Article pubs.acs.org/JPCB
Size Selectivity in the Confined Ternary Colloidal Mixtures: The Depletion in the Competition Zongli Sun,*,† Yanshuang Kang,‡ and Yanmei Kang§ †
Science and Technology College, North China Electric Power University, Baoding, 071051, China College of Science, Agriculture University of Hebei, Baoding 071001, China § University of International Relations, Beijing 100091, China ‡
ABSTRACT: Based on classical density functional theory, we study the size selectivity for ternary colloidal mixtures in the presence of a Gauss barrier. The competition between the external potential and the depletion potential is also investigated. The effects of bulk fraction of each species, the size asymmetry, and the strength and width of the Gauss barrier on the selectivity of the big species are calculated and analyzed in detail. The results in different conditions of bulk fraction suggest that the larger the bulk fraction for the small species, the stronger selectivity of big particles. On the contrary, increase of bulk fraction for the big species leads to a reduction in selectivity. In addition, results under different conditions of size asymmetry suggest that the medium particles can also be selected by the Gauss barrier when they are sufficiently large in comparison to the small particles. We also demonstrate the effect of barrier geometry on the selectivity of the big species and the competition between the depletion and the external potential.
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INTRODUCTION Size selectivity in asymmetric fluid mixtures arises from a purely steric effect.1 Since the pioneering work by Asakura and Oosawa (AO)2 on the theoretical prediction of depletion forces, this mechanism has invoked a rich variety of investigations on the phase behavior of biological or colloidal systems, including crowding of macromolecules,3−5 crystallization of proteins,6,7 selectivity of specific geometry,8−10 and self-assembly and self-diffusion in colloidal systems.11−14 In particular, the size selectivity has attracted considerable attention due to its importance in both experimental and theoretical aspects. Experimentally, studies on the size selectivity in biological ion channels help in understanding the physiological functions of the membrane proteins.15−18 On the theoretical side, the fact that the size selectivity usually occurs to increase the configurational entropy demonstrates the importance of entropy in the theory for liquid state and other complex condensed matter.19 Recently, it has been shown that when a binary mixture is exposed to a potential barrier with a finite hight, the big particles can be selected by the barrier.20,21 This phenomenon stems from the loss of ability for the small particles to enter the gap between big particle and the barrier, which results in the imbalance of the pressure on the two sides of the big particle. Unlike the original AO conditions, in which the external potentials are usually provided by the hard geometry, the finite potential barrier is soft. In this condition, not all the particles but only part of them are expelled out of the region where the barrier reaches. Accordingly, on the determination of the equilibrium of the big species, the barrier has two competitive effects. One leads to an attraction for the big particles toward its © 2014 American Chemical Society
center, which is responsible for the selectivity of the big species; the other expels the big particles away from its center. Besides, it has been frequently shown that for the binary asymmetrical mixtures in the confining geometries, the size selectivity occurs as long as the size ratio between the species is large enough.22−24 Inspired by the better understanding about the size selectivity in the binary mixtures, we are interested to understand how it will respond to the introduction of a third species, which may make the steric effect more complex. Theoretically, both the small and the medium species may contribute to the depletion potential on the big species.25 Accordingly, we expect that the ternary mixture may provide more insight and information about the size selectivity in colloidal systems. Thus, this work mainly concerns the following two issues: (i) the size selectivity in a ternary mixture, and (ii) the competition between the depletion and external potentials. On the theoretical aspect, density functional theory (DFT)26−28 provides a framework for determining the structural and thermodynamic properties of a wide variety of inhomogeneous fluids starting from the Hamiltonian of the system. In the DFT language, the intrinsic Helmholtz free energy functional is an unique functional of the singlet density and its form does not depend on the external potential.29 Minimizing the grand potential functional with respect to the singlet density can readily determine the equilibrium density profile, which is essential to determine other properties. Received: May 20, 2014 Revised: September 15, 2014 Published: September 26, 2014 11826
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Actually, in its application to a particular model fluid, proper approximate functional is required because no exact expression for the excess Helmholtz free energy or the first-order direct correlation function (DCF) exist for the realistic systems. The most representative approximations include functional perturbation expansion approximation (FPEA),30−32 bridge density functional approximation (BDFA),33−35 weight density approximation (WDA) 36,37 and fundamental measure theory (FMT).38 Most of these approximations ignore long-range fluctuation effect, which makes the prediction deviating from the exact solution of the given Hamiltonian. Therefore, the efficacy of the given DFT approximation is usually tested by making comparisons with the computer simulation results. DFT represents a powerful tool in the study of the size selectivity because the selectivity of a particular species can be observed through comparison of its density profile with that of the other species, which are the immediate outputs of the DFT. By using DFT, Goulding et al.39 studied the size selectivity of the hard-sphere mixtures in different confining geometries. Their results showed that for certain ratios of the solute and solvent to pore diameters, the excluded volume alone can lead to very strong selectivity of the pores. They also concluded that the selectivity is strongest for spherical cavities while it is least pronounced in the slit geometry. Within the framework of DFT, González et al.40 investigated a simple model consisting of a binary hard-sphere mixture in a narrow cylindrical pore. They concluded that for large pore radii the selectivity is driven by an enhancement of the depletion force at the cylinder walls, whereas for the narrowest cylinders excluded-volume effects lead to a shift of the effective chemical potential of the particles in the pore. In all these studies, the microscopic image of size selectivity can be clearly observed and well understood by calculating the density of each component within the DFT framework. In the present work, a ternary colloidal mixture is modeled by an additive hard-sphere mixture, whose Helmholtz free energy functional can be readily obtained from the modified version of FMT (MFMT).41,42 The density profiles and the size selectivity are calculated within the framework of DFT. This paper is organized as follows. The second section describes the ternary colloidal mixture subject to a Gauss barrier. The framework of DFT and the formula for MFMT are briefly reviewed. To interpret the microscopic nature of size selectivity, we define the excess chemical potential (EXCP) and the total effective potential. In the third section, the effects of bulk fraction of each species, geometry of the Gauss barrier and the size asymmetry on the size selectivity are calculated and analyzed. A short summary is presented in the last section.
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⎛ z − z 0 ⎞m Vconf, ν(z) = εconf ⎜ ⎟ ⎝ Lν ⎠
(1)
with ν = s,m,b distinguishing the species in the mixture and Lν ≡ L + σν/2. m = 20 is set in this work so that the confining potential can be regarded as that produced by a slightly soft slit pore with a separation h = 2L. εconf and z0 are respectively the strength and midpoint of the pore. In addition, an external barrier potential Vext(z), which is of the Gaussian geometry,20,21 is applied to the ternary mixture: ⎡ ⎛ z − ξ ⎞2 ⎤ ⎟ ⎥ Vext(z) = εext exp⎢ −⎜ ⎣ ⎝ w ⎠⎦
(2)
where εext and w are respectively the height and width of the Gauss barrier. ξ denotes the position of the Gauss barrier, which is set as ξ = z0 in this work. Accordingly, the total potential Vν(z) felt by the νth species is the sum of the two, that is, Vν(z) = Vconf,ν(z) + Vext(z). In the equilibrium of the mixture, the total potential Vν(z) is the most important factor because it is responsible for the inhomogeneity of density, which will in turn influence the size selectivity in the mixtures. DFT for the Mixtures. In the framework of the classical DFT,26−28 the grand potential functional Ω[ρν(r)] for mixtures can be formally expressed as Ω[ρν (r)] = -[ρν (r)] +
∑ ∫ ρν (r)[Vν(r) − μν ] dr ν
(3)
where μν is the chemical potential of a particle of the νth species in its bulk phase and -[ρν (r)] is the corresponding intrinsic Helmholtz free energy. As is well-known, Ω[ρν(r)] is minimized when the system is at equilibrium, and the equilibrium density profile ρν(r) satisfies the variation principle: δ Ω[ρν (r)] δρν (r)
=0 (4)
For the hard sphere fluids, -[ρν (r)] can be perturbatively split into contributions from the ideal gas and the hard sphere repulsion. That is -[ρν (r)] = -id[ρν (r)] + -hs[ρν (r)]
(5)
Expression for -id is exactly known as -id = kBT × ∑ν ∫ ρν (r)[ln(λν3ρν (r)) − 1]d r with λν denoting the thermal wavelength. Here kB and T are respectively the Boltzmann constant and absolute temperature. However, exact expression for -hs is unknown for most practical systems. Therefore, approximation for it is required to implement the calculations. Among all the DFAs, MFMT41,42 for the hard-sphere fluids has shown more satisfactory performance in predicting the structure, phenomena transitions, and thermodynamic properties of the inhomogeneous fluids. Therefore, in this work, MFMT is employed to express -hs[ρν (r)]. According to the MFMT for mixtures, -hs is given by
MODEL AND THEORY
Modeling of the Ternary Colloidal Mixture. In this paper, the system under consideration is a ternary colloidal mixture composed of small, medium, and big colloidal particles with diameters σs, σm, and σb, respectively. For simplicity, the additive hard-sphere potential is employed to model the interparticle interactions. That is, σij = (σii + σjj)/2 with i,j = s,m,b. The system under consideration is confined in an onedimension box, whose boundaries locate at z = 0 and z = H ≡ 2z0. The total confining potential from both boundaries of the box is given by20,21
β -hs =
⎧ ⎪
∫ d r⎨−n0 ln(1 − n3) + ⎩ ⎪
+
n1n2 − nInII 1 − n3
⎤ n 3 − 3n n 2 ⎫ n32 1 ⎡ 2 II ⎬ ⎢n3 ln(1 − n3) + ⎥ 2 36π ⎣ n33 (1 − n3)2 ⎦ ⎭ ⎪ ⎪
(6) 11827
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where β = (1/kBT) and nα (r) = ∑v∫ ρv (r′)ω(α) v (r − r′) dr′ (α = 0, 1, 2, 3, I, II) is the weighted densities, with ω(α) v denoting the density independent weight functions, whose explicit form can be found in the references.41−43 Given the particle number Nν and the external potential, the equilibrium density profiles ρν(r) can be numerically solved by minimizing the grand potential under the constraint ∫ ρν(r) dr = Nν. This leads to the following coupled integral equations: ρν (r) = Nν
β −1cν(r; ρν ) = β −1 ln(λν3ρν (r)) − μν + Vext(r)
On the other hand, in terms of the potential distribution theorem,46 −β−1cν can be interpreted as the change in the grand potential of the mixture when a test particle of the νth species is inserted. This has been pointed out by Roth et al.,47 who define the depletion potential of the big species as βWb(r) = cb(r → ∞ ; {μν ≠ b }) − cb(r; {μν ≠ b })
exp[−β(Vν(r)) + c(1)(r; ρν )]
∫ exp[−β(Vν(r)) + c(1)(r; ρν )] dr δ -hs[ρν (r)] δρν (r)
β Φtot ≡βWb + βVext
is the first order DCF of the
⎛ ρ (r) ⎞ ⎟⎟ + βVext(∞) = − lim ln⎜⎜ b μ b →∞ ⎝ ρ (∞) ⎠ b
νth species. Note that in this paper, the particle number Nν can be controlled by adjusting the volume fraction of each species, which is defined as ην = (1/6)πσ3ν ρbulk,ν with ρbulk,ν ≡ (Nν/H) the bulk density of the νth species. Energetics Characterization of the Selectivity. Once the density profile of each species is obtained, the size selectivity by the Gauss barrier can be immediately observed by comparing the densities in the region of the barrier. To further understand the microscopic nature of the size selectivity, the difference between the EXCP of the species can be calculated as done by Gillespie et al. in their work for the selectivity of Ca2+ in the channels.44,45 In the energetics landscape, the chemical potential is a constant for the system in equilibrium, that is, βμv ex ex ex = ln ρbulk,ν + βμex bulk,ν = ln ρv(r) + βμν (r), with μbulk,ν and μν are respectively the EXCP of νth species in their bulk and nonuniform phases. Defining the difference Δμν (r) ≡ μex ν (r) − μex bulk,ν, one can easily arrive at ⎛ ρbulk, ν ⎞ ⎟⎟ β Δμν (r) = ln⎜⎜ ⎝ ρν (r) ⎠
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RESULTS AND DISCUSSION We calculated the density profiles of the ternary mixture under different conditions of bulk density for each species, size asymmetry, and geometry of the barrier. Note particularly that under each condition, the role that the external potential plays is twofold. On the one hand, it expels all species out of its region because of its repulsive nature. On the other hand, the inhomogeneous density of small particles leads to an attraction for the big particles toward the center of the barrier. In what follows, the focus is mainly on these two effects on the size selectivity in the ternary colloidal mixtures. Test for the DFT Calculation. In all calculations, the MFMT is employed to construct the Helmholtz free energy functional and the DCFs for the mixtures. To valuate the results from our calculations, the density profiles for binary mixture, exposed to the same external potential as defined in eqs 1 and 2, are calculated and compared with those from the Monte Carlo (MC) simulation.21 In the calculation, the parameters are set as εconf/kBT = 10.0, εext/kBT = 2.0, z0/σs = 10.24, L/σs = 7.5, w/σs = 1.5, and σb/σs = 2.0, which are the same as those used in the MC simulation. Figure 1a−c respectively gives the results for the total volume fraction ηtot = ηs + ηb = 0.1, 0.2, 0.3 with ηs = ηb. The agreement of our results with those from simulation suggests that the DFT calculation can provide reliable prediction for the equilibrium structure in the given external potential. However, a slight deviation from the simulation results can be observed for the results near the walls. As pointed out by Kumar,21 the residual attractive tail is responsible for the deviation from the additive hard-sphere. Prior to the calculation for the system we have modeled, it is necessary to check the performance of our codes in the study of inhomogeneous ternary mixture confined by simple geometry. In the calculation, we consider a ternary hard sphere mixture near a hard wall. The system parameters are set as ηtot = ∑νην = 0.337, the mole fraction xν = (1/3) for each species and σs/σm = 0.4, σb/σm = 1.6, which are the same as those set in the MC
(8)
(9)
in which ν = s,m only. Combining with eq 8, eq 9 can be readily rewritten as ⎛ρ ⎞ ⎛ ρ (r) ⎞ bulk,b ⎟ + β ΔΔμ (r) ln⎜⎜ b ⎟⎟ = ln⎜⎜ ⎟ ν− b ⎝ ρν (r) ⎠ ⎝ ρbulk, ν ⎠
(13)
Note that for the ternary mixtures, eq 13 is applicable for each species because it is formally derived from the generalized potential distribution theorem, rather than for one particular species. Due to the inclusion of Vext in its expression, Φtot contains the information about the relation between its two constituents, especially the competition between them, which is crucial for understanding of the change in the structure of the mixture.
which indicates that Δμν(r) includes the information about both the external potential and the interparticles correlation. Note that in the work of Boda et al.,45 the difference is decomposed into four contributions, while is this work, only the hard sphere contribution is involved because of the additive hard sphere model that has been employed. To ultimately characterize the size selectivity by the Gauss barrier, the difference of difference is further defined as ΔΔμν − b (r) ≡ Δμν (r) − Δμ b (r)
(12)
Based on eqs 11 and 12, the total effective potential Φtot, including the depletion potential and the external potential, can be derived in the dilute limit of the big species:
(7)
which can be solved by using the Picard-type iterative method. In eq 7, c(1)(r; ρν ) = −β
(11)
(10)
Given the bulk fraction of each species, the magnitude of ΔΔμν−b(r) reflects the extent of the selectivity of the big species. Note that in eqs 9 and 10, no external potential contribution enters the expressions for ΔΔμν−b due to the facts that the confining walls provides only a short-ranged potential, which is too far from the barrier, and that the barrier provides each species with an identical potential, which is canceled during the calculation of ΔΔμν−b. Competition of the Depletion with the External Potential. In the DFT for nonuniform fluid mixture, the DCF of the νth species can be expressed as26 11828
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adsorption of both the small and the medium species decreases to an asymptotic value. To observe the pattern in which the species are selected, the equilibrium density profile of each species is calculated in different cases of bulk fraction. In the calculations, the system parameters are set as εconf/kBT = 10.0, εext/kBT = 5.0, z0/σs = 10.24, w/σs = 1.5, L/σs = 7.5, σb/σs = 4.0, σm/σs = 2.0. As illustrated in Figure 3a−i, most of the big particles gather in the region near the Gauss barrier, which shows obvious selectivity of the big species in the ternary mixtures. Simultaneously, selectivity of medium species by the confining walls is also observed. The reason why the medium and the big species are respectively selected by the wall and the Gauss barrier can be understood by the competition between the energy and entropy in the mixtures. Note that the special interest in this study is only in the size selectivity due to the Gauss barrier. Comparing the results in Figure 3, it is obvious that when the bulk fraction for a given species is modulated, the density profile of each species may respond differently, and some interesting phenomena can be observed. For the given ηm and ηb, the results in Figure 3a−c suggest that the increase of ηs enhances the selectivity of the big species near the Gauss barrier. In addition, as shown in Figure 3c, some medium particles are also selected by the Gauss barrier. This results from the depletion potential felt by the medium particles in this region. In Figure 3d−f, the increase of ηm leads only to a slight enhancement for the selectivity of the big species near the barrier. In the meantime, the medium particles near the walls and the barrier shift toward the bulk region, which leads to the reduction in its adsorption by the barrier. In Figure 3g−i, the increase of ηb leads to a decrease in its density near the Gauss barrier, but with a wider range, which is responsible for the reduction in its adsorption shown in Figure 2c. This arises mainly from the competition between the depletion interaction and the excluded volume, which weakens the depletion interaction between the big particles. From the point view of energetics, the difference in the EXCPs of the species, defined in eq 9, interprets the microscopic nature about the local advantage of one species over the other species, i.e., which species is favored and to what extent it is favored. The results of ΔΔμν−b(z) for the systems used in Figure 3a−i are respectively given in Figure 4a−i. In each figure, ΔΔμν−b(z) for ν = s,b are positive near the barrier, which indicates that the big species is favored because less work is needed when a test particle of the big species is inserted there. The positive ΔΔμν−b(z) is also necessary to overcome the number disadvantage of the big species in its bulk phase. Interestingly, unlike the results in Figure 4a−h, the result in Figure 4i shows a smaller value for the small species than that for the medium species, which suggests that, in this case, the small species is favored over the medium species. Effect of the Size Asymmetry. Theoretically, the size asymmetry is responsible for the occurrence of the depletion. For the ternary mixture in this work, the mechanism for the depletion is more complex than that for the binary mixture, because both the small and the medium species may contribute to the depletion potential felt by the big particles. In the meanwhile, the entropy-induced attraction can also emerge between the medium particles because the gap between them is inaccessible for the small particles. In Figures 5−7, the calculations are performed under the conditions of εconf/kBT = 10.0, εext/kBT = 16.0, z0/σs = 10.24, L/σs = 7.5, w/σs = 1.5, and ηs = ηm = ηb = 0.08. Figure 5 plots the excess adsorption of
Figure 1. (a−c) Reduced density profiles of the binary mixture along the z-direction. The symbols correspond to the MC simulation results from ref 21. (d−f) Equilibrium density profiles of the ternary hard sphere mixture near a hard wall. The symbol is the MC simulation result from ref 48.
simulation in ref 48. Comparison in Figure 1d−f suggests that our results agree very well with those from the simulation. Effect of the Bulk Fractions. The excluded volume is the key factor determining the equilibrium density profile of the particles with a hard core. It is also the excluded volume which is responsible for the emergence of the depletion interaction and size selectivity. To quantitatively characterize the effect of the excluded volume on the selectivity, the excess adsorption of each species, which is defined as ν Γ ex =
∫ (ρν (r) − ρbulk,ν ) dr
(14)
is calculated. Note that the integral is performed in a symmetrical region with its width being the same as that of the Gauss barrier. The results plotted in Figure 2 suggest that
Figure 2. Excess adsorption of each species under different conditions of bulk fractions.
when ηs is increased, the adsorption of both the big and the medium species are enhanced. In addition, the adsorption of big species tends to an asymptotic value because of its fixed bulk fraction. When ηm is increased, the adsorption of the big species is enhanced while that of the medium species decreases linearly. When ηb is increased, the curve for the adsorption of big species shows a maximum due to the competition between the depletion and the excluded volume. In the meanwhile, the 11829
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Figure 3. Reduced density profiles of the ternary mixture along the z-direction in different conditions of bulk fractions for each species.
Figure 5. Excess adsorption of each species under different conditions of size ratio.
Figure 4. Difference in EXCP of the small and medium species from that of the big species. The same system parameters as those in Figure 3 are set.
each species for σb/σs = 6.0. The results show that as σm/σs is increased, adsorption of the small and the big species remains respectively negative and positive, while that of the medium species undergos a negative−positive change. This can be well understood by the depletion felt by the medium particles due to the small species surrounding them. In calculations for Figure 6a,b, the size ratio is set respectively as σs:σm:σb = 5:6:30 and 1:5:6. Significant change in the density of the medium species is directly shown in the figures. In addition, results of ΔΔμν−b(z) in Figure 6c,d show that there is no difference between the curves for ν = s and ν = m when
Figure 6. Results in panels a and b show equilibrium density profiles of the ternary mixture along the z-direction under different conditions of size ratio. Results in panels c and d show ΔΔμν−b(z) for the systems with the same parameters as those in panels a and b, respectively.
their size ratio is small. However, when σm/σs is large enough, ΔΔμν−b(z) for ν = m decreases and approaches zero, which implies that the difference between the selectivity of the 11830
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both the small and the medium species decreases, while that for the big species shows a positive maximum, which implies that when εext/kBT is too large, selectivity of the big species is suppressed by the strong barrier. Moreover, when εext/kBT is large enough, excess adsorption of each species tends individually to an asymptotic value, which corresponds to the situation where all the particles of this species are expelled out of the barrier region due to the strong repulsion from the barrier. The density profiles under different conditions of barrier strength are shown in Figure 9a−d, which clearly demonstrates
medium species and that of the big species is reduced, as shown in Figure 5. Results for the total effective potential plotted in Figure 7b depict the competition between the depletion and the external
Figure 7. Profiles of the total effective potential for each species along the z-direction under different conditions.
potentials felt by the medium particles. Particulary in the curve for σm/σs = 5.4, a maximum reappears in the center of the barrier, which indicates that accompanied by the selectivity of the medium species, part of them are expelled by the barrier potential away from the central region of the barrier, as shown in Figure 6b. However, as shown by Figure 7c, this does not happen in the results for the big species because the barrier is not too strong. For the given barrier strength, the big particles win the competition for space due to the requirement of entropy production in the mixture. Effect of the Barrier Strength. The mechanism for the depletion between the big particles and the Gauss barrier is similar in nature to that between two big particles. When one big particle approaches the barrier, the inhomogeneous density of small particles surrounding it leads to an attraction toward the center of the barrier. Therefore, the features of the Gauss barrier, especially its strength, may influence the depletion and further the size selectivity of the big particles, because the inhomogeneity of the small particles responses sensitively to the barrier strength. In Figures 8−10, the effect of the barrier strength on the size selectivity of the big species is calculated under the conditions of εconf/kBT = 10.0, z0/σs = 10.24, L/σs = 7.5, σb/σs = 4.0, σm/σs = 2.0, w/σs = 1.5 and ηs = ηm = ηb = 0.08. Results in Figure 8 show that as εext/kBT is increased, the excess adsorption for
Figure 9. Results in panels a−d show equilibrium density profiles of the ternary mixture under different conditions. Results in panels e−h show ΔΔμν−b(z) for the systems with the same parameters as those in a−d, respectively.
the response of the selectivity of the big species to the barrier strength. Results in Figure 9a,b suggest that when εext/kBT is not too large, the selectivity of the big species is enhanced by the increase of εext/kBT. This can be understood by comparing the results of ΔΔμν−b(z) shown in Figure 9e,f, that is, the maximum in the latter is larger than that in the former. This implies that the big particles are easier to be selected by a stronger barrier, for two reasons. One is that the stronger barrier leads to a stronger depletion, and the other is that the stronger barrier expels the small particles to vacate space for the accommodation of more big particles. On the contrary, when εext/kBT is large enough, the big particles are also expelled away from the barrier center, as illustrated by the double-peak structure in Figure 9c,d. This again stems from the competition between the depletion and the external potentials for the big particles, as depicted in Figure 10c. Moreover, results in Figure 10c suggest that as εext/ kBT is further increased, the region where the external potential is dominant becomes wider, which is responsible for the departure of the two peaks from the center of the barrier. Effect of the Barrier Width. The depletion between the big particles and the Gauss potential relates not only to barrier strength, but also to its width. For the given barrier strength, the increase of its width may lead to the decrease in the spatial
Figure 8. Excess adsorption of each species under different conditions of barrier strength. 11831
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Figure 10. Profiles of the total effective potential of each species along the z-direction under different conditions.
gradient of the potential within its range. The weakened external force felt by particles leads to the tendency of homogeneous profile for the small species, which further results in the reduction of the depletion force felt by the big particles. On the contrary, the external force becomes stronger when the barrier turns narrower. In Figures11 and 12, the effect of the
Figure 12. Results in panels a−d show equilibrium density profiles of the ternary mixture under different conditions of barrier width. Results in panels e−h show ΔΔμν−b(z) for the systems with the same parameters as those in a−d, respectively.
competition with the depletion. With further increase of w/σs, the double-peak disappears because the depletion potential gains more advantage in the central region. In addition, the results show that with the increase of w/σs, the big particles diffuse away from the barrier center, which makes the layer of the big particles wider. However, some of them, especially those in the wing of the layer, are out of the barrier region, which leads to a decrease in the adsorption as shown in Figure 11. This can be verified by the results in Figure 12e−h, which shows that the maxima of ΔΔμν−b(z) is small if the barrier width is too small and too large.
Figure 11. Excess adsorption of each species under different conditions of barrier width.
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barrier width on the size selectivity of the big species is calculated under the conditions of εconf/kBT = 10.0, εext/kBT = 16.0, z0/σs = 10.24, L/σs = 7.5, σb/σs = 4.0, σm/σs = 2.0, and ηs = ηm = ηb = 0.08. In Figure 11, it is depicted that as w/σs is increased, Γsex and Γbex respectively exhibit a minimum and a maximum, while Γm ex shows an oscillation. The results for the small species can be first understood as follows by a two-stage process. In the first stage, during which w/σs is small, the strong external force remains dominant over the interparticle force. In this case, the wider the barrier, the more small particles are expelled away. In the second stage, however, the value of w/σs is large enough so that the external potential near the barrier tends to be uniform. As a consequence, the small particles prefer to return the barrier region to form a uniform phase. For the same reason, the medium particles behave in a similar manner at first. However, when w/σs is increased beyond a value of 7.2, Γm ex is shown to decrease. This is due to the fact that in order to meet the requirement of entropy production in the mixture, the small particles gain the absolute advantage in its competition for space with the medium particles. The density profiles for each species under different conditions of w/σs are plotted in Figure 12a−d. For w/σs = 0.5, the double-peak structure in curve for big species appears again due to the dominance of the external potential in its
CONCLUSION Within the framework of the classical DFT for hard-sphere mixtures, the size selectivity of the big species is investigated for the ternary colloidal mixture subject to a Gauss barrier. To characterize quantitatively the size selectivity, the excess adsorption of each species is calculated. Moreover, the profiles of density and the EXCP are also calculated to understand the microscopic nature of the selectivity. In addition, by calculating the total effective potential for each species, the competition between the external Gauss barrier and the depletion arising from it is first observed and analyzed. The competitive mechanism shown in the confined mixture provides new insight about understanding the nature of the phase equilibrium and the size selectivity. For more systematical knowledge about the selectivity and its relation to the system parameters, the calculations are performed under different conditions of bulk fraction and size asymmetry. The results in different cases of bulk fractions suggest that the increase of fraction for the small and the medium species enhances the selectivity of the big species, while the increase of that for the big species leads to a reduction of its selectivity. The results in different cases of size asymmetry show that the medium species can also be selected by the barrier only if its size is large enough with respect to the small 11832
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species. However, part of the selected medium particles are expelled away to increase the entropy of the mixture, which can be verified by the competition between the two constituents of the total effective potential. Further calculations are performed under different conditions of barrier geometry. On one hand, the results in different cases of barrier strength interpret well the competition between the depletion and the external potential. The maximum shown in the excess adsorption of the big species suggests that the selectivity is suppressed when the barrier is too strong. On the other hand, for the given barrier strength, the results show that as the barrier width is increased, the excess adsorption of the big species shows a maximum, while that of the small species shows a minimum. The difference in the trend of change stems from the fact that, as the barrier width is increased, the potential it provides near its center tends to be uniform, which makes the homogeneous bulk preferable for each species. In conclusion, the results obtained in this work provide new insight and useful clues to understand the selectivity in the colloidal mixtures. It is expected that these results can be used to explain a rich variety of phenomena in fluids and fluid mixtures.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Z.S. would like to acknowledge A. V. Anil Kumar for his useful suggestions and his generosity to send the simulation results. This work is financially supported by the Fundamental Research Funds for the Central Universities (No.13MS105) of China and the Technology Research and Development Program of Hebei (No.13213704) .
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