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Fluid Phase Equilibria 362 (2014) 288– 299

Contents lists available at ScienceDirect

Fluid Phase Equilibria

jou rn al h om epage: www.elsev ier .com/ locate / f lu id

comprehensive framework for surfactant selection and designor emulsion based chemical product design

ichele Matteia,b, Georgios M. Kontogeorgisb, Rafiqul Gania,∗

Computer Aided Process-Product Engineering Center (CAPEC), Department of Chemical and Biochemical Engineering, Technical University of Denmark,øltofts Plads, Building 229, DK-2800 Kgs. Lyngby, DenmarkCenter for Energy Resources Engineering (CERE), Department of Chemical and Biochemical Engineering, Technical University of Denmark, Søltofts Plads,uilding 229, DK-2800 Kgs. Lyngby, Denmark

r t i c l e i n f o

rticle history:eceived 23 July 2013eceived in revised form 11 October 2013ccepted 16 October 2013vailable online 25 October 2013

eywords:roduct designmulsionsroperty predictionloud point

a b s t r a c t

The manufacture of emulsified products is of increasing interest in the consumer oriented chemical indus-try. Several cosmetic, house-hold and pharmaceutical products are in the emulsified form when soldand/or they are expected to form an emulsion when used. Therefore, there is a need for the developmentof a methodology and relevant tools in order to spare time and resources in the design of emulsion-based chemical products, so that the products can reach the market faster and at a reduced cost. Theunderstanding and modeling of the characteristic behavior of emulsions and their peculiar ingredientsis consequently necessary to tackle this problem with computer-aided methods and tools. A compre-hensive framework for the selection and design of surfactants, the main responsible for the formationand the stability of emulsions, is presented here together with the modeling of the cloud point, a key-property of nonionic surfactants, with a group-contribution model. The mathematical formulation of astandard product design problem is presented, together with the list of both the pure component prop-erties (related to nonionic surfactants) and the mixture properties (relevant to the overall products as

an emulsion) needed for the solution of the design algorithm. These models are then applied togetherwith established predictive models for pure component properties of ionic surfactants and for standardmixture properties such as the density, the viscosity, the surface and the interfacial tension, but also thetype of emulsion expected (through the hydrophilic–lipophilic balance), and its stability (through thehydrophilic–lipophilic deviation), forming a robust chemical product design tool. The application of this

for t

framework is highlighted

. Chemical product design

Recently, a substantial shift is observed from materials valuedor their purity, to materials sold for their performance behav-or [1,2]. To meet these challenges, on a global and local scale,

hile remaining profitable and maintaining sustainable growth,here has been an increasing interest in the formulation and solu-ion of product design problems [3]. The chemical product treehown in Fig. 1 gives an idea of the size of this shift: the rootsf the tree consist in a limited number of Raw Materials whichre processed to obtain the commodity products (Basic Products).pecialty chemicals (Intermediate Products) are then manufacturedrom the commodities and finally the leaves of the tree repre-

ent a large portfolio of higher value products (Refined Chemicals

Consumer Products) obtained by processing and/or combininghe chemicals of the previous product classes. As one ascends the

∗ Corresponding author. Tel.: +45 45 25 28 82; fax: +45 45 93 29 06.E-mail address: rag@kt.dtu.dk (R. Gani).

378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.fluid.2013.10.030

he design of some emulsion based chemical products.© 2013 Elsevier B.V. All rights reserved.

product tree, the number of products belonging to each categorygrows exponentially from around 10 for the raw material class, upto almost 30,000 in the last class of higher value added products.This last class is composed of formulations, devices and technologybased consumer goods. Formulated products include pharmaceut-icals, paints, food, cosmetic, detergents, pesticides, in which 5 tomore than 20 ingredients are usually present, representing a widerange of chemical compounds such as polymers, surfactants, solids,solvents, pigments, and aromas [4]. The common practice, in thedevelopment of such products, is still the experiment-based andtrial-and-error approach. However, a systematic procedure, able todesign a higher added value product with enhanced product qual-ities, represents an efficient alternative, with respect to time andresources, speeding up the product development.

1.1. Formulation design

Many chemical-based personal care products of everyday lifesuch as sun lotions, shower creams, and insect repellents are liquidformulations, while examples of non-personal care products are

M. Mattei et al. / Fluid Phase Equi

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needed. Necessarily, some of the models applied for the definitionof the target properties might differ when considering an emulsi-

Fig. 1. The chemical product tree: classification of chemical-based products [3].

aints, pesticides and drugs. These can be classified as “consumer-riented products” since their needs, on the basis of which they areesigned, are defined by the consumers. Therefore these productseed to satisfy multiple needs of the consumers [5]. A sunscreen

otion, for example, must provide protection against sunburns andkin cancer, but must also prevent skin aging, and be, for example,ong lasting, safe, easily applicable, with good sensorial propertiescolor and odor) [6]. Because a single chemical is unlikely to satisfyhese multiple needs, a blend of several chemicals is usually sought.

formulation may then contain materials from different classes ofhemicals, such as polymers, surfactants, solvents, pigments andromas. These classes of chemicals are usually classified as follows5]:

Active ingredients: these chemicals are the most important ones inthe formulation, because they satisfy the main needs of the prod-uct, thus defining the function of the product itself. For example,the function of a sunscreen lotion is to protect the skin againstthe UV radiation.Solvent mixture: it is usually present in high concentration in theformulation and has the function of dissolving the active ingredi-ents and other chemicals in the formulation, ensuring the productto be a single liquid phase and to be properly delivered. The sol-vent mixture must evaporate after application.Additives: these chemicals are usually present in low concen-tration and they satisfy the secondary needs of the product,enhancing the end-use product properties. Examples are pig-ments and aromas, to enhance the sensorial properties of theformulation.

The presence of several classes of chemicals to be included inhe formulation design leads to the necessity of a step-by-stepierarchical design methodology, in order to avoid any combi-atorial explosion due to the high number of possible candidate

ormulations to be generated and screened, systematically and effi-iently, while at the same time excluding “blind” trial-and-errorolutions. Several methodologies and frameworks have been devel-ped, in order to address the need for the solution of a formulation

libria 362 (2014) 288– 299 289

design problem, with the aid of adequate property models andcomputer-aided tools. Raman and Maranas [7] addressed the prob-lem incorporating topological indices for correlating the necessaryphysico-chemical properties, while Chemmangattuvalappil et al.[8] applied combined property clustering and group-contributiontechniques. Teixeira et al. [9] directed their attention toward struc-tured products (more specifically microencapsulate perfumes fortextile application), while Charpentier [10] focused on the multi-scale problem generated by the introduction in the methodologyof economic, social and environmental constraints. Fig. 2 showsthe work-flow diagram of the computer-aided design/verificationstage, based on “define target – match target” paradigm as pre-sented by Conte et al. [5], highlighting input, output and tools usedfor each step. The methodology employs the “reverse design” tech-nique. The defined target properties of the product are then theknown variables and input for the property models. Appropriateproperty models are needed to estimate the target properties of thecandidates so that they are evaluated and then accepted or rejected.At the same time the mixture compositions that satisfy the productconstraints are determined, using suitable mixture property mod-els as well as phase stability algorithms. As shown in Fig. 2, if in anytask a solution is not found, it is possible to return to a previoustask to refine the problem definition.

1.2. Emulsion design

Formulations can also have other physical forms [11]: sus-pensions containing insoluble chemicals dispersed in the liquidwith the help of a dispersant; emulsions where solid constituentshave been emulsified through selected emulsifiers together withsolvents and additives; solid products such as pharmaceuticaltablets or soap bars. In chemical product design, Cussler and Mog-gridge [2] distinguish between commodities, chemical devices,molecular products and microstructured products, where the term“microstructure” refers to a chemical organization on the scale ofmicrometers, belonging to the colloidal domain and incorporatingpolymer solutions, foams, gels and emulsions. The performances ofsuch products are related not only to the presence of active ingre-dients and additives in the formulation, but also to the product’sstructural and material properties [12]. Among the microstructuredproducts, emulsified products are the most relevant, particu-larly in the food and cosmetic industries. Emulsions are definedas mixtures of two normally immiscible liquids, kinetically sta-bilized by emulsifiers (most often surface active agents, betterknown as surfactants) that lie on the interface between the twophases. Active ingredients and additives are usually dissolved inthe continuous and/or dispersed phases, according to the needsof the products. Bernardo and Saraiva [13] proposed a simul-taneous approach to address product and process design, withspecial attention to cosmetic emulsions, while Bagajewicz et al. [14]extended a generic approach [15] to consider price-competitivemarkets. Recently, a systematic procedure, which is applicable tothe design of emulsified formulated products, has been proposedby Mattei et al. [16] and it is further extended in this work (seeFig. 2).

For emulsion-based chemical product design, the solvent(s)design task (Task 3a) provides as output two non-miscible liquidphases and an additional task (Task 3b: Surfactant(s) design) is

fied product, rather than a homogeneous formulation. In particular,surfactants are key chemicals in most emulsified formulations anda wide range of peculiar properties need to be considered whendesigning or selecting chemicals such as surfactants [17].

290 M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299

ided

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Fig. 2. Work-flow diagram for the computer-a

. Surfactant selection and design

A surfactant design problem does not differ from any otherolecular and/or mixture/blend design problem and it can there-

ore be described through the following generic mathematicalepresentation [3]:

OBJ = max{CT y + f (x)} (1)

i(x) = 0 (2)

2(x) = 0 (3)

3(x) = 0 (4)

1 ≤ g1(x) ≤ u1 (5)

2 ≤ g2(x) ≤ u2 (6)

3 ≤ By + Cx ≤ u3 (7)

In the above equations, x represents the vector of continuousariables (such as mixture compositions), y the vector of binarynteger variables (such as compound identity), hi(x) is a set of equal-ty constraints (related to molecular structure generation, chemicaleasibility rules, etc.), gi(x) is a set of inequality constraints (relatedo environmental constraints and/or special property constraints)nd FOBJ is the objective function (to be maximized) on the basis ofhich the design choices are taken. In the specific case of surfac-

ant design, the numerical constraints presented in the equationsbove depend on the problem definition, such as on the need forhe surfactant system into the final product. In fact, the main appli-ation of this category of chemical is the emulsion stabilization,hile surfactants can also be found in commercial products to

orm microemulsions (particularly common in the pharmaceuti-al industry) or as active ingredients (as previously defined) inetergents. Other formulated products which are not in the emulsi-

ed form have key-properties strongly influenced by the surfactantontent: surfactants are usually added to formulations with theoal of preventing the re-deposition of dispersed chemicals (as inoothpastes) or of reducing the surface tension of the product (as innhanced oil recovery fluids). Each of the various needs generate aifferent product design problem and while it can be mathemati-allydescribed with the same set of equations presented above, the

stage of the formulation design methodology.

form of hi(x), gi(x) and FOBJ is different and the property modelsneeded are also different in each case.

2.1. Surfactant property models

Surfactants are characterized by an amphiphilic nature, whichmeans that a part of them is hydrophilic (water-like), while anotherpart is hydrophobic, or lipophilic (oil-like). In order to describetheir behavior in relation to two non-miscible phases and the rangeof temperatures at which they are active, some non-conventionalproperties, such as cloud point and critical micelle concentration,are needed. It should be noted, however, that model-based prod-uct design methodologies for emulsified products have not beenyet developed to a level as those for homogeneous formulations,and so predictive models for some of the needed properties aremissing. However, although many of these properties are strictlymixture properties since they refer to the mutual behavior of sur-factants in water mixtures, they can be modeled as pure componentproperties, depending only on the molecular structure of the sur-factant involved, since either the temperature or the compositionare usually kept constant. Hence these properties are modeled asprimary properties and can be, as a first approximation, estimatedusing group-contribution methods. The set of properties neededfor surfactant selection and design in an emulsion-based chemicalproduct design problem and the models used for their estimationare summarized in Table S1 [18,20–26] for primary properties andin Table S2 [5–6,27–29] for secondary properties, in Supplementarymaterial.

The models used to estimate the properties may be classified, foreach class of properties, into those that are predictive by nature andthose that are not. For example, estimating properties only frommolecular structural information involves predictive models, suchas the group-contribution methods, while estimating propertiesfrom compound specific coefficients involves the use of correla-tion which are not predictive by nature. In chemical product design,both types of models are needed. During the evaluation of candidateproducts, the models need to be predictive and computationallyfast and cheap, while during the verification of a small numberof candidates, correlative models may be used, if the correlationcoefficients are available. During the evaluation stage, the modelsneed to be, at least, qualitatively correct, while during the veri-fication stage, the models also need to be quantitatively correct.

Group-contribution methods are extensively considered here sincethey only need the molecular structure of the pure component andthey exhibit a good accuracy together with a wide range of applica-bility. In the Marrero and Gani group-contribution model [18], the

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M. Mattei et al. / Fluid Phas

roperty estimation is performed at three levels, and the propertyrediction model has the form reported below:

(X) =∑

i

NiCi +∑

j

MjDi +∑

k

OkEk (8)

here f(X) is a function of the property X and it may containdjustable model parameters depending on the property involved.n (8), Ci is the contribution of the first-order group of type-i thatccurs Ni times. Dj is the contribution of the second-order group ofype-j that occurs Mj times. Ek is the contribution of the third-orderroup of type-k that occurs Ok times. The definition and identifi-ation of the second-order groups has a theoretical basis, wherehe principle of conjugation has been employed [19]. The criteriased for the identification of third-order groups are analogous tohose used for second-order groups except for the types of com-ounds intended to be represented [18]. The role of the secondnd third-order groups is to provide more structural informationbout the portions of the molecular structure of a compound,here the description through the lower-order groups is insuffi-

ient. Through this addition, obstacles as partial description of theroximity effects and the lack of distinction between isomers cane overcome. The ultimate objective of this multilevel scheme iso enhance the accuracy, reliability and the range of applicationf the model [18]. For determination of contributions Ci, Dj andk, Marrero and Gani [18] suggested a multi-level approach. Asn alternative to the step-wise regression method, a simultaneousegression method can be applied, in which the regression is per-ormed by considering all the terms containing first, second andhird-order groups in a single regression step.

When addressing the reverse problem with group-contributionodels, computer-aided tools are necessary in order to reliably

nd quickly identify the chemical satisfying the target propertiesefined. It is also necessary, then, to provide connectivity rules sohat the groups collected can be combined in all the possible legalombinations. When applying to surfactants, this stage becomesery relevant since it is not only necessary to make sure to con-ect groups so that no free attachments are left in the moleculesenerated, but also that two distinct moieties (the hydrophilic andipophilic parts) are generated. This introduces a number of extra-onstraints to be satisfied in order to generate acceptable surfactantandidates.

.2. Mixture properties

The design and selection of surfactants for chemical productesign need pure component properties as well as mixture proper-ies. In fact, the interaction of the chosen surfactant with the otheronstituents of the product needs to be evaluated; in the specificase of emulsified products, particularly, the interaction of surfac-ants with both the water and the oil phases are of great importance.roper mixture property models which are able to relate the for-ulation properties to those of the single ingredients are needed.

comprehensive list of the models used is provided in Table S329–32] in Supplementary material.

As listed in Table S3, for some mixture properties linear mixingules can be used to predict the property. For the generic mixtureroperty �, the mixture property model based on a linear mixingule is described by the following equation:

=∑

i

xi�i (9)

here xi is the composition of compound i in the mixture, and �ihe pure component property.

Linear property models give good predictions of mixture prop-rties for chemical systems characterized by negligible excess

libria 362 (2014) 288– 299 291

properties of mixing. For those chemical systems having largeexcess properties of mixing, more detailed models are needed.Emulsions are systems far from the ideality, and therefore dedi-cated models for the estimation of some of the mixture properties(listed in the right column of Table S3) are necessary. Since thesemodels are quite complex, they can hardly be employed as modelsfor screening of alternative designs, but they should be rather usedin the verification stage, or on a second step, when the search spacefor the candidates has been reduced by the application of properconstraints on other mixture properties.

3. Modeling of the cloud point of surfactants with agroup-contribution method

One of the surfactant-related pure component propertiesconsidered necessary for the development of a model-basedmethodology for surfactant design is the cloud point, sometimescalled the cloud temperature. This property is specific of mix-tures between water and nonionic surfactants and it is definedas the temperature at which the mixture starts to phase sep-arate and two phases appear, thus becoming cloudy [33]. Thisphenomenon is of particular relevance for surfactants contain-ing polyoxyethylene chains (thus nonionic surfactants), exhibitingreverse solubility versus temperature behavior. This is clearly visi-ble in Fig. 3, where a standard phase behavior of an aqueous mixtureof a polyoxyethylene-based nonionic surfactant is reported.

In Fig. 3, different regions can be recognized: L1 identifies anaqueous surfactant solution where the surfactant is organized inordinary or reverse spherical micelles; W represents a very dilutedsurfactant solution (around the critical micelle concentration); Sindicates the presence of solid surfactant; L˛, H1 and V1, instead,are regions where the surfactant is aggregated in “uncommon”structures such as, respectively, lamellar, normal hexagonal and bi-continuous cubic structures. The last three regions are sometimesgrouped together as “viscous” meso-phases, since their rheologi-cal properties and behavior are substantially different from thoseof the ordinary and reverse spherical micellar solution. The linedetermining the W + L1 two-phase area is also known as the cloudpoint line. It is evident that, as a mixture property, the cloud pointdoes not depend only on the system considered, but it is influ-enced by the surfactant content. However, it is common practiceto define as cloud point the numerical value assumed by the cloudpoint curve at a surfactant weight percentage of 1%, measured byvisual observation method: the temperature at which the visiblesolubility changes to cloudy over a range of 1 ◦C or less is taken ascloud point [35].

Several efforts have been made in order to develop a modelto predict the cloud point, only on the basis of the molecularstructure of the surfactant involved and the quantitative structure-activity relationship (QSPR) models have been extensively applied[35–37]. The authors, on the other hand, are not aware of any group-contribution based method developed in relation to this property.These methods, however, apply very well to chemical process andproduct design because they can provide accurate predictions with-out being computationally demanding. Moreover, they can be usedin computer-aided molecular design because they employ the samebuilding blocks for molecular representation [24]. In this work wehave applied the Marrero and Gani GC-method [18] to this property.

3.1. Data-set

An original data-set consisting of 86 nonionic surfac-tants have been collected from different sources [35–38].The data set contains linear alkyl, branched alkyl, alkylphenyl ethoxylates, carbohydrate-derivative ethoxylates, alkyl

292 M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299

0

20

40

60

80

100

1009080706050403020100

Tem

pera

ture

[°C]

Surfactant weight %

W + L1

L1

H1

V1

Lα

S

F surfac ◦

w

S

pamT

sagstgstoatita

3

s8ostlea

nOoo

C

wi

order groups, instead. A step-by-step systematical data-error anal-ysis as in [24] has been performed, generating 5 new third ordergroups taking care of complex structures peculiar of the available

0

20

40

60

80

100

120

0 20 40 60 80 10 0 120

Calc

. Clo

ud P

oint

[°C

]

ig. 3. Phase diagram of an aqueous mixture of a polyoxyethylene-based nonionic

ater and dodecyl-esaethylene oxide.

ource: Data are taken from [34].

olyoxyethylene–polyoxypropylene copolymers and ethoxylatedmides. All experimental data are measured by visual observationethod in 1% aqueous surfactant solutions and are reported in

able 1, divided in different classes.Before applying the Marrero and Gani CG-method to the data-

et chosen for the parameter estimation step, it is necessary tonalyze the matrix of group occurrences to make sure that eachroups describes at least two of the surfactants presents in the data-et. A single occurrence would actually distort the performance ofhe model, leading to a perfect match for the compounds with thoseroups, providing uncertain extrapolation capabilities. Moreover,ome of the data of Table 1 are excluded from the data-set sinceheir experimental value for the cloud point is inconsistent withther values and are therefore identified as outliers. The outliersre identified as they are inconsistent with the assumption thathe cloud point of linear alkyl ethoxylates increases with increas-ng length of the ethoxylated chain and with decreasing length ofhe carbon chain. These surfactants whose CP values are excludedre highlighted in gray in Table 1.

.2. Model development

In order to determine the most suitable form of f(X) of the con-titutive equation of the Marrero and Gani method, as in Equation, it is necessary to observe the trend of the experimental dataf the property to be estimated as a function of the main repre-entative groups of the chemicals under investigation. Consideringhe largest family of nonionic surfactants: the linear alkyl ethoxy-ates, the trend of the cloud point as a function of the number ofthoxylate groups (CH2CH2O) in the hydrophilic chain is analyzed,s shown in Figs. S1 and S2 in the Supplementary material.

As seen in Fig. S1, the dependence of the cloud point on theumber of ethoxylate groups of linear alkyl ethoxylates is not linear.n the other hand, Fig. S2 shows that the dependence of the squaref the cloud point is linear. This justifies then the choice of the formf f(X), as in following Eq. (10):

P2 =∑

N C +∑

M D +∑

O E (10)

i

i i

j

j j

k

k k

here the cloud point is expressed in K. Given the equation above,n order to represent the remaining 72 compounds, 13 first order

tant over the temperature range 0–100 C Data are relative to the system between

groups and 1 second order group are needed, according to the orig-inal set of parameters by Marrero and Gani [18]. The results ofthe parameter estimation step performed through the step-wiseregression method are illustrated in the parity plot of Fig. 4.

The results in Fig. 4 indicate that the accuracy of the Marrero andGani GC-methods using only first and second order groups is notsatisfactory, as quantified in Table S4 in Supplementary material.

In particular, the maximum absolute errors (column AADmax)are too high for many categories of surfactants considered to con-sider the model reliable enough to be implemented in the productdesign methodology. It is therefore considered necessary to includenew third order groups in the set of parameters, in order to improvethe performances of the method, as described in [24,39], in partic-ular relation to those compounds for which the correlation indiceswere poor: branched alkyl ethoxylates and carbohydrate-derivateethoxylates primarily. According to Marrero and Gani [18], in fact,the second order groups are strictly defined and one cannot arbi-trarily add new second order groups as one can do with third

Expt. Cloud Point [°C]

Fig. 4. Parity plot relative to the correlation of 72 data-points regarding cloud point(in ◦C) of nonionic surfactants using the Marrero and Gani GC-method with onlyfirst and second order groups.

M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299 293

Table 1List of the original data-set of experimental cloud point (in ◦C) of nonionic surfactants (1% weight percentage). Distinction in classes is based only on the molecular structure,each of which is described. Compounds highlighted in gray have been excluded from the parameter regression step.

Code CPexp (◦C) Code CPexp (◦C) Code CPexp (◦C)

Linear alkyl ethoxylatesCnEm CnH2n+1O(C2H4O)mHC4E1 44.5 C9E6 75 C12E10 95.5C5E2 36 C10E4 19.7 C12E11 100.3C6E2 0 C10E5 41.6 C13E5 27C6E3 40.5 C10E6 60.3 C13E6 42C6E4 63.8 C10E7 75 C13E8 72.5C6E5 75 C10E8 84.5 C14E5 20C6E6 83 C10E10 95 C14E6 42.3C7E3 27.6 C11E4 10.5 C14E7 57.6C8E3 7 C11E5 37 C14E8 70.5C8E4 38.5 C11E6 57.5 C15E6 37.5C8E5 58.6 C11E8 82 C15E8 66C8E6 72.5 C12E4 6 C16E6 35.5C8E8 96 C12E5 28.9 C16E7 54C8E9 100 C12E6 51 C16E8 65C8E12 106 C12E7 64.7 C16E9 75C9E4 32 C12E8 77.9 C16E10 66C9E5 55 C12E9 87.8 C16E12 92

Branched alkyl ethoxylatesICnEm (C(n−2)/2Hn−1)2CHCH2O(C2H4O)mH (n = 6, 10)

(C(n−1)/2Hn)2CHO(C2H4O)mH (n = 13)TCnEm (C(n−1)/3H(2n+1)/3)3CO(C2H4O)mHIC6E6 78 TC10E7 22 TC13E9 34IC10E6 27 IC13E9 35 TC16E12 48

Phenyl alkyl ethoxylatesCnPEm CnH2n+1C6H4O(C2H4O)mHTC8PE9 64.3 C9PE8 34 C12PE9 33C8PE7 22 C9PE9 56 C12PE11 50C8PE9 54 C9PE10 75 C12PE15 90C8PE10 75 C9PE12 87C8PE13 89 C9PE13 89

Alkyl polyoxyethylene–polyoxypropylene copolymersCnEmPk CnH2n+1C6H4O(C2H4O)m(C3H6O)kHC12E4P5 22.1 C12E3P6 10.6 C12E5P4 29.8

Carbohydrate-derivate ethoxylatesCnCOOEmC CnH2n+1COO(C2H4O)mCH3

CnCOOEm CnH2n+1COO(C2H4O)mHC9COOE7C 44 C9COOE12 74 C11COOE8 53C9COOE10C 65 C11COOE6 54

Ethoxylated amidesCnGEm CnH2n+1NHCH2COO(C2H4O)mHCnAEm CnH2n+1NHCHCH3COO(C2H4O)mHCnSEm CnH2n+1NCH3CH2COO(C2H4O)mHC12GE2 78 C12GE4 75 C12SE3 44C12GE3 46 C12AE3 22.5

Alkyl branched ethoxylatesAGM-n(3) CnH2n+1CH(O(C2H4O)3H)2

S

difoG

3

teTirmt

AGM-7(3) 34 AGM-11(3)

ource: Data from [35–38].

ata-set. The introduction of structural parameters, in order to takento account the peculiar molecular assemblies of the nonionic sur-actants considered, as third order groups is necessary since theriginal sets of first and second order groups of the Marrero andani method cannot be arbitrarily modified.

.3. Results and discussion of results

Once the new set of groups has been identified, a final parame-er regression is performed, where all the group contributions arestimated simultaneously. The parameter values are reported inable 2, while the performances of the improved methods are given

n Fig. 5, as a parity plot and in Table S5 in Supplementary mate-ial as statistical indices. The simultaneous approach gives betterodel performances, compared to the step-wise approach, and it is

herefore preferred. Obviously, when this approach is chosen, the

30 AGM-13(3) 29

absolute values of the third order contribution might be compara-ble or even exceed those of the first and second order groups.

It has to be noted that the only second-order group used in thiswork, AROMRINGs1s4, has a zero contribution since the regres-sion performed with the simultaneous method did not find anynon-zero value for it. Fig. S3 in Supplementary material highlightsthe effect of the addition of dedicated third-order groups in thereduction of the absolute error of the model. In particular, it canbe noticed that only 2 calculated values clearly differ of more than10 K after the introduction of the third order groups, while in theregression with only first and second order groups 11 correlationsshow high absolute errors.

By comparing the results before and after the addition of the newdedicated third order groups, it can be seen that evident improve-ments have been achieved. In particular, after the addition of thethird order groups, linear and branched alkyl ethoxylates show

294 M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299

Table 2Marrero and Gani group definition and contributions after the regression based on 72 experimental data of cloud point.

1st order group (i) Ci (K2) 2nd order group (j) Dj (K2) 3rd order group (k) Ek (K2)

CH3 6.4351e+04 AROMRINGs1s4 0 (CH2)n (OCH2CH2)m (m = 3, n < 8) and (m = 4, 5, n > 8) −1.1108e+04CH2 −2.2149e+03 (CH2)n (OCH2CH2)m (n = 5) −6.7595e+03CH −6.5736e+04 (CH2)m CO (OCH2CH2) (m = 8) −2.1595e+04C −1.4320e+05 (CH2)m C6H4 (OCH2CH2) (m = 8) 6.0521e+03aCH −5.8171e+03 ((CH2)n)mCOC2H4 (n > 2, m > 1) −2.4357e+04aC-CH2 0OH −3.0249e+03CH2COO −2.7706e+03CH3O 3.7198e+04CH2O 8.9104e+03aC-O 0CH2NH 0OCH2CH2OH 3.3508e+04

Table 3Statistical indices of performances relative to the correlation of 72 data-points regarding the cloud point of nonionic surfactants using Marrero and Gani method before andafter the addition of 3rd order groups, compared with 3 different QSPR models [35–37].

Model Data-points for theregression

SD AAD AADmax

Marrero and Gani method without 3rd order groups 73 8.91 7.65 25.90Marrero and Gani method with 3rd order groups 73 5.65 4.62 15.83

iba[

gebflotw

4

etp

Fod

QSPR model [35] 81

QSPR model [36] 68

QSPR model [37] 78

mproved statistical indices, and in general the absolute errors haveeen strongly reduced. These results represent an improvementlso if compared with those obtained with different QSPR methods35–37], as shown in Table 3.

An example of the application of the model developed here isiven in Table A1 in the Appendix. The availability of more reliablexperimental data regarding cloud points or nonionic surfactantselonging to other families (such as alkanediols, ethers, esters anduorinated linear ethoxylates) will broaden the application rangef the model. However, it can already be safely applied in the surfac-ant design methodology considering the limited maximum error,ith basic limitation the molecular structures available.

. Case studies

The framework presented in Fig. 2, together with the prop-rty models reported in Sections 2.1 and 2.2 have been appliedo two different surfactant design and selection problems, as aart of emulsion-based chemical product design. Also the newly

0

20

40

60

80

100

120

120100806040200

Calc

. Clo

ud P

oint

[°C]

Expt. Cloud Point [°C]

ig. 5. Parity plot relative to the correlation of 72 data-points regarding cloud pointf nonionic surfactants using the Marrero and Gani GC-method after the addition ofedicated third order groups.

9.31 7.09 50.25.89 4.69 17.987.46 3.13 52.8

developed group-contribution models for the prediction of thecloud point and the critical micelle concentration [24] have beenapplied as a comprehensive set of group-contribution modelsregarding nonionic surfactants, forming a reliable and consistenttool for the surfactant design. Table 4 lists the properties neededby the methodology and the models used in this work.

All the property models reported in Table 4 have been testedagainst experimental values in order to verify their reliability. Thevalues relative to the coefficients of determination of the differ-ent models are high, except for the Hansen solubility parametersand the toxicity parameter. However, there are no other predictivemodels with comparable correlation performances, therefore theyare applied for screening of alternative designs, and more accurateand complex models are employed only a few alternatives are avail-able, to verify the reliability of the predictions. Once these modelshave been verified, they have been implemented in the method-ology for chemical product design, as described in the previouschapters.

The first case study presented is about the design of an emul-sified UV sunscreen, where the surfactant system is necessary asemulsifier that is to stabilize the formulation. The second casestudy, on the other hand, is about the design of an emulsified hand-wash, where the surfactant system acts simultaneously as activeingredient, as a main function, and as emulsifier. This contribution,however, is not intended to give more than an overall picture of anemulsion-based product design problem, while it focuses on thesurfactant design and selection as a part of the above-mentionedframework. A comprehensive description of the whole step-by-step methodology is given by Mattei et al. [16].

4.1. Surfactant design as emulsifier – emulsified UV sunscreen

As seen from Fig. 2, in the step-by-step methodology for emul-sified product design, the surfactant system is designed after theactive ingredients and both the continuous and the dispersed

phases have been selected. Table S6 in Supplementary material pro-vides the output of steps 2 and 3a, that is the chemicals chosen asactive ingredients and continuous and dispersed solvents, togetherwith consumer assessments and the target properties responsible

M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299 295

Table 4List of the surfactant property applied in the case studies reported together with the model used and statistical indices.

Surfactant property Model used Coefficient of determination

Cloud point Group-contribution method [this work] R2 > 0.94Critical micelle concentration Group-contribution method [previously developed [24]] R2 > 0.99Hansen solubility parameters Group-contribution method [21] R2 > 0.78Hydrophilic–lipophilic balance Definition [24] –a

Hydrophilic–lipophilic deviation Definition [32] –a

Krafft temperature QSPR model [26] R2 > 0.94Open cup flash point Group-contribution method [29] R2 > 0.96

ethod

devia

fo

tsi

abTTttcmiOff

4h

nascrdmmdtcaehptet

TT

Surface tension reduction QSPR model [26]

Toxicity parameter Group-contribution m

a Statistical indices for hydrophilic–lipophilic balance and hydrophilic–lipophilic

or their design (solvents) and selection (active ingredients), as theutput of the step 1 of the methodology.

The surfactant to be designed, as previously mentioned, needso act as emulsifier, ensuring the formation of an oil-in-water emul-ion, stable under the range of temperature in which the products supposed to be used.

With relation to the set of Eqs. (1)–(7), then, proper boundariesre set so that the designed surfactant is safe, non-toxic, not affectedy the presence electrolytes and able to generate a stable emulsion.hese targets are translated into target properties, as in Table 5.hrough a computer-aided molecular design (CAMD) technique, allhe above-mentioned constraints have been applied, thus reducinghe search space to a limited number of candidate surfactants. Thehoice of the most advantageous between the candidates is takeninimizing the cost connected to its use, which means the min-

mum cost at which the critical micelle concentration is reached.ctyl esaethylene oxide has been designed as optimum surfactant

or this purpose and Table 5 provides the results of the models usedor the design.

.2. Surfactant design/selection as active ingredients – emulsifiedand wash

There are some classes of products which contain surfactantsot only to keep the formulation in the emulsified form, but also toct as active ingredients. These are detergents in general, whereurfactants are responsible for the wetting of the surface to beleaned, for the dissolution of the dirt and for not allowing thee-deposition of the dirt itself. When these products need to beesigned, then, surfactants are included at the second step of theethodology of Fig. 2, before any other ingredients of the for-ulation. The consumer assessments to be satisfied might then

iffer from those listed in the previous paragraph, as different arehe target properties and the relative boundaries set on them. Aomprehensive list of the consumer assessments for surfactants asctive ingredients, together with the property constraints and mod-ls used is reported in Table 6. Compared to the previous case-study,ere the surfactant system needs to satisfy the main needs of the

roduct. Between them, high foam-ability and non-irritability ofhe skin. The foam-ability has been qualitative modeled by Pandeyt al. [40] as a function of the surface tension of the system andhe critical micelle concentration of the surfactant used. More

able 5arget properties, relative physico-chemical properties and models needed, constraints a

Target properties Properties considered and models use

Oil-in-water emulsion desired Hydrophilic–lipophilic balance [25]

Thermal stability Cloud point

Safety Flash point [29]

Non-toxicity Toxicity parameter [22]

Not influences by electrolytes Nonionic surfactants preferred

Stability as an emulsion Hydrophilic–lipophilic deviation [32]– Critical micelle concentration [24]

R2 > 0.99[22] R2 > 0.77

tion cannot be provided since relative experimental values cannot be found.

precisely, the lower the surface tension, the higher the foam-ability,and the lower the critical micelle concentration, the higher thefoam-ability. The non-irritability of the skin, on the other hand, isestimated through Hansen solubility parameters [31]. If the Hansensolubility parameters of the designed ingredient are compatiblewith the parameters characterizing the proteins of the skin, thenthat ingredient is likely to partly dissolve the protein layer, con-sequently irritating the skin. Therefore, proper boundaries on theHansen solubility parameters are set in order to qualitatively satisfythe constraint of non-irritability. Given the above, the foam-abilityis usually related to the presence of ionic surfactants (since ionicsurfactants are characterized by low surface tension as well as lowcritical micelle concentration), while the skin-care is usually con-nected to the use of nonionic surfactants, due to the values of theirsolubility parameters, usually far from those of the proteins of theskin. Therefore, a mixture of ionic and nonionic surfactants is usu-ally chosen, the stability of which needs to be checked once theother ingredients of the formulation have been chosen. Numeri-cal constraints on the properties, as reported in Table 6, are appliedand through a CAMD techniques a restrict number of candidates aregenerated. Between these, octyl esaethylene oxide is designed asthe most advantageous nonionic surfactant to be used as a nonionicsurfactant, while Sodium Laureth Sulfate is selected with rule-based selection criteria as the most advantageous ionic surfactantto be chosen. Calculated values for the key-properties consideredin the case-study are reported in Table 6.

The stability of the two surfactants in the formulation, though,needs to be ensured after all the other ingredients have beenselected and the most advantageous overall composition of theproduct has been chosen. Table S7 in Supplementary material liststhe ingredients chosen, together with the relative consumer assess-ments and target properties to be satisfied.

The overall composition of the formulation is needed in order toqualitatively estimate the stability of the emulsion by calculatingthe HLD-value of each of the two active-ingredients selected at thesecond step of the procedure, that are expected to form a stableemulsion.

5. Future perspectives: surfactant-related phase behaviors

The description of binary (water-surfactant and oil-surfactant)and ternary (water–oil-surfactant) phase behaviors with a

nd modeled value for surfactant design for an emulsified UV sunscreen.

d Constraints Modeled value

HLB > 12 13.4CP > 55 ◦C 73 ◦CTf > 55 ◦C 341 ◦C−log(LC50) > 3.16 mol/m3 3.97 mol/m3

– – HLD /= 0 –0.7

– 0.009 mol/L

296 M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299

Table 6Target properties, relative physico-chemical properties and models needed, constraints and modeled value for surfactant design for an emulsified hand-wash.

Target properties Properties considered and models used Constraints Calculatedvalue nonionicsurfactant

Calculatedvalue ionicsurfactant

Foam-abilitySurface tension [26] � < 33 mN/m 31 mN/m 27.8 mN/mCritical micelle concentration [24] CMC < 0.01 mol/L 0.009 mol/L 0.008 mol/L

Non-irritability Solubility parameters [21]ıD /= 22.4 ± 1.5 Mpa0.5 16.9 –a

ıP /= 9.8 ± 1.5 Mpa0.5 9.2 –a

ıH /= 11.9 ± 1.5 Mpa0.5 13.7 –a

Cleaning performancesSurface tension [26] � < 29.3 mN/m 31 mN/m 27.8 mN/mHydrophilic–lipophilic balance [25] HLB > 10 13.4 40

Safety Flash point [29] Tf > 55 ◦C 341 ◦C 94 ◦C

Non-toxicity Toxicity parameter [2] −log(LC50) > 3.16 mol/m3 3.97 mol/m3 LD50 = 5 g/kgb

Thermal stabilityCloud point [this work] CP > 55 ◦C 73 ◦C –Krafft temperature [26] TK < 20 ◦C – 16 ◦C

Stability as an emulsion Hydrophilic–lipophilic deviation [32] HLD /= 0 −0.9 −1.5

a Hansen solubility parameters are not available, but the surfactant (Sodium Laureth Sulfate) is known to be soluble in water, so it is unlikely to dissolve the proteins oft

rovide

thtbfctpitpf

5

s(tpp

pddtlSeetspamootder

he skin, given their Hansen solubility parameters.b LC50 is not available, so a value for LD50, another toxicity parameter, has been p

hermodynamic model, instead of the adoption of simplifiedeuristic, is considered as a major progress to be implemented inhe product design methodology. This way, it is possible to identifyoundaries in terms of temperature and, especially, compositionor a surfactant to generate a stable emulsified formulation. Thishapter will highlight some perspectives for future development ofhe above-mentioned analysis and a few preliminary results; to thisoint, however, the application of surfactant-related phase behav-

ors is limited to the availability of experimental data relative tohe systems of interest. Therefore this approach can be applied forroduct analysis as well as for verification of the design obtainedrom the methodology.

.1. Binary systems: water-surfactant and oil-surfactant

Water-surfactant phase diagrams are fundamental when theurfactant is expected to be mainly dissolved in the water-phasehigh values of the hydrophilic–lipophilic balance), which leads tohe formulation of oil-in-water emulsions. On the other hand, thehase diagrams between oil and surfactants are to be consideredrimarily when a water-in-oil emulsion is desired.

As an example of water-surfactant phase behaviors, Fig. 3resents the phase diagram of the system between water andodecyl-esaethylene oxide. For emulsion-based chemical productesign, therefore, it is relevant to define the boundaries in terms ofemperature and concentration, so that the designed formulationies in the area defined as L1, where the models presented in Tables1–S3 in Supplementary material can be safely applied. Mitchellt al. [34] and Sjöblom et al. [41] provide a satisfactory amount ofxperimental data as well as theoretical explanations for the forma-ion of the different meso-phases, relatively to aqueous surfactantolutions. The methods proposed, however, cannot be used for therediction of these phase boundaries and therefore they cannot bepplied in the framework for chemical product design. Approxi-ate predictions of these phase behaviors, based on the analysis

f several phase diagrams in parallel with the molecular structuref the species involved, are however expected to be possible by

he authors and they are considered to be a potential importantevelopment on the way for a fully model-based methodology formulsion-based chemical product design. The adoption of such cor-elations can lead to the calculation of simplified water-surfactant

d instead.

phase behaviors as described in Fig. S4 in Supplementary material,related to the system reported in Fig. 3.

When a water-in-oil emulsion is wanted, on the other hand,the phase behavior of surfactant mixtures with oil is rele-vant. Fig. S5 [42] in Supplementary material shows an examplethe phase behaviors between oils and surfactants, in terms ofliquid-liquid miscibility boundaries. The reference surfactant ishexyl-pentaethylene oxide, while four different n-alkanes are con-sidered as the oil-phase.

In Fig. S5, only the miscibility curve as function of surfactantconcentration and temperature is reported, while the formationof micellar solutions of standard appearance or of viscous meso-phases with different self-assemblies of the surfactant in absence ofwater is debatable [43]. For use in emulsion-based chemical prod-uct design, it is necessary that the designed formulation lies abovethe line of the miscibility gap. It is easy to determine in Fig. S5 atrend of the miscibility curves as a function of the number of carbonatoms of the n-alkane considered. This leads to the considerationthat a correlation based on the molecular structure of the chemi-cals involved in the phase equilibrium can approximately describethese curves and therefore a correlative model may be developedand applied for use in emulsion-based chemical product design.

In relation to both water-surfactant and oil-surfactant phasebehaviors, however, experimental measurements are needed inorder to define numerical boundaries on the composition of thedesired formulation, since model-based generation of data is notconsidered yet reliable.

5.2. Ternary systems: water–oil-surfactant

The understanding of the behavior of ternary water-oil-surfactant systems is also considered to be crucial, in order todetermine temperature and composition boundaries for a stableemulsion. These type of phase envelops can be represented inseveral ways, since many variables are involved; in relation toemulsion-based chemical product design, the most useful alterna-tive is represented by the use of the so-called Kahlweit’s fish phasediagram [42]. Here, ternary water–oil-surfactant data are drawn in

an X–Y diagram, where the surfactant content (usually in weightpercentage) is in the X-axis, while the temperature is on the Y-axis.These diagrams represent a valid tool for emulsion-based prod-uct design since different types of products can be recognized and

M. Mattei et al. / Fluid Phase Equi

290

300

310

320

330

340

350

360

0% 20% 40% 60% 80%

Tem

pera

ture

[K]

Surfactant weight %

3φ

2φ - O/W

1φ

2φ - W/O

Fig. 6. Calculated fish-diagram of the system water-tetradecane-2-butoxyethanol;w

S

tfiAw

wcWfdededilmripdesdppbWtoosattavaavstdtm

Appendix.

Appendix B. Supplementary data

ater–oil ratio: 7.03.

ource: Data taken from [44].

he possibilities for the formation of each of them are easily identi-ed, given the temperature and the composition of the formulation.n example of such as “fish-type phase diagram” for the systemater-tetradecane-2-butoxyethanol is shown in Fig. 6.

In Fig. 6 different areas are can be identified: the region definedith the symbol 1ϕ represents the area where a micro-emulsion

an be formed, the region defined with the symbol 2ϕ (both 2ϕ –/O and 2ϕ – O/W) is the area where an emulsified product can be

ormulated, while the region identified by the symbol 3ϕ is a hybridomain, where an emulsion and a microemulsion may coexist. Themulsion domain (2ϕ) consists of two areas: one above the hybridomain, described by the symbol 2ϕ – W/O, where a water-in-oilmulsion can be formed and another below the hybrid domain,escribed by the symbol 2ϕ – O/W, where an oil-in-water emulsion

s favored instead. Consequently, the region of the emulsion domainocated at intermediate temperatures between the two above-

entioned areas represents an unstable region where it is notecommended to design an emulsified product, since its life times expected to be limited. As a part of an emulsion-based chemicalroduct design procedure, then, it is necessary to make sure that theesigned formulation lies in the 2ϕ – W/O area when a water-in-oilmulsion is desired and in the 2ϕ – O/W if an oil-in-water emul-ion is wanted. This type of diagrams can be used both during theesign of the surfactant and during the verification of the designedroduct. The authors are not aware of any reliable model for therediction of such phase equilibria, and therefore this analysis cane performed up to now only when experimental data are available.hen predictions are necessary, the hydrophilic–lipophilic devia-

ion (HLD) approach [32] is applied instead. This method consistsf the application of an experimental-based correlation for eachf the surfactant in the formulation, considering several variablesuch as the presence of electrolytes, the nature of the oil as wells of the water phases, the temperature, and the molecular struc-ures of the surfactant. If the calculated HLD-value is zero, thenhe formulation is located in the 3ϕ domain of Fig. 6 and thereforen unstable system is expected. On the other hand, if a positivealue is obtained, then a water-in-oil emulsion is favored, while if

surfactant is characterized by a negative value of its HLD, thenn oil-in-water emulsion may be formed. The higher the absolutealue of the HLD of the surfactant is, the more stable the emul-ion formed is expected to be, since it is located further away fromhe unstable region identified by the hybrid domain. This method

oes not have the thermodynamic basis of the representation of theernary phase diagram, but it can be used as a qualitative predictive

odel when the needed experimental data are not available.

libria 362 (2014) 288– 299 297

6. Conclusion

A comprehensive framework for surfactant design and selectionfor emulsion-based chemical product design has been presented.The necessary properties for the whole procedure have beenlisted and the need of predictive models for a reliable solutionof the reverse problem has been highlighted. Moreover, a group-contribution model, based on the Marrero and Gani method hasbeen developed for the correlation and further prediction of thecloud point of nonionic surfactants. The addition of new dedicatedthird order groups, in order to take into account the peculiar struc-tures of the surfactants considered, has been necessary in order toachieve acceptable performances. Compared to the existing QSPRmodels, the group-contribution model here developed performedbetter given the standard deviation and the average absolute devi-ation as statistical indices and the number of different families ofsurfactant considered, as an indicator of the range of application ofthe model.

The application of the methodology has been then illustratedusing two conceptual case-studies. In the first, the surfactant sys-tem to be designed was needed in order to keep the formulationin a stable emulsified form. So, the surfactant needs to be chosenonce the active ingredients and solvents were known. Boundarieswere set on the basis of effectiveness (related to the ingredientspreviously designed and/or selected), safety and toxicity, while thefinal decision has been performed by minimizing the cost of theingredient. In the second case-study, instead, the surfactant sys-tem was supposed to act as emulsifier and active ingredient at thesame time. The surfactant is then designed before any other ingre-dient of the formulation, and in order to satisfy the main needsof the product, a surfactant mixture of ionic and nonionic surfac-tants was needed. The stability of the surfactants in relation to theother chemicals present in the designed product is ensured after theoverall composition has been selected. Further perspective for theapplication of rigorous thermodynamic descriptions of the phasebehavior of surfactant-related system, both in binary mixture withwater and oil, and in ternary mixture with both an oil- and a water-phase, is considered a relevant improvement that can help leavingheuristics and rules of thumb which are usually applied to esti-mate the phase stability of emulsified products. The applicationrange of the methodology is wide, in the sense that other simi-lar products can be designed, once the needs-property relationsare established. Further works will focus on the development ofgroup-contribution methods for the pure component properties ofionic surfactants (adequate new first order groups are needed) andthe development of predictive, reliable models able to efficientlydescribe both the binary and the ternary phase diagrams involvingsurfactants.

Acknowledgments

Financial support from the Technical University of Denmark isgreatly acknowledged.

Advice given by Professor Michael Hill, from the Departmentof Chemical Engineering of Columbia University, New York, in thedevelopment of one of the case studies is greatly appreciated.

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.fluid.2013.10.030.

298 M. Mattei et al. / Fluid Phase Equilibria 362 (2014) 288– 299

Table A1Calculation of the cloud point of Nonyl phenyl octaethylene oxide after the introduction of new third order groups. Calculated value is compared with experimental valueand with the calculations with the GC-model without 3rd order groups and three QSPR models.

Nonyl phenyl octaethylene glycol Molecular structure

Molecular formula: C33H60O10

HOO

OO

OO

OO

O

O

First order groups Occurrences Group contribution (K2)

CH3 1 6.4351e+04CH2 16 −2.2149e+03aCH 4 −5.8171e+03aC-CH2 1 0CH2O 7 8.9104e+03aC-O 1 0OCH2CHOH 1 3.3508e+04

Second order groups Occurrences Group contribution

AROMRINGs1s4 1 0

Third order groups Occurrences Group contribution

(CH2)m C6H4 (OCH2CH2) (m = 8) 1 6.0521e+03

(CP2)calc =∑

NiCi +∑

MjDj +∑

OkEk = 107576.3 K2, (CP)calc = 327.99 K GC-model without 3rd order groups: (CP)calc = 322.04 K, QSPR models from Table 8:

(

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

i j k

CP)calc = 320.65 K [35], 326.33 K [36], 328.45 K [37], (CP)exp = 329.15 K.

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