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Fluid of discs with competing interactionsRoland Roth aa Institut fr Theoretische Physik , Universitt Erlangen-Nrnberg , Staudtstr. 7 , 91058Erlangen , GermanyPublished online: 12 Oct 2011.
To cite this article: Roland Roth (2011) Fluid of discs with competing interactions, Molecular Physics: An InternationalJournal at the Interface Between Chemistry and Physics, 109:23-24, 2897-2905, DOI: 10.1080/00268976.2011.615765
To link to this article: http://dx.doi.org/10.1080/00268976.2011.615765
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Molecular PhysicsVol. 109, Nos. 2324, 10 December20 December 2011, 28972905
Fluid of discs with competing interactions
Institut fur Theoretische Physik, Universitat Erlangen-Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany
(Received 5 July 2011; final version received 12 August 2011)
We study within the framework of density functional theory the behaviour of a two-dimensional fluid with acompeting attraction at short distances and repulsion at longer distance in addition to a hard core. The soft partof the interaction is treated by a first-order thermodynamic perturbation theory, which for a homogeneous bulkfluid predicts that thermodynamic quantities reduce to that of the hard-disc reference system. Despite thisseemingly simple thermodynamic result, the behaviour of this fluid with competing interactions is rich andintriguing. In this work we follow the insight and inspirations by Reatto and co-workers, who performedcomputer simulation studies of the same system.
Keywords: density functional theory; competing interaction; cluster formation; lamellar structure
A variety of interesting fluid behaviours can beobserved in model systems that interact via pairpotentials with strong, hard-core repulsion at shortdistances and a soft attraction 
Vr 1, r5 2R,Vsoftr, otherwise:
Here R is the radius of the hard-core interaction. In thecase of simple liquids, where the soft part of theinteraction is a purely attractive potential, phaseseparation into a low density gas and a high densityliquid can be found at sufficiently low temperatures.Simple fluids are, of course, well studied in theory andexperiment.
Very different behaviour follows if the soft inter-action is given by competing attraction at shortdistances and a longer ranged repulsion of theform 
with "a4 0 and "r4 0, and Ra5RR. The hard-corediameter of the disc 2R. This potential was firstintroduced by Sear et al. [2,3] to study the formation ofclusters, or micro-phase separation, of nanoparticles(quantum dots) trapped at a waterair interface.That rich and interesting equilibrium behaviour
should occur was recognized and studied by Reattoand co-workers. The model given by Equation (2)in dimension d 2 was studied by Imperio and Reatto. A slightly modified version of this model in d 3,where the exponential functions were replaced byYukawa interactions, was studied Reatto andcolleagues  and also inspired studies by Archeret al. [11,12] and by Li et al. .
Here we study, inspired by Reattos insights andfindings, the two-dimensional system with pair inter-actions given by Equation (2) within the framework ofclassical density functional theory (DFT). By doing sowe hope to add a somewhat different perspective to theresults already reported on this intriguing system. Inparticular we make use of the fact that within DFT it ispossible to access the structure and the thermodynam-ics of a system on an equal footing.
The paper is organized as follows. In Section 2we describe the theoretical framework of this study.The main ideas of DFT are outlined for the particularapplication. The system of discs with competinginteractions is studied first in the absence of anyexternal field in Section 3. In Section 4 a singleplanar hard wall is added to the system. One point ofparticular interest here is the fact that the wallcontact theorem imposes a constraint on the densityprofile. In Section 5 we add a second parallel hardwall and study the discs in a simple confined geometry.We conclude with a discussion of our results inSection 6.
ISSN 00268976 print/ISSN 13623028 online
2011 Taylor & Francishttp://dx.doi.org/10.1080/00268976.2011.615765
2. Density functional theory
Within the framework of density functional theory for
classical systems [14,15] there exists a functional of the
average one-body density (r) which upon minimiza-tion yields the grand canonical potential:
O F Z
d2rrVextr , 3
where F  is the intrinsic Helmholtz free energyfunctional, Vext(r) is the external and the chemicalpotential.
The intrinsic Helmholtz free energy functional can
be split into the exact ideal gas part
F id 1Z
d2rr ln 2r 1
where this is written in dimension d 2. As usual, 1/(kBT) with Boltzmann constant kB and absolutetemperature T. is the thermal wavelength. The excess(over the ideal gas) free energy F ex
F F id F ex 5
contains all the information about the inter-particle
interactions and for most systems of interest is only
known approximately. Once the functional of the
excess free energy is constructed it can be employed to
study the system in an arbitrary external field described
by Vext(r). By minimizing the functional of the grand
potential, Equation (3), one can obtain the generally
inhomogeneous equilibrium structure, 0(r), and thethermodynamics of the system through the relation
[0(r)], where is the grand potential of thesystem. In the following we will suppress the index 0,
which indicates the fact that an equilibrium structure is
used.Since the excess free energy functional for a system
of discs interacting with competing interactions is not
known we follow a standard perturbation theory
approach in which we use a fluid of hard discs (HD)
as the reference system [16,17] described by
fundamental-measure theory (FMT) [18,19] and a
perturbation term for the soft interaction:
F ex FHDex 1
Zd2r0r0 ~Vsoftjr r0j:
In FMT the excess free energy of a hard-core system is
written as [18,19]
F ex Z
where the excess free energy density is a function ofweighted densities n. The form of used here and theweight functions are given in .
A first-order perturbation theory of the form (6) isknown to underestimate the correlation in the system.Therefore the soft interaction part in the pair potential(2), Vsoft(r), is empirically changed to ~Vsoftr inEquation (6), and is defined by
~Vsoftr Vsoftr 2R, r5 2R,Vsoftr, otherwise,
where the attraction is extended into the hard core, i.e.to r! 0.
Following Imperio and Reatto [4,5] the lengthscales Ra and Rr in the soft interaction, Equation (2),are set to Ra 2R and Rr 4R. In addition a relationbetween the depth of the attraction "a and the strengthof the repulsion "r is established by imposing thecondition that Z
d2r ~Vsoftr 0: 9
This is slightly different from the condition imposed onthe interaction potential Vsoft(r) within simulations byImperio and Reatto [4,5] where
Rd2r Vsoft(r) 0 leads
to "a "r. Here, the condition (9) leads for the presentchoice of length scales to