Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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23 Statistical Theory and Molecular Dynamics

of Diﬀusion in Zeolites

Reinhold Haberlandt

23.1 Introduction

That part of theoretical physics where the macroscopic behaviour of systems

is derived from their microscopic structure and from the properties of the

particles is called Statistical Physics.

In order to evaluate the structural, thermodynamic or transport behaviour

of macroscopic systems, statistical physics gives the starting quantities –

radial distribution functions, partition functions or correlation functions –

and the interrelations between them. The foundations will be sketched in the

next section. One ﬁnds more detailed descriptions in the literature [1–3].

The motion of systems with diﬀerent numbers of particles can be discussed

within three levels

– mechanical: for small particle numbers (N<3), allowing an analytical

solution of the equation of motions,

– statistical theory: for large numbers of particles (N ≈ 10

), by applying

methods of statistical physics on the basis of theoretical mechanics and

probability theory,

– computer simulations: in a wide range up to some thousand particle sys-

tems, which are already representative of macroscopic systems in many

properties (N<10

). This last level completes the treatment of systems

by use of computers and concepts of mechanics, statistics and probability

theory.

In general, however, there is some lack of knowledge in the statistical the-

ory itself (especially for irreversible processes) as well as in structural details,

such as, e. g., inter- and intramolecular potentials. Computer simulation can

help to overcome these diﬃculties in evaluating the properties one is inter-

ested in.

Figure 23.1 (left) shows the double role of such simulations in understand-

ing the behaviour of macroscopic objects following the detailed microscopic

motion of the particles belonging to the macroscopic system. For instance,

Fig. 23.1 (right) shows a snapshot of the diﬀusion of CH

molecules through

diﬀerent cavities of Linde type A (LTA) zeolites using special models for struc-

tures and potentials, respectively (more details will be given in Sect. 23.4).

916 Reinhold Haberlandt

Comparison Comparison

Modeling Model

Real

System

Experiment

Computer

Simulation

Approx.

Theory

Exper.

Result

f. Model

Theoret.

Prediction

Test

Model

Test

Theory

Fig. 23.1. Left: Role of computer simulations. Right: Snap shot of methane mole-

cules (small light grey spheres) in LTA zeolites. Large cavities are represented by

grey isopotential surfaces. The occupation number is I = 5 molecules. The initial

state is represented by small dark grey spheres in the lower left cavity.

The validity of the models used is checked by comparing the experimental

results with the simulations.

On the other hand, one can test the approximation of the theory by com-

paring its results with the simulations. Thus, computer simulations can give a

more detailed understanding of complex processes. This will be demonstrated

by the case of zeolitic diﬀusion in Sect. 23.4. Some features of computer sim-

ulations [4–13], especially molecular dynamics (MD), are given in Sect. 23.3.

23.2 Some Notions and Relations of Statistical Physics

In this section, the radial distribution function g(r) and the correlation func-

tion K(r, t), as two fundamental quantities of statistical thermodynamics and

non-equilibrium theory, are introduced and correlated with the observables

and system properties that can be evaluated.

23.2.1 Statistical Thermodynamics

General Mechanical Background

The mechanical state of a system of f degrees of freedom can be illustrated

geometrically in a so-called gas conﬁguration space (Γ -space) using the gen-

eralized coordinates q

and the generalized velocities ˙q

23 Statistical Theory and Molecular Dynamics of Diﬀusion in Zeolites 917

,...,q

, for short: q, ˙q

, ˙q

,..., ˙q

, for short: ˙q, (23.1a)

where the ˙q

are related to the generalized momenta p

∂L

∂ ˙q

i =1, 2,...,f, (23.1b)

L = L(q

, ˙q

,t) is the Lagrangian function, connected with a Legendre-Trans-

formation to the Hamiltonian function

H(q

,t)=



˙q

− L(q

, ˙q

,t). (23.2)

The mechanical motion of the systems is governed by equations of motion,

e. g. Lagrange’s equations of motion or Hamilton’s (canonical) equations of

motion, respectively

∂L

∂ ˙q

−

∂L

∂q

=0; ˙q

∂H

∂p

, ˙p

= −

∂H

∂q

(23.3a)

and – as an important starting point for molecular dynamics – Newton’s

equations of motion (Newton’s second law) for the radius vector r

of the ith

particle (in Cartesian coordinates)

= F

, with F

= −

∂U

∂r

, (23.3b)

with m

: particle masses, F

: force components, U(r

):potentialenergy.

The whole system is considered as a single super-molecule with f general-

ized coordinates and f generalized momenta. Following Gibbs, a large number

of physically equivalent systems – the Gibbs ensemble – is considered. This

ensemble can be visualized in the gas conﬁguration space by a cloud of phase

points distributed with the particle density ρ(q,p,t). Its motion will charac-

terize the mechanical state of the system. With respect to diﬀerent physical

situations there exist diﬀerent ensembles, e. g. the microcanonical ensemble of

isolated systems (energy E,volumeV ,particlenumberN) and the canonical

ensemble of closed systems in contact with a heat bath for thermalization

(temperature T ,volumeV ,particlenumberN ). Initially, molecular dynam-

ics (MD) was conducted in the microcanonical ensemble and Monte Carlo

(MC) in the canonical ensemble. Today both MD and MC can use several

ensembles [5–8]. For macroscopic systems – in the thermodynamic limit with

N →∞and N/V = const – all ensembles yield equivalent results, diﬀering

in ﬂuctuations only, which are not important for our current purposes.

Distribution Functions

Considering an ensemble of ν systems with f degrees of freedom each (total

number of degrees of freedom F = ν ×f in Γ -space) then ρ

(N)

(q, p, t)dqdp is

918 Reinhold Haberlandt

the number of systems in the range q...q+dq,p...p+dp of Γ -space. Intro-

ducing the (normalized) phase space density or the N -particle distribution

function, respectively, ρ

(N)

(q, p, t)by

(N)

(q, p, t)=

(N)

(q, p, t), (23.4)

(N)

(q, p, t)dqdp is the probability to ﬁnd the system – consisting of N par-

ticles – in q...q+dq,p...p+dp of the Γ -space. More often it is suﬃcient to

use so-called reduced distribution functions for ﬁnding k particles (k ≤ N)

only, deﬁned by a k-particle distribution function

(k)

(q, p, t)=



(N)

(q, p, t)dq

(N−k)

. (23.5a)

Particularly important are the two-particle distribution function (k =2)and

the one-particle distribution function (k = 1). Integration over the momen-

tum space gives

(2)

(q)=



(2)

(q, p, t)dp, n

(1)

(q)=



(1)

(q, p, t)dp. (23.5b)

(2)

)dq

is the probability to ﬁnd a particle i in q

...q

+dq

and

another particle j in q

...q

+dq

. n

(1)

)dq

is the probability to ﬁnd a

single particle i in q

...q

+dq

. For N  1, one can introduce a distribution

function g(q

)by

(2)

)=n

(1)

)g(q

). (23.6)

g(q

) tends towards one for q

= |q

−q

|→∞. The deviation from one

is a measure for the correlation of the pairs of molecules (pair distribution

function). The total potential energy U

is assumed to be pair-wise additive:



i<j

u(q

In isotropic systems u(q

), g(q

) are functions of the distance q

= r

only. The thus deﬁned radial distribution function g(r)isveryimportant

to evaluate structural and thermodynamic data and to compare them with

experimental data as well. As examples, some relations are given for the

internal energy U, the pressure p, the chemical potential µ and the static

structure factor S(k) – particularly important to compare theoretical data

with experiments (see, e. g. , Chaps. 3, 18, 20 and [14]):

U =

T +

∞



u(r)g(r, n, T )4πr

dr, (23.7a)

23 Statistical Theory and Molecular Dynamics of Diﬀusion in Zeolites 919

p = k

Tn−

∞



(r)g(r, n, T )4πr

dr, u



(r)=∂u/∂r, (23.7b)

=ln(nλ



∞



u(r)g(r, ξ)4πr

drdξ, λ=



2πmk



1/2

(23.7c)

with ξ as insertion parameter (0 ≤ ξ ≤ 1), the Boltzmann constant k

,and

the de Broglie wavelength λ [6]. The structure factor

S(k)=1+n



{g(r) − 1}exp (ik · r)dr =1+nh(k) (23.7d)

plays an important role in the comparison between theoretical and exper-

imental data (scattering data: neutron, X-ray; see, e. g., Chaps. 3, 13, 15

and [15]). There are: n –density;h(k) – Fourier transform of {g(r) − 1}.The

limit for k → 0gives

lim

k→0

S(k)=1+n



{g(r) − 1}dr. (23.7e)

23.2.2 Statistical Theory of Irreversible Processes

Irreversible processes are driven by generalized forces X and characterized by

transport coeﬃcients L [2,15,16]. It is very important to determine transport

coeﬃcients – e. g. using the famous linear response theory (Kubo [17]) – via

correlation functions [18, 19]. The correlation functions are as powerful in

non-equilibrium as partition functions or radial distribution functions are in

equilibrium (see, e. g., (23.7)).

Transport Coeﬃcients and Correlation Functions

In the linear approximation, the transport coeﬃcients L

are deﬁned by

relations between the ﬂux densities J

and their corresponding generalized

forces X

J = L · X +(higher terms) J



, (23.8)

with the Onsager-Casimir reciprocity relations L

= L

– valid near equi-

librium. For instance, in the case of diﬀusion the particle transport is caused

by concentration gradients:

= −D ·

∂n

∂z

+(higherterms). (23.9)

Using the ensemble average in the Γ -space

920 Reinhold Haberlandt

A



A(q(t),p(t))ρ(q(t),p(t)) dqdp (23.10)

with the equilibrium phase space density (cf. (23.4)), classical time correlation

functions

(t)=A(t)B(0)



···



A(q, p,t)B(q, p, 0)ρ(q, p)dqdp, (23.11)

for the phase space functions A{q(t), p(t)} = A(t)andB{q(t), p(t)} = B(t)

are deﬁned by (23.11). With A = B, K

(t) is called autocorrelation func-

tion, and for A = B = v (velocity): velocity autocorrelation function K

(t).

The spectral density f (ω), which is important for comparisons with experi-

mental data, is a Fourier transform of the velocity autocorrelation function

f(ω)=

∞





v(0) · v(t)





v(0)



cos(ωt)dt. (23.12)

Linear Response Theory – Green-Kubo and Einstein Relations

The linear response theory [17] is a bridge between microscopic theory and

experiment. It can be outlined as follows

– The linear response theory assumes near-equilibrium situations so that

linear relations are suﬃcient (e. g. also to describe relaxations back to

equilibrium).

– This linear response is determined by ﬂuctuations from equilibrium.

The linear response theory yields relations between the irreversible behav-

iour of the systems and the observables to measure, but no instruction to

determine the transport properties. Here methods of statistical physics – in-

cluding molecular dynamics – and measurements come into play. Generalized

susceptibilities (transport coeﬃcients) are Fourier transforms of correlation

functions

σ(ω)=

∞



exp(−iωt)

A(t)

B(0)

dt. (23.13a)

In the limit of long waves (ω → 0,k→ 0) – suﬃcient for transport coeﬃcients

–itis

σ =

∞





A(t)

B(0)

dtL

∞





(t)

(0)

dt. (23.13b)

In the long-time limit (t →∞) for autocorrelation functions, (23.13b) can be

substituted by Einstein relations of the kind

23 Statistical Theory and Molecular Dynamics of Diﬀusion in Zeolites 921

2tσ = (A(t) − A(0))

. (23.13c)

A further important quantity in comparing theoretical and experimental data

is the dynamic structure factor (see Chaps. 3, 13, 16, 20, and [14])

S(k,ω)=

2π

∞



−∞

F (k,t)expiωt dt. (23.14a)

Here F (k,t) is the – numerically as well evaluable – scattering function

(Chaps. 2, 13 (cf. (13.1)))

F (k,t)=

ρ(k,t)ρ(−k, 0), (23.14b)

which is related to the so-called Van Hove correlation function G(k,t)by

F (k,t)=



G(r,t)exp(−ik · r

)dr (23.14c)

(cf Chap. 3, (3.10)) and can be determined by ensemble averaging ··· over

the Fourier transform ρ(k,t) of the local one-particle density as well:

ρ(k,t)=



exp (ik · r

(t)) =



ρ(r,t)exp(ik · r

)dr. (23.14d)

In the limit ω → 0,k→ 0 one has the usual static structure factor (see

(23.7d)).

As an example of the discussed general theory of irreversible processes

(see Table 23.1) here only relations for the (self-) diﬀusion coeﬃcient are

given:

D =

∞



v(t) · v(0)dt;2tD =

(r(t) − r(0))

. (23.15)

Table 23.1. Correlation functions – theoretical and experimental quantities

Observable Correlation Function Equation

spectral density v(0) · v(t) (23.12)

structure factor ρ(k,t)ρ(−k, 0) (23.14a)

transport coeﬃcient 

(t)

(0)

(23.15)

922 Reinhold Haberlandt

23.3 Molecular Dynamics

Due to the lack of analytical solutions, the motion of guest molecules in

zeolites must be evaluated numerically by computer simulations (molecular

dynamics (MD), Monte Carlo (MC)). Computer simulations are reviewed in

many text books [4–9], thus here only a few principles of MD simulations are

sketched.

23.3.1 General Remarks

The procedure of MD simulations may be subdivided into several steps:

– Analysis of the physical situation

– Derivation of the equations of motion (e. g. (23.3b))

¨r

= −

∂U

∂r

(23.16)

– Selection of the interaction potential U(r).

From the great variety of (empirical) potential functions (e. g. [1]), Figs. 23.2

and 23.3 display only two examples:

a) Square well potential

U(r)=∞ r<σ

U(r)=−σ

<r<σ

(23.17)

U(r)=0 r>σ

U(r)

−

Fig. 23.2. The square well potential.

23 Statistical Theory and Molecular Dynamics of Diﬀusion in Zeolites 923

Fig. 23.3. Typical representative of the potentials used in MD simulations

(Lennard-Jones-potential U(r)).

b) Lennard-Jones potential

U =4







−







(23.18)

with  denoting the minimum value of the potential energy and σ

deﬁned by U(σ)=0.

These and other potential functions [1] are often used, while quantum

chemical calculations of potential surfaces are restricted to systems being

not too complex.

– Selection of the computational model (algorithms, time steps (l), duration

(l × τ ))

– Solution of the equations of motion (trajectories, propagators)

– Evaluation of the desired dynamical data via time averages of a single

system

A

n=l



n=1

A(n∆t) (23.19)

or ensemble averages A

(cf. (23.10)) and correlation functions K

(t),

(cf. (23.11)), respectively

– Discussion of the results.

23.3.2 Procedure of an MD Simulation

Starting with the ﬁrst MD papers for hard sphere systems in 1957 [10] many

investigations and applications have been published using several diﬀerent