Langmuir 1996,11, 3510-3515
3510
A Surface Equation of State Based on Clustering of Surfactant Molecules of Insoluble Monolayers Eli Ruckenstein" and Buqiang Li Chemical Engineering Department, State University of New York at Buffalo, Amherst, New York 14260 Received February 6, 1995. In Final Form: June 5, 1995@
Insoluble monolayers of nonionic or ionic surfactants at the airlwater interface are assumed to be made up of singly dispersed molecules and clusters of different sizes. A surface equation of state is derived combiningthe Gibbs adsorptionequation with an expression for the chemical potential derived on the basis of the above model. The equation can be used to describe the changes with nonhorizontal lines in the n-A isotherms. The equation is predictive, since expressions are proposed for all the free-energy contributions to the free energy offormation ofthe clusters. The surface equation of state is used to interpret experimental data regarding the behavior of pentadecanoic. hexadecanoic, and octadecanoicfatty acids as well as the behavior of sodium octadecyl sulfate.
Introduction
It is well-known that numerous surface pressurearea isotherms of insoluble monolayers exhibit changes which are not first-order with nonhorizontal phase transitions, since the latter involve horizontal lines. The behavior of surfactants is directly related to the interactions between the hydrophobic tails of the surfactant molecules and between their polar headgroups.6 However, a predictive equation of state is still lacking. The most commonly used two-dimensional equation of state is of the van der Waals kind,' but even its improved forms provide only a first-order phase transition, for which the horizontal line linking the two phases in thermodynamic equilibrium is determined via the Maxwell construction. A possible explanation for the existence of nonhorizontal changes is due to Langmuir,s who suggested that the transition from a liquidcondensed to a liquid-expanded state occurs via the breaking down of the liquid-condensed monolayer into fragments or molecular groups. In recent years, some experimental results provided evidence for the existence of surface molecular cluster^.^,^^^-'^ However, no theoretical model was proposed until recently when RuckenAbstract publishedinAdvanceACSAbstracts, August 15,1995. (1)Goddard, E.D.Monolayers; Advances in Chemistry Series 144; American Chemical Society: Washington, DC, 1975. (2)Birdi, K.S.Lipid and Biopolymer Monolayers at Liquid Interfaces; Plenum: New York & London, 1989. (3)Yue, B. Y.; Jackson, C. M.; Taylor, J. A. G.; Mingins, J.; Pethica, B. A. (a) J . Chem. Soc., Faraday Trans. 1 1976,72,2685;Ibid. 1982, 78,323. (4)Miller, A.;Mohwald, H. J . Chem. Phys. 1987,86,4258. ( 5 ) Florsheimer, M.; Mohwald, H. Colloids Surf. 1991,55,173. (6)Adamson, A. W. Physical Chemistry of Surfaces, 3rd ed.; Wiley: New York, 1976. (7)Weinberg, W. H. Surface Efects in Crystal Plasticity; NATO Advanced Study Institute Series E ; Applied Science: New York, 1977; No.17. (8)Langmuir, I. J . Phys. Chem. 1933,1 , 756. (9)Albrecht, 0.; Gruler, H.; Sackmann, E . J . Phys. (Paris) 1978,39, 301. (10)Chi, L. F.;Anders, M.; Fuchs, H.; Johnston, R. R.; RingsdorfH. Science 1993,259, 213. (11)Sankaram, M. B.;Marsh, D.;Thompson,T. E.Biophys. J . 1992, 63,340. (121Zhu, J.; Lennox, R. B.; Eisenberg, A. Langmuir 1991, 7,1579. (13)Zhu, J.; Eisenberg, A.; Lennox, R. B. Macromolecules 1992,25, 6547. (14)Li, S.;Hanley, S.; Khan, I.; Varshney, S. K.; Eisenberg, A.; Lennox, R.B. Langmuir 1993,9,2243. @
stein and Bhakta15 derived an adsorption isotherm in which the effect of clustering on adsorption was taken into account. A somewhat similar treatment regarding the behavior of insoluble monolayers was proposed by Israelachvili.16 The goal of the present paper is to develop a predictive treatment for the behavior of insoluble monolayers. A surface equation of state is derived by considering that the monolayer is made up of molecular clusters of different sizes and by calculating the free energy of formation of the clusters in a way similar to that employed by Nagarajan and Ruckensteinl7-lg for micellar solutions. Finally, a comparison with experimental data is made.
Thermodynamics of Clustering The insoluble monolayer of surfactant molecules at the airlwater interface is regarded as a two-dimensional solution which is made up of singly dispersed molecules, clusters of various sizes, and water molecules. The clusters are considered as distinct chemical species, and the interactions between the various species are assumed negligible. The standard states of the clusters and the singly dispersed molecules are taken to correspond to infinitely dilute conditions. Thus, the chemical potential for the clusters containing i surfactant molecules, pi, can be expressed as
(1) where pf and Ni are the standard chemical potential and the number of clusters containing i surfactant molecules, respectively; pf also involves the interaction between the cluster and the water molecules. Nt stands for the total number of chemical species of the two-dimensional solution and is given by (15)Ruckenstein, E.;Bhakta, A. Lungmuir 1994,10, 2694. (16)Israelachvili, J. N. Langmuir 1994,10, 3774. (17)Nagarajan, R.;Ruckenstein, E . J . Colloid Interface Sci. 1977, 60,221;Ibid. 1979,71,580. (18)Nagarajan, R. Adv. Colloid Interface Sci. 1986,26,205. (19)Nagarajan, R.; Ruckenstein, E. Lungmuir 1991,7,2934.
0743-746319512411-3510$09.00100 1995 American Chemical Society
Langmuir, Vol. 11, No. 9,1995 3511
Surface Equation of State of Insoluble Monolayers lmax
Nt = N ,
+x N i i=l
where N , is the number of water molecules. For the two-dimensional solution containing various clusters as components, the chemical potential per surfactant molecule at equilibrium must be the same, regardless of the size of the cluster, and equal to that of the singly dispersed molecules, i - - - + -kT I n N= p lo i i c N t
Pi-PF
+ k T l n Nl Nt
(3)
The above relation provides the following equation for the size distribution of clusters:
xi = xiKi
(4)
largest number of molecules assumed in a cluster. For different cluster sizes and for different locations in a given cluster, the surface area per molecule is not necessarily the same. One expects the moleculeslying at the boundary of a cluster to occupy a different area than those in its interior. However, for sufficiently large clusters, it is expected that the molecular area will vary slightly with the size of the cluster. Here we assume that regardless of the size of the cluster, the molecular area is constant. For surfactants with straight hydrocarbon tails and small polar headgroups, the molecular area is assumed to be equal to the cross-sectional area of the aliphatic hydrocarbon chain. The commonly used value for the crosssectional area (21 A2) will be employed. The molecular cross-sectional area of water is estimated from its molar volume (V,)to be A i = (VJN0)y3= 9.6 A2,where NOis Avogadro's number. Dividing the two sides of eqs 2, 9, and 10 by Nt and combining with eq 4 give
where xi stands for the molar fraction of the clusters containing i surfactant molecules and x l is the molar fraction of the singly dispersed molecules. Ki is an equilibrium constant given by
i=l lmar
(12) and
where
1 = x&, n
(6) is the standard free-energy change per molecule for the transfer from the singly dispersed state to that in a cluster containing i surfactant molecules. Surface Equation of State Gibbs adsorption equation can be used to establish a relationship between the surface pressure (x)and the twodimensional molar concentration (r)or molar surface area (A). For the monolayer containing a single surfactant and at constant temperature,
im,
+ CiKixiAo
I t
(13)
i=l
where
rt = Nt At
stands for the total number of chemical species per unit area. Combining eqs 11through 13 leads to ,i
Using eq 1for the chemical potential, the above equation becomes
i=l
Inserting the above equation into eq 8, the differential equation describing the insoluble monolayer becomes In order to integrate the above equation, a relationship between x1 and r is required. Denoting by N and At the total number of surfactant molecules and the total surface area, we have
im,
ZiKp'; i=l
&=kT i=l
(9)
and
The surface pressure can be readily calculated by integrating eq 16. One obtains imax
hnal
A, = N&:
+ciNAo
(10)
i=l
where A i and Ao stand for the areas of a water molecule and a surfactant molecule, respectively, and i,, is the
Ruckenstein and Li
3512 Langmuir, Vol. 11, No. 9, 1995 Equation 15 can be solved numerically to provide the molar fraction of the singly dispersed molecules (XI) as a function of r, which, introduced in eq 17, allows us to obtain the JG-Aisotherm through its numerical integration. The solution of eqs 15 and 17 requires an expression for Ap; for the calculation of the equilibrium constants Ki,as well as the areas per water and surfactant molecules &,and AO).
The Free Energy of Formation of Clusters The standard free-energy change contains a bulk term proportional to i and a line contribution which is proportional to iu2. Let us denote by R the radius of the cluster, which can be expressed in terms of Ao via
(18) Consequently,
iAp: = iA&
+ yi u 2
(19)
surface ofwater with the polar headgroup in contact with water. Hence, the hydrocarbon tail in the compact cluster does not behave like a molecule in a liquid hydrocarbon, while the hydrocarbon tail of a singly dispersed molecule still has the freedom to behave almost like a gas molecule. Acorrection term which takes into account this constraint on the chain conformation must be included in the freeenergy change. Secondly, hydrocarbon tails-water, as well as hydrocarbon tails-air, interfaces are generated which provide positive contributions to A&. Thirdly, the formation ofa cluster results in the presence of polar headgroups at the water surface. Since the area occupied by the headgroupis excluded for the translational motion of the other molecules, a steric repulsion among the headgroups is generated. Consequently, this steric interactions should be included. Furthermore, electrostaticinteractions among the polar headgroups are present for ionic surfactants. These interactions are repulsive and provide a positive contributions to A&. Taking into consideration all the above contributions, A& can be written as
where
y = 2 n 1/2( A0 )u 2a,
(20)
a, being the line tension. Assuming only nearest-neighbor
interactions among the molecules in the cluster,
(21) where c is the interaction energy between two neighboring molecules and z is the number of nearest neighbors. The line tension represents the excess free energy when the interactions with some molecules of the cluster are replaced by the interactions with molecules of air. Here the latter interactions are neglected. If /3 represents the fraction of the nearest neighbors of a surfactant molecule in the same line layer, one molecule in the line layer will have, for large clusters, (/3 (1- /3)/2)2 nearest neighbors. Hence, the line tension is given, for large clusters, by
+
where A&,, Apfkric,and represent the change in the standard free energy of the hydrocarbon tails due to its transfer from a gas to a liquid hydrocarbon phase, the conformational free energy change of the hydrocarbon tail, the hydrocarbon tails-air and hydrocarbon tails-water interfacial free energies, the headgroup steric interaction free energy, and the headgroup electrostatic interaction free energy, respectively.
Calculation of Different Contributions to A& Phase Change Free Energy of the Hydrocarbon Tail, ApL. For aliphatic hydrocarbons, the standard free-energy change from an ideal gas state at 1 atm pressure to their pure liquid state can be calculated from vapor pressure data using the following relation Ap; = kT In P
where P is the vapor pressure. The temperaturedependence contributions of the methylene and methyl groups to the free-energy difference have been estimated from vapor pressure-temperature experimental data. For the methylene and methyl groups, the fitted temperaturedependence relations arelg
and
Considering B
% V3,
eq 23 becomes
Y
0 -APB
(26)
(24)
Calculation of A& The predictive method developed by Nagarajan and Ru~kenstein"-~~ for the free energy of formation of aggregates in a three-dimensional micellar solution is extended here to the estimation ofthe free-energychange for the transfer of a surfactant molecule from the singly dispersed state at the interface to the two-dimensional cluster. Firstly, the transfer involves the environmental change from air to a liquid hydrocarbon experienced by the hydrocarbon tail ofthe surfactant molecule. This freeenergy change can be estimated from independent experimental data regarding the transfer of a hydrocarbon molecule from the gas to the liquid hydrocarbon phase. However, the molecule is constrained to remain at the
(%?gA,methylene
= 2.24 In T - 430lT - 10.85 -
0.0056T (27)
for the methylene group and
kT g/l,methyl = -16.46 In T - 3297/T+ 98.67 + (@) 0.02595T (28) for the methyl group, where the temperature Tis expressed in Kelvin. Considering that the CH2 group adjacent to the polar group lies in the hydration sphere of the headgroup and does not provide a contribution to the phase change free
Surface Equation of State of Insoluble Monolayers
Langmuir; Vol. 11, No. 9, 1995 3513
energy,z0the free energy for the transfer of a hydrocarbon tail of length ncfrom the singly dispersed state to a liquid hydrocarbon can be estimated from the relation (29)
Conformational Free Energy of the Hydrcarbon Tail, A,&,. The conformational free energy of the hydrocarbon tails at the two-dimensional surface is calculated in a manner similar to that used by Nagarajan and Ruckensteinlg for the conformational free energy of a surfactant tail in a lamellar aggregate. For a hydrocarbon tail consisting of N segments placed on a lattice whose sites have a size L , the local deformation free energy can be written as
where r stands for the location of the nth segment from the water surface, ro is that of the free end (the methyl terminal), (dnP2Lis the unperturbed end-to-enddistance of a chain containing dn segments, and E(r,ro) = drldn is the local chain deformation function. L is taken to be 4.6 A, and N is calculated using the expression N = (n, 1113.6. Denoting by G(r0)d r o the number of tails in the cluster whose free ends lie between ro and ro d r o and by h the thickness of the layer of hydrocarbon tails, the conformational free energy of a cluster is given by
+
using a group contribution of 1.265 A for the methylene group and 2.765 A for the methyl group. "he Interfacial Free Energy, 4&t.The formation of a cluster is accompanied by the generation of both a hydrocarbon tails-water interface and a hydrocarbon tails-air interface. The free-energy contribution can be taken as the product of the interfacial area and the macroscopic interfacial tension:
+ u,,(AO
UOIW.
Headgroup Steric Interaction Free Energy, The area occupied by the headgroup at the surface of water is excluded for the translational motion ofthe surfactant molecules in the cluster. This generates a steric repulsion among the headgroups and results in a positive contribution to A&. The commonly used relation is
(g) steric
which gives
= - In( 1 -
4)
(37)
For fatty acids, the effective cross-sectionalarea of the head oup (-COOHI is estimated to be approximately 19.7 For sodium octadecyl sulfate, the cross-sectional area ofthe sulfate headgroup has been estimated to be 17
E.
A2.19
Headgroup Electrostatic InteractionFreeEnergy, For surfactant molecules which have a charged headgroup, the electrostatic interactions among the headgroups must be included in the free energy of formation of a cluster. Assuming complete dissociation and that the ions are monovalent, the electrostatic free energy is given by the expressionz1
.L$&,
where 6 is the 6 function. The local chain deformation function E(r,ro) corresponding to the above G(r0)can be obtained from the following relation: (33)
(36)
where 00 and u0lw represent the hydrocarbon tailsair and hydrocarbon tails-water interfacial tensions, respectively,Ao is the surface area per surfactant molecule in a cluster, and A, is the cross-sectional area of the polar group. The value of 25 mN1m is used for the hydrocarbon tails-air interfacial tension a0 and 50 mN1m for the hydrocarbon tails-water interfacial tension
+
For the present case, the surface area per surfactant molecule in a cluster is assumed to be the cross-sectional area of the hydrocarbon chain. This corresponds to the particular situation in which the surfactant molecules in the cluster are completely extended and all the free ends are located at a distance equal to the length of the hydrocarbon tail lo; that is, h = lo. Consequently,
- A,)
kT
= 2{ ln[u
+ (1 + u 2 Y 1 (38)
where (39)
E(r,ro)= L3 A0
(34)
Substituting the above expressions of E(r,ro)and G(r0) in eq 31, the following expression for the conformational free energy is obtained:
e is the protonic charge, E is the dielectric constant of the solution near the surface, and K is the reciprocal Debye length. The latter quantity is related to the ionic strength of the solution via
(40) (35) where the length of a hydrocarbon chain is calculated
where n is the number of counteri~nslcm~ in the bulk solution. The dielectric constant is taken to be that of
(20) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1973. Tanford, C. J . Phys. Chem. 1974, 78, 2469.
(21)Hunter,R. J.Foundation ofCoZloid Science; Blarendon: Oxford, 1987.
3514 Langmuir, Vol. 11, No. 9, 1995
Ruckenstein and Li
Table 1. Values of the Various Contributions to 4; a surfactants
T,"C
solution
sodium octadecvl sulfate
14 14 14 14 14 25 30 30 25
0.001 M NaCl 0.01 M NaCl 0.01 M NaCl 1.0 M NaCl 5.0 M NaCl 0.01 M HC1 0.01 M HC1 0.01 M HC1
pentadecanoic acid hexadecanoica cid octadecanoic acid a
- 18.96 -18.96 -18.96 -18.96 -18.96 -12.82 -12.39 -13.56 -16.24
4uln
4u$
7.86 7.86 7.86 7.86 7.86 6.31 6.36 6.78 7.56
1.66 1.66 1.66 1.66 1.66 2.78 2.78 2.78 2.78
&:leet
&:tenc
1.83 1.83 1.83 1.83 1.83 1.43 1.41 1.41 1.43
10.06 7.78 5.54 3.44 2.16 0.0 0.0 0.0 0.0
44 2.45 0.16 -2.07 -4.18 -5.45 -2.30 -1.84 -2.58 -4.64
The values of Ap0 are given in kT units. 30.0
20.0
- : Theoretical
* : Experlmental(24)
-
c-
4-
* : Experimental(P3)
20.0
e,
E!
g
- : Theoretical
-
15.0
t0
10.0
i
5 fn
8
10.0
5.0
I
0.9
I
30.0
50.0
70.0
90.0
0.0
10.0
~
50.0
30.0
70.0
90.0
Moisculaf Area (A')
Figure 1. Effect of,,i on the pentadecanoic acid n-A isotherm at a water-air interface at 25 "C.
Figure 2. Effect ofi,, on the sodium octadecyl sulfateisotherm at a 1.0 M NaCl aqueous solution-air interface at 14 "C.
pure water whose dependence on temperature is givenbylg E
= 87.74 exp[-0.0046(T - 273)l
(41)
Results and Discussions In this section, the above equations are used to interpret the x-A isotherms of sodium octadecyl sulfate monolayers at the airLNaC1aqueous solution interface for different salt concentrations and those of pentadecanoic, hexadecanoic, and octadecanoic fatty acids. Sodium octadecyl sulfate is considered to be completely dissociated, and the fatty acids are considered to be undissociated. Table 1provides the values of the various contributions to A&. Effect of imm.,,i is the upper limit of the size of the clusters assumed in the calculations. The predicted x-A isotherm depends on the value of.,,i As in the previous theory on adsorption isotherms15and the surface equation of state proposed recently by Israelachvili,16 the model can predict a first-order phase transition with an horizontal line, without using a Maxwell construction. It occurs for a large value ofi,, (imm4-1. In fact, the value of 1000 fori,, is sufficient to predict the first-order phase transition. When,,i = 1, a simple surface equation of state for a one-phase x-A isotherm is obtained. For intermediate values of i, (1 i, < lOOO), the theory predicts phase transitions characterized by nonhorizontal lines. As examples, Figures 1and 2 show how the x-A isotherms depend on,,i for a pentadecanoic acid monolayer and for a sodium octadecyl sulfate monolayer at a 1.0 M NaCl aqueous solution-air interface, respectively. For the simple case of i, = 1,eq 17.can be integrated to give the following surface equation of state:
Equation 42 contains two molecular area parameters (Ao, area of the water molecules (A): is omitted, reduces to the Frumkin and Volmer equation22
d)and, when the cross-sectional kT
n=-lnA'
A A-A'
(43)
For sufficiently large molecular areas, the ideal surface equation of state is obtained: d = k T
(44)
Equation 42 is compared in Figure 3 with experimental data for the gaseous surface monolayer of sodium octadecyl sulfate at a 0.01 M NaCl aqueous solution-air interface for two different temperatures. As expected, as the molecular area decreases, the deviations between experiment and eq 42 become important. At the other extreme, i, -, a first-order transition occurs. As shown in Figure 4, where XI is plotted against ,,i for various values of+:, assymptotic values ofzl are Since the attained for sufficiently large values of.,,i surface pressure depends on x1 (eq 171, it is clear that it becomes independent of i, at sufficiently large values of the latter quantity. Sodium Octadecyl Sulfate Monolayer. The surfactant sodium octadecyl sulfate has a low solubilty in water and generates almost insoluble monolayers at sufficiently high salt concentrations. Figure 5 presents
-
(22) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic: New York and London, 1963.
Surface Equation of State of Insoluble Monolayers
Langmuir, Vol. 11, No. 9, 1995 3515
3.0
lo Symbols : Experimental(23) - : Theoretical # , curve 1 : C(NaCl)=O.OOlM l(imax.lOOO) 0 , curve 2 : C(NaCl)-O.OlM 0 , curve 3 : C(NaCl)-O.lM x , curve 4 : C(NaCI)-l .OM A , curve 5 : C(NaC1)-5.0M
Symbols : Experlmental(1) - :Theorellcal i, curve 1 :at 20% 0 ,curve 2 :at 5%
5 2.0 E
e
a
1
5
1.0
v)
0.0
0.0 1u.u
Molecular arm ... - .. lI h,
Figure 3. Gaseous monolayer of sodium octadecyl sulfate at a 0.01 M NaCl aqueous solution-air interface.
- 1: h V k T - 2 . 3 M d X;4.1 -2 h 0 1 k T r 2 . 7and x;io.067 -3 &0/kT=-3.0 and X;=Oo.05 ls
'
L
G
30.0
I
-
*
I
50.0
Molecular area (A')
70.0
90.0
Figure 5. Isotherms of sodium octadecyl sulfate monolayers at a NaCl aqueous solution-air interface at 14 "C.
-
cs
E,
8
t
15.0
Symbols : Expedmental - : Theoretical 0 ,curve 1: pentadecanoic acM, 30%(24) * , curve 2: pentadecanolc acM, 25%(24) x ,curve 3: hexadecanolcacld, 30%(24) 0 ,curve 4: ocladecanoic acid, 25%(6)
f Pz
5.0 0.0
1
0.0
'
'
200.0
I 400.0
6CQ.O
8CQ.O
1wo.0
lmax
Figure 4. Mole fraction of the singly dispersed surfactant molecules (xl) against imm. According to eq 3, the assymptotic value of xl(x,) is given by x; = exp(&$KT).
a comparison with experimental n-A isotherms for five NaCl concentrations ranging from 0.001 to 5.0 M. The present model predicts a first-order phase transition for the monolayer on a 5 M NaCl aqueous solution. At salt concentrations of 0.001, 0.01, and 0.1 M, the theoretical predictions are consistent with the experimental n-A isotherms. For the monolayer at a 0.1 M NaCl solution, the equation predicts a weak first-order phase transition, which is also compatible with experiment. One may note that Figure 2 shows that for a salt concentration of 1M, satisfactory agreement between theory and experiment is obtained when the largest clusters in the monolayer contain only five surfactant molecules. Deviations between the theoretical and experimental curves occur in the region with molecular areas close to the limiting molecular area (Ao). This is because in that region, the interactions between clusters are significant and the theory has neglected them. Fatty Acid Monolayers. For fatty acid monolayers, two transitions were observed.' The first is a first-order phase transition from a gaseous film to the so-calledliquid-
I 10.0
0.0
I 0 X . x
30.0
61
I
50.0
A .
Molecular area (A*)
70.0
90.0
Figure 6. Isotherms of fatty acid monolayers at a water-air interface.
expanded film, and the second is a transition from the liuqid-expanded film to the liquid-condensed film. However, the second transition does not exhibit in all cases a horizontal n-A isotherm consistent with a first-order phase transition. The predicted and experimental results are compared in Figure 6 for three fatty acids. The theory predicts the existence of a single (calledabove the second) transition. In that region of the n-A isotherm, the agreement with experiment is only qualitative because the interactions among clusters have been neglected, The first transition is not predicted by the theory because the latter interactions have been ignored. LA9500843 (23) Shinoda, K.; Nakagawa, T.; Tamamushi, B.-I.; Isemura, T. Colloidal Surfactants; Academic: New York and London, 1963. (24) Pallas, N. R.; Pethica, B. A. Langmuir, 1986,1 , 509.
