Modeling adsorption on energetically heterogeneous surfaces with an extended SAFT-VR approach

Accepted Manuscript Title: Modeling adsorption on energetically heterogeneous surfaces with an extended SAFT-VR approach Authors: Johannes Kern, Monika Johannsen PII: DOI: Reference:

S0896-8446(17)30201-2 http://dx.doi.org/doi:10.1016/j.supflu.2017.07.014 SUPFLU 3981

To appear in:

J. of Supercritical Fluids

Received date: Revised date: Accepted date:

22-3-2017 14-7-2017 14-7-2017

Please cite this article as: Johannes Kern, Monika Johannsen, Modeling adsorption on energetically heterogeneous surfaces with an extended SAFT-VR approach, The Journal of Supercritical Fluidshttp://dx.doi.org/10.1016/j.supflu.2017.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modeling adsorption on energetically heterogeneous surfaces with an extended SAFT-VR approach

Johannes Kern, Monika Johannsen* Institute of Bioprocess and Biosystems Engineering, University of Technology Hamburg, Germany, Denickestr. 15, 21073 Hamburg *corresponding

author: [email protected]

Abstract Adsorption equilibria of methanol/carbon dioxide mixtures on plain silica particles at 308 K and pressures of 10.1 MPa, 12.7 MPa, 15.1 MPa and 20.1 MPa were measured with the frontal analysis by characteristic points (FACP) method using supercritical fluid chromatography. The data show a bimodal adsorption energy distribution and cannot be sufficiently modeled with a two-dimensional SAFT-VR approach. Therefore, an extension of SAFT-VR is proposed to model adsorption on heterogeneous surfaces and applied to the presented data. It can be described by the extended model with very low ARDs between 1.0 % and 2.4 %.

Keywords: adsorption, SAFT-VR, equation of state, carbon dioxide, heterogeneous surface

1 Introduction Adsorption equilibria are the fundamental basis of many industrial separation processes. Therefore, modeling adsorption equilibria is a key tool for process design to minimize experimental effort and for developing suitable simulation tools. In chromatography, it is common to make use of empirical or semi-empirical adsorption equations to model adsorption. More specific, in supercritical fluid chromatography (SFC), apart from adsorption equilibria, fluid properties like density and viscosity are crucial for successful process design [1]. Equations of state can be used to calculate densities and active research is being done to implement other thermodynamic frameworks into equations of state to be able to also calculate viscosity [2,3]. It would be intriguing to be able to use a unified approach, where one equation of state is used to calculate fluid properties as well as model adsorption without the need for an empirical adsorption isotherm equation. This study is working towards that goal by investigating the possibility to model adsorption systems interesting for supercritical fluid chromatography with an approach based on the statistical associating fluids theory (SAFT). The SAFT formalism, as first described in these terms by Chapman and coworkers [4] in 1989, has since found widespread adoption especially in chemical engineering applications. Subsequently, several adaptions of SAFT have emerged, e.g. simplified SAFT [5], PC-SAFT [6], CK-SAFT [7,8] and SAFT-VR [9]. While SAFT is mostly used to calculate fluid properties or phase equilibria of bulk phases, SAFT-VR was applied to two-dimensional phases (SAFT-VR-2D) by Martinez et al. [10]. With this approach, adsorption of gases could be modeled [10,11], which can be used to predict the adsorption of their mixtures [12,13]. Because of its predictive qualities and the absence of need for an isotherm equation, SAFT-VR-2D was investigated here.

Another aspect of adsorption is that the surface of the adsorbent is often energetically heterogeneous resulting in a more complex adsorption behavior that has to be taken into account [14]. In this work, we propose a small extension to the SAFT-VR-2D model, which enables a description of the adsorption of methanol from supercritical carbon dioxide on an energetically heterogeneous adsorbent with high accuracy.

2 Material and methods 2.1 Apparatus For this study a self-assembled adsorption apparatus was used, that was described in detail elsewhere [15]. It was used with two Teledyne-ISCO 260D syringe pumps for delivering supercritical carbon dioxide and a Shimadzu LC-20AD Nano as a modifier pump. Breakthrough curves of methanol have been detected with a Knauer Smartline 2600 UV/VIS detector.

2.2 Column The adsorption experiments have been carried out with a 250x4.6 mm column packed with Kromasil SIL-100 plain silica particles with a particle diameter of 5 µm and a pore diameter of 100 Å. The mass of particles in the column was determined by weighing in the empty column and weighing out after packing and thorough washing with supercritical CO2. Particle mass was determined to be 2033 mg. Overall porosity of the column for hold-up time calculations was determined with a weight difference method [16] to εp = 0.775. The surface area of the particles was reported by the supplier as 316 m2/g.

2.3 Chemicals Carbon dioxide, 99.995 vol.-% purity, was supplied by YARA GmbH, Dülmen, Germany. Methanol Rotisolv 99.9 vol.-% UV/IR grade was purchased from Carl Roth, Karlsruhe, Germany.

2.4 Frontal analysis by characteristic points To measure adsorption isotherms, the frontal analysis by characteristic points method (FACP) was applied. In this method a single injection with high concentration is performed and the dispersed part of the resulting peak is analyzed. The injection of methanol is created with a modifier pump to produce a wide, rectangular injection profile. Compared to using fixed volume injection loops, this has the advantage of preventing distorted injection profiles caused by injections of pure solvent into the CO2 stream and better control of the height of the injection profile. This way, the whole column is in equilibrium with the mobile phase when the modifier pump is switched off. The adsorbed methanol is then displaced by carbon dioxide and the resulting disperse elution profile can be used to calculate the slope of the adsorption isotherm by 𝑑𝑞 𝑡𝑅 (𝑐) − 𝑡𝑀 − 𝑡𝑖𝑛𝑗 = . 1 − 𝜀𝑝 𝑑𝑐 𝑡𝑀 𝜀 𝑝

(1)

Numerical integration for c then yields the adsorption isotherm. Since carbon dioxide and methanol can be separately metered into the system, the mass fraction of methanol can be exactly set, in contrast to other plateau experiments, in which one component has to be dissolved in another [17]. A calibrated detector is needed for this kind of measurement, which is why the UV/VIS detector was calibrated with a series of staircase profiles. Due to the high nonlinearity of the detector response in the observed concentration range, a 7th order polynomial of the form

7

𝑐(𝑝⃑, 𝐷) = ∑ 𝑝𝑘 𝐷𝑘

(2)

𝑘=0

was used as calibration function. To allow for fitting of eight calibration parameters, 20 concentration plateaus have been used for calibration. Each point of the calibration curve was averaged from at least four measurements. Calibration was carried out with the column in bypass and separately for each pressure investigated. NIST Refprop 9.0 was used to convert mass fractions of the CO2/methanol mixture into volumetric concentrations. The initial equilibrium concentration of methanol in carbon dioxide in the column was set to 5 wt.-% for each measurement, which still allowed reliable calibration.

3 Theory 3.1 Adsorption energy distribution Surfaces in technical adsorption systems are most often not energetically homogenous, but are covered with adsorption sites that differ in their adsorption energies. That means a general adsorption isotherm can be written as an integral over all sites [18] 𝑏

𝑞(𝑐) = ∫ 𝑓(𝜀)𝜃(𝜀, 𝑐)𝑑𝜀

(3)

𝑎

with a and b being the lowest and highest energy sites, the adsorption energy ε and the local adsorption model 𝜃. In this study a simple Langmuir-type local model of the form 𝜃(𝑘, 𝑐) =

𝑘𝑐 1 + 𝑘𝑐

(4)

was used with k as an energy-like parameter that is related with but not identical to the adsorption energy ε. To obtain the distribution function f (ε), the ExpectationMaximization-Algorithm [19] is used on experimental adsorption data [18,20]. For this, Equ. ( 3 ) is used in discrete form and with the Langmuir-parameter k:

𝑁𝐺

(5)

𝑞(𝑐𝑗 ) = ∑ 𝑓(ln 𝑘𝑖 ) 𝜃(𝑘𝑖 , 𝑐𝑗 )Δlnk 𝑖=1

This equation gives the relationship between the adsorbed amount and the bulk phase concentration when the distribution function is known. It can be calculated with the iteration equation 𝑁𝐷

𝑓 𝑛+1 (ln 𝑘𝑖 ) = 𝑓 𝑛 (ln 𝑘𝑖 ) ∑ 𝜃(𝑘𝑖 , 𝑐𝑗 ) ∆lnk 𝑗=1

𝑞𝑒𝑥𝑝 (𝑐𝑗 ) 𝑞𝑐𝑎𝑙 (𝑐𝑗 )

.

(6)

As initial value for the distribution function, the total ignorance guess 𝑓 0 (ln 𝑘𝑖 ) =

𝑞𝑚𝑎𝑥 − 𝑞𝑚𝑖𝑛 𝑁𝐺 Δlnk

(7)

is used to ensure bias free estimation of the adsorption energy distribution. The distribution function was evaluated in the interval ln 𝑘𝑖 ∈ [−11.5,11.5], which was chosen such that the distribution function converged to zero at the interval borders for all data sets. The energy space was divided into 1000 grid points. As stop criterion a loop count of 106 iterations was used [21].

3.2 bi-Langmuir adsorption model The simplest, non-linear adsorption model for surfaces with two adsorption sites is the bi-Langmuir model, which represents an extension of the original Langmuir equation to surfaces with two adsorption sites [22]. It can be written as 𝑞(𝑐) = 𝑞 (1)

𝑘 (1) 𝑐 𝑘 (2) 𝑐 (2) + 𝑞 , 1 + 𝑘 (1) 𝑐 1 + 𝑘 (2) 𝑐

(8)

where the superscript (𝑛) denotes values referring to adsorption site 𝑛.

3.3 SAFT-VR for modeling adsorbed phase equilibria 3.3.1 SAFT-VR for three-dimensional fluids The “statistical associating fluid theory for chain molecules with attractive potentials of variable range” (SAFT-VR) equations have been proposed by Gil-Villegas et al. [9]. The

SAFT equations are based on the work by Wertheim on perturbation theory [23–26] but the term SAFT has first been used by Chapman and coworkers [4]. In this study, a square well potential is used for the mixture, which in the SAFT-VR formalism is written as [27] ∞, 𝑢𝑖𝑗 (𝑟𝑖𝑗 ) = {– 𝜀𝑖𝑗 , 0,

𝑟𝑖𝑗 ≤ 𝜎𝑖𝑗 𝜎𝑖𝑗 < 𝑟𝑖𝑗 ≤ 𝜆𝑖𝑗 𝜎𝑖𝑗 𝜆𝑖𝑗 𝜎𝑖𝑗 < 𝑟𝑖𝑗

(9)

with the distance 𝑟𝑖𝑗 of the particle centers of species i and j, the depth of the energy well 𝜀𝑖𝑗 , the potential range 𝜆𝑖𝑗 𝜎𝑖𝑗 and the mixture parameter 𝜎𝑖𝑗 . The cross-parameters can be obtained with the following combining rules 𝜎𝑖 + 𝜎𝑗 , 2

( 10 )

𝜀𝑖𝑗 = (𝜀𝑖 𝜀𝑗 )1⁄2

( 11 )

𝜎𝑖𝑗 =

and 𝜆𝑖𝑗 3 = 1 + 𝜎𝑖𝑗−3 [(𝜆𝑖 3 − 1)(𝜆𝑗 3 − 1)𝜎𝑖3 𝜎𝑗3 ]

1⁄2

.

( 12 )

The Helmholtz energy is written as the sum of the contributions derived from the molecular model of chains of spherical monomers: 𝐴 𝐴𝑖𝑑𝑒𝑎𝑙 𝐴𝑚𝑜𝑛𝑜 𝐴𝑐ℎ𝑎𝑖𝑛 = + + . 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇

( 13 )

And correspondingly for the chemical potential 𝜇𝑖 𝜕𝐴 = 𝑘𝐵 𝑇 𝜕𝑁𝑖

( 14 )

𝜇𝑖 𝜇𝑖,𝑖𝑑𝑒𝑎𝑙 𝜇𝑖,𝑚𝑜𝑛𝑜 𝜇𝑖,𝑐ℎ𝑎𝑖𝑛 = + + . 𝑘𝐵 𝑇 𝑘𝐵 𝑇 𝑘𝐵 𝑇 𝑘𝐵 𝑇

( 15 )

follows

The equations for the respective contribution terms can be taken from the literature [27].

3.3.2 SAFT-VR for two-dimensional fluids The SAFT-VR formalism for two-dimensional fluids was developed analogue to the approach for three-dimensional fluids, as described in section 3.3.1, by Martinez et al. [10] for pure fluids and later for mixtures by Castro et al. [12]. The fluid is modeled as a mixture of solid discs interacting with the wall over a wall-particle potential given as ∞, 𝑢𝑝𝑤 (𝑧) = {– 𝜀𝑤𝑖 , 0,

𝑧≤0 0 < 𝑧 ≤ 𝜆𝑤 𝜎𝑖 𝜆𝑤 𝜎𝑖 < 𝑧

( 16 )

The overall Helmholtz energy of the adsorbed phase is the sum of the contributions of the two-dimensional phase and the wall: 𝐴𝑎𝑑𝑠 𝐴2𝐷 𝐴𝑤𝑎𝑙𝑙 = + . 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇

( 17 )

𝐴𝑤𝑎𝑙𝑙 𝜆𝑤 𝜎 = − ln ( ) − 𝜀𝑤 𝑁𝑘𝐵 𝑇 Λ

( 18 )

With the term for the wall

and the Helmholtz energy of the two-dimensional phase 𝐴2𝐷 𝐴2𝐷,𝑖𝑑𝑒𝑎𝑙 𝐴2𝐷,𝑚𝑜𝑛𝑜 𝐴2𝐷,𝑐ℎ𝑎𝑖𝑛 = + + . 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇

( 19 )

The chemical potential is given as the derivative of Equ. ( 17 ): 𝜇𝑖,𝑎𝑑𝑠 𝜇𝑖,2𝐷 𝜇𝑖,𝑤𝑎𝑙𝑙 = + 𝑘𝐵 𝑇 𝑘𝐵 𝑇 𝑘𝐵 𝑇

( 20 )

𝜇𝑖,𝑎𝑑𝑠 𝜇𝑖,2𝐷 𝜀𝑤𝑖 = − ln 𝜆𝑤 − 𝑘𝐵 𝑇 𝑘𝐵 𝑇 𝑘𝐵 𝑇

( 21 )

3.4 SAFT-VR-2D for heterogeneous surfaces To use the SAFT-VR equations for two-dimensional fluids for the adsorption of substances dissolved in supercritical carbon dioxide on surfaces covered with multiple adsorption sites, the model is slightly modified using the following assumptions:



Adsorption sites are evenly distributed across the surface



The adsorption energy of a site is not influenced by the proximity of another adsorption site



Only monolayer adsorption is allowed



Adsorption of carbon dioxide is considered to be energetically homogeneous

To characterize the surface of the adsorbent, a surface fraction is defined as 𝑠

(𝑛)

𝑆 (𝑛) = ∑𝑘 𝑆 (𝑘)

( 22 )

with 𝑆 (𝑛) being the area covered with adsorption sites of type (𝑛). All molecules adsorbed on one type of adsorption site are grouped together to a virtual adsorbed phase. In thermodynamic equilibrium the chemical potentials of the virtual adsorbed phases and the bulk phase need to be equal: (1)

(2)

(𝐾)

𝜇𝑖,𝑏𝑢𝑙𝑘 = 𝜇𝑖,𝑎𝑑𝑠 = 𝜇𝑖,𝑎𝑑𝑠 = ⋯ = 𝜇𝑖,𝑎𝑑𝑠 .

( 23 )

The difference in adsorption energy for each virtual phase is accounted for by a specific (𝑛)

𝜀𝑤 for each adsorption site. To keep the count of adjustable parameters of the model low, the range parameter in the adsorbed phase 𝜆𝑎𝑑𝑠 was assumed to be independent of the adsorption sites. In this work the range parameter of the wall is set to 𝜆𝑤 =0.8165 to only allow monolayer adsorption, as proposed by Martinez et al. [10]. For the case of two adsorption sites, the adsorbed amount can be calculated as (1)

(1)

(2)

(2)

𝑞𝑎 = 𝑠 (1) 𝑞𝑎 (𝜀𝑤 ) + 𝑠 (2) 𝑞𝑎 (𝜀𝑤 ) (𝑛)

( 24 )

(𝑛)

where 𝑞𝑎 (𝜀𝑤 ) can be calculated by solving Equ. ( 23 ).

3.5

SAFT-VR model parameters

The parameters for the pure compounds in the case of carbon dioxide have been taken from the literature [12], as well as the wall-particle potential for carbon dioxide/silica [11]. Parameters for methanol have been obtained by fitting to vapor pressure data

between 180 K and 500 K acquired online from NIST [28]. The used parameters are listed in Tab. 1. Tab. 1 Pure component parameters for the SAFT-VR model.

CO2 Methanol

𝑚 [-]

𝜎 [Å]

𝜆 [-]

𝜀 ⁄𝑘𝐵 [𝐾]

𝜀𝑤 ⁄𝑘𝐵 [𝐾]

2.00 1.94

2.79 2.11

1.526 1.480

179.3 405.0

1075.7 -

4 Results and discussion 4.1 Experimental The measured adsorption isotherms of methanol on plain silica are shown in Fig. 1. While the number of data points from the FACP method is only limited by the acquisition rate of the detector, the number has been reduced to 20 equidistant points in the concentration space per isotherm to allow for faster calculations in the phase equilibria modeling. Measurements have been performed at least twice for each isotherm to ensure reproducibility of the results.

Fig. 1 Experimental data of the excess adsorption of methanol in carbon dioxide on silica versus bulk phase concentration of methanol at 308 K and 10.1, 12.7, 15.1 and 20.1 MPa. Solid lines represent the bi-Langmuir model.

The pressure dependence of the isotherms is only little pronounced for the isotherms at 12.7 MPa, 15.1 MPa and 20.1 MPa, where the adsorption behavior is almost identical for each pressure. The adsorption at 10.1 MPa, however, is always larger compared to the other pressures at the same concentration. This is coherent with the common observation, that the adsorbed amount of a substance dissolved in supercritical carbon dioxide decreases with increasing pressure. This is especially pronounced with nonvolatile compounds, where the density of the fluid is the major governing factor of phase behavior. In the case of more volatile compounds as methanol it is reasonable that the pressure dependence of adsorption is less pronounced, since methanol and carbon dioxide are completely miscible in the investigated pressure and temperature regime. The influence of pressure only becomes apparent at higher isothermal compressibility at lower pressures closer to the critical point of the mixture and hence larger changes in

the fluid density. All isotherms are concavely shaped and increasing monotonously with increasing methanol concentration but apparently do not show any inflection points, which indicates adsorption isotherm type I according to IUPAC classification. It has been reported for supercritical adsorption systems, that the adsorbent presents more than one adsorptions site, and the interactions of the adsorbing molecules with these adsorption sites are characterized by different adsorption energies [21,29]. The Expectation-Maximization (EM) algorithm can be used to calculate the adsorption energy distribution (AED) of the adsorbent-adsorptive interaction from isotherm data. This in turn can be used to identify the number of adsorption sites when the local adsorption mechanism is known. Since there is no indication of multilayer formation in the adsorption data, a simple Langmuir-type model was used for the local adsorption kernel in the EM algorithm (see Equ. ( 4 )). It has to be noted, that the EM algorithm was applied to the raw adsorption isotherm data. As was stated by Samuelsson et al. [30], the EM algorithm can be directly applied to the slope data calculated using Equ. ( 1 ), leading to improved convergence and smaller error. Since, however, the AED was calculated to show energetic heterogeneity in the data used for modeling and compare the energies of the found adsorption sites, it made more sense to apply both EM algorithm and hetSAFT-VR to the same data set in this case. The calculated AED show two distinct peaks for each isotherm at different pressures, as can be seen in Fig. 2. This shows, that the adsorbent is energetically heterogeneous and two adsorption sites are present.

Fig. 2 Adsorption energy distributions for the adsorption of methanol in carbon dioxide on silica at 308 K and 10.1, 12.7, 15.1 and 20.1 MPa.

4.2 Modeling As a measure for the model error, the average relative deviation was used. It is calculated as 𝑁

|𝑞𝑒𝑥𝑝,𝑖 − 𝑞𝑐𝑎𝑙𝑐,𝑖 | 1 𝐴𝑅𝐷 = 100% ∙ ∑ . 𝑁 𝑞𝑒𝑥𝑝,𝑖

( 25 )

𝑖=1

A bi-Langmuir isotherm was fitted to each isotherm, since it is the appropriate model considering the results from the AED calculations and that would be the isotherm model of choice, if isotherm-based modeling was desired. The best-fit parameters are given in Tab. 2.

Tab. 2 Best-fit parameters for the bi-Langmuir model for adsorption of methanol on plain silica at 308 K and 10.1 MPa, 12.7 MPa, 15.1 MPa and 20.1 MPa.

P [MPa] 𝑞1 [mmol mg-1] 𝑘1 [ml mmol-1] 𝑞2 [mmol mg-1] 𝑘2 [ml mmol-1] ARD [%] 10.1

0.0024

0.9344

0.0012

41.6535

1.4

12.7

0.0029

0.5216

0.0011

26.1375

0.3

15.1

0.0029

0.4889

0.0011

20.9755

0.6

20.1

0.0027

0.5219

0.0011

20.9057

0.1

Since the bi-Langmuir equation has the same number of adjustable parameters as the later used het-SAFT-VR model, it is sensible to compare both models. The bi-Langmuir isotherm gives a very good representation of the data, as can be seen by the low model errors, and the Langmuir-parameters expectedly behave as suggested by the AED calculations. The model is plotted against the experimental data in Fig. 1. The excellent fit of the model confirms the surface heterogeneity and makes this data suitable to test the extended het-SAFT-VR approach. The equation of state based modeling of the adsorption data was first carried out using the regular SAFT-VR model for adsorbed phases. For simplicity’s sake and since methanol concentrations are comparably low, self-association of methanol was not considered in this work. The best-fit parameters are listed in Tab. 3. As can be seen in Fig. 3a, the representation of the experimental data is very poor with ARDs between 40.8 % and 48.3 %.

Tab. 3 Best-Fit parameters for the SAFT-VR and het-SAFT-VR model for the adsorption of methanol on plain silica at 308 K and 10.1 MPa, 12.7 MPa, 15.1 MPa and 20.1 MPa.

(1)

(2)

𝜆𝑎𝑑𝑠 [-]

𝜀𝑤 ⁄ 𝜀 [-]

𝜀𝑤 ⁄ 𝜀 [-]

𝑠 (1) [-]

ARD [%]

10.1

1.055

4.93

-

-

48.3

12.7

1.055

4.78

-

-

43.9

15.1

1.054

4.72

-

-

40.8

20.1

1.053

4.78

-

-

42.2

10.1

1.078

6.47

4.05

0.141

2.4

12.7

1.073

6.35

4.34

0.127

1.3

15.1

1.069

6.22

4.31

0.123

1.2

20.1

1.077

6.25

4.30

0.125

1.0

P [MPa] SAFT-VR

het-SAFT-VR

a)

b)

Fig. 3 Adsorption of methanol on silica at 308 K and 10.1, 12.7, 15.1 and 20.1 MPa modeled with a) SAFT-VR for two-dimensional phases b) het-SAFT-VR for adsorption on heterogeneous surfaces

From the AED calculations it is clear that the adsorption energies of the respective sites are very different in magnitude. This cannot be addressed by SAFT-VR, since the model

relies on a single interaction potential between the adsorbed molecules and the wall. In het-SAFT-VR, a second interaction potential is introduced to account for the surface heterogeneity of the adsorbent.

When applied to the experimental data, the

representation is greatly improved (Fig. 3b), with the highest ARD of 2.4 % at 10.1 MPa and the lowest ARD of 1.0 % at 20.1 MPa. Compared to the bi-Langmuir equations, model errors are somewhat higher, however errors for the bi-Langmuir model are extremely low, with values between 1.4 % and 0.1 %. These low errors further emphasize the correct assumption of the underlying adsorption mechanism. The introduction of another potential parameter and the surface fraction also increases the number of adjustable parameters, which one could argue, improves the description of the data in itself. Notably, with four adjustable parameters the model uses the same number of parameters as the bi-Langmuir isotherm. To investigate if the improvement is simply due to an increase in the degrees of freedom of the model or an actual better representation of the underlying thermodynamics of the adsorption process, one has to check that the model delivers physically meaningful parameters. An interesting outcome is that the square well range parameters 𝜆𝑎𝑑𝑠 are very small with values between 1.069 and 1.078. This means that methanol molecules are behaving like hard-discs in the adsorbed phase, with only little contribution of the attractive forces. This is unexpected for associating molecules like methanol. One might argue, that the very polar nature of the surface leads to a strong orientation of the molecules with the hydroxyl group towards the adsorbent. This way, methanol molecules would mostly interact with their non-polar methyl group, similar to methane molecules, which are also reported with somewhat lower range parameters in the adsorbed phase of 1.2 by Castro et al. [12]. Another possibility is that the reality of the methanol molecules is not fully attributed to the pure component parameters by fitting only to vapor pressure data. For this study, the aim was to keep the amount of experimental data for parameter fitting low but for

further research the influence of the data used for pure component parameters could be taken into account. For densities towards zero, the well-depth 𝜀𝑤 is equal to the adsorption energy, which is equivalent to the energy of a chemical bond. The adsorption energy for infinitesimally small densities for methanol, as calculated from het-SAFT-VR, varies only little with pressure and lies between 21.0 kJmol-1 at 15.7 MPa and 21.8 kJmol-1 at 10.1 MPa. This is strikingly close to the bond energy of hydrogen bonds of the configuration O-H:O of 20.5 kJmol-1 [31]. Because of the polar nature of silica due to its hydroxyl groups on the surface, it is to be expected for methanol to form hydrogen bonds with the surface. The energies of the lower energy site range between 13.6 kJmol-1 at 10.1 MPa and 14.6 kJmol-1 at 12.7 MPa, which is still in the range of polar interactions. While it is not possible to calculate any adsorption energies from the AEDs, the energy-difference between the sites can be calculated as [32] 𝑘 (2) 𝜀 (2) − 𝜀 (1) = 𝑅𝑇 ln ( (1) ) 𝑘

( 26 )

where 𝑘 (𝑛) is the value of the Langmuir parameter at the maximum of the respective peak in the AED. With this, an average difference in the adsorption energies of 10.9 kJmol-1 can be calculated, which is comparable to the one obtained from het-SAFTVR, where the average difference between the respective high and low energy sites is 7.0 kJmol-1. The results presented here are promising for future applications for example in SFC. Carbon dioxide/alcohol-mixtures are very common mobile phases in SFC and it was shown that the het-SAFT-VR model has the ability of describing methanol adsorption over a wide pressure range with very high accuracy. In practice, modifier concentrations can far exceed 5 wt.-% and while the model is applicable to the whole concentrations range, the accuracy remains to be shown. For higher concentrations, with pair interactions of methanol gaining importance, self-association can probably no longer be

neglected and needs to be implemented into the model. Further, more complex adsorption behavior like multilayer adsorption and chemisorption need to be considered. Also an equation-of-state-approach comes with an extra cost in terms of numerical complexity. However, the isotherm-based models most often used in chromatography are defined for single component adsorption, which in adsorption from mixtures is virtually never the case. While this still allows for accurate description of adsorption isotherms at fixed physical conditions in many cases, interpolation between experimental conditions or application of pure component data to mixtures is not possible. SAFT-VR on the other hand only uses pure component parameters, which can theoretically be used for arbitrarily complex mixtures at different physical conditions. The need to introduce binary interaction parameters into the combinations rules depends on the complexity of the mixture and needs to be evaluated in further studies. Nonetheless, this makes it interesting for modeling chromatographic separations where ternary mixtures of two mobile phase components and one analyte contained in a sample mixture need to be considered. This can help to significantly reduce the number of experiments needed for method development.

5 Conclusions In adsorption from supercritical fluids, the adsorbing compounds often interact with more than one adsorption site on the adsorbent surface, making the adsorption process energetically heterogeneous. Extending the SAFT-VR model for two-dimensional fluids by another potential function for a second adsorption site dramatically increased the performance of the model when describing the heterogeneous adsorption of methanol on silica in a large pressure interval and at supercritical conditions for carbon dioxide, introducing only two more adjustable parameters. This makes this model an interesting candidate for modeling heterogeneous adsorption, when the gains of an equation-of-

state-type model are needed, like the possibility to calculate thermodynamic properties of the system or thermodynamically sound interpolation between a limited amount of experimental conditions.

6 Acknowledgement The authors want to thank Prof. Alejandro Gil-Villegas for the help with the SAFT-VR model and Prof. Brett Stanley for the help with programming the EM algorithm. The financial support from the DFG under grant number Jo-9-1 is greatly appreciated.

7 Nomenclature Latin letters 𝐴

Helmholtz energy [J]

𝐴𝑤𝑎𝑙𝑙

contribution of the wall to the Helmholtz energy [J]

𝐴2𝐷

Helmholtz energy of the two-dimensional phase [J]

𝑐

concentration [mmol mL-1]

𝐷

detector signal [mAU]

𝑓(𝜀)

adsorption energy distribution function

𝑘

isotherm parameter [mL mmol-1]

𝑘𝐵

Boltzmann constant [J K-1]

𝑚

SAFT-VR chain length

𝑁

number of molecules

𝑁𝐷

number of data points

𝑁𝐺

number of grid points in AED

P

pressure [MPa]

𝑝⃑

vector containing detector calibration coefficients 𝑝𝑘

𝑞

volume specific adsorption [g mL-1]

𝑞𝑎

area specific adsorption [mmol m-2]

𝑟𝑖𝑗

particle-particle distance [m]

𝑅

universal gas constant [J mol-1 K-1]

𝑆

area [m2]

𝑠 (𝑛)

area fraction

𝑇

temperature [K]

𝑡𝑅

retention time [min]

𝑡𝑀

hold-up time [min]

𝑡𝑖𝑛𝑗

width of injection profile [min]

𝑢𝑖𝑗

potential function [J]

𝑧

particle-wall distance [m]

Greek letters ∆𝑙𝑛𝑘

grid spacing in AED

𝜀

well-depth particle-particle potential [J]

𝜀𝑤

well-depth particle-wall potential [J]

𝜀𝑃

porosity

𝜃

local adsorption isotherm

Λ

thermal De Broglie wavelength [Å]

𝜆

SAFT-VR range parameter

𝜆𝑎𝑑𝑠

SAFT-VR range parameter in adsorbed phase

𝜆𝑤

SAFT-VR range parameter of the wall potential

𝜇

chemical potential [J]

𝜎

SAFT-VR segment diameter [Å]

Subscripts i

related to species i

j,k

running integers

ideal

related to the SAFT ideal contribution

mono

related to the SAFT monomer contribution

chain

related to the SAFT chain contribution

exp

related to experimental values

calc

related to calculated values from a model

Superscripts (𝑛)

related to adsorption site (𝑛)

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