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This is to certify that the
dissertation entitled
"Rare Gas Atoms and Clusters of Atoms
Inside Microporous Solids"

presented by

BOYONG CHEN

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Physics

 

AW

Major professor

S.D. Mahanti

 

Dme 5/22/96

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

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MSU I. An Affirmative Action/Equal Opportunity Inflation

 

RARE GAS ATOMS AND CLUSTER OF
ATOMS INSIDE MICROPOROUS SOLIDS

By

Boyong Chen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1996

ABSTRACT

Rare Gas Atoms and Cluster of Atoms Inside Microporous Solids
By

Boyong Chen

Physical properties of both quantum and classical rare gas atoms and cluster

of atoms confined inside microporous solids have been studied in this thesis.

In microporous media characterized by pore size in the range 5 ~ 15 A, mass
transport of He atoms at low temperatures is quantum mechanical in nature. By
solving the one-particle Schrodinger equation, we investigate the effective mass
m“/m of He atoms (bare mass m) moving inside one-dimensional tubular chan-
nels consisting of cylindrical cages connected by necks of different diameters and
lengths. We find that m/m“ is a highly nonlinear function of the geometrical
parameters characterizing these channels. We also find that in the presence of
an attractive potential produced by positive ions located on the channel wall, the

atoms are trapped near the wall, resulting in a drastic reduction in m/m‘.

Low lying excitations of 4H e and 3H e atoms confined inside zeolite cages have
been modelled by Bose-Hubbard and Mott-Hubbard rings with strong intrasite
repulsion and finite intersite attraction. Calculated temperature and concentra-
tion dependence of the heat capacities agree well with the experiment except at

very low temperatures. We argue that the discrepency with experiment is due

to the disorder effects which exist in the system. After the inclusion of the ef-
fect of tunnelling disorder, we obtained theoretical results which in much better

agreement with experiment. results.

Thermal properties and dynamic properties of X 83c cluster and X e60 AM — a
cluster assembled material confined inside L-zeolite cages have been investigated
using extensive Monte Carlo (MC) and molecular dynamics simulations. By mon-
itoring the temperature (T) dependence of the average energy < E > and the
bond length fluctuation (6), we find that unlike a free Xes cluster which starts to
‘melt’ at a reasonably well defined T, the confined X eac cluster for similar lengths
of MC runs shows a series of ‘melting’ temperatures. However, the average of
many long MC runs gives a relatively smooth variation of < E > and 5 with
temperature. One can still identify a ‘melting’ transition but at a much higher
temperature than that seen in the free cluster. On the other hand, X 850 AM sys-
tem shows a well defined ‘rnelting’ transition. The molecular dynamics simulation

demonstrates that X e50 AM has a much more stable structure than X ego.

The percolation properties of Ar and He atoms in the mixed ion pillared lay-
ered silicate clay systems [C'r(e'n)§+],,[C'o(en)‘.3Jr — (en)]1_,- fluorohectorite(FH T),
where (en) is an ethylenediamine ligand, at room temperatures have been investi-
gated using continuum MC simulation method. We find that the adsorptive and
diffusive properties depend sensitively on the size of the diffusing species and the
concentrations a: and (1 — 23) of the intercalants. Ar adsorption studies in the
above FHT system shows a percolative response when 2: reaches 0.79. Using sim-
ple geometrical models to describe these microporus media, along with computer

simulation, we can understand the a: = 0.79 percolation threshold.

ACKNOWLEDGEMENTS

I would like to express my sincere thanks to my thesis advisor Professor S.
D. Mahanti for his careful and patient guidance. Without his constant support,
understanding, encouragement, and physical insights, this work would not have

been possible.

I also would like to thank Dr. M. Yussouff for his cooperation, helpful discus-
sions and suggestions and his patience and concern. I thank Dr. Bhattacharya
Aniket for allowing me to cooperate with him. I learned a lot from his work. I
am also grateful to Professor M. F. Thorpe for his valuable lectures, support and

discussions.

I would like to thank Professor B. Golding, Professor D. Stump and Professor
T. J. Pinnavaia for serving on my Ph.D guidance committee and their valuable

suggestions.

I want to thank Professor J. S. Kovacs, Ms. S. Holland, Ms. J. King, and
Ms. K. Cords for their concern, help, and assistance, Professor T. Kaplan for his
concern and encouragement. I also want to thank all my friends in this department

for their valuable discussions and help in preparation of this thesis.

I would like to express my deepest thanks to my husband Shixi and my son
Philip for their love, understanding and support. I also thank my parents and my

family for their love and encouragement.

Financial support from the Natural Science Foundation and the Center for

Fundamental Materials Research of MSU is also greatfully acknowledged.

iii

Contents

Abstract ..................................

Acknowledgments ............................

1 Introduction
1.1 Characterization of porous media ..................
1.1.1 Zeolites as examples of microporous (MP) solids ......
1.1.2 Pillared layered silicates (pillared clays) ...........
1.2 Hubbard model ............................

1.3 Contents of different chapters ....................

2 Single Quantum Particle in 1D MP Channels
2. 1 Introduction ..............................
2.2 System and method ..........................
2.3 Results for the ground state .....................

t

2.3.1 Effect of geometry on the energy spectrum and m / m

2.3.2 Analytical fits to m/m" ...................

iv

iii

10

12

12

20

22

22

29

2.3.3 Effects of attractive potential due to adsorbed ions on the

cage surface and silicate framework ............. 34
2.4 Excited state ............................. 40
2.5 Summary ............................... 43

Low-Temperature Thermodynamic Properties of Helium Clus-

ters in K-L Zeolite 44
3. 1 Introduction .............................. 44
3.2 Physical system ............................ 46
3.3 Experimental results ......................... 48
3.4 Models of the system ......................... 50
3.5 Results for Bosons (B) and Spinless fermions (SF) ......... 52
3.5.1 Difference between B and SF due to statistics ....... 52
3.5.2 Difference between B and SF due to mass ......... 54
3.5.3 Comparison with experiments ................ 57
3.5.4 Spin effect ........................... 60
3.6 Conclusion ............................... 66

Effect of Tunneling Disorder on the Low Temperature Heat Ca-

pacity of 3H e and 4H e in Zeolite Channels 67
4.1 Introduction .............................. 67
4.2 Model with disorder ......................... 70
4.3 Results and discussions ........................ 71

V

4.4 Summary ............................... 75

5 Thermal Excitations of Xenon Clusters Inside a Model L-Zeolite 80

5. 1 Introduction .............................. 80
5.2 Method and the physical system ................... 82
5.3 Results ................................. 85
5.3.1 MC simulation ........................ 85
5.3.2 MD simulation ........................ 101
5.4 Discussion ............................... 105

6 Percolation and Diffusion in 2D Microporous Media: Pillared

Clays 110
6.1 Introduction .............................. 110
6.2 Physical system and the geometrical model ............. 112
6.3 Review of experimental situation .................. 116
6.4 Results of computer simulation ................... 119
6.4.1 Lattice model ......................... 119

6.4.2 Continuum diffusion ..................... 122

6.4.3 Comparison with experiment and discussion ........ 128

6.5 Summary ............................... 132

7 Conclusion 134

vi

A Estimation of nearest neighbor He-He interaction in K-L Zeolite136

vii

List of Figures

1.1

1.2

1.3

2.1

2.2

A [001] view of the framework of L-zeolite. .............

A unit cell of L-zeolite (also called cage) showing the main channel

viewed along the channel axis .....................

An idealized 2:1 layered silicate ....................

A typical channel structure of Cs+(180rown6)2e‘ electride. The
ring structures are (18Crown6); and spheres are Cs ions, each Cs

ion is sandwiched between two (18Crown6) molecules ........

The channels of Cs+(18Crown6)2e" electride (It is the negative
picture of the previous figure ) at 0.54 A from the molecular van
der Waals surfaces. Major channels are about 1.9 A by 4 A in cross
section and 2—3 A long, with a pronounced ”pinch” in the channel
center caused by hydrogen atoms of the crown center. Those major
channels are connected by narrow channels which are m 1.5 A in

diameter and z 4 A long. ......................

15

16

2.3 A channel structure of L-zeolite which is formed by silicate framework 17

viii

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

The channels of L—zeolite (It is the negative picture of the previous
figure) at 2.4 A from the channel surfaces. The porosity is about
57 % ...................................

(a). Schematic picture of a one-dimensional tubular channel in
K-L zeolite; (b). Geometrical parameters describing the one di-

mensional model channel .......................

A typical energy band structure in the one-dimensional tubular MP

channel ................................

The c—dependence of m/m‘ for three different neck diameters in the

2D case. Dashed lines are guide to the eyes. ............

The c-dependence of m/m“ for two different neck diameters in the

3D i.e. tubular channel case. Dashed lines are guide to the eyes. .
The c—dependence of Em,“ for 2D and 3D case ............
Square of the wave function in 2D case ...............
Effective mass as a function of neck diameter d for 3D case . . . .
Effective mass as a function of neck diameter d for 2D case . . . .

Ground state energy as a function of a. The insert is a log-log plot

which shows Em," ~ a". Dashed lines are guide to the eyes. . . .

m/m“ as a function of a which shows that the effective mass ra-
tio does not depend on a (b/a and c/a. are fixed). The observed

variations with a represents the inaccuracies in our calculation of

18

19

23

25

26

27

28

30

31

32

 

2.15 Comparison between fitted (varying a and fixing n) and calculated

2.16 Comparison between fitted (varying n and fixing a) and calculated

2.17 The azimuthal distribution of He - K-L zeolite potential with or
without K + ions. (a) and (b) show the potential along the center
of the channel wall. (c) shows the potential along the wall of neck.
OK”, 4K + and 6K + correspond to 0, 4 and 6 K + ions silicated on

the cage wall, respectively. ......................

2.18 Angular dependence of square of the wave function at k = 0 with

(a). p22, (b). p=3 ..........................

2.19 Square of the wave function at k = 0 with (a). V20, (b). V = 2.0eo,
(C). V = 5.080 .............................

2.20 Energy bands for the ground state, first excited state and second
excited state for 2D case with a = b = 10 A, c = 7.5 A and d = 5.0

3.1 Schematic picture of a cross section of K—L Zeolite channel showing

the silicate framework and ring arrangement of He adsorption sites.

3.2 The experimental results of heat capacity of He atoms as a function

of T for fixed He concentration. ...................

3.3 Lowest 20 (out of 70) energy states of Spinless fermions (SF) and

bosons (B) for particles in a 8-site ring ................

35

36

37

39

41

47

55

3.4

3.5

3.6

3.7

3.8

3.9

4.1

4.2

4.3

4.4

4.5

Temperature dependence of the molar heat capacity for 8-site n-

particle spinless fermion and boson systems. ............

The molar heat capacity for the (8,4) system with bose statistics
and different t, V=-24K. .......................

Heat capacity for the spinless fermions as a function of < n > (a)

and as a function of T (b) .......................

Heat capacity for the spinless fermions as a function of T (shows

the discrepacies between experiment and theory). .........

Heat capacity for the bosons as a function of < n > (a) and as a

function of T (b) ............................

Comparison of the heat capacity between SF and F for (8,2) and
(8,3) systems. t=14K, V=-24K. ...................

Comparison of the density of states with and without disorder for

4 site 2 particle case. .........................

Comparison of the heat capacity with and without disorder for 4

site 2 particle system ..........................

Heat capacity vs. temperature for fixed helium atom concentration

for 3H e with and without disorder. .................

Heat capacity vs. helium atom concentration < n > for fixed tem-

peratures for 3H6 with and without disorder .............

Heat capacity vs. temperature for fixed helium atom concentration

for 4He with and without disorder. .................

56

58

61

62

63

65

73

74

76

77

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.11

Single-particle potential energy V, for Xe as a function of ‘I> in a
plane passing through the center of the L-zeolite cage in units of

K, r is the radial distance from the cage axis. ...........

Singleparticle potential energy V, for Xe as a function of <I> in a
plane passing through the center of the L-zeolite cage in units of

K, r is the radial distance from the cage axis. ...........

Single-particle potential energy V, as a function of 2 inside the L-
zeolite cage in units of K, r is the radial distance from the cage

axis. The azimuthal angle ()5 is chosen to be zero. .........
Average energy < E > vs T for a single Xe atom inside L-zeolite.

Positions of the six Xe atoms(large circles) in the ground state of

Xeac and the framework Oxygen atoms(small circles) in the 2:0

The ground state of a free Xea cluster ................
Melting of a X eec cluster inside the L-zeolite; ...........

The same as in the previous figure excepting here each run consists

of 10° MC steps and the number of MC run was 3. ........
Melting of a free X 63 cluster: the average energy < E > vs T

Melting of a free X as cluster: average bond length fluctuation 5 vs.

T ....................................

Average energy < E > vs T for XeGCAM, a cluster assembled ma-

terial consisting of a periodic array of Xee clusters inside L—zeolite

5.12 r2 vs. time for the X eec system ...................

xii

87

88

89

91

92

93

95

96

98

99

100

102

5.13

5.14

5.15

5.16

5.17

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

1'2 vs. time for each Xe atom along with the result of the averaged

one for the X 83.: system ....................... 103
1*” vs. time for the X 63,: system at three different temperatures . . 104
1'2 vs. time for the X 66C AM system ................. 106
1'2 vs. time of run2 with 2:2, yz, 22 also for the X6663”; system . . 107
Diffusion rate D(t) vs. time for the X 860 AM system ........ 108
Argon atoms diffusing in the annealed pillared clay ........ 117

The amount of Argon adsorbed by (003+(en)3)1_,(Cr3+(en)3)a, —

FHT as a function of x. ....................... 118
Lateral constraints on Argon atoms diffusing in the annealed system 120

The x dependence of the size of the largest cluster of A-A bonds
obtained from simulation of 100 x 100, 400 X 400 and 2000 x 2000

lattices. ................................ 123
Typical curves of R2 and D / Do vs. time steps ........... 126
D/Do vs. step length for Ar(x=1) and He(x=0,1) ......... 127

The relative diffusion constant of Argon atom (dA, : 4.0 A) diffus-

ing in FHTD as a function of x. ................... 129

The relative diffusion constant of Helium atom (the = 2.6 A) dif-

fusing in FHTD as a function of x. ................. 130

The ratio DH°(:c = 0)/DH‘(:c = 1) as a function of the diameter of
Helium atom when Helium gas diffuses in [C’r3+(en)3],[003+(en)2 +

(en)l1_z — FHT system. ....................... 131

A.1 A view of the geometrical arrangement of Helium atoms and K +

ions of K-L zeolite cage. .......................

A2 (a) Shows the interaction between one Helium atom and K + ion.

(b) shows the interaction between two Helium atoms and K+ ion.

137

Chapter 1

Introduction

Physical properties of molecular or atomic systems are known to be dramatically
modified by confinement in restricted geometry. However, making explicit and
unambiguous connections between the geometry of the confining space and the
molecular properties have proven to be difficult in general, and the connections,
when made, are frequently controversial [1]. Nevertheless, behavior of these con-
fined systems is of considerable fundamental and practical interest. From a fun-
damental point of view, confinement reduces the phase space accessible to the sys-
tem. For example, the system of molecules / atoms can behave as a d-dimensional
system with d < 3 and we know that physical properties such as phase transitions,
transport and electronic properties are profoundly affected by the dimension d.
In addition, confinement can introduce inhomogeneities and disorder produced by
the confining medium which can also modify the physical properties of the imbibed
system. From a practical point of view, a wide class of porous confining media
such as zeolites, vycors, aerogels etc. are used commercially in such diverse fields

as chromatography, oil recovery, catalysis and membrane separation technology.

Many experimental and theoretical studies have been carried out to investigate

adsorption, diffusion and thermodynamic and dynamic properties of molecules
trapped inside porous media [1]-[11]. In addition, computer simulations have been
extremely useful in elucidating the behaviour of molecules confined in different
types of porous media [12H23]. One of the major tasks of theoretical condensed
matter physics is to model the physical system possible and to explain the observed
electronic, magnetic, structural, and dynamical properties of the confined system

and furthermore, to predict the properties of new but related systems.

During the years of my research in condensed matter theory in the group of
Prof. Mahanti, I have been focusing my attention on understanding the general
physical properties of rare gases trapped inside microporous materials, such as
pillared clays and zeolites. I have studied the behavior of He systems which behave
quantum mechanically at low temperatures and classically at room temperature
and Xe, Ar which are of practical interest. Because of their heavy mass, Xe and Ar
can be treated classically. In this chapter, I will give a general introduction to the
confining media of interest to me, namely zeolites and pillared layered silicates.
In addition, I will give a brief description of the Hubbard model which I have
used to represent the quantum mechanical behaviour of He atoms trapped inside

zeolites. I also give a brief description of the contents of each following chapter.

1.1 Characterization of porous media

Systems characterized by spatial restrictions and low dimensions, such as zeolites,
membranes, polymers, and porous glasses and minerals, are called porous media.
According to their pore size, porous media can be divided into three classes [24].

The first class includes macroporous media, where the length scale of the pore

 

is greater than 500A. Mesoporous media constitute a second class in which the
pores are larger than ~ 20A and smaller than 500A. A third class, with which
we are concerned in the thesis, embraces microporous (MP) media, with pore
sizes less than 20A. Zeolites are the most common examples of the MP media
[25]. Furthermore, depending on the pore structure, the effective dimensionality
of the physical system can be different. For example, parallel—walled slit pores,
such as those thought to occur in some partially graphitized carbon blacks [26],
graphite intercalation compounds [27] and pillared clays [28], restrain the particles
to move in two-dimensions, i.e., the system has one finite dimension out of 3D.
An example of restrictions in two finite dimensions is provided by the cylindrical
pores in porous vycor [29, 30]. Zeolites are mostly 3D constriction but with
periodic structures [25]. Aerogels and vycor glasses have much more complicated
pore structures which make the effective dimensionality of the system become a

non-integer number, namely their geometry is fractal [31].

1.1.1 Zeolites as examples of microporous (MP) solids

Zeolites are microporous inorganic compounds [25, 32]. Their crystal structure
contains large pores and voids which are usually regular and the effective pore
sizes are in the range from 3A to over 10A which are sufficient to permit the
diffusion of small organic molecules — a feature which gives rise to many of the
important applications of these materials, such as heterogenous catalysts and

molecular sieves. Most zeolite networks are aluminosilicates.

There are many kinds of zeolites. One of the simpliest zeolites with which
we are interested in our research is L-zeolite. Fig. 1.1 is a view along the [001]

direction of the aluminosilicate framework of L-zeolite [33, 34]. The idealized

composition per unit cell of L- zeolite is KaNaa[(A102)9(3i02)27]21H20. The
structure of the Linde type L-zeolite[34, 35] consists of a series of one-dimensional
channels which form hexagonal crystal structure and the c axis is along the channel
direction. The main channel has twelve framework oxygen atoms bounding the
aperture which is a nearly regular dodecagon. Six atoms form a ring of radius of
4.92A and the other six form a ring of radius 5.25A. The main channel opens up to
a diameter of 14.43A and then narrows back down in a unit cell length of 7.5A. Fig.
1.2 gives another view of the main channel seen along the c-axis. As can be seen
in fig. 1.1 the main channel is surrounded by six symmetrically spaced secondary
channels that connect the main channels of adjacent unit cells. We will refer to
these secondary channels as 8-channels because of the elongated octagon shape of
the channel when viewed along an axis parallel to the main channel. According
to this same viewpoint, there are six 4-channels which surround the main channel
and separate the 8-channels, and adjacent to each 4—channel on the opposite side
from the main channel is a 6-channel. A unit cell of this zeolite contains one
main channel, two 6-channels, three 8-channels, and six 4-channels. The shape
of the unit cell is that of a rhomboid with side length 18.4A, an interior angle
of 60 degrees, and a depth of 7.52A. The symmetry of the crystal is P6/mmm
as described in ”The International Table for X-ray Crystallography” [36]. The
total number of framework atoms in this unit cell is 108, with 72 Oxygen and 36
T-atoms (Si or A1).

K-L zeolite has the same overall structure as L—zeolite but with nearly nine K +
ions in each cage, for example there are 8.94 K + ions and 0.61 N (1+ ions / unit cell.
There are four of these nearly nine K + ions sitting near the wall of the cages while

the rest of them are distributed in the aluminosilicate framework. The presence

 

Figure 1.1: A [001] view of the framework of L—zeolite.

Active Y

 

Figure 1.2: A unit cell of L—zeolite (also called cage) showing the main channel
viewed along the channel axis. Only oxygen atoms of the framework are shown.

of K + ions near the wall of the cage and inside the silicate frame in K-L zeolite
provide additional attractive potential on the adsorbed atoms and also introduce

disorder in the system.

1.1.2 Pillared layered silicates (pillared clays)

Pillared clays are intercalation compounds in which stable pillaring ions of various
sizes are intercalated into the galleries of layered silicates. The layered silicates
are usually alumina—silicate minerals or clay minerals such as vermiculiate, fluoro-
hectorite, and montmorillonite. Fig. 1.3 is a schematic illustration of an idealized
2:1 layered silicate which has two tetrahedral sheets fused to a central octahedral
sheet [37, 38]. In ideal case, the layers are neutralized. But in most cases, there
are net negative charges in the layers because of the random replacement of Si4+
by Al“. Usually, these charges are neutralized by small positive ions , such as
N a+, in between the layers. Using special techniques, one can replace these small
ions by some large cations(called pillars) into the galleries of the layers to obtain a
new material which has large voids or inter lamellar microporous space accessible

to guest molecules.

Because of the nature of the layers and large pores (about 20A), pillared clays
are excellent two-dimensional microporous materials inside which large molecular

species can diffuse into active surface sites [33] and undergo catalytic reaction.

1 .2 Hubbard model

Hubbard model [39, 40], one of the simplest models describing an interacting

many-body system, was proposed by Gutzwiller, Hubbard and Kanamori in early

 

1" {trzoinm'

' “—5 «ea-“i"w-vo-e’
V..— . ‘ _.. h

or;

 
  
 

 

Figure 1.3: An idealized 2:1 layered silicate. Open circles are oxygens, small closed
circles are tetrahedral site cation Si, Al. Large closed circles are octahedral site
cation Al, Mg, Fe, Li, vacancy.

sixties. This model has been applied to study a great variety of physical systems,
electrons in solid [39], liquid 3H e [41], Bose condensation of 4He in disordered
media [42] etc. Recently, this model and its generalizations have been found to
be quite promising to study the strong correlation physics in pristine and doped
Mott insulators, the latter showing high temperature superconductivity, colossal

magneto-resistance etc.

The simplest Hubbard model for fermions with spin is usually written as fol-
lows:

H = —t Z CLCJ', + UanTnu (1.1)

<ij>c i

where cL(c;,) are creation (annihilation) fermion operators associated with site i
and spin 0'; 11,-, = glad-a is the number operator. Attention is normally restricted

to hopping (or tunnelling) between neighbouring atoms, denoted by < ij >.

The first term corresponds to chemical bonding and is known as ‘hopping’
[39, 40]. This term is a single particle term and represents a transfer of a particle
from one site to a nearest neighbor site with hopping matrix element t. This con-
tribution favours itinerancy or delocalization by reducing the one particle energy

(also refered to as the kinetic energy).

The second term corresponds to the Coulomb repulsion between two particles
when they occupy the same localized site and is therefore a two particle interaction
term. The long range contributions to the electron-electron interaction is assumed
to be screened and only the interaction when both the particles are on the same
site is retained, yielding a repulsive energy of U. The model can be extended to

take care of both longer-range hopping (t —) t;,-) and longer-range interaction.

10

For fermions (electrons or 3He) the Hubbard model is also refered to as the Mott-
Hubbard model because Mott realized the significance of the repulsive U term in
the problem of metal-insulator transition. If we apply the model to Bosons (such
as 4H e), then c’s become Boson operators and the model is called Bose-Hubbard

model.

There are some basic properties of the Hubbard model. Two of them are
important in the current study, namely (i) n, = 2;, c[,c,-,, commutes with H,

therefore it is a good quantum number, and (ii) there is particle-hole symmetry.

1.3 Contents of different chapters

We will next briefly describe the contents of each chapter.

In chapter 2, we investigate the effective mass m‘ / m of He atoms (bare mass m)
moving inside one-dimensional tubular microporous channels consisting of cylin-
drical cages connected by necks of different diameters and lengths by solving
the one-particle Schrddinger equation numerically. These channels are simplified
models of L—zeolite channels. From the relation between the effective mass, wave
function and the geometrical parameters, we can get some insight into the single
particle transport property. Most of the contents in this chapter were published in
Phys. Rev. B. [43]. In chapter 3, we discuss the low-lying excitations of He clus-
ters inside K-L zeolite using extended Mott—Hubbard and Bose-Hubbard models.
The results of the heat capacities are compared with experiments. The content of
this chapter was published in Phys. Rev. Lett. [44]. In chapter 4, we extend our
Mott-Hubbard and Bose-Hubbard models introduced in chapter 3 by including

disorder effect into the models. We study in detail the effects of disorder in the

11

tunnelling on the low temperature thermal properties. This modification has led

to a much better agreement with experiments especially at very low temperatures.

In chapter 5, thermodynamic and dynamic properties of an isolated Xenon
cluster confined inside L—zeolite (denoted as X em, 2 S n S 6) and Xenon clusters
confined inside L—zeolite as a cluster assembled material (denoted as X enc AM,
n = 6) are studied by Monte Carlo and molecular dynamic simulation meth-
ods. The differences between free X e3 cluster, X 63,; and X 660 AM are explored.
A manuscript based on the contents in this chapter is going to be published
[21]. In chapter 6, we use continuum diffusion simulation to probe the per-
colative properties of Argon and Helium atoms in the heteroionic pillared clay
[Cr3+(en)3],[Co3+(en)2 + (en)]1-,—fluorohectorite(FHT), respectively. The con-
tents of this chapter was published in J. Chem. Phys [45]. Finally, we give a brief

conclusion in chapter 7.

 

 

Cl

M

2.]

Chapter 2

Single Quantum Particle in 1D
MP Channels

2.1 Introduction

It is known that the physical properties of helium atoms (3H8, 4He), when mov-
ing in restricted geometry, differ dramatically from those in the bulk, particularly
at low temperatures, due to the influence of the topology of the confining medium
and the adsorption potentials exerted by the medium on the He atoms. Typi-
cal examples are: helium adsorbed on graphite [46, 47] and alkali metal [48, 49]
substrates, inside porous glass such as vycor [50, 51], aerogel [31] or microporous
media such as zeolites [52]-[60]. In fact, recent sorption measurements [61] of he-
lium in fullerite crystals and films show that helium atoms are mobile within the
microporous space of these crystals, thus opening up the possibility of realizing a

new type of three-dimensional quantum fluid.

One of the fundamental quantities that not only controls the mobility (trans-
port), but also other physical properties such as heat capacity and magnetic sus-
ceptibility (through density of states and interaction effects) in a confining geom-

etry is the effective mass (m‘/m) of He atoms. Depending upon the strength of

12

13

the substrate potential and the nature of the microporous geometry, m"/m can
vary enormously. In particular, if this effective mass ratio is much larger than
1, any small perturbation such as coupling to the vibrational degrees of freedom
of the confining medium, inter-particle interaction or static disorder will tend to
localize the particles, thus strongly affecting their transport and thermodynamic
properties. Therefore, before one attempts to understand the effects of disorder
(both structural and thermal) and inter—particle interaction, one must understand

the effect of periodic geometry on the motion of a single helium atom.

In this chapter, we address the question of the effective mass of a single helium
atom moving in one-dimensional microporous channels found in K-L zeolite [57,
58], immogolite [62] and similar one dimensional zeolites. We focus on K-L zeolite
for which we will construct simple geometric models. Our results are however quite
general. We also present some results about a few low-lying excited states. We
study these systems primarily for two reasons. First, one-dimensional systems are
relatively simple and second, extensive studies of low-T thermodynamic properties
of 3H e and 4H e atoms in K-L zeolite have been made [57] which we can address
theoretically. As discussed in chapter 1, K-L zeolite consists of one-dimensional
channels of about 14.43 A diameter modulated by constrictions (necks) of about
7.5 A in diameter. Other one-dimensional zeolites in which helium adsorption

studies have also been made are ZSM-23 [59, 60], whose diameter is ~ 5.5 A.

Before going to the zeolite, we would like to point out that such physical
systems for which our calculations of m / m" are also relevant are electrons moving
inside the narrow channels of electrides [63] where the channels are formed in

the space between large organic cage-like molecules encapsulating positive alkali

14

ions. Channels in a typical electride are shown in fig. 2.1 and fig. 2.2 [64].
Electrides are an exciting new class of systems which exhibit a wide variety of
electronic properties governed by the channel geometry. We also show in fig.2.3
and fig.2.4 [64] the channels in L—zeolite for comparison. Also recently electronic
energy band structures have been obtained by Lent et.al. [65] for electrons moving
inside periodically modulated channels, but these authors have not focussed on

the question of the effective mass which is of major interest here.

The atomic structure of an actual K-L zeolite as we have discussed before
is quite complicated. However we can analyze the essential physics of quantum
transport along the channel axis by using a simple model of the microporous
geometry. In a later chapter where we deal with classical systems such as Xe, we
will use a realistic atomic model of zeolite. Fig. 2.5a gives a schematic picture
of K-L zeolite, whose channels consist of cages connected by neck regions. The
boundary walls are formed by Si-O-S'i networks. In K-L zeolite, some of the S'i4+
ions are replaced by Al"+ ions and in addition there are charge compensating K +
ions. Some of these K + ions are attached to the interior walls of the cage and the
rest are embedded in the silicate framework. The K + ions on the wall provide
an attractive potential on the helium atoms. Thus in addition to the geometrical
confinement effects, the attractive potential produced by the K + ions and the
silicate framework can also affect m / m‘ dramatically, particularly when the neck

becomes narrow.

15

 

 

 

Figure 2.1: A typical channel structure of Cs+(18Crown6)3e‘ electride. The ring
structures are (18Crown6): and spheres are Cs ions, each Cs ion is sandwiched
between two (18Crown6) molecules.

16

 

 

 

Figure 2.2: The channels of Cs+(18Crown6)2e‘ electride (It is the negative pic-
ture of the previous figure ) at 0.54 A from the molecular van der Waals surfaces.
Major channels are about 1.9 A by 4 A in cross section and 2-3 A long, with a
pronounced ”pinch” in the channel center caused by hydrogen atoms of the crown
center. Those major channels are connected by narrow channels which are z 1.5
A in diameter and z 4 A long.

17

 

 

 

Figure 2.3: A channel structure of L-zeolite which is formed by silicate framework

18

 

 

 

Figure 2.4: The channels of L-zeolite (It is the negative picture of the previous
figure) at 2.4 A from the channel surfaces. The porosity is about 57 %.

19

(a)

 

 

 

 

(b)

Figure 2.5: (a). Schematic picture of a one-dimensional tubular channel in K-
L zeolite; (b). Geometrical parameters describing the one dimensional model
channel

20

2.2 System and method

Fig. 2.5b describes the geometrical parameters used in our simple one-dimensional
model of the channels in K-L zeolite: A given channel is assumed to be cylindrical
and periodic along the z-axis with periodicity a. The channel consists of cages
(chambers) of diameter b and length 0 S c S a, connected by necks of diameter
d and length a — c. Transport properties of a quantum particle in such a channel
are influenced both by the geometry and the potential. For simplicity, we assume

that the boundary is a hard wall where the wave function vanishes.

The single particle Schrodinger equation inside the cages is:

h
-§;V2¢+V¢=E¢ (2'1)

We solve the single particle Schriidinger equation by setting the eigenfunctions
as Bloch functions, ¢n,1,k(1',¢, z) = un,l,k(r, qS, z)e““'. We then solve the equation
for the cell periodic part um”. and eigenvalues enl(k). The numerical method used
is the optimized iterative method for eigenvalue problems [66] for the ‘1D’ case,
i.e. 1])(1', 45, z) —) 1M2), with appropriate generalization to our problem. It includes

the following steps
1. Divide the unit cell into n, x n, x 124, mesh.
2. Give an initial guess of the wave function ¢° which is normalized properly.

3. The energy of the system is then given by

E =< ¢|H|¢ >= / (1V v’ 21) + v¢'¢)dr (2.2)

where it = 1,00, V = V/eo and so is the energy unit we have used in this chapter

(so = hz/(2mA’), m is the mass of the particle).

21
4. Modify the wave function as following:
1/2’ =(1— Haw"
where dt is a fake time interval. Then, normalize it’.

5. Repeat step 3 and 4 until [AB] 2 [EH1 — E‘] S e , where e is an appropri-
ately chosen convergence parameter. Thus, we get the ground state energy and

the wave function for a fixed wave number It.

To get the first excited state, the only thing we have to do is, after step 4,
not only normalize the wave function but also orthogonalize it to the previously

obtained ground state wave function.

Using the above method we can get the eigenvalues and corresponding eigen-
vectors for the ground state and several low lying excited states. To probe the
effect of the dimensionality of the lateral (perpendicular to the tube axis) confine-

ment, we have also studied the two dimensional case where 1M3, y, z) = 1/2(z, 2).

Although we have obtained excited states in some case, here we discuss only
the lowest energy band for a fixed overall size of the channel i.e. fixed a, b shown
in fig. 2.5b, but with different cage size c and neck diameter d. Fig. 2.6 shows the
typical ground state energy band structure. Using this band structure and the
effective mass concept, i.e. assuming that E = hzkz/(2m') = (m/m")k2(h2 / 2m)
near the bottom of the energy band, we have calculated m / m‘. The unit of energy
is h2 / (2m A”). For 3H e this turns out to be 8K. One can easily see that m/m’ = 1
when c = 0 and/ or d = b which correspond to the propagation of a free particle
along the z-axis. Thus, if m/m‘ 2 1, the particle can move freely along the
channel axis and if m/m‘ < 1, the effective mass of the particle becomes large,

i.e. the particle becomes heavier so that it is harder to move along the channel.

22

As m/m“ —> 0, the particle has a tendency to get localized in the presence of
structural or thermal disorder and/ or interparticle interaction. In addition to
m/m’, we have obtained the wave function densities (uz) for the k = 0 state to
show what happens to the unit cell wave functions as m/m‘ —> 0. Finally, to see
the effect of the K + ions, we have compared the difference between m / m‘ for two

cases, i.e. with and without the attractive potential produced by the K + ions.

2.3 Results for the ground state

2.3.1 Effect of geometry on the energy spectrum and
m/m"

To examine exclusively the dependence of the m/m‘ on the geometry, we
assume that the potential inside the channel is zero (i.e. V=0 in eqn. 2.2). The
mesh for the results we present in this section is 40x40 for the 3D case and
80x80 for the 2D case (in 3D case, we can integrate out the angular dependence

analytically when there is no extra potential).

We will first present our results for fixed values of a and b and different values
of cage length (c) and neck diameter (d). For simplicity, in this paper we choose
a = b = 10 A, we do not expect much qualitative change from our present results
when we take a. 76 b. We will discuss the a dependence later. We find that the

effective mass has a very complex geometry-dependence.

In fig. 2.7 we give the c-dependence of m/m' for three different values of d
(= 3.5 A, 5.0 Aand 6.5 A) for the 2D case, and in fig. 2.8, similar results are
given for the tubular channel (3D case) for two different values of d (= 3.66 A,

23

 

 

 

 

J

’ <> 0 t
0.22 r .
, <> 0 .

A o 0

A _ a

g: 0.20
N [ <> 0

v L

\

"x:

V 0.18 7 O 0 '
>5 ’ .
an
L.

g 0 0
En?!
0.16 - 0 0 .
i <> <>
0 o *
O O 0 i
0.14 t .4
—0.2 0.0 0.2
k (Ac—1)

Figure 2.6: A typical energy band structure in the one-dimensional tubular MP
channel

24

5.12 A). The general trend is as follows: For small c, m/m" ~ 1. It decreases with
increasing c and attains a minimum for c ~ d. Then it increases with increasing c.
The decrease is quite sharp for the tubular channel when d is small. In this case,
there is a large region of c for which m/m‘ << 1. Finally, when c = a and d < 10
A, m/m‘ is still less than 1 because in this limit the particle still feels the effect
of geometry even if the neck length —> 0. In fact m/m" —+ 1 only when d —) b, in

which case the particle moves freely inside a tube of uniform cross section.

Fig. 2.9(a) and (b) are the corresponding ground state energy curves. As we
can see, the ground state (k=0) energy curves are relatively simple and they go

to the minimum when c = a.

To understand the physical origin of the sharp drop in m/m“ ie. enhanced
tendency to localize as we increase the cage length c, we plot in fig. 2.10(a) (c = 2
A) and fig. 2.10(b) (c = 6 A) the square of the ground state wave function (k = 0)
for the neck size (1 = 3.5 A for the 2D case. For c = 2 A< d, m/m" 2: 1 and the
probability of the particle inside the neck region is quite high, wheras for c = 6
A> (I, most of the probability is in the cage with almost vanishing probability in
the neck. In the latter case when we make k 75 0, the k-dependence of the energy
comes from small inter-cage tunnelling process. This leads to a rather low value

for m/m‘ which for these parameter values is 2 0.19.

One can also study the above tendency to ‘localize’ by changing the neck
diameter d for fixed values of the cage length c. In fig. 2.11, we give m/m‘ as a
function of d for three different values of c (= 2.5 A, 5.0 A, 7.5 A) for the tubular
channel. Similarly, in fig. 2.12 we give m/m‘ as a function of d for the 2D case.

For small values of d, m/m" —-> 0 but as we increase d, there is a rapid increase

1.0L;

o.a
*
E o 6
\
E
0.4
0.2
0.0 '

25

 

 

 

 

...,....,....,....,...A
b Wgfi+** 4’
i \ \lSl ~+‘*“"+~+—-4>-+—+—-'+"‘*“+
L
i- \ hm Bid
, \
. \b at: a’D/
: \ \GBg—D’B
_ \ l-
' \ l‘
I it ¢
r- I q
_ 0d=3.5A° \ ,4” —
* \q 9 .
Cld=5.0A° \ o/O’
— + d=6.5A° so“). 9 4. 20’0” _
l r .l r l - . l .
o 2 4 6 a 10
0
C(A )

Figure 2.7: The c-dependence of m/m‘ for three different neck diameters in the
2D case. Dashed lines are guide to the eyes.

26

 

 

 

 

_ IT I ' l I
i
1 0? B <51 _
. sf".
\
. ‘ \\ ]
_. ' \ _ o _
0.8 § \ <> d—3.66A
[ m ‘
' ’ ‘ c1 d=5 12A° ‘3
l \ ' E
1* . l‘ \ L
\ I
' | \ I
\ l \ I .
E \ \ d
. \ \ l’ .
\ <>
0.4 " b [‘3‘ ’2) .1
’ l \ .2, I
l t! / i
\ \ ’2’ 'i
_ \ \GL S EJJJ’ #
0.2 _ “ ‘El—D—El' / _
i \ X"
b 6
l /
[ \ 0. ‘° ’°
0 0 WWW—‘91 A l A A A A
' o 2 4 e a 10
o
C(A )

Figure 2.8: The c-dependence of m/m’ for two different neck diameters in the 3D
i.e. tubular channel case. Dashed lines are guide to the eyes.

27

 

 

 

 

 

1.5 — 0 (a) ‘
. o .
° 0: d=3.66A°
1.0 ~ ° .
. o .
O

A O
N i o .
0 0.5 - <> 0 O .
< r O O O [

E . <> 0 O o.
N .
\ , ,
AG (9,0 : i : :

Ci o o (b)
.'—‘ > O 00 J
v 0.8 ~ 00 0: d=3.5A° .

t O .
c. > D: d=5.0A°
.H , O
E . 4»: d=6.5A° .
m 0-6 ‘ <> ‘
<> 1
r [j [j *
0.4 + DDDDDD ° o +
1:: o .
D D O o l
[ '4’ + + + + .¢. + o
t + + + DE] 0
0.2 L’ + + 4.. D 1:] § 0 o -
: + + D+D+D
l
0.0 1 l r A I g L
O 2 4 6 10

Figure 2.9: The c-dependence of Em," for 2D and 3D case. (a). 2D. (b). 3D
tubular case.

28

    
 

(b)

Figure 2.10: Square of the wave function (14’) in a unit cell of a particle moving in

3.5 A. For these parameters,

b=10 A, (a). c=2.0A andd
cage length is smaller than the neck diameter and m/m‘ for this case 2 0.83 2 1.

(b). c = 6.0 A and d

a channel for k = 0 corresponding to the 2D case. The channel parameters of the

unit cell are: a

3.5 A. For these parameters, cage length is larger than

the neck diameter and m/m‘ 2 0.19 < 1.

29

in m/m“ to 1. The sharpness of this increase depends on c, the increase being
sharpest for small c. The value of d at which m/m‘ = 0.5, initially increases with

c but then tends to saturate at about 6 A.

After discussing the geometry dependence of the effective mass, we would like
to briefly discuss the a dependence of the energy, particularly the ground state
energy, E, and the effective mass ratio. We have calculated Em,“ and m/m‘ by
changing the overall length scale (a) but keeping all the ratios b/ a, c/ a, d/ a same
and we find that Emgn scales as inverse square of a (see fig. 2.13, the insert is a
log-log plot) while the effective mass ratio m/m' is unchanged (see fig. 2.14), the

small differences are due to numerical inaccuracies.

2.3.2 Analytical fits to m/m*

In order to express the geometry-dependence of m/m‘ in a simple analyti-
cal form, we have attempted to express m/m‘ as a function of the scaled vari-
ables b/a,c/a,d/a. In our calculation, we have fixed b/a = 1 and changed
0 < c/a,d/a < 1. When d/a is small and c/a is large, one knows that m/m" —> 0
exponentially as d/ a —> 0 due to tunnelling between the cage states. But to
capture the d/ a, c/ a dependence over a broad parameter space, we have used a
different functional form. In fact, a two parameter function that fits m/m‘ data
reasonably well is given by

m [1 _ Jas(1—Jé)]a(1—JE)
71—1.: 1 + (3%)”

 

where E E c/a, cf E- d/a and the two parameters are a and n.

30

 

. ' V Y ' l V V W W l ' r V ' I ' Tfif ' I V ' I '
1.0_ e 44‘3’9'4

 

 

 

or

i 9’ i

. of .

0.8— f,’ 9/ _

p I 1

. / .

an: t I, )1 .

E 0.6— ¢ 5 -
\ I
I

E 7 I 4

0.47 I ‘

i. I I, 4

. , 41 o c=2.44A° ,

02— I“ o A

- b I (I D c=4.88A 1

. é [Ff + a .

_ I g c=7.31A ,

0,01I—o—I—IJHI—‘521--A-‘--A-l--AA’

D 2 4 B 10

Figure 2.11: m/m" as a function of neck diameter d for three specific values of
the cage length c for the tubular case. Dashed lines are guide to the eyes.

31

 

 

 

 

f T W I I ' v Y I v I
r
1.0 _ 0 ‘04,..0—0-
/d /?l
[21’
. 6’ /)" .
0.8 — / a, _
' ’ ll ‘
I’ II
. 7’» I?
an:
S 0.6 - I f _
[' I -i
\ _ I é .
Ei ' l
i ' .
o I I,
4 i— 4, _
L 7‘
L I ’1 _ o
I I] 0 0—2.5A
I f]
P I I = o
0 2 _ I ,/m D c 5.0A _
v- I
, r,’ + c=7.5A°
I I
p d I! # 4
(rob—MEI: - I - ' - . '
0 2 4 6 8 10
O
d(A )

Figure 2.12: m/m‘ as a function of neck diameter d for two specific values of the
cage length c for 2D case. Dashed lines are guide to the eyes.

32

 

 

2.0 [- ? A
I 200<§""' ""'”¥
I El \ o; d/a=0.49
L I 100: \:\ 1
’ | \ \ D: d/a-O.73 J
I 0.50 - \ \>\ q
I I In \ i
\

In
OI
I
I
/
1
I

 

 

 

 

 

 

I 0.20
A E" h\ ,
N I \Q
° 0 04°? ax 1 .
E (l I WE ‘
005- °-
02 " ’ 8‘
\ [ 'l I I I 4 - -1 --1-
x: ,I 5 7 10 20 30
1.0’ ‘\ -I
.E I‘ .
I‘ +
E I \‘ .
12:1 \ ‘
\ \
\ \
0.5“ \ b ..
\ \
\
El \
\
\ \ ’
\ ‘9.
\SZ\\
[ ‘ ‘232228
0.0 l ‘ ‘ 1
O 10 20 30
o
a (A )

Figure 2.13: Ground state energy as a function of a. The insert is a log-log plot
which shows Em," ~ a". Dashed lines are guide to the eyes.

33

 

 

 

 

1.0 f ‘17 T
]
J
l
0.8 t ..
0: d/a=0.49
D: d/a=0.73
l
0.6 ' a
an:
E
0.4 r ]
Cl
[3 D o
P O D O
0 i
0.2 _ e
0.0 L A A a l - A A A l . A A L L A A A 4 L A - . ‘_
0 5 10 15 20 25
O
a (A )

Figure 2.14: m/m“ as a function of a which shows that the effective mass ratio
does not depend on a (b/a and c/a are fixed). The observed variations with (1
represents the inaccuracies in our calculation of m/m".

34

In fig. 2.15 we compare the results of Eqn. 2.3 with the numerically calculated
values of m/m‘ as a function of E for two different values of (I (=0.366, 0.512). We
see that the parameter (1 controls the sharpness of the rapid decrease in m / m" as
5 increases from 0. For large values of E, m/m‘ is relatively insensitive to a. In
fig. 2.16, we give similar results but fix a and vary 12.. Clearly the m/m’ values
depend sensitively on n as E —> 1 and there is hardly any n—dependence for E < 0.4.
The parameters which fit our calculated results well are a = 18 and n = 4. One
can obviously fit the data better by increasing the number of parameters but one

does not necessarily gain any additional insight.

2.3.3 Effects of attractive potential due to adsorbed ions
on the cage surface and silicate framework

In many zeolites, there are positive ions distributed inside the structure which
can provide additional attractive potential to the atoms moving in the channels.
The nature of the attractive potential V(r) produced by the positive ions on
the channel wall and silicate framework depends sensitively on the location and
number of positive ions inside the zeolite cage. As an example, let us discuss the
case of K-L zeolite. Depending on the number of K + ions, there will be several
minima in V(r) as one goes around the wall of the cage. Fig. 2.17 shows the
azimuthal potential distribution of K—L zeolite as the number of K + ions changes.
To represent the general features of this ion-induced attractive potential, we add
to the channel in fig.2.5 a potential of the form

—%Zwr—2-jsin2(p¢), 0 < z < c

0, otherwise

V(r,¢,z) = [ (2-4)

35

 

 

 

 

 

Figure 2.15: Comparison between the two—parameter function (Eqn.2.3) for m / m“
and the calculated m/m‘ as a function of c/a for different values of d/a; varying
parameter a and fixing n.

1.0L " V,

36

0
I

 

0.8 . q.
*8 0 6 L
\

 

: d/a=0.366

a :d/a=0.512
— —: a=18., n=3
—. a=18., n=4

a=18.. n=5

 

 

/
/
/
/
/
/
/ I
/
/
OAT “ , .
° /
/ I
P ‘ / /
/ ' //'
/ I /
. 3 / a // 04
0.2 D D " / /
\‘J/ /
/
/
0 / Ox/
/ /
’ o
00 l . owe ‘Ar .3 0.91 O .0 . I a A
0.0 0.2 O. 0.6 0.8 1.0
c/a

parameter n and fixing a.

Figure 2.16: Comparison between the two-parameter function (Eqn.2.3) for m/m"
and the calculated m/m" as a function of c/ a for different values of d/a; varying

37

(4) FOR Ho R-5.oo z- 0.00 A:
I v I I I v r ' v V v

 

      
  

 

6k"

V(K)

 

 

(b) FOR Ho R-5.00 z- 0-00A‘
0 . . I . - . I 1 1 .

- 1| 2 III[ q
Al [11 Ill Ill .“

 

V(K)

'IIIIT'VII'II

 

 

7%
.

(c) FOR H. R-2.50 z- 3.76 A‘
o 1 - . - r v . l . . r

6 K+
41¢

V(K)

IIIVVII'I'llill II

-400

 

_5°°....l. I

Figure 2.17: The azimuthal distribution of He - K-L zeolite potential with or with—
out K + ions. (a) and (b) show the potential along the center of the channel wall.
(c) shows the potential along the wall of neck. 0K+, 4K + and 6K + correspond
to 0, 4 and 6 K + ions silicated on the cage wall, respectively.

38

In the above, 1' is the radial distance of the quantum particle, for example a He
atom, from the axis of the tube, V}, and p characterize, respectively, the strength
and the angular periodicity of the attractive potential. Here we have assumed, for
simplicity, that the K + ions do not affect the helium atoms when they are inside
the neck region. In a more realistic model, one may have to relax this simplifying

approximation.

In fig. 2.18, we plot the square of wave function for a particle moving inside a
tubular channel as a function of the angle along the cage wall when the attractive
potential in Eqn. 2.4 is present. In fig. 2.18a, we choose p=2 and V}, = 560. In fig.
2.18b, we choose p=3 and V0 = 580 but set two of the potential peaks zero. One
can see very clearly that the quantum state of the particle depends very strongly

on the potential shape.

To understand the full effect of the attractive potential, we have chosen as
beforea=b= 10 A.Wefixc=6A,d=5A,takep=2andvaryVoinEqn. 2.4.
We find that when V0 = 0, m/m‘ 2: 0.19, when V0 = 2a., (co = hz/ZmAz), m/m“ 2
0.06 and when V0 = 5.080, m/m‘ = 0.03. Thus, an attractive potential produced
by the cations located on the cage wall tends to increase the effective mass. The
underlying physics is quite simple. When V0 = 0, m/m‘ << 1 indicating that
inter-cage tunnelling rate is small. As we turn on V3, the particles get attracted
towards the wall thus decreasing their inter-cage tunnelling probability and hence
decreasing m/m‘. As VI) becomes sufficiently strong, the particles get trapped
near the cage wall and do not contribute to the mass transport along the tube
axis. However, these trapped particles can dominate the low-T thermodynamic

properties, particularly at low He concentrations as we will discuss in the following

39

 

 

 

 

 

 

0.025 I .
I \ D
II I\ ’A‘ n i
I
I ‘ I \ I
0.020 - , I I ‘\ I I l \ .
0.015 - I ‘| i I , ‘ I I j
I I I
I l ' i : l‘ I I
0 010 ~ I “ ,' I I I I “ q
. I ' ' \ I ‘ I
0 005 I \ I ‘ I \ I ‘ ‘
’ \ , \ l \ \
N O 00 I - \ / \ * / f\ __ / x
9 0202.9, - -
'I \ /‘ '\ J
I I .
I |
0 020 ~ I , I . ' ‘, ' '. 1
l I l l I. | l I ‘
I I I ‘
I I I .
0.015 . : I I I : ', : . ~
I | '
» . I . . ' ‘. . '.
. I .
0.010 7 ,' I f I‘ I I ,' I 1
I I l I \ I l‘ l \\
. I , ‘ I I
0.005 _-. ‘\ , , I \\ I \\ -
I \I ‘ I \I
. \ I \\
i "- / n ‘ 44
0.000 4 ‘ * m
0 2 4 6

Figure 2.18: Square of the wave function (11.2) at k = 0 in the center of the channel
as a function of azimuthal angle 4’ with a fixed r. The channel parameters are:
a = b = 10 A, c = 6.0 Aand d = 5.0 A; (a). with attractive potential p=2 and
Va 2 5.060 (b). with attractive potential, V0 2 5.060 and p=3 but two of the
peaks in the potential were set to zero.

40

two chapters.

The square of wave function u.2 E 1,02 for k = 0 is plotted in fig. 2.19 for the
three above values of V0. When V0 2 0, the cage size is large enough such that the
particle spends most of the time inside the cage (m/m“ = 0.19). However, u2 is
reasonably large near the tube axis indicating an appreciable inter-cage tunnelling
probability. As we increase V0, u2 near the cage axis tends to decrease and for
V0 2 5.080, the particle spends most of the time away from the tube axis, i.e. they
are trapped near the cage walls. In this case, inter-cage tunnelling is practically

zero. The excitations of these particles then come from tunnelling around the

cage wall, an intra—cage process.

2.4 Excited state

It is also interesting to investigate the excited energy bands and the wave functions.
Since the method we used is good for only low-lying excited states (because of
numerical accuracy), here we only present our results for a small number of these
excited states. We show, in fig. 2.20, the ground state, the first and the second
excited state energy bands for the 2D case with a = b = 10 A, c = 7.5 Aand
d = 5.0 A, and without the attractive potential. The unusual feature is the near
absence of dispersion of the first excited state. Physically it means that if we can
excite the particle from the ground state to the first excited state, the particle

will get localized.

41

 

Figure 2.19: Square of the wave function (uz) at k = 0. The channel parameters
are: a = b = 10 A, c = 6.0 Aand d = 5.0 A; (a). without the additional attractive
potential (Vo = 0); (b). with attractive potential, V}, = 2.060; (c). with attractive
potential, V0 = 5.0eo. Here co = hz/2mA2.

42

 

 

 

 

 

I <3
o o l
0.5" 4
. o <>
<> 0 l
l o 0 I
‘2?" 0.4? 0 -
a <> <> 0 i
N I
c0\ 03 I
v
>~I
an .
I I
cu o o 0000 000 o o
c: 02- ° ° «
EL] .
i
0.1" .
0000 00000 ,
I <>°<><><><><><><><><>° ,
0.0 ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ *
—O.4 —O.2 00 0.2 0.4-

Figure 2.20: Energy bands for the ground state, first excited state and second
excited state for 2D case with a = b = 10 A, c = 7.5 A and d = 5.0 A.

43

2.5 Summary

In summary, we have investigated the effects of geometrical confinement on the
effective mass m‘/m of quantum particles moving along tubular channels. We
find that m.“ / m is a highly nonlinear function of the geometrical parameters such
as cage size c/a and/ or neck diameter d/a. There are large regions of parameter
values where the m‘ / m is quite large. Consequently any small perturbations like
defects, scattering from thermal vibration of the wall or inter-particle interaction
will have a strong effect on the mass transport along the channel axis. The effect
of positive ions embedded in the channel walls can also be very significant. In
particular, in the low helium atom concentration regime, helium atoms will be
trapped in states near the wall with an extremely low probability of inter-cage
motion. These states will not contribute to mass transport but will show up in

low-temperature thermal excitations [44].

Chapter 3

Low-Temperature

Thermodynamic Properties of
Helium Clusters in K-L Zeolite

3. 1 Introduction

Properties of 4He and 3H6 atoms confined to move in restricted geometry have
been of considerable interest during the last several decades. Examples of con-
fining media are Vycor [50], aerogels [67], zeolites [54]-[60], fullerites [61] and the
surface of graphite [68]. Some of the basic questions that have attracted both
theoretical and experimental attention include the nature of Bose-Einstein con-
densation in porous media (singly and multiply connected pore structure) [69]-[72],
quantum transport through microporous channels [73], and the ground and ex-

cited states of quasi one-dimensional bosonic and fermionic quantum liquids, the

so called Luttinger-liquids [74, 75, 76].

In chapter 2, we have discussed the nature of single particle states and the
effective mass of transport (m‘) through a simplified 1-dimensional zeolite chan-

nel. We argued that in the presence of a strong attractive potential produced by

44

45

the ions located near the cage wall, the particle gets bound near the wall and the
effective mass associated with intercage motion is large (m‘/m —> 00). In this
case the particles are trapped inside the cages and the motion is primarily along
the cage walls perpendicular to the channel axis. In this Chapter, we discuss the
results of our theoretical studies on the low-lying excitations of small numbers
of 4He and 3H8 atoms (Helium clusters) tunneling inside a single cage of K-L
zeolite along the cage wall. These low—lying excitations have been obtained using
Bose-Hubbard (BH) and Mott-Hubbard (MH) models [44] with strong intrasite
repulsion and finite intersite attraction. For small number of particles and at low
enough temperatures, these systems can be represented by one-dimensional BH
or MH rings containing a finite number of sites. Differences between ‘He and 3H8
arise due to mass and statistics peculiar to the finite number of sites on the ring.
We show how, in principle, these two effects can be disentangled. Calculated tem-
perature and concentration dependences of the heat capacity agree qualitatively
with experiment, but a quantitative comparison suggests that effects of disorder
are extremely important at very low temperatures which will be discussed in the

next chapter.

In section 3.2, we give a detailed description of the physical system and what
is interesting to us. In section 3.3, we give some results from experiment done by
Kato et.al. [58]. In section 3.4, we give the models that we have used based on
the physics that we have described. Then, we give our numerical results in section
3.5 and compare our results with the experiments of Kato et.al. In section 3.6,

we summarize our work.

46

3.2 Physical system

In chapter 2, we have shown a schematic picture of a typical K-L zeolite chan-
nel. In fig.3.1, we give a slightly different schematic picture of the cross-section
of the one-dimensional channels of K-L zeolite. In this figure we also show a
few He atoms trapped near the cage wall. The K-L zeolite crystal has a hexag-
onal lattice structure with lattice constants of a = 18.4A and c = 7.5A, and it
has one-dimensional channels along the c axis [58]. Each channel is composed of
cages, about 13A in diameter and 7.5A in length. These cages are interconnected
through apertures of diameter about 7.4A. On the cage walls there are K + ions
(large circles in fig.3.1) which exert an attractive potential [43] on the He atoms in
addition to the potential produced by the Si (Al) and O atoms of the zeolite net-
work. Also the K + ions block the pathway for the He atoms connecting different
channels. The potential produced by the zeolite cage gives rise to binding sites
(in this case eight) for the He atoms; these binding sites (and associated localized
cage states) are arranged in a ring geometry along the cage wall shown as small
circles in the figure where we also indicate that some of these binding sites are
occupied by He atoms (medium circles). Our aim is to develop a suitable theo-
retical model for this system and to understand how the difference in statistics
(bosons and fermions) shows up both in the ground and excited states when the
atoms are confined to move inside the cage in a ring geometry. But before that,

we would like to briefly summarize the main experimental observations.

47

 

Figure 3.1: Schematic picture of a cross section of K—L Zeolite channel showing
the silicate framework and ring arrangement of He adsorption sites (small circles).
Large and medium circles are respectively K + ions along the channel walls and
adsorbed He atoms.

48

3.3 Experimental results

Kato et.al. [58] have made detailed measurements of the low—T (T < 5 K) heat
capacity of He atoms adsorbed inside K-L zeolite. Fig. 3.2(a) and (b) show
their results of the heat capacities as a function of He concentration for fixed
temperatures (fig. 3.2(a) is for 4He and fig. 3.2(b) is for 3He). Their results can
be broadly divided into two regimes; regime 1 where < n >, the number of He
atoms/ cage 5 nc, regime 2, < n > > nc. In regime 1, they find the following:
(i) for fixed T 3 2K, the heat capacity/atom, C vanishes at both < n >= 0,
and < n >= nc = 8 and is a relatively flat function of < n. > between these
limits. In this regime C is nearly symmetric about nc / 2 for 3H6 but not so for
4H e. (ii) For a given < n >, C is a monotonically increasing function of T. (iii)
At same T and < n >, C for 3H6 is larger than that for 4He by more than
25 %. A careful analysis of the experimental data suggest that in regime 1, He
adatoms are trapped near the wall of the cage (we will denote it as cage state).
The number of cage states is 8 per cage. In regime 2, where nc << n >< 12, the
excess He adatoms over nc, i.e., < n > - nc, move along the channel axis in the
presence of ne He atoms bound in the potential minima of the cage, and behave
as a one—dimensional quantum liquid [74]-[76]. When < n > is increased above
about 15 atoms/ cage, the heat capacities become small for both 3H8 and 4H6.
This indicates the freezing out of thermal motion of the He adatoms which form

a solid-like structure.

In this chapter we study the thermal-excitations of He atoms in regime 1 where
only the cage states which are localized along the cage walls are thermodynami-

cally relevant.

49

 

 

 

 

 

 

 

 

 

4He in K-L Zeolite c vs n
5 b T= const.
1. _ 1.4. K _
2 . Iatom/coge °
g . 13K = 0.217 mmoI
éB— . 1. K r
0 ° 0. K '
2 b , ' 0.6 K I
1‘ ”13 f: . . 0.4K - . "
:3 _ ;;.;4$1_3h._:__:_
00 5 10 1 20 2‘5
(ot0m5/coge)
b r I I I I
5+- 3He in K—L Zeolite C vs n _
L.Ki T=COHSL
SE 4” 1. , "*
2, I. K I atom/cage
E 3.. - ' =0.217 mmol fl
V 0.8 K
o ' . ‘ '
2' /.‘ - 06 -*
1 r “I % . 0'4 . ' _ .4
" “’“ (3; =A§_§ ;-
O0 5 10 15 0 25
n (atoms/cage)

Figure 3.2: The experimental results of heat capacity of He atoms as a function

of T for fixed He concentration < n > by Kato

et.al. [58]

50

3.4 Models of the system

As discussed in the previous section, at low temperatures (T < 10K) and for
sufficiently small concentration n, He atoms are bound near the wall of the K-
L zeolite cage. The probability of He atoms going from one cage to another is
extremely small because the barrier for this inter-cage motion is estimated to be
about 150K from the isosteric heat measurements [58]. The dominant mode of
thermal excitation at low-T is therefore tunneling from one binding site to another
inside a single cage. These bound states are referred to as the cage states [43].
When n is increased beyond a critical value 116, n.6 He atoms fill up all the cage
states and the additional n — nc atoms move in the region near the cage axis
and go from one cage to another. These states will be referred to as the channel
states. Of course, at higher temperatures, the atoms trapped in the cage states
get thermally excited to the channel states and undergo intercage motion. As
has been stated above, our interest is to study the thermal-excitations of low-
temperature cage states. Our model is designed to study this low temperature (T
< 10K) behavior of 3H6 and 4He in the concentration regime 0 S n S nc. At
these temperatures, the channel states are practically unoccupied and one has to

deal with the statistical mechanics of cage states only.

The excitations of the 4H8 system in the manifold of cage states can be de-
scribed by an extended Bose-Hubbard mode1[44, 71, 72]. The Hamiltonian for

this model is given by

N,“ U Nm Nm
H = —t 2(bfbpr1 + h.c.) + 3 217.502.,- — 1) + VanngH, (3.1)

3:1 i=1 3:1

where Nm is the number of binding (localized) sites which equals to 8 for the

51

present case, b.- (b,+ ) destroy (create) a boson (B) at the ith localized site, U is the
repulsive energy between two He atoms occupying the same binding site and V is
the attractive energy between two atoms occupying neighboring sites. The boson
creation and destruction operators satisfy the usual commutation rules and the

number operator for the ith site 17.,- has eigenvalues 0,1,2, etc.

The 3H6 system is similarly described by an extended Mott-Hubbard model
[40, 44] and the corresponding Hamiltonian is given by
N". Nm

Nm
H = —t Z(f$fi+la + hue) + [12114an + Vanng+1 (3.2)

.3: mi
Here the fermion destruction (fic) and creation (fit) operators associated with
state i and spin 0' satisfy the usual anti-commutation rules and ng, can have
eigenvalues 0 or 1. The total number of fermions at site i is n,- = ngT + nu. Since
the spin exchange energy is quite small (about t2 / U ~ 0.01K), we first ignore
the spin degrees of freedom of the 3H6 atoms and treat them as spinless fermions
(SF’s) and later we will discuss the effects of including spin on the many— particle
energy spectrum in the limit of large U. In fact, He—He repulsion is quite strong
when two atoms occupy the same binding site, we therefore let U —+ 00. Then,

n.- can have eigenvalues 0 or 1 for both bosons and fermions. In this limit both B

and SF systems can be described by a single Hamiltonian.

N". Nm
H = —t 2(c2'qH + h.c.) + Vanng+1 (3.3)
i=1 i=1
where c,- = b.-(f,-) and 11,-: ci'c; = 0 and 1.

This hamiltonian can be diagonalized exactly numerically for different values of
Nm
Nm and n 2 Eng. The heat capacity is then calculated by using the fluctuation-

i=1

52

dissipation theorem:

1
C(n, T) 2 Wk E(n,T)2 > — < E(n, T) >2), (3.4)
where
EEK-fig”
J'
< E(n,T) >= w,
j
and
ZEfe’fiE"
< E(n,T)2 > j

= —Ze—fiEj
J'
The summation goes over all the energy levels of the n particle system.

3.5 Results for Bosons (B) and Spinless
fermions (SF)

3.5.1 Difference between B and SF due to statistics

Although the Hamiltonians for B and SF have the same form, the differences
between B and SF cases lie in the commutation properties of the operators and
the transfer energy t arising from the 3H6 and 4He mass difference. For the same
t and V, the spectrum of bosons and spinless fermions are identical for an open
chain [77]. For the ring geometry, the statistics induced differences between B and
SF systems can be seen by writing down the Hamiltonian matrix in localized (on

the ring sites) representation. We denote the system of Nm sites and 11 particles

53

as (Nm,n). For the same t and V, one gets identical matrices for the two systems
when n is odd [77]. Therefore, the corresponding energy spectra and C are also
identical. However, when n is even, they differ considerably. For example, the

hamiltonians for 4 site 2 particle systems are:

 

 

 

 

(V000tt\
OVOOtt
OOVOtt

HB‘ OOOVtt
ttttOO

\ttttOO/

whereas
{V000t—t\
OVOOtt
OOVO—tt
HSF‘ OOOV—t—t
tt—t—tOO
\—ttt—t00)

Thus, in addition to the mass difference, the different statistics can give rise to
differences in the heat capacity of the 3H e and 4He systems in this ring geometry
due to the difference in energy spectrum for even-n values. In fact we will argue
that the difference in the statistics can explain the observed [58] trend in C whereas

the mass difference goes the other way.

An example of the energy spectrum is shown in fig.3.3 for the system (8,4). It
has 8! / 4!x4! = 70 states. Ift is finite and V=0, then the SF states can be obtained
simply by singly occupying the one particles states k,- = (1r /4)(0, :lzl, :l:2, :l:3,4)
with energy —2tcosk,-. But if V 76 0, one must diagonalize the 70 X 70 matrix.
The lowest 20 states for both V = 0 and V at 0 are shown in fig.3.3. It shows that

54

the energy spectrum depends sensitively on both the statistics and the intersite
attraction. One characteristic feature of these results is that for SF, each energy
level is 2n-fold degenerate whereas for B, the levels can be both 2n and 2n+1~fold

degenerate.

In fig.3.4, we show the T—dependence of the molar heat capacity N AC (n, T) / n
for the Nm = 8 system for different even values of 11. Here, N A is the Avogadro’s
number. Bosons are found to have much smaller heat capacity than spinless
fermions. The reason for this result is that for the same t and V, the lowest
energy gap for the bosons is found to be considerately larger than that for the
spinless fermions whereas the ratio of the degeneracy of the first excited state to
the ground state is 2 for both the systems. (For odd n, B and SF systems have
identical heat capacity). We suggest that one should be able to see this difference
between even and odd-n systems experimentally. In fact, one can use the odd
11 results to extract the effect of mass difference (through t) and in principle
disentangle the effects of mass and statistics on the heat capacity. In practice this
is not easy because due to fluctuations in n from one cage to the other [58], one

usually measures C as a function of average 11.

3.5.2 Difference between B and SF due to mass

To see the difference between B and SF due to mass, we choose boson statistics
and vary the t parameter(which depends on the mass). Fig. 3.5 shows the heat
capacities for a (8,4) system with bose statistics and two different values of the
tunnelling parameter t. It can be seen clearly that when t is small, i.e. when the
mass is large, the heat capacity is large. The same qualitative results are obtained

using fermion statistics. This means that if only mass difference is taken into

55

 

t=14K, V20 t=14K, v=—24K
—50
* SF bosons SF bosons .
— 2 "
_ — — 4
20 — 4 1 - —60
I
—— a _ 2 l
— 1 — 2 .
J _
A _ __ 2 [ 70
M 8 __ .
v—40— —— 2 3
>»
00 —2 — 3 ‘ -80
b — 4
2
m 2 __—-— g
- -90
-60— — 4 1
— 2 —— 2 — -100
1 -—— 1
-80 -110

 

 

 

 

Figure 3.3: Lowest 20 (out of 70) energy states of spinless fermions (SF) and
bosons (B) for particles in a 8-site ring. The hopping parameter t = 14K and the
intersite attractive interaction V is OK and -24K, U —> 00. The numbers besides
the energy levels indicate their degeneracy.

c (J/K/mol)

56

 

1.0

 

: spinless fermion —

: boson

   
   

 

 

T (K).

Figure 3.4: Temperature dependence of the molar heat capacity for 8-site n-
particle spinless fermion and boson systems. The parameter values are t = 14K
and V = —24K.(For a discussion of the parameter values, see text)

57

consideration, the heat capacity of 4He should be larger than the heat capacity
of 3H8.

3.5.3 Comparison with experiments

To compare our theoretical results with experiments of Kato et.al. [58] we note
that C was measured as a function of < n >, where < n > is the mean occupation
number of atoms / cage. In order to obtain a uniform distribution of He atoms
inside different cages, Kato et.al. [58] heated the system to a temperature To in
order to facilitate a nearly uniform density of He atoms throughout the system and
then measured C(< n >,T) vs T at T < T,. At T < 2K where C was measured,

the intercage equilibration is extremely slow. We therefore assume that

C(,=<n>T) ZO(nT)P T) (3.5)
n=0
where
s
< n >= ZnP(n,To),
n=0
and
_ Z(n,To)z”
P(n’T°) _ L(z,To) ’

L(z,To) = Z Z(n,To)z"

n=0

2 : efiou : efl/kBTo.

where p is the external chemical potential. In other words, the distribution P(n,T.,)

of He atoms in different cages is governed by the quenching temperature To (E’

58

 

0.6 — — t=14K / _.
— — t=12K I ‘
/
I
.I
/
/ I
'3 0.4 — l -—
E /
\ /
ix:
\ /
3
>
U

0.2 —

 

 

 

 

Figure 3.5: The molar heat capacity for the (8,4) system with bose statistics and
different t, V=-24K.

59

20K in the experiment of Kato et.al. [58]) whereas C is measured at temperatures
T << To. At these T, different cages have different number of He atoms, this

distribution corresponds to the higher temperature To.

As regards the parameter values, we have estimated (see appendix A) V to
be about -20K to -25K. A large component of this attraction comes from the
3-body He—K+-He interaction. The hopping parameter t was chosen such that
the heat capacity for both the He systems is of the same order as the experiment
(2 0.6J/K/mol at T 2 1.5K). We have fixed t=14K and V=-24K for our numerical
calculation of C. The bandwidth W=2t is of the same order as the magnitude
of V. With the same parameter values, i.e. when the mass effect is not taken
into account, we calculate C for both 3H6 and 4H e systems. The effect of mass

difference will be commented upon later.

For comparison with the experimental results [58], the computed theoretical
values of C (heat capacity / atom) multiplied by a constant, which the experiment
gives as 1 atom/cage=0.217mmol. Fig.3.6(a) gives the theoretically calculated
values of C(< n >,T) as a function of < n > and T for three different T values
for the spinless fermions. As expected, C = 0 when < 1:. >= 0 and 8 when the
sites are either all empty or full. The rapid increase in C as one moves away from
these two limits and a relatively flat structure of C for 2 << n >< 6 is reproduced
nicely in our calculations. The theoretical values agree well with experiment at
1.5K but are about 30 % smaller at lower temperatures. Fig. 3.6(b) gives the heat
capacity vs. T curve for fixed < n >. The insert is the experimental results. Our
calculated results have captured the main features of this sytem although we still

have some discrepancies at very low-T. Fig. 3.7 shows the discrepacies between

60

experimental and our results.

For the bosons, we also see a similar n—dependence but with the choice of the
same parameter values, the theoretical values of C are much too small compared
with the experiment (see fig. 3.8). Also, for the bosons, a small particle hole
asymmetry is observed [58] in the experiment whereas our model has in-built
particle-hole symmetry. Our calculated C drops much faster with T compared
to the experiment and for T < 0.75K, the theoretical values are practically zero.
A plausible reason for this may be that the value of t is too large. Since 4He is
heavier than 3H8, we expect the value of t to be smaller for 4He. This will reduce
the excitation energy and hence increase the low-T heat capacity. For example,
for the system (8,4), if we choose t=12K instead of 14K, the lowest energy gap
(A) changes from 10.3K to 7K. This reduction in A increases the heat capacity
at 1K from 0.015 to 0.189 J/K/mol, an increase by nearly a factor of 10. We find
that for < n >= 1.67, our theoretical value is 0.344 J /K/mol compared to the
experimental value of 0.4 J / K / mol. However, the theoretical values are still too
small at temperatures less than 0.5K. From the above discussions, we see that (i)
the difference in statistics leads to a larger heat capacity for the spinless fermions
whereas the mass difference, through a smaller excitation energy gap, leads to a
larger heat capacity for the bosons, and (ii) theoretical values are smaller than

experiment at very low T.

3.5.4 Spin effect

In the above sections, we have neglected the spin of 3H6 and treated them as
spinless fermions. In this section, we will discuss the effect of including spin of 3H6

on the low-T heat capacity. Inclusion of spin for 3H6 dramatically increases the

61

 

 

0.5 _
<> : T=1.4K
O4”- ———$:T=1.0K
+ : T=0.6K

 

 

 

 

 

(b)

 

 

 

c (J/K/mol)

 

 

 

 

Figure 3.6: (a).Heat capacity for the spinless fermions as a function of < n >,
average number of particles per cage, at three different temperatures. The symbols
are experimental values for 3H6. Theoretical values have been obtained using
t = 14K and V = —24K. (b).Temperature dependence of the heat capacity for
spinless fermions for three different values of < n >, average number of particles

per cage. In the insert we give the experimental results for 3He obtained by Kato
et.al. [58]

62

 

C (J/K/mol)

 

 

 

 

Figure 3.7: Temperature dependence of the heat capacity for spinless fermions
for three different values of < n >, average number of particles per cage. which
shows the differences between our calculated results and the experiments by kato
et.al. [58].

63

 

 

 

 

 

 

 

0.5
. ----- <> - T=1.4K
0.4 _. — — — *9 T=1.0K
A L 4» T=0.5K
M : <>
\ 0.3 _ O
E I
. <>
» o &«
v 0.2 f- , ----- n _ _______ 4; _. ..... a
U : ./' o Is \'
0.1 L -/ o *P + + + + ‘.
[f i + o ‘.
/ , __________________ \
00 f/m n n i l l m l .1.\.
0 2 4 6
<I1>
0.6 — *i' : <n>=1.'7
r_‘ b - _ + : <n>=3.3
g — - — <> : <n>=6.7
\ 0.4 -
ix: .
\ _
“"0 .
0.2 -
C.) .
0.0
0.0

Figure 3.8: (a).Heat capacity for the bosons as a function of < n >, average
number of particles per cage, at three different temperatures. The symbols are
experimental values for 4He. Theoretical values have been obtained using t =
14K and V = —24K. The values for T=0.5K are very small to be plotted.
(b).Temperature dependence of the heat capacity for bosons for three different
values of < n >, average number of particles per cage. The symbols give the

 

experimental results for 4He obtained by Kato et.al. [58]

 

 

64

dimension of the Hilbert space. For example, if we include spin in the system (8,4),
the manifold of states we have to consider is 1120 instead of 70 for the spinless
fermions. But, if the spin and translational degrees of freedom had decoupled [78]
(spin-charge separation) as in the case of an open chain geometry or an infinite
system (both with U = 00), then the T-dependence of C would have been identical
for the SF and fermions excepting for a delta function at T = 0 for the latter;
the total number of states associated with this peak being 24 z 24 for the open
chain with 4 particles. On the other hand, for the ring geometry, the spectrum
of SF and fermions differ from each other and in principle the heat capacities
should be different. Furthermore, for fermions, one has to take into consideration
the thermal equilibration between states differing in total spin quantum number
as in the classic ortho and para hydrogen (H2) problem [79, 80]. Computations
in the (8,2) and (8,3) systems suggest that, after taking into account the above
thermal equilibration (or lack there of) between manifold of states with different
total spin quantum numbers, the heat capacities for the SF and fermion systems
are comparable. In fig.3.9, we see that for the (8,2) and (8,3) systems, the T-
dependence of the heat capacities have similar behavior for SF and F and, C(2,T)
for F is smaller but C(3,T) for F is larger compared to the corresponding SF
values. Thus we expect that inclusion of spin and averaging over 11 will not
change C (< n >,T) very much. However we have not done this averaging for the
spinful fermions because of our lack of any detailed knowledge of the equilibration

problem (as in the case of otho and para hydrogen).

C (J/K/mol)

Figure 3.9: Comparison of the heat capacity between SF and F for (8,2) and (8,3)

. . I . I T T I I3.
I . ‘ /
1.0 _ —— : spinless fermion ’v ' - I 9—
.. ” p ‘
— — : fermion ,, X7
<>/ ‘23
. l, 0’ /
0 8 — p/ / n
. ’ 0 11:2 0/ l -I
D {1:3 0 ¢ d
/ / .
. d 1
0.6 _ / _
. " / /
9 P
/ /
/ /
. ¢ d '3 .
0.4 “' 0 / / —
b d/ // u
/ l3
h 0 / '1
0.2 — / _
L ¢ Ff a
/ / 1
4, JD '3 1
o a
/ - 5 )3/ "‘
0.0 - -‘-' " 2 _ 2 "—-"
0.0 0.5 1.0 1.5

65

 

 

systems. t=14K, V=-24K.

 

66

3.6 Conclusion

In summary, we have shown that the low-T thermodynamic properties of 3H6
and 4H6 atoms trapped inside the cages of K-L zeolite can be modeled by Mott
(Bose)-Hubbard rings. The ring geometry brings out nicely the differences be-
tween bosons, spinless fermions and fermions. This behavior should be character-
istics of He atoms trapped inside other zeolitic cages such as Na-Y zeolite [55, 56].
The disagreement between theory and experiment at very low temperatures ( both
for SF and B, particularly for the later) suggests that at very low T, effects of

disorder might be important, we therefore explore the effect of disorder next.

Chapter 4

Effect of Tunneling Disorder on
the Low Temperature Heat
Capacity of 3He and 4He in
Zeolite Channels

4. 1 Introduction

In the last chapter, we argued that the thermal excitation inside K-L zeolite of
He system within the manifold of the cage states (for n < 12.6) can be described
by Bose-Hubbard (for 4He) or Mott-Hubbard (for 3He) model with binding sites
localized on a ring. As pointed out before, the K + ions located close to the cage
walls and the silicate network provide the potential in which the He atoms are
bound. But the K + ions are not necessarily situated symmetrically around the
ring. As a matter of fact, there are six equivalent sites available for the K +
ions and a maximum of only four K + ions are available per cage to occupy these
sites randomly. In addition there is disorder in the silicate framework because
some of the silicon ions are randomly replaced by aluminum ions and K + ions

not located near the cage wall are also randomly distributed inside the silicate

67

68

framework. This random distribution of the ions can give rise to disorder in
the physical parameters of the Hubbard-type models describing the excitations of
the He system, namely the single—site binding energy, inter-site tunnelling matrix
elements (also called hopping parameters), etc. Consequently the low temperature
heat capacity will be affected by the disorder. In this chapter, we introduce the
effects of disorder in our model by treating the hopping parameters as random
variables and computing the average heat capacity of the He system using different

probability distributions.

The presence of disorder in the model, as noted in chapter 3, is expected to
improve agreement of the computed results with experiment. This is so because
the system without disorder has low-lying energy levels which are discrete (due
to the small size) and separated from one another. Suppose the lowest energy
gap is A. Then, the heat capacity of the system must exhibit Schottky-type of
anomaly [81], leading to an exponentially decreasing heat capacity at tempera-
tures T << A/kB. In the present case, the discrete energy levels of the system
calculated in chapter 3 lead to A/kg _>_ 5K. Hence one gets vanishingly small heat
capacity at temperatures below 0.5K. In the presence of disorder the low lying
excitation spectrum should be drastically modified and we expect an increase in
the density of low energy excitations arising primarily from the region of smaller
hopping parameters. This should result in an enhancement of the heat capacity
at low temperatures when a proper averaging over the various disorder states is

performed.

Before discussing our model and results, we would like to briefly review the

models of disorder which have been proposed to explain the enhanced heat capac-

69

ities seen in glasses (disordered insulators) at low temperature. A rather simple
model was proposed by Kaplan, Mahanti, and Hartmann (KMH) [82] within the
context of the standard Hubbard model with its three characteristic parameters;
single-site energy (6), tunnelling matrix element (t), and intra—site Coulomb in-
teraction (U). In the KMH model, the single-site energies (5;) are assumed to be
random, t = 0, U = constant. For a continuous distribution of a; one obtains a
heat capacity linear in T at low T. In the context of Hubbard-type models, our
present work is an extension of the KMH model taking into account a finite but
random t, constant 6;, U = 00, and a finite inter-site Coulomb interaction V.
The most general case is however when all the parameters of the Hamiltonian are

random.

Although mathematically our present model is an extension of the KMH model
[82], physically it is much closer to the one proposed by Anderson, Halperin, and
Varma (AHV) and [83] by Phillips [84]. This model was proposed to explain
the observed low-temperature specific heat varying linearly with T in a variety
of insulating glasses. The essence of the model was the hypothesis that in any
glass system there should be a certain number of atoms (or group of atoms) which
can sit more or less equally in two or more equilibrium positions. The thermal
excitation between the associated localized ‘tunnelling levels’ leads to a Schottky-
type heat capacity with a characteristic energy A. A statistical distribution P(A)
of these localized tunnelling levels (LTL’s) gives a linear heat capacity at low
T. Although this model describes the correct physics of the observed large low-T
heat capacity in insulating glasses, a microscopic picture of these LTL’s is not easy
to come by [85]. Our model of Helium atoms trapped inside zeolitic cages with

random tunnelling matrix elements indeed gives one possible microscopic picture

70

of these LTL’s.

It is in general quite difficult to compute thermodynamic quantities by intro-
ducing disorder in the Hamiltonian and then averaging over the disorder. The
ring geometry with a small number of sites in the present problem leads to some
simplifications, but even then, the computations are quite involved. Therefore,
only some calculations illustrating the important trends at low temperatures are
presented below. Nevertheless, the numerical evaluation of the effect of disorder
by introducing randomness and computation of the average heat capacities is an
important feature of the present work. It must be emphasized that the aim of our
work is to show the improvements in the computed low temperature specific heat
vis-a-vis experiment by including disorder rather than fitting the experimental

results accurately.

4.2 Model with disorder

Disorder in the system can arise due to various reasons. One reason could be the
random occupation of K + ion sites by the ions. Distortions, imperfections etc
would give rise to additional disorder. In terms of the Hamiltonian, there can
be randomness in the single-site energies, tunneling parameter t, the attractive
potential V and the intrasite repulsion U. We consider only the disorder in the
tunneling parameters and take single site energies to be zero. Then, the Hubbard

Hamiltonian in the limit U —> 00 may be written as
' Nm Nm
HB(SF) = - Z t;(c,?Lc,-+1 + hue) + V2(n,-n,~+1), (4.1)
i=1 i

where the tunneling parameters are random numbers which follow some distribu-

tion function f(t,-,a'), with a' defining the width of the distribution. Note that the

71

number operator n.- has eigenvalues of 0 or 1 in the case of both bosons (B) and

spinless fermions (SF) since the intra-site repulsion energy U z 00.

The heat capacity for a set of (ti, i=l,...,Nm} can be obtained as before through
C(n, T, {t;}) = (< E2 > — < E >’)/kT2. (4.2)

Here, the thermal averaging is done with the help of the exact partition function.
Next, one must perform the configurational averaging for disorder. We have chosen
both uniform and gaussian distributions but will report the results for the latter.

The gaussian distribution for the random tunneling parameters is giving:

f(t.-, 0) = e“““°”/"'- (4-3)

 

0' 7r

These distributions are symmetric around t,- = to. The heat capacity for the
canonical ensemble may be configurationally averaged over the disorder and the

concentration distributions as follows:

C(n, T) = g: f: f: C(n,T,{t;}P(n,To,{t;})f(t1,o')...f(tNm,a)dt1...dtNm(4.4)

For simplicity, in the numerical calculations we have assumed that P(n, To, {t.-}) =
P (n, To, {to‘v’i}). In this way, we can seperate these two averages. The integration
is performed numerically by using 6-point Gaussian quadrature. Finally, C(n,T)
of Eq. 4.4 is averaged over the distribution in number of particles/ cage in the

same way as discussed in chapter 3 using P(n, To, {toVi}).

4.3 Results and discussions

Before presenting our results for the 8—site system appropriate to the He atoms

inside K—L zeolite let us discuss a simpler system consisting of 4 sites and two

72

spin-less 3H3 atoms. This will help us elucidate the nature of the tunnelling
states in this system. Let (i,j) denote a state when two particles are located at
site i and j, respectively. In the absence of tunnelling (t=0), the ground state is
4-fold degenerate corresponding to the bound pairs (we take V=-24K following
chapter 3), (1,4), (3,4), (2,3), and (1,2). The excited state is 2—fold degenerate
corresponding to the unbound pairs (1,3) and (2,4) since site 1 and 3 and 2 and
4 are next nearest neighbors. The energy splitting between the ground and the
excited state is 24K, quite large. In the presence of tunnelling the ground state
degeneracy is lifted partially. The spectrum now consists of three doublets which,
for t=-7K, have energies -30.44K, -24K, +6.4K respectively. The lowest energy
gap reduces dramatically from 24K to 6.44K. One can physically think of this
splitting as resulting from tunnelling of a bound pair from one configuration to
another. This lowest energy gap depends on t in a nonlinear fashion, the three
values of the gap corresponding to t=-7K, -14K,and -20K are respectively 6.44K,
18.46K, and 29.76K. Thus for a distribution of t, one expects to see smaller energy
gaps and hence an increased heat capacity at low-T. Fig. 4.1 gives the comparison
of the density of states with and without disorder for the spinless fermions (fig.
4.1(a)) and for bosons (fig. 4.1(b)) for the 4 site 2 particle system. The energy
and the disorder parameter a' are in unit of to. The dashed lines correspond to the
system without disorder. One can easily see that the density of states is shifted to
the lower energy in the presence of disorder. Fig. 4.2 gives the comparison of the
corresponding heat capacity changes. The disorder increases the low temperature

heat capacities.

Next we present the numerical results for the 8—site system. Since in chapter 3

we concentrated on the spinless fermion model, we will discuss in detail the effects

73

 

 

 

 

 

 

 

 

 

0.6 *(a) I I -
SF, (4,2) ' '
0.4 - J
Q 0.6 —(b) I , -
Boson, (4,2) I I

0.4 — -

0.2 - -

0.0 AAAA A .A i . [A MA A A

O 1 2 3 4 5 6

Figure 4.1: Comparison of the density of states with (solid line, <7 = 0.5) and
without (dashed line) disorder for the 4 site 2 particle case. Energy and a in unit
to, V=0. (a) Spinless fermions. (b)Bosons.

74

 

(a) SF

 

C(J/K/mol)

 

 

 

 

Figure 4.2: Comparison of the heat capacity with (dashed line, a' = 0.5) and
without (solid line) disorder for the 4 site 2 particle system. Temperature (T) and
a' are in unit to, V=O. (a) Spinless fermions, (b)Bosons.

75

of disorder for this system. In our calculation, we kept the inter-site attraction
parameter V the same as in the non disorder case (V=-24K) and varied the mean
value of the tunnelling parameter to and the width of the gaussian distribution a
to get an optimal fit with the experiment for different values of < n > and T. The
parameter values for which we present our results are: to=-17K, V=~24K, and
0:7.OK. The ratio of the disorder parameter to the mean band width is about
0.2 which looks quite reasonable. Fig. 4.3 gives the results of the temperature
dependence of the heat capacity for 3H6 treated as spinless fermions without
(fig.4.3(a)) and with (fig.4.3(b)) disorder. As we can see, the effect of disorder is to
enhance the low-temperature heat capacity as expected. Fig.4.4 gives the results
of helium concentration (< n >) dependence of the heat capacity again without
(fig.4.4(a)) and with (fig.4.4(b)) disorder. It is obvious that the results with
disorder agree much better with experiments especially at very low temperatures.
But, there is still a large discrepancy between our calculations and experiment
for < n > larger than 6. We do not understand the reason for this rather large

discrepancy.

We have carried out the same calculations for bose system. As we can see from
fig. 4.5, there is a dramatic increase in the very low temperature heat capacity in

the presence of disorder.

4.4 Summary

In summary, we have investigated the effect of disorder on the low-T heat capac-
ity of small He clusters inside K-L zeolite by extending the model developed in

the previous chapter by taking a random distribution of tunnelling parameters,

76

 

 

C (J/K/mol)

 

 

 

 

Figure 4.3: Heat capacity vs. temperature for fixed helium atom concentration for
3H e. Symbols are experimental results and lines are our corresponding calculated
results. (a) without disorder, =-14K, V=-24K. (b) with disorder, t=—17K, V=-
24K, 0 = 7.0K.

77

 

 

 

 

0.5
------- o ; T=1.4K t=—14K, V=-—24K
0 4 — — — — 'i' : T=1.0K 0=O
{ o : T=O.6K
I
0.3 b . ..........
. <> 0 o
o ”
0.2[- % +9 + .
F 4% ________
/ / — — — '— \ \ '
A 0.1 — / o O O \
m : // O \ .-
\ ,,-/ \ 1
'—.~; *’ 1 1 N
06 . . i i i . i i 1!; i i . i i n ’
é
_ ------- <> : T=1.4K t=—17K, V=—24K
U 0.4 — — — — e : T=1.0K 0:7.OK

 

 

 

 

Figure 4.4: Heat capacity vs. helium atom concentration < n > for fixed temper-
atures for 3H e. Symbols are experimental results and lines are our corresponding
calculated results. (a) without disorder, t=-14K, V=-24K. (b) with disorder, t=-
17K, V=-24K, a' = 7.0K.

78

 

0.6 —

0.4 —

 

0.0,

c (J/K/mol)

0.4 r

0.2 —

 

 

 

 

0.0 0.5 1 O 1.5

T(K)'

Figure 4.5: Heat capacity vs. temperature for fixed helium atom concentration for
4He. (a) without disorder, t=-14K, V=-24K. (b) with disorder, t=-14K, V=-24K,
0‘ = 8.0K.

79

computing the energies and averaging the heat capacity using different probability
distributions. This approach to a quantitative evaluation of the effects of disorder
on the low-T heat capacity yields results in much better agreement with experi-
ment although detailed quantitative agreement is still lacking. Our model gives a
microscopic picture of disorder induced low energy excitation spectrum associated

with the correlated tunnelling motion of cluster of He atoms.

Chapter 5

Thermal Excitations of Xenon
Clusters Inside a Model
L-Zeolite

5. 1 Introduction

As we have discussed in earlier chapters, an important feature of the intracrys-
talline space of zeolites is its inhomogeneous nature. To see how the thermody-
namic properties of clusters of atoms or molecules confined inside these micro-
porous media depend on their internal inhomogeneity we have investigated the
physical properties of small rare gas clusters confined inside the cages of L-zeolite.
In the previous chapters (chapter 2, 3, 4), we discussed the case of interacting
helium particles (‘H e, 3H e atoms) by modeling the internal inhomogeneity of the
confining medium and the inter-particle interaction through Hubbard-like models.
In this chapter and the next one, we focus our attention on confined classical sys-
tems instead but take into account the effect of the microporous hosts in a more

realistic fashion [21].

We have chosen cluster of Xe atoms inside L—zeolite, because these atoms are

80

81

excellent N MR probes and their physical properties can be experimentally studied
using NMR experiments. It is also possible to directly probe the structure of the

confined cluster by either X-ray or neutron diffraction measurements.

Thermodynamic and dynamic properties of rare gas atoms, such as Ar and
Xe, trapped inside intracrystalline cavities of zeolites (referred to as hosts) have
attracted considerable attention in recent years.[86]-[92] These trapped atoms (re-
ferred to as guests), depending on their number density and the relative strengths
of the guest-guest and the guest-host interactions, form clusters whose structure
and physical properties can differ dramatically from those of their free counter-
parts. One of the interesting features is the so called gas / liquid phase coexistence
[90] resulting from a rapid exchange between the gas phase atoms inside zeo-
lite cages with those adsorbed on the intracrystalline surface. Depending on the
thermodynamic state of the adsorbed atoms one may think of this as gas/ solid
coexistence instead. To develop a microscopic understanding of the above men-
tioned coexistence, Li and Berry carried out preliminary studies of the dynamics
of six Ar atoms trapped inside model cavities.[90] Their model cavity consisted of
a spherical shell with the guest-host interaction modeled by Morse potential, with
comparable strengths of the guest-guest and guest-host potentials. They observed
that the confinement strongly altered the geometry i.e. instead of 2 minima and
4 saddles seen in Are in free space they found 5 minima and 8 saddles. The over-
all melting characteristics were however quite similar for both the free and the
confined Are clusters. Our interest here is to introduce a more realistic host and

study the above mentioned properties.

In addition to isolated confined clusters, there is also a great deal of current

82

interest in cluster assembled materials(CAM), where a solid is formed out of clus-
ters. A simple example of CAM is Can solid. Physical properties of CAM’s are
expected to be dramatically different from their atomic counterparts. A simple
way to fabricate CAM’s is by trapping the clusters inside the intra—crystalline

space of inhomogeneous hosts such as zeolites, layered silicates etc..

5.2 Method and the physical system

We have concentrated on the thermodynamic and dynamic properties of two types
of confined Xe clusters, (1) a single 6 atom Xe cluster inside an L-zeolite cage
(denoted as X egg) and (2) a periodic array of X 63c which we identify as a cluster
assembled material and denote as X 850 AM. For the thermodynamic properties,
we have used Monte Carlo (MC) method, the details of which can be found in the
earlier work of Etters and Kaelberer who were the first ones to study the ‘melting’
of free Lennard-Jones clusters [93]-[95]. For the dynamics studies we have used

classical constant temperature molecular dynamics simulation [96].

The structure of the L-zeolite has been shown in fig. 1.1 and fig. 1.2. Most of
the zeolites normally contain a mixture of Si and Al atoms in the silicate frame-
work and for overall charge neutrality there are a number of charge compensating
cations which occupy different positions inside the system. For simplicity, in the
present study we assume the framework contains no Al ions and therefore we do
not have to worry either about the disorder in the positions of framework anions
or the charge compensating alkali (K and Na) cations as in the L and K-L zeo-
lites. We expect the basic physics underlying both melting and dynamics at high

temperatures not to be greatly affected by this approximation.

83

Before performing the simulation with many Xe atoms, the potential energy
surface for a single Xe atom was analyzed to get an idea of how the single par-
ticle potential V, varied inside the cage of the main channel and whether it was
possible for a single Xe atom to go from one main channel to another through
the subsidiary 4- and 8-channels. Following Vernov et.al. [88] the interaction
between the Xe and Si atoms has been replaced by an effective Xe-O interaction
potential and by readjusting the Oxygen Lennard-Jones parameters. Hence the
cage consists of only a frame work of Oxygen atoms. The Lennard Jones pa-
rameters used in this calculation are: a'x,_o=3.3A, ex._o=151K, ax,-x,=4.1A,
and ex,_x¢=221K. We also compared the potential calculated using the effective
Xe-O parameters with that obtained using more realistic Xe—Si and Xe—O param-
eters [97] and the two results agreed very well. It is found that the energy barrier
along the transverse direction (perpendicular to the channel axis which we denote
as the z axis) is extremely high, of the order 150,000 K for a rigid framework.
For the temperature range in which we are interested in this paper (room tem-
perature or less), the probability of Xe atoms escaping from one main channel
to other neighboring channels in this transverse direction is therefore extremely
small. Therefore, periodic boundary conditions are applied only along the 2 di-
rection of the cell. The position of the cage is taken symmetrically to be about
220. Hence each time the potential produced by the framework atoms on a Xe
atom is required, the coordinate of the latter is folded back to the central cage.
This ensures the calculation of the potential due to all the Oxygen atoms within

a given cutoff radius.

We have chosen a cut off radius of 4-50'xe-0, where O'Xe—o is the length pa-

rameter associated with the Xe-O Lennard Jones potential. A cage consisting of

84

five unit cells along the z direction, and parts of the neighboring unit cells along
in the x—y plane are used to satisfy the cutoff requirement from any point inside
the central unit cell. The total number of Oxygen atoms inside the cut-off radius

turns out to be 1464.

Before analyzing the interplay of Xe—cage and Xe—Xe interactions we first in-
vestigated the thermal properties of a single Xe atom and then starting from the
ground state configuration of the Xegc cluster, we slowly heated up this system.
‘Melting’ of the Xegc cluster is monitored by recording the average energy (< E >)
and average bond length fluctuation (6) which typically show a characteristic rapid
change even for small clusters.[90, 93, 94, 95]. For the Xegc system, we have taken
two sets of MC runs. A set of 11 runs, each consisting of 105 MC step/ particle
(MCS / p) at each temperature; the other set consists of 3 runs where the number
of MC steps is increased by a factbr of 10. We will discuss the comparison of
these results later. For the XewAM system, we do the same steps as for the X66¢
system. We have taken 3 runs and each run takes 105 MCS/p for the heating
process and at least 2 x 10" MCS / p for the cooling process. The step length is
0.165 (in unit of cow“) for heating and 0.033 for cooling. We will discuss the

step length problem later.

The molecular dynamics simulation method we have used to probe the dynam-
ics is the well-known Nosé dynamics [96] which generates a canonical ensemble (or
constant temperature ensemble). In this method, an additional degree of freedom
3 is introduced, which acts as an external system. The interaction between the

physical system and s is expressed via scaling of the velocities of the particles,

V; = 31",'. (5.1)

85

The Lagrangian of the extended system of particles and s is given by

L = Z: 1%321‘3 — ¢(r) + g-z — (f +1)kT,qln(s). (5.2)

where the third term and the fourth term are the kinetic energy and the potential
energy associated with the dynamic variable 8, respectively. The quantity f is the
number of degree of freedom of the physical system and Q has the dimensions of

energy. Tea is the required equilibrium temperature.

The equation of motion for the particles are

1 043 — Eh. (5.3)

mgsz 0r; 3

 

 

The equation of motion for s is

_ (f + mu.

8

Q5 :2 2min? (5.4)

To make sure that the system is at equilibrium, we first run MC simulation
upto 105 time steps/particle at the temperature we want. Then, we read the
configuration and use this to initiate the MD run. The time interval is 10‘5ns

and the total time period that we followed is typically about 1.6ns.

5.3 Results

5.3.1 MC simulation
A single Xe atom inside L-zeolite channel

Figures 5.1 and 5.2 show the potential energy lanscape for the z=0 plane for

different values of the azimuthal angle <I> and r, the distance from the center as

86

one approaches the cage wall (please see fig. 1.1 and fig. 1.2 for the L-zeolite
structure). The potential has nearly 12-fold azimuthal symmetry as we approach
the cage wall, the symmetry is slightly broken because the 12 oxygen atoms,
although at the same distance from the center, are not uniformly distributed in
multiples of 30 degrees along Q. In addition, there are 6 pairs of oxygen atoms
further away which are oriented parallel to the z-axis, and are separated by 60
degrees intervals along ‘1’. These oxygens atoms also break the l2-fold degeneracy.
The primary minima, with energy -2391K, are located at r=4.45A. The difference
in the energy values of the primary and secondary minima is about 80K. As r is
reduced or the particle moves away from the cage wall, the potential shows nearly
6-fold symmetry. We see from the figure that the <I>-barrier goes from about 350K
for r=4.4A to a few K for r=3A. As regards the barriers along the channel axis,
we show in fig. 5.3 how the potential changes as we change z starting from one of
the (1)-minima, for different values of r. For r = 0, i.e along the channel axis, the
potential is minimum near the aperture (neck) and maximum near the cage center,
the energy barrier to move along the channel axis being about 250K. Therefore
the most likely path taken by a Xe atom in going from a primary minima near
the wall to that near the neck is to first move towards the cage center and before

actually making it to the center move towards the neck.

In fig. 5.4 we give the results of MC simulation for a single Xe atom in the
temperature range 5K< T <150K. Starting from the ground state where the
particle occupies one of the primary minima, as we heat the system, the average
energy < E > initially increases linearly with a slope 1.5, consistent with that
for a 3-dimensional harmonic oscillator. In the temperature interval 10-70K, the

slope rises and then changes back to 1.5 after about 70K. The physical origin

87

 

 

 

 

 

 

 

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Figure 5.1: Single-particle potential energy V, for Xe as a function of <I> in a
plane passing through the center of the L-zeolite cage in units of K, r is the radial
distance from the cage axis.

88

 

 

 

 

 

 

 

—14oo:— r=4.6A° - I ’ -f
—1600 ~ i
—1800 _f
—2000 _j
—2200‘ 1
—24oof 4 . + . .4 - . . o. g . m - n.
. r=4.7A 1
—1000 7 _‘
i 1
—1500 _- _‘
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v 0 .* -
m —500 :- _2
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2000 [" r=4.9A° I I _
1000 :- .3
0 E -
—1ooo :- .1
[ 1
_2000 I A A L 1 ‘ ‘ - 1 + - A - L
0 Zn

Figure 5.2: Single-particle potential energy V, for Xe as a function of Q in a
plane passing through the center of the L-zeolite cage in units of K, r is the radial
distance from the cage axis.

89

 

 

 

 

 

 

 

--lOOO —

 

 

 

VS (in K)

 

 

 

 

 

—15OO _ —
[ r=3.5A°
‘ U
L 4
—2000 — I‘=4~.OAo —
l l l . l . l I
—4 -2 O 2 4

z (in A")

Figure 5.3: Single-particle potential energy V, as a function of 2 inside the L-
zeolite cage in units of K, r is the radial distance from the cage axis. The azimuthal
angle 43 is chosen to be zero.

90

of this temperature dependence of the slope can be attributed to the existence
of six secondary minima separated from the primary minima by an energy A of
about 80K. The particle still behaves like a harmonic oscillator as regards r and z
degrees of freedom. On the other hand the partition function for the Q degree of
freedom can be approximated as Z4, = 6(1 + exp(-A/kT))Zo,c which gives a cross
over region in T dependence of < E > at kT near 0.5A. This is in accordance

with our MC simulation results.

X85¢ cluster

At first, the ground state structure of Xe,“ clusters was determinated by Monte
Carlo quenching (some times referred to as MC docking). Because of the strong
one particle potential, the Xe atoms occupy essentially the primary minima as
long as nS6 per cage. In fact the distance between two Xe atoms occupying the
two nearest primary minima is about 4.45A whereas the distance between two
isolated Xe atoms at their LJ minimum is 4.60A at which distance the Xe-Xe
pair energy is -221K. Since the Xe—Xe pair energy at a separation 4.45A is about
-210K, the Xe atoms prefer to form a nearly commensurate structure with the
zeolite single particle potential. The ground state of the Xesc cluster is therefore a
ring where the Xe atoms are located very close to the six primary minima as shown
in the fig. 5.5. In contrast, the free Xee cluster has the usual bi-pyramid structure
(see fig. 5.6). Thus the guest-host interaction strongly affects the structure of

confined Xeac.

The melting of the XC3c system is studied by slowly heating the system in

temperature intervals of 8K. As mentioned in the method section we carried out

 

91

 

 

 

 

-140....,....,....,...T,..+fir.r..
O
O
O
O
A L 0
IX!
LO , 0 T
‘—'l 0
H—c 0
O » <>
4..)
OS 0
:3—15.o— O —
S: O
”—1 r J
V O
r
/\ <>
[if-l * <>
V—15.5— —
O
i...l.r..li...l...mlrl..ll...
O 25 5O '75 100 125 150

<T> (in K)

Figure 5.4: Average energy < E > vs T for a single Xe atom inside L—zeolite.

92

Active

0.
oo
o o o o
0“ oo “o
O .0. O
.00 Go.

.00 0..
o coo o
0.. oo ..o
o o o o
0.

Figure 5.5: Positions of the six Xe atoms(large circles) in the ground state of Xegc
and the framework Oxygen atoms(small circles) in the z=0 plane.

93

 

Figure 5.6: The ground state of a free Xea cluster

94

two sets of MC runs, one with 105 MCS / p and the other with 10" MCS/p. Fig.
5.7(a) shows the average energy < E > as a function of temperature for the first
set of runs. Fig.5.7(b) gives the corresponding value of the average bond length
fluctuation. In this set of runs, for each T, the number of MC steps were 100,000,
following 20,000 discarded sweeps for thermalization. The line in each figure is the
average taken over all the 11 runs. Fig. 5.8 shows the same functions as fig. 5.7
except in this case the number of MC steps is 10 times more than in the previous
set. At low T the excitations of the 6-atom ring are those of a harmonic oscillator
and hence one expects to see a slope of 1.5 in the < E > vs T curve. The observed
slope in fig. 5.7 and in fig. 5.8 is 1.4. We find that ‘melting’ of the cluster occurs
when one of the six ring Xe atoms is dislodged and moves towards the center of
the cage and then towards the neck region. The movement of the remaining five
Xe atoms between the ring sites becomes much more likely and this process can be
looked at as a vacancy moving from one ring site to another. We will refer to this
excited state structure as the (5—1) cluster. The average bond-length fluctuation 5
increases rapidly in the neighborhood of 120K which may be close to the ‘melting’
temperature of the confined Xeac cluster. We also note that the value of 5 above
the melting temperature is larger for the longer MC runs. This results from the

intercage diffusive motion of the excited Xe atom.

The energy needed to excite one Xe atom from one of the primary minima (ring
sites with energy about ~2400K) to the minimum energy position of the ‘odd’ atom
of the (5-1) cluster is about 1150K (see fig. 5.3, r=1-1.5A). The T=0 energy of
the (5—1) cluster differs from the true ground state (6-atom ring) by about 1000K.
Thus the rapid increase in 5 vs T near 120K must be due to reasons other than

simple energy considerations. One contributing reason for this rapid change is the

<E> (in unit 151K)

<6>

95

 

_85 _
-90].

r
_95;

—100'

 

 

0.20}
0.15}
0.10}

0.05 L

 

0.001 I m 14 l A L4 1 A 1 m l L n L A l n A 1

 

J

llellLlLll

 

 

ICC 125
<T> (K)

150

 

Figure 5.7: Melting of a X 636 cluster inside the L-zeolite; (a) the average energy
< E > vs T; (b) average bond length fluctuation 5 vs T. The solid line is the
average of 11 sets of run, each consisting of 105 runs for each temperature.

96

 

 

 

 

 

 

 

l l l m 7 fl l l
[ 1
A —85 — _
i=4 .
H _
LO
u—l
-o—> —90 — _
”—4 b
a q
S .;
E —95 - _
~ 1
A '1
m I-
V —-100 _ _
[I l
1 Vi
0.4 _— _
0.3 L ;
/\ : j
00 , ,
V 0.2 _— _d
0.1 L _.
:13: zit: :;_: :3: :35: :3g. .. ;
(10’l ‘ h . ‘ Le - - . l . . 1 . . 1 . . 1 . ‘
50 75 100 125 150 175
<T>

Figure 5.8: The same as in the previous figure excepting here each run consists of
10" MC steps and the number of MC run was 3.

97

higher degeneracy of the (5—1) cluster. Another contributing factor could be the
rather large phase space available to the ‘odd’ atom in the (5—1) cluster compared
to the very tightly confined atoms near the primary minima. To see how zeolite
confinement alters the thermal properties of a Xea cluster , we show in figs. 5.9
and 5.10 the variation of < E > and 5 as a function of T for a free X65 cluster.
Compared to Xegc in L-zeolite, the fluctuation in < E > is smaller for a free Xea
cluster and also the melting temperature of the free Xee cluster is much lower,

about 11K.

Melting of XeacAM solid—a cluster assembled material

XewAM corresponds to an one-dimensional solid made out of Xea rings and with
one ring/ cage. The ‘melting’ curve is obtained by gradually heating the system
from 15.1K and at a interval of 8K. We run up to 105 MC steps for each tempera-
ture point and the step length is 0.165A. We have averaged over three sets of MC
runs. The cooling curve is much harder to get. It takes at least 2 x 10" MC time
steps and the step length has to be reduced to 0.033A. This is because there exist
some metastable states in this complex potential which make it very difficult for
the system to relax to its equilibrium configuration. Fig. 5.11 shows the average
energy/ cage vs. T curves for both heating and cooling processes. Comparing to
the XCac system, the ‘melting’ of XewAM system is much more well defined, but
the melting temperature is quite close to but slightly larger than that for a single
X 83¢ cluster. This reduction in the fluctuation in going from X 65., to X eGCAM is
clearly due to the inter-cluster interactions. This inter-cluster interaction effects
also clearly show up in the difference of < E > for X 36,: (shown as a solid line)

and X 860 AM in the same figure.

98

-11.0 . 1 ' T

 

(a)

-11.5

I

-12.0 _

-12.5

<E>(in units of 221K)

 

 

 

 

-13.0 ‘ ‘ ‘ ‘ ‘ '
0.00 0.05 0.10 0.15
T(in units of eulx)

Figure 5.9: Melting of a free X as cluster: the average energy < E > vs T; The
solid line is the average of 6 sets of runs, each consisting of 1.6X10" runs for each

temperature. < E > and T are expressed in terms of Lennard-Jones parameter
ex, = 221 K.

99

 

0.25 . . - i

Orun#1 (b)‘
Drun#2

Grun#3

0.20 ~

0.15

I

0.10 -

0.05

U

 

 

 

 

0.00 ‘ ‘ ‘ ‘ ‘
0.00 0.05 0.10 0.15
T(in units of (EU/K)

Figure 5.10: Melting of a free X es cluster: average bond length fluctuation 5 vs
T. The solid line is the average of 6 sets of runs, each consisting of 1.6X10" runs
for each temperature. T is expressed in terms of Lennard-Jones parameter axe/k3

= 221 K.

100

 

   

 

 

 

_85 .- .4
A ’
if] [ D: heating U.55
L0 —90 ~ 0: cooling 83 -
v-1 3.
0’
’H B
O E
:4: .8!
£3
:3 —95~ ~
Cl
0H
v
Cl)
DD
(U
Q —100 ~ .
/\ [:1 *
m [3‘3
V r DD
DC]
[ CID .
—105 " CID -
O 50 100 150 200 250

T(K)

Figure 5.11: Average energy < E > vs T for XeBCAM, a cluster assembled material
consisting of a periodic array of X63 clusters inside L-zeolite. Solid line is the
averaged energy of X eac for comparison.

101

5.3.2 MD simulation

To study the dynamics of Xe atoms in L-zeolite, we performed molecular dynamics
simulation at about room temperature using Nosé dynamics [96]. As we have
stated before, we run MC first to make sure that the system is in equilibrium and
then observe the diffusion of Xe atoms using MD simulation. Fig. 5.12 is the
averaged 7'2 vs. time for the X 65c system. Results for three runs are shown in the
figure, each with a different initial configuration. As can be seen from the figure,
the path depend sensitively on the initial condition. This is due to the complexity
of the system potential. The plateau in one of the curve means that there are no
inter-cage motions while the sharp increase in the other means some particle (or
particles) is undergoing inter-cage motion. Fig. 5.13 is the 1'2 vs. time for each
individual Xe for the run 3 which gives evidence for the above statements (note:

the cage length is about 7.5A.).

We next study the temperature dependence of the inter- and intra—cage dy-
namics. In fig. 5.14 we give the averaged 1'2 vs. time for X123c system at three
different temperatures starting from the same initial configuration. The diffusion
are quite different in the three runs and for the three temperatures that we have
chosen. From our limited number of runs we find that the diffusion rate does not
necessarily increase with temperature. Our limited simulation studies indicate
that one needs much longer runs and averaging over many more initial configura-

tions to probe the dynamics in this highly inhomogeneous medium accurately.

For the X 83c AM system, we also take three runs for 1‘2 vs. time at temperature
302K. As we can see in fig. 5.15, the diffusions is much slower than that of X 66,:

system shown in fig. 5.12 (note the scale differences between the two figures).

162.05102)

102

 

 

 

 

 

150 t T U I r l ' T 7 I 1 V I T l
' T=302K
- : run 1
t - — -: run 2 ‘
125: ---- : run 3 ,1:
, .
.l’
100 — , _
’ I
I.
. .l
75 ~ ./ _
. I 1
. /.
r /-
50 — _, ~ ,. ’ r
f, -— / I“ ~ ‘_ i_ _._
/ J " “' f /
/
/
O b 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l
0 5000 10000 15000

t(in unit of 1.0e—138)

Figure 5.12: 1‘2 vs. time for the X 6% system

”'1

103

 

 

 

 

150 ' ' * ' r w v T . I . l' ,7, I
1 , ,
1 —: r2 of run 3, T=302K ,
' J
’-— :r2 of each Xe ! ‘
125 * 1 ‘
> I <
,
I
100 _ [If (”A’V‘ -
/'\ [J 4~~/‘ w-v“ ‘\
N A 'r/ r/ ”(r *G'CVN’
J/ 11 *
O ~\~/ / f
< 75 ’- u/f " 2 ri‘”~ I
V / I“ I"
N VNQ. kh\ I /
S-1 1 If, \ Ir / v"
' JJ/ \‘alm l I f \ ’
5O ” /‘%‘/ “\frl“ ”x I A ‘/ \x‘
. / :0 p1, / \ .1 f f A ,
// ll “(r \ / \ If /’ \
/ v "
i (Ill / \ \ / IV ‘74/ \‘*
i l // * fi/fl \ \ .4
25 ” ///I// " \\ / \ / «_
1/ // \ ’/ I
D // ~
, /
O A 1 1 - l 1 A m 4 l 1 4 1 l
0 5000 10000 15000

t(in unit of 1.0e—138)

Figure 5.13: '1'2 vs. time for each Xe atom along with the result of the averaged
one for the X 63¢ system

104

 

 

 

 

 

150[ . . . . r . . . . , . . . . ,
* :T=272K
* - — -: T=302K
1 — — :T=332K ,
125 — ,a*—
_ I .
1’
. / .
100-— ,’ —
E f ‘
_.. _ /
02
o ‘ / ‘
<3 75 — / —
v f / 1
m - /
L: > ”’1‘:/\f.__._.- ,“1
50 _ // “'fi/"’ 7
/ / ~» I
1- / / ..
/
25.5 1;. I
O P 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l
0 5000 10000 15000

t(in unit of 1.0e—138)

Figure 5.14: r2 vs. time for the X eac system at three different temperatures

105

Also, there are no inter-cage motion occuring during the time period that we
followed which means that it is much harder for Xe atoms to diffuse through
the channel in the X 6504M system. This is due to the strong particle-particle
interactions between the thermally excited atoms in nearest neighbor cages. Fig.

3, 232, y2 and .2:2 vs. time in run 1. It shows that there is almost no

5.16 gives r
intercage motion along the channel axis(i.e. in the z direction). Fig. 5.17 is the
effective diffusion rate D(t) vs. time for the three runs. Clearly the dynamics is

not diffusive since D(t) decreases ( perhaps approches zero) with time.

5.4 Discussion

Monte Carlo studies of confined (in L-zeolite) and free Xee clusters clearly brings
out the dominant role played by the guest-host interaction. The inhomogeneous
nature of the single particle potential produced by the zeolite cage profoundly
alters both the single particle and many particle potential energy surfaces and
consequently their thermal properties. Unlike the free cluster where the melting
process is rather smooth, i.e. 100,000 MC steps / atom give reasonably converged
results for the T dependence of < E > and 5, the fluctuation effects are large
in the case of the confined cluster. A careful inspection for the two sets of runs
indicates that melting process shifts to a lower temperature in fig. 5.8 which
consists of runs 10 times longer than for those shown in fig. 5.7. The reason
could be traced back to the large potential energy barriers present in the zeolite
systems. The energy barriers for particles close to the wall is estimated to be
~ 1200 K. This corresponds to an exponential factor emp(—fl =I= AE) ~ 10'“ for

temperature ~ 100 K. Since the transition is triggered by the dislodging of one of

T2(A02)

4O

30

20

10

106

 

l

l

 

 

 

,' :runl
—--:run2
—'--::run3

 

m L L 1 1 l - 1 l

 

2000 4000 6000

t(in unit of 2.0e—138)

Figure 5.15: 1'2 vs. time for the X 650 AM system

107

 

w v v ' v v

ofrun.2,T=802K
r—-:X ofrunZ
30. —'-Iy ofrun2
""22 ofrunZ

_—ZI'

lemma)

 

 

 

O 1 1 1 1 n 1 1 1 1 I m .

 

0 2000 4000 6000

t(in unit of 2.0e—188)

Figure 5.16: '1'2 vs. time of run2 with 1:2, y”, 22 also for the X 636 AM system

D(t)(in unit of 10—5cm2/s)

108

 

 

21) if . r . , . . .¥1., .gfi . . I s
I 4
', T=302K ]
t] :run 1 *
'[ —'--:run.2

 

 

 

 

. . . . . . I1 . 1
0 2000 4000 6000

t(in unit of 2.0e—138)

 

0.0 1 l 1 l l 1 4 1

Figure 5.17: Diffusion rate D(t) vs. time for the X 63c“; system

 

109

the six ring atoms it is possible to miss this transition for small MC steps / particle.
But with increasing number of MC steps the chance of accepting the move which
would dislodge the particle becomes more probable thereby causing the melting
temperature to shift to a lower value. For increasing number of runs the ‘melting’
asymptotically becomes ’a continuous process as it should be but with a rapid
change (with T) in 5 determined purely by the equilibrium canonical distribution.
If one can relate the MC steps to the actual time scale in which a real time
experiment is performed, then depending upon the waiting time one should see
different transition temperature as observed in our simulation. In addition if one
experimentally probes the dynamics of Xe atoms, say through NMR, one should
see two distinct type of dynamics: one associated with the correlated hopping
of Xe atoms from one ring site to the other and the second associated with the
intercage motion of the excited ‘odd’ atom of the (5-1) system. X 6604M system
shows a very well defined ’melting’ curve. This is because that the system has
a rather stable structure. The inter-cluster interaction reduces the flunctuations
and its effects are seen clearly when we study the diffusion of the system , namely

the inter-cage diffusive motion is dramatically suppressed in the X 830 A M system.

Chapter 6

Percolation and Diffusion in 2D
Microporous Media: Pillared
Clays

6.1 Introduction

Although adsorption and diffusion of gases in porous media have been the subject
of investigation since the beginning of the century, it is only in recent years that
any fundamental understanding of the underlying physical processes have begun
to emerge[98]. As we have already discussed in the earlier chapters, a simple
way to incorporate the effects of a porous medium on the dynamical properties of
particles moving through it is through geometrical constraints. These constraints
can not only alter these dynamical properties drastically but their thermodynamic
properties as well. For example, in a macroporous medium one sees capillary
condensation of the adsorbed gas to a liquid phase at a pressure which is less than
the bulk saturated vapor pressure, a result of the shifted gas-liquid transition
temperature caused by the confining geometry [99]. The diffusion rates both in
the gas and the liquid phases are also known to be profoundly affected by the

porosity leading, in certain cases, to the phenomena of percolation [100].

110

111

Until now we have been looking at one-dimensional microporous systems such
as zeolites. Now we shift our attention to two-dimensional microporous system.
A novel class of porous media that has been also of great interest during the last
decade or so are pillared clays and related complex layered oxides. In this sys-
tems the guest particles diffuse between host layers held apart by laterally spaced
pillaring agents. Pillared clays, in general, are intercalation compounds obtained
by intercalating large pillaring ions into the galleries of layered oxides such as
vermiculite [101], fluorohectorite [28], and montmorillonite [101]. Intercalation
implies the insertion of guest species into a layered material without disturbing
the main structural features of the host [102]. Intercalation compounds have been
of interest for many years [6, 7, 20, 27, 103, 104]in part because they provide a
two dimensional(2D) arena for both theoretical and experimental studies of many
physical properties such as diffusion, percolation, commensurate-incommensurate
structures, 2D—melting on corrugated substrates, 2D magnetic phase transition
and so on. In clay intercalation compounds (CIC’s), the host layers are highly
rigid towards transverse distortion. In fact, one needs a large transverse layer
rigidity to form good MP medium, otherwise the host layers will collapse around

pillaring ions thereby precluding gallery access.

In almost all the CIC’s there are fixed negative net charges in the layers which
are compensated by counter ions in the intralamellar gallery space. These clays
have the ability to be intercalated by guest ions or intercalants through ion-
exchange mechanism [38, 105]. When the guest ions are of nanoscopic dimen-
sions the resultant pillared clay is characterized by host layers that are propped
apart by these laterally spaced guest ions, sometimes called pillars. The lateral

separation between these guest ions can range from a few A to as large as 10-20

112

A(considerably larger than their lateral size) thus giving a well characterized 2D
MP system with varying degree of microporosity. The specific sites where these
intercalants reside depend on the charge of the intercalants, their lateral size and
the commensuration energy associated with the interaction of the intercalants

with the atoms of the host layers.

In this chapter, we model a real physical system representing such a 2D MP
system (section 6.2), and then briefly discuss the experimental results on adsorp-
tion [38] and diffusion [28] carried out in this system (section 6.3). We introduce
in section 6.4, microscopic lattice models with different levels of physical reality to
characterize the microporosity, probe diffusion and percolation of particles mov-
ing in this MP medium using simulation techniques and compare our results with
experimental measurments wherever available. Finally we give a brief summary

of this work in section 6.5.

6.2 Physical system and the geometrical model

Mixed ion (or Heteroionic) clays with different size intercalants occupying spe-
cific gallery sites are excellent models of 2D MP media. Examples of such
systems in which adsorption and diffusion measurements can be performed are
ternary intercalated layered silicates A3314 — Y, where Y is the host layer; A
and B represent the gallery cations which have different sizes [28, 38, 104, 105].
One usually studies the diffusion of a neutral guest species C (for example He,
Ar) inside this 2D MP medium [28, 103]. An example of such MP medium is

[C"r(en)§+],.,[C’o(en)‘,’+ — enli—a - F H T, where FHT stands for fluorohectorite, a

layered silicate, and the intercalants are Cr and Co complexes with en ligands.

113

The geometrical characteristics of the mixed-ion systems are the following: The
precise positions and the structure of the intercalants inside the gallery of FHT are
not known. However, from the X—ray measurement of the in-plane packing of the
intercalants (pillars) in smectic clays as a function of relative pillar charge (pillar
ion charge / clay surface charge ratio), there is strong evidence that the intercalants
form a triangular lattice [106]. From a physical point of view it is reasonable to
assume that the intercalants form an incommensurate triangular lattice. The size
of the intercalants C'r(en)g+ and 000212.):+ are ~5.4A° compared to the lattice
constant a ~ 5.2A", where a is a measure of the size of the hexagonal cage of the
substrate Kagomé network of O-atoms. In fact, the Ar and N3 adsorption data
[38] shows that the lateral size of these complexes are slightly larger than 5.4A".
Thus, one expects the substrate commensuration energy to be rather small. This
is in contrast to the well known alkali-graphite intercalation compounds [27, 107].
Ifwe neglect, in the leading order, the effect of substrate corrugation, then we have
a system of (3+) ions interacting via unscreened Coulomb interaction. The ground
state structure of such a system is known to be a triangular lattice. Consequently,
we can model the [C'r(e'n)§3,+],.,[C'o(en)‘2’+ — en]1_z — FH T system as a triangular
net whose sites are randomly occupied by C"r(sn)§+ and Co(en)g+ ions, the lattice

constant being 10.22A".

In order to characterize the microporosity, we will briefly review the discussion
presented by Cai et.al.[103]. Assuming that the intercalants(to be denoted as
cations for simplicity) A, B and the diffusing molecule C can be represented by

cylinders with (diameter,height) given by (dA,hA), (£13,113), and (dc,hc), one

114

can define a set of porosity parameters by the following:

 

 

 

.91 = 0[a;CdA]

1, = 0["_‘":,:—d”/21
13 = 0[“;c""]

1, : (NZ—3

.15 = 0[:—:-

where the function 0(3) 2 1 if a: > 1, 0(3) = 0 if a: < 1, and a is the
distance between two nearest neighbor(nn) cations which are assumed to form
a triangular lattice. Although the ideas presented here are not restricted to a
triangular lattice, we will argue later that for the physical systems we have studied,
a triangular lattice is the appropriate model. If C can pass between two nn A
cations laterally, then .91 = 1, otherwise 31 = 0. Similar definitions can be given
for 32 and 33 which are the other two lateral porosity parameters. The parameters
34 and .95 describe the vertical porosity of the MP medium. We however note that
the transverse rigidity of the host layers can significantly alter the definition of
these vertical porosity parameters. For a given system, the diffusion of the guest
particle C is controlled by the above set of five porosity parameters. Depending
on the values of sg’s, one can have no diffusion, complete diffusion, one or two
percolation thresholds [103] in the system A2314 — Y as one changes x from 0
—+ 1. Our physical system was modeled with the assumption that the heights of
the intercalants and the guest species do not influence the diffusion in the plane,

ie. we take .94 = 1 and s5 = 1. Later we will argue that this is reasonable

115

for the systems under study. Thus there are only three relevant lateral porosity

parameters 8 = (31,32,133) left to deal with.

Rare gas adsorption [38] and diffusion [28] experiments were done in het-
eroionic pillared clay [Cr3+(en)3],[Co3+(en)3l1-z — FHT, where en is ethylene-
diamine ligand. This system has the following properties: At room temper-
ature 000211.):+ and C'r(en)§+ have the same diameters and height ~5.4 A.
They prop up the gallery of FHT to a height of ~5.4 Aand the average dis-
tance between two adjacent intercalants is about 10.22 A. This allows atoms
like Argon(dc E (1,4,. = 4.0 A[28]) and Helium (dc E (13, = 2.6 A[25])
to pass through the space between AA, BB and AB pairs of intercalants, ie.
s=(1,1,1). When the system is heated to a temperature somewhere between
1000 to 1500, 003+(en)3 complex breaks into C03+(en)2 + (en), which has
smaller vertical dimension (about 3.8 A) and larger lateral dimension(> 6 A)
than the parent cation. The Cr3+(en)3 cations remain unchanged during heat-
ing. So if one takes a system [Cr3+(en)3],[003+(en)3]1_, — FHT and heats
upto a temperature between 100°C and 150°C, the resultant system becomes
[Cr3+(en)3],[003+(en)g + (en)]1—. — FHT. Using the values of the lattice con-
stants and diameter of the pillars and the diffusing particles, we find that for
Argon atoms s=(0,0,1), whereas for Helium atoms s=(1,1,1) and remains un-
changed. This implies that Argon atoms can pass only through two nn Cr3+(en)3
(see fig.6.1), and therefore in this case there exists a threshold 23.: such that the
Argon atoms can not diffuse through the system at all when a: < 2,, ie. one should
see one percolation threshold as observed in adsorption measurements [38]. Z. X.
Cai [108] used lattice model simulation and I have used continuum diffusion sim-

ulation to probe this percolative property. In section 6.4, I will introduce both

116

the models and compare the simulation results with experiments.

6.3 Review of experimental situation

Before we discuss the details of our simulations, we briefly review the exper-
imental situation. Two types of studies have been done [28, 38]. (1) Mass
adsorption measurement were carried out [38] where the mass uptake of Ar-
gon in the system [Cr3+(en)3]e[Co3+(en)g + (611)]1_,B — FH T was measured(see
fig.6.2) as a function of x at room temperature. The measurements show that
for 0 < z < 0.8, the system does not take up Argon except for some background
effects whose cause is not completely known. As x exceeds 0.8, there is a rapid
increase in the mass uptake indicating that the system begins to open up to Ar-
gon which can diffuse through the system and get adsorbed at available sites.
(2) Diffusion experiments were carried out [28] where the diffusive property of
Argon and Helium gas was measured at room temperature in both unannealed
C 03+(en)3—FH T and annealed C 03+(en)2+(en) —FH T films. These two systems
have the same effective properties as the x=1 and x=0 limits of the mixed system
[Cr3+(en)3],[003+(en)2 + (en)]1_, — FH T, because the unannealed 003+(en)3
has the same dimensions as Cr3+(en)3. Helium gas was found to diffuse through
both the systems but with different saturation time (t,) [28], 308. and 1800s.
respectively [109]. The corresponding diffusion constants(D) were estimated to be
~ 10‘°cm2/s and 10‘8cm2/s). On the other hand Argon gas could diffuse only
through the unannealed system, i.e. when x=1, with t, = 26008, and D ~ 10‘8

emf/s. No Argon diffusion was detected in the annealed system i.e. when x=0.

117

Interc alants

 

Figure 6.1: Argon atoms diffusing in the annealed [003+(en)3]1_,[Cr3+(en)3]. —
F H T. Single circles represent Cr3+(en)3. Concentric circles represent C03+(en)3
where smaller circles are for the ions before annealing and larger circles are for
the ions after annealing.

118

 

 

 

 

1
. a
so- °
0
r
A i o a
Q , an
°‘ 0
3401 °
‘ D
a i
’0' i '0:
i111 .~ °°
5 ’ n°o°°
n
5 ’ .
D a a
D ' a %
O L11nn1L---+ln---L1-mlm-nxlmn
0 20 40 60 80 100
3(2)

Figure 6.2: The amount of Argon adsorbed by [Co3+(en)3]1-3[Cr3+(en)3], — FH T
as a function of x. [38]

119

6.4 Results of computer simulation

To understand the adsorption, diffusion and percolation in the above system, we
have constructed models with different levels of sophostication and have stud-
ied the diffusion and percolation properties using different computation tech-
niques. The simulations were done on a triangular lattice where the Cr3+(en)3
and 003+(en)3 + (en) cations were randomly distributed on the triangular lattice

sites with probability x and l-x respectively.

6.4.] Lattice model

A simple lattice model was constructed by Cai and Mahanti [108] to see if one
can understand the sharp rise in the Ar adsorption near about 80% concentration
of Cr3+(en)3 intercalants. The Cr3+(en)3(A) and 003+(en)2 + (en)(B) complexes
were assumed to occupy the triangular lattice sites randomly. They further as-
sumed that only AA bonds are large enough so that Argon atoms(see fig.6.3) can
diffuse through it and get adsorbed near these bonds. Thus if we look at the
dual honeycomb lattice(DHL), a bond is open or closed depending on whether
two sites of the triangular lattice on the two ends of this bond have A atoms or
not. We assume that when the open bonds on the DHL percolate then only there
is significant Ar adsorption. The rationale behind the assumption is that one
needs access before adsorption in macroscopic scale can occur. However as will
be discussed later, this assumption may not be completely correct and one sees
observable macroscopic adsorption even below this threshold. Here we consider

this type of adsorption as background effect.

For the A381.c system when a: < 0.5, one does not have an infinitely perco-

120

 

 

 

 

 

 

 

 

Cr3+(en)3 003+(en)2 + (en)

Figure 6.3: Lateral constraints on Argon atoms diffusing in the annealed system
[003+(en)3]1_.[0r3+(en)3]. — FH T. The dashed lines are closed bonds and the

solid lines are open bonds of the honeycomb lattice on which the Argon atoms is

diffusing.

 

121

lating A cluster. For 0.78 > a: > 0.5, although we have a infinite percolating A
cluster, the concentration of AA bonds is not large enough for the corresponding
bond percolation to occur on the DHL. In fig.6.4, we plot the fraction of AA bonds
i.e. the fraction of open bonds(P) on the DHL belonging to the infinite percolating
cluster for different lattice sizes. Near a: z 0.78 one sees the usual rapid increase
of P with x and we estimate the percolation threshold :cc to be 0.78 < 2: < 0.80.

For the following discussion we take 22,, = 0.79.

Now let us compare the pure bond percolation on the honeycomb lattice with
the above percolation threshold. Associating the sites of the triangular lattice
with a variable 0’,- such that a,- = 1, if the site i is occupied by an A atom, and
a“; = —1, if it is occupied by an B atom, we can define a parameter bij associated
with the bond between site i and site j (or equivalently the bond on the DHL).
The parameter bi,- = 1, if the bond is open, Ibij = 0, if the bond is closed. The

relation between 6;,- and ag’s is
1
bi,- = 10+ 01+ 01' + 01°01], (6-1)
and
< 055 >= 232 (6.2)
where we have used < 0 >= (22: — 1), x being the concentration of A atoms, and

< 030’,- >=< 0’; >< O'j >.

Since the bond percolation threshold for a honeycomb lattice is 0.6527 [110],
from Eqn. 6.2 we find that 1:: = 0.6527, which gives me = 0.8079. This is slightly

larger than our result 0.79 for 23¢. Equivalently, the latter gives a lower percolation

threshold (~ 0.6241) for the DHL. The reason for this is that the bonds on the

122

DHL are not statistically independent but are correlated instead. The correlation

between two adjacent bonds 5;,- and by, is found to be:
< 0,305]. > — < 0.3 >< 0;], >= 33(1 —- 2:) (6.3)

This correlation is positive and is relatively large near a: z 0.75 ie. if a bond (ij)
is open then (ik) has a slightly more probability of being open than when the
bonds are randomly open or close. This results in the percolation threshold being

smaller by ~ 4% compared to the random case(0.6527).

The percolation threshold at me = 0.79 is close to the concentration a: a: 0.8
where one sees a sudden increase in the Ar adsorption experiment. The uncer-
tainty in the experimental x values is much larger than 4% to distinguish between
correlated and random bond percolation. However we believe that we have cap-
tured the essential microscopic picture underlying the adsorption process in these
2D miroporous systems. In the following section we confirm this picture by study-

ing the diffusion of Ar and He atoms using a slightly more realistic model of the
MP medium.

6.4.2 Continuum diffusion

Recently, a novel approach has been introduced to study the diffusion and trans-
port properties of physical systems where the true continuum structure is main-
tained [111]. The random walk algorithm is used to calculate the diffusion constant
in this approach and we belive that this is a more realistic approach to probe the
microporosity of the pillared lamellar systems which have been studied experi-
mentally. We apply this method to calculate D for the system shown in fig.6.1.

We allow a walker to land at a random location on the system. If the walker

 

123

 

1.0 + . . - 1 1 ~ - a:
L g“.
‘5
“a
U
’ x 100x100 111’" i
013' w
. 0 400x400 .

O 2000x2000

0.6 - -
1 092
0.. 9< 1
1 0° 1
0.4 - X ~
X
0.2 - X 0° .

 

 

 

(L9 IMO

Figure 6.4: The x dependence of the size of the largest cluster of A-A bonds
obtained from simulation of 100 x 100, 400 x 400 and 2000 x 2000 lattices (averaged
over 10 different configurations for each lattice).

124

lands on a pillar, it dies and is removed, if it lands on the remaining part of the
system, it begins to diffuse. The effective sizes of the pillars are taken to be either
((1,; + do) or (d3 + dc) and the walker is assumed to be a point object. The step
length (I) of the walker is fixed to be 0.1 in the unit of the lattice parameter(a),
but the direction of the step is picked randomly. The walker moves one step along
that direction. If the walker hits a pillar during its diffusive motion, it stays at its
original place, but the clock advances one step forward. The diffusion constant D
is then obtained from the distance(R) that the walker travels in a time t via the

relation
It:2 2 4Dt (6.4)

The diffusion constant for a particle freely diffusing in 2D in the absence of any

pillar is simply

D0 = fizz (6.5)
We have calculated the relative diffusion constant D / Do for different Cr3+(en)3
concentrations x in the system [Cr3+(en)3]c[Co3+(en)3 + (en)]1_., — F H T for both
Argon and Helium atoms. Because the diameter of He(~ 2.6 A) is much smaller
than that of Ar(~ 4.0 A), the diffusion property of He is quite different from
that of Ar. We chose a 50 x 50 triangular lattice, and our results were obtained
by averaging over 50 configurations of the A281”, — F H T system for a given 1:
and 20 random walkers in each configuration for Argon and 50 walkers in each
configuration for Helium. The number of steps taken was typically ~ 10°. We
have taken the diameter of both Cr3+(en)3 and 003+(en)3 to be 5.4 AMA), and

the diameter of 003+(en)2 + (en) to be 7.2 A(d3). Since the lattice parameter

125

of the triangular lattice is a : 10.22 A[28], Ar atoms can only pass through
AA bonds while He atoms can pass through all the bonds as suggested from the
measurements of Zhou [28]. Fig. 6.5 shows the typical curves for R2 and D/Do

vs. the number of time steps
We obtain the following results from our simulation.

(1). The relation between the relative diffusion constant of Argon atom( d A, =
4.0 A) and the concentration x of Cr3+(en)3 in [Cr3+(en)3]z[Co3+(en)2+(en)]1_.,—
FH T. Note that as the porosity of the system becomes smaller, more paths with
narrow necks (such as BB bond) contribute to D and therefore the step length
should be chosen smaller to properly access these narrow neck regions. But this
will consume enormous computer time. We have checked the dependence of D / Do
on I and find that D / Do increases with decreasing I and saturates for small enough
value of I (see fig. 6.6). As a compromise between small I (hence long computer
time) and large I (less accuracy of D / Do), we have chosen I = 0.1 (in unit of the
lattice parameter). This value of I is sufficient to give accurate information about
D/Do for He (within 5 ~ 10%) when x is near 1, whereas for Ar and for He near
2: ~ 0 one can make appreciable error. To check this we studied D / Do for Ar
when a: = 1 by taking several values of I from 0.1 to 0.01. We find that D/Do
increases by about 30% as we decrease I from 0.1 to 0.01. Thus we estimate of
D / Do for Ar is reliable up to 30%. Clearly one needs a more careful calculation of
D / Do by either choosing a smaller value of I or by using a completely independent
method. Fig. 6.7 gives our calculated values of D / Do vs. a: ( expressed in %). We

clearly see the percolation threshold near 77% is consistent with the me = 0.78.

 

126

 

 

 

 

 

400 - ~ + . ta . ~ . - - , - f -
. <‘>
300 . o°°°q
. 00000 l
0 4
. °o°°°°° :
N 200 . 00000 .4
D: : 000 ‘
. 000°
. 0000
100 - <><> -
. 00°
00
o°°°
. 000°
03109-- : hr. 4
0.008
+ .
o 0.006:- i
Q . i
O 0.004 - «
IWW
+ .
0.002 f -
0.000W .- ‘~* ' ‘4‘
0 20000 40000 60000 80000 100000
N step

Figure 6.5: Typical curves of R2 and D / Do vs. time steps for Ar atoms diffusing
in 50 x 50 [Cr3+(en)3]z[003+(en)g + (en)]1_,, — FHT lattice. a: = 1. Step length
is 0.1(in unit of the lattice parameter). Total step is 105. We averaged over 50
configurations and 50 random walkers.

D/ Do

127

 

 

 

 

1.0 - - . r
0.8 l .
C] D
Cl
C]
0.6 ’
D
o
o
o
0.4 *
<>
0 : Ar (x=1)
D : He (x=1) o
0.2 t
0.0 ‘ L ‘ ‘
0.005 0.010 0.020 0.050 0.100 0.200

Step Length(in unit of lattice constant)

Figure 6.6: D/Do vs. step length for Ar(le) and He(x=0,1)

128

(2). The relation between the relative diffusion constants of Helium atom(
d3, = 2.6 A) and the concentration x of Cr3+(en)3 in [Cr3+(en)3],[003+(en)2 +

(en)]1_, — FHT. This is shown in fig.6.8.

(3). Because the dimension of Helium atom varies a lot from literature to litera-
ture, we have also calculated the Helium size dependence of DH‘(2: = 0)/DH°(1: =
1). This result is plotted in fig.6.9. As we can see in this figure, for Helium diam-
eter larger than ~ 3.02 A, there is no diffusion in the z = 0 system and there is a

rapid decrease of DH‘(z = 0)/DH‘(2 = 1) near this critical diameter.

6.4.3 Comparison with experiment and discussion

From fig.6.7, we see that in order for Argon atoms to pass through the annealed
[Cr3+(en)3],[C’o3+(en)2 + (871)]1-9; - F H T system, the concentration of 0r3+(en)3
must be greater than 0.78, which is in good agreement with the lattice simulation

results (:0: ~ 0.79) and experiment(~ 0.8).

The diffusion results shown in fig.6.8 show that for this particular Heluim size
(d3. = 2.6 A), Helium gas can diffuse through this system for all concentrations
(x) with DH°(2 = 0)/DH‘(:c = 1) z 0.35. But experimental estimate gives this
ratio to be almost ~ 10". This large discrepancy is difficult to understand. One
possibility is because of the large value of Bye for a: = 1, the equilibration time is
so short that there may be a problem with estimating the experimental value of
D accurately for this concentration. Secondly, because of the very small porosity
for x=0, there may be quantum effects which make the effective size of Helium
atom larger than 2.6 A, thereby suppressing diffusion near narrow passage ways.

Furthermore, the size of the Helium is in fact not a well defined quantity. Thus

129

 

 

 

 

0.4 ""]"YVTfivvarfiw—vlvyiv—T—VVY
<>
0.3— .4
O
O i
Q L 4
l
O
0.1— _
0
O
O
O
0‘0...Ll.e..l.iiiliii.l....li.4.
7O 75 80 85 90 95 100
x(%)

Figure 6.7: The relative diffusion constant of Argon atom (d4, = 4.0 A) diffusing
in [Cr3+(en)3],[003+(en)3 + (en)]1_, — F H T as a function of x, the percolation
threshold 3,; ~ 0.78.

130

 

 

 

 

0.6 s . . . , . . . . j . . . . I . . . . I
0.5 ~ 2
h 0
0.4 — _
i O
O
Q 0 3 — o _
Q
0
0.2 - ..
4?
0.1 — _
00 . L i i i l i . . . l . . i . l i i i L l i . . . 1
0 20 40 60 80 100
X(%)

Figure 6.8: The relative diffusion constant of Helium atom (CIR: = 2.6 A) diffusing
in [Cr3+(en)3],[003+(en)2 + (en)]1_, — FH T as a function of x. Helium gas can
diffuse through the system for all x.

131

5.000

 

1.000 *

1)

0.500 * <>

0

0) /D(X

0.100 * 0

0.050 ' 0

D(X

0.010~ °

 

 

0.005 ‘ ‘ ‘
1.5 2.0 2 5 3.0

C1He (A0).

 

Figure 6.9: The ratio DH‘(2: = 0)/DH°(2 = 1) as a function of the diameter of
Helium atom when Helium gas diffuses in [Cr3+(en)3]z[003+(en)2 + (en)]1_, --
FH T system.

132

there is a possibility that the actual size of Helium atom is larger than what
we have used. From fig.6.9, we can see that the value of d3, is in the range
where the ratio DH‘(z = 0)/DH‘(z = 1) decrease rapidly with dye. Even a small
change in (13,, or by increasing the size of the big pillars slightly (from 7.2 A—)
7.6 A), one can cause a very large change in this ratio which can then explain the
experimental observation. What it means physically is that the system containing

all large pillars is very near the critical threshold for He diffusion.

Finally there is another phenomena which needs to be addressed. It was found
experimentally that Bye for z = 0 and DA, for z = 1 were of the same order of
magnitude and D342: 2 0) was slightly larger than DA,(2: = 1). However, our
simulation gives D342: 2 0)/Do ~ 0.19 which is smaller than DA,(:c = 1)/Do (~
0.32). The experimental observation can be understood by taking into account the
mass difference between Argon and Helium atoms i.e. M A, / M He z 10. In reality,
there is a 1 /M1/2 dependence of D [33] which has not been taken into account in
our simulation and its inclusion can explain the experimental observation. This
mass dependence can be indirectly incorporated through D0 which is then different
for He and Ar atoms. However the precise mass dependence of Do needs further

study.

6.5 Summary

In summary, we have used two simulation methods to get some insight into the
percolative properties of heteroionic clays. According to the above discussions, we
suggest that more detailed experiments should be done for Helium gas diffusing in
the annealed [C'r3+(en)3],[Co3+(en)2 + (en)]1_, - F H T system for different values

133

of x. Also an improved method is needed to calculate D, particularly when the
porosity of the system is small. It may be necessary to incorporate quantum effects
on diffusion because they may possiblely be important near narrow constrictions.
Finally, we would like to make some remarks on the large background observed
in the adsorption measurements. There appears to be a rapid increase in the
adsorption near 2: = 0.5 (see ref.[38]) where, as we discussed in sec.IV A, one
should have an infinite percolating cluster of A (Cr3+(en)3) atoms. This may
open up some gallery space where Ar atoms can get adsorbed although they
cannot diffuse through the entire system. A more realistic model of adsorption

phenomena is needed to check this idea.

Chapter 7

Conclusion

In the abstract, I have given a detailed summary of my thesis project. Here I
briefly discuss the main projects and what further work needs to be done. As
summarized in the abstract, I have investigated the thermodynamic and dynamic
properties of both quantum and classical rare gas atoms and clusters confined in-
side microporous media. I have compared my theoretical results with experiments

whenever possible and in most cases find a qualitative agreement.

For the mass transport of quantum particles inside K-L zeolite channels, single
particle Schréidinger equation has been solved for simplified model of the channel
geometry using finite difference iterative method for eigenvalue problems. For a
more realistic potential model, this method is not suitable because of the compli-
cated boundary conditions. In this case, quantum Monte Carlo simulation method

may be more appropriate.

Extended Mott-Hubbard and Bose-Hubbard models have been used to probe
the low temperature thermal excitations of helium clusters (bosons for 4He and
spinless fermions for 3H e) confined inside K-L zeolite cages and the effect of

tunnelling disorder has been investigated. However, spinful fermion problem still

134

135

needs to be solved more accurately and the effect of disorder in single-site energies
on the thermal properties should be explored. Furthermore, the dynamics of
confined quantum clusters should be very interesting because they can be probed

by inelastic neutron scattering measurements.

To study the ‘melting’ and dynamics of Xe clusters inside L-zeolite, classical
Monte Carlo and MD simulations have been employed. Monte Carlo studies of
melting bring out the difference between a single cluster and a cluster assembled
material. However, for a better understanding of the dynamics, more extensive
simulations have to be carried out. Finally, percolative properties of He and Ar
atoms inside a model 2D microporous medium pillared heteroionic silicate clay
have been studied using continuum classical MC simulations. It is possible that
He atoms even at room temperature can show quantum effects on the diffussion
rate because when they go through narrow constrictions, their de Broglie wave

length may be comparable to the pore width.

Appendix A

Estimation of nearest neighbor
He-He interaction in K-L Zeolite

We use the realistic geometry of the K-L zeolite cage to estimate the interaction
energy between two helium atoms trapped inside K-L zeolite. There are nine K +
ions in a unit cell of K-L zeolite and four of them are believed to be sitting near
the wall of the cage while others are located inside the aluminosilicate framework.
The four K + ions which are near the wall of the cage can interact with helium
atoms and contribute to the binding energy of the He atoms in what we refer to
as cage states. Experiment suggests that there are eight binding sites per-cage
(or cage states) available to helium atoms. The schematic arrangement of the He
atoms and K + ions is shown in fig. A.1. Fig. A.1(a) is the cross section of cage
center in K-L zeolite. Fig. A.1(b) shows the arrangement of two He atoms bound

near a K+ ion.

He—He interaction can be affected by the presence of K + ions. The interaction
between two helium atoms include two parts: (i) He—He direct L-J interaction:
V1 (ii) He-He indirect interaction through K +2 Vz, i.e. one He atom polarize

the K + ion which then interacts with another He atom. We estimate the two

136

137

(b)

 

K ”K“

Figure A.1: A view of the geometrical arrangement of Helium atoms and K + ions
of K-L zeolite cage.

O—O
He K+ He K+ He

(a) (b)

Figure A2: (a) Shows the interaction between one Helium atom and K + ion. (b)
shows the interaction between two Helium atoms and K + ion.

   

contributions V1, V2 to the He-He using the following procedure:
(1) Estimation of V1:

In K—L zeolite, the diameter of the cage d 2 13 A. The diameter of a He atom
is d3. 2: 2.6 A. Putting 8 He atoms as show in fig.A.1(a), one estimate the average
nearest neighbor seperation between the He atoms (133-3, 9: 2.7 A. Also, the L-J

parameters for He-He are: e = 146 — 16erg, a' = 2.56 A. So, we estimate

a 12 a

 

 

V1 = 4€[( )6] :3 —9K

dHe—He _ dHe-Hc

(2) Estimation of VZ:

138

To estimate this potential, we first have to have the He-K+ effective L—J pa-
rameters. This can be estimated using the known results from quantum chemical
calculation by Rao et.al. [112] for He — K+ and He — K“L - He arrangements
shown in fig.A.2(a), (b) (Fig. A.2(a) shows one He atom interacting with one K +
ion. Fig. A.2(b) shows the interaction between two He atoms and one K + ion).
The known results from quantum chemical calculations of binding energy 61, and

equilibrium bond length I5 are:
ef“x+ = —12.0mev

151"“ = 3.11A"

_ +_
851‘ K H“ = —24.9mev

zfe-m-He = 3.18A°

Using the above results for the He-K+ interaction, we can estimate the effective
L-J parameters. The He—K+ interaction includes two parts. One is the K +-He

polarization energy and the other is the K +-He L-J energy.

2
K+—H _ aHee aHe—K+ 12 aHe—K+ 6
V e — _2(r'1,qe_x+ )4 + 45[(rfiqe_x+) "’ Tiara—1“) l:

where age = 0.205A"3 is the He atom polarizability. This potential energy
VK+ "H“ should be equal to the known value ef‘_K+ when rife—I“ = lye—I“
and its derivative should be equal to zero. From these relations, we obtained the
parameters

01 = 450115I2e-K+ = 8.83e — 17.9tatcouI2A"11

‘72 = “Ugh—In = 4.298 — 20.9ta.tco'u.lzz’l°11

139

From these parameters, we can calculate He -— K + —- He interaction energy
at If‘"K+—H‘ = 3.18A" which is VHc_K+.Hc = —23.7314mev. This should be

the binding energy of this system if there are no three-body effects. But the real

He—K+—He

binding energy 5,, = —24.9mev The difference between VH,_K+ _H, and
€f°_K+ "He is the indirect He—He interaction through polarizable K + ion which is

V2. Thus we obtain
V2 = 53°"K+"H° — VH¢_K+_H, 2 —1.17me'v 2 —14K

Combining the direct and indirect He-He interactions, we get the total nearest

neighbor interaction energy

V=V1+V22—23K

The interaction between He—He atoms in real K—L zeolite is much more compli-

cated. The two contributions we have estimated are perhaps the most significant

OIICS .

Bibliography

[1] J. M. Drake and J. Klafter, Phys. Today, May, 46 (1990).

[2] J. K arger, G. Seiffert and F. Stallmach, J. Magnetic Resonance, A102,
327 (1993).

[3] H. Pfeifer and J. K iirger, J. Mol. Liq. 54, 253 (1992).

[4] G. B. Woods, A. Z. Panagiotopoulos and J. S. Rowlinson, Molecular
Physics, 63(1), 49 (1988).

[5] P. T. Callaghan, A. Coy, D. MacGowan and K. J. Packer, J. Mol. Lid.,
54, 239 (1992).

[6] P. A. Politowicz, J. Phys. Chem., 92, 6078 (1988).

[7] P. A. Politowicz, L. B. S. Leung and J. Kozak, J. Phys. Chem., 93, 923
(1989).

[8] L. M. Schwartz and D. L. Johnson, Phys. Rev. B, 30, 4302 (1984).

[9] J. H. Page, J. Lin, B. Abeles, H. W. Deckman and D. A. Weitz, Phys.
Rev. Lett., 71, 1216 (1993).

[10] B. Abeles, L.F. Chen, J. W. Johnson and J. M. Drake, Mat. Res. Soc.

Extended Abstract (EA-22), 83 (1990).
140

141
[11] R. Blumenfeld and D. J. Bergman, J. Appl. Phys., 62(5), 1616 (1987).
[12] G. B. Woods and J. S. Rowlinson, J. Chem. Soc., 85(6), 765 (1989).
[13] J. M. Shin, K. T. No, and M. S. Jhon, J. Phys. Chem., 92, 4533 (1988).

[14] S. D. Pickett, A. K. Nowak, J. M. Thomas,B. K. Peterson, J. F. P. Swift,
A. K. Cheetham, C. J. J. den 0uden, B. Smit and M. F. M. Post, J.
Phys. Chem., 94, 1233 (1990).

[15] S. D. Pickett, A. K. Nowak, J. M. Thomas and A. K. Cheetham, Zeolite,
V9, 123 (1989).

[16] J. B. Nicholas, F. R. Trouw, J. E. Mertz, L. E. Iton, and A. J. Hopfinger,
J. Phys. Chem., 97, 4149 (1993).

[17] A. Nakano, L. Bi, R. K. Kalia and P. Vashishta, Phys. Rev. Lett., 71, 85
(1993).

[18] I. Bitsanis, J. J. Magda, M. Tirrell and H. T. Davis, J. Chem. Phys.,
87(3), 1733 (1987).

[19] A. Loriso, M. J. Bojan, A. Vernov and W. A. Steele, J. Phys. Chem., 97,
(1993).

[20] M. Sahimi, J. Chem. Phys., 92(8), 5107 (1990).

[21] W. R. Hammond, B. Y. Chen, Aniket Bhattacharya, and S. D. Mahanti,
To be published.

[22] A. V. Vernov and W. A. Steele, J. Phys. Chem., 97, (1993).

[23] B. P. Feuston and J. B. Higgins, J. Phys. Chem., 98, 4459 (1994).

142

[24] J. Karger and D. M. Ruthven, Difl'usion in Zeolites and other Microp-
orous Solids, Wiley, New York, 1993.

[25] D. W. Breck, Zeolite Molecular Sieves (Wiley, New York, 1974), p.636
[26] Carbon black.

[27] H. Seong, S. Sen, T. Cagin and S. D. Mahanti, Phys. Rev. B, 45,
8841(1992); Graphite Intercalation Compounds, V1, Structure and Dy-
namics, ed. by H. Zabel and S. A. Solin, Springer Series in Topics in

Current Physics, Springer (1990).

[28] Ping Zhou, Ph.D. Thesis, M.S.U., 1991. Ping Zhou and S. A. Solin (un-
published).

[29] P. C. Ball and R. Evans, Langmuir 5, 714 (1989).
[30] R. Evans, J. Phys. Condensed Matter 2, 8989 (1990).

[31] R. Jullien and R. Botet, Aggregation and Fractal Aggregates, World Sci-

entific, Singapre, 1987.

[32] C. R. A. Catlow, in Modelling of Structure and Reactivity in Zeolites, p1,

1992.

[33] R. M. Barrer, Zeolites and Clay Minerals as Sorbents and Molecular
Sieves, 1978. T. J. Pinnavaia, Science, 220, 365 (1983).

[34] R. M. Barrer and H. Villiger, Zeitschrift fiir Krystallographies, Bd, 128,
352 (1969).

143

[35] P. A. Wright, J. M. Thomas, A. K. Cheetham, and A. K. Nowak, Nature,
318, 611 (1985).

[36] International Tables for X-ray Crystallography, Vol 1, The Kynoch Press,
Birmingham, England, 1952.

[37] C. J. B. Mott, Catalysis Today, 2, 199 (1988).
[38] H. Kim, Ph.D Thesis, Michigan State University (1991)

[39] M. W. Long, in Series on Advances in Statistical Mechanics-V7: The
Hubbard Model Recent Results (ed. by M. Rasetti, World Scientific, 1991).

[40] E. Fradkin, in Field Theories of condensed Matter Systems (Addison-
Wesley, Redwood, CA, 1991), Chap. 2.

[41] D. Vollhardt, Rew. of Mod. Phys. 56, 99 (1984).

[42] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys.
Rev. B 40, 546 (1989).

[43] B. Y. Chen, S. D. Mahanti and M. Yussouff, Phys. Rev. B 51, 5800
(1995).

[44] B. Y. Chen, S.D. Mahanti and M. Yussouff, Phys. Rev. Lett., 75, 473
(1995).

[45] BY. Chen, H. Kim, S. D. Mahanti, T. J. Pinnavaia and Z. X. Cai, J.
Chem. Phys., 100(5), 3872 (1994).

[46] J. G. Dash, Films on Solid Surfaces (Academic Press, New York, 1975),
Chap.8; F. J. Milford and A. D. Novaco, Phys. Rev. A, 1136 (1971).

144
[47] J. Yuyama, J. Low Temp. Phys. 47, 1 (1982);

[48] E.Cheng, M.W.Cole, W.F.Saam, and J. Treiner, Phys. Rev. Lett. 67,
1007 (1991)

[49] E. Cheng, M. W. Cole, J. Dupont-Roc, W.F. Saam, and J. Treiner, Rev.
Mod. Phys. 65, 557 (1993) and references therein.

[50] R. H. Tait and J. D. Reppy, Phys. Rev. B 20, 997 (1979).

[51] B. C. Crooker, B. Hebral, E. N. Smith,Y. Takano and J. D. Reppy, Phys.
Rev. Lett. 51, 666 (1983).

[52] N. Wada, T. Ito, and T. Watanabe, J. Phys. Soc. Japan 53, 913 (1984).

[53] N. Wada, H. Kato, H. Shirataki, S. Takayanagi, T. Ito and T. Watanabe,
in proceedings of the 17th International Conference on Low Temp. Phys.,
U. Eckern, A. Schmid, W. Weber and H. W iihl, eds. (North Holland,

Amsterdam, 1984), p. 521.

[54] H. Kato, N. Wada, T. Ito, S. Takayanagi and T. Watanabe, J. Phys. Soc.

Japan, 55, 246 (1986)

[55] N. Wada, Y. Yamamoto, H. Kato, T. Ito and T. Watanabe, in New
Developments in Zeolite Science and Technology, Y. Murakami, A. Iijima
and j. w. WARD, eds. (Kohdansha, Tokyo, 1986), p. 625.

[56] N. Wada, K. Ishioh, and T. Watanabe, J. Phys. Soc. Japan, 61, 931
(1992).

145

[57] N. Wada, H. Kato, S. Sato, S. Takayanagi, T. Ito and T. Watanabe,
in proceedings the 17th International Conference on Low Temp. Phys.,
U. Eckern a. Schmid, W. Weber and H. W iihl, eds. (North Holland,
Amsterdam, 1984), p. 523.

[58] H. Kato, K. Ishioh, N. Wada, T. Ito and T. Watanabe, J. of Low Temp.
Phys. V68, 321 (1987).

[59] H. Deguchi, K. Konishi, K. Takeda and N. Uryu, Physica B, 16585166,
571 (1990).

[60] K. Konishi, H. Deguchi and K. Takeda, J. Phys.: Condens. Matter 5,
1619 (1993).

[61] C. P. Chen, S. Mehta, L. P. Fu, A. Petrou, F. M. Gasparini, and A.
Hebard, Phys. Rev. Lett., 71, 739 (1993).

[62] V. 0. Farmer, M. J. Adams, A. R. Fraser and F. Palmieri, Clay Minerals,
18, 459 (1983).

[63] M.J.Wagner, R.H.Huang, J .L.Eglin, and J .L.Dye, Nature 368, 726 (1994)

and references therein.

[64] J. L. Dye, M. J. Wagner, G. 0verney, R. H. Huang, T. Nagy, and D.

Tomanek, to be published in J. American Chem. Society.
[65] C. S. Lent and M. Leng, Appl. Phys. Lett. 58, 1650 (1991).
[66] S. E. Koonin, Computational Physics, 1986.

[67] S. B. Kim and M. H. W. Chan, Phys. Rev. Lett. 71, 2268 (1993).

146
[68] A. D. Novaco and F. J. Milford, Phys. Rev. A 5, 783 (1972).
[69] I. Doi, J. Phys. Soc. Jpn. 58, 1312 (1989).
[70] T. Minoguchi and Y. Nagaoka, Prog. Theor. Phys. 78, 552 (1987).

[71] M. P. A. Fisher,P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys.
Rev. B 40, 546 (1989).

[72] D. K. K. Lee and J. M. F. Gunn, Phys. Rev. B 46, 301 (1992).
[73] Z. Q. Zhang and P. Sheng, Phys. Rev. E49, 3050 (1994)

[74] G. Batrouni, R. T. Scalettar, and G. T. Zimanyi, Phys. Rev. Lett. 65,

1765 (1990).
[75] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981).
[76] J. Voit, J. Phys. (Cond. Matt.), V5, 8305 (1993).

[77] For a open chain, when U —> 00, two particles, whether bosons or
fermions, cannot interchange their positions. Therefore they have identi-
cal symmetry properties. For closed rings, when n is odd, an even number
of permutations is needed to rearrange the many particle states, and as
a result bosons and fermions behave identically. For even 11, an odd num-
ber of permutations is needed instead, leading to the defferences between

bosons and fermions.
[78] M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 (1990).

[79] D. M. Dennison, Proc. R. Soc. London Sect. A 115, 483 (1927).

147

[80] R. K. Pathria, Statistical Mechanics (Pergamon Press, New York, 1978),
p. 167.

[81] C. Kittel, Introduction to Solid State Physics, 5“ Edison, John Wiley

and Sons, Inc, p454.

[82] T. A. Kaplan, S. D. Mahanti, and W. M. Hartmann, Phys. Rev. Lett.
27, 1796(1971).

[83] P. W. Anderson, B. I. Halperin, and C. M. Varma, Philos. Mag. 25, 1
(1972).

[84] W. A. Phillips, J. Low Temp. Phys., V7, 351(1972).

[85] N. E. Cusack, The Physics of Structurally Disordered Matter: An Intro—
duction, IOP Publishing, Ltd: Bristol.

[86] B. F. Chmelka, D. Raftery, A. V. McCormic, I. C. de Menorval, R. D.
Levine, A. Pines, Phys. Rev. Letters, 66, 580 (1991).

[87] R. L. June, A. .T. Bell, and D. N. Theodorou, J. Phys. Chem. 94, 8232
(1990).

[88] A. V. Vernov, W. A. Steele, and L. Abrams, J. Phys. Chem. 97, 7660,
(1993)

[89] A. Loriso, M. J. Bojan, A. Vernov, and W. A. Steele, J. Phys. Chem. 97,
7665, (1993).

[90] Feng Yin Li and R. Stephen Berry, Z. Phys. D 26, 394, (1993).

[91] Feng Yin Li and R. Stephen Berry, J. Phys. Chem. 99, 15557 (1995).

148
[92] Feng Yin Li and R. Stephen Berry, J. Phys. Chem. 99, 2459 (1995).
[93] R. D. Etters and J. Kaelberer, Phys. Rev. A, 11, 1068, (1975).
[94] J. Kaelberer and R. D. Etters, J. Chem. Phys, 66, 3233, (1977).
[95] R. D. Etters and J. Kaelberer, J. Chem. Phys, 66, 5112, (1977).
[96] s. Nosé, Molecular Phys., 52, 255 (1984).

[97] J. B. Nicholas, F. R. Trow, J. E. Mertz, L. E. Iton, and A. Hopfinger, J.
Phys. Chem. 97, 4149, (1993).

[98] S. J. Greg and K. S. W. Sing, Adsorption, Surface Aera and Porosity
(Academic, New York, 1982)

[99] N. J. Wilkinson, M. A. Alam, J. M. Clayton, R. Evans, and S. G. Usmar,
Phys. Rev. Lett., 69, 3535 (1992)

[100] R. Zallen, Percolation Structures and Processes(NY, 1983)
[101] R. E. Grim, Clay Mineralogy, 2nded.,(McGraw-Hi]l Book, 1968)

[102] M. S. Whittingham, Intercalation Chemistry (Academic Press, NY
1982),p1.

[103] Z. X. Cai, S. D. Mahanti, S. A. Solin, T. J. Pinnavaia, Phys. Rev. B42,
6636 (1990)

[104] I. D. Johnson, T. A. Werpy, T. J. Pinnavaia, J. Am. Chem. Soc. 110,
8545 (1988)

[105] H. Kim, W. Jin, et al, Phys. Rev. Lett., V60, 21, 2168(1988).

149
[106] F. Tsvetkov and J. White, J. Am. Chem. Soc. 110, 3183 (1988).
[107] G. Reiter and S. C. Moss, Phys. Rev. B 33, 7209 (1986).

[108] Z. X. Cai, Ph.D thesis, 1991.

[109] The measurements were carried out by putting the diffusing gas on one
side of the porous film with a fixed pressure (> 95 torr) and then mea-
suring the time in which the partial pressure on the other side of the
film(which is pumped to keep the pressure at ~ 10'7 torr) approach its

saturation or steady state value.
[110] D. Stauffer, Introduction to Percolation Theory, 1985
[111] L. M. Schwartz and J. R. Banavar, Phys. Rev. B39, 11965 (1989).

[112] Rao et.al, unpublished.

 

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