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UMI

ELECTROCHEMICAL MEASUREMENTS AND THERMODYNAMC PROPERTIES

OF

ALKALI FULLEEUDES

BY

JOON HONG KIM, M. S.

A Thesis

Subrnitted to the School of Graduate Studies

in Partial Fulfillrnent of the Requirements

for the Degree

Doctor of Philosophy

McMaster University

O Copyright by Joon Hong Kim, September 1997

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EMF MEASUREMENTS AND THERMODYNAMICS OF ALKALI: FULLEFUDES

DOCTOR OF PHnOSOPKY (1997)

(Materials Science and Engineering)

McMaster University

Hamilton, Ontario

TITLE: Electrochemical Measurements and Themodynamic Properties of Alkali

Fuilerides.

AUTHOR: Joon Hong Kim, M.S. (Stevens Institute of Technology)

SUPERVISOR: Professor A. Petnc

NUMBER OF PAGES: xiv, 238

Abstract

The systems Na - Csa, K - C60 and Sr - Ai were studied by the Electromotive

force (EMF) technique using electrochemical cells which were cornposed of beta-alurnina

tubes glass-sealed to a-alumina lids. EMF measurements of NaxCso at 599 K indicated

solid solution regions at 1.7 < x < 3 and 3.3 < x 4 2 . (A sudden drop in EMF in the range

of 3.3 < x < 3.7 allows the possibility of a nearly stoichiometric phase

compounds were observed in the composition range of O < x <

measurements, the Gibbs energies of mixing of f Na3& and $. Na&so I+

in this range). No

1.7. From EMF

rere determined as

85 and 81 kJ/rnol, respectively. From EMF measurements of KXCm at 572 EC, the Gibbs

energies of mivng of & KK,Cso were found to be 83, 1 17, 120 and 12 1 IdIrno1 for x = 1,

3, 4 and 6, respectively. The ideal interstitial solution model of the Na - Cso system

indicates that tetrahedral sites are favored by Na. The difference between NaxCso and

u60 seems to be related to the ionic size effects. EMF measurements of the Sr - Al

system using a Sr beta-alumina solid electrolyte were unsuccessful. Thermodynamic

consideration suggests that Sr beta-alurnina is not compatible with pure Al or Sr.

A model for the intermolecular interactions between Cso molecules, based on the

effective Lennard-Jones interaction centers and local charge distribution, was proposed

and tested for the PU? structure of pure Cm by both regular solution and cluster variation

methods. As the effective Lemard-Jones interaction centers move fiom carbon atoms to

the centers of adjacent double bonds of the same rnolecule, the fa3 structure in which

the pentagons of a molecule face double bonds of nearest rnolecules becomes more stable

relative to the P ~ T structure in which the hexagons face double bonds. By assigning

charges to carbon atoms and to the centers of bonds, we can model the increase in

activation energy for the jump between 22 and 82 degree orientations.

Extemal vibrational frequencies of simple cubic (SC) C h 0 and (M = K., Rb)

were computed by the harmonic approximation, using the model for the intermolecular

interactions between C6* molecules. The computed fiequencies of phonons are quite

consistent with expenments. However, for librons there are some differences between

computed and observed hequencies, which suggests that the harmonic approximation is

not adequate for librons. By using group theory, the extemal vibrational modes of SC C60

at r, C, A and A are labeled and the corresponding symmetry adapted vectors are

obtained.

Thermodynarnic properties of SC Ca were computed using the canonical

partition function. The lanice energy and configurationai entropy terms are estimated by

the regular solution model. The dispersion curves were approximated by the Debye model

for acoustic modes and the Einstein model for optical modes. The Debye characteristic

temperature and optimum GrIineisen constant were determined as 54.14 K and 7.5,

respectively. The contribution of the 82 degree orientation of the molecules to the

constant volume heat capacity was found to be significant in the temperature range of 20

to 100 K.

ACKNO WLEDGMENTS

I should like to acknowledge the people who helped me in studying at McMaster

University: Dr. Petrk, my supervisor, for his kindness and helpful instructions; Tadek and

Gord for their technical help. I already miss the friends at the school: Ron, Jason, Pash,

Jinhyung, Heekwon, Dongkyun, Sungkyu and many others. I cannot thank my family too

much for their patient support and encouragement. This thesis is dedicated to my parents,

wife Iunghye and precious daughter Patricia.

Table of Contents

Abstract

List of Figures

List of Tables

1. fntrodliction

II. Literature review

II. 1. solid electrolytes

iI.2. beta-alurnina

II.2.a. crystal structure

II. 2 .b. phase relationships between Na B- and p"-duminas

I1.2.c. powder preparation

II.2.d. forming beta-alurnina tubes

II.2.e. sintenng

11.2.f. ion exchange

ïï. 3. alkali fillerides

II.3.a. pure Cso

n.3 .b. alkali fullendes

n.4. electrochemistry of solids

II.4.a. electrochemical potential

X

xiii

1

6

6

9

9

12

II.4.b. transport in solids

II.4.c. thermodynamic measurements

II. S. thermodynamics of solids

II. 5 .a. statist ical thermodynamics

II.5.b. lattice statistics

II.6. lattice dynarnics

II. 6. a. external vibrational modes

11.6. b. group theory in lattice dynamics

III. Expenrnents

III. 1 . fabrication of Na beta-alumina cells

III. 1 .a. slip-casting

IU. 1 .b. sintering

III. 1 . c. glass-sealing

III. 1 .d. ce11 assembly

III.2. fabrication of K beta-alurnina

ILI.2.a. K ion exchange

III.2.b. direct synthesis of K beta-dumina

Ln.3. EMF measurements

m.3.a. Na - Ca system

III.3.b. K - Ca system

III.3.c. Sr - Al system

IV. Results and discussion

IV. 1. slip-casting

IV .2. sintering

IV.3. K ion exchange

IV. 4. glass-sealing

1V.5. Na - C60

IV.5.a. XRD

IV. 5 .b. EMF measurements

IV.5 ,c. ideal interstitial solution

IV.6. modeis for Ca

IV.6.a. the symmetry of simple cubic Cso

N.6.b. models for the intermolecular interactions between

C60 molecules

IV.6.c. lattice statistics of Cm

IV.6.d. orientational order-disorder transformation in CsO

IV.7. lattice dynarnics of Cao

IV 7.a. SC Cao

IV.7.b. MîCao

N .7 .c. quantum mechanical consideration of

angular vibration of SC Ca

IV.8. group theory of C ~ O

viii

IV.9. thermodynarnics of Cs0

IV.9.a. the equation of state for Cso

IV.9. b. the bulk modulus and the thermal expansion

coefficient of C60

IV.9.c. the heat capacity of Cs0

IV.lO. K - Cao

W.11. Sr -Ai

IV. 1 1 .a. EMF measurements

IV. 11 .b. compatibility of Sr P-alumina with pure Al

IV. I 1 .c. conclusion

V. Conclusions

Appendices

A. Eulenan angles

B. Orthogonal matrices for proper rotations

C. Construction of the unitary multiplier representation at

the symmetry point r for SC Cao

D. The numencal method for calculation of the Debye fùnction

Refer ences

List o f Figures

Figure 11.2.1. Schematic projection of beta-alumina ont0 the b-c plane.

Figure 11.2.2. Conduction plane of beta-alurnina.

Figure II.3.l. (a) The icosahedron projected fiom 2-fold axis and

The standard orientations (b) The truncated icosahedron

Figure IL3.2. The molecule at standard orientation A projected

from [ l 1 11 direction.

Figure 11.3.3, The 12 nearest neighbors around the molecule at the origin.

Figure 11.3 -4. Proposed phase diagram of Mx& (M = K, Rb).

Figure 11.4.1. Relationship among diffusion coefficients, mobility and

conductivity of ions.

Figure 11.4.2. The equivalent circuit of mixed ionic-electronic conductors.

Figure n.5.1. Order-disorder transformation (Landau expansion) for

unsymmetrical case.

Figure III. 1.1. (a) The ceIl assembly. (b) The expenmental apparatus for

EMF measurements. 97

Figure N.4.1. Thermal expansion of K and Na p-aluminas. 111

Figure IV. 5.1. X-ray difiaction pattern of pure Cso. 114

Figure IV.5.2. The accurate determination of the lattice parameter of pure C60. 1 14

Figure IV.5.3. EMF versus x in Na&* at 599 K.

Figure IV.5.4. EMF versus x in Na,Cso in the dilute solution region at 599 K.

Figure IV.5.5. Proposed phases in M,Cso.

Figure IV. 5.6. The activities of Na and Cs0 in Na,Cso.

1 Figure IV.5.7. The formation energy of - Na,Cso.

x+l

Figure IV.5.8. EMF of Na,C60 computed by equation L1.5.2 1. The ratio of

occupied tetrahedral sites to Cno molecules.

Figure iV.6.1. Lu's and Lamoen's models for SC C6o.

Figure 1V.6.2. The effects of shifi x of effective interaction centers to adjacent

double bonds on the Le~ard-Jones potentials in SC Cao.

Figure N.6.3. The effects of local-charge distribution on the Coulomb

potentials in SC C6().

Figure IV.6.4. The mode1 for the intermolecular interactions in SC Cao

proposed by this work.

Figure IV.6.5. The formation energy of defects in hypothetical

SC C60 (y = 68 degrees).

Figure N. 6.6. Cluster of 10 molecules in SC Cso.

Figure N.6.7. Computed thennodynamic properties (X, A, E, TS) of SC C6o.

Figure IV.6.8. Fused interaction centers infcc Cso.

Figure IV.6.9. Computed fkee energy curves versus temperature for

Figure IV.7.1. Dispersion curves for SC Cso along A, Z and A.

Figure IV.7.2. The probability densities of the first 10 energy levels for

angular vibrations of SC Cao.

Figure I V 9 1 . (a) Lattice parameter and @) Xp of SC C60

Figure IV.9.2. Bulk moduius of SC Cao.

Figure IV.9.3. Thermal expansion coefficient of SC Cso.

Figure IV.9.4. Constant volume heat capacity of SC C ~ O .

Figure IV.9.5. Constant pressure heat capacity of SC Cao.

Figure IV. 1 O. 1. EMF of Kx& at 572 K.

Figure IV. 10.2. EMF of &Ca for dilute solution region at 572 K.

Figure IV. 1 1 . 1 . Schematic diagram of Na-ALO.

Figure N. 1 1.2. EMF of pseudo-temary systerns in Sr-4-0 .

Figure A.A. 1 . Transformation of coordinates from (x, y, z) to (x', y', z').

L

Figure A.D. 1. f (2) = - approximated by equation (A.D.3) for 1.16 < z e' - 1

and by equation (A.D.5) for O < z < 1.16.

Figure A.D.2. Debye function.

xii

List of Tables

Table IL2.1. Lattice parameters of beta-alurninas.

Table 11.3.1. The coordinates of C in C60 with standard orientation A.

Table IV.2.1. Density of XB2-SG.

Table IV.2.2. Density of well-crystallized powder tubes sintered at 1585 O C .

Table IV.3.1. Weight gain of K-ion exchanged p-alumina (well-crystallized).

Table IV.3.2. K-ion exchange of XB2-SG p-alumina tubes.

Table I V 5 1. The X-ray difiaction of pure Cso.

Table.IV.5.2. The reported phases in alkali-fullerides.

Table IV.6.1. The position and the rotation axes of 4 molecules in

a basis of SC C60.

Table IV.6.2. Four sets of parameters for the mode1 of intermolecular

interactions between Cao molecules.

Table IV.6.3. Parameters used in cluster variation method.

Table N.7.1. Extemal vibrational fiequencies of SC Ca at r.

Table IV.7.2. The extemal vibrational fiequencies of MZCso (M = K, Rb) at T.

Table IV.8.1. The character table for Th.

Table N.8.2. The character table of C2,,('y).

Table N.8.3. The exponent term in equation (II.6.5 lb) for A.

xiii

Table 1V.8.4. The character table of Cs(-).

Table IV.8.5. The exponent term in equation (IL6.5 1b) for L.

Table IV.8.6. The character table of C3.

Table IV.8.7. The exponent term in equation (IL6.5 1 b) for A.

Table IV. IO. 1. The formation energy of KxC60 at 572 K.

Table A.D. 1. The coefficients in equation (A.D.6).

1. Introduction

Solid electrolytes have many potential applications in devices such as sensors,

pumps, batteries and ion-selective membranes [1,2,3]. Such devices would operate on

simple and straightfonvard electrochemical principles which rhould allow simple setup,

control and measurement of themochemical properties. However, the technique is

materials dependent (or limited). Developrnent of new electrol ytes as well as

improvements in their processhg have become prominent research topics.

Beta-alumina is a well-known solid electrolyte which has been studied for

decades. The ideal stmcture of Na P-alumina consists of altemathg spinel-like blocks

(Al& ,&' and conduction planes (Nd20)" [4,5]. The high sodium ionic conduaivity is

due to both a low energy bmier for jumps between sites and the number of sites available

which is greater than the number of Na ions. As Na beta-ahmina can be ion-exchanged

with monovalent and multivdent ions [6,7], it is used as a host for preparing a variety of

ionic conductors. This thesis dernonstrates the use of beta-alurnina for thermodynamic

measurements of the Na - CM and K - & binary systems as one of the electrochemical

applications.

The recent discovery of & fullerene, the third major form of pure carbon,

opened a new era in chemistry [8,9,10]. It is a very attractive material due not only to its

symmetry but also to its potential applications. The stable GO molecule can form a vanety

of compounds [ I l ] including the alkali metal doped among which K, Rb and Cs

fullerides exhibit superconductivity with critical temperatures as high as 41 K

[12,13,14,15]. Because this family of superconductors has a simple intercalated face-

centered-cubic stmcture [16,17] with three dimensional superconductivity, they may offer

a window into the still mystenous mechanism of superconductivity [9 ] . Indeed, a simple

correlation between lattice parameters and Tc's of fullendes has been proposed [17].

Though the absence of superconductivity in the smaller alkali (Li, Na) fuilerides could be

explained by the disproportionation of A& into A2Cso and &Cao [15], the phase

diagrams of these systems are not clear. Difficulties in synthesizing some fullendes (Li,

Na, Cs, and their mixtures) [14,15,17] and codicting structures of have been

reported [14,15]. When we consider the large size of interstitial sites (1.12 A for

tetrahedral sites and 2.07 A for octahedral sites) [18] in the pristine C6(), the slow rate of

reaction in alkali - Cs0 is unusual. The high degree of intemal Freedom [19], especially

rotational or orientational, should play an important role in these compounds. The

structural difference [15] between light alkaii (Li, Na) and heavy alkaii (Y Rb) fullerides is

also interesting. Themodynamic data wiil provide fundamental information about the

intercalation process in these systems. In this paper, a simple mode1 based on ideal

interstitial solid solution has been developed to explain the phase relations in NaxCso which

are obtained by the EMF measurements.

The thermodpamic properties of Ca have many interesting features. Above 400

K., the molecules, cubic-close-packed, rotate continuously [20]. That is to Say, they are

restricted to solid-like freedom in translation but exhibit liquid-like freedom i i i rotation.

Below 260 K, the molecules are ordered in orientation [21]. However, they are ratcheting

between two orientations of which one is more favorable in energy by about 10 meV 2221.

An understanding of the origins and magnitudes of intemolecular forces, and their

dependence on molecular properties and intemolecular separation and orientation, is

essentid for understanding rnany properties of molecular crystals [23]. In this paper, a

model for intermolecular interactions between C6() molecules has been developed based on

van der Waals and local-charge interactions and has been used to compute some

thennodynamic properties of pure alkali Ca. This model can be used to predict structures,

lattice energies and the relative stability of proposed new Cao compounds.

In terms of statistical mechanics [24,25], the thermodynamics of solids may be

separateci by the static part and dynamic part. The static part consists of the lattice energy

and configurational entropy tems. The dynamic part contains the thermal energy terms

which are detemineci by the dispersion curves, the relation between the fiequency and

wave vector of phonons. Thus, the partition functions for the static part and dynamic part

can be treated separately and from the resulting Helmholtz kee energy we can derive

thermodynamic properties using the relations between them [26]. The new model for

intemolecular interactions has been used to compute the lattice energy and the vibrational

fiequencies by harmonic approximation [27]. To complete dispersion curves, the

frequencies should be label4 by group theory [28]. The purpose of computing properties

of Cao is not only to predict the properties of CbO but also to test the model by cornpanson

with experimental results.

This work consists of three parts; 1) preparation of solid electrolyte cells, 3)

EMF measurements of alkali fullerides, and 3) modeling of the systems. The background

on solid electrolytes, beta-dumina and alkali fullendes is reviewed in section II. 1 to 11.3.

Principles of electrochernistry related to themodynamic measurements are discussed in

section 11.4 which includes the concept of electrochernical potential, definitions of

diffision coefficients and Onsager's equation. Thennodynarnic properties and their

relations are given in section II. 5. Section II. 5.a introduces the statistical mechanics of

solids, the quasi-harmonic model and the Griineisen constant. In section II. 5 .b, the Ising

model, lattice gas model, regular and quasi-regular solutions, order-disorder

transformation and cluster variation method are discussed. Section 11.6 provides

formulations to obtain dispersion curves related to the dynamic part of thermodynamics of

solids. In section II.6.q the dynamical matnx is expressed by Bom-von Karman constants

and the Ewald Keiierman method is introduced to treat Coulomb interactions. Applicatica

of group theory to lattice dynamics is discussed in section II.6.b. Experimental procedure

is given in chapter III. The fabrication of beta-alumina tubes and the assembly of cells are

describeci in section IV. 1 to 3. The electrochernical measurements of Na-Cso and K-C6* are

given in IV.5 and IV. 10. The static part of thermodynarnics of Ca is discussed in section

IV.6 which contains models for intermolecular interactions between Ca moIecules and

lattice statistics of SC andfcc Cm. Lattice dynarnics and group theory of Cso are presented

in section IV.7 and IV.8. Using the results of section IV.6 to IV.8, the thermodynarnic

properties of Cs0 have been computed in section IV.9. The cornpatibility of Sr beta-

alurnina and pure Al is discussed in section IV. I I .

11. Literature review

IL 1) Solid electrolytes

Solid electrolytes are materials with high ionic conductivity and low electronic or

hole conductivity (< 1%) [29]. These materials are also named supenonic solids or fast

ionic conductors. The electrical conductivities of usehl solid electrolytes are in the range

of 0.01 - 1 il-lcrn'l. A list of eleancd conductivities of various materials zt room

temperature is shown below for cornparison.

Solid electrolyte; Na- P -alumina [9] a = 2 x 10" ~ ' l c rn - '

RbAg415[30] a = 2.7 x IO-'

Aqueous solution; 0. l M NaCI [3 11 a = 1.1 x 10-'

Metal; Copper [3 11 O = 6 x 10'

Semiconductor; Ge (pure) [3 21 a = 2.5 x IO-'

hsulator; Si02 giass [32] a -; 10-l4

The total eIearical conductivity of a material is the sum of the conductivities

contributed by each species in the matenal. The transference number (t) of the species j is

dehed by the equation [32];

t , = Q//Q~

where a, is the conductivity due to the migration of the species j and a, is the total

conductivity. For good solid electrolytes, the transference numbers of conducting ions are

almost unity and those of electrons and holes are less than 104. An electronic band gap

larger than T/300 eV is required for low electronic conductivity, less than W4 ohm-'cm"

~ 9 1 .

Solid electrolytes can be classified by the matenal type as crystalline materials

[33], glasses [34],polymers [ 3 5 ] , and composites [36,37]. The transport mechanisms are

described classically by the vacancy, interstitisl, and interstitialcy models [3 11. However,

the mechanisms are unique to each material, although still not clearly identified in many

cases. The ionic conductivity can be related to the chernical diffusion coefficient D of the

ideal solution by Nernst-Einstein equation [102];

CT = Z 2 e 2 c ~

(II. 1.2) kT

Generally accepted common structural features charactenstic of solid electrolytes are [34];

1) a highly ordered structural array in the form of t u ~ e l s , layers, or three dimensional

array s

2) a highly disordered complementary sublattice in which the number of equivalent sites is

greater than the number of available ions to fil1 them.

For amorphous materials, the first point should be replaced by a more general statement:

3) a open fiamework with an abundance of physicaiiy interconnected and accessible sites

and relatively large windows connecting these sites.

The following are cornmon mobile ions in solid electrolytes:

1) protonic ( R, a', H(H20)x' etc.) [38]

2) anionic ( O", F, Cr) [39,40]

3) alkaline ( ~ i ' , Na', K', Rb +) [29,4 11

4) IB elements ( ~ g ' , Cu') [30,42]

5) multivalent ( ca2', ~ a * ' , cd2') [43]

Solid electrolytes have many applications [L,2,3]. Because galvanic sensors with

solid electrolytes can be used for direct and rapid measurements of concentrations in gases

and liquids, they are suitable for automated process controi. Solid electrolytes can also be

used in energy storage and conversion systems. Solid state primary and secondary

batteries, fuel cells, and load-leveling storage batteries are major applications that offer

advantages such as high efficiency and low pollution. Solid electrolytes have many other

applications such as ionic or atomic pumps, thermoelectric devices, high temperature

mixed conductors, electrochromic devices, etc.

KI.2) Beta-alumina

IL2.a) Crystal structure [44]

The Na beta-alurninas are composed of alternating spinel-like blocks and conduction

planes. The spinel-like blocks consist of four close-packed oxygen layers with Al3' ions in both

octahedral and tetrahedral interstices (figure II.2. l). The conduction planes are loosely packed

with conduchg cations (Na+ for Na beta-alumina) and oxygen ions. The oirygen ions in the

conduction planes form A-0-Al bonds which hold the adjacent blocks apart. Beta-alumina

may refer to either p- or pt'-duminas. wereafter, beta-alumina refers to both P- and Pu-

alumina phases unless otheFIJise specified.)

p-alurnina containing a two-fold screw axis has a hexagonal structure (space group

P63/mmc) with the ideal formula Na209 1 1&0, (NaAi1 10 i7 ) . The unit ceU consists o f two

spinel-iike blocks. The condudon plane perpendiailar to c is a mirror plane and the

conduding cations are in the middle of the conduction planes. There are three types of sites for

conduhg ions: one Beever-Ross, one anti Beever-Ross and three mid-oxygen sites per .

conduction plane in a unit cd (figure 1.2.2). p"-alumina with a three-fold screw axis has a

rhornbohedral structure ( ~ 3 m ). The unit ce11 contains three spinel-ke blocks. The ideal

formula of Pt'-alumina is NaO5.33AI-h ( N a 2 A l l i . v ~ O ~ ~ ) . The conduction plane is not a mirror

plane but staggered. Beever-Ross and anti Beever-Ross sites are not distinguishable in v- alumina.

The dEerence in the structures of the two phases cornes fiom the sequence of close-

packed oxygen layers.

P-alumina: Ce-ABCA-B'-ACBA-C'

p"-alumina: Ce-ABCA-B'-CABC-A'-BCAB-C'

where primed characters represent conduaion planes in which only one fourth of sites are

occupied by oxygen ions. The sequence of close-packed oxygen layers is k e cubic close

packing. However, the sequence in the B-alumina is modified by the mirror image in the

conduction planes.

Figure IL2.1. Schematic projection of t I Cr beta-alumina onto the b-c plane [4].

A The horiiontal lines 4 B, C, and A 0- @ O ~4 C I c3 represent the sequence of close

1 8 packed oxygen layers. Alurninurns

11.3 A

î and sodiums are solid and open

circles, respectively. The oxygens in a b 8 02 84 81 z the conduction plane are hatched

A

I m 3 circles.

Figure II.2.2. Conduction plane of

beta-alumina [ 5 ] . The base layer

consists of close packed oxygens (al1

A sites in figure II.2.1) In the

conduction plane oxygens are placed

at one fourth of B sites (B3 in figure

II.2.1). Sodiums are at the Beever-

Ross sites (C2). The anti-Beever-

Ross site is marked as b (Ai). The

n id oxygen sites (md) are vacant B

sites.

II.2.b) Phase relationships behveen Na f3- and p"-aluminas

The phase diagram of this syaem is not clear. The beta-aluminas are non

stoichiometric phases of the compositions between Na20.5.33Ai203 and Na20-9A.1203 where

both P- and B" phases coexist. In the undoped Na20-M203 system, it is suggested that P"-

alurnina is metastable above 1585 O C and that P and P" phases coexist at temperatures below

this, where the p phase may be metastable [45]. Doping of beta-alumina with cations such as

Li' or M ~ ~ ' stabilises the P" phase at high temperatures TEM study showed that beta-alutninas

fonn a syntactic intergrowth of B and P" structures [44] in a single phase, which explains the

sluggish kinetics of equilibnum in this system.

Newsarn [46] calculated the upper k t of the sodium content in p-alurnina

conside~g two charge compensation mechanisrns.

1) Nal,Ai110i7w (x<O.57 1,-Na20-7A1&). Excess Na' ions occupy rnid oxygen sites, which

requires the adjacent Beever-Ross site to be empty. Additional oxygen ions charge-

compensating for excess Na' ions reside at interstitial sites in the conduaion planes. The

additional oxygen ions are bonded to the two adjacent spinel-like blocks by an Al3'-O*'-AI)'

bridge where the two ~ l ~ ' ions are fiom Al sites in the spinel-like block leavhg two vacant

Ai sites.

2) Nai+,,MgyAl i,O i7 0.<0.667,-Na20*0.4MpO*6Al2G). The excess charge is compensated by

dopants which have lower valency than Al. The dopants substitute Al in spinel-like blocks.

However, if another charge compensation mechanism is available, the sodium content will

exceed the above limit in Na p-durnina. Ions with a radius < 0.97 (Li' and ~~~1 cm

substitute for ~ l " ions in the spinel-like blocks [45].

The end composition Nal n,M~niAl iop~0~7 of B-alurnina is also the ideal composition

for p"-alutnina [44]. At that composition, a maximum in mid-oxygen site ocaipancy and

Beever-Ross site vacancy seems to rnaxhnh the electric conduciivty (vacancy diffusion).

IL2.c) Powder preparation

Polycry stalline beta-durnina is fab ricated by high t ernperature readon (sint ering) of

powder compacts which range fiom M y crystaked beta-alumina to a mixture of raw

materiais. The powder preparation consists of rniving of raw materials and reaction (calcination

or fusion) of the mixture foliowed by milling. The powder prepared should be suitable for the

subsequent forrning processes. The properties of the siitered body so strongly depend on the

properties of the powder used that without close control of powder preparation one can not

produce good ceramics.

M i h g of fùsed ka-dumina is the most direct way to prepare the beta-dumina

powder. The fùsed material is crushed and then d e d in a stainiess steel bali ma. M W c

irnpurities are rernoved by leaching with HCl which dso removes any sodium aluminates [47].

Sol-gel processes yield very homogeneous mixtufes on the ionic or microcrystalline

(400 A) scaie 1481. The cornponents are mixed in liquid form as either coiloicial sols or

aqueous solution and are subquently converted into a solid gel by either chernical or physical

means without segregation [45]. The intirnate mWng and fine particle size results in very

homogeneous fine microstructures.

Slurry solution spray drying [48] produes powder suitable for isostatic pressing.

Non-soluble raw materials siich as a-alutnina or boehmite (AlOOH) are rnixed with a solution

of sodium and lithium hydroxides. The slurry is spraydried resulting in a chemically

homogeneous, flowable, sofi powder. The beta-alurnina phases are formed d u ~ g siintering.

Calcined beta-durnina powder is çpray-dried in order to get agglornerates of powder suitable

for isostatic pressing.

The conventionai and most simple way is to miil raw matenals in wet or dry medium-

Milling provides both mWng and size reduction of raw materials which are in the form of

oxides (aluminum oxide, Lithium oxide), carbonates (sodium carbonate, lithium carbonate) or

nitrates (sodium nitrate). In order to avoid reaction during milling, organic liquids such as

acetone and aicohol are used as a medium.

Calcination is employed before sintering because it wiil reciuce the evolution of a gas

phase, shrinkage due to phase transformation and tirne for chernical reactions d u ~ g the

sintering. However, calcination is not necessary when above effects are not signifiant (e.g.,

slurry solution spray drying).

The zeta process is developed for making lithium s t a b ' i Na pu-alumina [49]. The

process uses zeta phase (Li&&) as a source of lithium. nie LiAIJOI is sniiied with a-ahmina

and sodium carbonate and then caicined. In another route, the LiAlrOc is rnilled with a

precalcined mixture of alumina and sodium carbonate and, without calcining, &ed. Due to the

good distribution of lithium compareci with other conventional methods, alrnost complete

convasion to De'-alumina and a small grain s i i (-50 pm) can be attained.

II.2.d) Foming beta-alumina tubes

Ln isostatic pressing, the green body is formed by cornpressing the powder which is

contained between a concentric metal mandrel and a tubular polymer bag through which the

pressure is transmitted to the powder. Compareci to die pressing, isostatic pressing can form

more complicated shapes with uniform densiîy [Ml. Isostatic pressing can compact alrnost aii

the powders with l e s restrictions and is suitable for mas production. Spray drying is a

prerequisite for isostatic pressing. It provides uniform f f i g of the mould (flowabiiity) and high

compaction with small pores (sofbiess).

In electrophoretic deposition, electridy charged beta-alumina powder suspended in

an organic iiquid (e.g., amyi alcohol) deposits in an elecüic field on either anode or cathode

[50,51]. The powder can be charged negativey or positively depending on the rnechanism of

charging .

beta-alumina + (betay + ~ a ' (dissociation mode)

beta-alumina + HA + (H/beta)' + A' (absorption mode)

The exceIlent sinterab'ity of elecüophorePcally deposited greenware r d t s f h m

very fine pores in the green articles (-1 pm compare to 5-10 pm in isostatic pressing).

However, a prerequisite for the powder is the ability to fom a stable suspension. The

propdes of the suspension depend on the el&d double layer between the partide and the

liquid vehicle. Well-crystallizôd particles have a pater tendency to be charged properiy.

Adequate mobility (mobility of particles in an electric field), low drying shnnkage and ease of

removing the greenware &om the mandrel are the factors for ~uccesstùl forming.

In slip-casting, the beta-alumina powder in suspension deposits on the porous mould

as the mould absorbs the liquid kom the slip. For even dense deposits, high stability and low

viscosity of the slip are cmcial requirements which limit the powders available for this process.

In order to reduce the shrinkage during drying, a higher ratio of powder to liquid is used than

in electrophoretic deposition. The DLVO (Deryaguin, Landau, Venvey and Overbeck) theory

explains the stability of suspensions in ternis of the London-van der Waals attraction and the

coulornbic repulsion associateci with charged particles [52]. 'The attraction between particles is

mainly van der Waals which is proportional to rd (short rangeci). The repulsion due to the

double layer is roughly proportional to the surface potential and ranges over the thiclaiess of

the double layer. The thickness of the double layer is a good measure of the stability of the slip.

The thickness (K") of the electncd double layer of an f i t e flat interface is given as [52]

where the parameters are defined as foilows

D : the dielectric constant of the iiquid

k , T : the Boltzmann constant and absolute temperature

ni ,Z, : the concentration and the valency of the ionic species i in the Liquid

e : the elementary charge.

1124 Sintering

Sintering is the most diEcult step io cuntrol in the fabrication of beta-alurnina. The

final products should have dense, uniform, fine microstnictures for good electncal conductivity

and high mechanid strength.

The high volatility of sodium oxide results in a high rate of loss during sintering. The

loss of sodium wüî reduce the siierabiiity and electncal conductivity. Beta-alumina crucibles,

platinum lining, Mgû crucibles and a-alumina crucibles treated with sodium oxides are useû to

prevent the reaction between beta-duminas and crucibles [53]. Packing sodium rich powders

around the greenware reduces the sodium loss during firing. Zone siintering (passing beta-

alurnina through a srnall hot zone) also minimises the sodium los.

Another problem is duplex microstructure: fine grain matrùc (4 O pm) and isolated

large grains (-100 pm). Duncan [54] observed in lithium doped Na beta-alumllia that the

overgrown grains were p"-alumina. It was suggested that the poor distribution of iithium

shouid cause this problem Zeta processed products [49] which rnust have more uniform

distribution ofiithium showed an increased proportion of fl"-aiumina and well-controkd grain

s o i distriution.

Two peak sintering (presintering, annealing and cooling down foliowed by final

sintering at higher temperaîure) is a another route to cuntml the microsenichire [54].

Moderately coarse (20 p) but d o m microstructures have high mechanid strength

For the sintering of stabüized ~ " - a l u ~ where a high ratio of P" to P phase is

desi i le for high conductiviiy, the densification seerns much fister than the conversion to the

p"-alumina, The postsintenng annealing [55] was employed to promote the conversion while at

the sarne t h e inhibiting exaggerated grain growth.

The seeding catalysis (addition of a smd amount of p"-alurnina crystals (<5 pm) into

the geen c e d c ) 1561 can enhance the phase conversion to pu-alurnina and even eliminate the

need for pre- or postsintering anneating.

Hot pressing produces the best beta-duminas with almost full density and extremely

fine (-1 pm) uni fon microstructures. However, this method is iimited to simple shapes such

as disks and rods.

IL24 Ion exchange

Na beta-duminas can be ion-exchanged with monovalent (K', Li', Rb', 43') [6] and

rnuitivaient (Cd", ca2', sZ', pb23 [7] cations. Ion exchange is performed in the melts or

vapour phases of salts. During ion exchange the dierence in the lattice parameters between

dEerent beta-aluminas produces internai stresses which rnay lead to rrachg of the produa.

The lattice parameters of ka-duminas are given in the Table II.2.1.

Ion archange of polycrystalline beta-aluminas in molten dts c m cause corrosive

attack which leads to a considaable increase in the porosity [571. Two types of stresses can be

formed in Na-K ion exchange [58]. The macrostresses are developing due to the rnaaoscopic

compositionai gradient and the rewlting uneven lattice expansions between grains.

Microstresses are caused by the anisotropy of the lattice expansion due to both ion exchange

and the subsequent cooling.

Table n.2.1. Lanice parameters [A] of beta-durninas [44,48,59]

A phase transition frorn P" to f3 phase was observed in divalent ion exchange of Li-

stabilized Na beta-durninas [60]. The divalent ion exchange causes extreme stresses to the

spinel-iike blocks. Lidoped rnaterials minimise these stresses with reorientation of the structure

to the f! phase.

Tan et al. [61] studied the strength controlling flaws in Na beta-duminas which were

isostaticaiiy p r d and zone sintered. The rnicrostnichires were a fine grainecl rnatrix (1.5-2.5

pm) cuntaining a mail proportion of larger se~ndary grains, up to 200 W. For smaii

samples, the M r e initiating flaws were swfke irregularities and large crystai grains or a

combination of these. The strengths of larger sarnples were governed by larger aystai grains,

by regions of high porosity or by irnpunty areas associated with large grains.

In order to prwent the damage due to ion exchange, one can reduce the stresses or

strengthen the rnaterials in the foUowing ways:

Na B"(Li doped)

K D''(Li doped)

Rb p"(Li dcped)

Na B

K f i

Li

Ag P

c axis

33.539

34.078

34.344

22.530

22.729

22.642

22.498

one block

11.180

11.359

1 1.448

1 1.265

1 1.365

11.321

1 1 .249

a axis

5.617

5.619

5.613

5.594

5.596

5.593

5.594

ion size

0.95

1.33

1.48

0.95

1.33

O. 60

1.26

1) High temperature employed can enhance the diffusion in the beta-alurnina and may induce

stress relieving processes such as creep.

2) Usiig vapour phase of salts reûuces the ion exchange rate at the interface relative to molten

salts.

3) Salts of mixed cations (e.g., Na and K salts for K ion exchange) can reduce the

compositiod difference between the surface and the interior of beta aluminas.

4) Beta-aluminas which have the lattice parameters similar to those of final ion exchanged beta-

alumina c m be used as a preairsor matenals.

5 ) For high mechanid sûength, homogeneous and fine grain microstnictures are desirable.

6) Strengthening mechanisms such as zirconia second phase can be employed.

IL3) Alkali Fuiierides

lI.3.a) Pure C60

In 1985, Kroto et al. [62] discovered the stable species during experiments

involving laser-vaporizing graphite in a pulsed jet of helium. The truncated icosahedral

molecule (figure II.3.1) was named buckminsterfullerene or buckyball for short. The

stability of Ca over planar graphite fragments is due to the absence of dangiing bonds

[63] . In 1990, Kràtschmer et al. [64] found that the Ca molecules formed by graphite-arc

discharge in low pressure helium. Due to the hexagonal carbon rings in the molecule,

can be extracted from the carbon soot by arornatic solvents such as benzene and toluene.

The separation of Cd0 fTom mal has also been reported [65].

The shape of C60 is-a tmncated icosahedron (figure II.3.1 b). The icosahedron has

i,, (or mj) symmetry with 12 vertices, 30 edges and 20 triangular sides (figure II.3.h).

There are 12 5-fold axes at each veriex, 15 2-fold axes each of which bisects 2 edges, and

10 ? -fold axes each of which cunnects the centers of 2 triangular sides. Mer tmncation it

has 60 vertices, 12 pentagons, 20 hexagons and 90 edges; 30 of them are between

hexagons (double bonds) and 60 of them are between a hexagon and a pentagon (simgle

bonds) (figure 11.3.2). The lengths of double and single bonds are about 1.40 and 1.45 A,

respectively [66]. The mean ball diameter and baii outer diameter are 7.10 and 10.34 A,

respectively .

@)

Figure 11.3.1 (a) The icosahedron projected nom 2-fold axis. The standard orientations are

related by the 90 degree rotation about any coordinate axis (x, y, z) which is aligned with a

2-fold axis. (b) The truncated icosahedron. Mer truncation, the vertices and sides become

pentagons and hexagons, rcspectively.

Figure 11.3.2. The C60 molecule at standard orientation A projected from [1 1 11 direction. The double bonds (thick line) are between hexagons, the single bonds are between hexagon and pentagon. The dashed lines represent the direction of the lattice vectors.

Figure II.3.3. The 12 nearest neighbors around the molecuie at the origin in SC C&. Each of them are rotated about fout different cl 1 I> directions. The rotation a i s of the molecule at the origin is perpendicular to the paper.

Pure C60, fiillerite, has a first-order phase transition at a temperature between 249

K [67,68] and 260 K [69]. In both phases the molecules are cubic-close-packed. Above the

transition temperature the structure is face-centered-cubic with ~ m 3 m syrnrnetry [2 11. In

this phase the molecules are orientationally disordered. Above 400 K., the molecule exhibits

continuous rotation [20]. Between 260 and 400 K there are some orientational correlation

between adjacent molecules which seem to have rapid-ratcheting motions (jump

reorientation between orientations) rather than constant angular momentum rotations [70].

The rotational reorientation time is about 9 to 12 ps at room temperature [22]. Below the

transition temperature, the development of orientational order causes the transition to

simple cubic structure with ~ a 5 symmetry [21]. In the low temperature phase, the

molecules are still 'ratchethg [71]. However, the rotational reorientation time is much

slower (2 ns at 250 K) than in the high temperature phase. Below 90 K the reorientational

motion is fiozen although a small amount of static disorder still persists [69]. The values of

enthalpy and entropy of transition are estimated as 6.7 IdIrnole and 28 J/mole K.,

respectively [72].

When three perpendicular 2-fold axes of the molecuIe are digned in three

Cartesian coordinates, the molecule is said to be at one of two standard orientations (figure

II.3.la); those are related to each other by 90 degree rotation about any Cartesian

coordiate. Note that four out of twelve ?-fold axes of the molecuie which is at one of the

standard orientations are aligneci in four difEerent <1 1 1> directions. The structures of pure

& [68] can be attained from the cubic-close-packed molecules which are onentationally

ordered with any one of the standard orientations (hereafier, if not specified, the standard

orientation A will be used). M e n the orientation of the molecules is disordered, the

structure is fcc ( ~ m j m ) , the high-temperature fom (above 260 K). The structure of the

low-temperature phase (SC. PU^) consists of a bais of four molecules each of which is

rotated about four different <1 t 1> axes through an equal angle from the standard

orientation (figure II.3.3). The structure has energy minima at rotation angles of about 22

and 82 degrees dock-wise [69] from standard orientation A. At about 22 degrees (see

figure ;J.3.2), the electron-rich double bonds face the centers of electron-poor pentagonal

sides of the nearest neighbors (absolute minimum). At about 82 degrees, the electron-rich

double bonds face the centers of hexagonal sides of the nearest neighbors (local minimum).

The coordinates of three C atoms in standard orientation A are given in table

O

II.3.1 where the distance between C atom and the center of the rnoIecule is about 3.55 A

O

and the lengths of single and double bonds are 1.45 and 1.40 A , respectively. The whole

set of coordinates cm be obtained by cyclic permutations of (x, y, z) and the combinations

of positive and negative signs (b, &y, y,).

The standard enthalpy of formation of crystalline C60 was measured by

combustion calorirnetry: A ~ H : (298.1 5 ~ ) = 2422 f 14 kJlmole [73]. The vapour pressure

of & was measured over the temperature range 730 to 990 K by the torsion e&sion and

Knudsen effusion methods [74]:

log P = (828+.020) - (9145 f 150) / T

where the unit of P is kPa.

The Cso molecule has 174 vibrational modes and 46 distinct mode frequencies

+ 7H. (5)

where the subscript g

the parenthesis is the

and u denote even and odd parity, respectively

degeneracy of the type of mode. Among them

8HJ and 4 infiared TI.)

[76,77]. The non-active

dculated [79,8O].

fundamental frequencies were observed by

(11.3.2)

and the number in

10 Raman (2& +

Bethune and Frum

frequencies were observed by neutron scattering [78] or

Table 11.3.1. The coordinates of C in & with the standard orientation A [66].

J s + i 4s-i where C, = - , C,=- , D, = 1.45 A and D, = 1.40 A .

4 4

LI.3. b) Alkali fullerides

The routine way to synthesize Arcao (A = alkali) is the direct reaction of Cm with

alkali-metai vapor in sealed Pyrex tubes ai temperatures in the range of 200 - 500 K, for

several hours to several days. &Cao (A = K., Rb, Cs) was reported to be an alkali saturated

forrn of fùlleride [15,81,82]. Difficulties in synthesking Na and Cs containhg fullerides

were overcome by changing the starting materials [ 1 5,821. Some superconducting phases

could be metastable [ 141.

The electrical conductivities of alkali doped Ca films (Li, Na, K, Rb, Cs) were

reported by [83]. A,& is insulating for x = O or 6, and metallic for x = 3 [84,85].

However, Na and Li fullendes could be insulators across the entire range of composition

[86]. The transitions to the superconducting state for K3C60 [12] at 18 K, RblCm [13] at .

28 K., Cs3& [87] at 30 K, and other mixed aikali doped C60 have been reported. AU of

them have fcc structures with alkali metals in completely filled octahedral and tetrahedrai

sites [16,17,8 1,821. Superwnductivity at 8.4 K in Ca& waç aiso found [18]. A simple

cubic structure with multiple occupancy at octahedral sites was proposed for this

compound. The effect of pressure [88,89] and isotope [go] on superconductivity were aiso

studied. The critical temperature for the superconductivity of alkali fullendes increases

monotonically as the unit-celi size increases 117,861.

A phase diagram of the Rb,& system at room temperature was proposed [82].

The phase included an fcc structure (rock salt structure, x < l), another fcc line phase (full

occupancy of tetrahedral and octahedral sites, x = 3), a bct structure (x = 4) [91], and a

bcc structure ( 5 < x 5 6) [8 11 as determined by X-ray diffraction. Kx& and C~CMI

systems are similar to RbxCso except for the absence of C S ~ C ~ . The proposed diagrams are

shown in figure II.3.4 [92]. A polymeric cross-linking between the molecules was reported

for both KICm and Rb& [93], which have a pseudo-body-centered orthorhombic

stmcture and transform reversibly on heating to the rock-salt phase [22].

The phases of NaxCsa (x = 1, 2, 3, 6, 11) were reported [22]. In this system, the

molecules are cubic-close-packed regardless of the composition. Stable Na& and NaiCso

phases showed a phase transition to a simple cubic structure [94, 951 like pure Cm

fullerite. Na3C6~ exlubited disproportionation into Na2Cm and Na,& [22, 941. In Na&,

each tetrahedral site is occupied by a Na atom and each oaahedral site accommodates 4

Na atoms in a tetrahedral fom [15]. For Naii&, 9 Na atoms are expected to go into an

octahedral site [96]. The L i F a system was studied by the electrochernical technique [97].

Three domains were observed; O to 0.5, 2 to 3, and 4 to 12. Intercalation appeared

reversible up to x = 3. The structures of the phases were not determined.

The Lemard-Jones type interaction between the molecules tends to order the

orientation of the molecule as in pure 194, 951 whereas the repulsion between the

alkali metal and the molecule tends to align the moleniles in one of the standard

orientations. Na2&, due to the s m d repulsion, has a phase transition to PUT due to the

orientational order as in pure &. For heavy alkalis such as K and Rb, the repulsion causes

a merohedrai disorder (random distribution between two equivaient orientations, &Cm)

or order in the orientation of the molecule [66, 941.

C a 1 2 3 4 5 6 K Composition

Figure 11.3.4. (a) proposed phase diagram of MX& (M = K., Rb) [94]. RS and M I

represent rochait and monoclinic structure, respectively. @) Proposed phase diagram of

KxCso pz].

IL4) Electrochernistry of solids

There are two approaches to electrochemistry: a traditional (thermodynarnic)

one and a modem (physical) one [98]. Traditional electrochemistry is derived from the

concept of equality between chernical and electrical energies in the electrode reaction.

This approach emphasizes chemical and energetic aspects of the problem, but the physical

picture is certainly obscure. The modem approach is focused on a description of the

molecular interaction at the interfacial region. Energetic aspects are less evident.

However, it gives a generd picture with insight beyond the traditional barriers. Although

the interfacial phenornena such as the electrical double layer are not treated in this section,

we need to know the concepts of electrochemical potential and transport in solids in order

to understand the electrochernistry of not only ionic but aiso rnixed ionic-electronic

conductors.

I.4.a) EIectrochemical potential

The definition of electrochemical potential of component i, ji, , is;

ii, = v, + Z,F4 (rr.4.1)

where p, , ZI , F , and 4 are chemical potential, charge of i, Faraday constant, and

electrical potential, respectively. The electrochemical potentiai is the sum of the energy

increase in the phase by adding an infulltesirnally s m d amount of i ( w , ) and the work

necessary to bnng charge 2, F from a vacuum at infinity into the phase (2, F4 ). Thus

electrochemical potential is not a material constant but depends on the size of the phase

and the properties of the interface. In equation (IL4.1) 4 is the electrical potential inside a

phase (the imer potential or Galvani potential) and is the sum of the outer potential y

and the surface potential x .

~ = w + x (n.4.2)

The outer potential for a phase in vacuum is defined as the elcctrostatic work donc to a +1

charge from infinity to a point just outside the phase. As the electncal potential is fairly

constant over a wide region (IO-' to IO-' cm), the exact position of the outer potential is

not important [99]. Whereas the outer potential is proportional to the excess charge, the

surface potential is related to the dipolar layer at the interface. The dipolar layer may be

due to the spreading of electrons into vacuum (metal-vacuum interface) or the orientation

of dipole moments of molecules (solution-vacuum interface) [98]. In either case, the

electrical potential is related to the distribution of charges by Poisson's equation

1 v2,$ = --p (n.4.3) E

where s and p are the permittivity and the electnc charge density.

When two phases (1 and 2) are at equilibnum, it is not the chernical potentials

but the electrochemical potentials which are equal.

Fi (1) = Pi (2) (IL4.4)

It should be noted that for neutrd species the electrochemical potential is equal to the

chemical potential and that the electrochemical potential of the charged species is related

to the chemical potential of the neutral species by

LY

v , +Z,F2 = CI,' (U.4.5)

where 1, 2 and x denote ion, electron and neutral species, respectively.

In most good solid electrolytes, the chernical potential of the mobile ions is

essentially constant throughout the bulk electrolyte because of the high ionic disorder.

There is no major potential drop within the electrolyte in this case. Also, an electncal field

may not build up within the electrodes because of high electronic conductivity. Thus the

electrostatic potential drop occurs nearly exclusively at the interfaces between the

electrodes and the electrolyte [100]. A half-ceIl or electrode is said to be reversible if the

rates of its forward and reverse reactions are rapid compared to the rate at which

conditions are changed. The electrode is polarizable when the reaction rate of oxidation or

reduction is so slow that charges can accumulate at the interface.

IL4.b) Transport in solids

Some usehl equations are su- below [ 100,lO 1,1021.

j = - ~ V C (Fick's lirst law, chernical diffusion coefficient 6 )

(Fick's second law)

(mechanicd rnobility or absolute rnobility B,)

D, = B,kT (component diffusion coefficient Dk)

(n.4.6)

(11.4.7)

(IL 4.8)

0, = fo, (tracer diffusion coefficient D, and correlation factor j) (II. 4.1 2)

D = D , w (Wagner factor or thermodynamic factor W) (11.4.13)

B = B,kT (Nernst-Einstein relation for ideal solutions) (11.4.14)

ô9 i = -0 - = -4 le 9 (conductivity o and electrical mobility id) & ax

Thus, the particle flux in general case is expressed as

The interrelations arnong diffusion coefficients and mobilities and conductivity are surnrnarized

in figure K4.1 [IO 11.

DiBision is an irreversible process in which the total entropy increases [103, 1021.

The scope of this review is limited to hear transport processes and near-equilibnum situations

with rnicroscopic reversibility (local equilib rium) . nie local equilibnum means that the

thermodynamic state functions have the same values as they would have in an equilibrium

situation.

The entropy production is the sum of the products of general forces X, and flues J,

where 8 is the entropy increase per unit volume per tirne.

tracer diffusion cwff icient

s e l f diffusion r---- coefficient I I 1

absolute mobility H conductivity B(cm2s-'erg" ) a( Q -'cm-' j

1

diffusion cwfficien t absolute mobility mobility of dcfect of defect u(cm2s-IV-' )

Dd(cm2s-l ) ~~(cm~s"erg - ' 1 ,

C : ionic concentration Cd : defect concentration

rate constant of diffusion controlled chernical reactions

Figure II.4.1. Relationship among diffuson dc ients , mobüity and conductivity of ions

[101]. The self diffusion coefficient (or cornponent clifhion coefficient in equation II.4.11) is

used in a sense that the jump is random.

In a contmuous adiabatic system, the entropy production due to the transfer of heat, mass and

electric charge at constant pressure is given as

where d p , is the change in the chernical potential only due to the composition gradient

The entropy production lads to a dissipation of energy.

Onsager's equation in the isothermal case is

where L, is a phenomenological coefficient, independent of the forces.

Onsager's reciprocity relates the coefficients in the foilowing way:

4 = L,i

Omger's equation can explain many irrevenible processes such as

Du four effect: flux of heat causeci by a concentration gradient

Thermal diffusion: flux of rnatter caused by a temperature gradient

Soret &kt: t h d diision in liquids

Peltier effect: flux of heat caused by a electficai gradient

Seebeck &ect (or thennoelectnc effect): flux of charge caused by temperature

gradient.

For isothemal diffusion with three mobile species, Orisager ' s equations are

*l 2 a3 % j , = L,, F + L , , -+ L , , + 4,- at ax ax

a c r l a2 * 3 4) j, = L,, + L,, - + L,3 - + LM -

& & dx

acr , & 3 0 j4 = L,, - + Ld2 -+ Ld3-+ La- & ax ax ar

where J , , j, are the fluxes of ions, J , is the flux of electron and J4 is the electric airrent.

On the other hand, we have

where by cornparison we can determine the phenomenological coefficients and see the

reciprocity of the coefficients. The flwes are coupleû since L,, = L,, Ë O . From equation

@.4.30b), we have

where r, = transference number (equation II. 1 . 1 ) . Thus equation (11.4.27) is rewritten as

dlna,, Z,alna,. k,. 1, . (1 - t,)-- t , -+-j4

8 ln c,. ~ 4 n c , . I ùx Z,e

where the superscript x denotes neutral species.

For chernical diffusion, j, = O because of the charge neutmlity. We define the

chernical dfision coefficient as

Thus, f?om equation (iI.4.32b)

where the general Wagner factor y [IO01 is

It is not only the component dfision coefficient but aiso the Wagner factor that determine the

chernical dfision coefficient. For metals t , = 1 and in an ideal solution, the chernical dfis ion

coefficient is qua1 to the component difision coefficient. For good solid electrolytes i , 2 l

the chemical diffusion coefficient is almost r d . Foc good electrodes, t , should be close to one

and the derivative of the activity with respect to the concentration should be large, which can

be attained in nonstoichiometric compounds with a narrow range of stability.

IL4.c) thennodynamic measurements

The electrochernical techniques have general advantages over other methods

11 041;

1) Electrical quantities are readily measurable with high precision.

2) The fundamental thennodynamic and kinetic quantities are directly transduced into cell

voltages and electncal curent.

3) The composition may be precisely changed in-situ.

4) The experimental arrangement is generally very simple.

Sotid electroiytes are the essential part of this technique. General requirements

for solid electrolytes are [IOSI;

1) high ionic conductivity

2) extremely low electronic conductivity

3) structural stability over a wide temperature interval

4) chemical stability and cornpatibility with electrode materiais

5) ease of fabrication of thin and dense membranes

6) mechanical strength.

p -dumina is one of few materials which satisfj above requirements.

Thermodynamic approach

When t , = 1, the transfer of one mole of component i through the electrolyte

accompanies the d e r of Z p moles of electrons through the extemal circuit. The electrical

work done by the transfer of electrons is less than or qua1 to the change in the Gibbs Free

energy .

Z,FE 5 -Ap,= (II.4.36)

The revenible potential is defineci as

The limiting value of E measured as the m e n t goes to zero is called the electro-motive

force of the 4 (the ceU E W ) and is quai to the reversible ceii potential E, . EMF can be

expressed in the form of the Nernst equation.

-=- 2.3 O3 RT ap ,,, log- = wO- 2303RT

ZIF ZIF l W l .

R

wtiere the abscripts R and W denote the reference and the working electrodes, respectively.

Kinetic approadi

From equations (II.4.5) and v.4.30)

where subscript 1 and 2 represent ions and 3 denotes electrons.

Since for the open circuit, i = j 4 = 0.

At the interface, the electrochemical potentials of electrons on both phases are equal. As

the compositions of the leads to the extemal circuit are identical, the difference in the

electrochemical potentids of electrons is the same as in the electrical potentials.

where the mean transference number is defined as

Thus, the EMF is the sarne as in the thermodynamic approach only when the mean

transference number is equd to unity.

2) stoichiometiic compound MISI Xi

Using the GibbsDuhem relation, 12, ldp,, + Z,dp,, = O, equation v.4.4 1) is written as

Thus the EMF is constant in the muiti-phase region and it will change rapidly as the

composition of the solution changes.

Equivalent circuit

The equivalent circuit of the solid electrolyte cell is show in figure 11.4.2 where

Ei represents the reversible ce11 potential due to ion i . The sum of the potential difference

dong a closed loop is zero.

E, + r,i, + r,(il + i,) = O

E, + r,i2 + r,(il +i,) = O

Thus,

(n. 4.4 5)

(II. 4.46)

Equgions @.4.37), (U.4.41) and m.4.47) indicate that the EMF is the sum of - - - -

- - - - - - - - - - - - - - - - - - - - - - -

the produas of the transference number times the chernical potential of the neutral

species. If the transference number of the component A in the sotid electrolyte is unity,

then the EiW between phase 1 and 2 is given as

The following mixture properties can be derived h m l35 measurernents; fiom equation

(II.4.48), when phase 1 is pure component A,

Gy = R T l n a , =-Z,F*EMF (II.4.49)

cA = FT(a2w/a2 T ) x,.~

From Gibbs-Duhem integration [106], if phase 2 is a binary alloy,

we c m derive partial mixture properties of B.

The integral property Y, is given as

Y, = X,Y* + X,YB

Figure 11.4.2. The equivalent circuit of rnixed ionic-electronic conductors. The subscnpts

1 and 2 are for ions and 3 is for electrons. e, is given by equation (11.4.37).

IL5) Thermodynamics of solids

In thermodynamics the change in the interna1 energy is given as the sum of the

products of the intensive variables and the conjugate extensive variables.

wtiere 6 and X / are the generalized force and the conjugate extensive variable [107]. Using the

Legendre Wonnation, state fundons are dehed as a funaion of a combination of

variables.

U = U ( S , V ; = . , N , )

where the unspecified variables can be expressed as the partial derivative of correspondmg

au d~ conjugate variables. For example, T = - and S = - - as ar '

For a complete desaipion of the thmodynamic propedes of a substance, it is

diciait to have one of the thermodynamîc potentials. The fiindion A(T, Y) is most

cornrenient because it is associatexi with the structural modd of a body in the siiplest way [26].

I fA = A(T, Y) is known, we can e d y derive other pro@- with the heip of the relation

However, the e d y meamrable propedes are C, , a, P, and P, where

me auioiiary equaîions can be used to derive these properties fiom the fùnction A(T, Y).

Thus, if the proposed model gives the fiee energy A(T,V), we can cornpute other properties.

Convemly, ifwe masure C, , a and p , other properties cm be curnputed.

The change in the intenial energy due to the change in state f?om (Ti, Vl, P l = O) to

(T2, VI, P2 = O) is aven as

dW=cW=CpdT

(E) Y) ,dV , the wark of expansion is defined as Since dU(T,V) = CvdT + -

If the intemai mergy can be repnsented by the sum of the lattice energy u(V) and the

vibrational energy Pb(T,v), the inaement of the lattice energy can be computed with

r wa2 The work of expansion = -dT = u - u, + AEd

P r

IL5.a) Statistical Thennodynamics

Suppose that the macrosystem is mechanically and chemicaiiy isolated, but in thmal

equiîiirium with its environment at T. The canonical aisanbie partition hction Q is defined as

where j , E, and !2 are a microstste (or an eigenstate), the energy (the eigenvalue) and the

degeneracy, respectively [24], The probability that the microsystern is in the state j is given as

And the thermodynamic fiindons are Wntten as

dA and = - are used to dmve A and correspondhg properties.

The extended version of the canonical partition hction is &en as the geneml

partition hction [107]. Assume that the system has (n+l) independent variables. In the

canonid partition hction, the system is speafied by T and n intensive variables Q, .

Q= Q K Q , ; . - , Q r ,a+, ;.-,Q,> (11.5.22)

where it should be noted that Q and Q, are differentiated. The gened partition function Zr is

defined by (rt 1) extensive variables 6, and (n-r) intensive variables.

where A is the space of the microstates cr compatible with (Q,,, ; *-, Q,) .

i3 ln Zr Q, =< Q, (a) >= kT -

gr

pr+l ton)

(II.5.26)

(II. 5.27)

The con£igurational degrees of Beedom can be rqarded as a separate system.

The relative fluctuation of the interna1 energy is given as

When the system consists of 1 mole of atorns, the relative fluctuation is extremely s m d (on the

order of 10"3. There is an overwhelming probability that the system wili be in one of a set of

1 S= k M 2 = kln-

PW

Thus, entropy is the measure of disorder. For large number of particles, the partition funaion

can be expressed by the maximum-tm (the maximum terni method).

In the h m o n i c approximation, the lanice component of the fiee energy of a crystal

may be written as

where u is the lattice energy [log, 261. Since the intermoleailar vibrations can be separated

fiom the intramolecular ones,

where A is the intmolecular part of the fiee energy and a d is the intramoleailar part.

Neglecting the zero-point energy,

where k and s represent the wave vedor and the branch of the dispersion surfaces. Thus if we

know the lattice energy and the spectnim of the normal modes as functions of the volume, any

thermodyiamic property can be dadated.

L A

Introducing the density of the normal modes,

where &)do is the number of normai modes between o and do in the fkst Brillouin zone.

If a crystal has N unit d s and each unit teil consists of I atorns and m nonlinear

molecules, there are N(31+6m) normal modes of extemal vibrations in the first Brillouin zone

and N wave vecton for each of (3l-c6m) branches [26]. In the Einstein model, the dispersion of

the fiequencies is represented by orie fiquency. Hereafter in this section, we restrkt our

disaission to a crystal with W n o d modes.

In the Debye modei, the dispersion is appmhted by a parabolic equation.

3

= 6N and v. = the average sound velocity . 81t 3

Introducing the Debye fundion

(II. 5.49)

mu . The characteristic temperature is determined where the characteristic temperature O = - k

to fit the experimental results to the model.

From equation (IT. 5.1) and @. 5.4) and the relation U = A + TS, we obtain

If the viirational spectnim is constant, O

Cv and thermal expansion does not ouw - - - - - - - - - - - - - - -

is a constant. Such a model is called h o n i c ; Cp =

- - - - - - - - - - - - - - - - - - -

In the quasi-hannonic mode1 [26], the characteristic temperature is a function of

volume per unit cd.

0 = @(Y)

In this modei the U, S and Cv are the same as in the hamionic model.

Thus, the equation of state for a aystal is

Other properties can be cornputed by the equalities @.5.5 to D.5.7).

The Griineisen constant [108] is defined as

v where y, = -- a ,(k) av

Then, the thermal expansion coefficient can be expressed as

In the Debye mode1 the Grùneisen constant is written as

The experimental c haraaerist ic temperature depends on w hich thermodynamic

function is used to determine it; u d y Cv or S is used (equation ( I I .5.52) or (11.5.5 1)).

Unfortunately the experimental characteristic temperature is not only a fùnction of the volume

but also of the temperature [26].

O =@(Y, T)

This is the result o f the anharmonicity of the vibrations.

Consider a lattice in which each site has two states [log]. If the aates are 'spin up'

and 'spin down' as in a magnétic system, we have an Ising model. In the lattice gas mode4 we

interprete the states as 'full' or 'empty' which is suitable for intercalation compounds. A binary

alloy can be modeled by lattice sites with 'atom A' and 'atom B' .

Ising model

Permanent magnetism is due to die exchange: energy between unpaired spins. In the

Iwig modd [25, 1071 the intemai energy due to the alignments of spins is &en as

where the sign is n&e for ferromagnetism and positive for anti-fmornagnetism, J is

positive, (n-n..) means the swnmation is for nearest neighbor only and a is one and negative

one for spin up and spii down, respectiveiy. The magnetic enthalpy is given as

where H and M are the magnetic field and the magnetization, respectively. The general grand

partion funaion for the ferromagnetism is written as

Ifwe apply the mean-field approximation to the Ising modei,

M = N c o , >=Nm

where N and Z represent numbers of sites and coordinates, respectively.

Lattice gas mode1

In the lattice gas model (or c d model) [107, 1091, the continuous domain V is

replaced by a regular lanice of N sites or N qua1 ceiis. Assume that there is not more than one

moleaile in each ce11 and that the interaction energy between two moleailes depends on the

relative positions of the cells, not on their locations inside the cds. The enthalpy for the grand

canonical distribution (T, p, N) is given as

wfiere N, is the number of the molecules. If site i is ocaipied cr, is one. Othenvise, it is

negative one.

Thus, the lattice gas model is quivalent to the Ising mode1 for fmomagnetism.

Substitutionai solid solutions

The mode1 has twa restrictions: (1) the volume per site is canstant (v,, = 5) and

(2) every lattice site is ocaipied. As in the Ising model only nearest neighbor interactions will

be considered [1071. The numbers of A atoms, B atoms, AA bonds, BB bonds and AB bonds

are expressed by O , which is one if site i is ocaipied by atom A and negative one if by atom B.

(II. 5.68)

where E = 2cAs - E, - cgg

The enthalpy for the distribution ( T , p , - p, , N ) is wrinen as

H,(P, -P,,N)=E-(P* -) ls)N.. ,

It is equivalent to the ferromagnetic or antiferromagnetic Ishg model depending on the sign of

Using mean-field approximation, the correlation between adjacent sites is ignored.

With the zeroth-order approximation (randomized approximation) and maximum-tm method,

we have the regular solution model [Ml.

V.5.79)

For the systerns in which the deviation fiom randornness is considerable, the short-

range order parameter a, is introduced 111 O].

where * means the value in a random distribution. For complete randornness

E = O and a, = O . For association of unlike atoms E < O and a, < O . For clustering of like

atoms E > O and a, > O. In the first-order approximation (quasi-chernical approximation)

[106], the distribution of bonds between atoms is considerd rather thm the distribution of

'ZN! SN, 'NA, = h

2

N* ! NB, ! [(NA, 1 2) ! l2

Thus, the Helmholtz en= is a hction of ( I : Y. NA, Ng, NAB). The equilibnum state is

(II. 5.83)

where N, is Avogadro's number [ 1 1 11.

For order-disorder transformation, assume that A atoms tend to order themselves on

type a sites and B atoms tend to order themselves on the other type P sites [107]. Some

parameters are given as

where N, i and j represent number, atom i and site j, respective1y.

The long-range order parameter 0 is dehed as

For the disorderd state, 0 = O and XA. = XAP = XA. For the ordered state, 8 > O and XAa > XA >

&. Using the zeroth-order approximation, the dierence fiom a regular solution in the

Helmholtz energy is indu& as

NkT M(e)=l[x[X,(l +m)ln(l+ SB) + X,(1- ySB)ln(l -ySB) + SX, (1 - 8)ln(l- 8)

(II.5.88)

where y = X, /X, and Zj is the number of site j around site i. In Landau expansion M(0) is

(D.5.89)

For qua1 sublattices (S = 1)

M@)= b,e2 + b,B4 + b,e6 + * a . (n.5.90)

where 0 is determineci to minimize the energy. Because bn > 0, for b> O, 0 = 0 is the only

solution, that is, the system is disordered. For < O, there are two solutions (ie # O). Thus

the system is ordered. The transformation is secondader.

For S;t 1

AA(8)= b,B2 + b,e3 +b,04 + - - *

For fixed XA, we need to solve the foiiowing equations.

a u M(B) = O and (=) = O

XA

The solution wiil be 0 = O or 8, and T = T, . For T < r , minimum bA yields ordered phase

(8 # O). For T > T, , the phase is disorderd (0 = O). The resulting lransformation is k t o r d e r

(see the figure 11.5.1).

*A d

Figure KI.5.1. (a) Free energy-order parameter isotherms for the unsymmetncal case (S # 1).

Curve (3) corresponds to the transition temperature. Temperature increases with the

number of the curves. @) Free energy-mole-hction isotherm showing cunvex envelope.

Point C corresponds to curve (3). (c) Conjugate phases in the mole-fiaction-temperature plane

[107].

An exteoded version of the quasi-chemical method is the cluster variation method

[107]. In the Lst-order approximation, energies and weights for the microsystems on a cluster

or group of sites are treated exactly, but the correlations due to sites shared between groups

are handleâ by plausible assumptions. The lattice is divided into N, basic groups of sites in such

a way that each nearest neighbor pair belongs to only one group and the sites in a group are

quivalent. Some ternis are dehed as

N, = the nurnber of groups

v = the number of nearest neighbor pairs in a group

p = the number of sites in a group

s = the number of species in the system

o, , = the number of configurations of type k

p* = the nurnber of sites ocaipied by species i

vI = the probability that a group has a &c configuration of type k

where k refers to an energy level 6 , with multiple configurations.

The relations between them are

Note that each site belongs to 2 groups. 2 v

The first-order approximation is

ZP -- I Zv

k

where the denorninator is the correction Eictor which makes the degeneracy exact for randorn

distribution. The configurational enthalpy for the distribution ( V , p, , N ) is

(-II. 5 99)

USing equatiom (Il.5.95) and v.5 .%), at equilibrium

where a is independent of v, . Though the solution is not exact, we can sinipllfy it by the

relation

where y, and X are numerically determinai.

An approximation of the nonconfigurational entropy terms is attempted here.

Assume that the interaction is painvise and additive and that the potential energy is spherical.

The potential energy of atom A rnay be written as a hction of displacement r and

ZN* where P, =- ZN,

Pt is the probabiity that a site next to atom i is ocaipied by atom j.

Ai hi& temperatures the vibrationai partition fimction is

1 3 3 - -

2 m : k T d 2 Z U , where q; = ( ) ( ) ( the iibrational partition fundion of pure A )

By analogy, for atom B

Thus, the nonconfigurational excess fiee energy is

Since a, cc 1 , we use the approximation

Cornb ig the nonconfigurational term to reguiar solution model results in the quasi-reguiar

solution [ 1 061.

The m d i e d quasi-chernical solution rnodel wiil be

L

Interstitial solid solutions

There sems to be some confusion or difficulty in the choice of the reference state in

interstitial solutions [112]. The difference between the substitutional and the interstitial solid

solutions is that in the former, the end compositions of the solution are the pure cornponents

and aistomarily the standard state is chosen as the reference state. However, in the latter, the

end composition (completeiy £lied interstices) is not always experimendy accessible and the

Henrian standard state is sometimes more convenient. Shce the change in the

nonconfigurational entropy t e m of the interstitial solutions is more important than in the

substitutional solutions, that should be considered in the model.

In the regular interstitial solution model, the assumptions are the same as in the

regular substitutional solution model. The following ternis are dehed.

NM = the number of solvent atoms M

Ns = the number of solute atoms S

NI = the total number of interstial sites

Nv = the number of vacant interstitial sites

Using the painvise nearest neighbor interactions

E = N w ~ w + NWcw + NSca + + N W g W

Usiig mean-field theory

where Zy is the number of the nearest j sites around a t m i .

After rearranging the ternis,

The subscripts V and I represent the phase with completely vacant interstices and the phase

with fùiiy occupied interstices, respectively.

The zeroth-order approximation gives

Thus, the interstitial solid solution corresponds to the substitutional soiid solution in the

pseudo-blliary system which consists of two building blocks;

M atoms and t vacant

M atoms and 1 S atom at

where the last equation is the same as in the lattice-gas mode1 [log].

We introduce the arbitrq reference state,

For the Raoultian standard state,

kTlny, = -& + c l P - k ~ l n ( P ~ , - x,) for O I X, 5 - P + l

(II.5.128)

For the Hennan standard state, as Xs approaches zero, the advity coefficient goes to one.

Now consider the nonconfgurational teams. The treatments are S i a r to those in the quasi-

regular solution model. The potential energy is a funaion of displacement and composition.

u, (r ) = UMM (7) + U, @)Ys @.S. 13 1)

4 ('1 = U, (r ) + W r ) y s @.S. 132)

The partition fundons are given as

4; = the vibrational partition fiinaion of the solcte in dilute solutions.

Note that ai in interstitial solutions is not necessarily small compared to unity.

3 G--& X, In q; + Xs In p; - -[x, 2 ln(i +a&) + X, ln(1 +a,~,)]}

@S. 135)

Thus, the sum of the configurational terms and nonconfiguraiional t e m results in

For dilute solutions, the chernicai potential of solute is in the fom of

lI.6) Lattice dynamics

U6.a) Erternal vibrational modes

In molecular or complex ionic crystals it is advantageous to separate the ngid-

body motions of the well-bound groups of atoms (external modes) fiorn the relative

motions of atoms in the group about its center of mass (intemal modes). Such a situation

is characterized by 1271:

1) close similarity in the vibrational frequencies of the isolated group and those of the

group in the crystal

2) large differences between the scale of extemal vibrational frequencies and that of

intemal vibrational fiequencies.

The behavior of the normal modes of a one-dimensionai lattice with a bais is discussed by

Ashcroft [108], which shows that when the force constants between atoms in a group are

much larger than those betxveen atoms in dEerent groups, the modes can be separated as

intemal and extemal ones. Suppose that the primitive unit celi contains p single atoms or

ions and v noniiiear well-bound groups. There are 3p + 6v extemal branches; three of

them are acoustic branches and the rest of them behave like optic branches. If there are n

atoms in the primitive cell, the nurnber of intemal branches equals 3n - (3p + 6v) which

have the character of optic branches. In this section the dynamics of external vibrations

will be discussed mainly following Venkataraman [27,113].

When the displacements, both linear and angular, of vibrating units are referred

to a set of Cartesian axes fixed to the crystal, the fiame of reference is referred to as the

fixed fiame. Alternatively, when the displacements are referred to a set of local axes which

coincide with the principal axes of the moment of inertia of the unit in question, this is

known as the principal-axis fiame. Hereafter the dynamics will be treated in the fixeci

Both linear and angular displacernents cm be represented by vectors (the anguiar

displacements are supposed to be infinitesimal). Vectors for linear displacements change

sign under inversion. However, vectors for angular displacements do not. Thus, under an

orthogonal transformation S, those vectors transfomi in matriv representation as

where the superscnpt t and r denote linear and angular displacements, respectively [114].

The determinant of S is itl depending on whether S is a proper or improper rotation.

The equilibriurn position of the kth atorn in the th unit in the lth ce11 is given as

X(hck) = X(I) + X(K) + X(k) (II. 6.3)

where m, = the mass of the unit K (for ai 1 K )

I(K) = the moment of inertia of the unit K(K = p + 1 to p + v).

In the hamonic approximation the potential energy can be expanded in a Taylor senes of

the displacements as

@do + O , +a,

The first-order terni is obviously zero and the second-order term is

where the coupling coefficient between units is

Thus, the Hamihonian is, in the harmonic approximation, given as

The equation of motion for K = 1 to (p + v) is

for K = (pi- 1) to (p+ v)

The solution may have the wavelike form [28] of

u: (W = (+i) exp m. X ( I ) - (qltl}

where q is the wave vector.

Substituting it in the equation of motion, we obtain in rnatrix notation [27]

6, ' (q)mU(q) = B W U ( ¶ )

where the diagonal submatnces of m are

mn(wc)= m, 1, (1, = unitary rnatrix in three dimensions)

mm (KK) = I(Y)

and the offdiagonal submatrices of m are nuil.

U(q) which has (3p + 6v) elements may be written as

The ~rresponding scherne of the dynamid matrix B(q) is

where the elernent of the submatrix is

The eigenvalues o (q) are obtained by solving the secular equation

( ~ ( d - ' (dm1 = 0

We rewrite the amplitudes of the vibration in (n.6.13) as

clé, si) = a(ql)e:, (+ü-)

or

U(W) = a(ql)e(ql)

Thus, e(q) is the eigenvector of the eigenvalue o (q) .

The normalization constraint is

e' (ql)me(qj) = M (II.6.18)

where subscnpt + represents the Hermitian conjugate and M is the total mass of a

primitive unit c d . We can constma e(ql]'s such that

e' (qj)me(qjt) = O ÿ # j' orthogonality) (11.6.19)

Define a matrix e(q) b y

e ( d = [e(ql), e(q2), 2e**'e(q(3~ + 6v))l (U.6.20)

Then, froni (II.6.18) and (II.6.19)

e+ (drne(q) = M13p+6v

Combining (II.6.11) and (II.6.2 1)

e+ (q)B(q)e(q) = M W 0

where Rr (q) = o (q)6 Y.

Thus, the matriv e(q) brings both rn and B(q) into diagonal fonn.

The general solution of the equation of motion is the sum of the displacements

(11.6.10) due to harrnonic vibrations of al1 available q and o (q) .

where the A(ql)'s are referred to as the normal coordinates.

We can invert @.6.23b) to express A(qj).

where superscript * means the complex conjugate.

With (II.6.24) the vibrational Hamiitonian (II.6.8) can be written as

where the (0, t e m has been suppressed. The Hamiltonian (13.6.25) is just the sum of

those of the independent simple harmonic osciiiators of which the allowed energies are

given in quantum mechanics as

In the Born-von Karman treatment, the potential energy is expaiided in a Taylor

senes (n.6.5) of the displacements of atoms rather than those of units. The second-order

where 4, (ld; I 'K 'k ') = 8 0

au, (IKk)d21p (k 'k ')

The displacements of the kth atom in the i ah unit in the Ith ce11 is due to the linear and

angular displacements of the e h unit in the celi.

The cross product can be expresseci by the Levi-Civita symbol eDbr .

where the symbol E* is defined as

E* = O if any two of@, p, y) are equal

= 1 if(% p, y) corresponds to a cyclic order of (qy,z)

= -1 $(a, p, y) corresponds to a noncyclic order of (%y,@.

Cornparhg (II.6.27) to @.6.6) with the help of m.6.29) and (II.4.30), we obtain the

expression for the couphg coefficient (I1.6.7) in terms of the Born-von Karman constants

(II.6.28) [27,ll3].

(II.6.3 la)

If we know the interaction potential of an atom pair, we can compute the coupling

coefncient (the atorn-atom potential method) [26,115]. The interatomic forces depend on

the type of the bonds. For van der Waals interactions, the Lennard-Jones (I1.6.32a) or

Buckingham potential (II.6.32b) is used.

In the covalent binding not only the stretching of the bonds but also the bending of the

bond angle should be considered. In order to be rigorous about metaibc binding,

allowance must be made for electrostatic repulsion of the ion cores (screened by the

electron gas), van der Waals attraction of ion cores and their overlap repulsion, binding

due to incomplete inner sheiis (important in transition elements), and electron correlations

within the electron gas [116]. For ionic crystals the evaluation of the electrostatic

contribution to the dynarnical matrix is dBcult due to the long-range character of the

Coulomb potential. The Coulombic contribution to the Born-von Karman constant in

w.6.28) is given by

where X = X ( l t d k ' ) - X ( M ) and kk I ' K ' ~ ' .

Using equations (IL6.3 1) and (11.6.33) we can cornpute the Coulombic contribution to the

dynarnical matrix (11.6.1 5)

However, uniike van der Waals interactions where nearest or second nearest neighbor

interactions are enough, in electrostatic interactions the surnmation in (iI.6.15) rarely

converges. Thanks to Ewald and Kellerman [27,113] the dynarnical matrix is given by

for K ZK'

where the superscript c represents Coulombic contribution to the dynamical matrix

The reciprocal space series and real space series are defined as

(II.6.35b)

where v = volume of the primitive ce11 and y = ~ x ( z ' ) - X I . The parameter tl can be

chosen such that both the reciprocal space senes and the real space series converge

rapidl y.

IL6.b) Group theory in lattice dynamics

The general terms and definitions in group theory are summarized From

[117,118]. A set G is said to be a group if it has the following four properties.

1) There exits multiplication operation whkh associates with ewry pair of elements

T and T' of G another T" of G: if T, T' E G, then n' = T" and T" E G .

2) The elements obey a law of association: if T, T'and T" E G, then T(T 'TW) = (m')Tu .

3) There is an identity element E: E E G such that ïE = ET = T for any T E G .

4) Every element has an inverse element: T' E G such that T' = T' T = E for any

TeG.

The number of elements in a group is said to be the order of the group. If every pair of

elements is commutative, the group is called Abelian: TT' = TT. If the number of

elements is finite, the group is called a finite group. Othenvise, it is caiied as a Lie group.

A subseWof a group G that is itself a group with the same multiplication operation as G

is cailed a subgroup of G. An element Tt of G is conjugate to another element T of G if,

for an X of G, T' = XTX-' . A class of G is a set of mutually conjugate elements of G. A

subgroup9of a group G is invariant if XTX-' €9' for every S €9' and every X E G .

The coset of a subgroup9with respect to any fixed element T of G is defined as

Right coset = (ST; S = every element of*

Left coset = (TS; S = every element of*

Factor group G / Y is defined as the set of the nght cosets of an invariant subgroup9of a

group G.

G / 9 = { ~ , 5 4 p , a , ~ , . - . )

G = 9 + 9 p + 9 q + Yt+ ...

where p, q and t are the elements of G.

If$ is a mapping of a group G ont0 another group G' such that 4(T, )+(T,) = +(q T , ) for

aii Tl and TI of G, then the mapping 4 is homornorphic. If 4 is a one-to-one mapping of a

group G ont0 another group G' such that 4(T, )4(T, ) = 4(q T, ) for al1 Tl and Tt of G,

then the mapping $ is isomorphic. If there exits a homomorphic mapping of a group G

ont0 a group of non-singular d x d matrices T(T) for elernent T of G, with matrix

multiplication as the group multiplication operation, then the group of matrices T(T) forms

a d-dirnensionaf representation r of group G. If S is any d x d non-singular rnatrix, the

similarity transformation is defined as

r y T ) = S-T(T)s

and r and T' are said to be equivalent representations of group G. A unitary

representation of a group G ia a representation r in which the matrices T(T) are unitary

for every T of G, that is ï(7)' = T(T)'' where the superscnpt + represents the Hermitian

conjugate (the complex conjugate of the transpose). Representation r(T) of a group G is

reducible if it is equivalent to a representation r(7) of the group G that has the fom

A representation ï(7) is irreducible if it can not be transformed by a similarity

transfomation to the reduced form. For al1 T of G, if the representation T(7) is equivalent

to the completely reduced form P ( T ) where aii the diagonal submatrices are irreducible

and alî the offdiagonal submatrices are null, the representation T(T) is completely

reducible.

where T ' T ) = irreducible. The completely reducible representation is equivaient to the

direct sum of T ' T )

r =rl, c ~ r ~ ~ e * ~ e r ~ (11.6.40)

The orthogonality theorem for matrix representation says that if P and P are two unitary

irreducible representations of a finite group G, then

where g and d, are the order of G and the dimension of l? , respectively.

The characters of a representation are defined as

x ( T ) = trace of T(T) (11.6.42)

The character systern of a representation is the set of characters corresponding to the

representation. A necessary and sufficient condition for two representations of a finite

group to be equivalent is that they must have an identical character systern. If xP(7) and

xq(T) are characters of two irreducible representations of a finite group of order g, then

The number of tirnes c, that an irreducible representation r (or a representation equivdent

to it) appears in a reducible representation ï of a finite group is

The symmetry of space about a point can be described by a collection of

symmetry elements called a point group. It is convenient to use five symmetry elements for

point groups; inversion through a point, rotation about an axis, reflection at a piane,

rotation-reflection and identity. In a point group, the aiiowed combination of symmetry

elements laves at least one point unchanged. Only 32 crystdbgraphic point groups are

compatible with the translational symmetry of crystals. The mathematically allowed

combinations of the symmetry elements of crystals are called the space group of crystals.

The symmetry operation Sm of a space group can be represented in the Seitz notation [28]

by

s, = (q v(s) + ~ ( m ) ) (11.6.45)

where S is 3 3x3 real orthogonal rnatrix representation of one of the proper or irnproper

rotations of the point group of the space group, V(S) is a vector which is smaller than any

primitive translation vector of the crystal, and X(m) is a translation vector of the crystal.

When the operation is applied to the position vector X(k) of the a h unit in the Ith unit

cell, the operation transforms it according to the mle

(s[v(s) + x(m))x(k) = S X ( k ) + V(S) + X(m) = X ( M ) (II.6.46)

To express the fact that IC is carried into K by S, = ( ~ v ( s ) + ~ ( m ) ) , we write

Note that X(m) can not affect the relation berneen r and K and only the rotational

element S of Sm is required to speciQ the relationship because when S is deterrnined the

vector V(S) is specified. Space groups for which V(S) is zero for every rotation S are

called syrnmorphic. Al1 other space groups are calied nonsymrnorphic in which non-zero

V(S')'s are associated with glide planes or screw axes.

The application of group theory to the eigenvalue problem [27]

(0 ' (q)mU(q) = B(q)U(q) m.6.11)

not only enables us to classify the eigenvalues and the eigenvectors using group theoretical

labels, but also simplifies the eigenvdue problem. The basic principle is to construct a set

of matnces each of which describes the effects of the synunetry element and cornmutes

with both B(q) and m.

Defined as G(q) = (RJ, the space group of the wave vector q , is the subgroup

of the space group G = (Sm), which is cumposed of space group elernents R, = (RI V(R)

+ X(m)) whose rotationai part R has the property [28]

R q = q + G (11.648)

where G is a translation vector of the reciprocal latticc. Note that G vanishes if q lies

inside the first Brillouin zone and it can be nonzero only if q lies on the boundary of the

zone. We construct unitary matrices I'(q;Rm) whose elements are given by

r, (m '(q;~,) = & w , F o (K ';RI) exp{iq *{x(d - R,x(K')]} (II.6.49)

We see that those matrices cornmute with both B(q) and m and provide a unitary

representation of the space group of the wave vector q [27].

r ( q ~ ~ m ) B ( q ) = wq)r(q; R,) (ïT.6.5Oa)

r(q;R,)m = mr(q;&) (II.6.50b)

m ; R m )Th; RL) = r(q;RmRk ) (II.6.50~)

The purely rotational elements R in G(q), the space group of wave vector q,

comprise a point group G(q) called the point group of the wave vector q [28]. We

associate a unitary matrix T(q;R) with each element R of the point group Go(@.

T(q; R) = exp[iq * (V(R) + X(m)]r(q;~,) (n.6.5 la)

or

Ta, (KK 'Iq; R) = Rao6(ic, Fo (K '; R)) exp[iq - (X(K) - RX(K ')] (11.6.5 1b)

We can see that T(q;R) comrnutes with both B(q) and m and the set of them furnishes a

unitary multiplier representation of Go(q) [27].

Th; WWq) = B(q)T(q; RI

T(q; R)m = mT(q; R)

T(q; R)T(q; R') = 4(q; R, R')T(q; =')

where the multiplier is given by

((4; R,Rt) = exp{ i [g - ~ ' ' ~ 1 - v(R' ) )

Using equation (II.6.48),

+(q; R, R') = exp(iGe v(R')) (II.6.53b)

IF q lies entirely w i t h the Bdlouin zone, the reciprocal lattice G is zero and the multiplier

equds unity. Altematîvely, if the crystai belongs to a symmorphic space group, V(R') is

zero and the multiplier is also qua1 to unity. In each of two cases the set (T(q;R))

provides an ordinary representation of the point group Go(q).

The sarne operation can be c d e d out with an alternative treatrnent known as

the factor group method.

G / p = (x @,a ,54,...) (P7q,~- E G) (II.6.36a)

Lf we take as the invariant subgroup 9 the set of the infinite number of translational

elements, then for symmorphic crystals the factor group is identical with the point group

GO(^)^ and for nonsymmorphic crystals the factor group is isomorphic with the point

g r 0 U P Go(cl) 181.

9 = {{qx(m))) (II.6.54a)

where {R, ) E Go (4) .

Using equation p.6.44) and (II.6.5 l), we obtain cg, the number of tirnes the sth

irreducible multiplier representation T(''(~) of Go(q) whose dimension is f, occurs in the

reducible multiplier representation T(q).

where h is the order of the group Go(q)

ci; R) = Trace of T(q; R) (II.6.56a)

x( ' ) (q; R) = Trace of r (q; R) (II.6.56b)

Thus, there are c, eigenvalues each of which is frfold degenerate. That is, c, sets of

eigenvectors are associated with c, eigenvalues and each set hasf, eigenvectors. In other

words, if there are (3p + 6v) coordinates, c, x f , coordinates belong to the T(')(~)

multiplier representation and c, x f, coordinates divided by c, sets, associated with c,

eigenvalues, contninj. coordinates.

To simplify the eigenvaiue problems the projection operator is

constmcted

where k could be any one of (1, 2, ... ,f,) [28,118]. Applying this operator systematically

on any set of (3p + 6v) orthogonal vectors which have the dimension of (3p + 6 4 , we

project out cs orthogonal vectors; there will be only c, independent vectors. These after

normalkation may be labeled [(qslX), 6(qs2X)), -... ., c(qsc,h). The partnen of these

vectors may be generated by applying to them the operator

Thus, we have c, orthonornid sets of symmetry-adapted vectors.

Applying the same procedure to ail irreducible multiplier representations we h d y have

(3p + 6v) symmetry-adapted vectors {c(qsoh)} which have the property of

unless s = s' and k = h'

Thus, by grouping the symrnetry-adapted vectors with the same s and A together, we cm

block-diagonalize B(q) and m [27].

where subscnpt D represents the block-diagonalized form.

t(q) = transpose of

For each s, the block-diagonal form hasl, diagonal blocks with the dimension of c,.

Under special circumstances, time-reversal symmetry can produce extra

degeneracies in the lattice vibrational freguencies in addition to those due to spatial

syrnmetry. If the point group of the crystal contains a rotational element S- such that

S - q=-q+G (11.6.62)

the space group, which is the sum of the space group G(q) plus the coset

(s_lv(s-))~(~) is designated by the symbol G(q;q) [28]. Note that G(q) is the

where & is the complex conjugate operator.

The purely rotational elements {R) and {S-R) in G(q;-q) form a point group

Go(q;-q) in which Go(q) is an invariant subgroup. With each element S-R an anti-unitary

matrix operator T(q;S-R) is associated.

T ( g S R ) = K , ~ X ~ ( - ~ ~ - [ V ( S - R ) + X ( ~ ) ] ) T ( ~ ; (s-R[V(S-R)+x(~))) (11-6.65)

Introducing R to denote the element {R) and (S-R} in the point group Ga(q;-q), we c m

see that T(q;R) comrnutes with both B(q) and m and the set of them fumishes a

multiplier representation of Go(q) [27].

T(q; Q B ( ~ ) = ~ol)T(q;R)

T ( ~ ; R ) ~ = ~ T ( ~ ; R ) .

~ ( q ; R ) T ( ~ ; F ) = 4(q; R , F ) T ( ~ ; R R ' )

where the multiplier is given by

for R = R

+(q; R,F) = exp{i[q - R - ' ~ ] v(F))

for R=s-R

(II. 6.66a)

(Ti.6.66b)

(11.6.66~)

If either q lies entirely within the Brillouin zone or the space group G(q;-q) of the crystal

is symrnorphic, the multiplier is unity.

Consideration of the transfomation of the eigenvectors (II.6.61) by the

additional operator m.6.65) results in the criterion for the existence of the additional

degeneracy due to the time-reversai symmetry given [28] by

~ ~ ( q ; ~ , ~ , ~ , ~ ) X ( s ) ( q ; ~ , ~ ~ , ~ ) = h (noadditionaldegeneracy) (11.6.68a) R

= - h or O (25 degeneracy) (II.6.68b)

where & is an arbitrary element of the coset S-Go(q).

IIL Experiments

IIL 1) Fabrication of Na beta-alumina ceiis

IL1.a) slipcnsting

Three kinds of powders were used to fonn green tubes: calcined Na fhlumina

powder (Alma XBZSG), calcined Lidoped p"-dumina powder (Ceramatec), and fused and

then ground Na p-alumina powder (hereafter called well-crystalIized powder). The fiised P-

alumina rwived as crushed nuggets (1-3 cm in size) was M e r cmshed and ground in a

v i i r o d for several days. The powder was separated and smaii pdcles (<50 p) were

coiiected by sieving.

Those powden were dned at temperatures higher than 4ûû°C overnight. A mixture

of 45% powda to 55% ethano1 by weight was bail-milled for 3 to 5 days. The slip was

degassed under w u m to remove air bubbles and then slipcast. Coarse a-dumina powder

moulds were used. The residence t h e of the slip in the rnoulds was 10 to 15 minutes. After

three to five days drying, the green tubes were aufacefinished with abrasive papas to remove

a-dumina powder on the s u h a . The typical s i i of the green tubes was 6 cm in length, 1.3

cm in outer diameter and 1.5 mm in thiclaiess.

IlL1.b) sintering

p-aiumina tubes were siitered at temperatures between 1550 and 1620°C for 2 to 12

hours. An a-dumina crucible and iid were used for sintering. The tubes were packed in excess

(3-alumina powder to reduce the ioss of Na. pu-durnina tubes were sinterd at 1550 to 1600 O C

for 5 minutes and then annealed at 1475 OC for 45 minutes to promote the conversion to P"-

aiumina whiie suppressing the grain growth (two peak sintering). To minimw the loss of Na,

the green tubes were pushed into the hot zone at the sintering temperature in 25 minutes (rapid

heating).

IIL 1.c) glass-seding

Dense beta-durnina tubes (He gas-tight) were giass-ded to the a-dumina iids. a-

alurnina lids were formed by diepressing reactive alumina powder (Alma A16), prefiring at

800°C, pefiorathg a 3 mm diameter hole and then siintering at 16W°C for two hours.

Aluminosilicate glas was used as a sealant. The glas was ground and mixed with giycerol in a

m o m and applied to the joint. The tubeglass-lid sealant was heated to 1 3 1 5 O C to melt, held

for 5 minutes, then annealed at 760 O C for 45 min during the cooling stage.

IIL1.d) ce11 assembly

A helium leak test was employed to select tubes with a good seal, i.e., those

havhg a leak rate less than 104 cm3/s. After Uing each cell with the working eiectrode or

reference electrode wmponent in an argon atrnosphere, a tantalum cane was fixed to the

a -alumina iid by a compression seai (figure m. 1. la). This provided a hermetic closure for

the ceil as well as an electncal feed-through. Thereafter, the expenment was operated in

an argon atmosphere glove box.

lU.2) Fabrication of K beta-alumina cek

mL2.a) K ion exchange

Na beta-durnina tubes were ion exchanged in KCI vapour at temperatures between

950°C and 1100°C for two days and then immersed repeatedly into KCl melts at 750 to 8ûû°C

or K N a melts at 400 to 450°C for a day untii the ion exchange was completed. At each step

fiesh salts were used. The tubes were washed in water. The sealing procedure was the same as

for Na beta-alumllias.

llL2.b) Direct synthesis o f K beta-aiumina

- a-alumina ( h a A16) or boehmite (AiûûH), lithium carbonate and potassium

carbonate were used to produce K1.d7Lb3iAlia.&i7. AU chernicals were dned before weighmg

at 300 to W C . Boehmite was added as either caicined (700 to 11ûû O C ) or raw. The

materials were mixed in a baU mil1 with waîery edianol or acetone and calcineci at 1200 to 1450

O C for 2 hotus. The calcined powder was b d - d e d in a liquid medium (ethanol or acetone)

for 3 days and then slipuist in a powder mould foliowed by sintering at 1550 to 1650 OC up to

4 hom.

/ Beta-aiumina tube

/ Electrode components

r'- \

Galvanostat Voltmcttr

I Mctal Bath

Figure m. 1.1 (a) The ceIl assembly. @) The experimental apparatus for EMF rneasurements.

III.3) EMF measurements

ILL3.a) Na - Ca system

The experimental apparatus used for the electromotive force @MF)

measurements is shown in figure (III. Llb). The ce11 consisted of a reference electrode

(RE), a counter electrode (CE), and a working electrode (WE) which can be represented

as

(RE) (CE) W)

Ta 1 Bi-17 at.% Na ( Na B-alumina 1 Na-Pb-Bi 1 Na p-alurnina 1 Na.Cso 1 Ta

where the RE (Bi-17 at% Na) was calibrated against a pure sodium electrode using a

similar ce11 configuration. AU EMF readings are reported with respect to the pure sodium

reference. The CE had a eutectic composition of Pb-Bi with a smali amount of sodium (<

1 wt %). The role of the CE was to provide both a source of sodium for coulometric

titration and a electric contact between the RE and CE. The WE was initially charged with

about 50 mg of fùllerene and 200 mg of nickel powder. Nickel powder, which is

essentiaiiy urneactive to both sodium and carbon, provided the electrical contact between

the O-durnina and the tantalum lead. The composition of the WE was changed in precise

increments by coulometric titration at low current (< 0.1 mA).

(III. 3.1)

where n, i and t are the number of moles transferred, current and tirne, respectively. Mer

each interval of cuulometric titration, the galvanostat was switched off and the EMF was

monitored by a chart recorder. At 675 K it took a few days to several weeks to stabilize

the EMF. Equilibrium was verified by cycling the temperature between 475 and 725 K and

checking that the EMF retumed to the initial value.

m.3. b) K - CHI system

The expenmental procedure is similar to that of the Na system. The ceil

configuration is as follows:

(RE) (CE) W)

Ta 1 Pb-Bi-K 1 K P-alurnina 1 K-Pb-Bi 1 K p-alumina 1 K& 1 Ta

The reference electrode contained 1 gram of the eutectic composition o f Pb-Bi and 3.3 wt

% of K. The counter electrode also contained the eutectic composition o f Pb-Bi and small

arnount of K (< 1 wt %).

IIL3.c) Sr - Al system

Sr p-alumina was used as a sotid electrolyte. The counter electrode was an open

crucible containhg 700 g of Sn with 1 wt % Sr. The reference electrode contained 0.7 g

of Sr with 0.37 wt % of Al. In the test range of 660 - 740 O C , thk is a Liquid-solid two phase

mixture with unit d t y for solid Sr. The working electrode was filied with about 1 g of

99.999 % pure Al at the start of the acperiment.

W) (CE) (WEI

Ta 1 Sr-0.37 wt % Al 1 Sr p-durnina 1 S n 4 wt % Sr 1 Sr p-durnina 1 Ai-Sr 1 Ta

IV. Results and discussion

IV. 1) slip-casting

The most important factor in this process is to control the amount of the moisture in

the slip. A small amount of water in the slip increases the viscosity and tends to cause

f ldat ion, resulthg in a poor green density d e r dryuig. This point was disaissed by Rivier et

ai. for P-alumina-methan01 [119] and Whiteway et al. for rnapesia&ol [120] systerns. A

trace of water in the alcohol seerns to adsorb on the surfkm of the particles and fom gels.

Rivier et al. claimed casting should be done only in a dry rwm with relative humidity less than

30 %. Whiteway et al. clairned 0.5 % of water was the maximum that muld be tolerated.

ûtherwise, the cast articles were soft and slow to dry. In order to dry ethanol, type 4A

m o l d a r sieves (bead) were used. The drymg power of mol& sieves was reported as

follows [121]: with 5% WN desiccant loading for 24 hours, the water content of ethanol (initial

mer content 1500ppm) was reduced to 401 pprn Moleailar sieves were activated at 250 to

320 O C . Extremely dry h o 1 was obtained simply by batchwise drying over moleailar siwes

[122]; allowing 500 cc of ethano1 to stand over about 20 g of sieves for minimum 24 hours and

changing molecuiar sieves 5 tirnes. In spite of such an effort to keep the slip dry, sornetimes the

slip properties were poor espeaaUy on humid days. Water d d be absorbeci during the bail-

milling or fiorn the powder mould during d i e . Na p-aluminas (XB2-SG and weli-

c r y d i powders) showed better slip properties than Na p"-alumina (Cerarnatec). The PM-

dumina was more sensitive to moimire. The difFerence between B and P" powders should be

related to the suTf=dce structures. The slips were decreasingly stable in the order: weii-

crystallized XB2-SG and Ceramatec powder. The more stable slip was less viscous and less

sensitive to water. DuMg slipcasting, a high deposition rate (>2 rndmin) due to high water

content resulted in soft and low density green tubes, sometimes cracking on drying.

Drying should be slow enough not to cause cracks due to shnnkage. There were two

types of cracks: parailel or perpendicular to the length of the green tubes. The parailel cracks

seemed related to the high shrinkage resulting 6om poor slip properties. However, the

perpendicular cracks seemed due to the high dryuig rate or to both factors. The drying

shnnkage was not measurable (c 0.1 %) in most cases.

N.2) Sintering

The loss of Na drastidy changed the densification rate d u h g firing. The a-alurnina

cnicibles were not closed adequatdy to prevent the loss of Na The final density was dependent

on the total amount of beta-aIumina used as packhg powder.

Green XB2-SG tubes were packed in fiesh (never used More) XB2-SG powder

wtiich was coarsened at 12ûû°C for one hour (Eesh-powder-pack). Mer sintering, not only the

tubes but also the powder pack shrunk by about 18 %. The &bct of Na los was so serious that

when fnir tubes were sintered at on- the density was lower than when one tube was sinterd

alone. Because the shrinkage of the powder pack left open space dong the wall of the cnicile,

as the number of tubes inmeased, the diffusion length to the open space decreased. The tubes

with highest density was obtained at 1 576 OC for 4 hnurs (Table N.2.1 Set A). nie densi ty was

measured by the Archimedes method in kerosene. Optical microscopy showeû duplex

rnicrostnia~res; the m e l y large (> 1 0 p) and isolated grains (overgrown grains) were

distriiuted in a fhe grain rnatrix. This stnicture is typical for beta-aluminas. The number and

size of overgrown grains increased with the sintering time and temperature. Though the density

was high (-98 %), the duplex structure caused dltsdties in the subsequent glas-sealing or ion-

exchange because of a higher tendency to frachire.

The same green tubes were buried in a powder which had been used as packing

powder several tirnes (used-powder-pack). This powder was very coarse and the powder pack

showed very linle shrinkage (-1 %) after firing. The siiteting behaviour of XBZSG tubes in a

used-powder pack was quite diierent fiom that in a fiesh-powder pack The densification rate

was lower and constant with respect to time (Table IV.2.1 Sn B). In a used-powder pack,

higher siintering temperatures and longer thes were required for a dense material than in a

fksh-powder pack. The type of powder pack (fresh or used) may affect the sintering

mechanism. The fksh powder maintains high Na pressure, especially at the initial stage of

sintering, which may resuit in liquid phase sintering. Meanwhile, the used powder provides just

enough Na vapour pressure to prevait the conversion to a-alumina The slow and steady

densification suggests ody soiid state siintering ocain, with used-powder packs. The

rnicrosbr~ctures ofthe latter were her and more homogeneous than the former (the overgrown

grains were rare). However, the control of siintering was not easy b w s e too low Na vapour

resulted in low density or second phase a-alumina.

Table IV.2.1. Density of XB2-SG

Set A. fiesh powder pack (apparent density I bulk density)

Set B. used powder pack @uk density)

6 hours I -

-

3.21 / 3.17

-

Anothef p-alurnina powder (weliuystabed, < 50 pn) was b d - d e d for 5 days,

siipcast and then sinterd in a usxi powder pack at temperatures between 1550 to 1600 OC.

The densities of tubes, givm in Table TV.2.2, were measured by the Archimedes method in

water. The full density (= 3.26) was determined by measuring a piece of aystal (as-rdved, -1

an in size). The high apparent density of balumina tubes 3.26) was due to the presence of

a-aiumuia (d=3.98). A srnafi grain size (-10 p) for palumina and second phase a-alumina of

the same size were observed by SEM. The a-aiumuia rnay have been introduced by extensive

4 hours

3.22 / 3.18

3.23 13.19

3.23 / 3.20

3.22 13.17

L

1596 O C

1586 O C

1576 O C

1566 OC

2 hours

-

3.22/3.18

3.21 / 3.17

-

8 hours

3.16

3.17

1585 "C

1596 "C

4 hours

3.12

3.09

2 hours

2.80

-

6 hours

3.14

3.16

p u l v e ~ n g processes and the loss of Na during f i ~ g . No overgrown gains were obsewed.

The a-alumina particles seemed to cause severe problems, such as cracking in ion exchange.

The criticai faaor in siintering ka-dumina is the Na vapour pressure [48]. A good

Table N.2.2. density of well-crym-powder tubes sinterd at 1585 OC

seai on the cruaile and a consistent powder pack should be used; otherwise, the results wiii not

be reproduciile. A second Eictor is the heating rate. Rapid heating is essentiai to avoid loss of

shrinkage (O?)

18.4 ,

18.5

18.5

sodium before reaching the sintering range. The Na20 vapour pressure of p-dumina has been

buik density

3.27

3.27

3.29

4 hour

5 hou

6 hour

dekmined by EMF measurements [l23].

apparent density

3.28

3.29

3.30

log p(atm) = 6520 - - 19866 (1300 to 190; K) (TV.2.1) T

The maximum rate of vaporization of a substance is given by the Hertz-Langmuir expression

where JN,, : the evaporation rate of sodium oxide

P , ~ , : the vapour pressure of sodium oxide

M : molecular weight of sodium oxide

R, T : the gas constant and absolute temperature

Thus at 1600 OC, the vapour pressure of sodium oade is about 1 0 ~ atm and the evaporation

rate is 0.13 mol I m2sec ( = 8 g 1 dsec). Ifthere is a O. 1 mm gap between the cnicible and lid

with 5 cm diameter, the rate of evaporation is about 0.5 g 1 hr, about 7 % of the sodium oxide

per hour with 100 g batch of p-alumina Thus the loss of Na is significant. Two points should be

noted. The above vaporization rate is the maximum possible rate under vacuum. Thus the real

v~rization rate of p-alurnina in air could be much lower. On the other hand, the green

materials are not in equilibrium. They are neither completely crystallized nor c h e m i d y

homogeneous. Thus the vapour pressure of Na could be hi& than above estimate especidly

at the f h t stage of sintering.

IV.3) K ion exchange

K-ion exchange of P-alumina tubes of weiiuystallized powder sintered at 1585 OC

for 4 hours is shown in Table N.3.1. In set 4 the tubes were ion exchanged in KCI vapour

(VP) with 0.73 : 1 = KCI : p-dumina by weight at 1050 O C for 18 hours, then in molten KCl

(LP) with 2 : 1 = KCI : p-alumina five times at 1 0 O C , one day each t h e . The variation of

weight gain is very narrow. After LP4, the variation is inaeased, which suggests some tubes

had chipped or cfacked. After LP5, al1 5 tubes wae cracked. The prolonged (>5 &y$ ion

=change at high temperatures (> 800 O C ) causeci damage which rnight be related to chernid

attack by KCI salt at high temperatures and to stms, that is, the subcritic. crack growth

ocamhg under stress assisted by the salt. The cracks ocairrd at the fimi stage (OhEX > 99)

and at high temperatures where the crack growth should be hi&. In set B, KN03 was used as a

molten 4 t . The final weight gains are very consistent (WG 4.0298) which suggests the

composition of p-dumina is Na&-10&O3. The presence of a - a i ~ m h a (-1O0/o) causes a lower

weight gain. KCl vapour archange at 1050 OC and subsequent one or two batches of KN03

liquid exchange is the best route for the K-ion exchange of this B-alumina (weU-crystalW)

ceramic. However, the K-ion exchanged tubes al l cracked during the glass sealing.

The XB2-SG tubes were K ion-exchanged and the results are given in table N .3.2.

For those tubes sinterd in the fiah-powder pack (set A), the variation of the weight gains in a

batch of tubes was large and the tubes 10s weight as the exchange proceeded. Mer vapour

archange the surface of the tubes became rough reveaiing the overgrown grains in the duplex

microstructure. It was dficult to rernove the sdt f?om the tubes because the salt resided in the

pores and microcraçks. It was not possible to detamine the % exchange by weight gain as

previously obsexved by Br& et al [571. There was indication of chipping due to residual

stress. W y He gas-tight tubes became petmeable due to pores and microcracks devdoped

during the ion archange. The XB2-SG tubes sintered in the used-powder pack (set B) were

more reliable in K ion exchange than those sintered in the fiesh-powder pack. No chipping was

observed. A f h ion archange, the surface was smooth and the sait was easy to remove. The %

exchange is based on chernid analysis (7% Na20 by weight corresponds to WG = 1.038). The

NOTE TO USERS

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UMI

The daerence between the fresh-powder and used-powder-packed tubes resuits £iom

the dBerence in the microstructures. The former has a duplex mcture with very overgrown

grains (>1ûû p). The fkacture initiating flaws, as mentioned by Tan et al [61], are surface

irregularities and large crystal grains which are abundant in duplex structures. From table U.2.1,

the K ion exchange of Na f3-alumina causes the c axis and a axis to increase by factors of

i -0088 and 1.0004, respectively. Thus ifthe a axis of a crystal is paralle1 to the c axis of another,

a main of 4 . 2 ~ 10-~ could be deveioped. Taking the grain boundary energy as 1 ~ r n ' ~ [6 11,

Young's modulus as 2 1 0 GPa and the qs ta l dimensian as 21, the limiting condition is [ 1 241;

whence spontaneous cracking would ocw with a grain s o i of -12 jun. Thus, with duplex

structures one can not ion archange beta-alurninas satisfactorily.

a-alumina second phase particles in well-crystallized powder tubes also cause stresses

during the ion exchange. Because of its smali size (-10 p), the effect of a-dumina particles is

less serious than that of overgrown grairis. However, the cracks observed in overtreated tubes

would be related to the a-alumina particles.

IV.4) glass-sealing

For Na beta-dumina, the alumho-silicate glas (Coming 1720) was a good sedant for

al1 three kinds of Na beta-aluminas. The sealing was strong and He gas-tight. However, the

seaihg ofK beta-alUrninas was not so satisf'actory. Two kinds of cracking were observed. One

was due to the difference in the t h e d expansion coefficients betwaen a and p-aluminas and

the other was failure between glass and a-alumina lid. The thermal expansion coefficients of Na

and K f3-durninas were rneasured (fig. N.4.1): 7.6~10" for Na and 8 . 5 ~ 1 0 ~ (on cooling) and

8.9~ 104 OC' (on heating) for K. In the K p - a l u w the expansion due to the stress relaxation

was observed on heating at temperatures higher than lûûû OC. The thermal expansion

coefficient of K P-alurnina tube is 0.5~10' OC" higher than that of a-alumina ( 7 . 9 ~ lo4 OC-' , <

540 OC), which will cause tension in the tube on cooling. On the other hand, Na 6-alurnina has a

t h e d expansion coefficient 0 . 3 ~ 1 0 ~ OC' less than a-alumina, resulting in compression in the

tubes on cooling. Another factor is the mechanid degradation due to ion exchange. The

second type of faiure seemed related to bubbte formation in the glass, dong the edge between

the tube and Gd. the K p-alumina seai, more bubbles were - than for Na p-alumina. The

residual salt in the tube &a ion exchange rnay cause this problem. Prolonged washing of K P-

alumina tubes enhanced the success rate.

The dumino-silicate gia.ss is ion exdiangeable and its thermai expansion coeficient is

very sensitive to the amount of alkaline oxides [125]. Moreover, it is easy to be devitrified. Thus

during the glas sealing, the t h e at hi& temperatures should be limited to suppress

devitrification. However, a agh rate of m h g may result in cracks due to thermal stress.

Figure N.4.1. T h e d expansion of (a) K and (b) Na

IV.5.a) XRD

The X-ray difiaction pattem of pure Cso is shown in figure I V 5 1 . The pattern

is consistent with fcc as reported [94]. The X-ray scattering factor of the Ca molecule,

SG is used as a first approximation with the assumption that the 60 C atoms are randomly

distributed on a sphere of radius R [94].

where fc is the atornic fom factor of C . a d G is the reciprocal lattice vector. Shce R =

a

3 S2 A and the lattice parameter a is 14.17 A at room temperature, Sc is close to zero

for G = @,O, 2n) where n is an integer.

Thus, peaks for (0, 0, 212) planes are rarely seen though they are allowed by the fcc

selection rules (Table IV.5.1). Ca is so large a molecule, 10 A in diameter, that Sc

decreases rapidly as 28 increases. Thus the intensities of the peaks in the high 28 region (>

35 degrees) are too low to observe. Only 8 peaks are used to detennine the lattice

In difiactometers the displacement of the specimen fiom the difiactometer axis

is usually the largest single source of error which is given by

M - D cos2 0 _ - - - d R sine

(IV. 5.2)

where D is the specimen displacement parallel to the reflection-plane normal and R is the

diffractometer radius [126]. Since the lattice parameter a is proportional to d in cubic

cos2 0 systems, the calculated parameters are extrapolated against - (figure IV.5.2).

sin 0

a

By least-square approximation, a* is determined as 14.18 A .

Table IV.5.1. The X-ray diffraction of pure Cso

1

2

3

4

5

6

7

8

28 (degree)

10.837

t 7.719

20.795

2 1.737

27.438

28.110

30.900

32.839

h k l

1 1 1

2 2 O

3 1 1

2 2 2

3 3 1

4 2 0

4 2 2

3 3 3

ao(A i

14.129

14.146

14.155

14.152

14.158

14.185

14.165

14.160

Figure N.5.1. X-ray difiaction pattern of pure Cm

Figure IV.5.2. The accurate determination of the lattice parameter a of pure G. The error is assumed to be minimum at 0 = 90 degrees.

W.5.b) EMF measurements

It will be instructive to denve the M F in terms of the electrochemical potential.

For the ce11 configuration

Ta ( Na alloy / Na - P - Alumina 1 Na.Cso 1 Ta

(L) (Phase 1) (electrolyte) (phase 2) (R)

with the assumption that the transference number of Na in the solid electrolyte is unity, we

need 6 equations to describe equilibrium; at the interfaces, from equation (II.4.4),

&) = P . (1) ( I V . S . ~ ~ )

in phases 1 and 2, fiom equation (II.4.9,

P N O ('1 = ZNa+ ('1 + Z, ('1 (IV.S.S~)

&a (2) = (2) + F a (2) (IV.5 Sb)

Thus

IJ Na (2) - P N . (1) = c, @) - Pd (LI (IV. 5.6)

Since the chernical compositions of the right and left lads are the same, the chemicai

potentiais of electrons in both Ta leads are equal. Thus,

P, (2) - P, (1) = -@ Pl +F4 (Ll = -FM

Equations (IV.5.4a to c) indicate that the electrolyte and the Ta leads are chemically inert

to the electrode rnaterials. Equations (IV.5.5a and b) imply that the phases 1 and 2 are

chemically homogeneous.

The resuits of the EMF measurements are s h o w in figure IV.5.3. The

eiectrochemical potential (equation IV.S.7) relates EMF to the chernical activity of aodiuni

by the equation

where R, T, F, and a ~ . are respectively the gas constant, absolute temperature, Faraday

constant, and chernical activity of sodium in the WE with respect to pure Na. For dilute

solutions, EMF obeys the Nernst law

(IV. 5.9)

where the sodium atom fiaction XN. is calculated with respect to Na atoms and Cs0

molecules. (The Cso molecule is treated as an elementary unit in the solution since the

covalent bonds between the s k t y carbon atoms remain intact throughout the ionization

process.) Figure N.5.4 shows this relation at 599 K. The theoretical value of the siope

(RTIF) is 5 1.6 mV, and based on this, the best fit of EMF' is 950 mV. The solubility of

Na in Ca is less than 1 mole % (- 0.7 %) at 599 K.

Figure N.5.3. EMF versus x in Nax& at 599 K.

Figure IV.5.4. EMF in the dilute solution region. The theoretical Nernst slope is shown in the plot (see equation 11.4.38).

In figure IV.5.3, the large plateau in EMF (O < x < 1.7) represents a two phase

region between fcc 1 and fcc II at 599 K. In the two-phase region the Na& in the WE is

a rnixnire of Na,lc60 and where x l z O and x2 z 1.7. As the total amount of

sodium increases the ratio of two phases changes. However, the chemical compositions of

the two phases are fixed. Thus, the chernical activity of sodium is constant in this region,

whereas, in a single-phase region (solid solution region), the chemical activity of sodium

changes as a funaion of the composition. The region of gradua1 decrease in M F (1.7 < x

c 3) represents the domain of the fcc II phase. Another two phase region (3 < x < 3.3)

exists between fcc II and the following phase. The phase relations at higher concentrations

(x > 3.3) have not been determined. A sudden drop in EMF in the range of 3.3 < x < 3.7

leaves the possibility that a line phase may exist at some stoichiometry between 3.3 and

3.7. The fcc III stmcture forms a large solution region beyond x = 3.7 (or 3.3). The

existence of sodium atom clusters in the octahedrai site suggests a range of compounds

NaxCa, 3.7 < x < 1 1. This is similar to Li.& where a solution range 3 < x < I 1 has been

observed [97]. However, it is possible that the fcc IIX phase has a miscibility gap: a lower

x phase with four Na atoms per cluster and a higher x phase with nine Na atoms per

duster.

The phases in alkali-fullendes are listed in Table IV.5.2. The proposed phase

equilibna in the NaxCso system are shown in figure N.5.5. Since phases are not in

complete agreement with the data from the iiterature, we have made arbitrary choices in

some cases. It appears that the size of the alkali ion and the related orientational ordering

of the Ca molecule are major factors in determining the phase relations.

Table IV.5.2. The reported phases in alkali-fullendes - - - -

Composition

bct

bec

ïernperature (K) Remarks

xientatiodly ordered wvith two

prefcrred orientations [7 1)

aniso tro pic dynamic disorder (7 O]

isotropie dynarnic disordcr [20]

the orientational order is the same as

in pure C& (SC) 194,951

merohedd disorder [ 127,1281

ordered orientation, Na clusters

[lB, 129)

[SZl

two-phase rcgion (921

disordered orientation, rock-salt

stnicture [130]

[ W

i93 1 unique to K [130]

except Cs, merohedral disorder

[66,82,127]

[821

[821

Figure IV.5.5 Proposed phases in MxCso at (a) high temperatures (T > 425 K) and (b)

room temperature. rs and ortho represent rocksalt and orthorhombic, respectively.

From equation (IV.5.8), the activity of Na was calculated From the EMF data.

The activity of Ca was derived by Gibbs-Duhem integration (equation 11.4.54) using the a.

function (figure IV. 5.6).

(IV.5.1 Ob)

The Gibbs energy of mixing (AG,) was calculated from the measured activity of Na

(equation IV.5.8) and the calculated activity of Cd0 (equation IV.5.l Ob) and is shown in

1 1 figure IV.5.7. The Gibbs energy of mixing of -Na& and -Na+& are evaluated as 85

4 7

and 8 1 kJ/mol, respediveiy.

Figure N.5.6. The activities of Na and &. The activities of Na were obtained from EMF

measurements (equation IV.5.8) and the activities of &J were calculated from Gibbs-

Duhem relation (equation IV.5.10).

1 Figure IV.5.7. The formation energy of -Na,C@. The solid fine is from the EMF

x+l

measurements. The dashed line is for fcc II (equation IV.5.15) and the dotted line is for

fcc 1 (equation IV.5.29).

W.5.c) Ideal intentitial solution

The simplest mode1 of the system may be the ideal interstitial solution [106,

1 121. The assumptions are

1) the energies of the aikalis (A) and the b ~ k y balls (B) are unaffected by their respective

positions on the lattices.

2) the distribution of the A atoms are random.

3) every host site is filled with B; no vacancy.

Here, the complication is that the alkalis can be accornrnodated by both octahedral and

tetrahedrai interstitial sites.

If we consider the following reaction:

xNa + C, = Na$, C/cc II)

the Gibbs energy of each species can be written as

GA = - k~ ln(q;)dY + xNUA

G, = - k ~ ln(q$ + M I ,

(IV5 I l )

where 4 is the vibrational partition function ofj in site i, N is the number of C6() and Nf

is the number o f j in site i. The relation between thern is given as

N r + N ' = x N

From assumption 1 above, the energy of the mixture is given by

(IV. 5.12d)

NU,,, - xNU, - NUB = N F E , + N A E ,

From assumption 2 and the merohedral diçorder of Ca, the degeneracy n(x) is

il@) = g . 2N

N, ! N , ! g = IV,"' ! (N, , - N;;')! N F !(IV, - N F ) !

where Ni is the total number of i sites. They are related by

Thus, the Gibbs energy of mking at constant temperature is given in the fom of

AG, = AN: + BNF + CN - kTlng (IV5 1 Sa)

where A = - k ~ ln(q=' 14;) + E, (N.5.15b)

c = - k ~ ln@ lq;) - k~ ln 2 (IVS. I ~ d )

The condition for the equiiibnum distribution of A in two distinct interstitials is

A - B M = exp(- -

kT 1

The equilibrium distribution is given, if M = 1, as

and otherwise as

The chernical potential of A is defined as

where ~4 is Avogadro's number. Using equation (N.5.18), we have

(IV. 5.17)

(IV. 5 . 1 8a)

Using equation (IV.5.8) and (IV.5.20), the EMF is related to the chernical potential by

where the nurnber of A in tetrahedral sites is given by equation (N.5.18).

We can determine the parameters A and B for which equation (IV.5.2 1) gives the

best fit to the measured values. A plot of this relation, taking A and B as -1.34 and -1.14

eV, is show in figure IV.5.8. Note that in Na&, about 90 % of tetrahedral sites are

occupied; a higher occupancy of tetrahedral sites was expenmentally observed at this

composition [95, 1281.

In order to complete equation ( W . 5 . lSa), we need to detemine the parameter

C. The main dinerence between fcc 1 and fcc II may originate from the type of

orientational disorder in molecules; though it may not be the case, it will show the

importance of the orientational disorder. Assume that

-k~i+aiqé) = -k~ln(q,iq,) (IV.5.22)

where qbj is due to the rotational libration of the molecule in Na,Cso and q, is due to the

fiee rotation of the molecule in pure Ca and the changes in the intramolecular and

intermolecuiar vibrational fiequencies are ignored. The rotational partition fùnction [24] is

given by

h2 where O, = -

8x21k

(IV. 5.23 a)

The symmetry number o is 60 and the moment of inertia I is 9.7 x 104 kg m2. The

vibrational partition function [24] is given as

At 599 K., the parameter C in equation ( I V 5 1 Sd) is about 0.26 eV, where the librational

fiequency is estimated as that of &Cm (26 cm-') [131]. The resulting plot of equation

(IV5 1 5 4 is shown in figure IV.5.7.

Consider the reaction:

xNa + CM = Na&* (fcc 1) (IV. 5.28)

Assuming that the number of Cao hindered in free rotation is proportional to the number of

Na in interstitial sites, equation (TV5 15a) is rnodified as

AG, =AN: + BN* +C(aN* +PN,") -kT lng ( N . 5 B a )

= A'Nf + B'N,? kTlng (IV. 5.29b)

Equation (W.5.29b) is show in figure IV.5.7 with A' = -0.65 and B' = -0.95 eV, which

are determined h m figure N.5.3.

The miscibility gap in A& where A is alkali can be partly explained by above

model; it has only two or three parameters. The orientational disorder must play an

important role in alkali fullendes as in the pure fullente, which could be related to the size

effects of alkalis.

Figure IV.5.8. EMF (solid line) calculated from equation (11.521). The parameters were determineci by the fitting to the data. The dashed line is the ratio of the number of Na ions

( y ) , caiculated by equation in tetrahedral sites to the total number of C60 molecules, -

W.6) Models for Ca

N.6.a) The symmetry of simple cubic Cm

The symrnetry elements of PUS [132], the fow-temperature form of C60, are

given as

where the symmetry elements are represented in the Seitz notation (equation II.6.45) and

S, C: and U denote 120,240 and 180 degree rotation, respectively, about the axis which

is given as the superscript following them. The orthogonal matrices for the proper

rotations can be derived by the Eulerian angles (Appendix A) and are listed in Appendk

B.

As mentioned in section II.3, SC & has a basis with 4 molecules, each of which

is rotated about 4 dxerent <1 1 1> directions through an qua1 angle y which is not fixed

by the symmetry of the crystal (see figure 11.3.3). XRD showed that the rotation angle y

was about 22 degrees clock-wise (orientation a) from the standard orientation A and there

were orientationai defects at about 82 degrees dock-wise rotation (orientation P) fiom the

standard orientation A [69] (see figure II.3.2). In cubic-close-packing, a & molecule has

12 nearest neighbors; 6 of them are on the plane normal to its rotation a i s (plane A) and

rest of them are on planes (plane B and C ) adjacent to plane A. Note that for orientation a

and p, a molecule has 6 double bonds facing 6 nearest neighbor molecules on plane A and

6 pentagons (orientation a) or 6 hexagons (orientation p) facing 6 nearest neighbor

molecules on plane B and C. Thus, there are four motifs for the two adjacent molecules;

(1) a pentagon of a molecule with 22 degrees rotation facing a double bond of the adjacent

molecule with 22 degrees rotation (Pa-Da),

(2) a pentagon of a molecule with 22 degrees rotation facing a double bond of the adjacent

molecule with 82 degrees rotation (Pa-DP),

(3) a hexagon of a molecule with 82 degrees rotation facing a double bond of the adjacent

molecule with 22 degrees rotation (HP-Da),

(4) a hexagon of a molecde with 82 degrees rotation facing a double bond of the adjacent

molecule with 82 degrees rotation (HP-DP).

When we take the [1 1 11 direction as the rotation axis for the molecule at the

origin (O, 0, O), then the positions and rotation axes of the remaining 3 molecuIes are

,- \

determined by the symmetry of the crystai. For example, applying U ' --O to the i 1;; } molecule at the origin produces a moiecule at (a& a/2, O) with rotation axis [ITT],

which can be shown by the operations of

The positions and the rotation axes of 4 molecules are given in table IV.6.1.

Table IV.6.1. The positions and the rotation axes of 4 molecules in a basis.

IV.6.b) Models for the intemolecular interactions behveen Ca molecules

K

1

2

3

4

Several atternpts to mode1 the interactions between molecules b y the Lennard-

Jones potential and Coulombic interactions fkom local-charge distributions have been

reported [ W , 134,13 5,13 6,13 71. Because the Le~ard-Jones interactions between C

atoms can not explain the 22 degree configuration of SC fuiiente, they introduced many

types of interaction centers at the centers of the double and single bonds as well as the C

atoms [M, 13 61. However, as the types of interaction centers increase, the number of

panuneters to be determined increases geometncally. For example, the number of

X(K)

(0, 090)

(V2, u2, O)

(112, O, 1/21

(O, 1f2, 1/21

Rotation axis 1

11 111

[iif]

[ï 1 11

[i i il n

parameters for 1, 2 and 3 types of interaction centers is 2, 6 and 12, respectively. On the

contrary, only 12 interaction centers of a type were used to simulate the symrnetry of the

icosahedron [134]. For Coulombic interactions, point charges were introduced at the C

atoms, the centers of the bonds [135] and the hexagonal or pentagonal faces [W].

However, the assignrnent of positive charges at the centers of the bonds or faces is not

physical. A simpler and more physical model has been developed in this section.

The philosophy to develop a model is to minirnize the number of parameters so

far as the model is physical and reflects the expenmental results. In this section the

foiiowing results of expenments for the low-temperature f o m of pure Ca were used as

conditions to develop a model and determine the parameters:

1) Accordhg to the XRD data [69], the PU^ structure with rotation angle y = 22 degrees

is the ground state.

2) The molecule with orientation a is more stable than the molecule (onentationd defect)

with orientation p by about 12 meV/molecule, which is the formation energy of the

defect Md (= 6(E4 - El), see equation IV.6.10).

3) The NMR data showed that the activation energy for the jump between two

orientations (a and p) is about 250 meV (E.) [138].

Condition (2) is employed to predict the population of orientationai defects in the

temperature range of90 to 260 K and condition (3) is chosen to explain the fieezing of the

reorientational motion below 90 K.

The potential energy between two molecules is the sum of the Lennard-Jones

(LJ) potential and the Coulomb potential.

where Z, K, i and j are the cell, the molecule, the U interaction center and the Coulomb

interaction center, respectively. In this calculation 1.45 and 1.40 A were taken as the

lengths of the single and double bonds, respectively [66]. The positions of C atoms and

other interaction centers of a molecule with the rotation angle y from the standard

orientation A were computed from the table II.3.1 using the Eulerian angles (equation

kA2) and the symmetry of the crystal (equation IV.6.1). The results of two previous

models [M, 1361 are shown in figure fV.6.1. Lu et al. [135] (figure IV.6.l .a and b) used

C atoms as the U interaction centers and introduced effective charges, positive q and

negative -2q for single and double bonds, respectively. in figure IV.6.1 a, the LJ potential

fàvors y = 87 degrees and the Coulomb potential makes y = 22 degrees as the ground state

phase. In figure IV.6. lb, the 22 degree phase has point defects (8 = 82) which cause an

energy increase of 200 meV per defeçt and the activation energy for the jump between

two orientations is about 800 meV. Though A& = 200 and E. = 800 meV are much

dïerent fkom experimental results, this mode1 is qualitatively correct. Lamoen et al. [136]

introduced as LJ interaction centers single bonds and double bonds as weli as C atorns

(figure IV.6.l c and d). This model is good not only qualitatively but also quantitatively;

Ed = 10 meV and E. = 150 meV. The only drawback of this model is the low activation

energy E. and the large number (12) of parameters that need to be fixed.

The model proposed here substitutes the LJ interaction centers at C atoms with

the effective U interaction centers which are initiaily located at the C atoms and begin to

move toward the center of the adjacent double bonds. When the SM x is zero, the y = 87

degree phase is the rnost stable (figure IV.6.2a) and A& is negative (figure IV.6.2b). As

the shift x increases, the 22 degree phase becoines more stable relative to 87 degree phase

and the formation energy of the defect Md increases. At about x = 0.06 4 the LJ

potential which is close to that of Lamoen almost satisfies both conditions (1) and (2).

However, E, = 150 meV is too small compared to condition (3). Thus, the effective

Coulomb interaction centers are introduced at C atoms and at the centers of single and

double bonds. For charge neutraIity, the negative charges imposed on the centers of single

(qs) and double (qo) bonds were compensateci by the positive charge (qc) at C atoms.

The behavior of the Coulomb potential is shown in figure IV.6.3a and b. When O < qs <

a the Coulomb potential increases E. and A& simultaneously and it stabilizes another

phase (y = 90 degree) over the y = 22 degree phase. Thus, when choosing parameters such

that E, > 200 meV,

< q ~ . Four sets of

conditions (1) and (2) can not be satisfied at the same tirne for O < gs

parameters are show in table IV.6.2, which are chosen with some

arbitrarîness. o is taken from graphite data. The behavior of the mode1 is shown in figure

IV.6.4. The contribution of Coulomb potential to the lattice energy is less than 2 %.

However, it increases E. by 30 to 110 meV (20 to 40 %). Set 1 satisfies the three

conditions best (figure IV.6.4a and b). In set 2, the parameters are chosen intentionally to

obtain not the 22 degree phase but the 68 degree phase as the ground state (figure

N . 6 . 4 ~ ) . However, at a certain temperature the presence of the defects (orientation f3)

with srnall formation energy Md (about 12 meV) in the 22 degree phase makes the 22

degree phase more stable than the 68 degree phase, owing to the configurational entropy;

the 68 degree phase also has defects (orientation a) with formation energy A& larger than

100 meV (figure IV.6.5). It shows that the 22 degree phase is not necessarily the ground

state. In set 3, qs is chosen as zero, which seems more physicai than q,/q, larger than

unity. However, E, at only 180 meV is too small compared to condition (3). Set 4 is

determineci to consider the buik modulus and the lattice energy of pure Ca (see section

N.7) in addition to the above three conditions. The behavior of set 4 is the same as set 1

except that the bulk modulus (17.7 GPa) and the cohesive energy (1.86 eV) of set 4 are

more acceptable [139,140,141,142] than those of set 1 (23.4 GPa and 2.4 eV). However,

the activation energy E, of set 4 is much smaiier than condition (3).

Contrary to the belief that the most important factor which stabilizes the 22-

degree phase is the Coulomb potential due to the local charge distribution and that the

stabilization cornes frorn the tendency for electron-rich double bonds to face electron-poor

pentagons [69,133,135,137], this mode1 shows that the LJ potential plays the most

important role in stabilizing the 22-degree phase, which agrees with Lamoen 11361. In set

1 and 2, the choice of a / q , larger than unity seems unphysical. However, it was

required to obtain an Eu larger than 200 meV. The result can be explained in two ways.

One is that the large Eu may be due to other types of potential such as charge-transfer

interactions. The other is that the tme value of E, may not be as high as 250 meV.

Another NMR study [143] concluded that Eu = 180 * 52 meV which is too low to cause

the Freezing of the reorientational motion below 90 K. The answer to this problem could

be the set 2 type parameters, that is, transition to another structure which has high

activation energy for the reorientational motion.

Table N.6.2. Four sets of parameters which satisfy three conditions.

*energiesin meV, length in A, angles in degret and charges in elementary charge. For qEI-E4) and 6AE see equation (N.6.7). Lattice parameter a = 14.06 A was used.

Figure IV.6.1 (a), (c) the whesive energy U [eV/molecule] of SC & as a hnction of y. (b), (d) the potential increase due to the rotation 8 of a moleaile whose 12 neighbors are k e d at orientation a (y = 22 degrees). (a), (b) are f?om Lu's model [13 51 and (c), (d) are fiom Lamoen's model 11361. Dashed, doad and solid hes are for Coulomb potential Uq, Le~ard-Jones potential (lu and overall potential U, respectively.

0 (degrees) @)

Figure IV.6.2 (a) The Lennard-Jones 0 potential (Iu [eV/molecule] of SC C6() as a fùnction of y. The shift x of effective interaction centers towards the center of the double bonds is 0, 0.03 and 0.06 À for solid, dasiied and dotted lines, respectively. @) the LJ potential increase due to the rotation 0 of a rnolecule whose 12 neighbors are k e d at orientation a (y = 22 degrees). The shift x is 0.04, 0.05 and 0.06 A for solid, dashed and dotted lines, respectively.

y (degrees)

(a)

60 a0

0 (degrees)

Figure IV.6.3 (a) The Coulomb potentid Uq [eV/molecule] of SC & as a hinction of y. @) The Coulomb potentid inaease due to the rotation 8 of a moleaile whose 12 neighbors are fixed at orientation a (y = 22 degr=). The ratio of qs to q~ is 0, 0.5, 1.0 and 1.5 for solid, dashed, dotted and mixed lines, respedvely .

Figure IV.6.4 (a), (c), (e) the cohesive energy U [eV/molecule] of SC C6() as a function of y. @), (d), ( f ) the potential increase due to the rotation 8 of a molecule whose 12 neighbors are fixed at orientation a (y = 22 degrees). (a), @) are fiorn Set 1 parameters, (c), (d) are from Set 2, (e), ( f ) are fiom Set 3 parameters. Dashed, dotted and solid lines are for Coulomb potential LI,, Le~ard-Jones potential (lu and overall potential U, respectively.

O 20 40 60 BQ 100 120

8 (degrees)

Figure IV.6.5. The potential increase due to the rotation 8 of a molecule whose 12 neighbors are tixed at orientation y = 68 degrees. Set 2 parameters were used for the calculation. Dashed, dotted and solid lines are for Coulomb potential U,, Le~ard-Jones potential and overall potential U, respectively.

IV.6.c) Lattice statistics of C60

Mode1 1. Regular solution mode1

In SC Ca, the molecules have one of two orientations; orientation a (y = 22

degree) and fi (y = 82 degree). In this sense it is like ferromagnetism. However, the

intemal energy t e n is not so simple as in the Ising mode1 because of the 4 motifs between

2 adjacent molecules, which are defined in section IV.6.a. Assume that there are no

vacancies and that every molecule at orientation a or p is randornly distnbuted. Because

of the overall charge neutrality and the high symrnetry of the molecule, the Coulomb

interaction due to the local charge distribution decays as R-" at long distance [135]. Thus,

it would be sufficient to consider oniy the nearest neighbor interactions since both the Van

der Waals interaction and the Coulomb interaction are short-ranged. The number of bonds

Ni berneen molecules with motifi (i = I to 4) is given as

where N = total number of molecules and 4 = fiaction of the molecules with orientation j

0' = a or p). Thus the energy of the lanice and the configurational degeneracy are given

as

where E, = the energy per bond between two molecules with the motif i (rvhich is

computed fiom equation N.6.3) and N, = the number of molecules with orientation j .

Thus, the fiee energy is written as

where AE = E, + E, - (E, + E,)

The equilibrium condition is given as

If we know the energy terms, we can compute A and X, as a function of temperature

(unfominately the solution is not exact). Conversely, equation (IV.6.8) can be solved as a

fiinction of X,.

The entropy term is obtained by

The results of equation (IV.6.6.b) and (N.6.9a to c) are shown in figures IV.6.7a to d,

where Ei was cornpute- f?om equation (IV.6.3) with the first set parameten in table IV.6.2.

Since the Lennard-Jones interactions are short-ranged, E, = E, and E, = E,

could be a good approximation. In this case, L 5 in equation (N.6.8b) can be ignored.

By fitting equation (IV.6.10) to the experimental data [69], 6(E4 - El) i s determined as

about 12 meV, which is used to detennine the parameters in the models for the

intennolecular interactions between C& moIecules.

Mode1 2. Cluster variation method

The cluster variation method [107] was presented in the section on substitutional

solid solutions (II.5.b). The cluster variation method has the advantage over the regular

solution mode1 in that it reflects the short-range order and provides better estimation of

the entropy.

A cluster of 10 molecules has been used as a group ( f iyre IV.6.6). A mo[ecule

has Z (= 12) nearest neighbors. There are v (= 24) bonds and p (= 10) sites in a group.

For N molecules, there is N, (= N/4) groups. Each site belongs to 5/2 (= pN'W groups.

Table N.6.3 shows some of the parameters used in the computation where k refers to an

energy b e l ~k (k = 1 to 240) with multiple configurations, ot = the number of configurations

of type k, = the number of sites ocaipied by orientation P and Ni = the number of bonds

with motif i in a goup. The energy is given as

where E was computed fiorn equation (IV.6.3) with set 1 of parameters in table IV.6.2.

For substitutional solid solutions, the cluster variation method shows that at

quilibrium, equations @S. 96), (II.5.98), @.S. 10 1) and @S. 102) hold.

whera i y k = the probability that a group hiis a specific configuration of type k, 11, = the chernical

potential of species i and s = the number of species in the system.

For & the situation is a Little bit dierent. That is, Xi is not fixed and = at

We can compute X, and y, at given T, V and N by iteration, repeating the foilowing

5 steps until Xa converges.

(1) X, =X, andX, = 1 - X,

Witb the results of iteration (figure N.6.7a), we can cornpute the Helmholtz energy f?om

equation (N.6.12) (figure N .6.7b). The intenial energy (figure N.6.7~) is given as

The entropy term given by quation (1V.6.9~) is shown in figure (XV.6.7d).

Figures N.6.7. a to d show that die results of regular solution and cluster variation

method are s i i a r to each other, especialiy in the hi& temperature region where randorn

mixing is more likely. The dserence in the energy t e m between the two methods is very

srnall because the d e f a forniaton energy is only about 10 meV. One may initially suspect the

result that, at temperatures lower than 100 K, the entropy tenn (-TS) for the cluaer variation

rnethod is lower than that for the regular solution (figure IV.6.7d) since the randorn rnixing

approximation used in the regular solution model will always have the lower entropy term (-

TS) than any other method if the system is defined in t e m of (T, V, X;). However, for the &O

system, X, is not hed. Since the cluster variation method reflects the short-range order better

than the regular solution m~deî., in the dusta variation method the introduction of a given

number of defects into the system gains less interna1 energy than in the regular solution model

and the randorn mhhg approximation outweighs the entropy term less as the solutions b m e

diiute. Thus, in the temperahire range b e e n 25 and 75 Y the cluster variation method has

more defeas than the regular solution model (figure IV.6.7a) while the interna1 energy of the

cluster variation method is qua1 to or even lower than that of the regular solution model

(figure W.6.7.c).

Table IV.6.3. Parameters used in cluster variation method

+There are 240 k's. Only fist 25 are given in the table.

Figure IV.6.6. Cluster of 10 molecules in SC &. Line c o ~ e c t s nearest neighbors. F and B

represent face (hexagon or pentagon) and double bond, respectively. There are ody 4

motifs for a pair of molecules (see the text).

Figure IV.6.7. (a) The fraction of molecules at orientation a. @) Helmholtz free energy of

SC (&. (c) The lattice energy. (d) The entropy term (-TS). Solid and dashed lines are for

cluster variation and regular solution models, respectively.

N.6 .d) Orientational order-disorder transformation in Ca

It is assumed that in the high temperature f o n of Cs0 the interaction centers in a

molecule are randomiy distributed on a sphere of radius R (= 3.5485 A} [22] and that the

molecules are rotating freeiy. Thus the potential energy between two molecules at the

distance r (figure IV.6.8) is given as

Using equation ( IV.S.23) and (IV.S.27), the fiee energies of the high temperature @c)

and low temperature (SC) forms are compared with each other in figure IV.6.9.

where Vf i and are the cohesive energies computed from equations (1V.6.15) and

(IV.6.3) with the parameters fiom set 4 in table N.6.2. The ai's are the libron fiequencies

computed by a pseudo harmonic approximation (see section IV.7, Lattice dynamics of

&) The difference in the translational vibrations between two phases is ignored. In the

computation, X, = 0.63 was used and the lattice parameters of fcc and SC phases were

fixed to 14.16 and 14.10 q respectively. In this mode1 the transition temperature is about

245 K which is close to the experimental results of 249 K [67,68] or 260 K (691.

Figure IV.6.8. Fused interaction centers. See equation (IV.6.15).

Figure IV.6.9. The free energy curves for SC (dashed line) and fcc (solid line) Ca,

computed by equations (IV.6.16) and (IV.6.17).

W.7) Lattice dynarnics o f C60

IV.7.a) SC Ca

The dispersion curves of extemal modes of sc Cso were computed by the pseudo-

hamonic approximation. AU the molecules were assumed to be at the 22 degree

orientation. The intermolecular interaction mode1 with parameters of set 4 (table IV.6.2)

was used and only the nearest neighbor interactions were considered. The dynarnic matrix

was computed by a Fortran prograrn and the eigenvahes were obtained by Matlab.

Since the molecule is sphencal top (1, = I' = I, = I ) , the fiame of reference is

fixed to the lattice vectors and the rnass-weighed or inertia-weighed displacements [26]

are used instead of linear and angular displacements in equation (II.6.1 and 2).

t t (k) = (ZK) (IV.7.l a)

t 0 (ZK) = &: (lic) (IV.7.1 b)

where subscript a represents coordinates (a = x, y, z) and superscript t and r denote hear

and angular displacements, m is the mass of the molecule, Z and K (r = 1 to 4) represent [th

c d and a h molecule, respectively. Thus, equation (I1.6.16) is rewritten as

- 0, (q)q = 0 (N.7.2)

where the element of 24x24 rnatrix B(q) is given as

BG(q,mc') = x4$(Ou; hc') exp[iqo X(Zt)] ( i = r or t and a$ = x, y or z) (II.6.15) f'

Since the mas-weighed or inertia-weighed displacernents are chosen, the

cou piiig coefficients (equation II.6.7) are modifiecl as

Thus, the coupling coefficients given in tems of the Born-von Karman constants by

equations (11.6.3 la to d) are also rewritten as

a2 where $M(OKk;l f~'k ' ) =

aU, (Oick)&, (PIC 'k ')

gDBr = O if any two of (a, B, y) are equal

= 1 if (a, p, y) corresponds to a cyclic order of (xyj)

= -1 $(a, p, y) corresponds to a noncyclic order of (XJJ)

k = Lennard-Jones and local charge interaction centers.

In pair-wise interactions the potential energy 0 is writtcn as

where h, p and s represent the hth ceii, pth molecule and sth interaction center,

respectively and

The potential between two interaction centers are given as

O a v(r(@q hf p'sl)) = ) ' - ( ) '1 i, interaction)

r ( h p , h'p's') r ( h v , h' p's')

- - QI% (Coulomb interaction) r (h~w; h' 's')

The first derivative of the potential4 with respect to the displacement is given as

Thus the Born-von Karman constant is written as

(1) for lidc = PK'K

(2) for ZK = PK' and k + K

(3) for k + PK'

The second derivative of pair-wise interaction V(r) with respect to displacements is given

(N.7.1 Oc)

The resulting B(q) has the following scheme.

(TV.7.1 la)

where the element of the dynamic matrix is given by equation (11.6.15).

The rnass- or inertia-weighed displacement vector is given as

(IV. 7.12a)

where the element is given by equation (IV.7.1).

The dynamic matrix is block-diagonalized by the symmetry-adapted vectors

(equation 11.6.59).

B, (q) = 5+ ( (w(q)wl) (II.6.6la)

where subscript D represents a blockdiagonal maüix and the symmetxy-adapted vectors

are obtained from section group theory of & ( 1 ' 3 ) . The use of Bo(q) makes not only

the computation of eigenvalues easier but also the classification (labeling) of eigenvalues

possible. The results of equation (IV.7.2) in which Bo(q) is replaced for B(q) are shown in

figure W.7.1 and table N.7.1. In figure IV.7. I two features should be noted. One is that

two modes of the same symmetry cannot have dispersion curves that cross (anti-crossing).

The other is that sorne wave vectors normal to the plane of the surface of the Brillouin

zone have non-zero gradient of dispersion curves at the zone boundary. It is typical for

non-symmorphic crystals that the fiequencies of pairs of branches for some zone boundary

wave vectors become degenerate, with the gradient of the upper branch equal to the

negative of the gradient of the lower branch [115]. Experimental evidence of 3 phonon

modes at 39, 40 and 54 cm-' and 4 libron modes at 15, 19, 22 and 32 cm-' was observed

by an inelastic neutron scattering study [144] at 200 K. When we consider that the lattice

parameter is about 14.08 A at 200 K [69], the mode1 seems to predict the phonon modes

quite accurately. The large difference between the cornputed and reported libron

fieyencies may be due to the fact that the potential well is flattened from the parabolic

f o m and that the vibrational amplitude is too large to apply the harmonic approximation

(see section IV.7.c).

Table IV.7.1. Extemal vibrational frequencies [cm'L] of SC Cm at r (q = 0).

* The first 5 rows with subscript u are phonons and the next 5 with g are librons. The

degeneracies of 4 E and T are 1, 2 and 3, respectively and a is the lattice parameter. The

fiequencies in the last column observed at 200 K [144] are arranged to compare with

those computed and not necessarily belong to the labels in the first colurnn.

2n Figure IV.7.1. (a) Dispersion curves for SC &. q = -(O

a

and mixed lines are for Ai, A2, Bi and BI, respectively.

0.4 0.5

q O). Solid, dashed, dotted

2x Figure IV.7.1. @) Dispersion curves for SC &. q = -(q O). Solid and dashed lines

4

are for A ' and A " , respectively.

t!'

2n Figure IV.7.1. (c) Dispersion curves for SC C6& q = -(q q q). Solid and dashed lines

a

are for A and E, respectively.

W.7.h) MJCa

The extemal vibrational frequencies of M3Cso (M = JC, Rb) at r (q = (O O 0))

were computed. The potential energy is assumed to be the sum of the LJ potential for C-C

interactions and the Coulomb potential (CL) for Mt-Ml, CM''- ~ 6 0 . ~ and M+'-C~O.~ and

the Born-Mayer-type repulsion (BM) for M"-M", @c&~'~*. Thus, the Born-von

Karman constants in equation (IV.7.4) are given as

4"' 4 - - 4 ~ i * (LJ) +~$(cL)++$(BM) (IV.7.13)

(1) 4:&w This has been computed by the same mode1 used for SC Ca (equation N.7.5 and

IV.7.7). The effects of net charge (-&e) to this type of interaction were ignored.

(2) $$, (CL)

Since the Coulomb interaction is long-range, it is not appropriate to approxirnate

the potential by the sum of the nearest and second-nearest interactions. Ewald and

Kelierman' s formalism is employai as in equations (II. 6.34) and (I1.6.3 5).

(3) ~ $ ( B M )

The Born-Mayer type repulsion [145] is given as

For ionic solids p is about 0.345 A and h has a form of

where 2, is the valence and ni is the number of the valence electrons and ri is the radius of

ion i. The parameters used for this calculation are given as

PK-K = 15 1 mm01 hKX = 0.337 A [L46]

pwfi = 33 1.4 MJ/mol h ~ ~ ~ b = 0.335 A

p ~ c = 293 MJlm~l A, = 0.27 À [146]

p ~ w = 426 MJIrnol hRK =0.27A

The parameters for M-C (M = K, Yb) were obtained by fitting the cornputed bulk modulus

to the reported values [66].

The calculated fkquencies are given in table N.7.2. There are three K and one

in a unit cell. Each K contnbutes to the extemal modes three degrees of fieedom and

each six degrees of freedom. Thus, there are 15 (= 3 x 3 + 6 x 1 ) vibrational modes.

Table IV.7.2. The frequencies computed at r

1

*AU of them have a degeneracy of three. The lattice paramete of 14.17 and 14.30 A were used for K and Rb fillerides, respectively.

N.7.c) Quantum mechanical consideration of angular vibration of SC Ca

The angular vibration of a molecule at the origin about the [L 1 1) axis (figure

II.3.3) has been studied in SC Cso. The 12 nearest molecules are assumed to be fixed at

orientation a (y = 22 degrees). Only nearest neighbor interactions are considered.

The Schrodinger equation is given as

where I is the moment of inertia of the molecule. Using equation (N.6.3), the potential

energy of the molecule at the origin is computed as a fùnction of rotation 0 with the set A

parameters in table IV.6.2 (figure N.6.4.b). The computed potential energy is

approximated by a Fourier senes of

2 x where G = 3m (m = integer) since the periodicity of the potentiai is - . The coefficients

3

UG (- 12 < m < 12) are determined by least-squares fitting.

Hannonic approximation

The potential energy in equation IV.7.17 is rewritten as

where 8, is 22 degreeq the second term on the right side is zero and in the third term

VL(0) = -G'U,~~ . The solution of equation (IV.7.16) is G

E,, = ho()? ++)

Thus, the vibrational frequency is 12.2 cm-'.

Central equation

The wavefunction may be expressed as a Fourier series surnrned over al1 values

of the wavevector permined by the boundary conditions [147].

Using the boundary condition of ~ ( 8 ) = ~ ( 8 + 2 x ) , we obtain k = an integer. Thus,

substituting equations (N.7.17) and (N.7.2 1) into equation (W.7.16) results in

($ - E) C,, + u~,, ,c ,~~ = O (- 12 < rn < 12 and n = any integer) (IV.7.22) m

which is called the central equation. Equation (IV.7.22) forms an uifinite matrix, a portion

of which is given as

It is sufficient to set equal to zero the deteminant of a portion of the mat& (400x400).

We obtain 400 eigenvalues E and 400 corresponding eigenvectors (C,) from equation

(rV.7.22). The first ten energy levels and their probability densities for k (= n) = O are

show in figure IV.7.2. The energy difference between successive levels is about 2.7 meV

I Ac which corresponds to a vibrational frequency of 21.7 cm-' since - = - (c = the velocity

h Ac

of light).

The vibrational fiequency computed by the central equation is 1.8 times that

cornputed by the hamonic approximation, which suggests that the harmonic

approximation is not sufficiently ngorous for angular vibrations. The failure of the

harmonic approximation is due to the fact that the potential weli around 0 = 22 degrees is

flattened from the parabolic form and that the amplitude of the vibration is sornewhat large

(about 7 degrees for the 9th level, see figure IV.7.2).

Figure IV.7.2. The probability densities of the first 10 energy levels for angular vibrations

about [L 1 11 axis. The energy of each level is in rneV from the ground state.

19.8

18.0

17.1

15.2

14.4

12.3

9.5

6.8 .

4.0

1.3

1 1 l I

-60 O 60 120

(degrees)

iV.8) Group theory of Ca

Group theory was applied to the eigenvalue problem (equation IV.7.2). The

symmetry-adapted vectors for T, 4 A and C were obtained and the extemal vibrational

modes of SC Cm were labeled. The dispersion curves, given in figure IV.7.1, were used for

thennodynamic calculations in section IV. 9.

The procedure to obtain the symmetry-adapted vectors which will block-

diagonalize the dynanÿc matrix (equation II.6.61a) is

(1) to determine Go(q), the point group of the wave veaor q by the equation

R q = q + G , (11.6.48)

(2) to construct a unitary m a t h T(q;R) associateci with each element R of the point

group Go(q) (equation II.6.5 1)

T(q; R) = exp[iq (V(Q + ~(m)]I'(q; R, ) (n.6.5 la)

where T@(nc'[q;R) =&.,S(u,F,(~';R))exp[iq~(X(w) - RX(d)], (II.6.5 lb)

(3) to obtain c, by equation p.6.55)

(4) to project out synimetry-adaptai vectors using the projeaion operators (equation

II.6.57 and 58)

f. P:" (q) = - C (q; R)' ~ ( q ; R) R,,,

( 5 ) and finally, to block-diagonalize the dynarnic matrix (equation 11.6.6 la).

B. (q) = c ' (q)B(s)W (11.66 la)

The dispersion curves are shown in figure rV.7.1.

The symmetry elements of ~ a ? [132], the iow-temperature form of Csa, are

given by equation (IV.6.1). If we take as the invariant subgroup Y t h e set of the infinite

number of translational elements, then for nonsymmorphic crystal ~ a ? , the factor group

is isomorphic with the point group Th (see equation II.6.54). The character table of Th is

given in table N.8.1.

Table XV.8.1. The character table for Th [117].

Since q = O, every rotational part R of space group ~d satisfies the equation

(11.6.48). Thus, the point group of the wave vector q = O is Th and the character table is

given in table IV.8.1. The unitary matrices T(OJ) are given in Appendix C.

Using equation (IL6.55), we have

CA,= i CEg = 1 Crg = 3

CA^ = 1 = I cru = 3 (IV.8.1)

Thus

ï = A , + E g + 3 T , + A , + E . + 3 T u (IV.8.2)

Among them one of the Tu corresponds to the acoustic modes. The other two Tu are IR

active and d of A,, E, and Tg modes are Raman active [148].

Using equations (II.6.57 and 58), the projection operator P'" is constructed and

applying this operator on 24 orthogonal vecton, c, times f , independent vectors are

projected out. The 12 symmetric modes with respect to I have the following symrnetry-

adapted vectors.

O 1 0

0 0 1

o i o

o o ï

1 0 0

0 0 1

- 1 O 0

0 1 0

o ï o

0 0 1

O 1 O

o o i

1 0 0

o o l

1 0 0

0 1 0

(IV. 8.3)

where O = (O O 0) and the coordinates of the syrnmetry-adapted vectors are

( U t ( i < = 1 ) U f ( I C = l ) u ' ( K = ~ ) u r ( I C = 2 ) u ' ( K = ~ ) u f ( I C = 3 ) U t ( I C = 4 ) U r 0 ( = 4 ) 1

(IV.8.4)

The 12 antisymrnetric modes with respect to I have symrnetry-adapted vectors which are

the sarne as those for symrnetnc modes except that the vectors for linear displacements

and angular displacements are switched. For exarnple,

Thus we obtain the rnatnx 5 of symmetry-adapted vectors in equation (11.6.58) given as

where semicolon and superscript t represent the end of rows and the transpose of the

Finally, we can block-diagonalize B(q) by equation 01.6.6 1 a)

The block-diagonalized form is

where A's and E's are one by one matrices and T's are three by three matrices and off-

diagonal elernents are zero.

From equation (IL6.48) the point group of the wave vector q is determined as

C&). The character table is given in table N.8.2. The exponent t e m in equation

QI.6.5lb) is given in table IV.8.3. The method to obtain unitary matrices is the same as for

q = O except it is necessary to rnultipiy by the factors in table IV.8.3.

where ody non-zero elernents are shown.

where Ai and Bi are 6 by 6 matrices.

Table IV.8.2. The character table of C2&) [117].

A

O, (= ICk) 1

1

- 1

1

- 1

4 (= ICk)

1

-1

- 1

1

C ~ Y

1

1

- 1

- 1

I

Al

Az

Br

B2

E

1

1

1

1

IV.8.c) Z [ q=-(11 0 q O)]

Table IV.8.3. The exponent tenn in equation (11.6.5 lb).

The point group of the wave vector q is c,(z). The character table is given in

table N.8.4. The exponent t e n in equation (II.6.51b) is given in table N.8.5.

oV.8.12)

4

ci'

1

v* 1

*P = exp(mq1

3

1

CL

I

Ct

2

P*

1

cc' I

, K = 1

2

3

4

K' = 1

1

P

1

P

where only non-zero elements are shown.

where Ai is 12 by 12 matrix.

Table IV. 8.4. The character table of c,(z) [117].

Table IV.8.5. The exponent term.

The point group of the wave vector q is CJ. The character table is given in table

N.8.6. The exponent term in equation (U.6.5 1 b) is given in table IV. 8.7.

4

Z W

O

1

1

*v = exp(Qq) and w = e x p ( q )

3 e

O

W

I

1

2

v

1 f l

W

Z W

K = l

2

3

4 J

K' = 1

1

v

a

O

( E ~ ( is the same as (E' 1 except that @ and $* are switched.

where only non-zero elements are shown.

where subrnatrices are 8 by 8.

Table IV.8.6. The character table of c&) [117].

Table IV. 8.7. The exponent term.

N . 9 ) Thermodynamics of Cm

The static (section IV.6) and dynamic (section IV.7) pans of free energy are

combined to compute the thermodynamic propenies of such as the lattice parameter,

the bulk modulus, the thermal expansion coefficient and heat capacity which are easy to

masure. Oniy the intemolecular parts of the thcmodynarnic properties were computed.

The main object of this section is to evaluate the model by comparing it with the

experiments and to see how those properties are interrelated with one another. Or we may

evaluate experiments by comparing them with the results of others.

N.9.a) the equation of state for Ca

From equations (IV.6.7a) and @. 5.3 9) the fiee energy of sc & is written as

where AE = E, + E , - (E, + E,) (IV.6.7b)

Since in the acoustic modes the 6equencies seem to be proportional to the wave vector and in

the optical modes the fiequencies seem to be l e s dependent of the wave vector (see figure

N.7.la to c), the dynamic part of the eee energy is approxhated by the Debye model for

acoustic modes and the Einstein model for the optical modes. Thus, tiom equations (IV.6.6),

(II. 5.46) and @.5.47), the intermoleailar part of the fiee energy is

and the Debye knction D(x) is numtxically approxhated in Appendix D.

With the assumption that the fkquencies are independent of X,, the relation between

T and X, is given in equation (TV.6.9a).

At the equilibrium state of (T, V, N, X,) the state equation is given as

where Griineisen constants are defined as

and assumed to be independent of V.

The quiiibnum state at P = O atm was computed f?om equation (TV.6.9a) and

(TV.9.4). The Debye velocity v and Debye characteristic temperature O were determined as

2.55 x 1 o3 mkec and 54.14 K from the dispersion m e s in figure W.7.l a to c. For optical

modes the frequencies at 14.05 A in table N.7.1 were used. Pohl et ai. reported that the

Einstein model with the characteristic temperature & = 35 K was much better agreement with

the specilic heat and thermal conductivity rneasurements than the Debye model and that from

the low temperature spedc heat they determined the Debye velocity and Debye temperature

as 2.39 x 1 o3 m k c and 80 K [ 1561. David et al. found that a Debye-only model gave a

sipficantly poorer fit to the lattice parameter data than an Einstein-ody model and that a

single Debye + single Einstein model gave an improved fit (the denved Debye and Einstein

temperatures were 52 K and 93 K) [157]. The Grüneisen constants were assumeci to be

independent of both the volume of the crystal and the type of 6equency. The procedure for the

computation was

(1) to obtain the energy t e m E,(Q and their fint and second derivatives with respect to the

volume at a fixed lattice parameter a (that is, fixed state of (V , N)) f?om equations (lV.6.3) and

(IV.6.6) with parameters of set 4 in table IV.6.3.

(2) to compute T at a given (V, N, X,) fiom equation (IV.6.9a)

(3) to compute P at a given (T, Y, N, X.) fkom equation (IV.9.4)

(4) to compare P calculated in (3) with zero and repeat (2) to (4) with v w g Xa until P = O

(5) to go back to (1) for a d8erent lanice parameter a.

By the least-squares method the Griineisen constant was determineci as 8.73 fiom table IV. 7.1.

However, to fit the r d t s to the reporteci values of the lattice parameters [69], thermal

expansion coefficient [[66 i d buk modulus [140,141,149], the Grüneisen constant is

detexminai to be 7.5 as an optimum value. The resulting equilibnum state of (T, V, N, X, P =

O) is given in figure (IV.9. la and b).

Figure N.9.1 (a) Lattice panuneters computed at P = 0. @) The computed fiaction of

molecules at orientation p. Solid circles are from David et al. [69].

XV.9.b) the bulk modulus and the thermal expansion coenicient o f CM

From equation m.5.7) and (IV.9.2) the buk modulus Br (= I I P r ) is written as

where

Frorn equation (N.9.6a) the bulk moldlus Br was computed at a given (T, V, N, X,, P = 0)

which is determined by equations (IV.6.9a) and (N.9.4). The results plotted in figure lV.9.2

show that the vibratiod part of the buik modulus is very smaii (about 2 % at 200 K) and

increases with temperature. The reporteci values of the bulk modulus are widely scattered

[140,l4 1,1491. There are large dEerences between the mode1 and the experirnents whicb niay

be due to the rigid molecule assumption or the assumption that the Gnuieisen constant is

independent of volume and X,.

From quation v.5.6) the thermal expansion coefficient is written as

From equation (IV.9.2) the derivative of entropy with respect to volume is given as

where (3 rN is given in equation (N.9.6~) . The results at given (T, V , N, X, P = O) is

show in figure IV.9.3. The anornaious behavior in the temperature range b e ~ n 20 and 80

K is due to the rapid increase in the population of molecules with P orientation (XD) (se figure

N.9.1 b).

rV.9.c) the heat capacity of Ca

From equation (II. 5.4) and (IV.9.2) the heat capacity at constant volume is given as

Note that the intrarnolecular part of the kat capacity is ornitted in the above equations.

Using equation (II.9.5) the heat capacity at constant pressure (P = O) is computed (figure

The contribution of the lattice en- part to Cv is the most interesthg feahlre in SC

&, which is due to the increase in the population of moleailes with 0 orientation (Xip),

signifiant in the low temperature range (20 to 100 K) in figure IV.9.4a. Since the optical

modes outnumber the amustic modes, Cvdecreases as exp - - 3.

Figure N.9.2. Computed buk modulus of sc &. Dashed line is for V (2) - r . Ckde, square

and cross are f?om reference [140], [141] and [149], respectivdy.

Figure N.9.3. Computed linear thermal expansion d c i e n t . Circle is f?om reference [66].

The inset is for low temperatura.

Figure N.9.4. Computed constant volume heat capacity. Inset is for constant volume heat

capacity at low temperatures. Dashed, dotted and mixe. lines are for heat capacity due to the

lattice energy, intermolecular optical modes, acoustic modes. Solid line is the sum of three.

Figure IV.9.S. Computed constant pressure heat capacity (P = O). Doned line is for constant

volume heat capacity. Dashed line is for the contribution of the anharmonicity of the vibrations

to the heat capacity.

TV.10) K - C m

Ion-exchanged and direct-synthesized K B-alumina tubes were used for EMF

measurements. The green density of tubes synthesized from bohemite as the source of

alumina was low due to the low stability of the slip. Ion-exchanged tubes were more likeiy

to crack afler sealing. The direct-synthesized tubes with a-alurnina as the starting material

showed better mechanical strength.

Cell measurements were made between a reference electrode (Pb : Bi : K = 3.3 :

41.6 : 55.1 by weight) calibrated against pure K and the working electrode containing Co*

intercalated with K. EMF versus x at 572 K is shown in figure IV. 10.1. Four plateaux are

seen at 1720 (O < x < 1), 1563 (1 < x < 3), 1354 (3 < x < 4) and 1271 mV (4 < x). Figure

IV.10.2 displays the EMF in the dilute solution region. The Nernst siope (dashed line) is

shown for cornparison. The phases are listed in table N.5.1. The formation energy

of KxC60 has been computed, assuming stoichiometric compounds (table IV. IO. 1). The

reaction is given as

Table IV. IO. 1 . The formation energy of KXCa at 572 K

(IV. 1 o. 1)

Figure IV. 10.1. EMF of KAdo at 572 K.

-5 -4 -3 -2 - 1 O

log XK

Figure N. 10.2. EMF of KxGO for dilute solution region. Dashed line is the guide for the

theoretical Nernst dope.

IV.ll) S r - At

Sr is an important addition to aluminum cast alloys containing Si because it

modifies the acicular structure of the AI4 eutectic. Sr beta-alumina is an obvious

candidate material for an electrochemical sensor to rnonitor the concentration of Sr in Al

in the range 0.00 1 to 0.1 ?G by weight. Ion-exchanged or direct synthesized S r P - or P " -

alumina tubes with good themal shock resistance were developed and tested by

Kirchnerova and Pelton [150]. Three reference electrodes (Al - Sr, SrO6FqOJ Fe20s/air

and SrCl, - AgCl/ Ag) were tested in molten Al baths. IIowever, the EMF results were

found to be scattered over a 100 mV range about expeded values. Interfacial effeds were

suspected to be responsible either due to poor wetting of the ceramic by liquid Al or due

to interfacial reaction resulting in a non-conducting phase. The objective of this work was

to test the different probe assemblies in a high-purity Ar atmosphere in order to elirninate

the cause of the problem and obtain good EMF results.

W.1l.a) EMF measurernents

Three experirnents were run, each with an independent working electrode, but

with the same reference electrode and counter electrode. In each case, titration resistance

at 720 O C was high (approxhately 1o3 R), somewhat lower for p"-alumina, and

equilibration wu slow. In fact, we never estabfished a stable reading until &er a month of

equilibration.

The first working electrode, Cefi I was made from Sr P -almina and was titrated

from pure Al to Ai-0.5 ppm Sr. NO Nernst slope was observed in this cell. The semi-stable

EMF reading was in the range of 485 mV. As the Sr content was increased up to 26 ppm,

the EMF increased and after a week, reached a value near 660 mV. EMF in this ce11 was

hundreds mV lower than expected values.

Ce11 II was made from Sr P" -alumina. The EMF at Al- 1 ppm Sr was in the

vicinity of 1010 mV, close to the evpected value, but continuously drifiing downward.

Al-1 wt % SrBr, was used for Ce11 III in order to eliminate the possible

interfacial problems, such as wetting of ceramic by molten Al. The EMF of CeIl III was

between those of Ce11 1 and II and tended to decrease with time. Eventually, after a month,

al1 three cells drifted to a common voltage near 390 mV.

W.1l.b) eompatibility of Sr Falumina with pure Al

EMlF values were inconsistent fiorn ce11 to cell and drifted with time. No Nernst

dope was observed. The proposed cause of the problem is due not to the interface but due

to thermodynamics. One of the requirements for good solid electrolytes is the

compatibility with electrode mateds. The compatibility of Sr beta-alumina with Al and Sr

can be calculated. Thermodynarnic data used in these calculations are given below.

2 A l + 1 0 2 =Ai,O, AG, =-1676000+320T [151] (IV.ll.l)

Na,O +xAl ,O , =Na20-xAl,O, AG, = -3l4OOO+68.92T [152] (IV.11.2)

2Na,, + f O = Na20 AG3=-420015+146.O22T [152] (IV.11.3)

Sro, + 4M,, = AiJr AG4 = -225000 + 107.5T [153] (IV. 1 1.4)

Sro, + 2M,, = Ai,Sr AG,=-166500+86.7T [153] (IV.11.5)

Sr ++O, = S r 0 AG, = -618300+ lOOT [ I 541 (IV. 1 1.6)

Though the formation energy of Sr B-alumina is not reported, it can be

estimated by the cornparison with Na B -ahmina. The reported values of the formation

energy of Na fl -almina are very consistent [152,155]. The formation of each compound

is given by the following reactions:

2Na+fO, +xAl,O, =Na20~xAl ,0 , AG, = AG, + AG, (IV. 1 1.7)

S r + i O , +yA1203 =SrO-yAl,O, (IV. 11 .8)

Consider the Born-Haber cycle of reactions (IV. 1 1.7) and (IV. 1 1.8) and assume that x =

Y .

2NaN = 2Na, = 2Na' + 2e- (IV. i 1.9)

Sr2' +(xA120, +20)'- = Sa-xA120, (IV.11.13)

Here we consider the change in enthalpy only. Reactions (IV. 1 1.9) and (IV. 1 1.10) are the

surn of the cuhesive energy and ionization energy. Thus the difference of (IV. 1 1.10) and

(IV.11.9) is 5.9 eV [= 1.72 + 16.72 - 2(1.113 + 5.14)] [147]. Reaction (IV.11.11) is

comrnon for both materials. In reactions (IV. 11.12) and (IV. 11.13) the surn of attractive

Coulomb interactions between Sr ions and spinel blocks may be close to that for Na ions

and spinel blocks since the Sr ion has twice the charge of the Na ion and the concentration

of Na ions is twice that of Sr ions. However the sums of repulsive Coulomb interactions

between conducting cations for Sr and Na p-aluminas are quire different from each other.

When we consider that the average distance between Sr ions on the same conduction

plane is fi tirnes that between Na ions, the repulsion between Sr ions is stronger than

that between Na ions. Thus Na P -alumina appears to be more stabie than Sr P -dumina,

AG, < AG, .

The compatibility of Na P -alutnina with Al can be determined by the following

reaction.

Na,O~xAl,O, +f Al =2Na +(x+f)Al ,O, AG9 (IV. 11.14)

AG,=(x+f)AG,-(AG7+xAG1)=fAG,-AG7=70Wat7000C (IV.11.15)

Thus, Na P -dumina is compatible with Al (figure IV. 1 1.1).

For Sr B -dumin%

SrO-yAl,O, +?Al = Al& +(y +*)Ai,O, AG1 O (TV.11.16)

AG,, = 1 AG, - AG, + AG, (IV.11.17)

Since AG, < AG, and the 1st term in (N. 11.17) is -120 kJ at 700 O C , Sr p-alumina rnay

decompose in the presence of Al. This can be seen from the proposed ternary phase

diagrarn (figure TV.11.Z) which shows that no tie line exists between Al and Sr B-alurnina

phase.

S r û - Al ,O, system has several intermediate compounds. Let's assume that the

formation of these compounds are suppressed with the exception of SrO.

SrO.yAi,O, + 4ySr = yA1,Sr + (3y + 1)SrO AGI 1 (iV.11.18)

AG,, = (3y + 1)AG, + yAG, - yAG, - AG, = -303 5 [W] - AG, < O at 700 O C

(IV. 1 1.19)

wherey = 9 is used in (W. 11.19). Thus Sr P -alutnina is not compatible with Sr.

The EMFs of the temary system at 700 OC are indicated on figure IV. 11.2. In

calculating these EMFs, the compounds are treated as point phases. For example, in Sr B-

alurnina - a-alumina - &Sr system, the chernical potential and EMF of each component

cm be calculated as

(IV. 1 1.20)

whether

Li A

Although the system is not thermodynamically stable, we should also consider

the reaction rnay be kinetically inhibited. If the kinetics is slow enough, a stable

EMF could be obtained on the measuring time scale. The direct-synthesized Sr f3 -dumina

may have an appreciable fiaction of a-alumina. In this case, the energy barrier for the

reaction (IV. 11.16) can be easily overcome at 700 O C . The initial low EMF in working

Celi I may result fiom the Sr P -ahmina - a -durnina - AI system. The initial high EMF in

ion-exchanged Sr P -alurnina may be due to the absence of a -alurnina. The k a t treatment

for the glass-sealing may introduce more a -alumina. Slow degradation in the reference

electrode can also be explained by the energy barrier for the formation of the oxides.

W. 1l.c) Conclusion

Sr -ahmina is not thermodynarnicaiiy stable with respect to both Al and Sr. If

the operating temperature of the electrochernical cells is so high that the system can

overcome the energy banier for the decomposition of Sr P -durnina, the Sr P -alumina

rnay not be an adequate probe for Al and Sr electrodes. (And this appears to be the case.)

One possible solution is to reduce the measuring temperature as low as possible.

Altematively, a sample of Ai-Sr alloy wuld be rnixed with a predetermined amount of

metd such as Sn or Pb to reduce the chernical potential of Ai. Development of new solid

electrolytes or introduction of separator could be yet another solution.

Figure IV. 1 1.1. Schematic diagram of Na-Ai-O. We assumed that the formation of oxides

that do not appear in the diagram was prohibited.

Figure IV. 11.2. EMF of pseudo temary systems in Sr-Ai-O where the formation energy of

Na P-alumina is used for Sr p-alumina.

V. Conclusions

Conclusions are as follows.

1) Dense green beta-alumina tubes were successfiilly formed by slip-casting, using ethanoi

as a liquid vehicle. The most important factor in this process is to control the arnount of

the moisture in the slip. Batchwise drying over molecular sieves is effective to obtain

extremely dry ethanol. The stability of the slips decreased in the order of well-crystallized,

XB2-SG and Cerarnatec powder.

2) Dense (>98 %) beta-alumina tubes were obtained by sintering. Green tubes were

packed with coarse beta-alumina powder in an a-aiümina cmcible covered with an a-

alurnina lid. The critical factor is the sodium oxide vapour pressure. The compositions of

not ody tubes but also packing powder should be controlled carehliy to attain fine

microstructures.

3) K beta-alumina tubes were prepared by ion-exchanging Na beta-alumina tubes. Ion-

exchange in a vapor phase (KCl) and subsequent liquid phase is effective. To

avoid mechanical degradation, the size of grains should be smaii (< 10 pm).

4) Na and K beta-dumina tubes were successfully sealed to a-alurnina lids with

aluminosilicate glas. The sealing was strong and impermeable to He (< lod cm3/s).

5) EMF measurements of Na& system at 599 K showed solution regions of 1.7 < x < 3

and 3.3 < x 4 2 . (A sudden drop in EMF in the range of 3.3 < x < 3.7 leaves the

possibility that a line phase may exit in this range). No compounds were found in the

composition range of O < x 4.7. From EMF measurements, the Gibbs energies of rnixing

of S N a & ) and $. N&C60 are determined as 85 and 81 kJ/rnol, respectively.

6) Na,& was modeled by an ideal interstitial solution. By fitting the parameters in the

model to experimental data, the model showed that tetrahedral sites are favored by Na (90

% occupancy of tetrahedral sites at Na2C60, figure IV.5.8).

7) From EMF measurements of the KxC60 system at 572 the Gibbs energies of mixing

1 of - KZC60 are determined as 83, 117, 120 and 121 kJ/mol for x = 1, 3, 4 and 6,

x+l

respectively .

8) A model for the intemolecular interactions between Cs0 molecules is proposed. As the

effective Lennard-Jones interaction centers move fiom C atoms to the centers of adjacent

double bonds, the 22 degree phase in the PUT stmcture of pure C60 becornes stabilized

relative to the 87 degree phase (figure IV.6.2). By assigning charges to C atoms and the

centers of bonds, we can increase the activation energy for the jump between 22 and 82

degree orientations (figure N 6.3).

9) The ~ a 3 stmcture of SC Ca is rnodeled by both regular solution and cluster variation

methods. In this model, four motifs between two adjacent molecules are considered. Using

the rnodel for the intennolecular interactions between Ca molecules, themodynarnic

properties of se C60 were computed (figure IV.6.7). The difference between regular

solution and cluster variation methods results from the short-range order reflected by

cluster variation method.

10) For the PUT stmcture of SC CdO, the 22 degree phase is not necessanly the ground

state. The presence of 82 degree oriented molecules with a srna11 fornation energy (-12

meVlmolecule) makes the 22 degree phase more stable at sorne temperature, owiiig to the

configurational entropy.

11) External vibrational frequencies of SC &J were computed by the harmonic

approximation (table IV.7.1 and figure IV.7. l), using the model for the intermolecular

interactions between C60 molecules. The computed fiequencies of phonons are quite

consistent with expenmental results. However, the difference between calculated and

expenmental results for librons suggests that harmonic approximation is not adequate for

!ibrons.

12) Using group theory, the extemal vibrational modes of SC Go at r, Z, A and A are

labeled and the correspondhg symmetry adapted vectors are obtained.

13) Using the results of (8) to (1 1), thermodynarnic properties of SC Cso were computed

(figure IV.9.1 to IV.9.5). The dispersion curves were approximated by the Debye model

for acoustic modes and by the Einstein model for optical modes. The Debye characteristic

temperature and optimum Griineisen constant are determined as 54.14 K and 7.5,

respective1 y.

14) The large difference between the computed and reported bulk moduli may be due to

the failure of the rigid molecule model (figure N.9.2).

15) The contribution of the formation of an 82 degree orientation among molecules to the

constant volume heat capacity is significant in the temperature range of 20 to 100 K

(figure IV.9.4).

16) The failure to obtain reproducibility in EMF measurements of Sr - Al alloys using Sr

beta-alumina relatcd to the stability

beta-alumina. Themodynamic considerations suggest that Sr beta-alurnina is not

compatible with pure Al or Sr (figure N. 11.2).

Appendices

Appendix A. Eulerian angles

Assume that an object is at the origin of a Cartesian coordinate system (x. y. z ) .

As the object rotates through angle y dock-wise about an axis z', the coordinate @, q, r)

on the object will move to the new coordinate @', q', J). Here, the new coordinate will be

derived as a function of the rotation angle y and the direction (8, 4) of the rotation axis z'

using Eulerian angles (see figure A.A. 1).

The rotation of a coordinate system through an angle 0 counter-dock-wise about

the x, y or z axis is given by three matrices.

(A. A. 1)

The rotation of the object at the origin through an angle 8 counter-dock-wise about x, y

or z axis corresponds to the above matrices with negative 8 instead of 0 .

When the axis of the rotation z' is given by (0, $1, the first step is to align the

new coordinate system (x', y', 2') in such a way that the z' axis and the axis of the rotation

coincide; thus, rotate the coordinate system by R40) and then by RJ4). The second step is

to rotate the object through y clock-wise about the 2 axis: R&). The final step is to rotate

the coordinate system (xt , y', 2') back to the original system (x, y, 2): RA-$) and then R k

0). Thus the whole procedure is given as

R(Y ;u) = T-' (%+Y, ( ~ ) T ( e , g ) (A. A. 2a)

(A.A.2b)

(A. A. 2c)

where R(y; 8,b) and T(B, 4) denote the rotation through angle y dock-wise about axis (O,

4) and the transformation of the coordinate system from (x, y, r ) to (Y, y', y),

respectively.

If the rotation axis i is given as [u v w], then instead of-using (8,4) we can

W u substitute COS# = - , sin( = ,/-, = - v

and s i n o = - into r r sin/ r sin4

equation (kA2).

Figure A.A. 1. Transformation of coordinates fom (x, y, z) to (Y, y', i).

Appendix B. Orthogonal matrices for proper rotations

R,, = R(US) =

-1 O 0' R,, = R ( U z ) =

O O 1,

R,, = R ( U Y ) =

Appendix C. Construction of the unitary multiplier representation at the symrnetry point r

for SC CbO.

Since q = 0, the multiplier (equation II.6.53a) is zero and the exponent tem in

equation (II.6.51b) is zero. The symmetry elements of the space group ~a'cr? and the

character table of the factor goup Th are given in section IV.8 (equation IV.6.1 and table

IV.8.1). The orthogonal matrices for proper rotations are given in Appendk B.

Set mat& K which represents the positions of 4 molecules in a cell.

where each column represents the i ah position in a ce11 (see table IV.6.1).

The unitary matrix for the element R wiii be in the form of

where w ' represents 4x4 matrix of B ( K , F* (K '; R)) in equation (LI.6.5 1 b) and R and fR

in the second matrix represent the orthogonal matrices for linear and angular coordinates,

respectively. The sign, is positive for proper rotation and negative for improper rotation

(see equation II.6.1 and 2).

1 0 0 0 E O O O

T(0; E ) = R(E)] O O E O

0 0 0 1

where the last matrix shows the definition of the operation 8 and 6 x 6 matrix E is given

Thus K' = [ 1 4 231

where each number K represents the colurnn vector for th position.

set K" =

set K" = 0 0 0 1

1 0 0 0

Appendix D. The numencal method for calculation of the Debye hnction

The Debye fûnction is defined as

where f ( 2 ) = L- e' - 1

(1) for large x

thus by partial integration we have

With ml = 16, the error is less than lod in the range of x > 1. However, as x goes to zero,

we nced more terms.

(2) for small x

Az) has been approximated by the polynomial senes.

For m2 = 6 the coefficients a,, have been detemined by the least-square method in the

range of O < x < 12 (table A.D. 1). In that range the error is less than 10?

The derivative of Debye function is given as

The fi) is shown

(A.D.7)

in figure A.D. 1 where the solid line (O< x 4 . 1 6 ) is for

equation (A.D.5) and the dashed line (1.16 < x) is for equation (A.D.3). In figure A.D.2

the solid line is for Debye hnction D(x) given by equation (A.D.6) for O < x < 1.16 and

equation (A.D.4) for 1.16 < x. The dashed line is for its derivative given by equation

(A.D.7).

Table A.D. 1. The coefficients in equation (A.D. 6)

&

Figure AD. 1. f (2 ) = - approximated by equation (AD.3) for 1.16 < x and by ex - 1

equation (A.D.5) for O< x < 1.16.

Figure AD.2. Debye function approximated by equation (A.D.4) for 1.16 < x and by

equation (A.D.6) for O < x < 1.16.

References

[ l] R A . Huggins, High Conductivity Solid Ionic Conductors: Recent Trends and

Applications, ed. by T. Takahashi, Worid Science, NJ, (1988) 664

[2] J. Maier, Science and Technology of Fast Ion Conductors, ed. by H.L. Tuller and

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