Estimation of the Thermodynamic Properties of Per- and Polyfluoroalkyl Substances Open Access

Per- and polyfluoroalkyl substances (PFAS) are a large group of manufactured chemicals that have been used in manufacturing processes, in the production of heat- and water-resistant materials for paper, packaging, textiles, leather, and carpet goods; in firefighting foams; and in non-stick coatings [1,2] since the 1940s. These compounds are characterized by a hydrophobic alkylated chain saturated with fluorine atoms, typically attached to a hydrophilic head [3]. Their beneficial properties in manufacturing have led to their widespread use and persistence in the environment and have thus made them ubiquitous. They have been detected throughout the globe, in soils, plants, surface water, and groundwater [4]. These compounds are bio-accumulative [5] and toxic and have been linked to liver disease, cancer, cardiovascular disease, and immune system disorders [6–9].

The presence of PFAS compounds in drinking water has resulted in limits imposed by both the US Environmental Protection Agency (USEPA) and individual states, requiring the treatment for the removal of PFAS from drinking water. Currently, the predominant method of removal is liquid phase adsorption onto granular activated carbon (GAC) media filters. Once the GAC has been expended, it is often regenerated (i.e., reactivation). Several methods are available for regeneration; however, the most common and mature process is the thermal method. The spent GAC is placed in a kiln, and the temperature is raised to 800–900 °C. In a study by Xiao et al. [10], mineralization to fluoride ions, of more than 80% of perfluorooctanoic acid (PFOA) and perfluorooctane sulfonate (PFOS) on the GAC was achieved at temperatures greater than 700 °C.

Although thermal regeneration of GAC is the most common, limited studies on the fate of PFAS compounds, especially in the gaseous phase and at elevated temperatures exist. Removal of PFAS by GAC and the subsequent regeneration is rapidly increasing. A better understanding of the fate of these compounds in thermal processes, operating at or near combustion temperatures, is required and could be enhanced by the application of computer simulations. However, it requires that the thermodynamic properties be determined.

As with the drinking water treatment, wastewater treatment has also created a need for understanding the consequences of exposing PFAS to combustion temperatures. Municipal wastewater treatment plants receive wastewater with PFAS from industry, landfills, and domestic sewage [11]. The PFAS is subsequently discharged either in the treated effluent or in the residuals, often referred to as biosolids, produced by the treatment process. In the United States, approximately 56% of the residuals produced, which are rich in nutrients and carbon, is land applied [12]. The presence of PFAS in the residuals has resulted in concerns over the environmental and health risks associated with this practice. The States of Maine and Connecticut have banned the land application of biosolids (NEBRA 2024). An estimated 13.8 million dry tons of biosolids are produced annually in the United States [13], so restrictions in disposal options present a major obstacle to the utilities and private companies operating these facilities.

The result has been an industry-wide reevaluation of the incineration of sewage sludge and the introduction of gasification and pyrolysis, but with an extremely limited application using sewage sludge as a feedstock. Gasification and pyrolysis produce biochar and gaseous fuels that are subsequently combusted in a thermal oxidizer for recovery of the thermal energy. Removal of PFAS compounds from the biochar has been demonstrated [14–16] and thus provides the option for use as a soil amendment due to the remaining carbon and nutrients. However, the fate of PFAS in the gaseous stream combusted in the thermal oxidizer is unknown.

Although incineration, pyrolysis, and gasification are technologies that remove the PFAS from the ash and biochar produced, the fate of these compounds in the exhaust gases must be understood before acceptance as viable technologies that remove the risk of releasing PFAS into the atmosphere. The speed and ease at which computer simulations can be applied to increase our understanding and thus decrease the time required to determine the efficacy of these technologies is important. The critical first step in this process is to determine the thermodynamic properties of PFAS compounds over a wide range of temperatures.

All the PFAS molecules are large and unbranched molecules. There are several types of PFAS, perfluorocarboxylic acids (PFCAs), perfluoroalkane sulfonates (PFSAs), perfluoroalkyl ether carboxylic acids (PFECAs), and so on. PFCA is characterized by having a carboxyl group (–COOH) attached to a fully fluorinated carbon chain. The most notable PFCA is PFOA, which has been widely used in industrial applications, including in the production of fluoropolymers like Teflon, firefighting foam, and water-repellent fabrics. PFSA is characterized by its fully fluorinated carbon chains attached to a sulfonate group (–SO_3⁻). One well-known PFSA is PFOS, which was widely used in products like firefighting foams, stain repellents, and coatings before being phased out due to environmental and health concerns. PFECAs are characterized by ether linkages (–O–) between fluorinated alkyl groups and a carboxylic acid (–COOH) functional group. One well-known example is GenX (Perfluoro-2-methyl-3-oxahexanoic acid), which is used as a replacement for PFOA in the manufacturing of fluoropolymers like Teflon [17].

The thermodynamic properties of PFAS throughout a broad temperature range are essential for determining the states of nonequilibrium evolution of removing these molecules, especially in combustion processes. The thermodynamic properties are also needed to determine chemical equilibrium composition and simulate the chemical kinetics of complicated reacting gas-phase systems. The main issue is that the data of PFAS for thermodynamic properties are lacking especially for the experimental data.

Several models have been applied to determine the properties of hydrocarbon, fluorinated hydrocarbon, and PFAS. He et al. [18,19] established a simple model to assess the sensible thermodynamic properties of unbranched and branched hydrocarbon over a wide temperature range. Wang et al. [20] used the computational quantum chemistry method (ab initio and density functional theory methods) to find thermodynamic properties of C₁–C₄ fluorinated hydrocarbon. Snitsiriwat et al. [21] also used the computational quantum chemistry method (ab initio and density functional theory methods) to find thermodynamic properties of fluorinated carboxylic acids. Ram et al. [22,23] from the North Carolina State University (NCSU)'s research group utilized computational quantum chemistry and ideal-gas statistical mechanics to predict thermodynamic properties of C₂–C₈ PFCAs and C₂–C₈ perfluorinated sulfonic acids (PFSAs) with extrapolation to C₁₆. The data of thermodynamic properties are reported in the form of NASA as used in the Cantera formats.

The vibrational mode and frequency are crucial for calculating thermodynamic properties. Certain studies employed various methodologies to assess or compute the vibrational frequency. Al-Kahtani et al. [24] utilized Raman spectroscopy, an experimental method to study the C–C single-bond stretching vibrational frequency of ethane. Subramanian and Sloan [25] measured Raman and infrared spectra of a molecule (ethane, isobutane, and n-butane) trapped in clathrate hydrate cages to obtain C–C single-bond stretching vibrational frequency. Kerkeni and Clary [26] used the computational quantum chemistry method to evaluate C–C and C–H single-bond stretching vibrational frequencies of ethane. Lewis and Pace [27] studied the vibrational spectra of the α-crystailline of hexafluoroethane(C₂F₆) to determine vibrational frequencies of all modes, which contain C–F single-bond stretching vibration. De Maré and Panchenko [28] utilized the computational quantum chemistry method to evaluate all vibrational frequencies in a different mode of hexafluoroethane. Martín Santa Daría et al. [29] used the GENIUSH–Smolyak approach and the potential energy surface to coverage vibrational modes of the formic acid (HCOOH) molecule, which includes vibrational frequencies of C–O single-bond stretching vibration, C–O double-bond stretching vibration, and O–H hydrogen stretching vibration.

This study aims to create an analytical model with a minimal set of input parameters to ascertain the thermodynamic properties (enthalpy and heat capacity) of PFCAs and PFECAs across a broad temperature range. As mentioned earlier, there are several methods to calculate the thermodynamic properties of PFCA. Consequently, a comparison of the thermodynamic properties of PFCA in this study with prior studies will be conducted. However, there is a lack of prior studies to assess the thermodynamic properties of PFECA. Therefore, the thermodynamic properties of PFECA will be computed in this study. A specific category of parameters, namely, the characteristic vibrational temperatures and their corresponding vibrational frequencies of PFCA and PFECA are challenging to measure or calculate directly. In this study, the vibrational frequencies of PFCA and PFECA can be estimated as the vibrational frequencies of hydrocarbon, fluorinated hydrocarbon, and carboxylic acid in the same mode. The outcome of this work can serve as a subroutine in computational models.

Additional properties may also be calculated utilizing the same methodology. The total number of degrees-of-freedom of a species containing n atoms is 3n. Nonlinear molecules have three translational degrees-of-freedom and three rotational degrees-of-freedom, whereas linear molecules only have two rotational degrees-of-freedom. Therefore, the number of vibrational degrees-of-freedom is 3n − 5 for linear molecules and 3n − 6 for nonlinear molecules. The vibrational degrees-of-freedom can be categorized into stretching, bending, and internal rotational modes. They also can be divided into heavy particles (carbon, oxygen, or fluorine) and hydrogen atoms. Then the heavy particle stretching modes may be divided into three modes associated with single, double, and triple bonds in the molecule. The heavy particle single bonds might be classified into C–C, C–F, and C–O single bond. The hydrogen single bonds might be classified into C–H and O–H single bond.

Table 1 lists the degrees-of-freedom for several modes of PFCA and PFECA molecules, including their corresponding symbols and values utilized in this model. The degrees-of-freedom for translation are three. There are three degrees-of-freedom for rotation if the molecular is nonlinear and two degrees-of-freedom for rotation if the molecular is linear. X represents the number of heavy particles, which is equal to the sum of a number of carbon, oxygen, and fluorine atoms. The value of degrees-of-freedom for heavy particle stretching vibration corresponds to the number of bonds, which is associated with two heavy particles. The degrees-of-freedom for heavy particle stretching vibration are X − 1. The bonds can be further divided into single, double, and triple bonds. The single bond can be much further divided into C–C associated with single bond, C–O associated with single bond, and C–F associated with single bond. The relationship of degrees-of-freedom about heavy particles is listed in Table 1. The degrees-of-freedom for heavy particles bending vibration are X − 2. The degrees-of-freedom for heavy particle internal rotation vibration are X − 3. EH is the number of hydrogen atoms. It equals the number of bonds, which is associated with a hydrogen atom and a heavy particle, and it also equals the degrees-of-freedom for hydrogen stretching vibration. The bonds about hydrogen can be further divided into O–H associated with a single bond and C–H associated with a single bond. HRR represents the degrees-of-freedom for hydrogen-restricted rotation vibration. The degrees-of-freedom for hydrogen bending vibration can be calculated as 2EH-HRR. The relationships of degrees-of-freedom about hydrogen are listed in Table 1. For example, PFOA is a molecule, which belongs to the class of PFCA. The structure of PFOA is shown in Fig. 1. The chemical formula of PFOA is C₈HF₁₅O₂. The number of atoms is 26. The total degrees-of-freedom are 78 $(Ntot=3n)$⁠. There are three degrees-of-freedom for translation and three degrees-of-freedom for rotation. The remaining 72 degrees-of-freedom are the vibrational degrees-of-freedom. Sixty-nine of them are heavy particle vibration (XVIB = 69) and three of them are hydrogen stretching vibration (HVIB = 3). For the heavy particle vibration, 24 of them are heavy particle stretching vibration (XSTR = 24), 23 of them are heavy particle bending vibration (XBND = 23), and 22 of them are heavy particle internal rotation vibration (XROT = 22). For heavy particle stretching vibration, seven of them are C–C single-bond stretching vibration (CB1 = 7), one of them is C–O single-bond stretching vibration (OB1 = 1), 15 of them are C–F single-bond stretching vibration (FB1 = 15), and one of them is C–O double-bond stretching vibration (STR2 = B2 = 1). For the hydrogen vibration, one of them is O–H bond stretching vibration (HSTR = OHSTR = EH = 1), one of them is hydrogen bending vibration (HBND = 2EH − HRR = 1), and one of them is hydrogen-restricted rotation vibration (HRR = 1).

For the stretching and bending vibrations whatever heavy particle or hydrogen, the Einstein model is used. For the hydrogen-restricted rotation vibrations, a sinusoidal potential is used.

The C–C single-bond stretching vibrational frequency and C–H hydrogen stretching vibrational frequency of PFAS can be assessed as the same mode frequency of ethane (C₂H₆). The value of a C–C single-bond stretching vibration frequency of PFAS can be estimated as 1000/cm⁻¹, which is the approximate value of the frequency of ethane from Al-Kahtani et al.'s experimental results [24] and Kerkeni and Clary's computational results [26]. The value of C–H hydrogen stretching vibrational frequency is 2960 cm⁻¹, which is the average value of six C–H hydrogen stretching vibrational frequencies from Kerkeni and Clary's computational results [26]. The C–F single-bond stretching vibrational frequency, heavy particle bending vibrational frequency, and heavy particle internal rotation vibrational frequency of PFAS can be assessed as the same mode frequency of hexafluoroethane (C₂F₆). The frequency data of hexafluoroethane are all obtained from Lewis and Pace experimental results [27] and De Mare and Pancheko computational results [28]. The value of the C–F single-bond stretching vibrational frequency can be estimated as 1265 cm⁻¹, which is the average value of six C–F stretching vibrational frequencies of hexafluoroethane. The value of the heavy particle bending vibrational frequency can be estimated as 558 cm⁻¹, which is the average value of six heavy particle bending of hexafluoroethane. The value of the heavy particle internal rotation vibrational frequency is estimated as 243 cm⁻¹, which is the average value of five heavy particle internal rotations of hexafluoroethane. The C–O single-bond stretching vibrational frequency, the C–O double-bond stretching vibrational frequency, and O–H hydrogen stretching vibrational frequency of PFAS can be assessed as the same mode frequency of formic acid (HCOOH). The frequencies of formic acid are all acquired from Martín Santa Daría et al.'s computational results [29]. Thus, the value of C–O single-bond stretching vibrational frequency, C–O double-bond stretching vibrational frequency, and O–H hydrogen stretching vibrational frequency of PFAS can be estimated as 1140 cm⁻¹, 1816 cm⁻¹, and 3765 cm⁻¹, respectively. The corresponding characteristic temperatures can be calculated by Eqs. (9)–(11). The values of the characteristic temperatures $θHRR$ and $θRB$ are $27/σ$ and 500 K. Additional details about these two characteristic temperatures were discussed by He et al. [18]. Table 3 presents the symbols and summarizes the value of the frequencies and characteristic temperatures.

The electronic contribution from the excited states is excluded in Tables 2 and 3. If the excited electronic energies are known, they might be incorporated utilizing the equations presented by He et al. [18].

Thus, the total sensible Gibbs free energy can be obtained by Eq. (7). The degrees-of-freedom $Ni$ for different modes (translational, rotational, and vibrational) can be determined by Table 1. The sensible Gibbs free energy $μi$ for each mode can be evaluated by equations listed in Table 2 and the corresponding characteristic temperature listed in Table 3.

Table 4 presents the equations for the dimensionless enthalpy and heat capacity associated with the various modes.

The model has been employed to determine the properties of various PFAS. Figures 2–6 compare the dimensionless heat capacity of PFCAs with computational results from NCSU's research group [22]. The names, chemical formulas, and structures of some PFCAs are presented in Table 5. The computational results from NCSU's research group are in well agreement with this model when the temperature is lower than 1000 K. At elevated temperatures, the results from NCSU's research group are no more than 3% inferior to this model. The difference between these two outcomes increases as the temperature rises.

Figures 7–11 compare the dimensionless enthalpy of PFCA with computational results from NCSU's research group [22]. The computational enthalpies of perfluorobutanoic acid (PFBA) and perfluoropentanoic acid (PFPeA) from NCSU's research group are in well agreement with this model [22]. The computational enthalpies of perfluorohexanoic acid (PFHxA), perfluoroheptanoic acid (PFHpA), and PFOA from NCSU's research group are no more than 5% higher than this model. The largest error occurs at the lowest temperature. This might be because the properties become highly sensitive to thermodynamic temperature as it falls below the characteristic temperature.

The dimensionless heat capacity and enthalpy of four PFECAs are shown in Figs. 12 and 13. The names, chemical formulas, and structures of some PFECAs are presented in Table 6. The thermodynamic properties of these molecules have been neither calculated nor measured.

A model, based on statistical thermodynamics, has been developed that necessitates a minimal number of parameters to estimate the thermodynamic properties such as Gibbs free energy, specific heat, enthalpy, and entropy of PFAS molecules. This model can be utilized to estimate properties of PFAS without measurements or comprehensive computations. For PFCAs, the model's conclusions align well with current computational results. The model is also capable of calculating the thermodynamic properties of PFECAs that were not assessed in prior research.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.