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Heat Capacity of Transfer of (Ethylene oxide)13-(propylene oxide)30-(ethylene oxide)13 from Water to the Aqueous Anionic Surfactant Solutions at 298 K. A Quantitative Treatment R. De Lisi, G. Lazzara, S. Milioto,* and N. Muratore Dipartimento di Chimica Fisica “F. Accascina”, Universita` degli Studi di Palermo, Viale delle Scienze, Parco D’Orleans II, 90128 Palermo, Italy Received April 28, 2004. In Final Form: July 30, 2004 Heat capacities of transfer (∆Cpt) of unimeric (ethylene oxide)13-(propylene oxide)30-(ethylene oxide)13 from water to the aqueous surfactant solutions as functions of the surfactant concentrations (mS) were determined at 298 K. The surfactants investigated are sodium hexanoate, sodium heptanoate, sodium octanoate, sodium undecanoate, and sodium dodecanoate. For short alkyl chain surfactants, the profiles of the ∆Cpt versus mS curves show maxima and minima; for long alkyl chain surfactants, the maximum becomes sharper and moved to lower mS values whereas the minimum tends to disappear. These experimental trends are different from those of the enthalpy in agreement with the fact that heat capacity, being the derivative of enthalpy with respect to temperature, reflects additional terms generated by temperature change on the equilibria in solution. On the basis of a thermodynamic model recently proposed by us for properties first derivatives of Gibbs free energy, a quantitative treatment of the experimental data was done. Such an approach assumes that even in the dilute surfactant region monomers of surfactant associate with unimeric copolymer generating surfactant-copolymer aggregation complexes and, whenever the surfactant achieves the conditions for the micellization, the formation of copolymer-micelle mixed aggregates takes place. The equation derived for the heat capacity of transfer is more complex than that for the enthalpy because it contains five additional terms due to the shift of the equilibria induced by the temperature change. It turned out that these contributions, evaluated by using the equilibrium constants and the associated enthalpies, cannot be neglected for a quantitative treatment of the experimental data. The minimizing procedure provided the heat capacity changes for the formation of the surfactant-copolymer aggregation complexes and the copolymer-micelle mixed aggregates.
Introduction Poly(ethylene oxides)-poly(propylene oxides)-poly(ethylene oxides) (PEO-PPO-PEO) are water soluble amphiphilic polymers of low toxicity which have widespread industrial applications in emulsifying, wetting, coating, solubilizing, stabilizing, and so forth. It was reported1 that on the different applications of these copolymers in medical and pharmaceutical industries, over 1000 articles were published. Therefore, PEO-PPO-PEO are very important and promising molecules in colloidal science. They are surface active, adsorbing onto the solid/ liquid2 and air/liquid2 interfaces whereas in the bulk phase they can form self-assembled structures of various shapes and sizes.3 The self-organizing phenomenon is due to the solvophobicity and solvophilicity of the constituent blocks, and, therefore, in aqueous solutions the micelles are formed by the PPO core and the hydrated PEO shell. These macromolecules were studied in water4-7 and in the presence of components such as electrolytes,7-9 apolar * To whom correspondence should be addressed. E-mail: milioto@ unipa.it. (1) Chu, B. Langmuir 1995, 11, 414. (2) Chen, C.; Even, M. A.; Chen, Z. Macromolecules 2003, 36, 4478. (3) Loh, W. In Encyclopedia of Surface and Colloid Science; Hubbard, A. T., Ed.; Marcel Dekker: New York, 2002; p 802. (4) Yang, L.; Alexandridis, P.; Steyler, D. C.; Kositza, M. J.; Holzwarth, J. F. Langmuir 2000, 16, 8555. (5) Senkow, S.; Mehta, S. K.; Douhe´ret, G.; Roux, A. H.; RouxDesgranges, G. Phys. Chem. Chem. Phys. 2002, 4, 4472. (6) De Lisi, R.; Milioto, S. Langmuir 1999, 15, 6277. (7) De Lisi, R.; Milioto, S. Langmuir 2000, 16, 5579. (8) Guomin, M.; Sukuraman, S.; Beaucage, G.; Saboungi, M. L.; Thiyagarajan, P. Macromolecules 2001, 32, 552 and references therein.
compounds,10,11 polar solutes,6,12,13 and surfactants14-17 by means of different techniques. Ionic surfactants are mixed with copolymers to stabilize the colloidal systems and to control their rheology. The most used thermodynamic techniques for studying the aqueous surfactant/copolymer mixtures are based on calorimetry (isothermal titration calorimetry14,17 and differential scanning calorimetry, DSC)16,18 because of the high sensitivity of the enthalpy and the heat capacity to the nature of the interactions. The DSC method evidences transition processes induced by temperature variations. For instance, Hoffmann and Hecht16 monitored the destruction of polymeric micelles upon the addition of surfactants which occurred in a more or less wide range of temperatures depending on the amount of the surfactant employed. Recently, other data based on volume,7,19,20 enthalpy,21,22 and isothermal heat (9) Su, Y.; Wei, X. F.; Liu, H. Z. J. Colloid Interface Sci. 2003, 264, 526. (10) Hurter, P. N.; Alexandridis, P.; Hatton, T. A. In Solubilization in Surfactant Aggregates; Christian, S. D., Scamehorn, J. F., Eds.; Marcel Dekker: New York, 1995; p 191. (11) Kozlov, M. Y.; Melik-Nubarov, N. S.; Batrakova, E. V.; Kabanov, A. V. Macromolecules 2000, 33, 3305. (12) Yang, L.; Alexandridis, P. Macromolecules 2000, 33, 5587. (13) Ivanova, R.; Lindman, B.; Alexandridis, P. Langmuir 2000, 16, 9058. (14) Li, X.; Wettig, S. D.; Verrall, R. E. Langmuir 2004, 20, 579. (15) Wettig, S. D.; Verrall, R. E. J. Colloid Interface Sci. 2001, 244, 377. (16) Hecht, E.; Hoffmann, H. Langmuir 1994, 10, 86. (17) Thurn, T.; Courdec, S.; Sidhu, J.; Bloor, D. M.; Penfold, J.; Holzwarth, J. F.; Wyn-Jones, E. Langmuir 2002, 18, 9267. (18) Li, Y.; Xu, R.; Courdec, S.; Bloor, D. M.; Wyn-Jones, E.; Holzwarth, J. F. Langmuir 2001, 17, 183. (19) De Lisi, R.; Milioto, S.; Munafo`, M.; Muratore, N. J. Phys. Chem. B 2003, 107, 819.
10.1021/la048943n CCC: $27.50 © 2004 American Chemical Society Published on Web 10/09/2004
Heat Capacity of Transfer of L64
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Table 1. Thermodynamic Properties of Sodium Alkanoates in Water at 298 Ka NaHept KM N 0 ∆Hm,S BCp CCp CpM,S Cp0S
NaDecb
NaOct
0.11 ( 9 ( 1c 11.5 ( 0.4e 47 ( 8
(2.2 ( 0.9) × 18 ( 2d 11.6 ( 0.2e 39 ( 12
108 ( 4 492 ( 2
224 ( 4 591 ( 3
0.05c
7.6 × 20 10 -70.37 40.15 302 771.8
103 d
1015
NaU
NaL
(1.0 ( 1.3) × 20 ( 4c 10 ( 3c
1019 c
424 ( 7 877 ( 17
(4 ( 3) × 1025 c 20 ( 3c 9.67f 423 ( 7g 937 ( 8g
0 Units are KM, kgN-1 mol1-N; BCp, J K-1 kg mol-2; CCp, J K-1 kg3/2 mol-5/2; ∆Hm,S , kJ mol-1; Cp0S and CpM,S, J K-1 mol-1. b From ref 29. d e f g From ref 23. From ref 22. From ref 27. From ref 31. From ref 30.
a
c
capacity19,20 were produced. Heat capacity, being a second derivative of Gibbs free energy, may reveal phenomena to which enthalpy and volume are not sensitive. On this basis, we determined accurate isothermal heat capacity data of aqueous sodium alkanoate/(ethylene oxide)13-(propylene oxide)30-(ethylene oxide)13 (L64) mixtures and modeled them with a thermodynamic approach recently proposed22 for the properties first derivatives of Gibbs free energy (enthalpy and volume). The combination of the present results with various thermodynamic properties22,23 (free energy, enthalpy, entropy, and volume) for different aqueous copolymer/surfactant systems adds new insights into the effect of the surfactant hydrophobicity on the mechanism of L64/surfactant interactions. Experimental Section Materials. Sodium hexanoate (NaHex), sodium octanoate (NaOct), and sodium dodecanoate (NaL), Sigma products, were used as received. Sodium heptanoate (NaHept) and sodium undecanoate (NaU) were prepared by neutralizing their acids (Aldrich), solubilized in absolute ethanol (Merck), with an ethanolic solution of sodium hydroxide (Fluka). The products were crystallized three times from ethanol and dried in a vacuum oven for 1 week at 323 K. The purity of such compounds was checked by using the procedure described elsewhere.24 The aqueous surfactant solutions gave pH ≈ 9. L64 was a kind gift from BASF. Its nominal molecular weight is 2900 g mol-1. L64 was employed as received because the apparent molar volumes determined in the dilute region agree well with the literature values5-7 obtained by using both purified and nonpurified copolymers. All solutions were prepared by mass using degassed conductivity water, and their concentrations were expressed as molalities. Heat Capacity Measurements. The experiments consisted of determining the relative difference between the heat capacity per unit volume of the water + surfactant + copolymer ternary mixture (σ) and water (σ0), that is, (σ - σ0)/σ0, by means of a Picker flow microcalorimeter (Setaram) at 298.426 ( 0.001 K. Using a flow rate of about 0.01 cm3 s-1 and a basic power of 19.7 mW, the temperature jump was approximately 0.5 K. The reproducibility of the specific heat capacity measurement is 1 × 10-4 J K-1 g-1. The specific heat capacity (cp) of a solution of density d is given by
cp ) cpwdwd-1{1 + (σ - σ0)/σ0}
(1)
where cpw and dw stand for the specific heat capacity and density of the reference solvent, that is, water. The values of cpw25 and (20) Senkow, S.; Roux, A. H.; Roux-Desgranges, G. Phys. Chem. Chem. Phys. 2004, 6, 822. (21) De Lisi, R.; Milioto, S.; Muratore, N. Macromolecules 2002, 35, 6075. (22) De Lisi, R.; Lazzara, G.; Milioto, S.; Muratore, N. J. Phys. Chem. B 2004, 108, 1189. (23) De Lisi, R.; Lazzara, G.; Milioto, S.; Muratore, N. Macromolecules 2004, 37, 5423. (24) De Lisi, R.; Lazzara, G.; Milioto, S.; Muratore, N. J. Phys. Chem. B 2003, 107, 13150.
dw25 used are 4.1799 J K-1 g-1 and 0.996 971 g cm-3, respectively. The densities of the solutions were taken from the literature.23 The apparent molar heat capacity (CΦ,P) of L64 at mP ) 5 mmol kg-1 in the given solvent was calculated as
CΦ,P ) Mcp +
103(cp - cp0) mP
(2)
where M is the molecular weight of the copolymer and mP represents the moles of the copolymer per kilogram of the water + surfactant mixture; cp and cp0 are the specific heat capacity of the water + surfactant + copolymer and the water + surfactant mixtures, respectively. The cp0 values were measured. The heat capacity of transfer of L64 from water to the aqueous surfactant solution (∆Cpt) was calculated as difference between CΦ,P and the apparent molar heat capacity of the copolymer in water. Water-Surfactant Binary System. The apparent molar heat capacity of the surfactant in water (CΦ,S) was calculated by means of eq 2, where M stands for the surfactant molecular weight, mP was replaced by the surfactant molality, and cp and cp0 are the specific heat capacities of the water + surfactant and water, respectively. A mass action model26 based on a simple one-step association process was assumed for the micellization
KM )
mS - [m0]
(3)
N[m0]N
where KM is the equilibrium constant, mS is the stoichiometric surfactant concentration, N stands for the aggregation number of the micelles, and [m0] is the monomer surfactant concentration. The model was applied to CΦ,S of NaHept, NaOct, and NaU to evaluate the partial molar heat capacity of the surfactant in the standard (Cp0S) and the micellar (CpM,S) states together with the solute-solute interaction parameter (BCp). For the minimizing procedure, eq 25 of ref 26 was used where the literature values22,23 of KM, N, and the standard enthalpy of micellization23,27 (∆H0m,S) were introduced (Table 1). The heat capacity of micellization was calculated26 as
∆Cpm,S ) CpM,S - Cp0S - 28.95[m0]1/2 - BCp[m0] CCp[m0]3/2 (4) where 28.95 (J K-1 mol-3/2 kg1/2) is the Debye-Hu¨ckel limiting law coefficient for heat capacity28 and CCp is the triplet interaction parameter. The present and the literature29,30 data required to (25) Kell, G. S. In Water-A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1972; Vol. 1. (26) Desnoyers, J. E.; Caron, G.; De Lisi, R.; Roberts, D.; Roux, A.; Perron, G. J. Phys. Chem. 1983, 87, 1397. (27) De Lisi, R.; Milioto, S.; Muratore, N. Langmuir 2000, 16, 4441. (28) De Lisi, R.; Ostiguy, C.; Perron, G.; Desnoyers, J. E. J. Colloid Interface Sci. 1979, 71, 147. (29) Yamashita, F.; Perron, G.; Desnoyers, J. E.; Kwak, J. C. T. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986. (30) De Lisi, R.; Inglese, A.; Milioto, S.; Pellerito, A. J. Colloid Interface Sci. 1996, 180, 174.
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Figure 1. Heat capacity of transfer of L64 from water to the aqueous NaHex solutions as a function of the surfactant concentration.
Figure 3. Heat capacities (filled symbols) of transfer of L64 from water to the aqueous solutions of NaU (a) and NaL (b) as functions of the surfactant concentration. Open symbols are the enthalpies of transfer (ref 23).
Figure 2. Heat capacities (filled symbols) of transfer of L64 from water to the aqueous solutions of NaOct (a) and NaHept (b) as functions of the surfactant concentration. Open symbols are the enthalpies of transfer (ref 23). calculate ∆Cpm,S for the surfactants investigated are collected in Table 1 where the values22,23,29 of KM and N together with those23,27,31 of ∆H0m,S are also reported.
Results and Discussion Qualitative Analysis of the Experimental Heat Capacity of Transfer. The plots of ∆Cpt as functions of mS are presented in Figures 1-3. The profiles of the ∆Cpt versus mS curves for L64 in aqueous solutions of NaHex and NaHept are similar; namely, a maximum at about 0.5 mol kg-1 is observed for both the surfactants and minima at about 1.8 and 0.9 mol kg-1 are exhibited by NaHex and NaHept, respectively (Figures 1 and 2). The L64/NaOct system shows the same profile as that of the inferior homologues, presenting a maximum and a minimum at about 0.2 and 0.45 mol kg-1, respectively (Figure 2). By increasing the surfactant hydrophobicity, the (31) Milioto, S.; Causi, S.; De Lisi, R. J. Colloid Interface Sci. 1993, 155, 452.
maximum becomes sharper and is located at lower mS values whereas the minimum tends to disappear (Figure 3). These curves are consistent with those19 for L64/sodium decanoate (NaDec) available at some unassociated copolymer compositions. Heat capacities of transfer of some PEO-PPO-PEO in sodium dodecyl sulfate (NaDS) were recently reported.20 For unimeric copolymers, the shape of the ∆Cpt versus mS curve depends on the nature of the copolymer; accordingly, upon increasing the macromolecule hydrophobicity, the maximum becomes more evident and thereafter ∆Cpt steeply decreases. The L64/ NaDS and L64/NaL systems exhibit identical profiles of the ∆Cpt versus mS trends in agreement with the comparable surfactant hydrophobicities. The understanding of the plots in Figures 1-3 may be assisted by the knowledge23 of the volumetric and the enthalpic behaviors which themselves are very similar. As the examples in Figures 2 and 3 show, ∆Ht sharply increases with the surfactant concentration up to a given mS and thereafter it slowly decreases. The maximum is more pronounced upon the surfactant increasing hydrophobicity, and it is located at a mS value nearly equal to the critical micellar concentration in water (cmcw). Volumes of transfer (∆Vt) exhibited similar trends.23 The ∆Vt versus mS curves of PEO-PPO-PEO in NaDS20 are monotonic when the PPO block is small and show maxima for the more hydrophobic copolymers. It was shown that ∆Vt and ∆Ht can be simulated quantitatively by using a thermodynamic model22 based on the coexistence of the following equilibria
zS + P / C
(5)
NS / M
(6)
wP + M / D
(7)
The approach assumes that even in the dilute surfactant region, z monomers of surfactant (zS) associate with one unimer of copolymer (P) leading to the formation of
Heat Capacity of Transfer of L64
surfactant-copolymer aggregation complexes (C) and, whenever micelles are present (M), they can interact with w copolymer molecules generating the micelle-copolymer mixed aggregates (D). On the basis of such a model, the maximum in the ∆Ht versus mS curves (Figures 2 and 3) was ascribed to the transformation of the aggregation complexes into the copolymer-micelle mixed aggregates enhanced by the surfactant hydrophobicity. This argument may not explain the maximum in the ∆Cpt versus mS plots because heat capacity is the derivative of enthalpy with respect to temperature thereby reflecting the contributions generated by temperature changes on the equilibria 5-7. Such terms are expected to be relevant for the present systems because of the large values of the enthalpies23 associated with the equilibria 5 and 7. On this basis, the difference between the profiles of the ∆Cpt versus mS curves and those of the enthalpy is justified. Quantitative Approach to the Experimental Heat Capacity of Transfer. Madan and Sharp32 stated that heat capacity is the most complex of the four main thermodynamic properties (free energy, enthalpy, entropy, and heat capacity) to describe hydrophobic and ionic solvation. This assessment is supported by the successful description of the heat capacity of various aqueous mixtures such as those containing surfactants33 and polymers.34 Concerning the micellar solutions, this property is important because, apart from the ability to evidence the micellization process, it detects post-micellar transitions.35-37 Desnoyers and Quirion36 explained the post-micellar transition of hexadecyltrimethylammonium bromide by invoking the counterion binding variations, which cause the loss of structural hydration cosphere to which the volume (first derivative of Gibbs free energy) is not sensitive. As a consequence, heat capacity is very attractive for studying aqueous copolymer-surfactant mixtures, although from the earlier analysis one expects that such a property is very difficult to be straightforwardly understood. With this in mind, efforts were addressed to the quantitative analysis of ∆Cpt data on the basis of the above cited thermodynamic model22 successfully applied to the properties first derivatives of Gibbs free energy. For the enthalpy of transfer, the following equation was derived22
∆Ht ) 2BH,PSxP[m] + xC∆HC + xD∆HD + Eshift∆Hm,S (8) The first term on the right-hand side of eq 8 takes into account the interactions between monomers of copolymer and surfactant where BH,PS is the interaction parameter, [m] is the monomer surfactant concentration in the presence of the copolymer, and xP is the fraction of free copolymer. xC∆HC is the contribution for the formation of the surfactant-copolymer aggregation complex, xC being the fraction of the copolymer-surfactant aggregation complex and ∆HC being the variation of the enthalpy. xD∆HD is the contribution for the formation of the (32) Madan, B.; Sharp, K. J. Phys. Chem. 1996, 100, 7713. (33) Desnoyers, J. E.; Perron, G.; Roux, A. H. In Surfactant Solutions: New Methods of Investigation; Zana, R., Ed.; Marcel Dekker: New York, 1987. (34) Aucouturier, C.; Roux-Desgranges, G.; Roux, A. H. J. Chem. Thermodyn. 1999, 31, 289. (35) De Lisi, R.; Milioto, S.; Triolo, R. J. Solution Chem. 1989, 18, 905. (36) Quirion, F.; Desnoyers, J. E. J. Colloid Interface Sci. 1986, 112, 565. (37) Roux-Desgranges, G.; Roux, A. H.; Viallard, A. J. Chim. Phys. 1985, 82, 441.
Langmuir, Vol. 20, No. 23, 2004 9941
copolymer-micelle mixed aggregates, the fraction of which is xD, and ∆HD is the enthalpy change. Both xC and xD are given by z
xC ) KCxP[m]
(xPmP)wmMKD xD ) mP
(9)
where KC and KD are the equilibrium constants and mM is the micelles concentration in the presence of the copolymer. The last term on the right-hand side of eq 8 is the shift of the micellization equilibrium induced by the copolymer being Eshift ) {[m0] - [m] - zxCmP]/mP, and ∆Hm,S is the enthalpy of micellization.23 By remembering that ∆Cpt is (δ∆Ht/δT)P, eq 8 was derived with respect to temperature and the following one was obtained
∆Cpt ) 2BCp,PSxP[m] + xC∆CpC + xD∆CpD +
{
}
δxP δ[m] + xP + δT δT δxD δEshift δxC ∆HC + ∆HD + ∆Hm,S (10) δT δT δT
Eshift∆Cpm,S + 2BH,PS [m]
where BCp,PS is the copolymer-surfactant interaction parameter whereas ∆CpC and ∆CpD are the changes in the heat capacity due to the formation of the surfactantcopolymer aggregation complex and the copolymermicelle mixed aggregates, respectively. Compared to eq 8, eq 10 contains five additional terms which are the relaxation contributions due to the displacement of the equilibria induced by temperature changes. If they are negligible, eq 10 becomes formally identical to eq 8. The large number of unknown parameters in eq 10 makes it impossible to perform a minimizing procedure of the heat capacity data. Fortunately, it can be reduced because of the availability of the equilibrium constants (KM, KC, and KD) at a given temperature and the related standard enthalpies (∆H0m,S, ∆HC, and ∆HD), which allowed the evaluation of the relaxation terms. According to eq 10, ∆Cpt, corrected for the relaxation contributions, may be treated like a property first derivative of Gibbs free energy. In this case, the fitting parameters reflect also the errors of the relaxation terms due to the uncertainties on the equilibrium constants, the enthalpy changes, and BH,PS. Fits of the Experimental Heat Capacity of Transfer. The relaxation contributions were calculated as follows. The KM temperature dependence was computed through the van’t Hoff equation by using the ∆H0m,S value at 298.15 K and assuming that it is temperature independent as a result of the small value of the heat capacity of micellization. By using KM at 298.15 K, the variation of [m0] with temperature was evaluated by supposing that N does not change within the interval of temperature investigated. The temperature shift of the equilibria of both the surfactant-copolymer aggregation complex and the copolymer-micelle mixed aggregates formation were also evaluated by using KC and KD together with ∆HC and ∆HD data23 at 298.15 K (Table 2). Because the experiments were carried out upon the temperature jump of 0.5 K, the KC and KD values at the final temperature of 298.65 K were calculated by stating that ∆HC and ∆HD are both temperature-dependent. This was done because the copolymer behavior is sensitive to temperature changes.
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Table 2. Thermodynamic Properties for L64-Sodium Alkanoate Binding in Aqueous Solutions at 298 Ka BCp,PS ∆CpC BH,PSg KCg zg ∆HCg ∆CpD KDg ∆HDg b
NaHept
NaOct
NaDecb
-1.4 ( 0.9 -18 ( 1 87 ( 4 3.6 ( 0.4 3.7 220 ( 12 2(1 130 ( 60 136 ( 12
-4 ( 3 -10 ( 2 98 ( 20 39.2 ( 1.7 3.1 178.9 ( 1.6 3(3 380 ( 80 119 ( 12
16 ( 36 ( 55 ( -13 ( 2;c,f -13 ( 2d,f -14 ( 2e,f 227 ( 36 3600 ( 600 3.1 191 ( 3 -7 ( 5;c,f -4 ( 7;d,f -2 ( 6e,f 520 ( 110 26 ( 14 15;c,f
15;d,f
14e,f
NaU
NaL
-7.3 ( 0.7 300 ( 20 2800 ( 150 2.2 143 ( 2 -8 ( 2 1680 ( 800 20 ( 20
98 ( 22 -7.8 ( 0.8 710 ( 30 (1.9 ( 1.6) × 104 2.4 130 ( 1 -11 ( 3 700 ( 90 -1 ( 4
a Units are B -1 mol-2 kg; heat capacity, kJ K-1 mol-1; B -2 kg; K , kgz mol-z; enthalpy, kJ mol-1; K , kg mol-1. Cp,PS, kJ K H,PS, kJ mol C D The literature data were taken from ref 22. c mP ) 15 mm. d mP ) 25 mm. e mP ) 35 mm. f Present work. g From ref 23.
The mass balances for both the surfactant and the copolymer were written as
mS ) KDKMN[m]N(xPmP)w + KCxPmPz[m]z + KMN[m]N + [m] (11) mP ) KDKMw[m]N(xPmP)w + KC xPmP[m]z + xPmP (12) At a given temperature, the nonlinear system (eqs 11 and 12) was solved through the Newton-Raphson method extended to two dimensions providing the [m] and xP values. Then, xC and xD were calculated from eqs 9. Moreover, their temperature slopes were computed by assuming z and w independent of temperature. For all the systems, w ) 1 was used.22,23 The minimizing process was carried out through a program based on a nonlinear least-squares fitting method which provided the BCp,PS, ∆CpC, and ∆CpD parameters collected in Table 2. In the fitting procedure, the contributions of the relaxation equilibria were also involved because of the assumed dependence of ∆HC and ∆HD on temperature. Concerning the L64/NaHex system, the best fit generated parameters affected by large errors due to both the small equilibrium constant values and the errors on the relaxation contributions. Finally, eq 10 was also applied to the L64/NaDec data19 available at mP ) 15, 25, and 35 mm. Figure 4 illustrates the ∆Cpt versus mS curves along with the best fits through eq 10 for the NaDec/L64 mixtures. The agreement between the experimental points and the calculated values is quite satisfactory. The fitting parameters, within errors, are independent of mP, corroborating the volume and the enthalpy results.22 However, it is to be pointed out that the ∆CpD values are affected by large errors becoming greater with mP because, within the fitting range, the xD variation is small and becomes smoother with increasing mP. Figures 5 and 6 show the dependence on mS of the relaxation contributions and ∆Cpt corrected for them (∆Cpt,corr) for the NaOct/L64 and NaL/L64 mixtures, respectively. As can be seen, the match between the experimental points and the computed values by means of eq 10 is good. The alkyl chain length of the surfactant determines the shape of such curves: for NaOct, the profile is S-shaped to about 0.35 mol kg-1, at which the property starts to increase, whereas for NaL the decreasing curve exhibits a change in the slope at about 0.04 mol kg-1. The magnitude of the relaxation contributions as well as their dependence on mS are functions of the surfactant hydrophobicity. For the surfactant with a short tail (Figure 5), the shift of the micellization equilibrium with temperature causes a very small effect whereas that of the process for the copolymermicelle mixed aggregate formation is slightly negative. The most significant are the relaxation terms of both the
Figure 4. Heat capacities of transfer of L64 from water to the aqueous NaDec solutions as functions of the surfactant concentration. Lines are the best fits according to eq 10. The data were taken from ref 19.
surfactant-copolymer aggregation complex formation and the interaction process between free surfactant and unimeric copolymer: the former is largely positive and shows an S-shaped profile to about 0.3 mol kg-1, and thereafter it decreases; the second is negative and exhibits a trend with a minimum at about 0.4 mol kg-1. Similar considerations can be done for the NaHept/L64 system. Concerning NaL/L64, the (δxC/δT)∆HC as a function of mS trend exhibits a sharp maximum at about 0.02 mol kg-1 and a minimum at about 0.05 mol kg-1. The term of the micellization shift is null to 0.03 mol kg-1 (cmcw of NaL); thereafter it decreases with mS whereas (δxD/δT)∆HD is negligible. The changes in the interactions between surfactant monomers and unimers of copolymer induced by the temperature variation allow negative values in the entire range of the surfactant studied with an extremum at 0.03 mol kg-1. The surfactants with a long alkyl chain behave like NaL. To show the importance of the several contributions to ∆Cpt, the experimental points and the various terms present in eq 10 were plotted against mS for the NaU/L64 system (Figure 7). The curve given by the sum of all the relaxation terms takes into account the sharp maximum observed in the dilute surfactant region. The addition of the contributions for the micellization shift, and the interactions between the copolymer and the surfactant
Heat Capacity of Transfer of L64
Figure 5. Aqueous L64/NaOct system: (a) variation with the surfactant concentration of the experimental heat capacity corrected for the relaxation terms (b) and the computed values according to eq 10 (line); (b) dependence on the surfactant concentration of the relaxation contributions related to the surfactant-copolymer aggregation complex formation (1), shift of the micellization equilibrium (2), copolymer-micelle mixed aggregates formation (3), and free surfactant-copolymer interactions (4).
Figure 6. Aqueous L64/NaL system: (a) variation with the surfactant concentration of the experimental heat capacity corrected for the relaxation terms (b) and the computed values according to eq 10 (line); (b) dependence on the surfactant concentration of the relaxation contributions related to the surfactant-copolymer aggregation complex formation (1), copolymer-micelle mixed aggregate formation (2), free surfactant-copolymer interactions (3), and shift of the micellization equilibrium (4).
influence such a curve beyond the minimum moving it toward slightly larger values. By further adding the term for the surfactant-copolymer aggregation complex formation, a curve shifted to lower values is obtained exhibiting a minimum at 0.08 mol kg-1 which reflects also the decomplexation process. The increasing part of the latter is mainly canceled out by the negative contribution for the copolymer-micelle mixed aggregate formation. The analysis of literature20 ∆Cpt of PEO-PPO-PEO in NaDS solutions would provide additional tools to confirm the reliability of eq 10 and to obtain new insights into the effect of the macromolecule hydrophobicity on
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Figure 7. Heat capacity of transfer of L64 from water to the aqueous NaU solutions as a function of the surfactant concentration. Curve c represents the sum of all the relaxation terms; curve b is the sum of the curve c and the shift of the micellization equilibrium term; curve a is the sum of the curve b and the free surfactant-copolymer interactions contribution; curve d was obtained by adding xC∆CpC to the curve a; curve e is the sum of the curve d and xD∆CpD.
the copolymer-surfactant interactions. Unfortunately, the fitting process was prevented by the lack of enthalpy and equilibrium constant data which are necessary to evaluate the relaxation terms. On the other hand, because of the similarity between such literature data and ours one expects that the relaxation terms are not negligible also for the copolymer/NaDS systems. The above discussion has demonstrated that heat capacity of transfer is a very complex property to analyze because the importance of a certain term with respect to another is strictly correlated to the nature of the surfactant. L64/Surfactant Interactions Evidenced by Heat Capacity. The quantitative fit of the heat capacity of transfer supports the validity of the model, but the generated quantities (BCp,PS, ∆CpC, and ∆CpD) may reflect the approximations and the large number of parameters involved (eq 10). Nevertheless, some general considerations can be made. ∆CpC is negative and nearly independent of nc whereas ∆CpD decreases with nc. Previous studies23 evidenced that both processes are controlled by the hydrophobic desolvation and the conformational changes of the copolymer induced by the attachment of the surfactant molecules; as a consequence of the conformational effects, hydrogen bonds between the EO units and water are formed. Heat capacity is expected to be sensitive to these effects according to Pyda’s38 report, following which, for flexible polymers, such a property mainly reflects the variations of the conformational motion originated by internal rotations. The desolvation of the copolymer and the surfactant as well as the loss of conformational freedom causes negative heat capacity whereas the formation of the hydrogen bonds between the EO units and water allows positive values. In the case of ∆CpD, the desolvation contribution of the surfactant molecules is absent because the binding involves the micelles but, in this case, quite important conformational effects23 are involved. For the copolymer-micelle mixed aggregate formation, another effect due to the changes of the degree of ionization of the micelles has to be taken into account. On the basis of the polymer-surfactant system studies,7,39 counterions are released and, thereby, the heat capacity change increases.36 (38) Pyda, M. Macromolecules 2002, 35, 4009. (39) Rodenas, E.; Sierra, M. L. Langmuir 1996, 12, 1600 and references therein.
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Conclusions proposed22
for the volume and The mass action model, the enthalpy, was applied to the heat capacity of transfer by demonstrating that it is a powerful tool for the thermodynamic investigation of aqueous copolymersurfactant systems. The theoretical approach allowed an equation which correlates ∆Cpt to mS in terms of several contributions, and none of them, as shown earlier, can be neglected. Such an equation has five more terms compared to that of the enthalpy, from which it was generated, and the high number (10) of unknown parameters makes performing a fitting procedure impossible. This limit can be encompassed by reducing the number of the variables, and this can be done by carrying out independent measurements addressed to the determination of the enthalpies and/or the equilibrium constants. If the values of the latter are correct, then the fit is quantitative. In our case, we used some quantities (BH,PS, z, w, ∆HC, ∆HD, KC,
De Lisi et al.
and KD) obtained from the quantitative simulation of the enthalpy of transfer data23 so that the equation for ∆Cpt reduced to a three-unknown-parameters equation (BCp,PS, ∆CpC, and ∆CpD). The quantitative fit obtained for the systems under investigation corroborates not only the validity of the model but also the reliability of the experimental enthalpy of transfer and the parameters produced by means of eq 8. Acknowledgment. Financial assistance provided by the Ministry of Instruction, University and Research is gratefully acknowledged. The authors thank BASF for kindly providing the copolymer. Supporting Information Available: Table of the apparent molar heat capacities of sodium alkanoates in water and L64 in aqueous sodium alkanoate solutions. This material is available free of charge via the Internet at http://pubs.acs.org. LA048943N
