Solvation forces between molecularly rough surfaces
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Journal of Colloid and Interface Science 362 (2011) 382–388
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Solvation forces between molecularly rough surfaces Kan Yang a,b, Yangzheng Lin b, Xiancai Lu a, Alexander V. Neimark b,⇑ a b
State Key Lab for Mineral Deposit Research, School of Earth Sciences and Engineering, Nanjing University, 22 Hankou Road, Nanjing 210093, PR China Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, NJ 08854, USA
a r t i c l e
i n f o
Article history: Received 25 April 2011 Accepted 23 June 2011 Available online 30 June 2011 Keywords: Solvation pressure Surface roughness QSDFT Adsorption and Adhesion
a b s t r a c t Surface heterogeneity affects significantly wetting and adhesion properties. However, most of the theories and simulation methods of calculating solid–fluid interactions assume a standard thermodynamic model of the Gibbs’ dividing solid–fluid interface, which is molecularly smooth. This assumption gives rise to a layering of the fluid phase near the surface that is displayed in oscillating density profiles in any theories and simulation models, which account for the hard core intermolecular repulsion. This layering brings about oscillations of the solvation (or disjoining) pressure as a function of the gap distance, which are rarely observed in experiments, except for ideal monocrystal surfaces. We present a detailed study of the effects of surface roughness on the solvation pressure of Lennard-Jones (LJ) fluids confined by LJ walls based on the quenched solid density functional theory (QSDFT). In QSDFT, the surface roughness is quantified by the roughness parameter, which represents the thickness of the surface ‘‘corona’’ – the region of varying solid density. We show that the surface roughness of the amplitude comparable with the fluid molecular diameter effectively damps the oscillations of solvation pressure that would be observed for molecularly smooth surfaces. The calculations were done for the LJ model of nitrogen sorption at 74.4 K in slit-shaped carbon nanopores to provide an opportunity of comparing with standard adsorption experiments. In addition to a better understanding of the fundamentals of fluid adsorption on heterogeneous surfaces and inter-particle interactions, an important practical outcome is envisioned in modeling of adsorption-induced deformation of compliant porous substrates. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Interactions between solid surfaces separated by thin liquid layers are characterized in terms of disjoining, or solvation pressure, which depends on the state of fluid in the gap between the surfaces and the gap width. The notion of disjoining pressure was first introduced by Derjaguin in 1936, who found that two mica surfaces in water repel each other, and this ‘‘disjoining’’ effect increases as the gap width decreases from 1 to 0.04 lm [1]. Later, with the progress in experimental techniques and the development of the modern surface force apparatus and atomic force microscope, the spatial resolution was extended down to the sub-nanometer scale [2,3]. It was established that the pressure exerted by fluid films onto confining solid surfaces is not always positive; it may be negative or co-joining, depending on the specifics of solid–fluid intermolecular interactions. In this respect, the term ‘‘solvation pressure’’ is better suited for the following discussion, since it does not bear a connotation that thin films should always ‘‘disjoin’’ the confining surfaces. Moreover, Israelachvili and Pashley [4] showed that the solvation pressure between mica surfaces ⇑ Corresponding author. E-mail address: [email protected] (A.V. Neimark). 0021-9797/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2011.06.056
in aqueous solution oscillates at separations 1.5 nm with a mean periodicity of 0.25 ± 0.03 nm, roughly the diameter of water molecule. They proposed that the origin of these oscillations is primarily due to the ordered layering of water molecules at mica surfaces. Similar oscillations of the solvation pressure as a function of the film thickness have been found for various polar and non-polar solvents and complex fluids [5–13]. The oscillating behavior of fluid density and solvation pressure in thin layers was confirmed in various thermodynamic models and molecular simulations performed assuming ideal ‘‘mathematically’’ smooth solid surfaces [14–17]. This traditional approach stems from the Gibbs definition of dividing solid–fluid interfaces, which provides zero excess of solid density [18]. The density oscillation in confined films is caused by the hard-core repulsion between the fluid molecules and between the fluid molecules and the surface that facilitates packing of fluid molecules near the surface in layers similarly to the layering of billiard balls compressed between two parallel bars. This effect was studied extensively in the works of Henderson [19] and others. It was shown that introducing soft solid–fluid interactions [19–21] or imposing various defects or irregularities on the surface [22,23], the layered order can be frustrated and, thus, the density oscillations damped. At the same time, as shown by Schoen and Dietrich [24] and
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The density q(r) of the adsorbate, confined in a pore at given chemical potential l and temperature T, is determined by the minimization of the grand thermodynamic potential: X½qðrÞ ¼ R F½qðrÞ l m qðrÞdr. Helmholtz free energy F[q(r)] is a functional of the fluid density q(r) as prescribed in Rosenfeld fundamental measure theory [43]. Equilibrium density profile qðr; l; V; TÞ is determined from the solution of the Euler equation l ¼ @@Fq [36]. In NLDFT model, the adsorption stress is calculated by as a derivative of the grand thermodynamic potential with respect to the pore width at given external thermodynamic parameters, ra ¼ A1 @@HX T;l;A , where A is the surface area and H is the pore width [17]. The density profiles at saturation conditions are presented in Figs. 1 and 2 respectively for commensurate and non-commensurate pores of different widths. Commensurate pores are defines as the pores that possess maximally dense packings of integer numbers of layers. These pores correspond to maxima of the average fluid density as a function of the pore width, Fig. 3. Minima of the average density are achieved in non-commensurate pores, where the molecular packings are loose. The density profiles in commensurate pores show pronounced oscillations from layer to layer with almost equal distance between the maxima that can be associated the mean layer positions. Thus defined inter-layer distance constitutes 0.95rff in the commensurate pores, which accommodate 2 layers and more. The distance between the first layer position and the wall is 0.45rff. The density profiles in
3
ρσff
Henderson et al. [25], the surface corrugations or grooves of the supra-molecular scale (larger than 5–10 fluid molecular diameters) do not obscure the molecular layering. The solvation pressure oscillations induced by supra-molecular corrugations on crystal surfaces, scale of which is comparable with the size of the AFM tip, can be probed with AFM as considered in by O’Shea et al. [26]. A detailed discussion on how the size of surface corrugations and tiling affects the pressure oscillating was given by Fink and van Swol [23]. The conclusion that confined fluid layers of molecular thickness possess an ordered layered structure causing the oscillating behavior of the solvation pressure has been widely adopted in the literature. However, it is worth noting that all experiments showing oscillating solvation pressures were performed with molecularly smooth surfaces like mica, so that the ordering of adsorbed molecules was not frustrated by chemical or geometrical surface heterogeneities of the molecular scale. At the same time, most of the non-crystalline surfaces of practical interest, such as amorphous oxides and carbons, are molecularly rough as shown by molecular simulations [27–29] and directly by XRD experiments [30–33]. In particular, the density of amorphous silica at the surface decreases from its bulk value to null within a certain layer of varying density called ‘‘corona’’. The corona thickness is of the order of molecular diameters, and depending on the material, it may extend from fractions to a few nanometers, accounting for the molecular level roughness of the solid surface [30,33,34]. In this work, we analyze the effects of molecular level roughness, drawing on the example of Lennard-Jones (LJ) fluid confined by parallel LJ walls. We study the dependence of the fluid density and the solvation pressure on the pore width and the degree of surface roughness, as well as, the variation of the solvation pressure in pores during the process of adsorption and capillary condensation. The latter problem has practical implications in the phenomena of adsorption-induced deformation of adsorbents, which is discussed also. For these purposes, we adopted the quenched solid density functional theory (QSDFT) that was recently introduced to model adsorption at the rough surfaces [35]. The rest of the paper is structured as follows. In Section 2, we set a stage for the analysis of surface roughness effects by presenting a detailed account of fluid behaviors in the pores with smooth walls given with the non-local density functional theory (NLDFT). This data serves as a reference for the QSDFT studies of the damping of the density and solvation pressure oscillations as the surface roughness increases, which are presented in Section 3. Special attention is paid to the investigation of adsorption in slit-shaped micropores and mesopores, which shows a variety of different non-monotonic solvation pressure isotherms. The conclusions are summarized in Section 4.
26 24 22 20 18 16 14 12 10 8 6 4 2 0
δ= 0 σff
0.73σff (0.26 nm) 1.82σff (0.65 nm) 2.78σff (0.99 nm) 3.75σff (1.33 nm) 4.71σff (1.67 nm) 5.66σff (2.01 nm) 6.62σff (2.35 nm) 7.57σff (2.69 nm) 8.52σff (3.02 nm) 9.50σff (3.37 nm) 10.45σff (3.71nm)
0
2
4
6
8
10
z, σff Fig. 1. Density profiles in commensurate pores with smooth walls. Curves are shifted by 2qr3ff .
22 δ= 0 σff
20
2. Confined fluid density and solvation pressure in pores with smooth walls
18
1.27σff (0.45 nm)
16
2.30σff (0.82 nm)
3
To demonstrate the specifics of the density distribution and solvation pressure obtained with the models of smooth pore walls, we employ the conventional NLDFT approach that is widely used to study the fluid density distribution in nanopores [17,36–39]. In the NLDFT approach, the adsorption and desorption isotherms in pores are calculated based on the intermolecular potentials of fluid–fluid and solid–fluid interactions. Fluid–fluid interactions are modeled by a pair-wise LJ potential in the WCA approximation [40], and fluid–solid interactions are modeled by an external Steele potential [41]. We consider slit pores and employ the fluid–fluid and fluid–solid LJ interaction parameters suggested for N2 adsorption on carbons, eff/kB = 95.77 K, rff = 0.3549 nm; esf/kB = 150 K, rsf = 0.269 nm [42].
ρσff
14
3.27σff (1.16 nm)
12
4.20σff (1.50 nm)
10
5.19σff (1.84 nm)
8
6.14σff (2.18 nm) 7.10σff (2.52 nm)
6
8.05σff (2.86 nm)
4
9.01σff (3.20 nm)
2
9.98σff (3.54 nm)
0 0
2
4
6
8
10
Pore width, σff Fig. 2. Density profiles in non-commensurate pores with smooth walls. The width of non-commensurate pore is determined by the average of pore widths of adjacent commensurate pores. Curves are shifted by 2qr3ff .
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1.4
1000 solvation pressure average fluid density
1.0
400 300
0.8
200 100 0.6
0
0.73σff (0.26 nm)
16
1.82σff (0.65 nm)
14
2.78σff (0.99 nm)
12
3.75σff (1.33 nm) 4.71σff (1.67 nm)
3
600
18
ρσ ff
1.2
700 500
δ= 0.56 σff
20 3
800
Solvation pressure, MPa
22
smooth surface P/P0 =1
Average fluid density, ρσff
900
10
5.66σff (2.01 nm)
8
6.62σff (2.35 nm)
6
7.57σff (2.69 nm)
4
8.52σff (3.02 nm)
2
9.50σff (3.37 nm) 10.45σff (3.71nm)
0
-100
0
-200 0
1
2
3
4
5
6
7
8
9
10
non-commensurate pores display less pronounced oscillations compared to those in commensurate pores. The layered structure is leveled due to packing disordering in the pore center beyond the first 2–3 ordered layers at the pore walls. The solvation pressure as a function of the pore width exhibits oscillation correlated with the oscillation of the mean density. That is the solvation pressure maxima corresponded to the most ordered states in commensurate pores and minima – to the least ordered states in non-commensurate pores. This behavior is typical for any theories of solvation pressure in the gap between smooth surfaces [2,17,20,44,45]. 3. The effect of surface roughness on confined fluid distribution and solvation pressure Using the NLDFT results for smooth surfaces as a reference, we demonstrate the effects of surface roughness by means of the QSDFT model [35,42]. The QSDFT model is based on the multicomponent density functional theory, in which the grand thermodynamic potential X is defined as a functional of the solid qs(r) and fluid q(r) densities
6
8
10
Fig. 5. Density profiles in commensurate pores with rough walls of the same roughness parameter d = 0.2 nm (0.56rff). Curves are shifted by 2qr3ff . Compare with Fig. 1.
X½qs ðrÞ; qðrÞ ¼ F id ½qðrÞ þ e F ex ½qs ðrÞ; qðrÞ
Z Z 1 drdr0 qðrÞqðrÞ0 uff ðjr r0 jÞ 2 Z Z þ drdr0 qðrÞqs ðrÞ0 usf ðjr r0 jÞ Z l drqðrÞ þ
ð1Þ
Here, e F ex ½qs ðrÞ; qðrÞ ¼ F ex ½qs ðrÞ; qðrÞ F ex ½qs ðrÞ; qðrÞ 0 is the excess Helmholtz free energy of the reference hard sphere fluid, Fid and Fex are the ideal and excess components of the Helmholtz free energy, l is the fluid chemical potential, and uff(|r r0 |) and usf(|r r0 |) are the attractive parts of the fluid–fluid intermolecular potential and the solid–fluid intermolecular potential. Both the solid–fluid and fluid–fluid interactions are split into hard-sphere repulsive and mean-field attractive parts with the WCA approximation similarly to the fluid–fluid interactions in NLDFT. Hard-sphere interactions are treated with the multicomponent fundamental measure theory (FMT) functional for quenched-annealed systems [43]. The equilibrium density distribution of fluid is determined by minimizing the grand thermodynamic potential with respect
6
A
solid density fluid density
5
Pore width=5σff, δ=0 nm
4
4
3
3
ρσff3
ρσff3
pore edge
6
Pore width, σff
Fig. 3. Plot of average densities and solvation pressure at smooth surface condition.
5
4
0.4 11
Pore width, σff
solid density fluid density
2
2
2
1
1
B
pore edge
Pore width=5σff, δ=0.2nm
0
0 0.0
0.5
1.0
1.5
Distance, σff
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Distance, σff
Fig. 4. Density profiles in the pore of width H = 5rff at the saturation conditions for the LJ model of N2 adsorption on carbon at 77.4 K. (A) Smooth pore walls, (B) molecularly rough pore walls with the roughness parameter d = 0.2 nm (0.56rff). The solid density profile is given by Eq. (3). The pore diameter is defined from the Gibbs condition of zero excess solid density that implies the equality of the shaded areas separated by the vertical line denoting the ‘‘theoretical’’ pore edge.
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pore wall edge, and, as such, to consider the molecular level surface roughness in the form of corona. In our calculation, the surface roughness is modeled as a corona of linearly reducing density (Fig. 4), which is described by the following equation [42]:
1.5 1.4 1.3 δ=0.56 σff
8 0 q 0 6 z < h0 > > < s 0 qs ðzÞ ¼ 0:75q0s 1 zh h0 6 z < h0 þ 2d 2d > > : 0 z P h0 þ 2d
δ=0.45 σff δ=0.37 σff
3
1.1
δ=0.25 σff
1.0
δ=0.17 σff
0.9
δ=0 σff
δ=0.08 σff
0.8 2
4
6
8
10
Pore width, σff Fig. 6. Pore width dependence of average density at P/P0 = 1 in pores with different roughness. Curves are shifted by 0:1qr3ff .
Solvation pressure, MPa
1000 800 δ=0.56 σff
600
δ=0.45 σff δ=0.37 σff
400
δ=0.25 σff δ=0.17 σff
200
δ=0.08 σff δ=0 σff
0 -200 0
2
4
6
8
10
Pore width, σff Fig. 7. Pore size dependence of solvation pressure at P/P0 = 1 in pores with different roughness. Curves are shifted by 100 MPa.
to the fluid density q(r) keeping the solid density qs(r) fixed (this is where the term ‘quenched’’ stems from). This yields the Euler– Lagrange equation,
dX½qs ðrÞ; qðrÞ=dqðrÞ ¼ 0
ð2Þ
that is solved in an iteration fashion [42]. Thus, in contrast with NLDFT, QSDFT accounts for the fluid–solid interactions through pair wise intermolecular potentials rather than through an effective external potential. This approach allows one to take into account the in homogeneous solid density at the 0.9
ff
1.82σ (0.65 nm)
ff
0.8 0.7 0.6 3
0.9
0.73σ (0.26 nm)
A δ=0 σ
0.5 0.4 0.3 0.2 0.1 0.0 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
P/P0
0.1
1
ff 2.78σ (0.99 nm) ff 3.75σ (1.33 nm) ff 4.71σ (1.67 nm) ff 5.66σ (2.01 nm) ff 6.62σ (2.35 nm) ff 7.57σ (2.69 nm) ff 8.52σ (3.02 nm) ff 9.50σ (3.37 nm) ff 10.45σ (3.71 nm) ff 1.27σ (0.45 nm) ff 2.30σ (0.82 nm) ff 3.27σ (1.16 nm) ff 4.20σ (1.50 nm) ff 5.19σ (1.84 nm) ff 6.14σ (2.18 nm) ff 7.10σ (2.52 nm) ff 8.05σ (2.86 nm) ff 9.01σ (3.20 nm) ff 9.98σ (3.54 nm) ff
B δ=0.37σ
0.73σ (0.26 nm) ff
1.82σ (0.65 nm)
ff
0.8
ff
2.78σ (0.99 nm) ff
3.75σ (1.33 nm) ff
0.7
4.71σ (1.67 nm) ff
5.66σ (2.01 nm) ff
3
0
ð3Þ
Here, qs = 0.114 Å3 is the density of bulk solid; h0 is the thickness of the solid wall assumed to be equal to h0 = 2 0.34 nm, in accord with [34]. The roughness parameter d, defined as the half-thickness of the corona, is used to characterize the surface roughness. Pore diameter is defined from the Gibbs condition of zero excess solid density (Fig. 4B). A typical example of calculated density profiles with NLDFT and QSDFT models is presented in Fig. 4. Even with such small roughness parameter as d = 0.2 nm (0.56rff), the density fluctuations are significantly leveled, compared to the reference density between smooth walls. Fluid penetrates into the outer part of the corona and its distribution across the pore is, in overall, more homogeneous. The destruction of layering reflects a disordered packing of molecules confined by rough walls. The density profiles in commensurate pores with rough walls of the same roughness parameter d = 0.2 nm (0.56rff) are presented in Fig. 5. They should be compared with those given in Fig. 1. The density oscillations are damped in all pores that accommodate two and more layers. Note that the surface roughness effect destructs layering to a lager extent than the pore size mismatch, Fig. 2. In Figs. 6 and 7, we present the dependences of the mean fluid density and the solvation pressure on the pore width for different roughness parameters at the saturation conditions. The correlated mean density and solvation pressure oscillations characteristic for the pores with smooth walls, are gradually damped as the roughness parameter increases. The oscillations of the solvation pressure disappear for the roughness parameter exceeding d = 0.37rff (0.13 nm). It is worth noting that this value of the roughness parameter was used earlier in [42] to model adsorption on carbons. It was shown that the adsorption of N2 and Ar on non-porous carbon Vulcan is adequately described by the QSDFT model with d = 0.13 nm. When pore wall roughness d P 0.37rff (0.13 nm), the solvation force at the saturation conditions monotonically decays with the pore width. The solvation pressure is disjoining and, as such, the porous sample should exhibit swelling upon saturation with the adsorbate. However, as shown below, the solvation pressure may vary nonmonotonically in the process of adsorption–desorption cycles.
1.2
0.6
6.62σ (2.35 nm)
0.5
8.52σ (3.02 nm)
ff
7.57σ (2.69 nm) ff ff
9.50σ (3.37 nm) ff
10.45σ (3.71 nm)
0.4
ff
1.27σ (0.45 nm) ff
2.30σ (0.82 nm)
0.3 0.2 0.1 0.0 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
1
ff 3.27σ ff 4.20σ ff 5.19σ ff 6.14σ ff 7.10σ ff 8.05σ ff 9.01σ ff 9.98σ ff
(1.16 nm) (1.50 nm) (1.84 nm) (2.18 nm) (2.52 nm) (2.86 nm) (3.20 nm) (3.54 nm)
P/P0
Fig. 8. (A and B): Equilibrium adsorption isotherms in pores with smooth (d = 0) and rough (d = 0.37rff) walls. The pore sizes are given in the right panels.
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A
800
δ=0 σff
0.73σff (0.26 nm) 1.27σff (0.45 nm)
600
1.82σff (0.65 nm)
500
2.30σff (0.82 nm)
400
B
800
Solvation pressure, MPa
Solvation pressure, MPa
700
2.78σff (0.99 nm)
300 200 100 0 -100
700
0.73σff (0.26 nm)
600
1.27σff (0.45 nm)
500
1.82σff (0.65 nm)
400
2.30σff (0.82 nm)
300
2.78σff (0.99 nm)
δ=0.37 σff
200 100 0 -100 -200
-200
-300
-300 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
1
P/P0
0.1
1
P/P0
Fig. 9. Solvation pressure variation in the process of adsorption in pores of different size with smooth (A) and rough (B) walls.
1.0
A
120
H=1.82σff (0.65 nm)
100
δ=0 σff δ=0.08 σff
80
Solvation pressure,MPa
0.8
0.6 δ = 0 σff
3
B
δ = 0.08 σff
0.4
δ = 0.17 σff δ = 0.25 σff δ = 0.37 σff
0.2
δ = 0.45 σff
1E-6
1E-5
1E-4
1E-3
0.01
0.1
δ=0.25 σff
40
δ=0.37 σff
20
δ=0.45 σff
0
δ=0.56 σff
-20 -40 -60 -80 -100
δ = 0.56 σff
0.0 1E-7
δ=0.17 σff
60
-120 1
H= 1.82σff (0.65 nm)
1E-7
1E-6
1E-5
P/P0 1.0
C
120
H= 2.30σff (0.82 nm)
100
Solvation pressure, MPa
3
D
0.6 δ = 0 σff δ = 0.08 σff
0.4
δ = 0.17 σff δ = 0.25 σff δ = 0.37 σff
0.2
δ = 0.45 σff
1E-4
0.1
1
0.01
0.1
1
δ=0.08 σff
1E-3
0.01
0.1
P/P0
δ=0.17 σff
60
δ=0.25 σff
40
δ=0.37 σff
20
δ=0.45 σff
0
δ=0.56 σff
-20 -40 -60 -80 -100
δ = 0.56 σff
1E-5
0.01
δ=0 σff
80
1E-6
1E-3
P/P0
0.8
0.0 1E-7
1E-4
-120 H= 2.30σff (0.82 nm) 1
1E-7
1E-6
1E-5
1E-4
1E-3
P/P0
Fig. 10. Equilibrium adsorption isotherms and variation of solvation pressure in micropores: of size 1.82rff (0.65 nm) (A and B) and 2.30rff (0.82 nm) (C and D) with different surface roughness.
3.1. Dependence on the vapor pressure, different regimes in micropores and mesopores 3.1.1. Micropores – no capillary condensation In Fig. 8A and B, we present the equilibrium adsorption isotherms in commensurate and non-commensurate pores with
smooth (d = 0) and rough (d = 0.37rff) walls. The pore widths were varied from 0.73rff to 10.45rff (0.26–3.71 nm). This range of pore sizes includes micropores and small mesopores, in which the experimental adsorption isotherms of N2 at 77.4 K are reversible and do not reveal hysteresis [46]. The calculated equilibrium isotherms in pores wider than 4rff (1.42 nm) possess a step that
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10
A
0
4
δ=0 σff
2
B
δ=0 σff
-20
3.27σff (1.16 nm)
-30
4.23σff (1.50 nm)
Solvation pressure,MPa
Solvation pressure, MPa
0
-10
3.75σff (1.33 nm) 4.71σff (1.67 nm)
-40
5.19σff (1.84 nm) 5.66σff (2.01 nm)
-50
6.14σff (2.18 nm)
-60
6.62σff (2.35 nm)
1E-4
1E-3
-4 -6
7.57σff (2.69 nm)
-8
8.05σff (2.86 nm)
-10
8.52σff (3.02 nm)
-12
9.01σff (3.20 nm)
-14
9.50σff (3.37 nm)
-16
7.10σff (2.52 nm)
-70
-2
0.01
0.1
-18 1E-4
1
9.98σff (3.54 nm) 10.45σff(3.71 nm) 1E-3
0.01
0.1
1
P/P0
P/P0
Fig. 11. Variation of the solvation pressure in the course of adsorption in mesopores with smooth walls. The values of pore width are given in the graphs.
0.8
A
2
H= 9.5σff (3.37 nm)
H= 9.5σff (3.37 nm)
0.332
0.324
0.316
3
0.320
0.4
δ = 0 σff
0.312 0.308 0.504
0.506
0.508
0.510
δ = 0.08 σff
0.512
δ = 0.17 σff δ = 0.25 σff
0.2
δ = 0.37 σff δ = 0.45 σff
Solvation pressure, MPa
0.6
-2 -4 δ=0 σff δ=0.08 σff
-6
δ=0.17 σff δ=0.25 σff
-8
δ=0.37 σff
-10
δ=0.45 σff
δ = 0.56 σff
0.0
δ=0.56 σff
-12 0.0
B
0
0.328
0.2
0.4
0.6
0.8
1.0
P/P0
0.0
0.2
0.4
0.6
0.8
1.0
P/P0
Fig. 12. Effect of the surface roughness of equilibrium adsorption isotherm and solvation pressure in the mesopore of width H = 9.5rff (3.37 nm). The roughness parameter varies from 0 to 0.56rff.
corresponds to the capillary condensation transition. This transition was determined from the condition of thermodynamic equilibrium between the adsorbed film and the filled pore given as the equality of the respective grand thermodynamic potentials. The main qualitative difference between the adsorption isotherms in smooth and rough pores is that the effect of surface roughness eliminates the layering transitions that are characteristic to smooth pores larger than six molecular diameters. This artifact of the smooth wall pore models has been discussed in the literature with respect to the problem of pore size calculations from adsorption isotherms [47,48]. The variation of the solvation pressure in the course of adsorption is presented in Fig. 9 for micropores of width smaller than 3rff (1.06 nm). Two types of solvation pressure behavior depending on the pore size can be identified. In super-micropores that can accommodate only one dense layer of guest molecules, the solvation pressure monotonically increases. Due to the extreme confining effect, adsorption of guest molecules is necessarily associated with pore expansion. In micropores, wider than one molecular diameter, the solvation pressure decreases at low gas pressures and increases at high pressures. The decrease of the solvation pressure is associated with attractive solid–fluid interactions, which take place when the packing of guest molecules is loose. With the further increase of adsorption, the packing is densified and the hard-core repulsion interaction become dominant that lead
to the increase of the solvation pressure. This behavior is typical for microporous carbon [49,50] and other microporous materials like zeolites [50–52]. The effect of the surface roughness of the adsorption isotherm and solvation pressure is presented in Fig. 10 for two micropore sizes, commensurate pore of H = 1.82rff and non-commensurate pore H = 2.3rff. It is worth noting that the roughness affects differently the solvation pressure in commensurate and non-commensurate pores. As the roughness parameter increases, the solvation pressure decreases in the commensurate pore and increases in the non-commensurate pore. This means that the molecular level roughness leads to a frustration of ordering in commensurate pores and a densification of packing in non-commensurate pores. Behavior of the solvation pressure in mesopores is drastically different from that in micropores due to the importance of capillary forces. Fig. 11 shows the variation of solvation pressure in the course of adsorption in smooth-wall mesopores of widths from 3.27rff to 10.45rff. The solvation pressure drops step-wise as the capillary condensation occurs. After the capillary condensation, when the pore is filled the solvation pressure increases for all pore sizes. However, the behavior of the solvation pressure prior to the capillary condensation depends on the pore size. It may either decrease or increase that may lead either to adsorbent contraction or swelling. The solvation pressure isotherms with the step-wise drop at the capillary condensation transition and subsequent relaxation
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are well documented in the experimental literature and can be explained by a competition between the disjoining and capillary pressure; for an extended review and discussion see Gor and Neimark [53]. Due to such characteristic behavior of solvation pressure in mesopores, the surface roughness effects happen to be secondary and do not change much the adsorption and solvation pressure. It is illustrated in Fig. 12 drawing on the example of the mesopore of width H = 9.5rff. Also, the increase of the surface roughness eliminated layering characteristic the adsorption isotherm in smooth-wall pore prior to the capillary condensation, the variation of the roughness parameter from 0 to 0.56rff caused only marginal changes in the solvation pressure, which would be hardly distinguishable in experiments. 4. Conclusions The main conclusion of this work is that the surface roughness, even of the magnitude comparable with the size of adsorbed molecules, significantly affects the adsorption behavior and damps the oscillations of confined fluid density and solvation pressure as the functions of the gap size. This effect is caused by a frustration of the ordered molecular layering characteristic to the models with smooth surfaces. Also, the fact that the layering adsorption transitions, which are observed experimentally only in case of ideally smooth monocrystal surfaces like mica, are artifacts of the theoretical ‘‘smooth-wall’’ models has been adopted in the literature [47,48,54], the roughness effect on the solvation pressure has not been fully appreciated. The results presented in this work call for re-consideration of the theories of adhesion and adsorption on real surfaces of amorphous solids with accounting for their inherent molecular level roughness. The QSDFT model, which treats the surfaces roughness in terms of varying solid density at the solid–gas boundary, provides a rational for accounting for the roughness effects. In addition to a better understanding of the fundamentals of fluid adsorption on heterogeneous surfaces and inter-particle interactions, an important practical outcome is envisioned in modeling of adsorptioninduced deformation of compliant porous substrates. QSDFT allows one to calculate the solvation pressure exerted by the confined fluid on the pore walls that is proportional to the elastic deformation of the solid matrix [55]. In particular, the authors have successfully applied the QSDFT model employed in this work for analyses of the deformation of coal induced by adsorption of methane and CO2 at geological conditions that is one of the important unresolved problems at CO2 sequestration in coal seams [56,57]. Acknowledgments This work was supported in parts by the NSF ERC ‘‘Structured Organic Particulate Systems’’, the National Science Foundation of China (Grant #40973029), and the Blaise Pascal International Research Chair fellowship to A.V.N. References [1] B. Derjaguin, E. Obuchov, Acta Physicochim. URSS 5 (1936) 1.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Solvation forces between molecularly rough surfaces Kan Yang a,b, Yangzheng Lin b, Xiancai Lu a, Alexander V. Neimark b,⇑ a b
State Key Lab for Mineral Deposit Research, School of Earth Sciences and Engineering, Nanjing University, 22 Hankou Road, Nanjing 210093, PR China Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, NJ 08854, USA
a r t i c l e
i n f o
Article history: Received 25 April 2011 Accepted 23 June 2011 Available online 30 June 2011 Keywords: Solvation pressure Surface roughness QSDFT Adsorption and Adhesion
a b s t r a c t Surface heterogeneity affects significantly wetting and adhesion properties. However, most of the theories and simulation methods of calculating solid–fluid interactions assume a standard thermodynamic model of the Gibbs’ dividing solid–fluid interface, which is molecularly smooth. This assumption gives rise to a layering of the fluid phase near the surface that is displayed in oscillating density profiles in any theories and simulation models, which account for the hard core intermolecular repulsion. This layering brings about oscillations of the solvation (or disjoining) pressure as a function of the gap distance, which are rarely observed in experiments, except for ideal monocrystal surfaces. We present a detailed study of the effects of surface roughness on the solvation pressure of Lennard-Jones (LJ) fluids confined by LJ walls based on the quenched solid density functional theory (QSDFT). In QSDFT, the surface roughness is quantified by the roughness parameter, which represents the thickness of the surface ‘‘corona’’ – the region of varying solid density. We show that the surface roughness of the amplitude comparable with the fluid molecular diameter effectively damps the oscillations of solvation pressure that would be observed for molecularly smooth surfaces. The calculations were done for the LJ model of nitrogen sorption at 74.4 K in slit-shaped carbon nanopores to provide an opportunity of comparing with standard adsorption experiments. In addition to a better understanding of the fundamentals of fluid adsorption on heterogeneous surfaces and inter-particle interactions, an important practical outcome is envisioned in modeling of adsorption-induced deformation of compliant porous substrates. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Interactions between solid surfaces separated by thin liquid layers are characterized in terms of disjoining, or solvation pressure, which depends on the state of fluid in the gap between the surfaces and the gap width. The notion of disjoining pressure was first introduced by Derjaguin in 1936, who found that two mica surfaces in water repel each other, and this ‘‘disjoining’’ effect increases as the gap width decreases from 1 to 0.04 lm [1]. Later, with the progress in experimental techniques and the development of the modern surface force apparatus and atomic force microscope, the spatial resolution was extended down to the sub-nanometer scale [2,3]. It was established that the pressure exerted by fluid films onto confining solid surfaces is not always positive; it may be negative or co-joining, depending on the specifics of solid–fluid intermolecular interactions. In this respect, the term ‘‘solvation pressure’’ is better suited for the following discussion, since it does not bear a connotation that thin films should always ‘‘disjoin’’ the confining surfaces. Moreover, Israelachvili and Pashley [4] showed that the solvation pressure between mica surfaces ⇑ Corresponding author. E-mail address: [email protected] (A.V. Neimark). 0021-9797/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2011.06.056
in aqueous solution oscillates at separations 1.5 nm with a mean periodicity of 0.25 ± 0.03 nm, roughly the diameter of water molecule. They proposed that the origin of these oscillations is primarily due to the ordered layering of water molecules at mica surfaces. Similar oscillations of the solvation pressure as a function of the film thickness have been found for various polar and non-polar solvents and complex fluids [5–13]. The oscillating behavior of fluid density and solvation pressure in thin layers was confirmed in various thermodynamic models and molecular simulations performed assuming ideal ‘‘mathematically’’ smooth solid surfaces [14–17]. This traditional approach stems from the Gibbs definition of dividing solid–fluid interfaces, which provides zero excess of solid density [18]. The density oscillation in confined films is caused by the hard-core repulsion between the fluid molecules and between the fluid molecules and the surface that facilitates packing of fluid molecules near the surface in layers similarly to the layering of billiard balls compressed between two parallel bars. This effect was studied extensively in the works of Henderson [19] and others. It was shown that introducing soft solid–fluid interactions [19–21] or imposing various defects or irregularities on the surface [22,23], the layered order can be frustrated and, thus, the density oscillations damped. At the same time, as shown by Schoen and Dietrich [24] and
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The density q(r) of the adsorbate, confined in a pore at given chemical potential l and temperature T, is determined by the minimization of the grand thermodynamic potential: X½qðrÞ ¼ R F½qðrÞ l m qðrÞdr. Helmholtz free energy F[q(r)] is a functional of the fluid density q(r) as prescribed in Rosenfeld fundamental measure theory [43]. Equilibrium density profile qðr; l; V; TÞ is determined from the solution of the Euler equation l ¼ @@Fq [36]. In NLDFT model, the adsorption stress is calculated by as a derivative of the grand thermodynamic potential with respect to the pore width at given external thermodynamic parameters, ra ¼ A1 @@HX T;l;A , where A is the surface area and H is the pore width [17]. The density profiles at saturation conditions are presented in Figs. 1 and 2 respectively for commensurate and non-commensurate pores of different widths. Commensurate pores are defines as the pores that possess maximally dense packings of integer numbers of layers. These pores correspond to maxima of the average fluid density as a function of the pore width, Fig. 3. Minima of the average density are achieved in non-commensurate pores, where the molecular packings are loose. The density profiles in commensurate pores show pronounced oscillations from layer to layer with almost equal distance between the maxima that can be associated the mean layer positions. Thus defined inter-layer distance constitutes 0.95rff in the commensurate pores, which accommodate 2 layers and more. The distance between the first layer position and the wall is 0.45rff. The density profiles in
3
ρσff
Henderson et al. [25], the surface corrugations or grooves of the supra-molecular scale (larger than 5–10 fluid molecular diameters) do not obscure the molecular layering. The solvation pressure oscillations induced by supra-molecular corrugations on crystal surfaces, scale of which is comparable with the size of the AFM tip, can be probed with AFM as considered in by O’Shea et al. [26]. A detailed discussion on how the size of surface corrugations and tiling affects the pressure oscillating was given by Fink and van Swol [23]. The conclusion that confined fluid layers of molecular thickness possess an ordered layered structure causing the oscillating behavior of the solvation pressure has been widely adopted in the literature. However, it is worth noting that all experiments showing oscillating solvation pressures were performed with molecularly smooth surfaces like mica, so that the ordering of adsorbed molecules was not frustrated by chemical or geometrical surface heterogeneities of the molecular scale. At the same time, most of the non-crystalline surfaces of practical interest, such as amorphous oxides and carbons, are molecularly rough as shown by molecular simulations [27–29] and directly by XRD experiments [30–33]. In particular, the density of amorphous silica at the surface decreases from its bulk value to null within a certain layer of varying density called ‘‘corona’’. The corona thickness is of the order of molecular diameters, and depending on the material, it may extend from fractions to a few nanometers, accounting for the molecular level roughness of the solid surface [30,33,34]. In this work, we analyze the effects of molecular level roughness, drawing on the example of Lennard-Jones (LJ) fluid confined by parallel LJ walls. We study the dependence of the fluid density and the solvation pressure on the pore width and the degree of surface roughness, as well as, the variation of the solvation pressure in pores during the process of adsorption and capillary condensation. The latter problem has practical implications in the phenomena of adsorption-induced deformation of adsorbents, which is discussed also. For these purposes, we adopted the quenched solid density functional theory (QSDFT) that was recently introduced to model adsorption at the rough surfaces [35]. The rest of the paper is structured as follows. In Section 2, we set a stage for the analysis of surface roughness effects by presenting a detailed account of fluid behaviors in the pores with smooth walls given with the non-local density functional theory (NLDFT). This data serves as a reference for the QSDFT studies of the damping of the density and solvation pressure oscillations as the surface roughness increases, which are presented in Section 3. Special attention is paid to the investigation of adsorption in slit-shaped micropores and mesopores, which shows a variety of different non-monotonic solvation pressure isotherms. The conclusions are summarized in Section 4.
26 24 22 20 18 16 14 12 10 8 6 4 2 0
δ= 0 σff
0.73σff (0.26 nm) 1.82σff (0.65 nm) 2.78σff (0.99 nm) 3.75σff (1.33 nm) 4.71σff (1.67 nm) 5.66σff (2.01 nm) 6.62σff (2.35 nm) 7.57σff (2.69 nm) 8.52σff (3.02 nm) 9.50σff (3.37 nm) 10.45σff (3.71nm)
0
2
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z, σff Fig. 1. Density profiles in commensurate pores with smooth walls. Curves are shifted by 2qr3ff .
22 δ= 0 σff
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2. Confined fluid density and solvation pressure in pores with smooth walls
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1.27σff (0.45 nm)
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To demonstrate the specifics of the density distribution and solvation pressure obtained with the models of smooth pore walls, we employ the conventional NLDFT approach that is widely used to study the fluid density distribution in nanopores [17,36–39]. In the NLDFT approach, the adsorption and desorption isotherms in pores are calculated based on the intermolecular potentials of fluid–fluid and solid–fluid interactions. Fluid–fluid interactions are modeled by a pair-wise LJ potential in the WCA approximation [40], and fluid–solid interactions are modeled by an external Steele potential [41]. We consider slit pores and employ the fluid–fluid and fluid–solid LJ interaction parameters suggested for N2 adsorption on carbons, eff/kB = 95.77 K, rff = 0.3549 nm; esf/kB = 150 K, rsf = 0.269 nm [42].
ρσff
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9.98σff (3.54 nm)
0 0
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Pore width, σff Fig. 2. Density profiles in non-commensurate pores with smooth walls. The width of non-commensurate pore is determined by the average of pore widths of adjacent commensurate pores. Curves are shifted by 2qr3ff .
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1.4
1000 solvation pressure average fluid density
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non-commensurate pores display less pronounced oscillations compared to those in commensurate pores. The layered structure is leveled due to packing disordering in the pore center beyond the first 2–3 ordered layers at the pore walls. The solvation pressure as a function of the pore width exhibits oscillation correlated with the oscillation of the mean density. That is the solvation pressure maxima corresponded to the most ordered states in commensurate pores and minima – to the least ordered states in non-commensurate pores. This behavior is typical for any theories of solvation pressure in the gap between smooth surfaces [2,17,20,44,45]. 3. The effect of surface roughness on confined fluid distribution and solvation pressure Using the NLDFT results for smooth surfaces as a reference, we demonstrate the effects of surface roughness by means of the QSDFT model [35,42]. The QSDFT model is based on the multicomponent density functional theory, in which the grand thermodynamic potential X is defined as a functional of the solid qs(r) and fluid q(r) densities
6
8
10
Fig. 5. Density profiles in commensurate pores with rough walls of the same roughness parameter d = 0.2 nm (0.56rff). Curves are shifted by 2qr3ff . Compare with Fig. 1.
X½qs ðrÞ; qðrÞ ¼ F id ½qðrÞ þ e F ex ½qs ðrÞ; qðrÞ
Z Z 1 drdr0 qðrÞqðrÞ0 uff ðjr r0 jÞ 2 Z Z þ drdr0 qðrÞqs ðrÞ0 usf ðjr r0 jÞ Z l drqðrÞ þ
ð1Þ
Here, e F ex ½qs ðrÞ; qðrÞ ¼ F ex ½qs ðrÞ; qðrÞ F ex ½qs ðrÞ; qðrÞ 0 is the excess Helmholtz free energy of the reference hard sphere fluid, Fid and Fex are the ideal and excess components of the Helmholtz free energy, l is the fluid chemical potential, and uff(|r r0 |) and usf(|r r0 |) are the attractive parts of the fluid–fluid intermolecular potential and the solid–fluid intermolecular potential. Both the solid–fluid and fluid–fluid interactions are split into hard-sphere repulsive and mean-field attractive parts with the WCA approximation similarly to the fluid–fluid interactions in NLDFT. Hard-sphere interactions are treated with the multicomponent fundamental measure theory (FMT) functional for quenched-annealed systems [43]. The equilibrium density distribution of fluid is determined by minimizing the grand thermodynamic potential with respect
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A
solid density fluid density
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Pore width=5σff, δ=0 nm
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4
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ρσff3
ρσff3
pore edge
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Fig. 3. Plot of average densities and solvation pressure at smooth surface condition.
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solid density fluid density
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Fig. 4. Density profiles in the pore of width H = 5rff at the saturation conditions for the LJ model of N2 adsorption on carbon at 77.4 K. (A) Smooth pore walls, (B) molecularly rough pore walls with the roughness parameter d = 0.2 nm (0.56rff). The solid density profile is given by Eq. (3). The pore diameter is defined from the Gibbs condition of zero excess solid density that implies the equality of the shaded areas separated by the vertical line denoting the ‘‘theoretical’’ pore edge.
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pore wall edge, and, as such, to consider the molecular level surface roughness in the form of corona. In our calculation, the surface roughness is modeled as a corona of linearly reducing density (Fig. 4), which is described by the following equation [42]:
1.5 1.4 1.3 δ=0.56 σff
8 0 q 0 6 z < h0 > > < s 0 qs ðzÞ ¼ 0:75q0s 1 zh h0 6 z < h0 þ 2d 2d > > : 0 z P h0 þ 2d
δ=0.45 σff δ=0.37 σff
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Pore width, σff Fig. 6. Pore width dependence of average density at P/P0 = 1 in pores with different roughness. Curves are shifted by 0:1qr3ff .
Solvation pressure, MPa
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0 -200 0
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Pore width, σff Fig. 7. Pore size dependence of solvation pressure at P/P0 = 1 in pores with different roughness. Curves are shifted by 100 MPa.
to the fluid density q(r) keeping the solid density qs(r) fixed (this is where the term ‘quenched’’ stems from). This yields the Euler– Lagrange equation,
dX½qs ðrÞ; qðrÞ=dqðrÞ ¼ 0
ð2Þ
that is solved in an iteration fashion [42]. Thus, in contrast with NLDFT, QSDFT accounts for the fluid–solid interactions through pair wise intermolecular potentials rather than through an effective external potential. This approach allows one to take into account the in homogeneous solid density at the 0.9
ff
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ff
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A δ=0 σ
0.5 0.4 0.3 0.2 0.1 0.0 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
P/P0
0.1
1
ff 2.78σ (0.99 nm) ff 3.75σ (1.33 nm) ff 4.71σ (1.67 nm) ff 5.66σ (2.01 nm) ff 6.62σ (2.35 nm) ff 7.57σ (2.69 nm) ff 8.52σ (3.02 nm) ff 9.50σ (3.37 nm) ff 10.45σ (3.71 nm) ff 1.27σ (0.45 nm) ff 2.30σ (0.82 nm) ff 3.27σ (1.16 nm) ff 4.20σ (1.50 nm) ff 5.19σ (1.84 nm) ff 6.14σ (2.18 nm) ff 7.10σ (2.52 nm) ff 8.05σ (2.86 nm) ff 9.01σ (3.20 nm) ff 9.98σ (3.54 nm) ff
B δ=0.37σ
0.73σ (0.26 nm) ff
1.82σ (0.65 nm)
ff
0.8
ff
2.78σ (0.99 nm) ff
3.75σ (1.33 nm) ff
0.7
4.71σ (1.67 nm) ff
5.66σ (2.01 nm) ff
3
0
ð3Þ
Here, qs = 0.114 Å3 is the density of bulk solid; h0 is the thickness of the solid wall assumed to be equal to h0 = 2 0.34 nm, in accord with [34]. The roughness parameter d, defined as the half-thickness of the corona, is used to characterize the surface roughness. Pore diameter is defined from the Gibbs condition of zero excess solid density (Fig. 4B). A typical example of calculated density profiles with NLDFT and QSDFT models is presented in Fig. 4. Even with such small roughness parameter as d = 0.2 nm (0.56rff), the density fluctuations are significantly leveled, compared to the reference density between smooth walls. Fluid penetrates into the outer part of the corona and its distribution across the pore is, in overall, more homogeneous. The destruction of layering reflects a disordered packing of molecules confined by rough walls. The density profiles in commensurate pores with rough walls of the same roughness parameter d = 0.2 nm (0.56rff) are presented in Fig. 5. They should be compared with those given in Fig. 1. The density oscillations are damped in all pores that accommodate two and more layers. Note that the surface roughness effect destructs layering to a lager extent than the pore size mismatch, Fig. 2. In Figs. 6 and 7, we present the dependences of the mean fluid density and the solvation pressure on the pore width for different roughness parameters at the saturation conditions. The correlated mean density and solvation pressure oscillations characteristic for the pores with smooth walls, are gradually damped as the roughness parameter increases. The oscillations of the solvation pressure disappear for the roughness parameter exceeding d = 0.37rff (0.13 nm). It is worth noting that this value of the roughness parameter was used earlier in [42] to model adsorption on carbons. It was shown that the adsorption of N2 and Ar on non-porous carbon Vulcan is adequately described by the QSDFT model with d = 0.13 nm. When pore wall roughness d P 0.37rff (0.13 nm), the solvation force at the saturation conditions monotonically decays with the pore width. The solvation pressure is disjoining and, as such, the porous sample should exhibit swelling upon saturation with the adsorbate. However, as shown below, the solvation pressure may vary nonmonotonically in the process of adsorption–desorption cycles.
1.2
0.6
6.62σ (2.35 nm)
0.5
8.52σ (3.02 nm)
ff
7.57σ (2.69 nm) ff ff
9.50σ (3.37 nm) ff
10.45σ (3.71 nm)
0.4
ff
1.27σ (0.45 nm) ff
2.30σ (0.82 nm)
0.3 0.2 0.1 0.0 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
1
ff 3.27σ ff 4.20σ ff 5.19σ ff 6.14σ ff 7.10σ ff 8.05σ ff 9.01σ ff 9.98σ ff
(1.16 nm) (1.50 nm) (1.84 nm) (2.18 nm) (2.52 nm) (2.86 nm) (3.20 nm) (3.54 nm)
P/P0
Fig. 8. (A and B): Equilibrium adsorption isotherms in pores with smooth (d = 0) and rough (d = 0.37rff) walls. The pore sizes are given in the right panels.
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K. Yang et al. / Journal of Colloid and Interface Science 362 (2011) 382–388
A
800
δ=0 σff
0.73σff (0.26 nm) 1.27σff (0.45 nm)
600
1.82σff (0.65 nm)
500
2.30σff (0.82 nm)
400
B
800
Solvation pressure, MPa
Solvation pressure, MPa
700
2.78σff (0.99 nm)
300 200 100 0 -100
700
0.73σff (0.26 nm)
600
1.27σff (0.45 nm)
500
1.82σff (0.65 nm)
400
2.30σff (0.82 nm)
300
2.78σff (0.99 nm)
δ=0.37 σff
200 100 0 -100 -200
-200
-300
-300 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
1
P/P0
0.1
1
P/P0
Fig. 9. Solvation pressure variation in the process of adsorption in pores of different size with smooth (A) and rough (B) walls.
1.0
A
120
H=1.82σff (0.65 nm)
100
δ=0 σff δ=0.08 σff
80
Solvation pressure,MPa
0.8
0.6 δ = 0 σff
3
B
δ = 0.08 σff
0.4
δ = 0.17 σff δ = 0.25 σff δ = 0.37 σff
0.2
δ = 0.45 σff
1E-6
1E-5
1E-4
1E-3
0.01
0.1
δ=0.25 σff
40
δ=0.37 σff
20
δ=0.45 σff
0
δ=0.56 σff
-20 -40 -60 -80 -100
δ = 0.56 σff
0.0 1E-7
δ=0.17 σff
60
-120 1
H= 1.82σff (0.65 nm)
1E-7
1E-6
1E-5
P/P0 1.0
C
120
H= 2.30σff (0.82 nm)
100
Solvation pressure, MPa
3
D
0.6 δ = 0 σff δ = 0.08 σff
0.4
δ = 0.17 σff δ = 0.25 σff δ = 0.37 σff
0.2
δ = 0.45 σff
1E-4
0.1
1
0.01
0.1
1
δ=0.08 σff
1E-3
0.01
0.1
P/P0
δ=0.17 σff
60
δ=0.25 σff
40
δ=0.37 σff
20
δ=0.45 σff
0
δ=0.56 σff
-20 -40 -60 -80 -100
δ = 0.56 σff
1E-5
0.01
δ=0 σff
80
1E-6
1E-3
P/P0
0.8
0.0 1E-7
1E-4
-120 H= 2.30σff (0.82 nm) 1
1E-7
1E-6
1E-5
1E-4
1E-3
P/P0
Fig. 10. Equilibrium adsorption isotherms and variation of solvation pressure in micropores: of size 1.82rff (0.65 nm) (A and B) and 2.30rff (0.82 nm) (C and D) with different surface roughness.
3.1. Dependence on the vapor pressure, different regimes in micropores and mesopores 3.1.1. Micropores – no capillary condensation In Fig. 8A and B, we present the equilibrium adsorption isotherms in commensurate and non-commensurate pores with
smooth (d = 0) and rough (d = 0.37rff) walls. The pore widths were varied from 0.73rff to 10.45rff (0.26–3.71 nm). This range of pore sizes includes micropores and small mesopores, in which the experimental adsorption isotherms of N2 at 77.4 K are reversible and do not reveal hysteresis [46]. The calculated equilibrium isotherms in pores wider than 4rff (1.42 nm) possess a step that
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K. Yang et al. / Journal of Colloid and Interface Science 362 (2011) 382–388
10
A
0
4
δ=0 σff
2
B
δ=0 σff
-20
3.27σff (1.16 nm)
-30
4.23σff (1.50 nm)
Solvation pressure,MPa
Solvation pressure, MPa
0
-10
3.75σff (1.33 nm) 4.71σff (1.67 nm)
-40
5.19σff (1.84 nm) 5.66σff (2.01 nm)
-50
6.14σff (2.18 nm)
-60
6.62σff (2.35 nm)
1E-4
1E-3
-4 -6
7.57σff (2.69 nm)
-8
8.05σff (2.86 nm)
-10
8.52σff (3.02 nm)
-12
9.01σff (3.20 nm)
-14
9.50σff (3.37 nm)
-16
7.10σff (2.52 nm)
-70
-2
0.01
0.1
-18 1E-4
1
9.98σff (3.54 nm) 10.45σff(3.71 nm) 1E-3
0.01
0.1
1
P/P0
P/P0
Fig. 11. Variation of the solvation pressure in the course of adsorption in mesopores with smooth walls. The values of pore width are given in the graphs.
0.8
A
2
H= 9.5σff (3.37 nm)
H= 9.5σff (3.37 nm)
0.332
0.324
0.316
3
0.320
0.4
δ = 0 σff
0.312 0.308 0.504
0.506
0.508
0.510
δ = 0.08 σff
0.512
δ = 0.17 σff δ = 0.25 σff
0.2
δ = 0.37 σff δ = 0.45 σff
Solvation pressure, MPa
0.6
-2 -4 δ=0 σff δ=0.08 σff
-6
δ=0.17 σff δ=0.25 σff
-8
δ=0.37 σff
-10
δ=0.45 σff
δ = 0.56 σff
0.0
δ=0.56 σff
-12 0.0
B
0
0.328
0.2
0.4
0.6
0.8
1.0
P/P0
0.0
0.2
0.4
0.6
0.8
1.0
P/P0
Fig. 12. Effect of the surface roughness of equilibrium adsorption isotherm and solvation pressure in the mesopore of width H = 9.5rff (3.37 nm). The roughness parameter varies from 0 to 0.56rff.
corresponds to the capillary condensation transition. This transition was determined from the condition of thermodynamic equilibrium between the adsorbed film and the filled pore given as the equality of the respective grand thermodynamic potentials. The main qualitative difference between the adsorption isotherms in smooth and rough pores is that the effect of surface roughness eliminates the layering transitions that are characteristic to smooth pores larger than six molecular diameters. This artifact of the smooth wall pore models has been discussed in the literature with respect to the problem of pore size calculations from adsorption isotherms [47,48]. The variation of the solvation pressure in the course of adsorption is presented in Fig. 9 for micropores of width smaller than 3rff (1.06 nm). Two types of solvation pressure behavior depending on the pore size can be identified. In super-micropores that can accommodate only one dense layer of guest molecules, the solvation pressure monotonically increases. Due to the extreme confining effect, adsorption of guest molecules is necessarily associated with pore expansion. In micropores, wider than one molecular diameter, the solvation pressure decreases at low gas pressures and increases at high pressures. The decrease of the solvation pressure is associated with attractive solid–fluid interactions, which take place when the packing of guest molecules is loose. With the further increase of adsorption, the packing is densified and the hard-core repulsion interaction become dominant that lead
to the increase of the solvation pressure. This behavior is typical for microporous carbon [49,50] and other microporous materials like zeolites [50–52]. The effect of the surface roughness of the adsorption isotherm and solvation pressure is presented in Fig. 10 for two micropore sizes, commensurate pore of H = 1.82rff and non-commensurate pore H = 2.3rff. It is worth noting that the roughness affects differently the solvation pressure in commensurate and non-commensurate pores. As the roughness parameter increases, the solvation pressure decreases in the commensurate pore and increases in the non-commensurate pore. This means that the molecular level roughness leads to a frustration of ordering in commensurate pores and a densification of packing in non-commensurate pores. Behavior of the solvation pressure in mesopores is drastically different from that in micropores due to the importance of capillary forces. Fig. 11 shows the variation of solvation pressure in the course of adsorption in smooth-wall mesopores of widths from 3.27rff to 10.45rff. The solvation pressure drops step-wise as the capillary condensation occurs. After the capillary condensation, when the pore is filled the solvation pressure increases for all pore sizes. However, the behavior of the solvation pressure prior to the capillary condensation depends on the pore size. It may either decrease or increase that may lead either to adsorbent contraction or swelling. The solvation pressure isotherms with the step-wise drop at the capillary condensation transition and subsequent relaxation
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are well documented in the experimental literature and can be explained by a competition between the disjoining and capillary pressure; for an extended review and discussion see Gor and Neimark [53]. Due to such characteristic behavior of solvation pressure in mesopores, the surface roughness effects happen to be secondary and do not change much the adsorption and solvation pressure. It is illustrated in Fig. 12 drawing on the example of the mesopore of width H = 9.5rff. Also, the increase of the surface roughness eliminated layering characteristic the adsorption isotherm in smooth-wall pore prior to the capillary condensation, the variation of the roughness parameter from 0 to 0.56rff caused only marginal changes in the solvation pressure, which would be hardly distinguishable in experiments. 4. Conclusions The main conclusion of this work is that the surface roughness, even of the magnitude comparable with the size of adsorbed molecules, significantly affects the adsorption behavior and damps the oscillations of confined fluid density and solvation pressure as the functions of the gap size. This effect is caused by a frustration of the ordered molecular layering characteristic to the models with smooth surfaces. Also, the fact that the layering adsorption transitions, which are observed experimentally only in case of ideally smooth monocrystal surfaces like mica, are artifacts of the theoretical ‘‘smooth-wall’’ models has been adopted in the literature [47,48,54], the roughness effect on the solvation pressure has not been fully appreciated. The results presented in this work call for re-consideration of the theories of adhesion and adsorption on real surfaces of amorphous solids with accounting for their inherent molecular level roughness. The QSDFT model, which treats the surfaces roughness in terms of varying solid density at the solid–gas boundary, provides a rational for accounting for the roughness effects. In addition to a better understanding of the fundamentals of fluid adsorption on heterogeneous surfaces and inter-particle interactions, an important practical outcome is envisioned in modeling of adsorptioninduced deformation of compliant porous substrates. QSDFT allows one to calculate the solvation pressure exerted by the confined fluid on the pore walls that is proportional to the elastic deformation of the solid matrix [55]. In particular, the authors have successfully applied the QSDFT model employed in this work for analyses of the deformation of coal induced by adsorption of methane and CO2 at geological conditions that is one of the important unresolved problems at CO2 sequestration in coal seams [56,57]. Acknowledgments This work was supported in parts by the NSF ERC ‘‘Structured Organic Particulate Systems’’, the National Science Foundation of China (Grant #40973029), and the Blaise Pascal International Research Chair fellowship to A.V.N. References [1] B. Derjaguin, E. Obuchov, Acta Physicochim. URSS 5 (1936) 1.
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