II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude...
Transcript of II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude...
![Page 1: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/1.jpg)
![Page 2: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/2.jpg)
c].ectricnl conductivity C()'!Jrynrf•ble to those of liquid
el'.,ctrolytes. These ·nn t!•r1.els are Ill so termed as "sunerionic
solids" or "ff' st ion con(!uctors". Typically a sunerionic
solid hv!! the following cht:rf;lcteristics (Ch<1ndra 1 , to be
oublished) :-
( i)
(11)
(iii)
crystal bon•Ung is ionic
~aectricol conductivity is high
( -1o·1 -10·4 ohm-1 cm-1 )
Prine ill" 1 chtl rge CF rri.ers ~Jre ions 1.1hi.ch
me1ms thet the ionic trvnsference nu':llber (t10n) is almost
equal to 1. Here t 10n refers to the frt~ction or charge
trc"lsferred by all tyoes of ions ta>ten together 'Aith
reference to the total charge trc n5ferred.
(iv) '!"hH electronic conductivity is s'!!all.
Gener<-lly '1111 ter ie.ls 1ofi th electronic tre nsference numbo:r (te)
less than 10-4 are considererl satisfectory superionic solids.
'l'he values r)f electrical c::mductivity of a few
ionic and su:oerionic solids ~lT!:! sho>-rn 1!1 Figure 1.1. 'l'he
highest corductivity at roont tem'1orature obtained so fer
is for RbAg<IS 11hich is 0.27 ohl!l-1 cm-1 • This is mFny orders
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J ,f: ( c ;·
II l ,:: ' "
,I ,~ .. ; I
f' I ' d ' I i '~! I .
f I ~-q ; !/ ; -II r.· ;; i it .~~ /
I ,')
.¥ "
' ;
i i .-;
' .;" ·t" ~i-' f;
' I ' {: j f f I ' ~ ' I '
" i ' ... f r{_]
.,
/ ,)
11'
I
' .'
I ' '
' .
l ' ~ .,, -l [·
l .. .u ;
,, ,, t
~ ' '_+.' l ,-.: p, \ ' .:._·, )
''J I ,, . / i}) /. .,
::·; , .. ,1.~ '
.,,
·''
~: -·l,
';;.; -~""<
(''1
,I)
., fJ-'
' il.' 1/! :)
~· .:. . .:)
1.},\ ~ ;/> ;:;:;
' ~;.. i',,,J ' ,,
;; .....
'~3 ,. ~4<>•
~' '--:~ v"'1
:..\: , .. .C '"'
~~ ,., r:n . ' '\.~- rj
, :n '1-1;
"' ',' t .. t"1 ·:s :.:J :0 ·"' .I'J-
•:! ' 1:· ~ 5;.., r.::;_·
;:~ "' ,, ~
:.~ . ...... , .. t.. .:: ~
" .~ C:I.J L .... ~-~ >
![Page 4: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/4.jpg)
of rnrenitude hir.hor ttwn those of the more com"lonly knolill
ioPiC solids KC1 1 l-'E,Cl otc. 1hich hr,ve room te'll'lerature
conductivity"-' lo-12 -lo-16 ohm-1 C'll-1• It rncy be noted
that most of the S'.m·rionic <1nteriols vttain high elG<'trical
conductivity r:fter undergoing a "lhasa transition at c, certain
tem'·>er~ture wl1ich !D1lY or 'lll>y not be WE!ll defined.
(1.1) Typ_g s of ionic solid~
f'rom the cryst!: llogra ohic point of view a ne:rfect
crystel of on ionic com;)~'lllnd would be an insulator (Lidit'rd 2).
~'he nr·esencc or defect or disorder is a necessity t·J sustr: in
mcr.:ningful ionic trE:n:1port. It is notural to cla ssi:f'y the
ionic solids tccor•Ung to the tyoos of defect or disorder
resnonsible for ionic transoort. '"hese solids f.re prinCi'Ofilly
of t;vo tyoos (Cht"ndra 1 , to be publ:ished):
(1) Point defect tyoo
(2) ?~olten subl!"ttice type
In the 'noint defect type' solids, tho tr&ns l)>:>rt is through
~ren!{el or '·ichottky defect cmirs which Pre th,.;rmally
generated. f,s & result, the nu.'Tiber of' defects (and hence
charge carriers) ore D function or tem'?'"r' ture. 'l'he acti-
vation energy is genernlly high rvl eV or :nora. 'Point
defect tyoe' solids c& r. further be subdivided according to
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defect concentration dun:;ity as follows a
!)1] ute:
(b) ConcentrRtadl l•:xnm::>la - stabilised .~irCOl'it'
or Hafnia, l'Al· 2 etc.
In 'molten suU.nttice ty:'le 1 solids, the nu·~ber of
&. X•rticul: r ion io less ti1rm the nu'llb-:•r of sitas flWJilt ble
for it in its sublattice. f. s :;. result, ell these ions cen
'ho~• or ·~ova like a free-ion' from one position to another.
inca a 11 these ions o.rc avr.iie 1Jle for tr~msport, the
conrluctivity 1s l;o\rgc~ and ~:ctiwtion energy is low. 'I'hese
<itt terials/ions possess an 1n veragc structure 1 rether than
fl rigid structure.
···ice & Roth 3 hns suggested another scheone of
cla ssHication of ionic solids lm sed on the '"'Srameters of
his 'free-ton like theory' of super ionic solids wr.ich is
esr:ef'tit.lly the snme as 1.: bovG. f,ccording to this scheme,
ionic soli'ls <J:rs of three ty·1es :-
":yoe I: ~hese ere conventionPl ionic solids
in 11hich the defect concentntion is low. The nQ'Ubcr or
~obile defects isrvlo18 /cm3 or less. This is the c~se of 'dilute noint defect type' ionic solid discussed above.
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Type III They contain a large concentration
of defects which hllve COf;lesced into coherent substructure
or 'exterded defects' of sub~icroscooic dimension. These
are the smne as 'concontreted point defect type' solids.
'l'he de fact concentration is "'1020 /cm3 •
'!'yl:le III a This is the case of 'liquid like
molten subl1'ttice 1 in ·,thich nractict.lly v.ll the ions in a
sublnttice ere available for :-~ovement. 22 :l mobile charge carriers is ,v 10 /crtr.
The number of
These are generally
mar!ted by a channelled or layered structure.
Types II & III a.re sunerionic solids or solid
electrolytes while Type I are essentially Insulating Ionic
.3ol1ds like alkali halides.
(1.2) Insulating Ionic Polids
They possess conductivities lower than lo-10 ohm-1cn)1
at roO!Il te'llpersture. A great majority of the ionic solids
belong to this class viz. alkali halides, silver halides
(t,gel and tgl2r) etc.
In an ideally perfect ionic crystal the movement
of ions under the action of an electric field is not
oossible if we r. ssume thct trt nsoort or charge could only
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take nlr ce vio <·nion - Cl·tion exchange. This nrocess involves
r very high onorgy of th::J order of 15 eV i.e. in one gram
molecule auch nn event would ::~ccur only once in 1030 years.
Prom extons~ ve series of 'llef.· sUI'e'"1er:t s of ionic conductivity
cs 11 function of te~·1orntur"', t:.vo f<,cts stand out :Jlrinly;
(i) ~hG existence of two main regions in
conductivity curve r,s n function of
(11) In both the regions tho log (di') vs T-l
is e. :>.,rox1mt1 tely linear.
'Phe high t<l'n'lerature conductivity 12bove the transition
rl'!gi::m or ';mae' .in genefal is an intrinsic prooerty of
the crystal nnd ·~easure;'!lents in this region ere quite
reproducible. On the other hand, low te:nnerature conducti
vity (o:r extrinsic conductivity) dis'Olf·ys a smaller slope
-1 in log o-'i.' vs T curve but it denends on the nurity of
the scmnle. 'i'o ~xnlllin the lou but finite te"!'>8rature
denendent ionic conductivity the existence ~f thermally ,.,_ I!L s ~ e-'TV
gener~1ted noint defect ,..invoked.
In general electrical conductivity is given by
cr = (1.1)
where ni is the nu~'ber of chorge carriers, 'i is the
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charge and p1 is the :nobility of the churge carrier of
ith soccies. It is no11 wall recognized th!!t rrodomiMnt
tyt)os of chB.rge Clorriers in ionic solids ere either Frenkel
defects or Schottky defects.
ln case rJf T'renkel defect, an ion (cation or
onion) from 11 norrnttl lnttice stte is activsted to a
ne~rby interstitial site, cfoat1ng an interstitial ion
llnd &!" ion 1n1cancy. Jf this 'Jair of defect becomes thermelly
dissoci~ted to beyond the range of their Debye - Huckel
charge clouds, they lofill migrEtte indeoondently in an
electric field, in general, with different mobilities. If
the te'!lnerature is high enough to o ssure ion defect
equilibrium (and impurity effects be ignored), there will
be a fixed and equal concentration of the two types of
defects given by the following expression.
. 1/2 gf (NN') exp (- 2Kr) , (1. 2)
\'there gr is the energy required to form a. thermally disso
ciated Frenkel defect pair, N is the number or normal
lattice sites !lnd r~· is the number of interstitial
sites per unit volume. Silver bromide and chloride are
the classic examnles of this class.
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:;ohottky defects (cation and 11n1on vacenc1es)
are '' ssumed to bo introduced in ooirs into the crystal
le. tt ice frorr the surface sites or from vr C!l ncy sinks
such es dtslocvtions or grl>in bound~>ries. Fixed and
eC'u.al concentrr.tions of the tt~o ty:Jes of vac&ncies result
ct ench tenmercture assuming ionic ec,uilibrium and negli•
giblo impurity effects. i-. go in dissociH ted defects may
migrate indeoendontly in an electric field. In general
'chottky defects occur in crystals with Co'lloorable
cHtionic end f>nionic rr1di1. The number of vacancies (ns)
of ef:.ch tyoe is given by
= gs
N exo ( - 2kT ) , (1.3)
where g8
is the amount or energy required to form n
~~chottky defect ne ir.
Ionic mobility in such solids occurs as a result
of random, tharmr,lly activated jumps, either of an inter
stitiel ion into an adj~cent unoccupied interstitial site,
or of an ion on u norm[,l lcttice site into on adjacent
vace.ncy for th[·t t-ype of ion. Another migration mechanism
is that in which a.n interstitiHl ion replaces a~ike ion on
normnl lattice site, pushing it into on unoccuoied edje.cent
interstitial site. The lttter is known as 1 interstitialcy 1
mechanism. In eny event, each jumn nrocess is ~ssumed to
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involve &n activrtion energy of L'l.g, such th£t the jumo
fro' uency 1s given by
w= 1 exp ( - ) , (1.4)
where y is the v1brationel frenuency. The above exnress1on
1s the ju:np freouency vthen there is no a nplied electric
field. , :unpose an uniform electric field B acts along
one of the sxes (say x - axis). The aryplied electric field
ccn be rcgr.rded as adding a term -eEx in the potential
energy of the interstitial. The net effect is that a
jump in the direction of field would take olace with an
increased probability
W' = '1 exp -1 -- ( kT
E - ! ea.E:) 2
and a jump in a direction opposite to the O.'Jplied field
takes nlace with & reduced prot~; bility
If ea.E (<( k1:', this would result in a current density
(L1diard 2 )
y exp (-.LlE ) ItT '
(1.5)
(1. 7)
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The mobility ( Fil ir then given by
= ea 2 A& = kT- Y exp ( - kT ) '
!fence the Of•rtial conductivity (fro111 Eq. 1.1) due to
mechanism consHered ~Jb.:>ve is
( 6.& ) v exp -- , \<T
(1.8)
(1.9)
where e is the charge on :_ r.~.rticulnr ion oartic1nat1.ng
in conduction 'll<~chanis~ and n is given by Bq. 1. 2 or
Ect. 1.3. Hence a n1ot bet~;een log cr •.r and l/T would
be linear if one type of defect mechanis~ is resoonsible
for the conductivity.
In ~ctuel experiment, broadly speaking, the log
a- T vs l/l' plot shows t~ro distinct regions. The high
temperature conductivity is indeoondent or purity of the
crystal (if the i~purity concentr&tion is not too large)
but the low temnerature conductivity is strongly impurity
sensitive. The former is ;::no1r1n as intrinsic conductivity
&n'l the l~}er e s the extrinsic. Intrinsic conduct! vity
is due to thermally generated defects while the extrinsic
conductivity is due to vac~ncies created by imnurity doning
(nliovelent cation or anion). J. few '!lore veri&tions in
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slOJ)e cen be seen in the conductivity curva due to associa
tion and nrocipitc tion of hnurities. For alkali ht•lidos,
it is now well established tho.t both the positive and
negative ion vncencies nrc 11obile. The transnort oora-
meters cc n be obtainod either by folloving Chandra and
l'olfe's 4 1roccdure (from electrical conductivity measure
~ent on oure and divalent cation and enion doned samples)
or by the method of Fuller et al 5 (fro11 diffusion and
electrical conductivity 'lll!lt;sure•nents on nure and divalent
cation doned sn"lloles).
In general both ce.tions e1:1d enions can move in
a solid lattice. 'l'he cations have reltetively small ionic
r!'ldii as co:noored to anions. t·or e.xa11ple, the Pfluling 0
ionic radii of rlknli metal ions are : Li+ (0.60 A), +, 0 + 0 + 0
ra i0.95 /.), K (1.33 A) and Rb (1.48 !>.)while the ionic 0 0
radii of halt de ions are : F'- (1. 3ti l\), Cl- (1.86 /,), 0 0
Br- (1.95 A) and I- (2.16 /\ ). f.•s such the number of
conducting solids vrith alkeli metal ions l'.re expected to
be more. The nossibility of halide ion conductors is much
less except tiwt for the s:nsllest 1dn F-. In fact, the
materiels aVt?ilable so far confir'1l this sttJtement.
The more cO''l"lOn ions showing high mobility are
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i + r + + + + - d o2-. L , Jo , K , A g , Cu , F an ~~ost Of the CO'IIpOunds
of the latter two ions (1·'- t-nd o2-) show high conductivity
only nt high te'l!)'lerntures. Ler.ving Oxygen ion conductors,
most of the sunerionic solids inv·)lve D monovalent ion,
There ۥre no di- 11nd trivnlent ions reoorted showing high
mobility though renorts for S111lll Ionic conduction exist.
'T'hls 1s soMewhat axooctad since the coulombic energies
involved in jumoing/hopping of di- or trivr•lent ions would
be more thnn those for monov~; lent ions. !IOI<Tever, the
obove descrl~tion ia an oversimplification or the actual
sitUP.tion. The structure is probably the most important
f~ctor ( cJiedersich & Geller5~. Structural princiPles
governing the formation or solids showing fast ion
transport has not yet been studied in. detr.11 though the
beginning has been made by Phillips 6 and Paleigh 7 •
Table 1.1. gives some of the irnnortant high
conductivity solid electrolytes or sunerionic solids.
bmongst these, the highest conductivity has been reported
for silver ion conductors out or which F.bAg4 I 5 is the
best. ~his thesis is concerned with silver ian
conductors and hence some of the salient features of
silver ion conductors are reviewed below.
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Te ble 1.1 :o:~~e 1mnortent ~:uner1on1c :;olids
(s~c Chnndrs 1 , to be published)
--------------------------------------------------------------------;.Jaterinl ~~obile
ion t:onducti- ?o:nnoratura i'.ctivntion vity _1 _1 (oC) energy (ohm em l (aV)
--------------------------------------------------------------------I II III IV v
-------------------------------~------------------------------------
Alkali metal iop conductors:
Lithium p-alu'!linn
··odium ~-nlu'rlina
Dod!U'll ~-gallate
:C:otassium f-4'lumim
Silver ~on conductors~
' '<!. g4 !5
u•
-1"" ,, Li+
,.~~ +
,.,,.,.., + dU
~~a+
,.+
1.2xl0-5
1.5xl0 -3
1.3xl0-4
1.4xlo-2
5.2xl.0-4
3.oxi.o-2
6.5x10 -5
0.21
0.21 (pellet)
0.27 (crystal)
0.19
25 0.43
400 o.ca
25 0.37
25 0.16
300 -300 0.27
300 0.29
22 0.1.
22 0.1
22 0.1
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Table 1.1 (contd)
-------------------------------------------------------------------I II III IV v
·------------------------------------------------------------------q ..t.gl
Copper ion conductors:
··cu " ·. 4 ... 5
CX: -cu2 ;e
oxrgen ion conductors:
Th02 : 8;i Y 2o3
C I) 7 ·. ~ 1 e 2 : ... r ..
Fluorine ion cQnductor§a
+ ·g
' + •'· g
~·•
1.0
0.02
0.09
0.09
0.6
5.5xl.o-2
-3 4.B:xJ.O
1.lxl.O-l
1
6:xl0-4
4xlo·2
4:x:lo·3
150
25
450
280
150
1000
1000
1000
500
200
700
100
0.16
0.11
0.19
0.05
1.10
1.12
o.e 0.26
0.2
0.38
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(1.4) .;..LUyor Iop Conductors
·uver ion conductors llre the most extensively
studied croup of su'mrionic solids and hnvo beon very
'"!Jll revietrad by :tedersictl ti Geller 6 , 7't•iwhnshi 8 ;
J·unke 9 1 hnhi 10 ond Ch1.ndra 1 (to be :mbl1shed).
"'hn enrliest good high ternnerntu:re silver ion conductor
"ltOb<'lbly rol'lorted in litor~'ture ls silver sulfide,
r g2:>, by f'r,rr,dny 11 ·.mo had also Slll'Tllised the ~rcser,ce
of many othel' such compounds -- "t.here is no other body
'•lith \·rhich I rm PCqWlinted., that like sul,buret or silver cz;n compere 1.rith metr.ls in conducting pot.,er for
eleot:r1c1ty when hot, b~t which •••• during cooling
lose in nower • • • • Probably, hot/ever, l!lllny others may,
Hhcn sought for, be found". 1 g2se and Ag21'e, compounds
similar to l'g2u, lu:ve been lr.ter found to have high
conduct tvity at high te:nrJerr. tures (Tubandt 12).
f{ovrever, tl1e ~e<'sure:nent of conductivity in silver 13 halides by 'uoondt ''- r,o:renz F~&y ~-tell be taken as
the starting no1nt ror the uresent day effort to
underst~:nd <:n.d se:>rch for mnr silver ion conduction.
'rhey renorted th~:t the conductivity of fo.gi abruntly
increases by more than 3 orders of 'ilagnitude at l47°C.
This high conductivity Ph11se is known e.s O{•Agi (cubic)
nnd the low conductivity obese is f3-!.gi (wurtzite).
C)1nce then, a lr,rge nu:nber of silver ion conducting
![Page 17: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/17.jpg)
solids ht•Ve be'ln disc11vnred.
''o~Jt of tho llilver ion conductors ere b;. ;_.od on
i ei. "'hcse conductive nhn~~cs hove been obtr-ined by
rx: rt!e lly :
( 1} + ubstitutin.<; different cntions for f,g
(11) 'ubstltutin;: different anions for x- or
(iii) oUbstit·uting both Cf-tions and anions
'"'he fir t>t re.,:rrtod ro•)'IJ! twl'l ,ero ture good silver ion
conductor is cetion substltuted end >1as obteined by
solid solution }H + 4 ;;g I ('" = ~, Hb, ~rrr4 ) independently
by <r£ldley ._-. ;;reant'l 14 and -~v1ens <;; .\rguo 15• ':'he
compounds '·l!;g4I 5 are known to decompose ~ccording to the
fol101~ing solid st£te disnr'1'Jortionation reection 16 (owens ) belo'" o certain tomnerature :-
7 /;g I
':'he dis'Orooortionation te1!:Jeraturcs for
1'ifg4 I 5 , H~\g4I5 Rnd ~H4r.g4r5 are 37, 28, 32°c
tesnectively. ~'he nresenca of water vapour seems to
eatalyse the disproportionation reaction rate.
Chcndra & :~ohabey 17 hr.ve studied the e.bove reaction by
observing the rP.te of growth of t gi exciton neak.
'rhese results are givAn in Cha;,ter VI es a subsidif;ry
matter of this thesis.
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Ion trnns'lOrt apd dywmicsh!!vt been :Jtudied in
silver ion conctuctors usinl': ~ \·tide V!•riety of techni<'uas.
's n r<"sult of these studios, more or lass a common
nicture is e·noreing out for the ~~ gl- ty'le su:>erionic
solids (l·unl<o 9 ; Chnndra 1 to be nubl1shed).
(a) '!'he c1 tions ~.J.re structurally disordered
e.r•d tho cntion sublrtticc is "liouid-lilte". The nurnber
of c1 tioTls is less tlm·~ tho nucnlJ.::lr of ~itos (voids)
avr• ilfcble for thecn nnd those site;; ~;~rc OOCU;'lied at
(b) '"he cnions vre arrenged in such a \·my
that the l•:>eal potenti&ls are rather flat r.long certain
lines whtch intercor~ect neighbourine sites. Along
these lines the difference in notentiul energy is of the
order of thermal energy. i ceording to Geller 13 "the
common structural motiff in the conductive comryounds
is the existence of pcssege 1<1ays of face•sherod iodide
ion polyhadr!'! n.
(c) f1s a conSG1;uenca of (a) and (b) all or
e. large frnction of the crctior:s cDn move from one site
to anothe:r site ~lith a. very low activation energy of the
order of ther:ne.l em!rgy and thus, in genore.l, partici!Ja•
ting in the cation diffusion nrocess. The diffusion ryath
is channel like cni not e_:xrctly liquid-like.
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(d) The disordering process may be first-
orlier or Mcond-order. ".t the first-order transition
t;e;.,ere.ture the conductivity changes nbruptly and generally
involves o chnng'3 in J.ntt1co symmetry G l!tant hent. Jn
s0cond order ptwse tr11nsltion, there is no Abruot change
in conductivity -- only the slone changes slightly.
i1cnerally, thore i!:l no or S'nall chEmge in lattice
sym'l!etry and is c·c~omut.nied by & power le.w divergence in
soecific heat.
(e) Ion-Ion correlr>t1on is en imoortnnt
f&ctor in brin,:;ing !'bout disorder (e.g. Uubel'!/IS.n 18;
Pice et al 19 i ''ardee & t~aha.n 20; ;fohnn 21 ; vergas et al
~iolomon 23; :eloh & Dienes et al 24 ; Lederman et al 25 ).
(f) The ion transr.>ort is -oossibly by a
ju'!lp·diffusi<m n:rocess on which an additional cn tion
local motion is superimposed (diffUsive and/or overda~,ad
oscillatory local motion).
Unusual struc1lure is tho single ~ost imnortent
factor responsible for high l~g+ ion trfinsport. There are
:nore sites avl'ilable than the nll'llber of ions par unit
cell giving the ions e 1feee-ion like state•. ~he
nU'llber of available sites end nu'llber of ions ner unit
cell ere given in Table 1.2 to illustrate this point.
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Table 1.2
SUperionic Solld
I
o<' -.Agi
CCHs)4N 2.Agl.3ILS
<<:siisNH) Agsis
RbAg415
r/ -Ag3~U
f3 -Ag3
SI
Ag3
::mr
v<. -Ag2S
o( -Ag2
se
«-Ag2Te
Structural parameters or some Ag ion conductors
structure
II
b.e.c.
HeXagonal
HIOO'lgonal
CUbic
b.c.c. (disordered)
b.c.e. (ordered)
b.c.c. {ordered)
b.e.c.
b.c.e.
r.c.c.
Lattice constant
Ao
III
5.06
a 12.77 c 26.54
a 11.97 c 7.41
11.24
4.99
4.90
4.81
4.88
5.0
6.57
lbmbfllr or .Ag pl'!r unit call
No. or .Ag s1 tes available
Stab111 ty range/ Rmarks
IV v VI
2 12• or 147-555° 42
*J>ref'erl"ntial
39 123
lD 34
16 56
3 12
3 3
3 3
4 6
4 -8 *4 are at ~ah('!dral 179°C
sites and the rP.maining mixf!ld conductor 4 distributf!ld are 20 posi-tions 3-,4- and G-coordin~ t~1.
l~
c
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(1.5) Superionic pht,se trrnsitions
It bE• s been found durinr. the electrical
conductivity Vs tem))erpturc studic s for most SU:JCrionic
solir'ls thllt the high ionic conductivity is ottninad only after
the mstariDl has undergone s001e phase trcnsition. 'f\~O tynicol
e:xemples ore givan in figure (1.2) for Rbllg4I5
l!.nd Agi. It
is seen thr.t at T1
the conductivity rises by order of mBg
nitudc • 'i'hhJ tr: nsition is rcferr()d to FlS Class I (or Insu
lc.toY'-Electrolyte phase trensition). '?he w lues of T1 for
RbAg4
I 5 and Agi respectively E:re -151 °c end l4?0c. There is
yet another type of transition in which the conductivity does
not cilf·nee repidly but at the trvnsition only the slope of
()Vs"f curve changes. The temper" ture T2 (•-64°C) merked
in Figure 1.2 for Rbllg4t5
refers to such a trrnsition. l.ft;tny
theo1•ies have been :out forward to account for tho above two
types of trt·nsitions outlined below.
A phenomenologiccl model was proposed aL~ost simul
taneousi;Jin 1974 by Huberman18 and Rice, Stressler and Tocxnbs19
(hereafter referred as RST model). <4elch & D1enes24 have
suggested a generd phenomenological model or which Huber-
man's and RGT models come out as a speclrl case. It is well
kn01m thlct the ionic conductivity dcpends U')On the number of
t'renkel or Schott'{' defect p~\irs which is an exnonential
function of the ectivrtion energy for their formetion. In
![Page 22: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/22.jpg)
IE 0
Agl
F,t~ 1'2: ELECTHICA L CONDUCTIVITY OF RbAq4
15
AND A.:jl AS A FUNCTION OF TEMPERATURE
SHOWtrJG THE VARIATION OF CONDUCTIVITY
AT PHASE TRANSITION. AT T1 , THE COND\..'CI-·
IVITY CHANC,ES ABRUPTLY WHILE AT T2 ,THE
CHANGE IS SLOW AND ONLY THE SLOPE CHANGES.
![Page 23: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/23.jpg)
the phenomenologicnl mod9ls, it is assumed that tho acti•
v: tion energy for formation is ooncentr1, tion deoondent due
to defect interactions. \ofithout going into thO deteils of
defect interaction, rlOlch & Dienas24 WTote the concentration
(c) denendent ter~ for frue energy per ion, F(o), as under •-
F(c),. B(c) - kT £ -2o ln c - (1-c) ln (1-o)
< o( -c> 1n col-e> ~lno(s-Tsv1b<c> c1.1o>
Where E (c)-,
~s the concentra.tion dependent enere;y required
pushing an otam into an 1nterst1t1el site;
o( is the rttio of nWilber of interstitial
"cells" to the nUIIlber of ions;
svib is the vibrPtionnl contribution to the
entropy.
for
delch & Dienes24 considered t~•o e pproximate forms
for E(c). In the 'Quttdrl tic form•,
B(c) = E1c - ~2c2 0~ (1·1i)
or' 0 c • El-2E 2c J
2 and, svib (c) ·~ c -.LJ.s2c
In thi3 approximation, Welch & Dienes model reduces to
Huberman or RST models. welch & Dienes formulation is
quite arbitrcry while Huberman & RST arrived at their
equations using & physiaFl model for defect interaction.
Huberli!B.n assumed an •attre.ctive intGraction• between an
interstial ion and a vacancy while RST considered the
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interaction of intHrstitinl C(ltion defects with the str:.in
field they indue A,
One more form or ~:(c) other than eqtmt1on 1.11
hE•s been tlltmn by ·telch & :)ienes \-lhich is bt sod on 1!4ott
Littleton tyne• o::>lculf: tions. This is ~~iven o s,
(1.12)
The ecu111br1uln Jei'ect concentrr tion (on whase
numb<n.' the conductivity del<mds) cun he obt[1n:Jd by minim!-
Sill_; F{c) with NSJ.l<lCt to c, i.e.
~(~0 (30
where F(c) is ,siven by equution 1.10 under approxillWtion of
equation 1.11 or equation 1.12. Theoretical CB1culations
show th&.t both class I (1'emper:·ture T1 of Fig. 1.2) or
cl:rcss II (Te:npe:rature T2 of Fig. 1.2) transitions or.n be
simulated by t&king different strength of interaction
terms (i.e. valt.es or E1 , E2, f2' etc.).
Lettice g:, s theories form another approach to
understr-nd the superionic phflse trDnsition. In these
theories it is ~:.cssumed th~t the mobile ions within the
suporionic solids can be considered as a 1lattice-gas'
hopping from one htticc point to another. 'l.'he hopping
'lntice-gas1 p&rticles interDct with each other and modify
![Page 25: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/25.jpg)
the ion diffusion or tr£•ns1'lort act1vnt1on energy. This is
referred to fis 1 cooptJrl·t1vo hopping' of 1ntcrnct1ng ions.
:>o:Jto & Kikuchi26 did fir:;t str t15t1ct.l c:.lcUV-·tion for
{?> -alumim• using the Pc.th Prob: bility !{et:hod. Moro
recently Murch &: Thorn<:? reported the above:: cclcubtion
using a '-!onto Cc rlo Method. Both the above cnlculr• tions
could simulate cln ss II type phE--se tr£,nsitions success-
fully.
f..nother interesting !?.pproech has bo'm suggo::tod
by Ptn•dee & l{e.han20 and Mahen21 which is referr:d as Ionic
Polsron Theorz. It. is considerad thvt en ion nolarises
the host crystal, and when hopping must cerry this polari
sing cloud with it cr,llcd 1 1on1c polaron•. This is
similar to the theory or conduction by hopping electrons
(small polaron) developed for metf ls and semiconductors.
The ionic poleron int•eraction hns been considered by
Mahan2l. quantum meeh&nim<lly through explicit use of
phonon co-ordinate and introducing coupling parameter
for the coupling or ion hop to the surrounding lattice
vibrut1ons. Ionic polaron thaory could also simulate
class II transitions but fails to account for class I
trf'nsit1ons.
(1.6) D'>Jlll:J!lics of ion transport
'l'he v•rious microscooic theories
proposed for the ion dynamics in super1onic solids can
broadly be clDssified e.sJ
![Page 26: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/26.jpg)
(1) !roo-ion !IIOOGl
(2) Dalll in modol
(3) :i )!)'>in•; or jump Lli:'l'uaion Dl•>dol
In frtjo-ion modol 1 it is ,,s:;umod th: t en ion is
thormslly excited fr'Jill a locnlis•-d st. tu to '' !>t: te in which
tho ion mov"s tr,nsln tionally tllrou~_;h tho aolid to anothor
loorlisod st1.V1. i!istoricnlly1 this WbS the first thoory
proposod by nico & noth3 but it is no more acc:N':ltt~c1 (see
aaas2B). It has been shown th[tt it is not n<Jcessnry to
invoke 'free-ion theory' &nd tho s<mo rosults cnn be obtained
by convvntionnl 'hopping l!IO(i;Jls 1 •
Van 3ool29 end Van Gool & iJottelberghs30 have
given a 'domain-model'. As nointed out er:rliar, the high
conductivity in suporionic solids is marked by low values
of activations. 'I'here are two imnortr:.nt possibilities for
low VF-•luos of activction energya (a) coonorrtiva jump or
a number of ions es tCtken by Sato & Kikuchi26 or (b) move
ment of walls between domains or orderad configurations.
The 90ssibility (b) ~s considered by Van Gool et a130• In
most superionic solids, the number of sites or l~ttice
position is much larger than number of ions avr. ilable. It
is assumed that in one domain ions are ocupying one type of
positions while in an adjacent domain another set of
positions is occupied. Considering the domain as a. "defect",
the energy per ion neces~Wry to creste the "defect•• has been
![Page 27: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/27.jpg)
found to be low in nn iderlised calculr.tion by Vnn Gool
et a130 for ~-eluminr-. 'l'hi.'l rtecounts for lou nctivr-tion
enorey or ion mL;rl·ti ·::m in tham. .'lo fr r, no '!X"'lcri-nentcl
ev1.iencA hf s been re'lorted ei th.:r to nrove c>r i!is'>rove
1 domain modal I.
It is no-.r wall rucognis0d thet d-~-'u":r nnd new
insi ~ht into ft> st diffusion in sun·~rionic conrluct·:)rs could
be gc:ined by invcst1g;-tirw the full dynmnics of i.onic motion
includin•::: the host lattice (see G; .;.ndr81 , to b•~ publishod).
':.'hG ~irst microscoryic theory in this direction l>'f s atte'!lpted
by Huberm€!n and l>en31 in which the mobil(l ions e:re assumed
to have two basic degree13 of freedom {1) an oscillfl.tory
motion in the hr.rmonic ,otentiel providod by the rip;id
lattice r:.nd (11) a random-wHlk 'roeass through which they
c~n diffuse throughout the crystcl. The ju~n process was
assU!'Ied as occuring "1nstante~neously" and heine; uncorNlated
with the oscillatory motion. This theory could successfully
oredict the oef:k in conductivity e.t 60 Clll•l for f:S·alumina
but det~iled structure could not be rJxolt;.inad. Racontly in
a series of uapers from
(Jrueseh et ~132, Fulde
Brown Baver! nose:,rch Gentre Group 33 34 et e.l , :t;eller et al , Bruesch
et a135) have investigc:.ted the Brownian motion in a nerio-
die lattice including the effects of pol~risability or lett ice and correle ted jum:;s of ions relevrnt to the problem
of super1on1c conductors.
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Various physical mechanisms for diffusion are
shown in Fig.l.3. The physical jump diffusion model shovn
in Fig.l.3(b) has been proposed by Clemen & Funke36, Funke9
to explain the microwave conductivity data of superionic
conductors. The# ion jumps are characterised by a mean
duration or the jumps <'1i ) and a mean residence time (·'""( 0 ) •
In classical jump diffUsion L.1 ~ <. l.'0 but for superionio
solids the experimental results indicate that L 1 -;::; 't0 •
So, a super~position of jump diffusion and a local tyne 3t.i motion we s considered by Clemen & l''unke as a good first,
approximation. The concept developed by Clemen & Funke to
understa.nd physically the microwave conductivity data 1s
interesting and deserves some mention though it does not
give the microscopic detail of ion dynamics. During conduc
tivity measurements, the applied elactrienl field is assumed
to give rise to two effects referred to as "start eff'ect" and
"acceleration effect". The "start effect" can be visualised
as in the conventional theory of conduct! vity of alkali halides.
The electrical field slightly supports or hinders the 'start'
of jumps in or a.geinst the instantaneously preferred direction.
At low frequencies ( '1:.1 .( ,Z 1/l.o.l ) the phase of the filld
is practically constant while an ion is performing a jump
giving a constant value of c-- (_o) or o- Cw) at low frequ
encies. At slightly higher frequencies, a cation starting
jump with the help or the electrical field is still on its way
while the field 1s already changing its sign and this would try to
![Page 29: II - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/41751/6/06_chapter 1.pdf · of rnrenitude hir.hor ttwn those of the more com"lonly knolill ioPiC solids KC1 1 l-'E,Cl otc.](https://reader033.fdocuments.in/reader033/viewer/2022050512/5f9ca7dcafca730c25749e83/html5/thumbnails/29.jpg)
j
lll l ',!MI'l \ \ill fl~l',\UN 1~1 1\qU\P
{b) I OCI\l 1'<10TIOI'.1 \ .ilH>1P tllfrU SION
( 1''o "'~ l; ·:::.: \ o 1\ s e, . )
( C ) JUMP 1) \ F F II'> I 0 N ( 1· ( < j 0 )
FIG-. i) VARIOU'.i '-'OSSIBLE: i)IFFUSION
MECH/\N1c:Y1
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reverse the ion jump direction consequently lending to
decrebse in conductivity. In the high frequency limit
, the applied field cnnnges its sign many
times during the flight of an ion &nd the overall conducti•
vity tends to zero. This exPlain~the decr~asing
experimental micrownve conductivity with the increase in
frequency but an "accelerotion effect" has to be invoked
to explain the conductivity pe~:k vl cm·1• During flight
or the !on, friction in expectod because the coulomb
1nternct1ons with the other moving cetions ceuse rspid
fluctuations of the local potenti~ls. Funke9 assumed that
the jump lengths of the ions ~re fixed, being given by the
geometry of the anion lattice and not being affected by the
electrical field. The jump distances being constrnt, the
effect of acceleration or retr,rdat!on of the moving ions
by the field changes their times of flight but cannot give
any contribution to (E) (0) , as this does not cho.nge the
jump rates.
The complex conductivity is given by,
"' 1\ ()' (w) = q • l(w,t)/(&
0e·iwt)
A
where I is the complex ionic flux 1n the d1roct1on of the
electric field &0
exp (•!tift), Clemen & Funke36 gave a
datE, iled expression for I = I start + lac eel • The flux
Isttrt was a function or change in ion number densities
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while Incccll deoend unon velocity Vl'lriation Ett differant times.
Tho cnlculf ted vv luc s i',H ve " good fit with experi•nental cW ta
confirming thn t the .1ump diffusion r.todel sunerimpos,Jd by
locnl 'llOtion is 1' good phono'ller,ological descriotion of ion
dynr.mics in supcrionic conductors.
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r:orerencos
1 nolievtl.>ns". (to be nublished by .Tohn :11ey & ..:ons).
2. I.idihrd, / .•• : in .tndbuch dar flhysil<, Vol. 201 p. 246;
edited by .• l'luc~e (:''1ringer - verlog, 1157).
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