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CU-Boulder PHYS 7450 - Density-functional Theory

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Density-functional theory for structure and freezingof star polymer solutionsBenito Groha)FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The NetherlandsMatthias SchmidtInstitut fu¨r Theoretische Physik II, Heinrich-Heine-Universita¨tDu¨sseldorf, Universita¨tsstraße 1,D-40225 Du¨sseldorf, Germany共Received 10 October 2000; accepted 21 December 2000兲We use the soft fundamental measure theory 共SFMT兲 to investigate a system of classical particlesinteracting with the pair potential of star polymers in solution. To that end we calculate liquid andsolid structural properties, as well as freezing, solid-to-solid, and remelting phase transitions. Evensubtle physical effects, like deviations from Gaussian crystal peaks and an anomalous peakbroadening upon increasing density as well as a reasonable vacancy concentration are capturedcorrectly. Good overall quantitative agreement with simulation data is found, however, with atendency to overestimate the structural correlations. Furthermore, we demonstrate that all recentdevelopments of its hard core counterpart can be incorporated systematically into SFMT. © 2001American Institute of Physics. 关DOI: 10.1063/1.1349092兴I. INTRODUCTIONThe understanding of classical many-body systems hasreceived a boost by the development of density functionaltheory 共DFT兲.1The density functional of a given system is anextremely powerful object, from which a complete under-standing of an equilibrium system can be gained. The ther-modynamics and correlation functions up to an arbitrary or-der are accessible in principle. Moreover, this is not only truefor the bulk but also for situations where an arbitrary influ-ence that can be modeled by an external potential energy, isacting on the system. Apart from externally caused spatialinhomogeneities, DFT also accounts for self-sustaineddensity-waves that are present in a crystal. Thus, it is able todescribe the liquid and solid phases on an equal footing, andhence gives a physical explanation of the existence of thefreezing phase transition.As the free energy density functional 共DF兲 is such apowerful object, it may become obvious that it is unknownfor most realistic systems. To construct an approximation tothe exact DF, the common strategy is to require that theapproximative DF yields the correct behavior in situationswhere one can solve the system, at least approximatively.The more conventional approach uses the homogeneous liq-uid phase as this starting point, and requires that the approxi-mative DF reproduces known results from liquid statetheory, like the equation of state and correlation functions.These quantities can be considered as input to the theory.A newer approach utilizes situations of reduced spatialdimensionality as limiting cases that are captured correctly.There one has the advantage that the system can be solvedexactly in dimensions as low as one or even zero, so noapproximations enter at that stage. The Rosenfeld hard-sphere functional2can be derived in this way,3and improvedversions of it can be systematically obtained,4,5as well asfunctionals for parallel hard cubes.6,7The approximation onehas to do is to construct a ‘‘functional interpolation’’5be-tween spatial dimensions. The fundamental measure func-tionals yield the Percus–Yevick direct correlation functionand equation of state for the bulk hard sphere liquid, giveexcellent results for the coexistence densities and describethe crystal structure up to close-packing excellently,8as wellas the vanishingly small vacancy concentration.9We notethat recently a similar approach was used to find a DFT foradhesive hard spheres.10The idea that a three-dimensional functional can be con-structed by imposing its correct dimensional crossover tolower dimensions is not limited to hard interactions. It can beapplied to penetrable spheres,11,12the Asakura–Oosawacolloid-ideal polymer mixture,13and has been exploited toderive a DFT for arbitrary soft pair interactions14,15and ad-ditive mixtures.16This so-called soft fundamental measuretheory 共SFMT兲 was demonstrated to predict the properties ofthe homogeneous liquid phase. The fluid equation of stateand pair correlation function are an output of the theory.In this work we apply the SFMT to a system of starpolymers in a good solvent, which has attracted a lot ofrecent interest.17–22,24,25The logarithmic pair interaction17present in this system leads to an anomalous liquidstructure18and to a rich phase diagram19,20with various solidphases and reentrant melting upon increasing density. Pair21and triplet22interactions have been investigated. Besidescomputer simulations, liquid integral equations17,18andEinstein-crystal perturbation theory19,20have been employed.It is of great interest to investigate the system from the uni-fying viewpoint that DFT provides. In addition, because ofthe richness of physical phenomena, star polymers provide asevere test to any DFT.a兲Current address: Fachbereich Physik, Bergische Universita¨t Wuppertal,D-42097 Wuppertal, Germany.JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 12 22 MARCH 200154500021-9606/2001/114(12)/5450/7/$18.00 © 2001 American Institute of PhysicsDownloaded 09 May 2003 to 198.11.27.12. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jspOur results show that the SFMT stands this test. In par-ticular, the predicted bulk pair correlations are in good agree-ment with simulations over the whole range from hard-sphere-like to ultrasoft behavior. The DFT yieldsthermodynamically stable face-centered cubic 共fcc兲 andbody-centered cubic 共bcc兲 crystals and reentrant melting. Wefind that the lattice peaks have broader wings than Gauss-ians. A peculiar decreasing of the Lindemann parameterupon increasing the density is captured correctly.In Sec. II the SFMT functional is described. We alsogive its refinements according to the latest developments inFMT for hard spheres, and discuss briefly its properties. Sec-tion III defines the theoretical model for star polymer solu-tions and gives explicit expressions for the quantities in-volved in SFMT. In Sec. IV we present results for the liquidand solid structure as well as the phase diagram. The presentapproach is discussed in the concluding Sec. V.II. THE DENSITY FUNCTIONALA. DefinitionThe SFMT is a weighted density approximation. It em-ploys a set of weight functions which are independent of thedensity profile.


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