PHYSICAL REVIEW E VOLUME 64 011405 Sedimentation profiles of systems with reentrant melting behavior J Dzubiella H M Harreis C N Likos and H Lo wen Institut fu r Theoretische Physik II Heinrich Heine Universita t Du sseldorf Universita tsstra e 1 D 40225 Du sseldorf Germany Received 26 January 2001 published 20 June 2001 We examine sedimentation density profiles of star polymer solutions as an example of colloidal systems in sedimentation equilibrium that exhibit reentrant melting in their bulk phase diagram Phase transitions between a fluid and a fluid with an intercalated solid are observed below a critical gravitational strength Characteristics of the two fluid solid interfaces in the density profiles occurring in Monte Carlo simulations for are in agreement with scaling laws put forth in the framework of a phenomenological theory Furthermore we detect density oscillations at the fluid gas interface at high altitudes for high gravitational fields which are verified with density functional theory and should be observable in surface scattering experiments DOI 10 1103 PhysRevE 64 011405 PACS number s 82 70 Dd 64 70 p 61 25 Hq I INTRODUCTION Colloidal particles in a suspension under gravitational influence show spatial inhomogeneities due to the symmetry breaking induced by the gravitational field The problem of sedimentation of particles in the presence of gravity has long been of scientific interest The simplest approximation is the one of noninteracting particles valid in the limit of dilute solutions This approach leads to an exponential sedimentation density profile which was observed by Perrin for a calculation of Boltzmann s constant in 1910 1 Taking into account particle interactions at higher concentrations will yield corrections to the exponential density profile For very small gravitational strength a local density approximation LDA of density functional theory DFT is justified 2 3 In this case there is a one to one correspondence between the sedimentation density profile and the isothermal equation of state This fact was exploited to extract the hard sphere equation of state experimentally by investigating sterically stabilized colloids 4 Furthermore within the LDA a change in the height z corresponds to a local change of the chemical potential of the bulk system This implies that in the limit of small gravity the phase behavior becomes visible as a function of height z a feature that has also been exploited to estimate the hard sphere freezing transition 4 Surprisingly comparison with Monte Carlo MC simulations show that the LDA is even reliable for relatively strong inhomogeneities or gravitational strengths 2 This was further confirmed by comparing LDA against the exactly soluble hard rod model in one spatial dimension While the LDA yields a monotonic decaying density profile z a layering shows up near the hard wall of the container bottom Even crystallization can be induced by the bottom wall 5 As shown recently 6 details of this surface induced crystallization may be significantly influenced by a periodic wall pattern Indeed pure colloidal crystals can be grown from sedimentation on a patterned substrate 7 9 In this case the gravitational field acts as an external force enforcing and accelerating heterogenous nucleation and growth Other fascinating phenomena in a gravitational field relevant for col Email address joachim thphy uni duesseldorf de 1063 651X 2001 64 1 011405 9 20 00 loidal suspensions are phase transitions such as wetting 10 surface melting 11 as well as dynamical effects as shocklike fronts 12 metastable phase formation 13 long range velocity correlations 14 stratification 15 and crystal growth 16 While the equilibrium sedimentation of hard sphere suspensions is well understood 2 4 5 17 18 charged suspensions are much more subtle as they reveal an apparent mass that is smaller than the bare mass at least for intermediate heights 4 19 21 In this paper we study a third kind of effective interaction between colloids namely a very soft core as realized for star polymer solutions 22 The qualitative new feature of those solutions as compared to the traditional hard sphere and Yukawa interactions is that their phase diagram exhibits a reentrant melting behavior for increasing density 23 In fact our analysis holds for any system with a reentrant melting behavior but we will mainly focus explicitly on star polymers Star polymers consist of f linear polymer arms attached to a central common core The complete bulk phase diagram for star polymers in a good solvent was calculated in Ref 23 and exhibits several unusual solid lattices as well as reentrant melting As will be discussed in detail in the following sections due to the reentrant melting behavior unusual density profiles featuring interesting effects arise and a wealth of scaling laws can be established The paper is organized as follows In Sec II results of computer simulations of a system of star polymers interacting by means of an ultrasoft pair potential 24 are presented In Sec III we present a phenomenological theory giving an account of the sedimentation profiles observed in the computer simulations Scaling laws are put forth Also in Sec III density functional theory in a simplified hybrid weighted density approximation HWDA is used to reproduce density oscillations at the fluid gas interface found in the simulation data Concluding remarks are contained in Sec IV II COMPUTER SIMULATION We performed canonical MC computer simulations keeping particle number N volume V and temperature T constant We used a simulation box with squared periodic boundary conditions in x y direction and semi infinite geom 64 011405 1 2001 The American Physical Society J DZUBIELLA H M HARREIS C N LIKOS AND H LO WEN PHYSICAL REVIEW E 64 011405 etry in z direction where the particles were confined only by the gravitational field for z 0 The bottom wall at z 0 was hard and interacting with the star polymers by means of an effective star polymer wall potential that is derived from the effective star polymer hard sphere interaction in the limit of a sphere with zero curvature The calculation was performed in Ref 25 It is of the following form V sw z f 3 2 2 ln 2z 4z 2 1 2 1 1 2 2 1 erf 2 z 1 erf z 0 z 2 else 1 With z we denote the distance from the center of one star polymer to the surface of the flat wall defines the socalled corona diameter of a star polymer which is related to its diameter of gyration g through 0 66 g see
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