Surface Tension and Tolman Length of Spherical Particulate in

May 27, 2008 - Surface Tension and Tolman Length of Spherical Particulate in Contact with Fluid. Yongjin He, Jianguo Mi* and Chongli Zhong. Department...
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J. Phys. Chem. B 2008, 112, 7251–7256

7251

Surface Tension and Tolman Length of Spherical Particulate in Contact with Fluid Yongjin He, Jianguo Mi,* and Chongli Zhong Department of Chemical Engineering, Key Laboratory for Nanomaterials of Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, China ReceiVed: December 12, 2007; ReVised Manuscript ReceiVed: March 03, 2008

The structure, surface tension and Tolman length of particulate-fluid interfaces were studied theoretically. Within the framework of density functional theory, the nonlocal, modified fundamental measure theory and direct correlation function from the first-order mean spherical approximation were incorporated. The theory accurately predicted the structure of fluid and the particulate-vapor surface tensions. The predictions of surface tensions for particulate-liquid interface and particulate in supercritical fluid are also reasonable. Especially, Tolman lengths for particulate-fluid interfaces were investigated systematically. The correct prediction of surface tension from Tolman length indicates that our analysis is reliable. Furthermore, Tolman length as a function of spherical particulate diameter, particulate-fluid interaction energy, and the properties of the fluid is fully discussed. 1. Introduction Vapor-solid or liquid-solid surface tension is a fundamental property for a fluid-solid system and plays a key role in a variety of interfacial phenomena, and the calculation of the interfacial surface tension is relevant to study wetting problems of considerable practical applications, such as emulsions, foams, dispersions, adsorption-based separations, and heterogeneous chemical reactions.1–4 Unlike a liquid droplet or gas bubble in vapor-liquid nucleation, which exist in the vapor-liquid equilibrium two-phase system, wetting of solid substrates is a one-phase system with the solid merely present as a “spectator phase”. Compared with the stable and regular vapor-liquid interfacial density profile of nucleation,5 the structure of a fluid around a solid surface is more complicated, since it is relevant to the solid geometry, solid-fluid interaction strength, and thermodynamic properties of the fluid.6–8 In recent years, wetting of planar substrates has been investigated systematically with the methodologies of statistical mechanics theory and molecular simulation, and a significant understanding of solid wetting phenomenology has been achieved.9–11 It is known that for fluid adsorbed on a planar solid substrate, a change of the fluid-substrate interaction strength can induce drying and wetting transitions, and the validity of Young’s equation as a description of the wetting behavior of a fluid adsorbed on a planar substrate, as well as indicating the order of wetting and drying transitions, have been tested. Simulation studies show that the wetting transition is first-order, whereas density functional theory (DFT) predicts drying transitions of second order.12 In general, the interfaces do not assume planar shape, but rather, exhibit some curvatures,13–15 whereas wetting of a curved surface presents several fundamental differences with respect to the wetting of a planar surface. Curvature inhibits the first-order wetting transition observed in the planar case.16,17 On the other hand, the magnitude of the surface tension can be affected by interfacial curvature. Such a curvature dependence of the surface tension can be characterized by the Tolman length. Similar to a liquid droplet,the Tolman length (δ∞) for a fluid in contact with a spherical particulate * Corresponding author. E-mail:[email protected].

can be written as a simple function of the radius of the surface of tension, Rs; that is,

γ(Rs) ) γ∞(1 - 2δ∞ ⁄ Rs) + O(R-2 s )

(1)

where γ∞ denotes the planar surface tension and δ∞ represents the distance between the equimolar surface, Re, and the surface of tension, Rs; that is, δ∞ ) Re - Rs. For a curvature solid and fluid system, δ∞ is difficult to obtain by molecular simulation due to its asymmetry, and theory has its advantage in this aspect. With DFT, Stecki and co-workers18,19 showed that for a fluid in contact with a hard wall, the leading order Tolman correction, determined from a plot of γ(Rs) as a function of 1/Rs, is small and either positive or negative in sign, depending on the bulk density and temperature. For the system of a hard sphere fluid near a hard wall, Bryk et al.20 obtained a negative Tolman length, which they showed to be in good agreement with results using “scaled particle theory”. Blokhuis and Kuipers21 compared the Tolman length for a hard wall with a liquid droplet and showed that the Tolman length for a fluid in contact with a hard wall can be determined from the fluid density profile in contact with a planar wall, just as it can in the case of a planar liquid-vapor interface. However, surface tension and Tolman length for systems with a strong fluid-wall interaction, relevant to the understanding of wetting on real nanomaterials, has attracted less attention. Hadjiagapiou22 considered an attractive compact spherical substrate immersed in a bulk vapor and calculated the wetting transition. Bresme23 employed integral equation theory to analyze the surface tension of one colloidal particle immersed in a vapor and low-density supercritical phase. The calculated colloid-vapor surface tensions are in good agreement with the simulation data. While in supercritical state, the corresponding calculated results deviated obviously from the simulation data, and the deviations increased with an increase in the fluid densities, and in the liquid phase, the theory is not valid any more. More important, Tolman length and its variation have not been covered. These emphasize the need for further work to find accurate theories to cope with the high-density regime. The main purpose of this work is to analyze asymptotic Tolman length systematically for various particulate-fluid

10.1021/jp711692j CCC: $40.75  2008 American Chemical Society Published on Web 05/27/2008

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He et al.

interfaces using DFT. We discuss the theory with a nonlocal density functional method, in which modified fundamental measurement theory (MFMT)24 and the direct correlation function (DCF) from the first-order mean spherical approximation (FMSA)25 are incorporated. As well-addressed before, the theory has been tested to give an accurate description of the structure for fluids near hard walls, inside silt pores, and around colloid particles.25,26 When taken by input of particulate-fluid potential for the external force in DFT, the resulting density distribution is the fluid structure around the particulate. This study could reveal the general capability of today’s theories for various situations. On the basis of the MFMT + FMSA, we investigated in detail the fluid structure, surface tension, and Tolman length of the particulate as a function of size, particulate-fluid interaction strength, and thermodynamic properties of the fluid. In particular, we compared the predicted surface tension from the Tolman length with simulation data to validate our theoretical model. In addition, we compared the results for a nanoparticulate with those of a particulate of infinite radius; that is, the planar wall limit. 2. Theory For the fluid in contact with a spherical particulate, the system is always in equilibrium, and the radius of curvature is simply varied asaboundarycondition.Therangeandstrengthofthefluid-particulate interactions are independent of the particulate radius and act on the fluid through an external potential (Vext(r)). For simplicity, the Vext(r) and the fluid-fluid interaction can be given by the same Lennard-Jones/spline (LJ/s) potential form,6,27

{

distribution. The grand potential Ω[F(r)] can be expressed in the following general form of the density functional of the Helmholtz energy to describe the density profile:

Ω[F(r)] )

∫ dr F(r)[ln(F(r) Λ3) - 1] + Frep[F(r)] + Fatt[F(r)] + ∫ dr [F(r)(Vext(r) - µ)]

where F(r) denotes the density distribution with configuration r, Frep[F(r)] stands for the hard-sphere reference system, Fatt[F(r)] accounts for the attractive interactions, Vext(r) is the external potential, and µ is the chemical potential in the ensemble. The hard-sphere functional has been well-given by MFMT,23 which gives more accurate density profiles in a number of inhomogeneous systems. It writes

Frep[F(r)] )

[

∫ dr -n0 ln(1 - n3) +

(

σf 12 σf 6 0 < r - s < rs,ij r-s r-s (2) aij(r - s - rc,ij)2 + bij(r - s - rc,ij)3 rs,ij < r - s < rc,ij rc,ij < r - s 0

where r is the distance between particles; σf is the diameter of the fluid particles; εij represents the potential depth for interaction between species i and j; and s ) (σp - σf)/2, where the subscript “p” and “f” represent the particulate and fluid, respectively; and σp ) 2R is the diameter of the solid particulate. The remaining variables are given by rs,ij ) (26/7)1/6σf, rc,ij ) (67/48)rs,ij, aij ) -(24 192/3211)(εij/rs,ij2), and bij ) -(387 072/610 09)(εij/rs,ij3). The reduced temperature is defined as T* ) kBT/εff, where kB the Boltzmann constant, and the reduced density, F* ) Fσf3. The FMSA is a general solution to the Ornstein-Zernike equation, with particular analytical simplicity for exponential potential function.28 To extend the FMSA to LJ/s potential, we first map the potential with the two Yukawa potentials, as done before by Tang et al.29 for the Lennard-Jones potential. With the given potential in eq 2, the mapping potential can be expressed by

u(r)map ) -k1ε

exp[-z1(r - σ)] exp[-z2(r - σ)] + k2ε r⁄σ r⁄σ

]

(5)

∫ ∆F(r1) dr1∫ cb(r1 - r2) ∆F(r2) dr2

(6)

∫ ∆F(r1) dr1 -

Fatt[F(r)] ) Fatt(Fb) + µ

and a curved surface.

r

∫ r1 ∆F(r1) dr1 -

Fatt[F(r)] ) Fatt(Fb) + µ 1 2

r

r

∫ r1 ∆F(r1) dr1∫ r12 cb(r1 - r2) ∆F(r2) dr2

(7)

In eqs 6 and 7, the subscript “b” represents the bulk fluid, µ is the chemical potential of bulk fluid, ∆F(r) ) F(r) - Fb, and cb(r) is the DCF of the bulk fluid given by

cb(r) )

{

{

-βuff(r)

r>σ c1(T*1, z1, r) - c2(T*2, z2, r) r e σ

(8)

in which β ) 1/kBT, T*1 ) T*/k1, T*2 ) T*/k2, c1, and c2 is defined as follows:

rci(T, z, r) )

(3)

where k1, k2, z1, and z2 are the mapping parameters, and the values are given by k1 ) 7.7481, k2 ) 7.7952, z1 ) 5.8690, and z2 ) 8.8010. The calculation of the surface tension and Tolman length require estimation of the free energy. For inhomogeneous fluids, the essential task of the DFT is to derive an analytical expression for the grand potential, Ω[F(r)], or equivalently the intrinsic Helmholtz free energy, F[F(r)], as a functional of the density

)

where nR(r) (R ) 0, 1, 2, 3), V1, and V2 are the scalar and vector-weighted densities from the fundamental measure theory.30 For the attractive part, the much improved and explicit analytical DCF from the FMSA provides an accurate description for a planar

1 2

[( ) ( ) ]

n1n2 - nV1nV2 + 1 - n3

n23 n32 - 3n2nV1nV2 1 n3 ln(1 - n3) + 36π (1 - n3)2 n33

uijLJ⁄s(r, s) ) 4εij

(4)

with

e-z(r-σ) T e-z(r-σ) 1 × 4 6 2 T (1 - η) z Q (z) T S2(z) e-z(r-σ) + 144η2L2(z) ez(r-σ) 12η2[(1+2η)2z4 + (1 - η)(1 + 2η)z5]r4 + 12η[S(z) L(z) z2 (1-η)2(1 + η ⁄ 2)z6]r2 24η[(1+2η)2z4 + (1 - η)(1 + 2η)z5]r + 24ηS(z) L(z)}

r>σ

reσ (9)

Spherical Particulate in Contact with Fluid

Q(t) )

J. Phys. Chem. B, Vol. 112, No. 24, 2008 7253

S(t) + 12ηL(t)e-t (1 - η)2t3

(10)

S(t) ) (1 - η)2t3 + 6η(1 - η)t2 + 18η2t - 12η(1 + 2η) (11) π L(t) ) (1 + η ⁄ 2)t + 1 + 2η, η ) Fσ3 (12) 6 An important fact in eqs 6 and 7 is that so far, only FMSA can provide the DCF analytically, as well as the reliable bulk physical properties. With the above expression for the intrinsic Helmholtz free energy, the density distribution can be obtained by minimizing the grand potential,

(

F(r) ) Fb exp βµb,ex - β

δ(Frep[F(r)] + Fatt[F(r)]) δF(r) βVext(r)

)

(13)

Equation 13 can be calculated by use of a Picard iteration. The iterations terminate when the maximum difference between two subsequent density profiles is smaller than 10-6. With the obtained density profile, the grand potential can be obtained by eq 4. The surface tension, γs, can be calculated with the grand potential and the pressure of the fluid (P).

γs )

∆Ω Ω + PV ) A A

(14)

For planar surface, it is written as

γs )

∫0∞ [f[F(z)] - F(z)µ + F(z) Vext(z) + P] dz

(15)

and for curved surface,

γs )

∫R∞ ( Rr )

2

[f[F(r)] - F(r)µ + F(r) Vext(r) + P] dr

(16)

With the surface tension, Tolman length can de determined with Eq.(1). However, this determination would introduce errors from surface tension. An alternative way of calculating the Tolman length δ(R) is to use the Gibbs-Duhem equation for the surface variables together with the surface adsorption31

Γs )

2

∫R∞ Rr 2 [F(r) - Fb] dr

(17)

where Γs denotes the surface adsorption. Following Tolman’s original derivation,32 there is an exact relation,

(

Γs δ(R) 1 δ2(R) + ) δ(R) 1 + ∆F Rs 3 R2 s

)

(18)

For a solid-fluid system, ∆F ) -Fb, and the location of the surface of tension, Rs, is assumed to be the radius of the particulate. Combining eqs 17 and 18, we have

(

δ(R) 1 +

)

δ(R) 1 δ2(R) 1 + )Rs 3 R2 Fb s

2

∫R∞ Rr 2 [F(r) - Fb] dr

(19)

This choice ensures the values of δ(R) to extrapolate correctly to the planar limit (δ∞). Note that δ∞ was calculated by an independent route under the condition of Rs . δ∞.

δ∞ ) -

1 Fb

∫0∞ [F(z) - Fb] dz

(20)

The corresponding equimolar surface in eq 1 can be determined with

R3e ) -

3 Fb

∫0∞ [F(r) - Fb]r2 dr

(21)

3. Results and Discussion Under the framework of DFT, the structures of fluid around the particulate were obtained. With the density profiles, the surface tensions and Tolman lengths were predicted for various particulate-fluid interactions. 3.1. Fluid Structures around the Particulate. We have considered one particulate with variable size and potential immersed in vapor, liquid, and supercritical states. When dealing with a specific potential, the only way to test theories is to compare them with molecular simulation, which can in principle represent exactly the behavior of the system. To validate our theory, the density profiles were calculated and compared with simulationdata.23 Figure1summarizestheresultsforparticulate-vapor density profiles as a function of particulate size and interaction strength. From the Figure, one can see that each density profile is well-structured with two adsorbed monolayers, and the density profile of large particulate shows a small decrease in the local density profile defined by the first adsorption layer. The DFT accurately predicts the two-layer structures and their changes, and the agreement with the simulation results23 is remarkable. In addition to the vapor state, we have predicted the adsorbed liquid at the same temperature. Figure 2 shows the results for this state as a function of particulate size and interaction strength. The particulate-liquid profiles expand more than two layer structures. Compared with molecular simulation,23 the DFT theory also accurately reproduces the peaks of the profiles and the thickness of the first layer. The adsorption of fluids on small spherical substrates at supercritical conditions is an important subject of current interest.33,34 Accordingly, we obtained our results for the particulate immersed in supercritical fluids with Fb* ) 0.3. The supercritical temperature we have chosen is slightly higher than the critical temperature of the LJ/s fluid (T* ≈ 0.91).23 Figure 3 depicts the structure of a supercritical fluid with different particulate-fluid potential. From the Figure, we find that the theory yields the same good results as those at low temperature states. In summary, all the above calculations indicate that our model is capable of describing the structure of fluid adsorbed on an attractive, spherical particulate. Here, we want to emphasize that our comparison is fully quantitative. 3.2. Surface Tensions. With the accurate expression of density profiles, the chemical potential, Helmholtz free energy, and grand potential were constructed using our model, and the surface tensions were calculated. In Figure 4, we compare the particulate-vapor surface tensions with simulation data. The agreement is excellent, whereas in Figure 5, the particulate-liquid surface tensions as a function of the inverse of particulate size and interaction strengths are plotted. It shows slight deviations between the predicted results and the simulation data.23 Although the deviations increase as the increase in the particulate-liquid interaction and the decreasing in the particulate radius, the predicted solid-liquid surface tension is still reasonable. These comparisons show that our theory is a good approach to calculate surface tensions for both solid-vapor and solid-liquid. Further calculations of surface tension of a particulate immersed in supercritical fluid were performed as shown in Figure 6. In comparison, the results by integral equation theory23 and molecular simulation23 are also plotted. Although good agreement was found for density profiles between our DFT method and the integral equation theory method,23 it is evident that

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Figure 1. Particulate-vapor density profiles as a function of particulate size (1, 5, and 10σf) and interaction strength (1.50εff) with T* ) 0.75 and Fb* ) 0.045. The open symbols represent simulation data,23 and the lines, the DFT results.

Figure 2. Particulate-liquid density profiles as a function of particulate size (1, 5, and 10σf) and interaction strength (1.25εff) with T* ) 0.75 and Fb* ) 0.675 . The open symbols represent simulation data, and the lines, the DFT results.

Figure 3. Particulate-fluid density profiles as a function of interaction strength and fluid with supercritical states (T* ) 1.0, Fb* ) 0.3) and σp ) 10σf. The open symbols represent simulation data, and the lines, the DFT results.

integral equation theory overestimates surface tensions at high density and high temperature, whereas our method yields much

He et al.

Figure 4. Particulate-vapor surface tensions as a function of the inverse of the particulate size and interaction strengths, εpf ) 1.0, 1.25, and 1.50εff, with T* ) 0.75 and Fb* ) 0.045. Solid symbols represent simulation data,,23 and lines, DFT results.

Figure 5. Particulate-liquid surface tensions as a function of the inverse of the particulate size and interaction strengths, εpf ) 1.25, 1.50, and 1.75εff, with T* ) 0.75 and Fb* ) 0.675. Solid symbols represent simulation data,,23 and lines, DFT results.

better results. We think the main reason lies on the different routes to calculate the chemical potential. In our method, the calculation is based on the grand potential, in which the chemical potential of the bulk fluid, the local chemical potential, the local density profile, and the external potential were included and solved simultaneously. In integral equation theory, the density profile was solved only by an OZ equation, and the chemical potential expression was deduced by using the coupling parameter integration method, which depends on the integration path. Consequently, it is not unique. The expressions for the chemical potential and its reference term are quite different from DFT. On the other hand, the reference term is different. In integral equation theory, the hard sphere is used, whereas in density functional theory, the MFMT applied. So although the similar density profiles are obtained by both methods, the different deductions and expressions for chemical potential may lead to dissimilar results. Similar to the simulation results, the vapor-solid surface tensions predicted by our model are always negative in the range of sizes and interactions studied. In contrast, the solid-liquid surface tensions have both positive and negative values, whereas in supercritical states, the values are mostly positive. We agree

Spherical Particulate in Contact with Fluid

Figure 6. Surface tensions of representative supercritical states as a function of density with σp ) 5σf, εpf ) εff, and T* ) 1.0, 1.5, and 2.0, respectively, from bottom upward. The solid symbols represent simulation data;23 the solid lines represent the DFT results; and the dashed lines, the integral equation results.23

Figure 7. Comparison of the predicted particulate-vapor (Fb* ) 0.045) and particulate-liquid (Fb* ) 0.675) surface tensions from Tolman length δ∞ at εpf ) 1.25εff,T* ) 0.75, with the simulation data.7,23

with Bresme’s conculsion23 that negative surface tensions could be interpreted in principle as a tendency of the particulate to disperse in the fluid and positive values would normally indicate the tendency of the particulates to aggregate, because this would minimize the interfacial surface area. Therefore, decreasing the particulate size and increasing the particulate-fluid interaction energy would be favorable to the dispersion of particles in fluid. 3.3. Tolman Lengths. The good prediction of surface tensions for solid-vapor, solid-liquid, and solid-supercritical fluid systems encouraged us to further investigate the Tolman length. Inspection of Figures 46 reveals an approximate linear dependence of the surface tensions on the inverse of the particulate diameter. This dependence can be used to directly determinate the values of the Tolman length with eq 1. However, this method would introduce errors from surface tensions, especially for particulate-liquid system. On the other hand, the adsorption route is free from the problem. Thus, Tolman lengths in this work were calculated with the adsorption route as given by eqs 1720. To validate our numerical calculation, we first used thus obtained δ∞ to predict the corresponding surface tension and

J. Phys. Chem. B, Vol. 112, No. 24, 2008 7255

Figure 8. Determination of Tolman length δ∞ (in units of σf) as a function of bulk density by the DFT model with below and above critical temperature (Tc* ) 0.91) and εpf ) 1.25εff. The vertical dot-dash lines represent the coexisting vapor and liquid phases.

Figure 9. Determination of Tolman length δ∞ and δ(R) (for σp ) 20σf) as a function of bulk density by the DFT model with below and above critical temperature (Tc* ≈ 0.91) and εpf ) 0.05εff. The vertical dot-dash lines represent the coexisting vapor and liquid phases.

compared with the simulation data. As shown in Figure 7, the successful prediction indicates that our Tolman length calculations are reliable. Figure 8 depicts δ∞ as a function of the bulk density with a strong particulate-fluid interaction (εpf ) 1.25εff). Below the critical temperature, fluid exhibits the coexisting vapor and liquid phase (vertical dot-dash lines in the Figure). δ∞ for the vapor branch is always negative and a diameter more than a molecule in size. On the liquid branch, however, δ∞ has both positive and negative values and a diameter of less than a molecule. In the vapor branch, the δ∞ decreases to the coexistence density with an increase in the vapor density, whereas in the liquid branch, the δ∞ increases up to the coexistence density with a decrease in the liquid density. An interpretation for the divergence of the δ∞ was provided by Evans.13 They showed a “wetting” layer of vapor is formed between the liquid and particulate when the coexistence density is approached on the liquid side. Above the critical temperature, the absolute value of δ∞ decreases with an increase in the fluid density down to a minimum, very close to zero at the density Fb* ) 0.4. This minimum is often relative to the adsorption maximum. The adsorption maximum above the critical region for gas absorbed on a solid surface was predicted by both theory35 and experiment.36 As the density increases continuously,

7256 J. Phys. Chem. B, Vol. 112, No. 24, 2008

He et al. solid particulate, the corresponding surface tension, and Tolman length. Compared with the simulation data reported,23 the model yields a quite accurate structure of the fluid and solid-vapor surface tension. The obtained solid-liquid surface tensions are reasonable and much better than those from other theoretical models. Most important, the Tolman length has been investigated systematically. The predicted surface tension from the Tolman length for both the solid-vapor and solid-liquid interfaces are in good agreement with simulation data, indicating that our numerical calculation from the adsorption route is reliable. Especially, our Tolman length at infinite limit (δ∞) is quite analogous to the latest molecular simulation results for a large droplet and a bubble. On the basis of δ∞, the δ(R) were analyzed as a function of the particulate size, particulate-fluid interaction energy, and thermodynamic properties of a fluid.

Figure 10. Determination of the Tolman length, δ(R), as a function of the particulate diameter using the DFT model with εpf ) 0.05εff and T* ) 0.75.

Acknowledgment. The financial support of the NSFC (20576006, 20676004 and 20725622) is greatly appreciated. References and Notes

Figure 11. Determination of the Tolman length, δ∞, as a function of the temperature using the DFT model with εpf ) 1.25εff.

the absolute value of δ∞ increases. Figure 9 exhibits δ∞ with the same two temperatures (solid lines and dashed line), as in Figure 8, but the particulate-fluid potential is decreased to εpf ) 0.05εff. Comparing the two corresponding results, we can find that all values of δ∞ are changed to positive in a weak potential. The results are analogous to Lei and co-workers’ simulation data for large droplets and bubbles.37 To give insight into the Tolman length, δ(R) for large particulate size (σp ) 20σf) was calculated as an example, as shown in Figure 9 (short dotted lines and dashed line). Compared with each δ∞ at the same state, δ(R) is slightly reduced in magnitude over a large density range. The results suggest that the Tolman length is not a constant with given properties of fluid. Further investigation of δ(R) as a function of the particulate size was carried out as shown in Figure 10. The results show clearly that Tolman lengths for both solid-vapor and solid-liquid interfaces are size-dependent. We found that δ(R) for particulate-vapor interface increases slightly with a decrease in the particulate size, but for a particulate-liquid interface, the above tendency is reversed. We also noted that δ(R) is a function of the fluid temperature. Figure 11 indicates that, with an increase in the temperature, δ(R) for the particulate-vapor and the particulate-liquid interface approach a common value. 4. Conclusions With the MFMT and FMSA, the accurate nonlocal DFT model was extended to describe the structure of fluid around a

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