3108
J. Phys. Chem. 1996, 100, 3108-3114
Phase Transition from a Liquid Expanded to a Liquid Condensed Surfactant Monolayer Eli Ruckenstein* and Buqiang Li Department of Chemical Engineering, State UniVersity of New York at Buffalo, Amherst, New York 14260 ReceiVed: May 25, 1995; In Final Form: October 19, 1995X
Liquid monolayers of insoluble surfactants at the air/water interface are treated as a two-dimensional solution made up of molecules with disordered chains and clusters of molecules with ordered chains. The FloryHuggins equation is used to calculate the activities of the species. On this basis, the phase transition from the liquid expanded to the liquid condensed state is predicted. Depending on the value of the interaction parameter between disordered and ordered molecules and the extent of clustering, the theoretical π-A isotherms exhibit either a first-order phase transition or a nonhorizontal, second-order, phase transition. For the tetradecanoic, pentadecanoic, hexadecanoic, and heptadecanoic fatty acids, the theoretical π-A isotherms are in agreement with experimental data. It should be emphasized that the theory is predictive since all the parameters are calculated on the basis of the molecular structure.
Introduction For the prediction of the phase transitions which occur in an insoluble surfactant monolayer or a bilayer membrane, numerous theoretical models have been suggested.1-15 The models have been applied to the phase transition from a liquid expanded to a liquid condensed monolayer and also to the solid-fluid melting transition. Two classes of models have been suggested. The first class1-10 considers that the phase transition is due to both the intramolecular and intermolecular interaction energies. The former interaction is a result of the gauche rotations of the bonds in the hydrocarbon chains and contributes to the conformational energy and entropy of the chains. As concerns the intermolecular interactions, various assumptions have been made. For monolayers with close-packed hydrocarbon chains, the intermolecular interactions were taken as van der Waals anisotropic interactions.6-10 Assuming the presence of vacancies, Nagle4 used a density-dependent van der Waals interaction in the mean field approximation. Scott et al.5 accounted for the hydrogen-bonding interactions between the head groups and proposed an expression containing the fraction of hydrocarbon chains in the trans configuration as a parameter. The interchain steric hindrance repulsion was also taken into account.9,10 In this class of models, the results obtained for the transition from the liquid expanded to the liquid condensed state are only in qualitative agreement with experimental data.16 The second class of models involves a lattice description in which only intermolecular interactions are taken into account.11-15 The surfactant molecules are assumed as rods with two or three orientation states, which occupy either the same number or a different number of sites on the lattice. No molecular details are taken into account. A typical equation for this class of models was suggested by Bell at al.11 For a phospholipid monolayer at the oil-water interface, that equation provides a second-order phase transition from a liquid expanded to a liquid condensed state. The model fits the experimental data only qualitatively.16 Recently, the phase transition corresponding to a nonhorizontal region in the π-A isotherm was explained on the basis of the formation of molecular clusters. Ruckenstein and Bhakta17 presented a model for the adsorption isotherm which takes into account this clustering, and Ruckentein and Li18 * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-3108$12.00/0
developed on its basis a predictive model for the behavior of an insoluble surfactant monolayer. The insoluble surfactant monolayer at the air-water interface was assumed to be an ideal solution made up of singly dispersed surfactant molecules, clusters of surfactant molecules, and water molecules exposed to air. The hydrocarbon chains of the singly dispersed molecules were considered to behave as in a gaseous monolayer, while those in the interior of a cluster were considered to have an all-trans parallel conformation. A similar qualitative discussion was also presented by Israelachvili.19 Using the clustering model, the phase transition corresponding to the nonhorizontal lines in the π-A isotherms can be explained. The slope of the nonhorizontal lines decreases with the extent of clustering of the surfactant molecules. In this paper we examine the phase transition from a liquid expanded to a liquid condensed monolayer. The surfactant molecules are assumed to be of two kinds: namely, with disordered and ordered hydrocarbon chains (they will be called in what follows disordered and ordered molecules). A twodimensional nonideal solution theory is developed which involves disordered molecules and ordered ones, with the latter generating clusters. Finally a quantitative comparison with experimental data for the tetradecanoic, pentadecanoic, hexadecanoic, and heptadecanoic fatty acids at the air-water interface is made. Theory The liquid monolayer of an insoluble surfactant is regarded as a two-dimensional solution, free of water, made up of disordered molecules, singly dispersed ordered ones, and clusters of ordered molecules. No vacancies are allowed between the hydrocarbon chains; that is, the liquid monolayer is assumed close-packed. The disordered molecules have at least one gauche rotation in the hydrocarbon chain. The ordered molecules have an all-trans configuration and are oriented normal to the interface. The phase transition which occurs in the liquid monolayer is due to the change from the disordered to ordered molecules and the formation of clusters from the ordered molecules. In the two-dimensional solution, the disordered molecules act as a solvent (continuous phase) and the clusters as solutes (dispersed phase). The chemical potentials can be expressed © 1996 American Chemical Society
Phase Transitions in Surfactant Monolayers
J. Phys. Chem., Vol. 100, No. 8, 1996 3109
by the equations20
µd ) µd0 + kT ln(ad) - Ad0σ
(1)
for the disordered molecules and
iNiA0
φi )
i ) 1, 2, ... imax
jmax
(9)
NdAd0 + ∑jNjA0 j)1
µi ) µi0 + kT ln(ai) - iA0σ
i ) 1, 2, ... imax
(2)
for the clusters containing i ordered molecules. In eqs 1 and 2, µ0 is the standard chemical potential, a is the activity, σ is the surface tension, and A0d and A0 are the surface areas per molecule occupied by the disordered and ordered molecules, respectively. A0 is assumed to be a constant. Each ordered molecule is assumed to occupy the same surface area regardless of the cluster size, and therefore the surface area of a cluster of size i is iA0. imax represents the largest number of molecules in a cluster. At equilibrium, the chemical potentials per surfactant molecule must be the same regardless of whether the species are disordered or ordered. Therefore, from the equality µ1 ) µd, one obtains that the surface pressure is given by
π ) σw +
µ10 - µd0 Ad0 - A0
+
()
a1 kT ln Ad0 - A0 ad
(3)
where π ) σw - σ, σw being the surface tension of pure water. From the equality µi ) iµ1, one obtains
ai ) a1i
i ) 1, 2, ... imax
(4)
To use eq 3 for the calculation of the surface pressure, the difference of the standard chemical potentials between the ordered and disordered molecules (∆µ0 ) µ01 - µ0d), the surface areas of the molecules in the disordered and ordered states (A0d and A0), and the activities (a1 and ad) are required. They are calculated in what follows. Activities and Size Distribution. The Flory-Huggins equation is extended to the two-dimensional solution. For the disordered and ordered molecules, the activities are given by
φd ln(ad) ) ln(φd) + 1 - + ln(γrd) xd
(5)
and
ln(ai) ) ln(φi) + 1 -
φi + ln(γir) xi
i ) 1, 2, ... imax
(6)
where φ and x stand for the surface area fraction and mole fraction, respectively. The first three terms on the right-hand sides of eqs 5 and 6 represent the combinatorial contribution due to the mixing entropy of the two-dimensional solution. The last term, ln(γr), stands for the residual contribution resulting from the mixing enthalpy of the two-dimensional solution. The area fractions and the mole fractions are given by
NdAd0
φd )
jmax
(7)
NdAd0 + ∑jNjA0 j)1
xd )
Nd + ∑Nj j)1
for the disordered molecules, and by
(8)
Ni
i ) 1, 2, ... imax
jmax
(10)
Nd + ∑Nj j)1
for the clusters of size i. In eqs 7-10, Nd and Ni stand for the numbers of disordered molecules and clusters of size i, respectively. The summations are for the clusters; jmax or imax represent the largest number of molecules included in a cluster. For the calculation of the residual contributions (ln(γr)), an expression for the mixing enthalpy (∆Hmix) is required, which is derived below. Taking into account only the nearest neighbor interactions, the mixing enthalpy depends on the contacts between any two different species and the corresponding interaction exchange energy. A two-dimensional lattice model is used to represent the two-dimensional solution. An ordered molecule is considered to occupy a lattice site surrounded by z0 nearest neighboring sites. A disordered molecule or a cluster occupies a larger number of sites and has a larger number of nearest neighboring sites. The number of nearest neighboring sites zd and zj of a disordered molecule and a cluster of size j is assumed proportional to their perimeter pd and pj, respectively. Assuming a circular cross section, the following equations can be written:
( )
Ad0 zd pd ) ) 0 z0 p0 A
1/2
(11)
for the disordered molecules, and
z j pj ) ) j1/2 z0 p0
j ) 1, 2, ... jmax
(12)
for the clusters containing j ordered molecules, where p0 stands for the perimeter of an ordered molecule and z0 is the number of its nearest neighboring sites, which is equal to 6 for a hexagonal close-packed lattice. Since the clusters contain only ordered molecules and only the interactions between the nearest neighbors are taken into account, the interaction exchange energies between any two different size clusters should be 0. Indeed, ∆Wij ) Wij - (Wii + Wjj)/2 is 0 because in the nearest neighbor approximation Wij ) Wii ) Wjj, where ∆Wij and Wij are the interaction exchange energy and the interaction energy between i and j clusters, respectively. Therefore, the mixing enthalpy of the twodimensional solution depends only on the interactions between the disordered molecules and the clusters. Each disordered molecule has zd nearest neighboring sites, and the probability for one of these sites to be occupied by a cluster of size i is given by
yi )
Nd jmax
xi )
Nipi jmax
i ) 1, 2, ... imax
(13)
Ndpd + ∑Njpj j)1
In this expression, we assume that the local concentrations around each species are the average concentrations in the twodimensional solution. Consequently, ∆Hmix can be written as
3110 J. Phys. Chem., Vol. 100, No. 8, 1996
Ruckenstein and Li
jmax
∆Hmix ) Ndzd∑yj∆Wdj
(14)
j)1
Here ∆Wdj stands for the interaction exchange energy per contact between the disordered molecules and cluster j. Regardless of the size of the cluster j, the interaction exchange energy is the same, since only the nearest neighbor interactions are taken into account. Hence,
Wdd + Wtt ∆Wdj ) ∆Wdt ) Wdt 2
j ) 1, 2, ... jmax (15)
where N stands for the total number of surfactant molecules in the liquid monolayer. Dividing the two sides of eq 22 by the total surface area leads to
1
Γ)
)
A
φd Ad0
jmax
φj
j)1
A0
+∑
(23)
where Γ is the number of surfactant molecules per unit area (surface density) and A is the average surface area per surfactant molecule. Introducing the area fractions in eq 13, one obtains
where subscript t stands for the ordered molecules with the hydrocarbon chains in an all-trans configuration and Wdt stands for the interaction energy per contact between the disordered and ordered molecules. Using eq 15 and taking into account that
φd
yd ) φd +
( )∑ Ad0
1/2 jmax
A0
(24) φj
1/2 j)1 j
where
jmax
y d + ∑ yj ) 1
(16)
jmax
φd + ∑φj ) 1
j)1
eq 14 becomes
(25)
j)1
∆Hmix ) Ndzd(1 - yd)∆Wdt
(17)
Combining eqs 23 and 25, one obtains for the area fraction φd of the disordered molecules the expression
Inserting eq 11 into eq 17, the following equation is obtained:
( )
φd )
0 1/2
∆Hmix Ad ) 0 kT A
Nd(1 - yd)χ
(18)
where χ ) z0∆Wdt/kT represents the interaction parameter between the disordered and ordered molecules. Using eq 18, one can rewrite eqs 5 and 6 in the form
( )
φd Ad0 ln(ad) ) ln(φd) + 1 - + 0 xd A ln(ai) ) ln(φi) + 1 -
1/2
(1 - yd)2χ
φi + i1/2yd2χ xi
(19)
(20)
Combining eqs 4 and 20, the following expression for the equilibrium size distribution of clusters is obtained:
φi ) φ1i exp(i-1) exp[(i-i1/2)yd2χ]
i ) 1, 2, ... imax (21)
The size distribution depends on two factors. The first, φ1i exp(i-1), arises from the mixing entropy of the twodimensional solution. It opposes the formation of clusters, because the entropy is larger when the number of single ordered molecules is larger. The second factor has its origin in the mixing enthalpy of the two-dimensional solution. It favors the formation of clusters because the number of contacts between the disordered and ordered molecules is decreased by the formation of clusters and in the clusters the interactions are stronger. Of course, the number of disordered molecules and the number of ordered molecules should satisfy the following mass balance relation: jmax
N ) Nd + ∑jNj j)1
(22)
Ad0A0 Ad0 - A0
(Γ0 - Γ)
(26)
where Γ0 ) 1/A0 is the surface density of the liquid condensed monolayer. As expected, the area fraction of the disordered molecules becomes 0 when Γ approaches Γ0. Obviously, the total area fraction of the ordered molecules is 1 - φd. By solving numerically the system of eqs 21 and 23-25, the size distribution of clusters, corresponding to a given composition of the two-dimensional solution, can be obtained. Thus, the activities of the single ordered and disordered molecules (a1 and ad) which are needed for calculating the surface pressure (eq 3) can be obtained. Interaction Parameter (χ) and Molecular Areas (A0 and 0 A d). Interaction Parameter. Using the geometric rule for the interaction energy between the disordered and ordered molecules, Wdt ) (WddWtt)1/2, eq 15 becomes
1 ∆Wdt ) (Ct1/2 - Cd1/2)2 2
(27)
where Ct ) -Wtt and Cd ) -Wdd. The dispersive interaction energy between two parallel, alltrans saturated hydrocarbon chains is proportional to the number of carbons, and for the hexagonal close-packed lattice, the following expression is available (ref 16, p 70):
Wtt )
W0ttnc [2(A0/3)1/2]5
(28)
where W0tt ) -8.62 × 10-11 erg Å5/molecule, nc is the number of carbons of the hydrocarbon chain and A0 is the surface area per molecule in Å2. The interaction energy per contact between two disordered molecules is considered to depend on their average height, hd, and is approximated by the interaction energy between the alltrans hydrocarbon chains with the carbon number corresponding to the average height of the disordered molecules. Since the
Phase Transitions in Surfactant Monolayers
J. Phys. Chem., Vol. 100, No. 8, 1996 3111
liquid monolayer is close-packed, the average molecular height (or thickness of the liquid monolayer) and the average molecular surface area satisfy the following simple relation:
A0h ) V
(29)
where the volume V is equal to the volume of a surfactant molecule, A0 is the surface area (A0d or A0), and h is equal to hd for the disordered and h0 for the ordered molecules. The carbon number nd of the corresponding all-trans hydrocarbon chain for the average height of the disordered molecules is therefore given by nd/nc ) hd/h0 ) A0/A0d. The interaction energy per contact between two disordered molecules can be therefore expressed as in eq 28:
Wdd )
W0ttA0nc/Ad0 [2(A0/3)1/2]5
(30)
Inserting eqs 28 and 30 into eq 27, the interaction parameter between the disordered and ordered molecules becomes
χ)
z0(-Wtt0)nc
[1 - (A0/Ad0)1/2]2 2kT[2(A0/3)1/2]5
(31)
In the above equation, z0 is taken equal to 6, as for a hexagonal close-packed lattice. Molecular Area. For the ordered molecules with all-trans parallel hydrocarbon chains and with the molecular axis normal to the surface, the surface area per molecule (A0) is equal to the cross-sectional area of the hydrocarbon chain. The commonly used value of 21 Å2 is employed. For the disordered molecules, an estimate is obtained as follows: Equation 29 shows that the surface area of the disordered molecules is related to their average height. The latter will be considered equal to the average end to end distance of a random chain that is oriented normal to the surface. For a surfactant molecule which has nc carbons in the hydrocarbon chain and one small head group, the following equation, which accounts for the hindered internal rotations, can be used to calculate the mean square end to end distance (h2):21
h2 ) ncl2
(1 + η2)cos R + 2η 1 + cos R 1 + η - 2l2 1 - cos R 1 - η (1 - cos R)2(1 - η)2 (32)
Here l is the bond length (l ) 1.53 Å), cos R ) 1/3 for a hydrocarbon chain (R being the supplement of the skeletal bond angle), and η ) 〈cos φ〉 (φ being the internal rotation angle and 〈 〉 indicating the thermodynamic average). In eq 32, the first bond connecting the head group and the hydrocarbon chain is considered as a normal C-C bond. According to the rotational-isomeric approximation, each internal bond has three possible states (trans, gauche+, and gauche-), corresponding to φ ) 0°, 120°, and -120°, respectively. Consequently, the thermodynamic average η can be calculated via the following relation:
( ) ( )
g 1 - exp kT η) g 1 + 2 exp kT
(33)
where g is the energy per gauche bond relative to a trans bond. The value g ) 800 cal/mol21 is selected in the present calculations. Considering that a terminal methyl group has a volume almost twice as large as that of a methylene group and that the volume of the head group is approximately equal to that of a methylene group, the surface area of the disordered molecules becomes
Ad0 )
(nc + 2)VCH2
(34)
(h )
2 1/2
where VCH2 ) 27 Å3 stands for the volume of a methylene group. The Standard Free Energy Change (∆µ0). The method used in refs 18 and 22 is employed to calculate the difference between the standard free energies of the disordered and ordered molecules. For a nonionic surfactant, the standard free energy change from a disordered to an ordered molecule has four contributions: (1) the change of the van der Waals interaction energy among the chains (∆µ0vdW); (2) the change in the hydrocarbon tails-air and hydrocarbon tails-water interfacial free energies (∆µ0interface); (3) the change in the head group steric interactions (∆µ0steric); and (4) the conformational free energy change of the hydrocarbon chains (∆µ0con). Consequently, ∆µ0 is expressed as
∆µ0 ) ∆µ0vdW + ∆µ0interface + ∆µ0steric + ∆µ0con (35) The Interaction Energy Change ∆µ0vdW. The interaction energy change from the disordered to the ordered state provides a negative contribution to ∆µ0. Considering only the interactions between the nearest neighbors, one can readily obtain, by combining eqs 11, 28, 30, and 31, that
[ ( )]
∆µ0vdW z0 Wtt zd Wdd A0 ) ) -χ 1 kT 2 kT 2 kT Ad0
1/2 -1
(36)
where the factor 1/2 was introduced to avoid double counting. The Interfacial Free Energy Change, ∆µ0interface. The change from the disordered to the ordered state in a close-packed liquid monolayer is accompanied by the reduction of both the hydrocarbon tails-air and hydrocarbon tails-water interfacial area and this provides a negative contribution to ∆µ0, given by
(σo + σo/w)(Ad0 - A0) ∆µ0interface )kT kT
(37)
where σo and σo/w are the hydrocarbon tails-air interfacial tension and the hydrocarbon tails-water interfacial tension, respectively. Head Group Steric Interaction Free Energy Change, ∆µ0steric. The head group steric interaction free energy arises because the area occupied by a head group is excluded for the translational motion of the surfactant molecule at the surface. The change from the disordered to the ordered state makes the surface more crowded, and this leads to an increase of the head group steric interaction. The steric interaction free energy can be calculated using the expression
( ) ( ) Ap µ0steric ) -ln 1- 0 kT Ad d
for the disordered molecules and the expression
(38)
3112 J. Phys. Chem., Vol. 100, No. 8, 1996
( )
Ruckenstein and Li
( )
Ap µ0steric ) -ln 1- 0 kT t A
(39)
for the ordered molecules, where Ap stands for the effective cross-sectional area of the head group. For fatty acids, Ap has been estimated to be approximately 19.7 Å2. From eqs 38 and 39, the steric interaction free energy difference from the disordered to ordered molecules is given by
[( )( )]
Ad0-Ap A0 ∆µ0steric ) ln kT A0-Ap Ad0
(40)
Conformational Free Energy Change, ∆µ0con, of the Hydrocarbon Chain. The hydrocarbon chains of ordered and disordered molecules are constrained to remain with the polar groups in contact with water and to form close-packed lamellar aggregates. This constraint affects differently the intramolecular interactions in the hydrocarbon chains of the disordered and ordered molecules, because these molecules occupy different areas on the interface. Consequently, a conformational free energy change occurs at the transition between the two states. For the intramolecular interactions in the hydrocarbon chains of a lamellar aggregate, an expression was derived,22 on the basis of a lattice model, which can be written as
( )
4 µ0con 3 NcL ) kT d 2 (A 0)2
Figure 1. Effect of the interaction parameter between the disordered and ordered molecules on theoretical π-A isotherms. The values of the calculated parameters are given in Table 1.
(41)
d
for the disordered molecules and
( )
4 µ0con 3 NcL ) kT t 2 (A0)2
(42)
for the ordered molecules. In these equations, L ) 4.6 Å is the size of a segment which can be located on a site of the lattice and Nc ) (nc + 1)/3.6 represents the number of segments of the hydrocarbon chain of length nc. Subtracting eq 41 from eq 42 leads to
[ ( )]
∆µ0con 3 NcL4 A0 1 ) kT 2 (A0)2 Ad0
Figure 2. Comparison between theoretical and experimental π-A isotherms for the C14-C17 fatty acids.
2
(43)
An Expression for the Surface Pressure. Combining eqs 3, 35-37, 40, and 43, one obtains
[ ( )]
π(Ad0 - A0) A0 ) -χ 1 kT Ad0
1/2 -1
+
[( )( )] [ ( )] ( )
A0d-Ap A0 (σw - σo - σo/w)(Ad0 - A0) + ln kT A0-Ap Ad0 4
A0 3 N cL 10 2 2 (A ) Ad0
2
+ ln A0
+
a1 (44) ad
where a1 and ad are given by eqs 19 and 20, and d by eq 34. The difference σw - (σo + σo/w), which appears in the second term on the right-hand side of eq 44, represents the spreading coefficient of a liquid hydrocarbon containing nc carbons. The hydrocarbon molecules with nc ) 13-16, corresponding to the C14-C17 fatty acids which are of interest here, have spreading coefficients close to 0 (for C16, it has the value of only -1.3 mN/m, (ref 16, p 14)). For this reason, σw - (σo + σo/w) is taken to be 0 in the calculations.
Figure 3. Comparison between theoretical and experimental π-A isotherms for the C14 fatty acid.
It is important to note that all the parameters in eq 44 are predicted by the model. Results and Discussions Effect of the Interaction Parameter. Figure 1 presents the effect of the interaction parameter between the disordered and ordered molecules for imax ) 1000 (larger values do not affect
Phase Transitions in Surfactant Monolayers
J. Phys. Chem., Vol. 100, No. 8, 1996 3113
Figure 4. Comparison between theoretical and experimental π-A isotherms for the C15 fatty acid.
Figure 5. Comparison between theoretical and experimental π-A isotherms for the C16 fatty acid.
the results, see Figure 6). It clearly shows that the characteristics of the calculated π-A isotherms depend strongly on the interaction parameter (χ). When the interaction parameter is 0, no phase transition occurs in the π-A isotherm (curve 1) and the surface pressure increases with the surface density. This is as expected. For values of the interaction parameter between 1.2 and 1.6, phase transitions do occur: nonhorizontal phase transitions for smaller values and first-order phase transitions for larger ones. It should be noted that the calculated π-A isotherm that exhibits a first-order phase transition (curve 4) is very similar to those provided by the van der Waals three-dimensional pressure-volume relation. The Maxwell construction is required to determine the equilibrium points. There is a critical point, for a critical value of the interaction parameter, at which both the first and the second derivatives of π with respect to A are 0. A first-order or a nonhorizontal phase transition (second-order) occurs when the interaction parameter is larger or smaller than the critical value.
Comparison of the Model with Experiment. For four fatty acids (from C14 to C17), the predicted and some measured values (refs 16 (p 113), 24, 28) are presented in Figure 2. More detailed comparisons with several sets of experimental data, obtained by different authors, are made in Figures 3-5. The values of the interaction parameter (χ), calculated from eq 31, the surface areas of disordered molecules (A0d), and the transition surface pressures are listed in Table 1. Figure 2 shows that for the four fatty acids the model and experiment are in agreement and provide either a first-order or a nonhorizontal phase transition. The increase of the length of the hydrocarbon chain changes the phase transition from a nonhorizontal to a firstorder one. For the tetradecanoic and pentadecanoic fatty acids, the predicted π-A isotherms exhibit nonhorizontal phase transitions, while for hexadecanoic and heptadecanoic fatty acids, first-order phase transitions. The pentadecanoic acid has an interaction parameter, 1.46, almost equal to the critical value 1.465 and a π-A isotherm with a nearly horizontal region. A coexistence curve of liquid expanded and liquid condensed monolayers for a first-order transition is predicted which starts from the heptadecanoic acid and ends with the pentadecanoic acid. In addition, according to our model, for a surfactant with a short hydrocarbon chain, no first-order phase transition should occur. For the C15-C17 fatty acids, the predicted results are in agreement with the experimental data of refs 16 (p 113) and 24. For the tetradecanoic fatty acid28 the agreement is less good. The predicted and experimental transition surface pressures are in good agreement (Table 1). For the C14-C16 fatty acids, the experimental data obtained by different authors are compared in Figures 3-5; there is only qualitative agreement among them. For the C14 acid (Figure 3), three sets of experimental data exhibit a nonhorizontal phase transition, as predicted by the model. The model fits qualitatively the experimental data of Smith.28 A small change, from 1.33 to 1.28, in the value of the interaction parameter improves the agreement. For the C15 and C16 fatty acids, either a firstorder or a second-order phase transition with a nonhorizontal line is indicated by various experimental results. The present model fits well the experimental data of Pallas and Pethica.24 The experimental π-A isotherms obtained by them for the two fatty acids exhibit first-order phase transitions. The absence of horizontal lines in the experiments of other authors is attributed by them to the presence of impurities. The other two sets of experimental data25,27 for C16 appear to display phase transitions at larger surface areas. One may observe from the figures that there are somewhat larger deviations between the theoretical and experimental isotherms for the liquid expanded monolayers. This is, however, expected. Indeed, in a liquid expanded monolayer there are vacancies between the hydrocarbon chains, because the hydrocarbon chains are not completely flexible to become closely packed, which were not included in the model. This additional effect leads to a higher two-dimensional compressibility in the liquid expanded monolayer. Effect of Clustering. Figures 6 presents the effect of clustering on the phase transition in the liquid monolayer. The
TABLE 1: Interaction Parameters, Molecular Surface Areas, and Transition Surface Pressuresa surfactant
χ
A0d (Å2/molecule)
πtr,exp (mN/m)
πtr,pred (mN/m)
transition type
tetradecanoic acid pentadecanoic acid hexadecanoic acid heptadecanoic acid
1.33 1.46 1.61 1.76
38.41 38.67 39.02 39.42
14.6(28) 7.2(24) 3.0(24) 0.5(16)
10.0b 7.5 3.9 0.4
second order first order at the critical point first order first order
a
πtr,exp and πtr,pred stand for the experimental and predicted transition surface pressures, respectively. b πtr,pred corresponds in this case to the molecular surface area of 28.0 Å2/molecule (approximately the middle point of the transition region).
3114 J. Phys. Chem., Vol. 100, No. 8, 1996
Ruckenstein and Li involves disordered and ordered surfactant molecules, the latter molecules generating clusters. A two-dimensional FloryHuggins equation is employed to express the chemical potentials of the disordered and ordered species. The standard chemical potential change from disordered to ordered molecules was calculated considering four contributions: (1) the change of the van der Waals interaction energy among the chains; (2) the change in the hydrocarbon tails-air and hydrocarbon tailswater interfacial free energies; (3) the change in the head group steric interactions; and (4) the conformational free energy change of the hydrocarbon chains. The equality between the chemical potentials of the disordered and ordered molecules provides an expression relating the surface pressure to the molecular surface area. A predictive equation is obtained which is compared with experimental data.
Figure 6. Effect of the largest number of ordered molecules included in a cluster (imax) on theoretical π-A isotherms.
Figure 7. Size distribution of clusters
extent of clustering is reflected in the value of imax, the largest number of molecules in a cluster. One can see that by increasing imax, the slope of the π-A isotherm in the nonhorizontal region decreases. For the limiting case of imax ) 1, the π-A isotherm still exhibits a phase transition, but with the largest slope in the transition region. The slope decreases with increasing imax. For large values of imax, whether a first-order or a nonhorizontal phase transition occurs depends on the value of the interaction parameter χ (Figure 2). When imax is large (for example larger than 15 in Figure 6), the calculated π-A isotherms become independent of imax. For not too large values of imax, the mole fraction of clusters decreases with the cluster size, and for large values of imax, the mole fractions of larger clusters is negligible (see Figure 7). Conclusion The transition from a liquid expanded to a liquid condensed insoluble monolayer is considered on the basis of a model which
References and Notes (1) Caille, A.; Rapini, A. Phys. Lett. 1976, A58, 357; Caille, A.; Rapini, A.; Zuckermann, M. J.; Cros, A.; Doniach, S. Can. J. Phys. 1978, 56, 348; Caille, A.; Pink, D. A.; deVerteuil, F.; Zuckermann, M. J. Can. J. Phys. 1980, 58, 581. (2) Georgallas, A.; Pink, D. A. J. Colloid Interface Sci. 1982, 89, 107. (3) Smith, T. AdV. Colloid Interface Sci. 1972, 3, 161. (4) Nagle, J. F. J. Chem. Phys. 1973, 58, 252; 1975, 63, 1255; Ann. ReV. Phys. Chem. 1980, 31, 157. (5) Scott, H. L., Jr. J. Theor. Biol. 1974, 46, 241; Scott, H. L., Jr.; Cheng, W.-H. J. Colloid Interface Sci. 1977, 62, 125. (6) Marcelja, S. Biochim Biophys. Acta 1974, 367, 165. (7) Bothorel, P.; Belle, J.; LeMaire, B. Chem. Phys. Lipids 1974, 12, 96. (8) McCammon, J. A.; Deutch, J. M. J. Am. Chem. Soc. 1975, 97, 6675. (9) Doniach, S. J. Chem. Phys. 1978, 68, 4912. (10) Jahnig, F. J. Chem. Phys. 1979, 70, 3279. (11) Bell, G. M.; Mingins, J.; Taylor, J. A. G. J. Chem. Soc., Faraday Trans. 1978, 74, 223. (12) Firpo, J-L.; Dupin, J. J.; Albinet, G.; Bois, A.; Casalta, L.; Baret, J. F. J. Chem. Phys. 1978, 68, 1369. (13) Dupin, J. J.; Firpo, J-L.; Albinet, G.; Bois, A.; Casalta, L.; Baret, J. F. J. Chem. Phys. 1979, 70, 2357. (14) Boehm, R. E.; Martire, D. J. Chem. Phys. 1977, 67, 1061. (15) Kaye, R. D.; Burley, D. M. J. Phys. 1974, A.7, 843. (16) Birdi, K. S. Lipid and Biopolymer Monolayers at Liquid Interfaces; Plenum: New York & London, 1989. (17) Ruckenstein, E.; Bhakta A. Langmuir 1994, 10, 2694. (18) Ruckenstein, E.; Li, B. Langmuir 1995, 11, 3510. (19) Israelachvili, J. N. Langmuir 1994, 10, 3774. (20) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans, Green and Co.: London, 1966. (21) Volkenstein, M. V. Conformational Statistics of Polymeric Chains; Wiley-Interscience, New York, 1963. (22) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (23) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (24) Pallas, N. R.; Pethica, B. A. Langmuir 1985, 1, 509. (25) Harkins, W. D. The Physical Chemistry of Surface Film; Reinhold Publishing Co.: New York, 1952. (26) Moore, B. G.; Knobler, C. M.; Akamatsu, S.; Rondelez, F. J. Phys. Chem. 1990, 94, 4590. (27) Gershfeld, N. L. J. Colloid Interface Sci. 1982, 85, 28. (28) Smith, T. J. Colloid Interface Sci. 1967, 23, 27.
JP9514638
