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Grand Canonical Monte Carlo Method for Predicting Hydrogen Adsorption
Tyllianakis Emmanuel 1* , Klontzas Emmanuel 2 , Dimitrakakis K. Georgios 2 , Mpourmpakis Giannis 2 and Frudakis George E. 2
1 Material Science and Technology Department, University of Crete, P.O. Box 2208, Heraklion, Crete, 71003, Greece
2 Department of Chemistry, University of Crete, P.O. Box 2208, Heraklion, Crete, 71003, Greece
*tilman@materials.uoc.gr
A lot effort has been devoted for the study of interaction between hydrogen and nanostructures, in an effort for the hydrogen
uptake to be calculated. From theoretical point of view, the only way to study this hydrogen adsorption is the Grand
Canonical Monte Carlo method, since it permits the total number of particles adsorbed to fluctuate, whereas the chemical
potential, μ, the cell volume, V, and the temperature, T, are kept constant i.e. (μ, V, T) ensemble. In this way, the pressure
and temperature experimental conditions can be simulated, since the chemical potential is explicitly calculated for the
specific thermodynamic state.
Three types of motions occur during the simulation i.e. particle displacement, creation, and destruction. These different
steps are executed with equal frequency. In a creation step, the position of the new particle is chosen randomly within pore
volume V and its potential energy U is calculated. Finally these steps are accepted with the probability min(1; P) with
⎛ U − μ ⎞ V
⎛U
− μ N
P = exp
⎜−
⎟
P
creation
destruction
⎝ kBT
⎠ N + 1
k
BT
⎟ ⎞
⎛ U1
−U
2
⎞
= exp
⎜
P =
⎝ ⎠ V
⎜−
⎟
displacement
exp
⎝ k
BT
⎠
Prior to the GCMC simulation, the exact value of chemical potential for the specific thermodynamic condition is needed, as
referred earlier. For that purpose the Widom test particle method [1] is implemented for bulk hydrogen, at densities that
correspond to that condition. The basic idea of this method is to try to insert at regular intervals, a particle at a random
position. The particle is assumed to have no difference on the rest of the system, but it has an energy, the average of which is
used for the calculation of chemical potential.
The correct form of the function that describes the non-bonded interactions is the central point of this method and to that
direction a lot of classical effective potentials have been used. Among them the Lennard-Jones is the one, most widely used.
According to this potential, the interaction between a pair of particles i and j separated by a distance r is given by the equation
12
6
⎡⎛
σ
⎤
ij
⎞ ⎛σ
ij
⎞
V ⎢⎜
⎟ − ⎜ ⎟ ⎥
ij
= 4 ⋅ε
ij
⎢
⎥
⎣⎝
rij
⎠ ⎝ rij
⎠ ⎦
where i and j denote hydrogen or carbon particles. ε describes the potential depth and σ is the size potential parameters.
Besides this potential, some other functions have also been used like the Silvera-Goldman [2], the Crowell-Brown [3] etc.
The main disadvantage of these potentials is the use of uniform parameters for a given pair of heteroatoms and hence the
absence of varying these values according to the particularities of the system under examination. For that purpose, Monte
Carlo method has often to corporate with accurate ab initio methods for the exact value of these parameters to be calculated.
In addition to these potential some other terms are used also to describe other kind of interactions like electrostatic
interactions, or the contributions of quantum effects which have been proven to contribute significally [4] especially at lower
temperatures.
The application of GCMC methods allows us to compare the efficiency of different nano-structures to adsorb hydrogen and
study the contributions of some factors to that uptake. In that sense, a variety of simulations on different systems has been
performed, including from graphene sheets and carbon nanotubes, that were the first systems to attract the scientific interest
concerning hydrogen uptake, to more novel materials proposed like carbon nanoscrolls and metal organic frameworks. Some
representative results from these calculations can be seen at Figures 1-5.
Figure 1: GCMC calculations for (9-9) nanotube arrays with 0.7 nm intertube distance at pressure of 10 MPa and
temperatures of 77, 175 and 293 K.
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