SCIENCE CHINA Chemistry. Signatures of shear thinning-thickening transition in steady shear flows of dense non-brownian yield stress systems

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1 SCIENCE CHINA Chmstry ARICES do: /s do: /s Sgnaturs of shar thnnng-thknng transton n stady shar flows of dns non-brownan yld strss systms Wn Zhng, Yu Sh & Nng Xu * CAS Ky aboratory of Soft Mattr Chmstry; Hf Natonal aboratory for Physal Sns at th Mrosal; Dpartmnt of Physs, Unvrsty of Sn and hnology of Chna, Hf 3006, Chna Rvd Sptmbr 9, 014; aptd Otobr 3, 014 Stady shar flows of dns athrmal systms omposd of soft dsks ar nvstgatd va non-qulbrum molular dynams smulatons, from whh w sort out lnks among th strutur, dynams, and shar rhology. h systms at rst ar jammd pakngs of frtonlss dsks wth a nonzro yld strss. Drvn by low shar rats, th flows shar thn du to th prsn of th nonzro yld strss, but transt to shar thknng abov a rossovr shar rat. At, w obsrv th strongst strutural ansotropy n th par dstrbuton funton, whh srvs as th strutural sgnatur of th shar thnnng-thknng transton. W also obsrv dynamal sgnaturs assoatd wth th transton: At, salng bhavors of both th man squard dsplamnt and rlaxaton tm undrgo apparnt hangs. By prformng a smpl nrgy analyss, w rval an undrlyng ondton for th shar thknng to our: dln /dln g wth g th knt tmpratur. hs ondton s onfrmd by smulatons. shar thknng, shar thnnng, strutural ansotropy, dynams, knt tmpratur 1 Introduton *Corrspondng author (mal: nngxu@ust.du.n) Complx fluds suh as ollods, gls, mulsons, foams, and granular matrals xhbt ntrgung and omplatd rhologal phnomna whn thy ar subjt to shar. Unlk Nwtonan fluds, whos shar vsosty s ndpndnt of th shar rat, omplx fluds an undrgo shar thnnng or thknng undr propr ondtons; ths ar haratrzd by th dras or nras of th vsosty as th shar rat nrass [1 5]. Shar thnnng and thknng ar attratv rsarh tops wth mportant applatons [,6]. In prat, w always wsh pants to shar thn, whras shar thknng s dsrd n th dsgn and manufatur of smart matrals suh as soft body armor. h undrlyng mhansms of shar thnnng and thknng hav bn dbatd for dads. Early smulatons suggstd that th layrng of partls along th drton of th shar flow ontrbuts to shar thnnng [7,8]; suh rsults hav bn susptd as artfal of profl-basd thrmostats [9 1] and hav bn hallngd by most rnt studs [13,14]. A rnt masur of mrosop snglpartl dynams has proposd that shar thnnng rsults from th dras of th ntrop for ontrbuton [3]. Shar thnnng an also b boostd by th prsn of yld strss whh mpds th mrgn of shar thknng [15]. Our undrstandng of shar thknng s vn poorr than our undrstandng of shar thnnng; ths s largly baus fwr systms xhbt shar thknng, whh usually happns at hgh shar rats whr probng s dffult. Shar thknng has bn obsrvd n both Brownan and non-brownan suspnsons [3,15,16 18] and n smulatons wth and wthout takng hydrodynams nto aount [,8, Sn Chna Prss and Sprngr-Vrlag Brln Hdlbrg 015 hm.shna.om lnk.sprngr.om

2 Zhng t al. S Chna Chm Aprl (015) Vol.58 No ,19]. Multpl mhansms (.g., th ordr-dsordr transton and formaton of hydrolustrs) [,8,0] hav bn proposd to xplan shar thknng. hs mhansms an orrtly prdt th rtal shar strss or shar rat of th onst of shar thknng. Howvr, thy fal to xplan th ourrn of dsontnuous shar thknng (DS), whh s an xtrm shar thknng phnomnon n whh th shar strss jumps dsontnuously wth th shar rat; t usually happns at hgh pakng fratons [5]. Rnt studs that hav unvld a possbl lnk btwn DS and jammng of onsttunt partls du to dlaton [5,16 18,1 3] hav proposd frton as th nssary lmnt [4,5]. All of ths xplanatons, whh ar basd on dffrnt physs, hav thr strngths and waknsss. Whthr shar thknng an or annot orgnat from a unvrsal mhansm for varous systms rmans a st of opn qustons. W studd th rhology of planar shar flows of dns non-brownan systms wthout th ntrfrn of hydrodynams va non-qulbrum molular dynams smulatons. h systm at rst s a jammd pakng of frtonlss sphrs wth a nonzro yld strss. Du to th prsn of th yld strss, th systm shar thns nstad of bng Nwtonan at low shar rats [15]. Abov a rossovr shar rat th systm shar thkns. h xstn of nabls us to xplor th strutural and dynamal sgnaturs assoatd wth th shar thnnng-thknng transton; to our knowldg, nformaton about ths sgnaturs s stll lakng n th ltratur. W obsrvd that th shar thnnng-thknng transton was aompand by th strongst strutural ansotropy and apparnt hangs n th salng of dynamal quantts. Furthrmor, w found that n ordr for th shar thknng to our th nras rat of th knt tmpratur g wth rspt to th shar rat, dln g /dln, must xd, basd on th nrgy balan btwn nput and dsspaton. hs ondton was onfrmd by smulatons and was not lmtd to th modl systm dsussd hr. Smulaton dtals Our systms ar two-dmnsonal squars onsstng of dsks wth an dntal mass m. Half of th dsks hav a damtr th othr half hav 1.4. s-edwards boundary ondtons [6] wr appld wth th shar bng mposd n th x drton and shar gradnt n th y drton. SOD quatons of moton assumng a lnar vloty profl [7] wr mployd: dr ˆ yx (1) dt d 1 F ˆ y x () dt m j whr r x, y and x, y ar th loaton and random vloty of partl, d / d t s th shar rat wth th shar stran, F s th for atng on partl by partl j, and th sum s ovrall partls j ntratng wth partl. h for F nluds two parts, th last for F V and dampng for F ˆ m y x [8], whr V,, and y ar th ntraton potntal, rlatv random vloty, and y dstan btwn partls and j that ar n ontat wth ah othr, and s th dampng offnt. Wks-Chandlr-Andrson (WCA) potntal (.., rpulsv nnard-jons potntal) s appld: 1 6 V 1 whn th sparaton btwn partls and j, r 7 r r, s smallr than th sum of thr rad, and zro othrws. W usd, m, and as th unts of lngth, mass, and nrgy. h unts of tmpratur and tm ar / kb and m /, rsptvly, whr k B s th Boltzmann onstant. W appld Gar prdtor-orrtor algorthm to ntgrat Eqs. (1) and () at a onstant pakng fraton 0.85 abov th = 0 jammng transton at 0.84 [9 3]. h systms at rst ar jammd solds obtand from -BFGS nrgy mnmzaton ( northwstrn.du/~nodal/lbfgs.html). Data wr olltd and avragd ovr tm aftr th systms wr shard for a muh longr tm than th haratrst tm sal 1/ and stady shar flows wthout th mmory of thr ntal stats wr ahvd. W guarant that th obsrvaton tm was suffntly long, so that th statsts ar good nough and th rsults do not hang quanttatvly vn whn a longr tm s appld. W hav also vrfd that our rsults do not dpnd on ntal ondtons. h shar strss Σ s alulatd from whr d x N N1 N m 1 x y xf y 1 1 j1 d (3) and x ar th dal gas strss and xss strss from partl ntratons. Manwhl, th shar vsosty an b alulatd from: d x d x (4) W vard th numbr of partls N from 56 to 4096 to

3 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 3 vrfy that our rsults do not show sgnfant systm sz dpndn. Hr w only show rsults for Strutural and dynamal sgnaturs Fgur 1 shows an xampl of th flow urv (.., th shar-rat dpndn of th shar strss or vsosty) alulatd at 0.1. h shar strss drasd whn drasng th shar rat and approahd a platau at low shar rats, ndatng that th shar strss n th 0 lmt (.., th yld strss) was nonzro, as markd by th arrow n Fgur 1. At low shar rats, th flow urv ould b wll fttd wth th Hrshl-Bukly formula, y A, whr y s th yld strss, and A and ar fttng paramtrs. For th as shown n Fgur 1, 0.53, onsstnt wth th 1/ obsrvd n ollodal xprmnts [15]. Du to th xstn of th yld strss, 1 1 / y A y / at low shar rats. Apparntly th vsosty drass as th shar rat s nrasd (Fgur 1) untl a rossovr shar rat s ahvd; hghr than ths, th Hrshl-Bukly formula no longr works and th vsosty nrass wth shar rat nstad. hrfor, marks th transton from th shar thnnng flow to th shar thknng flow. As wll b dsussd blow, our shar thknng flows ar assoatd wth th hgh-tmpratur and hgh-prssur gaslk stats ndud by vry hgh shar rats, whh ar n th sam rgm as prvous smulatons [9 1] but n a dffrnt rgm from rntly rportd DS flows that aros from th unjammng-jammng transton at low shar rats [16 18]. o hk f our rsults wr du to th s-edwards boundary ondtons, w studd systms onfnd btwn two rough walls at onstant volum wth th top wall movng n th x drton at a onstant spd and obsrvd smlar rsults. It s thus ntrstng to ask f any xprmntally assbl sgnaturs ar assoatd wth th shar thnnngthknng transton obsrvd hr. Fgur (a ) shows th snapshots of th shar flows at dffrnt shar rats. h systm undrgos apparnt strutural hangs from shar thnnng to shar thknng. In th shar-thnnng rgm (Fgur (a)), th systm s roughly Fgur 1 Shar-rat dpndn of th shar strss and vsosty wth th dampng offnt =0.1. h dashd ln s th fttng to th low shar-rat flow urv wth th Hrshl-Bukly formula: h arrow ponts to th yld strss 4 y Fgur (a ) Snapshots of shar flows at =0.1 and 0.01, 0.1, and 1 wth 0.1; (d f) par dstrbuton funton g(x, y) for systms shown n (a ). h olor bar quantfs th valu of g(x, y).

4 4 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 unform n spa; whras n th shar-thknng rgm (Fgur ()), partls tnd to form nstantanous lustrs wth larg ovrlaps and lav bhnd mor vods. W wr thrfor nsprd to sarh for possbl strutural sgnaturs of th shar thnnng-thknng transton. Rnt studs hav attmptd to brdg shar rhology and mrostrutur [13,33 35], although t has also bn argud that thr s no nssary onnton btwn mrostrutur and th ourrn of shar thknng [5]. o haratrz strutur, w alulatd th par dstrbuton g r g x, y funton for larg partls, r x, y r, wth spal attnton to ts N / j ansotropy. h sums ar ovr all larg partls and. dnots th tm avrag. h ontour plots of gr shown n Fgur (d f) ndat that s asymmtr gr n th vnty of th shar thnnng-thknng transton; th ontours ar longatd n th drton of y=x and omprssd n th prpndular drton. In ordr to quantfy th asymmtry, w show n th nsts of Fgur 3 th par dstrbuton funtons paralll and prpndular to y=x, rsptvly dnotd as g (r) and g r g r s hghr than that. h frst pak of of g () r. Morovr, r s r s, whr r s and r s dnot th lngths at whh g r and g () r start to b gratr than zro. hs faturs mply that whn th strutur s asymmtr, partls tnd to hav largr and mor unform ovrlaps n th drton prpndular to y x. W thrfor dfn r s / rs to haratrz th asymmtry. Fgur 3 shows that wth th nras of th shar rat frst nrass and thn drops aftr a maxmum. h maxmum mrgs approxmatly at. h shar thnnng- thknng transton s thus sgnfd by th most pronound strutural ansotropy. Strutural hangs may also ndu dynamal hangs at th shar thnnng-thknng transton. o aptur ths possbl hangs, w alulatd th y omponnt of th man squard dsplamnt (MSD) for larg partls, y, whr. dnots th partl and nsmbl avrag. As shown n th nst of Fgur 4(a), thr s a ballst moton at short tms ( y ~t ) followd by dffuson at long tms ( y ~t ). Apparntly, th rlaxaton tm drass as th shar rat nrass. Hr, s dtrmnd from y () ~10. Whn w plottd th MSDs aganst t/, as shown n Fgur 4(a), th MSDs n th shar thnnng rgm ollapsd nly onto th sam urv and th ballst parts of th shar thknng urvs rmand apart. h salng ol- gr Fgur 3 Ansotropy of valuatd by at varous shar rats for shar flows at =0.1. Inst: Par dstrbuton funtons paralll and prpndular to y=x, g r and g r, at 0.01 (sold), 0.1 (dashd), and 1 (dot-dashd). Fgur 4 (a) Man squard dsplamnt y n th shar-thnnng (magnta sold symbols) and -thknng (grn mpty symbols) rgms at =0.1. h blak rls ar at. h shar rat nrass from th rght to th lft n th nst. In th man panl, t s sald by th rlaxaton tm ; (b) shar-rat dpndn of th rlaxaton tm at =0.1. h sold lns show powr law bhavors n th shar-thnnng and -thknng rgms.

5 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 5 laps of th shar-thnnng urvs mpls that partls mov ballstally to approxmatly th sam dstan bfor dffusng. In th shar-thknng rgm, howvr, partls mov ballstally to a longr dstan as th shar rat s nrasd. As shown n Fgur (), partls form nstantanous lustrs n shar-thknng flows and lav many vods. If lustrd partls mov olltvly at short tms, th vods allow thm to mov ballstally to a longr dstan, a pross that may dpt th short-tm dynams of shar-thknng flows. Furthr analyss of nstantanous lustrs s nssary to vrfy our onjtur and gan a bttr undrstandng of shar thknng. h dstnt dynams btwn shar-thnnng and -thknng flows an also b dntfd from th shar-rat dpndn of th rlaxaton tm. As shown n Fgur 4(b), 0.8 n th shar-thnnng rgm, n agrmnt wth a rnt xprmnt [36], whras.5 n th sharthknng rgm. h dstnt shar-rat salng of th rlaxaton tm, togthr wth th brakdown of th salng ollaps of th short-tm MSDs, thus srv as dynamal sgnaturs of th shar thnnng-thknng transton. 4 Domposton of th shar vsosty hus far w hav rvald th strutural and dynamal sgnaturs of th shar thnnng-thknng transton for our modl systms, whh ar assbl to xprmnts on ollodal suspnsons or granular matrals. Nxt w show anothr undrlyng ondton for shar thknng to our, whh an b obtand from th domposton of th shar vsosty basd on th onsdraton of th nrgy balan. In a stady shar flow, nrgy njtd nto th flow by th shar for s dsspatd by th dampng for, from whh w obtan for our modl: mv (5), j whr th sum s ovr all ntratng partl pars. For th modl systms studd hr, smpl alulatons of Eq. (5) lad to th ddy vsosty [9]. whr (6) C 1 0 m NZ ( ) k (7) b j B g, j C m j, j, j (8) m 1 1 y x (9) m 0 y (10), j On th rght-hand sd of Eq. (7), z b s th avrag oordnaton numbr and N m g dnots th knt NkB 1 or granular tmpratur (.., fftv tmpratur at short tm sals) [8,37]. Whn ntrodung g n Eq. (7), w N 1 assum that th random part of th knt nrgy, m, 1 s qually parttond to all partls. h domposton of th shar vsosty by Eqs. (4) and (6) nabls us to fnd th atual sour of th shar strss n ontrol of th rhology. Eq. (4) s th mrosop xprsson of th vsosty from vral thorm, whras Eq. (6) s straghtforwardly drvd from th nrgy balan. In homognous stady flows, w would xpt that. Fgur 5(a) shows flow urvs at four dffrnt dampng offnts rangng from 0.1 to 1, whh all xhbt th shar thnnng-thknng transton at th dampng offnt dpndnt rossovr shar rat. Whn salng x y by, w not that ~. Salng ollaps at hgh shar rats n th shar thknng rgm s also obsrvd whn / s plottd aganst /, whh approxmatly obys a powr law salng. hs salng s dffrnt from th Bagnoldan bhavor ( ) [16 18,38], whh arss from rgd partl ontats. Emprally, shar thknng wth th hang of th vsosty wth rspt to th shar rat fastr than Bagnoldan an b tratd as DS [5]. Apparntly, th DS obsrvd n our modl systms should rsult from soft partl ntratons and s onsstnt wth th nrasd partl ovrlaps at hgh shar rats as dsussd abov, whh mak t dffrnt from th DS ndud by th jammng transton of frtonal partls. As ompard n Fgur 5(a), as xptd, xpt n th vnty of whr. tnds to nras wth. W vrfd that >0 s not a transnt bhavor by obsrvng no tm voluton. In th shar-rat rgm whr s apparntly gratr than 0, th knts show vsbl ansotropy (.g., ), whh s onsstnt wth th mrgn of th strutur ansotropy n th sam shar-rat rgm dsussd abov. h ansotropy should thus aount for th nonzro. Although and ar not ompltly qual, thy xhbt th sam ; thrfor, ths nqualty dos not afft our rsptv dsussons about th shar thnnng-thknng transton usng Eqs. (4) and (6). In Fgur 5(b), w ompar d wth x. At low shar rats, th random moton of partls s so slow that

6 6 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 Fgur 5 (a) Vsosty alulatd from Eq. (4) aganst th shar rat at =0.1 (blak rls), 0. (rd squars), 0.5 (blu damonds), and 1.0 (grn d trangls). h sold urvs ar ddy vsosty alulatd from Eq. (6); (b) shar-rat dpndn of th dal gas vsosty (blu squars) and xss vsosty (rd rls) at=0.1. h blak urv s th total vsosty; () shar-rat dpndn of th four omponnts of th ddy vsosty, x C (rd rls), (blu squars), 1 (grn damonds), and 0 (magnta trangls) at =0.1. h sold urv s th total ddy vsosty; (d) sharrat dpndn of th rat of granular tmpratur nras at th sam dampng offnts as (a). Inst of (d): omparson of and for a thrmostatd m systm at y 0.1. Nk B x >> d d. Wth nrasng shar rat, grows and vntually bats. d = x at a rossovr shar rat g x. Whn g, th systm bhavs fftvly as a hgh-tmpratur and hgh-prssur gas, wth 1 and 1 as th domnant tm sals; ths may b th aus of th hgh shar-rat salng ollaps shown n Fgur 5(a). x also xhbts th shar thnnng-thknng transton laggng slghtly bhnd. h shar thnnng- thknng transton obsrvd hr s thus dtrmnd by th omptton btwn th shar thnnng-thknng transton n x and th growth rat of d wth th shar rat. In ontrast to Eq. (4), th domposton of th shar vsosty usng Eq. (6) s mor usful to rval an undrlyng thrmodynam-lk ondton for shar thknng to our. In Fgur 5(), w show all th four vsosty omponnts n Eq. (6). In th whol shar-rat rgm studd hr, and ar small and nglgbl, 0, C C and 0. At low shar rats, s smallr than but omparabl to, and shows smlar shar-rat dpndn to. Nar, howvr, boms sgnfantly largr than C. Shar thknng s thrfor manly dtrmnd by. Assumng that th avrag oordnaton numbr z b dos not vary largly wth th shar rat, whh s atually tru for our systms, Eq. (7) mpls that f th knt tmpratur g vars mor stply than, shar thknng would our. In athrmal shar flows wth a onstant, g nrass wth th shar rat. o stmat th rat of th g nras, d lng w dfnd a quantty and plottd t aganst dln th shar rat (Fgur 5(d)). At low shar rats, 1.5, 1.5 whh ndats that g. hs salng law stops workng nar ; at ths pont, whn and, shar thknng happns xatly as xptd from our analyss of Eq. (6).

7 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 7 Havng stablshd a drt lnk btwn th knt tmpratur and shar thknng for our modl systms, w thn askd f ths lnk was spf to our systms. Frst of all, w hav vrfd that systms wth harmon, Hrtzan, and nnard-jons ntratons am to th sam onluson. Nxt, w trd to fnd out f > was also th ondton of shar thknng n othr typal modl systms. Othr than our athrmal modl, typal modls for th study of shar flows nlud th foam modl [39 41], angvn dynams [4,43], and thrmostatd shar flows [7,9 1,37]. For th foam modl dsrbd by th quaton of moton, F, whr s quvalnt to m n our j modl, smpl alulatons rsultd n. hus w an mmdatly s that shar thknng happns d ln whn, quvalnt to > xpt that partl dln nrta s gnord. Whn nrta s nludd, > an b straghtforwardly obtand as wll. For angvn dynams, a random for R t should b addd to th quaton of moton for th foam modl. Baus and R ar unorrlatd [43], w would xpt th sam ondton for shar thknng to our. For thrmostatd shar flows, w alrady knw that whn g s fxd shar thknng annot happn [7], whthr or not t s an artfat of thrmostat [6]. Evn wth mor ralst thrmostat, t has bn rportd that g s muh hghr than th bath tmpratur [6]. hs obsrvatons may ndat that shar thknng s also rlatd to th knt tmpratur n thrmostatd systms. In th nst of Fgur 5(d), w show both and aganst th shar rat for a thrmostatd shar flow govrnd by d 1 F ˆ ˆ y x y y, whr mantans a dt m j onstant knt nrgy n th y drton [37] but lavs th knt nrgy n th x drton and g unonstrand. For ths modl, nrgy balan rsultd n, y from whh > annot b rognzd. Intrstngly, > stll sgnfs shar thknng for th as shown hr. In ths papr, w manly onntrat on th athrmal modl dsrbd by Eqs. (1) and (). In ordr to vrfy whthr > s a robust ondton of shar thknng for thrmostattd systms, ntnsv studs at varous bath tmpraturs and dnsts and wth dffrnt ntraton potntals and thrmostats would b rqurd. For foam modl and angvn dynams, > s a straghtforward onsqun of th nrgy balan rqurd for shar thknng to our. For our athrmal modl, ths onsqun s not obvous baus th prondton s that s sgnfantly largr than th othr thr trms n Eq. (6), whh turns out to b tru. For thrmostatd shar flows, > s totally unprdtabl. Our rsults and analyss suggst that th ourrn of shar thknng s strongly oupld wth th bhavor of th knt nrgy, rgardlss of modls. hrfor, > may b a robust thrmodynam-lk ondton for shar thknng to our. 5 Conlusons In stady shar flows of dns non-brownan systms onsstng of soft partls, w obsrvd th shar thnnngthknng transton. hs transton was sgnfd by th strongst strutural ansotropy and apparnt hangs n th salng of dynamal quantts suh as man squard dsplamnt and rlaxaton tm. h shar-thknng flows obsrvd hr ar hgh-tmpratur and hgh-prssur gaslk stats wth larg partl ovrlaps, whh should b mpossbl n hard-sphr systms that prohbt partl ovrlaps. Not that our systms ar jammd solds at rst, whras n most of th ollodal xprmnts th pakng fraton s blow, th rtal pakng fraton of th jammng transton. hrfor, th shar thknng obsrvd hr must hav bn ndud by a dstnt mhansm from th unjammng-jammng transton proposd for unjammd systms [5]. Baus w dd not nlud hydrodynams n our modl, th lubraton for [] s not ssntal thr. Furthrmor, jammd solds ar dsordrd, whh may alrady xlud th ordr-dsordr transton [0] as th undrlyng aus of shar thknng. Although th shar thknng obsrvd hr annot b smply xpland by any of th thr popular thors dsussd abov, t may b too arly to prolam th xstn of nw mhansms of shar thknng basd smply on our obsrvatons. Aftr all, our systms ar n dffrnt dnsty rgms from thos that ar ommonly onrnd. Whthr shar thknng an happn n xprmntal jammd systms omposd of dformabl partls suh as PNIPAM ollodal systms s thus th ky to vrfyng our major fndngs. For shar thknng to b vdnt, th partls should b dformabl wth hard ors, n ordr to aus partl ovrlap, and should also supply suffntly larg onfnng strss to nsur vsbl shar thknng [5]. Jammd solds hav nonzro yld strss, whh boosts shar thnnng at low shar rats and mpds th ourrn of shar thknng. In shar flows of dlut ollodal suspnsons, a Nwtonan rgm an somtms b obsrvd aftr shar thnnng [,44], whh was unsurprsngly absnt n our systms. Whn th yld strss s turnd on for unjammd systms (for nstan, by addng attraton) and s small, a Nwtonan rgm an our aftr shar thnnng rgm at low shar rats. h Nwtonan rgm shrnks as th strngth of th attraton nrass and vntually ds-

8 8 Zhng t al. S Chna Chm Aprl (015) Vol.58 No.4 appars whn th yld strss s so larg that th sharthnnng rgm drtly nountrs th shar-hknng rgm. What w hav obsrvd n our modl systms should thrfor mrg, although th pakng fraton s low; ths hypothss, of ours, rqurs furthr nvstgaton. Furthrmor, by applyng a smpl analyss of th shar vsosty basd on th nrgy balan, w fnd an undrlyng thrmodynam-lk ondton for shar thknng to our: dln /dln g. W suggst that ths ondton may b robust to varous modl systms, whh stll nds to b vrfd spally for systms n whh th ondton annot b straghtforwardly drvd. In any as, ths ondton may provd us wth a possbl rfrn to manpulat th ourrn of shar thknng by tunng th proprts of onsttunt partls. hs work was supportd by th Natonal Natural Sn Foundaton of Chna (135418, ), th Natonal Bas Rsarh Program of Chna (01CB81500), th CAS 100-alnt Program ( ), and th Fundamntal Rsarh Funds for th Cntral Unvrsts ( , , ). 1 Vrmant J, Solomon MJ. Flow-ndud strutur n ollodal suspnsons. 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