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Comparison of various models describing the adsorption of surfactant molecules capable of interfacial reorientation
Journal of Colloid and Interface Science 261 (2003) 180–183 www.elsevier.com/locate/jcis
Comparison of various models describing the adsorption of surfactant molecules capable of interfacial reorientation V.B. Fainerman,a S.A. Zholob,a E.H. Lucassen-Reynders,b and R. Miller c,∗ a Medical Physicochemical Centre, Donetsk Medical University, 16 Ilych Avenue, Donetsk 83003, Ukraine b Mathenesselaan 11, 2343 HA Oegstgeest, The Netherlands c Max-Planck-Institut für Kolloid und Grenzflächenforschung, 14424 Potsdam/Golm, Germany
Received 24 July 2002; accepted 19 December 2002
Abstract The thermodynamic model for describing the adsorption of surfactant molecules in different adsorption states, the reorientation model, is reconsidered on a more rigorous level. The resulting model equations are used to describe experimental surface pressure data published in the literature. The new model proposed contains three physical parameters and describes the experimental dependencies Π(c) for oxethylated alcohols very accurately. 2003 Elsevier Science (USA). All rights reserved. Keywords: Adsorption isotherm; Surface equation of state; Surface reorientation; Oxethylated nonionic surfactants; Surface tension isotherms
1. Introduction For asymmetric surfactant molecules, which are able to adsorb in two or more states, a self-regulation mechanism leads to a preferential accumulation of states with large partial molar surface area at low surface pressure [1,2]. At high surface pressures, on the other hand, the majority of adsorbed surfactant molecules can occupy only a minimal molar area. Equations of state and adsorption isotherms for such systems were proposed earlier [2–4]. The hydrophilic part of oxethylated surfactants exhibits an adsorption activity at low surface layer coverage. The reorientation theory satisfactorily describes the behavior of oxethylated alcohols Cn EO8 at the water/air and water/oil interfaces [5–7]. At low concentrations the area occupied by these molecules is three to four times larger than that characteristic for higher concentrations. This feature of oxethylated alcohols results in an extremely high surface activity at low concentrations and an unusual shape of the surface tension isotherms not compatible with the Langmuir model. Although the reorientation model can very successfully be applied to various surfactants, there is one aspect requiring additional analysis. In [2–4] the equations of state and * Corresponding author.
E-mail address: [email protected] (R. Miller).
adsorption isotherms of the reorientation model were derived from the Butler equation with some simplifications. The object of this work is to give a more rigorous derivation of the model equations and to compare experimental data for molecules that can occupy a number of adsorption states with various theoretical models.
2. Results and discussion From the equations for chemical potentials µsi and µαi of the components of a mixture of different substances in the surface layer (Butler’s equation [8]), s s µsi = µ0s i + RT ln fi xi − γ ωi ,
(1)
and in the solution bulk α α µαi = µ0α i + RT ln fi xi ,
(2)
the most general equations of state for the surface layer and the adsorption isotherm for the ith component can be obtained [2,9]. For ideally dilute solutions RT s ln x0 + ln f0s , ω0 fis xis /fi0s ωi s ln ln x0 + ln f0s . = α Ki xi ω0 Π =−
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00226-6
(3) (4)
V.B. Fainerman et al. / Journal of Colloid and Interface Science 261 (2003) 180–183
Here R is the gas law constant, T is the temperature, fi are the activity coefficients standardized at fi = 1 at xi = 1, xi = Ni / Ni are the molar fractions, and Ni are the numbers of moles of the ith component, Π = γ0 − γ is the surface pressure, γ0 and γ are the surface tension of solvent and solution, respectively; Ki is the adsorption equilibrium constant defined at infinite dilution where Π = 0. The subscript 0 refers to the solvent, the superscript 0 denotes the standard state, and the superscripts s and α indicate the surface layer and bulk phase, respectively. It should be noted that the condition ω0 = const (or γ00 ω00 − γ ω0 = Πω0 ) is used to obtain Eqs. (3) and (4). We assume that the adsorption isotherm not only describes the bulk/surface equilibrium for a mixture of i components but also the equilibrium state for the mixture of different states of the same substance in the surface layer, i.e., the equilibrium between the molar fraction of the dissolved component in the bulk phase and the molar fractions of i different states of this component in the surface layer. These states can differ from each another, i.e., in the molar areas. It is assumed that the probability distribution for the states is well-defined. According to first order for nonideality of enthalpy and entropy we find for the solvent imax Γi ωi + 1 ln f0s x0s = ln 1 − imin
− ω0
imax imin
Γi + a
i max
2 Γi ωi
.
(5)
imin
Here the first three terms stem from the entropy of mixing [10], and the last term is the Frumkin correction for the enthalpy, assuming all states have the same value of the Frumkin constant, a. Similarly, we find for each state j of the molecule in the adsorption layer [10,11] imax s s s ωj (ω0 Γi ) 1+ ln fj xj /fj 0 = ln(ω0 Γj ) − ω0 imin 2 i max ωj +a (6) (Γi ωi ) − 1 . ω0 imin
The surface equation of state is obtained by substituting Eq. (5) into Eq. (3): Πω0 = ln(1 − ΓΣ ω) + ΓΣ ω − ΓΣ ω0 + a(ΓΣ ω)2 . (7) RT Here ΓΣ = 11 Γi is the total adsorption in all n states, and ω is the average molar area of all states calculated from the relationship i max ωi Γi ΓΣ , ω= (8) −
imin
with imin = ωmin /ω0 , imax = ωmax /ω0 , and n = imax − imin . The adsorption isotherm for each state is obtained by substituting Eq. (6) into Eq. (4) and is expressed in terms
181
of the constant bi = (ω0 Γi /c)Π=0 rather than Ki :
Γi ω0 ωi exp − (2aΓ ω) . (9) bi c = Σ (ωi /ω1 )α (1 − ΓΣ ω)ωi /ω0 ω0 However, the adsorption constants, bi , of the different states cannot be measured separately. The only measurable quantity is ΓΣ making up the total adsorption and, therefore, only the sum b = bi can be defined as a measurable parameter. We assume that the adsorption constant increases with increasing molar area ωi , according to a power law with a constant exponent α. We can now relate all bi to the state 1 (with ω1 = ωmin ) characterized by the minimum partial molar area
α ωi , bi = b1 (10) ω1 where ωi = ω1 + (i − 1)ω, and therefore the increment of area is defined by ω = (ωmax − ω1 )/(n − 1) [2]. Note that the variation of the molar area for adjacent [ith and (i + 1)th] states of the molecule in the surface layer is assumed to be discrete with the increment equal to ω0 . The total adsorption amount in Eqs. (7) and (9) can be expressed via the adsorption in state 1 with ω1 = ωmin imax
ωi α ΓΣ = Γ1 (1 − ΓΣ ω)(ωi −ω1 )/ω0 ω1 imin
ωi − ω1 × exp (11) (2aΓΣ ω) . ω0 The distribution of adsorptions over the states is then given by the expression Γi = ΓΣ ωi α
1 (2aΓΣ ω) (1 − ΓΣ ω)(ωi −ω1 )/ω0 exp ωiω−ω 0 × i ω α
. max i (ωi −ω1 )/ω0 exp ωi −ω1 (2aΓ ω) Σ imin ω1 (1 − ΓΣ ω) ω0 ω1
(12) Assuming that a = 0, one can rewrite Eqs. (7) and (9) as Πω0 = ln(1 − ΓΣ ω) + ΓΣ ω − ΓΣ ω0 , (13) RT Γ1 ω0 b1 c = (14) , (1 − ΓΣ ω)ω1 /ω0 where we assumed i = 1 in Eq. (14) for definiteness, and b1 = b exp(ω1 /ω). Earlier, another method for the solution of Eqs. (3) and (4) for reorienting surfactant molecules was proposed [4,6]. It was assumed that the molar area of the surfactant is equal to the average molar area, i.e., ω0 = ω = (ω1 Γ1 + ω2 Γ2 )/ΓΣ . In this case the contribution of the nonideality of entropy vanishes. Then, the adsorption isotherms for molecules existing in two states (1 and 2) in an ideal (with respect to enthalpy) surface layer become −
Πω = ln(1 − ΓΣ ω), RT Γ1 ω b1 c = . (1 − ΓΣ ω)ω1 /ω −
(15) (16)
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Table 1 Model parameters Cn EOm (models) C10 EO8 (1) C10 EO8 (2) C10 EO8 (3) C10 EO8 (4) C12 EO8 (1) C12 EO8 (2) C12 EO8 (3) C12 EO8 (4) C14 EO8 (1) C14 EO8 (2) C14 EO8 (3) C14 EO8 (4) C12 EO6 (1) C12 EO6 (2) C12 EO6 (4)
ω0 (105 m2 /mol)
ω1 (105 m2 /mol)
ω2 (105 m2 /mol)
3.7 1.0
3.94 3.7 3.5
10.2 11.1 7.0
3.3 1.0
3.42 3.3 3.1
10.0 9.9 10.0
3.0 1.0
3.0 3.0 2.8
13.4 10.5 10.0
1.0
3.2 2.5
9.2 4.5
α
5.6 3.0 2.5 4.82 2.75 1.5 4.62 2.76 1.7 3.85
Fig. 1. Dependence of equilibrium surface pressure on the concentration of C10 EO8 . (1) Data from [12] at 25 ◦ C; (E) data from [13] at 25 ◦ C. Theoretical curves were calculated from Models 1 (thick dashed line), 2 (thick continuous line), 3 (thin dashed line), and 4 (thin continuous line). For parameters, see Table 1.
Here the ratio of adsorptions in the state with minimum (ω1 ) and maximum (ω2 ) molar areas is expressed by a relation which follows from Eq. (12):
α ω2 − ω1 Π(ω2 − ω1 ) Γ2 ω2 = exp exp − . (17) Γ1 ω ω1 RT It was noted above that the choice of ω0 = const is incompatible with Eq. (1); however, for practical purposes this may be justified when the difference ω2 − ω1 is small as compared to ω. Let us compare now the results of calculations from theoretical models with experimentally measured isotherms of oxethylated alcohols Cn EOm . The theoretical models employed were: Model 1—Szyszkowski–Langmuir equation, where ω0 = ωL and a = 0, ΠωL = ln(1 − Γ ωL ) = − ln(1 + bc), RT Γ ωL . bc = (1 − Γ ωL )
−
ωL (105 m2 /mol)
(18) (19)
1.5
Fig. 2. The same as in Fig. 1 for C12 EO8 . (1) Data from [12] at 25 ◦ C; (E) data from [14] at 25 ◦ C.
Model 2—Eqs. (15) and (16), where a = 0 and ω0 is assumed to equal the average molar area of two states, i.e., ω0 = ω = (ωmin Γ1 + ωmax Γ2 )/ΓΣ ; Model 3—Eqs. (12)–(14), with a = 0 and ω0 assumed to be the minimum area of the surfactant molecule, i.e., ω0 = ω1 = ωmin ; and Model 4—Eqs. (12)–(14), with a = 0 and ω0 estimated from the molar area of a bulk-phase water molecule, e.g., ω0 = 105 m2 /mol. Model 4 is the most general, as it accounts for constant unequal values of ω0 , ωmin , and ωmax . In Fig. 1 the experimental data from the literature for C10 EO8 solutions [12,13] are compared with values calculated from Models 1–4, with optimum values of the model parameters given in Table 1. The literature data obtained for C12 EO8 [12,14] are compared with the Models 1–4 in Fig. 2. The best agreement with the experiments is given by Models 2 and 4, while the deviations for Models 1 and 3 are quite large, not surprisingly perhaps in view of the cruder nature of the latter models. For C14 EO8 , only Models 2 and 4 provide good agreement with the experiments [12], as shown in Fig. 3. In Fig. 4 the experimental data for C12 EO6 solutions [15–18] are
V.B. Fainerman et al. / Journal of Colloid and Interface Science 261 (2003) 180–183
183
surface pressure isotherms. From the approximate Models 2 and 3, only Model 2 agrees with the experiment (even better than Model 4), but this model involves the extra parameter α.
3. Conclusions
Fig. 3. The same as in Fig. 1 for C14 EO8 . (1) Data from [12] at 25 ◦ C.
Several reorientation models are proposed, in which the molar area of the solvent and the nonideality of entropy of mixing in the surface layer are taken into account in different ways. Experimental surface pressure data for nonionic Cn EOm surfactants reported in literature were analyzed by the different models and it was shown that the best agreement with the experiment is given by models in which the molar surface area of the solvent is either constant at a value estimated from the molar area of bulk water or equated to the nonconstant average value of the two states of the Cn EOm molecule in the surface layer.
Acknowledgments The work was financially supported by projects of the ESA (FAST and FASES), the DFG (Mi 418/11), and the Ukraine SFFR (03.07/00227). Fig. 4. The same as in Fig. 1 for C12 EO6 . (1) Data from [15] at 25 ◦ C; (E) data from [16] at 25 ◦ C; (P) data from [17] at 20 ◦ C; (!) data from [18] at 25 ◦ C. Theoretical curves were calculated from Models 1, 2, and 4. For parameters, see Table 1.
compared to the surface pressures calculated from the Models 1, 2, and 4, again with best results for Models 2 and 4. Several authors have noted that the Frumkin model— i.e., ω0 = ωL as in Model 1, but a = 0—leads to a better agreement with the experimental data for negative values of the Frumkin constant a [4,6,17]. The theoretical model proposed in [15] is based on the same first-order model for nonideal entropy and enthalpy we have used here. This treatment differs from the present model in two aspects: it does not assume reorientation into different areas for the surfactant, and it does not assume a zero value for the Frumkin constant a. This model used three parameters (ω1 , b1 , and a) and produced very good agreement with static and dynamic experiments on C10 EO6 solutions for a negative value of the Frumkin constant. The present reorientation models avoid negative values of the Frumkin constant by assuming a = 0. The most general model is Model 4, which describes the static experimental results for oxethylated alcohols very well. Because for this model the parameter α = 0, only three parameters (ωmin , ωmax , and b) are necessary to describe the experimental
References [1] P. Joos, Dynamic Surface Phenomena, VSP, Utrecht, 1999. [2] V.B. Fainerman, E.H. Lucassen-Reynders, R. Miller, Colloids Surf. A 143 (1998) 141. [3] V.B. Fainerman, R. Miller, R. Wüstneck, A.V. Makievski, J. Phys. Chem. 100 (1996) 7669. [4] V.B. Fainerman, R. Miller, R. Wüstneck, J. Phys. Chem. B 101 (1997) 6479. [5] M. Ferrari, L. Liggieri, F. Ravera, J. Phys. Chem. B 102 (1998) 10521. [6] V.B. Fainerman, R. Miller, E.V. Aksenenko, Langmuir 16 (2000) 4196. [7] R. Miller, V.B. Fainerman, H. Möhwald, J. Colloid Interface Sci. 247 (2002) 193. [8] J.A.V. Butler, Proc. R. Soc. Ser. A 138 (1932) 348. [9] V.B. Fainerman, E.H. Lucassen-Reynders, Adv. Colloid Interface Sci. 96 (2002) 295. [10] E.H. Lucassen-Reynders, Colloids Surf. A 91 (1994) 79. [11] V.B. Fainerman, E.H. Lucassen-Reynders, R. Miller, Adv. Colloid Interface Sci., submitted for publication. [12] M. Ueno, Y. Takasawa, H. Miyashige, Y. Tabata, K. Meguro, Colloid Polym. Sci. 259 (1981) 761. [13] H.C. Chang, C.-T. Hsu, S.-Y. Lin, Langmuir 14 (1998) 2476. [14] S.-Y. Lin, R.-Y. Tsay, L.-W. Lin, S.-I. Chen, Langmuir 12 (1996) 6530. [15] E.H. Lucassen-Reynders, A. Cagna, J. Lucassen, Colloids Surf. A 186 (2001) 63. [16] S. Ozeki, T. Ikegava, S. Inokuma, T. Kuwamura, Langmuir 5 (1989) 222. [17] J. Lucassen, D. Giles, J. Chem. Soc. Faraday Trans. 71 (1975) 217. [18] B.V. Zhmud, F. Tiberg, J. Kizling, Langmuir 16 (2000) 2557.
Comparison of various models describing the adsorption of surfactant molecules capable of interfacial reorientation V.B. Fainerman,a S.A. Zholob,a E.H. Lucassen-Reynders,b and R. Miller c,∗ a Medical Physicochemical Centre, Donetsk Medical University, 16 Ilych Avenue, Donetsk 83003, Ukraine b Mathenesselaan 11, 2343 HA Oegstgeest, The Netherlands c Max-Planck-Institut für Kolloid und Grenzflächenforschung, 14424 Potsdam/Golm, Germany
Received 24 July 2002; accepted 19 December 2002
Abstract The thermodynamic model for describing the adsorption of surfactant molecules in different adsorption states, the reorientation model, is reconsidered on a more rigorous level. The resulting model equations are used to describe experimental surface pressure data published in the literature. The new model proposed contains three physical parameters and describes the experimental dependencies Π(c) for oxethylated alcohols very accurately. 2003 Elsevier Science (USA). All rights reserved. Keywords: Adsorption isotherm; Surface equation of state; Surface reorientation; Oxethylated nonionic surfactants; Surface tension isotherms
1. Introduction For asymmetric surfactant molecules, which are able to adsorb in two or more states, a self-regulation mechanism leads to a preferential accumulation of states with large partial molar surface area at low surface pressure [1,2]. At high surface pressures, on the other hand, the majority of adsorbed surfactant molecules can occupy only a minimal molar area. Equations of state and adsorption isotherms for such systems were proposed earlier [2–4]. The hydrophilic part of oxethylated surfactants exhibits an adsorption activity at low surface layer coverage. The reorientation theory satisfactorily describes the behavior of oxethylated alcohols Cn EO8 at the water/air and water/oil interfaces [5–7]. At low concentrations the area occupied by these molecules is three to four times larger than that characteristic for higher concentrations. This feature of oxethylated alcohols results in an extremely high surface activity at low concentrations and an unusual shape of the surface tension isotherms not compatible with the Langmuir model. Although the reorientation model can very successfully be applied to various surfactants, there is one aspect requiring additional analysis. In [2–4] the equations of state and * Corresponding author.
E-mail address: [email protected] (R. Miller).
adsorption isotherms of the reorientation model were derived from the Butler equation with some simplifications. The object of this work is to give a more rigorous derivation of the model equations and to compare experimental data for molecules that can occupy a number of adsorption states with various theoretical models.
2. Results and discussion From the equations for chemical potentials µsi and µαi of the components of a mixture of different substances in the surface layer (Butler’s equation [8]), s s µsi = µ0s i + RT ln fi xi − γ ωi ,
(1)
and in the solution bulk α α µαi = µ0α i + RT ln fi xi ,
(2)
the most general equations of state for the surface layer and the adsorption isotherm for the ith component can be obtained [2,9]. For ideally dilute solutions RT s ln x0 + ln f0s , ω0 fis xis /fi0s ωi s ln ln x0 + ln f0s . = α Ki xi ω0 Π =−
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00226-6
(3) (4)
V.B. Fainerman et al. / Journal of Colloid and Interface Science 261 (2003) 180–183
Here R is the gas law constant, T is the temperature, fi are the activity coefficients standardized at fi = 1 at xi = 1, xi = Ni / Ni are the molar fractions, and Ni are the numbers of moles of the ith component, Π = γ0 − γ is the surface pressure, γ0 and γ are the surface tension of solvent and solution, respectively; Ki is the adsorption equilibrium constant defined at infinite dilution where Π = 0. The subscript 0 refers to the solvent, the superscript 0 denotes the standard state, and the superscripts s and α indicate the surface layer and bulk phase, respectively. It should be noted that the condition ω0 = const (or γ00 ω00 − γ ω0 = Πω0 ) is used to obtain Eqs. (3) and (4). We assume that the adsorption isotherm not only describes the bulk/surface equilibrium for a mixture of i components but also the equilibrium state for the mixture of different states of the same substance in the surface layer, i.e., the equilibrium between the molar fraction of the dissolved component in the bulk phase and the molar fractions of i different states of this component in the surface layer. These states can differ from each another, i.e., in the molar areas. It is assumed that the probability distribution for the states is well-defined. According to first order for nonideality of enthalpy and entropy we find for the solvent imax Γi ωi + 1 ln f0s x0s = ln 1 − imin
− ω0
imax imin
Γi + a
i max
2 Γi ωi
.
(5)
imin
Here the first three terms stem from the entropy of mixing [10], and the last term is the Frumkin correction for the enthalpy, assuming all states have the same value of the Frumkin constant, a. Similarly, we find for each state j of the molecule in the adsorption layer [10,11] imax s s s ωj (ω0 Γi ) 1+ ln fj xj /fj 0 = ln(ω0 Γj ) − ω0 imin 2 i max ωj +a (6) (Γi ωi ) − 1 . ω0 imin
The surface equation of state is obtained by substituting Eq. (5) into Eq. (3): Πω0 = ln(1 − ΓΣ ω) + ΓΣ ω − ΓΣ ω0 + a(ΓΣ ω)2 . (7) RT Here ΓΣ = 11 Γi is the total adsorption in all n states, and ω is the average molar area of all states calculated from the relationship i max ωi Γi ΓΣ , ω= (8) −
imin
with imin = ωmin /ω0 , imax = ωmax /ω0 , and n = imax − imin . The adsorption isotherm for each state is obtained by substituting Eq. (6) into Eq. (4) and is expressed in terms
181
of the constant bi = (ω0 Γi /c)Π=0 rather than Ki :
Γi ω0 ωi exp − (2aΓ ω) . (9) bi c = Σ (ωi /ω1 )α (1 − ΓΣ ω)ωi /ω0 ω0 However, the adsorption constants, bi , of the different states cannot be measured separately. The only measurable quantity is ΓΣ making up the total adsorption and, therefore, only the sum b = bi can be defined as a measurable parameter. We assume that the adsorption constant increases with increasing molar area ωi , according to a power law with a constant exponent α. We can now relate all bi to the state 1 (with ω1 = ωmin ) characterized by the minimum partial molar area
α ωi , bi = b1 (10) ω1 where ωi = ω1 + (i − 1)ω, and therefore the increment of area is defined by ω = (ωmax − ω1 )/(n − 1) [2]. Note that the variation of the molar area for adjacent [ith and (i + 1)th] states of the molecule in the surface layer is assumed to be discrete with the increment equal to ω0 . The total adsorption amount in Eqs. (7) and (9) can be expressed via the adsorption in state 1 with ω1 = ωmin imax
ωi α ΓΣ = Γ1 (1 − ΓΣ ω)(ωi −ω1 )/ω0 ω1 imin
ωi − ω1 × exp (11) (2aΓΣ ω) . ω0 The distribution of adsorptions over the states is then given by the expression Γi = ΓΣ ωi α
1 (2aΓΣ ω) (1 − ΓΣ ω)(ωi −ω1 )/ω0 exp ωiω−ω 0 × i ω α
. max i (ωi −ω1 )/ω0 exp ωi −ω1 (2aΓ ω) Σ imin ω1 (1 − ΓΣ ω) ω0 ω1
(12) Assuming that a = 0, one can rewrite Eqs. (7) and (9) as Πω0 = ln(1 − ΓΣ ω) + ΓΣ ω − ΓΣ ω0 , (13) RT Γ1 ω0 b1 c = (14) , (1 − ΓΣ ω)ω1 /ω0 where we assumed i = 1 in Eq. (14) for definiteness, and b1 = b exp(ω1 /ω). Earlier, another method for the solution of Eqs. (3) and (4) for reorienting surfactant molecules was proposed [4,6]. It was assumed that the molar area of the surfactant is equal to the average molar area, i.e., ω0 = ω = (ω1 Γ1 + ω2 Γ2 )/ΓΣ . In this case the contribution of the nonideality of entropy vanishes. Then, the adsorption isotherms for molecules existing in two states (1 and 2) in an ideal (with respect to enthalpy) surface layer become −
Πω = ln(1 − ΓΣ ω), RT Γ1 ω b1 c = . (1 − ΓΣ ω)ω1 /ω −
(15) (16)
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Table 1 Model parameters Cn EOm (models) C10 EO8 (1) C10 EO8 (2) C10 EO8 (3) C10 EO8 (4) C12 EO8 (1) C12 EO8 (2) C12 EO8 (3) C12 EO8 (4) C14 EO8 (1) C14 EO8 (2) C14 EO8 (3) C14 EO8 (4) C12 EO6 (1) C12 EO6 (2) C12 EO6 (4)
ω0 (105 m2 /mol)
ω1 (105 m2 /mol)
ω2 (105 m2 /mol)
3.7 1.0
3.94 3.7 3.5
10.2 11.1 7.0
3.3 1.0
3.42 3.3 3.1
10.0 9.9 10.0
3.0 1.0
3.0 3.0 2.8
13.4 10.5 10.0
1.0
3.2 2.5
9.2 4.5
α
5.6 3.0 2.5 4.82 2.75 1.5 4.62 2.76 1.7 3.85
Fig. 1. Dependence of equilibrium surface pressure on the concentration of C10 EO8 . (1) Data from [12] at 25 ◦ C; (E) data from [13] at 25 ◦ C. Theoretical curves were calculated from Models 1 (thick dashed line), 2 (thick continuous line), 3 (thin dashed line), and 4 (thin continuous line). For parameters, see Table 1.
Here the ratio of adsorptions in the state with minimum (ω1 ) and maximum (ω2 ) molar areas is expressed by a relation which follows from Eq. (12):
α ω2 − ω1 Π(ω2 − ω1 ) Γ2 ω2 = exp exp − . (17) Γ1 ω ω1 RT It was noted above that the choice of ω0 = const is incompatible with Eq. (1); however, for practical purposes this may be justified when the difference ω2 − ω1 is small as compared to ω. Let us compare now the results of calculations from theoretical models with experimentally measured isotherms of oxethylated alcohols Cn EOm . The theoretical models employed were: Model 1—Szyszkowski–Langmuir equation, where ω0 = ωL and a = 0, ΠωL = ln(1 − Γ ωL ) = − ln(1 + bc), RT Γ ωL . bc = (1 − Γ ωL )
−
ωL (105 m2 /mol)
(18) (19)
1.5
Fig. 2. The same as in Fig. 1 for C12 EO8 . (1) Data from [12] at 25 ◦ C; (E) data from [14] at 25 ◦ C.
Model 2—Eqs. (15) and (16), where a = 0 and ω0 is assumed to equal the average molar area of two states, i.e., ω0 = ω = (ωmin Γ1 + ωmax Γ2 )/ΓΣ ; Model 3—Eqs. (12)–(14), with a = 0 and ω0 assumed to be the minimum area of the surfactant molecule, i.e., ω0 = ω1 = ωmin ; and Model 4—Eqs. (12)–(14), with a = 0 and ω0 estimated from the molar area of a bulk-phase water molecule, e.g., ω0 = 105 m2 /mol. Model 4 is the most general, as it accounts for constant unequal values of ω0 , ωmin , and ωmax . In Fig. 1 the experimental data from the literature for C10 EO8 solutions [12,13] are compared with values calculated from Models 1–4, with optimum values of the model parameters given in Table 1. The literature data obtained for C12 EO8 [12,14] are compared with the Models 1–4 in Fig. 2. The best agreement with the experiments is given by Models 2 and 4, while the deviations for Models 1 and 3 are quite large, not surprisingly perhaps in view of the cruder nature of the latter models. For C14 EO8 , only Models 2 and 4 provide good agreement with the experiments [12], as shown in Fig. 3. In Fig. 4 the experimental data for C12 EO6 solutions [15–18] are
V.B. Fainerman et al. / Journal of Colloid and Interface Science 261 (2003) 180–183
183
surface pressure isotherms. From the approximate Models 2 and 3, only Model 2 agrees with the experiment (even better than Model 4), but this model involves the extra parameter α.
3. Conclusions
Fig. 3. The same as in Fig. 1 for C14 EO8 . (1) Data from [12] at 25 ◦ C.
Several reorientation models are proposed, in which the molar area of the solvent and the nonideality of entropy of mixing in the surface layer are taken into account in different ways. Experimental surface pressure data for nonionic Cn EOm surfactants reported in literature were analyzed by the different models and it was shown that the best agreement with the experiment is given by models in which the molar surface area of the solvent is either constant at a value estimated from the molar area of bulk water or equated to the nonconstant average value of the two states of the Cn EOm molecule in the surface layer.
Acknowledgments The work was financially supported by projects of the ESA (FAST and FASES), the DFG (Mi 418/11), and the Ukraine SFFR (03.07/00227). Fig. 4. The same as in Fig. 1 for C12 EO6 . (1) Data from [15] at 25 ◦ C; (E) data from [16] at 25 ◦ C; (P) data from [17] at 20 ◦ C; (!) data from [18] at 25 ◦ C. Theoretical curves were calculated from Models 1, 2, and 4. For parameters, see Table 1.
compared to the surface pressures calculated from the Models 1, 2, and 4, again with best results for Models 2 and 4. Several authors have noted that the Frumkin model— i.e., ω0 = ωL as in Model 1, but a = 0—leads to a better agreement with the experimental data for negative values of the Frumkin constant a [4,6,17]. The theoretical model proposed in [15] is based on the same first-order model for nonideal entropy and enthalpy we have used here. This treatment differs from the present model in two aspects: it does not assume reorientation into different areas for the surfactant, and it does not assume a zero value for the Frumkin constant a. This model used three parameters (ω1 , b1 , and a) and produced very good agreement with static and dynamic experiments on C10 EO6 solutions for a negative value of the Frumkin constant. The present reorientation models avoid negative values of the Frumkin constant by assuming a = 0. The most general model is Model 4, which describes the static experimental results for oxethylated alcohols very well. Because for this model the parameter α = 0, only three parameters (ωmin , ωmax , and b) are necessary to describe the experimental
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