Modelling the phase behaviour of polymer-solvent mixtures

and surfactant systems with the SAFT-γ Mie EoS

Mariana Leitão da Silva Santos

Thesis to obtain the Master of Science Degree in

Chemical Engineering

Supervisors: Dr. Thomas Lafitte, Prof. Dr. Eduardo Morilla Filipe

Examination Committee

Chairperson: Prof. Dr. Carlos Henriques

Supervisor: Prof. Dr. Eduardo Morilla Filipe

Member of the committee: Dr. Pedro Morgado

20th October 2016

2

Acknowledgements

I cannot find words to express my gratitude to my Supervisor Dr. Thomas Lafitte for his support and

valuable knowledge – it has truly been an honour to work with you. I would also like to thank Dr. Vasileios

Papaioannou, for his (much needed) help and availability.

This thesis would not have been possible without my Professor (and also Supervisor) Dr. Eduardo Filipe

– I am forever indebted for your support and for giving me this chance. I also owe my deepest gratitude

to Process Systems Enterprise Ltd., in particular to Dr. Costas Pantelides, for the opportunity to enrol in

this Internship.

A very special thank you to all my journey companions and colleagues, who enriched this experience:

Tom, Su, Alhandra, Eleni, Matteo, Federico, Renato,Artur, Mariana, Francisco, Maria, Pierre, Sancho,

and Leonor. To my shorter-term buddies – Alex, Luisa, and Josh –, I am grateful for the thoughts and

experiences we shared.

To my parents: no words can describe how much I appreciate you unconditional love and support, I love

you both very much. I would also like to thank my little brother, for looking up to me – you made me try

harder, even without knowing. To the rest of my family: thank you for being there; particularly to my

grandma, for sheltering me and for her endless patience (and comfort food), and to my grandparents

for their love and support.

Last but far from least, to my fiancé João: I will be forever grateful for the lovely years we shared so far,

and for the amazing last few months that made me certain this is only the beginning of something great!

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Resumo

A capacidade preditiva é uma das características mais procuradas nas ferramentas de modelação de

propriedades termodinâmicas, sendo estas continuamente desenvolvidas no sentido de diminuir a

necessidade de incluir dados experimentais na determinação dos parametros moleculares dos

modelos.

Aplicando o software gSAFT® na plataforma gPROMS®, foi avaliada a capacidade predictiva da

equação de estado SAFT-γ Mie (contribuição de grupos, GC) na descrição do equilíbrio de fases em

sistemas complexos, como polímero-solvente (poliestireno, poli(acetato de vinilo), polietileno,entre

outros) e misturas do surfactante não-iónico n-alkyl polyoxyethylene ether (CiEj).

Um dos maiores desafios na modelação destes sistemas consiste na escassez de dados experimentais

de equilíbrio líquido-vapor (ELV) para componentes puros, visto estes serem normalmente utilizados

na determinação dos parametros moleculares do modelo. Esta situação tende a ser contornada usando

dados experimentais relativos a misturas binárias e/ou multicomponente. A contribuição de grupos

permite caracterizar sistemas-objectivo após a determinação dos parâmetros relativos aos seus grupos

funcionais (podento resultar do estudo de dados experimentais de componentes mais simples e

facilmente disponíveis, incluindo ELV de compostos puros).

A maioria dos sistemas polímero-solvente estudados foram previstos correctamente, dispensando a

implementação de dados experimentais de polímeros na estimação de parametros. O ELV de

surfactantes puros e de misturas binárias surfactante-água é facilmente reproduzido pelos modelos

desenvolvidos (e frequentemente tranferíveis para sistemas semelhantes). Misturas binárias CiEj -

alcanos são muito difícies de modelar e misturas ternárias água- CiEj -alcano, impossíveis de prever.

A concentração micelar crítica (CMC) dos sistemas água- CiEj foi também modelada, e a dependência

desta em relação ao comprimento da molecula de surfactante foi bem reproduzida.

Palavras-chave: equação de estado, SAFT-γ Mie, polímeros, surfactantes, equilíbrio de fases,

imiscibilidade líquido-líquido

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Abstract

One of the most desired features in thermodynamic tools is a predictive capability, hence

thermodynamic modelling frameworks have been developed towards decreasing the need for

experimental data for the determination of molecular model parameters.

Using gSAFT® software within gPROMS® platform, an assessment of the predictive capability of the

SAFT-γ Mie EoS (a group contribution (GC) approach) is carried regarding the phase equilibrium of

complex materials. In this project are studied: polymer-solvent systems (PS, PVAc, PVME, PE, among

others) and non-ionic (n-alkyl polyoxyethylene ether, CiEj) surfactant mixtures.

The paucity of pure-component vapour-liquid experimental (VLE) data poses a challenge in the

modelling of these systems, as these are typically used to develop the required model parameters. This

is usually tackled by using experimental data for fluid-phase behaviour of binary and/or multi-component

mixtures. The GC formalism advantageously allows the characterisation of target systems once the

parameters for the functional groups that describe them have been obtained (e.g. by studying simpler

data typically available, including pure-component VLE).

The modelling of polymer-solvent systems was very successful - nearly all polymer systems studied

were accurately predicted without using binary polymer data to adjust the interaction parameters. Pure-

surfactant VLE and CiEj-water binary mixtures are easily correlated to by the developed models, and

sometimes even transferable to similar systems, but CiEj-alkane binaries are very difficult to reproduce,

and ternary water-CiEj-alkane mixtures are non-replicable. The critical micellar concentration (CMC) of

CiEj-water systems was also modelled, and the dependency of the CMC on surfactant chain length was

well captured.

Keywords: equation of state, SAFT-γ Mie, polymers, surfactants, phase behaviour, closed-loop

immiscibility.

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Contents

Acknowledgements ................................................................................................................................. 2

Resumo ................................................................................................................................................... 3

Abstract.................................................................................................................................................... 4

List of Tables ........................................................................................................................................... 7

List of Figures .......................................................................................................................................... 9

Nomenclature ........................................................................................................................................ 14

Introduction .................................................................................................................................... 17

1.1 Motivation .............................................................................................................................. 17

1.2 State of the art ....................................................................................................................... 18

1.2.1 Polymer-solvent systems ............................................................................................... 18

1.2.2 Surfactant mixtures ........................................................................................................ 20

Theoretical Background ................................................................................................................. 21

2.1 Intermolecular interactions .......................................................................................................... 21

2.2 Evolution of Equations of State (EoS) ................................................................................... 22

2.2.1 Ideal Gas Law ................................................................................................................ 23

2.2.2 Van der Waals and Cubic EoS ...................................................................................... 23

2.2.3 Statistical associating fluid theory (SAFT) ..................................................................... 24

Modelling of phase behaviour of polymer-solvent mixtures and surfactant systems using gSAFT

………………………………………………………………………………………………………………35

3.1 gSAFT .................................................................................................................................... 35

3.2 Polymer-solvent systems ....................................................................................................... 36

3.2.1 SAFT-γ Mie Group Parameter Matrix ............................................................................ 36

3.2.2 Refinement of SAFT-γ Mie interaction parameters ....................................................... 42

3.3 Surfactant mixtures ................................................................................................................ 56

3.3.1 SAFT-γ Mie Group Parameter Matrix ............................................................................ 59

3.3.2 Refinement of SAFT-γ Mie interaction parameters ....................................................... 69

3.3.3 Modelling the CMC implementing the SAFT-γ Mie EoS on gPROMS® ........................ 92

Conclusions and Future Work ....................................................................................................... 97

References ............................................................................................................................................ 99

Appendix A – Functional Groups: Name and Description ................................................................... 110

Appendix B – CiEj Pure-component VLE Experimental Data .............................................................. 111

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Appendix C – Modelling Simple Binaries with the Original SAFT-γ Mie Group Interaction Parameters

............................................................................................................................................................. 113

C.1 Water-Alkane ............................................................................................................................ 113

C.2 Water-Ether ............................................................................................................................... 113

C.3 Water-Alcohol ........................................................................................................................... 114

C.4 Ether-Alkane ............................................................................................................................. 115

C.5 Alcohol-Alkane .......................................................................................................................... 115

7

List of Tables

Table 1 – SAFT-VR pure component parameters for the square-well potential (57). ............................. 30

Table 2 – SAFT-VR combining rules for unlike intermolecular parameters; both parameters 𝛾𝑖𝑗 and 𝑘𝑖𝑗

are adjustable to a default value of zero. .............................................................................................. 30

Table 3 - SAFT-γ Mie parameters used to fully describe a functional group 𝑘. .................................... 32

Table 4 - SAFT-γ Mie combining rules for unlike intermolecular parameters; * - in case of associating

compounds (85). ...................................................................................................................................... 34

Table 5 – Polymer solvent systems and the corresponding missing interactions prior to this project –

when simulating these systems, the listed interactions’ parameters are automatically estimated as a

combining rule. ...................................................................................................................................... 39

Table 6 – Dispersive energy estimation results for of both aCH:COO and aCCH3:COO unlike

interactions. ........................................................................................................................................... 43

Table 7 – Dispersive energy estimation results for of both aCH:C=O and aCCH3:C=O unlike interactions

............................................................................................................................................................... 52

Table 8 – Summary of the estimated parameters while studying polymer-solvent systems. ............... 54

Table 9 – Group interaction parameters determined prior to this work; in this table only the interactions

of each group with alike groups are displayed. Since the OH group is the only one which can associate

with itself (hydrogen-bonding), it is the only one with defined association parameters. ....................... 60

Table 10 – Group unlike interaction parameters determined prior to this work. In this table only the

interactions of each group with unlike groups are displayed. Since further along this project the

interactions with H2O will also be taken into account, this group has been included in this table. [1]

Hydrogen-bonding between the ‘H’ in the OH group and the ‘O’ in H2O, [2] Hydrogen bonding between

the ‘O’ in the OH group and the ‘H’ in H2O. ........................................................................................... 60

Table 11 – Ternary water-surfactant-alkane systems for which experimental data was found in the form

of a fish diagram and success or failure in modelling the existence of a third liquid phase when using

the original SAFT-γ Mie model. ............................................................................................................. 67

Table 12 – H2O:cO parameter estimation results by regression to C6E2+Water LLE experimental data.

............................................................................................................................................................... 70

Table 13 – Results of both parameter estimations fitting to binary C6E5+n-dodecane LLE experimental

data. ....................................................................................................................................................... 72

Table 14 – Parameter estimation best result from fit to H2O + C6E2 + n-dodecane experimental data at

0.1MPa and 293 K. ................................................................................................................................ 75

Table 15 – Parameter estimations of the like and unlike interactions of the CH2 surf group, fitting only to

pure surfactant VLE data. The number of parameters was sequentially increased in order to achieve the

best fit possible to the experimental data with the least possible parameters being estimated – the like

parameters which are not estimated have been fixed as equal to the CH2 group parameters, while the

unlike interactions not estimated are calculated automatically by the program as per combining rule. 77

Table 16 – First attempt for the parameter estimation of the like and unlike interactions of the CH2surf

group, using both pure surfactant VLE data and surfactant-alkane LLE data (C7E5+n-dodecane). The

unlike interactions not estimated are calculated automatically by the program as per combining rule. 79

8

Table 17 – Second parameter estimation of the like and unlike interactions of the CH2 surf group, using

both pure surfactant VLE data and surfactant-alkane LLE data (C7E5+ n-dodecane). The unlike

interactions not estimated are calculated automatically by the program as per combining rule. .......... 82

Table 18 – Parameter estimation of the unlike interaction CH2 surf with water, using surfactant (C4E1) +

water LLE data Parameterization 6.1 is based on Parameterization 6 results, while Parameterization 7.1

is based on Parameterization 7. ............................................................................................................ 85

Table 19 - Parameter estimation of the interaction CH2 surf with water and of cO and water, using

surfactant (C4E1)-water LLE data. Parameterization 6.2 is based on Parameterization 6 results, while

Parameterization 7.2 is based on Parameterization 7. ......................................................................... 87

Table 20 – CH2CH2O group results of the parameter estimation of all like and relevant surfactant-alkane

binary unlike interactions, fitted to pure surfactant VLE data and to binary surfactant-alkane data. The

hydrogen bond (HB) was defined between the O in CH2CH2O and the H in water. ............................ 89

Table 21 – Parameter estimation results for the CH2CH2O:H2O interaction, fitting to C4E1+H2O binary

LLE data; the association interaction happens between the Oxygen in the CH2CH2O group and the

Hydrogen in water. ................................................................................................................................ 91

Table 22 – CMC modelling results and experimental data for 22 CiEj + water systems using the original

model; no value is presented in case the model failed to find the CMC. .............................................. 96

Table Appendix A 1 – List of SAF-γ Mie functional groups; name and description. ........................... 110

Table Appendix B 1 – Experimental data for liquid phase density at 0.1 MPa of small pure CiEj

surfactants. .......................................................................................................................................... 111

Table Appendix B 2 - Vapour Pressure experimental data for small pure CiEj surfactants. ............... 112

9

List of Figures

Figure 1 – General intermolecular potential energy curve.. .................................................................. 21

Figure 2 – The Lennard-Jones potential and the relation of the parameters to the features of the curve.

............................................................................................................................................................... 22

Figure 3 – Representation of an associating chain-like molecule in SAFT.. ......................................... 25

Figure 4 – Individual contributions of a SAFT EoS ............................................................................... 26

Figure 5- SAFT-VR square-well potential. ............................................................................................. 27

Figure 6 – Two spheres with association sites in the SAFT-VR EoS ................................................... 28

Figure 7 – Representation of a fused heteronuclear model, here depicted for n-hexane comprising two

methyl (CH3 group – in grey) and four methylene groups (CH2 – in red). (85) ....................................... 31

Figure 8 – Group parameter matrix: complete list of the functional groups (plus their like and unlike

interactions) developed prior to this project. .......................................................................................... 37

Figure 9 – Schematic representation of the polymers and solvents studied in this project and their

subdivision into functional groups. ........................................................................................................ 38

Figure 10 – Comparison between predictions using only parameters developed prior to this work and

experimental data for polymer-solvent binary VLE data as a function of the weight fraction of the solvent

in the liquid phase. Successful predictions. ........................................................................................... 40

Figure 11 – Comparison between predictions using only parameters developed prior to this work and

experimental data for polymer-solvent binary VLE data as a function of the weight fraction of the solvent

in the liquid phase. Unsuccessful initial predictions. ............................................................................. 41

Figure 12 – Binary VLE data based in which both εaCH:COO and εaCCH3:COO have been estimated;

comparison between the prior set of parameters (continuous blue line), the refined set of parameters

(continuous yellow line), and the experimental data: ● – pressure 760mmHg (98) and ▲ – pressure

101.33 kPa (99). ...................................................................................................................................... 43

Figure 13 - Poly(vinyl acetate) + benzene VLE comparison between experimental data (43) and the model

including εaCH:COO. .................................................................................................................................. 44

Figure 14 - Binary VLE data based in which εaCCH:COO has been estimated; comparison between the prior

set of parameters (continuous blue line), the refined set of parameters (continuous yellow line), and the

experimental data from: ● – pressure 97.3 kPa (100) and ■ - (101) ........................................................... 45

Figure 15 - Poly(styrene) + propyl acetate VLE comparison between experimental data (43) and the new

model including both εaCH:COO and εaCCH:COO. ......................................................................................... 45

Figure 16 - Binary VLE data based in which εCH:aCCH3 has been estimated; comparison between the prior

set of parameters (continuous blue line), the refined set of parameters (continuous yellow line), and the

experimental data: ● – 2-propanol+p-xylene (103) and ● -2-propanol+m-xylene (102). ........................... 46

Figure 17 - Poly(vinyl acetate) + toluene VLE comparison between experimental data (43) and the new

model including εaCH:COO, εaCCH3:COO, and εCH:aCCH3. ................................................................................ 46

Figure 18 - Poly(vinyl methyl ether) + toluene VLE comparison between experimental data (43) and the

new model including only one parameter previously missing: εCH:aCCH3. ............................................... 47

10

Figure 19- Binary VLE data based in which εCH3COCH3 COO has been estimated; comparison between the

prior set of parameters (continuous blue line), the refined set of parameters (continuous yellow line),

and the experimental data: ● –53.33KPa (104); ○ –101 KPa (105). ........................................................... 47

Figure 20 – Poly(vinyl acetate) + acetone VLE comparison between experimental data (43) and the new

model including the interaction previously missing, εCH3COCH3 COO. ........................................................ 48

Figure 21 - Binary VLE data based in which εC:aCH was estimated; comparison between the prior set of

parameters (continuous blue line), the refined set of parameters (continuous yellow line), and the

experimental data: ● – 283.15 K (107); ▲–318.15 K (108). ....................................................................... 49

Figure 22 - Poly(n-butyl methacrylate) + benzene VLE comparison between experimental data (43) and

the new model including εaCH:COO and εC:aCH. ......................................................................................... 50

Figure 23 - Binary VLE data based in which εaCCH2:COO has been estimated; comparison between the

prior set of parameters (continuous blue line), the refined set of parameters (continuous yellow line),

and the experimental data: ● – 707mmHg (109). .................................................................................... 50

Figure 24 - Poly(n-butyl methacrylate) + ethylbenzene VLE comparison between experimental data (43)

and the new model including εaCH:COO , εC:aCH , and εaCCH2:COO. .............................................................. 51

Figure 25 – Acetone pure-component VLE. Continuous line – model predictions for acetone subdivided

into (CH3)-(C=O)-(CH3); scatter - experimental data (112). .................................................................... 52

Figure 26 – Binary VLE data based in which εaCH:C=O has been estimated; comparison between the prior

set of parameters (continuous blue line), the refined set of parameters (continuous yellow line), and the

experimental data: ● –318.15 K (110); ▲–330.15 K (111). ........................................................................ 52

Figure 27 – Binary VLE data based in which εaCCH:C=O has been estimated; comparison between the

prior set of parameters (continuous blue line), the refined set of parameters including εaCH:C=O

(continuous yellow line), and the experimental data: ● – 387.15 K (113). ............................................... 53

Figure 28 - Poly(styrene) + 2-butanone and Poly(styrene) + 3-pentanone VLE comparison between

experimental data (43) and the new model including one of the interactions previously missing: εaCH:C=O.

............................................................................................................................................................... 54

Figure 29 – Modelling polymer-solvent binary VLE data (previously poorly described), as a function of

the weight fraction of the solvent in the liquid phase, using the set of refined group interactions, including

all the estimated unlike parameters. ...................................................................................................... 55

Figure 30 - Temperature-composition diagram for a hypothetical binary mixture ................................. 57

Figure 31 - The temperature-composition diagram for water and nicotine, which has both upper and

lower critical temperatures..................................................................................................................... 57

Figure 32– Type VI of phase behaviour in the SVK classification, shown in the form of a P-T projection.

............................................................................................................................................................... 58

Figure 33 – SAFT-γ Mie functional groups determined prior to this project which are comprised in the

CiEj surfactant molecules: light blue – methylene (CH2); red – ether (cO); blue – hydroxyl (OH). ....... 59

Figure 34 – Comparison between the prior gSAFT model predictions (density and vapour pressure) and

experimental data for pure small surfactants. Vapour pressure in logarithmic scale. The continuous line

11

represents the model predictions; scatters represent the experimental data from: ■ - (125), ● - (126), ▲ -

(127), ○ - (128). ........................................................................................................................................... 62

Figure 35 – Comparison between the original gSAFT interactions model predictions (density and vapour

pressure) and experimental data for the LLE of binary mixtures water + surfactant, at a fixed pressure

(specified on top). The continuous line represents the model predictions; scatters represent the

experimental data from: ■ – (129), ● – (130), ▲ – (55), ○ – (131), ♦ - (132). ....................................................... 64

Figure 36 - (continuation) Comparison between the original gSAFT model predictions (density and

vapour pressure) and experimental data for the LLE of binary mixtures water + surfactant, at a fixed

pressure (specified on top). The continuous line represents the model predictions; scatters represent

the experimental data from: ■ – (129). ..................................................................................................... 65

Figure 37 – Ternary systems phase behaviour representations for a CiEj + oil + water mixture. According

to the Gibbs Phase Rule (62), in order to completely represent such systems a four-dimension coordinate

system would be necessary (for instance T, P and two composition variables), so more convenient

diagrams are typically applied. A – “ABC-type” phase diagram (where A, B, and C are the mixture’s

components), consists of a 2-dimension representation, at constant T and P. B – “fish diagram”, consists

of a 2-dimensional cut in a ternary ABC diagram at constant P, in such a way that the proportions of two

of the components is fixed, while increasing the amount of a third component (usually the amphiphile)

and simultaneously changing T - in the “fish head” three phases are in coexistence, the “fish tail”

corresponds to only one phase, and the surrounding area is the 2-phase region (133). ........................ 66

Figure 38 – Ternary diagrams of water-surfactant-alkane systems; the continuous blue line represents

the prediction of gSAFT, while the red-dotted line represents the experimental data and the light-blue

lines are tie-lines. The triangular zones delimited by the large circles are the three-phase regions, and

the circles are the compositions of the three phases in equilibrium. a) H2O+C4E2+n-octane, at 283K and

0.1MPa; b) H2O+C6E2+n-dodecane, at 293K and 0.1MPa; c) H2O+C4E2+n-octane, at 303K and 0.1MPa;

d) H2O+C6E2+n-dodecane, at 303K and 0.1MPa. Experimental data: C6E2 and C4E2. ........................ 68

Figure 39 – Model correlation to C6E2-Water experimental data used to fit H2O:cO attraction parameters;

the continuous yellow line represents the refined model predictions, the continuous blue line is the

original model, and the scatter is the experimental data. ...................................................................... 70

Figure 40 – Verification of the model transferability to other CiEj-Water systems; continuous yellow lines

represent the refined model predictions and the scatters are the experimental data. .......................... 71

Figure 41 – Refined interaction parameters model test on ternary systems, using parameters only fitted

to the C6E2-Water binary system: a) water+C6E2+n-dodecane, at 293K and 0.1MPa; b) water+C6E2+n-

dodecane, at 303K and 0.1MPa; weight fractions, blue-lined scatter – experimental data, yellow

continuous line – model. ........................................................................................................................ 71

Figure 42 – Model correlation to C6E5 + n-dodecane experimental data used to fit CH2:cO and/or cO:OH

interaction parameters; the continuous pink line represents model in which only CH2:cO changed, the

continuous green line is the model where both interactions were fitted, and the scatter is the

experimental data (57). ............................................................................................................................ 72

12

Figure 43 – Verification of the model transferability to other CiEj-alkane systems; the continuous pink

line represents model in which only CH2:cO changed, the continuous green line is the model where

both interactions were refined, and the scatter is the experimental data (57) (52). ................................... 73

Figure 44 – Verification of the model validity for pure CiEj systems; the continuous pink line represents

model of Parameterization A (only CH2:cO changed), the continuous green line is the model of

Parameterization B (both CH2:cO and cO:OH fitted), continuous blue line is the prior set of parameters

prediction, and the scatter is the experimental data. ............................................................................. 74

Figure 45 – Correlation of the new H2O:cO interaction to the ternary data to which it was fitted; blue-

lined scatter – experimental data, continuous yellow line – best model fit to data achieved for the system.

............................................................................................................................................................... 75

Figure 46 – New division of CiEj molecules into functional groups: light blue – methylene (CH2); red –

ether (cO); yellow – hydrophilic chain methylene (CH2surf); blue – hydroxyl (OH). ............................. 76

Figure 47 – Vapour pressure and density predictions of pure small surfactants using different models

resulting from the parameter estimation using only those pure components experimental data. Vapour

pressure in logarithmic scale. The model details are listed in Table 15. When Parameterization 2 cannot

be seen, it is concealed by Parameterization 3. The continuous line represents the original model

predictions; scatters represent the experimental data. ......................................................................... 78

Figure 48 – Testing CH2surf fit to pure component only Parameterizations 3 to 5 to the C7E5 + n-dodecane

LLE; the scatter represents experimental data at 0.1MPa (52), the model matches this pressure. ....... 79

Figure 49 – Vapour pressure and density predictions of pure small surfactants using two models

resulting from the parameter estimation using either only those pure components experimental data or

both that and alkane-surfactant LLE. The model details are listed in Table 16 and Table 15. Scatters -

experimental data. ................................................................................................................................. 80

Figure 50 – Comparing CH2surf parameter estimations: Parameterizations 5 and 6 applied to the system

C7E5 + n-dodecane. Scatter - experimental data (57).............................................................................. 81

Figure 51 – Verification of Parameterization 6 model transferability to other CiEj-alkane systems; scatter

- experimental data (57). .......................................................................................................................... 81

Figure 52 – Vapour pressure and density predictions of pure small surfactants using Parameterizations

6 and 7 (estimations fitted to both pure surfactant VLE and alkane-surfactant LLE data, but considering

different parameters); scatters - experimental data. ............................................................................. 83

Figure 53 – Comparing CH2surf parameter estimations’ Parameterizations 6 and 7 applied to the system

C7E5 + n-dodecane; scatter - experimental data. .................................................................................. 84

Figure 54 – Comparison between Experiment 6 and 7 model transferability to other CiEj-alkane systems;

scatter - experimental data. ................................................................................................................... 84

Figure 55 – Comparing CH2surf parameter estimations obtained in Parameterizations 6.1 and 7.1 applied

to the system C4E1 + water used to fit the parameters; scatter - experimental data. ............................ 85

Figure 56 – Comparison between Parameterization 6.1 and 7.1 model transferability to other CiEj-water

systems, at 10MPa to avoid VLE; scatter - experimental data. ............................................................. 86

13

Figure 57 – Parameterizations 6.2 and 7.2 model’s correlation to C4E1+H2O and their transferability to

other CiEj - water systems and its comparison with Parameterizations 6.1 and 7.1, at 10MPa to avoid

VLE; scatter - experimental data. .......................................................................................................... 87

Figure 58 – Comparing Parameterization 6.2 and 7.2 regarding ternary water-surfactant-alkane

systems. The triangular zones delimited by the large circles are the three-phase regions, and the circles

are the compositions of the three phases in equilibrium. Ternary systems: a) and c) - water+C4E1+n-

decane, at 313K and 0.1MPa; b) and d) water+C6E2+n-dodecane, at 293K and 0.1MPa. Pink continuous

line – Parameterization 6.2 (plots a) and b)); purple continuous line – Parameterization 7.2 (plots c) and

d)); continuous blue line – prior group interaction parameters; red-dotted line – experimental data;

dashed lines – three-phase region. ....................................................................................................... 88

Figure 59 – New division of CiEj molecules into functional groups: light blue – methylene (CH2); green

– oxyethylene (CH2CH2O); blue – hydroxyl (OH). ............................................................................... 89

Figure 60 – Correlation of Parameterization 8 to the experimental data used to fit this model’s

parameters. ........................................................................................................................................... 90

Figure 61 – Verification of Parameterization 8 model transferability to other CiEj-alkane systems; scatter

- experimental data; continuous green line - model predictions. ........................................................... 91

Figure 62 – Parameterization 8.1 correlation to the experimental data used (C4E1 + H2O) and model’s

transferability to similar CiEj + H2O systems; continuous green line – model predictions, scatter –

experimental data. ................................................................................................................................. 92

Figure 63 – Spherical micelle structure layout, for a surfactant in aqueous solution.(137) ..................... 93

Figure 64 – CMC modelling results for nineteen CiEj + water systems using the pre-existing groups

parameters. ........................................................................................................................................... 95

Figure Appendix C 1 – Water + n-hexane VLE gSAFT model predictions applying the SAFT-γ Mie group

interaction parameters determined prior to this project. Scatter – experimental data (112). .................. 113

Figure Appendix C 2 – Water + diisopropyl ether (DIPE) LLE gSAFT model predictions applying the

SAFT-γ Mie group interaction parameters determined prior to this project. Scatter – experimental data

(145) ....................................................................................................................................................... 113

Figure Appendix C 3 – 1-butanol + water gSAFT model predictions applying the SAFT-γ Mie group

interaction parameters determined prior to this project. Scatter – experimental data (112). .................. 114

Figure Appendix C 4 – 1-heptanol + water LLE gSAFT model predictions applying the SAFT-γ Mie group

interaction parameters determined prior to this project. Scatter – experimental data (112). .................. 114

Figure Appendix C 5 – Diethyl ether + n-hexane VLE gSAFT model predictions applying the SAFT-γ Mie

group interaction parameters determined prior to this project. Scatter – experimental data (112). ....... 115

Figure Appendix C 6 – 1-butanol + n-hexane VLE gSAFT model predictions applying the SAFT-γ Mie

group interaction parameters determined prior to this project. Scatter – experimental data (112). ....... 115

14

Nomenclature

𝐴 Helmholtz free energy

𝑎, 𝑏 Fluid-specific parameters of the Van der Waals EoS

𝑎𝐻𝑆 Helmholtz free energy of the hard-sphere reference system

𝑎𝐻𝑆′ Dimensionless contribution to the hard-sphere free energy per segment

𝑎𝑗,𝑘𝑙 Pairwise interactions between groups 𝑘 and 𝑙

𝑎𝑚𝑜𝑛𝑜𝑚𝑒𝑟 Helmholtz free energy per monomer segment

𝑎1, 𝑎2,, 𝑎3 Perturbative terms associated with the attractive energy

𝑎𝑀 Activity of micelle surfactants (in “idealized state”)

𝑎𝑆 Activity of free surfactants in solution

𝑎𝑆𝐶𝑀𝐶 Activity of free surfactants in solution at CMC

𝐶 Mie potential factor that ensures the minimum of interactions is – 𝜀

𝑑, 𝑟0 Intermolecular distance at which the potential is zero

𝑑𝑘𝑘 Effective temperature-dependent diameter

�̅�𝑖𝑖 Effective hard-sphere diameter

𝐸𝑘 Kinetic energy of a system

𝑓𝑎,𝑏 Mayer 𝑓-function of the a-b site-site association potential

𝑔𝑚𝑜𝑛.,𝑆𝑊 Radial distribution function (RDF) at contact (SAFT-VR)

ℋ Hamiltonian operator

ℎ Planck constant

𝑖 Quantum microstate; component

𝐾 Bonding volume of the interaction

𝑘𝐵 Boltzmann constant

𝑘𝑖𝑗 Deviation from the Berthelot geometric mean

𝐾𝑘𝑘𝐻𝐵

Bonding volume of the like association interaction

𝐾𝑘𝑙𝐻𝐵

Bonding volume of the unlike association interaction

𝑚 Mass of a particle

𝑚𝑖 Number of segments per molecule/chain 𝑖 (SAFT-VR)

𝑁, 𝑛 Number of classical particles/ molecules in a system

𝑁𝐴 Avogadro constant

𝑁𝑐 Number of components in a mixture

𝑁𝑖 Number of molecules of component 𝑖 in a mixture

𝑛𝑘,𝑎 Number of sites of type a on a group 𝑘

15

𝑁𝑠 Total number of spherical segments in a chain (SAFT-VR)

𝑁𝑆𝑇,𝑘 Number of different site types in a given group 𝑘

𝑃 Pressure

𝑟𝑎𝑏 Range of interaction of associating sites on Wertheim’s TPT1

𝑟𝑐 Intermolecular distance for which the well is minimum (LJ); cut-off range of directional

(association) interaction (SAFT-VR)

𝑟𝑘𝑙 Distance between the centres of a site in a segment 𝑘 and a site in a segment 𝑙

𝑟𝑘𝑙𝑐 Cut-off range of interaction for a site in a segment 𝑘 and a site in a segment 𝑙

𝑟𝑘𝑙𝑑 Position of the association site, distance from the centre of the segments

𝑅 Ideal gas constant

𝑠 Number of associating SW sites of type a in a molecule

𝑆𝑘 Shape factor of the group 𝑘

𝑇 Temperature (absolute)

𝑈 Potential energy of a system

𝑢 Pair potential between tangentially bonded monomers

𝑉 Hard-sphere potential; system volume

𝑊 Number of microscopic configurations

𝑋𝑎 Fraction of molecules not bonded at site 𝑎

𝑥𝑖 Mole fraction of component 𝑖 in a mixture

𝑋𝑖,𝑘,𝑎 Fraction of molecules of component 𝑖 that are not bonded at a site of type 𝑎 on group 𝑘

𝑥𝑆𝐶𝑀𝐶 Mole fraction of free surfactants in solution at the CMC

𝑥𝑠,𝑘 Fraction of segments of a group of type 𝑘 in a mixture

𝑦𝑚𝑜𝑛.,𝑆𝑊 Cavity distribution function of the monomer fluid (SAFT-VR)

𝛾𝑆𝐶𝑀𝐶 Activity coefficient of a free surfactant at CMC

Δ𝑎,𝑏 Strength of the association between sites a and b on different molecules

𝜀 Depth of the monomer-monomer potential well

𝜀𝑎𝑏 Energy of interaction between associating sites on TPT1

𝜀𝐻𝐵 Attraction energy/ potential depth of hydrogen-bonding (HB) association sites

𝜀𝑘𝑘 Energy of interaction between the segments of a group 𝑘

𝜀𝑘𝑙 Energy of interaction between the segments of two groups – 𝑘 and 𝑙

𝜀𝑘𝑘𝐻𝐵 Association energy of two segments 𝑘

𝜀𝑘𝑙𝐻𝐵 Association energy of a site in a segment 𝑘 and a site in a segment 𝑙

16

𝜀𝑖 Energy of a system in a quantum state 𝑖

𝜖�̅�𝑖 Average interaction energy

Λ de Broglie thermal wavelength

𝜆𝐴 , 𝜆𝑅 Attractive (A) and repulsive (R) exponentials of the Mie potential

𝝀𝒌𝒌𝑨, 𝜆𝑘𝑘

𝑅 Repulsive and attractive exponents of the potential between segments of a group 𝑘

𝝀𝒌𝒍𝑨, 𝜆𝑘𝑙

𝑅 Repulsive and attractive exponents of the potential between segments of groups 𝑘 and 𝑙

𝜆̅𝑖𝑖 Average exponents of the potential

𝜆𝜎 Cut-off range of interaction (SW, SAFT-VR)

𝜇𝑀 Chemical potential of the reference state of surfactant molecules included in micelles

𝜇𝑀0 Chemical potential of the reference state of surfactant in micelle-state

𝜇𝑆 Chemical potential of monomeric surfactant molecules in solution

𝜇𝑆0 Chemical potential of the reference state of surfactant monomers in solution

𝜐𝑘∗ Number of identical segments forming a group 𝑘 (SAFT-γ Mie)

𝜐𝑘,𝑖 Number of occurrences of a group of type 𝑘 on component i

𝜌𝑖 Number density of component 𝑖 in a mixture

𝜎 Monomer segment diameter (at which the potential is zero); hard-sphere limit (SAFT-VR)

𝜎𝑘𝑘 Diameter of the segments of a group 𝑘

𝜎𝑖𝑖 Average molecular effective size

Φ𝐿𝐽 Lennard-Jones (LJ) potential

Φ𝑀𝑖𝑒 Mie potential

𝜙𝑆𝑊 Square-well potential (SAFT-VR)

ϕ𝑘𝑙𝐻𝐵 Square-well association potential of a site in a segment 𝑘 and a site in a segment 𝑙

𝜙1 Singlet energy (only ≠ 0 if in presence of an external field) contribution to 𝑈

𝜙2 Pair energy contribution to 𝑈

𝜙3 3-body energy contribution to 𝑈

𝜙2𝑒𝑓𝑓 , 𝜙 “Effective” pair potential; intermolecular potential

17

Introduction

Motivation

Thermodynamic modelling frameworks are being continuously developed and improved to meet the

need for efficient and robust property prediction. In the chemical industry, predictive approaches have

become increasingly important for process design, which can be considerably affected by the accuracy

of the description of the thermodynamic properties. Since one of the most desired features in

Thermodynamic tools is a predictive capability, frameworks have been developed towards decreasing

the need for experimental data for the determination of molecular model parameters.

Group contribution (GC) approaches are an important class of highly predictive thermodynamic

methodologies, developed based on the assumption that the properties of a given compound can be

determined from the chemically distinct functional groups which are included in the molecule.

The Statistical Associating Fluid Theory (SAFT) is an advanced molecular-based equation of state (EoS)

which allows for the prediction of thermodynamic properties of complex materials using physically-

realistic models of molecules. Namely, molecules are described as chains of tangentially bonded

spherical segments. This family of EoS is believed to be more versatile than the more widely used

engineering methodologies such as cubic equations of state or activity coefficient models.

Process Systems Enterprise’s gSAFT software is a commercial implementation of one of the most

recent SAFT-based EoS: SAFT-γ Mie EoS, developed by Imperial College London. This version of SAFT

is a Group Contribution approach, and its use within gPROMS® process modelling platform is expected

to open up new ways to address industrially relevant processes involving complex materials.

The purpose of this work is to provide an assessment of the predictive capabilities of the SAFT-γ Mie

EoS by focusing on the fluid-phase equilibrium of complex materials: polymer-solvent systems (involving

poly(styrene), poly(ethylene), poly(vinyl acetate), poly(vinyl methyl ether), among others) and non-ionic

(n-alkyl polyoxyethylene ether, CiEj) surfactant mixtures.

18

State of the art

1.1.1 Polymer-solvent systems

Since polymers are generally synthesized in one- or two-phased systems with a mixture of solvents,

unreacted monomers, and additives that must be separated from the product, understanding vapour-

liquid equilibrium and liquid-liquid equilibrium is key to polymer production (for instance in the design

and optimization of polymerization reactors, and separation and devolatilization equipment) (1).

Several thermodynamic models have been proposed in the literature to describe the phase behaviour

of polymer systems, such as activity coefficient models and equations of state (EoS). More traditionally

applied EoS, such as cubic equations of state, accurately describe simple nearly-spherical molecules

(alkanes, CO, etc.) but they have been proven to poorly describe polymer systems (1). Thus, recently,

significant efforts have been focused on the statistical associating fluid theory (SAFT); in its many

variations, SAFT has been successfully applied to a wide range of fluids and their mixtures (1) (2). Given

the fact that SAFT describes molecules as chains of tangentially bond spherical segments, it appears

to be a more suitable alternative when it comes to the description of polymers.

Many versions of the SAFT EoS have been applied to describe the phase behaviour of such systems.

In 1990, Huang and Radosz (3) proposed a version of the SAFT EoS that has been applied to a wide

range of polymers – particularly to polyolefins (4) – (7). In 2000, Sandowski et al. developed PC-SAFT, the

perturbed chain SAFT equation, which successfully described the phase behaviour of a multiplicity of

polymer systems (8) – (17). The SAFT-VR approach, proposed by Gil-Villegas et al. in 1997 (18) (19),

successfully described a wide range of systems (20) – (31), including the absorption of light hydrocarbons

in poly(ethylene) (32) (33) and the cloud-point curves of poly(ethylene) solutions (34).

In 2007, Clark applied the SAFT-VR approach to investigate the fluid phase equilibria of water-soluble

polymers, in particular those of aqueous solutions of polyethylene glycol (PEG) (35). Here, a fully

transferable intermolecular potential model is presented, which allows one to study any aqueous PEG

system with the molecular weight of the polymer as sole input; this model also achieves an excellent

prediction of LLE.

One of the main problems when modelling polymer systems would be characterizing branching and

heterogeneity (functional groups of different molecular composition) in the molecule structure. With this

in mind, several heterosegmented versions of SAFT have been developed, allowing the model chain to

comprise segments of different size and/or energy (36) – (39).

Another challenge in modelling polymer systems with EoS is determining the model parameters, since

pure-component parameters are generally determined by regression to experimental pure-component

VLE data. In case of polymers, pure-component data is scarce. Thus, identifying pure-component

parameters for polymers is more difficult, and paired with a higher degree of uncertainty compared to

other more volatile substances (10) as VLE data is not available.

19

A first alternative would be to obtain the parameters that characterise the pure polymer from PVT data,

i.e. single-phase liquid densities at specified temperature and pressure. However, and specific to SAFT-

type theories, the parameters obtained in this way for the segment size and the dispersion energy have

been shown to result to poor predictions of the fluid-phase behaviour of mixtures including polymers (1).

Alternatively, polymer parameters can be regressed from correlations based on the parameters for

smaller molecules in a homologous series. This has been shown to work well for polyolefins by

extrapolation of n-alkane parameters to high molecular weight (32) (40) (41). However, such an approach as

shown to fail when applied to other polymers where the effects of other polymer functional groups is not

appropriately taken into account (42).

In an attempt to tackle the challenge of developing parameters for polymeric systems, several

approaches have been proposed to obtain parameters for different types of polymers within a

heterosegmented SAFT model (using copolymer-SAFT (37) and PC-SAFT (38)). Nonetheless, in those

approaches it is required to include experimental data for mixtures including polymers – reducing the

predictive capabilities of the equation. A final alternative, is the recast of SAFT theories within the group

contribution formalism. Such theories can obtain parameters for functional groups from simpler

substances for which pure-component VLE experimental data are typically available, and transfer the

determined parameters to polymeric systems. Such theoretical approaches, due to the relevance to the

presented work, are presented in the following paragraph.

One of the first applications of the GC formalism within a SAFT EoS is the work of Lora et al. (152) in the

context of a study of the fluid phase behavior of polyalkylacrylates in ethylene and carbon dioxide. Here,

the contributions that the CH3, CH2, CH, and the acrylate groups, make to the overall molecular chain-

length and segment size parameters for hydrocarbon systems were determined. However, this method

lacks full predictive capability, as the energy parameters are not treated at the GC level. Afterwards,

Vijande et al. (153) have described the chain length, size, and dispersive energy parameters, within a

complete GC context by using the PC-SAFT EoS for the pure component VLE of hydrofluoroethers

(including CH3 CH2 CH, and ether –O- groups).

Lymperiadis et al.(154) presented the first heterosegmented SAFT group contribution (SAFT-gamma)

where a satisfactory prediction of the phase behaviour of polyethylene with n-pentane was shown, with

parameters obtained from short alkanes (ethane to n-decane).

Peng et al. (2010), who did the most extensive work on the subject, demonstrated the ability of GC-

SAFT-VR (a group contribution approach) to predict the VLE and LLE of polymer solutions and

accurately capture the effects of polymer molecular weight and molecular topology on phase behaviour

without the need to fit to experimental polymer mixture data (43). Here, the molecules are described by

heterosegmented chains in which each type of segment represented a functional group comprised in

the molecules – an approach similar to SAFT-γ Mie.

20

1.1.2 Surfactant mixtures

The non-ionic n-alkyl polyoxyethylene ether, or CiEj, surfactants, when in aqueous solution, exhibit a

region of liquid-liquid immiscibility (denoted by closed-loop or cloud curve) which is bounded by an upper

critical solution temperature (UCST) and a lower critical solution temperature (LCST). This closed-loop

region has been studied experimentally by various authors, such as Mitchell et al. (44), Corti et al. (45),

and Schubert et al. (46).

Hirschfelder et al. (47) were the first to postulate that directional forces such as hydrogen bonding were

responsible for the LCST. Since the SAFT EoS accounts for this type of directional interactions, it is

particularly suited to model the phase behaviour of aqueous surfactant solutions.

Garcia-Lisbona et al. (48) used the SAFT-HS approach to model the closed-loop curves for a range of

CiEj surfactants, where the molecules were assumed to have three association sites per oxyethylene

ether in the surfactant chain. Herdes et al. (49) used the soft-SAFT EoS to compare results obtained with

those of Monte Carlo simulations, studying the influence of the location of the association sites on the

surfactant chain. Evans et al. (50) compared the perturbation and the Flory-Huggins theory to reproduce

the cloud-curves of five CiEj surfactants.

Besides the LLE in aqueous surfactant solutions, the liquid-phase behaviour of microemulsions of water,

CiEj and alkanes has been studied experimentally by Kahlweit et al. (51), Sassen et al. (52), and Hu et al.

(53), and modelled by Andersen et al. (54) using the SAFT-HS approach, Rudolph et al. (55) who used the

Peng-Robinson EoS and the UNIQUAC gE-model, and by Browarzik et aI. (56) who used the Flory-

Huggins Theory.

The CiEj binary and ternary mixtures have also been studied by Schreckenberg (57), using the SAFT-VR

EoS. Here a transferable model is proposed which can be used to represent the closed-loop curves of

aqueous CiEj surfactants ranging from 5 < i < 18 and 1 < j < 8. The binary mixtures of water + n-alkane

and n-alkane + CiEj were also analysed and models able to quantitatively represent the experimental

data for these binaries have been presented – besides, they were used to model ternary water + n-

alkane + CiEj mixtures, showing qualitative agreement with the experimental data.

21

Theoretical Background

Any rigorous statistical mechanical treatment of associating fluids should be based on a well-defined

intermolecular potential model which is suitable for the representation of the specific interactions

between the molecules. Thus, first of all, an introduction regarding intermolecular interactions is provided

in the following section.

2.1 Intermolecular interactions

When molecules are forced to meet, the nuclear and electronic repulsions and the rising electronic

kinetic energy begin to dominate the attractive forces – a generic representation of this behaviour is

shown in Figure 1.

Figure 1 – General intermolecular potential energy curve. At long range the interaction is

attractive, but repulsion dominates at short range (58) .

One greatly simplified approximation of the resulting potential energy is the hard-sphere potential, in

which it is assumed that the potential energy rises steeply to infinity as soon as the particles have an

intermolecular distance d (Equation 1) (58).

𝑟 ≤ 𝑑 → 𝑉 = ∞ , 𝑟 > 𝑑 → 𝑉 = 0 (1)

The well-known Lennard-Jones potential Φ𝐿𝐽 (Figure 2) represents the intermolecular interactions and

London-dispersion attractions, and is often written as

Φ𝐿𝐽(𝑟) = 4𝜀 [(

𝑟0𝑟

)12

− (𝑟0𝑟

)6

] (2)

where 𝜀 is the depth of the well and 𝑟0 is the separation at which Φ𝐿𝐽=0; the well minimum occurs at an

intermolecular distance of 𝑟𝑐 = 21/6𝑟0. In this potential, the first term represents repulsion and the second

term represents attraction. However, the Lennard-Jones potential has sometimes been found to poorly

describe the repulsive potential (58).

22

Figure 2 – The Lennard-Jones potential and the relation of the parameters to the features of the

curve. The green and purple l ines are the contributions (attraction and repulsion). (58)

Amore versatile option would be the Mie intermolecular potential, Φ𝑀𝑖𝑒, a generalized Lennard-Jones

potential with adjustable attractive and repulsive exponents (𝜆𝐴 and 𝜆𝑅) that characterize the

softness/hardness and the range of the interaction. The form of the pair interaction energy between two

monomeric segments is given as a function of the intersegment distance 𝑟 by (59)

Φ𝑀𝑖𝑒(𝑟) = 𝐶𝜀 [(

𝜎

𝑟)

𝜆𝑅

− (𝜎

𝑟)

𝜆𝐴

] (3)

Where 𝜎 is the segment diameter (at which the potential is zero), 𝜀 is the depth of the potential well, 𝜆𝑅

and 𝜆𝐴 are the repulsive and attractive exponents, and the factor 𝐶 ensures that the minimum of the

interactions is – 𝜀:

𝐶 = 𝜆𝑅

𝜆𝑅 − 𝜆𝐴(

𝜆𝑅

𝜆𝐴)

𝜆𝐴

𝜆𝑅−𝜆𝐴

(4)

The Mie potential is base for the key Equation of State of this project: SAFT-γ Mie, and thus it will be

further mentioned in Section 2.2.3.2.

2.2 Evolution of Equations of State (EoS)

Equations of State have been continuously improved ever since the ideal gas law was developed, so as

to meet the increasingly demanding needs of the more and more complex systems under study.

Because the simplest EoS provide a correlation obtained from a region of fit to experimental data, they

are limited to the experimental state conditions and become even less useful for systems with little

experimental data.

On the other hand, molecular-based EoS (such as the SAFT approach) connect the microscopic and

the macroscopic worlds, allowing to relate the effect of different types of interactions with fluid phase

23

equilibria - thus enabling the modelling of the behaviour of more complex molecules. Furthermore, they

allow thermodynamic behaviour predictions away from the region of experimental fit, with reasonable

accuracy.

In this section a brief explanation of some of the more relevant EoS is provided, focusing on the SAFT-

γ Mie EoS which is key to this project.

2.2.1 Ideal Gas Law

The ideal gas law is the simplest equation of state. It concerns the behaviour of monomeric gaseous

systems and is given by

𝑃𝑉 = 𝑛𝑅𝑇 (5)

where 𝑃 is the pressure of the gas, 𝑉 its volume, 𝑛 the number of gas molecules, 𝑅 the ideal gas

constant, and 𝑇 the temperature of the gas. This model is based on two assumptions: the gas molecules

are dimensionless (point-mass particles) and there are no intermolecular interactions of any kind

whatsoever (the intermolecular potential is zero, whichever the intermolecular distance).

2.2.2 Van der Waals and Cubic EoS

The Van der Waals (vdW) equation of state accounts for spherical repulsions (excluded volume) and

attractions between particles, therefore being one step ahead of the ideal gas law, and may be written

as

𝑃 =

𝑛𝑘𝐵𝑇

𝑉 − 𝑁𝑏−

𝑛2𝑎

𝑉2 (6)

where 𝑎 and 𝑏 are fluid-specific parameters, 𝑃 is the pressure of the fluid, 𝑉 its volume, 𝑛 the number of

molecules, and 𝑇 the temperature (62).

This form of the equation enables visualising the repulsive and attractive contributions to the pressure

proposed by vdW. In this model, the molecules are seen as rigid spheres with a fixed diameter.

According to this assumption, van der Waals stated that the intermolecular attraction decreased the

pressure of the fluid and was proportional to the number of molecules in the system and inversely

proportional to the volume of the system. To account for those interactions, vdW introduced the

correction 𝑎 in Equation 6. The total volume available to the molecules is also reduced by their excluded

volume (a sphere of radius equal to the molecular diameter), which is accounted for by the correction 𝑏

in Equation 6.

The vdW EoS provides more realistic predictions than the ideal gas law; in fact, it was the first theory to

describe the transition between vapour and liquid phases. however, when deriving the vdW Helmholtz

free energy (vide (35)), the potential energy of a molecule with all others is approximated as a mean-field,

which means that one assumes that the molecules are randomly distributed – again, this is only

24

applicable to the gaseous phase, and fails to meet reality in liquid systems. Thus, it is a very approximate

description of a fluid and it is only reasonably accurate when dealing with gaseous systems.

Despite so, this is a very popular EoS, and also provided the basis for numerous cubic equations of

state – mostly empirically derived extensions of the vdW equation, and so-named since the volume can

be expressed in cubic form for a given 𝑇 and 𝑃 –, which offer solutions for some of the issues of the

vdW EoS. Among these cubic EoS are the Peng-Robinson and the (Soave-)Redlich-Kwong (SRK).

Although the goals of modelling fluid phase equilibria have not changed greatly since the time of vdW,

the systems of interest have gradually increased in complexity. For instance, complex molecules as

surfactants (e.g. n-alkyl polyoxyethylene ether (CiEj)), polymers (e.g. polystyrene, polypropylene),

hydrogen-bonding molecules (e.g. water), and polyfunctional molecules (e.g. amino acids and peptides)

are now commonly considered, as this project exemplifies.

Thus, developing a thermodynamic modelling framework that considers anisotropic interactions (as in

Hydrogen-bonding fluids), molecular anisotropy (chain-like molecules), and electrostatic interactions

(Coulombic, dipolar, quadrupolar) becomes increasingly necessary. In fact, one of the main motivations

for the development of the Statistical Associating Fluid Theory (SAFT) in the late 1980s (63) (64) was the

need for a reliable EoS for the thermodynamic properties and phase equilibria of associating fluids,

which could not be accurately described with the traditional cubic EoS available at the time.

Therefore, Wertheim’s work on associating and polymeric fluids (65) – (68) and its implementation as an

EoS in the SAFT has constituted a major advancement towards modelling these complex fluids (69).

2.2.3 Statistical associating fluid theory (SAFT)

The basis of the statistical associating theory (SAFT) family of EoS is obtained by combining Wertheim’s

first-order thermodynamic perturbation theory (TPT1) for associating fluids with an accurate description

of the structural and thermodynamic properties of the monomer fluid forms (59).

In Wertheim’s approach (65) – (68), molecules are seen as hard-core spheres with directional embedded

associating sites – square-well, off-centre bonding sites – which are placed at a distance 𝑟 of the centre

of the sphere, with a range of interaction 𝑟𝑎𝑏. When two molecules are within a distance 𝑟𝑎𝑏 from each

other they will interact, being the energy of interaction equal to −𝜀𝑎𝑏.

It is noteworthy that this theory rests in several assumptions, such as: multiple bonding in the same site

is not permitted, there is no steric hindrance between association sites, no double bonding of molecules

is allowed, bond cooperativity is not taken into account, and association is independent of bond angle

(70). However, a number of recent developments have been made in order to increase Wertheim’s theory

general applicability – the theory has now been extended to deal with ring-like structures (29) (71) – (75),

double bonding (75) (76), and the effects of cooperative bonding on the association interactions’ energy (77)

(78) (79).

Wertheim’s formalism was delivered both as an integral-equation theory and as a first-order

thermodynamic perturbation theory (TPT1), the latest being particularly convenient for developing

25

algebraic EoS for associating fluids and now firmly rooted in the SAFT family of equations of state. SAFT

has become a standard in both academia and industry as an equation of state for associating (hydrogen

bonding) fluids.

In the SAFT approach, molecules are modelled as chains formed from spherical monomeric segments

(either with hard cores – e.g. square-well -, or soft cores – e.g. Lennard-Jones, vide Section 2.1

Intermolecular interactions) which are not allowed to overlap but are fused to each other forming chains

of spheres; association happens between bonding sites with short-range interactions placed in those

monomeric segments (Figure 3). Consequently, the SAFT approach comprehends two types of

asymmetry: the chain contribution for the nonsphericity of molecules, and the association contribution

for anisotropic interactions (80).

Figure 3 – Representation of an associating chain-l ike molecule in SAFT. This molecule consists

of 5 monomers; Hydrogen-bond-l ike, directional attractive forces are i l lustrated as association

sites on the terminal spheres (adapted from (57)).

A general form of the SAFT theory may be presented in terms of the perturbation in the Helmholtz free

energy, being

𝐴

𝑁𝑘𝐵𝑇=

𝐴𝑖𝑑𝑒𝑎𝑙

𝑁𝑘𝐵𝑇+

𝐴𝑚𝑜𝑛𝑜𝑚𝑒𝑟𝑖𝑐

𝑁𝑘𝐵𝑇+

𝐴𝑐ℎ𝑎𝑖𝑛

𝑁𝑘𝐵𝑇+

𝐴𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛

𝑁𝑘𝐵𝑇 (7)

where:

- 𝐴𝑖𝑑𝑒𝑎𝑙 corresponds to the free energy of the ideal polyatomic gas system;

- 𝐴𝑚𝑜𝑛𝑜𝑚𝑒𝑟𝑖𝑐 is the residual free energy due to the formation of spherical monomeric segments

(includes repulsive and attractive/dispersive interactions);

- 𝐴𝑐ℎ𝑎𝑖𝑛 refers to the energy due to the formation of a chain of tangentially bonded monomeric

segments;

- 𝐴𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 is the free energy due to the association between molecules.

- 𝑁 is the total number of molecules, 𝑘𝐵 is the Boltzmann constant and 𝑇 is the absolute

temperature.

Both chain and directional association contributions can be obtained from Wertheim’s theory for

associating parties. In Figure 4 the individual contributions of a SAFT EoS are represented.

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Figure 4 – Individual contributions of a SAFT EoS; 1 – ideal gas, 2 – molecules as atomised

spherical segments exhibit ing repulsive forces (excluded volume, no overlap), 3 – attractive

forces between segments (which perturb the fluid of hard -spheres, represented as dashed

circles), 4 – spheres are fused together to form chain molecules, 5 – contributions due to

directional forces (adapted from (57)).

This convenient division of the SAFT theory when it is written in terms of Helmholtz free energy

translates into a versatility which has led to many different variations in the theory, most involving just a

change in the monomer term. The common point of these is the presence of both an association and a

chain term in the free energy, both originating from Wertheim’s theory.

The original SAFT was developed by Chapman et al. (63), who used a Lennard-Jones reference fluid

within the monomer term, thus modelling repulsive and attractive forces of pure components with a

Lennard-Jones potential – given by Equation 2 in Section 2.1 Intermolecular interactions. Several other

versions of SAFT have since been developed (69), for instance:

- SAFT-HR (3) : Developed by Huang and Randosz (HR), in this theory the mean field dispersive

attractions perturbation term is obtained from the power series of Alder, fitted to PVT, internal

energy and second virial coefficient data for Argon;

- SAFT-HS (81) (82): Green and Jackson’s theory includes a hard-sphere (HS) reference term and

a perturbed mean field vdW attractive term that was extended to mixtures;

- SAFT-VR (18): developed by Gill-Villegas et al., this version uses a reference fluid of hard-

spheres with perturbed attractions of variable range (VR) that has been extended to mixtures.

In this project, this SAFT version was applied in testing phase equilibrium models developed in

gPROMS® by comparing the results with Schreckenberg’s work (57), and thus a deeper

explanation is given in Section 2.2.3.1;

- PC-SAFT (83): considering a hard-chain instead of a hard-sphere as the reference system, it is

particularly suited to polymer systems, the EoS developed by Gross and Sandowski takes into

account the chain-length dependence on attractive interactions (as opposed to previous

versions which focus on the segment-segment attractions in a reference monomer fluid);

- SAFT-γ Mie (84): Laffite et al. have used a modified version of SAFT-VR, the SAFT-VR Mie,

incorporating the Mie potential (Section 2.1 Intermolecular interactions). PC-SAFT and SAFT-

VR (with SW and LJ potentials) fail to yield good second-order derivative properties (e. g.

isothermal compressibilities, speed of sound); by optimizing the repulsive exponent of the

potential (𝜆𝑅) it is shown that an accurate description of VLE and derivative properties for the n-

alkane series can be obtained. Eventually this evolved into a Group Contribution (GC) approach,

27

SAFT-γ Mie (85), which is the key SAFT EoS for this project, and so it will be further explained in

Section 2.2.3.2

Other SAFT approaches include the soft-SAFT (86), the treatment of heteronuclear chain molecules (21)

(37) (39) (87), the critical region (22) (88), and electrolytes (89). Other group contribution approaches include the

GC-SAFT-VR, which has been applied to polymers by Peng et al (44).

2.2.3.1 SAFT-VR

SAFT-VR is a variant of SAFT in which the associating chain molecules formed from hard-core

monomers have attractive potentials of variable range (VR). Compared with previous approaches (like

SAFT-HS), SAFT-VR is a step further improving the monomer contribution and provides additional

parameters that characterize the range of the attractive part of the monomer-monomer potential.

In the work of Schreckenberg (57), mentioned in Section 1.1.2, the SAFT-VR with square-well (SW)

potential, 𝜙𝑆𝑊, was used. This SW potential has three different regions (Figure 5), separated by the

hard-sphere limit 𝜎 and a defined cut-off range at 𝜆𝜎

𝜙𝑆𝑊(𝑟) = {

+∞ if 𝑟 < 𝜎−𝜖 if 𝜎 ≤ 𝑟 < 𝜆𝜎

0 if 𝑟 ≥ 𝜎 (8)

Figure 5- SAFT-VR square-well potential . A single sphere (m=1) and a polymer with four fused

spheres (m=4) are considered: in the l imit of the hard sphere the repulsive potential energy of

two overlapping spheres is infinite, between the contact distance and the range of attraction the

two spheres interact with a constant energy ( −𝜖𝑖𝑗), i f the distance between spheres is bigger than

the cut-off range the spheres do not interact (57) .

The directional attractive forces are modelled with association sites, which have a defined range of

attraction given by a cut-off range; two sites (a and b) on two different molecules will interact with a

constant attraction energy 𝜀𝐻𝐵 if the distance between them (𝑟) is up to the cut-off range 𝑟𝑐 (Equation 9,

Figure 6).

28

𝜙𝐴𝑆𝑆𝑂𝐶 (𝑟) = {

−𝜖𝐻𝐵 if 𝑟 ≤ 𝑟𝑐0 if 𝑟 > 𝑟𝑐

(9)

Figure 6 – Two spheres with association sites in the SAFT-VR EoS; 𝑟 is the distance between the

two interacting sites. (Adapted from (57))

For the SAFT-VR approach, the free energy of the ideal gas in Equation 7 for a mixture (90) is given by

𝐴𝑖𝑑𝑒𝑎𝑙

𝑁𝑘𝐵𝑇= (∑ 𝑥𝑖ln(𝜌𝑖 ∧𝑖

3)

𝑁𝑐

𝑖=1

) − 1 (10)

where 𝜌𝑖 = 𝑁𝑖/𝑉 is the number density of component i, with 𝑁𝑖 being the number of molecules of

component i and 𝑉 the volume of the system, 𝑥𝑖 is the mole fraction of component i, and Λ the thermal

de Broglie wavelength (Λ =ℎ

√2𝜋𝑚𝑘𝑩𝑇, 𝑚 the mass of the particle and ℎ the Planck constant) incorporating

the kinetic contributions (translational, vibrational and rotational) to the partition function of the molecule,

so that ∧𝑖3 is the thermal de Broglie volume. The summation is over all of the components 𝑁𝑐 present in

the mixture (so that 𝑁 = ∑ 𝑁𝑖𝑁𝑐𝑖=1 ).

The monomer-monomer interactions contribution term can be written as

𝐴𝑚𝑜𝑛𝑜𝑚𝑒𝑟

𝑁𝑘𝐵𝑇= (∑ 𝑥𝑖𝑚𝑖

𝑖=1

)𝐴𝑚𝑜𝑛𝑜𝑚𝑒𝑟

𝑁𝑠𝑘𝐵𝑇= (∑ 𝑥𝑖𝑚𝑖

𝑖=1

) 𝑎𝑚𝑜𝑛𝑜𝑚𝑒𝑟 (11)

where 𝑚𝑖 is the number of segments per molecule/chain i, 𝑁𝑠 is the total number of spherical segments

and 𝑎𝑚𝑜𝑛𝑜𝑚𝑒𝑟 is the Helmholtz free energy per monomer segment. 𝑎𝑚𝑜𝑛𝑜𝑚𝑒𝑟 is treated as the second

order Barker-Henderson high-temperature perturbation expansion (91) (92) (93) with a hard-sphere

reference, so that for a fluid composed of chains of 𝑚 segments of diameter 𝜎 :

𝑎𝑚𝑜𝑛𝑜𝑚𝑒𝑟 = 𝑎𝐻𝑆 +𝑎1

𝑘𝐵𝑇+

𝑎2(𝑘𝐵𝑇)

2 (12)

Where 𝑎𝐻𝑆 is the Helmholtz energy for the hard-sphere fluid given by a Boublík-Mansoori expression

(94), and 𝑎1 and 𝑎2 are the first two perturbative terms associated with the attractive energy (69).

29

The chain contribution is given by

𝐴𝑐ℎ𝑎𝑖𝑛

𝑁𝑘𝐵𝑇= − (∑ 𝑥𝑖(𝑚𝑖 − 1)

𝑁𝑐

𝑖=1

) ln (𝑦𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎)) (13)

being 𝜎 the monomer segment diameter, and 𝑦𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎) is the cavity distribution function of the

monomer fluid, related to the radial distribution function (RDF) at contact 𝑔𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎), and to the pair

potential between tangentially bonded monomers 𝑢(𝜎), by

𝑦𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎) = 𝑔𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎) exp (

𝑢(𝜎)

𝑘𝐵𝑇) (14)

Association interactions may also be included in the model, via square-well bonding sites – the

contribution due to association for s sites on a molecule is

𝐴𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛

𝑁𝑘𝐵𝑇∑ (ln(𝑋𝑎) −

𝑋𝑎2

)

𝑠

𝑎=1

+ 𝑠

2 (15)

where the sum is over all 𝑠 sites of type 𝑎 in a molecule and 𝑋𝑎 the fraction of molecules not bonded at

site 𝑎, obtained from the mass-action equation

𝑋𝑎 =

1

1 + ∑ 𝜌𝑋𝑏Δ𝑎,𝑏𝑠𝑏=1

(16)

where Δ𝑎,𝑏 characterizes the strength of the association between sites a and b on different molecules,

being

Δ𝑎,𝑏 = 𝐾𝑓𝑎,𝑏𝑔𝑚𝑜𝑛𝑜𝑚𝑒𝑟,𝑆𝑊(𝜎) (17)

with 𝐾 being the bonding volume of the interaction and 𝑓𝑎,𝑏 the Mayer 𝑓-function of the a-b site-site

association potential given by

𝑓𝑎,𝑏 = exp (

−𝜀𝐻𝐵

𝑘𝐵𝑇) − 1 (18)

in which 𝜀𝐻𝐵 is the potential depth of the interaction (69).

In the SAFT-VR framework, the parameters for the square-well potential required to describe a pure

component 𝑖 are presented in Table 1; the depth of the square-well is commonly given in the form 𝜀/𝑘𝐵

so it comes in units of K, as well as the association energy which is usually given as 𝜀𝐻𝐵/𝑘𝐵.

30

Table 1 – SAFT-VR pure component parameters for the square -well potential (57) .

Symbol Description

𝝈 Hard-sphere diameter / diameter of a monomer (m)

𝝀 Range of square-well interaction

𝜺 Depth of the monomer-monomer potential well (J)

𝒎 Number of spheres/segments in a chain

𝒔 Number of association sites of each type

𝒓𝒄 Range of interaction at an associating site (m)

𝜺𝑯𝑩 Association energy / depth of the site-site potential (J)

For mixtures of different species, a method is needed to obtain cross-interaction (unlike) parameters.

Every cross-interaction parameters could be adjusted to optimise the description of experimental

mixture data, but usually this is not done, since a large number of cross-interaction parameters can lead

to a large number of unnecessary degrees of freedom and fitting to experimental data reduces the

predictive capability of the EoS.

Thus, in mixtures the unlike intermolecular parameters are introduced using combining rules to facilitate

the study of binary and multicomponent systems. These rules are summarised in Table 2 the cross-

interaction square-well parameters for two hard-sphere segments (𝑖 and 𝑗). The unlike cross-interaction

association parameters 𝑟𝑐 and 𝜀𝐻𝐵 are generally specified for each bonding type individually (57) by

adjusting to experimental data, and 𝑘𝑖𝑗 is the deviation from the Berthelot geometric mean that is also

fitted to experimental data.

Table 2 – SAFT-VR combining rules for unlike intermolecular parameters; both parameters 𝛾𝑖𝑗 and

𝑘𝑖𝑗 are adjustable to a default value of zero.

Unlike parameter Combining rule

Interaction distance 𝜎𝑖𝑗 = (𝜎𝑖𝑖 + 𝜎𝑗𝑗)/2

Range of attraction 𝜆𝑖𝑗 = (1 − 𝛾𝑖𝑗)(𝜎𝑖𝑖𝜆𝑖𝑖 + 𝜎𝑗𝑗𝜆𝑗𝑗)

𝜎𝑖𝑖 + 𝜎𝑗𝑗

Dispersion energy 𝜀𝑖𝑗 = (1 − 𝑘𝑖𝑗)√𝜀𝑖𝑖𝜀𝑗𝑗

2.2.3.2 SAFT-γ Mie

The SAFT-γ Mie is a generalization of the SAFT-VR Mie (introduced by Laffite et al.. (84)), formulated

within the framework of a group contribution approach.

Group contribution approaches are based on the assumption that the properties of a given compound

can be determined as a function of the functional groups within it, supposing that the contribution of

each group is independent of its position in the compound. Thus in this version molecules are modelled

as comprising distinct functional groups based on a fused heteronuclear molecular model (Figure 7).

31

The interactions between those functional groups are described the Mie potential of variable attractive

and repulsive range.

Figure 7 – Representation of a fused heteronuclear model, here depicted for n-hexane comprising

two methyl (CH3 group – in grey) and four methylene groups (CH2 – in red). ( 85)

The Mie potential, 𝜙𝑀𝑖𝑒, has been briefly mentioned in Section 2.1 Intermolecular interactions as a

general form of the well-known Lennard-Jones potential, and it was defined for the pair interaction

between two segments as a function of the intersegment distance (𝑟) in Equation 3. As previously stated,

the repulsive and attractive exponents of the potential, 𝜆𝐴 and 𝜆𝑅 , which characterize the

softness/hardness and the range of the interaction, are allowed to vary freely.

Additional short-range sites can be used to model the association interactions (like hydrogen-bonds,

HB) in some polar compounds. These are square-well sites (similar to the ones in SAFT-VR, Equation

8) and are placed on the segments in such a way that the association interaction of a site in a segment 𝑘

and a site in a segment 𝑙 is:

ϕ𝑘𝑙

𝐻𝐵(𝑟𝑘𝑙) = { −𝜀𝑘𝑙

𝐻𝐵 if 𝑟𝑘𝑙 ≤ 𝑟𝑘𝑙𝑐

0 if 𝑟𝑘𝑙 > 𝑟𝑘𝑙𝑐 (19)

where 𝑟𝑘𝑙 is the distance between the centres of both sites, −𝜀𝑘𝑙𝐻𝐵 the association energy, and 𝑟𝑘𝑙

𝑐 is the

cut-off range of both sites.

When dealing with a GC-based theory as SAFT-γ Mie, all model parameters are group-specific. Thus,

no parameter specific to the molecule is used, apart from the knowledge of how many groups appear in

that molecule.

Within the SAFT-γ Mie approach a group may comprise several identical segments (95); thus the

parameters that fully describe a functional group 𝑘 are: 𝜐𝑘∗ (number of identical segments forming the

group), 𝜎𝑘𝑘 (diameter of the segments of the group), 𝜀𝑘𝑘 (energy of interaction between the segments

of the group), and 𝜆𝑘𝑘𝐴 and 𝜆𝑘𝑘

𝑅 (repulsive and attractive exponents of the potential, determine the form

of the interaction potential). The shape factor, 𝑆𝑘, translates the extent to which the segments of a given

group 𝑘 contribute to the overall molecular properties – so this also a key parameter in the methodology.

The parameters used to fully describe a functional group (𝑘) are summarised in Table 3.

32

Table 3 - SAFT-γ Mie parameters used to ful ly describe a functional group 𝑘.

Symbol Description