A perturbation density functional theory for the competition between inter and intramolecular association

Transcription

1 petubatin density functinal they f the cpetitin between inte and intalecula assciatin Bennett D. Chapan a leand J. Gacía-Cuélla b and Walte G. Chapan a a Depatent f Cheical and Bilecula Engineeing Rice Univesity 600 S. Main Hustn Texas b Depatent f Mechanical Engineeing Tecnlógic de Mnteey v. Eugeni Gaza Sada 50 Mnteey N.L Méxic bstact Using the faewk f Wethei s thedynaic petubatin they we develp the fist density functinal they which accunts f intalecula assciatin in chain lecules. T test the they new Mnte Cal siulatins ae pefed at a fluid slid inteface f a 4 segent chain which can bth inta and inteleculaly assciate. The they and siulatin esults ae fund t be in excellent ageeent. It is shwn that the inclusin f intalecula assciatin can have pfund effects n intefacial ppeties such as intefacial tensin and the patitin cefficient. Keywds Density functinal they Wethei s they inhgeneus fluids cplex fluids statistical echanics lecula siulatin uth t wh cespndence shuld be addessed Eail: bennettd980@gail.c

2 Intductin: Hydgen bnding assciatin plays an integal le in u eveyday lives. F the eakable ppeties f wate t the flding f pteins hydgen bnding is key t u vey existence. Mdeling assciating fluids is cplicated by highly diectinal asyetic inteactins; f this easn the develpent f accuate statistical echanical based theies f assciating fluids lagged behind that f siple fluids with spheically syetic ptentials. In the 980 s Wethei 3-7 develped a they capable f accuately descibing assciating fluids by intducing the highly diectinal inteactins at an ealy pint in the they. By intducing a ulti-density falis whee each bnding state f a lecule is teated as a distinct species Wethei was able t ewite the statistical echanics f assciating fluids in a f which was vey aiable t appxiatin. One such appxiatin Wethei s thedynaic petubatin they TPT has pven eakably successful. In TPT the change in fee enegy due t assciatin is btained as a petubatin t a had sphee efeence fluid. TPT is typically used as a fist de petubatin they TPT and pvides a basis f the SFT 8 9 equatin f state; SFT has fund widespead use in bth industy and acadeia. 0 One key appxiatin intduced in TPT is the neglect f all gaphs with ings f assciatin bnds. F st systes this appxiatin will intduce a sall nnexistent e hweve lecules such as glycl ethes shw a significant degee f intalecula assciatin which affects the thedynaics f the syste. T accunt f the pssibility f intalecula assciatin Sea and Jacksn dified TPT by adding a ing gaph t the fundaental gaph su. In a sepaate appach Ghnasgi and Chapan 3 4 develped a they t accunt f intalecula assciatin; thei they was fund t be in excellent ageeent with lecula siulatins.

3 In additin t hgeneus systes TPT has pven vesatile and accuate in the desciptin f inhgeneus systes. 5 By letting the assciatin enegy bece infinitely lage cplex plyatic lecules can be cnstucted allwing the develpent f plye density functinal theies 6- DFT s in the faewk f TPT. In additin if se assciatin enegies ae allwed t eain finite DFT s capable f descibing assciating plyatic lecules can be develped. -4 In these assciating DFT s the pssibility f intalecula assciatin has been neglected. If we ae t develp an accuate DFT f the desciptin f intefacial systes invlving glycl ethes t accuately descibe ptein flding the pssibility f intalecula assciatin ust be accunted f. In this wk we will develp a DFT capable f descibing lecules which can bth inta and inteleculaly assciate. We will fllw Sea and Jacksn and intduce a ing gaph in the fundaental gaph su t accunt f intalecula assciatin. With this fee enegy functinal we will cnstuct and iniize a gand ptential which will allw us t btain the inhgeneus density pfiles. s a test f the they we pef new Mnte Cal siulatins f a 4-e chain which can bth inta and inteleculaly assciate nea a had wall. The they is shwn t be in excellent ageeent with siulatin. We shw that intefacial ppeties such as intefacial tensin and the patitin cefficient ae stngly affected by intalecula assciatin. 3

4 II: They In this sectin we will intduce the type f lecules we want t study the ptential del and develp the Helhltz fee enegy and segent densities. Hee we will cnside linea fully flexible lecules f length cnsisting f had sphees segents whee each lcatin n the chain is ccupied by a cetain species f segent. Each segent has tw assciatin sites ed and B geen as shwn in Fig.. The inteactin ptential between segents β and γ is given as the su f a had sphee and assciatin ptential HS B B The ntatin epesents the psitin and ientatin f a spheical segent is the distance between the segents and is the had sphee ptential HS 0 HS whee site 9 is the css species diaete. The assciatin ptential B is that f a cnical i i B i c ; i c; B c B i 0 thewise 3 Whee i is the angle between the vect f the cente f segent i t site and the vect i c is the cutff angle beynd which assciatin is nt allwed and c is the cutff adius which is the axiu sepaatin between tw segents whee assciatin can ccu. Thee is 4

5 i i n assciatin between sites f the sae type that is 0. T ceate the chain the BB liit f cplete assciatin is taken i B f all assciatin bnds intenal t the chain while leaving the assciatin enegy between the assciatin site n segent and the B assciatin site n segent finite and adustable. s illustated in Fig. bth intelecula and intalecula assciatin is allwed. In Wethei s they each bnding state f a lecule is teated as a distinct species. The density f species β bnded at a set f sites α at lcatin in the fluid is. F the site fluid the ttal density f cpnent β will be the su f the segents which ae bnded at bth sites and B thse bnded at sites B and thse which ae nt bnded 4 B B whee is the ne density. We will als use a set f density paaetes 5 B B In TPT the change in fee enegy due t assciatin is Wethei ln Q d c 6 Equatin 6 is witten f lecules with fixed bnd angles between assciatin sites. T allw f distibutins f bnd angles bnd angles α ae teated as intenal vaiables and bnd angle 5

6 distibutin functins can be intduced. 7 In the fully flexible liit nn adacent segents n the chain can velap and / at which pint Eq. 6 is ecveed. Intducing the bnd angle celatin functins will nt change the f f the esults s f ntatinal siplicity we will nt use this fality. The f f the final equatins is valid f fully and sei flexible chains ne siply needs t enfce any bnd angle cnstaints. F a tw site fluid Q is given as Q B B 7 The fundaental gaph su c f this type f lecule can be witten as the su f cntibutins f chain fatin ing fatin and intelecula assciatin plyeizatin c c c c chain ing ply 8 Whee c chain and c ply ae given by 7 c chain B d d 9 ply c d d 0 ply B 6

7 i i i The te y F whee i y is the inhgeneus cavity i i i celatin functin f the efeence syste and F exp f cntibutin due t intalecula assciatin is given by Sea and Jacksn s ing gaph HS B. The c ing ~ Ring... d Whee ~ Ring ~... Chain... ing and ~ Chain We iniize the fee enegy with espect t ne densities t btain Ring B ~... d 4 Nw iniizing with espect t the and B f chain fing assciatin sites B B d 5 7

8 d 6 F site n segent and site B n segent ply d 7 B B ply d 8 Using Eqns. 6 thugh 8 the Helhltz fee enegy can be witten as Wethei ln B d B d 9 Equatin 9 was iginally deived by Sea and Jacksn in the develpent f a bulk equatin f state hweve Eq. 9 is geneal f inhgeneus systes. Witing Eq. 4 f segent B I ing 0 Whee 8

9 Ring ~ I... d ing Cbining Eqns. 7 and 0 B I ing ply d Dividing each side f Eq. by we btain X ply ing X 3 Whee X is the factin f cpnent nt bnded at site X B / siilaly f site B n segent X /. The ing factin is the factin f species which is bnded intaleculaly t species B ing ing I ing 4 and X ply is the factin f cpnent nt bnded at site if nly intelecula assciatin wee pssible X ply X B ply d 5 and the equivalent factin f site B n segent 9

10 X ply B X ply d 6 Nw we wish t take the liit f cplete assciatin f all chain fing sites. In the liit f cplete assciatin f chain fing sites ne densities bece sall s the secnd te n the left hand side f Eqns. 5 and 6 can be neglected. Equatins 5 and 6 ae nw i used t ecusively eliinate the density paaetes and sites in Eq. 4. The esulting segental densities ae i B assciated with chain fing ~ X Ring ~ ply... chain... X d ply B d 7 k We will find it necessay t eply a tw pint chain density k ; t btain this quantity we fist nte that the densities in Eq. 7 can be btained thugh functinal deivatives f a geneating functinal ~ c ~ c 8 The functinal ~ c is given by c~ X ~ ~ ply Ring chain X ply B d d 9 0

11 Using the geneating functinal ~ c we can als btain the tw pint chain density as 30 Evaluating Eq Cpaing Eqns. 7 and 3 it is clea that 3 T cplete the they we ust btain equatins f the unknwn s. T deteine these densities density functinal they will be eplyed. In density functinal they we define a gand ptential at fixed cheical ptential μ vlue V and tepeatue T subect t an extenal field V ext 33 Whee ~ k c k k k k k... ~... ~ d d X X k k k Ring k ply B chain ply k k k d k k }] [{ }] [{ V d ext

12 d 34 is the density integated ve all ientatins; thee is a siila elatin f. Miniizatin f the gand ptential with espect t the segent densities yields the set f Eule Lagange equatins. [{ }] V ext 35 The slutin f this set f equatins will yield the needed ne densities. The Helhltz fee enegy is given as [{ }] [{ }] [{ }] [{ id Wethei HS }] 36 id Wethei HS whee [{ }] [{ }] and [{ }] ae the cntibutins f ideal gas and excess cntibutins due t chain fatin / assciatin Eq.9 and had sphee epulsins. The functinal deivatives f the ideal gas te is knwn exactly ideal [{ }] ln 37 F the had sphee te we use Rsenfeld s fundaental easue they 5 HS [{ ex hs d n }] [{ }] 38

13 3 T btain the cntibutin due t }] [{ Wethei we take the functinal deivative f Eq. 6 and enfce the liit f cplete assciatin f chain fing sites we btain 39 whee 40 The intelecula assciatin stength ply is given as 6 4 The tw pint density ing is given by 4 We can elate the tw pint ing density t the ttal segent density and ing factin 43 ln Wethei c d Ring ing... ~ ing ing d ing ing ply ply B d d d d X X d d y c ln ln ln g f B B ply

14 4 Using Eqns we can slve f the ne densities 44 whee 45 Using Eq. 44 t eliinate the ne densities in Eq. 7 we btain 46 Whee k k M is the lecula cheical ptential I Ring is given by 47 F fully flexible chains the chain integal I chain is facted and evaluated with ecusin elatins 48 exp ext HS V c d I Ring Ring ] exp[... ~ ] exp[ Ring chain M I I chain I I I

15 The I ' s ae evaluated using the fllwing ecusin elatins I X ply 49 and f > I exp I d 50 i i Hee y i i i 4 i i whee is the infinitely lage agnitude f the chain fing Maye functins as will be seen this te cancels exactly with an identical te in the cheical ptential. Siilaly f the I ' s I X ply B 5 and f < I exp I d 5 ll that eains is the appxiatin f the efeence syste celatin functins. Kielik and Rsinbeg 7 appxiated the inhgeneus had sphee pai celatin functin as a fist de functinal Tayl Seies in density aund the hgeneus esult. If we tk this path we wuld fist be equied t slve f the tw pint density Eq. 3 and then integate thugh Eq. 5

16 6 3 t btain the segent densities. siple appach 0 8 which has pven t yield accuate esults is t appxiate the efeence pai cavity celatin functin as the aveage f the ptential f ean fce 53 Whee the y ae evaluated by using the bulk esult at an aveage density 54 Using Eqns and assuing a syetic lecule we can ewite Eq. 40 as 55 Equatin 55 cpletes the density functinal they f the cpetitin between inte and intalecula assciatin in assciating chain fluids. Unftunately the ing integal in Eq. 47 is ieducible and cannt be factized. F lage flexible ings diect nueical evaluatin f this integal by quadatue will be cputatinally ipactical. One pssible eslutin wuld be t evaluate the ing integal by single chain Mnte Cal siulatin. 3 3 This will be the subect f ln ln y y y d ln ln d y X d y c

17 a futue study and will nt be discussed futhe hee. The appendix gives a detailed discussin f ethds t evaluate the ing integal. F the systes studied in the pape the Calculatinal ethd is as fllws. bulk density ρ is specified and the bulk X is calculated by 3 X 3 ply ing ply ply 0 X X 56 Using this X the bulk ply X can be calculated using Eq. 5 which allws f the calculatin f the bulk thugh Eq. 3. We can slve f the excess cntibutin t the cheical ing ptential due t chain fatin and assciatin Wehei ; the esult f a hnuclea lecule is Wehei ln X ln X ln y ln y X ln ln ply ln y 57 whee is the ttal segent density. The last te in Eq. 57 cntaining the assciatin stength f chain fing bnds Λ gives an infinite cntibutin hweve this te cancels exactly with the Λ s cntained in the chain fing May functins. Nw Eqns. 46 f the density pfiles and an additinal equatin f the ing factin ing I ing I I ing chain 58 ae slved using a Picad iteatin whee the initial guess f the density and ing factin pfiles ae the bulk values at each pint in the dain. 7

18 III: Siulatin T test the they DFT calculatins and lecula siulatins will be cpaed f the classical case f a fluid in a plana slit pe f width H with walls lcated in the xy plane subect t the extenal ptential V ext z 0 if z 0 z H thewise 59 We will use the lecula del f Ghnasgi and Chapan 3 wh cnsideed 4 tangentially bnded had sphee segents with assciatin sites lcated n the fist and futh segents. The assciatin sites ae aanged such that the vect f the cente f the assciating segent t the assciatin site is always at a 90 angle t the vect which pints f the cente f the assciating segent t the cente f the neighbing segent n the chain see Fig. 3. The chain lecules inteact with the ptential given by Eq. with the cutff paaetes chsen as c. and c 7. Mlecula siulatins ae pefed in the NVT enseble using the geneal ethd descibed in ef [3]. ttal f 87 chain lecules wee siulated in a bx with tw had walls n ppsite sides. F the the fu sides peidic bunday cnditins wee applied. Maxiu displaceent and angle change paaetes ae adusted in each siulatin un t allw f an veall 30-40% ate f acceptance. The siulatins whee caied ut f 0 6 cycles whee a cycle cnsists f an attept t displace and eient all lecules nce. The esults f the density pfiles and bnding factins wee btained afte the lecula cnfiguatins wee sufficiently equilibated. The syste was said t be equilibated nce the 8

19 factins f cpnent bnded intaleculaly z and inteleculaly z had achieved steady values thughut the pe. t the high assciatin enegies / kt 7 and 8 and at a packing factin f η = 0.3 z and z did nt stabilize sufficiently thughut the ing entie pe ve the length f u siulatins. inte ing inte 9

20 IV: Results In this sectin we cpae density functinal they DFT calculatins t the Mnte Cal MC siulatins discussed in sectin III f the case f an assciating 4 e chain nea a had wall. When the 4-e chain self assciates int a ing the sht ange epulsins f the lecule will keep the assciated ing in a nealy plana cnfiguatin. Hence t a gd appxiatin we can appxiate the ing integal as that f a plana ing; see the appendix f appxiatin f I ing f this case. ll calculatins pefed in this sectin ae f the case ε inte = ε inta = ε and all density pfiles ae scaled by the aveage density f a segent in the pe ρ ave. Thee ae tw segent types in this lecule; the end segents with assciatin sites will be called type and the iddle segents will be called type. Figue 4 cpaes MC and DFT density pfile calculatins at an aveage packing factin in the pe f η =0.. t this lw density bth end and iddle segents ae depleted f the wall due t a lss f cnfiguatinal entpy nea wall cntact with the wall cntact value f the end segent always lage than that f the iddle segent. s assciatin enegy inceases the density f segent in cntact with the wall eains appxiately cnstant while that f segent deceases. The decease in the wall density f segent is the esult f a lss f cnfiguatins whee this segent can be nea the wall when assciatin int ings lnge es ccus. Figue 5 cpaes MC and DFT density pfile calculatins f an aveage pe factin η = 0.. Like the η = 0. case inceasing the assciatin enegy esults in a decease in the density wall cntact value f segent. The they is vey accuate in pedicting the density pfile f the assciating segent while it is less accuate f the iddle segents. In geneal these types f petubatin density functinal theies will be st accuate f end type segents due t the fact that the density pfile f an end segent is clse t that f the efeence had sphee fluid than that f a iddle segent. 7 0

21 Figue 6 gives density pfiles f η = 0.3. t this packing factin had sphee packing effects esult in an enhanceent in density at wall cntact. s the assciatin enegy is inceased t 6kT the density cntact value f segent deceases while that f segent tw eains cnstant. In additin t density pfiles we can als calculate the factin f segent type bnded assciated intaleculay ing and the factin f segent type bnded inteleculaly χ inte ; Fig. 7 cpaes DFT and MC calculatins f these quantities f aveage syste packing factins f η = 0. and 0.. In geneal the factins bnded intaleculay shw a axiu aund z = σ and appxiately btain thee bulk value at wall cntact while the factins bnded inteleculaly shw a steady decease as the wall is appached. Intelecula assciatin is hindeed nea wall cntact due t the fact that thee ae less ways that tw chains can psitin and ient theselves such that assciatin ay ccu. The situatin f intalecula assciatin is quite diffeent. The degee f intalecula assciatin depends n the pbability that the tw ends f the chain ae psitined such that assciatin can ccu. t wall cntact appxiately half f the chain cnfiguatins which can lead t intalecula assciatin in the bulk will be available hweve when segent is in cntact with the wall nly half as any chain cnfiguatins in ttal as cpaed t the bulk will be available; hence the ati f these quantities at wall cntact shuld appxiately yield the bulk esult giving a cntact value f ing nealy that f the bulk fluid. The MC and DFT pedictins ae in excellent ageeent. The thedynaics f the syste depends n the factin f cpnent nt bnded X z. Figue 8 cpaes MC and DFT slid lines calculatins f X z at packing factins f η = and 0.3. F cpaisn we have included DFT calculatins dashed

22 lines whee the pssibility f intalecula assciatin was neglected. We see that the cuent DFT is in excellent ageeent with siulatin while DFT s which d nt include the pssibility f intalecula assciatin unde pedict the aunt f assciatin in the syste. F systes whee intalecula assciatin can ccu the cuent DFT is clealy supei t pevius vesins f DFT. With the cuent DFT we can study hw the cpetitin between inte and intalecula assciatin affects patitining at a slid / fluid inteface. Figue 9 pesents patitin cefficients at packing factins f η = 0. and 0.. t η = 0. when nly intelecula assciatin is cnsideed the patitin cefficient cntinually deceases as assciatin enegy is inceased T deceased due t the fact that the chain lecules ae assciating int lnge es which excludes assciated clustes f the wall. Hweve when intalecula assciatin is accunted f we see a iniu in the patitin cefficient nea ε / kt = whee the patitin cefficient begins t incease with assciatin enegy. The iniu in the patitin cefficient esults f the fact that at lw densities and high assciatin enegies lw T intalecula assciatin dinates 3 ; beaking intelecula assciatin bnds t f intalecula bnds esults in salle assciated clustes which can e easily appach the wall esulting in an incease in the patitin cefficient. t a packing factin f η = 0. this iniu disappeas. Inceasing density futhe t η = 0.3 des nt change the qualitative dependence f the patitin cefficient n assciatin enegy bseved in the η = 0. case. Inteestingly the esults in Figue 9 lk vey siila t the cpessibility facts calculated by MC siulatins by Ghnasgi and Chapan 3 ; they studied the bulk behavi f this syste. The link between the patitin cefficient and the bulk cpessibility fact is the

23 wall cntact thee which states that the bulk pessue is equal t the wall cntact value f the density P kt 4 i z 0 i 60 Using Eq. 6 we calculated the cpessibility fact Z P and cpaed the esults t kt the siulatins f Ghnasgi and Chapan 3 at a bulk packing factin Fig. 0. We see that the MC and DFT calculatins ae in gd ageeent. The iniu in the cpessibility fact esults f tading intelecula assciatin bnds f intalecula assciatin bnds. This esults in salle clustes f assciated 4 es and a cespnding incease in the cpessibility fact. ls iptant in any applicatins is the intefacial tensin γ f the slid / fluid inteface whee the intefacial tensin is calculated as the suface excess gand ptential pe aea f inteface bulk bulk 6 Figue pesents DFT calculatins f at packing factins f η = 0. and 0.. t η = 0. inceasing assciatin enegy deceasing T esults in an incease in γ at all enegies cnsideed. This incease in γ esults f attactins between the lecules becing e significant s e enegy is equied t sepaate the lecules t f the inteface; the lwe γ btained when intalecula assciatin is accunted f stes f the fact that lecules which ae assciated int ings have n attactins t the the lecules in the syste. t η = 0. thee is 3

24 still a cntinues incease in γ as assciatin enegy is inceased when intalecula assciatin is neglected hweve when intalecula assciatin is accunted f thee is a axiu nea ε / kt = 8 and then γ begins t decease. This behavi is analgus t that bseved in the patitin cefficient hweve the axiu in γ is lcated at a lwe enegy than the iniu K suggesting that the intefacial tensin is e affected by ing fatin than the patitin cefficient. Inceasing density futhe t η = 0.3 des nt change the qualitative dependence f γ n assciatin enegy bseved in the η = 0. case. 4

25 Cnclusins: We have develped the fist density functinal they f chain lecules capable f intalecula and intelecula assciatin. s a test we pefed NVT Mnte Cal siulatins f a 4 e in a slit pe. The they was shwn t be in excellent ageeent with siulatin esults. It was shwn that inclusin f intalecula assciatin can esult in dastic qualitative changes t ppeties such as intefacial tensin and the patitin cefficient; this behavi cannt be captued with pevius vesins f DFT. cknwledgeents The financial suppt f this wk was pvided by the Rbet. Welch Fundatin Gant N. C-4 and by the Natinal Science Fundatin CBET

26 Refeences:. Z. Feng. Byaste C. Ebsky D. Ballal B. Mashall K. Gng. Gacia K. R. Cx and W. G. Chapan Junal f Statistical Physics C. N. Pace B.. Shiley M. McNutt and K. Gaiwala The FSEB unal M. Wethei Junal f Statistical Physics M. Wethei Junal f Statistical Physics M. Wethei Junal f Statistical Physics M. Wethei Junal f Statistical Physics M. Wethei The Junal f Cheical Physics W. Chapan K. Gubbins G. Jacksn and M. Radsz Fluid Phase Equilibia W. G. Chapan G. Jacksn and K. E. Gubbins Mlecula Physics E.. Mülle and K. E. Gubbins Industial & engineeing cheisty eseach R. L. Binkley and R. B. Gupta Industial & engineeing cheisty eseach R. Sea and G. Jacksn Physical Review E D. Ghnasgi and W. G. Chapan The Junal f Cheical Physics D. Ghnasgi V. Peez and W. G. Chapan The Junal f Cheical Physics C. P. Ebsky Z. Feng K. R. Cx and W. G. Chapan Fluid Phase Equilibia D. Ca and J. Wu The Junal f Cheical Physics S. Jain. Dinik and W. G. Chapan The Junal f Cheical Physics E. Kielik and M. Rsinbeg The Junal f Cheical Physics B. Mashall and W. G. Chapan The Junal f Physical Cheisty B S. Tipathi and W. G. Chapan The Junal f Cheical Physics Y. Yu and J. Wu The Junal f Cheical Physics Byaste and W. Chapan The Junal f Physical Cheisty B G. J. Gl G. Jacksn F. J. Blas E. M. Del Rí and E. de Miguel The Junal f Cheical Physics Malievský P. Byk and S. Sk wski Physical Review E Y. Rsenfeld Physical eview lettes Byaste and W. G. Chapan The Junal f Physical Cheisty B E. Kielik and M. Rsinbeg The Junal f Cheical Physics S. Jain. Dinik and W. Chapan The Junal f Cheical Physics Byaste S. Jain and W. Chapan The Junal f Cheical Physics

27 30. S. Jain P. Jg J. Weinhld R. Sivastava and W. Chapan The Junal f Cheical Physics P. Byk and L. G. MacDwell The Junal f Cheical Physics D. Ca T. Jiang and J. Wu The Junal f Cheical Physics

28 ppendix: Calculatin f I z ing In this appendix ethds t evaluate the ing integal I z in plana D systes will be discussed. We begin with the intalecula assciatin stength aveaged ve segent ientatins ing ing inta inta f B g C B y F Whee 0 c thewise and C is a nalizatin fact. When evaluating the ing integal I z we ae essentially cunting the nube f cnfiguatins the ing can take with segent at z and the ing lcated in the field ceated by the the lecules in the fluid and the extenal ptential; f each ing ing cnfiguatin the intalecula assciatin stength cntls if segents and lcated at lcatins and in the fluid assciate t f a ing. s witten Eq. 47 is f a feely inted ing whee nn adacent segents alng the ing can velap. F plana systes with inhgeneities in the z diectin the density is a functin f z nly and we can ewite the ing integal as ing 8

29 inta I Ring z C BF Ding z... z z... z exp[ z ] dz 3 The functin z...z is a puely geetic quantity given by z... z... i d i 4 z...z can be efeenced t the lcatin f a segent such that it is independent f the abslute z psitin in the pe. The integal D ing z... z is the pduct f cavity celatin functins. F flexible 4 e chains the hgeneus z...z is pefed nce and sted f use. ing is knwn 3 ing F inta D 5 Whee D and η is the bulk packing factin. Nalizing the ing integal t this hgeneus esult we find the cnstant C f a 4 segent chain C B y DV dz... dz 4 z... z 4 7 Whee V is vlue and y is the bulk cavity celatin functin. 9

30 Nw as a test we cpae DFT calculatins t MC siulatins f ing factins f a fluid which can nly intaleculaly assciate; Fig. shws these esults. The they and siulatin ae in fai ageeent. The they pedicts gd ing factin cntact values hweve it unde pedicts the ing peak lcated nea z = σ and des nt captue the dips nea z = σ. These deficiencies aise f the fully flexible teatent f the ing integal. The fully flexible teatent shuld be sufficient f lage ings hweve the self aviding assciated 4 e ing is sufficiently igid that the fully flexible teatent f the ing integal will incu e. One slutin is t evaluate bth the ing and chain integals such that n inta lecule segent velap is allwed self aviding. n altenative slutin which is cputatinally siple and faste than the self aviding case is t teat the ing integal as igid with segents and 4 bnded at cntact. That is igid z... z i d i 8 Evaluating the ing integal this way will unde pedict the nube f lecula cnfiguatins that can lead t ing fatin due t the fact that the actual ing has flexibility and assciatin ccus within a shell f thickness c ; t cect f this fact we siply include the pbability W in segents can assciate ing f the pbability that the chain is in a cnfiguatin whee the tw end ing inta 4 C B y 4 F W 9 30

31 Whee W is the pbability that in a syste with chains f length = 4 and bulk density ρ that if we anch segent at a psitin that segent 4 will be in a psitin whee intalecula assciatin can ccu; that is 4 and 4 c. We will appxiate this quantity as W 4 chain ef 4 4 c 4 chain ef all cnfiguatins 4 4 d 4 d 4 4 chain ef 4 4 c chain ef 4 d 4 I I shell chain ef chain ef 0 is the tw pint chain density f the nn-assciating fully flexible 4 The functin chain ef chain efeence syste. 4 4 chain ef nnassciating F the D syste the integal ve the bnding shell f a segent sphee at psitin in the fully flexible efeence fluid is I shell chain ef z Dchain z... z 4 z... z 4 exp[ chain ef z] dz 4 Whee z...z 4 is given by Eq. 4 and D chain is the pduct f 3 cavity celatin functins. SinceW z is a functinal f the chain efeence syste density pfile we can say ln ing k z z z z 4 z4 ln y z k 4 3 3

32 We have calculated W z / W f a feely inted 4-e chain nea a had wall f bulk packing factins f η = and 0.3. These esults ae pesented in Fig. 3. t each density we see a distinct axiu lcated nea z = σ and at wall cntact the pbability is appxiately equal t its bulk value. We nte that f η = 0.3 the functin W z / W has an dd cuvatue in the egin σ / < z < σ. In lecula siulatin tw segents can be cnsideed bnded even if the assciatin enegy is ze. Figue 4 pesents siulatin esults f ing factins in the nn assciating chain efeence syste at a packing factin f η = 0.3. s can be seen the dd shape pesent in W z is als pesent in this quantity shwing that this is indeed a featue f the chain efeence syste. s a test we calculated ing factins f a 4 e chain which can nly intaleculaly assciate ε inte = 0 and cpaed these esults t lecula siulatin; the esults can be seen in Fig. 5. F packing factins f η = 0. and 0. the theetical esults ae in excellent ageeent with siulatin. F η = 0.3 the theetical esults ae in gd ageeent with the siulatin data ve st f the dain hweve the peak in the theetical calculatins nea z ~ σ has an dd shape. This dd shape is the esult f the cuvatue f the efeence syste W z at this density as discussed abve. Oveall the ageeent with siulatin is uch bette than the fully flexible case. integal T btain ipved esults at η = 0.3 we can estict the chain integal 3 bulk k I chain and ing k I ing such that n intalecula velaps ae allwed self aviding. The chain integal will n lnge be able t be facted Eq. 48 will n lnge be valid and additinal ulti diensinal integals will need t be pefed. Since we will nt assue that the assciated ing is igid with segents and 4 bnded at cntact we will n lnge need the efeence

33 syste pbability Wz. The ethds t develp I k chain z and I k ing z f the self aviding case is siila t the develpent f the fully flexible ing integal Eq. 3 except nw additinal cnstaints ae added. In the inteest f bevity these equatins will nt be deived hee. F η = 0. and 0. nealy the sae esults ae btained f ing factins as the igid case. Figue 6 shws the esults f ing factins f an intaleculaly assciating fluid an aveage packing factin f η = 0.3. The esults ae in excellent ageeent with siulatin. The self aviding ethd gives the st accuate esults at high density; hweve the igid ing ethd is cputatinally faste. F this easn we will eply the igid ing ethd t study the cpetitin between inta and intelecula assciatin. 33

34 Figue Captins: Figue : Fatin f assciating chain lecules f length f spheical building blcks. Figue : Intelecula and intalecula assciatin f assciating chain lecules Figue 3: Diaga f assciating 4 e Figue 4: Density pfiles f assciating 4 e with an aveage packing factin η = 0.. Cuves ae theetical calculatins slid assciating end segent dashed nn assciating cente segent and sybls give Mnte Cal esults ed iddle segent black end segent Figue 5: Sae as Figue 4 with η = 0. Figue 6: Sae as Figue 4 with η = 0.3 Figue 7: Cpaisn f DFT cuves and MC sybls calculatins f the factins f segent type bnded inteleculaly χ inte and the factin f segent type bnded intaleculay χ ing. Figue 8: Cpaisn f cuent DFT slid lines DFT with neglect f intalecula assciatin dashed lines and MC sybls calculatins f factin f segent type nt bnded X z f aveage pe packing factins f η = and 0.3 Figue 9: Patitin cefficient z 4 / bulk K dz f packing factins f 0. tp and btt. Cuves give theetical pedictins ed cuent DFT blue neglecting intalecula assciatin and sybls give MC siulatins ed squaes bth inta and intelecula assciatin blue tiangles intelecula assciatin nly 34

35 Figue 0: Cpessibility fact calculated thugh the wall cntact thee cpaed t the MC siulatins f Ghnasgi and Chapan 3. Cuves and sybls have sae eaning as in Fig. 9 Figue : Intefacial tensin f slid / fluid inteface at packing factins f η = 0. and 0.. Cuves have sae eaning as Fig. 9 Figue : Factin f segent bnded intaleculaly χ ing cuves DFT sybls NVT siulatin f a fluid which is nly allwed t intaleculay assciate n intelecula assciatin and the ings ae feely inted Figue 3: Calculatin f Wz f η = and 0.3 Figue 4: Factin f cpnent nt bnded in the nn assciating efeence syste calculated by NVT siulatin f a packing factin f η = 0.3 Figue 5: Sae as Fig. except assciated ings ae assued igid and plana Figue 6: Sae as btt panel f Fig. f η = 0.3 except chains and assciated ings ae self aviding 35

36 Figue : 36

37 Figue : 37

38 Figue 3: 38

39 Figue 4: 39

40 Figue 5: 40

41 Figue 6: 4

42 Figue 7: 4

43 Figue 8: 43

44 Figue 9: 44

45 Figue 0: 45

46 Figue : 46

47 Figue : 47

48 Figue 3: 48

49 Figue 4: 49

50 Figue 5: 50

51 Figue 6: 5