J . Phys. Chem. 1992, 96, 8683-8686
Another Approach to a Surface Equation of State Klaus Lunkenheimer* Max-Planck-Institut fiir Kolloid- und Grenzfldchenforschung, Rudower Chaussee 5, D-0- 1 1 99 Berlin- Adlershof, Germany
and Rolf Hirte Projektgruppe WIP "Umweltgerechter Kunststoffeinsatz" bei der Universitdt Potsdam, 0-0-1530 Teltow-Seehof, Kantstrasse 55, Germany (Received: May 8, 1992; In Final Form: August 18, 1992)
A new approach for describing the adsorption isotherms of soluble amphiphiles at fluid interfaces is proposed. This approach assumes the adsorption to occur in two thermodynamically discriminable states in the surface layer for the same amphiphilic species such that the first one is realized at the lower and the second one at the higher bulk concentrations. There is an interval at medium concentrations, characterized by a gradually occurring transition from state I to state 11. This transition is described by a polynomial. This approach succeeds in describing the entire adsorption isotherm of any amphiphile in terms of known thermodynamic assumptions with an accuracy unknown so far. This is demonstrated by using three different amphiphiles which cannot be described satisfactorily by known surface equations of state.
Introduction Surface active agents have been used in various fields of practical application and basic research for a long time. Although 'the accelerated tempo of interest and research in the field of surfactants in solution is going to continue",l it is rather strange that there are still serious difficultiesin formulating the most basic feature of surfactants, Le., their surface equation of state. Besides Traube's empirical equation, valid at sufficiently low concentrations UO - U, = KC (1) the equations of LangmuirZ and of Szy~zkowski,~ which are equivalent r = r,c/(a + C) (2) uo - u, = R i T , In (1 + c / a ) (3) and that of Frumkin4 u0 - ue = - R i T , In (1 - r/r,) - a'(I'/I'm)z (4a) are the mostly applied ones. Here, K is an empirical constant. uo and u, stand for the surface tension of the pure solvent and the equilibrium surface tension of surfactant solution, respectively, l' and r, for the surface concentration and saturation surface concentration, a for the surface activity parameter or the bulksurface distribution coefficient, and a' for the surface interaction parameter which, according to refs 6 and 7,can be correlated with the partial molar free energy of surface mixing of surfactant and solvent at infinite dilution Hs by a'=
r,P
(5)
To apply the Frumkin equation to the experimentally accessile variables of bulk concentration c and surface tension u,, eq 4b has to be applied additionally c=
ar/(r- - r) exp(-2HS/RT(r/rm)J
(4b)
Lucassen-Reynders and van den Tempe1 proved that these equations can be derived thermodynami~ally.~-'They correspondingly discriminated between "ideal" (eqs 2 and 3) and 'regular" ( q s 4) surface behavior. Joos extended this model to mixed monolayers and introduced the possibility to consider very different saturation adsorption values.* Ross and Morrison criticized the thermodynamic derivation of Lucassen-Reynders and van den Tem~el.~JO They examined the basic assumption of Langmuir's adsorption equation and concluded that already the use of this equation, "however well it describes observations, necessarily harms the theoretical model by introducing improbable elements".I0 0022-3654/92/2096-8683303.00/0
Recently Lin, McKeigue, and Maldarelli have put forward two new equations of state, derived from phenomenological models, which they called the 'general Frumkin" and/or 'phase transition" model." Although Ross's and Morrison's critical scrutiny reveals the rigorous thermodynamic implications of eqs 1-4, the question still remains open whether and for which amphiphilic system each of these equations can be used to describe an experimentally determined adsorption isotherm exactly throughout the entire concentration interval.
Necessary Experimental Prerequisites To decide upon the validity of these equations, first of all, one should consider the necessity of providing reliable adsorption isotherms of amphiphiles. As we have shown, this presupposes not only the application of an appropriate measuring techniquetzt3 but also the use of a particular degree of purity of the surfactant solutions called "surface chemical p ~ r i t y " . ' ~ " On the basis of these requirements we have determined equilibrium surface tension (ue)-concentration (c) isotherms of amphiphiles experimentally by applying convenient measuring'*J3 and purification techniques18 including relevant criteria to judge the purity of the s01ution.l~~'~ As we have found, new insight into the adsorption properties of homologous series of surfactants has been obtained following these requirements. Thus, for example, the values of the limiting surface area demand per molecule adsorbed, A ~reveals , a distinct dependence on the carbon chain length within the homologous series.'920 Hitherto, it was found for medium-chain length fatty acids, alkanols, and alkyl sulfates that the Aminvalues remain constant within a homologous series and are much higher than the values of the corresponding cross-sectional areas of geometrical models.5~2'-26
Findings and Problems Exploiting the precise adsorption isotherms of soluble n-alkanols and n-alkanoic acids by using Frumkin's equation, we did not succeed in obtaining a satisfactory best fit when we included the overall adsorption isotherm. This is illustrated for ndecanoic acid in 0.005 M hydrochloric acid in Figure 1, which shows (i) the measured surface tension values a,,(ii) the values uF calculated from the best fit by using the Frumkin equation (4), and (iii) the u, values obtained from our approach, in dependence on the concentration. This figure also contains (iv) the difference between the experimental values a, and the values uF,calculated by using the Frumkin equation (4). The misfit illustrated by (iv) clearly reveals the systematic error involved in the best fit indicating a phenomenological physical feature of the adsorption isotherm, which obviously cannot solely be described by the Frumkin 0 1992 American Chemical Society
Letters
8684 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 n-decanolc acld in
160
10
1
0 005 n HCI
-I
'\
10-5
10.6
El3
b
, , , , \ , __
.
io-
Figure 1. Equilibrium surface tension (a,)-concentration ( c ) isotherm of aqueous solutions of n-decanoic acid in 0.005 M HCl at 295 K: (0) measuring values; (+) best tit data obtained by using eq 4; dashed line, difference between surface tension values measured ( P P ) and values calculated by the best fit of eq 4 (aF);solid line, best fit by using our new
approach. equation as it stands. The maximum deviations amount to 0.54.7 mN/m. (It was already Frumkin himself who emphasized that his equation could not be considered the exact equation of state of the adsorption layer, in particular in the region of condensation phenomena.") When we attempted to describe the adsorption isotherm of longer chain dimethyl-n-alkylphosphineoxides, it turned out that these compounds revealed "ideal surface behavior" according to refs 5-7 if the concentration interval of lower surface pressures was excluded. However, the best fit was not satisfactory if the entire concentration interval was included. Neither could it be improved by applying eq 3. Hence, we again excluded that part of the isotherm with the lowest surface pressure to obtain approximately consistent surface parameter^.'^ There we argued that the adsorption equilibrium might not have been established yet for these very low concentrations. If this were true, the equilibrium surface tension values of the lowest concentrations would have to be still lower. This, however, would only deteriorate the best fitting of the isotherms. These facts made us assume that some basic thermodynamic information must be hidden in the adsorption isotherms, preventing the surface equations of Langmuir-Szyszkowski and/or Frumkin from matching the complete adsorption isotherms satisfactorily.
Basic Idea of f3ur New Approach The basic idea of our approach is to regard the adsorption isotherm as consisting of two distinct regions, either of which being describable by its individual, thermodynamically reasonable expressions. Accordingly, two different sets of surface parameters are to be attributed to either region, reflecting two different standard states in the surface. These two states can be thought of as representing two separate courses indicated by the dashed lines in Figure 2. As eqs 1-4 can be derived reasonably in terms of surface thermodynamic^,^^^^^ whatever the physical content of the surface parameters will we tentatively used these equations, assuming the two surface states to be subject to the following conditions: be,599J0
c1 ICII
> a11 AGO" < AGO1 ,I
n - d e c a n o i c acid In 0 005 M H C I
I
(6)
(7a) (7b)
according to AGO = RT In a
with AGOrepresenting the standard free energy change per mole of surfactant adsorbed. We assumed that state I was described by Traube's law, eq 1, whereas state I1 was described by eq 3 or eq 4. Then we evaluated
io+
10-6
g,
io-'
-
Figure 2. Experimentally determined ae-c isotherm of n-decanoic acid in 0.005 M HCl (see Figure 1): (-) best fit obtained by our approach; (- - -) values extrapolated from the surface parameters of state I and/or state I1 of this approach.
the experimentally determined isotherms by using a two-state nonlinear regression model containing a transition function a to describe the transition in the following way: Qe = 4 Q I ) + (1 - .)(an) (8) uI and uIIrepresent two different state functions. The transithn function is defined by (Y = 0.5[1 - p ( X , ) ] (9) with X , = (In c - In c,)//3 where c, stands for the concentration of the transition point and a for the width of the transition interval. Various transition functions have been applied in the literature, for example tan (c,j3), tanh (Q), or polynomials.*' We have applied the latter. Equation 9 represents a nonlinear problem of minimization which was solved by an iteration procedure.26
Results and Discussion This approach turned out to be very successful. The improvement in the mathematical approximation of the adsorption isotherms measured was remarkable. The residual deviation s of the best fit is given by s
IC[cevai(Ci) i
- ~exp(~r)I~/fgI"~
(10)
where fg is the degree of freedom of the system, given by f,=m-n+l (11)
m and n denote the number of measuring points and the number of parameters, respectively, used in the best fit approximation. The residual deviations were generally between f O . l and f 0 . 2 mN/m, sometimes even better, whereas the measuring accuracy of the ring tensiometer by means of which the surface tension was measured amounted to 10.1 mN/m. The measuring error was about f0.2 mN/m. The resulting surface parameters became physically reasonable. This improvement shall be illustrated by three different amphiphiles: The fmt one is n-decanoic acid (cf. Figure I), the sccond one is dimethyldodecylphosphineoxide (DDP0),19 and the third one is 1,Zdodecanediol. Figure 3 shows the experimentally determined isotherms together with the corresponding curves of the best fit obtained by our approach, eq 8, with eq 1. The adsorption data obtained by applying on the one hand eq 3 and/or eq 4 and on the other hand our approach, eq 8, are compiled in Table I. As one can see, the best fit quality obtained is very good. No systematic error has been observed. The deviations do not exceed the measuring accuracy.
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8685
Letters
TABLE I: Surface huntion of State hta' substance
C9COOH DDPO
C12(OH)2
K
B
model eq4 eq8 eq3 4 4
1.23E5'
1.5 1E-5
0.55
eq8
7.50E5
2.30E-6
0.56
eq4 eq8
2.68E4
9.308-6
0.53
Ctr
r,
a 9.44E-5 1.46E-5 5.74E-6 6.32E-6 5.35E-6 8.72E-5 3.9lE-5
6.07 6.68 4.38 4.21 4.29 5.60 6.83
"Dimensions of the parameters: K,m N / m dm3 mol-'; us,mN/m; a, c , ~mol , dm"; mN/m; j3,dimensionless. 'Read as 1.23 X lo5.
r,,
I-P
S
3.62 2.55 0 0.60 0 5.36 2.31
f0.37 h0.17 f0.53 f0.53 f0.27 f0.68 f0.20
A4G -1.43 -2.36 6.80
mol cm-2; Hs, AA&, kJ/mol; s (residual deviation),
Applying this to the Henry region as surface standard, eq 17 becomes Apo = RT In l/K1
(18)
This procedure can likewise be applied to the second surface state, characterized by eq 12. Thus, a quantitative thermodynamical difference A& = &I1 between the two postulated surface states can reasonably be established, which is related to experimentally accessible surface parameters, resulting in AApos = Ap,"-
Ap,' = RT In K ' a / R l T ,
(19)
According to the basic principles of thermodynamics AApt 30 10.'
10.~
10-6
% dm
10.' C -
Figure 3. ue-c isotherms of aqueous solutions of dimethyl-n-dodecylphosphine oxide (O), n-decanoic acid in 0.005 M HC1 (t),and 1,2-dodecanediol (A) at 295 K. Solid lines represent the best fit by our approach.
It is interesting to note that an improvement is also observed with DDPO,although the best fit is obtained by applying eq 3 instead of eq 4 for the second term of the sum of eq 8. This fact also points to the existence of two different states in the adsorption layer. With respect to thermodynamics our approach means that the constant K1,which is separately calculated according to eq 1 from the measured isotherm within the Henry region only, is different from that one which is obtained from the best fit by using eq 3 and/or eq 4 and by evaluating the measured isotherm with the exclusion of the Henry region. Following the latter procedure, the Henry constant is then given by
K" = R l T , / a
for c/a
-,0
(12)
As a matter of fact, it was found that always holds
K' < K" (13) These experimentally proven findings enable us to check our hypothesis within the thermodynamical meaning of the above assumed discriminable standard states in the surface. As there are various possibilities of defining standard chemical potentials, we only have to apply the same definition to either state. Conveniently, we derive from Betts and Pethica's definition of surface fugacity
+ RT In
ps = pOs
T*
(14)
where **A = TPA = RT
(15) with ps,f", and 7 Aue denoting surface chemical potential, surface activity coefficient,and equilibrium surface pressure, respectively. Using the ordinary chemical (bulk) potential
+ RT lnfbc
pb = pob
(16)
the standard free energy of adsorption is given by AGO Aw0 = gos- pOb= RT In r*/fbc
(17)
