A Comparison of the Adsorption Transitions of Methane and Krypton

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A Comparison of the Adsorption Transitions of Methane and Krypton on Graphite at Sub-Monolayer Coverage Han Zhang, Shiliang Johnathan Tan, Lumeng Liu, Duong Dang Do, and David Nicholson J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00535 • Publication Date (Web): 27 Mar 2018 Downloaded from http://pubs.acs.org on March 30, 2018

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The Journal of Physical Chemistry

A Comparison of the Adsorption Transitions of Methane and Krypton on Graphite at Sub-Monolayer Coverage by Han Zhang1,2, Shiliang (Johnathan) Tan2, Lumeng Liu2,3, D. D. Do2* and D. Nicholson2 1

Key Laboratory of Coalbed Methane Resources and Reservoir Formation Process of Ministry of Education, China University of Mining and Technology, Xuzhou, Jiangsu, 221006, China 2

School of Chemical Engineering University of Queensland, St. Lucia, QLD 4072, Australia 3

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China

Abstract Simulation of gas adsorption on graphite is commonly carried out using a model that assumes that graphite has an energetically homogeneous surface, constant interlayer spacing, and isotropically polarizable carbon atoms. This simple model fails to describe experimental isotherms and isosteric heats for many gas/graphite pairs. In this paper, we investigate the adsorption of krypton and methane on graphite, using a recently developed molecular model for graphite, that has been shown to improve the description of experimental isotherms and isosteric heats for nitrogen and argon1-3. Although the collision diameters of krypton and methane are almost the same, their isotherms and heats are significantly different. With the aid of detailed microscopic analysis, we establish the mechanism underlying the transitions in adsorbate loading as well as the origin of the spike in the experimental isosteric heat versus loading for methane and krypton. The 1st adsorbate layer of both adsorbates exhibits the transition from a 2D-fluid to a commensurate (C) state; but only krypton shows a subsequent transition to incommensurate (IC) packing before the onset of the 2nd layer. However, the first adsorbate layer of the methane isotherm does also undergo this transition, but only after the second layer has been formed at higher temperatures (T>87K). This is explained by the difference between the inter-molecular spacing in the Cpacked lattice and in the IC-lattice. The difference is smaller for methane than for krypton, making it more difficult for methane to achieve IC-packing and it is also reflected in the isosteric heat versus loading (heat curve). The heat curves for both adsorbates exhibit an increase in the isosteric heat at sub-monolayer coverage and reach a maximum at the coverage corresponding to C-packing at temperatures far below the triple point. As the temperature is increased, the isosteric heat also increases to a maximum at a loading less than the Cpacking, followed by a cusp due to the onset of the second layer, prior to the spike at the C-packing. The difference between the two adsorbates is shown by the appearance of an additional spike for krypton when the C-IC transition occurs. These heat spikes for Kr due to C-packing and IC-packing, shift to higher loading and have smaller magnitudes at higher temperatures because of the contributions from higher layers.

* Author to whom all correspondence should be addressed. E-mail: [email protected]

1 Environment ACS Paragon Plus

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1. Introduction Graphitized thermal carbon black (GTCB) is an ideal material for the study of the intrinsic interaction between an adsorbate and the graphene basal plane. Although it is atomistically corrugated

4-6

, they are commonly modelled as an energetically homogeneous surface in

simulation, and the solid-fluid interaction energy that is calculated using the molecular parameters suggested by Crowell and Steele (CS) 7,8 for carbon atoms in a graphene layer (σ = 0.34nm and ε/kB = 28K) and a graphene interlayer spacing of 0.3354nm. This model is adopted in many molecular simulation studies because of its simplicity when the fine details of the first adsorbate layer are not required. However, the availability of high resolution experimental data has prompted us to re-evaluate this molecular model.

Low Energy

Electron Diffraction (LEED) and Surface-Extended X-ray Absorption Fine Structure (SEXAFS)

9,10

results have shown that the distance of the first adsorbate layer from the

graphite surface is smaller than that calculated with the CS molecular parameters. This suggests that the carbon atom in the first graphene layer has a smaller collision diameter than 0.34nm, and consequently the spacing between the first and second layers is smaller than 0.3354nm, which is confirmed by TEM studies of highly graphitized thermal carbon black 11. The interlayer spacing between layers beneath the 1st layer remains at 0.3354nm, as determined by extensive measurements with XRD

12,13

. The recent paper by Ustinov

1

proposed a smaller collision diameter (0.26nm) for the carbon atom for all layers and showed that this choice gives a better account of the adsorption isotherm of nitrogen, particularly of the sub-step in the isotherm and the spike in the plot of the isosteric heat versus loading at 77K. Recently, we have further modified this model 2,3. Details are given in Section 2.2, and our modified model is hereafter referred to as the corrugation and anisotropy (CA) model.

To test the modified graphite model, we examined the fine details of the 2D transition in the 1st adsorbate layer at temperatures around the 3D-triple point and the 3D-boiling point. We selected krypton and methane as adsorbates, because the experimental isotherms for krypton show three transitions in the sub-monolayer coverage region: gas to commensurate state (GC) and commensurate to incommensurate (C-IC) states, while the methane isotherms exhibit only the gas – fluid – C transitions. In the commensurate (C) state on a graphitic surface molecules are localized at the centres of the carbon hexagons, but in an incommensurate (IC) state the adsorbed molecules pack as a hexagonal 2D-solid.


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Krypton adsorption on graphite has a rich phase diagram and is one of the most widely studied systems in the literature for the evolution of the isotherm 19-23

structure

14-18

and the adsorbate

with temperature. Krypton is also used as a probe molecule for testing energetic

homogeneity of a graphite surface 24. Putnam and Fort

25

measured adsorption isotherms at

various temperatures and calculated the isosteric heat versus loading (from the ClausiusClapeyron equation) for krypton adsorbed on graphitised carbon black and reported spikes coinciding with the phase transitions.

Bockel et al. studied the gas-liquid-solid transitions in the isotherms of methane/graphite system 26, while Inaba et al. used a calorimetric method to study the evolution of the isosteric heat with temperatures in the range between 65K and 110K 27. The evolution of the isosteric heat for methane with temperature is similar to that observed for argon 2,28,29.

The phase diagram of coverage versus temperature for methane has been measured on a single graphite crystal using LEED, on grafoil by neutron diffraction and by calorimetry 30-34 as well as simulation

35,36

. The experimental works show the existence of 3 solid phases

occurring below the monolayer coverage: commensurate, expanded and compressed 2D structures (the latter is referred to as IC-solid in this paper). In addition, an expanded (low density) IC-solid above the monolayer coverage has been found to exist even at very low temperatures. There is little consensus about the precise location of the phase boundaries, possibly because the density in the phase diagrams is the overall density, so that monolayer coverage in fact refers to a statistical monolayer and includes molecules in higher layers. These different phases have been the subject of many simulation studies: Kim and Steele

35

studied the effects of corrugation of the graphite potential energy parallel to the surface on the adsorption of methane. They found it necessary to increase the inter-site energy barriers empirically by a factor of 1.5 to observe the C-packing. Jiang et al.

37

, also increased the

corrugation in their grand canonical Monte Carlo (GCMC) simulation to show the C-solid of methane. To study the IC solid they resorted to a canonical Monte Carlo scheme 37, in which the pressure of IC solid formation is not known. Most studies in the past have neglected the anisotropy in polarizability of carbon in the graphite surface, and led to an empirical factor to increase the corrugation energy barrier. By the same token, it comes as no surprise that the homogeneous solid model does not capture the C-packing 28,38 since there are no distinct sites on the surface.



Page 4 of 27

A molecular model for graphite with correct molecular parameters should be able to discriminate between the subtle differences in behaviour for adsorbed krypton and methane even though the collision diameters of these adsorbates are very close to each other (0.363nm 1/6 and 0.373nm for krypton and methane, respectively) and their IC lattice spacings 2 σ ff are

close to the commensurate

(

)

3 × 3 R 30° lattice spacing.

In this paper, we use grand canonical Monte Carlo (GCMC) to simulate krypton and methane adsorption on graphite, with a modified molecular model for graphite, that accounts for the corrugation in the adsorption potential parallel to the surface, the anisotropy in the polarizability of carbon, and which incorporates the revised spacing between the graphene layers: 0.2987nm between the first and second layers, 0.3354nm for the remaining layers2,3, which agree with the spacings observed experimentally9-13.

An important modification

introduced in this model is the variation in corrugation with loading in the sub-monolayer coverage region. We tested this modification by running simulations of adsorbed nitrogen and argon, which are not as sensitive to this variation of the corrugation with loading as krypton and methane. Details are provided in Section 3.4.

2.

Theory

2.1 Fluid-Fluid Potentials Krypton and methane are both modelled as a single dispersive/repulsive sites, with intermolecular energy given by the Lennard-Jones-12-6 (LJ-12-6) equation and parameters optimised for isotropic liquid simulaton. Argon and nitrogen are revisited as well to test the validity of graphite model proposed. The molecular parameters are listed in Table 1.

Table 1 Molecular parameters of adsorbates Molecule LJ-Atom σ (nm) ε/kB (K) Krypton39 Kr 0.363 162.6 Methane40 CH4 0.373 148.0 Argon39 Ar 0.3395 116.79 Nitrogen41 N 0.331 36.0 *positive charge is located at the mass centre of molecule

q -0.482/+0.964*


Bond Length (nm) 0.11

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2.2 Graphite models and their Solid-Fluid potentials The graphene surface is a tessellation of hexagons of discrete carbon atoms, with a carboncarbon (C-C) bond length dC −C = 0.142nm . To enforce the periodic boundary condition when the first adsorbate layer is commensurate (one adsorbate molecule for every three hexagons) the linear dimensions of a graphene layer must satisfy:

Lx = N x × aGr

Ly = Ny × 3aGr

(1)

where aGr = 3dC −C = 0.246nm is the hexagon centre to centre spacing of the graphite lattice, and N x , N y are multiples of 3. For krypton, we used N x = N y = 30 , and for methane

N x = 45 and N y = 12 ; these values were chosen to give both the C-packing, and a perfect IC1/6 packing when molecules are close packed with a separation distance of 2 σ ff ,

corresponding to the pair separation at the LJ minimum. With these dimensions, 300 and 360 molecules are required to form a commensurate monolayer of krypton and methane, respectively.

We considered 3 models for graphite: (1) a homogeneous solid model, (2) a model with potential corrugation and anisotropy (CA) and (3) a modified CA model (mCA).

2.2.1. Homogeneous model The parameters defining the homogeneous solid model are: (1) the molecular parameters for a carbon atom in a graphene layer are σ = 0.34nm and ε / k B = 28 K , (2) the interlayer spacing is ∆ = 0.3354nm and (3) the surface density of carbon atom is ρ =38.2 nm-2 (Figure 1a). The solid-fluid (SF) potential energy is calculated with the 10-4-3 equation:

ϕsf = 2π ( ρσ

2 sf

)

 2  σ sf 10  σ sf 4 1  σ sf4  ε sf   − −    3  5  z   z  3 ∆( z + 0.61∆) 

(2)

where εsf and σsf are the cross molecular parameters between an LJ site and a carbon atom in a graphene layer. 2.2.2. Corrugated and Anisotropy (CA) model 2,3 The CA model is a modified version of the model suggested by Ustinov 1. It accounts for the corrugation of potential energy parallel to the surface, the anisotropy of the carbon



polarizability in the uppermost graphene layer and the variation in the interlayer spacing. It is defined by the following specification:

1. the molecular parameters of a carbon atom in the outermost graphene layer are

σ ss = 0.28nm and ε ss / kB = 35K . The collision diameter was derived from LEED measurements of the distance between the first adsorbate layer and the graphite surface for adsorbed noble gases, and the well depth of the interaction energy was derived by matching the Henry constant against the experimental data 42. 2. The interlayer spacing between the first and second layers is 0.2987nm, derived from Monte Carlo simulation and confirmed with TEM measurements 11. 3. For the second and lower layers, we used the CS molecular parameters, i.e.

σ ss = 0.34nm and ε ss / kB = 28K . The interlayer spacing of graphene layers in the bulk graphite is 0.3354nm 12,13. 4. The polarizability of a carbon atom in a graphene layer normal to the surface is greater than that parallel to the surface, modelled with the equations derived by Carlos and Cole 43,44.

Figure 1 Illustration of (a) the homogeneous 10-4-3 model with the Crowell and Steele’s molecular parameters and (b) the corrugated anisotropy (CA) model accounting for the difference of the uppermost layer, compared to the bulk graphite.


Page 6 of 27

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Instead of summing the pairwise interaction energies between an LJ site and all the carbon atoms in a graphene layer (which is very computer-intensive), to give the the SF potential energy, Steele summed the energies in each graphene layer and exploited the periodicity of the adsorbent crystal to express the potential variation in the surface plane as a Fourier expansion 8:

ϕsf = 2π ( ρσ

2 sf

)

 2  σ sf 10  σ sf 4  ε sf    −   + ϕcorr ( r )  5  z   z  

(3)

where the first term is the 10-4 Crowell equation, describing the spatially averaged potential, and the second term accounts for the corrugation, and is written as:

ϕcorr ( r ) = ϕg ( z ) f ( x, y )

(4)

The function ϕ g ( z ) accounts for the decay of ϕcorr ( r ) normal to the surface: 2 2  16π5   σ sf 5  σ sf 5  σ σ 2  sf   sf  ϕ g ( z ) = 2π ( σ ρ ) ε sf       K 5 ( u ) − 4π   K 2 ( u )  (5)   30   d 0   z    d0   z  2 sf

where u = 4πz / d 0 and d0 = aGr 3 = 0.426nm , which is the lattice spacing of the commensurate

(

)

3 × 3 R30° structure. The functions K5 and K2 are Bessel functions of the

second kind. This equation was used by Kim and Steele

35

in their computer simulation of

methane adsorption on graphite. The second function f ( x , y ) in eq. (4) accounts for the 2D-corrugation of the periodic assembly of carbon hexagons

20

(the Cartesian coordinates and the positions of the carbon

atoms are shown in Figure 2a):

{

}

f ( x, y ) = −2 cos ( 2πb1 ) + cos ( 2πb2 ) + cos  2π ( b1 + b2 ) 

(6a)

where b1 and b2 are function of x and y:

b1 =

1 2y ; aGr 3

b2 =

y  1  x−   aGr  3


(6b)


Page 8 of 27

Figure 2 (a) A contour map of the f(x,y) distribution and (b) the area of each f(x,y) at intervals of 0.1. The carbon hexagon is shown by the dashed lines.

The function f ( x , y ) is a measure of the (x,y)-energy contour, and varies between -6 (when an adsorbate atom is at the centre of the hexagon) to +3 (when it is above a carbon atom). An ensemble average < f > can be calculated as follows:

< f >=

1 M

 1  ∑ K =1  N K M

NK

∑ f ( x, y ) i =1

i,K

  

(7)

in which M is the total number of configurations, NK is the number of molecules in the first layer, and f ( x , y ) i , K is the value of f ( x, y) for molecule i in the Kth configuration.

Eqs. (4) and (5) developed by Steele do not account for the difference between the carbon polarizability parallel to the graphite surface and that normal to the surface. Consequently, the surface energy is not sufficiently corrugated for the first adsorbate layer of methane to form a stable C-packing; for this reason, Kim and Steele 35 introduced an empirical constant λ in eq. (4) to increase the corrugation:

ϕcorr ( r ) = λ ϕg ( z ) f ( x, y )

(8)

and found that a choice of λ = 1.5 gave a sufficiently strong corrugation for methane to form a commensurate layer.


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Bonino et al.

45

, in a study of graphite structure, were the first to consider the effects of

anisotropy of polarizability on the interaction potential energy, but found that this made no substantial difference compared to calculations made with the Lennard-Jones potential. Subsequently, in a study of He scattering from a graphite surface, Carlos and Cole [36, 37] found that the modification in potential energy corrugation made a significant contribution in matching calculation with experiment, and replaced eq. (5) by:  16 π 5   σ sf  5  σ sf  5 ϕ ( z ) = 2 π ( σ ρ ) ε sf       K 5 ( u ) + γ R g R ( z )   30   d 0   z  2 2  σ sf   σ sf   −4π 2      K 2 ( u ) + γ A g A ( z )    d0   z   ' g

2 sf

(9a)

where u = 4πz / d 0 and d0 = aGr 3 = 0.426nm . The functions γ R g R ( z ) and γ A g A ( z ) account for the effect of the anisotropy in polarizability on the attraction and repulsion, with g R ( z ) and g A ( z ) given by: g R ( z ) = K 5 (u ) −

u K 6 (u ) 10

(9b)

g A ( z ) = K 2 (u ) −

u K 3 (u ) 4

(9c)

The attraction parameter γA was given as 0.4, based on the dielectric data 44. The repulsion parameter γR has no firm theoretical basis. It was given values of - 0.54 for helium 1.05 for nitrogen

47

46

and -

to show the best fit to scattering and adsorption data, respectively. The

introduction of this modification makes the potential parallel to the surface plane more corrugated as illustrated in Figure 3, where we compare the solid-fluid potential energy of Kr, for the anisotropic (CA) model with γ R = −0.54 to that of the Kim and Steel model with

λ=1.5 (in eq. 7). It is clear that the CA model, which has a physical basis, gives the same degree of corrugation as the Kim-Steele model with an empirical constant. The CA model can describe all the features observed in the experimental isotherms and isosteric heats for nitrogen and argon 2,3.

The 3-body (and higher order) interactions among neigbouring molecules and between adsorbate and C atoms in the graphite would be expected to reduce the interactions between adsorbate molecules. Here we propose a modified corrugation and anisotropy (mCA) model, which we describe in the next section.



0

(a) S A SP

-2

(b)

-2

Solid-Fluid Energy (-)

0

Solid-Fluid Energy (-)

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

-7.0

-4 -7.5

-8.0

-6 -8.5

-7.0

-4 -7.5

-8.0

-6

-8.5

∆U=0.6

-8

-9.5

-10.0 0.28

0.30

0.32

0.34

0.36

-9.5

-10.0 0.28

0.38

-10

∆U=0.61

-9.0

-9.0

-8

Page 10 of 27

0.30

0.32

0.34

0.36

0.38

-10 0.2

0.4

0.6

0.8

1.0

0.2

0.4

Z-Distance (nm)

0.6

0.8

1.0

Z-Distance (nm)

Figure 3 The Kr graphite potential energy profile using the CA model with (a) the new parameter (0.28nm/35K) and (b) the Steele parameter (0.34nm/28K) and corrugation enhanced by 1.5 f(x, y), Calculated at points directly on top of the C-hexagon centre (S), a carbon (A) and between two carbon (SP).

2.2.3 Modified Corrugation and Anisotropy (mCA) Model In the mCA model, we propose an empirical modification to account for the presence of the (repulsive) many-body interactions among a number Nn, of neighbouring molecules, by introducing a factor λ in eq. (4). We assume this factor to be a linear function of the number of the nearest neighbours in the first adsorbate layer (Figure 4):

λ = 1− α Nn

(10)

Figure 4 Modification of the corrugation energy barriers with respect of number of neighbours. The barrier height is decreased as the number of neighbours is increased.


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When there are no neighbouring molecules, λ=1, and an adsorbed molecule feels the full extent of the intersite barriers in the graphene layer. When there are 6 nearest neighbours the barriers are reduced by a factor λmin, with α given by:

α = (1 − λmin ) / 6

(11)

Because the Bessel functions in eq. (5) decay rapidly, the potential corrugation comes mainly from the uppermost layer of the graphite. The minor perturbation to the potential energy from the underlying graphene layers was therefore ignored and the potential energy from these layers was modelled with the 10-4-3 equation.

2.3 Simulation method We used two molecular simulation schemes: Monte Carlo with a Metropolis importance sampling algorithm

48

and kinetic Monte Carlo (kMC) with entropic sampling

29,49-51

; both

were run in the grand canonical ensemble. The simulation results from these two schemes were identical, confirming the validity of the simulation techniques. From the kMC scheme we can make accurate determinations of the chemical potential in any region to check whether a simulation has reached equilibrium 52,53. To generate an isotherm, the simulation was started from an empty box, and 1×108 configurations were used for both the equilibration and sampling stages. configuration was then used as the initial configuration for the next pressure point.

3.

Results and Discussion

3.1 Isotherms 3.1.1 Homogeneous model


The final


12

12

b)

2

Surface excess (µmol/m )

a)

10

10

8

8

6

6

4

4

Homogeneous Model CA Model mCA Model (λm in=0.1)

2

Homogeneous Model CA Model mCA Model (λmin=0.01) Experimental Data

2

Experiment

0

0

-2

-2

-4

-4

0.01

0.1

1

10

100

1000

0.1

1

Pressure (Pa)

10

100

1000

Pressure (Pa)

Figure 5a and b present the simulated isotherms for krypton and methane adsorption at 77K on a homogenous graphite model in the sub-monolayer coverage region. These isotherms serve as a reference for comparison with those obtained with the CA and mCA models. 12

12

b)

2


a)

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 12 of 27

10

10

8

8

6

6

4

4

Homogeneous Model CA Model mCA Model (λm in=0.1) Experiment

2

2

0

0

-2

-2

-4

-4

0.01

0.1

1

10

100


1000

0.1

1

Pressure (Pa)

10

100

1000

Pressure (Pa)

Figure 5 Isotherms (solid lines) and the corresponding factor (dashed lines) of Kr (a) and CH4 (b) at 77K fordifferent models for graphite. The two horizontal short dashed lines show the theoretical commensurate (C) and incommensurate (IC) densities. Experimental data for krypton are taken from 14 and methane from 26. Note that the experimental data for methane take commensurate density as full coverage of 1st layer.

The isotherms for both adsorbates show a number of transitions from dilute fluid to dense fluid, then solid in the sub-monolayer coverage region before the onset of the second layer. Although the transitions in these simulated isotherms are qualitatively in agreement with the experimental data for krypton

14

and methane

26

, the densities at these transitions do not

match the experimental results, which clearly show the commensurate state at the theoretical density of 10.56 µmol/m2, for both adsorbates.

3.1.2 Corrugation and Anisotropy (CA) Model


Page 13 of 27

12

12

b)

2


a)

10

10

8

8

6

6

4

4

Homogeneous Model CA Model mCA Model (λm in=0.1)

2


2

Experiment

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


0

0

-2

-2

-4

-4

0.01

0.1

1

10

100

1000

0.1

1

Pressure (Pa)

10

100

1000

Pressure (Pa)

Figure 5 includes the simulated isotherms for krypton and methane at 77K in the submonolayer coverage for the CA model. To our surprise, although the isotherms show the correct commensurate density of 10.56µmol/m2 for both adsorbates, this model fails to exhibit the observed liquid-like state of the adsorbate; instead it shows a first order transition from the gas-like state to C-packing. However, the C-packing remains stable over a very wide range of pressure, and even after the 2nd layer has been formed, in contradiction to the experimental data. The stability of the C-state is reflected in the plot of the ensemble average of commensurability factor < f > (eqs. 6-7) of the first adsorbate layer in Figure 5Error! Reference source not found..

Before discussing the mCA model, we present a calculation to show why molecules are trapped in the C-packing for the CA model. We assumed that along the z-direction normal to the surface, molecules reside at the minimum in the SF potential (Figure 6a). Let NC and NIC be the numbers of molecules required to form C-packing and IC-packing, respectively. The theoretical densities of the IC-state depend on the collision diameter, and by assuming a twodimensional hexagonal packing of adsorbate with a separation distance of 21/6 σ ff the area occupied by one molecule is ( 21/6 σ ff

)

2

3 / 2 , from which we calculated the theoretical IC

densities for krypton and methane as 11.54µmol/m2 and 10.99µmol/m2, respectively. Thus, the transition from the C-packing to the IC-packing depends on the difference between their respective lattice spacings, 3dC −C and 21/6 σ ff . This difference is 0.0188nm and 0.0073nm for krypton and methane, respectively.

The transition from the C-packing to IC-packing involves two steps:



Page 14 of 27

Step 1: Molecules move from the centres of the hexagons of carbon atoms to get close 1/6 to their neighbours at the separation distance of 2 σ ff as shown in Figure 6e. This

results

in

an

increase

in

the

SF

interaction

energy

as

(1) SF ∆U SF = 0.5 × N C × (ϕ IC − ϕ CSF ) > 0 and a decrease in the FF interaction energy by (1) FF ∆U FF = 0.5 × N C × (ϕ IC − ϕ CFF ) < 0 .

Step 2: Additional molecules then adsorb into the first adsorbate to give an IC close packed

layer.

This

decreases

the

SF

and

the

FF

energies

by

(2) SF (2) ∆U SF = 0.5 × ( N IC − NC ) × ϕIC < 0 and ∆U FF = 0.5 × ( N IC − NC ) × ϕICFF < 0 .

The transition from the C-packing to the IC-packing can only occur when the change in energy for Step 1, which is proportional to the number of molecules added to give C-packing, is negative. The molecular SF energy ϕ CSF for C-packing is calculated by eqs. (3), and that at the IC-packing, ϕ ICSF , can be approximated by the SF energy of a homogeneous surface which is given by eq. (2).

Figure 6 Schematic illustration of adsorption on a graphite surface: (a) two adsorbate molecules at the Cpacking (on top of the centres of the hexagons of carbon), (b) the pairwise fluid-fluid potential energy, showing the IC lattice spacing 2 σ ff smaller than the C lattice spacing, 3dC-C; (c) the minimum SF potential energy 1/6

along the direction from the centre of the hexagon to a carbon vertex; (d) the SF potential energy at the centre of the hexagon with respect to z, corrugation and anisotropy surface (black) and homogeneous surface (blue); (e) the two steps to transition from C to IC packings


Page 15 of 27 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


Table 2 Molecular potential energy in a simplified 2D model. Energies are reduced by k B ε ff of the adsorbate.

Kr CH4

ϕCSF

ϕ ICSF

∆ ϕ SF

ϕ CFF

ϕ ICFF

∆ϕ FF

∆ϕ (1)

-9.04 -9.76

-8.50 -9.24

0.54 0.52

-6.22 -6.61

-6.74 -6.74

-0.52 -0.13

0.02 0.39

Table 2 shows the molecular SF and FF energies for the C and IC configurations. As an example, if krypton is moved from C- to IC-packing, the increase in the molecular SF energy is ∆ ϕ SF = ϕ ICSF − ϕ CSF = 0.54 , while the decrease in the molecular FF energy is ∆ ϕ FF = ϕ ICFF − ϕ CFF = − 0.52 , giving a net positive energy change, which is the barrier to

accomplish Step 1. For methane, the situation is worse because the decrease in the FF energy is even smaller, giving a corrugation energy barrier ( ∆ϕ (1) = 0.39 ) which prevents methane from forming a layer with IC-packing.

3.1.3 Modified Corrugation and Anisotropy (mCA) Model The simulated isotherms for krypton adsorption on the homogeneous graphite in Figure 5, show transitions to two solid states in the first adsorbate layer, but do not give the correct densities of the C- and IC-packings, which is a failure of the homogeneous model. The CA model, on the other hand, does give the correct commensurate density, but the high intersite barriers prevent a transition from C-packing to IC-packing. These results for these two models have led us to suggest a model having sufficient corrugation at low loading to yield commensurate packing, but where the amplitude of the corrugation decreases when the density of the 1st adsorbate layer is approaching full packing, allowing the transition from Cto IC-packing to occur. This decrease in corrugation is given by eqs. (10-11). We have found that with λmin equal to 0.1 for krypton, and 0.01 for methane it is possible to describe the transition from gas to fluid, fluid to C-packing and C-packing to IC-packing (Figure 7). Experimentally there is no liquid state for krypton at 77K, and this might be caused by the nonuniform environment system in adsorption. The C-IC transition for methane occurs only after the second layer has been formed and for temperatures greater than 87K. This mCA model, used in the grand canonical Monte Carlo simulations, gives the transitions in the correct pressure range and at the correct densities across the transitions. It should be noted, however, that the transition from the C-packing to the IC-packing in krypton occurs in two stages while only one stage is observed experimentally, which we attribute to the finite size of the simulation box.



12

12 (b) 10

2

Surface Excess(µmol/m )

10

2

Surface Excess(µmol/m )

(a)

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 27

Kr 77K

8 6 4 2

6 4 2

0

0

-2

-2

-4 0.01

87K Total Isotherm 87K Layer1

8

-4 0.1

1

10

100

0.1

1

Pressure (Pa)

10

100

1000

10000

Pressure (Pa)

Figure 7 Isotherms of Kr at 77K (a) and methane at 87K (b) using the mCA model and the corresponding factor . The effect of choosing a different value for λmin is shown in Error! Reference source not found.Error! Reference source not found.

3.2 Local Properties Analysis 3.2.1 The 1st adsorbate layer transition The local density distribution (LDD) and radial density distribution (RDD) in the first adsorbate layer have been studied. We chose 4 points on the krypton isotherm as shown in the inset of Figure 8a to represent the four different states of the first adsorbate layer: A (2D fluid), B (C solid), C (IC1 solid) and D (IC2 solid). The LDD in Figure 8a confirms that only molecules in the 1st adsorbate layer contribute to these transitions, as the number of molecules in the 2nd layer is negligible. The structure of the 1st adsorbate layer across the transitions is shown by the 2D radial density distributions in Figure 8b. The RDD at Point A, before the formation of the commensurate solid (black solid line), has a fingerprint characteristic of a liquid-like structure. The RDD of the solid structures at Points B, C and D show three modifications as the first adsorbate layer is densified: (1) the distances between adjacent peaks become closer, (2) all peaks increase in height, and (3) a shoulder due an ordered array of distant neighbours, appears in the second and third peaks. signatures of a solid phase.


These are

Page 17 of 27

400

12.0

(a)

11.5

Surface excess (µmol/m2)

3

Local Density Distribution (kmol/m )

D 300

11.0

C 10.5

B

10.0

g(r)

9.5

200

A 9.0 0.1

1

10

100

Pressure (Pa)

2D-Fluid Commensurate-Solid Incommensurate-1-Solid Incommensurate-2-Solid

100

0 0.2

0.3

0.4

0.5

0.6

0.7

Z-Distance (nm) 5

5.0

(b)

I

Ι

4.8

4.6

4

4.4

4.2

3 4.0 0.38

g(r)

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


0.39

0.40

0.41

II

0.42

III

2

1

0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

Distance (nm)

Figure 8 Local density distribution (a) and radial density distribution (b) corresponding to the points marked in the insert of (a). The insert in (b) shows the peak at I.

For methane, the IC-packing only occurs in the 1st adsorbate layer after the 2nd layer has formed and at temperatures greater than 87K (i.e. there is sufficient kinetic energy for molecules in the 1st adsorbate layer to overcome the inter-site energy barriers). The transition from the C-packing to the IC-packing is not observed experimentally because of contributions from adsorbate molecules in higher layers, and therefore the transition in the 1st adsorbate layer is masked by molecules in the 2nd layer. The local properties for methane are similar to those of krypton (see Error! Reference source not found. Error! Reference source

not found.in supporting information). 3.2.2. 2D structure of the 1st adsorbate layer



Page 18 of 27

The structure of the 1st adsorbate layer can be probed by a histogram of the number of molecules as a function of f ( x, y ) given by eq (6). The range of f(x,y) from -6 to +3 is divided into bins, and number of molecules in each bin is determined. If the histogram is presented as < N j > / < N > , i.e., the ensemble average of molecules falling into the jth bin over total ensemble average number of molecules, there is a misleading representation of localisation of the adsorbate on the carbon hexagons because the fractional area of the graphene surface varies with f(x,y) as shown in Figure 2b. To compensate for this variation, we defined the following area-corrected number histogram as a function of f(x,y):  1 < Nj >  1  < Fj >=  /  α < N >   α  j    1

(12)

where α j is the area corresponding to the jth bin, and α 1 is the area of the first bin corresponding to f(x,y) = - 6. In the limit when all molecules reside at the centres of the carbon hexagons, = 1 and = 0 for all j > 1 which corresponds to a perfectly commensurate 1st adsorbate layer. A plot of the ensemble averaged area-corrected histogram versus f(x,y) is shown in Figure 9.

At low pressures, the 1st adsorbate layer is gas-like and the ensemble average factor f

is -

0.6 (Figure 7). The area-corrected histogram in Figure 9 shows a higher concentration at f(x,y)=-6, which indicates that molecules tend to favour locations at the centres of the carbon hexagons. Since the 1st adsorbate layer is in a fluid state, the ensemble average in Figure 7 is approximately zero, as confirmed by the rather flat area-corrected histogram in Figure 9, indicating that there is no preference for any specific sites on the surface. The sudden drop of at the transition to C-packing in Figure 7 indicates that molecules have concentrated towards the centres of the carbon hexagons, and this is also reflected by a higher concentration at f(x,y)= -6 in the area-corrected histogram in Figure 9. When the transition to IC-packing has occurred, the area-corrected histogram is similar to that of a 2D fluid state, showing a random distribution of molecules.


Page 19 of 27

1.0 (b)

(a) 2D Gas 2D Liquid Commensurate Solid Incommensurate Solid

3

0.8

2D Gas 2D Liquid Commensurate Solid Incommensurate Solid

0.6

10

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


0.4

0.2

0.0 -6

-4

-2

0

2

-6

-4

-2

f(x,y)

0

2

f(x,y)

Figure 9 Area corrected histogram of number of molecules in each f(x,y) bin for (a) krypton at 77K

and (b) methane at 87K of certain phases. The second stage of IC solid of krypton is omitted for clarity given that the pattern is almost the same as the first stage IC.

3.3 Isosteric heat Our previous extensive simulation studies of argon adsorption on graphite

2,28,29,38

show that

the plot of the isosteric heat versus loading (the heat curve) exhibits a maximum at an incommensurate density and at temperatures far below the triple point of bulk argon. As the temperature is increased, the heat curve evolves into a local cusp shaped minimum, before rising steeply to a sharp maximum (called a spike) and the spike shifts to higher loading with temperature.

When the temperature approaches the bulk triple point (87.3K) the spike

disappears. Recent work on the evolution of the heat curve for methane adsorbed on a homogeneous graphite surface exhibits a similar trend

54

.

However, the homogeneous

surface cannot account for the existence of commensurate densities, since there are no potential minima parallel to the surface plane.

Next, we consider the evolution of the heat curve over a range of temperatures for the mCA model introduced here.

1.

T Tb

T>Tb

65

110K

120K

50 T

A Comparison of the Adsorption Transitions of Methane and Krypton

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