Thermodynamic perturbation theory for self assembling mixtures of multi - patch colloids and colloids with spherically symmetric attractions

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1 Thermdyamic erturbati thery fr self assemblig mixtures f multi - atch cllids ad cllids with sherically symmetric attractis eett D. Marshall ad Walter G. Chama Deartmet f Chemical ad imlecular Egieerig Rice Uiversity 600 S. Mai Hust, Texas bstract I this aer we exted ur revius thery [. D. Marshall ad W.G. Chama, J. Chem. Phys. 39, (03)] fr mixtures f sigle atch cllids ( cllids) ad cllids with sherically symmetric attractis (s cllids) t the case that the cllids ca have multile atches. The thery is the alied t the case f a biary mixture f bi-fuctial cllids which have a ad tye atch ad s cllids which are t attracted t ther s cllids ad are attracted t ly atch the cllids. This mixture reversibly self assembles it bth cllidal star mlecules, where the s cllid is the articulati segmet ad the cllids frm the s, ad free chais cmsed f ly cllids. It is shw that temerature, desity, cmsiti ad relative attractive stregths ca all be varied t maiulate the umber f s er cllidal star mlecule, average legth ad the fracti f chais (free chais + s) which are star s. uthr t whm crresdece shuld be addressed beettd980@gmail.cm

2 I. Itrducti I recet years there has bee a raid icrease i ur ability t sythesize atchy cllids. Patchy cllids are cllidal articles which have discrete attractive atches lcated their surface. This results i aistric rietati deedet attractive tetials betwee the cllids ad ecdes the cllids with a defied valece. The valece ad aistric tetials ca be maiulated by varyig the umber, size 3-7, shae 8, iteracti rage 9 ad relative lcati 0, f attractive atches the surface f the cllid. Thrugh the tailred desig f these atch arameters these cllids ca be rgrammed t self assemble it re-determied structures such as cllidal mlecules, the Kagme lattice 3 ad diamd structures fr htic alicatis 4 ; as well as frm fluid hases such as gels 5-7, emty liquids, 8 ad fluids which exhibit reetrat hase behavir 9. These atchy cllids have bee sythesized by glacig agle desiti 0,, the lymer swellig methd ad by stamig the cllids with atches f sigle straded DN. With the raid advace i ur sythetic ability it seems certai that, i the ear future, sythetic chemist will be able t rduce these atchy cllids t a high degree f recisi fr secific desig secificatis. Recetly, researchers sythesized mixtures f atchy ad sherically symmetric s cllids by bidig DN t the surfaces f the cllids. The cllids had a sigle sticky atch termiated with tye sigle straded DN sticky eds ad the s cllids were uifrmly cated with DN termiated with cmlemetary tye sigle straded DN sticky eds. The DN tyes were chse such that there were attractis but r attractis. That is, s cllids attract the aistric cllids, but s cllids d t attract ther s cllids ad cllids d t attract ther cllids. It was shw that this mixture wuld reversibly self

3 assemble it clusters where a sigle s cllid wuld be bded t sme umber f cllids (cllidal star mlecules csistig f s f legth e). Subsequetly, Marshall ad Chama 3 develed a erturbati thery t describe the thermdyamics ad self assembly f this mixture usig the tw desity statistical mechaics f Wertheim 4, 5. The challege f develig such a thery is the iheret higher rder ature f the attracti betwee the atches the cllids ad the s cllids. Wertheim s tw desity frmalism takes a simle frm if it ca be assumed that each atch is ly sigly bdable. This is the sigle bdig cditi. T allw fr a atch t bd times, + bdy crrelatis must be accuted fr. If the sigle bdig cditi hlds, ly air crrelatis are required. Fr the ad s cllid mixture csidered 3, the s cllid, which has sherically symmetric attractis, culd bd t a maximum f thirtee cllids. Clearly, this is well beyd the sigle bdig cditi. Thrugh the itrducti f cluster artiti fuctis, Marshall ad Chama 3 where able t devel a relatively simle thery t mdel this mixture. y cmaris t simulati data, this ew thery was shw t be very accurate fr the redicati f the reversible ad temerature deedet self assembly f the system it cllidal star mlecules ad the resultig thermdyamic rerties. The thery discussed abve is limited by the fact that the cllids ly have a sigle atch. What abut the may atch case? Fr istace, csider the case (case I) where the cllids have tw atches (a atch ad atch lcated site les f the cllid) ad the s cllids are ly attracted t the tye atch the cllid. s befre, there are attractis but r attractis. The simle additi f this extra atch the cllid sigificatly icreases the cmlexity ad richess f the behavir f the s ad cllid mixture, as cmared t the e atch case. Nw there will be a cmetiti betwee the cllids 3

4 assciatig it free chais f cllids ad the cllids bdig t the s cllids t frm cllidal star mlecules. Ulike the e atch case, w the umber distributi f cllidal star mlecules as well as legth distributi will vary with temerature, cmsiti ad desity. Sice there are attractis, tw s cllids ca be cected by a ath f assciati bds (tw s searate star mlecules cat frm a assciati bd). This leads us it case II. Case II is idetical t case I with the ly chage beig we allw attractis. Fr this case the s cllids will w act as juctis which seed etwrk frmati. Fr lw s cllid mle fracti this mixture wuld likely frm a emty liquid as the distace betwee s cllid juctis icreases. These s ad cllid mixtures rereset a ew class f materials with a ermus tetial. I this aer we devel a simle erturbati thery t mdel mixtures f this tye. We will devel the thery i Wertheim s multi desity frmalism fr multi-site assciatig fluids. 6 The aistric attractis betwee cllids will be treated i stadard first rder erturbati thery (TPT) 7-9 while the attractis betwee ad s cllids are treated i a mdificati f the results f ur revius aer 3. The cllids are allwed t have ay umber f atches ad thrughut the aer we csider a hard shere referece system. T iclude attractis betwee s cllids e simly eeds t use a arriate referece system. Oce develed, we aly the thery t case I discussed abve; case II will be csidered i a future aer. We shw that this s ad cllid mixture self assembles it a mixture f free cllidal chais csistig f ly cllids ad cllidal star mlecules which csist f a s cllid articulati segmet ad cllid s. We shw that the fracti f chais which are star s, average umber f star s er s cllid, average legth f star s, fracti f cllids bded k times ad resultig thermdyamics ca be maiulated by varyig 4

5 temerature, desity, cmsiti relative attractis. s a quatitative test, we cmare theretical redictis t ew mte carl simulati results. The thery is shw t be accurate. I secti II the ew thery will be develed fr the geeral case i which the cllid is allwed t have a arbitrary umber f atches. I secti III we secialize the thery t the tw atch case I discussed abve ad csider the lw temerature limit f this mdel. I secti IV we will discuss the simulati methdlgy used ad i secti V we study the self assembly f the case I mixture i detail. Fially, i secti VI we give cclusis ad discuss ther theretical araches t mdel mixtures f this tye. 5

6 II. Thery I this secti we derive the thery fr a tw cmet mixture f atchy ad sherically symmetric s cllids f diameter d. We chse the case f equal diameters fr tatial simlicity, extesi f the geeral arach t mixtures f differet diameters is straight frward. The cllids have a set,, C... f shrt rage attractive atches whse sizes are determied by the critical agle L L which defies the slid agle f atch L as cs c. The s cllids are thught f as a cllid with a sigle large atch f critical agle s c 80 c. diagram f these tyes f cllids ca be fud i Fig. fr the case that attractis are mediated by grafted sigle straded DN with sticky eds. This is a geeralizati f the e atch case studied theretically i ur revius aer 3 ad exerimetally by Feg et al. The tetial f iteracti betwee tw cllids is give by the sum f a hard shere (, ) tetial r ad rietati deedat attractive atchy tetial HS,, (, ) r HS () The tati r, reresets the siti r ad rietati f cllid ad r is the distace betwee the cllids. Here we fllw Ker ad Frekel 30 wh emlyed a tetial fr cical assciati sites 7, 3 t mdel the atchy attractis,,, r rc ad c ad c 0 therwise () 6

7 which states that if cllids ad are withi a distace r c f each ther ad each cllid is rieted such that the agles betwee the site rietati vectrs ad the vectr cectig the tw segmets, fr cllid ad fr cllid, are bth less tha the critical agle, fr c ad fr, the tw sites are csidered bded ad the eergy f the system is decreased c, by a factr. The iteracti betwee ad sherically symmetric s cllids is similarly defied as s, s, ( s, ) r HS (3) Where the attractive tetial is give by 3 s, s,, r rc ad c 0 therwise (4) which states that if s cllid ad cllid are withi a distace r c f each ther, ad the cllid is rieted such that the agle betwee the site rietati vectr ad the vectr cectig the tw segmets is less tha the critical agle csidered bded ad the eergy f the system is decreased by a factr cllids are assumed t iteract with hard shere reulsis ly, that is c, the tw cllids are s,. Lastly the s s, s s, s r HS (5) 7

8 We devel the thery i the multi desity frmalism f Wertheim 6, 9, 3 where each bdig state f a cllid is assiged a umber desity. The desity f secies k {, s} bded at the set f atches is give by itrduced the desity arameters k k. T aid i the reducti t irreducible grahs, Wertheim k k (6) k where the emty set is icluded i the sum. Tw table cases f Eq. (6) are k k ad k k k ; where is the ttal umber desity f secies k ad k is the desity f cllids k t bded at ay atch (mmer desity). I Wertheim s multi desity frmalism the chage i free eergy due t assciati is 6, 3, 33 give by Vk S T k k l k k Q k k c ( ) / V (7) where V is the system vlume, T is temerature ad k Q is give by Q k k k c k k k (8) The term () c is the assciative ctributi t the fudametal grah sum which ecdes all assciati attractis betwee the cllids, ad k c is btaied frm the relati 8

9 c k c ( ) / V k k (9) where i Eq. (9). The attractive fudametal grah sum () c is decmsed as c c c ( ) ( ) ( ) s (0) Where () c accuts fr the attractis betwee cllids ad () c s accuts fr the attracti betwee ad s cllids. We treat the iteracti betwee cllids i first rder erturbati thery (TPT) givig () c as 3 c ( ) / V L M, L LM f LM M () where LM L M cs cs c / 4 is the rbability that tw cllids are rieted such that atch L cllid ca bd t atch M cllid ad c r d y d 4d () c HS is the itegral f the hard shere referece cavity crrelati fucti ver the bd vlume. Lastly, the term,, / k T f LM ex LM is the magitude f the assciati Mayer fucti. The ctributi () c s cat be btaied i TPT due t the fact that we must mve beyd Wertheim s sigle bdig cditi 6 which restricts each atch t bdig ly ce. The s cllids exhibit sherical symmetry meaig they ca bd multile times. The maximum 9

10 umber f bds is simly the maximum umber f cllids max which ca ack i the s cllids bdig shell. We ca rewrite () c s as max ( ) c s c (3) where c is the ctributi fr atchy cllids bded t a sigle s cllid. We arximate c i a geeralizati Wertheim s sigle chai arximati 9, 3 ad csider all grahs csistig f a sigle assciated cluster with atchy cllids bded t a s cllid with assciati bds. We the simlify the results as described i ur revius aer 3 t btai c s / V! (4) where y HS d L s, L f L LL (5) ad is the secd rder crrecti t the first rder suersiti f the may bdy crrelati fucti fr the assciated cluster. This term is evaluated usig the brached TPT sluti f Marshall ad Chama 34 as 0

11 3 4 fr 4 3 fr (6) The term where is the ackig fracti. 35 Fially, the terms are the cluster artiti fuctis which are ideedet f temerature, desity ad cmsiti ad were evaluated i ur revius aer. 3 Nw that calculated thrugh the relati 6 () c has bee fully secified the desities f the varius bdig states ca be k k P c k (7) where P is the artiti f the set it -emty subsets. Fr examle, the desity C is give by c c c c c c c c c c C C C C C C. Sice the sherically symmetric cllid is a sigle atch, Wertheim s thery ly assigs tw desities; the desity f s cllids s s t bded ad the desity f s cllids which are bded s. The desity b s is b btaied frm Eq. (7) as s b s ( ) c / V s max s (8) s where is the desity f s cllids bded times which we idetify as s c V fr 0 (9)

12 Turig ur atteti t the cllids we te frm Eqs. (9) ad (0) that c 0 fr which results i the fllwig rule frm Eq. (7) k k k k (0) Equati (0) leads t the fllwig relati fr the fracti f cllids t bded at atch / as c () as well as the relati fr the mmer fracti () We btai c frm Eqs. (9) ad (0) as c, M M f M max s, M ( s)! y HS d f (3) Sice c 0 fr we btai frm Eq. (8) Q c (4)

13 which whe cmbied with Eq. () gives Q (5) Fr the sherically symmetric s cllids Eq. (8) simly gives Q s s s (6) We ca w write ( c ) V as / c V ( ) c max s s (7) The term s s s / is the fracti f sherically symmetric cllids bded times btaied frm Eq. (9). Cmbiig these results we btai the free eergy Nk S T x l s s s s s max x l (8) where N is the ttal umber f cllids i the system ad Usig the relati s x is the mle fracti f s cllids. max 0 s (9) ad the defiiti f the average umber f bds er s cllid 3

14 max 0 s (30) Eq. (8) ca be further simlified as Nk S T x l s s s x l (3) Where the fractis are btaied by slvig Eqs. () i cjucti with the relati s max! (3) which was btaied usig Eq. (9). With Eq. (3) we cclude this secti. The chemical tetial, ressure P ad excess iteral eergy E are all calculated i the aedix. 4

15 III. licati t atch cllids Here we csider the case, case I discussed i the itrducti, where the cllid has a tye ad tye atch where,, 0. Fr attractis betwee the ad s cllid we set s, C, (33) s, 0 where the cstat C is defied by Eq. (33). The restrictis s,, 0 will suress the frmati f a etwrk, meaig a brach emaatig frm a s cllid cat termiate ather s cllid. This situati is deicted i Fig.. This system reresets a mixture f cllids which exhibits the reversible ad temerature deedat self assembly it cllidal star mlecules ad free chais, where the s cllids are the articulati segmets fr the star mlecules ad the cllids make u the s f the star mlecules ad the segmets f the free chais. Frm Eqs. (3), (33) ad (30) c, f s (34) ad c, f (35) 5

16 Equatis (34) (35) cmbied with () give the fractis t bded f, (36) ad f, s s (37) T cmare t simulatis we will use the fracti f cllids bded k times which we k btai frm Eqs. (0) - () as (38) We will lace the cllids it tw classes. The first class csists f cllids which are art f a chai which emaates frm a s cllid, we will call these star cllids with a cllid desity. The secd class csist f cllids which are t i a bded etwrk which icludes a s cllid, we will call these free cllids with a cllid desity (te the free mmer desity is icluded i free ). Ufrtuately, the desities ad are t free directly accessible with the curret arach; hwever, the desities f free chais (icludig 6

17 mmers) chai ad star s free star s are kw. quatity which will rvide isight it the cmetiti betwee self assembled star mlecules ad free chais is the fracti s s (39) Where is the fracti f chais (free chais icludig mmers + star s) which are star s. The last quatity we wuld like t determie is the average legth f the star s L. T determie this legth (i a strict way) the desities ad wuld have t be kw. free They are t. s a alterative, a arximati f ttal desity f chais t icludig free mmers is L ca be cstructed as fllws. The. Fr the star s, the chais termiate e side with a cllid (this cllid is bded at bth atches ad ) bded t a s cllid ad the ther side with a cllid ly bded at atch. T estimate assume that the average umber f duble bded cllids i a chai (t icludig free L we will mmers i this average) is equal fr free chais (t icludig mmers) ad star s. With this we ca arximate L as fllws L (40) We will fialize this secti with a discussi f the lw T limit f these quatities. Ulike a ure cmet tw atch fluid which has the lw T limits 0 ad T 0 T 0, there will be, i geeral, a zer 7 i the curret case f a mixture f s T 0

18 cllids ad tw atch cllids. Usig Eqs. (36) (38), with 0, we btai the T 0 fllwig limitig frms fr ad s T 0 T 0 T 0 (4) s T 0 T 0 Cmbiig Eqs. (39) (4) we btai fr Ψ ad L 0 T (4) ad L s T 0 T 0 (43) These equatis state that i the limit T 0 all cllids will be assciated as star s ad there will be free chais f cllids. s cllid Equatis (4) ad (43) deed the lw T limit f the average umber f star s er T 0. Ufrtuately, there des t aear t be a simle relati fr this quatity. Frm Equatis (9), (30) ad (3) we btai 8

19 T 0 max! max! T 0 T 0 (44) Nw we slve fr T frm Eqs. (36) ad (4) as 0 T 0 4d r d c s T 0 ex, C k T (45) Equatis (44) (45) rvide a clsed sluti fr T 0, which must be evaluated umerically. Oce is btaied, the ther lw T rerties ca be evaluated thrugh the simle equatis T 0 (4) (43). Fr the case C = the exetial i (45) becmes uity ad there is temerature deedece i the lw T limit. s will be shw, fr this case, the thery reduces t a stable limitig result at temeratures which are t t lw. 9

20 IV. Simulatis T rvide a quatitative test f the ew thery we erfrm ew mte carl simulatis fr the case discussed i secti III where the ad atches are lcated site les f the cllids. We use the tetial arameters r c. d ad 7 such that ly c c sigle bdig f a cllid will ccur. Cstat NVT (umber f cllids, vlume, temerature) simulatis were erfrmed usig stadard methdlgy. 36 Each NVT simulati was allwed t equilibrate fr trial mves ad averages where take fr ather trial mves. trial mve csists f a attemted relcati f a s cllid r a attemted relcati ad rerietati f a cllid. Fr each simulati we used a ttal f N = 864 cllids. Cstat NPT (umber f cllids, ressure, temerature) simulatis were erfrmed i the same maer as the NVT simulatis with the additi f a attemted vlume chage each N trial mves. 0

21 V. Results I this secti we cmare thery ad simulati results fr the case that the cllid has tw atches as discussed i sectis III IV ad illustrated i Fig... Deedece cmsiti I this subsecti we cmare thery ad simulati fr the case that the assciati, eergy betwee atchy cllids is set t k T = 7 with the assciati eergy rati i / 3 Eq. (33) set t C =. We csider bth lw desity d 0. ad high desity 0.7 cases. Figure 3 gives the fracti f cllids bded k times, Eq. (38), versus mle fracti f s cllids s x. Fr x s 0the fluid is a ure cmet fluid f cllids with beig the dmiat fracti fr the lw desity case ad beig dmiat fr the high desity case. Itrducig s tye cllids it the system, icreasig x s, results i a decrease i ad with a icrease i. The decrease i ad icrease i is a result f the fact that lger free chais are beig sacrificed t frm star s. Sice C = there is eergetic differece betwee a bd betwee tw cllids ( bd) ad a bd betwee s ad cllids (s bd); hwever, the ealty i decreased rietatial etry fr frmig a bd is duble that which is aid fr a s bd. Thery ad simulati are i excellet agreemet. Figure 4 shws calculatis fr the average umber f s (bds) er s cllid, fracti f chais which are star s ad average legth f star s L. Fr s x 0 the s cllids are dilute. Sice there are a abudace f cllids available t slvate the s cllids, it is i this realm where is a maximum ad is a miimum. It is als i this regi

22 where s L is maximum. I fact, fr x 0, L L free where L free is the average legth f free chais. Icreasig s x results i a decrease i ad L as there is w less cllids t slvate the s cllids ad the itrducti f s cllids breaks lger chais f cllids. s exected, icreasig s x results i a icrease i as there are w mre s cllids t seed star s. s x s, there are w few cllids t slvate the s cllids ad the rbability f frmig bds becmes very small. I this limit 0, ad L. ccrdig t these results, if e desired t create a small umber f cllidal star mlecules with may lg s, this is best achieved fr small s x. O the ther had, if e wished t create a larger umber f stars with a few shrt s, this wuld be best achieved fr larger cases fr lw ad high desity we see that icreasig desity icreases, ad L. s x. Cmarig the Overall the thery ad simulati are i excellet agreemet; hwever, the thery redicts t be t small fr s x 0 at high desity. This is the result f the arximati f the + bdy cavity crrelati fuctis as discussed i r revius aer. 3 Figure 5 shws the excess iteral eergy E E Nk T ad cmressibility factr Z S / = P / k T fr these same cditis. Iitially, fr x s 0, icreasig s x results i a decrease i bth E ad Z as assciati i the system is icreased. O the ther extreme, x s, cllids are limitig ad icreasig s x decreases assciati i the system. This results i a icrease i bth E ad Z. ssciati is maximized at the lcati that E ad Z shw distict miimums, abut s x 0. fr 0. 7 ad s x 0. fr 0.. Thery ad simulati are i excellet agreemet fr E. T rvide a quatitative test fr the theretical redictis f Z we erfrmed NPT simulati fr varius s cllid mle fractis s x as a

23 fucti f. These results ca be fud i Fig. 6. s ca be see, thery ad simulati are i gd agreemet.. Deedece ε at fixed cmsiti with C = Nw we will aalyze the effect f assciati eergy fluid rerties whe cmsiti remais cstat. Secifically we will csider the case where s x ;, s, agai, we will kee the eergy f a bd equal t a s bd. We erfrm calculatis fr bth lw 0. ad high 0. 7 desity cases. Figure 7 shws the fractis f cllids bded k times versus fr this case. Fr small, the mmer fracti is the dmiat ctributi, due t the fact that the etric ealty f bd frmati utweighs the eergetic beefit fr this lw. Icreasig results i a icrease i as the eergetic beefit f frmig a sigle bd utweighs the etric ealty. t arud = 5 fr 0. ad = 3 fr 0. 7 the fracti begis t have a sigificat icrease with icreasig. This icrease i results i a maximum i ear = 7.5 fr 0. ad = 5.5 fr Evetually the eergetic beefit f beig fully bded utweighs the etric ealty ad becmes the dmiat fracti. Overall thery ad simulati are i excellet agreemet, althugh fr the desity 0. 7, the thery des slightly verredict ad uderredict. fr large Figure 8 shws calculatis fr the average umber f s (bds) er s cllid, fracti f chais which are star s ad average legth f star s L 3 ; while Fig. 9 gives

24 the results fr the excess iteral eergy E E Nk T. Icreasig, r equivaletly S / decreasig T, results i a icrease i,, L ad E as assciati i the system icreases. Overall thery ad simulati are i excellet agreemet fr each quatity at lw desity. Fr the high desity case thery ad simulati are i gd agreemet, althugh the thery uderredicts ad verredicts L fr large. The simulati results fr the fracti seem t surt the redicti give by Eq. (4), that at lw T there are few free chais ad mst cllids are assciated it star s. C. Deedece assciati eergy rati C Nw we csider the secific effect f the rati C defied by Eq. (33). Fr C < the attractis are strger tha s attractis, while fr C > the site is true. Figure 0 shws calculatis fr the average umber f s (bds) er s cllid, fracti f chais which are star s ad average legth f star s L versus at a desity f 0. ad cmsiti s x 0.05 fr ratis C = 0.6, 0.8,,.. First we fcus the case C =.. Fr this case the s attractis are greater tha attractis. Sice s attractis are favred, is large, ~ 7. 3 at = 0, ad icreases steadily as is icreased. Fr the studied cases, is a maximum at C =.. I ctrast, the average star legth L is a miimum fr this case. This is a direct result f the large. Sice there are a large umber f s ad a fiite umber f cllids, the s must be shrtest fr this case. Iitially L icreases with as assciati icreases i the system, ges thrugh a maximum ad the decreases as bds are traded fr s bds. We als te that is a maximum fr this case. 4

25 Nw csiderig the case C = 0.6, where attractis are greater tha s attractis, we see that iitially icreases with, ges thrugh a maximum as s bds are traded fr bds ad the decreases as. It is fr this C that is a miimum, ~ at ~ 0, ad L is a maximum, L ~ 4. 5 at ~ 0. Icreasig further t ~ 0 we fid ~ ad L ~ 550, meaig the s cllids exist rimarily as mmers ad chai eds fr lg chais f cllids. Fr the case C = 0.8 we see similar behavir t the case C = 0.6, althugh less ruced. Fially csiderig the case C =, we see that there are maximums ad bth L ad reach a lw temerature limitig value arud ~. Frm these results it is clear that the attracti rati C ca be tued t achieve a rage f cllidal star mlecules. I the t ael f Fig. 0 we iclude the lw T limit f btaied thrugh Eqs. (44) (45). Fr C =,. the lw T limit is reached ear ~ with the lw T limit givig reasable redictis fr 0. Whe C the lw T limit is attaied at a higher. It shuld be exected that the larger the rati C 0, the wider the rage f alicability f the lw T limit. D. Lw T limit fr the case C = s a last case, we csider the lw T limits f the cllid bdig fractis ad, average umber f s (bds) er s cllid ad average legth f star s L as a fucti f s cllid mle fracti x s. We use the limits develed i secti III. These results are reseted i Fig.. We csider the case that 0. ad C =; hwever, the lw T results are very weakly deedet desity. I the limit f a ure fluid f cllids we see 5

26 T 0 0 ad T 0. I this limit eergy dmiates ad there is a sigle ifiitely lg chai f cllids. s we itrduce s cllids, icreasig s x, sme f the cllids assciate it star s which breaks aart this ifiitely lg chai. This results i a decrease i T 0 ad icrease i T 0. This tred ctiues util becmes uity ad T 0 T 0 vaishes as x s ad iteractis betwee cllids becme rare. The average umber f s er s cllid is at a maximum max 3 T 0 T 0 whe x s 0ad all cllids ca be assciated it a sigle cllidal star. Iitially, icreasig T 0. Fr istace, icreasig s x slightly t s s x results i a raid decrease i x ~ 0.0 results i a decrease f the average umber f star s er s cllid t ~ This behavir is als see i the term L T 0 T 0 which is ifiitely large fr s x 0 ad decreases t ~ 3 at s x ~ 0.0. These limitig results rvide guidace the tyes f cllidal stars which will frm fr a give mle fracti s x. Csiderig the results f Fig. 8, we see that this system had early reached it s lw T limit at the reduced temeraturet / , ad the lw T limit rvides a reasable estimate f the average umbers ad legths fr eve higher temeratures. Thrughut this aer, fr cveiece f resetati, we have ly csidered the average umber f bds er s cllid. Hwever, the fracti f s cllids bded times (with star s) s is readily available thrugh Eq. (9). We shw the distributi f these fractis i the lw T limit, with C =, fr three s cllid mle fractis i Fig.. Fr the very dilute case s 5 x 0 the distributi is sharly eaked ad asymmetric with the ly tw sigificat fractis beig fr = ad =. Icreasig s cllid mle fracti t 6 s x = 0.0

27 the distributi is symmetric ad eaked at = 7 with sigificat ctributis betwee = 5 ad = 9. Lastly, fr the equimlar case s x = 0.5, the distributi is highly asymmetric with the dmiat ctributis cmig frm = 0 ad ad sigificat ctributis frm = ad = 3. iterestig feature f this system is that eve i the lw T limit where eergy dmiates there is still a etric ctributi which stems frm the umber f times a give s cllid ca bd. 7

28 VI. Cclusis We have develed a ew thery t mdel biary mixtures f multi atch cllids ad cllids with sherically symmetric attractis (s cllids). We develed the thery i Wertheim s 6, 3 multi desity frmalism fr assciatig fluids usig mdificatis f the grahs develed i ur revius aer 3 fr the case f a mixture f sigle atch cllids ad s cllids. We alied the thery t the case f a mixture f bi-fuctial cllids, csistig f a atch ad atch lcated site les f the cllid, ad s cllids which ly attract the atch f the cllid. There were ly attractis betwee the atches the cllid, ad there were attractis betwee s cllids. This system was shw t self assemble it a mixture f free chais ad cllidal star mlecules. The average legth f star mlecules, rati f free chais t star s ad average umber f s er cllidal star ca be maiulated by varyig desity, temerature, cmsiti ad the rati f assciati eergies. The thery was shw t be accurate i cmaris t mte carl simulati data. We will csider the case II discussed i the itrducti i a future aer. I the develmet f the thery we assumed there were attractis betwee s cllids which allwed us t write the thery as a erturbati t a hard shere referece fluid. T iclude attracti betwee the s cllids e wuld eed t use a arriate referece system fr the s cllids. esides the defiiti f the referece system, the geeral results reseted here wuld still be valid. The erturbati thery develed i this aer gives a simle ad accurate thery fr the redicti f the self assembled structures ad thermdyamic rerties f mixtures f s ad cllids. Hwever, there is ifrmati abut fluid structure f i the thery. Fr this we eed t g t itegral equati theries. Iterestigly, the mdel reseted i this wrk shares 8

29 may similarities with the thery f highly asymmetric electrlyte slutis which are cmsed f large highly charged lyis ad small cuter is with lwer charge. Like the case csidered i this wrk, the umber f times the lyi (s cllids) ca bd is urestricted ad the cuter is ( cllids) are restricted t bd a maximum f k times. Multi desity itegral equati theries 37-39, which draw isirati frm Wertheim s itegral equati thery sluti 40, 4 f the Smith Nezbeda 4 mdel f assciatig fluids, have rve t be accurate i mdelig highly asymmetric slutis. Extesi f these mdels such that the cuteri ( cllid) has a atchy rietati deedet tetial (istead f sherically symmetric as is the case fr iic iteractis) with multile atches will allw fr the redicti f the structure f s ad cllid mixtures f the tye studied i this aer. 43 ckwledgmets The fiacial surt f The Rbert. Welch Fudati Grat N. C 4 is gratefully ackwledged. The authrs wuld like t thak Y. V. Kalyuzhyi fr useful discussi. 9

30 edix: Calculati f thermdyamic quatities eergy I this aedix we calculate the chemical tetial, ressure P ad excess iteral S E. The simlest way t calculate the chemical tetial is t use the Euler Lagrage equati sulied by Wertheim 6 k k T k HS k T l k c k / V () where k is the hard shere referece chemical tetial fr cmet k. Frm Eq. () we HS btai fr the s cllids s k T s HS l k T s s l y HS s d max s l y d s l s s HS () ad fr the cllids HS k T k T s l y HS d l max s s l l y d HS (3) With the chemical tetials kw the ressure is easily calculated thrugh the relati P j j j / V (4) 30

31 3 Lastly we btai the excess iteral eergy as (5) Secifically fr the bi fuctial atch case csidered i III with C = we have the derivatives (6) Where f ), ( ad s max 0. The remaiig derivatives are the calculated as (7) where f l l ), (. s s s s S S x x N N E l l ), ( x x x x f s s s s l s l

32 Refereces:. G.-R. Yi, D. J. Pie ad S. Sacaa, Jural f Physics: Cdesed Matter 5 (9), 930 (03).. E. iachi, J. Larg, P. Tartaglia, E. Zaccarelli ad F. Scirti, Physical review letters 97 (6), 6830 (006). 3.. Giacmetti, F. Lad, J. Larg, G. Pastre ad F. Scirti, The Jural f Chemical Physics 3, 740 (00). 4. Y. Kalyuzhyi, H. Dcherty ad P. Cummigs, The Jural f Chemical Physics 33, (00). 5. Y. Kalyuzhyi, H. Dcherty ad P. Cummigs, The Jural f Chemical Physics 35, 0450 (0). 6.. D. Marshall, D. allal ad W. G. Chama, The Jural f Chemical Physics 37 (0), (0). 7.. D. Marshall ad W. G. Chama, The Jural f Chemical Physics 38, (03). 8. F. Rma ad F. Scirti, Nature Cmmuicatis 3, 975 (0). 9. F. Rma, E. Saz ad F. Scirti, The Jural f Chemical Physics 3, 8450 (00). 0.. D. Marshall ad W. G. Chama, Physical Review E 87, (03)... D. Marshall ad W. G. Chama, The Jural f Chemical Physics 39, (03).. Y. Wag, Y. Wag, D. R. reed, V. N. Mahara, L. Feg,. D. Hlligswrth, M. Weck ad D. J. Pie, Nature 49 (74), 5-55 (0). 3. Q. Che, S. C. ae ad S. Graick, Nature 469 (7330), (0). 4. Z. Zhag,. S. Keys, T. Che ad S. C. Gltzer, Lagmuir (5), (005). 5. D. de Las Heras, J. M. Tavares ad M. M. T. da Gama, Sft Matter 7 (), (0). 6. D. de Las Heras, J. M. Tavares ad M. M. T. da Gama, Sft Matter 8 (6), (0). 7. F. Scirti, The Eurea Physical Jural 64 (3-4), (008). 8. D. de las Heras, J. M. Tavares ad M. M. T. da Gama, The Jural f Chemical Physics 34, (0). 9. J. Russ, J. Tavares, P. Teixeira, M. M. T. da Gama ad F. Scirti, The Jural f Chemical Physics 35, (0) Pawar ad I. Kretzschmar, Lagmuir 4 (), (008).... Pawar ad I. Kretzschmar, Lagmuir 5 (6), (009).. L. Feg, R. Dreyfus, R. Sha, N. C. Seema ad P. M. Chaiki, dvaced Materials 5 (0), (03). 3.. D. Marshall ad W. G. Chama, The Jural f Chemical Physics 39, (03). 4. M. Wertheim, Jural f Statistical Physics 35 (), 9-34 (984). 5. M. Wertheim, Jural f Statistical Physics 35 (), (984). 6. M. Wertheim, Jural f Statistical Physics 4 (3), (986). 7. W. G. Chama, G. Jacks ad K. E. Gubbis, Mlecular Physics 65 (5), (988). 3

33 8. G. Jacks, W. G. Chama ad K. E. Gubbis, Mlecular Physics 65 (), -3 (988). 9. M. Wertheim, The Jural f Chemical Physics 87, 733 (987). 30. N. Ker ad D. Frekel, The Jural f Chemical Physics 8, 988 (003). 3. W. l, Mlecular Physics 45, 605 (98). 3. M. Wertheim, Jural f Statistical Physics 4 (3), (986). 33. W. G. Chama, PhD. Thesis. 988, Crell Uiversity: Ithaca, NY D. Marshall ad W. G. Chama, The Jural f Chemical Physics 38, 7409 (03). 35. S. Pha, E. Kierlik, M. Rsiberg, H. Yu ad G. Stell, The Jural f Chemical Physics 99, 536 (993). 36. D. Frekel ad. Smit, Uderstadig mlecular simulati: frm algrithms t alicatis. (cademic ress, 00). 37. Y. V. Kalyuzhyi, M. Hlvk ad V. Vlachy, Jural f Statistical Physics 00 (-), (000). 38. Y. V. Kalyuzhyi ad V. Vlachy, Chemical hysics letters 5 (5), 58-5 (993). 39. Y. V. Kalyuzhyi, V. Vlachy, M. Hlvk ad G. Stell, The Jural f Chemical Physics 0, 5770 (995). 40. Y. V. Kalyuzhyi ad I. Nezbeda, Mlecular Physics 73 (3), (99). 4. M. Wertheim, The Jural f Chemical Physics 88 (), 45 (988). 4. W. R. Smith ad I. Nezbeda, The Jural f Chemical Physics 8, 3694 (984). 43. Persal cmmuicati with Y. V. Kalyuzhyi, (03). 33

34 Figures: Figure : Illustrati f sherically symmetric ad tw atch cllids 34

35 Figure : Examles f assciated clusters which ca be btaied frm a s ad cllid mixture whe the cllids have tw atches ad ad,, s, 0 35

36 Figure 3: Fractis f cllids bded k times versus mle fracti f s cllids desities 0. (t) ad 0. 7 (bttm) ad a assciati eergy / T = s x at (, ) k T = 7 with C = s,, / =. Curves give theretical redictis ad symbls give / simulati results: (slid curve thery, squares simulati), (shrt dashed curve thery, circles simulati), (lg dashed curve thery, crsses simulati) 36

37 Figure 4: verage umber f s (bds) er s cllid (t), fracti f chais which are star s (middle) ad average legth f star s cllids s L (bttm) versus mle fracti s x at 0. (dashed curve thery, circles simulati) ad 0. 7 (slid curve thery, squares simulati). ssciati eergy is (, ) = T = 7 with C = s,, / = / k 37

38 Figure 5: Same as Fig. 4 excet the excess iteral eergy E E Nk T is the deedet variable (t) ad cmressibility factr Z (bttm) S / 38

39 Figure 6: Cmressibility factr Z versus desity at a assciati eergy 7 with C = s,, / =. Curves give theretical redictis ad symbls give NPT (, ) T = / k ( s) ( s) simulati results fr x (lg dashed curve thery, triagles simulati), x ( s) (slid curve thery, crsses simulati), x (shrt dashed curve thery, circles ( s) simulati) ad x 0. 6(dashed dtted curve thery, squares simulati) 39

40 Figure 7: Fractis f cllids bded k times versus assciati eergy desities 0. (t) ad 0. 7 (bttm) ad a s cllid mle fracti = x (, ) T at s / k with C = s,, / =. Curves give theretical redictis ad symbls give simulati results: (slid curve thery, squares simulati), (dtted curve thery, circles simulati), (lg dashed curve thery, crsses simulati) 40

41 Figure 8: verage umber f s (bds) er s cllid (t), fracti f chais which are star s (middle) ad average legth f star s 4 L (bttm) versus assciati eergy (, ) = T at 0. (dashed curve thery, circles simulati) ad 0. 7 (slid curve / k thery, squares simulati). Mle fracti f s cllids is s x with C = s,, / =

42 Figure 9: Same as Fig. 8 excet the deedat variable is w excess iteral eergy E S / Nk T E = 4

43 Figure 0: verage umber f s (bds) er s cllid (t), fracti f chais which are star s (middle) ad average legth f star s L (bttm) versus assciati eergy (, ) = T. Desity is ( s) 0., with a s cllid mle fracti x fr varius values / k f assciati eergy betwee the ad s cllids s,. Red curves begiig at C, 5 t ael give lw T limit f thrugh Eqs. (44) (45) 43

44 Figure : Lw temerature limits f the fracti f cllids bded e r tw times (t), average umber f s er s cllid (middle) ad average legth f star s (bttm) versus mle fracti f s cllids. ssciati eergy rati is C = ad the desity is fixed at 0. 44

45 Figure : Lw T limit f fractis s at a desity f 0. ad C = s,, / =. Symbls give theretical redictis ad lies are simly meat t guide the eye 45