Site-averaging in the integral equation theory of interaction site models of macromolecular fluids: An exact approach

2 Pages·0.08 MB·English

by  V. Krakoviack

Site-averaging in the integral equation theory of interaction site models of
macromolecular fluids: An exact approach
∗
V. Krakoviack
Department of Chemistry, University of Cambridge,
Lensfield Road, Cambridge CB2 1EW, United Kingdom
Asimple”trick”isproposed,whichallowstoperformexactlythesite-averagingprocedurerequired
3
when developing integral equation theories of interaction site models of macromolecular fluids. It
0
showsthatnoapproximationisinvolvedwhenthenumberofOrnstein-Zernikeequationscouplingthe
0
site-sitecorrelation functionsisreducedtoone. Itspotentialpracticalinterestforfuturetheoretical
2
developmentsis illustrated with a rederivation of theso-called molecular closures.

n
a
J In the recent years, the application of liquid state in- one finds that Eq. (1) reduces to the single scalar equa-

7 tegralequationmethods1 to soft matter systems allowed tion
2 significant advances in our understanding of their prop-
h(r)=ω∗c∗[ω+ρNh](r), (2)
erties. One of the most prominent developments in this
]
t fieldistheformulationofthePolymerReferenceInterac- with
f
o tion Site Model (PRISM), extending to macromolecular
N
s systems the Reference Interaction Site Model (RISM) of 1
t. Chandler and Andersen.2 Mostly applied to polymeric ω(r)= N X ωij(r). (3)
a fluids,3 it has recently been consideredto study fluids of i,j=1
m
platelet-like colloids.4 ω(r) is the total intramolecular distribution function.
-
d It is usually derived from the RISM theory as follows. Using a perturbation theory approach, Curro and
n Consider a one-component fluid with molecules made of Schweizer6 haveshownthatthebestchoicesforh(r)and
o N identical sites, i.e. the interaction potentials between c(r) are the site averagedvalues,
c any pair of sites on different molecules are the same.
[ These sites are in general non-equivalent (except those 1 N
1 related by obvious symmetries) and, at the pair level, h(r)= N2 X hij(r) (4)
v the structure ofthe fluid has thus to be characterizedby i,j=1
4 a matrix of N × N site-site total correlation functions and similarly for c(r), thus demonstrating that PRISM
1
h (r), where indices i and j refer to types of sites on
5 ij involvesapre-averagingofchainend(orplateletedge)ef-
different molecules. The RISM approach relies on the
1 fectsandevidencingthepracticalrelevanceofthetheory,
generalized Ornstein-Zernike matrix equation
0 sincethedoublysummedquantityofEq.(4),towhichit
3 gives access, appears naturally in the calculation of the
0 N scattering properties or of the thermodynamics of the
t/ hij(r)= X ωik∗ckl∗[ωlj +ρhlj](r), (1) system.
a
k,l=1 InthepresentNote,Iwanttoputforwarda“trick”for
m
the derivation of PRISM from RISM equations, showing
- where ρ is the molecular number density and ∗ denotes that the reductionof the N×N matrix equations of the
d
n convolutionintegrals. cij(r)isthedirectcorrelationfunc- latter to the scalar equation of the former can in fact be
o tionbetweensitesoftypesiandj ondifferentmolecules, done without invoking any approximation.
c and ωij(r) the intramolecular distribution function of The basic idea of this derivation is that, if all sites on
v: sites of types i and j. To solve Eq. (1), it has to be a molecule are identical in term of their interaction po-
i supplemented by some appropriate closure. In the case tential, then there is a permutation symmetry between
X
of hard-sphere sites of diameter d, it can be the Percus- them. To take this symmetry into account, one can ar-
ar Yevick approximation hij(r) = −1 if r < d, cij(r) =0 if tificially decompose each molecule into two objects: a
r >d.1,5 backbone made of N sites, labelled 1 to N, in which are
PRISMisobtainedbymakingthesimplifyingassump- embedded the geometry and the intramolecular interac-
tion that all N sites are equivalent. This is true for ring tions of the original molecule, and a set of N identical
moleculesforinstance,butitamountstoaneglectofend “atoms”,labelledˆ1to Nˆ, carryingthe intermolecularin-
effectsforlinearmoleculesorofedgeeffectsforplatelets. teractions and occupying at random the backbone sites
Then, using h (r) ≃ h(r) and c (r) ≃c(r) for all i,j’s, suchthatallpermutationsofN atomshaveequalproba-
ij ij
bility. Clearly, this delocalisation of the interaction sites
does not correspond to any real physical process and is
only introduced for formal consideration.
∗Present address: Laboratoire de Chimie, E´cole Normale Sup´e- One can write RISM equations for the mobile atoms,
rieuredeLyon,46,All´eed’Italie,69364LyonCedex07,France which have the usual form (1), now with hatted indices.
2
These simply are formaldefinitions of the direct correla- Starting with the classical RISM approach, c (r) for
ij
tionfunctions c (r), applicabletoanymoleculewithina r > d is often given the interpretation of a renormal-
ˆiˆj
site-site description. ized intermolecular site-site interaction potential. In the
Inthissystem,allinteractionsitesarebyconstruction backbone plus delocalized atoms system, it can thus be
equivalent. Thus, for allˆi and ˆj, one has h (r) =h′(r), considered as an effective interaction between backbone
ˆiˆj
c (r) = c′(r), and Nˆ ω (r) = ω′(r). In addition, sites. It is effective because the backbone sites do not
ˆiˆj Pˆj=ˆ1 ˆiˆj directlyinteractandthe interactionis actuallymediated
it is obvious that the total distribution functions of the
by the delocalized particles. Consider a configuration in
present model coincide with the site-averagedquantities
which particle kˆ on one molecule and particle ˆl on an-
obtained within the conventional RISM approach, i.e.
h′(r) = h(r) and ω′(r) = ω(r). It immediately follows other molecule occupy respectively the backbone sites i
and j. Using the simplest superposition approximation,
from Eq. (1) with hatted indices that Eq. (2), with c re-
placed by c′, is exactly obeyed. Note that nothing can thecontributiontocij(r)oftwoparticles,mˆ onthesame
be said a priori on the relation between c and c′ and molecule as kˆ and nˆ on the same as ˆl, can be approx-
in particular there is no reason to think they should be imated as ωkˆmˆ ∗ cmˆnˆ ∗ ωnˆˆl(r). Summing over all mo-
equal. bile particles and averaging over all their permutations,
So,whatdidwelearn? Fromapracticalpointofview, one eventually finds cij(r) ≃ ω ∗c′ ∗ ω(r) for r > d.8
not much: within the present approach,the reduction of Then, using a so-called atomic closure for c′(r), Eq. (2)
RISM to PRISM has been made a tautology and, owing with c replacedby c′, andthe impenetrability constraint
totheoddnatureofthebackboneplusdelocalizedatoms h(r)=−1ifr <d,weareleftwiththesameequationsas
system,thefinalsituationissomewhatequivalenttosim- those proposed in Ref. 7, but with a somewhat different
plytakingEq.(2)asaformaldefinitionofthetotaldirect physical content. This points to possible improvements
correlation function. The problem can be considered as ofthemolecularclosures: indeed,ithasforinstancebeen
only being displaced to the question of the non-trivial shownthatinsomecases,simplyconvolutingbyω isnot
relation between the c ’s of the usual RISM approach the best way to give an account of the macromolecular
ij
and the c ’s of the present fictitious system. However, connectivity.9
ˆiˆj
the fundamental point remains: the reduction from N2 Finally, like in PRISM, equivalence of all sites on a
to 1 of the number of Ornstein-Zernike equations does moleculeisoftenassumedintheframeworkofthedensity
not involve any approximationand, at variance with the functionaltheory,inordertoderiveequationsformacro-
usual claim, nothing like end or edge effects is lost when molecular fluids from theories of heteronuclear site-site
going from the RISM level of description to the simpler fluids.10Onecouldthusthinkofusingthepresent“trick”
PRISM one. As usual with integral equation theories, if in this context. However, since it is already known
PRISM,comparedtoa fullRISM treatment,canappear that the excess free-energy functional can exactly be ex-
inadequate in some case, it is only because of our inabil- pressed in terms of the total monomer density11 (which
itytoguessacorrectexpressionforthedirectcorrelation can be thought of as a site-averaged quantity for inho-
function. mogeneous systems), here again the proposed approach
More interesting is probably the fact that the present wouldprobablynotleadtosignificantpracticalprogress,
picturecouldprovidenewinsightsonthePRISMformal- but only help clarifying the formal relation between the
ism and thus lead to improvements of the theory. To il- site-specific and site-averagedlevels of description.
lustratethispoint,Iproposeaderivationoftheso-called TheauthoracknowledgessupportfromtheEPSRCun-
molecular closures7 developed for models where the in- der grantnumber GR/M88839andthanks J.-P.Hansen,
termolecular site-site interaction consists of a hard core A. A. Louis, R. Allen, and D. G. Rowan for useful dis-
part of diameter d plus a soft tail potential. cussions.
1 Seee.g.J.P.HansenandI.R.McDonald,TheoryofSimple 7 A.Yethirajand K.S.Schweizer, J. Chem. Phys. 97, 5927
Liquids, 2nd Ed. (Academic Press, London,1986). (1992); K. S. Schweizer and A. Yethiraj, J. Chem. Phys.
2 Forareview,seeD.Chandler,inStudiesinStatistical Me- 98, 9053 (1993).
chanics, editedbyE.MontrollandJ.L.Lebowitz(North- 8 Note that cij(r) is found independent of i and j in this
Holland, Amsterdam,1982), vol. 8, p. 275. approximation.
3 For a review, see K. S. Schweizer and J. G. Curro, Adv. 9 V. Krakoviack, J.-P. Hansen, and A. A. Louis, Europhys.
Chem. Phys. 98, 1 (1997). Lett. 58, 53 (2002).
4 L. Harnau, D. Costa, and J.-P. Hansen, Europhys. Lett. 10 See for instance J. P. Donley, J. G . Curro, and J. D.
53, 729 (2001). McCoy, J. Chem. Phys. 101, 3205 (1994); A. Yethiraj, J.
5 D.ChandlerandH.C.Andersen,J.Chem.Phys.57,1930 Chem.Phys.109,3269(1998);A.Yethiraj,H.Fynewever,
(1972). andC.-Y.Shew,J.Chem.Phys.114,4323(2001);T.Sumi
6 J.G.CurroandK.S.Schweizer,J.Chem.Phys.87,1842 and F. Hirata, J. Chem. Phys.118, 2431 (2003).
(1987). 11 C. E. Woodward, J. Chem. Phys. 94, 3183 (1991).