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# How to parameterise the association term in SAFT models? Insights from the I-PC-SAFT Equation of State

Haziq Ridwan Bin Asmuni<sup>a,b</sup>, Romain Privat<sup>a,* ID</sup>, Saifuddin Ahmed<sup>b</sup>, Marc Bonnissel<sup>b</sup>, Jean-Noël Jaubert<sup>a,* ID</sup>

## ARTICLE INFO

Keywords:

PC-SAFT

Parameterisation

Association

Association schemes

Alcohols

Amines

Acids, Water

## ABSTRACT

An extended version of the I-PC-SAFT equation of state (EoS) which incorporates the association term is presented, and a parameterisation protocol is proposed for water and three well-known homologous series of self-associating compounds: linear alcohols (from C₁ to C₁₈, except C₁₄), linear carboxylic acids (from C₂ to C₁₇) and linear amines (from C₁ to C₁₂). The protocol follows the same constraints used in the original non-associative I-PC-SAFT (NonA-I-PC-SAFT), i.e., an exact reproduction of the critical temperature and pressure, acentric factor as well as the saturated liquid density at a reduced temperature of 0.8.

Seven (1A, 2A, 2B, 3A, 3B, 4A and 4B) and eight classical association schemes (1A, 2A, 2B, 3A, 3B, 4A, 4B and 4C) were examined for the three homologous series and water respectively, in addition to the reference variant without the association term (NonA-I-PC-SAFT). For each component, the optimal reduced association parameters ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$) were determined by minimising an objective function that prioritises the vapour pressure, followed by liquid density and finally thermal properties.

Compared to the NonA-I-PC-SAFT EoS, the association term substantially improved predictions for strongly associating fluids. For alcohols, the mean absolute percentage error (MAPE) in vapour pressures decreases from ~15 % to ~2 %, heat capacity and heat of vaporisation errors decrease from ~16 % and ~10 % to <5 %. Similar improvements are observed for carboxylic acids: from ~8 % to less than ~1 % for the vapour pressures, from ~14 % to 9 % for the heat of vaporisation, while the errors of the heat capacities were barely reduced from ~8 % to ~7 %. For water, the association term reduces MAPE for vapour pressures from ~5 % to ~1 %, for heat of vaporisation from ~4 % to ~2 % and for heat capacity from ~24 % to ~4 %. In contrast, minimal improvements are observed for the amines that are weakly associating. The liquid densities errors remained close to ~5 % for all systems.

In the case of alcohols, amines and water, all association schemes gave comparable results. The optimal association scheme was found to be 2B for both amines and alcohols, 1A for the acids and 4B for water. These results demonstrate that the association term is essential for strongly associating alcohols, carboxylic acids and water, whilst the NonA-I-PC-SAFT EoS offers a simpler alternative for amines without significantly compromising the accuracy of the predictions.

## 1. Introduction

Accurate description of the behaviour of complex fluids relies upon a robust equation of state (EoS), capable of capturing both dispersion forces and specific directional interactions, such as hydrogen bonding or polar interactions. The EoS inspired by the Statistical Associating Fluid Theory (SAFT) [1] have emerged in the late 1980s as a versatile

framework in which molecules are represented as chains of spherical segments, whilst the residual energy is decomposed into repulsive, dispersive, and the associative contributions accounting mainly for hydrogen bonding or $\pi$-$\pi$ interactions.

Despite the proven success of SAFT models for a range of non-associating molecules [2–8], their optimal parameterisation for associating molecules remains an open challenge. This issue concerns the

<footnote><sup>a</sup> Université de Lorraine, CNRS, LRGP, F-54000 Nancy, France</footnote>

<footnote><sup>b</sup> Gaztransport & Technigaz (GTT), 1 route de Versailles, 78470 Saint-Rémy-lès-Chevreuse, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: romain.privat@univ-lorraine.fr (R. Privat), jean-noel.jaubert@univ-lorraine.fr (J.-N. Jaubert).</footnote>
selection of the underlying association scheme, and the determination of association parameters.

In this study, a systematic evaluation is performed on several association schemes. The Industrial-Perturbed-Chain SAFT (I-PC-SAFT) parameters are determined for water and three homologous series of self-associating compounds: alcohols, amines, and carboxylic acids. The parameterisation procedure follows that used for the original non-associative I-PC-SAFT (NonA-I-PC-SAFT) proposed by Privat et al. [9] This model is a volume-translated [10,11] version of the classical PC-SAFT EoS, in which the four input parameters (m, σ, ε and the volume-translation parameter c) are determined to exactly reproduce the experimental critical temperature and pressure, acentric factor, as well as the saturated liquid density at a reduced temperature of 0.8. By comparing the non-associative (NonA) and associative versions of the I-PC-SAFT model, the influence of the association term can be isolated and its impact on key thermodynamic property predictions can be quantified. Whilst certain association schemes yield substantial improvements in predicting vapour pressures, heat capacities and heat of vaporisation, in some cases the gain over the baseline model (NonA-I-PC-SAFT) is modest.

These results highlight two important points. Firstly, the inclusion of the association term in the EoS can meaningfully enhance the predictive accuracy of strongly hydrogen-bonded species. Secondly, for moderately or weakly associating components such as amines, the added complexity of an explicit association term may offer limited benefit relative to its computational and parameterisation overhead. The parameterisation methodology is described in detail in the following sections and subsequently, the influence of the association term is also examined before comparing the performance of each association scheme to that of the simpler NonA-I-PC-SAFT EoS.

This paper focuses only on pure components. The modelling of binary systems, such as those included in the reference database [12] developed by the team, will be addressed in future publications.

## 1.1. PC-SAFT with the association term

As an alternative to the well-established cubic EoS [13–15], the Statistical Associating Fluid Theory (SAFT) EoS refers to a class of thermodynamic models that has gained significant traction within the academic community. SAFT models are inspired by molecular simulation models and adopt the same molecular potential parameters. In this framework, each molecule is treated as a chain of m spheres (also named segments) that are often all identical, characterised by a size parameter σ (the temperature independent diameter of segment) and an energy parameter ε reflecting dispersive interactions.

The Perturbed-Chain SAFT (PC-SAFT) [5,16] is one of the variants of the SAFT-type EoS that can be found in the existing literature. As with other SAFT-type EoS, it consists of the residual-(T, v) Helmholtz energy (i.e., the difference of the Helmholtz energy of the real fluid and that of the perfect gas at the same temperature and molar volume) and is expressed as a sum of different Helmholtz energy contributions:

$$ \frac{a^{\text{res}}}{RT} = \frac{a^{\text{hs}}}{RT} + \frac{a^{\text{hc}}}{RT} + \frac{a^{\text{disp}}}{RT} + \frac{a^{\text{ass}}}{RT} + \frac{a^{\text{dip}}}{RT} + \dots \qquad (1) $$

where $\frac{d}{RT}$ is the molar Helmholtz energy of the contribution i, the superscripts represent the residual term (res), the hard segment term (hs), the hard chain term (hc), the dispersion term (disp), the association term (ass), and the dipolar term (dip). Unlike other SAFT models [17–19], the dispersion term is added to the hard chains in PC-SAFT rather than to the hard segments [20].

The first three terms in Eq. (1) represent the non-associating, non-polar contributions. Conversely, the association and dipolar terms account for the directional interactions. The present study does not include the dipolar term; rather, it considers solely the first four contributions.

$$ \frac{a^{\text{res}}}{RT} = \frac{a^{\text{hs}}}{RT} + \frac{a^{\text{hc}}}{RT} + \frac{a^{\text{disp}}}{RT} + \frac{a^{\text{ass}}}{RT} \qquad (2) $$

As previously mentioned, in the PC-SAFT EoS, the non-associating terms are expressed as a function of three molecular parameters: the number of segments in a chain m, the temperature independent diameter of segment σ, and the depth of the potential well ε characterising dispersion forces between molecules.

The term “association” generally refers to intermolecular bonding, which is stronger than Van der Waals interactions but weaker than covalent or ionic bonds [2,21]. It is mostly used to model the hydrogen bonding between molecules. The process of hydrogen bonding occurs when a hydrogen donor site (typically a hydrogen atom), interacts with a hydrogen acceptor site (typically a lone pair of electrons). The strength of this bond, with a magnitude of approximately 20 kJ/mol [2], is roughly 20 times weaker than the strength of chemical bonds e.g., covalent and ionic bonds (~500 kJ/mol) [2], but around 20 times stronger than the strength of physical bonds such as Van der Waals bonds (<1 kJ/mol) [2]. It is important to recognise the existence of alternative form of associations, such as the electron transfer observed in benzene stacking [21,22]. However, these mechanisms are not considered in the present study.

The association term remains generally unchanged in most SAFT-type EoS, with only a minor variation in the functional expression of the strength of association $\Delta^{AB}$. It should be noted that the following expressions refer to a pure component; accordingly, the component index i has been omitted for clarity. The molar Helmholtz energy of association is expressed in Eq. (3).

$$ \frac{a^{\text{ass}}}{RT} = \frac{1}{2} M + \sum_{\text{all sites A}} \left[ \ln X^A - \frac{X^A}{2} \right] \qquad (3) $$

where M represents the total number of sites on the molecule, and $X^A$ represents the molar fraction of the non-bonded (free) association sites A on the molecule.

The molar fraction of the non-bonded association sites A, $X^A$, is expressed in Eq. (4). This expression is implicit and therefore requires an iterative numerical method, such as successive substitution, to determine its value.

$$ X^A = \left[ 1 + N_{\text{av}} \sum_{\text{all sites } B \neq A} [\rho X^B \Delta^{AB}] \right]^{-1} \qquad (4) $$

where $\Delta^{AB}$ represents the association strength between sites A and B, $\rho$ represents the molecular density and $N_{\text{av}}$ is Avogadro’s constant.

The strength of the association bond between sites A and B, $\Delta^{AB}$, is defined in Eq. (5) [19,23,24] and has the dimension of a volume. Some authors [16,18,25] replace the temperature-dependent segment diameter d in Eq. (5) with the temperature-independent segment diameter σ, thereby yielding Eq. (6).

$$ \Delta^{AB} = d^3 \cdot g^{hs}(d) \cdot \kappa^{AB} \cdot \left[ \exp\left(\frac{\epsilon^{AB}}{k_b T}\right) - 1 \right] \qquad (5) $$

$$ \Delta^{AB} = \sigma^3 g^{hs}(d) \cdot \kappa^{AB} \cdot \left[ \exp\left(\frac{\epsilon^{AB}}{k_b T}\right) - 1 \right] \qquad (6) $$

where $\kappa^{AB}$ is the dimensionless volume of association, $\epsilon^{AB}$ is the energy of association, $k_b$ is the Boltzmann’s constant, $g^{hs}(d)$ is the radial distribution function of the hard sphere fluid, and d is the temperature-dependent segment diameter. The energy of association is typically written as $\epsilon^{AB}/k_b$ and has the dimension of a temperature.

A total of $2n + 1$ parameters are needed to model the association of a given associating component with n hydrogen bonds: 1) n volumes parameters, 2) n energy parameters, 3) total number of sites M. However, it is commonly assumed that all hydrogen bonds within a component
exhibit identical volumes and energies of association. Under this approximation, the number of independent association parameters reduces to three: one volume parameter, one energy parameter, and one number of sites parameter. Therefore, excluding the total number of sites *M*, the model requires only five parameters to characterise a pure component: three non-associating parameters (*m*, *σ*, *ε*), and two associating parameters (*κ*<sup>AB</sup>, *ε*<sup>AB</sup>).

The formation of hydrogen bonds is only possible between sites of differing nature, i.e., between a hydrogen donor and a hydrogen acceptor site. Consequently, certain combinations are not permitted. The use of an association scheme provides a formalised framework for enumerating the possible hydrogen bond formation. The same scheme is employed for both the SAFT-type EoS and the cubic-plus-association (CPA) [26–28] EoS. Following the terminology introduced by Huang & Radosz [18], it is denoted by a number and a letter, for instance, ‘1A’. The number indicates the total number of sites on the molecule, whereas the letter represents the assumptions regarding the nature of those sites. There are three distinct types of site groups in the association schemes for associating components.

* **HD**: hydrogen donor sites; sites containing one or more hydrogen atoms. They can bond with the **HA** or **HAHD** sites.

* **HA**: hydrogen acceptor sites; site containing one or more lone pairs. They can bond with the **HD** or **HAHD** sites.

* **HAHD**: ambivalent sites; sites that can act as both hydrogen donor and acceptor. They can bond with all **HA**, **HD** or **HAHD** sites.

Huang & Radosz proposed eight association schemes [18], and novel schemes have been introduced in recent years. Table 1 provides an overview of all classical association schemes and several novel schemes [29–31]. It should be noted that the choice of the letter in the association scheme nomenclature is arbitrary and bears no relation to that used in other association schemes. For example, there is no link between the classical 4C scheme and the novel 2C scheme.

According to the available open literature, selecting the appropriate association scheme is essential for accurately predicting the thermodynamic properties of associating fluids in SAFT-type and CPA EoS. Initially, the parameterisation relied heavily on empirical approach, with an emphasis on reproducing vapour-liquid equilibria (VLE) data. More recently, advances in infrared spectroscopy [23], quantum-chemical calculations, and molecular simulations [32], have prompted a reconsideration of the parameterisation procedure, thereby providing a firmer physical foundation.

For small alcohols such as methanol and ethanol, the rigorous 3B scheme is often selected because it reproduces measured monomer fractions [23,33], e.g. the well-known experimental data from Luck

<table><caption>**Table 1**
List of classical and novel association schemes for associating components.</caption><thead><tr><th></th><th>Nomenclature</th><th>Description (types of site group)</th><th>Reference</th></tr></thead><tbody><tr><td rowspan="10">Classical association schemes</td><td>1A</td><td>1 HAHD</td><td>[18]</td></tr><tr><td>2A</td><td>2 HAHD</td><td>[18]</td></tr><tr><td>2B</td><td>1 HA + 1 HD</td><td>[18]</td></tr><tr><td>3A</td><td>3 HAHD</td><td>[18]</td></tr><tr><td>3B</td><td>1 HA + 2 HD</td><td>[18]</td></tr><tr><td></td><td>or</td><td>[18]</td></tr><tr><td>4A</td><td>2 HA + 1 HD</td><td>[18]</td></tr><tr><td>4B</td><td>1 HA + 3 HD</td><td>[18]</td></tr><tr><td></td><td>or</td><td>[18]</td></tr><tr><td>4C</td><td>3 HA + 1 HD</td><td>[18]</td></tr><tr><td></td><td>2 HA + 2 HD</td><td>[18]</td></tr><tr><td rowspan="4">Novel association schemes</td><td>2C</td><td>1 HA + 1 HAHD</td><td>[29]</td></tr><tr><td>3C</td><td>1 HA + 1 HD + 1 HAHD</td><td>[31]</td></tr><tr><td>6D</td><td>4 HA + 2 HD</td><td>[30]</td></tr><tr><td>7D</td><td>5 HA + 2 HD</td><td>[30]</td></tr></tbody></table>

et al. [34]. For larger alcohols where one acceptor site becomes sterically hindered, many groups switch to the simpler 2B scheme [16,18, 23]. It is noted that the 2C scheme was later introduced as an alternative to the 2B and 3B as the former underestimates and the latter overestimates the association, respectively, in mixtures containing alcohols and modelled with the sPC-SAFT EoS [29].

Furthermore, carboxylic acids must account for the formation of cyclic dimers, which involve two simultaneous hydrogen bonds. Modern parameter sets therefore employ the 2B scheme [5]. Whilst the 1A association scheme does not explicitly represent two simultaneous hydrogen bonds [18], its single high-energy site can mimic the net effect of cyclic dimerisation in a schematic sense.

For amines, the choice mostly reflects substitution: primary amines use the 3B scheme, secondary amines the 2B scheme, and tertiary amines the 1A scheme [18,35].

More recently, specific association schemes have been developed for non-self-associating components – those that possess either hydrogen acceptor or hydrogen donor sites, and therefore can participate only in an induced association (i.e., association driven by the interaction of a donor site of one component and an acceptor site of another) – such as acetone and chloroform. Notable examples of such schemes include the N and P association schemes [36]. However, this article only considers the association between self-associating pure components.

## 2. Corresponding states principle

The Van der Waals (VdW) corresponding states principle (or law) stipulates that all fluids exhibit the same thermodynamic properties, when compared at the same values of reduced variables $T_r$, $P_r$, $v_r$, defined as:

$$T_r = \frac{T}{T_c}, \quad P_r = \frac{P}{P_c}, \quad v_r = \frac{v}{v_c} \qquad (7)$$

where $T_c$, $P_c$, $v_c$ are the critical temperature, pressure and volume, respectively. In other words, one can derive an expression that links the three previously stated reduced variables and define a so-called universal EoS.

$$P_r = f_1(T_r, v_r) \qquad (8)$$

“Universal” here means that the EoS can be applied to any species as it does not involve any component-specific parameter. The application of the equation of state at the critical point, together with the imposition of the critical constraints, leads to an expression for $v_c$ as a function of $T_c$ and $P_c$. Consequently, in VdW’s theory, the EoS of a pure fluid can be written as Eq. (9).

$$P = f_2(T, v, T_c, P_c) \qquad (9)$$

Eq. (9) suggests that two parameters ($T_c$, $P_c$) should be sufficient to describe the behaviour of a pure fluid. This is indeed the case for the Van der Waals cubic EoS (see Eq. (10)):

$$P = \frac{RT}{v-b} - \frac{a}{v^2} \begin{cases} a = \frac{27 R^2 T_c^2}{64 P_c} \\ b = \frac{1 RT_c}{8 P_c} \end{cases} \Leftrightarrow P_r = \frac{8T_r}{3v_r - 1} - \frac{3}{v_r^2} \qquad (10)$$

In the last quarter of the 20th century, most cubic EoS formulations were based on an extended version of the corresponding states law involving not two parameters but three. This law is now referred to as the “three-parameter corresponding states principle” and follows the general expression given in Eq. (11):

$$P_r = g_1(T_r, v_r, \omega) \qquad (11)$$

Here, the introduction of the acentric factor, which accounts for the molecular size effects, significantly improves the accuracy of the EoS. This demonstrates that two parameters (i.e., $T_c$ and $P_c$) are insufficient to
accurately describe the properties of a pure fluid, and that a third parameter, such as the acentric factor or the critical compressibility factor $z_c$, must be included.

## 2.1. PC-SAFT sans the association term and the corresponding states principle

PC-SAFT, sans the association term, obeys the three-parameter corresponding states principle. The reduced pressure $P^*$ – not to be confused with $P_r$ – can be expressed as a function of the reduced temperature $T^*$, the reduced volume $v^*$ and the number of segments $m$, which is a size parameter as is the acentric factor $\omega$ (see Eq. (12)).

$$P^* = g_2(T^*, v^*, m) \text{where: } T^* \equiv \frac{T_k}{\epsilon}, P^* \equiv \frac{P\sigma^3}{\epsilon}, v^* \equiv \frac{v}{N_{av}\sigma^3} \quad (12)$$

where $k_b$ is the Boltzmann constant, $N_{av}$ is the Avogadro constant, $\sigma$ and $\epsilon$ are PC-SAFT size and energy parameters, respectively. Like a cubic EoS, PC-SAFT requires knowledge of three parameters ($m, \sigma, \epsilon$) to predict the properties of a pure fluid.

The reduced pressure is derived from the reduced molar residual Helmholtz energy, which is itself a function of the three reduced variables listed above. It is important to note that the reduced variables are denoted by the superscript $^*$, for instance, $P^*$.

$$a^{\text{res}^*} = \frac{a^{\text{res}}}{\epsilon N_a} = h_1(T^*, v^*, m) \quad (13)$$

$$P^{\text{res}^*} = P^* - \frac{T^*}{v^*} = -\left(\frac{\partial a^{\text{res}^*}}{\partial v^*}\right)_{T^*} \quad (14)$$

Furthermore, different reduced properties can be derived from the reduced molar residual Helmholtz energy. Therefore, all of these properties are also functions of $T^*, v^*$, and $m$. Following is a non-exhaustive list of the reduced properties.

* Reduced molar residual entropy: $s^{\text{res}^*} = \frac{s^{\text{res}}}{k_b N_a} = -\left(\frac{\partial T^{\text{res}^*}}{\partial T}\right)_{T^*}$

* Reduced molar residual internal energy: $u^{\text{res}^*} = \frac{u^{\text{res}}}{\epsilon N_a} = a^{\text{res}^*} + T^*(s^{\text{res}^*} - 1)$

* Reduced molar residual enthalpy: $h^{\text{res}^*} = \frac{h^{\text{res}}}{\epsilon N_a} = a^{\text{res}^*} + P^* v^* + T^*(s^{\text{res}^*} - 1)$

* Reduced molar residual isobaric heat capacity: $c_p^{\text{res}^*} = \frac{c_p^{\text{res}}}{k_b N_a} = \left(\frac{\partial h^{\text{res}^*}}{\partial T}\right)_P$

* Reduced molar residual isochoric heat capacity: $c_v^{\text{res}^*} = \frac{c_v^{\text{res}}}{k_b N_a} = \left(\frac{\partial v^{\text{res}^*}}{\partial T}\right)_P$

Eventually, the corresponding states principle applied to the PC-SAFT EoS can be expressed as follows: in the scenario where two components have identical $T^*, v^*$ and $m$, they exhibit the same aforementioned reduced properties in single-phase state. Moreover, in the multi-phase states, they have the same reduced saturated vapour pressure $P^*_{\text{sat}}$, reduced saturated liquid and vapour volume $v^*_{\text{sat,liq}}$ and $v^*_{\text{sat,gas}}$, reduced critical properties, $T_c^*, P_c^*$ and $v_c^*$.

The present study addresses a volume-translated version of the PC-SAFT EoS, termed I-PC-SAFT. This model sans the association term (hereafter referred to as Non-A-I-PC-SAFT [9]) follows the three-parameter corresponding states principle [37]. It has been

demonstrated that, for the Non-A-I-PC-SAFT EoS, the reduced critical temperature $T_c^*$, the reduced critical pressure $P_c^*$ and the acentric factor $\omega$ are all only correlated with the number of segments $m$. These correlations are denoted as $f(m)$ in Eq. (15).

$$P_{c,EoS}^*, T_{c,EoS}^*, \omega_{EoS} = f(m), \text{ with: } \left\{ \begin{array}{l} \omega = -1 - \log_{10} \left( \frac{P_{\text{sat}}^* (T^* = 0.7 T_c^*)}{P_c^*} \right) \\ T_c^* = \frac{T_c}{\epsilon/k_b} \\ P_c^* = \frac{P_c \sigma^3}{\epsilon} \end{array} \right. \quad (15)$$

The correlations $\omega_{EoS}(m)$, $T_{c,EoS}^*(m)$, and $P_{c,EoS}^*(m)$, established by Privat et al. [37], are used to determine the parameters ($m, \epsilon/k_b, \sigma$) of the Non-A-I-PC-SAFT EoS that ensure strict reproduction of the experimental critical temperature, critical pressure and acentric factor.

$$\text{System to be solved: } \left\{ \begin{array}{l} \omega_{\text{exp}} = \omega_{\text{EoS}}(m) \\ T_{c,\text{exp}}^*(T_{c,\text{exp}}, \epsilon/k_b) = T_{c,\text{EoS}}^*(m) \\ P_{c,\text{exp}}^*(P_{c,\text{exp}}, \epsilon/k_b, \sigma) = P_{c,\text{EoS}}^*(m) \end{array} \right. \quad (16)$$

Unknowns: ($m, \epsilon/k_b, \sigma$)

Readers are referred to Polishuk et al. [20] for further details on the application of the corresponding states principle to various SAFT models, and to Privat et al. [37] for its specific application to the Non-A-I-PC-SAFT model. It should be noted that the PC-SAFT model uses the dimensionless packing factor $\eta$ as natural variable for describing the fluid density. For consistency, this reduced variable replaces the reduced volume $v^*$.

In a previous study, the Non-A-I-PC-SAFT EoS was used to model self-associating components, i.e., associating components were treated as non-associating components [7]. As stated in the introduction, the present study aims: (i) to examine the benefits and limitations of incorporating an association term in I-PC-SAFT in order to model the phase-equilibrium and thermal properties of self-associating molecules and (ii) to develop a universal parameterisation strategy of the EoS. In the following section, the corresponding states principle is examined in detail for the I-PC-SAFT model with the association term in order to develop a universal parameterisation procedure, enforcing strict reproduction of the experimental coordinates of the critical point in the $(T, P)$ plane and the acentric factor.

## 3. PC-SAFT with the association term and the corresponding states principle

Initial testing is conducted on the association term to ensure that the PC-SAFT, and subsequently I-PC-SAFT EoS, still obey the corresponding states principle when the association term is included. The previous section and the studies by Polishuk et al. [20] and Privat et al. [37] have demonstrated that, without the association term, the I-PC-SAFT EoS obeys the three-parameter corresponding states principle. As previously discussed, the association term adds three parameters per self-associating component: the number of associating sites, the association volume and association energy.

To adopt a procedure analogous to that of Privat et al. [37] for the parameterisation of the Non-A-I-PC-SAFT EoS, the reduced association term ($a^{\text{ass}}/RT$) is expressed in a universal form, i.e., solely, as a function of reduced (dimensionless) variables. This allows for the determination and clear identification of the total number of reduced variables. In Eqs.
(3) to (6), the reduced variables are as follows: 1) the reduced temperature-dependent diameter of a segment $d/\sigma$, 2) the radial distribution function of the hard sphere fluid $g^h$, 3) the reduced molecular density ($\rho\sigma^3$), 4) the reduced strength of association $\Delta^{AB}/\sigma^3$ and 5) the molar fraction of non-bonded sites $X^A$.

As indicated in Eq. (17), the reduced diameter $d/\sigma$ depends solely on the reduced temperature $T^*$ whereas both the radial distribution function $g^h$ and the reduced molecular density ($\rho\sigma^3$) depend on three dimensionless variables: $m, \eta$ and $T^*$. Full details can be found in Polishuk et al. [20]

$$ \left\{ \begin{aligned} \frac{d}{\sigma} &= \left[ 1 - 0.12 \exp\left(-\frac{3}{T^*}\right) \right] = f_1(T^*) \\ g^h &= f_2[\zeta_3 = f_3(m, \eta, T^*), (\zeta_2\sigma) = f_4(m, \eta, T^*)] = f_5(m, \eta, T^*) \\ (\rho\sigma^3) &= \frac{6}{\pi}\eta \left[m\left(\frac{d}{\sigma}\right)^3\right]^{-1} = f_6(m, \eta, T^*) \end{aligned} \right. \quad (17) $$

By combining Eq. (5) and the first and second sub-equations of Eq. (17), the reduced strength of the association bond $\Delta^{AB}/\sigma^3$ can be expressed as a function of five reduced parameters: $m, \eta, T^*, \kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$ as shown in Eq. (18).

$$ \frac{\Delta^{AB}}{\sigma^3} = \left(\frac{d}{\sigma}\right)^3 g^h \kappa^{AB} \left[\exp\left(\frac{\varepsilon^{AB}}{\varepsilon}\right) - 1\right] = f_7(m, \eta, T^*, \kappa^{AB}, \varepsilon^{AB}/\varepsilon) \quad (18) $$

In certain models (see Eq. (6)), the strength of the association bond is proportional to $\sigma^3$ instead of $d^3$. Consequently, the corresponding reduced variable becomes simpler yet remains independent of $\sigma$, as expressed in Eq. (19).

$$ \frac{\Delta^{AB}}{\sigma^3} = g^h \kappa^{AB} \left[ \exp\left(\frac{\varepsilon^{AB}}{\varepsilon}\right) - 1 \right] = f_{7,2}(m, \eta, T^*, \kappa^{AB}, \varepsilon^{AB}/\varepsilon) \quad (19) $$

Next, the molar fraction of the non-bonded association sites $X^A$ can be expressed as a function of the same five dimensionless parameters: $m, \eta, T^*, \kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$.

$$ X^A = \left[ 1 + N_{av} \sum_{\text{all sites } B \neq A} \left[ \left( \rho\sigma^3 \frac{\Delta^{AB}}{\sigma^3} X^B \right) \right]^{-1} \right] = f_8(m, \eta, T^*, \kappa^{AB}, \varepsilon^{AB}/\varepsilon) \quad (20) $$

Finally, the reduced molar Helmholtz energy of association $a^{ass}/RT$, which, for a pure component is a function solely of $X^A$ and $M$, becomes a function of six reduced parameters: $m, \eta, T^*, \kappa^{AB}, \varepsilon^{AB}/\varepsilon$ and $M$.

$$ \frac{a^{ass}}{RT} = \frac{M}{2} + \sum_{\text{all sites } A} \left[ \ln X^A - \frac{X^A}{2} \right] = f_9(m, \eta, T^*, \kappa^{AB}, \varepsilon^{AB}/\varepsilon, M) \quad (21) $$

Since the number of sites $M$ is fixed by the association scheme, the PC-SAFT EoS (with the association term) requires five dimensionless variables to model pure components, i.e., to express the reduced quantity $a^{res}/RT$.

1. Reduced temperature: $T^*$

2. Packing fraction: $\eta$

3. Dimensionless chain length: $m$

4. Dimensionless association volume: $\kappa^{AB}$

5. Reduced association energy: $\varepsilon^{AB}/\varepsilon$

According to Gibbs' phase rule, the five-parameter corresponding states principle applied to the PC-SAFT EoS stipulates that four

parameters ($T^*, m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon$) are required to describe liquid-vapour equilibrium, whilst three parameters ($m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon$) are sufficient to determine the critical point of a pure compound.

## 4. Preliminary step for PC-SAFT parameterisation: search for the acceptable EoS parameters intervals based on the study of the critical points of pure components

The previous section has demonstrated that PC-SAFT – for a specified association scheme – can predict the critical point of a pure component from only three reduced parameters. Therefore, the reduced critical temperature $T_c^*$, the reduced critical pressure $P_c^*$ and the acentric factor $\omega$ are all functions of:

1. The dimensionless chain length: $m$

2. The dimensionless association volume: $\kappa^{AB}$

3. The reduced association energy: $\varepsilon^{AB}/\varepsilon$

The influence of the association parameters on the reduced critical temperature and pressure, as well as the acentric factor, is examined separately for each association scheme. The purpose of the present study is to identify the possible PC-SAFT parameters that ensure a strict reproduction of the experimental values of the critical point coordinates $T_{c,exp}$, $P_{c,exp}$ and acentric factor $\omega_{exp}$.

The reduced association parameters $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$ are fixed over a wide interval. For each pair ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$), the acentric factor $\omega$, the reduced critical temperature $T_c^*$ and the reduced critical pressure $P_c^*$ are calculated for different values of $m$. This method is inspired by the NonA-I-PC-SAFT EoS parameterisation approach by Privat et al. [37]

For certain pair ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$), peculiar points (hereafter referred to as features) and variation are observed in these three properties. The coordinates of these features are plotted in a contour plot in the ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$) plane for a simpler visual analysis.

The following sections provide a comprehensive overview of the 1A association scheme. The features are analogous across all association schemes. The only difference is found in their numerical values. Consequently, the same line of reasoning remains applicable to 2A, 2B, 3A, 3B, etc., schemes. The results for the other association schemes are provided in the supporting materials (the reader is referred to the PDF File: "SM1_Peculiar points of each association scheme.pdf").

The following conclusions can be drawn. Across different pairs ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$), two distinct variations of the acentric factor with respect to $m$ are observed. Additionally, the reduced critical temperature $T_c^*$ and pressure $P_c^*$ exhibit a single consistent variation with respect to $m$. They are illustrated in Figs. 1 and 2. The same variations are observed for all examined association schemes. The sections below describe the variations on $m$ and the features of each function. These features are also illustrated in Figs. 1 and 2.

### 4.1. Acentric factor $\omega$

Depending on the pair of parameters ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$), the evolution of the acentric factor $\omega(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ with respect to $m$ (hereafter simply referred to as $\omega(m)$) may exhibit either of the two behaviours illustrated in Figs. 1(a) and 2(a).

In form 1 of $\omega(m)$ (see Fig. 1(a)), only a global minimum – located at ($m_{\omega_{min,g}}, \omega_{min,g}$) – is observed. In form 2 (see Fig. 2(a)), two local extrema are present, in addition to the global minimum: a local maximum ($m_{\omega_{max,1}}, \omega_{max,1}$) and a local minimum ($m_{\omega_{min,1}}, \omega_{min,1}$). It should be noted that form 1 is qualitatively identical to what was observed with the
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Prediction from the PC-SAFT EoS of the acentric factor ω and the reduced critical temperature $T_c^*$ and pressure $P_c^*$ with respect to m at $κ^{AB} = 0.05$ and $ε^{AB}/ε = 10.0$ using the 1A association scheme.</figcaption></figure>

(a) Acentric factor ω. A global minimum $m_{ω_{min,g}}$ occurs at the $m \approx 0.3$. The bottom-left inset highlights the behaviour near this global minimum (grey-shaded area). For $m > m_{ω_{min,g}}$, the function increases monotonically.

(b) Reduced critical temperature $T_c^*$. At least two critical points exist for each m. The smaller $T_c^*$ curve (blue) represents the fictitious liquid-liquid reduced critical temperature loci, an artefact of the PC-SAFT model [37–39]. It is quasi-invariant with respect to m, except at small values of m. The greater $T_c^*$ curve (black) represents the ordinary liquid-vapour reduced critical temperature loci. A global minimum $T_{c,min}^*$ can be found at $m \approx 0.7$ before the function increases. A salient point at $m_{salient} \approx 100$ limits the function beyond which two distinct reduced critical temperatures emerge.

(c) Reduced critical pressure $P_c^*$. It is continuously decreasing function, with a steep slope at the small m values and a gentler slope at large m values.

NonA-I-PC-SAFT EoS [37]. The non-monotonic dependence of ω on m of form 2 is problematic, as a single value of ω may correspond to up to three different values of m, whereas the objective is to infer m from ω. Section 4.4 explains how this issue can be addressed.

To make this section as clear as possible, only the conclusions regarding the dependence of the coordinates ($m_{ω_{min,g}}$, $ω_{min,g}$), ($m_{ω_{max,l}}$, $ω_{max,l}$) and ($m_{ω_{min,l}}$, $ω_{min,l}$) on the variables ($κ^{AB}$, $ε^{AB}/ε$) are provided here. The supporting elements for these conclusions are included in Appendix 1.

The $m_{ω_{min,g}}$ corresponding to the global minimum – in both form 1 and form 2 of $ω(m)$ – is quasi-invariant with respect to the reduced association parameters ($κ^{AB}$, $ε^{AB}/ε$) and has always a value much lower than unity. However, in a limited interval of these reduced association parameters, the acentric factor $ω_{min,g}$ can change from a negative to a

positive value. As the majority of components are observed to take on values $m > 1$ (with a few exceptions that are found in certain quantum fluids[8]), this global minimum will most likely be disregarded.

Form 2 of the function $ω(m)$, observed for certain pairs ($κ^{AB}$, $ε^{AB}/ε$), is characterised by the presence of two local extrema on $ω(m)$. These phenomena appear at low $κ^{AB}$ values and moderate to high values of $ε^{AB}/ε$. In this form, the local maximum occurs at $m_{ω_{max,l}}$, followed by the local minimum at $m_{ω_{min,l}}$, with $m_{ω_{min,g}} < m_{ω_{max,l}} < m_{ω_{min,l}}$ (see Fig. 2(a)). Both local extrema display a strong dependence on the reduced association parameters. Notably, the local minimum $m_{ω_{min,l}}$ can reach values as high as 4.5 with a corresponding acentric factor value $ω_{min,l}$ close to unity for the 1A association scheme.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Prediction from the PC-SAFT EoS of the acentric factor $\omega$ and the reduced critical temperature $T_c^*$ and pressure $P_c^*$ with respect to $m$ at $\kappa^{AB} = 0.0003$ and $\epsilon^{AB}/\epsilon = 15.0$ using the 1A association scheme.</figcaption></figure>

(a) Acentric factor $\omega$. A global minimum $m_{\omega_{\text{min},t}} \approx 0.3$ is observed. A local maximum $m_{\omega_{\text{max},t}} \approx 1.5$ and minimum $m_{\omega_{\text{min},t}} \approx 3.5$ are found near the global minimum. The bottom-left inset illustrates these extrema (grey-shaded area). For $m > m_{\omega_{\text{min},t}}$, the function increases monotonically.

(b) Reduced critical temperature $T_c^*$. Same description found in Fig. 1(b).

(c) Reduced critical pressure $P_c^*$. Same description found in Fig. 1(c).

## 4.2. Reduced critical temperature $T_c^*$

This section is devoted to studying the function $T_c^*(m, \kappa^{AB}, \epsilon^{AB}/\epsilon)$, denoted hereafter as $T_c^*(m)$ for simplicity. For a given pair of parameters $(\kappa^{AB}, \epsilon^{AB}/\epsilon)$, the same number of segments $m$ can yield at least two different reduced critical temperatures $T_c^*$—referred to as *fictitious* and *ordinary/physically realistic* critical temperatures [37]. These two types of critical points can be found in Fig. 1(b) and 2(b). Similarly, only the conclusions of the study of the dependence of $m_{T_c^{\text{min}}}$ and $T_c^{\text{min}}$ with respect to $(\kappa^{AB}, \epsilon^{AB}/\epsilon)$ are presented here. Further details can be found in Appendix 1.

In contrast to the acentric factor, the dependence of $T_c^*$ on $m$ is independent of the reduced association parameters and has the same trend as that observed with the NonA-I-PC-SAFT equation [37].

The physically realistic reduced critical temperature is plotted as a function of $m$ and the yielding black curve $T_c^*(m)$ is visible in Figs. 1(b) and 2(b). A minimum $T_{c,\text{min}}^*$ is observable along these curves. The function $T_c^*(m)$ decreases for $m < m_{T_c^{\text{min}}}$ and increases for $m > m_{T_c^{\text{min}}}$ before reaching a salient point at $m_{\text{salient}}$. Beyond this point, two reduced critical curves emerge. Consequently, for $m > m_{\text{salient}}$, three reduced critical temperatures may be obtained: one fictitious critical locus and,

two *physically realistic* critical loci. Both the $m_{\text{salient}}$ and the reduced critical temperature $T_{c,\text{salient}}^*$ associated to the salient point are quasi-invariant with respect to $(\kappa^{AB}, \epsilon^{AB}/\epsilon)$, at approximately 100 and 4.5 respectively.

In addition to the ordinary reduced critical temperature, PC-SAFT models predict the low-valued fictitious reduced critical temperature $T_c^*(m)$. Fictitious critical loci, plotted in blue, correspond to extremely high pressures and very low temperatures (far below the ordinary vapour-liquid critical point). These involve nearly incompressible phases and, whilst predicted by most SAFT EoS, have never been observed experimentally. They are therefore regarded as artefacts of the EoS [37].

Therefore, when performing calculations using the PC-SAFT EoS, it is crucial that the fictitious critical curve remains sufficiently distant from the domain of application where the EoS is used and does not interfere with the ordinary critical curve. In other words, the difference between these two curves must remain large enough to ensure that the fictitious critical temperature does not affect VLE calculations. For the NonA-I-PC-SAFT EoS, the ratio $T_{c,\text{fictitious}}^*/T_{c,\text{ordinary}}^*$ ≈ 0.3 indicates that $T_{c,\text{fictitious}}^* \ll T_{c,\text{ordinary}}^*$ [37]. For PC-SAFT with the association term, the ratio is even smaller at approximately 0.25 for all $(\kappa^{AB}, \epsilon^{AB}/\epsilon)$ values (see Appendix 1). It is therefore highly unlikely that a numerical procedure would
converge towards $T_c^{\text{ficititious}}$ when determining the ordinary reduced critical temperature, $T_c^{\text{c.ordinary}}$.

### 4.3. Reduced critical pressure $P_c^*$

As previously, the function $P_c^*(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ is now denoted as $P_c^*(m)$ for simplicity. For a given pair of parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$, the function $P_c^*(m)$ is represented graphically in Figs. 1(c) and 2(c). Similarly to the reduced critical temperature, the reduced critical pressure exhibits a single trend as a function of $m$. Furthermore, the resulting curve is almost universal, i.e., unaffected by the reduced association parameters. The reduced critical pressure $P_c^*(m)$ decreases monotonically with $m$, with a steep slope at low $m$ and a gentler slope at high $m$. Its behaviour is consistent with that obtained using the NonA-I-PC-SAFT model [37]. No notable feature is observed for this function.

### 4.4. Constructing acceptable functions of the acentric factor $\omega$ and reduced critical temperature $T_c^*$ with respect to $m$, for given association parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$

To implement a parameterisation analogous to that used for the NonA-I-PC-SAFT, it is essential that several features of the functions $\omega(m)$ and $T_c^*(m)$ are discussed. Firstly, the acentric factor function $\omega(m)$ must be bijective with respect to $m$ for any pair $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$, thereby ensuring that a unique value of acentric factor $\omega$ corresponds to an $m$ value. As the value of $m$ is deduced from equating the experimental and the calculated acentric factors, the bijectivity of the function $\omega(m)$ is crucial. Secondly, $T_c^*(m)$ and $P_c^*(m)$ must be well-defined functions so that a unique pair of $T_c^*$ and $P_c^*$ values correspond to each specified $m$ value. As $P_c^*(m)$ is a monotonically decreasing function, this criterion is already satisfied.

To guarantee the bijectivity of $\omega(m)$, a lower bound is imposed on $m$. Two possible candidates for this lower bound are considered depending on the form of the function. When $\omega(m)$ exhibits a single global minimum, the corresponding $m_{\omega_{min,g}}$ value is intuitively chosen as the lower bound. However, as the global minimum $m_{\omega_{min,g}}$ takes very small values and can be difficult to determine, a fixed lower boundary of $m = 0.7$ is instead used. When the function has both a local maximum and minimum in addition to the global minimum, the local minimum $m_{\omega_{min,l}}$ is selected as the lower bound. Ultimately, the lower bound applied to $m$ that considers both cases is defined as: $m_{inf} = \max(m_{\omega_{min,l}}, 0.7)$.

An upper bound on $m$ is also required to ensure that a one-to-one $T_c^*(m)$ function is obtained. The possibility of the interference of the fictitious critical temperature loci has been addressed in Section 4.2, therefore eliminating one of the possible candidates of $T_c^*$. For the ordinary critical temperature loci, two values of $T_c^*$ may occur for $m \ge m_{salient}$. The salient point $m_{salient}$ could thus be considered as the theoretical upper bound. In practice, however, $m$ rarely exceeds 30, and thus an upper bound of $m = 30$ can be adopted.

<caption>**Table 2**</caption>

Acceptable interval of $m$ for a given pair of association parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ for a safe $T_c^*$ and $P_c^*$ determination, knowing the acentric factor $\omega_{exp}$.

<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>Acceptable range of <i>m</i>:</td><td><i>m</i> &isin; [<i>m</i><sub>inf</sub>, <i>m</i><sub>sup</sub>]</td></tr><tr><td>Lower bound of <i>m</i>:</td><td><i>m</i><sub>inf</sub> = max(<i>m</i><sub>ω<sub>min,l</sub></sub>, 0.7) where <i>m</i><sub>ω<sub>min,l</sub></sub> is <i>m</i> value corresponding to the local minimum of the <i>ω</i>(<i>m</i>) function</td></tr><tr><td>Upper bound of <i>m</i>:</td><td><i>m</i><sub>sup</sub> &asymp; 30</td></tr></tbody></table>

### 4.5. Conclusion: acceptable $m$ interval for the identification of PC-SAFT parameters

In summary, for a given pair of association parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$, the key step prior to component parameterisation (i.e., determining $m$, $T_c^*$ and $P_c^*$, from the experimental acentric factor $\omega_{exp}$) is to define an acceptable interval of $m$. It is recommended that $m$ is chosen between $m_{inf} = \max(m_{\omega_{min,l}}, 0.7)$ and $m_{sup} &asymp; 30$. Table 2 summarises the definitions of different terms found in constructing the $m$ interval.

## 5. Parameterisation procedure for the I-PC-SAFT EoS with association term

### 5.1. I-PC-SAFT EoS, a volume-translated variant of PC-SAFT

In a previous publication, the PC-SAFT model–without the association term but combined with a universal and transparent parameterisation procedure–was evaluated against the NonA-I-PC-SAFT model, a volume-translated variant of PC-SAFT [6]. This comparison involved 1,800 fluids, encompassing both associating and non-associating compounds.

In the classical approach, the PC-SAFT parameters are fitted by minimising the deviations between the predicted and the experimental properties of interest, typically the vapour pressures $P_{sat}$ and the saturated liquid densities $\rho_{liq,sat}$. Whilst this strategy results in excellent predictions of these properties, it generally leads to an overestimation of critical pressure. If the aim is to reproduce the critical point in the $(P, T)$ plane, this approach is therefore not advisable.

An alternative strategy is the volume translation approach, widely used for the cubic EoS parameterisation. The NonA-I-PC-SAFT model adopts this approach by determining the parameters to strictly reproduce the experimental critical point in the $(P, T)$ plane and the experimental acentric factor. With this strategy, all property prediction, except the saturated liquid densities, remains comparable to those obtained by the classical approach. As observed for cubic EoS parameterisation, the predictions of the saturated liquid densities become less accurate.

To mitigate this drawback, an additional volume translation parameter $c$ is introduced and adjusted such that the experimental liquid density at the reduced temperature $T_r = 0.8$ is strictly reproduced by the EoS. This distributes the liquid density errors over the full temperature range. It should be noted, however, that this procedure does not guarantee an accurate prediction of the experimental critical volume $v_c$.

Table 3 summarises the mean absolute percentage errors (MAPE) for different properties of interests, allowing a quantitative comparison between the two parameterisation strategies.

### 5.2. Determination of $(m, \sigma, \varepsilon/k_b)$ of pure components given $T_{c,exp}$, $P_{c,exp}$ and $\omega_{exp}$ for a specified set of $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ parameters

The parameterisation of the I-PC-SAFT EoS with an association term relies on the function $\omega(m)$ being bijective over the admissible interval, ensuring that a unique value of $m$ can be determined from the $\omega_{exp}$, and that this $m$ in turn defines unique values of $T_c^*$ and $P_c^*$. Parameters $\varepsilon/k_b$ and $\sigma$ are then straightforwardly determined from $T_c^*$ and $P_c^*$. As previously stated, each pair of association parameters leads to three new functions $\omega(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$, $T_c^*(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ and $P_c^*(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$. To simplify the parameterisation procedure, they are approximated using polynomials in $m$, whose coefficients are expressed as polynomials in $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$. Their general expression is given by Eq. (22). The coefficients $b_{jk}$ are provided in the supporting materials (the reader is referred to the Excel file "SM2_Numerical coefs values for the developed correlations. xlsx"). It is important to emphasise that they are specific to each association scheme.
<caption>**Table 3**</caption>

Comparison of the deviations between experimental and calculated values for different properties as predicted by the PC-SAFT and NonA-I-PCSAFT. The compared values are taken from Ramírez-Vélez et al [6].

<table><thead><tr><th>EoS</th><th>MAPE on P<sub>sat</sub>(1800 fluids)</th><th>MAPE on v<sub>liq,sat</sub> (1800 fluids T<sub>r</sub> &lt; 0.9)</th><th>MAPE on ΔH<sub>v</sub> (1536 fluids)</th><th>MAPE on c<sub>p,liq,sat</sub> (890 fluids)</th><th>MAPE on T<sub>c</sub> (1800 fluids)</th><th>MAPE on P<sub>c</sub> (1800 fluids)</th><th>MAPE on v<sub>c</sub> (1800 fluids)</th><th>Global average deviation over the 306,700 available data points</th></tr></thead><tbody><tr><td>PC-SAFT sans the association term</td><td>1.18 %</td><td>0.95 %</td><td>3.25 %</td><td>4.18 %</td><td>2.28 %</td><td>20.8 %</td><td>4.38 %</td><td>2.2 %</td></tr><tr><td>NonA-I-PC-SAFT</td><td>2.08 %</td><td>4.60 %</td><td>4.00 %</td><td>4.10 %</td><td>0 %</td><td>0 %</td><td>12.4 %</td><td>3.6 %</td></tr></tbody></table>

$$ \text{For } X \in \left\{ \omega_{EoS}, T_{c,EoS}^*, \frac{1}{P_{c,EoS}^*} \right\}: \quad \left\{ \begin{array}{l} X = f_X(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon) = \sum_{i=0}^{N} a_i \cdot m^i \\ \text{with } : a_i = \sum_{j=0}^{5} \sum_{k=0}^{j} b_{jk} (\kappa^{AB})^j \left(\frac{\varepsilon^{AB}}{\varepsilon}\right)^{j-k} \end{array} \right. \qquad (22) $$

Where N is the degree of the polynomial in m.

For a pure component, the non-associating parameters ($m, \sigma, \varepsilon/k_b$) are determined so as to exactly reproduce the experimental critical temperature and pressure and the experimental acentric factor for a specified set of association parameters ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$). The following procedure has to be followed step by step:

### 1. Fix reduced association parameters

Specify the values of $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$

### 2. Determine m

Solve the equation $\omega_{exp} = f_\omega(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ with respect to m, i.e., use the polynomial $f_\omega(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ to determine the parameter $m \in [m_{inf}, m_{sup}]$ that strictly reproduces the experimental acentric factor $\omega_{exp}$.

### 3. Determine $\varepsilon/k_b$

Knowing m, calculate the reduced critical temperature $T_{c,EoS}^*$ from the polynomial $f_{T_c^*}^*(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$. Imposing $T_{c,EoS}^* = T_{c,exp}^* \equiv \frac{T_{c,exp}}{\varepsilon/k_b}$, the parameter $\varepsilon/k_b$ can readily be determined as:

$$ \frac{\varepsilon}{k_b} = \frac{T_{c,exp}}{T_{c,EoS}^*} \qquad (23) $$

### 4. Determine $\sigma$

Knowing m, calculate the reduced critical pressure $P_{c,EoS}^*$ from the polynomial $f_{P_c^*}: P_{c,EoS}^* \approx f_{P_c^*}(m, \kappa^{AB}, \varepsilon^{AB}/\varepsilon)$. Imposing $P_{c,EoS}^* = P_{c,exp}^* \equiv \frac{P_{c,exp}}{\varepsilon^3}$, the parameter $\sigma$ can readily be determined as:

$$ \sigma = \sqrt[3]{\frac{\varepsilon \cdot P_{c,EoS}^*}{P_{c,exp}}} \qquad (24) $$

### 5. Determine $\varepsilon^{AB}/k_b$

Calculate the non-reduced association energy parameter $\varepsilon^{AB}/k_b$ from $\varepsilon^{AB}/\varepsilon$ and the energy parameter $\varepsilon/k_b$ as:

$$ \varepsilon^{AB} / k_b = \frac{\varepsilon^{AB}}{\varepsilon} \frac{\varepsilon}{k_b} \qquad (25) $$

### 6. Determine c

Calculate the volume translation c such that the experimental saturated liquid density at a reduced temperature of 0.8 is strictly reproduced by the I-PC-SAFT EoS.

### 5.3. Determination of the optimal association parameters ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$) for a given pure component

The previous section demonstrated that the values of the parameters ($m, \sigma/k_b, \sigma$) and the volume translation c can univocally be determined once a pair of reduced associating parameters $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$, has been specified. The remaining step is, therefore, to determine the optimal values of the association parameters $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$.

This question is addressed through two tasks:

* Task 1 – evaluate the impact of the association scheme and examine the evolution of the non-associating parameters ($m, \varepsilon/k_b, \sigma$) with the association parameters $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$.

* Task 2 – assess the accuracy of the I-PC-SAFT EoS in predicting the target properties: vapour pressures, saturated liquid densities, and the thermal properties (i.e., isobaric heat capacities and heats of vaporisation) for different association parameters. For each property, the mean absolute percentage error (MAPE) defined in Eq. (26) was used to quantify the prediction quality.

$$ \text{MAPE}_{\text{PROP}} = \frac{1}{N_{\text{points,PROP}}} \sum_{i}^{N_{\text{points,PROP}}} \left| \frac{\text{PROP}_{i,\text{exp}} - \text{PROP}_{i,\text{EoS}}}{\text{PROP}_{i,\text{exp}}} \right| \qquad (26) $$

where PROP$_i$ denotes the property i of the pure component, e.g., the saturated vapour pressure, the saturated liquid densities etc., the subscripts exp and EoS refer to the experimental and the predicted properties, respectively. Each property is evaluated using 50 pseudo-experimental points $N_{\text{points,PROP}}$, obtained from DIPPR correlations within the temperature range $T \in [T_{\text{min,correlation}}, T_{\text{max,correlation}}]$. $T_{\text{min,correlation}}$ and $T_{\text{max,correlation}}$ represent lower and upper boundaries of the validity of the correlation.

No unique set of association parameters $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$ simultaneously minimises deviations across all predicted properties. An objective function, therefore, must be defined. The “optimal” set of parameters is the one that minimises this function, with the additional constraint that its value must not exceed that obtained using the NonA-I-PC-SAFT.

To ensure that the global minimum of the objective function is found, two methods can be used to determine the optimal set of parameters:

* **Brute force grid search method**

Contour plots are generated in the ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$) plane. The optimal parameters correspond to the minimum objective function value. The accuracy of the parameters greatly depends on the parameter step size.

* **Numerical minimisation**

A solver is used to minimise the objective function directly. This approach identifies optimal parameters ($\kappa^{AB}, \varepsilon^{AB}/\varepsilon$) more efficiently. Multiple initialisations are performed to search for the “global” minimum.

Following Ramírez-Vélez et al. [7], vapour pressures are prioritised followed by the liquid densities, and then the thermal properties. Eq. (27) defines the weighted objective function.

$$ F_{\text{ob}} = \frac{3\text{Error}_{p_{\text{sat}}} + 2\text{Error}_{p_{\text{li,sat}}} + \text{Error}_{c_{p,\text{li,sat}}} + \text{Error}_{\Delta H_{\text{vap}}}}{7} \quad \text{where} \\ \left\{ \begin{array}{l} \text{Error}_{\text{prop}} = \frac{1}{N_{\text{points}}} \sum_{i}^{N_{\text{points}}} \left| \ln \left( \frac{\text{prop}_{i,\text{EoS}}}{\text{prop}_{i,\text{exp}}} \right) \right|, \text{prop} = P_{\text{sat}} \\ \\ \text{Error}_{\text{prop}} = \frac{1}{N_{\text{points}}} \sum_{i}^{N_{\text{points}}} \left| \frac{\text{prop}_{i,\text{EoS}} - \text{prop}_{i,\text{exp}}}{\text{prop}_{i,\text{exp}}} \right|, \text{prop} \neq P_{\text{sat}} \end{array} \right. \qquad (27) $$
<figure><img src="image_4.png" /><figcaption>**Fig. 5.** The plots of the predicted properties of 1-octanol using the 3A scheme and the pseudo-experimental points from DIPPR. Blue curve: I-PC-SAFT EoS prediction. Red hollow circles: the experimental points from DIPPR correlations. Filled circles: critical properties (blue = predicted, red = experimental).</figcaption></figure>

(a) Saturated vapour pressure in logarithm as a function of the inverse of temperature.

(b) Temperature as a function of saturated liquid density.

(c) Liquid heat capacity as a function of temperature.

(d) Heat of vaporisation as a function of temperature.

The numerical method to obtain the best accuracy. Apart from the experimental critical point and the acentric factor, ensured by the parameterisation, all optimal sets of parameters yield acceptable property predictions.

Fig. 4 summarises the MAPEs of all predicted properties ($P^{sat}$, $ρ_{liq,sat}$, $c_{P,liq,sat}$, $ΔH_{vap}$) for 1-octanol. When compared to the NonA-I-PC-SAFT EoS [7], the present model – regardless of the association scheme – demonstrates superior predictions of properties, except liquid density. Variations across association schemes are modest, nevertheless, the 3A scheme slightly outperforms the others (global MAPE ~2.12 %), with prediction errors ~1.0 % (vapour pressure), ~4.0 % (liquid density), ~1.5 % (heat capacity), and ~2.9 % (heat of vaporisation).

Fig. 5 plots the predicted properties of the 1-octanol using the 3A association scheme, together with the pseudo-experimental points for comparison. In Fig. 5(a), the experimental critical temperature and pressure are exactly reproduced; the saturated vapour pressures are also accurately predicted. Furthermore, Fig. 5(b) shows that the saturated liquid density is precisely reproduced at the reduced temperature $T_r = 0.8$ via the additional volume translation parameter c. For $T_r > 0.8$, the liquid density is overestimated by the I-PC-SAFT whilst for $T_r < 0.8$, it is underestimated. As previously mentioned, this parameterisation procedure does not ensure the experimental critical volume. Plotted in Fig. 5(c), the heat capacities are also underestimated by the EoS. Fig. 5 (d) demonstrates that the heat of vaporisation is predicted reasonably well at lower temperatures, with improved agreement as the temperature approaches the critical temperature.

<figure><img src="image_5.png" /><figcaption>**Table 4**
List of tested components and information on experimental data.</figcaption></figure>

<table><tbody><tr><td>Tested alcohols</td><td>Methanol to 1-octadecanol (except 1-tetradecanol)<br/>Small alcohols<br/>- Methanol to 1-propanol<br/>Large alcohols<br/>- 1-butanol to 1-octadecanol</td></tr><tr><td>Tested carboxylic acids</td><td>Acetic acid to n-heptadecanoic acid<br/>Small acids<br/>- Acetic acid to n-hexanoic acid<br/>Large acids<br/>- n-heptanoic acid to n-heptadecanoic acid</td></tr><tr><td>Tested amines</td><td>Methylamine to n-dodecylamine<br/>Small amines<br/>- Methylamine to n-butylamine<br/>Large amines<br/>- n-pentylamine to n-dodecylamine</td></tr><tr><td>Other tested component</td><td>Water</td></tr><tr><td>Tested association schemes</td><td>Alcohols/Acids/Amines<br/>- 1A, 2A, 2B, 3A, 3B, 4A, 4B<br/>Water<br/>- 1A, 2A, 2B, 3A, 3B, 4A, 4B, 4C</td></tr><tr><td>Experimental database</td><td>DIPPR correlations</td></tr><tr><td>Temperature range</td><td>$T \in [T_{min,correlation}, T_{max,correlation}]$</td></tr><tr><td>Pressure range</td><td>$P \ge 1 \text{ mbar}$</td></tr><tr><td>Properties</td><td>$P_{sat}(T), ρ_{liq,sat}(T), c_{P,liq,sat}(T), ΔH_{v}(T)$</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>**Fig. 6.** Bar chart of average property predictions for linear alcohols (methanol to 1-octadecanol) using the optimal parameter set across different association schemes. Further details are provided in Fig. 4 caption. The number of considered components is also indicated above the global MAPE.</figcaption></figure>

In the next section, multiple self-associating components of different sizes – alcohols, carboxylic acids, amines and water – are parameterised using the previously mentioned numerical minimisation procedure to ensure the highest accuracy.

## 6. Performance of the I-PC-SAFT EoS estimated over three families of self-associating compounds and water

The performance of I-PC-SAFT is evaluated using numerous associating components in the homologous series of linear alcohols (from methanol to 1-octadecanol, except 1-tetradecanol), linear carboxylic acids (from acetic acid to n-heptadecanoic acid), and linear amines (from methylamine to n-dodecylamine) and water. Details about the experimental database can be found in Table 4.

For each associating component, a variety of association schemes are tested. The deviations obtained with the NonA-I-PC-SAFT EoS are also included for comparison [7]. Similarly to the 1-octanol, the parameters of the I-PC-SAFT model of different association schemes are determined using the numerical method. The L-BFGS-B algorithm is implemented to minimise the objective function expressed in Eq. (27). Note that the MAPEs yielded by the NonA-I-PC-SAFT model are different than those

<figure><img src="image_7.png" /><figcaption>**Fig. 8.** Bar chart of average property predictions for large linear alcohols (1-butanol to 1-octadecanol) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.</figcaption></figure>

documented in Ramírez-Vélez et al. [6] as they have been re-evaluated over a wider pressure range ($P \ge 1$mbar).

A family-specific analysis is then performed to investigate: 1) the influence of the association term (comparison with NonA-I-PC-SAFT), and 2) the influence of the association scheme. The MAPE between calculated and experimental properties is evaluated for each component and family-specific mean MAPEs are subsequently represented in a bar chart to facilitate comparison between the association schemes. The bar chart for the alcohols, amines and acids are presented in Figs. 6, 11 and 16 respectively.

The influence of the chain length is also studied. For each family, a cut-off value is set specific for each family to distinguish between small and large components, such that the behaviours (i.e., the properties) within the subgroups are consistent. Similarly, bar charts corresponding to the small and large alcohols, amines and acids are illustrated in Figs. 7 and 8, Figs. 12, and 13, Figs. 17 and 18, respectively.

For each association scheme and component considered, the numerical values of the I-PC-SAFT parameters, along with the corresponding MAPE for $P_{sat}(T)$, $\rho_{liq,sat}(T)$, $\Delta H_{vap}(T)$ and $c_{p,liq,sat}(T)$ are listed in the Excel file supporting materials "SM3_Ass. scheme and corresponding I-PC-SAFT parameters.xlsx".

### 6.1. Estimation of alcohols using I-PC-SAFT EoS

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Bar chart of average property predictions for small linear alcohols (methanol to 1-propanol) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.</figcaption></figure>

Fig. 6 shows the improvement of I-PC-SAFT compared to the NonA-I-PC-SAFT model for alcohols. The improvement is observed for all properties, except for the liquid densities. The MAPE of vapour pressures is reduced from ~15% to ~2% across the different association schemes, and the MAPE of the heat of vaporisation decreases from slightly above 10% to just under 5%. A notable improvement is also observed for the isobaric heat capacity, MAPE of which is reduced from ~16% (NonA-I-PC-SAFT) to an average of 4% for the different association schemes.

When the different association schemes are compared with each other, the difference is however small. All association schemes give similar global MAPE (~3.00%). The individual MAPEs are also similar across the association schemes. Nonetheless, the 2B association scheme provides the lowest observed MAPE at 2.99%, corresponding to the MAPEs of 1.2% for vapour pressures, 5.0% for liquid densities, 4.0% for heat of vaporisation and 2.5% for heat capacities. The scheme is consistent with the trend commonly adopted in the literature.

A similar analysis is also performed onto the two alcohol subcategories: the small alcohols (from methanol and 1-propanol), shown in Fig. 7, and the large alcohols (from 1-butanol to 1-octadecanol, except 1-tetradecanol), shown in Fig. 8. For the large alcohols, the 2B still
<figure><img src="image_9.png" /><figcaption>**Fig. 9.** The plots of the predicted properties of ethanol using the 2B scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.</figcaption></figure>

outperforms the other schemes, with a global MAPE of 3.02 % and individual property MAPEs are similar to those previously observed in Fig. 6. In contrast, the best performing association scheme for the small alcohols is 3A, with a global MAPE of 2.79 %, corresponding to the MAPE of ~1.0 % (vapour pressures), 4.0 % (liquid densities), 3.5 % (heat of vaporisation) and 5.0 % (heat capacities).

Nevertheless, the differences between the various association schemes, for both large and small alcohols, are negligible, particularly for vapour pressures, liquid densities, and heats of vaporisation.

The predicted properties are compared against the experimental points using the parameters obtained for the 2B scheme. Figs. 9 and 10 illustrate the influence of the association term on the properties of ethanol and 1-undecanol, respectively.

In both cases, the predicted vapour pressures closely match the experimental values over the entire temperature range. The heat of vaporisation of ethanol is also well reproduced whilst that of 1-undecanol is slightly less accurate, particularly at higher temperatures. The saturated liquid densities follow the same behaviour as 1-octanol, i.e., they are overestimated at $T_r < 0.8$ and underestimated at $T_r > 0.8$. The critical volume (or density) of both components is also not ensured by the model. Finally, the heat capacities of both ethanol and 1-undecanol are reasonably well reproduced across the temperature range.

## 6.2. Estimation of amines using I-PC-SAFT EoS

Fig. 11 illustrates the individual MAPEs (for $P_{sat} \cdot \rho_{liq,sat} \cdot \Delta H_{vap} \cdot c_{P,liq,sat}$) and the global MAPE defined as: $$ \left( 3MAPE_{P_{sat}} + 2MAPE_{\rho_{liq}} + MAPE_{c_{P,liq}} + MAPE_{\Delta H_{vap}} \right) / 7 $$ when different association schemes are used to model amines using I-PC-SAFT and NonA-I-PC-SAFT. When the association

term is included in the I-PC-SAFT model, only marginal improvements are observed for linear amines. The global MAPE decreases modestly from 4.62 % to an average of 3.50 %. Examining the individual properties reveals that only vapour pressure and the heat capacity show clear improvement, albeit modestly. The errors of vapour pressure decrease from 5.0% to approximately 3.0 %, and the error of the heat capacity from slightly above 4.0 % to approximately 2.8 %. In contrast, the heat of vaporisation and the saturated liquid density prediction show little to no improvement, with the errors being approximately 4.5 % and 3.8 % respectively.

As with alcohols, no single association scheme clearly outperforms the others; both the global MAPE and the individual MAPEs are similar across all the association schemes. Nonetheless, the 2B association scheme yields the lowest global MAPE (3.31 %), corresponding to 3.0 % for vapour pressure, 3.5 % for liquid density, 4.5 % for heat of vaporisation and 2.8 % for heat capacity.

Dividing the linear amines into small amines (methylamine to n-butylamine), in Fig. 12 and large amines (n-pentylamine to n-dodecylamine), in Fig. 13, reveals two observations. Firstly, for small amines, the association term improves of all property predictions across all association schemes, most notably for vapour pressures (error decreased from slightly under 6.0 % to on average 1.5 %), and the heat capacities (from slightly above 6.0 % to an average of 2.0 %). Slight improvements are seen for the heat of vaporisation, whose MAPE is reduced from slightly under 3.0 % to approximately 2.0 %. The liquid density exhibits negligible improvement. The 4A scheme slightly outperforms other schemes for the small amines, giving MAPEs of 1.2 % (vapour pressures), 3.0 % (liquid densities), 2.0 % (heat of vaporisation), 2.5 % (heat capacities) and a global MAPE of 2.05 %.

As for the large amines in Fig. 13, the association term brings little improvement; the individual MAPEs and the global MAPE of all
<figure><img src="image_10.png" /><figcaption>Liquid heat capacity as a function of temperature
$C_{P, liq, sat} = f(T)$</figcaption></figure>

<figure><img src="image_11.png" /></figure>

**Fig. 10.** The plots of the predicted properties of 1-undecanol using the 2B scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.

<figure><img src="image_12.png" /><figcaption>Heat of vaporisation as a function of temperature
$\Delta_{v}H = f(T)$</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

**Fig. 11.** Bar chart of average property predictions for linear amines (methylamine to n-dodecylamine) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.

<figure><img src="image_15.png" /></figure>

**Fig. 12.** Bar chart of average property predictions for small linear amines (methylamine to n-butylamine) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.

association schemes are similar and are comparable to those of the NonA-I-PC-SAFT. Inspection of supporting materials (the reader is referred to the Excel file: "SM3_Ass. scheme and corresponding I-PC-SAFT parameters.xlsx") shows that for the larger amines, the reduced association parameters are negligible. Consequently, the non-associative parameters ($m, \epsilon/k_b, \sigma$) are exactly the same as those found in the NonA-I-

PC-SAFT, hence the similar results for both models in Fig. 13. In both amines subcategories, the difference resulted between the various association schemes are minimal.

Since the I-PC-SAFT model (with association term) requires solving of implicit site-to-site interaction balances, its computational cost is substantially higher. For amines, whose hydrogen bond is comparatively
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Bar chart of average property predictions for large linear amines (n-pentylamine to n-dodecylamine) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.</figcaption></figure>

weak, these extra iterations yield only marginal gains. The simplified NonA-I-PC-SAFT variant is, therefore, a more attractive choice, offering near-identical predictive power at a fraction of the computational time.

Figs. 14 and 15 illustrate the properties of n-propylamine and n-nonylamine, respectively, using the parameters obtained for the 2B scheme. The vapour pressures of both components are accurately

predicted. The heat of vaporisation of n-propylamine is well captured; noticeable deviations occur only at lower temperatures. However, the heat of vaporisation of n-nonylamine shows deviations at both low and high temperatures. The saturated liquid densities follow the same trend observed for other associating components: they are slightly overestimated at low temperature and underestimated at higher temperature. The prediction of the heat capacity of n-nonylamine is qualitatively correct, whereas that of n-propylamine deteriorates at high temperatures. Whilst that of n-nonylamine is not represented as it is unavailable in DIPPR database, the critical volume of n-propylamine is relatively accurate.

### 6.3. Estimation of carboxylic acids using I-PC-SAFT EoS

Fig. 16 illustrates the individual MAPEs and the global MAPE of the linear carboxylic acids yield by the I-PC-SAFT and the NonA-I-PC-SAFT models. As for alcohols, the association term improves most properties, particularly the vapour pressures (from slightly above 8.0 % to less than 1.0 %), and heat of vaporisation (from slightly above 14.0 % to an average of 9.0 %). Only marginal improvements are observed for liquid density (from slightly below 6.0 % to an average of 4.0 %), and heat capacity (from 8.0 % to 7.0 %). The global MAPE is approximately halved across the association schemes, averaging at 4.00 %.

When comparing association schemes, clear differences can be observed for certain properties. Vapour pressures and liquid densities are similar across the association schemes, whereas the thermal properties vary considerably. Simpler association schemes (1A, 2A, 2B) predict the heat of vaporisation more accurately but yield slightly poorer

<figure><img src="image_17.png" /><figcaption>Liquid heat capacity as a function of temperature
$C_{P, liq, sat} = f(T)$</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 14.** The plots of the predicted properties of n-propylamine using the 2B scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Heat of vaporisation as a function of temperature
$\Delta_v H = f(T)$</figcaption></figure>

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /><figcaption>Liquid heat capacity as a function of temperature
$C_p(\text{liq, sat}) = f(T)$</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Temperature as a function of liquid density
$T = f(\rho_{liq, sat})$</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Heat of vaporisation as a function of temperature
$\Delta_v H = f(T)$</figcaption></figure>

<figure><img src="image_24.png" /></figure>

Fig. 15. The plots of the predicted properties of n-nonylamine using the 2B scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.

<figure><img src="image_25.png" /></figure>

Fig. 16. Bar chart of average property predictions for linear carboxylic acids (acetic acid to n-heptadecanoic acid) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.

heat capacities prediction. The opposite trend is observed for the more complex association schemes (4B, 4A, 3B), which reproduce heat capacity more accurately but provide less satisfactory the heat of vaporisation. The 1A association scheme performs best overall, a finding consistent with the approach typically adopted in various association EoS.

Dividing the acids into small (acetic acid to n-hexanoic acid) and

<figure><img src="image_26.png" /></figure>

Fig. 17. Bar chart of average property predictions for small linear carboxylic acids (acetic acid to n-hexanoic acid) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.

large (n-heptanoic acid to n-heptadecanoic acid) subcategories, illustrated in Figs. 17 and 18 respectively, reveals two distinct observations. Both figures confirm the necessity of the association term. Similar trends as observed in the general carboxylic acids are observed for small acids: simpler association schemes give small MAPE for heat of vaporisation and high MAPE for heat capacity, whilst the more complex schemes give small MAPE for heat capacity and high MAPE for heat of vaporisation.
<figure><img src="image_27.png" /><figcaption>**Fig. 18.** Bar chart of average property predictions for large linear carboxylic acids (n-heptanoic acid to n-heptadecanoic acid) using the optimal parameter set across different association schemes. See Fig. 6 caption for details.</figcaption></figure>

The vapour pressures and the liquid densities are also comparable across the schemes. The smallest global MAPE (5.10 %) is also calculated for the 1A association scheme, about 50 % lower than that of the NonA-I-PC-SAFT. For the larger acids, the association term improves the prediction remarkably: MAPE of vapour pressures drops from 10.0 % to under 1.0 %, and the MAPEs of heat of vaporisation and heat capacity decrease

from 7.0 % and 9.0 % respectively to roughly 4.0 %. Here, the choice of the scheme is unclear, they all produce similar individual MAPEs and global MAPE. The smallest global MAPE, 2.54 % is obtained for the 4A association scheme.

Upon further examination of the MAPE of the thermal properties (the heat of vaporisation and the heat capacities) within the subcategories, the mean of the MAPE for the small acids exceeds 10 % for all association schemes. Even the 1A association scheme, which gives the lowest deviation, yields a mean of ~12 %. Such behaviour was not observed with alcohols and amines, suggesting that current association schemes do not accurately describe the association phenomena that occur in small acids. A literature review led to the conclusion that cyclic dimerisation, which is not considered by the classical association schemes, indeed occurs in small acids. To overcome this limitation, a more sophisticated perturbation theory was proposed by Sear and Jackson [40], which can describe the cyclic dimerisation. It was later used by Janecek and Paricaud [41,42] to construct the Doubly-Bonded Dimer (DBD) model. In contrast, the mean of the MAPE of the thermal properties of the larger acids remains excellent at ~4 % across the different association schemes. This confirms the absence or the negligible influence of the cyclic dimerisation of the larger acids.

Figs. 19 and 20 illustrate the properties of n-butanoic acid and n-dodecanoic acid, respectively, using the parameters obtained for the 1A scheme. For both components, the vapour pressures are accurately represented. The heat of vaporisation is also well represented for n-dodecanoic acid but less accurately for n-butanoic acid, particularly at lower temperatures. The liquid densities follow the same trend observed previously: they are overestimated at $T_r < 0.8$ and underestimated at

<figure><img src="image_28.png" /><figcaption>Liquid heat capacity as a function of temperature
$c_{P, liq, sat} = f(T)$</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 19.** The plots of the predicted properties of n-butanoic acid using the 1A scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Heat of vaporisation as a function of temperature
$\Delta_v H = f(T)$</figcaption></figure>

<figure><img src="image_31.png" /></figure>
<figure><img src="image_32.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(d)</figcaption></figure>

**Fig. 20.** The plots of the predicted properties of n-dodecanoic acid using the 1A scheme and the pseudo-experimental DIPPR points. See Fig. 5 caption for detailed description.

<figure><img src="image_36.png" /></figure>

**Fig. 21.** Bar chart summarising the quality of the property prediction of water obtained for the optimal set of parameters for different association schemes. See Fig. 6 caption for details.

$T_r > 0.8$. The heat capacities of both acids are quantitatively acceptable, with slightly better agreement for n-dodecanoic acid are slightly more accurate, whose deviation appears only at higher temperatures.

### 6.4. Estimation of water using I-PC-SAFT EoS

Unlike the previously tested components, the 4C association model

was also examined, as it is one of the typically used schemes to model water. Consequently, the water compound is tested using eight association schemes and the corresponding global MAPE and the MAPE of the individual properties are illustrated in Fig. 21.

The best results are obtained for the 4B association scheme, yielding a global MAPE of approximately 3.15 % versus 8.35 % for NonA-I-PC-SAFT. This scheme reduces the MAPE of vapour pressures from slightly below 5.0 % to slightly below 1.0 %, the heat of vaporisation from 4.0 % to under 2.0 % and the heat capacities from 24 % to approximately 4 %. The MAPE of the liquid density is comparable in both models, approximately 6.0 %. Surprisingly, the 4C association scheme typically used model for water in the literature, ranks only third best, with a global MAPE of 3.88 %, following the 3B association scheme (3.17 %) and the 4B scheme (3.15 %).

Fig. 22 presents the predicted properties of water using 4B scheme. The vapour pressures and the heat of vaporisation are well predicted whereas the liquid densities and the heat capacities show higher deviation from the experimental data.

## 7. Conclusion

This study extends the previously established I-PC-SAFT EoS by integrating the association term without compromising the ability of the model to reproduce the critical temperatures and pressures as well as the acentric factors. A universal and transparent parameterisation procedure was applied across seven association schemes for three homologous series of self-associating compounds: linear alcohols, amines, and carboxylic acids; and eight association schemes for water. This comprehensive analysis provides several key insights:
<figure><img src="image_37.png" /><figcaption>Saturated vapour pressure as a function of temperature
$\ln(P_{sat}) = f(1/T)$</figcaption></figure>

(a)

<figure><img src="image_38.png" /><figcaption>Liquid heat capacity as a function of temperature
$C_{P, liq, sat} = f(T)$</figcaption></figure>

(c)

<figure><img src="image_39.png" /><figcaption>Temperature as a function of liquid density
$T = f(\rho_{liq, sat})$</figcaption></figure>

(b)

<figure><img src="image_40.png" /><figcaption>Heat of vaporisation as a function of temperature
$\Delta_v H = f(T)$</figcaption></figure>

(d)

**Fig. 22.** The plots of the predicted properties of water using the 4B scheme and the pseudo-experimental DIPPR data points. See Fig. 5 caption for detailed description.

1. Strongly associating families, namely alcohols, acids and water, benefit most from the explicit consideration of association. For these compounds, the average MAPE of vapour pressures and heats of vaporisation is reduced by roughly a factor of two, whilst errors in heat capacities decrease by more than a factor of two compared to the non-associating baseline.

2. Weakly associating molecules such as amines barely shows limited improvement from the explicit association term. This suggests that for these compounds, the NonA-I-PC-SAFT model (without the association term) provides sufficient accuracy without adding unnecessary additional calculation complexity.

3. In the case of the alcohols, amines and water, the choice of association scheme has a smaller impact than initially anticipated. The smallest global MAPE is obtained with the 2B association scheme, for both alcohols and amines, whereas the 1A and 4B schemes offer the best prediction of properties for the carboxylic acids and water, respectively.

In general, the newly developed parameter sets have been shown to reduce mean property MAPEs for the vast majority of components, to below 5% for the VLE data, and below 10% for the thermal properties. The extensive test of various association scheme establishes a rational selection roadmap while preserve the elegance of corresponding-state-

based fitting. Future studies should aim to provide a comprehensive analysis of vapour-phase dimers in small acids, a subject that has received insufficient attention in the existing literature.

Furthermore, the team is comparing the performance of the I-PC-SAFT volume translation model with that of the classical PC-SAFT parameterisation approach, i.e., regression of the model parameters to fit the selected thermodynamic properties, both with the association term.

### CRediT authorship contribution statement

**Haziq Ridwan Bin Asmuni:** Writing – original draft, Conceptualization, Methodology. **Romain Privat:** Writing – review & editing, Resources, Project administration, Conceptualization. **Saifuddin Ahmed:** Resources, Project administration. **Marc Bonnissel:** Resources, Project administration. **Jean-Noël Jaubert:** Writing – review & editing, Resources, Project administration, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
# Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.fluid.2025.114649.

## Appendix 1: Peculiar points on the acentric factor and the reduced critical temperature functions

This appendix illustrates how the parameters ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$) influence the quantities $m_{\omega_{min,g}}$, $\omega_{min,g}$, $m_{\omega_{min,1}}$, $\omega_{min,1}$, $m_{\omega_{max,1}}$, $\omega_{max,1}$, $m_{T'_{c,min}}$, $T^*_{c,min}$ and $T^*_{c,fictitious}/T^*_{c,ordinary}$ that are all defined in Section 4 and illustrated in Figs. 1 and 2.

Coordinates of the global minimum of the $\omega(m)$ function: $m_{\omega_{min,g}}$ and $\omega_{min,g}$

Fig. 23 illustrates how the coordinates of the global minimum of the $\omega(m)$ function $-m_{\omega_{min,g}}$ and $\omega_{min,g}$, defined in Section 4.1. and shown in Figs. 1(a) and 2(a)–vary with the parameters ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$).

<figure><img src="image_41.png" /></figure>

Fig. 23. Contour plot of $m_{\omega_{min,g}}$ and $\omega_{min,g}$ for different values of the reduced association parameters using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\epsilon^{AB}/\epsilon$ values range between 1.0 and 20.0. The black curve forming a droplet-like zone delimits the zone for which pair ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$) produces the local extrema described in Fig. 2(a).

(a) $m_{\omega_{min,g}}$. The m values are quasi-invariant across the studied range at approximately $m_{\omega_{min,g}} \approx 0.25$.

(b) $\omega_{min,g}$. The $\omega_{min,g}$ values vary from negative to positive despite $m_{\omega_{min,g}}$ values remain invariant. The positive values are only found in the droplet-like zone.

Fig. 24 illustrates how the coordinates of the local minimum of the $\omega(m)$ function $-m_{\omega_{min,1}}$ and $\omega_{min,1}$, defined in Section 4.1. and shown in Figs. 1(a) and 2(a)–vary with the parameters ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$).

<figure><img src="image_42.png" /></figure>

Fig. 24. Contour plot of $m_{\omega_{min,1}}$ and $\omega_{min,1}$ for different values of the reduced association parameters using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\epsilon^{AB}/\epsilon$ values range between 1.0 and 20.0. The black curve forming a droplet-like zone delimits the zone for which pair ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$) produces the local extrema described in Fig. 2(a). The white zone shows the pair ($\kappa^{AB}$, $\epsilon^{AB}/\epsilon$) for which no local extremum is found.

(a) $m_{\omega_{min,1}}$. Inside the droplet-like zone, the $m_{\omega_{min,1}}$ ranges approximately from 1.0 to 4.5.

(b) $\omega_{min,1}$. Inside the droplet-like zone, the $\omega_{min,1}$ ranges from approximately 0.01 up to 0.8.
Coordinates of the local maximum of the ω(m) function: $m_{\omega_{max,l}}$ and $\omega_{max,l}$

Fig. 25 illustrates how the coordinates of the local maximum of the ω(m) function $-m_{\omega_{max,l}}$ and $\omega_{max,l}$, defined in Section 4.1. and shown in Figs. 1(a) and 2(a)–vary with the parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$.

<figure><img src="image_43.png" /></figure>

Fig. 25. Contour plot of $m_{\omega_{max,l}}$ and $\omega_{max,l}$ for different values of the reduced association parameters using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\varepsilon^{AB}/\varepsilon$ values range between 1.0 and 20.0. The black curve forming a droplet-like zone delimits the zone for which pair $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ produces the local extrema described in Fig. 2(a). The white zone shows the pair $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$ for which no local extremum is found.

(a) $m_{\omega_{max,l}}$. Inside the droplet-like zone, the $m_{\omega_{max,l}}$ ranges approximately from 0.7 to 4.0.

(b) $\omega_{max,l}$. Inside the droplet-like zone, the $\omega_{max,l}$ ranges from approximately 0.01 up to 0.8.

Fig. 26 illustrates how the coordinates of the minimum of the $T_c^*(m)$ function $-m_{T_c^{\text{min}}}$ and $T_{c,\text{min}}^*$, defined in Section 4.1. and shown in Figs. 1(b) and 2(b)–vary with the parameters $(\kappa^{AB}, \varepsilon^{AB}/\varepsilon)$. For clarity, it is recalled that $T_c^*$ is here the ordinary reduced critical temperature (see the black curve in Figs. 1(b) and 2(b)).

<figure><img src="image_44.png" /></figure>

Fig. 26. Contour plot of $m_{T_c^{\text{min}}}$ and $T_{c,\text{min}}^*$ for different reduced association parameters values using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\varepsilon^{AB}/\varepsilon$ values range between 1.0 and 20.0.

(a) $m_{T_c^{\text{min}}}$. The maximum value, shaded in burgundy, can be found along a curve. However, the value varies little throughout the test interval, at an average of 0.7.

(b) $T_{c,\text{min}}^*$. The lowest value of $T_{c,\text{min}}^*$, close to 1.1, is obtained for small values of $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$, whereas the highest value, around 1.8, is observed for large values of $\kappa^{AB}$ and $\varepsilon^{AB}/\varepsilon$.
Coordinates of the salient point of the $T_c^*(m)$ function: $m_{salient}$ and $T_{c,salient}^*$

Fig. 27 illustrates how the coordinates of the salient point of the $T_c^*(m)$ function–$m_{salient}$ and $T_{c,salient}^*$, defined in Section 4.1. and shown in Figs. 1(b) and 2(b)–vary with the parameters ($\kappa^{AB}$, $\varepsilon^{AB}/\varepsilon$). For clarity, it is recalled that $T_c^*$ is here the ordinary reduced critical temperature (see the black curve in Figs. 1(b) and 2(b)).

<figure><img src="image_45.png" /></figure>

**Fig. 27.** Contour plot of $m_{salient}$ and $T_{c,salient}^*$, found along the ordinary critical curve, for different reduced association parameters values using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\varepsilon^{AB}/\varepsilon$ values range between 1.0 and 20.0.

(a) $m_{salient}$, value of m corresponding to the salient point. This value is quasi-independent of the association parameters and is approximately 96.5.

(b) $T_{c,salient}^*$, value of the reduced critical temperature corresponding to the salient point. This value is quasi-independent of the association parameters and is approximately 4.38.

Fig. 28 illustrates how the fictitious liquid-liquid reduced critical temperature ($T_{c,fictitious}^*$)-to-ordinary reduced critical temperature ($T_{c,ordinary}^*$) ratio, both defined in Section 4.1. and shown in Figs. 1(b) and 2(b) vary with the parameters ($\kappa^{AB}$, $\varepsilon^{AB}/\varepsilon$).

<figure><img src="image_46.png" /></figure>

**Fig. 28.** Contour plot of the ratio $T_{c,fictitious}^*/T_{c,ordinary}^*$ for different reduced association parameters values at $m_{ref} = 4.0$ using the 1A association scheme. The interval of $\kappa^{AB}$ values considered ranges from $10^{-4}$ and 0.1 whilst the $\varepsilon^{AB}/\varepsilon$ values range between 1.0 and 20.0. The ratio can be considered constant (independent of the association parameters) and take a value of on average 0.23.

## Appendix 2: EoS parameters and property prediction of 1-octanol for different association schemes

This appendix illustrates how the different non-associative parameters $m$, $\sigma$, $\varepsilon/k_b$ and the normal energy of association $\varepsilon^{AB}/k_b$ evolve with the reduced associative parameters, for each tested association scheme. For each pair ($\kappa^{AB}$, $\varepsilon^{AB}/\varepsilon$), the mean absolute percentage error (MAPE) of the vapour pressure $P_{sat}$, liquid density $\rho_{liq,sat}$ and the two thermal properties, the heat of vaporisation $\Delta H_{vap}$ and the isobaric saturated liquid heat capacity $c_{p,liq,sat}$ are calculated. Subsections 10.1 to 10.4 present the parameters whilst subsections 10.5 to 10.8 focus on the MAPE of the properties.
Number of segments *m*

<figure><img src="image_47.png" /><figcaption>(a)</figcaption></figure>

Number of segments *m*

<figure><img src="image_48.png" /><figcaption>(c)</figcaption></figure>

Number of segments *m*

<figure><img src="image_49.png" /><figcaption>(e)</figcaption></figure>

Number of segments *m*

<figure><img src="image_50.png" /><figcaption>(g)</figcaption></figure>

Number of segments *m*

<figure><img src="image_51.png" /><figcaption>(b)</figcaption></figure>

Number of segments *m*

<figure><img src="image_52.png" /><figcaption>(d)</figcaption></figure>

Number of segments *m*

<figure><img src="image_53.png" /><figcaption>(f)</figcaption></figure>

**Fig. 29.** Contour plot of the number of segments *m* for 1-octanol for different reduced association parameters ε<sup>AB</sup>/ε and κ<sup>AB</sup> using different association schemes. The tested κ<sup>AB</sup> and ε<sup>AB</sup>/ε intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.
<figure><img src="image_54.png" /><figcaption>Diameter of segment σ</figcaption></figure>

**Fig. 30.** Contour plot of the diameter of the segment σ for 1-octanol for different reduced association parameters ε<sup>AB</sup>/ε and κ<sup>AB</sup> using different association schemes. The parameter – coupled with the other EoS parameters – allow the exact reproduction of the experimental acentric factor and the critical point. The tested κ<sup>AB</sup> and ε<sup>AB</sup>/ε intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.
<figure><img src="image_55.png" /><figcaption>**Fig. 31.** Contour plot of the depth of the potential well $\epsilon/k_b$ for 1-octanol for different reduced association parameters $\epsilon^{AB}/\epsilon$ and $\kappa^{AB}$ using different association schemes. The parameter – coupled with the other EoS parameters – allow the exact reproduction of the experimental acentric factor and the critical point. The tested $\kappa^{AB}$ and $\epsilon^{AB}/\epsilon$ intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.</figcaption></figure>
Energy of association $\epsilon^{AB}/k_b$

<figure><img src="image_56.png" /><figcaption>**Fig. 32.** Contour plot of the energy of association $\epsilon^{AB}/k_b$ for 1-octanol for different reduced association parameters $\epsilon^{AB}/\epsilon$ and $\kappa^{AB}$ using different association schemes. The parameter – coupled with the other EoS parameters – allow the exact reproduction of the experimental acentric factor and the critical point. The tested $\kappa^{AB}$ and $\epsilon^{AB}/\epsilon$ intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.</figcaption></figure>
<figure><img src="image_57.png" /><figcaption>**Fig. 33.** Contour plot of the MAPE of the vapour pressures $P_{sat}$ for 1-octanol for different reduced association parameters $\epsilon^{AB}/\epsilon$ and $\kappa^{AB}$ using different association schemes. The tested $\kappa^{AB}$ and $\epsilon^{AB}/\epsilon$ intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.</figcaption></figure>
<figure><img src="image_58.png" /><figcaption>**Fig. 34.** Contour plot of the MAPE of the liquid densities $\rho_{lsat}$ for 1-octanol for different reduced association parameters $\epsilon^{AB}/\epsilon$ and $\kappa^{AB}$ using different association schemes. The tested $\kappa^{AB}$ and $\epsilon^{AB}/\epsilon$ intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.</figcaption></figure>
<figure><img src="image_59.png" /><figcaption>MAPE of the heat capacities $c_{P,\text{liq}}$ for 1-octanol</figcaption></figure>

**Fig. 35.** Contour plot of the MAPE of the isobaric liquid heat capacities $c_{P,\text{liq}}$ for 1-octanol for different reduced association parameters $\epsilon^{AB}/\epsilon$ and $\kappa^{AB}$ using different association schemes. The tested $\kappa^{AB}$ and $\epsilon^{AB}/\epsilon$ intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.
<figure><img src="image_60.png" /><figcaption>**Fig. 36.** Contour plot of the MAPE of the heat of vaporisation ΔH<sub>vap</sub> for 1-octanol for different reduced association parameters ε<sup>AB</sup>/ε and κ<sup>AB</sup> using different association schemes. The tested κ<sup>AB</sup> and ε<sup>AB</sup>/ε intervals vary depending on the association scheme. The black zones show instances where the methodology failed to exactly replicate the experimental acentric factor and critical point. The red dot represents the optimal parameters that minimise the chosen objective function. (a) 1A scheme; (b) 2A scheme; (c) 2B scheme; (d) 3A scheme; (e) 3B scheme; (f) 4A scheme; (g) 4B scheme.</figcaption></figure>
# Data availability

The authors do not have permission to share data.

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