Adsorption Thicknesses of Confined Pure and Mixing Fluids in

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Article Cite This: Langmuir 2018, 34, 12815−12826

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Adsorption Thicknesses of Confined Pure and Mixing Fluids in Nanopores Kaiqiang Zhang,† Na Jia,*,† and Lirong Liu‡ †

Petroleum Systems Engineering, Faculty of Engineering and Applied Science and ‡Institute for Energy, Environment and Sustainable Communities, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

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S Supporting Information *

ABSTRACT: In this paper, adsorption thicknesses of confined pure and mixing fluids in nanopores are quantitatively determined and their influential factors are specifically evaluated. First, a new analytical formulation is developed thermodynamically to calculate the adsorption thicknesses. Second, a new generalized equation of state (EOS), which considers the confinement effect-induced phenomena, is developed analytically for calculating the thermodynamic confined fluid phase behavior. Third, the modified model based on the generalized EOS and coupled with the parachor model is applied to calculate the vapor−liquid equilibrium (VLE) and fluid adsorptions for the pure CO2, alkanes of C1−C10, and two mixtures of CO2−C10H22 and CH4−C10H22 in nanopores. Finally, the following five important factors are studied to evaluate their effects on the adsorption thickness: temperature, pressure, pore radius, wall-effect distance, and feed gas-to-liquid ratio (FGLR). The proposed modified EOS is found to be accurate for the VLE and adsorption isotherm calculations. The adsorption thicknesses of confined pure or mixing alkanes are increased with the increasing carbon number but decreased with the temperature increase. For the alkanes of C1−C10, the degree of temperature effect is strengthened with the carbon number increase. Moreover, the adsorption thicknesses are significantly decreased with the pore radius increase until rp = 50 nm, after which they have slight changes or are even constant at any pore radii. On the other hand, the wall-effect distance (δp) increase causes the adsorption thickness to be linearly increased at δp/rp ≥ 0.02. In addition, the effects of the FGLR and pressure on the adsorption thicknesses at the nanoscale are found to be negligible.

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INTRODUCTION Numerous physical, chemical, physiochemical, or biological processes take place at the boundary of the two phases, such as the surface or interfacial tensions (IFTs),1,2 wettability,3,4 fluid adsorptions,5,6 etc. The fluid adsorption is referred to be the corresponding concentration changes of a given fluid at the interface in comparison with the other contacting phase.7,8 In this case, the term “fluid” denotes the gas or liquid contacting the boundary surface of the solids/walls. In theory, the adsorption system is defined as an equilibrium separate phase containing the adsorbent that contacts with the bulk fluids and the interfacial layer that consists of the part of fluid residing under the force field of the solid surface and the surface layer of the solid.9 In thermodynamics, the adsorbate−adsorbent (adsorption) system is considered to be a two-dimensional fluid or a solution where the adsorbate is the solute and vacancies act as the solvent.10 Although various qualitative and quantitative parameters can be determined for describing the fluid adsorption, the adsorption isotherm, which correlates the quantity of the adsorbed materials with the pressure/ concentration/temperature in the bulk fluid phase, has always been treated as an important concept in the adsorption study.7,11 A number of theoretical models,12,13 numerical simulations,14,15 and experimental methods16,17 have been conducted © 2018 American Chemical Society

to determine the adsorption isotherms. More specifically, some common simplified models (including the equations of state) and density functional theory (DFT) are two main theoretical approaches for calculating the adsorption isotherms. The Langmuir model is the most famous adsorption model, which states the fluid adsorbs onto a flat surface on the basis of the kinetic point of view.18 The Langmuir model is extended to be the multicomponent Langmuir model19 or coupled with the ideal adsorbed solution theory20 for calculating the mixture adsorptions on the basis of the pure substance adsorption isotherm. In the classical Gibbs thermodynamic approach, the volume and pressure in bulk phase are replaced by the area and spreading pressure, on the basis of which various twodimensional adsorption isotherm equations, such as the Henry, Volmer, Hill−de Boor, and Fowler−Guggenheim, are developed.21−23 Another model, the vacancy solution model, which assumes the surface of an adsorbent to be treated as a solvent and the adsorption process to be the formation of the adsorbate−adsorbent solution, is also widely used to calculate the adsorption isotherms.24 Some cubic equations of state (EOS), such as the van der Waals (vdW) and Peng−Robinson Received: August 28, 2018 Revised: October 2, 2018 Published: October 9, 2018 12815

DOI: 10.1021/acs.langmuir.8b02925 Langmuir 2018, 34, 12815−12826

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Table 1. Recorded Critical Properties (i.e., Temperature, Pressure, and Volume), van der Waals Equation of State (EOS) Constants, and Lennard-Jones and Square-Well Potential Parameters of CO2 and C1−C1041−43 component CO2 CH4 C2H6 C3H8 n-C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

Tc (K)

Pc (Pa)

304.2 190.6 305.4 369.8 425.2 469.6 507.5 543.2 570.5 598.5 622.1

× × × × × × × × × × ×

7.38 4.60 4.88 4.25 3.80 3.37 3.29 3.14 2.95 2.73 2.53

Vc (m3/mol) 6

10 106 106 106 106 106 106 106 106 106 106

9.40 9.90 1.48 2.03 2.55 3.04 3.44 3.81 4.21 4.71 5.21

× × × × × × × × × × ×

−5

10 10−5 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4

a (Pa m6/mol2) 1.02 6.34 1.54 2.33 3.84 5.27 6.89 8.52 1.04 1.24 1.46

× × × × × × × × × × ×

−48

10 10−49 10−48 10−48 10−48 10−48 10−48 10−48 10−47 10−47 10−47

b (m3/mol) 7.12 7.14 1.08 1.59 1.94 2.41 2.91 3.38 3.82 4.49 5.07

× × × × × × × × × × ×

10−29 10−29 10−28 10−28 10−28 10−28 10−28 10−28 10−28 10−28 10−28

(ε/k)LJ (K) 294 207 155 120 118 145 199 206 213 220 226

σLJ (m) 2.95 3.57 3.61 3.43 3.91 3.96 4.52 4.70 4.89 5.07 5.23

× × × × × × × × × × ×

10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−10

(ε/k)sw (K) 179 105 174 193 211 213 223 231 237 238 245

Table 2. Measured44,45 and Calculated Bubble-Point Pressures for the N2−n-C4H10, C10H2−CH4, and C8H18−CH4 Mixtures at Different Temperatures and Pore Radii system (mol %) 5.40 N2−94.60 C4 5.01 N2−94.99 C4 90.00 C10−10.00 C1

T (°C) 26.0 51.0 26.0 51.0 38.0 52.0

90.00 C8−10.00 C1

38.0

rp (m) 5.0 × 10

3.5 3.7 3.5 3.7 3.5 3.7

× × × × × ×

−9

10−9 10−9 10−9 10−9 10−9 10−9

Pb‑BPa (kPa)

Pbb (kPa)

changec (%)

Pb‑EvdWd (kPa)

AD (%)

3390.6 3429.5 3115.1 3189.6 2579.0 2579.0 2717.0 2717.0 2503.0 2503.0

4268.5 4105.8 4035.5 3883.8 1669.0 2034.0 2220.0 2468.0 1765.0 2186.0

25.89 19.72 29.55 21.76 35.28 21.13 18.29 9.16 29.48 12.66

3967.1 3517.1 3823.7 3215.7 1988.9 1856.3 2642.3 2099.5 1901.7 1810.4

7.06 14.34 5.25 17.20 19.17 8.74 19.02 14.93 7.74 17.18 13.06 19.17

AAD (%) MAD (%) a

Measured bubble-point pressure in bulk phase. bMeasured bubble-point pressure at the pore radius of rp. cChange of the measured bubble-point pressure in bulk phase and porous medium in percentage. dCalculated bubble-point pressure at rp from the generalized equation of state.

EOS, have also been applied to study the confined fluid adsorptions in a simple and accurate manner.25,26 On the other hand, the DFT is usually applied to study the phase behavior in nanopores with the fluid adsorptions.27 Accordingly, some simplified forms of the DFT, such as the simplified local density28 and multicomponent potential theory of adsorption29 models, are extensively used in the mixing fluid adsorption modeling. However, the DFT requires the integral equations to be solved by the numerical iterations to calculate the fluid density profiles, which can be complicated and timeconsuming. Recently, numerical simulations, especially molecular simulations, are emerging for the confined fluid adsorption modeling and calculations, such as the molecular dynamics (MD),30 grand canonical Monte Carlo,31 Gibbs ensemble Monte Carlo,32 and configurational-bias Monte Carlo33 simulations. Molecular simulations are methods of tracking the phase trajectories of molecular systems via intermolecular potentials, which can be used to investigate some microscopic fluid properties, such as the equilibrium states or nonequilibrium motions.34,35 It is certain that the molecular simulations perform well to provide a direct route in terms of the microscopic details to macroscopic properties of a system and model the vapor−liquid equilibrium (VLE) at a wide range of temperatures and pressures from an atomistic viewpoint.36 However, some quantity of input data is required prior to the simulations, especially when the adsorbent has a wide pore size distribution (PSD). Also, the simulation results

are always limited to certain adsorbents and adsorbates in a specific small area.37 Experimental methods are always considered to be the most straightforward and accurate method for studying the fluid adsorptions, which are commonly conducted in two different ways: gravimetric and volumetric methods.38,39 However, most experimental methods are time-consuming, in particular, for some measurements at a nanometer scale, plus most of the existing nanoscale apparatus are expensive and intolerant to high pressures.40 Although numerous studies have been conducted with regard to the fluid isotherms, almost no research has been found in the public domain to specifically study the confined fluid adsorption thickness and analyze its influential factors in nanopores. On the other hand, the previously developed EOS usually partially considers the confinement effect-induced phenomena but no generalized EOS exists. In this study, first, an analytical formulation of the adsorption thickness for the pure and/or mixing fluids in nanopores is developed thermodynamically. Second, a new generalized EOS, which considers the pore radius effect, intermolecular interaction, and wall effect, is developed in the analytical formulation for calculating the thermodynamic phase behavior of confined pure and mixing fluids in nanopores. Third, the modified model based on the generalized EOS and coupled with the parachor model, which is fully capable of modeling the abovementioned confinement effect (also including capillary pressure and shifts of critical properties), is applied to calculate the VLE and fluid adsorptions in nanopores. The adsorption 12816

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Figure 1. Schematic diagram of the nanoscale pores, configurational energy, and fluid adsorption in nanopores.47

and/or mixing fluids in nanopores is derived by taking account of the confinement effects, which include the effects of pore radius, shifts of critical properties, molecule−molecule (m−m) and molecule−wall (m−w) interactions, and the interface curvature. Suppose that a closed system, as shown in Figure 1, consists of a pure component (1) in two phases (α denotes the free fluid, and β is the adsorbed fluid), the corresponding Gibbs free energy as an interfacial excess quantity is given by46

thicknesses of the confined pure and mixing fluids in nanopores are quantitatively calculated by using the developed formulation coupled with the modified EOS. Finally, the following five important factors are comprehensively studied to evaluate their effects on the adsorption thickness: temperature, pressure, pore radius, wall-effect distance, and feed gas-toliquid ratio (FGLR).

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EXPERIMENTAL SECTION

G(p , T ) = γA + μ1α N1α + μ1β N1β = U + PV − TS

In this study, pure CO2 and a series of alkanes from C1−C10 are used, whose critical properties (i.e., temperature, pressure, and volume), van der Waals (vdW) EOS constants, and Lennard-Jones and square-well potential parameters are summarized41−43 and listed in Table 1. The pressure−volume−temperature (PVT) tests for the N2−n-C4H10 system were conducted by using a conventional PVT apparatus connected to a high-temperature and pressure container with a shale coreplug at the temperatures of T = 26.0 and 51.0 °C.44 It should be noted that the shale coreplug was hydrocarbon-wetting and its dominant pores were around 5 nm, which is applied for the subsequent calculations in this study. The purities of N2 and n-C4H10 used in the experiments equal 99.998 and 99.99%, respectively. In addition, the PVT tests of the C10H22−CH4 and C8H18−CH4 systems were conducted at T = 38.0 and 52.0 °C and the pore radii of rp = 3.5 and 3.7 nm (i.e., silica-based mesoporous materials SBA-15 and SBA16), respectively.45 The measured bubble-point pressures of the above-mentioned systems are summarized and listed in Table 2.

(1)

where γ is the IFT between the vapor and liquid phases, A is the surface area of the interface, μ is the chemical potential, Nj is the mole number of the jth component, U is the internal energy, P is the system pressure, V is the volume, T is the temperature, and S is the entropy. Legendre transformation of the internal energy potential energy gives dU = T dS − P dV + γ dA + μ1α dN1α + μ1β dN1β

Given the fact that

(Nα1

+

Nβ1)

(2)

is constant, eq 2 is rewritten as

dU = T dS − P dV + γ dA + (μ1α − μ1β )dN1α

(3a)

dU = T dS − P dV + γ dA + (μ1β − μ1α )dN1β

(3b)

or

■

THEORY Adsorption Thickness. In this study, an analytical formulation for the adsorption thickness of the confined pure

In eq 3a or 3b, (μα1 − μβ1)dNα1 or (μβ2 − μα2 )dNβ2 is referred to be the chemical potential changes due to the phase transition. 12817

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Langmuir Physically, (μα1 − μβ1)dNα1 or (μβ2 − μα2 )dNβ2 represents the change of chemical potentials of the free and adsorbed fluids. Then, subtracting (γA + μα1 Nα1 + μβ1Nβ1) from the full differentiation of U + PV − TS yields48 −S dT + V dP − N1α dμ1α − N1β dμ1β = A dγ

where Pci is the critical pressure of component i, Tci is the critical temperature of component i, and ωi is the acentric factor of component i. Then, the Rachford−Rice equation is applied to calculate xi and yi N

(4)

∑

Both sides of eq 4 are divided by A

i=1

−s dT + δ dP − n1α dμ1α − n1β dμ1β = dγ

ÄÅ θ−1 ÅÅ B θ i by Å Z L3 − (1 + BL )Z L2 + ÅÅÅÅAL − L (NA ·εsw )jjj1 − zzz (1 − Fpr) ÅÅÅ RT V{ k Ç ÄÅ ÉÑ θ−1 2 ÅÅ ÑÑ B θ i by Å Ñ (1 − e−NAεsw / RT )ÑÑÑÑZ L − ÅÅÅÅAL BL − L (NA·εsw )jjj1 − zzz (1 − Fpr) RT V{ k ÅÅÅÇ ÑÑÑÖ ÉÑ ÑÑ Ñ (1 − e−NAεsw / RT )ÑÑÑÑ ÑÑÑ Ö

(6)

In the literature, the chemical potentials for each phase are found to be equivalent at the phase equilibrium48 so that μα1 = μβ1 = μ1. Also, the total mole number of the component 1 is constant so nα1 + nβ1 = n1. Accordingly δad =

dμ dγ + n1 1 dP dP

dγij dP

+

∑ nij

(7)

dμij (8)

dP

(12a)

=0

(12b)

where ZL and ZV are the respective compressibility factors of the liquid and vapor phases; Ä É P ÅÅÅ bP i c c yÑÑÑ AL = 2 L 2 ÅÅÅÅa−2εLJσLJ3·jjj 1 + 2 zzzÑÑÑÑ, BL = L , A {ÑÑÖ RT R T ÅÅÇ k A Ä É Å Ñ P ÅÅ bP i c c yÑÑ AV = 2 V 2 ÅÅÅÅa−2εLJσLJ3·jjj 1 + 2 zzzÑÑÑÑ, B V = V A {ÑÑÖ RT R T ÅÅÇ k A

Equations 7 and 8 are the analytical formulations for the adsorption thickness of the confined pure and mixing fluids in nanopores, which can be easily calculated when the IFT and chemical potential are given. Modified Equation of State. A generalized EOS for the confined fluid in nanopores is developed, which considers the above-mentioned confinement effects. Figure 1 shows the schematic diagrams of the nanopore network model, configuration energy, and fluid adsorptions in nanopores. The detailed derivations for the generalized EOS are specified in the Supporting Information, whose analytical formulation is shown as follows

Constants of a and b are obtained by applying the van der Waals mixing rule a=

É ÅÄ i c c yÑÑÑ nRT n2 ÅÅ P(N , V , T ) = − 2 ÅÅÅÅa − 2εLJσLJ3·jjjj 1 + 2 zzzzÑÑÑÑ V − nb A {ÑÑÖ V ÅÅÇ k A i nbθ yi nb zy zz + (nNA ·εsw )jjj 2 zzzjjj1 − V{ k V {k

=0

ÄÅ θ−1 ÅÅ B θ i by Å Z V3 − (1 + B V )Z V2 + ÅÅÅÅAV − V (NA ·εsw )jjj1 − zzz (1 − Fpr) ÅÅÅ RT V { k Ç ÅÄÅ ÑÉÑ θ−1 2 Å Ñ θ B i by Å Ñ (1 − e−NAεsw / RT )ÑÑÑÑZ V − ÅÅÅÅAV B V − V (NA ·εsw )jjj1 − zzz ÅÅÅ ÑÑÑ RT V { k Ç Ö ÑÉÑ Ñ Ñ (1 − Fpr)(1 − e−NAεsw / RT )ÑÑÑÑ ÑÑÑ Ö

In eq 7, δad is the distance between two phases of the single component, which is also denoted as the adsorption thickness. The adsorption thickness for a confined mixing fluid in nanopores is shown as δad =

(11)

where β is the vapor fraction. The compressibility of the liquid or vapor phase can be determined as follows

(5)

where s = S/A, δ = V/A, nα1 = Nα1 /A, and nβ1 = Nβ1/A. At a given constant temperature, eq 5 can be rearranged as α β dγ α dμ1 β dμ1 + n1 + n1 δ= dP dP dP

zi(K i − 1) =0 1 + (K i − 1)β

∑ ∑ xixjaij i

b=

j

∑ xibi i

θ−1

(1 − Fpr)(1 − e−NAεsw / RT )

(13a)

(13b)

where aij represents the binary interaction of component i and component J, aij = (1 − kij) aiaj ; kij is the binary interaction coefficient of component i and component j; kij = kji and kii = kjj = 0. Minimum Gibbs free energy is applied to select roots of the compressibility factors for the liquid and vapor phases.48 The liquid and vapor phases are assumed to be the wetting phase and nonwetting phase, respectively.41 Thus, the capillary pressure (Pcap) is

(9)

where R is the universal gas constant, a and b are the EOS constants, εLJ is the molecule−molecule Lennard-Jones energy parameter, σLJ is the molecule−molecule Lennard-Jones size parameter, c1 = 3.5622, c2 = −0.6649,49 Fpr is the fraction of the randomly distributed fluid molecules in the square-well region of the pores, θ is the geometric term, NA is the Avogadro constant, and εsw is the molecule−wall square-well energy parameter. The newly developed generalized EOS in nanopores from eq 9 is applied to calculate the VLE in this study. The initial K value of each component can be estimated from Wilson’s equation41 ÄÅ É Pci ij Tci yzÑÑÑÑ ÅÅÅ Ki = expÅÅ5.37(1 + ωi)jj1− zzÑÑ ÅÅ P (10) k T {ÑÑÖ Ç

Pcap = PV − PL

(14)

where PV is the pressure of the vapor phase and PL is the pressure of the liquid phase. On the other hand, the capillary pressure can be expressed by the Young−Laplace equation Pcap = 12818

2γ cos ϕ rp

(15) DOI: 10.1021/acs.langmuir.8b02925 Langmuir 2018, 34, 12815−12826

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■

where ϕ is the contact angle of the vapor−liquid interface with respect to the pore surface, which is assumed to be 30° according to the experimental results in the literature.50 Therein, the IFT is estimated by means of the Macleod− Sugden equation, which will be specifically introduced in the next section. The fugacity coefficient of a mixture is

Article

RESULTS AND DISCUSSION Model Verification. The proposed VLE calculations based on the generalized EOS are applied to calculate the phase behavior for three mixtures, the N2−n-C4H10, C8H18−CH4, and C10H22−CH4 mixtures, at different conditions, whose results are compared with and verified by the measured data. Figure 2a,b shows the measured45 and calculated pressure−

2 ∑ nij nbi ji n RT zyz z+ − ln fiL = lnjjj i j V − nb zz V − nb RTVL L k L { ÄÅ ÉÑ ÅÅ Ñ ÅÅa−2ε σ 3 ·jij c1 + c 2 zyzÑÑÑ − NAεsw (1 − F )(1 − e−NAεsw / RT ) zÑ ÅÅ LJ LJ j pr j A A z{ÑÑÑÖ RT ÅÅÇ k ÄÅ ÉÑ ÅÅ Ñ ÅÅ(1− nb )θ − 1· nbiθ · 2VL − nb(θ − 1) ÑÑÑ ÅÅ ÑÑ 2 ÅÅ ÑÑ VL VL − nb VL (16a) ÅÇ ÑÖ

2 ∑ nij nbi ji n RT zyz z+ − ln fiV = lnjjj i j V − nb zz V − nb RTVV V k V { ÄÅ ÉÑ ÅÅ Ñ ÅÅa−2ε σ 3 ·jij c1 + c 2 zyzÑÑÑ − NAεsw (1 − F )(1 − e−NAεsw / RT ) zÑ ÅÅ LJ LJ j pr j A A z{ÑÑÑÖ RT ÅÇÅ k É ÅÄÅ Ñ θ−1 Ñ ÅÅji nbiθ 2VV − nb(θ − 1) ÑÑÑÑ nb zyz ÅÅjj ÑÑ ÅÅÅjj1 − V zzz · 2 · ÑÑ VV − nb ÅÅk VV V{ ÑÑÖ (16b) ÅÇ

The VLE calculations based on the modified EOS require a series of iterative computation through, for example, the Newton−Raphson method. IFT Calculations in Nanopores. The parachor model is most commonly used by the petroleum industry to predict the IFT of a liquid−vapor system.51 Macleod first related the surface tension between the liquid and vapor phases of a pure component to its parachor and the molar density difference between the two phases52 σ = [p(ρL −ρV )]4

(17)

where σ is the surface tension and p is the parachor. At a later time, eq 17 was extended to a multicomponent mixture, which is the so-called Macleod−Sugden equation53 ij γ = jjjjρL j k

yz zz ∑ xipi − ρV ∑i= 1 yp i iz zz i=1 { r

4

r

(18) Figure 2. Measured45 and calculated pressure−volume curves for the CH4−C10H22 system at the pore radii of rp = 3.5 and 3.7 nm and (a) T = 38 °C and (b) T = 52 °C.

where xi and yi are the respective mole percentages of the ith component in the liquid and vapor bulk phases; i = 1, 2, ..., r; r is the component number in the mixture; pi is the parachor of P MW P MW the ith component. Since ρL = ZL RTL and ρV = ZV RTV , eq 18 L

is rewritten as

ij P MW L γ = jjjj L j Z LRT k

r

P MW ∑ xipi − V V Z VRT i=1

volume (P−V) curves for the CH4−C10H22 systems at the pore radii of rp = 3.5 and 3.7 nm and temperatures of T = 38 and 52 °C. The calculated P−V curves are in good agreement with the measured results in the two figures, where the volume is increased with the pressure decrease. The pressures at rp = 3.5 nm are always lower than those at rp = 3.7 nm at the same volume due to an enhanced confinement effect. Obviously, a temperature increase leads the bubble-point pressures to be increased in comparison with the lower-temperature case. Moreover, the calculated pressure differences between the 3.5 and 3.7 nm cases are reduced at a higher temperature. This may be attributed to the large temperature increase from 38 to 52 °C, which dominates the phase behavior change so that the effect of the relatively smaller reduction of the pore radius becomes negligible.

V

yz zz ∑i= 1 yp i iz zz {

4

r

(19)

where MWL is the molecular weight of the liquid phase and MWV is the molecular weight of the vapor phase. The parachor model coupled with the generalized EOS is applied for calculating the confined fluid IFTs in nanopores. Therefore, the chemical potential and surface tension or IFT are determined, which are substituted into eq 7 or 8 to calculate the adsorption thicknesses of the confined pure or mixing fluids in nanopores. 12819

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Langmuir The calculated bubble-point pressures for the N2−n-C4H10, C10H22−CH4, and C8H18−CH4 mixtures in nanopores are calculated from the proposed modified EOS, which are compared with the measured data44,45 and listed in Table 2. More specifically, the measured Pb of the two N2−n-C4H10 mixtures is increased from 3390.6 to 3429.5 kPa and from 3115.1 to 3189.6 kPa with the temperature increase from 26.0 to 51.0 °C in bulk phase, whereas the measured Pb at the pore radius of 5.0 nm is decreased with the same temperature increase. It is found that the calculated Pb for the two mixtures from the modified EOS agrees well with the measured ones, whose percentage absolute deviations (AD%) with the measured results are in the range of 5.25−17.20%. The measured Pb in bulk phase and nanopores for the 90.00 mol % C10H22−10.00 mol % CH4 and 90.00 mol % C8H18−10.00 mol % CH4 mixtures is increased with the temperature increase and also listed in this table. The respective overall percentage average absolute deviation (AAD%) and mean absolute deviation (MAD%) between the measured and calculated Pb for the three mixtures at different temperatures and pore radii are 13.06 and 19.17%, which means the modified EOS is capable of accurately calculating the phase behavior of confined mixing fluids in nanopores. In addition to the above-mentioned model verifications in terms of the confined fluids phase behavior, the modified EOS is also applied to calculate the adsorption amounts in different activated carbons with different pore size distributions (PSDs) at different temperatures. Figure 3a,b shows the measured54,55 and calculated adsorption amounts of CH4 in the sample 4 K (δp = 0.71 nm and εp/k = 1136 K) and ACB-5 (δp = 0.59 nm and εp/k = 1394 K) at T = 24.85 °C and CO2 in the ACB-5 at T = 39.85 and 59.85 °C. The calculated adsorption amounts are found to agree well with the measured data for both cases. Therefore, the modified EOS is proven to be accurate for calculating the adsorptions of confined fluids in nanopores. Adsorption Thickness. In the theory section, the adsorption thickness (δad) is defined as the partial derivative of the surface tension/IFT (γ) with respect to the pressure (P) and the chemical potential (μ) with respect to the pressure at a dγ dμ constant temperature, i.e., δad = dP + n dP . In this study, the adsorption thickness is obtained by using the forward finite difference approximation of the partial derivatives of the surface tension/IFT with respect to the pressure and the chemical potential with respect to the pressure at a constant Δγ Δμ temperature, i.e., δad = ΔP + n ΔP . It should be noted that although the sign of δp can be negative or positive, the physical adsorption thickness has to be positive.56 Hence, the absolute values of the calculated adsorption thicknesses are reported to better present the results and cause no confusion. Figure 4a−c shows the calculated adsorption thicknesses of the pure CO2 and alkanes of C1−C10 and five different mixing fluids, the CH4−C4H10, CO2−C8H18, CH4−C8H18, CO2− C10H22, and CH4−C10H22, at the pore radius of rp = 10 nm and temperature of T = 53.0 °C, with the wall-effect distance of δp = 1 nm. It is seen from the figures that the adsorption thicknesses of different confined pure or mixing fluids are different, but overall they are slightly increased with the pressure at low pressures but remain almost constant afterward. More specifically, the adsorption thicknesses of alkanes are increased with the carbon number increase, which means the C10H22 adsorption thickness is the largest whereas that of the CH4 is the smallest among the alkanes of C1−C10.

Figure 3. Measured54,55 and calculated adsorption amounts of (a) CH4 in the sample 4 K and ACB-5 at T = 24.85 °C and (b) CO2 in the ACB-5 at T = 39.85 and 59.85 °C.

Furthermore, the adsorption thicknesses are substantially increased from CH4 to C2H6, moderately increased from C2H6 to C3H8, but slightly increased from C5H12 to C10H22 at the same conditions. The CO2 adsorption thickness is larger than that of C2H6 but smaller than that in the C3H8 case. The adsorption thickness is certainly affected by the gas and liquid compositions. For example, the vdW force among the alkanes coupled with the CO2 was found to be weakened with the pressure decrease.57 It is found from Figure 4c that the adsorption thicknesses of the CH4−C4H10, CH4−C8H18, and CH4−C10H22 are slightly increased in the range of 0.15−0.18 nm. On the other hand, given that the liquid phase remains unchanged, the adsorption thickness in the C8H18 case roughly increases from 0.17 to 0.25 nm and in the C10H22 cases increases from approximately 0.18 to 0.26 nm by changing the gas phase from CH4 to CO2. Hence, it is concluded that the gas-phase composition affects the adsorption thickness to a larger extent in comparison with the liquid phase. Temperature Effect. Temperature is always considered to be an important factor affecting the confined fluid adsorptions in nanopores.58 Figure 5a−f shows the calculated adsorption thicknesses of the CO2, CH4, n-C4H10, C10H22, CO2−C10H22, and CH4−C10H22 systems at the pore radius of rp = 10 nm and 12820

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Information, the calculated adsorption thicknesses of the C2H6, C3H8, C5H12, C6H14, C7H16, and C8H18 at the same conditions are plotted versus the pressure. Obviously, the temperature effects perform differently on various confined pure or mixing fluids. For the alkanes of C1−C10, the degree of temperature effect is strengthened with the carbon number increase. More specifically, at the temperature variations of T = 15.6−116.1 °C, the adsorption thicknesses of the CH4, C2H6, C3H8, nC4H10, C5H12, C6H14, C7H16, C8H18, and C10H22 have approximate increases of 0.05, 0.08, 0.10, 0.11, 0.11, 0.12, 0.12, 0.13, and 0.13 nm, respectively. The CO2 adsorption thicknesses with the temperature variations are similar to those in the C2H6 case. The temperature effects on the adsorption thicknesses of the confined mixing fluids are different from those in the pure fluid cases because the liquid and gas phases perform dissimilarly with the temperature variations. It is found from the figures that the respective adsorption thicknesses of the CO2−C10H22 and CH4−C10H22 mixtures have increases of approximately 0.11 and 0.08 nm when the temperature is increased from 15.6 to 116.1 °C. The adsorption thickness increases of the mixtures are closer to those of the pure CO2 and CH4 cases, which means the temperature effect on the gas phase dominates the mixture results. The adsorption thicknesses of the CO2, CH4, n-C4H10, C10H22, CO2−C10H22, and CH4−C10H22 systems at rp = 10 nm, five different temperatures, and three different pressures of P = 1.0, 8.5, and 25.0 MPa are calculated from eqs 7 and 8, which are compared with the measured adsorption thicknesses at the same conditions and shown in Figure 6a−f. It is found from Figure 6a,b,e,f that the calculated adsorption thicknesses of the CO2, CH4, CO2−C10H22, and CH4−C10H22 systems agree well with the measured data.59 The accuracies of the thermodynamic formulation of adsorption thickness and proposed modified EOS are validated once again. Figure S3a−f in the Supporting Information shows the adsorption thicknesses of the C2H6, C3H8, C5H12, C6H14, C7H16, and C8H18 at the same conditions. It is obvious that the adsorption thickness is decreased with the temperature increase. In this way, different temperature effects on different fluids are more apparent, which have been specifically stated in the previous paragraph. Another important point should be noted that the pressure effect on the adsorption thickness is almost negligible. Three typical pressures, P = 1.0, 8.5, and 25.0 MPa, are purposely selected. In Figures 6a−f and S3a−f, the adsorption thicknesses of any fluid at the three pressures are equivalent. Effects of Pore Radius and Wall-Effect Distance. Apparently, effects of the pore radius and wall-effect distance on the adsorption in nanopores are important but cannot be evaluated by means of the original EOS. The developed thermodynamic formulation of the adsorption thickness coupled with the modified EOS, which considers the confinement-induced effects of pore radius, shifts of critical properties, and molecule−molecule and molecule−wall interactions, is capable of evaluating the effects of pore radius and wall-effect distance on the adsorption thickness. The calculated adsorption thicknesses of the CO2, CH4, n-C4H10, C10H22, CO2−C10H22, and CH4−C10H22 systems at T = 53.0 °C and six different pore radii of rp = 1, 5, 10, 50, 100, and 1000 nm are shown in Figure 7a−f. In addition, the calculated adsorption thicknesses of the C2H6, C3H8, C5H12, C6H14, C7H16, and C8H18 at the same conditions are shown in Figure S4a−f in the Supporting Information. Overall, the adsorption

Figure 4. Calculated adsorption thicknesses of the (a, b) pure CO2 and alkanes of C1−C10 and (c) the five mixing fluids at the pore radius of rp = 10 nm and temperature of T = 53.0 °C, with the wall-effect distance of δp = 1 nm.

five different temperatures of T = 15.6, 30.0, 53.0, 80.0, and 116.1 °C. Overall, the adsorption thickness is always decreased with the temperature increase, which is in good agreement with the literature results.58 In Figure S2 in the Supporting 12821

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Figure 5. Calculated adsorption thicknesses of the (a) CO2, (b) CH4, (c) n-C4H10, (d) C10H22, (e) CO2−C10H22, and (f) CH4−C10H22 systems at the pore radius of rp = 10 nm and five different temperatures of T = 15.6, 30.0, 53.0, 80.0, and 116.1 °C.

Figure 6. Measured59 and calculated adsorption thicknesses of the (a) CO2, (b) CH4, (c) n-C4H10, (d) C10H22, (e) CO2−C10H22, and (f) CH4− C10H22 systems at the pore radius of rp = 10 nm, three different pressures of P = 1.0, 8.5, and 25.0 MPa, and five different temperatures of T = 15.6, 30.0, 53.0, 80.0, and 116.1 °C.

thicknesses of the confined pure and mixing fluids are significantly decreased with the pore radius increase until rp = 50 nm, after which the adsorption thicknesses have slight changes or are even constant with the variations of the pore radii. Moreover, the effect of pore radius can be different for different fluids. For example, the CH4 adsorption thicknesses increase from 0.05 to 0.31 nm whereas those of the C10H22 increase from 0.22 to 0.68 nm when the pore radius is reduced

from 1000 to 1 nm. The effects of pore radius on the adsorption thicknesses are weaker at larger pore radii but become strengthened with the reduction of the pore radius. Accordingly, the differences of the adsorption thicknesses of different fluids are smaller at larger pore radii. Thus, it is concluded that the adsorption thickness is sensitive to the pore radius at small nanopores (rp ≤ 10 nm) but becomes insensitive at rp > 50 nm. As aforementioned, the effect of 12822

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Figure 7. Calculated adsorption thicknesses of the (a) CO2, (b) CH4, (c) n-C4H10, (d) C10H22, (e) CO2−C10H22, and (f) CH4−C10H22 systems at the temperature of T = 53.0 °C and six different pore radii of rp = 1, 5, 10, 50, 100, and 1000 nm.

Figure 8. Calculated adsorption thicknesses of the (a) CO2, (b) CH4, (c) n-C4H10, (d) C10H22, (e) CO2−C10H22, and (f) CH4−C10H22 systems at the temperature of T = 53.0 °C, three different pressures of P = 1.0, 8.5, and 25.0 MPa, and six different pore radii of rp = 1, 5, 10, 50, 100, and 1000 nm (i.e., δp/rp = 1, 0.2, 0.1, 0.02, 0.01, and 0.001, respectively).

In Figures 8a−f and S5a−f, the calculated adsorption thicknesses of the CO2, CH4, n-C4H10, C10H22, CO2−C10H22, and CH4−C10H22 systems at T = 53.0 °C, six different pore radii, and three different pressures of P = 1.0, 8.5, and 25.0

pore radius on the gas phase also dominates the liquid−gas mixture results. However, at an extremely small pore radius like rp = 1 nm, the gas composition effect on the adsorption thickness is moderately weakened. 12823

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Langmuir MPa are plotted versus the pore radius. It is worthwhile to mention that although δp is equal to 1 nm, the δp/rp varies to be 1, 0.2, 0.1, 0.02, 0.01, and 0.001, with the different pore radii of rp = 1, 5, 10, 50, 100, and 1000 nm, respectively. As mentioned above, the adsorption thicknesses in Figures 8a−f and S5a−f are depicted to be linearly deceasing with the pore radius increase until rp = 50 nm in a clearer way. The effect of the wall-effect distance (δp) on the adsorption thickness is also demonstrated by plotting the calculated adsorption thickness versus δp/rp. It is found that the wall-effect distance affects the adsorption thicknesses of different fluids equally. More specifically, at δp/rp ≤ 0.02, the effect of the wall-effect distance on the adsorption thickness is almost negligible. Otherwise, the adsorption thickness is linearly increased with a stronger effect of the wall-effect distance (i.e., an increased δp/ rp). At any pore radius, the calculated adsorption thicknesses at the three selected pressures are almost equivalent, which agrees well with above-mentioned results and means that the pressure has no effect on the adsorption thickness. Effects of Feed Gas-to-Liquid Ratio. Feed gas-to-liquid ratio (FGLR) has been found to affect some fluid behavior, such as the saturation pressure, IFT, minimum miscibility pressure, etc.60−63 However, so far few studies have been found to analyze the FGLR effect on the fluid adsorptions. Figure 9a,b shows the calculated adsorption thicknesses of the CO2− C10H22 and CH4−C10H22 systems at the T = 53.0 °C, rp = 10 nm, and five different FGLRs of 0.9:0.1, 0.7:0.3, 0.5:0.5, 0.3:0.7, and 0.1:0.9 in mole fraction. In these two cases, the adsorption thicknesses are easily found to slightly increase with the pressure initially and then become almost constant. This kind of increase is so small (less than 0.05 nm) that the pressure effect on the adsorption thickness is considered to be weak or even negligible. It is seen from the figures that the adsorption thicknesses always remain the same at any FGLR. Therefore, it is concluded that the adsorption thicknesses of the confined mixing fluids like CO2−C10H22 and CH4−C10H22 are independent of the FGLR in nanopores.

Figure 9. Calculated adsorption thicknesses of the (a) CO2−C10H22 and (b) CH4−C10H22 systems at a temperature of T = 53.0 °C, pore radius of rp = 10 nm, and five different feed gas-to-liquid ratios of 0.9:0.1, 0.7:0.3, 0.5:0.5, 0.3:0.7, and 0.1:0.9 in mole fraction.

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CONCLUSIONS The following seven major conclusions can be drawn from this work:

•

• An analytical formulation for calculating the adsorption thickness of confined pure and mixing fluids in nanopores is developed thermodynamically for the first time. In addition, a new generalized equation of state (EOS), which considers the pore radius effects, intermolecular interactions, and wall effects, is developed in the analytical formulation for calculating the thermodynamic phase behavior of confined pure and mixing fluids in nanopores.

•

• The modified model based on the generalized EOS and coupled with the parachor model, which is fully capable of modeling the above-mentioned confinement effects (also including capillary pressure and shifts of critical properties), is found to be accurate for vapor−liquid equilibrium (VLE) and adsorption calculations in nanopores.

•

• The adsorption thicknesses of alkanes are increased with the carbon number increase at the same conditions. More specifically, the adsorption thicknesses are substantially increased from CH4 to C2H6, moderately

• 12824

increased from C2H6 to C3H8, but slightly increased from C5H12 to C10H22 at the same conditions. The adsorption thickness of confined fluids is always found to decrease with the temperature increase. The temperature effects are different on various confined pure or mixing fluids: For the alkanes of C1−C10, the degree of temperature effect is strengthened with the carbon number increase. The adsorption thicknesses of the confined pure and mixing fluids are significantly decreased with the pore radius increase until rp = 50 nm, after which the adsorption thicknesses have slight changes or are even constant with the variations of the pore radii. Moreover, the adsorption thickness is sensitive to the pore radius at small nanopores (rp ≤ 10 nm) but becomes insensitive at rp > 50 nm. Once the wall-effect distance become small (i.e., δp/rp ≤ 0.02), the effect of the wall-effect distance on the adsorption thickness is almost negligible. Otherwise, the adsorption thickness is linearly increased with a stronger effect of the wall-effect distance (i.e., an increased δp/rp). The adsorption thicknesses of confined mixing fluids, e.g., CO2−C10H22 and CH4−C10H22, are independent of DOI: 10.1021/acs.langmuir.8b02925 Langmuir 2018, 34, 12815−12826

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the feed gas-to-liquid ratio in nanopores. Furthermore, the pressure effect on the adsorption thickness is found to be negligible.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b02925. Derivations of the generalized equation of state; schematic diagram of the molecule−molecule and molecule−wall potentials; calculated adsorption thicknesses of the C2H6, C3H8, C5H12, C6H14, C7H16, and C8H18 substances at different conditions (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 1-306-337-3287. Fax: 1306-585-4855. ORCID

Na Jia: 0000-0002-1641-2603 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the Petroleum Systems Engineering at the University of Regina. They also want to acknowledge the financial supports from Petroleum Technology Research Centre (PTRC) and Mitacs Canada.

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DOI: 10.1021/acs.langmuir.8b02925 Langmuir 2018, 34, 12815−12826