Brazilian Journal
of Chemical ISSN 0104-6632 Printed in Brazil Engineering www.abeq.org.br/bjche
Vol. 33, No. 03, pp. 515 - 523, July - September, 2016 dx.doi.org/10.1590/0104-6632.20160333s20150129
MOLECULAR THERMODYNAMICS OF MICELLIZATION: MICELLE SIZE DISTRIBUTIONS AND GEOMETRY TRANSITIONS
M. S. Santos1, F. W. Tavares1,2* and E. C. Biscaia Jr1
1Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, C.P. 68502, CEP: 21941-972, Rio de Janeiro, RJ - Brazil. E-mail: [email protected] 2Escola de Química, Universidade Federal do Rio de Janeiro, CEP: 21941-909, Rio de Janeiro - RJ, Brazil.
(Submitted: February 26, 2015 ; Revised: June 10, 2015 ; Accepted: June 18, 2015)
Abstract - Surfactants are amphiphilic molecules that can spontaneously self-assemble in solution, forming structures known as micelles. Variations in temperature, pH, and electrolyte concentration imply changes in the interactions between surfactants and micelle stability conditions, including micelle size distribution and micelle shape. Here, molecular thermodynamics is used to describe and predict conditions of micelle formation in surfactant solutions by directly calculating the minimum Gibbs free energy of the system, corresponding to the most stable condition of the surfactant solution. In order to find it, the proposed methodology takes into account the micelle size distribution and two possible geometries (spherical and spherocylindrical). We propose a numerical optimization methodology where the minimum free energy can be reached faster and in a more reliable way. The proposed models predict the critical micelle concentration well when compared to experimental data, and also predict the effect of salt on micelle geometry transitions. Keywords: Micellization; Molecular thermodynamics.
INTRODUCTION separations in the petrochemical industry. Intensifica- tion of studies in this field is reinforced by the acces- Self-assembly is a broad term applied to spontane- sibility of critical micelle concentration data and its ous organization of substances under appropriate con- great importance for the physicochemical behavior of ditions and proportions. It is a reversible process and surfactant solutions. Furthermore, several thermody- represents a condition of thermodynamic equilibrium, namic and transport properties of surfactant solutions which is relevant to a large variety of phenomena and are affected by the size and shape of micelles (Iyer and systems, such as crystal formation, colloidal systems, Blankschtein, 2012). For example, the solubility of lipid bilayers, among others (Whitesides and Boncheva, micelles increase with their size (Rusanov, 2014), and 2002). When surfactants self-assemble, the structures micelle shape transitions significantly affect the vis- formed are called micelles. Micellization phenomena cosity of the solution (Kamranfar and Jamialahmadi, have been of great interest to both the academic com- 2014). All these facts contribute to the development munity and industry. It is involved in several applica- of models that can predict the behavior of surfactant tions and processes. We can mention its application in solutions and also can be used as a tool for designing personal care products, paints, processed food, and new surfactants.
*To whom correspondence should be addressed This is an extended version of the work presented at the 20th Brazilian Congress of Chemical Engineering, COBEQ-2014, Florianópolis, Brazil.
516 M. S. Santos, F. W. Tavares and E. C. Biscaia Jr
The molecular thermodynamics approach to de- the more stable geometry (spherical and spherocylin- scribe the micellization phenomena was first pro- drical) of micelles. Here, the stable state for a given posed by Tanford (Tanford, 1974). Tanford analyzed set of temperature (T), pressure (P) and composition the adequacy of the Gibbs thermodynamic equilib- (N), obtained by the minimization of total Gibbs free rium to describe the self-assembly of amphiphilic energy of the system, gives us information about the molecules. In Tanford’s work, the equilibrium state is most stable micelle geometry, micelles sizes and the calculated by imposing the necessary condition of a critical micelle concentration (CMC) of the solution. stoichiometric linear relation of chemical potential of all species. That implies that the sum of the Gibbs free energies of reactants (amphiphilic molecules) must be MODEL the equal to the sum of the Gibbs free energies of products (micelles). The model used to calculate the total Gibbs free Molecular thermodynamic models aim to be com- energy of the system is similar to the one proposed by pletely predictive. They describe the micellization Nagarajan and Ruckenstein (1991) and Moreira and phenomena only based on information about the Firoozabadi (2009). Given a global specification of T molecular characteristics of substances and conditions (temperature), P (pressure), NSA (total number of sur- of the medium, such as, for example, temperature, sur- factant molecules in the solution) and N (number of factant concentration, and ionic strength (Goldsipe w water molecules), the Gibbs free energy is calculated and Blankschtein, 2007). Nagarajan and collaborators as a sum of two contributions: free energy of for- made important contributions to the development of mation and free energy of mixture. molecular thermodynamic models used to describe the self-assembly of surfactant solutions. In their models (Nagarajan and Ruckenstein, 1991; Naga- oo o GNww N sAA1 Ng g g rajan, 1993; Nagarajan, 2002) they analyzed small g2 and monodisperse micelles and also the transition (1) between two micelle geometries. However, in those kT Nwwln X N14 A ln X N gg ln X works, only the necessary condition for equilibrium is considered. The Nagarajan model was improved by g2 Moreira and Firoozabadi (2009) when they started considering the minimum of Gibbs free energy as a o o o where w , 1A and g are the standard chemical condition for the thermodynamic equilibrium of the potentials of water, free surfactant and micelle with solution. However, Moreira and Firoozabadi (2009) aggregation number g, N is the number of micelles, k used the maximum term approach, which assumes g is the Boltzmann constant, and X , X and X are the that the micelle size distribution can be represented by w 1A g mole fractions of water, free surfactant and micelles. one characteristic micelle size and a single number of It is possible to reorganize the previous expression micelles formed. Therefore, Moreira and Firoozabadi by separating the terms that depend only on fixed (2009) did not take into account the fact that micelles variables: T, P, N , and N , and dividing the expres- can be formed with a large distribution of sizes. They SA w sion by kT: also assumed that only small micelles could be formed, with either a spherical or globular shape. In o G their subsequent works (Moreira and Firoozabadi, Ngg Nln X 2010; Moreira and Firoozabadi, 2012; Lukanov and kT g kT ww g2 Firoozabadi, 2014) those aspects were kept the same. (2) Therefore, to the best of our knowledge, the present NXln NX ln work is the only one that considers the minimum 11A Agg Gibbs free energy as the most stable state together g2 with the micelles’ size distributions and also analyzes the transition between small (spherical) and rod-like Equation (2) represents the expression to be mini- (spherocylindrical) micelles as a function of tempera- mized to predict the most stable state of the system. o ture and salt concentration. g In this work we present an approach where the mini- The free energy of micellization is defined as kT mum of total Gibbs free energy is obtained for aque- ous surfactant solutions (with and without electro- a sum of different contributions. These contributions lytes) considering the micelle size distributions and are detailed elsewhere by Nagarajan and Ruckenstein
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Molecular Thermodynamics of Micellization: Micelle Size Distributions and Geometry Transitions 517
(1991) and Moreira and Firoozabadi (2009). deterministic method Sequential Quadratic Program- When the micelles size distribution is narrow ming (SQP). After obtaining the concentration of enough, it is common to use the Maximum Term ap- surfactant added (NSA), we can calculate the critical proximation. when the Maximum Term approxi- micelle concentration using the number of water mation is used, Equation (2) becomes: molecules assumed in the simulation (NW).
o G Ngg Nln X kTg kT ww (3) NXNX11A lnAg ln g
METHODOLOGY
Critical Micelle Concentration (CMC)
Experimentally, the critical micelle concentration (CMC) is defined as the concentration of surfactant where a sharp change in any property of the solution is observed. This information is available in the literature for a wide variety of surfactants. Therefore we decided to validate our model by comparing our Figure 1: Determination of the CMC. Each point repre- predictive calculations with experimental CMC data. sents one minimization process of the Gibbs free en- To obtain the CMC, we perform different simulations ergy and the continuous line is the adjusted regulation of a solution containing an increasing amount of sur- function. CMC is obtained by parameter fitting of the factant. Then, for each solution with a different amount regularization function represented by Equation (4). of surfactant added we performed the minimization of the Gibbs free energy. This procedure permits one to Micelle Size Distribution relate the total number of surfactant molecules added (NSA) with the number of free surfactant molecules in To develop the methodology for the micelle size the solution (N1A), as is graphically presented in distributions, first we verify what kind of behavior Figure 1. The CMC is then defined as the concentra- this distribution can have. For that, we consider a set tion of surfactant added (NSA) where an inflexion of of optimization variables, where it is assumed that we the curve is observed. To obtain this point automati- could obtain micelles with an aggregation number cally, a regularization function is used which relates from 2 to 100. To construct the next figure, we define N1A as a function of NSA, according to Equation (4). as optimization variables the vector of the number of micelles formed for each aggregation number g. Thus, for this specific case we did not pre-set any shape for NN1aSA aNbcN SA SASA N (4) the micelle size distribution. Figure 2 shows the tendency obtained for the micelle size distribution for NNSA SA an aqueous solution of sodium dodecyl sulfate (SDS) 1 at a concentration ten times higher than its critical 2 NN 2 micelle concentration. It is interesting to observe that SA SA the problem converges to a micelle size distribution with a behavior similar to a Gaussian distribution. * where NSA , determined by non-linear regression, From now on in this work, a Gaussian shape is as- corresponds to the critical number of surfactant sumed for all the micelle size distributions. Therefore, molecules added to the solution under the conditions it is possible to define the total number of micelles T and P, and ξ is the regularization parameter, where formed, the mean (or expectation) aggregation num- 10 . The parameters a, b, and c are obtained ber of the distribution and its standard deviation as from the fitting of the data to the regularization important variables for calculating the minimum free function. energy of the system. To perform the parameter fitting we used the To validate our methodology for the micelle size
Brazilian Journal of Chemical Engineering Vol. 33, No. 03, pp. 515 - 523, July - September, 2016
518 M. S. Santos, F. W. Tavares and E. C. Biscaia Jr distribution, first we compared it with the maximum geometric transition is considered, we define as opti- term approach, both considering the minimum Gibbs mization variables the number of micelles formed free energy to define the most stable state. Because we (Ng), the number of surfactant molecules in the cy- calculate the critical micelle concentration, only small lindrical part of the micelle (gcyl), the radius of the micelles are considered (spherical and globular shapes spherical ends of the micelles (Rs), and the length of only). The geometrical relations used for this section the cylindrical part (Lc). The number of surfactant are similar to those presented by Nagarajan and Ruck- molecules in the spherical ends of the micelle is then enstein (1991). Using our methodology, three optimi- obtained from the geometric relations presented in zation variables, as mentioned before, are used to cal- Table 1, which assumes the form of: culate each minimum of Gibbs energy. Using the Maxi- mum Term Approximation, we have two optimization 18 32 2 variables: the number of micelles formed, N , and the gsph RHRH s 3 s (5) g 33 specific aggregation number, g. s
To reduce the optimization search region, we defined limits for the optimization variables. The maximum value of the radius, Rs, is assumed to be equal to the extended length of the surfactant tail, en- suring that there is no empty space inside the micelle. Besides that, Rs must be equal to or greater than the radius of the cylindrical part of the micelle to ensure that the amount of surfactant molecules in the spheri- cal ends is positive. Figure 3 presents a surface of the Gibbs energy, to be minimized, fixing values for Lc and Rs. We can observe that the function has a smooth behavior and a well-defined global minimum.
Numerical Method
Figure 2: Micelle size distribution for a solution of As presented in the previous section, calculating sodium dodecyl sulfate (SDS) at a concentration ten the critical micelle concentration involves repeating times its critical micelle concentration. Xg is the mole the Gibbs energy minimization several times for dif- fraction of micelles with the aggregation number g. T ferent surfactant concentrations. To reduce the com- = 25 ºC, P = 1 atm, and N w = 3400000. The micelle putational time to perform these calculations, we pro- size distribution converges to a Gaussian-like distribu- pose the following optimization procedure. First, we tion after the optimization procedure. define an amount of water to be considered and the temperature of the system. After that, we start our simu- Geometry Optimization lations with a small amount of surfactant (to guarantee that we are below the critical micelle concentration). As in all optimization problems, we must pay at- For this first simulation we use the Particle Swarm tention to the correct definition of the optimization Optimization method (PSO). The result is then used as variables to be used in the numerical algorithm. The an initial guess for the Sequential Quadratic Program- incorrect selection of variables, if they do not satisfy ming (SQP) deterministic method. Then, we make a some constraints or correspond to an over specifica- small increment in the amount of surfactant and per- tion, may lead to erroneous results and misinterpreta- form the optimization with the deterministic method. tions. It is even more important when using nondeter- The initial guess now, in this case, is the optimum ministic algorithms because the analyticity of the value of the optimization at the lower surfactant con- functions and equality constraints are not taken into centration. With this methodology, it is only necessary account. This can result in objective functions with to use the nondeterministic method once, which several local minima. largely reduces the computational effort. All simula- Analyzing the geometric relations presented in tions were performed with the software Matlab® Table 1, we can observe that, if the length of the cylin- version R2008a. More details about the PSO method drical part of the micelle (Lc) is zero, the geometry can be found elsewhere (Kennedy and Eberhart, 1995) reduces to a spherical micelle. For cases where the and for the SQP method in Byrd et al. (2000).
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Molecular Thermodynamics of Micellization: Micelle Size Distributions and Geometry Transitions 519
Table 1: Geometric relations for a spherocylindrical micelle.
Cylindrical part Spherical part (radius Rc, length Lc) (radius Rs) 1 2 2 2 Rc VRLggccs HRs 11 R s 3 8 Rs 2 2 ARLgagcc2 VHRHggss3 33 ARga2 ARRHga842 gc gss V 1 g s 2 Pf Ags 84RRHga s ARgc aR c 2 V g Pf Ag R s
Vg is the volume of the micelle, Ag is its surface area, Agδ is its area at a distance δ from the surface, Pf is the packing factor, and R is the radius of the micelle (s for the spherical part and c for the cylindrical part). Variable H is a geometrical parameter. The expression of the volume of the surfactant tail (υs), as well as molecular parameters for different surfactants are presented elsewhere (Nagarajan and Ruckstein, 1991).
micelle size distribution method were compared. We observe that both methodologies are in very good agreement and predict the critical micelle concentra- tions well. We expect that at low surfactant concentra- tion (close to the CMC) the micelle-size distribution tends to be narrower, compared with high surfactant concentration. For example, in Figure 5 we present the micelle size distribution for different surfactants at different concentrations. We observe relatively wide size distributions for all the cases presented in Figure 5, meaning that, for those cases, the Maximum Term method certainly is not the best approach. We expect wider distributions for high surfactant concentration or for microemulsion systems, in which case the methodology proposed here becomes more adequate Figure 3: Typical example of the Gibbs energy, to describe those systems. Another aspect that moti- GkT / , as a function of the number of micelles with vates the use of the size distribution approach pro- aggregation number g, Ng, and the number of surfac- posed here is because sizes (mean, variance and other tant molecules in the cylindrical part of the micelle, moments of the distribution) are important infor- gcyl. We fixed the values of the length of the cylindrical mation for the stability of emulsions. For example, part of the micelle, Lc, and the radius of the spherical when an electric field is applied to an emulsion, its ends of the micelles, Rs, to be equal to 2.89 nm and stability is a function of the diameter of the micelles 1.23 nm, respectively. to the fourth power. We also proposed a methodology to describe the micelle geometry transitions. The results obtained are RESULTS AND DISCUSSION presented in Table 2. The surfactant dodecyl trime- thylammonium bromide (C12TAB) forms spherical Figure 4 presents the critical micelle concentration micelles even at high surfactant concentration. How- (CMC) for different surfactants as a function of their ever, sodium dodecyl sulfate (SDS) presents a transi- tail length. To perform these calculations we assumed tion from spherical to spherocylindrical as observed that micelles are globular or spherically shaped. The experimentally (Victorov and Koroleva, 2014). An- performance of the maximum term method and the other result obtained is that the addition of electrolytes
Brazilian Journal of Chemical Engineering Vol. 33, No. 03, pp. 515 - 523, July - September, 2016
520 M. S. Santos, F. W. Tavares and E. C. Biscaia Jr to the surfactant solution causes the formation of sphe- the model predicts the formation of spherocylindrical rocylindrical micelles for tetradecyl trimethylammo- micelles even without electrolytes added to the solu- nium bromide (C14TAB), and hexadecyl trime- tion, but at higher surfactant concentrations. In general, thylammonium bromide (C16TAB). For the nonionic we are in good qualitative agreement with the trends surfactant, decyl(dimethyl)phosphine oxide (C10PO), observed by experiments.
Figure 4: Critical micelle concentration (CMC) for different surfactants in aqueous solutions obtained by the maximum term method (continuous lines), and the size distribution methodology (dashed lines). The squares are experimental data (Khan and Shah, 2008; Neves et al., 2007; Evans, 1956; Zhao et al., 2012, Velázquez and Lopez- Días, 2007). (a) Sodium nc-alkyl sulfate, (b) nc-alkyl pyridinium bromide, (c) nc-alkyl trimethylammonium bromide, (d) nc-alkyl(dimethyl)phosphine.
Figure 5: Micelle size distributions for different surfactants obtained by the proposed methodology. X g is the molar fraction of micelles with aggregation numberg : (a) 16.34 mM SDS solution; (b) 81.7 mM C12TAB solution; (c)
32.68 mM C14TAB solution; (d) 32.68 mM C16TAB solution. We defined T = 25 ºC, P = 1 atm, and N w = 3400000.
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Molecular Thermodynamics of Micellization: Micelle Size Distributions and Geometry Transitions 521
Table 2: Geometry transition between spherical and spherocylindrical micelles.
Surfactant Electrolyte Ng Geometry RS gsph
concentration (mM) concentration (M) 49.0 - 31 Spherical 1.54 65 81.7 - 52 Spherical 1.57 69 49.0 0.6 7 Spherocylindrical 1.67 92 gcyl = 333, SDS Lc = 17.27 nm 81.7 0.6 3 Spherocylindrical 1.67 92 gcyl = 1565, Lc = 81.56 nm 49.0 - 49 Spherical 1.23 34 C12TAB 81.7 - 92 Spherical 1.25 35 49.0 0.6 60 Spherical 1.35 46 81.7 2.0 14 Spherical 1.59 70 81.7 4.0 10 Spherocylindrical 1.62 79 C14TAB gcyl = 20, Lc = 1.56 nm 16.4 0.2 12 Spherical 1.76 82 81.7 0.2 64 Spherical 1.74 95 16.4 2.0 6 Spherocylindrical 1.80 97 gcyl = 69, C16TAB Lc = 4.99 nm 81.7 2.0 9 Spherocylindrical 1.80 98 gcyl = 457, Lc = 33.79 nm 9.8 - 6 Spherical 1.34 57 16.4 - 3 Spherocylindrical 1.34 62 C10PO gcyl = 201, Lc = 17.56 nm
CONCLUSIONS especially when external fields are applied. We sug- gest, for future work, adapting this model to consider Here we proposed a methodology to consider the a better description of systems with high electrolyte micelle size distribution and also an approach to concentration, where spherocylindrical micelles are predict micelle shape transitions, both considering the expected to be formed, and also to extend this ap- minimum of Gibbs free energy of the system. Com- proach by considering equations of state, which is a paring the predictive performance of calculated criti- very promising path. cal micelle concentrations with experimental data validated the model. Good results are obtained. The proposed methodology is able to predict quantita- ACKNOWLEDGEMENTS tively the transition between spherocylindrical and spherical in sodium dodecyl sulfate and qualitatively We thank the Brazilian Agencies CNPq (Conselho for n-alkyl trimethylammonium bromide, according Nacional de Desenvolvimento Científico e Tecno- to experimental data. Furthermore, the methodology lógico) and CAPES (Coordenação de Aperfeiçoa- used to obtain the critical micelle concentration mento de Pessoal de Nível Superior) for providing (CMC) significantly reduces the computational cost scholarships and for supporting this work. involved. This work pioneers a more detailed description of the micellization phenomena by Molecular Thermo- NOMENCLATURE dynamics simulation considering the minimum of Gibbs free energy. It can be extended to describe a Micelle surface area per surfactant (m2) systems where the micelle size distribution is relevant aδ Micelle surface area per surfactant at a and also where large micelles are formed. For exam- distance δ (m2) ple, the micelle size distribution is important to con- Ag Superficial area of the micelle of sider when studying the stability of emulsions, aggregation number g (m2)
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Agδ Superficial area of the micelle of thermodynamic model of nonionic surfactant aggregation number g at a distance δ (m2) micelles. Journal of Physical Chemistry, B, 116, p. a, b, c Regularization constants 6443-54 (2012). g Aggregation number Kamranfar, P. and Jamialahmadi M., Effect of surfac- G Gibbs free energy (J) tant micelle shape transition on the microemulsion H Geometric constant (m) viscosity and its application in enhanced oil recov- k Boltzmann constant (J · K-1) ery processes. Journal of Molecular Liquids, 198, Lc Length of the cylindrical part of the p. 286-291 (2014). micelle (m) Kennedy, J. and Eberhart, R., Particle swarm optimi- nc Number of carbons in the surfactant tail zation. Proceedings of IEEE International Confer- Ni Number of molecules of i ence on Neural Networks IV, p. 1942-1948 (1995). P Pressure (Pa) Khan, A. M. and Shah, S. S., Determination of Critical Pf Packing factor Micelle Concentration (CMC) of Sodium Dodecyl Rc Radius of the cylindrical part of the Sulfate (SDS) and the effect of low concentration micelle (m) of pyrene on its CMC using ORIGIN software. Rs Radius of the spherical part of the Journal of Chemical Society of Pakistan, 30(2), p. micelle (m) 186-191 (2008). T Temperature (K) Lukanov, B. and Firoozabadi, A., Specific ion effects Vg Volume of the micelle of aggregation on the self-assembly of ionic surfactants: A mo- number g (m3) lecular thermodynamic theory of micellization 3 vs Volume of the surfactant tail (m ) with dispersion forces. Langmuir, 30(22), p. 6373- X Molar fraction 6383 (2014). Moreira, L. A. and Firoozabadi, A., Thermodynamic Greek Letters modeling of the duality of linear 1-alcohols as co- surfactants and cosolvents in self-assembly of μ Chemical potential (J) surfactant molecules. Langmuir, 25(20), p. 12101- Δμ Free energy of micellization (J) 12113 (2009). Moreira, L. A. and Firoozabadi, A., Molecular ther- Subscripts modynamic modeling of droplet-type microemul- sions. Langmuir, 28(3), p. 1738-1752 (2012). w Water Moreira, L. A. and Firoozabadi, A., Molecular ther- SA Surfactant modynamic modeling of specific ion effects on 1A Free surfactant micellization of ionic surfactants. Langmuir, g Micelle of aggregation number g 26(19), p. 15177-15191 (2010). s, sph Spherical Nagarajan, R., Modelling solution entropy in the c, cyl Cylindrical theory of micellization. Colloids and Surfaces A, 71, p. 39-64 (1993). Nagarajan, R., Self-assembly: The neglected role of REFERENCES the surfactant tail. Langmuir, 18(01), p. 31-38 (2002). Byrd, R. H., Gilbert, J. C. and Nocedal, J., A trust Nagarajan, R. and Ruckenstein, E., Theory of surfac- region method based on interior point techniques tant self -assembly: A predictive molecular ther- for nonlinear programming. Mathematical Pro- modynamic approach. Langmuir, 7(3), p. 2934- graming, 89(1), p. 149-185 (2000). 2969 (1991). Evans, H. C., Alkyl sulphates. Part I. Critical micelle Neves, A. C. S., Valente, A. J. M., Burrows, H. D., concentrations of the sodium salts. Journal of Ribeiro, A. C. F., Lobo, V. M. M., Effect of Chemical Society, p. 579-586 (1956). terbium(III) chloride on the micellization proper- Goldsipe, A. and Blankschtein, D., Molecular-ther- ties of sodium decyl and dodecyl-sulfate solutions. modynamic theory of micellization of multicom- Journal of Colloid and Interface Science, 306, p. ponent surfactant mixtures: 1. Conventional (pH- 166-74 (2007). Insensitive) surfactants. Langmuir, 23(11), p. Puvvada, S. and Blankschtein, D., Molecular-thermo- 5942-5952 (2007). dynamic approach to predict micellization, phase Iyer, J. and Blankschtein, D., Are ellipsoids feasible behavior and phase separation of micellar solutions. micelle shapes? An answer based on a molecular- I. Application to nonionic surfactants. Journal of
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