Article pubs.acs.org/JPCC
Role of Solubility Parameters in Understanding the Steric Stabilization of Exfoliated Two-Dimensional Nanosheets by Adsorbed Polymers Peter May, Umar Khan, J. Marguerite Hughes, and Jonathan N. Coleman* School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland S Supporting Information *
ABSTRACT: In this paper we show that graphene, hexagonal boron nitride, and molybdenum disulfide can all be exfoliated and stabilized against aggregation in solvents that cannot alone exfoliate these materials, provided that dissolved polymers are present. In each case we demonstrate this steric stabilization for a range of polymers. To understand this, we have derived an expression for the free energy of adsorption of polymer chains onto the surface of nanosheets in a solvent environment. Critically, we express all energetic interactions in terms of the Hildebrand solubility parameters of solvent, polymer, and nanosheet. This allows us to predict the dispersed nanosheet concentration to display a Gaussian peak when plotted against polymer Hildebrand parameter. This is borne out by experimental data. The model correctly (within ∼2 MPa1/2) predicts the peak to occur when polymer and solvent solubility parameters match. In addition, the model describes both the peak width and the dependence of nanosheet concentration on polymer molecular weight. Because of the wide availability of solubility parameters for solvents, polymers, and many nanomaterials, this work is of practical importance for the production of polymer−nanosheet composite dispersions. However, more importantly it extends our understanding of the conditions required for steric stabilization and provides simple rules which define the required combination of solvent and polymer to best stabilize a given type of nanomaterial or colloid.
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INTRODUCTION Graphene and other layered compounds have generated renewed interest recently as the potential building blocks for a range of nanoenabled functional materials.1,2 For many applications, such as reinforcing composites3 or as thin films,4 graphene will be required in large quantities.5 For practical reasons and to facilitate processing, it is likely that liquid exfoliation of graphite to give graphene will be the production method of choice. To this end, much work has been reported on the dispersion of graphene oxide in water.6 Alternatively, ultrasonication can be used to exfoliate graphite in solvents7−11 or aqueous−surfactant solutions12−15 to give dispersed, defectfree graphene. More importantly for this work, graphene can be exfoliated in otherwise poor solvents using noncovalently attached small molecules16−19 or polymers3,20−24 as stabilizers. This last method is particularly useful, as graphene which is exfoliated in the presence of a polymer makes an ideal starting point for composite formation.20,22 Recently, it has been shown that inorganic layered compounds such as BN and MoS2 can also be exfoliated in solvents2,25−28 or by using surfactants.29 However, the demonstration of exfoliation of such materials using polymers as stabilizers has not been reported. This is a pity as both BN and MoS2 have been shown to be promising fillers in composites.28,30 Part of the reason that the polymer stabilization work reported for graphene has not been extended to inorganic layered compounds is that it is not clear what rules relate © 2012 American Chemical Society
solvent and polymer choice to the material being stabilized. Without such rules, it may not be straightforward to transfer what has been learned from graphene to other two-dimensional systems. In fact, very little work has been done to explore the physical chemistry of polymer stabilization for noncovalently functionalized graphene sheets. No comprehensive study has been carried out to determine which polymer−solvent combinations can be used to stabilize exfoliated graphene. Such knowledge would be important as it would greatly facilitate a range of applications such as solution processing of polymer−graphene composites. Of course, it is well-known that polymer stabilization of colloids is generally described by the steric stabilization mechanism.31−34 In this mechanism, polymer chains adsorb or are attached to the colloid to be stabilized such that the chains protrude into the solvent. When two polymer-coated colloids approach, the protruding polymer chains enter the same region of solvent, reducing the number of available conformations and so increasing the system's free energy. This results in an effective repulsion, stabilizing the colloid against aggregation. This mechanism has been well-studied for decades.32−37 However, the majority of papers focus on modeling the free energy of polymer-coated colloids as a function of separation.35−37 When polymer adsorption is Received: March 12, 2012 Revised: April 30, 2012 Published: May 2, 2012 11393
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studied, simple rules describing polymer/solvent choice are generally not available. The comparison with the solubility of solutes in solvents is useful. Solubility is maximized when the solubility parameters of solvent and solute match.38 This provides a simple framework to guide solvent choice. Unfortunately, no equivalent framework exists to inform the choice of solvent/ polymer combination to stabilize a given colloid. As we will show below, we can adapt the concept of solubility parameters to provide such a framework to describe steric stabilization. Interestingly, a number of authors have taken preliminary steps toward using solubility parameters or related concepts in the characterization or description of steric stabilization. For example, several authors have shown that there is a link between the difference between polymer and solvent solubility parameters and adsorption of polymers on surfaces.39−41 However, this work has not been extended to describe colloidal stabilization. Alternatively, authors have related the potential barrier associated with steric stabilization to solubility parameters of solvent and solute (more accurately they used the related concepts of the Flory−Huggins parameter or the second virial coefficient).31,34,42 However, this work has not been extended to link solvent, polymer, and colloid solubility parameters with the dispersed concentration to give general guidelines on system choice. As a result, in experimental papers, polymer/solvent combinations are generally found either by trial and error or serendipity.20,21,43−45 In this paper we study the exfoliation of nanosheets of graphene, BN, and MoS2 in two solvents using a range of polymers as stabilizers. We show that a number of solvent/ polymer combinations can be used to exfoliate graphene. We also demonstrate polymer stabilization of exfoliated BN and MoS2 for the first time, also using a range of solvent/polymer combinations. More importantly, we present a simple phenomenological model of the polymer adsorption process which results in an expression for the concentration of dispersed material as a function of polymer, solvent, and nanosheet solubility parameters. This model makes a number of predictions which are borne out by experiment.
Table 1. Solubility Parameters for the Four Dispersion Typesa nanosheet
solvent
δG (MPa1/2)
δS (MPa1/2)
graphene BN MoS2 graphene
THF THF THF CXO
21.25 21.25 22.5 21.25
18.6 18.6 18.6 20.3
a
Nanosheet solubility parameters taken from previously reported data.2,9 Solvent solubility parameters were taken from the Polymer Handbook.46.
Table 2. Polymers Used in the Study and Their Hildebrand Solubility Parametersa polymer
solubility parameter (MPa1/2)
graphene/ THF ⟨C⟩ (μg/mL)
BN/ THF ⟨C⟩ (μg/mL)
MoS2/ THF ⟨C⟩ (μg/mL)
graphene/ CXO ⟨C⟩ (μg/mL)
PBD PBS PS PVAc PVC PC PMMA PVDC CA
17.2 17.7 18.1 19.3 19.8 21.9 22.7 25.0 26.7
17 17 16 22 6 − 20
8 11 17 3 34 − 13 2 10
17 23 29 32 33 − 28 20 −
68 79 110 119 92 129 141
1
126
a
Also shown are the mean concentrations for each dispersion. Solubility parameters were taken from the Polymer Handbook.46.
Graphite flakes (Branwell Natural Graphite, grade 2369), powdered hexagonal boron nitride (BN) (Aldrich), and powdered MoS2 (Aldrich) were added at a concentration of 3 mg/mL to 1 mL of each polymer solution in THF (30 mg/ mL). Solvent (9 mL) was added so that samples had layered material and polymer concentrations of 0.3 and 3 mg/mL, respectively. In addition, graphite was also added in the same manner to each polymer dissolved in CXO. This results in a significant excess of polymer10:1 by mass. We expect that under these circumstances the polymer will reach an adsorption−desorption equilibrium such that the majority of polymer chains are free (i.e., not adsorbed). All samples were sonicated using a point probe sonic tip (VibraCell CVX: 30% of 750 W for 30 min per sample), using 10 mL of sample in a 14 mL vial. The sonic tip was used in pulsed mode (7 s on/10 s off) and cooling was employed to avoid solvent evaporation. After sonication the dispersions were centrifuged at 1500 rpm for 45 min (Hettich Mikro 22R) and the supernatants collected. The entire procedure was repeated for each polymer/layered compound/solvent combination to give at least two independent dispersions for each combination. Optical absorption measurements were taken with a Varian Cary 6000i (1 mm cells). The concentration, C, was found from the measured absorbance per cell length, A/l, using A/l = αC and taking αGra = 3620 mL/mg/m (550 nm), αBN = 592 mL/mg/m (600 nm), and αMoS2 = 3460 mL/mg/m (672 nm).2,9 We note that due to the presence of scattering effects,2,29 these coefficients are not intrinsic to the materials and may vary slightly with sample preparation conditions. However, they are accurate enough to give a good estimate of dispersed concentration. For the analysis used in this study, only relative concentrations are required so a high level of accuracy is not necessary. All reported concentrations are the average of at least two measurements. The error bar represents
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EXPERIMENTAL METHODS The polymers chosen for the study were polybutadiene (PBD), poly(styrene-co-butadiene) (PBS), polystyrene (PS), poly(vinyl chloride) (PVC), poly(vinyl acetate) (PVAc), polycarbonate (PC), poly(methyl methacrylate) (PMMA), poly(vinylidene chloride) (PVDC), and cellulose acetate (CA). The molecular weights were all close to 100 kg/mol with the exception of PBD which was considerably higher (400 kg/mol). These polymers were chosen based on their Hildebrand solubility parameters46 and the relationship between these parameters and those of the solvents and nanosheets used (see below for explanation and Tables 1 and 2 for full list of solubility parameters). Although the solvents used have Hildebrand parameters reasonably close to the nanosheets under investigation, they do not stably disperse any of the nanosheets alone.2,10 To ensure this was the case, the layered materials were sonicated in the relevant solvents in the absence of polymer and centrifuged to check that no nanosheets remained suspended. This confirmed that any dispersion of nanosheets was due to steric effects rather than solvent stabilization. All polymers (Aldrich) were obtained in granular form and initially dissolved in tetrahydrofuran (THF) or cyclohexanone (CXO) at concentrations of 30 mg/ mL. 11394
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dimensional objects with examples shown in Figure 2. In all cases the nanosheets appear to be exfoliated to a degree similar
the deviation between maximum and minimum values. Samples for TEM were prepared by pipetting a few milliliters of dispersion onto holey carbon grids (400 mesh). These were then analyzed using a Jeol 2100 operating at 200 kV. No attempt was made to remove excess polymer prior to TEM analysis. The Hildebrand parameters of graphene, BN, and MoS2 were estimated from previously reported data.2,9 We had previously quoted the Hildebrand parameter for graphene dispersed in solvents as 23 MPa1/2. This was calculated from the average solubility parameter for the solvents studied weighted by dispersed graphene concentration. However, since then we have found that better estimates are found by fitting a Gaussian envelope47 to the concentration−solvent Hildebrand parameter data. This method gives a value of ≈21.25 MPa1/2 for graphene which is used here. Similar methods gave values of ≈21.25 and ≈22 MPa1/2 for BN and MoS2, respectively.
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RESULTS AND DISCUSSION Initial Characterization. Three layered compounds, graphite, hexagonal boron nitride (BN), and molybdenum disulfide (MoS2), were sonicated in solutions of a range of polymers in the solvent tetrahydrofuran (THF) (both THF and cyclohexanone (CXO) in the case of graphite). The resultant dispersions were then centrifuged to remove any undispersed materials (see Experimental Methods for more details). It is well-known that when performed in the presence of suitable solvents or aqueous surfactant solutions, such a procedure results in the exfoliation and stabilization of two-dimensional nanosheets of graphene (Gra), BN, and MoS2.2,10,14,29 After centrifugation, dark-colored dispersions were obtained in all cases. However, it was clear, even to the naked eye, that the dispersed concentration varied from sample to sample (Figure 1). We estimated the dispersed concentration by optical
Figure 2. TEM images of nanosheets dispersed in polymer/solvent solutions. Each image is labeled using the following convention: nanosheet/solvent/polymer. In each row, the polymer solubility parameter increases from left to right. All scale bars are 500 nm.
to that observed in previous publications for solvent or surfactant exfoliation.2,9−11,13−15,29 Furthermore, the vast majority of objects were electron transparent. We note that the flake thickness can be estimated for graphene and BN using flake edge examination.2,11 However, this procedure is not quantitatively reliable for MoS2. Nevertheless, inspection of the flake edges, coupled with the observed contrast, suggested the dispersed nanosheets to be thin multilayers. This was confirmed in the case of graphene where detailed flake edge analysis showed the nanosheets to have flake thicknesses between 1 and 10 layers with a mean of 3−4, independent of polymer type (Supporting Information). In addition, the lateral flake size was ∼500−1000 nm, also invariant with polymer type (Supporting Information). Modeling Polymer Adsorption. It is important to understand what factors control the dispersed concentration of polymer-stabilized nanosheets. As described above, polymer stabilization of colloids is generally via the steric stabilization mechanism.31,32,48 Here, once chains are attached to the nanosheet, a repulsive potential will exist which will oppose aggregation. Chains can be attached either covalently (grafted) or by weak van der Waals forces.32 While polymer chains can be covalently attached to the basal plane of graphene,49 such attachment will perturb a monolayer’s electronic structure, resulting in significant property alteration. Thus, for graphene and probably other two-dimensional materials, noncovalent attachment is preferable. It has long been known that sonication of carbon nanotubes in a polymer solution results
Figure 1. Photograph of dispersions of graphene dispersed in THF by a range of polymers (after centrifugation). Note that the degrees of darkness may not exactly correlate with the numbers given in Table 2, and these numbers are an average over a number of batches while the photograph is from a single batch.
absorption spectroscopy.2,10 Over the range of polymers used, the measured concentrations varied from 1 to 22 μg/mL (Gra/ THF), from 3 to 34 μg/mL (BN/THF), from 17 to 33 μg/mL (MoS2/THF), and from 68 to 141 μg/mL (Gra/CXO). While it was apparent that layered material had been dispersed in the polymer solution, the exfoliation state was unknown. Transmission electron microscopy (TEM) measurements were carried out to analyze the exfoliation quality of the dispersions. The TEM images showed a large number of two11395
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respectively. That these energies must be similar implies that the nanosheets are best stabilized by adsorbed polymer when δS ∼ δG. This is the simplest condition for stabilization of nanosheets by adsorbed polymer and is roughly fulfilled in our case, since for THF and CXO, δS = 18.6 and 20.3 MPa1/2 while for graphene, BN, and MoS2, δG ≈ 21.25, 21.25, and 22.5 MPa, respectively (see Experimental Methods). However, this analysis gives no information about the requirements for the polymer solubility parameter. To properly describe the scenario where the polymer adsorbs on the surface of the nanosheet, it is necessary to analyze the energetics of polymer adsorption in more detail. We will consider a cube containing solvent molecules and one polymer chain and where one side of the cube is a nanosheet. We fill the cube with a cubic lattice of M × M × M sites. We model the polymer chain as a linear array of m lattice sites, each containing one monomer. All other lattice sites are occupied by solvent molecules. We consider two scenarios: in scenario 1 the polymer is free and surrounded by solvent molecules, while in scenario 2 it is partially adsorbed on the nanosheet with n lattice sites bound to the graphene and m − n sites protruding into the solvent. By considering the relevant solvent−solvent (SS), solvent−polymer (SP), nanosheet−solvent (GS), and graphene−polymer (GP) interaction energies in both scenarios, we will work out the difference in energies between scenario 2 and scenario 1. We consider only nearest neighbor “face to face” interactions, whose energies, εSS, εGS, εSP and εGP, we take to be negative. We find the energy difference is (see Supporting Information)
in exfoliation of the nanotubes and subsequent stabilization by adsorbed polymer.45,50,51 Recently, similar results were achieved for polymer-stabilized exfoliated graphene.20−24 However, it is worth noting that only partial adsorption of the polymer chains can occur if stabilization is to be successful. This means that the polymer may be anchored onto the surface at a number of sites but loops of the polymer chain must protrude into the solvent to provide steric stabilization.31,32,52 Here we aim to develop a simple model which describes when this partial adsorption can occur and so leads to rules which can inform the choice of successful polymer/solvent combinations. We note that most papers assume polymer adsorption and then calculate the free energy as a function of separation, for pairs of polymer-coated particles, to give information about the repulsive potential barrier.35−37 In this paper we take the opposite approach. We assume the presence of adsorbed polymer will lead to a potential barrier and so focus on the conditions for polymer adsorption. It is known that for a given surface, there is competition for adsorption between polymer and solvent.31,40 The resultant degree of adsorption and the conformation of the adsorbed polymer chains depend on the combination of surface, polymer, and solvent and the balance of the different intermolecular interactions. Partial polymer adsorption can only occur if it is favorable in terms of free energy. The free energy for adsorption of a polymer chain on a surface consists of energetic and entropic components.38 To estimate the energy of absorption, we need to consider the competing intermolecular interactions involved in such a process. We can approach this task using Hildebrand solubility parameters.48,53,54 We note that such parameters have been used to empirically characterize polymer adsorption onto surfaces39−41 but have not been used to theoretically calculate the energetics of polymer adsorption. The Hildebrand solubility parameter, δi, of a material is the square root of its cohesive energy density, Ec: δi = (Ec,i)1/2. Within a lattice model, it can then be shown that the intermolecular energy of interaction, εAB, between two lattice sites each containing a molecule or a portion of a molecule can be approximated as38
εAB
2v = − s δAδ B z
ΔE = nεGS − nεSS + nεSP − nεPG
(2)
Using eq 1, this can be rewritten as 2v ΔE = − s (δGδS − δGδ P + δ PδS − δSδS) n z 2vs = − (δS − δ P)(δG − δS) z
(3)
This equation predicts that the adsorption energy becomes more negative (adsorption becomes more favorable) as the difference between solvent and polymer solubility parameters increases. This agrees with previous empirical observations.39,41 Overall, this expression suggests that combinations of solubility parameters exist such that polymer chains can be driven from the bulk solvent to adsorb on the nanosheet in order to reduce their energy. However, it is necessary to consider the free energy of adsorption. If the adsorption energy is too high, many sites on the polymer chain will bind and the chain will be forced to adopt a compact conformation close to the surface at high entropic cost. Conversely, low adsorption energy means few binding sites resulting in many loops extending into the solvent. While this is entropically favorable, it will be less energetically favorable. Flory wrote the free energy per chain as the sum of an energetic and an entropic term. He then minimized this sum to give the free energy of adsorption in terms of the binding energy per site. Thus when the polymer chain adsorbs onto a surface, the free energy change per adsorbed chain is approximately given by38
(1)
Here we approximate the molecular size of A and B to be equal (molecular volume: vs) and z is the number of nearest neighbors per molecule. We note that when dealing with polymers, each lattice site is considered to contain one monomer. Equation 1 can be used to estimate the binding energy of both like and unlike molecules although it is most accurate where the intermolecular interactions are dominated by the dispersion (London) interaction.38,54 This expression is less valid for very polar or hydrogen bonding systems. In particular, we do not expect the following analysis to apply well to aqueous systems. In order for the steric interaction to occur, the polymer must partially adsorb onto the nanosheet surface such that part of the chain protrudes into the solvent. For this to occur the polymer−nanosheet (PG) and solvent−polymer (SP) interaction strengths must be similar: εPG ∼ εSP (we use the subscript G to represent the nanosheet). If this were not the case, we would expect the chain to remain completely in the solvent or to bond completely to the graphene. The energy associated with polymer−nanosheet and polymer−solvent interactions can be written as εGP ∝ −δPδG and εSP ∝ −δSδP,
⎛ ΔE /n ⎞2 ⎟ ΔF = kTN ⎜ ⎝ kT ⎠ 11396
(4)
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where N is the number of Kuhn monomers per chain. This means that the free energy change per adsorbed chain is ⎛ 2vs (δ − δ )(δ − δ ) ⎞2 S P G S ⎟ ΔF = kTN ⎜ z ⎜ ⎟ kT ⎝ ⎠
(5)
We can use these equations to develop rules of thumb for the relationship between polymer, solvent, and nanosheet for sterically stabilized dispersions. We expect adsorption to be favorable when ΔF is minimized, i.e., ΔF ≈ 0. With reference to eq 5, this is the case when either δS ≈ δP or δG ≈ δS. We note that this is consistent with the intuitive condition δS ∼ δG given above. By analogy with solvent and surfactant exfoliation of graphene and other two-dimensional materials,2,10,13,14,29 we make the assumption that sonication of the layered material in the polymer solution results in large-scale exfoliation. Then polymer adsorption is required to stabilize the dispersed nanosheets. We assume that, if favorable, polymer adsorption occurs quickly and reaches full coverage in all cases.31 In this scenario, we propose that the dispersed nanosheet concentration is controlled by the probability of polymer adsorption via the Boltzmann factor (i.e., C ∝ e−ΔF/kT) 2 ⎡ ⎛ 2vs (δS − δ P)(δG − δS) ⎞ ⎤ ⎢ C = A exp −⎜ N ⎟⎥ ⎠ ⎦⎥ z kT ⎢⎣ ⎝
(6)
Figure 3. Concentration of dispersed nanosheets, measured after centrifugation plotted as a function of Hildebrand solubility parameter of stabilizing polymer. The arrows indicate the Hildebrand parameters of solvent, δS, and nanosheet, δG. The dashed lines are fits of eq 6 to the data.
where A is a constant. We note that this equation predicts that C will display a Gaussian peak when plotted versus δP centered at δP = δS. However, when plotted against δS, a double peak will be observed with local maxima at δS = δP and δS = δG. Of course, when δG ≈ δP, this will appear as a single peak. (It is worth commenting that solvent−solute interactions are sometimes expressed in terms of the Flory−Huggins parameter, χ, which is related to the difference between solvent and solute solubility parameters.38 In this framework, eq 6 can be expressed as C ∝ exp[−4NχSPχGS/z2] where χSP and χGP are the Flory−Huggins parameters for solvent−polymer and nanosheet−polymer interactions, respectively.) Unfortunately eq 6 cannot predict the absolute concentration of dispersed nanosheets, only relative concentrations. The absolute concentration, i.e., the value of A, is most likely controlled by the details of the repulsive potential energy stabilizing the nanosheets. Comparing Experiment with Theory. We can easily test these predictions by plotting the dispersed nanosheet concentration as a function of polymer Hildebrand parameter, δP, as shown in Figure 3 for all four systems. We expect to see a peak in concentration when δS ≈ δP. In each case, we do indeed find a peak at polymer solubility parameter close to that of the solvent. However, in all cases, the peak was centered slightly above the solubility parameter of the solvent (shifted by 0.5− 2.7 MPa1/2). This indicates the limitations of our simple model and suggests that in practice it may be more appropriate to have δP closer to δG than δS in order to drive the polymer onto the nanosheet. We note that some scatter is observed which is to be expected as the polymers used do not have identical values of molecular weight (and so N). The main difference between these peaks is the width. The peaks for graphene and BN in THF are relatively narrow while those for Gra/CXO and MoS2/THF are much broader. We can explain this by noting
that the full-width at half-maximum of the Gaussian in δP as described by eq 6 is given by γ=
1.6zkT 2vS(δG − δS) N
(7)
We can test this by plotting the measured width, γ, versus kT/[vS(δG − δS)] as shown in Figure 4A. We find reasonable linearity, consistent with eq 7. From the slope of the linear fit, we can estimate a representative value of N, the number of Kuhn monomers per polymer chain. We obtain a value of 105 which is of the same magnitude as the values of 180 and 390 for the PMMA and PS used in this work. (N.B. we cannot estimate N for all the polymers used here as values of the Kuhn molecular weight are not readily available for all polymers.) In addition, eqs 6 and 7 predict that the dispersed nanosheet concentration should scale exponentially with N, i.e., with the polymer molecular weight. We have tested this by measuring the dispersed concentration of graphene in polystyrene/THF for a range of polystyrene molecular weights from 35 to 100 kg/mol. We then calculate N from the molecular weight taking the molecular weight of a Kuhn monomer to be 720 g/mol.38 We see a clear exponential dependence of concentration on N. By fitting this data, we find d ln C/dN = 5 × 10−3. We can also calculate the expected value from eq 7 using the data in Tables 1 and 2 and taking vs = 82 × 10−6 m3/mol and z = 6, finding d ln C/dN = 0.2 × 10−3. That both obtained values are of approximately the same magnitude gives support to our model and data. 11397
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Article 2 ⎡ ⎛ 2v (δ − δ P)(δG − δS) ⎞ ⎤ C ∝ exp⎢ −⎜ N s S ⎟⎥ ⎠ ⎥⎦ z kT ⎢⎣ ⎝
× exp[−[(δ P − δS)/3.5]2 ]
(10)
We have taken eq 10 and plotted the predicted nanosheet concentration in Figure 5 as a 2D contour map in δS and δP.
Figure 5. Contour plot of eq 10 calculated using δG = 21.25 MPa1/2, N = 200, vS = 10 × 10−5 m3/mol, and z = 6. The concentration (i.e., the color) has been normalized to a peak value of 1. The white lines represent the solubility parameter of graphene: δG = 21.25 MPa1/2.
Figure 4. (A) Measured Gaussian widths plotted as a function of kT/ [vS(δG − δS)]. Linear behavior is predicted by eq 6. (B) Dispersed concentration of graphene in PS/THF plotted as a function of the number of Kuhn monomers per PS chain, N.
While we accept that this map may be slightly shifted from its true position for reasons described above, we believe it captures the essence of the situation. The most salient feature of this graph is the prediction that the concentration will be maximized if δS ≈ δP ≈ δG. In addition, it predicts that concentrations will be reasonably high so long as δS ≈ δP even if neither are equal to δG. We note that the double peak mentioned above has been suppressed by the inclusion of the second term in eq 10. All that remain are two small lobes directly above and below the main peak. Finally, we note that Hildebrand solubility parameters represent a first-order approximation to the intermolecular interaction energies. As mentioned above, they only describe London interactions and ignore polar or hydrogen bonding interactions. It is well-known that better results can be obtained using Hansen solubility parameters.53 However, we have used Hildebrand parameters here because eq 1 (and so subsequent equations) can be expressed much more simply in this format than would be possible using Hansen parameters. We believe the loss of accuracy is more than compensated for by the simplicity of the system we have outlined.
This analysis ignores the requirement that the solvent must be capable of dissolving the polymer chains in an extended state to allow protrusion of adsorbed chains into the solvent. This can only occur in good solventsin poor solvents, the chain contracts to form relatively tightly packed globules.38 This contraction begins to occur when the Flory−Huggins parameter, χSP, exceeds 0.5.38 We can write χSP in terms of solubility parameters38 v χSP ≈ s (δ P − δS)2 (8) kT which means that the polymer chain can extend into the solvent only if
δ P − δS
