Rheological Model for the Study of Dilational Properties of Monolayers

dipalmitoyl phosphatidylcholine (DPPC) layer at the dichloromethane/water interface is characterized. ... the method is to follow, at constant tempera...
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Rheological Model for the Study of Dilational Properties of Monolayers. Comportment of Dipalmitoylphosphatidylcholine (DPPC) at the Dichloromethane (DCM)/Water Interface under Ramp Type or Sinusoidal Perturbations P. Saulnier,† F. Boury,† A. Malzert,† B. Heurtault,† Tz. Ivanova,‡ A. Cagna,§ I. Panaı¨otov,‡ and J. E. Proust*,† Inge´ nierie de la Vectorisation Particulaire, INSERM ERIT-M no. 0104, Bat. IBT, 10 rue A. Boquel, 49100 Angers, France, Biophysical Chemistry Laboratory, University of Sofia, James Bourchier str. 1, 1123 Sofia, Bulgaria, and Interfacial Technology Concept, Parc de Chancolan, 69770 Longessaigne, France Received November 27, 2000. In Final Form: September 6, 2001 The dilational properties of monolayers are analyzed using the classical linear approximation. In most cases, the observed interfacial behavior can be approached by a model corresponding to a two-dimensional viscoelastic solid. The monolayer is characterized by two dilational elasticity terms (Ee, equilibrium elasticity, and Ene, nonequilibrium elasticity) and by one relaxation time (τ). These three physical constants are obtained from the responses of a ramp type perturbation, or from the responses (as a function of the frequencies) after sinusoidal area variations. Using axisymmetric drop shape analysis experiments, a dipalmitoyl phosphatidylcholine (DPPC) layer at the dichloromethane/water interface is characterized. Measurements of the surface pressure variations as the response to linear or sinusoidal variations of surface area are performed. Identical rheological physical constants (equilibrium elasticity, nonequilibrium elasticity, and relaxation time) are obtained using both methods. Dilational behavior of DPPC monolayer can be attributed to the molecular diffusion between the DPPC layer and the adjacent phases.

Introduction The states of monolayers are often characterized by surface pressure-area isotherms obtained by means of different experimental devices. Traditionally, the aim of the method is to follow, at constant temperature, the various interfacial phases appearing during the compression, consequently to the different molecular interfacial rearrangements.1-4 The compression velocity seems to be a crucial experimental parameter that can influence dramatically the shape of the isotherm, since some interfacial phases can exist in a metastable state. The experimental surface pressure-area isotherms are often considerably modified by various nonequilibrium effects such as the Marangoni effect, weak dissolution or evaporation of amphiphilic molecules, orientation processes during the phase transition, etc. In a different way, it can be interesting to perform compression-expansion cycles in order to illustrate the dynamic hysteresis phenomena.4-9 This approach was very * To whom correspondence should be addressed. † Inge ´ nierie de la Vectorisation Particulaire, INSERM ERIT-M. ‡ Biophysical Chemistry Laboratory, University of Sofia. § Interfacial Technology Concept, Parc de Chancolan. (1) Albrecht, O.; Gruler, H.; Sackmann, E. J. Phys. 1978, 39, 301313. (2) Boury, F.; Gulik, A.; Dedieu, J. C.; Proust, J. E. Langmuir 1994, 10, 1654. (3) Xiao, Y.; Ritcey, A. M. Langmuir 2000, 16, 4252. (4) Tchoreloff, P. C.; Gulik, A.; Denizot, B.; Proust, J. E.; Puisieux, F. Chem. Phys. Lipids 1991, 59, 151. (5) Boury, F.; Olivier, E.; Proust, J. E.; Benoit, J. P. J. Colloid Interface Sci. 1993, 160, 1. (6) Makino, M.; Kamiya, M.; Nakajo, N.; Yoshikawa, K. Langmuir 1996, 12, 4211. (7) Makino, M.; Yoshikawa, K. Langmuir 1997, 13, 7125. (8) Krueger, M. A.; Gaver , D. P. J. Colloid Interface Sci. 2000, 229, 353.

useful in order to study the respreading properties of the lung surfactant.4,8,10 Unfortunately, a quantitative interpretation of the contribution of all the effects occurring in such experiments is difficult, especially for large hysteresis cycles where the kinetic effects are nonlinear.7 That is why the determination of rheological parameters (mainly interfacial elasticity and viscosity) of various interfacial systems has been commonly performed. It is done in order to study interesting dynamic properties (adsorption, desorption, interfacial organization, interfacial interaction) of amphiphilic molecules at different interfaces.8-16 2D rheology enables a good description of many processes which are controlled by time-dependent interfacial properties such as emulsification, foaming, detergency, and so on.17 It represents a powerful tool in many fields such as food technology18 or cell biology.19 The common idea of all the related experiments is to apply a controlled perturbation to the surface in order to (9) Cornec, M.; Narsimham, G. Langmuir 2000, 16, 1216. (10) To¨lle, A.; Meier, W.; Greune, G.; Ru¨diger, M.; Hoffmann, K. P.; Ru¨stow, B. Chem. Phys. Lipids 1999, 100, 81. (11) Mellema, M.; Clark, D. C.; Husband, F. A.; Mackie, A. R. Langmuir 1998, 14, 1753. (12) Miller, R.; Wu¨stneck, R.; Kra¨gel, J.; Kretzschmar, G. Colloids Surf. A 1996, 111, 75. (13) Wu¨stneck, R.; Enders, P.; Ebisch, Th.; Miller, R. Thin Solid Films 1997, 298, 39. (14) Murray, B.; Ventura, A.; Lallemand, C. Colloids Surf. A 1998, 143, 211. (15) Wu¨stneck, R.; Wu¨stneck, N.; Grigoriev, D. O.; Pison, U.; Miller, R. Colloids Surf. B 1999, 15, 275. (16) Li, J. B.; Kretzschmar, G.; Miller, R.; Mo¨hwald, H. Colloids Surf. A 1999, 149, 491. (17) Van den Tempel, M.; Lucassen-Reynders, E. H. Adv. Colloid Interface Sci. 1993, 18, 281. (18) Dikinson, E. Colloids Surf. B 1999, 15, 161. (19) Shchipunov, Y. A.; Kolpanov, A. F. Adv. Colloid Interface Sci. 1991, 35, 31.

10.1021/la001634m CCC: $20.00 © 2001 American Chemical Society Published on Web 11/30/2001

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simultaneously follow the related surface pressure variations. These surface perturbations can be very small (for example, thermal fluctuations20 or capillary waves21) when the interest is the characterization of dynamic behavior with a characteristic time between 10-6 and 1 s. On the other hand, the deformations can be more important (moving barrier, funnel method, drop deformation)12,18,22 when the viscoelastic characteristic time is on the order of 1-103 s. The dynamic response of a surface film to a dilational mechanical stress in the time scale of 1-103 s is studied by means of two experimental approaches. The first approach consists of a continuous and monotonic compression and expansion of a surface layer on a Langmuir trough16,23-26(or of a pendant or rising drop13,16) after the application of small disturbances. A complete description of the simultaneous motion of a monolayer and liquid substrate is presented in this paper. Because of the friction of the monolayer on the liquid substrate, the propagation is not instantaneous. This friction is accompanied by a counter flow in the subphase related to the Marangoni effect. A characteristic time of propagation, τM, along the length of the monolayer is determined.27-32 Other dissipation processes may occur in the monolayer, in addition to viscous dissipation in the bulk subphase. So, in a general pattern, the rheological analysis simultaneously involves the relaxation processes with characteristic relaxation time τ, and the propagation process with a characteristic time τM, since the relaxation processes accompany the Marangoni propagation all along the monolayer. But, when the reequilibration of the surface pressure gradients along the monolayer is faster than the surface relaxation time, one can consider the dynamic response as a whole, neglecting the Marangoni effect. In this case, a convenient theoretical model, corresponding to a solid viscoelastic body, has been developed and applied to many kinds of interfacial systems (phospholipids or polymers).23-26 One can find interesting rheological approaches12,13,16,33 related to continuous interfacial area deformations (slope, pulses) after spreading or adsorption of the active molecules. The case of the nonlinear regime has also been developed.6,7 The second approach seems very close to the common 3D rheology field. A sinusoidal interfacial

perturbation (drop tensiometer15,16,22,34,35or Langmuir balance14-16,36-38) is applied in order to follow the interfacial tension response. Relative area variation and surface tension are considered, respectively, as the input and the output of the interfacial system, from which it is possible to evaluate a transfer function (complex function) often called complex elasticity modulus E. These approaches have been applied to dilational perturbations34-37 but also to shear perturbations.38 The real part of this function found with various notations in the literature (′;14 r;34 E′;35 Ed;36,37 G′38) characterizes a conservative monolayer comportment. The imaginary part (′′;14 i;34 E′′;35 Ev;36,37 G′′38) characterizes a dissipative monolayer comportment and often defines the dilational (shear) monolayer viscosity with η ) ′′/ω (where ω ) 2πf and f is the frequency of the oscillations). Unfortunately, these functions depend not only on the monolayer comportment, but also on the experimental conditions for which they are evaluated; in particular, G′ and G′′ (and the corresponding parameters) are frequency dependent. In this way, it is difficult to compare all the experimental results rarely obtained in similar experimental conditions. In this paper we will show that it is possible to characterize a given interface with three physical constants: the relaxation time (τ), an interfacial conservative elasticity (Ee), and an interfacial dissipative elasticity (Ene). These constants represent extrapolated values corresponding to static conditions which are not dependent on the experimental conditions. Furthermore, we will propose a comparison between the continuous approach and the sinusoidal one by use of the pendant drop technique. We have applied both methods to a film of DPPC at the dichloromethane/water interface using a pendant drop tensiometer. The interfacial properties of phospholipids have been intensely studied in many fields (cell membranes, pulmonary surfactants, liposomes, etc.). Consequently, this phospholipid has been chosen as a model molecule.

(20) Earnshaw, J. C. In Polymer Surface and Interface II; Feast, W. J., Munro, H. S., Richards, R. W., Eds.; J. Wiley & Sons: New York, 1993; p 101. (21) Monroy, F.; Giermanska Kahn J.; Langevin, D. Colloids Surf. A 1998, 143, 251. (22) Benjamins, L.; Cagna, A.; Lucassen-Reynders, E. H. Colloids Surf. A 1996, 114, 245. (23) Panaiotov, I.; Ivanova, Tz.; Proust, J.; Boury, F.; Denizot, B.; Keough, K.; Tavena, S. Colloids Surf. B 1996, 6, 243. (24) Boury, F.; Ivanova, Tz.; Panaı¨otov, I.; Proust, J. E.; Bois, A.; Richou, J. Langmuir 1995, 11, 1636. (25) Boury, F.; Ivanova, Tz.; Panaı¨otov, I.; Proust, J. E.; Bois, A.; Richou, J. J. Colloid Interface Sci. 1995, 169, 380. (26) Doisy, A.; Proust, J. E.; Ivanova, Tz.; Panaiotov, I.; Dubois, J. L. Langmuir 1996, 12, 6098. (27) Panaiotov, I. Thesis, University of Sofia, 1987. (28) Dimitrov, D.; Panaiotov, I. Ann. Univ. Sofia, Fac. Chim. 1975/ 76, 70, 103. (29) Dimitrov, D.; Panaiotov, I.; Richmond, P.; Ter-Minassian Saraga, L. J. Colloid Interface Sci. 1978, 65, 483. (30) Panaiotov, I.; Dimitrov, D.; Ter-Minassian Saraga, L. J. Colloid Interface Sci. 1979, 72, 49. (31) Panaiotov, I.; Dimitrov, D.; Ivanova, M. J. Colloid Interface Sci. 1979, 69, 318. (32) Panaiotov, I.; Sanfeld, A.; Bois, A.; Baret, J. F. J. Colloid Interface Sci. 1983, 96, 315. (33) Joos, P.; Petrov, P.; Fang, J. J. Colloid Interface Sci. 1996, 183, 559.

Pendant Drop Tensiometer. The drop tensiometer (Tracker, IT Concept, France) allows the determination of the interfacial tension by analyzing the axial symmetric shape (Laplacian profile) of the pendant drop of the phospholipid dichloromethane solution (density ) 1.48) in water. The 7.5 × 10-3 mg/mL phospholipid concentration was chosen in order to reach rapidly an equilibrium surface tension. All the measurements were performed in triplicate on the same drop at controlled room temperature (20 ( 1 °C), but they were also repeated on two different drops characterized by the same surface area and the same 7.5 × 10-3 mg/mL phospholipid concentration. In that way, all the numeric values given in their related graph represent mean values calculated on six experiments.

Materials The DPPC (DL-dipalmitoyl phosphatidylcholine) was purchased from Sigma Chemical Co. (L’Isle d’Abeau, France). The water used was ultrapure grade from Milli-Q plus system (Millipore, Paris, France). Analytical-grade dichloromethane was obtained from Prolabo.

Methods

(34) Wu¨stneck, R.; Moser, B.; Muschiolik, G. Colloids Surf. B 1999, 15, 263. (35) Myrvold, R.; Hansen, F. K. J. Colloid Interface Sci 1998, 207, 97. (36) Rodriguez Nino, M. R.; Wilde, P. J.; Clark, D. C.; Rodriguez Patino, J. M. Ind. Eng. Chem. Res. 1996, 35, 4449. (37) Rodriguez Nino, M. R.; Wilde, P. J.; Clark, D. C.; Rodriguez Patino, J. M. Langmuir 1998, 14, 2160. (38) Brooks, C. F.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1999, 15, 2450.

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Saulnier et al. The related theoretical approach was intensely developed in previous papers23-26 considering the mechanical model represented in Figure 1c. τ was deduced from the experimental results related to the fast compression for t > tf with the following equation:

π(t) - π∞ t )π0 - π∞ τ

ln

(3)

In the same way we have shown that the addition of strains for the two parallel branches was represented by

∆π τ A ) Ee + Ene (1 - e-t/τ) Ut i t

Figure 1. (a) Schematic ramp type area variation of a drop of DPPC in DCM in water. (s) Fast compression when 0 < t < tf. For t > tf the drop area is maintained constant. (- - -) Slow compression when 0 < t < ts. For t > ts the drop area is maintained constant. (b) Schematic related surface pressure change ∆π versus time. (s) Fast compression. (- - -) Slow compression. (c) Mechanical model of the monolayer. Furthermore, for all the experiments it was possible to control, at different times, the Laplacian shape of the drop. This is strongly related to a true calculated surface tension value. Ramp Type Perturbation Approach. Theoretical Aspects. When the characteristic time of relaxation process τ is much larger than the characteristic time of propagation, one can consider the rheological response of the monolayer as a whole, neglecting the surface pressure distribution along the interface. Figure 1a,b presents theoretical results that could be related to fast compressions (during the time tf) and slow compressions (during the time ts) of a monolayer at the DCM/water interface. Figure 1a and Figure 1b correspond respectively to the relative area variation and surface pressure variations versus time. For both compressions, it is important to note that the relaxation occurs not only for t > tf or ts, but also during the compression (for t < tf or ts). But this effect is usually observed only at slow compression speed considering the nonlinear variations of the surface pressure for 0 < t < ts. One can remark in Figure 2b that the surface pressure does not reach its initial value at the end of the relaxation. In such case, it can be assumed that only a certain part of the interfacial energy has been lost during the experiment, and the other part is conserved. That is why we assume that during the relaxation, the total surface pressure change,

∆π ) π(t) - πi

(1)

can be represented as a sum of one equilibrium ∆πe and one nonequilibrium ∆πne contribution:

∆π ) ∆πe + ∆πne

(2)

(4)

Experimental Protocol. For the determination of Ee, Ene, and τ, one has to successively perform the following: i. Fast (d/dt ∆A(t)/Ai ) U/Ai typically higher than 0.005 s-1) compression (with ∆πmax typically lower than 2 mN/m) in order to determine more precisely the relaxation (t > tc, see Figure 1b). Using eq 3, the relaxation time τ is deduced. ii. Slow (d/dt ∆A(t)/Ai ) U/Ai typically lower than 0.003 s-1) monolayer compression (with ∆πmax < 2 mN/m) in order to determine the compression relaxation part (t < tc, see Figure 1b). The value of τ and eq 4 are used to determine Ee and Ene. Sinusoidal Perturbation Approach. Theoretical Aspects. The aim of this method is to interpret the response of the interfacial tension to several harmonic variations (characterized by their pulsation ω) of the interfacial area A (Figure 2a). For each pulsation, the relative variation of the interface area versus time is considered as the “input” of the system defined by the drop in its environment. In the same way, the variation of the interfacial surface pressure (π) versus time is considered as the output. We obtained the following situation: For each pulsation ω, and at every time t, an adequate harmonic analysis of these two signals allows the calculation of a complex transfer function (or complex elasticity) given by

dπ G h (ω) ) E h (ω) ) A h dA h

(5)

We have verified that this complex function is independent of t when the surface pressure π has reached its equilibrium value. This transfer function can be transformed in the following expression:

G h (ω) ) G′(ω) + iG′′(ω)

(6)

where G′(ω) and G′′(ω) correspond respectively to the real part and the imaginary part of the transfer function. So the graphs G′(ω) and G′′(ω) are characteristic of the rheological comportment of the interfacial layer. The dephasage angle φ is also given by

tg(φ) )

G′(ω) G′′(ω)

(7)

Considering that the interface follows a mechanical model (Maxwell theory) previously described (see Figure 1c), the conservative part of the transfer function (G′(ω)) is theoretically represented by the following equations:19

ω2τ2 G′(ω) ) Ee + Ene 1 + ω2τ2

(8)

Similarly, the imaginary part of the transfer function is written:

G′′(ω) 1 ) Eneτ ω 1 + ω2τ2

(9)

The characteristic theoretical graphs of these functions (presented in Figure 2b) were drawn considering eqs 8 and 9 with Ee ) 6 mN/m, Ene) 18 mN/m, and τ ) 10 s. Experimental Protocol. i. The relative area variation of the drop was chosen at 5% (for a drop of 12 mm2) corresponding to the linear regime. We have

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Figure 2. (a) Schematic sinusoidal drop area variation (s) and surface pressure variation (- - -) versus time. (b) Theoretical variations versus pulsation ω of G′ (eq 8) and G′′/ω (eq 9) where G′ and G′′ are respectively the real and the imaginary parts of the transfer function G h (ω) ) A h dπ/dA h ; the extrapolations allowing the determination of Ee, Ene, and τ are presented. verified that nonlinear phenomena appear when relative variations are larger than 10%. ii. The range of pulsation is 0.05 < ω < 3 rad/s corresponding to the apparatus limitations. iii. After the stabilization of the surface pressure, three sinusoids are performed for each pulsation. iv. An adequate Fourier analysis allows for each pulsation the determination of G, G′, and G′′. v. The validity of the mechanical model (Figure 1c) is verified considering the shape of the experimental graphs G′ and G′′/ω versus ω. vi. The physical constants Ee, Ene, and τ are extrapolated considering:

limG′(ω) ) Ee

(10)

lim G′(ω) ) Ene + Ee

(11)

ωf0

ωf+∞

lim ωf0

G′′(ω) ) Eneτ ω

(12)

Results and Discussion DPPC Adsorption Kinetic at the Dichloromethane/ Water Interface. We present in Figure 3 the experimental surface pressure variations versus time of a drop of DPPC in DCM for several concentrations (5 × 10-3, 7.5 × 10-3, and 10 × 10-3 mg/mL). To link the surface pressure variations only to the adsorption of DPPC at the dichloromethane/water interface, the volume and the area of the drop were maintained

constant for each concentration. One can see that the equilibrium surface pressure values (respectively 13, 14, and 15.5 mN/m) were reached after 200, 60, and 50 s, respectively, which represents the characteristic times of adsorption. The different values of the equilibrium surface pressures obtained for concentrations chosen around 7.5 × 10-3 mg/ mL indicate a nonsaturated state of the layer at these surface pressures. In conclusion, a fast equilibrium time and a nonsaturated monolayer state were the reasons for the choice of a DPPC concentration of 7.5 × 10-3 mg/mL in DCM. All the rheological comparisons were performed at this concentration. Ramp Type Measurements. In Figure 4 are presented the -t (a) and π-t (b) graphs for a slow ramp type compression (dashed line) (d/dt ∆A(t)/Ai ) U/Ai ) 0.0026 s-1) and a fast (full line) one (0.008 s-1). One can see in Figure 4b the development of the relaxation phenomena at the end of the fast compression (for t > tf) and at the end of the slow one (for t > ts) corresponding to the characteristic time necessary for the layer to reach a new equilibrium state. It is important to note that the layer relaxation still occurs during the compression. This can be observed in Figure 4b for the slow compression (t < ts) where the surface pressure variation versus time is not linear, in contrast to the area one (see Figure 4a). This is less visible for the fast compression.

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Figure 3. Experimental variations of surface pressure versus time for a drop (5 µL and area maintained constant) of DPPC in DCM immersed in water for three concentrations of DPPC (5 × 10-3 mg/mL; 7.5 × 10-3 mg/mL; 10 × 10-3 mg/mL).

Figure 4. Experimental ramp type rheological measurements for a 5 µL drop of DPPC in DCM (7.5 × 10-3 mg/mL) in water. (a) Relative absolute area variations versus time; (s) fast compression 0.008 s-1 followed by constant area asservissment; (- - -) slow compression 0.0026 s-1. (b) Related surface pressure variations versus time; (s) fast compression (t < tf) followed by relaxation (t > tf); (- - -) slow compression (t < ts) followed by relaxation (t > ts).

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Figure 5. Sinusoidal type rheological measurements. Area and surface pressure sinusoidal variations (ω ) 1.26 rad/s) versus time for a 5 µL drop in water of DPPC in DCM (7.5 × 10-3 mg/mL).

However, at the beginning of the fast compression, one can note in Figure 4a,b mechanical artifacts that could be little lengths of stroke of the syringe piston which were not able to simultaneously follow the motor. Sinusoidal Measurements. We have presented in Figure 5 typical sinusoidal type experiments for an ω ) 1.26 rad/s pulsation, which correspond to the simultaneous variations versus time of the area and the surface pressure. One can observe an important dephasing angle between the area variations and the surface pressure. The area variations do not strictly follow typical sinusoids at the top and bottom of the sinusoids, in relation with a little length of stroke of the syringe piston. These artifacts still exist for lower pulsation (results not shown) and cannot be attributed to nonlinear phenomena. Itis important to note that these artifacts are significant only in the fourth harmonic of the related Fourier development (results not shown) and do not significantly influence the values of G′ and G′′. The experimental variation of G′ and G′′ versus ω is presented in Figure 6. The full line corresponds to the theoretical approach after including in eq 8 and 9 the values Ee ) 6 mN/m, Ene) 18 mN/m, and τ ) 10 s, where these values were deduced from extrapolations of the experimental curve (see eqs 13, 14, and 15). One can observe a good accordance between these experimental points and this theoretical approach throughout the graph. In that way, this theoretical experimental correspondence confirms the mechanical model represented in Figure 1c. One can note that when nonlinearity prevails (too much important amplitude or pulsation), the experimental asymptotic comportment is broken (results not shown). Comparison of the Ramp Type and Sinusoidal Approaches. The experimental methodology described above, in the case of the ramp type approach, was developed on the basis of the results presented in Figure 4 for the determination of the relaxation time τ, considering the important related speed of compression (0.008 s-1). Applying eq 7 to the data for which t > tc, we obtain an average value:

τ ) 10 ( 1 s Considering this value and eq 6, Ee and Ene were

determined considering slow compression speed (0.026 s-1) beginning at π ) 14 mN/m. We obtained

Ee ) 6 ( 1 mN/m, Ene ) 18 ( 2 mN/m Considering the characteristic graphs of the sinusoidal approach (see Figure 6 for π ) 13.8 mN/m), linear extrapolations of G′(ω) and G′′/ω(ω) (eqs 10, 11, and 12) give the three physical constants:

Ee ) 6 ( 2 mN/m, Ene ) 19 ( 2 mN/m, and τ ) 9 ( 3 s One can see that both approaches give similar values for Ee, Ene, and τ after extrapolation to static conditions. One can note that the apparent dilatational elasticity, Ea, usually found on true π-A isotherms, can be extrapolated with

limG(ω) ) Ea

(13)

G(ω) ) xG′2(ω) + G′′2(ω)

(14)

dπ Ea ) -A dA

(15)

ωf0

where

Significance of Ee, Ene, and τ. Considering the monolayer as a whole interacting with the two adjacent fluid phases, the intrinsic monolayer cohesion should be strongly dependent on Ee in relation with interactions (hydrophobic as well as hydrophilic) between all the amphiphilic molecules in the monolayer. In that way, Ee describes all the lateral interactions between all the molecules in the interfacial zone, taking the role of the so-called Hooke constant in 3D mechanical fields. This energetic constant corresponds to a conservative elastic response of the monolayer induced by the rheological perturbation. On the other hand, Ene characterizes the dissipation of the rheological perturbation energy related to interactions between the interfacial phospholipidic molecules and the

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Figure 6. Experimental (+) and theoretical (s) variations of G′ and G′′/ω versus ω for a 5 µL drop of DPPC in DCM (7.5 × 10-3 mg/mL). Theoretical curves are deduced from eq 8 and eq 9 and from the values of the extrapolations of the experimental curve Ee ) 6 mN/m, Ene ) 19 mN/m, and τ ) 9 s.

bulk phase ones (chloroform and water). In this way, Ene strongly depends on the phospholipid HLB and solubility values. Finally, τ represents the necessary time for the interface to reach a new equilibrium energetic state. Hysteresis Studies. In the sinusoidal regime, we have presented in Figure 7 the evolution of the surface pressure versus the area for several cycles. One can remark that a well-established elliptic hysteresis is reproducible for other pulsation (results not shown). This elliptic shape denotes the absence of dynamic phenomena on the interface related to the propagation of the sinusoı¨dal perturbation (for example, Marangoni rearrangements). It was previously described that in the case of interfacial

perturbations related to the propagation, the elliptic shape can be strongly deformed. In the case of linear perturbations on a Langmuir through, Ivanova and co-workers39 established a relationship to quantify hysteresis when the relaxation is only attributed to a molecular diffusion between the interface and one of the adjacent phases (DCM). In Figure 7 we can measure ∆πh ) 0.7 mN/m (length of the double arrow). This value can also be calculated using

∆πh ) πmax(Amean) - πmin(Amean)

(16)

(39) Ivanova, Tz.; Panaiotov, I, Georgiev, G.; Launois-Surpas, M. A.; Proust, J. E.; Puisieux, F. Colloids Surf. 1991, 60, 263.

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agrees well with the experimental one. In conclusion, the hysteresis can be strongly attributed to the bulk diffusioncontrolled process of interchange of molecules between the monolayer and the DCM.

Figure 7. π-A hysteresis curves in a sinusoidal regime (ω ) 1.26 rad.s-1). The double arrow indicates the location of the calculated ∆πh.

Considering that Amean is reached at time ti ) 0,

∆πh ) {πmean + Aπ sin[ωti + φ]} -

{π

mean

[(

+ Aπ sin ω ti +

) ]} (17)

T +φ 2

T ) 2π/ω is the period of the sinusoids. Aπ represents the amplitude of the sinusoidal variation of the surface pressure. φ is the dephasing angle between the surface pressure variation and the area one and can be defined with the following expression:

sin φ )

G′′(ωh) G(ωh)

(18)

Finally we have

∆πh ) 2Aπ sin φ

(19)

We obtained ∆πh ) 0.72 mN/m. This theoretical value

Conclusion In this paper we performed two 2D rheological approaches (ramp type compression and sinusoidal area variation) for a given interface (DPPC at the DCM/water drop interface), to determine three physical constants: equilibrium elasticity Ee, nonequilibrium elasticity Ene, and relaxation time τ. These three interfacial constants are calculated under a mechanical model based on the Maxwell theory. The latter indicates that in linear conditions the interfacial response, after a moderated perturbation, comports a storage part represented by Ee, and a lost part represented by Ene and τ. In this way, Ee could be related to a long range organization of the interfacial monolayer, and consequently to the interactions between the interfacial molecules. Ene is related to their interactions with the adjacent liquid phases molecules. τ denotes the characteristic time of the dynamic process. Finally, τ represents the necessary time for the interface to reach a new equilibrium energetic state after the perturbation. We show that both approaches give the same results. We have verified that interfacial perturbations occur in a characteristic time that is very low compared to the relaxation expected. This verification is performed after the analysis of the π-A hysteresis curve. Consequently, no stress relaxation is detected, which could be related to the Marangoni molecular rearrangements just after the interfacial perturbation. Independently, we have shown that the relaxation can be attributed to the diffusion of DPPC in the adjacent liquid phases (surely DCM). In conclusion, sinusoidal type rheological experiments seem adequate in order to better understand the interfacial molecular organization. Ee, Ene, and τ characterize not only the interfacial layer, but also its interactions with the adjacent phases. Our main goal is the application of these methods to more complex interfacial systems such as proteins or polymers. LA001634M