J. Phys. Chem. B 2009, 113, 6311–6313
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Equation of State for Monolayers with Additional Phase Transition between Condensed Phases of Different Compressibility V. B. Fainerman† and D. Vollhardt*,‡ Donetsk Medical UniVersity, 16 Ilych AVenue, Donetsk 83003, Ukraine, and Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: January 15, 2009; ReVised Manuscript ReceiVed: March 5, 2009
An improved model for the equation of state for Langmuir monolayers proposed in J. Phys. Chem. B 1999, 103, 145 is introduced for the case where two or more phase transitions occur in the monolayer. The model allows the theoretical description of the phase transition between the two condensed phases under the realistic precondition that the phase transition is accompanied by the change of molecular compressibility parameters in the condensed phases. N-Alkyl-β-hydroxy-propionic acid amide monolayers undergo two phase transitions during the compression at low temperature, demonstrated experimentally both by the surface pressure-area (Π-A) isotherm and the results of grazing incidence X-ray diffraction (GIXD) for the C13H27 and C14H29 alkyl chain lengths. The theoretical Π-A isotherms obtained by the new model agree well with the experimental Π-A isotherms and with the data obtained from the GIXD experiments. Introduction General information about the phase behavior of amphiphilic monolayers can be obtained from the surface pressure area (Π-A) isotherms at different temperatures.1 The temperature region at which the Π-A isotherms show a plateau region, is of particular interest. The plateau region is attributed to the main phase transition from a low density fluid (gaseous, G, liquidexpanded, LE) state to a condensed (liquid-condensed, LC) state. The main phase transition in Langmuir monolayers is indicated by a discontinuity in the Π-A isotherm at the main phase transition point and behind it, a weak slope of the isotherm within the phase transition region. In the main phase transition region, condensed-phase domains are formed that are surrounded by nonaggregated molecules. Number and size of the condensed phase domains increase in the transition region as the area per molecule decreases. It is of special interest that, at low temperatures, the Π-A isotherms of some amphiphilic compounds reveal a striking second critical point at A < Ac indicating the existence of a second phase transition between two condensed phases. This phase transition between two condensed phases is accompanied by an abrupt change of the 2D lattice parameters obtained by grazing incidence X-ray diffraction (GIXD).2,3 In refs 4 and 5 attempts were made to generalize the model for the equation of state proposed in ref 6 to phase transitions for which the area per molecule in the condensed state exhibits either a linear decrease with increasing surface pressure or a sharp jump. For the theoretical description of a second phase transition between two condensed phases, the theory developed in ref 4 assumes two states of condensed phase with different parameters that characterize the molecular compressibility in each condensed state. The transition to these two states takes place from the fluid (LE) state of the monolayer. This theory also implies the condition of the thermodynamic stability of * To whom correspondence should be addressed. E-mail: dieter.vollhardt@ mpikg.mpg.de. † Donetsk Medical University. ‡ Max Planck Institute of Colloids and Interfaces.
the system, dΠ/dA e 0,7 which is a precondition to realize the transition between the two theoretical Π-A isotherms. The objective of the present work was to introduce a new generalized model that allows the theoretical description of the phase transition between the two condensed phases under the realistic precondition that the phase transition is accompanied by the change of molecular compressibility parameters in the condensed phases. It is shown that the new model agrees well with the experimental Π-A isotherms and with the data obtained from the GIXD experiments, and is capable of a much better description of the system in the range of the second phase transition than the previous model.4 Theory The equation of state for monolayers in the fluid (G, LE) state is represented by the Volmer type equation:4,5,8
Π)
kT - Πcoh m(A - ω)
(1)
where Π is the surface pressure, k is the Boltzmann constant, T is temperature, ω is the partial molecular area for monomers (or the limiting area of molecule in the gaseous or LE state), A is the area per molecule, Πcoh is the cohesion pressure, which takes into account the intermolecular interaction, and the m value allows for the association (m > 1) or dissociation (m < 1) degree of amphiphilic molecules in the monolayer, or the size of these molecules. The equation of state for monolayers in the fluid (G, LE)/ condensed (LC) transition region (A e Ac1) that takes into account the entropy nonideality of mixing of monomers and clusters on the basis of the Flory-Huggins theory9 was derived recently.8 As usually,4-6 it was assumed that the area per one molecule in the aggregate can differ from the area per free monomer molecule, and the condition that describes the equilibrium between the monomers and aggregates was formulated considering the free surface existing in the monolayer. The following equation of state in the transition region (A e Ac1) for the bimodal distribution (large clusters and monomers or small aggregates) takes the form:8
10.1021/jp900441a CCC: $40.75 2009 American Chemical Society Published on Web 04/10/2009
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J. Phys. Chem. B, Vol. 113, No. 18, 2009
Π)
Fainerman and Vollhardt
kTRβ 1 kT - (1 - Rβ) - Πcoh (2) m A - ω[1 + ε(Rβ - 1)] A
where ε ) 1- ωcl/ω is the relative decrease of the area per molecule in a cluster as compared with the area per molecule in the fluid monolayer, ωcl is the area per monomer in a cluster, and Ac1 is the area per molecule which corresponds to the onset of the first phase transition (i.e., at Π ) Πc1). The parameter R in eq 2 expresses the dependence of the aggregation constant on the surface pressure:
R)
[
]
Π - Πc1 A exp -ε ω Ac1 kT
(3)
and the parameter β is the fraction of the monolayer free from large clusters:
β ) 1 + ω(1 - ε)(R - 1) ⁄ A
(4)
The main difference between the model proposed herein and that described in ref 8 is the dependence of the parameter ε on the surface pressure that accounts for the existence of two phase transitions. For the first phase transition, that is, at Π g Πc1 or A e Ac1 the expression for ε is
ε1 ) ε01 + η1(Π - Πc1)
(5)
In the surface pressure range above the critical pressure of the second phase transition (i.e., for Π g Πc2 or A e Ac2) the ε value is expressed as
ε2 ) ε02 + ε01 + η2(Π - Πc2) + η1(Πc2 - Πc1)
(6)
In these expressions, ε0i is the relative jump of the area per molecule during ith phase transition, and ηi is the relative twodimensional compressibility of the condensed monolayer in the ith state, ηi ) -d ln ωcl/dΠ. Note that the transition from the condensed state 1 to the condensed state 2 at Π ) Πc2 (A ) Ac2) is thermodynamically possible, because in this case, the mechanical stability condition dΠ/dA e 0 is valid. Experimental Section The two tailored amphiphiles N-tridecyl-β-hydroxy-propionic acid amide, C13HPA (C13H27-NH-CO- CH2-CH2OH) and N-tetradecyl-β-hydroxy-propionic acid amide, C14HPA (C14H29NH-CO- CH2-CH2OH), different only by one methylene group in the alkyl chain, were synthesized and purified, as described elsewhere.3 The chemical purity of both amphiphiles (g99%) was checked by elemental analysis and HPLC. The used spreading solvent was chloroform (p.a. grade, Baker, Holland). Ultrapure water with a specific resistance of 18.2 MΩ used for the monolayer experiments was obtained from a Millipore desktop system. Equilibrium surface pressure (Π-A) isotherms recorded at a compression rate of e0.1 nm2/(molecule min) were measured in a experimental setup consisting of a self-made computerinterfaced film balance.3 Using the Wilhelmy method, the surface pressure was measured with a roughened glass plate with an accuracy of (0.1 mN m-1 and the area per molecule with (5 × 10-3 nm2. The measured Π-A isotherms were in complete agreement with those obtained previously.2,3 Results and Discussion In our paper,2 C14HPA was taken as an example of the system in which two phase transitions take place. For this amphiphile two sets of experimental data were obtained in ref 2: Π-A isotherms and GIXD data for the dependence of the C14HPA molecular area in the condensed state on the surface
Figure 1. Experimental Π-A isotherm of the C14HPA monolayer at T ) 10 °C (open points, from ref 2). Filled points and lines a and b correspond to the C14HPA molecule area in the condensed monolayer state on the surface pressure obtained from the GIXD data.2 The arrows 1 and 2 indicate the initial points of phase transitions. The theoretical curves (red and blue) were calculated from the model on the basis of eqs 1-6 with the values of parameters listed in Table 1.
TABLE 1: Parameters of C14HPA Monolayers at Different Temperatures °C
5 °C
10 °C
15 °C
20 °C
25 °C
ω, nm2 Ac, nm2 Πcoh, mN/m ε0i ηi, m/mN m
0.24 0.528 7.9 0.02/0.10a 0.003/0.001a 1.4
0.24 0.446 7.7 0.07/0.11 0.004/0.0015 1.4
0.23 0.38 7.7 0.14/0.03 0.004/0.001 1.3
0.22 0.342 7.4 0.14 0.003 1.3
0.21 0.31 7.3 0.16 0.003 1.2
a In a denominator the parameters of the second phase transition are presented.
pressure. The experimental data, newly measured and used in the present paper to verify the new theoretical model described by eqs 2-6, agree well with those of ref 2. Figure 1 shows the experimental Π-A isotherm at 10 °C of the C14HPA monolayer (open points) and the corresponding molecule area in the condensed monolayer state obtained by GIXD data (filled points connected by straight lines a and b), taken from ref 2. The positions of two phase transitions on the Π-Α isotherm are shown by arrows. The values of the projected area of the C14HPA molecule on the horizontal plane in the condensed state (Axy ) ωc) are strongly dependent on the surface pressure. It is seen that in the surface pressure range between the two phase transitions (indicated by the arrows 1 and 2 in Figure 1), the area per molecule in the condensed state decreases linearly with the increase in the surface pressure (line a in Figure 1). Above the second phase transition (line b) the molecular area of C14HPA in the condensed state decreases also, but here the slope is lower than in the region of the first condensed phase. At the inflection point of second phase transition of the Π-A isotherm in Figure 1, the area per C14HPA molecule in the condensed state changes drastically. The theoretical Π-A isotherms shown in Figure 1 were calculated using eqs 1-6. The red curve in Figure 1 corresponds to the fluid (G, LE) state and the first (LE-LC) phase transition of the monolayer. The theoretical curve for the second phase transition between the two condensed phases (LC-LC) is shown by blue line in Figure 1. It is seen that the theoretical dependence agrees well with the experimental data throughout the whole range after the second phase transition. It follows from the GIXD data, see Figure 1, that η1 ) -d ln Axy/dΠ ) 0.004 m/mN, η2 ) 0.001 m/mN (the slopes of
Improved Equation of State for Langmuir Monolayers
Figure 2. Experimental (black lines, from ref 2) and theoretical (red lines) Π-A isotherms of C14HPA in the temperature range between 5 and 25 °C. The values of the model parameters obtained on the basis of eqs 1-6 are listed in Table 1.
the straight lines a and b, respectively), and the area variation at the pressure value of 16.5 mN/m (the jump between the lines a and b) is ∆ε ) 0.13. These experimental values of the C14HPA monolayer compressibility parameters in the LC state are almost coincident with the corresponding parameters of eqs 5 and 6 used for the calculation of theoretical curves shown in Figure 1: η1 ) 0.004 m/mN, η2 ) 0.0015 m/mN, and ε02 ) 0.11. This coincidence corroborates the validity of the theoretical model proposed by this study. Figure 2 shows the experimental and theoretical Π-A isotherms for the C14HPA monolayers at different temperatures. The theoretical model parameters are listed in Table 1. It follows from Figure 2 that the second phase transition between the two condensed phases of the C14HPA monolayers occurs at temperatures of e15 °C. The surface pressure at the second phase transition within the range 16-17.5 mN/m is nearly independent of the temperature. Note the strong temperature dependence of the main fluid/condensed phase transition (see for example, Figure 2). The theoretical Π-A isotherms exhibit very good agreement with the experimental data. This demonstrates the essential progress of the new model compared with the model proposed earlier,4,5 where the agreement with experiment was more approximate, especially in the range after the second phase transition. Figure 3 presents the experimental (agreeing with the data reported in ref 10) and theoretical Π-A isotherms of C13HPA. The C13HPA molecule is only by one methylene group shorter than the C14HPA molecule. It is seen that the C13HPA Π-A isotherms of 5 and 3 °C exhibit two phase transitions, that is, the temperature within which the condensed/condensed phase transition of the C13HPA monolayer occurs is by ∼10 °C lower than that for C14HPA. At the same time, the surface pressure and area per one molecule at the point where the second phase transition commences are almost the same for C13HPA and C14HPA: 17-17.5 mN/m and 0.23-0.24 mn2, respectively. However, at higher temperatures and the values of area per one molecule of 0.23-0.24 mn2, neither of these amphiphiles does exhibit the second phase transition. Therefore it could be supposed that the surface pressure is the factor that governs the very existence of the second phase transition between the two condensed phases in the monolayers of N-alkyl-β-hydroxypropionic acid amides. It is seen from Figure 3 that the isotherms calculated from the model on the basis of eqs 1-6 agree well with the
J. Phys. Chem. B, Vol. 113, No. 18, 2009 6313
Figure 3. Experimental (black lines, from ref 10) and theoretical (red lines) Π-A isotherms of C13HPA in the temperature range between 3 and 20 °C. The values of the model parameters obtained on the basis of eqs 1-6 are listed in Table 2.
TABLE 2: Parameters of C13HPA Monolayers at Different Temperatures 2
ω, nm Ac, nm2 Πcoh, mN/m ε0i ηi, m/mN m
3 °C
5 °C
10 °C
15 °C
20 °C
0.22 0.39 7.3 0.08/0.07 0.004/0.002 1.4
0.22 0.364 7.3 0.1/0.1 0.004/0.003 1.4
0.205 0.314 7.2 0.15 0.004 1.3
0.195 0.284 6.6 0.17 0.004 1.3
0.19 0.253 6.1 0.19 0.004 1.3
experimental data. The parameters of theoretical model for C13HPA are listed in Table 2. These values are somewhat different from those reported in ref 10; this is because the ε dependence on surface pressure assumed in the present study is different from that used in ref 10, cf., eqs 5 and 6. It is seen that the values of parameters estimated for the C13HPA and C14HPA monolayers at the same temperature are quite similar. A significant difference exists only for the Ac values, which should be ascribed to the fact that the clusterization free energy for C13HPA is lower than that for C14HPA. To summarize, the model, introduced in the present work, that allows the theoretical description of the phase transition between the two condensed phases under the realistic precondition that the phase transition is accompanied by the change of molecular compressibility parameters in the condensed phases is capable of a good description of the Π-A isotherms with two phase transitions in Langmuir monolayers.. The theoretical Π-A isotherms obtained by the new model agree well with the experimental Π-A isotherms of N-Alkyl-β-hydroxy-propionic acid amide monolayers for the C13H27 and C14H29 alkyl chain lengths and with the data obtained from the GIXD. References and Notes (1) Vollhardt, D. AdV. Colloid Interface Sci. 1996, 84, 143. (2) Melzer, V.; Weidemann, G.; Vollhardt, D.; Brezesinski, G.; Wagner, R.; Struth, B.; Mo¨hwald, H. Supramol. Sci. 1997, 4, 391. (3) Vollhardt, D.; Wagner, R. J. Phys. Chem. B 2006, 110, 14881. (4) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2003, 107, 3098. (5) Vollhardt, D.; Fainerman, V. B. J. Phys. Chem. B 2004, 108, 297. (6) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (7) Rusanov, A. I. Phase Equilibria and Interfacial Properties; Khimija: Leningrad, 1967. (8) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2008, 112, 1477. (9) Flory, P. J. J. Chem. Phys. 1941, 9, 660;. 1942, 10, 51. (10) Vollhardt, D.; Fainerman, V. B. J. Phys. Chem. B 2008, 112, 10514.
JP900441A
