AGGREGATION AND ADSORPTION AT THE AIR-SOLUTION INTERFACE OF THE CETYLTRIMETHYLAMMONIUM TOSYLATE WITHTWO POLY(OXYETYLENE)-POLY(OXYPROPYLENE)- POLY(OXYETYLENE) BLOCK COPOLYMERS AQUEOUS MIXTURES
Maximiliano Brigante, Pablo C. Schulz
THE MODEL OF MIXED MICELLES BASED ON THE REGULAR SOLUTION THEORY AND THE MICELLISATION AS PHASE SEPARATION.
The regular solution theory has been widely used to model the thermodynamic nonidealities of mixed micelles; it has been shown to accurately model critical micelle concentration (CMC) values [1] and monomer-micelle equilibrium compositions [2] in surfactant systems exhibiting negative deviations from ideality. However, it must be pointed out that the theoretical validity of using regular solution theory to describing nonideal mixing in mixed surfactant micelles has been questioned [3]. Although this theory assumes that the excess entropy of mixing is zero, it has been demonstrated that in some surfactant mixtures this assumption is not true [4]. However, the pseudophase separation model of micellisation and regular solution theory combination remains as a very widely used and convenient method for analyzing experimental data[5-7].
In this model is considered a mixture of two different surfactants A and B form micelles with composition XmA and XmB, in equilibrium with solution monomers of composition A and B. These mole fractions are on a surfactant-only basis, so that
XMA + XMB = 1(1)
A + B = 1(2)
At the CMC [8]:
AfmA CMCA =XMAfMACMCmix(3)
BfmB CMCB =XMBfMBCMCmix(4)
where fmA and fMA are the activity coefficients of A in the intermicellar solution and in micelles, CMCA, CMCB and CMCmix are the CMC of pure surfactants A and B, and the mixture. Each surfactant monomeric form is assumed to be dilute enough to obey Henry’s law, i.e., based on the infinite dilution standard state, surfactant monomer activity coefficients are unity.
The CMCmix value can be calculated as a function of the composition by the expression:
CMCmix = _ A _ + _ B _-1(5)
fMACMCA fMB CMC2B
In the ideal approximation, fMA= fMB = 1, then Eq. (5) reduces to [5, 9, 10]:
CMCmix = _ A _ + _ B _-1(6)
CMCA CMCB
In applying regular solution theory to mixed micelles, the micellar activity coefficients are given by [11]:
fMA = exp (M XMB 2)
(7)
fMB = exp (M XMA 2)
where M is the dimensionless regular solution theory intramicellar interaction parameter (in kBT units), kB is the Boltzmann constant and T the absolute temperature. In Equations (7) pure surfactant micelles are considered to be at the standard state.
Equation (8) relating the mixture CMCmix), CMCA, CMCB, Ais solved for each A value to obtain XMA[12]:
F = (XMA)2 ln (A CMCmix/XMACMCA) - 1 = 0(8)
(1 – XMA)2 ln [(1 - A)CMCmix/(1- XMA)CMCB]
The ideal composition of mixed micelles may be found with Motomura and Aratono equation [13]:
XM,idA = _ ACMCB _(9)
ACMCB + BCMCA
Then, the interaction parameter M is obtained by equation [12]:
M /kBT = ln (A CMCmix/XMACMCA) (10)
(1 – XMA)2
The nature and strength of the interaction between two surfactants are measured by the value of the M parameter, which is a measure of the degree of non-ideality of the interaction in a mixed micelle. The larger the negative value of M, the stronger the attractive interaction between the two different surfactant molecules, and the greater is the probability of the existence of synergism between them [14]. Repulsive interactions yield a positiveM value and the possibility of antagonism, whereas M = 0 indicates an ideal mixture. Positive M values occur in mixtures of fluorocarbon-hydrocarbon surfactants [15].
The minimum error for M in a single determination is nearly 0.1kBT. The error is strongly increasing when one component in the micelle dominates[2].
Theoretically, M is independent on both temperature and composition of the micelle. However, in practice Mis temperature dependent [16-18] and it often depends on the micelle composition [20, 21], so an average value is commonly used. The nature and strength of the interaction between two surfactants are measured by the value of the Mparameter, which informs about the degree of non-ideality of the interaction in a mixed micelle, through a single number that can be easily compared among different pairs of surfactants. The larger the negative value of M, the stronger the attractive interaction between the two different surfactant molecules and the greater is the probability of the existence of synergism between them [14]. Repulsive interactions yield a positive M value and the possibility of antagonism, whereas m = 0 indicates an ideal mixture. Positive M values occur in mixtures of fluorocarbon-hydrocarbon surfactants [15].
The parameter m reflects the two main contributions to the excess Gibbs free energy of mixed micellisation. These are a free-energy contribution associated with the interactions between the hydrophobic groups of surfactants A and B in the micelle core, Mcore, and an electrostatic contribution Melec, associated with electrostatic interactions between the charged hydrophilic groups of the surfactants [19]:
M = Mcore + Melec(11)
It is noteworthy that Mcore is typically considered equal to zero for mixtures of two hydrocarbon based (or fluorocarbon based) surfactants [20, 21]. However, recently it was stated that this assumption is not true if the two chains are of different length and depends on the difference in number of carbon atoms nC [22, 23] and the interaction is attractive (i.e., Mcoreis negative if nC 0). The Mcorevalue is typically positive for a binary mixture of hydrocarbon and fluorocarbon surfactants due to the repulsive interactions in the micellar core [24-26].
The parameter M is related to the molecular interactions in the mixed micelle by [12]:
M = NA(WAA + WBB - 2WAB)(12)
where WAA and WBB are the energies of interaction between molecules in the pure micelle and WAB is the interaction between the two species in the mixed micelle. NA is the Avogadros’s number.
Typical values of M are +2.2 for lithium dodecylsulfate - lithium perfluorooctanesulfonate [27] and -3.9 for the system SDS- poly (oxyethylene) (4) dodecylether [12]. For typical anionic – cationic surfactants mixtures M is about –20 [28] or -13.2 for sodium decylsulfate-decyltrimethylammonium bromide [6]. However, small m values were found for some particular anionic – cationic surfactant aqueous mixtures, as CTAB-sodium deoxycholate (M = -2.7 [29]), CTAB - sodium cholate (M = -4.0 [30]), and dodecyltrimethylammonium bromide –disodium dodecanephosphonate (M = - 1.66 [30]). All of these systems have some structural characteristics that are different to the most commonly studied cationic-anionic mixtures.
From the values of M and XMA the activity coefficient of each surfactant (e.g. fmA for component A) in the mixed micelle may be obtained using equations (7). [31]:
It can be demonstrated within the regular solution theory that the chemical potential of mixing is given by
mixexcess = RT[XAln fMA + XBln fMB] = MRTXMAXMB(13)
where R is the gas constant. Negative values of mixexcess indicate the attraction between the two components in the micelles, most of which may result from a decrease in the electrostatic energy of the micelles. mixexcess is the difference between the partial molar free energy of the mixed micelles and that calculated according to the ideal behavior, as a function of the mixture composition. This energy is expected to depend much on the surface charge density of micelles and the ionic strength, and less on the size and shape of micelles [9, 32]. The mixexcess value does not take into account the change in the degree of association of the counterion upon surfactant mixing [16-18].
Synergism
Synergism in mixed micelle formation exists when CMCmix is lower that any of the components CMCs. Conditions for the existence of synergism are [33]:
- M must be negative(14)
- Mln (CMCA/CMCB)(15)
- S -M ln (CSA/CSB) - ln (CMCA/CMCB)(16)
where CSA and CSB are the concentrations of components A and B needed to produce a given surface tension value.
THE ADSORBED MIXED MONOLAYER AT THE INTERFACE AIR/SOLUTION
Two different procedures are available to analyze the mixed monolayer adsorbed at tha air/solution interface, both based on the regular solution theory.
To determine the composition of the monolayer adsorbed at the air/solution interface with the procedure proposed by Rosen and Hua [34], the following equation is numerically solved for XSA, the mole fraction of component A at the monolayer, without considering the solvent:
F = _ XSA2 ln (ACmix/XSACA) _ - 1 = 0(17)
(1-XSA)2 ln [(1- A)Cmix/(1-XSA)CB
where CA, CB and Cmix are the molar concentrations of the pure components A and Band their mixture at a given A value, respectively, needed to produce a fixed value of the surface tension, e.g., 40 mN.m-1. The interaction parameter in the adsorbed monolayer in kBT units is given by:
S = _ln (ACmix/XSACB)_(18)
(1- XSA)2
and the activity coefficient of each component in the monolayer is computed as:
fSA = exp (SXSB2)(19)
Alternatively, the interfasial behavior of the mixed system can also be treated by the extension of the pseudophase separation model for micelles, using a nonideal analog of Butler’s equation [35,36], giving [37]:
mix = RT ln(fMAXMA /fSAXSA) + A(20)
AA
in which AA is the area per mole of pure surfactant A at the interface [31], mix and A are the surface pressure at the CMC of the surfactant mixture and the component A, respectively; fSA and XSA are the activity coefficient and mole fraction of component A in the surface adsorbed state. When this equation was derived, the assumption thatAA does not change in surfactant surface mixtures was taken. The activity coefficient at the adsorbed monolayer is given by:
fSA = exp (SXSB2)(21)
fSB = exp (SXSA2)(22)
where S is a dimensionless parameter, interpreted as representing an excess free energy of mixing in the surfactant aggregate at the interface. Equations (20) to (22) together with the constraint that XSA +XSA = 1, and the measured values of A and AA, give the basis for iterative solution of the model, provided the micellar composition and activity coefficients were previously computed.
S can be viewed as empirically accounting for the free energy changes that occur in forming the mixed surfactant aggregate, including those due to any counterion effects, changes in molar areas on mixing and residual solvent effects at the interface. However, the significance of S as a proper measure of the magnitude of the excess free energy changes involved in the adsorption phenomenon is very uncertain [28].
Some literature reported S values of are -3.7kBT (decyl dimethylphosphine oxide (C10PO)- SDS in aqueous Na2CO3 1 mM); -3.0kBT (decyl methyl sulfoxide (C10MSO) - SDS in aqueous Na2CO3 1 mM); -0.3kBT (C10PO- C10MSO in aqueous Na2CO3 1 mM); -2.9kBT (tetraoxyethylene glycol monododecyl ether (C10E4) - SDS in aqueous Na2CO3 0.5 mM); -2.0kBT (C10E4 - dodecyl dimethylamine oxide (C12AO) in aqueous Na2CO3 0.5 mM); -7.2kBT (C12AO - SDS in aqueous Na2CO3 0.5 mM) and -19.7kBT (sodium decyl sulfate - decyl trimethylammonium bromide in 0.05 M NaBr) [37], and between -40kBT and -31kBT in some catanionic systems [28].
Synergism
Sugihara et al. [38,39] proposed an thermodynamic quantity to evaluate the synergism in the mixed monolayer at the air/solution interface, Gºmin, the Gibbs free energy for a given air/solution interface, defined as:
Gºmin = AminCMCNA(23)
where Amin is the minimum area per adsorbed molecule Amin = 1020/NAmax, CMC being the surface tension at the CMC and max the maximum surface excess (mol/m2). Gºmin is considered as the work needed to produce the surface unit per mol, or the free energy change associated to the transition of the solutions components from bulk to the interface. The lower the Gºmin value, the easier will be the formation of the surface and the more thermodynamically stable will be this surface.
Conditions for the existence of synergism in the air/solution interface, i.e., for the efficiency in the reduction of the surface tension in surfactant mixtures were also proposed as [40]:
- S must be negative(24)
- Sln (CSA/CSB)(25)
where CSA and CSB are the concentrations of components A and B needed to generate a given value of surface tension.
THE MICELLE IONIZATION DEGREEE
The micelle ionization degree () was determined with the equation:
α ≈ [(d)M/dC]/[(d)m/dC](26)
whered/dC is the slope of the specific conductivity dependence on the total concentration, M and m meaning the micellar and monomer regions. This equation is based on the assumption that micelles contribution to conductivity is negligible, although this assumption is not strictly true [41].
THE AREA PER ADSORBED SURFACTANT MOLECULE
The surface excess was computed by the Gibbs equation:
= - _ 1 _. _ _(27)
nRT ln C
where is the surface tension, C the total surfactant mixture concentration and n the number of species capable of adsorption at the air/solutions interface per molecule of surfactant. In Poloxamer –ionic surfactant mixtures we used n = 2αCTAT + (1- αCTAT). Then, the area per adsorbed surfactant molecule was computed as:
A0 = 1/NA(28)
NA being the Avogadro’s number. When is computed at the CAC, A0 is the minimal area per surfactant molecule. In mixtures, this is an average value. The ideal value was computed with:
A0,ideal = αCTATA0,CTAT + (1-αCTAT)A0,Pluronic(29)
and the value computed with the surface composition was computed with:
A0,computed = XsCTATA0,CTAT + (1-XsCTAT)A0,Pluronic(30)
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