Assessing the Quality of Solvents and Dispersants for Low

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Assessing the Quality of Solvents and Dispersants for LowDimensional Materials Using the Corresponding Distances Method Adam Hardy, and Henry Bock J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b09172 • Publication Date (Web): 12 Oct 2016 Downloaded from http://pubs.acs.org on October 24, 2016

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The Journal of Physical Chemistry

Assessing the Quality of Solvents and Dispersants for Low-Dimensional Materials Using the Corresponding Distances Method A. Hardy and H. Bock

∗

Institute of Chemical Sciences, Heriot Watt University, Edinburgh, UK E-mail: [email protected] Phone: +44 (0)131 451 8339

1

ACS Paragon Plus Environment


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Abstract We demonstrate that the Corresponding Distances Method is an accurate, highly ecient and simple method to assess the quality of solvents and dispersants for lowdimensional nanomaterials. It provides potential of mean-force curves at very high resolution from a single simulation using atomistic models with common simulation software. Applying the Corresponding Distances Method to a pair of (10,10) single-wall carbon nanotubes immersed in bromotrichloromethane, we nd that bromotrichloromethane is not a sbuying theolvent for carbon nanotubes. This assessment is in agreement with experimental results but contradicts predictions from Hansen solubility parameters. We argue that the reason for the false-positive prediction of Solubility Theory is that it does not capture the structural details of the adsorbed solvent layer which governs the solvent-mediated forces between the tubes.

Introduction Low-dimensional materials often have sets of unusual properties that make them interesting for applications ranging from ultra strong, ultra light materials perconductors

2

to transistors

3

to electrodes

4

1

to high temperature su-

amongst many others. The award of the 2010

Nobel Prize in Physics for groundbreaking experiments with graphene is a testament to the potential that these materials hold. Carbon nanotubes and graphene are the most prominent members of this material class. Here we focus on single-walled carbon nanotubes (SWCNTs) but the methods presented are equally applicable to any other low dimensional material. As manufactured, carbon nanotubes are in an aggregated (bundled) state. Unfortunately, they reveal their unique properties only as individual particles, i.e.

only individualised

tubes exhibit the unique record properties that form the basis of their desired applications. For example, a very small amount of CNTs can render an electrically insulating polymer conductive.

The resulting composite has many more applications compared to the native

2


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polymer, which is particularly important to enhance renewable bio-polymers.

5

However,

nanotube bundling dramatically increases the percolation threshold, blackens the material and often weakens it. The bundling tendency originates from strong tube/tube cohesion, e.g. approximately -40kT/nm for a pair of (10,10) tubes (Ø

= 1.36nm). 6

As the tubes can be centimetres long

7

this accumulates to very high energies which make the bundles very stable. Consequently, it is a major challenge to exfoliate individual tubes from a bundle and to preserve this individualised state. While some progress has been made

8

individualisation of low-dimensional nanoparticles

remains the key bottleneck for their broad application. Liquid phase processing is the most researched technology to individualise CNTs.

911

It has the advantage of being compatible

with many other materials processing steps and it is reasonable to hope that the solvent/tube interactions my be able to compete withthe strong tube/tube cohesion. There are two fundamentally dierent ways to maintain the individualised state of CNTs in a solvent: dispersion and dissolution. Dispersants stabilise the metastable individualised (or dispersed) state by separating it from the aggregated (thermodynamically stable) state by a free energy barrier that can't be overcome via thermal activation.

Many substances

provide such a barrier; surfactants are the most prominent example. However, dispersants have the critical disadvantage that the same energy barrier that stabilises the dispersion also makes unbundling of the raw CNT material much harder as it increases the activation energy of the exfoliation process. The un-zippering mechanism

12

provides a route from the aggregated to the individualised

state that avoids the barrier. If one end of a tube has been liberated from the bundle far enough, this end is in the individualised state. As the region where the tube contacts the bundle can now slide along the bundle thereby only changing the length of tube that is separated from the bundle, this sliding of the contact region provides a "route" between the bundled and the dispersed state that does not cross the barrier.

3



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While this solves most of the problem with the barrier, the process still requires the energy to overcome the free energy dierence between the (stable) aggregated and the (metastable) individualised state.

This means that energy has to be provided continuously during the

entire exfoliation process.

If at any point this energy is not provided, the reverse of the

un-zippering mechanism leads back to the bundled state - spontaneously and without a barrier.

Consequently, incompletely separated tubes always snap back into contact with

their bundles. Similar arguments apply to lateral sliding of the tubes. This naturally raises the desire for "thermodynamic solvents" in which the individualised state is thermodynamically stable. In such a thermodynamic solvent CNTs would unbundle spontaneously. If a barrier between the metastable bundled state and the thermodynamically stable individualised state remains, activation of the exfoliation process would be required but after that it would proceed spontaneously via the un-zippering process. This advantage motivates our search for CNT solvents rather than dispersants. The behaviour of CNTs in solvents has been studied extensively in theory, simulation and experiment.

8,10,1331

The most common dispersion systems are based on aqueous surfactant solutions. This is driven by water being intrinsically environmentally benign and by the success of surfactants in stabilising suspensions of spherical colloids, e.g. in paints.

32

A large number of surfac-

tants have been investigated, including sodium dodecyl sulphate (SDS), benzene sulphonate (SDBS)

1315

and triton X-100.

13,14

sodium dodecyl

13,16

The function of these systems is based on the amphiphilic molecules' ability to absorb onto the nanotube with their hydrophobic part(s) leaving the hydrophilic part(s) pointing away from the tubes, which renders the tube/surfactant complex eectively hydrophilic. As a result the hydrophilic complexes repel each other in aqueous solution which constitutes the energy barrier that prevents rebundling. While some surfactants have been shown to stabilise CNT suspensions over long periods of time,

15

they are generally found to be ineective in aiding the exfoliation process. Often

4


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dispersion yields are well under 10% of the raw material, with tube concentrations in the O(1) mg/mL range

1315,25

and the tubes are dispersed as small bundles rather than individually.

Computer simulations generally indicate that surfactants are dispersants rather than solvents. As they can incorporate an entire small bundle in an adsorbed micelle, many surfactants do not provide a driving force for individualisation.

17

In many cases the surfactants

not only crate a re-aggregation barrier but may also further stabilise the aggregated state

18,19

which is clearly unwanted. From a processing point of view surfactants have the disadvantage that they are dicult to remove

20

and if transferred into a composite, are detrimental for its properties.

Pure solvents are an attractive proposition for the dispersion of CNTs as they avoid some of the problems experienced with surfactant solutions. solvents have been experimentally tested,

8,21,22,30,31

and dimethyl-tetrahydro-2-pyrimidinone (DMPU)

8

A large number of small-molecule

with N-methyl-2-pyrrolidone (NMP)

21,22

being amongst the most prevalent. Typ-

ical CNT yield percentages are in the single-gure range with concentrations of the order of 0.1mg/mL,

8,22

which does not represent an improvement compared to surfactant solutions

and is far too low to be economically viable in most cases.

∗ Flory's theory of rigid rods can be used to estimate the concentration limit (C ) of an isotropic CNT dispersion. they calculated C

∗

33,34

For the CNT sample used in the experiments by Bergin et al.,

≈15mg/mL.

C

∗

8

is not a solubility limit, but is the upper bound before

the formation of a nematic phase; even so, the best solvent, 1-cyclohexylpyrrolidone (CHP), did not approach this isotropic limit, with the second best of the tested solvents performing over ve times worse. This suggests that large improvements over the tested solvents may be possible and that there is scope for more systematic research. One such approach is the determination of Hansen solubility parameters

35

for SWCNTs.

The hope is that the "like dissolves like" concept of Solubility Theory is extendable to CNTs. The Hansen parameters are split into three components: a dispersive term ( δd ), a polar term (δp ) and a hydrogen bonding term ( δh ), each quantifying it's namesake's characteristic of a

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solvent. To obtain solubility parameters for CNTs they are generally tted to match those of the best solvents in a solubility test. Interestingly, although several studies

8,22,24,25

use a number

of dierent solvents, they broadly agree that the dispersive parameter is most signicant and that it should have a value of about

δd ≈ 19,

while

δp

and

δh

showed a much larger

variability, which perhaps suggests that they are less important. The Hansen parameters are eective at ruling out potential solvents, i.e.

liquids that

do not have the correct Hansen parameters will not be CNT solvents. However, they are less predictive for molecules that do have the correct parameters, i.e. even with the correct parameters a liquid may be a non-solvent. Understanding why the Hansen parameters make false-positive predictions would be an important step to better understand CNT solvents. In a systematic theoretical study based on partition coecients Torrens

26,27

found that

CNTs preferred the tested organic solvents over water but the absolute solubilities were generally very small. Pure water as a CNT solvent has been the subject of several molecular dynamics simulation studies, often as control cases for studying aqueous solutions containing surfactants and/or ions.

3638

However, the results are conicting, predicting dierent solvent performance

for the model water. The contradictions may be related to the diculties in parameterising water models for this particular application.

39

Several computer simulation studies investigate the solvent structure around individual CNTs.

28,29

While structural information is important to understanding the solvation process,

free energies are ultimately required to estimate a molecule's potency as a CNT solvent. Free energy information is computationally expensive and typically obtained via the potential of mean force (PMF), which covers a continuous path between the dispersed and the aggregated state. In this manuscript we demonstrate that the Corresponding Distance Method

40

can be

used in conjunction with atomistic models to obtain very high resolution PMF curves at low

6


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computational cost in a single simulation. We show that it provides all necessary structural information to interpret the structure/function relationship for the solvent. Our test case is a false-positive prediction of the Hansen solubility parameters, bromotrichloromethane

CBrCl3 .

Theoretical Methods Potential of Mean Force

Tubes d

y

z

Solvent Layers d’

x

l

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Figure 1: Schematic of the two system setups used (top: parallel, bottom: CDM). The link 0 between the two systems is established by the corresponding distance, d = l + d

Figure 2: Schematic of a (10,10) armchair carbon nanotube. Highlighted is the double ring of carbon atoms that represents the smallest repeat unit along the tube axis.

The quantity that is commonly used to assess the quality of a solvent or dispersant is the potential of mean force,

w(d),

as it provides the entire tube/tube distance dependence

7



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of the free energy, which allows assessment of the thermodynamic stability of the bundled compared to individualised state and provides information of any barriers between the two. The PMF is dened as the negative integral of the mean force two nanotubes over the tube/tube distance

F (d)

acting between the

d. Z∞

w(d) = −

F (d)∂d

(1)

d In this manuscript

F (d)

and

d

d is always the surface-to-surface distance (top panel of Figure 1), and

have the same direction, i.e. a positive

F

points in the direction of increasing

In practice the PMF is obtained by numerically integrating distance

d∗

to

d,

such that for

F (d)

d.

from a far enough"

d ≥ d∗ F (d) = 0. Zd w(d) =

F (d)∂d

(2)

d∗ This makes the separated state

d∗

the reference state with

w(d∗ ) = 0.

Thus, the PMF represents the negative of the work required to move two immersed and parallel tubes from a nite distance

d

to a separated state with

d ≥ d∗ .

Equivalently this

means that apart from an additive constant, i.e. the free energy of the system at separation

d∗ ,

the PMF represents the free energy of the system.

The Corresponding Distance Method

The forces

F (d)

needed to calculate the PMF can be obtained from a series of simulations

with two parallel tubes at varying distance of the forces (parallel to

d)

d, where F (d) is the ensemble average of the sum

acting on all the carbon atoms on one of the tubes.

Due to the high symmetry of the nanotubes one would obtain the same force

length

per tube

for each of the periodic rings of carbon atoms (Figure 2) as for the entire tube,

although sampling would have to be increased proportional to the reduction in tube length

8


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considered. The general idea of the Corresponding Distance Method is to create a system where all tube/tube distances are present at the same time.

40

The simplest way to achieve this is to

use two tubes that cross at a small angle (bottom panel of Figure 1).

The forces on the

horizontal tube are calculated in the same way as for the parallel tubes but for each ring of carbon atoms separately. The distance to the other tube that is associated with each of the rings is determined as shown in Figure 1, i.e. it is given by the distance to the rst solvent layer added to the (shortest) distance from the rst solvent layer to the other tube. The CDM is clearly an approximation, however, it performs very well for coarse-grain models

40

and below we demonstrate its excellent performance for an atomistic model.

Simulation Molecular dynamics simulations were run using the GROMACS 5.1 simulation package. Initial box dimensions for each of the simulations with parallel tubes were:

y = 12.00nm, z = 24.59512nm,

z -axis

x = 12.00nm,

x, y = 0, z

plane

(Figure 1, top). Each of the tubes had

4, 000

with the two parallel tubes placed in the

and the tube axes oriented along the

4147

atoms. The number of solvent molecules varied between simulations (due to the increasing excluded volume of the tubes as they move apart) but was approximately total system size of circa

101, 000

18, 500,

giving a

atoms.

In the Corresponding Distance Method the two tubes are angled to one another (bottom panel of Figure 1). Here the axis of one of the tubes coincides with the one is rotated from this position around the the

x, y = 0, z

y -axis

by

10◦ .

z -axis while the other

Thus, the two tubes are still in

plane but cross in the centre of the box.

Complete periodicity is maintained for the horizontal tube, while the rotated tube has defects at the edges of the system, which are irrelevant for the force calculation. The initial box dimensions for the CDM simulation were:

x = 10.8480nm, y = 61.4716nm, z = 6.0000nm.

9



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The horizontal tube of the CDM simulation had atoms and there were

20, 612

10, 000 atoms,

solvent molecules, giving

123, 220

Page 10 of 35

the rotated tube had

10, 160

atoms in total.

Very similar simulation procedures were followed for both systems. Once the tubes have been placed in the box, the space is lled by overlaying repeat units of a pre-equilibrated solvent box.

All solvent molecules that overlap with or are located inside the tubes are

removed. After an initial energy minimisation step (EM), a short temperature equilibration step was performed (NVT), followed by a pressure coupled equilibration phase (NPT). During (NPT) equilibration the box dimensions were only allowed to deform in the dimensions that were not coupled to the tube lengths, i.e.

x

and

y

for the parallel simulations and only

y

for

the CDM simulation (Figure 1). The last step is the (NVT) production run to collect data for analysis. All simulations were conducted at atmospheric pressure (1 bar) and a temperature of 300 K. The remaining simulation parameters are collated in Table 1.

Model and Force Field

The forceeld used was GROMOS 53a6. Parameterisation of the solvent was done with the aid of the Automated Topology Builder

4951

and manually veried.

The tubes used in this study are (10,10) armchair CNTs with a diameter of The nanotubes were built with the aid of the Buildcstruct script

52

1.36nm.

and were given the same

non-bonded characteristics of a standard carbon atom in the GROMOS 53a6 force eld. The parameters for the solvent and nanotubes can be found in the Supporting Information. The CNTs are approximated as rigid rods, i.e. the CNT atoms are frozen" in place. In the simulations the tube atoms are treated normally, but their positions are never updated. In the solvent simulations no inter- or intra-tube C/C interactions are considered. Thus the direct tube/tube interaction contribution to the PMF is calculated separately using the

a Coupled

only to the solvent. which only the last 9ns is used for data collection. c Or until machine precision is reached.

b Of

10


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Table 1: Simulation Parameters Production Timestep (ps)

0.002

Ensemble

NVT

Integrator

velocity verlet

VdW and electrostatic cut-o (nm)

1.2

VdW scheme

Lennard Jones 12-6, potential shift

Electrostatic scheme

Reaction Field Zero, potential shift verlet

Thermostat

v-rescale

Thermostat coupling time/constant (ps)

0.1

VdW mixing rule

arithmetic mean

Integrator

Velocity verlet

Time (ns)

10

Machine precision

double

Neighbour list update frequency (steps)

20

Neighbour list cut-o (ns)

1.5

a

b

Pressure Coupled Equilibration Ensemble

NPT

Time (ns)

1

Barostat

berendsen

Barostat coupling constant ( ps) −1 Compressibility ( bar )

2.0

4·105

Temperature Coupled Equilibration Time (ps)

300

Ensemble

NVT

Energy Minimisation EM Type

conjugate gradient

nsteps

10000

c

Parameters for the energy minimisation and equilibration steps are the same as for the production step unless otherwise stated. Descriptions of the simulation parameters (integrator, non-bonded schemes etc.) can be found in the GROMACS manual.

11


48


same techniques with a single step MD simulation in a vacuum using the forceeld's base C-C potential for the inter-tube interactions. Like many other force elds, GROMOS 53a6 is a biomolecular forceeld in origin. It was parameterised to match free energies of solvation of amino acids (in water and cyclohexane) and to reproduce the properties (density and heat of vaporisation) of a range of small solvent molecules. These two criteria are in line with the goals of the work in this manuscript. The force eld was also shown to reproduce the tube-tube potential for (10,10) carbon nanotubes.

6

Thus we believe that the force eld is applicable for our purposes.

Results and Discussion Corresponding Distances Method

100

Force PMF

50

80 60

40

40

30

20

20

0

10

-20

F(d) / kT nm-2 (300K)

60

w(d) / kT nm-1 (300K)

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Page 12 of 35

-40

0 0

1

2

3

4

5

-60

d / nm

Figure 3: The solvent-mediated part of the potential of mean force associated mean force

F (d)

w(d)

(green) and the

(blue) as a function of the surface-to-surface distance

d

between

a pair of parallel (10,10) tubes obtained from individual simulations at various distances

d.

The potential of mean force (PMF) is a convenient tool to assess solvent quality as it quanties the work necessary to separate two CNTs immersed in a solvent. We compute the PMF from a tube-tube distance of

d = 4nm;

we shall call this the separated or individ-

ualised state, whereas in the bundled or aggregated state the tube/tube distance has a value near the minimum of the direct tube/tube interaction ( d

≈ 0.33nm).

In the applied model the solvent-mediated part of the potential of mean force (smPMF) and the direct tube/tube interaction are additive. While this additivity may not fully rep-

12


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resent reality, it provides excellent inside into the challenges of dissolving carbon nanotubes. The smPMF for a pair of (10,10) single-wall carbon nanotubes (SWCNTs) immersed in bromotrichloromethane

CBrCl3

is presented in Figure 3.

The simulations leading to Figure 3 are challenging: rstly, they must all be carried out at the exact same thermodynamic conditions, secondly, they are computationally demanding and also carry signicant administrative overheads for managing simulation les and directories, creating initial coordinates, validating completed simulations, post processing etc. and thirdly, it is not possible to

a priori know where data points are needed to obtain the correct

result. The latter point is made worse by the fact that a few missing data points can lead to an incorrect smPMF which may not be detectable by inspection (see Figure S1, Supporting Information). We have developed the Corresponding Distances Method (CDM)

40

that allows us to

extract the complete smPMF from a single simulation of crossed tubes.

By design this

guarantees that all distances correspond to the same bulk system and allows the smPMF to be computed at very high resolution. 120

40

CDM Parallel

100 80

30

60

25

40 20 0

(a)

-20 -40 -60

0

0.5

CDM Parallel

35

w(d) / kT nm-1

F(d) / kT nm-2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 15 10

(b)

5 0 1

1.5

2

2.5

3

3.5

-5

4

0

0.5

1

d / nm

w(d),

2

2.5

3

3.5

4

d / nm

Figure 4: The solvent-mediated force tential of mean force

1.5

F (d),

(a), and the solvent-mediated part of the po-

(b), as a function of tube/tube distance, demonstrating the

excellent performance of the Corresponding Distances Method. Solid lines represent results from the Corresponding Distances Method, while the data indicated by open symbols has been obtained from individual simulations of systems comprising pairs of parallel carbon nanotubes.

Although the method is an approximation, the results in Figure 4 demonstrate it's excellent performance. The smPMF curve is based on 400 individual data points compared to

13



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only 35 from parallel simulations. The individual data points are more clearly visible in the mean force (Figure 4(a)). The CDM requires less than a tenth of the computational resources needed to achieve the same by simulating individual systems of parallel tubes. The savings originate primarily from eliminating the need to equilibrate a large number of small systems, from reduction of the total bulk liquid that needs to be simulated and from reducing the total size of the systems at each distance because periodic boundary conditions don't apply. Perhaps even more important is the well over 20fold reduction in administrative overheads for the operator by setting up and analysing one simulation instead of 30 or more and the avoidance of errors related to do doing the latter. Thermodynamic consistency is guarantied in the CDM approach while it is extremely challenging to setup a large number of liquid systems such that they all have exactly the same (bulk) pressure and temperature. The Corresponding Distances Method represents a convenient method to routinely compute accurate potential of mean force curves at very high resolution.

Bromotrichloromethane,

CBrCl3

If carbon nanotubes (CNTs) behaved as normal solutes, solvents with similar solubility parameters should show similar performance. This is not the case. In solubility tests bromotrichloromethane turned out to be a non-solvent although it has the same solubility parameters as the best solvents in the study.

8

In the following we use the CDM approach to

determine the origin of this poor performance of In Figure 4(a) the

CBrCl3

CBrCl3 .

mediated forces between two (10,10) nanotubes are shown.

Three features stand out: at the smallest distances a large repulsive region can be observed, this is followed by some oscillations and then the force expectedly vanishes at large distances. Noise can be observed throughout the force curve which is due to averaging over only a few carbon atoms at the respective tube/tube distance.

However, because of the high

resolution of the force curve the noise is not inherited by the smPMF (Figure 4(b)).

14


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(a)

(c)

(b)

Figure 5: Local densities of carbon atoms at

d = 1.84nm, (a, b) and d = 1.34nm highlighting

solvent layering and the squeeze-out of a solvent layer from (b) to (c). Results form the CDM (b, c) and the simulation of parallel tubes are identical apart from better averaging for the latter.

The longer range oscillations between about

d = 0.8nm and 2.5nm are layering oscillations

which originate from the ordering of solvent molecules near the surface of an immersed object. This ordering of the

CBrCl3

molecules is clearly visible in the local densities of carbon atoms

shown in Figure 5. The gure also demonstrates that the structural information obtained from the CDM is identical to that calculated using parallel tubes. Between the tubes the solvent layers are under connement. At zero force a given number of layers are mechanically stable in the sense that if the tubes were allowed to move, the solvent-mediated forces would not move them.

As the tube/tube distance is reduced

from this point, the conned solvent gets compressed leading to a repulsive solvent-mediated force, while an increase of the distance would stretch the structure and result in an attractive force.

If the layered structure gets compressed too much, a layer is squeezed out

(compare Figure 5(b) and (c)). Layering in

CBrCl3

is not very intense beyond the rst layer

and only leads to comparatively small oscillations in the smPMF of only a few kT/nm. The rst solvent layer has a fairly high density suggesting comparatively strong adherence between the individualised state and the bundled state.

Consequently, this layer resists

connement and squeezes out with a far greater repulsive force than all other layers and the force acts over a larger range of distances. This leads to an extended region at small tube/tube separation where the smPMF is positive. In other words, the solvent pushes the tubes apart at all distances smaller than 0.88nm.

15



Solvent Contribution PMF CNT Contribution

40

w(d) / kT nm-1 (300K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 35

20

0

-20

-40 0

0.2

0.4

0.6

0.8

1

1.2

1.4

d / nm

Figure 6: The full potential of mean force (yellow) and its two additive contributions, the direct tube/tube interaction (green) and the smPMF (magenta), showing that, while the aggregated state has been weakened, it remains stable and that a barrier is formed between the aggregated and the dispersed states.

This repulsive part of the smPMF is decisive for the solvent and dispersant quality of a uid as it overlaps with the region of strong direct tube/tube cohesion. The full PMF and the two additive contributions, i.e.

the smPMF and the direct tube/tube interaction are

shown in Figure 6. It is immediately obvious that the solvent strongly reduces the stability of the bundled state from about -40kT/nm in vacuum to -12kT/nm in

CBrCl3

rated state), but it does not destabilise it completely. Thus

(compared to the sepa-

CBrCl3

is not a solvent in the

thermodynamic sense in which the tubes would separate spontaneously. A barrier appears at 0.58nm with a maximum of 11kT/nm. Such a barrier is characteristic for a dispersant as it separates the dispersed from the aggregated state. However, the barrier must also be overcome when separating the tubes from the bundled state. With 23kT/nm the barrier is twice as high in this direction. How signicant the barrier is depends on the exact debundling process. If the tubes slide laterally, this particular barrier is irrelevant. If they are driven apart perpendicularly as in the unzippering hypothesis

12

then this barrier

must be overcome during the activation of the process, while the rest proceeds barrier-less. As the tubes are very sti, it is likely that more than a few nm must be separated initially leading to absolute activation energies that are well beyond thermal activation. However, neither of these debundling processes will take place in

CBrCl3 ,

remains thermodynamically stable.

16


as the aggregated state

Page 17 of 35

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While the tubes are naturally parallel when they undergo a debundling process. They are more or less randomly oriented in the dispersed state. In general one expects the total barrier to decrease with decreasing contact area, making two tubes colliding at

90◦

the most

vulnerable case. As our tubes have a diameter of 1.36nm one estimates that the rebundling barrier in the

90◦

case will be somewhere in the 15kT range.

This is probably too low

to stabilise a dispersion, particularly considering that the barrier is 15kT only on

average.

Thermal uctuations of the adsorbed solvent layer will easily cause it to be lower temporarily.

Structure/Function Relationship

(b)

(a)

Figure 7: Tube carbon to solvent atom van der Waals interactions (a) and N(r), the number of solvent atoms as a function of distance from the surface of an individual nanotube (b). Part (b) indicates layering and intense orientational order of the solvent molecules in the rst layer.

We begin the analysis of the structure/function relationship by studying the solvent structure around an isolated tube. The number of solvent atoms versus distance to the tube surface curves, N(r), clearly visualise the aforementioned layering Figure 7(b).

While the

second and all following layers are unstructured, as is evident by the broad peaks in all three N(r) curves that are located at exactly the same position, molecules in the rst layer have a preferred orientation. For each carbon atom approximately one bromine atom and two chlorine atoms are located slightly closer to the tube than the carbon, while the fourth halogen atom points away from the tube. The atom pointing away is usually Cl not Br because the Br/C van der Waals interaction is stronger than the Cl/C interaction (Figure 7(a)). The distance between

17



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Page 18 of 35

(a)

Figure 8:

Snapshot (a) and local densities for all solvent atom types (b) visualising the

solvent structure around an individual carbon nanotube.

Shown is also a force map (b)

highlighting that most of the force between the tube and the solvent originates from the atoms closest to the tube. Stronger colours indicate higher density and force. In the force map repulsive forces are shown in red and attractive forces in blue.

Complete scales are

provided in Figures S2 and S3, Supporting Information.

this Cl-peak and the C-peak is 0.156nm which is is consistent with the expected C/Cl bond length of 0.176nm. Thus, the tetrahedral molecule adsorbs to the CNT with its triangular base (Figure 8(a)), which is consistent with the expectation that the molecules would tend to maximise the adsorption energy. to a much lesser extent.

We also observe the exact opposite orientation but

These particular orientations of the adsorbed molecules lead to

the double ring signature of the rst layer in the Br and Cl density maps and a single C ring in between (Figure 8). Note that the Cl atoms are located slightly closer to the tubes compared to the Br atoms because the Cl/C van der Waals interaction is slightly shorter ranged (Figure 7(a)). It is instructive to briey consider the interactions between the solvent and the nanotube. In Figure 8(b) the local solvent/nanotube force at the location of the solvent atoms is shown. A signicant part of the total force is repulsive, which is expected as the tube/solvent interface must sustain the external pressure. These repulsive interactions are shared between the Cl and Br atoms in the base of the adsorbed tetrahedron, while the atom pointing away from the tubes contributes only little. Note, that the interaction with the carbon atoms is relatively small, but contributes attractively and repulsively due to it's relatively long range

18


Page 19 of 35

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(Figure 7(a)). The particular orientation of the adsorbed molecules in the rst layer determines the range and strength of the forces leading to the repulsive region of the smPMF, which in turn determines the solvent and dispersant quality of the molecule.

Understanding this

structure/function relationship is essential for the rational design of CNT solvents and dispersants.

Figure 9: Snapshot of a thin slice of the CDM system at

d = 0.88nm.

Clearly visible is the

structuring of the conned solvent layer into two rows of molecules running along the tubes.

The repulsive region of the solvent-mediated (average) tube/tube force

d = 0.88nm,

F (d)

begins at

where the force is zero and the smPMF has a minimum (Figures 4 and 6). At

this particular distance a single layer of solvent molecules is sandwiched between the tubes as can be observed by following the ring structure in the local density (top row in Figure 10) and by inspection of the snapshot shown in Figure 9. We expect this layer to be largely unstrained, which is consistent with the observation that the molecules in the conned region retain the orientation they assume in the unconned part of the adsorbed layer (compare also to Figure 8(a)). However, in this conguration the atom at the top of the adsorbed tetrahedron can now interact with the other tube.

This is possible only at very specic positions, which

causes lateral structuring of the conned layer. The formation of two distinct rows of solvent molecules either side of the centre of the gap between the tubes is particularly obvious in the local densities of the carbon atoms (middle of top row in Figure 10). When the distance between the tubes is reduced, the conned layer is compressed and the two rows of molecules are pushed out of the gap (second row of Figure 10) which results

19



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Cl

Page 20 of 35

C

Br

d=0.88nm

d=0.74nm

d=0.54nm

d=0.41nm

d=0.14nm

0.5nm

Figure 10: Local densities of the solvent atoms at various distances

d

gure. Note that the Cl density has been scaled down by a factor of 3.

20


as indicated in the

Page 21 of 35

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in a repulsive restoring force.

Thus, the size of the repulsive region is determined by the

geometry of the solvent molecules.

y

x

1st adsorbed layer

z

Figure 11: Schematic introducing the wedge" model. The wedge is shown in purple. The arrows indicate the tube/wedge force (green) and the force with which the adsorbed layers push the wedge into the gap between the tubes (red). Only the top wedge is indicated.

To analyse the specic behaviour of the nanotube/ CBrCl3 system in the repulsive region of

F (d)

it is instructive to consider a simpler proxy (Figure 11).

At the point where the

adsorbed rst layers of the two tubes overlap and terminate, they push the two rows of molecules between the tubes like a wedge (magenta).

The responsible force (red arrows

in Figure 11) originates from the lateral adsorption pressure in the adsorbed layers. As the force contributions from the two layers are mirror symmetric with respect to the centre of the tube/tube contact, their

x-components

cancel, while their

y -components

Thus, the resulting force that pushes the wedge between the tubes only has a

add.

y -component.

At equilibrium this force must be balanced by the wedge/nanotube forces (green arrows in Figure 11). These forces also have Vitally, although the

x-components which cancel due to the same symmetry.

x-components

of the two wedge/nanotube forces cancel at the

wedge, they are counterbalanced by dierent tubes and therefore push them apart. is one contribution to

This

F (d).

In a very similar way the

x-components

of the forces pushing the wedge into the gap

between the tubes (red arrows in Figure 11) must also be counterbalanced by the tubes. These

x-components

cause the two layers to deform and/or shift slightly in the

−x

and

+x

direction for the left and right tube, respectively, which creates the required counterbalancing

21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

forces. These forces are the second contribution to

Page 22 of 35

F (d).

This means that much of the prole of the repulsive region of

F (d)

is governed by the

geometry of the tubes which determines the angle at which the adsorbed layers interact with the wedge (angle between red arrows in Figure 11). In addition to this, the geometry of the wedge, i.e. the exact geometry of the solvent molecules in the two adsorbed rows, makes a contribution via the orientation of the nanotube/wedge force (angle between green arrows in Figure 11). In general we expect that when the distance between the tubes is reduced the repulsive force decreases as the wedge moves out of the gap and the geometry becomes less favourable. Using this model we nd that at

d ≈ 0.74nm

the solvent-mediated force

F (d)

in Fig-

ure 4(a) is high because the two rows of molecules that from the wedge" are located deep in the gap between the tubs resulting in a large ers on the wedge and a large leads to the large value of

Figure 12:

x-component

y -component of the force from the adsorbed lay-

x-component of the wedge/nanotube force, which consequently

F (d ≈ 0.74nm).

of the local forces at

d = 0.74nm

showing the location of atoms

that interact with the tubes repulsively (red) or attractively (blue). The tubes are shown as a solid black line. The area marked by the dashed rectangle is enlarged in Figure 13.

At

d ≈ 0.74nm

d ≈ 0.88nm

the wedge molecules have increased their order slightly compared to

and are now oriented in such a way that the bromine atom is located at the

tip of the gap formed by the two tubes, while the three chlorine atoms face outward. Two of the Cl atoms are adsorbed on one tube with the third on the other tube (second row of Figure 10).

In this conformation the wedge formed by the molecules is too blunt for the

22


Page 23 of 35

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d=0.88nm

d=0.74nm

d=0.54nm

d=0.41nm

d=0.14nm

Figure 13: Same as Figure 12 but showing an enlarged area at various tube/tube distances corresponding to Figure 10.

C

Cl

Br

Figure 14: Same as gure 12 but showing an enlarged area for

d = 0.41nm

and split into

contributions from the three solvent atom types as indicated in the gure, showing that the the repulsive forces are sustained by Cl and Br, while C never comes close enough to the tubes to interact repulsively.

23



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Page 24 of 35

gap between the tubes. Consequently, the chlorine atoms sustain the repulsive force with the tubes (Figures 12 and 13), while the bromine atom dangles in the gap as can be seen by the rather broad Br spot in the local density in combination with quite sharp Cl and C spots (second row of Figure 10). As

d

a

is reduced further, the gap between the tubes becomes shallower.

This means

that eventually the Br atom will interact with both tubes and sustain most of the repulsive tube/wedge force. This is a more favourable geometry for a large value of

F (d),

as the Br

atom lies deeper in the gap than the Cl atoms ( Figures 10 and 14). However, simultaneously the wedge is being pushed out quickly with reducing

d,

which appears to overcompensate

the previous eect. Interestingly there is a second rather broad maximum around

d = 0.41nm.

This can only

be caused by the wedge molecules as the tubes don't have any non-monotonous geometrical features. At

d ≈ 0.41nm

the wedge has moved still further out of the gap as is evident from the

location of the carbon atoms (row 4 of Figure 10). However, the molecule has rotated such that two halogen atoms (including bromine) point into the gap and interact with both tubes simultaneously while the other two chlorine atoms point outward and each interact with a dierent tube. This amounts to 6 tube/halogen interactions as compared to 5 found in the conguration at

d ≈ 0.74nm.

It is likely that this increase drives the reorientation of

the molecules that form the wedge". (Note that the tube/atom interaction energy can be negative although the associated force is repulsive). In this orientation of the molecules the wedge is slightly more blunt than before, but also the gap between the tubes has become more blunt until at

d ≈ 0.41nm

the wedge is sharper

than the gap and therefore interacts repulsively with the tubes at is tip (Br, Cl) as opposed

a Although the C atom of CBrCl is well shielded by the halogen atoms, the force-eld assigns a repulsive 3 interaction with the tubes (see Figure 7(a) and Figure 12). This rather long range repulsion may be an artefact of the force-eld caused by using an LJ(12,6) for van der Waals type interactions for all atoms. However, the contributions of the C atom of CBrCl3 to the tube/tube force is consistent with that of the halogen atoms and does not aect the presented interpretation of the forces.

24


Page 25 of 35

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to the repulsive interactions at its heel at larger

d

(sequence in Figure 14). This moves the

interaction centre slightly further into the gap and to a more favourable geometry, which appears to be responsible for the slight increase in the solvent-mediated tube/tube force at

d ≈ 0.41nm. Decreasing the distance between the tubes pushes the wedge molecules further out of the gap between the tubes, which makes the geometry less favourable for strong solvent-mediated tube/tube repulsion. The atoms at the tip of the wedge continue to sustain the repulsive force while only one of the other two halogen atoms can continue to interact with a tube. From the discussion above it is clear that strong adhesion between the solvent molecules and the nanotubes is essential for a strong repulsive region in

F (d).

However this strong

adsorption should not lead to a deep rst minimum in the smPMF. Indeed,

CBrCl3

only has

a very shallow rst minimum (Figure 6). The reason for this is likely the strong asymmetry of its interaction with the tubes; when one layer is stable between the tubes four atoms interact with one tube while only one interacts with the other.

Conclusions In this manuscript we demonstrate the usefulness of the Corresponding Distances Method to investigate solvents and dispersants for low dimensional nanomaterials.

In particular

we show that it provides correct results for atomistic systems much faster than alternative methods and that it can be used with standard simulation packages. The Corresponding Distances Method has a range of signicant advantages:

•

It provides very high resolution potential of mean force curves.

•

It is more than 10 times faster than alternative methods because much less bulk needs to be simulated and equilibrated in total.

•

It is more than 20 times more ecient in terms of administration as only one simulation is required.

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

•

Thermodynamic consistency (same

P

and

T)

Page 26 of 35

between all distances is guaranteed as

they are simulated in the same system.

•

The method is also more suitable for High Performance Computing as only one big simulation with good parallel scaling is required.

We have applied the Corresponding Distances Method to investigate the solvent and dispersant quality of by Bergin et al.

8

CBrCl3 .

According to the solubility parameters of SWCNTs as predicted

this molecule should be a good solvent, but it was found experimentally to

be a non-solvent. Our results show that while

CBrCl3

reduces the stability of the bundled

state from -40kT/nm to -12kT/nm it can not destabilise it completely. It is therefore not a solvent in the thermodynamic sense.

CBrCl3

creates a barrier of 11kT/nm between the individualised and the bundled states

in the re-aggregation direction. However, this is for parallel tubes and the true barrier for the most vulnerable, i.e. the

90◦ ,

conguration is likely to lie below 15kT on average. This

is unlikely to be sucient to stabilise a CNT dispersion and might be reduced even further temporarily by thermal uctuations of the adsorbed solvent layers. Our nding that

CBrCl3

is neither a good solvent nor a good dispersant for SWCNTs is

consistent with experimental results.

8

From this work we can learn several lessons for solvent design. mediated PMF must be

>40kT/nm

Firstly, the solvent-

at the tube/tube equilibrium distance.

this energy must be raised by solvent adsorption from the liquid.

Ultimately

At the same time this

required strong adsorption must not lead to a strong free energy minimum at one solvent layer sandwiched between the tubes. It is this dependance of the behaviour of the solvent molecules on the tube structure that the solubility parameters can not capture. Secondly, the barrier between the bundled and the dispersed state should be removed for good solvents, which requires to shorten the range of the solvent-mediated repulsion to match the range of the direct tube/tube attraction. This in turn requires the molecule or fragment that is inserted between the tubes to be small and/or at.

26


Page 27 of 35

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Finally, it should be mentioned that nanotube properties such as diameter, chirality and metallic/semiconducting can be recognized by larger molecules wrapping around the tubes, as shown by numerous nanotube sorting studies.

5356

Thus, particularly when polymeric

additives are used, the specic tube properties may impact on the quality of solvents and dispersants. The notorious demand of these systems for large and long simulations makes the CDM approach particularly useful to study them.

Acknowledgements We gratefully acknowledge the provision of computational resources by the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), the EPSRC UK National Service for Computational Chemistry Software (NSCCS) at Imperial College London and by PRACE. The work was funded by EPSRC through the DTP programme.

Supporting Information Force eld parameters, pitfalls of using low resolution PMF curves, scale bars for density and force maps.

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Assessing the Quality of Solvents and Dispersants for Low

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