Disordered porous solids : from chord distributions to small angle scattering

Transcription

1 Dsodeed poous solds fom chod dstbutons to smll ngle sctteng P. Levtz D. Tchoub To cte ths veson P. Levtz D. Tchoub. Dsodeed poous solds fom chod dstbutons to smll ngle sctteng. Jounl de Physque I EDP Scences (6) pp < /jp >. <jp > HAL Id jp https//hl.chves-ouvetes.f/jp Submtted on 1 Jn 1992 HAL s mult-dscplny open ccess chve fo the depost nd dssemnton of scentfc esech documents whethe they e publshed o not. The documents my come fom techng nd esech nsttutons n Fnce o bod o fom publc o pvte esech centes. L chve ouvete pludscplne HAL est destnée u dépôt et à l dffuson de documents scentfques de nveu echeche publés ou non émnnt des étblssements d ensegnement et de echeche fnçs ou étnges des lbotoes publcs ou pvés.

2 poous solds fom chod dstbutons to smll Dsodeed sctteng ngle Les soldes poeux bphsques sont des exemples de mleux ntefcux complexes. Rdsumd. dffuson ux petts ngles (SAS) ddpend fotement des popdt s g om6tques de lntefce L le mleu poeux. Les pop6t6s de l d6v6e seconde de l foncton dutoco61ton pttonnt de denst6 d6flnt quntttvement le nveu de connecton ente l dffuson ux petts l connssnce de ces dstbutons de codes pemet de dst1tlgue cetns types de quo stuctuux. Une elton explcte ente le specte de dffuson ux petts ngles et les d6sodes de d6sode est dscut6e et les p dctons du modme comp6es ux 6sultts exp6mentux types P utlston du ttement dmges nous nous nt essons h tos types de dsponbles. co616 vec une ttenton ptcule pou le cs dun vee poeux (le Vyco) et enfn des sctteng stongly depends on the geometcl popetes of the nteml sufce ngle poous system. Popetes of the second devtve of the bulk utocoelton pttonng popetes of ths ntefce. A tctble expesson of ths second devtve nvolvng sttstcl poe nd the mss chod dstbuton functons ws poposed by Meng nd Tchoub (MT). the on the pesent possblty to mke quntttve connecton between mgng technques Bsed the smll ngle sctteng ths ppe tes to complete nd to extend the MT ppoch. We nd J. Phys. I Fnce 2 (1992) JUNE 1992 PAGE 771 Clssfcton Physcs Abstcts 61.10D 61.12D P. Levtz nd D. Tchoub Cente de Recheche su l Mtbe Dvsde CNRS lb ue de l Feollee Oldns Fnce (Receved 4 Mch 1992 ccepted16 Mch 1992) et les cct6stques sttstques de cette ntefce. Une expesson utlsble de cette ngles d v6e mplqunt les dstbutons de codes ssocdes h l phse mssque et u dseu seconde de poes fut popos6e p Meng et Tchoub (MT). Mettnt h poft l possblt6 ctuelle dune quntttve ente les technques dmgee et l dffuson ux petts ngles ce compson tente de compl te et d6tende lppoche MT. Dns un peme temps nous montons en ppe dstbutons de codes est los popos6e. Dns une tosbme pte lpplcton h dff6ents ddsode le mleu ldtoe de Debye pou ses popdtds h gndes dstnces le ddsode complexes ok des pop6t6s dnvnce d6chelle de longueu peuvent dte ognstons obsev6es. Abstct. Dsodeed bphsc poous solds e exmples of complex ntecl med. Smll functon quntttvely defnes the level of connecton between the smll ngle sctteng nd the fst dscuss how chod dstbuton functons cn be used s fngepnts of the stuctul dsode. An explct elton between the smll ngle sctteng nd these chod dstbutons s then In thd pt the pplcton to dffeent types of dsode s ctclly dscussed nd poposed. e comped to vlble expementl dt. Usng mge pocessng we wll pedctons consde thee types of dsode the long-nge Debye ndomness the coelted dsode wth specl emphss on the stuctue of poous glss (the vyco) nd fnlly complex stuctues whee length scle nvnce popetes cn be obseved.

3 poous solds ply n mpotnt ole n ndustl pocesses such s septon Dsodeed heteogeneous ctlyss ol ecovey glss nd cemc pocessngs [I]. The scence confnement nd the geometcl dsode of these systems stongly nfluence the dynmc o sctteng (SAS). It s well known tht the densty fluctutons e the mn ogn of the ngle In the cse of bphsc mtx these fluctutons e loclzed just t the shp sctteng. whch pttons the system. The smll-ngle sctteng consdeed s puely ntefce phenomenon [4] s then stongly dependent on the geometcl popetes of ths ntefcl between smll-ngle sctteng nd the sttstcl popetes of the ntefce [5 6]. connecton hs been known fo long tme lmost fo smooth nd convex ptcles. As shown by Ths cn be mde usng the concept of chod dstbuton. A chod s segment sctteng to the ptcle nd hvng both ends on the ntefce. It cn be consdeed s belongng only one chod dstbuton ws needed. Extenson to bphsc poous sold (genelly the mss utocoelton functon nvolves delcte sufce ntegl s shown by of et l. [5]. Meng nd Tchoub [12] hve poposed moe tctble expesson of Cccllo Sevel hypotheses e nvolved n the nlyss (I) the poous sold s consdeed s n uncoelton between djcent chods hs to be ssumed. At ths tme n the sxtes no Tchoub. In secton 2 we dscuss how chod dstbuton functons cn be used s utocoelton nvolvng the poe nd mss chod dstbutons s computed usng thee-dmensonl devton. We lso gve n explct expesson of the smll-ngle whch dectly depends on the Foue tnsfoms of chod dstbutons. In pt 4 sctteng 772 JOURNAL DE PHYSIQUE I N 6 1. Intoducton. pocesses whch cn tke plce nsde the poe netwok [2]. Ths ses the themodynmc poblem of descbng the mophology of these poous solds. A stuctul chllengng cn be hndled usng numbe of technques [3] dect obsevton of the mss nlyss by optcl o electon mcoscopy molecul dsopton dect enegy tnsfe. dstbuton Coeltons t dffeent length scles of the mss dstbuton e genelly pobed by smll oented ntefce. Ths sttement cn be qulttvely undestood f we consde dsodeed medum s completely defned ethe by ts mss dstbuton o by the oented poous septng the vod spce fom the mss pt of the mtx. Popetes of the ntefce second devtve of the mss utocoelton functon quntttvely defne the level of Gune [7] nd Pood [8] eltonshp between the shpe of convex ptcle nd ts lne pth whch coeltes two dstnct ponts of the ntefce. Mny ttempts hve been mde to develop ths concept futhe. Connectons between smll expnson of the chod dstbuton nd locl o sem-locl popetes of the ntefce such s the cuvtue the wee developed ndependently by Pood [8] Meng nd Tchoub [9] Wu nd ngulty [10 1II. These dffeent studes wee essentlly delng wth convex ptcles whee Schmdt non-convex stuctue) s moe complcted. The genel expesson of the second devtve ths second devtve nvolvng the poe nd mss chod dstbuton functons. bphsc ndom med (t) the devton consdes the dstbuton of mtte sotopc ndom lne (n one-dmensonl nlyss) ; () specfc type of ndomness whee long ws mde to gve n explct expesson of the smll ngle sctteng nvolvng chod ttempt functons nd ble to be checked by ndependent expementl dt. Bsed on the dstbuton possblty to mke quntttve connecton between mgng technques nd smllngle pesent sctteng ths ppe tes to complete nd extend the ppoch of Meng nd of the stuctul dsode. As exmples we gve nlytcl expessons nd lge fngepnts of chod dstbuton functons fo two types of ndom bny med (I) the expnsons ndomness nd (t) systems whee length scle nvnce popety cn be obseved Debye fo the bulk pt o the ntefce. In secton 3 the second devtve of the mss ethe

4 the pplcton to dffeent types of ndomness s ctclly dscussed nd pedctons e I BACKGROUND. A smple descpton of two-phse system ncludes two ssumptons. 2 ech phse s consdeed s homogeneous nd s chctezed by ts vege densty. Fst these two phses e septed by n del shp ntefce. As dscussed by Second [13] the el stuctue s then pobed wth cose gn sze lge thn the Cccello chod dstbuton functons [14]. The chod sze dstbuton n numbe (clled fo shot chod dstbuton) s elted to the condtonl pobblty of hvng chod sze between The chod sze dstbuton n length g() (g()) gves the pobblty densty to fnd (mss) chod hvng sze between nd + d nd pssng though pont M ndomly poe n the poe (mss) phse.these dstbutons e null fo negtve dstnces. The dstbuted elton between g nd f dstbutons s [12 14] j f() d m p (1) (3) ws fst poposed by Debye Andeson nd Bumbege [15 16]. As ecently system by Cccello [17] ndomness n Debyes sense s elted to the theoy of dscussed length sttng fom ndom pont M locted ethe n the poe (I p) o n the mss phse m). Assumng Debye ndom system the pobblty tht no collson occus n the (I ntevl of length + d eds s Q( + Q() ( p d) I p m (4) In ths equton two hypotheses e mplctly ssumed. Fst ny event occung on one N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 773 to vlble expementl dt. Usng mge pocessng we wll consde thee comped of dsode the long-nge Debye ndomness coelted dsode wth specl types on the stuctue of the vyco poous glss nd fnlly complex stuctues whee emphss scle nvnce popetes cn be obseved. length 2. Bphsc ndom medum nd chod dstbuton functons. tomc scle. In ths domn of ppoxmton t s possble to defne t lest two types of nd + d knowng tht the chod begns t specfc pont of the ntefce. Ths dstbuton wll be noted f() o f() whee the ndces p nd m stnd fo poe nd mss espectvely. j g() I wth j j f() d (2) o nd o An nteestng queston cn be sked t ths level concemng the possblty of usng chod functons s fngepnts of dffeent models of dsode. In the next two sectons dstbuton dscuss two of them Debye ndomness nd dsode nvolvng length scle nvnce. we 2.2 DEBYE RANDOMNESS. A theoy of smll-ngle sctteng fom bphsc ndom ndom functons [18]. Moe ptcully lnel nlyss of Debye ndom sttony s closely connected wth the model of the ndom telegph sgnl solved by Rce [19]. system Let Q;() denote the pobblty tht no collson wth the ntefce occus long segment of d) sde of the ndom lne gong though M s ndependent of ny events occung on the othe

5 sde. Second the pobblty denstes p nd p of httng the ntefce fom the poe o n the ntevl ( + d) s wtten s Combnng equtons (4) nd (5) we get P;() wth tself g() Q;() p d I p exp( p ) I p m. (5) p m (6) Usng equton (I) the two chod sze dstbuton functons n numbe ed f() I exp( /j) (8) /p; (8) exhbts postve vlue t the ogn nd negtve slope. Ths type of dsode equton stong ntefcl ngulty [ ]. Ths explns why the sctteng ntensty nvolves by Debye does not follow the symptotc behvo pedcted by Kste nd Pood pedcted fo &mooth nd cuved ntefces. [21] 23 DISORDER INVOLVING A LENGTH SCALE INVARIANCE. We consde dsodeed ntesecton ponts long dstnce R s (j) oz ( ( m p (1) 774 JOURNAL DE PHYSIQUE I N 6 the mss pt of the mtx e constnt. The pobblty tht the fst cossng pont ppes P;() d P() The chod sze dstbuton functons n length e computed usng the convoluton of HI exp( /l ( l)) exp( /l l) dl (7) wth j (9) As shown n secton 3 equton (8) pemts the etevl of the exponentl vton of the bulk utocoelton nd consequently the well known Debye expesson fo smll-ngle [16]. Howeve the negtve exponentl fom of f() nd f() ses some sctteng concemng the locl popetes of the ntefce. Fo completely smooth questons ntefce the smll expnson of the two chod dstbutons scles s. The (dffeentble) slope of ths lne eltonshp s dectly elted to the cuvtue popetes of the postve ntefce [10 20]. Debye ndomness cts n dffeent wy. Smll expnson of systems wth length scle nvnce popetes. Moe speclly we wll focus on the long- behvo of chod dstbuton functons. Let us consde bphsc medum hvng nge sml ntefce. The ntesecton of ths sufce wth ndom lne s lso self sml self set of ponts. Usng the ule of thumb concemng ntesecton of sets [22] the numbe of Ns(R) Fs R (10) whee ds s the fctl dmenson of the ntefce nd Fs shpe fcto. The vege chod fo the poe (o the mss) poton of ths mtx computed on chctestc sze R lengths gven by s

6 # f;() d (12) nd the long nge behvo of chod dstbuton functons cn be wtten s f;(r) # dr " fst consde the cse of mtx hvng self sml sufce nd compct (non-fctl) Let of the mss nd the poe netwoks [23]. P(R) nd P(R) e ndependent of R dstbuton nd we get f(r) oz ths cse the mss volume fcton scles wth R s Alp(R) ( m) m p. (14) chod functon hs well-defned fst moment nd must decy fste thn mss dstbuton vod logthmc dvegence t lge On I/ the conty the poe chod dstbuton to functon scles s oz (18) f(r) R (1s -1) gves the fctl dmenson of the ntesecton of the 3D ntefce nd ndom N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 775 whee P;(R) s the vege volume fcton of the phse I mesued on the length scle R. Fom equton (2) we hve R (j) 0 d(j)) (13) ds-1 A second nteestng stuton concems bphsc medum hvng fctl dstbuton of mss s found fo nstnce n dffuson lmted ggegtes cluste-cluste stuctues [24] In P(R) F R" (15) nd P(R) (16) whee F s shpe fcto. Fo mss fctl the two exponents d nd ds e equl [23] nd we obtn fom equtons (10) nd (11) F (17) The vege mss chod length s ndependent of the scle used n the computton The In the thee fome exmples lge expnsons of chod dstbuton functons exhbt specfc popetes of the stuctul dsode. It cn be obseved tht equtons (14) nd (18) cn be dectly used on ndom secton of the mtx In ths cse the exponent such s plne. 3. Fom chod dstbuton to smll-ngle sctteng. to clsscl theoy the smll ngle sctteng I(q) s elted to the 3D Foue Accodng of the fluctuton utocoelton functon tnsfom

7 nvolvng the devtve poe nd mss chod dstbuton functons hs been poposed Ths computton s bsed on sttstcl nlyss of the medum long [12]. ndom lne nd gves fo 0 w f() + f() 2 f() * f() + unt. G() s null fo connected wth fnte dscontnuty of1 l t thee-dmensonl computton. Let us stt fom the functonl expesson poposed by Cccello et l. [5] 16l ) ) defnes two hlf spces one fo dectons pontng to the poe netwok (noted Os +) nd one fo sold ngles pontng to the mss phse (noted l ). Equton (22) cn be l wtten _ d6(*s ds +) S 776 JOURNAL DE PHYSIQUE I N 6 ) ( do(p(o) >) (p(o + ) >) (19) whee p() s the mss (the densty) dstbuton nd p the volume vege of p(). As mentoned n the ntoducton smll-ngle sctteng s stongly dependent on the popetes of the ntefce septng the poe nd the mss pt of the mtx. geometcl of the second devtve of the mss utocoelton functon quntttvely defne Popetes the level of connecton between smll-ngle sctteng nd the sttstcl popetes of ths ntefce [5]. Fo n sotopc bphsc ndom medum n expesson of ths second (G() 2 &()) (20) )" wth G() + fm() * fp() * fm() + fp() * fm() * fp() (21) The convoluton poduct s symbolzed by st. Sv s the totl ntefcl e pe volume 0. The Dc dstbuton on the hs of equton (20) s dectly 0 (I.e. fnte vlue of Sv). Ths sngul of equton (20) detemnes the q ledng tem n the hgh q expnson of the smll pt sctteng (The Pood lw) nd the lne behvou of l) t vey smll. ngle The egul pt of 1 ()" noted [1 )"] nd equl to Sv G()/4 cn be eteved usng l (22) j ds dk("s k) ds( OS k) 3(SS k)j w s postve o null. S s the totl sufce. &s s the unt vecto pependcul t the sufce S n the pont to whch the dffeentl element ds efes. &s nd &s le outsde the mss netwok. (22) s mthemtclly sgnfcnt f the boundy S s such tht tngent plne cn Equton defned lmost eveywhee except fo set of sngul ponts hvng null mesue. be ound sttstcl pont Os belongng to the ntefce the ngul vege ove ll Lookng dectons b cn be splt nto two equl pts. The tngent plne to the sufce t possble )" j F+ (os ) + F-(os )> (23) wth " SS 6) The bckets stnd fo the totl sufce vege defned by the fst ntegl n equton (22).

8 one gets ds(&s< 6) P s lengthy functon nvolvng ptl devtves of ss wth espect to o nd o nd p p we hve wth ech ntesecton of the ntefce wth lne hvng the decton &. Fo l + hlf spce the fst ntesecton fom Os nvolves poe chod the second poe chod followed by x &) + x - ( 3(Rp ) + 3(Rp m ) l 3(R s the length of the poe chod whch goes fom Os to the fst ntesecton long the R &. Relted to the second ntesecton R decton s the totl length long the decton & of the poe chod followed by mss chod nd so on. be checked fo ech system. n ths cse we hve 8(R ld&(&s n+ ) &) Fnlly we get dp do sn (o) cos (o ) duf(u) n+ I d(&s. 6) 3(R ) gf() g * f() (30) JOURNAL DE PHYSIQUE -T 2 N 6.JUNE t N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 777 Let us consde locl othogonl coodnte system specfed by the tngent plne t the pont Os (x nd y xs) nd the decton &s (the z xs). Usng stndd dffeentl geomety dodwp(o w o w) (25) nd 3(8 8 w) 3(w w w) 3(ss ) (26) 3(ss ") whee (ss< W o) nd (p o) e the sphecl coodntes of Os nd & espectvely. W" Howeve fo o P(o w o w) sgn (&s< 6)R$ sn (o). (27) The second ntegl of equton (24) gves sees of contbutons espectvely ssocted mss chod nd so on. Usng equtons (24) (26) nd (27) we get F (Os ) ld&(ds ) + (28) The next step conssts n vegng equton (28) ove ll possble postons of Thee s no smple nd genel wy of obtnng ths vege. If djcent poe nd mss Os. e stongly coelted the computton hs to tke nto ccount dstbuton functons chods two thee... consecutve chods. Followng Meng nd Tchoub [12] we hve to nvolvng specfc type of ndomness whee sttstcl sotopy of the mtx nd consde uncoelton between djcent chods e ssumed. Ths s stong hypothess whch must 2 w w/2 2 o o j 3(u ) f() (29) nd ) g( f() + f() * f() f() * f() * f() + ) (31) (F (os ))

9 778 JOURNAL DE PHYSIQUE I N 6 ( + fm() fm() * fp() + fm() * fp() * fm() (32) smll-ngle sctteng I(q). Fo n sotopc medum we hve gl ) d (34) 4 q l) beng n even functon the fome equton cn be wtten wth the genel notton " ) the stndd popetes of the Foue tnsfom of dstbutons nd knowng tht Usng s lso n even dstbuton we get fom equton (20) l)" (q) j 2 q l II fm(q) + fp(q) 2 f(q) f(q) Fnlly usng equtons (35) (37) nd (39) the smll-ngle sctteng cn be wtten s gven n equton (8) gsv d I (l /m(q)) (1 fp(q)) (40) dq q2 f(q) /p(q) I(q) nd IF- (Os )) ( fm() + fp() 2 f() * f() + l )"I + fm() * fp() * fm() + fp() * fm() * fp() (33) esult s sml to the egul pt of equton (20). Ths us now ty to connect the chod dstbuton functons to n explct expesson of the Let I(q) l (Rel ( I 2(q)) (35) + w - w() exp(q) d (36) w sv Rel (I G(q)) (37) wth G() expq The modulus of /(q) o /(q) nges between 0 nd I. Lookng t the Foue tnsfom of the hs of equton (21) we fnd convegng geometcl sees nd d(q) eds fm(q) fp(q) l The well known Debye expesson cn be ecoveed usng the chod dstbuton functons 8 d (41) (1 + f q

10 whee bsc steeology gves two genel eltons Sv " bp) connecton wth smll-ngle sctteng elly descbed by equton (40)? In othe wods cn EXPERIMENTAL PROCEDURE AND IMAGE PRocEssNG. Fo n sotopc nd bphsc 4I medum the fluctuton utocoelton functon defned n equton (19) s one of ndom lnes. These e unfomly dstbuted long dffeent ndom dectons. To mnmze pllel effect the mxml vlue of s less thn 1/3 of the vege sze of the pctue. The smll- sze sctteng I(q) s computed fom equton (35). The chod dstbuton clculton s ngle n fou steps (I) defnton of ndom decton ; (t) loclzton long ths pefomed Ths p defnes the fst end of chod () estmton of the chod segment long mss). decton chosen. Ths chod cn belong to the poe o to the mss netwok nd (v) the pxels whee the dscete ntue of the dgtl mge nduces slght tefcts. dstbutons scle wth t smll dstnces. Just fte the mxmum of ech dstbuton exponentl fom s obseved. Ths s good exmple of long-nge Debye negtve In fgue 3 we compe the dect computton of I(q) fom the mge wth the ndomness. Comng). The mtx s peped by lechng phse septed booslcte glss nd ppes N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 779 I l (42) d p nd (43) 4. Applcton to dweent types of dsodeed med. Fom the expementl pont of vew two questons cn be sed. Fst e the chod dstbutons effcent to gve specfc nfomton on the stuctul dsode? Second s the we fnd dffeent types of dsodeed stuctues whee uncoelton between djcent chods cn be ssumed? Bsed on the pesent possblty to mke quntttve connectons between technques nd smll-ngle sctteng we ctclly dscuss possble pplctons to mgng types of ndomness. dffeent the few chctestcs tht emn unchnged whethe you obseve them two- o thee- [25 26]. In the followng dgtzed sectons of dffeent poous stuctues e dmensonlly nlyzed. Dect computton of l) s pefomed long sevel bundles of numeclly decton of p of neest neghbo pxels hvng dffeent vlues (0 fo the poe I fo the of the sze hstogm by teton of the fome steps. Ths lgothm gves the computton sze dstbuton n numbes nd ws checked on bsc fgues (the cede fo exmple) chod whee nlytcl expessons e known. A good geement s obtned except fo the fst 4.2 LONG-RANGE DEBYE RANDOMNESS. Fgue I shows dgtzed secton of dolomte fom [27]. The evoluton of the poe nd the mss chod dstbutons follows two dpted egmes (see Fg. 2). As s the cse fo smooth nd cuved ntefce these successve chod dstbuton fomlsm (Eq. (40)). On the sme scle one obseves good geement. 4.3 CORRELATED DISORDER. The smll-ngle sctteng of some poous solds shows pek coespondng to the exstence of eltvely well defned coelton length. We dscuss how chod dstbuton model cn succeed o fl to pedct the sctteng popetes of these coelted stuctues. Let us fst consde the cse of poous glss (Vyco 7930 lot Tdemk

11 E *f j - w fl j t z W ". "$ o " 780 JOURNAL DE PHYSIQUE I N 6 W t" O * # w 4# o* j 4 # f f. / #.. p j I. Dgtzed pctue of thn secton of dolomte dpted fom efeence [27]. The poe netwok Fg shown n blck The hozontl b gves n pxels the numecl esoluton. s o P lo 0 200

12 o 3 o o o o o - f > t. jff.. to length of The two chod dstbuton functons e shown n coespondng 6. A specfc mode ( pek) cn be obseved fo ech of them followed by exponentl fgue tl Computton of the smll-ngle sctteng fom equton (40) s n good geement wth N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS (Q) 1 0 o ) QlPxel 3. Smll ngle sctteng of poous sold hvng Fg. ndom secton s the one shown n fgue I open ccles e the dect computton fom the mge nd the sold lne s the clculted sctteng The usng the chod dstbuton model (Eq (40)) jjl t t.. - j T hq db<g 4 Dgtzed poe netwok of vey thn secton of the poous vyco glss obtned fom Fg. electon mcogph [3] tnsmsson dect estmte usng mge pocessng (see Fg 5). Moove the chod dstbuton model

13 782 JOURNAL DE PHYSIQUE I N 6 5. Smll-ngle sctteng of the poous vyco glss. The open ccles e the dect computton Fg. the mge. The sold lne s the clculted sctteng usng the chod dstbuton model (Eq. (40)). fom o 8 10 _ je 6 10" ( " 4 10 P of the fom q.? The esoluton of the dgtzed poe netwok typclly s n fct too 4 10 co () o o The full ccles e the mesued smll-ngle X sctteng. Ths cuve s nomlzed to the clculted sctteng usng equton (40). / 11 f m exponentl tls 2 10 o o R(A) 6. Poe nd mss chod dstbuton functons clculted fom the dgtzed mge 4 (the poous Fg. glss) Full sques the poe (p) Open ccles the sold (m). vyco fts the expementl coelton pek obseved by smll-ngle X y sctteng. Howeve two dscepnces cn be obseved. The model slghtly undeestmtes the left pt of the coelton pek nd exhbts q dependnce n the hgh q egme (bove 0. I h). Fo ths mtel the expementl symptotc behvou s chctezed by powe-lw dependence

14 to hndle possble locl oughness of the ntefce [3]. Nevetheless the hypothess low the uncoelton between djcent chods ppes to be cceptble n ths concemng nteestng exmple s dsplyed n fgue 7 showng thn secton of gnul Anothe bult fom smooth nd lmost convex ptcles. The mss chod dstbuton exhbts mtel 7. Dgtzed pctue of secton of gnul mtel dpted fom efeence [31]. The poe Fg. s shown n whte. The hozontl b gves n pxels the numecl esoluton. netwok 0 o N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 783 coelted poous medum. pek ound 100 pxel followed by n exponentl tl (see Fg. 8). The poe dstbuton o P E Fg. secton of gnul poous medum. Full sques the poe (p) ; Open cles the

15 deceses fom the ogn nd vey pdly evolves s negtve exponentl contnuously Ths tend eclls some geometcl popetes of ndom pckng of dentcl hd functon. cuve coectly nd gves negtves vlues t smll q Ths exmple shows clely sctteng lmtton of the chod dstbutons n pedctng the smll-ngle sctteng of stongly the Fg. 9. Smll ngle sctteng of poous sold hvng ndom secton s tht shown n fgue 7. The ccles e the dect computton fom the mge nd the sold lne s the clculted sctteng open the chod dstbuton model (Eq. (40)). usng 30]. Ths s mss fctl hvng fctl dmenson of 1.7. The mss chod dstbuton shows utocoelton functon. As shown n fgue12 the chod model fts the functon bulk whch ws computed dectly fom the mge. Two dstnct pts cn be obseved lq) 784 JOURNAL DE PHYSIQUE I N 6 It s known tht such ndom system exhbts stong coeltons spedng on sphees. shells of coodnton. Moove the poe chod dstbuton s shown to hve n sevel fom [28 29]. In fgue 9 we compe the dect computton of I(q) fom the exponentl wth the chod dstbuton fomlsm (Eq. (40)). The chod model does not ft the mge coelted system o (Q) 1 1 o o o - o Q(Pxel) 4.4 COMPLEX STRUCTURES WITH LENGTH SCALE INVARIANCE. Let us fst consde the bdmensonl stuctue shown n fgue 10 nd known s the dffuson lmted ggegte [24 mxmum followed by n exponentl tl (see Fg. I IA). At lge dstnces the poe chod dstbuton exhbts I/q" fom wth between 1.65 nd 1.70 (see Fg. I lb). These esults e n good geement wth nlytcl expessons nd lge expnsons of chod dstbutons of mss fctl (See Sect. 2 whee d I gves the fctl dmenson of the ntesecton of 3D mss fctl nd ndom plne). The DLA shown n fgue 10 does not mtch ny bdmensonl cut of 3D mtx. In tht sense we wll focus ou ttenton on the Pood egme t hgh q unnng s q nd self sml egme t smll q n good wth the mss fctl dmenson of 1.7. geement moe complex ognzton s exhbted n fgue 13 whee the dgtzed mge of thn A secton of cement (Hydted clcum slcte) s shown [31]. At lge dstnces the mss nd chod dstbutons evolve n sml wy (See Fg. 14) nd ppoxmtely decese s poe The smll-ngle sctteng computed fom equton (40) s shown n fgue15. The I/"

16 10. Bdmensonl DLA dpted fom efeence [30]. The sold s shown n blck. The hozontl Fg gves the numecl esoluton n pxels. b l 0" N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS P 0 o f R o R(Pxel) ll A) The mss chod dstbuton functons clculted fom the dgtzed mge 10 (the 2D Fg. B) The poe chod dstbuton functons clculted fom the dgtzed mge 10. The sold lne DLA). hs slope of 17.

17 f( R) l 0 ". ". Slope l o h 12. Computton of the Foue tnsfom of the fluctuton utocoelton functon Fg. n the cse of the DLA shown n fgue 10. The full sques e the dect computton fom the () usng ld Foue tnsfom of equton (19). The sold lne s the clculted sctteng usng the mge dstbuton model (Eq. 137)). chod cuve scles s I/q. fo smll q nd exhbts Pood lw bove q 786 JOURNAL DE PHYSIQUE I N o o 100 1oo0 R(Pxel) Fg. ll (contnued). o o o 2 o" o I o I o o I o Q(Pxel) 06 pxel It cn be obseved tht the chod dstbuton model gves less nosy esult thn dect computton I(q) fom the mge Accodng to the ltetue [32 33] the smll-ngle sctteng ppes of be elted to sufce fctl hvng fctl dmenson of 2.7. An the othe hnd the to lgebc evoluton of the two chod dstbutons cn be descbed by equton (14) sml gves fctl dmenson of A dffeent wy to check the possblty of fctl whch

18 (< 13. Dgtzed pctue of cement dpted fom efeence [31]. The poe netwok s shown n Fg. The hozontl b gves the numecl esoluton n pxels whte. o m 1 - loo R(Pxel) Fg 14. Poe nd mss chod dstbuton functons clculted fom the dgtzed mge 13. Full pctue ths ntefcl egon s defned s the set of pxels belongng to the poe (the the netwok nd hvng one neest neghbo pxel nsde the mss (poe) dstbuton. Ths mss) cn be descbed by usng densty whch s one n the ntefcl egon nd 0 eveywhee set Fgue 16 shows the condtonl utocoelton functon of ths densty. Ths functon else. N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS 787 W ; ; 0 $ sques the poe (p) Open ccles the sold (m). sufce s to nlyse the popetes of the hull septng the mss nd the poe netwok On

19 15. Smll-ngle sctteng of poous sold hvng Fg. ndom secton s tht shown n fgue 12. open ccles e the dect computton fom the mge nd the sold lne s the clculted sctteng The Fg. 16. Intecl slope fctl dmenson (n 3D) of Ths esult s n esonble geement wth the othe two 788 JOURNAL DE PHYSIQUE I N 6 Q Q (). 10" o o o o o o Q(Pxel) usng the chod dstbuton model (Eq (40)). o j (R). """ R(Pxel) condtonl utocoelton functon clculted fom the dgtzed mge shown n fgue13 (see text fo defnton). the vege densty of ntefcl stes t gven dstnce of n ogn pont locted gves the ntefcl egon. At lge dstnces one obseves I/. dependence elted to nsde detemntons.

20 fome secton shows tht the nlytc fom of chod dstbuton s senstve to specfc The of stuctul dsode. A close nspecton of the medum nd long-nge behvou of the type nd poe dstbutons povdes wy to dstngush between long nge Debye mss coelted dsode nd mss o sufce fctl systems. Computton of smll- ndomness sctteng bsed on the chod dstbuton model (Eq. (40)) pples to dffeent types of ngle bny med. Obvously equton (40) fls n pedctng the sctteng of stongly ndom system whee the level of dsode s stongly educed. The exmple of poous coelted shown n fgue 4 s n nteestng ntemedte cse whee coelton nd dsode glss connecton between mgng technques nd smll-ngle sctteng s vey Quntttve lmost fo sotopc systems. Ths compson povdes wy to clfy dffeent ppelng fetues nd to check the lkelhood of the mge. In ths espect chod dstbuton sctteng e not essentl but povde enough nfomton to be vluble stuctul tool n functons nteestng dscussons wth J. M. Dke nd S. K. Snh e gtefully cknowledged. Mny DLA numecl smulton ws kndly povded by Pete Ossdnk (Ref. [30]). The DULLIEN F. A. Poous Med Flud Tnspot nd Poe Stuctue (Acdemc Pess 1976). [1] KLAFTER J nd DRAKE J M. Molecul Dynmcs n estcted Geometes (Wley Intescence [2] 1989). LEVITz P. EHRET G. SINHA S. K. nd DRAKE J. M. J. Chem. Phys. 95 (1991) [3] CICCARIELLO S. COCCO G. BENEDETTI A. nd ENzO S. Phys. Rev. B 23 (1981) [5] KJEMS J. K nd SCHOFIELD P. Sclng phenomen n dsodeed systems R. Pynn A. Skjeltop [6] MERtNG J. nd TCHOUBAR-VALLAT D. C-R- Acd Sc. Ps 262 (1966) [9] Wu H nd SCHMIDT P W. J. Appl. Cyst. 4 (1971) [10] Wu H nd SCHMIDT P. W. J. Appl. Cyst. 7 (1974) [11] MERING J. nd TCHOUBAR D. J. Appl. Cyst. 1 (1968) [12] CtCCARIELLO S. J. Appl. Cyst. 21(1988) l [13] SERRA J. Imge nlyss nd mthemtcl mophology (Acdemc Pess 1982) p [14] [15] DEBYE P. nd BUECHE A. M. J. Appl. Phys. 20 (1949) N 6 DISORDERED POROUS SOLIDS FROM CHORDS TO SAS Concluson. coexst t the mesoscopc scle. the elboton of elble nd undestndble model of dsodeed poous systems. of these dstbutons s not estcted to smll-ngle sctteng. They ply Applcton ole n some tnspot pocesses such s dect enegy tnsfe nd Knudsen dffuson centl n poous medum [ ]. In ths egd dect connecton between mgng nd smll-ngle sctteng cn lso be consdeed s n nteestng wy to get technques descpton of these chod dstbuton functons. elble Acknowledgments. Refeences AUVRAY L nd AUROY P. Neutons X-Ry nd Lght Sctteng P. Lndne nd T. Zemb Eds. [4] Scence Publshes 1991) pp (Elseve Eds. (Plenum Pess) NATO ASI sees B 123 (1985) 141. GUINIER A. nd FOURNET G. Smll ngle sctteng of X ys (John Wley & son 1955) pp [7] POROD G. Smll ngle X y sctteng Sycuse 1965 H. Bumbege Ed. (Godon nd Bech [8] Scence Publ. 1967) pp

21 DEBYE P. ANDERSON H. R. nd BRUMBERGER H. J. Appl Phys. 28 (1957) [16] CICCARIELLO S. Phys. Rev. B 28 (1983) [17] YAGLOM A. M. Intoducton to the theoy of sttony ndom functons (Dove 1962). [18] RICE S. O. Bell System Tech. J. 23 (1944) [19] MANDELBROT B. B. The fctl geomety of ntue (Feemn 1982) p [22] KOLB M. nd HERRMANN H. J. Phys. Rev. Lett. 59 (1987) [23] JULLIEN R. nd BOTET R. Aggegton nd fctl ggegtes (Wold Scentfc 1987). [24] JOSHI M. Ph. D. thess Unv. of Knss (1974). [25] QIBLIER J. A. J. Collod Intefce Sc. 98 (1984) [26] SCHWARTz L. M. Dynmc n smll confnng systems J. M. Dke J. Klfte R. Kopelmn [27] Eds. Poceedngs of symposum M Fll meetng of the MRS (1990) pp DIXMIER M. J. Phys. Fnce 39 (1978) [28] PAVLOVITCH A. JULLIEN R. nd MEAKIN P. Physc A 176 (1991) [29] BALE H. D nd SCHMIDT P. W. Phys. Rev. Lett. 53 (1984) j32] TEIXEIRA J. On gowth nd fom H. E. Stnley N. Ostoswsky Eds. (M. Njhoff Publshes j33] pp ) LEVITz P. n pepton j34] DRAKE J. M. LEVITz P. KLAFTER J. TURRO N. J. NITSCHE K. S. nd CASStDY K. F. Phys. j35] Lett. 61(1988) Rev. 790 JOURNAL DE PHYSIQUE I N 6 LEVITz P. nd TCHOUBAR D. Submtted. j20] KIRSTE R. nd POROD G. Koll. Z. Z Polym. 184 (1962) 1-7. [21] OSSADNIK P. To ppe n Physc A. j30] VAN DAMME H. Cments Bdtons Pldtes Chx 782 (1990) j31]