ 10 nm), whose end-functionalized groups undergo attractive interactions with the spherical nanoparticle surface, e.g., via zwitterionic moieties placed at a terminal group of the chain.5,38 The appropriate parameters for PS in toluene are33,39 w = 0.0182 and a = 0.76 nm. Consequently, the overlap concentration ϕ*p ≈ 4 × 10−2. Figure 1 shows the distribution of the number of polymer chains per particle at three different nanoparticle radii: R/RG = 0.5, 1, and 5. The polymer and nanoparticle volume fractions are ϕNP = 10−3 and ϕp = 0.1, respectively. We have selected for the enthalpy gained per end-adsorbing group (in unit of kBT), ε = 10, which is approximately the binding strength of zwitterionic groups on mica.4,33,39 For the three nanoparticle sizes considered, the number of chains per nanoparticle is observed to be uniformly distributed around a maximum but decays rapidly to zero both at very low (n → 1) and extreme values (n ≫ n̅). The mean number of chains per nanoparticle coincides with the peak of the distribution. This observation is better illustrated in Figure 2, which shows a logarithmic plot of

nmax

n̅ =

nmax

∑ nxn/ ∑ xn n=1

n=1

(10)

Figure 3a shows the variation of n̅ with nanoparticle radius. Not surprisingly, the average number of chains increases with nanoparticle size. However, the mean number of chains is much lower than the maximum packing (dotted line in Figure 3a) as given by eq 9. The most numerically expensive aspects of the calculation involves the series summation in eq 8. Because of the nonlinear nature of the functional dependence of the free energy per chain ( fconf) on n, no simplifying integral approximation of the series is possible. It is however possible to define an optimal number of chains per NP by minimizing the distribution function, xn, as given by eq 7. After algebraic manipulation, we obtained the following transcendental expression for the optimal number of chains per nanoparticle at equilibrium: ⎡ ⎛ a 2n ⎞ opt ⎟ − βnopt1/2⎢log(1 + γnopt1/3) ε + log xp − log⎜⎜ 2 ⎟ ⎢⎣ ⎝ 4πR ⎠ +

⎤ ⎥−1=0 3(1 + γnopt1/3) ⎥⎦ γnopt1/3

(11)

Figure 3a also shows the variation of nopt with nanoparticle radius. We see that over the range of interest in Figure 3a nopt ≈ n.̅ Hence, for all purposes nopt can be considered as a good approximation to the average number of chains per nanoparticle. Following Ligoure and Leibler,33 we define a dimensionless grafting density, σ̅ = na̅ 2/(4πR2). Figure 3b illustrates the variation in σ̅ with nanoparticle radius at the same conditions as Figure 3a. Although the average number of chains per NP decreases with curvature, the mean grafting density, which quantifies the amount of chains per unit area, increases with decreasing particle size. The increase in grafting density with decreasing particle size is a result of the reduction in the entropic penalty of packing chains on curved interfaces.16 It is worthwhile to mention that, except for the last two terms, eq 11 is identical to that obtained by Ligoure and Leibler for the adsorption of end-functionalized polymer chains on a single spherical defect of radius R located on a flat surface.16 A major difference between the two models is that eq 11 explictly accounts for nanoparticle concentration (the Ligoure−Leibler model assumes infinite dilution) and is therefore valid at finite nanoparticle concentrations of interest in most experiments. Variation of the Degree of Polydispersity in the Grafting Density with Nanoparticle Size. Inspired by the observation that the fraction of nanoparticles with n chains attached is well described by a Gaussian probability distribution centered around the mean or optimal n, we expanded the distribution function, eq 7, around nopt. Ignoring third- and

Figure 2. Log plot of the normalized polymer distribution at four nanoparticle radii as a function of the deviation from average: R/RG = 0.5 (diamond), 1 (square), 2 (triangle), and 5 (circle). Lines are Gaussian fits to the nonzero interval of the distribution.

xn/ΨNP against the variation from the mean, n − n,̅ at four different nanoparticle radii (R/RG = 0.5, 1, 2, and 5). Figure 2 shows that large portions of the numerically generated data are well described by a Gaussian distribution centered around the mean n.̅ The breadth of the distribution, i.e., its standard deviation, increases with the size of the nanoparticle. Figure 1d illustrates the prediction of the Carvalho et al. model at the same polymer and nanoparticle concentrations and binding strength of the end-adsorbing group for nanoparticle radius R = RG. The distribution function is found to 7652


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Figure 3. Variation of (a) the average number of chains adsorbed and (b) the average grafting density, σ̅, with nanoparticle radius at fixed polymer and nanoparticle concentration, ϕNP = 10−3 and ϕp = 0.1. In (a), the mean number of chains (open circles) and the optimal number of chains (solid line) per nanoparticle are determined from eqs 10 and 11, respectively. The maximum number of chains per NP (dotted line) is estimated from the geometrical packing limit of Miracle and coworkers, eq 9.

Figure 4. Variation of (a) the standard deviation and (b) the polydispersity index (PDI) with particle radius of curvature at ϕNP = 0.001, N = 1250, ε = 10, and w = 0.0182.

Figure 5. Variation of (a) the fraction of unadsorbed chains (normalized to the total number fraction of end-functionalized polymer chains present) and (b) dimensionless grafting density with polymer concentration. Calculation was done using the full expression eq 7 at RNP = RG = 10 nm for three values of the binding strength of the end-adsorbing group: ε = 6 (solid), 8 (dashed), and 10 (dotted). The other input parameters are ϕNP = 0.001, N = 1250, and w = 0.0182 (which corresponds to the excluded volume parameter of polystyrene in toluene).

0.0182. The symbols were obtained from the Gaussian fit to the distribution of chains per nanoparticle while the line was determined from eq 12. In agreement with Figures 1 and 2, the standard deviation increases with nanoparticle radius. An experimentally relevant measure of the spread of the distribution is the polydispersity index, related to the mean and standard deviation via40 PDI = 1 + (Sn/n)̅ 2. Hence, the lower the index, the narrower the distribution of the grafting density of the polymer−nanoparticle hybrids. We see from Figure 4b that the polydispersity index increases as the nanoparticle size becomes smaller with the largest variation occurring when R < RG. This has important ramification for experiments: It suggests difficulty in obtaining monodisperse polymer grafted nanoparticles as their size becomes smaller. Effect of Polymer Concentration and the Binding Strength of the End-Adsorbing Groups on Adsorption

higher-order terms in the expression allows us to rewrite xn/ ΨNP as exp[−n(fconf + log σ − ε − log xp)] xn = ∞ ∑n = 0 exp[−n(fconf + log σ − ε − log xp)] ΨNP ⎡ (n − n )2 g ″(n ) ⎤ opt opt ⎥ ≈ A exp⎢ − ⎢⎣ ⎥⎦ 2 (12)

where the function g(n) = −n( f + log σ − ε − log xp) and g″(n) its second derivative with respect to n. The prefactor A = exp[−g(nopt)]/∑∞ n=0 exp[−g(n)]. The standard deviation Sn = 1/g″(nopt)1/2 is a measure of the breadth of the distribution. Figure 4a shows the variation of the standard deviation with nanoparticle size at ϕNP = 0.001, N = 1250, ε = 10, and w = 7653


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Properties. We now consider the effect of the polymer concentration and the strength of the end-adsorbing group on the adsorption of end-functionalized polymer chains onto spherical nanoparticles at fixed nanoparticle concentration. To simplify the discussion in this section, we shall restrict our analysis by fixing the polymer size, N = 1250, and focusing on a single nanoparticle radius selected so that R = RG = 10 nm. (As discussed in the previous section, the parameters of the polymer chains are selected so as to mimic that of polystyrene in toluene: w = 0.0182 and a = 0.76 nm.) Figure 5a shows the variation of the number fraction of the unadsorbed or singly dispersed polymer chains, normalized to the total number fraction of end-functionalized polymer chains present, with the polymer volume fraction at three binding strengths of the end-adsorbing group (in units of kBT): ε = 6 (solid), 8 (dashed), and 10 (dotted). We find that the fraction of unadsorbed chains varies nonlinearly with polymer concentration. It is characterized by a rapid or steep rise at low polymer concentration ϕp < 0.01 followed by a decrease in slope (plateauing) as the fraction of the unadsorbed chain approaches its peak value of unity. Hence, we see that at all concentrations, even though the grafted layer has not reached the maximum possible grafting density, equilibrium dictates that there will always be some free chains in the solution. Clearly, Figure 5a is indicative of the fact that the ratio between the amount of polymer chains in solution and those attached to the spherical interfaces depends strongly on the binding strength of the end-adsorbing groups. Once the fraction of unadsorbed chain xp has been determined, it is straightforward to compute the average number of chains attached to each nanoparticle, n̅ (using eq 10), and subsequently the average grafting density, σ̅. (Alternatively, we can compute the grafting density directly from nopt determined from the solution of eq 11.) Figure 5b illustrates the variation in σ̅ as a function of polymer volume fraction, ϕp, at the same conditions as Figure 5a. We see from the plot that the grafting density, σ̅, increases with polymer volume fraction, ϕp. This nonlinear dependence between σ̅ and ϕp (which, in principle, is not describable by a simple power law or scaling expression) also exhibits a sharp rise at low polymer volume fraction before leveling off at higher concentration in close similarity to Figure 5a. Another key observation from Figure 5b pertains to the increase in the number of adsorbed chain with increasing strength of the adsorbing group. For instance, increasing the strength of the adsorbing group by a factor of 2 at ϕp = 0.1 yields nearly a 5fold increase in the grafting density. The trends observed in Figure 5b are in agreement with the predictions of the theoretical model of Ligoure and Leibler.16 Concomitant with the increase in amount of chains adsorbed is a corresponding increase in the thickness of the grafted layer. This is clearly illustrated in Figure 6, which shows the variation in the average layer thickness, normalized to the unperturbed radius of gyration of the polymer chain, with polymer volume fraction at the binding strength, ε = 10. The functional dependence of the brush height with polymer concentration mirrors closely that observed in the variation in the grafting density with ϕp. Experimentally, the brush height can be monitored using techniques such as dynamic light scattering which provides a measure of the size of the bare particles and that of the spherical nanoparticle brush. We note that even at sufficiently low initial polymer concentrations, the grafted chains are extended in a brush-like conformation within the

Figure 6. Variation in the thickness (solid line) and the conformation free energy (dotted line) of the adsorbed layer with polymer volume fraction, ϕp, at fixed nanoparticle radius, R = RG = 10 nm, and concentration, ϕNP = 10−3. The degree of polymerization of the polymer chain, N = 1250, and the dimensionless excluded volume parameter, w = 0.0182.

nanoparticle corona as evidenced by h/Rg > 1. This observation lends credence to the suitability of the Daoud−Cotton model for the description of the conformational properties of the grafted layer, particularly when R ∼ RG. To understand why the slope of σ vs ϕp (and xp vs ϕp) decreases at high polymer volume fraction, it is instructive to consider the variation in the conformational free energy (fconf) of the adsorbed layer as a function of polymer concentration. Figure 6 depicts the variation of the conformational energy of the grafted chains, determined using eq 6, as a function of ϕp. Not surprisingly, fconf increases with the polymer concentration due to increase in number of adsorbed chains. Beyond a threshold polymer volume fraction ϕp ∼ 0.01, the conformational free energy becomes comparable to the thermal energy. Interestingly, this threshold volume fraction corresponds to the onset of the leveling off in the grafting density. To understand why this is the case, it is helpful to consider the kinetics of adsorption onto the curved interface. At zero or low grafting density, the construction of the grafted layer is essentially controlled by the diffusion of the chains to the curved interface in close similarity with adsorption onto planar interfaces.33 However, at sufficiently high grafting density, the grafted chains will begin to overlap, and as a result their presence on the curved nanoparticle surface will serve as a barrier to the incoming chains. (The barrier height is comparable to the conformational energy of the grafted layer.) Hence, as the grafting density increases, it becomes more and more difficult for the incoming chains to penetrate this energetic barrier. We expect the limiting grafting density to correspond to fconf ∼ ε. The role of kinetics will be further discussed in the next section. Effect of Nanoparticle Concentration on Adsorption Properties. We now consider the role of nanoparticle concentration on the adsorption of end-functionalized polymer chains onto highly curved interfaces. This treatment is noticeably missing in most theoretical studies on adsorption of polymer chains onto nanoparticles where the emphasis is often on very dilute particle concentration. Even when considered, it is not uncommon to find (particularly in experimental studies) the assumption that the final surface composition of the polymer chains on the nanoparticle is proportional to the ratio of their initial number fractions, i.e. n̅ ≈ 7654

Ψp ΨNP

=

ρ3 ⎛ ϕp ⎞ ⎜⎜ ⎟⎟ N ⎝ ϕNP ⎠

(13)


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10−4 (solid line in Figure 8), the effect of the nanoparticle curvature is more pronounced. However, as the nanoparticle concentration is increased, the effect of the increased nanoparticle volume fraction at higher curvature becomes more important. Depending on the initial polymer concentration, it is possible to arrive at a scenario where the average number of chains per NP actually decreases with curvature (dashed and dotted lines in Figure 8). This prediction further underscores the important role that nanoparticle concentration plays in the determination of the grafting density. Comparison with Experiments. We now compare the predictions of our model with experiments. Papke, Bartley, and Migdal considered the adsorption of poly(isobutenyl) bis(succinimide) dispersants onto calcium alkarylsulfonate colloidal dispersions in hexane.23 The succinimide dispersants were prepared by reacting polyisobutylene (ca. 2060 numberaveraged molecular weight) with maleic anhydride and a polyethylene polyamine to yield ABA copolymer chains (A = polyisobutylene and B = polyamine). The adsorption of dispersants on the calcium carbonate (CaCO3) nanoparticles (R ≈ 12 nm) occurs as a result of the attractive interaction between the polar polyamine midblocks and sulfonatestabilized colloidal particles. Four different types of ethylenepolyamines were considered in the study: pentaethylenehexamine (PEHA), triethylenetetramine (TETA), diethylenetriamine (DETA), and ethylenediamine (EDA). The binding strength of the adsorbing group is proportional to the number of amine groups in the polyamine midblocks. (Note that as a result of the linear ABA architecture two A blocks are adsorbed simultaneously.) The physical interactions between the dispersants and the colloidal nanoparticles were determined using ultracentrifugation/infrared-based techniques. Figure 9 illustrates the variation of the average grafting density with initial concentration of the poly(isobutenyl) bis(succinimide) dispersants in the solution. (The volume fractions of the nanoparticles, based on a 1/6 w/w concentration, is ∼5.6 × 10−3.) Symbols in Figure 9 are the experimental data, while lines are the fit to the data based on eq 11. In addition to the binding strength ε, the other unknown parameter in the model is the scaling prefactor β. The parameter γ ≈ 0.3 is determined from the knowledge of the unperturbed end-to-end radius of the polyisobutylene endchains in hexane and the radius of the CaCO3 nanoparticles. (From the given number-averaged molecular weight of the polyisobutylene segment, we estimate N ≈ 37. Using the appropriate value for the Flory interaction parameter of polyisobutylene in cyclohexane,41,42 w = 1 − 2χ = 0.12, yields an unperturbed radius of gyration of PIB in hexane, RG ≈ 3 nm.) Clearly, our model captures qualitatively the trends obtained in the experiments. Our fit to the data on PEHAfunctionalized dispersants (see Figure 9a) yielded β = 1.29 and ε = 12.2. The numerical prefactor β in eq 11 is the same for all the four cases considered in Figure 9 since it is independent of molecular details of the adsorbing groups. Using the extracted value of β = 1.29, we determined the binding strengths for the three remaining polyamine functional groups by performing a single parameter fit to the data in Figures 9b−d using eq 11: ε = 12.8 (TETA), 11.6 (DETA), and 6.62 (EDA). Consequently, our model predicts a contribution of roughly 2kBT per amine group present. Interestingly, our estimated values for the binding strength of the polar groups, ε ∼ 6−13 (in units of kBT), are in good agreement with the theoretical calculations of Milner39 and Ligoure and Leibler33 for the experiments of

Figure 7 shows the variation of the average number of adsorbed chains, n,̅ normalized to the ratio between the initial

Figure 7. Variation of the mean number of adsorbed chains, normalized to the ratio between the initial nanoparticle and polymer number fractions, with nanoparticle concentration at three nanoparticle sizes: R/RG = 1 (solid), 1.5 (dashed), and 2 (dotted).

nanoparticle and polymer number fractions at three nanoparticle radii: R/RG = 1, 2, and 5. If the empirical assertion (i.e., eq 13) is true, this ratio should equal one at all nanoparticle concentration. Figure 7 shows this is not the case: significant deviation from unity occurs at very low nanoparticle concentration. This result has important ramifications for experiments and synthesis. It highlights the importance of judicious selection of nanoparticle concentration on the final equilibrium coverage. The choice of the initial nanoparticle concentration can also affect the grafting density, specifically in regions where the nanoparticle number fraction exceeds the polymer volume fraction. This is best illustrated by considering the variation of the grafting density as a function of nanoparticle radius at three different nanoparticle volume fractions (Figure 8). For the NP

Figure 8. Variation of the grafting density as a function of nanoparticle radius at three different nanoparticle volume fractions: ϕNP = 10−4 (solid), 10−3 (dashed), and 10−2 (dotted).

volume fractions considered, Figure 8 indicates that the nature of the functional dependence is remarkably different at R < RG. This unusual observation can be rationalized as follows. Increasing the nanoparticle curvature at fixed nanoparticle volume fraction impacts our results in two ways. First of all, as already alluded to, the grafting density increases with nanoparticle curvature because the favorable reduction in the conformation entropy with curvature allows more chains to pack on the NP surface. Second, the nanoparticle number fraction increases with decreasing particle size as ΨNP ∼ ϕNP/ ρ3; hence, more nanoparticles are actually present at a given volume fraction when the radius is decreased. The balance of these two competing tendencies dictates the final surface coverage. For dilute nanoparticle concentration, e.g. ϕNP ≤ 7655


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Figure 9. Variation of the average grafting density with initial concentration of the poly(isobutenyl) bis(succinimide) dispersants in the solution for four different polar ethylenepolyamines midblocks: (a) pentaethylenehexamine (PEHA), (b) triethylenetetramine (TETA), (c) diethylenetriamine (DETA), and (d) ethylenediamine (EDA). The volume fractions of the nanoparticles ϕNP ≈ 5.6 × 10−3. Symbols are the experimental data, while lines are the fit to the data based on eq 11.

Taunton and co-workers5 on the adsorption of polystyrene functionalized with zwitterionic groups on mica.

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ρ* =

DISCUSSION

(5/3)Nσ 1/3w1/3 ≈ N (σw)1/3 ⎡ (5 / 3)N (σw)1/3 ⎤ log⎢⎣1 + ⎥⎦ ρ*

(14)

Hence, ρ* ∼ N and consequently (R/RG)* ∼ N2/5. Substituting N = 1250, we get (R/RG)* ≈ 17, which is in good agreement with Figure 3b. In general, (R/RG)* ∼ 10−100 for most polymers of practical interest. Several other theoretical studies have corroborated this finding.17−19 For instance, Wijmans, Leermakers, and Fleers studied the adsorption of diblock copolymers from a selective solvent (theta for A monomers but athermal good for the B monomers) on small spherical particles.17 They found that the fraction of copolymer adsorbed increases with curvature and the size of the A block (i.e., with the number of A monomers in contact with surface). Their model also shows that the lower the nanoparticle radius, the shorter the A block necessary to achieve the maximum coverage (measured as amount of polymer chains adsorbed per unit area), suggesting that a lower degree of stickiness is required at higher curvature to achieve the same grafting density. Further confirmation for the increase in grafting density with decreasing nanoparticle size comes from the experimental study of Singh and co-workers on the adsorption of iminiumterminated polystyrene chains onto model rough surfaces.21 Texturing on the surface is modeled via colloidal (silica) beads spin-coated on the surface of aluminum-coated silicon substrate. The degree of roughness is prescribed by varying the diameter of the silica beads. The authors compared the adsorption of PS of three different MW on the modeled rough surfaces with that of a flat surface. They found enhancement in the surface coverage (compared to planar interfaces) when the radius of the beads is comparable with the radius of gyration of the polymer. However, for colloidal bead radius much lower than RG they found reduction in the number of adsorbed chains. The authors attributed this suppression in surface coverage to steric hindrances. For very large particle, i.e., at

Variation of the Average Grafting Density with Nanoparticle Curvature. The primary objective of this study was to determine theoretically how the structure of the corona of polymer−hybrid nanoparticles formed as a result of adsorption of end-functionalized polymer chains depends upon the physicochemical properties of the polymers and nanoparticles in the solution. Specifically, we wanted to investigate how the distribution of the grafting density depends on the nanoparticle size and the molecular weight of the endadsorbing polymer chains. The prediction of our model for the variation of average grafting density with particle size is in good qualitative agreement with the model of Ligoure and Leibler16 developed to study the adsorption of end-functionalized polymer chains onto isolated spherical bumps on planar surfaces. Both models show that, for fixed polymer and nanoparticle concentrations and the strength of the binding group, the mean grafting density increases with nanoparticle curvature. Ligoure and Leibler attributed the increase in surface coverage with curvature to improved packing afforded the chain molecules on the curved surface; less conformational entropy is lost at the same graf ting density compared to a flat interface when the chains are attached to a spherical nanoparticle. In contrast to a flat interface, the volume available to each macromolecule increases as r2 as it travels a distance r from the center of the spherical particle.43 The crossover between highly curved to planar behavior occurs at particle radius when the conformational free energy of the spherical NP is comparable to that of the flat brush. The crossover radius can be estimated by equating the two free energies to yield16 7656


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roughly R ∼ 100RG, the obtained coverage of the endfunctionalized chains on the colloidal particles approaches that seen on a planar surface. Variation of the Distribution of the Grafting Density with Curvature. As alluded to in the Introduction, one of the main contributions of our studies is the elucidation of the role of curvature on the distribution of the grafting density. Prior to this study, very few attempts have been made to address this experimentally important issue. Our model predicts that the number of chains per nanoparticle is uniformly distributed around its average. The distribution is well described by a Gaussian centered at the mean value. Although the width of the distribution increases with particle size, the degree of polydispersity in the grafting density decreases with increasing nanoparticle radius. Interestingly, our prediction of Gaussian distribution has also been obtained in the study of micellization of polymeric surfactants. Goldstein shows that the equilibrium distribution of polymer chains in micellar aggregates is bimodal.44 The first maximum is due to the presence of unassociated or singly dispersed polymeric chains, while the second corresponds roughly to the average aggregation number. The distribution of aggregates around the second maximum was found to be Gaussian. To our knowledge only one other study has addressed this problem in the context of adsorption of end-functionalized polymer chains on to curved nanoparticle surfaces. Carvalho and co-workers developed a model to explain the results of their static light scattering experiment on the adsorption of endfunctionalized poly(ethylene−propylene) (i.e., hydrogenated polyisoprene) onto calcium carbonate nanoparticles in decane.22 The two types of end-adsorbing groups in the study are a tertiary amino group and a strongly polar sulfonate−amine zwitterionic moiety. The authors considered the adsorption of the end-adsorbing polymers in the limit when the adsorption energy of the end-functional group is not much larger than kBT. Their model ignored the effect of curvature on the conformational entropy of attached chains. Rather, the conformational entropy change of the grafted spherical layer was determined from the number of self-avoiding walks (SAW) for a single polymer chain attached to a planar interface. Within this formulation, the model predicts very low mean grafting density of polymer chains (usually below the mushroom-tobrush transition). It also predicts an exponential decay in the fraction of nanoparticles with n polymer chains attached to them (see Figure 1d), a result that is contrary to the Gaussianlike distribution obtained using our model. Furthermore, the model of Carvalho and co-workers also predicts that the fraction of adsorbed chains is independent of the absolute number fraction of polymers Ψp but depends rather on the number fraction of the nanoparticles, ΨNP, and the binding strength, ε. Because of its inadequate treatment of the conformation properties of the grafted chains, the model gives unphysical result for moderate to high binding strength. The limitations of the Carvalho and co-workers’ model was recognized early on. Its prediction of the independence of the grafting density with the polymer concentration is in direct contradiction with other experiments. For example, Papke and co-workers23 found in their study on the adsorption of poly(isobutenyl)succinimide dispersants on CaCO3 that the grafting density increases with dispersant concentration. The experimentally determined mean number of chains per NP is roughly 45, an order of magnitude higher than that predicted by Carvalho on similar systems. Further verification of the

limitations of the Carvalho et al. model comes from the theoretical studies on the conformational statistics of grafted polymer chains on spherical interfaces. Carignano and Szleifer have shown that the conformational properties of a polymer chain attached to a curved interface is remarkably different from those of a similar chain grafted to a planar.19 Their analysis, based on a single chain mean-field theory technique, specifically shows that the number of the self-avoiding walks increases with curvature. Bulk-like or planar behavior was shown to emerge at nanoparticle radii, R/RG > 10. The authors concluded that even at low surface coverage the adsorption process cannot be be properly treated without considering all the appropriate interaction in the spherical grafted layer. Limitations of Our Model. There are several limitations to the current model. In our formulation, we have ignored completely the role of the ability of the end-functionalized chains to form micellar aggregates in solution on the adsorption process. In the absence of nanoparticles, the end-functionalized polymers can lower their free energy by self-associating into micelles, whose size and shape depend on the solution properties of the polymeric surfactants. Several authors have studied the competition between adsorption and micellization.20,36 The general conclusion is that smaller colloids tend to favor adsorption over micellization. This is because the stretching energy of the polymer chains is lower on the surface of the spherical nanoparticles than in the corresponding micellar corona when the radius of the particles is smaller than the core radius of the micelles. Consequently, we only expect micellization to be important when the equilibrium size of the micellar core is lower than the radius of the nanoparticle. In such instances, the formation of micelle would lead to a reduction in the equilibrium grafting density of the nanoparticles. Another limitation of the current model is our choice of expression for the conformational free energy of the spherical brush layer. Our current analysis utilizes the scaling results of Daoud−Cotton. One particular drawback of this approach is that it fails to recover the correct expression for the configurational free energy of the grafted layer in the large particle limit, namely the celebrated result of Milner, Witten, and Cates.45 Instead, the free energy expression, eq 6, reduces to the classic Alexander−de Gennes expression.46,47 Consequently, our model does not reproduce the same expression as those of Ligoure and Leibler for the equilibrium adsorption of end-functionalized polymers on a flat interface.33 Although the trends are identical, our model predictions differ quantitatively. It is possible to introduce a more accurate treatment of the conformational properties, a la the variational approach of Li and Witten,48 but at the expense of significant complications in the analysis. Finally, the current model ignores the role of kinetics in determining the grafting density. Ligoure and Leibler have shown that there are two successive regimes during adsorption of end-functionalized chains on to planar interfaces.33 At short times, the adsorption process is governed by Brownian diffusion of the polymer chains to the planar surface. Beyond a characteristic time, further adsorption of the chain is impeded by the emergence of an “activation barrier” which appears as soon as the adsorbed chains begin to overlap. The brush construction time is found to depend exponentially on the binding strength and the degree of polymerization. We also expect similar trends to prevail in the adsorption of endadsorbing chains on to curved interfaces. Because of the 7657


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(7) Kim, J.; Van der Bruggen, B. Environ. Pollut. 2010, 158, 2335− 2349. (8) Voevodin, A. A.; Vaia, R. A.; Patton, S. T.; Diamanti, S.; Pender, M.; Yoonessi, M.; Brubaker, J.; Hu, J.-J.; Sanders, J. H.; Phillips, B. S.; MacCuspie, R. I. Small 2007, 3, 1957. (9) Huynh, W. U.; Dittmer, J. J.; Alivisatos, A. P. Science 2002, 295, 2425−2427. (10) Zheng, G.; Patolsky, F.; Cui, Y.; Wang, W. U.; Lieber, C. M. Nat. Biotechnol. 2005, 23, 1294−1301. (11) Mirau, P. A.; Naik, R. R.; Gehring, P. J. Am. Chem. Soc. 2011, 133, 18243. (12) Ferrari, M. Nat. Rev. Cancer 2005, 5, 161−171. (13) Peer, D.; Karp, J. M.; Hong, S.; Farokhzad, O. C.; Margalit, R.; Langer, R. Nat. Nanotechnol. 2007, 2, 751−760. (14) Balazs, A. C.; Emrick, T.; Russell, T. P. Science 2006, 314, 1107− 1110. (15) Zhao, B.; Zhu, L. Macromolecules 2009, 42, 9369−9383. (16) Ligoure, C.; Leibler, L. Macromolecules 1990, 23, 5044. (17) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 1331−1333. (18) Dan, N.; Tirrell, M. Macromolecules 1992, 25, 2890−2895. (19) Carignano, M. A.; Szleifer, I. J. Chem. Phys. 1995, 102, 8662. (20) Qiu, X.; Wang, Z.-G. J. Colloid Interface Sci. 1994, 167, 294−300. (21) Singh, N.; Karim, A.; Bates, F. S.; Tirrell, M.; Furusawa, K. Macromolecules 1994, 27, 2586. (22) Carvalho, B. L.; Tong, P.; Huang, J. S.; Witten, T. A.; Fetters, L. J. Macromolecules 1993, 26, 4632. (23) Papke, B. L.; Bartley, L. S.; Migdal, C. A. Langmuir 1991, 7, 2614−2619. (24) Jayaraman, A.; Schweizer, K. S. Langmuir 2008, 24, 11119− 11130. (25) Zhang, Z.; Horsch, M. A.; Lamm, M. H.; Glotzer, S. C. Nano Lett. 2003, 3, 1341−1346. (26) Glotzer, S. C.; Horsch, M. A.; Iacovella, C. R.; Zhang, Z.; Chan, E. R.; Zhang, X. Curr. Opin. Colloid Interface Sci. 2005, 10, 287−295. (27) Akcora, P.; Liu, H.; Kumar, S. K.; Moll, J.; Li, Y.; Benicewicz, B. C.; Schadler, L. S.; Acehan, D.; Panagiotopoulos, A. Z.; Pryamitsyn, V.; Ganesan, V.; Ilavsky, J.; Thiyagarajan, P.; Colby, R. H.; Douglas, J. F. Nat. Mater. 2009, 8, 354. (28) Ndoro, T. V. M.; Voyiatzis, E.; Ghanbari, A.; Theodorou, D. N.; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2011, 44, 2316. (29) Srivastava, S.; Agarwal, P.; Archer, L. A. Langmuir 2012, 28, 6276−6281. (30) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd ed.; John Wiley and Sons: New York, 1980. (31) Tanford, C. Science 1978, 200, 1012. (32) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (33) Ligoure, C.; Leibler, L. J. Phys. (Paris) 1990, 51, 1313. (34) Dukes, D.; Li, Y.; Lewis, S.; Benicewicz, B.; Schadler, L.; Kumar, S. K. Macromolecules 2010, 43, 1564−1570. (35) Daoud, M.; Cotton, J. J. Phys. (Paris) 1982, 43, 531. (36) Marques, C.; Joanny, J. F.; Leibler, L. Macromolecules 1988, 21, 1051−1059. (37) Miracle, D.; Sanders, W.; Senkov, O. N. Philos. Mag. 2003, 83, 2409. (38) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571. (39) Milner, S. T. Europhys. Lett. 1988, 7, 695. (40) Hiemenz, P.; Lodge, T. Polymer Chemistry; CRC Press: Boca Raton, FL, 2007. (41) Eichinger, B. E.; Flory, P. J. Trans. Faraday Soc. 1968, 64, 2061. (42) Brandrup, J.; Immergut, E. H.; Grulke, E. A.; Abe, A.; Bloch, D. R. Polymer Handbook, 4th ed.; John Wiley & Sons: New York. (43) Tambasco, M.; Kumar, S. K.; Szleifer, I. Langmuir 2008, 24, 8448−8451. (44) Goldstein, R. E. J. Chem. Phys. 1986, 84, 3367. (45) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610−2619. (46) Alexander, S. J. Phys. (Paris) 1977, 38, 983−987.

correlation of the barrier height with stretching energy of the grafted chains, we expect the size (i.e., radius) of the nanoparticle to affect the onset of the second kinetic regime. In addition to the molecular weight and the binding strength, we speculate that the characteristic time will also depend to a large extent on the degree of curvature of the nanoparticle. It is quite conceivable that the long-term adsorption behavior predicted by a kinetic model could be remarkably different than that obtained by the equilibrium treatment presented here.

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CONCLUSION We have developed a theoretical model to study the role of nanoparticle curvature and solution properties of end-functionalized homopolymer chains on their equilibrium adsorption onto spherical interfaces. Our model predicts that the surface coverage increases with the nanoparticle curvature and polymer concentration. Comparison between the prediction of our model and experiments on the adsorption of poly(isobutenyl) succinimide dispersants onto surfaces of CaCO3 nanoparticles yields good agreement. Increasing the nanoparticle concentration is found to result in a corresponding decrease in the areal density of the adsorbed chains. At very low nanoparticle concentrations, the average number of polymer chains adsorbed per nanoparticle is found to be lesser than the ratio between the number fractions of the polymer chains and the nanoparticles. A significant finding of this study pertains to the role of nanoparticle curvature on the distribution of the grafting density. Our model predicts that the grafting density is uniformly distributed around its mean value. The probability distribution function is Gaussian. Furthermore, the standard deviation from the mean is found to increase with nanoparticle radius. However, the polydispersity index, which is proportional to the ratio of the standard deviation to the mean number of chains per NP, is found to decrease with the size of the particles. This important finding suggests possible difficulty in obtaining monodisperse polymer grafted nanoparticles via the grafting-to approach as the nanoparticle size becomes smaller.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was performed while F. Oyerokun held the National Research Council Associateship Award at the Air Force Research Laboratory. The authors also thank the Air Force Office for Scientific Research for funding.

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REFERENCES

(1) Ober, C. K.; Cheng, S. Z. D.; Hammond, P. T.; Muthukumar, M.; Reichmanis, E.; Wooley, K. L.; Lodge, T. P. Macromolecules 2008, 42, 465−471. (2) Kumar, S. K.; Krishnamoorti, R. Annu. Rev. Chem. Biomol. Eng. 2010, 1, 37−58. (3) Vaia, R. A.; Maguire, J. F. Chem. Mater. 2007, 19, 2736. (4) Milner, S. T. Science 1991, 251, 905−914. (5) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Nature 1988, 332, 712. (6) Tchoul, M. N.; Fillery, S. P.; Koerner, H.; Drummy, L. F.; Oyerokun, F. T.; Mirau, P. A.; Durstock, M. F.; Vaia, R. A. Chem. Mater. 2010, 22, 1749−1759. 7658


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(47) de Gennes, P.-G. Macromolecules and Liquid Crystals: Reflections on Certain Lines of Research. In Solid State Physics; Liebert, L., Ed.; Academic Press, Inc.: New York, 1978; Vol. 14. (48) Li, H.; Witten, T. A. Macromolecules 1994, 27, 449−457.

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Distribution in the Grafting Density of End-Functionalized Polymer

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