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Tricritical point in the p-T phase diagram of Ag2HgI4
TRICRITICAL P O I N T IN ~[HEp-T P H A S E D I A G R A M O F Ag~Hg|4 ~og~aa VARANOWSKI Milan IFRIESEL and Arnold L U N D ] ~
Dc~rtmentcfp~ysi~.Chr
t ~nsiti~ ~cu~ betweentwo d i ~ o r ~ ~ a ~ The ~ b i l i t y of IHcd~l ~ m ~ in ~her ~
1. Ian~uclion Al~edy in 1935 Landau [ I ] p ~ d i ~ e d the exi~ t e n ~ of a "critical point" b e l w ~ n a fi~t- and a ~ o ond~xder p h a ~ t m n s i t i ~ This w ~ based on a n ~panslon o f the t h e ~ o d y ~ m i e potential (Gibbs t ~ energy, G) in a p ~ e r series of the order parame t ~ u p to the lhird p m ~ r . This fi~t t h ~ t l r a13p ~ a r h indicated for fi~t-order phase tr~sition at the temperature T a n d the p ~ p, that both the lr~sition enthalpy A H ~ d the i n ~ o f |he heat
capaeityanomaly~C~erep~portionahioApfr-p<. where p,~ is *he p ~ u ~ o f the tsicriti~l poim. I~ is ~markable that neither ~lationship was r ~nfronted with any ~ 1 system. At [cut t ~ reaso~ can be ~ n t i ~ c d as ~sponsible for this s i l ~ t i ~ : First o f all, L a n d a u s paper from 1935 was later ove~hado~dbyhisfamousmoregeneralpape~on p h ~ tramitlons [2]. Secondly, it rook a I ~ g sime before clear e v l d ~ was ~mblished o f Ihr exist ~ c e of such a ~[tiea] point, In f ~ t , it w a s f i ~ t l n 1970that Gfiffiths [3],taking the phase diagram o f a sHe-~He mixture as a n example, int reduced the ~ n ~ p t of a tricriti~l point as a n i n t e n t i o n o f dtree ~itlcal lines. Some y e a ~
wi~ a h ~ ionic ~ d u ~ l y
later this concept w ~ t~ated by the ~ m e author in a m o ~ general ~ d profound way for fluid mtxtu~s [4] and f ~ p h a s e d i a g t a m s o f ~ b i t ~ r y s y s t r [~]. The first example o f such a point in a liquid crystal system was found in 1973 [6]. This ObSe~lion by ~ optical m e t h ~ w ~ later r by calorimetsicme~u~ments [7]. S t u ~ that time ~ n y other examples h a ~ been found o f liquld c~,sta[s exhibifing t rierilJ~l points [ 8-1(}]. Other ~ m p l e s of s y s t e ~ suitable for the app ~ m n e e of tdctiti~l points a ~ fi~troel~tries [ 11131 and metamagnets [14]. T h e ~ are many examples in fluld mlxtures [ 15 ] where the t i n t eases were obse~ed a l ~ a d y in the ~ r l y sixties. Besides ~ m e ~ a m p l ~ in f e ~ l e e t r i c mlts, the most eelebntrd ~ s e of a tr~sision from a ~ n t i n . uous io adi~ontinuous t r ~ s f o ~ a t i o n foundin ~ inor~nic sallwas that of NH4C[, being l~ted around 0.15 G P a [ 16]. The l ~ s i o n o f thls r ~ ~ [ 1 as the q u ~ t i ~ of its tree n a l u n has been the task of many investigations. For this question ~[or. imetrir ~ e l t s should be eonsid~cd as d~isivr However. both the p ~ i o u s measu~ments [ 17.18 ] ~ d the most ~ c ~ t quansit ative ones b y differential ~anr~ng calorimetry, DSC [ 19], give clear e v i d ~
0 167-2738/88/$ 03.50 9 Elsevier S e i e ~ Publ~Sbe~ B.V. (North-Holland Physics Pub[ishlng Division)
way it should if it w e ~ a wic~ti~l poinL What actu~J]y is o b e y e d is a ~ i s t ~ c r at the transition of an enthalpyc h ~ g e a n d a lambda-shapcd heat ~ pactiyanomaly.When the hyd~slatieprocure isinc ~ d , a decreaseof the tr~sition~thalpy occurs, which flatt~s at higher pre~u~s, but n ~ vanishes.The~fom it is h a r d to ~ e p t the orlevtatlon~ o r d ~ d i ~ r d e r t r a n s f ~ a t i o n of NH4C1 as a n example tha: a trlcriti~l point ~ n exist in an inn~ ganic salt which is ~ i t h ~ fe~elecwie nor mel~agnetlo. The purpo~ of this p a p ~ i s to p ~ s ~ c ~ e ~ m p l e of snch a salt, namely silver mercury iodide, Ag2HgId. Clear olorimelde s v i d e n ~ o f a tricwii~l polnt will be gi~n, and its de[ailed chan~eristiss will be d i ~ ~ s e d in t e ~ s of Landau's theory. 2. E x ! ~ d ~ t a l M e a s u ~ e n t $ ~ r e ~ r t i e d out ~ d e r hydrostatic ~nditions using argon gas as p~ssure t ~ s m i t t i n g medium in a l w ~ t e p cylinder-piston high p ~ u ~ d e v i ~ (B~set-B~tagne-Loire). The transiti~ eharactewiti~, i.e. the tempemtu~ a n d the changes in ent halpy ~ heat capacity, w e ~ ~ d e d by ~ n s o f a ~ m m e r d a l DSC d e v i ~ ( R~lo0, Japan), which was of the e n e r g y ~ m p e n ~ t i o n type. Details o f the ~]orlmetdo ~ a n g e m e m , ineludin8 the p~eedu~ of calibration, h a ~ h e ~ d~cribed previously [20,211. Let us only ~ m r k that the whole high. pre~ure chamber was heated ~ d coaled, thus avoiding any t ~ p e ~ t u ~ gradi~t, which is easily introduced when working with internal heating. The pt e s m ~ w ~ d e t e ~ i R e d with a roenganirte gauge, Polyeryst alline ~ p l e s cf about 50 m$ of A$~HfJ4 w ~ kept in platinum capstfles, which were p l a i d in the DSC ~ m p l e holder, 3. Result~ ~ d d i ' ~ s l ~
3. I. Disorde,~d phases of/.gzHgl~ The phase diagram of Ag2HgI4 for p ~ s m ~ u p to I G P a and t e m p e r a t u ~ h e t ~ n 278 and 440 K is reported elsewh~e [22,23]. The r e g i ~ w h e ~ t h r
x x ,e0
o~
~ i. A ~ i o n o f the p y p ~ diagnm ~ A ~ H ~ I n the v~in. i:y oft~e znc~i~*l poi.i ~t r.,= 3 ~ K and #~=474 M ~ Solid line: fl~l~rd~ t ~a~lion;dashedllne: s e c o n d e r t~nsilion. is e v i d e n ~ fur a t dcwii~l point is s h o ~ in fry. 1. The indicted lines ~parate two d i ~ r d e m d g h a t s . There have been no h i g h - p ~ u ~ X-my diffraction studiesof Ag~HgL at t e m p e r a t u ~ above ambient, ~ d conclusions on the s t ~ e l u ~ oflhe phases called a a n d 5 have to rely on structure sludi~ at n o ~ l p~u~. ~ ~ t s the m l l . k n o ~ d i ~ r d e ~ d pha~-Ag2HgI~,withafevkididesublatti~anda eompIetely di~rdered (Ag +, Hg l+ and v a ~ n e [ ~ ) ~ t i ~ sublatti~. AS n ~ a l p ~ this p h a ~ is obtained at ~ o u n d 325 K due to a fi~t~rder t n n sitlon o f ~ ordered tetmgonal structure (I~-Ag2Hgl.) [22-24]. A t n o ~ a l p ~ s m aoAgaHgl, has a f ~ t ~ r d e r tmnsitkin at 425 K. A ~ t d i n 8 to Steger's study o f theAgl-Hgl~pha~diagram [25],theobtainedpha~ is the ~ l l - k n ~ disorde~d bee p h a ~ ~-AgI. Howe v ~ . Otsubo et at. [26] ~port that Ihe ~ ~ phase is Rot obtained until at a higher temperature. T h e ~ is a n i n t ~ e d l a t e range o f some 20 K. catIed the AgHgl~ phase, f ~ which the st~cture ~ s identiffed b y X-my d i f f ~ t kin to be hop, i.e. that of I~AgL (Fnrther support for the exist~ce o f the hop phase was obtainedfrom a ITfA studyof the syst~ Aga~2H~, [26].) For pure Agl the hop--bee I ~ transition oecu~ at 420 Kr a n d poAgI is a a or~ dered phase with only a small solid solubility ofHgI 2 [261. T h e solid solubility is small also for the two other orde~d phases of the AgI-Hgl2 syst ~ , namely tetragonal Hgl~ [26] and [$oAg~HgI4 [27], while it is wlde for the disordered a-Ag2Hgl~. A ~ r d i n g to
them, ~n accommodate between 21 and 75 mot% of A~I, which r be taken as a slro~ indi~lion that the ~fions ~ disordered in the AgHgIj pllas~. Although the iodide sublaRice thus is identical for the [~Agl sod A-gHgI a mgi~s, th~ is the difference that the cations ~e ordered in lhe s ~ but disordered in the second, It is of some i n f e s t for the fur-
lher d i ~ s s i o n that the ]atti~ p a ~ m e t e ~ of AgHgls, a = 4.49 A and c = 7.34 k, | 26[ a ~ ~ m e w h a l srna~er than those of [l-Agl, a=4.58 k a n d c=7.49 A [ 28]. A diffi~lly when working with Agl-HgI2 at high t ~ p e m t u ~ s is that its ~mposition ~ change gradually due to sublimation of H~2, For ~hr cornposition Ag~Hgl, lo~es have been ~ported of about
1% below 470 K but as much as 90% n ~ r the ~elting poinl [26]. In a n earlier stage of our invesdgati~n ~ i n t ~ d c d to d a t e l i n e the high-presw~ stsbility ranges o f the phases denoted AgHgls ~d ~.
AgI in I~f. [25]. Indi~linns o f transitions ~ indeed obtained, but the rvp~ducibhity ~ s not as good as desired, presumably due to changes in ~ m position ~ u ~ d by partial sublimation o f Heir. Thus, we d ~ i d c d to mark Ihe r e g ~ above the disordered a-Ag2Hgl, as me ti.pha~, withoul giving any tentative phase b o ~ d a d e s within this ~glon of our p T phase diagram [22,23 I. Although we cannot ~ y for ~ r t a i n whether the stmcmm above the ificdti~l point~ i-~g. 1, is boc or hop (iodide sublatti~ o f ~ A g I or of [$-AgI), it is important for the discussion below that the 8-pha~ is distal 9 in ei~h~ ~sc, 3.2. Evide~e for a tricntlcal poi~ t ]~or t h e ~ g i o n o f t h e p h a ~ d i a g r a m r e p ~ d u c ~ l i n fig. 1, a large number of DSC scans were made w h ~ o the t m p e m t u m was i n c r ~ c d at ~ t a n t p r e s s ~ , O v ~ l a r g e inte~als it is possible VJ express the t o n shinn t ~ p ~ t u r c as a l i n e r funetinn of the press ~ [23 ]. In fig. I the continuous line ~presents a first-order transition, while the dashed llne c ~ sounds to a ~ n d - o r d e r one. The intemetinn of these two lines is at 382 K a n d 474 MPa. We have previously [22,23] called it a singularily or a rain* imum p o i n t . O ~ more detailed ~nsid~ations ofexperiment and theow show that we ~ dealing with a tricritical point al which t h e ~ is a tmnshion from a flrSt~rder transfo~ation to a se~nd-order one.
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F~ 2.Tmmiti~ enth~l~y,~.ofthc e~ t~sitien ~ 1~ mnse 474 to 700 Mpa. ~e ba~ indicte t~ c~r ~0~,
The ~[ofimetfic p ~ f o f this slatemenl is given in fig. 2, which shows ~he linear d ~ s e o f the tran. sition ~ t h a l p y f m ~ 700 MPa d o ~ to Ihc tticfiti~l point The approach to nil at the trictltJ~l point was c o n f i ~ e d by addhional measurements in its vieinity. As is s h o ~ in fig. 6 of ~f. [23l. when the tri. critical polnI is approached, t h ~ is not only a c [ ~ r d ~ r e ~ of the W~sitinn enth~]py hut a l ~ a simult~s inc~ o f the specific heat anomaly, AC,, On the I ~ p r e s s u ~ slde o f th~ tric~tlc~fl point, i,r for ~ i n g the dashed line o f i~g. I, only d i ~ n t i n uiti~ of the heat ~tpacity m noticed for transhio~ from the a- to the 5-pha~. There a ~ more f~tures of inte~st on the fi~t~rder (high-pressu~) side of the t ricki~l point. When AC~ is plotted (in arbitrary units) ~ u s the i n ~ of Ap = p - p ~ a st~ight line is ohialncd, as ~s s h o ~ in fig. 3. This ~[ationship will be d i s c ~ e d bdow. Since we a ~ dealing With firsl~rdcr transfo~atlons, the Clausi~Clapeyron eq~fion ~ be upplicd. T h e d T I d p o f f i g l a n d t h e A H o f f i g . 2gtve us the ~ s p o n d i n g v o l u m changes A V shown in fig. 4. AS could by foemen ~.V tends to n~l when the t fietitical point is approached, just ~ the transition emhalpy A H d o ~ . These two tendencies are the quired ~ d i t i n n s for a change f ~ m fl~t- to ~ n d order transitions according to the traditional tinss l f ~ t i ~ . Regarding the s ~ o n d ~ r d ~ transition llne (dashed in fig. 1 ) only the d ~ n t l n u i t y o f the heat ~pacity d u n n g the t ~ s i f i o n was measttred direetly, as a l ~ a d y stated above. Since we did not evaluate this dis~nfinhity in a qu~titative way, a
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~r ~' --~t.~) ' ~ ~g. 4 vo]~e chan~ aV, at mr fl~t~mer Pha~ tm~tio~ ~lcuat~ bYm ~ ofthe Onesiu~Cla~y~equalled" c o , s p e n d i n g applimtiono f Enrenfest's equatlonis out o f q~stlon,
3.3. lmp~taa~ of cation di~rder for tdcriticallity What are the conditionsfor the ~ r o f a tricriticalpoint i n the system AgI-Hgh? Such a point has nr been f ~ u d i n the phase diagram o f AgI [ 29] n ~ i n that o f HgI~ [ 30]. As stated in ~efion 3.1, the ~Iid ~luMlltyis ~ 1 1 at ambient t ~ l x * ature for these two p u ~ salts as well as for the ~ m pound ~-Ag~Hgl~. The ~ m p o u n d is o f s o c i a l i~tccest s i n ~ its ionic ~nducrivityis small ha spite o f the fact that 25% o f the cationso f pure AgI have
repLamJ by ~ = n c i ~ . H e w e r , they ~nsri-
they ~ n n o t ~ n t rlbute to any ~ n d u a i v i t yuntilthe ~rions become d i ~ r d ~ d at the transition to aA-g~HgI.at 325 K. It is to be noticed that the anion sublattiee r ~ a i n s the ~ m e (f~e) during this lmnsltion,whilefor pure AgI t h ~ is no e x ~ o f ~ t i o n sites (in either hop ~-AgI or in fee *-Agl) untilafter the transitionat 42O IL when the iodide sublattice changes io beg. T h e i ~ i c conducti,dty is about 3 orde~ o f magnitude h l g h ~ for r thou for Ag2HgJ,; this diffe~nce is ~ t surprisings l n ~ 516 o f the tetrahedml cation s i t ~ a ~ ~ t in a-A#, while only 25% o f the ~ t [ o n silos are v a u n t i n aAg2Hg~, AS stated in ~ t i ~ 3.1 we do noz know i f the stmetu~ o f the disorde~d 8-phase c o m s p ~ d s to hop A ~ # 3 ~ ~ a - A ~ In lho f o ~ ~ ~ ~ u l d the order o f 25%, ~ d the c h ~ g e in electri~l ~ n duetivitywhen going from the a- to the ~/-pha~ shouldbr much s m i l e r in this ~sr than i f the t ~ sition g ~ s ~o a bec iodide s u b l a t t i ~ Thus a m n ductivitystudy appea~ to be an a l t c m a t i ~to X-my diff~ctlon for d e t a i n i n g the structure o f the ~p h a ~ at high p ~ u ~ s . The t r l c r i t i ~ l ~oint o f Ag~Hgl, is s l t ~ t c d bet~ two p h o n s which am di~rd~cd, i.e. liquidlike, i n r~pcr to the ~ l i o n s u b l a t t i ~ I f this is a n e c e m r y ~ n d i t i o n , the ~ c u ~ c c o f a tricrlti~l r is excluded f ~ pure Agl or H$1:, but a r to be quitelthclyf ~ saltS~lated to Ag:Hgl,.We have found that the transitionis s c ~ n d ord~ b e t w ~ aCa~HgI, ~ d anoth~ d l s o r d ~ d pha~ ~ thewhole m a t e f r ~ n n ~ s l prepare to I GPa, and thal the slope o f t his Innsitionlineis very slmilarto the sloFr o f the dashed line o f fig. 1 [ 31 ]. "s mightwellb~ a t d c t i t i ~ l point beyond m e p ~ s s u ~ range available with our device. Another interesting~ is the eut~tie ~ m p o s l t i n n o f the Ag~HgI,-Cu~HgI4 sy~ tern, ~ s p o n d i n g to a ~l~rpositiono f the two ~[ts [32]. For salts like AgI ~ HgI2 R might be possible to change the situationby applyingan externalelectiio al field,which~ u l d directlyinfluencethe order paramet~ofthecationsublatti~.Thismightst~ngly ~ h a n ~ the probabilityfor a ~ n d . o r d e r transitinn line and the ex,stenee e r a t r i c r i t i ~ lpoint. O f c o u ~
the ~ t ~ a l p ~ u r e shouldbe mai~tahled as a neces~ry experimentalvariable.An i n f i o e n ~o f an cxt~ e k ~ f i ~ fieldon ph~c t ~ t ~ o n s m ~ f i s ~dth a hlgh ~nduct Jvity is fo~ th~lefically [33,34 ]9 ~ d h~ ~ e n been o ~ s e ~ d ex~cHmeutally [ 35]. AS mentioned in station I, the f,~t theoretical t ~ a t m e n t o f a j o l n tpoint fl~fifst-and~ n d ~ v d e r t ~ s i t l o n sgoesback to a paper by Landau published i n 1935 [ 1]. It is qulte i m p ~ i v e that ~ ~ use ~ m r ~ u l t s o f this paper for the interpretationo f our ~sults. In Landau's opinio~ first-~d ~dord~ t~siti~s ~ u l d b= reafized i n diffc~nt ~ #ons by varyingthe p m s m ~ , especially i n syslems es,hibifing o r d e r - d i ~ r d ~ Iransfo~ations. Both these r ~ u l r e m ~ t $ are fulfilledi n our system. Let us extra= t ~ details from Landau's results which ~ n he d i ~ t l y compared with ~ r experimerits: The specific heat anomaly AC~ should be p~ po~ional to the imbrue& f f ~ n ~ b a ~ n the m u m ( p ) and thr t r i c r l t i ~ l p r ~ u r e ( p = ) , eq. ( I l ) i n re~
[ i ]: Furtbe~om, when the tricritical point is appreached, t he Wansifion enth~lpy AH shotdd be p ~ ourfional to the d i f f ~ e c i n tempe~ture A T ~ in p~ssure Ap, eq. ( l 6) i n n f . [ 1]:
AH=B(p-p~).
(2) Both ~nditions a ~ fulfilledi n our ca~, see figs. 3
and 2. O f ~ U ~ it ~ u l d be i n t ~ t a l g to make a closer ~mpafisono f our ~sults with Landau's or later theo m i ~ l t ~ t m ~ t s o f tt~criti~lpoints. Thls would, h o o v e r , requirea q ~ f i t a t i ~ k n ~ l e d g e o f the ~ d~p ....... hich appfm i n the p o ~ , 9 o f the f~e energy. In o ~ case it is quite llkelythat two order p~amete~ ~ u l d be mqeimd, describing the ~nfigllrationalorder o f the H g2+ ~ d Ag+ ions separately, It is hardly p ~ t h l e at present to propo~ a simple ~ p e r i m ~ t a l procedure which could fulfil the~ ~qui~ent~ In m o d e ~ t e m i n o l o g y[ 15] the trlcrfiicalpoint de~rlbed i n this paper is a symmetricalone. This m ~ n s that the p o ~ r ~pansion o f the ~ ~ e r g y shouldincludeonly t e ~ s o f e v ~ p ~ r i n the order parameter. The a p p ~ r a n ~ o f the t rlcriti~lpoint is then e h ~ c t e r i z e d by the simultan~es vanishingo f
the ~ t o f i c i ~ ts o f terms p ~ p o r fional to p o ~ r two and four in the order paramet~.
Adm~kdRe~at This investigationis supported by the Swedish Natural S~i~ce R e s e ~ h C o ~ c l l and by Ofiie och Elof Erics~ns Sfiftvl~.
R e ~ [ ] ] L ~ d a u , P h Y s i ~ n i ~ n a ( ] 9 3 5 ) ll3, [2l L ~ d a ~ PhYsik'Z ~ ~u~i~ I [ ([95~) 26, ~4~" [4] It,s. Gfimth~J.Ch~. PhYs.60 0974) 19S. DlmB. C.flmth~eb~ Rev.~t209~5) 34S. [ 6] P.H.K~s, H.T.West~ andW~. ~ e S , ~y~. Roy.~ t t m 31 097~) 62S [7] R, She~i~= and S ~ s n d ~ r , L Phy~ (~ns) 36 (]9?~) CI~9.
[s~ ~. a = ~ , , , , e. S , , d ~ , w. s l ~ t t = , ~ O.M.~ h = ~ , . sg(l~s~)Tg0. [ to] R' Shashidhar'Ph~i~ I ~WI~B (19gfi)~ " [[I ] ~& M~i[~, W. ~rlin~and J.C.S[~cz~ki, phl,s Roy. Lettm 25 (1970)734~ [ 12] G,A.Samara,Phys.Rev.B]7 {1978)3020. [ 13]~ ~rdo~ ~ysi~ 122B(1983)32L [ [4] i . ~ ~wfic and S. ~ a c h , in:p h ~ t ~ t i ~ $ andc~ti. ~I ~enome~a' V~ 9' ~S C ~ m b and LU ~ w i l z (A~6~mlcP r ~ NewYo~, 1984)p,l. [ 15] Ch.e. ~ b l ~ ~ d ~L. ~ml, m: Pha~ Imnsilio~ and efitlcaI p h e n o l , Vo].9,~s. C. Dumb~ d j . k ~ i t z ( A ~ m i c Press.NewYork.19~4}p+163. [ 16]/.F. NaOe~d J.C.~ , Ann.R~- ~W. ~ 2? [197fi) 29t. (1966) [171~~';;~; t" ~ [ 18]C.W,~rlandand j.D. ~1o~, phy~lev.Bi6(1977)331. [tg] M. w i . h t end G.M.S~hne~d~,$. CMm.Tn~o@o. 13 (tgsl) 70L [20] K'E"Mflla~d~' B"Samn~ski aad A' L " ~ ' Ph)s' Rev' B23098D 3770. [21] ~.E. MelIa~er, B. ~ m ~ ,n d A. ~nd~n, in; Hi# e ~ e ~ J. R9 andI n d i t e , F ~ . 81h~ ~nf.r~.eds.C.M. B~k~n, T.Johaunis~and L ~ r ( Insitut~of PhY~l Ch~ist~, Unive~ityof Up~Ia, Uppmle'1981) p"428' [ 22] B'~mn~ M' F n ~ l ~ d A' ~ ' Ph~i~ 1397l~B [231 B, Bmno~k L M. Fflv~land A, Lo,~n, p h i . R~. B~3 (1986)t753.
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Dc~rtmentcfp~ysi~.Chr
t ~nsiti~ ~cu~ betweentwo d i ~ o r ~ ~ a ~ The ~ b i l i t y of IHcd~l ~ m ~ in ~her ~
1. Ian~uclion Al~edy in 1935 Landau [ I ] p ~ d i ~ e d the exi~ t e n ~ of a "critical point" b e l w ~ n a fi~t- and a ~ o ond~xder p h a ~ t m n s i t i ~ This w ~ based on a n ~panslon o f the t h e ~ o d y ~ m i e potential (Gibbs t ~ energy, G) in a p ~ e r series of the order parame t ~ u p to the lhird p m ~ r . This fi~t t h ~ t l r a13p ~ a r h indicated for fi~t-order phase tr~sition at the temperature T a n d the p ~ p, that both the lr~sition enthalpy A H ~ d the i n ~ o f |he heat
capaeityanomaly~C~erep~portionahioApfr-p<. where p,~ is *he p ~ u ~ o f the tsicriti~l poim. I~ is ~markable that neither ~lationship was r ~nfronted with any ~ 1 system. At [cut t ~ reaso~ can be ~ n t i ~ c d as ~sponsible for this s i l ~ t i ~ : First o f all, L a n d a u s paper from 1935 was later ove~hado~dbyhisfamousmoregeneralpape~on p h ~ tramitlons [2]. Secondly, it rook a I ~ g sime before clear e v l d ~ was ~mblished o f Ihr exist ~ c e of such a ~[tiea] point, In f ~ t , it w a s f i ~ t l n 1970that Gfiffiths [3],taking the phase diagram o f a sHe-~He mixture as a n example, int reduced the ~ n ~ p t of a tricriti~l point as a n i n t e n t i o n o f dtree ~itlcal lines. Some y e a ~
wi~ a h ~ ionic ~ d u ~ l y
later this concept w ~ t~ated by the ~ m e author in a m o ~ general ~ d profound way for fluid mtxtu~s [4] and f ~ p h a s e d i a g t a m s o f ~ b i t ~ r y s y s t r [~]. The first example o f such a point in a liquid crystal system was found in 1973 [6]. This ObSe~lion by ~ optical m e t h ~ w ~ later r by calorimetsicme~u~ments [7]. S t u ~ that time ~ n y other examples h a ~ been found o f liquld c~,sta[s exhibifing t rierilJ~l points [ 8-1(}]. Other ~ m p l e s of s y s t e ~ suitable for the app ~ m n e e of tdctiti~l points a ~ fi~troel~tries [ 11131 and metamagnets [14]. T h e ~ are many examples in fluld mlxtures [ 15 ] where the t i n t eases were obse~ed a l ~ a d y in the ~ r l y sixties. Besides ~ m e ~ a m p l ~ in f e ~ l e e t r i c mlts, the most eelebntrd ~ s e of a tr~sision from a ~ n t i n . uous io adi~ontinuous t r ~ s f o ~ a t i o n foundin ~ inor~nic sallwas that of NH4C[, being l~ted around 0.15 G P a [ 16]. The l ~ s i o n o f thls r ~ ~ [ 1 as the q u ~ t i ~ of its tree n a l u n has been the task of many investigations. For this question ~[or. imetrir ~ e l t s should be eonsid~cd as d~isivr However. both the p ~ i o u s measu~ments [ 17.18 ] ~ d the most ~ c ~ t quansit ative ones b y differential ~anr~ng calorimetry, DSC [ 19], give clear e v i d ~
0 167-2738/88/$ 03.50 9 Elsevier S e i e ~ Publ~Sbe~ B.V. (North-Holland Physics Pub[ishlng Division)
way it should if it w e ~ a wic~ti~l poinL What actu~J]y is o b e y e d is a ~ i s t ~ c r at the transition of an enthalpyc h ~ g e a n d a lambda-shapcd heat ~ pactiyanomaly.When the hyd~slatieprocure isinc ~ d , a decreaseof the tr~sition~thalpy occurs, which flatt~s at higher pre~u~s, but n ~ vanishes.The~fom it is h a r d to ~ e p t the orlevtatlon~ o r d ~ d i ~ r d e r t r a n s f ~ a t i o n of NH4C1 as a n example tha: a trlcriti~l point ~ n exist in an inn~ ganic salt which is ~ i t h ~ fe~elecwie nor mel~agnetlo. The purpo~ of this p a p ~ i s to p ~ s ~ c ~ e ~ m p l e of snch a salt, namely silver mercury iodide, Ag2HgId. Clear olorimelde s v i d e n ~ o f a tricwii~l polnt will be gi~n, and its de[ailed chan~eristiss will be d i ~ ~ s e d in t e ~ s of Landau's theory. 2. E x ! ~ d ~ t a l M e a s u ~ e n t $ ~ r e ~ r t i e d out ~ d e r hydrostatic ~nditions using argon gas as p~ssure t ~ s m i t t i n g medium in a l w ~ t e p cylinder-piston high p ~ u ~ d e v i ~ (B~set-B~tagne-Loire). The transiti~ eharactewiti~, i.e. the tempemtu~ a n d the changes in ent halpy ~ heat capacity, w e ~ ~ d e d by ~ n s o f a ~ m m e r d a l DSC d e v i ~ ( R~lo0, Japan), which was of the e n e r g y ~ m p e n ~ t i o n type. Details o f the ~]orlmetdo ~ a n g e m e m , ineludin8 the p~eedu~ of calibration, h a ~ h e ~ d~cribed previously [20,211. Let us only ~ m r k that the whole high. pre~ure chamber was heated ~ d coaled, thus avoiding any t ~ p e ~ t u ~ gradi~t, which is easily introduced when working with internal heating. The pt e s m ~ w ~ d e t e ~ i R e d with a roenganirte gauge, Polyeryst alline ~ p l e s cf about 50 m$ of A$~HfJ4 w ~ kept in platinum capstfles, which were p l a i d in the DSC ~ m p l e holder, 3. Result~ ~ d d i ' ~ s l ~
3. I. Disorde,~d phases of/.gzHgl~ The phase diagram of Ag2HgI4 for p ~ s m ~ u p to I G P a and t e m p e r a t u ~ h e t ~ n 278 and 440 K is reported elsewh~e [22,23]. The r e g i ~ w h e ~ t h r
x x ,e0
o~
~ i. A ~ i o n o f the p y p ~ diagnm ~ A ~ H ~ I n the v~in. i:y oft~e znc~i~*l poi.i ~t r.,= 3 ~ K and #~=474 M ~ Solid line: fl~l~rd~ t ~a~lion;dashedllne: s e c o n d e r t~nsilion. is e v i d e n ~ fur a t dcwii~l point is s h o ~ in fry. 1. The indicted lines ~parate two d i ~ r d e m d g h a t s . There have been no h i g h - p ~ u ~ X-my diffraction studiesof Ag~HgL at t e m p e r a t u ~ above ambient, ~ d conclusions on the s t ~ e l u ~ oflhe phases called a a n d 5 have to rely on structure sludi~ at n o ~ l p~u~. ~ ~ t s the m l l . k n o ~ d i ~ r d e ~ d pha~-Ag2HgI~,withafevkididesublatti~anda eompIetely di~rdered (Ag +, Hg l+ and v a ~ n e [ ~ ) ~ t i ~ sublatti~. AS n ~ a l p ~ this p h a ~ is obtained at ~ o u n d 325 K due to a fi~t~rder t n n sitlon o f ~ ordered tetmgonal structure (I~-Ag2Hgl.) [22-24]. A t n o ~ a l p ~ s m aoAgaHgl, has a f ~ t ~ r d e r tmnsitkin at 425 K. A ~ t d i n 8 to Steger's study o f theAgl-Hgl~pha~diagram [25],theobtainedpha~ is the ~ l l - k n ~ disorde~d bee p h a ~ ~-AgI. Howe v ~ . Otsubo et at. [26] ~port that Ihe ~ ~ phase is Rot obtained until at a higher temperature. T h e ~ is a n i n t ~ e d l a t e range o f some 20 K. catIed the AgHgl~ phase, f ~ which the st~cture ~ s identiffed b y X-my d i f f ~ t kin to be hop, i.e. that of I~AgL (Fnrther support for the exist~ce o f the hop phase was obtainedfrom a ITfA studyof the syst~ Aga~2H~, [26].) For pure Agl the hop--bee I ~ transition oecu~ at 420 Kr a n d poAgI is a a or~ dered phase with only a small solid solubility ofHgI 2 [261. T h e solid solubility is small also for the two other orde~d phases of the AgI-Hgl2 syst ~ , namely tetragonal Hgl~ [26] and [$oAg~HgI4 [27], while it is wlde for the disordered a-Ag2Hgl~. A ~ r d i n g to
them, ~n accommodate between 21 and 75 mot% of A~I, which r be taken as a slro~ indi~lion that the ~fions ~ disordered in the AgHgIj pllas~. Although the iodide sublaRice thus is identical for the [~Agl sod A-gHgI a mgi~s, th~ is the difference that the cations ~e ordered in lhe s ~ but disordered in the second, It is of some i n f e s t for the fur-
lher d i ~ s s i o n that the ]atti~ p a ~ m e t e ~ of AgHgls, a = 4.49 A and c = 7.34 k, | 26[ a ~ ~ m e w h a l srna~er than those of [l-Agl, a=4.58 k a n d c=7.49 A [ 28]. A diffi~lly when working with Agl-HgI2 at high t ~ p e m t u ~ s is that its ~mposition ~ change gradually due to sublimation of H~2, For ~hr cornposition Ag~Hgl, lo~es have been ~ported of about
1% below 470 K but as much as 90% n ~ r the ~elting poinl [26]. In a n earlier stage of our invesdgati~n ~ i n t ~ d c d to d a t e l i n e the high-presw~ stsbility ranges o f the phases denoted AgHgls ~d ~.
AgI in I~f. [25]. Indi~linns o f transitions ~ indeed obtained, but the rvp~ducibhity ~ s not as good as desired, presumably due to changes in ~ m position ~ u ~ d by partial sublimation o f Heir. Thus, we d ~ i d c d to mark Ihe r e g ~ above the disordered a-Ag2Hgl, as me ti.pha~, withoul giving any tentative phase b o ~ d a d e s within this ~glon of our p T phase diagram [22,23 I. Although we cannot ~ y for ~ r t a i n whether the stmcmm above the ificdti~l point~ i-~g. 1, is boc or hop (iodide sublatti~ o f ~ A g I or of [$-AgI), it is important for the discussion below that the 8-pha~ is distal 9 in ei~h~ ~sc, 3.2. Evide~e for a tricntlcal poi~ t ]~or t h e ~ g i o n o f t h e p h a ~ d i a g r a m r e p ~ d u c ~ l i n fig. 1, a large number of DSC scans were made w h ~ o the t m p e m t u m was i n c r ~ c d at ~ t a n t p r e s s ~ , O v ~ l a r g e inte~als it is possible VJ express the t o n shinn t ~ p ~ t u r c as a l i n e r funetinn of the press ~ [23 ]. In fig. I the continuous line ~presents a first-order transition, while the dashed llne c ~ sounds to a ~ n d - o r d e r one. The intemetinn of these two lines is at 382 K a n d 474 MPa. We have previously [22,23] called it a singularily or a rain* imum p o i n t . O ~ more detailed ~nsid~ations ofexperiment and theow show that we ~ dealing with a tricritical point al which t h e ~ is a tmnshion from a flrSt~rder transfo~ation to a se~nd-order one.
I
o
~D
~,~
_ r .~]
,~
F~ 2.Tmmiti~ enth~l~y,~.ofthc e~ t~sitien ~ 1~ mnse 474 to 700 Mpa. ~e ba~ indicte t~ c~r ~0~,
The ~[ofimetfic p ~ f o f this slatemenl is given in fig. 2, which shows ~he linear d ~ s e o f the tran. sition ~ t h a l p y f m ~ 700 MPa d o ~ to Ihc tticfiti~l point The approach to nil at the trictltJ~l point was c o n f i ~ e d by addhional measurements in its vieinity. As is s h o ~ in fig. 6 of ~f. [23l. when the tri. critical polnI is approached, t h ~ is not only a c [ ~ r d ~ r e ~ of the W~sitinn enth~]py hut a l ~ a simult~s inc~ o f the specific heat anomaly, AC,, On the I ~ p r e s s u ~ slde o f th~ tric~tlc~fl point, i,r for ~ i n g the dashed line o f i~g. I, only d i ~ n t i n uiti~ of the heat ~tpacity m noticed for transhio~ from the a- to the 5-pha~. There a ~ more f~tures of inte~st on the fi~t~rder (high-pressu~) side of the t ricki~l point. When AC~ is plotted (in arbitrary units) ~ u s the i n ~ of Ap = p - p ~ a st~ight line is ohialncd, as ~s s h o ~ in fig. 3. This ~[ationship will be d i s c ~ e d bdow. Since we a ~ dealing With firsl~rdcr transfo~atlons, the Clausi~Clapeyron eq~fion ~ be upplicd. T h e d T I d p o f f i g l a n d t h e A H o f f i g . 2gtve us the ~ s p o n d i n g v o l u m changes A V shown in fig. 4. AS could by foemen ~.V tends to n~l when the t fietitical point is approached, just ~ the transition emhalpy A H d o ~ . These two tendencies are the quired ~ d i t i n n s for a change f ~ m fl~t- to ~ n d order transitions according to the traditional tinss l f ~ t i ~ . Regarding the s ~ o n d ~ r d ~ transition llne (dashed in fig. 1 ) only the d ~ n t l n u i t y o f the heat ~pacity d u n n g the t ~ s i f i o n was measttred direetly, as a l ~ a d y stated above. Since we did not evaluate this dis~nfinhity in a qu~titative way, a
b~
_Lr of h ( )AC e ~ h ~ whereAo~p~p~aC~is~Pr~ina~itmryunits, and6Pi, G~.The ba~ indicte ther mn~
{
1~
~r ~' --~t.~) ' ~ ~g. 4 vo]~e chan~ aV, at mr fl~t~mer Pha~ tm~tio~ ~lcuat~ bYm ~ ofthe Onesiu~Cla~y~equalled" c o , s p e n d i n g applimtiono f Enrenfest's equatlonis out o f q~stlon,
3.3. lmp~taa~ of cation di~rder for tdcriticallity What are the conditionsfor the ~ r o f a tricriticalpoint i n the system AgI-Hgh? Such a point has nr been f ~ u d i n the phase diagram o f AgI [ 29] n ~ i n that o f HgI~ [ 30]. As stated in ~efion 3.1, the ~Iid ~luMlltyis ~ 1 1 at ambient t ~ l x * ature for these two p u ~ salts as well as for the ~ m pound ~-Ag~Hgl~. The ~ m p o u n d is o f s o c i a l i~tccest s i n ~ its ionic ~nducrivityis small ha spite o f the fact that 25% o f the cationso f pure AgI have
repLamJ by ~ = n c i ~ . H e w e r , they ~nsri-
they ~ n n o t ~ n t rlbute to any ~ n d u a i v i t yuntilthe ~rions become d i ~ r d ~ d at the transition to aA-g~HgI.at 325 K. It is to be noticed that the anion sublattiee r ~ a i n s the ~ m e (f~e) during this lmnsltion,whilefor pure AgI t h ~ is no e x ~ o f ~ t i o n sites (in either hop ~-AgI or in fee *-Agl) untilafter the transitionat 42O IL when the iodide sublattice changes io beg. T h e i ~ i c conducti,dty is about 3 orde~ o f magnitude h l g h ~ for r thou for Ag2HgJ,; this diffe~nce is ~ t surprisings l n ~ 516 o f the tetrahedml cation s i t ~ a ~ ~ t in a-A#, while only 25% o f the ~ t [ o n silos are v a u n t i n aAg2Hg~, AS stated in ~ t i ~ 3.1 we do noz know i f the stmetu~ o f the disorde~d 8-phase c o m s p ~ d s to hop A ~ # 3 ~ ~ a - A ~ In lho f o ~ ~ ~ ~ u l d the order o f 25%, ~ d the c h ~ g e in electri~l ~ n duetivitywhen going from the a- to the ~/-pha~ shouldbr much s m i l e r in this ~sr than i f the t ~ sition g ~ s ~o a bec iodide s u b l a t t i ~ Thus a m n ductivitystudy appea~ to be an a l t c m a t i ~to X-my diff~ctlon for d e t a i n i n g the structure o f the ~p h a ~ at high p ~ u ~ s . The t r l c r i t i ~ l ~oint o f Ag~Hgl, is s l t ~ t c d bet~ two p h o n s which am di~rd~cd, i.e. liquidlike, i n r~pcr to the ~ l i o n s u b l a t t i ~ I f this is a n e c e m r y ~ n d i t i o n , the ~ c u ~ c c o f a tricrlti~l r is excluded f ~ pure Agl or H$1:, but a r to be quitelthclyf ~ saltS~lated to Ag:Hgl,.We have found that the transitionis s c ~ n d ord~ b e t w ~ aCa~HgI, ~ d anoth~ d l s o r d ~ d pha~ ~ thewhole m a t e f r ~ n n ~ s l prepare to I GPa, and thal the slope o f t his Innsitionlineis very slmilarto the sloFr o f the dashed line o f fig. 1 [ 31 ]. "s mightwellb~ a t d c t i t i ~ l point beyond m e p ~ s s u ~ range available with our device. Another interesting~ is the eut~tie ~ m p o s l t i n n o f the Ag~HgI,-Cu~HgI4 sy~ tern, ~ s p o n d i n g to a ~l~rpositiono f the two ~[ts [32]. For salts like AgI ~ HgI2 R might be possible to change the situationby applyingan externalelectiio al field,which~ u l d directlyinfluencethe order paramet~ofthecationsublatti~.Thismightst~ngly ~ h a n ~ the probabilityfor a ~ n d . o r d e r transitinn line and the ex,stenee e r a t r i c r i t i ~ lpoint. O f c o u ~
the ~ t ~ a l p ~ u r e shouldbe mai~tahled as a neces~ry experimentalvariable.An i n f i o e n ~o f an cxt~ e k ~ f i ~ fieldon ph~c t ~ t ~ o n s m ~ f i s ~dth a hlgh ~nduct Jvity is fo~ th~lefically [33,34 ]9 ~ d h~ ~ e n been o ~ s e ~ d ex~cHmeutally [ 35]. AS mentioned in station I, the f,~t theoretical t ~ a t m e n t o f a j o l n tpoint fl~fifst-and~ n d ~ v d e r t ~ s i t l o n sgoesback to a paper by Landau published i n 1935 [ 1]. It is qulte i m p ~ i v e that ~ ~ use ~ m r ~ u l t s o f this paper for the interpretationo f our ~sults. In Landau's opinio~ first-~d ~dord~ t~siti~s ~ u l d b= reafized i n diffc~nt ~ #ons by varyingthe p m s m ~ , especially i n syslems es,hibifing o r d e r - d i ~ r d ~ Iransfo~ations. Both these r ~ u l r e m ~ t $ are fulfilledi n our system. Let us extra= t ~ details from Landau's results which ~ n he d i ~ t l y compared with ~ r experimerits: The specific heat anomaly AC~ should be p~ po~ional to the imbrue& f f ~ n ~ b a ~ n the m u m ( p ) and thr t r i c r l t i ~ l p r ~ u r e ( p = ) , eq. ( I l ) i n re~
[ i ]: Furtbe~om, when the tricritical point is appreached, t he Wansifion enth~lpy AH shotdd be p ~ ourfional to the d i f f ~ e c i n tempe~ture A T ~ in p~ssure Ap, eq. ( l 6) i n n f . [ 1]:
AH=B(p-p~).
(2) Both ~nditions a ~ fulfilledi n our ca~, see figs. 3
and 2. O f ~ U ~ it ~ u l d be i n t ~ t a l g to make a closer ~mpafisono f our ~sults with Landau's or later theo m i ~ l t ~ t m ~ t s o f tt~criti~lpoints. Thls would, h o o v e r , requirea q ~ f i t a t i ~ k n ~ l e d g e o f the ~ d~p ....... hich appfm i n the p o ~ , 9 o f the f~e energy. In o ~ case it is quite llkelythat two order p~amete~ ~ u l d be mqeimd, describing the ~nfigllrationalorder o f the H g2+ ~ d Ag+ ions separately, It is hardly p ~ t h l e at present to propo~ a simple ~ p e r i m ~ t a l procedure which could fulfil the~ ~qui~ent~ In m o d e ~ t e m i n o l o g y[ 15] the trlcrfiicalpoint de~rlbed i n this paper is a symmetricalone. This m ~ n s that the p o ~ r ~pansion o f the ~ ~ e r g y shouldincludeonly t e ~ s o f e v ~ p ~ r i n the order parameter. The a p p ~ r a n ~ o f the t rlcriti~lpoint is then e h ~ c t e r i z e d by the simultan~es vanishingo f
the ~ t o f i c i ~ ts o f terms p ~ p o r fional to p o ~ r two and four in the order paramet~.
Adm~kdRe~at This investigationis supported by the Swedish Natural S~i~ce R e s e ~ h C o ~ c l l and by Ofiie och Elof Erics~ns Sfiftvl~.
R e ~ [ ] ] L ~ d a u , P h Y s i ~ n i ~ n a ( ] 9 3 5 ) ll3, [2l L ~ d a ~ PhYsik'Z ~ ~u~i~ I [ ([95~) 26, ~4~" [4] It,s. Gfimth~J.Ch~. PhYs.60 0974) 19S. DlmB. C.flmth~eb~ Rev.~t209~5) 34S. [ 6] P.H.K~s, H.T.West~ andW~. ~ e S , ~y~. Roy.~ t t m 31 097~) 62S [7] R, She~i~= and S ~ s n d ~ r , L Phy~ (~ns) 36 (]9?~) CI~9.
[s~ ~. a = ~ , , , , e. S , , d ~ , w. s l ~ t t = , ~ O.M.~ h = ~ , . sg(l~s~)Tg0. [ to] R' Shashidhar'Ph~i~ I ~WI~B (19gfi)~ " [[I ] ~& M~i[~, W. ~rlin~and J.C.S[~cz~ki, phl,s Roy. Lettm 25 (1970)734~ [ 12] G,A.Samara,Phys.Rev.B]7 {1978)3020. [ 13]~ ~rdo~ ~ysi~ 122B(1983)32L [ [4] i . ~ ~wfic and S. ~ a c h , in:p h ~ t ~ t i ~ $ andc~ti. ~I ~enome~a' V~ 9' ~S C ~ m b and LU ~ w i l z (A~6~mlcP r ~ NewYo~, 1984)p,l. [ 15] Ch.e. ~ b l ~ ~ d ~L. ~ml, m: Pha~ Imnsilio~ and efitlcaI p h e n o l , Vo].9,~s. C. Dumb~ d j . k ~ i t z ( A ~ m i c Press.NewYork.19~4}p+163. [ 16]/.F. NaOe~d J.C.~ , Ann.R~- ~W. ~ 2? [197fi) 29t. (1966) [171~~';;~; t" ~ [ 18]C.W,~rlandand j.D. ~1o~, phy~lev.Bi6(1977)331. [tg] M. w i . h t end G.M.S~hne~d~,$. CMm.Tn~o@o. 13 (tgsl) 70L [20] K'E"Mflla~d~' B"Samn~ski aad A' L " ~ ' Ph)s' Rev' B23098D 3770. [21] ~.E. MelIa~er, B. ~ m ~ ,n d A. ~nd~n, in; Hi# e ~ e ~ J. R9 andI n d i t e , F ~ . 81h~ ~nf.r~.eds.C.M. B~k~n, T.Johaunis~and L ~ r ( Insitut~of PhY~l Ch~ist~, Unive~ityof Up~Ia, Uppmle'1981) p"428' [ 22] B'~mn~ M' F n ~ l ~ d A' ~ ' Ph~i~ 1397l~B [231 B, Bmno~k L M. Fflv~land A, Lo,~n, p h i . R~. B~3 (1986)t753.
[261YI OBU~,~ ~i11~M , ~ k ~ [27] ~ u l e
y, ]warna ~ ~ Uckk b
n.d H.'mum++J. Pb?+ C b + SOlids42 (tgBI)
J~ (198?) J4., [331Y. G = m m aod V. + m + m ~ 1
271b ~ f c ~
NI.k. 2+6
[34]Y. G u ~ v ~
129]m+~ Mella.der+J.E. +ow]imnRd m m m + ~ , S ~ p m 2 2 (]+B0)S4L+
[3s IV. Valiukcnas,~ mali+s+ A. ~;.km, V, mmu1+mis and A. mkn]~ J, ptys. S m +ap+.49 (tP80) +57.
Phy++m
and Y ~ n t ~
~m
[28] ~H. Hglw~c, ~ , ~ndo~-~msl~n+ ~ M m w + ~ c ~nd
of ~ E