Understanding the Stabilization of Single-Walled Carbon Nanotubes

Article pubs.acs.org/JPCC

Understanding the Stabilization of Single-Walled Carbon Nanotubes and Graphene in Ionic Surfactant Aqueous Solutions: Large-Scale Coarse-Grained Molecular Dynamics Simulation-Assisted DLVO Theory Chih-Jen Shih,†,§ Shangchao Lin,†,‡,∥ Michael S. Strano,† and Daniel Blankschtein*,† J. Phys. Chem. C 2015.119:1047-1060. Downloaded from pubs.acs.org by AUCKLAND UNIV OF TECHNOLOGY on 01/28/19. For personal use only.

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Department of Chemical Engineering and ‡Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ABSTRACT: Understanding the dispersion of single-walled carbon nanotubes (SWCNTs) and graphene in surfactant aqueous solutions is essential for processing of these materials. Herein, we develop the first theoretical framework, which combines large-scale coarse-grained (CG) molecular dynamics (MD) simulations with the Derjaguin−Landau−Verwey−Overbeek (DLVO) theory and Langmuir isotherm, to inform the mechanisms of surfactant adsorption and induced colloidal stability. By carrying out large-scale CG-MD simulations (∼1 million CPU h), we successfully calculated the surface coverage of the widely used ionic surfactant sodium cholate (SC) on different carbon nanomaterials at experimentally realistic SC concentrations. This was accomplished by simultaneously simulating SC micellization and adsorption in one simulation box. Our theoretical framework further allows us to quantify the surface electric potential and the potential energy barrier height that maintains colloidal stability of dispersed SWCNTs or graphene sheets as a function of SC concentration, C, and radius of the SWCNT, r. We found that, for a specific carbon nanomaterial, there exists an optimal surfactant concentration, resulting from a trade-off between the increase in surfactant adsorption (i.e., surface charge density) and the increase in ionic strength (i.e., inverse Debye length) in the bulk aqueous phase. For the first time, we predict that small-radius SWCNTs promote higher surfactant adsorption than largeradius SWCNTs in the high SC concentration regime and, surprisingly, that the opposite behavior is exhibited in the low SC concentration regime. These findings suggest new experimental implications for the separation of SWCNTs based on their radius by tuning the SC concentration. In addition, we also predict that monolayer graphene exhibits better colloidal stability than multilayer graphene.

1. INTRODUCTION Carbon nanomaterials, such as single-walled carbon nanotubes (SWCNTs) and graphene, have attracted considerable research interest due to their outstanding optoelectronic, thermal, and mechanical properties.1,2 These properties, however, are associated with single-chirality, isolated, SWCNTs or graphene, which requires advanced postprocesses to debundle the SWCNTs, or to exfoliate graphene, for large-scale production. To this end, it is common to disperse these carbon nanomaterials in an aqueous surfactant solution,3−5 a method that has shown its superiority for further scaling-up, separation, and chemical functionalization.6−10 Indeed, this dispersion method represents the only viable route to sort SWCNTs by their chirality, following application of techniques like electrophoresis,11−14 density gradient ultracentrifugation (DGU),7,9 or gel chromatography6,15,16 to the solution. In addition, surfactant-induced stabilization and exfoliation of graphene becomes potentially important for the production of ABstacked bilayer and trilayer graphene.8,17,18 This method can also be used as a high quality alternative to chemical vapor deposition.19 In combination with the appropriate surface © 2014 American Chemical Society

chemistry on the target substrate, the interactions between the adsorbed surfactant molecules on the carbon nanomaterials and the functionalized substrate surface further enable the creation of device arrays for integrated circuits.20,21 The key challenge for the implementation of this method involves debundling/ exfoliating the appropriate carbon nanomaterials over a specific range of surfactant concentrations, capable of dispersing the resulting SWCNTs/graphene sheets in a stable manner.3 The widely recognized, ionic surfactant-aided dispersion process requires applying an external energy input (ultrasound) to separate the carbon nanomaterials at the bundled SWCNT ends (or at the graphite flake edges) by overcoming the strong van der Waals (vdW) attractions between them.3 Subsequently, the separated ends (edges) provide new adsorption sites for the hydrophobic tails of the surfactant molecules.22 As a result, the repulsive potential energy resulting from the electrostatic interactions between the hydrophilic, ionic heads of the Received: September 15, 2014 Revised: December 13, 2014 Published: December 19, 2014 1047

DOI: 10.1021/jp5093477 J. Phys. Chem. C 2015, 119, 1047−1060

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The Journal of Physical Chemistry C

surface coverage of surfactant molecules on a carbon nanomaterial as a function of C, as well as the effect of the carbon nanomaterial geometry, has not been investigated computationally. The main reason is that modeling experimentally relevant surfactant concentrations, typically in the cmc range, requires simultaneous simulation of surfactant micellization and adsorption in one simulation box, which is extremely computationally expensive. Indeed, using AA-MD simulations, one can only simulate a length scale of ∼10 nm, which is too small to demonstrate micelle formation, requiring length scales of the order of ∼50 nm. In addition, the time scale for capturing the dynamics of micelle formation is also very demanding (∼100 ns).55 As a result, most MD simulations can only be used to model/rationalize local structural features of self-assembled surfactant molecules on carbon nanomaterials under a “known” surfactant surface coverage (see, e.g., ref 43), while the dependence of the surface coverage on r and C, as well as the effect of micellization, remains undetermined. Consequently, the mechanism underlying the surfactant-aided stabilization of aqueous dispersions of carbon nanomaterials is still not sufficiently understood to permit a rational design of suitable surfactants for different carbon nanomaterials at the molecular level. With the above background in mind, in this paper, we develop the first theoretical framework, which combines CGMD simulations and the DLVO theory and Langmuir isotherm, to inform the mechanisms of surfactant adsorption and the induced colloidal stability, as well as their dependence on surfactant concentration, C, and radius of the SWCNT, r. In order to reduce computational expense, in addition to using CG simulations, the planar anionic surfactant sodium cholate (SC), which has a small micelle aggregation number (n = 2− 6),56 is considered. As a result, SC micellization and SC adsorption can be simulated simultaneously in one simulation box, thereby enabling simulation of graphene/SWCNT aqueous solutions at experimentally realistic concentrations (C = 1−200 mM). Specifically, we have carried out large-scale CG-MD simulations (∼1 million CPU h), and have successfully calculated the SC surface coverage of different carbon nanomaterials, including SWCNTs (6,6), (12,12), and (20,20), as well as monolayer, bilayer, and trilayer graphene (MLG, BLG, and TLG), as a function of SC concentration. The simulated surface coverage allows us to generate a continuum spectrum of the surface electric potential, Ψ0, as a function of r and C. Combining the simulation results with the DLVO theory and Langmuir isotherm, we have further quantified the potential energy barrier height required to maintain colloidal stability between two parallel identical SWCNTs or graphene sheets in aqueous SC solutions as a function of r and C. The theoretical methodology presented here allows us to rationalize many experimental findings, as well as to provide practical implications for separating different carbon nanomaterials from their isomers in a solution phase. Our investigation also provides fundamental insight on the molecular-level design of efficient surfactant or cosurfactant systems7,57 to enhance the dispersion and separation of SWCNTs and graphene in the solution phase.

adsorbed surfactant molecules further enhances this separation process.22 Eventually, individually isolated, surfactant-coated carbon nanomaterials are released to and dispersed in the bulk aqueous phase. The ionic surfactant molecules adsorbed on the nanomaterial surfaces generate a diffuse layer of counterions present in the bulk solution phase to form an electrical double layer (EDL).23 The diffusive nature of the counterions results in an effective surface charge, which can be quantified in terms of the ζ potential, the surface electric potential under an electrophoretic field.23 The interactions between nearby EDLs induce a repulsive potential energy, which offsets the attractive vdW potential energy, thereby stabilizing the suspended carbon nanomaterials. The competitive repulsive and attractive potential energies can be quantified using the Derjaguin− Landau−Verwey−Overbeek (DLVO) theory.23 In recent years, significant research effort has been devoted to (i) screen through many types of surfactants for the preparation of high weight fraction, aqueous solutions of SWCNTs, including ranking of surfactants in terms of their ability to solubilize suspended SWCNTs and graphene,18,24−29 and (ii) understand the self-assembled structures of the adsorbed surfactant molecules around SWCNTs and graphene,29−34 including rationalizing the surfactant-induced colloidal stability as a function of the surfactant concentration, C,18,35−42 and the SWCNT radius, r (note that, for graphene, r → ∞).36,43−45 For example, it has been experimentally shown that the widely used anionic surfactant sodium cholate (SC) preferentially stabilizes SWCNTs possessing smaller radii, 44 and is therefore particularly suitable for SWCNT separation via DGU.7 In addition, the stability of carbon nanomaterials can be optimized when the SC concentration, C, is in the range 10·cmc > C > cmc, where cmc is the critical micelle concentration of SC,18,27 although micellization is not a prerequisite to obtain stable colloidal dispersions.37 Indeed, these experimental findings clearly suggest that the adsorption of surfactant molecules highly depends on the geometry of the carbon nanomaterials and the surfactant concentration. In this respect, however, very little is known about the molecular details of the interactions between carbon nanomaterials and surfactant molecules, including correlating these interactions with the surfactantassisted colloidal stability. In particular, there is a lack of fundamental understanding about the adsorption isotherms that describe the behavior at high surfactant concentrations (above the cmc), where typical experiments of carbon nanomaterials dispersions are conducted. In order to bridge macroscopic variables (e.g., C) and the nanoscale self-assembly, molecular simulations have been used to elucidate the interactions between carbon nanomaterials and surfactants. For example, molecular dynamics (MD) simulations have been carried out to resolve the self-assembled structures of surfactants, including sodium dodecyl sulfate (SDS),43,46,47 single-tailed and double-tailed phosphatidylcholine,48−50 and SC,51,52 on SWCNTs and graphene, by using allatomistic (AA) or coarse-grained (CG) force fields. In addition, interactions between two surfactant-coated SWCNTs/graphene sheets were studied using MD simulations to elucidate the contributions from electrostatics and vdW interactions to the simulated potential of mean force (PMF) between two parallel SWCNTs or graphene sheets.43,51,52 The density differences of SC−SWCNT assemblies having different SWCNT radii have been also discussed.53,54 Despite significant progress on modeling/rationalizing local structural features of selfassembled surfactant molecules on carbon nanomaterials, the

2. SIMULATION METHOD We carried out MD simulations under the NPT (constant number of atoms, constant pressure of 1.0 bar, and constant temperature of 298.15 K) ensemble using the GROMACS 4.558 software package and the MARTINI-CG force field.59 The 1048


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Figure 1. Coarse-grained MD simulations of the stabilization of a carbon nanomaterial in a surfactant (sodium cholate, SC) aqueous solution accounting simultaneously for the phenomena of SC micellization and SC adsorption in one simulation box. (a) Chemical structure of SC showing the rigid, steroid-ring backbone of this facial surfactant; (b) AA molecular model of SC showing its hydrophobic and hydrophilic faces; (c) CG molecular model of SC showing the grouped beads used in our simulations. (d) Representative postequilibrium simulation snapshot (top) and the simulated concentration profile (bottom) of a SC-coated MLG sheet. Color code for AA model: red, oxygen; cyan, carbon; green, sodium; and white, hydrogen. Color code for the CG model: purple, PW.

The CG model of SC was adopted from Marrink,61 and the resulting CG structure is shown in Figure 1c. All the nonbonded interactions (LJ and Coulombic) were treated using the default values in the MARTINI library.59 The velocity-rescaled Berendsen thermostat was implemented to maintain the system at a constant temperature.62 The pressure was coupled to a semi-isotropic Berendsen barostat (Py = Pz = 1 bar for the SWCNT case, and Pz = 1 bar for the graphene case).63 The LJ and Coulombic interactions were treated with a cutoff distance of 1.2 nm. The equation of motion was integrated at a time step of 0.02 ps for the self-assembly process. For each simulation with a different carbon nanomaterial, we varied the number of surfactant molecules in the simulation box by distributing cholate ion monomers and sodium counterions randomly in the bulk phase. Subsequently, depending on the surfactant concentration considered, SC monomers can stay in the bulk, form micelles, or adsorb on the carbon nanomaterial surface freely. For each simulation, an ∼8 × 8 × 24 nm3 simulation box, which contains ∼12 500 coarsegrained water molecules, is used. For example, Figure 1d (top) shows a postequilibrium snapshot of a MLG-SC simulation, which is comprised of SC monomers, micelles, and a sheet of SC-adsorbed MLG. In the system considered, the density profile of cholate ions as a function of z was calculated, as shown in Figure 1d (bottom), and z = 0 corresponds to the location of the MLG. One can observe a peak on each side of the two graphene surfaces, which is associated with monolayer adsorption of the cholate molecules. The density profile of the cholate ions indicates that they prefer to distribute in a very

SWCNT and graphene species, including SWCNTs (6,6), (12,12), and (20,20) and MLG, BLG, and TLG, were modeled using the 3:1 mapping scheme proposed by Wallace and Sansom,49 and described by the uncharged SC359 LennardJones (LJ) spheres in the MARTINI force field. The target carbon nanomaterial was fixed in the middle of the simulation box with three-dimensional periodic boundary conditions. For the SWCNT simulations, the length of the simulation box along the x-axis, Lx, was kept constant in order to mimic an infinitely long nanotube along this axis, and similarly, for the graphene simulations, the lengths of the simulation box along the x-axis and y-axis, Lx and Ly, respectively, were kept constant to simulate infinitely large graphene sheets in the xy plane. In terms of the water model, it is noteworthy that we observed a serious freezing of CG water when the original MARTINI water beads or even antifreeze beads59 are used, particularly for graphene simulations due to flat-geometry-enforced ordering effects. Therefore, to avoid this artifact, the polarizable MARTINI water model is used,60 which models four water molecules with three CG beads. Sodium cholate, unlike conventional linear surfactants like SDS, is a rigid facial amphiphile, often referred to as a “two-faced detergent”. Parts a and b of Figure 1 show its chemical structure and all-atomistic molecular structure, respectively. The SC surfactant possesses a quasi-planar, slightly bent but rigid steroid ring with a hydrophilic face (the hydroxyl groups and the charged carboxylate group), and a hydrophobic face (the methyl groups and the tetracyclic carbon backbone) residing back-to-back. 1049


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Figure 2. Representative postequilibrium simulation snapshots (side and front views) for SWCNTs (a) (6,6), (b) (12,12), and (c) (20,20) at C = 10 (left) and 60 mM (right).

postequilibrium simulation snapshots of SWCNTs (6,6), (12,12), and (20,20) covered with cholate ions at two specific concentrations (C = 10 and 60 mM) are shown in parts a, b, and c, respectively, of Figure 2. Similar to the AA-MDsimulated self-assembly structure of cholate ions on SWCNTs and graphene,51,52 the cholate molecules form an adsorbed monolayer on the surface. An important feature in the lowconcentration (10 mM) simulations is that, instead of distributing uniformly on the SWCNT surfaces, the adsorbed cholate molecules prefer to self-assemble as clusters, exposing a fraction of the SWCNT surface to the aqueous phase, particularly for SWCNT (6,6). Interestingly, in the three configurations associated with C = 10 mM (see Figure 2a−c, all left), it appears that the extent of surface coverage for SWCNT (6,6) is much lower than those for SWCNTs (12,12) and (20,20). Since more surfactant adsorption generally implies a lower interfacial tension and stronger repulsive electrostatic interactions between the dispersed nanomaterials (for additional discussions, see section 3.2),3 this finding predicts that, at low SC concentration, large-radius SWCNTs and graphene (because r → ∞ for graphene) should exhibit higher colloidal stability. It further suggests that dispersing SWCNTs at low SC concentration potentially allows separation of large-radius SWCNTs from small-radius SWCNTs. To our knowledge, this method has not been proposed before, and we look forward to its experimental confirmation in the future. At low SC concentration, we believe that the preferential adsorption of cholate ions onto large-radius SWCNTs results from the rigid facial structure of the cholate ion. In general, adsorption of surfactant monomers from the bulk phase to the hydrophobic carbon nanomaterial surface is determined by the competition between enthalpic and entropic contributions. For a surfactant monomer, in order to minimize the contact area between its hydrophobic tail and water, there is an enthalpic driving force for adsorption onto the hydrophobic carbon nanomaterial surface. However, since SC is a planar surfactant, in order to cover the surface, it has to bend to some extent, depending on the local curvature of the surface, with a higher extent of bending resulting in a reduction in the enthalpic contribution, thereby making the SC adsorption less enthalpically favorable. On the other hand, when the surfactant monomer is adsorbed, it is confined on a two-dimensional surface, leading to a reduction in its entropy. In the case of adsorption onto large-radius SWCNTs (see Figure 2b,c, left),

compact manner within 1 nm on the graphene surface (where the density profile decays to the bulk concentration beyond z = ±1 nm). Note that, throughout the concentration range considered in this paper, we did not observe multilayer adsorption. Using the obtained concentration profile, the surfactant concentration in the bulk region, C (in units of mM), was therefore quantified by averaging the concentration of cholate ions in the bulk region (z > 1 nm or z < −1 nm), while the surface coverage of surfactants on carbon nanomaterials, Γ (in units of nm−2), was obtained by integrating the density of cholate ions along the axis normal to the nanomaterial surface (the z-axis in the case of graphene and the radial axis in the case of SWCNTs) in the adsorption region (−1 < z < 1 nm). It is noteworthy that in the low SC concentration regime, particularly in the case of graphene, almost all the inserted SC molecules are adsorbed, leaving only a few SC monomers in the simulation box. Consequently, the assumptions of constant concentration in the bulk phase, as well as of thermodynamic equilibrium of the system, may be less accurate. The accuracy of the simulations in the low SC concentration regime can be improved by considering a larger simulation box, which is more computationally expensive and beyond the scope of this paper. The simulations considered in this paper typically took at least 2−5 μs to reach equilibrium, by monitoring the calculated Γ as a function of time. Therefore, each simulated system was equilibrated for a total of 20 μs, and only the last 5 μs of simulation were used for data analysis. Considering the time scale and the MARTINI mapping schemes used in this paper, we estimate that the computational load is only about 1−5% of that using AA simulation methods. In order to carry out large-scale calculations (note that ∼1 million CPU h were used), we believe that the CG approach is still necessary. For the statistical analysis of micelle formation in the bulk region, we assume that surfactant molecules belong to the same micelle if the distance between any hydrophobic beads (SC1 and SC3 beads) within their structures is smaller than the cutoff distance of 7.5 Å.64 The large-scale CG-MD simulations allow us to investigate the effects of surfactant concentration and carbon nanomaterial geometry for the first time.

3. RESULTS AND DISCUSSION 3.1. Large-Scale CG-MD Simulations. 3.1.1. Surfactant Adsorption at Low Surfactant Concentration. Representative 1050


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of SC (∼10 mM),65 the solid line corresponds to C1 = C, and the blue circles correspond to the simulated C1. Recall that for an ideal micellization process, when the surfactant concentration exceeds the cmc, it becomes thermodynamically unfavorable for the added surfactant molecules to remain dispersed as free monomers. Accordingly, C1 should deviate significantly from the line C1 = C, reflecting micelle formation. In other words, when C > cmc, C1 should reach a plateau and remain constant at the cmc value.23 Figure 3b shows the simulated populations of micelles as a function of micelle aggregation number for the two SC concentrations considered (10 and 200 mM). For C = 200 mM, more than 50% of the population consists of micelles (n > 2), while for C = 10 mM, monomers are clearly the dominant species (>95%). However, the CG-MD-simulated C1 (the blue circles in Figure 3a) only exhibit a gradual deviation from the C1 = C line. As a result, it becomes challenging to infer the cmc from Figure 3a, since the monomer-to-micelle transition is too gradual. Two observations can be made to rationalize the behavior shown in Figure 3b: (i) The size of the SC micelles is inherently small (the reported experimental aggregation number n = 2−6, which is consistent with the simulated range of small n values in Figure 3b),56 because its facial structure does not allow the formation of large aggregates. Consequently, a gradual monomer-to-micelle transition is expected.23 (ii) The CG force-field parameters of water and SC do not allow a very accurate cmc prediction, based on the free monomer concentration.64 Several recent reports have suggested that in the case of linear surfactants like SDS, which form micelles having larger aggregation numbers (n ∼ 50),23 the CG-MD-simulated free monomer concentration strongly depends on the total surfactant concentration considered, and the extracted cmc values are often systematically underestimated.64,66 Observations (i) and (ii) are also reflected in the simulated aggregation number distributions (see Figure 3b), which do not show a peak aggregation number, with larger micelles becoming more favorable at higher surfactant concentrations.64 Nevertheless, to our knowledge, the results presented here represent the first CG-MD study to directly investigate the monomer-to-micelle transition near C = cmc. This was possible because the small aggregation number of the SC micelles significantly reduces the computational expense incurred at low SC concentrations. We hope that advances in the development of graphics processing units (GPU) based algorithms,64 as well as further refinements in the development of CG force fields, will facilitate simulating the micellization behavior of linear surfactants like SDS, which form micelles having large aggregation numbers. This will allow developing a better understanding of the mechanisms underlying micelle structural transitions at the molecular level. 3.1.3. Modeling SC Adsorption and Micellization in One Simulation Box. The various symbols in Figure 4a correspond to the simulated SC surface coverage, Γ (in units of nm−2), as a function of C for the three SWCNTs considered. First, as discussed in section 3.1.1, when C < 10 mM, the simulated Γ vs C for SWCNT (6,6) is only about half of those for SWCNTs (12,12) and (20,20) due to the large local curvature of SWCNT (6,6), which significantly reduces the enthalpy change associated with SC adsorption. Moreover, the simulated Γs for SWCNTs (12,12) and (20,20) are very close, suggesting that the curvature effect becomes negligible when r exceeds some threshold value. As discussed in section 3.1.1, we believe that understanding the SWCNT radius dependence of the SC surface coverage at low SC concentration may be useful for

the local curvature is small, such that the adsorption is more enthalpically favorable, and the enthalpic contribution dominates the entropic one, thereby driving more adsorption of SC molecules onto the SWCNTs. On the other hand, in the case of adsorption onto small-radius SWCNTs at C = 10 mM (see Figure 2a, left), the enthalpic contribution is diminished because the cholate molecules need to bend to a higher extent in order to cover the more curved surface of the small-radius SWCNT. As a result, the SC monomers prefer to stay in the bulk phase, and for the adsorbed SC molecules, they tend to form aggregates to diminish the enthalpic bending penalty. The same rationale can also be used to explain the SC adsorption at high SC concentration (C = 60 mM, see Figure 2a−c, right). The SC concentration increase in the bulk phase diminishes the entropic contribution, such that the surface is covered to a greater extent by the cholate molecules. On a small-radius SWCNT, there is an enthalpic bending penalty that hinders extensive SC adsorption, while the formation of SC aggregates on the surface, on the other hand, can stabilize the adsorbed SC molecules and reduce the enthalpic bending penalty. On the other hand, on a large-radius SWCNT, the enthalpic bending penalty is small, such that the adsorbed SC molecules tend not to aggregate on the surface. Additional discussion is presented in section 3.2. 3.1.2. Micellization in the Bulk Phase. As discussed in section 1, one of the most difficult challenges associated with simulating surfactant adsorption onto the surfaces of nanomaterials is to accommodate micellization and adsorption in one simulation box. In order to monitor micellization in the bulk phase, we statistically analyzed the population of SC micelles as a function of the total SC concentration, C. Figure 3a shows the simulated concentration of free SC monomers (not part of a micelle) at the cmc in the bulk phase, C1, as a function of C. The dashed vertical line corresponds to the experimental cmc

Figure 3. Micellization of sodium cholate molecules in the bulk phase. (a) Simulated free monomer concentration, C1, as a function of C (blue circles). The solid and dashed lines correspond to C1 = C and to the experimental cmc, respectively. (b) Simulated aggregation number distributions for C = 10 and 200 mM. 1051


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In order to model the dependence of the simulated Γ on SC concentration and SWCNT radius at high SC concentration, we utilize the Langmuir isotherm to fit the simulated Γ values for C > 20 mM. Specifically23 Γ(r , X ) = Γmax(r )

K ads(r )X 1 + K ads(r )X

(1)

where r is the radius of the SWCNT, Γmax(r) is the radiusdependent saturated SC surface coverage, X is the molar fraction of cholate in the bulk phase (i.e., X ∼ 18C/1000, where C is in units of M), and Kads(r) is the radius-dependent equilibrium constant, which can be calculated using the freeenergy change associated with the adsorption of a SC molecule from the bulk phase to the surface of a SWCNT of radius r, Gads(r), as follows: ⎛ − G (r ) ⎞ K ads(r ) = exp⎜ ads ⎟ ⎝ kBT ⎠

(2)

where kB is the Boltzmann constant and T is the absolute temperature. Note that, at constant temperature, the two parameters, Γmax(r) and Gads(r), depend solely on r. Note also that eqs 1 and 2 are based on the assumptions of (i) cholate monolayer adsorption and (ii) no interactions between the adsorbed cholate molecules.23 Assumption (i) is generally valid (see Figure 1d), while assumption (ii) is approximate, particularly at low SC concentrations, where, as discussed above, the adsorbed cholate molecules tend to self-assemble as clusters (see Figure 2a). Consequently, to carry out the required fitting, we only considered our simulation results at high SC concentration (C > 20 mM). Specifically, by leastsquares fitting eqs 1 and 2 to the MD simulation results for C > 20 mM, we extracted values for Γmax(r) and Gads(r) for each SWCNT considered, as shown in Table 1. The solid lines in

Figure 4. Adsorption of sodium cholate molecules on SWCNTs and graphene sheets. The various symbols denote the simulated SC surface coverages on single, isolated (a) SWCNT (6,6), SWCNT (12,12), and SWCNT (20,20), and (b) MLG, BLG, and TLG sheets as a function of C, and the various lines are the fitted adsorption isotherms using eq 1, where the colors used for each symbol and line match.

separating large-radius SWCNTs from small-radius ones. Note that, for C < 10 mM, almost all the SC molecules in the bulk phase are free SC monomers (see Figure 3a). Therefore, the increase in Γ with SC concentration shown in Figure 4a results from the equilibrium between free SC monomers and adsorbed SC molecules. Next, by further increasing C (C > 20 mM), the population of micelles increases gradually (see Figure 3a). The CG-MD simulated Γ for SWCNT (6,6) increases rapidly and exceeds those for SWCNTs (12,12) and (20,20) at C ∼ 20 mM, corresponding to the vertical dashed line in Figure 4a. In addition, the surface coverage for SWCNT (20,20) appears to increase more slowly, and begins deviating from that for SWCNT (12,12). In other words, relative to the simulations at low SC concentrations, the trend of radius dependence is reversed, that is, the small-radius SWCNTs exhibit higher SC surface coverage. More quantitative discussions will be presented in section 3.2. Our simulated SC surface coverage is consistent with the recent experimental finding that SC preferentially stabilizes SWCNTs with smaller radius at a high SC concentration (specifically, an ∼30 mM SC aqueous solution was used).44 As suggested in Figure 4a, we further predict that a higher SC concentration would be even more favorable to differentiate between SWCNTs with different radii. Considering the fact that a sufficiently high SC concentration can reduce the solubility of SWCNTs27 (additional mechanistic discussions are provided in section 3.2), based on our simulation results, we propose that a SC concentration of ∼40−80 mM (or 1.7−3.5 wt %) would be desirable to separate small-radius SWCNTs from large-radius ones. In combination with postseparation techniques (e.g., DGU7,9 and electrophoresis11−14), we believe that surfactant concentration is an useful parameter to tune the solubility of SWCNTs based on their radius, at least for the case of SC.

Table 1. Extracted Values of Γmax and Gads for the SWCNTs and Graphene Sheets Considered in Figure 4 SWCNT (6,6) Γmax [nm−2] Gads [kBT]

SWCNT (12,12)

SWCNT (20,20)

MLG

BLG

TLG

2.75

1.77

1.47

0.95

0.98

0.99

−8.59

−8.46

−8.39

−8.98

−8.37

−8.19

Figure 4a correspond to the fitted Langmuir isotherms for each SWCNT considered, where the colors of each line and symbols match. Figure 4a shows that each Langmuir isotherm can describe the SC adsorption behavior for C > 20 mM reasonably well, suggesting that (i) SC adsorption reaches equilibrium with the SC micelles and the SC monomers in the bulk phase, and (ii) interactions between the adsorbed cholate molecules are either negligible or weak relative to the SWCNT−SC interactions. Note, however, that the isotherms in Figure 4a, which are obtained by fitting the simulated Γ for C > 20 mM, do not accurately describe the SC adsorption at low SC concentration (C < 10 mM). In particular, they do not allow prediction of the preferential adsorption of SC on large-radius SWCNTs (see symbols in Figure 4a and section 3.1.1). Moreover, the adsorption isotherms fitted at high SC concentration tend to underestimate the SC surface coverage at low SC concentration, except for the small-radius SWCNTs (e.g., SWCNT (6,6) in Figure 4a). As discussed above, we believe that SC micellization in the bulk phase is playing an 1052


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The Journal of Physical Chemistry C important role. Nevertheless, in order to attain a better solubility of carbon nanotubes, typically, it is experimentally desirable to disperse SWCNTs and graphene above the cmc,3 as we described in section 1. Hereafter, we will utilize the fitted Langmuir isotherms to model the adsorption behavior. The various symbols in Figure 4b correspond to the CG-MD simulated Γ for MLG, BLG, and TLG. The Langmuir isotherms (the various lines having colors corresponding to those of the symbols) were obtained by fitting the simulated Γ for C > 20 mM. The extracted values of Γmax and Gads are shown in Table 1. Note that because graphene allows SC adsorption on both sides, the simulated surface coverages for graphene are inherently higher. The simulated SC surface coverages for MLG, BLG, and TLG do not change significantly with the SC concentration. Interestingly, as Figure 4b shows, the simulated SC surface coverages on graphene at high SC concentrations are significantly limited, as will be discussed later. In addition, the simulated Γ’s for MLG, BLG, and TLG are very close, indicating that SC adsorption is not strongly influenced by the substances adsorbed on the opposite side of the first graphene layer, including the adsorbed cholate molecules or the additional graphitic layers. This finding confirms that MLG is a “weakly transparent” material for the transmission of vdW interactions through it.67,68 As indicated above, Table 1 summarizes the extracted values of Γmax and Gads for the three SWCNTs and three graphene species considered. The extracted values of Gads are all very close, suggesting that the adsorption free-energy change is mainly determined by the interactions between the SC molecules and the carbon atoms on the nanomaterial surface, irrespective of the local curvature. Therefore, we can further simplify eqs 1 and 2 by assuming that Gads is a constant, which is independent of the radius r of the SWCNT (or graphene, corresponding to an infinite r value). Accordingly, the average value of Gads = −8.5 kBT (calculated by averaging the six Gads values reported in Table 1) will be used in all the calculations which follow. On the other hand, Table 1 shows that Γmax is r-dependent (for graphene, r → ∞), and more importantly, a small-radius SWCNT tends to have higher Γmax (the surface coverage when C → ∞), as stressed earlier. In order to understand the resulting radius dependence, in Figure 5b, the extracted Γmax values are plotted as a function of 1/r. Note that, for graphene, 1/r is equal to 0, and in order to make a proper comparison, the calculated Γmax for graphene should be divided by 2, since graphene potentially allows SC adsorption on both sides. Interestingly, we find that the extracted Γmax is linearly proportional to 1/r, suggesting a semiempirical relation for Γmax(r), which is given by Γmax(r ) = a +

b r

Figure 5. Preferential adsorption of cholate molecules on small-radius SWCNTs in the high SC concentration regime (C > 20 mM). (a) Schematic illustration of the reduction of electrostatic repulsions between the adsorbed SC molecules on a small-radius SWCNT. (b) Fitted values of saturated surface coverage, Γmax, from the CG-MD simulations as a function of 1/r for the carbon nanomaterials considered (symbols) and their linear fit (line).

the SC surface coverage increases to a great extent and the interactions between the charged hydrophilic heads of the adsorbed SC molecules become more significant. These interactions are dominated by electrostatic repulsions, which thermodynamically limits further adsorption of SC molecules. Figure 5a depicts schematically the adsorption of cholate molecules on small-radius (top) and large-radius (bottom) SWCNTs having the same surface coverage. Note that although the structure of the cholate molecule is relatively facial, following adsorption, the negatively charged carboxyl group (the cyan bead in Figure 1c) does not completely attach to the carbon nanomaterial surfaces (see Figure 2), due to steric effects. Therefore, in Figure 5a, the extension attached to each hydrophilic SC head group only represents part of its hydrophobic tail. As shown in Figure 5a, the small-radius SWCNT allows a larger distance between the charged head groups, thereby enabling a higher saturated SC surface coverage. The reverse is observed for the large-radius SWCNTs. In other words, the electrostatic repulsions between the adsorbed surfactant clusters increase with increasing r. Consequently, small-radius SWCNTs allow more SC adsorption at high SC concentration, where Γmax is inversely proportional to the radius of the SWCNT, as discussed above. Interestingly, recall that at low SC concentration there is an enthalpic bending penalty for the adsorption on small-radius SWCNTs (see section 3.1.1). However, it appears that at high SC concentration the electrostatic repulsion between the adsorbed SC head groups becomes more dominant. One can observe that the adsorbed SC molecules on a small-radius SWCNT tend to aggregate (see Figure 2), in order to gain more enthalpic reduction. By increasing the SC concentration in the bulk phase, more SC aggregates are formed, thereby

(3) −2

where a and b are fitting parameters (a = 0.947 nm and b = 0.725 nm−1 are obtained for the various nanomaterials considered in Figure 5b). Along with eqs 1 and 2, the extracted values of a, b, and Gads allow us to generate a continuous description of Γ as a function of r and C, based on the CG-MD simulated results. Note that, as discussed above, eqs 1−3 should provide more accurate predictions at high SC concentration (C > cmc). The mechanism that is responsible for the radius dependence in the high SC concentration regime is schematically illustrated in Figure 5a. Specifically, by increasing the SC concentration, 1053


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where e is the elementary charge. The exact solution of the Poisson−Boltzmann equation for a cylinder of radius r in a 1:1 electrolyte (where the electrolyte refers to SC) aqueous solution provides an analytical description to relate the surface electrostatic potential, Ψ0(r,C), to σ(r,C) as follows:69

stabilizing SC adsorption on small-radius SWCNTs. Consequently, we expect that the effect of the enthapic bending penalty is diminished and overwhelmed by the electrostatic repulsions between the adsorbed SC head groups. We can therefore conclude that the radii of SWCNTs and graphene determine the SC surface coverage through different mechanisms. In the low SC concentration regime, the SC surface coverage for small-radius SWCNTs is lower, since the cholate molecules need to bend to a greater extent in order to cover the increasingly curved surface. On the other hand, in the high SC concentration regime, the electrostatic repulsions between the adsorbed SC clusters increase with increasing r, resulting in a higher saturated SC surface coverage for smallradius SWCNTs. Because, experimentally, it is generally desirable to disperse SWCNTs and graphene above the cmc of SC, in section 3.2 we will utilize a simplified adsorption model (eqs 1−3) to describe Γ(r,C), which actually provides a better prediction of SC adsorption in the high SC concentration regime. 3.2. Simulation-Assisted DLVO Theory. 3.2.1. Modeling the Surface Electric Potential. The theoretical analysis presented so far provides a thermodynamic description of surfactant (SC) adsorption on an isolated SWCNT or graphene sheet. However, for practical purposes, in order to correlate the calculated Γ(r,C) to the resulting colloidal stability observed in actual experiments, additional modeling which accounts for the interactions between two surfactant-coated SWCNTs or graphene sheets is required. With this in mind, we next present a theoretical analysis that combines the Γ(r,C) results from our CG-MD simulations with a simulation-assisted DLVO theory23 of colloidal stability. This analysis will allow an overall understanding of the surfactant-assisted stabilization of SWCNT and graphene aqueous dispersions. In order to elucidate the colloidal stability of SWCNTs/ graphene sheets with different chiralities, in this section, we model the interactions between two parallel, identical, SCcoated SWCNTs or graphene sheets in water. The following assumptions were made to simplify the model: (i) The thickness of the adsorbed cholate molecules is negligible, implying that the adsorbed SC molecules are treated as effective surface charges uniformly distributed on the surface. (ii) The SC molecules in micellar form contribute to the solution ionic strength as the SC monomers do. This assumption is reasonable because of the following: (1) The simulated SC micelle size is generally small (n ∼ 2−6). As a result, the electric field generated by the negatively charged SC micelle is expected to be weak, resulting in small binding of positively charged sodium counterions. (2) The SC micelles are dynamic entities. (3) According to our CG-MD simulations, even at a very high SC concentration (C = 200 mM), more than 50% of the cholate molecules exist as monomers (see Figure 3b). We anticipate that assumption (ii) may overestimate the solution ionic strength at a very high SC concentration. (iii) The dominant interactions between two carbon nanomaterials (SWCNTs or graphene sheets) include (a) vdW attractions and (b) electrostatic repulsions resulting from the adsorbed SC molecules, consistent with the description underlying the DLVO theory. Assumption (i) above implies that the charge density on the carbon nanomaterial surface as a function of r and C, σ(r,C), is given by σ(r , C) = Γ(r , C) ( −e)

σ (r , C ) =

⎡ e Ψ (r , C ) ⎤ 2εε0κkBT sinh⎢ 0 ⎥ e ⎣ 2kBT ⎦ ⎤1/2 ⎡ β −2 − 1 ⎥ ⎢1 + cosh2(e Ψ0/4kBT ) ⎦ ⎣

(5)

where ε0 is the dielectric permittivity of vacuum, ε is the relative dielectric permittivity of water, and β is a shape (cylindrical) parameter given by

β=

K 0(κr ) K1(κr )

(6)

where Ki (i = 0 and 1) is the ith-order modified Bessel function of the second kind, κ = (e2ρ∞/(εε0kBT))1/2 is the inverse of the Debye−Hückel screening length, and ρ∞ is the SC number concentration in the bulk. Assumption (ii) above implies that ρ∞ = 2CNA, where NA is Avogadro’s number. It is noteworthy that eq 5 reduces to the well-known Goüy−Chapman solution23 of the Poisson−Boltzmann equation in the case of graphene (r → ∞). Indeed, limr→∞ β = 1, which reduces the last term in eq 5 to unity. In combination with the proposed adsorption model (eqs 1−3) and the three parameters a, b, and Gads deduced in section 3.1.3 (Gads = −8.5 kBT, a = 0.947 nm−2, and b = 0.725 nm−1), the surface electrostatic potential as a function of r and C, Ψ0(r,C), can be calculated by solving eq 5 numerically. A continuous spectrum of Ψ0, spanning a concentration and reciprocal radius space, can therefore be generated, as shown in Figure 6. The scale of Ψ0 ranges from −80 (blue) to −160 mV (red) (see the color scale bar in Figure 6). The reciprocal radii corresponding to SWCNTs (6,6), (12,12), and (20,20), and to graphene are shown as dashed vertical lines for reference (note that for graphene 1/r = 0, corresponding to the y-axis). For each carbon nanomaterial considered (along the dashed vertical lines for the three

Figure 6. Calculated surface electrical potential, Ψ0, as a function of 1/ r and C. The solid white line is the calculated optimal concentration maximizing Ψ0, Copt, as a function of 1/r, the dashed black lines correspond to the different carbon nanomaterials considered, and the dashed white line denotes the experimental cmc.

(4) 1054


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end, one needs to experimentally investigate ζ as a function of r and C, which, to our knowledge, has not yet been pursued extensively. In the future, in addition to developing new techniques to prepare large-scale, single-chirality SWCNT or graphene dispersions,6 we hope that a better understanding can be advanced to correlate the predicted Ψ0 to the experimentally measured ζ. 3.2.2. Modeling the Energy Barrier Height. In order to further understand the surfactant-assisted stabilization of dispersions of carbon nanomaterials, we next quantify the interactions between two identical, parallel SWCNTs or graphene sheets, as a function of the intertube or intersheet separation, d. In combination with the calculated Ψ0(r,C), the repulsive electrostatic potential energy between two identical, parallel SWCNTs per unit length, Φrep, as a function of r, C, and d can be obtained by considering the interactions between two EDLs,23 which is given by23

SWCNTs and the y-axis for graphene), there exists an optimal surfactant concentration which depends on 1/r, Copt(1/r), corresponding to the maximum value of Ψ0, as shown by the solid white line in Figure 6. Interestingly, the optimal concentration for graphene, Copt(0), corresponds to C ∼ 10 mM, consistent with the experimental cmc of SC.65 This finding is actually coincidental, since as we discussed above, our CG-MD simulations do not accurately describe the micellization of SC in the bulk aqueous phase. In fact, we found that our model (eqs 1−6) mathematically demands that the value of Copt(0) be equal to the molar fraction of ln(Gads) (i.e., Xopt(0) = ln(Gads)), by analytically determining the maximum of limr→∞ Ψ0(r,C). The value of Copt(1/r) slightly increases with 1/r due to the shape (cylindrical) factor in eq 5 (the term associated with β). The optimal concentration results from the competition between the SC adsorption and the ionic strength in the bulk aqueous phase. Qualitatively speaking, one can simplify eq 5 by considering the limits of graphene (β → 1) and of the Debye− Hückel approximation (eΨ0 ≪ kBT), which yield σ = εε0κΨ0.23 Combining this result with eq 4 and the definitions of κ and ρ∞ given above, it follows that

|Ψ0| ∝ Γ/ C

Φrep(r , C , d) =

(κr )1/2 (64πεε0(kBT /e)2 2 π tanh2[e Ψ0(r , C)/4kBT ]) exp( −κd)

(8)

As assumed above, the attractive vdW potential energy between two identical, parallel SWCNTs per unit length, Φatt, is not influenced by surfactant adsorption. Therefore, Φatt is independent of C, and can be obtained by integrating the Lennard-Jones (LJ) potential over the surfaces of two SWCNTs. This yields70

(7)

Indeed, along the dashed vertical line for graphene (1/r = 0) in Figure 6, one can clearly see the competition between Γ and √C. Specifically, at low SC concentration, as described by the Langmuir isotherm (see eq 1), Γ increases rapidly with C and dominates the √C term in the denominator of eq 7, resulting in the increase of |Ψ0| (more negative) with SC concentration. On the other hand, at high SC concentration (C > 10 mM), SC adsorption approaches its saturated surface coverage. Accordingly, the increase of Γ with SC concentration becomes much slower, and the √C term becomes dominant, thereby reducing |Ψ0| (less negative) with concentration. Although mathematically more complex in the case of SWCNTs, because of the finite curvature of the SWCNT surface and the nonlinearity of the Poisson−Boltzmann equation when β is not unity (see eq 5), a similar competition still exists between the SC adsorption (the Γ in the numerator of eq 7) and the ionic strength in the bulk aqueous phase (the √C in the denominator of eq 7). Theoretically, the predicted Ψ0 is closely related to the experimentally measured zeta potential, ζ, which corresponds to the electric potential in the EDL at the location of the slipping plane, residing a few κ−1 away from the charged surface.23 As a result, it is possible to relate ζ to the stability of dispersions of carbon nanomaterials.27,29 For example, a highly negative ζ typically suggests better colloidal stability, due to a stronger electrostatic repulsion between the dispersed nanomaterials.23 Therefore, if we assume that ζ ∝ Ψ0, Figure 6 clearly rationalizes the experimental findings that (i) at any given r the colloidal stability of SWCNTs and graphene is significantly reduced when C ≫ cmc (Ψ0 becomes more bluish or less negative), and (ii) there is an optimal SC concentration range, which maximizes colloidal stability, at around 1−10 times the cmc,18,27,37 corresponding to the reddish area in Figure 6. Furthermore, the theoretical predictions presented in Figure 6 also have important implications for the separation of SWCNTs based on their radius. Specifically, in addition to the separation of small-radius SWCNTs from large-radius SWCNTs at high C, as we suggested earlier, it is also possible to separate SWCNTs based on their radius using electrophoresis11−14 by tuning the surfactant concentration. To this

Φatt(r , d) =

3πσ 2 ⎛ 21B ⎞ ⎜ −AIA + IB⎟ 8r 3 ⎝ 32r 6 ⎠

(9)

where A and B are the attractive and repulsive Lennard-Jones constants (A = 15.2 eV·Å6 and B = 24.1 × 103 eV·Å12),70 σ = 4/ (√3a2) is the mean surface density of carbon atoms,70 and IA and IB are integrals given by70 ⎡ 2 ⎢⎣(cos θ2 − cos θ1) ⎤−5/2 + (sin θ2 − sin θ1 + d /r )2 ⎥ ⎦ dθ1 dθ2 2π

IA =

∫0 ∫0

2π

⎡ 2 IB = ⎢⎣(cos θ2 − cos θ1) 0 0 ⎤−11/2 + (sin θ2 − sin θ1 + d /r )2 ⎥ ⎦ dθ1 dθ2 2π

∫ ∫

2π

(10)

where θ1 and θ2 are angular coordinates, ranging from 0 to 2π. Therefore, the total potential energy between two identical, parallel SWCNTs per unit length, Φ, is given by23 Φ(r , C , d) = Φatt(r , d) + Φrep(r , C , d)

(11)

Note that the unit of Φ in eq 11 is energy per unit length, which is inherently higher for large-radius SWCNTs and diverges in the graphene limit (r → ∞). Therefore, graphene is studied separately and will be discussed below. Using eqs 8−10, with known Ψ0(r,C) (see Figure 6), one can calculate Φrep(r,C,d), Φatt(r,d), and Φ(r,C,d). For each SWCNT species considered (known r), the continuum spectrum of Φ as a function of C and d was generated. As an illustration, the 1055


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The Journal of Physical Chemistry C calculated total potential energy, Φ, between two parallel (6,6) SWCNTs per unit length as a function of the intertube separation, d, and the surfactant concentration, C, is shown in Figure 7a. The energy scale of Φ ranges from 0 (blue) to 27

scale of electrostatic influence in an electrolyte solution; i.e., a larger κ−1 results in a wider boundary layer thickness. Consequently, as we discussed earlier, because κ−1 decreases with C, Figure 7a clearly shows that the boundary layer thickness of Φ decreases with C. At a fixed surfactant concentration, we can also calculate the height of the energy barrier hindering the reaggregation of the SWCNTs, ΦBarrier, as the maximum potential energy attainable, corresponding to d = dmax. Accordingly, the following two equations are utilized to determine ΦBarrier: ⎛ ∂Φ ⎞ ⎜ ⎟ ⎝ ∂d ⎠r , C

= 0;

ΦBarrier (r , C) = Φ(C , r , dmax )

d = dmax

(12)

The calculated height of the energy barrier, as a function of SWCNT radius, r, and SC concentration, C, is shown in Figure 7b. The energy barrier height scales from 0 (blue) to 50 meV/ nm (red) (see the color bar scale in Figure 7b). Typically, a higher value of ΦBarrier implies better colloidal stability.23 However, again, the unit used for ΦBarrier (energy per unit length) magnifies the values for the large-radius SWCNTs. To interpret Figure 7b properly, one should compare the relative values of ΦBarrier for each SWCNT considered (along the vertical dashed lines in Figure 7b). Accordingly, for each SWCNT, we are able to clearly define an optimal concentration, corresponding to the highest energy barrier height, located at C ∼ 10 mM, which is close to that described by the Copt curve in Figure 6. Another important finding is that the energy barrier height of the SWCNTs at a very high SC concentration (C > 200 mM) is even lower (more bluish) than that at a low SC concentration (C ∼ 1 mM). This clearly suggests that stabilization of SWCNTs at a low SC concentration should be feasible, while a high SC concentration could lead to an even lower colloidal stability of SWCNTs, consistent with most experimental observations.37,38 Following a modeling procedure similar to the one discussed above in the SWCNT case, a simulation-assisted DLVO theory was developed to model the stabilization of graphene sheets in a SC aqueous solution. First, the repulsive electrostatic potential energy between two parallel graphene sheets per unit area, Φrep, as a function of SC concentration, C, and intersheet separation, d, is given by23

Figure 7. Simulation-assisted DLVO theory accounting for the interactions between two identical, parallel SWCNTs. (a) Calculated total potential energy between two SWCNTs (6,6) per unit length as a function of intertube separation, d, and C. (b) Calculated energy barrier height hindering the reaggregation of SWCNTs, ΦBarrier, as a function of r and C. The dashed white line denotes the experimental cmc.

meV/nm (red) (see the color scale bar in Figure 7a). Note that, in order to focus on the energy barrier height, the energy scale for Φ < 0 is not presented here (i.e., the color scale for Φ < 0 remains blue). At a fixed surfactant concentration (constant y, e.g., C = 10 mM), the total energy is zero when the two SWCNTs are far apart (d → ∞). By reducing d, the competition between Φatt and Φrep leads to an increase of Φ (repulsive) first, followed by a rapid decrease to negative (attractive), when the two SWCNTs are too close, as described by the classical DLVO theory.3,23 This is not surprising, since the vdW attractions are relatively short-ranged, and therefore, when the two SWCNTs are far apart, the repulsive electrostatic potential dominates, thereby kinetically stabilizing the SWCNTs. In other words, one can view the reddish region for d > 10 Å as the “boundary layer”, responsible for protecting the SWCNT from reaggregation, at a specific surfactant concentration. Since Φatt is independent of C (see eq 9), the thickness of the boundary layer directly relates to the Debye− Hückel screening length, κ−1, which characterizes the length

ΦRep(C , d) = 64kBTρ∞κ −1 tanh2[e Ψ0(∞ , C)/kBT ] exp( −κd)

(13)

where Ψ0(∞,C) was obtained by solving eq 5 in the limiting case β = 1. Note that in deriving eq 13 we have assumed that graphene is a continuous conductor that can perfectly screen the electric field exerted by the EDL on the opposite side of graphene. In other words, we have assumed that graphene is “opaque” to the electric field, and eq 13 is therefore independent of the layer number of the graphene sheet considered. On the other hand, the attractive vdW interactions can be partially transmitted through the graphene layers,67,68 suggesting that the layer number of graphene needs to be considered to formulate the attractive vdW potential energy per unit area between two parallel graphene sheets. Based on a model which accounts for the vdW interactions between two parallel MLG sheets,70 we have derived the expression of the vdW interactions between two parallel, N-layer and M-layer 1056


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The Journal of Physical Chemistry C graphene sheets per unit area as a function of d, Φatt,NM(d), as follows: ⎧ N−1 ⎡ M−1+i ⎛ ⎪ −A Φatt, NM (d) = πσ ⎨ ∑ ⎢ ∑ ⎜ ⎢ ⎪ i = 0 ⎣ j = i ⎝ 2(d + jd0)4 ⎩ 2

+

⎞⎤⎫ ⎪ B ⎟⎥⎬ 10 ⎥ 5(d + jd0) ⎠⎦⎪ ⎭

(14)

where d0 = 3.35 Å is the interlayer distance between two graphitic layers in an N-layer (N ≥ 2) graphene sheet. In the subsequent calculations, we consider the cases of N = M = 1 (MLG), N = M = 2 (BLG), or N = M = 3 (TLG). Finally, similar to the SWCNT formulation, the total potential energy between two identical, parallel, N-layer graphene sheets per unit area, ΦNN, is given by23 ΦNN (C , d) = Φatt, NN (d) + Φrep(C , d)

(15)

Following a procedure similar to the one discussed above in the SWCNT case, Φrep, Φatt, and Φ are obtained by solving eqs 13, 14, and 15, respectively. In addition, the height of the energy barrier hindering the reaggregation of two identical, N-layer graphene sheets, ΦBarrier,NN, can be determined using the following equations: ⎛ ∂ΦNN ⎞ ⎜ ⎟ ⎝ ∂d ⎠C

= 0;

Figure 8. Simulation-assisted DLVO theory accounting for the interactions between two identical, parallel graphene sheets. (a) Calculated total potential energy between two MLG sheets per unit area as a function of intersheet separation, d, and C. (b) Calculated energy barrier height hindering the reaggregation of the graphene sheets, ΦBarrier, for MLG, BLG, and TLG as a function of C.

ΦBarrier, NN (C) = Φ(C , dmax )

d = dmax

(16)

The calculated total potential energy between two identical, parallel, MLG sheets per unit area, Φ11, as a function of C and d is shown in Figure 8a, which exhibits characteristics which are very similar to those shown in Figure 7a. The energy scale of Φ11 ranges from 0 (blue) to 27 meV/nm (red). In particular, at a specific SC concentration, there exists a boundary layer that can hinder the reaggregation of MLG sheets, where the boundary-layer thickness decreases significantly with C, due to the increased ionic strength in the bulk phase. Figure 8b shows the calculated ΦBarrier,NN for MLG (N = 1), BLG (N = 2), and TLG (N = 3). Again, similar to the SWCNT case, we find that the colloidal stability of graphene sheets becomes very low when C > 200 mM, while at a relatively low SC concentration (C ∼ 1−10 mM), an adequate colloidal stability can still be imparted. Most importantly, in the graphene case, we predict that the colloidal stability significantly decreases with the layer number, a very interesting finding which has never been reported experimentally. According to our predictions, monolayer graphene can provide about a 40% higher energy barrier than multilayer graphene, thereby exhibiting better colloidal stability. It is noteworthy that our CG-MD simulation results suggest that SC adsorption is only weakly affected by the layer number (see Figure 4b). Therefore, it would appear that the reduced colloidal stability as the layer number increases results from an increase in the vdW interactions with layer number. Since research on the stability of graphene aqueous dispersions is still at an early stage, we hope that our results will stimulate experimental work to confirm their validity.

4. CONCLUSIONS The mechanism of surfactant-assisted stabilization of SWCNTs and graphene sheets in an aqueous solution of the surfactant SC was investigated using large-scale CG-MD simulations. In addition, a simulation-assisted DLVO theory which quantifies the energy barrier hindering the reaggregation of the carbon nanomaterials was proposed to elucidate their radius and surfactant concentration dependences. By carrying out largescale CG-MD simulations, we successfully calculated the SC surface coverage on SWCNTs (6,6), (12,12), and (20,20) and MLG, BLG, and TLG in SC aqueous solutions at experimentally realistic SC concentrations (C = 1−200 mM). Most importantly, the phenomena of SC adsorption and SC micellization were considered in one simulation box when C > cmc. According to our simulation results, for the first time, we predict that, in a SC aqueous solution in the low SC concentration regime (C < 10 mM), small-radius SWCNTs have lower SC surface coverages than large-radius SWCNTs, because their large local curvature can reduce the adsorption enthalpy change. On the other hand, in the high SC concentration regime (C > 20 mM), small-radius SWCNTs have higher SC surface coverages than large-radius SWCNTs due to the reduced electrostatic repulsions between the negatively charged cholate head groups of the adsorbed SC molecules. These findings provide new experimental implications to separate SWCNTs based on their radius by tuning the surfactant (SC) concentration. In order to describe the CG1057


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MD simulated surface coverages, we developed a new adsorption model which explicitly accounts for the nanomaterial radius, r, and the surfactant concentration, C. In combination with analytical solutions of the Poisson− Boltzmann equation, a continuum spectrum of the surface electric potential, Ψ0, as a function of r and C was generated, suggesting that separating SWCNTs based on their radius using electrophoresis may be feasible. Finally, we developed a simulation-assisted DLVO theory to quantify the potential energy barrier height required to impart colloidal stability to two identical SWCNTs or graphene sheets in a SC aqueous solution as a function of C and r. We found that, for a given chirality of SWCNT or graphene, there exists an optimal SC concentration, corresponding to the highest energy barrier height. This results from a competition between the increase in SC adsorption and the increase in the ionic strength in the bulk phase, consistent with previous experimental findings. We also found that monolayer graphene provides about a 40% higher energy barrier than multilayer graphene, thereby suggesting better colloidal stability for monolayer graphene dispersions in a SC aqueous solution. The mechanism and theoretical methodology presented here are expected to be useful in understanding the stabilization of SWCNTs and graphene sheets in surfactant aqueous solutions, as well as in allowing development of new separation methodologies aimed at preparing single-chirality carbon nanomaterials on a large scale. In addition, we hope that the molecular-level design of better surfactant or cosurfactant7,57 systems to disperse and sort SWCNT and graphene dispersions will be facilitated by the fundamental principles and theoretical methods presented here.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses §

C.-J.S.: Department of Chemical Engineering, Stanford University, Stanford, CA 94305. ∥ S.L.: Department of Mechanical Engineering and Materials Science & Engineering Program, Florida State University, Tallahassee, FL 32310. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful for the financial support from the National Science Foundation CBET-1133813. M.S.S. acknowledges funding from the 2009 U.S. Office of Naval Research Multi University Research Initiative (ONR-MURI) on Graphene Advanced Terahertz Engineering (GATE) at MIT, Harvard, and Boston University. C.-J.S. is grateful for partial financial support from the Chyn Duog Shiah memorial Fellowship awarded by Massachusetts Institute of Technology. S.L. is grateful for the startup funding from the Energy Materials Initiative at Florida State University.

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