J. Phys. Chem. B 2005, 109, 6397-6404
6397
Sedimentation Equilibrium of Colloidal Suspensions in a Planar Pore Based on Density Functional Theory and the Hard-Core Attractive Yukawa Model Shiqi Zhou* Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Wenhua Road, Zhuzhou City 412008, P. R. China
Hongwei Sun National Nature Science Foundation of China, Department of Chemical Sciences, 100085, P.R. China ReceiVed: August 19, 2004; In Final Form: NoVember 19, 2004
The sedimentation equilibrium of colloidal suspensions modeled by hard-core attractive Yukawa (HCAY) fluids in a planar pore is studied. The density profile of the HCAY fluid in a gravitational field and its distribution between the pore and uniform phases are investigated by a density functional theory (DFT) approach, which results from employing a recently proposed parameter-free version of the Lagrangian theorembased density functional approximation (Zhou, S. Phys. Lett. A 2003, 319, 279) for hard-sphere fluids to the hard-core part of the HCAY fluid, and the second-order functional perturbation expansion approximation to the tail part as was done in a recent partitioned density functional approximation (Zhou, S. Phys. ReV. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 061201). The resultant DFT approach is, thus, the first adjustable parameter-free DFT for HCAY fluids. The validity of the present DFT for HCAY fluids of reduced range parameter zred ) 1.8 under various external potentials is established in the first of the papers cited previously. The present DFT for HCAY fluids can predict the radial distribution function for the bulk HCAY fluid accurately in the colloidal limit (large value of zred), and in the hard-sphere limit, its prediction for the density profile of the hard-sphere fluid in a gravitational field is in very good agreement with the existing simulation data. The dependence of the density profile and distribution coefficient on the magnitude of the interparticle attraction, gravitational field, and degree of confinement is investigated in detail by the present DFT approach. Intuitive and qualitative analyses are also compared with the quantitative DFT calculational results.
I. Introduction During the past decade, nonuniform fluids have attracted much attention from theoretical and experimental researchers.1 One main feature of the fluids under confined conditions is their nonuniform density distribution, and the resultant consequence is the basis of many interesting phenomena,1,2 such as layering in the liquid, adsorption separation, capillary condensation, the first- and second-order wetting transition, a shift of freezing and melting points, and so forth. Previous investigations mainly focused on the inhomogeneity due to a planar wall, planar slit pores, large colloidal particles, or a spherical cavity. However, for a suspension of large colloidal particles with a size on the order of several hundred nanometers, the gravitational potential exerted on the fluid particles is comparable to the thermal energy kBT (kB is the Boltzmann constant, and T is the absolute temperature); therefore, the effect of the gravitational potential should not be ignored. Even for cases of protein particles of smaller sizes, where the effect of the gravitational potential is much smaller compared with that of the thermal energy kBT, in centrifuge separations, the potential energy exerted on the protein particles can be enhanced as if the acceleration of gravity were increased. In particular, the effect of the sedimentation on colloid crystallization was investigated experimentally,3 and it was found that the sedimental effect leads to a so-called nucleation-and-growth mechanism for crystallization, whose resultant consequence is a polycrystalline end state. In fact, it is difficult to grow colloidal crystals while avoiding the effect * E-mail:
[email protected].
of sedimentation.4,5 Furthermore, the method to grow colloidal crystals through sedimentation is often used in (photonic) applications.6-11 All of these phenomena originate from an external field and are associated with the nonuniform particle density distribution, which determines the Helmholtz free energy of the nonuniform system and eventually leads to the abovementioned phenomena. The problem of the density profile of colloidal suspensions in a sedimentation equilibrium inside slit pores was investigated by Rodriguez and Vicente,12 who treated the problem within the Ornstein-Zernike (OZ) integral equation formalism of inhomogeneous fluids, supplemented by the Percus-Yevick (PY) closure, to investigate the behaviors of a one-component system and a binary mixture of hard spheres (HSs) as a model of sterically stabilized colloids. Jamnik13 also investigated the same problem but with an adhesive hard-sphere (AH) potential as a representation of the competition between hard-core repulsion and short-range attraction existing in the practical colloidal systems, and Jamnik also investigated the problem with the OZ integral equation formalism of inhomogeneous fluids, which was supplemented by a hypernetted chain (HNC) approximation for a wall-fluid correlation. Only in a few cases and just for comparison, the PY closure was also used. Through a comparison of corresponding simulation data obtained in a grand canonical ensemble, it was found that the HNC closure is more accurate than the PY closure for the case of narrow pores (for example, 5σ), where the prediction from the PY closure is even qualitatively incorrect at the side of the upper wall, while for the case of wide pores (for example, 11σ and 21σ), both the HNC and PY closures become less reliable;
10.1021/jp0462512 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/26/2005
6398 J. Phys. Chem. B, Vol. 109, No. 13, 2005 the prediction from the HNC closure is even qualitatively incorrect at the side of the upper wall. From the point of view of the validity of the AH model potential as a representation of the colloidal interaction, some discussion is necessary. An existing investigation14 indicates that an analytical solution for the static structure factor of the AH system under a PY approximation can only describe the structure factor of a squarewell (SW) system adequately at low to moderate volume fractions (φ < 0.2618) when the range of the SW potential is less than 20% of the particle diameter and the magnitude of the reduced contact potential is less than 3, although the AH potential can be applied to even larger values of the reduced contact potential magnitude and to higher-volume fractions at low values of the width of the square well (less than 5% of the particle diameter). The domain of the validity of the AH potential prediction for the thermodynamic properties of the SW system, for example, the equation of state and the gas-liquid coexistence curve, is similar to that for the case of the static structure factor.15 However, the structure factor and thermodynamic properties are the global quantities; here, by “global quantities”, we mean that the structure factor is the result of the Fourier transformation of the real-space structure functions. Because of the infinitely small range of the attraction interaction potential, the AH model potential, especially the radial distribution function (RDF), whose nonuniform counterpart is exactly the so-called reduced density distribution profile, a topic of the sedimentation equilibrium investigation, is not useful as a representation of the real-space structure functions.. This can be seen clearly from the discontinuity of the RDF at distances that are multiples of the hard-sphere diameter as a consequence of the presence of a δ-function attraction on the surface of the sphere; such singular points never occur for the realistic colloidal systems whose interaction potentials are composed of the hardsphere repulsion and the continuous short-range attraction. Because of the failure of the AH model potential as model of realistic colloidal potential for the representation of the density distribution profile, it is very important and, therefore, necessary to employ a new model potential that can be a very good representation of the realistic colloidal potentials. Also, to overcome the low accuracy, and even the qualitative errors, in some cases of the OZ integral equation formalism of inhomogeneous fluids supplemented with the HNC or PY closure, we will propose a DFT approach free of adjustable parameters for the chosen model potential to investigate the sedimentation equilibrium of the chosen model potential system representative of colloidal suspensions in a planar pore. The organization of this paper is as follows. In section II, we will choose a hard-core attractive Yukawa (HCAY) model potential as the representation of the practical colloidal potential by comparing the HCAY potential with the DerjaguinLandau-Verwey-Overbeek (DLVO) potential, which can be a very good representation of the weakly charged and uncharged colloidal suspensions. We will propose a new DFT approach based on a recently proposed parameter-free version of the Lagrangian theorem-based density functional approximation (LTDFA) for hard-sphere fluids16 and a recent partitioned DFA17 for non-hard-sphere fluids, and we will point out that the present DFT approach in the hard-sphere limit performs far more accurately than the above-mentioned HNC- and PY-based OZ integral equation formalisms of inhomogeneous fluids under the influence of a gravitational field. Also, the RDF of the bulk HCAY fluid is predicted and compared with a recently proposed, accurate semianalytical expression used to confirm the validity of the present DFT approach in the colloidal limit (large value
Zhou and Sun of zred), which is exactly the case investigated in this paper. With an accurate DFT approach at hand, we will also employ the present DFT approach to study the sedimentation equilibrium of the HCAY potential system in the colloidal limit in a planar pore. Finally, we conclude this paper in section III with some discussion. II. DFT Approach and Calculational Results Recent studies18 indicate that the well-known DLVO potential can describe the metastable liquid-liquid coexistence reasonably well with respect to the stable solid-liquid equilibrium and cloud and solubility temperatures as a function of the salt concentration in a lysozyme solution. The DLVO potential can be written as the sum of a short-range attractive van der Waals term
VHA(r) ) -
( )]
[
A H d2 d2 r2 - d2 + 2 + 2 ln 2 2 12 r r -d r2
(1)
and a Debye-Hu¨ckel-like contribution.
VDH(r) )
[
]
2exp[-χ (r - d)] z pe 1 DH 4π0r 1 + χDHd/2 r
(2)
Here,
χDH ) x4πLB1000NAIs
(3)
where LB ) e2/(2π0rkBT) is the Bjerrum length, NA is Avogadro’s number, ionic strength Is is defined by Is ) (1/2)∑RMR, with MR standing for the molar concentration of ion R, and zp is the charge attached to one colloidal particle. To circumvent the singularity of the van der Waals term, a Stern layer thickness δ is introduced, which is related to the intrinsic size of the counterions that condense on the macromolecule surface; the resultant whole DLVO potential is
VDLVO(r) )
{
∝ VHA(r) + VDH(r)
r
