Laser-Induced Local Collapse in a Langmuir Monolayer - The Journal

fascinating collapse phenomena in nature, such as a massive star collapsing .... with measurable velocities reaching as far as 500−1000 μm from...
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J. Phys. Chem. B 2006, 110, 22160-22165

Laser-Induced Local Collapse in a Langmuir Monolayer† RM. Muruganathan and Th. M. Fischer* Department of Chemistry and Biochemistry, The Florida State UniVersity, Tallahassee, Florida 32306-4390 ReceiVed: February 8, 2005; In Final Form: May 10, 2005

Heating of a two-dimensional, methyloctadecanoate, Langmuir monolayer with a focused laser induces the local collapse of the monolayer. We observe the growth of a three-dimensional collapse aggregate that is fed by an inward flow of the two-dimensional monolayer surroundings. The experiments are explained with a hydrodynamic model describing the dynamics of the local collapse. From this theory we predict that local collapse can be induced if the collapse pressure of the monolayer decreases faster with temperature than with the surface tension of the pure air/water interface. Such conditions are fulfilled for lung surfactants, and it should therefore be possible to perform time-resolved local studies of the collapse of lung surfactants at those temperatures.

Introduction The collapse of an originally stable system under the influence of centrosymmetric attractive forces has intrigued scientists in various fields. There are many fascinating collapse phenomena in nature, such as a massive star collapsing into a black hole under the action of its own gravitational forces;1 in chemistry the sudden decrease of pressure due to a chemical reaction generally leads to the implosion in the system. In contrast to implosions in chemical systems, where initially the material is attracted toward the center of the implosion and when it reaches the center the materials are expelled to the outside, the collapsing material and energy are for ever consumed by the black hole, where it is destined to crush into a singularity of the space time continuum from where it can never exit into the outside universe. A two-dimensional example of the reversible collapse and reassembly of material that bears to be important for the proper function of the lung is the behavior of insoluble surfactant monolayers at the air/water interface, when being compressed beyond the collapse pressure.2 Monolayer collapse is a certain two-dimensional (2D) to three-dimensional (3D) transformation of a Langmuir monolayer occurring at air/water interfaces due to the over-compression of the densely packed, insoluble, amphiphilic film. The collapse pressure, πc, at which collapse appears, depends on the rate of compression and asymptotically approaches the equilibrium spreading pressure,3 πe, if compressed adiabatically with an infinitesimally slow compression rate. Surfactants in the alveoli of the lung are reducing the surface tension of these alveoli, where the uptake of oxygen into the blood stream occurs, to reduce the work load associated with the expansion and compression of their surface.4-9 The surfactants in the alveoli monolayer are at the collapse pressure in the entire breathing cycle. When we inhale, due to the compression of the alveoli monolayer, the sufactants fold into the aqueous subphase, and when we exhale, due to the expansion of the alveoli monolayer, the surfactants reassemble at the surface. It is the relevance of monolayer collapse in the breathing cycle that leads researchers to have a closer look at the structure of monolayer collapse. Earlier investigations revealed a rich †

Part of the special issue “Charles M. Knobler Festschrift”. * Corresponding author. E-mail: [email protected].

variety of collapse scenarios that differ in the way that surfactants get ejected from the monolayer into the subphase or on top of the monolayer. Such phenomena have been observed using Brewster angle,10 light scattering,11-13 phase contrast,14,15 fluorescence,16,17 electron,18 and atomic force microscopy.13 Those experiments prove that the collapsing behavior of monolayers is strongly governed by structural properties of the film, such as the order of the phase10 or coexisting phases12,16 prior to collapse. Collapse can be reversible or irreversible such that the monolayer can or cannot be recovered upon expansion. Reversible collapse includes the nucleation of 3D buds11 or vesicles,16 the buckling of the monolayer with a bilayer being expelled and folded on top of the monolayer.18 The monolayer may also fold into the subphase with small height differences between two coexisting phases serving as heterogeneous nucleation sites for the folds.2,16 Roughening of the monolayer upon collapse has been observed by Schief11 and by Hatta.14,15 Reversible elliptical folds in collapsing monolayers could be explained by the self-attraction of the deformable opposing monolayer surfaces that fold into the subphase. All those studies have been performed by homogeneously compressing the monolayer resulting in a collapse of the monolayer at random locations. The present work reports on monolayer collapse where the location of collapse is predetermined and therefore allows the detailed study of the collapse dynamics. We report on the local collapse of a two-dimensional Langmuir monolayer locally heated by an infrared laser. The collapsing material is collected in the focal region of the laser, where isotensile heating raises the surface pressure beyond the collapse pressure. The surfactants in the collapsing region exit from the two-dimensional air/water interface to form a three-dimensional assembly of material. This 3D assembly will not be expelled back to the surrounding two-dimensional interface as long as the monolayer is heated by the IR laser. As the IR-laser intensity can be easily controlled as a function of time, this technique allows us to study the time-dependent monolayer collapse, which was not possible with the techniques mentioned earlier.2,6-16 The technique described here requires a collapse pressure that decreases with temperature. This may appear counter-intuitive, as one may be used to characteristic transition pressures

10.1021/jp0506991 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/15/2005

Laser Induced Collapse in Langmuir Monolayers

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increasing with temperature; however, collapse pressures decreasing with temperature are observed in a variety of monolayer systems, including those of lung surfactants consisting of a mixture of dipalmitoylphosphatidylcholine (DPPC) and palmitoylglycerophosphoglycerol (POPG).16 Experimental Section Our setup consists of a home-built Langmuir trough with a fluorescence microscope that works with a 100× water immersion objective, numerical aperture 1.0, built into the bottom of the temperature controlled film balance. The same objective is used to focus an IR laser beam (λ ) 1064 nm, P ) 10 mW 4 W) onto the central imaging region of the monolayer. The main purpose of this laser is to locally heat the monolayer. The light is partially absorbed by the water (absorption coefficient of water at 1064 nm: R ≈ 0.1 cm-1) and converted into heat. The setup has been described in detail elsewhere,19 and we have made use of it previously to visualize cavitations20,21 and local phase transitions22 in monolayers that are caused by the heating. Methyloctadecanoate (from ICN Biomedicals, Inc., Germany) claimed to be 99+% pure has been used without further purification. BODYPY-FL C16 was used as a fluorescent dye which was purchased from Molecular Probes, USA. Surfactant solution containing 0.8% of fluorescent dye was spread from chloroform (p.a. Fisher) onto pure water (Millipore Milli-Q at 18 MΩ) contained in the Teflon trough.

accordance to eq 1

)

∂(π - σw) ∂T ∇Fs ) ∂π ∂Fs T

)

Fs

∇T

(2)

Generically this transient flow is from the hotter to the colder region (∇Fs/∇T < 0). Upon further heating, one eventually reaches the intersection point of the isotension with the collapse line. For temperatures higher than this intersection temperature the mechanical equilibrium is no longer possible since there are no equilibrium states beyond the collapse. Rather than further following the isotension, heating will partially convert the monolayer toward a three-dimensional collapsed phase. The system follows the collapse line (black line in Figure 2) with a surface tension gradient,

∇σ ) -

d(πc - σw) ∇T dT

(3)

Due to Gibb’s phase rule, the collapse pressure πc(T), unlike the surface pressure π(Fs,T), is a function of the temperature only. If dσw/dT > dπc/dT, the surface tension gradient points in the same direction as the temperature gradient and generates

Results Figure 1 displays pressure versus area isotherms of the methyloctadecanoate/fluorescent dye mixture at various temperatures. The pressure increases monotonically, beyond the equilibrium spreading pressure πe where the monolayer becomes metastable until it collapses at the compression rate dependent collapse pressure πc that decreases with increasing temperature. The phase diagram in Figure 2 displays the temperature dependence of the collapse pressure. The collapse pressure decreases with increasing temperature with a slope dπc/dT ≈ -0.7 10-3 N/mK, which is significantly smaller than the change of the surface tension of the bare air/water interface with temperature dσw/dT ) -0.17 10-3 N/mK. A monolayer experiencing a temperature gradient is in mechanical equilibrium, if the gradient in surface tension vanishes 0 ) ∇σ ) ∇σw - ∇π ) (dσw/dT)∇T - ∇π, which occurs if

∇π )

dσW ∇T dT

Figure 1. Π-A isotherms of methyloctadecanoate Langmuir monolayer containing 0.8% of fluorescence dye (BODYPY), at various temperatures. Compression speed: 50 cm2/min. Arrow indicates the collapse region.

(1)

where π and T denote the surface pressure and temperature and σw is the surface tension of the bare air/water interface. Mechanical equilibrium lines are therefore isotension lines that appear as lines with slope dσw/dT in the surface pressure versus temperature diagrams. If one starts in the liquid condensed monolayer phase and locally heats the monolayer, then the system locally moves from its original point in the phase diagram along an isotension toward higher temperatures. We assume that the surface pressure π(Fs,T) is a state function of the surface density Fs and the temperature T. The mechanical equilibration is achieved by a transient surface flow that generates a surface density gradient of surfactants ∇Fs in

Figure 2. Collapse pressure versus temperature for methyloctadecanoate, obtained from isotherms at a compression speed of 50 cm2/ min. The solid line represents linear fit for the data points. The dotted lines are lines of constant surface tension.

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Figure 5. Radius R of the aggregate as a function of the time after increasing the laser power to P ) 2.4, 2.8, 3.2, and 3.6 W at a temperature of T ) 37 °C. The solid lines are fits according to eq 11, with Req and τ as listed in Table 1. Figure 3. Fluorescence microscopy images of a methyloctadecanoate monolayer T ) 37 °C π j πc ) 33 mN/m showing the collapse of molecules at the laser focus (cross hair) at various times after increasing the IR laser to P ) 3.6 W, (a) t ) 0 s; (b) t ) 1 s; (c) t ) 2 s; (d) t ) 7 s. A bright fluorescence arising from the developing 3D aggregate can be seen in the hot spot as time progresses. The velocity profile of the monolayer can be measured by following characteristic points in the monolayer texture as time progresses. Video clip 1 shows the entire time sequence of the local collapse including the recovery of the monolayer after turning off the IR-laser (see Supporting Information).

Figure 6. Variation of the equilibrium radius Req of the aggregate and the compression ratio Ac/Aaggr - 1 of the aggregate with the laser power P of the IR laser.

TABLE 1: Variation of the Hydrodynamic Flow Parameters as Determined from Fitting the Experimental Data Displayed in Figures 4 and 5 Figure 4. Velocity profile -ur(r, t ) 0.06 s) of the monolayer after increasing the laser power to P > 2 W. The profile is determined from the velocity of various regions of the texture of the monolayer that are dragged along with the flow. The solid lines are fits according eq 4 with UR (t ) 0.06 s) as listed in Table 1.

a surface flow that points along the temperature gradient (from cold to warm) in a direction opposite that of the transient flow that occurs while staying away from the collapse line. Simply speaking, the direction of the surface flow of the monolayer is determined by its collapse line; that is, when the monolayer is at the collapse line and dσw/dT > dπc/dT, the surface flows from colder to hotter regions. On the other hand, when the monolayer is far away from the collapse line the flow is from hotter to colder regions. Figure 3 shows consecutive fluorescence microscopy images of a methyloctadecanoate monolayer at T ) 37 °C, π j πc ) 33 mN/m locally heated by the Nd:YAG laser (λ ) 1064 nm, P ) 3.6 W). The laser is focused on the central region of the image (cross hair), where part of the light is adsorbed and heats the surrounding water and the monolayer.

laser power [W]

UR (t ) 0.06 s) [µm2 s-1]

Req [µm]

τ[s]

Ac/Aaggr - 1

3.6 3.2 2.8 2.4

5500 4400 4400 4200

100 65 50 28

6.25 6.5 6 4

352.36 632.31 872.72 1028.46

Initially the entire monolayer is in the liquid condensed phase. Upon sufficient heating above a threshold laser power P > Pc ≈ 2 W, a radial flow of the surface toward the center sets in. The flow can be measured quantitatively by following the characteristic texture of the monolayer as a function of time. Surfactant material aggregates into a much denser23 threedimensional structure in the hot spot, visualized as highly fluorescent material. The collapsed 3-D material grows in radius due to a radial inward flow of the monolayer surrounding the aggregate with measurable velocities reaching as far as 5001000 µm from the hot spot. The velocity profile in the periphery is approximately inversely proportional to the distance from the laser, implying that the density of the monolayer in the periphery does not change (Figure 4). It is only at the edge of the aggregate

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where an abrupt jump in surface density from the monolayer collapse density 1/Ac toward the aggregate density 1/Aaggr occurs. The initial collapse velocity of the inward flow of the monolayer does not depend much on the laser power once the laser power is above the threshold laser power P > Pc. Within a few seconds the peripheral flow slows down and eventually stops when the aggregate has grown to a disk of equilibrium radius Req. The former liquid condensed phase collapses to less than 1% of the original area, implying that the average thickness of the 3-D aggregate amounts to more than 100 molecular layers of the surfactant. The growths of the aggregate toward its equilibrium radius at different laser powers are shown in Figure 5. At short times the radius of the aggregate increases linearly with time until it saturates and reaches the equilibrium radius Req.. Figure 6 shows that the equilibrium radius Req increases with the applied laser power.

power of the laser, w ≈ 1 µm is the focal width of the laser, R ) 0.1 cm-1 is the adsorption coefficient of water at the wavelength of our IR laser, κ ) 0.6 W/mK is the heat conductivity of water. From the equation of continuity,26 we find the growth rate R˙ of the aggregate to be related to the velocity U via

Theoretical

where the relaxation time, τ, is given by

De Koker and McConnell have solved the hydrodynamic problem of domain size equilibration of electrostatically stabilized domains in monolayers.24 Most of the description of the hydrodynamics of the collapse can be understood when following their description. The radial velocity profile ur(r) of the collapsing monolayer can be approximated by

UR ur(r) ) θ(r - R) r

)

(

)

2ηU 8R R - 4ηU ≈(5) K ln π π R + xAc xAc

where σ∞ is the surface tension far away from the hot spot, σc(R) ) σw - πc is the collapse tension corresponding to the temperature at location R, η ≈ 0.67 mPas (at T ) 37 °C) is the viscosity of water, z is the coordinate normal to the surface, K is the complete elliptic integral of the first kind, and Ac is the collapse area per molecule (xAc is a small cutoff length, where the incompressibility of the surface breaks down in the close vicinity of the aggregate edge). The flow will stop when the collapse surface tension corresponding to the temperature at the equilibrium radius of the aggregate equals the surface tension far away from the hot spot:

σc(Req) ) σ∞

(6)

Assuming a temperature profile that is governed by the dissipation of heat via heat conduction in the water κ∇2T ) RPwδ(r), not by convection, we find outside the illuminated region:

T(r) - Τ∞ )

RPw 2πκr

(8)

Ac/Aaggr - 1

where Ac is the collapse area per molecule and Aaggr the area per molecule of the 3-D aggregate. Expanding equations (46) around the equilibrium radius we obtain a rate equation for the domain radius (details are given in the Appendix):

1 d (R - R) ) - (Req - R) dt eq τ

τ)

4ηReq2κ(Ac/Aaggr - 1)

8Req

ln

RwP(dσw/dT - dπc/dT)

xAc

(9)

(10)

Equation 9 has the solution

R ) Req(1 - e-t/τ)

(11)

Inserting eq 11 into eq 8 and taking the limit tf0, we find that

∂u

∫R∞ η ∂zrdr

U

(4)

where R is the radius of the aggregate, U < 0 is the velocity of the monolayer crushing into the aggregate, and θ(x) is the Heaviside function. The flow described by eq 4 changes as a function of time through the dependence of both the aggregate radius R(t) and the aggregation velocity U(t) on time. The surface flow is accompanied by a viscous stress from the aqueous subphase25 that has to be balanced by the static surface tension difference

σc(R) - σ∞ )

R˙ ) -

(7)

Here T∞ is the temperature far away from the hot spot, P is the

(Ac/Aaggr - 1) ) -

UR τ2 lim 2 tf0 t Req

(12)

We fitted eq 4 to the data displayed in Figure 4 and eq 11 to the data in Figure 5. From the first fit we obtain UR/t at very short times, from the second fit we obtain Req and τ. In Table 1 we summarize those fits for the different laser powers used in the experiment. Once these three quantities have been determined, we are able to determine the compression ratio Ac/ Aaggr - 1 in the aggregate. All surfactants initially placed at a distance of less than xAc/Aaggr Req will finally end up in the aggregate. The rest of the surfactants, which stayed far away from the distance xAc/Aaggr Req, also flow toward the aggregate but the flow stops before this material reaches the aggregate and it therefore remains in the monolayer. One realizes that only two parameters, the equilibrium Radius Req and the compression ratio Ac/Aaggr - 1, significantly vary with the laser heating. The equilibrium radius Req and the compression ratio Ac/Aaggr-1 are displayed as a function of the laser heating in Figure 6. The threshold of the instability can be read off from the graph by extrapolating to the power where the equilibrium radius would vanish. From this we expect the threshold to be around Pc ) 2 W, which is consistent with what we observe in the experiment. It should be noted, however, that the threshold also depends on the surface pressure far away from the hot spot. The decrease of the compression ratio Ac/Aaggr 1 with the laser power may be explained as follows: Increasing the laser power leads to a faster growth R˙ of the aggregate, while the monolayer feels the same collapse tendency at the edge of the aggregate and collapses with approximately the same flow profile. As a consequence, less material can accumulate in the aggregate while the aggregate is growing, this is indeed what is observed in Figure 6. At lower laser power P ) 2.4 W, the monolayer collapses to roughly a 1000 layers in the aggregate, while at higher laser power the compression ratio drops to 400. We may rearrange eq 10 by separating the

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Muruganathan and Fischer

Figure 7. Collapse pressure vs temperature for a 7:3 DPPC/POPG mixture. Reproduced with permission from Gopal and Lee (reference 16). Reproduced with permission from ref 16, copyright 2001 American Chemical Society.

parameters characterizing the flow from those parameters that are not directly associated with the flow:

8Req δ

(Ac/Aaggr - 1)Req2ln τP

)

Rw(dσw/dT - dπc/dT) 4ηκ

(13)

The left-hand side of eq 13 contains all the nonequilibrium parameters characterizing the flow, while the right-hand side is either known transport properties of water or equilibrium properties, measured independent of the flow. Using R ) 0.1 cm-1, w ) 1 µm, (dσw/dT - dπc/dT) ) 0.53 × 10-3 N/mK, η ≈ 0.67 mPas, κ ) 0.6 W/mK, Αc ) 19Å2, and the data displayed in Table 1, we obtain reasonable agreement of the left and right-hand side independent of the laser power P which supports our model of the monolayer collapse. Basically our theory shows that the driving force for the aggregate growth is merely due to the heating from the IR laser. This theory is different from De Koker and McConnell’s theory that shows that electrostatic effects drive the domain equilibration in the liquid condensed-liquid expanded coexistence region. To facilitate our mechanism to work, the decrease in the collapse pressure while increasing the temperature must be faster than the reduction in the surface tension of pure water, i.e., dσw/dT > dπc/dT. Additionally the heating must be so strong that one reaches the intersection point of the isotension with the collapse tension (equation 5). Both predictions of the theory are verified by the experiment. For fatty acid monolayers, the typical threshold intensity of the laser for the collapse is on the order of Pc ≈ 2 W. The collapse is observed in all fatty acid esters (metyhy palmitate, methyl arachidate, and methyl stearate) they all exhibit a strong decrease in collapse pressure with temperature. An even stronger drop of collapse pressure with temperature is observed in lung-surfactant systems. Figure 7 shows the collapse behavior of a 7:3 DPPC/POPG mixture as determined by Gopal and Lee 2001.16 The collapse pressure, which occurs around 70 mN/m at temperatures between 20 and 28 °C, drops toward 60 mN/m at 29 °C and then further decreases to 45

mN/m when increasing the temperature to 40 °C (Figure 7). Here the slope of the collapse pressure with temperature is at least an order of magnitude steeper than the one measured for fatty acids (Figure 2). Time-resolved local collapse pressure experiments in lung surfactant systems should therefore be within experimental reach. Such experiments are currently under way at our laboratory. Observations of this kind will help to reveal the dynamic processes happening in the alveoli during the breathing cycle. Conclusion Locally heating a Langmuir monolayer with a focused laser induces the local collapse of the monolayer upon satisfying the following conditions: (i) the surface pressure is slightly below the collapse pressure and (ii) the collapse pressure of the monolayer decreases faster with temperature than with the surface tension of the pure air/water interface. Local collapse is associated with the growth of a 3-dimensional collapse aggregate that is fed by an incompressible inward flow of the two-dimensional monolayer surroundings. It should be possible to locally induce collapse in lung surfactant monolayers, which would enable time-resolved local studies of the monolayer collapse in the lung. Acknowledgment. We thank Ka Yee Lee and Ajay Gopal for useful discussion as well as for the kind permission to use their data in Figure 7. Appendix Here we derive the rate equation for the domain radius (eq 8). Taylor expansion of the temperature profile eq 7 around Req results in

T(r) - T(Req) ) -

RPw (r - Req) 2πκReq2

Expansion of eq 5 with the use of eq A1 leads to

(A1)

Laser Induced Collapse in Langmuir Monolayers

(

J. Phys. Chem. B, Vol. 110, No. 44, 2006 22165

)

dσc RPw 2η 8Req (R - Req) ) - Uln 2 dT π 2πκReq xAc

(A2)

The temperature dependence of the collapse tension can be expressed by the temperature dependence of the bare air/water surface tension and the collapse pressure:

dσc dσw dπc ) dT dT dT

(A3)

Now we insert eq A3 into eq A2 and express the aggregation velocity U with the speed of the aggregate edge (R˙ - R˙ eq) by the use of eq 8 to find

(

)(

)

dσw dπc RPw (R - Req) ) dT dT 2πκR 2 eq 8Req 2η d(R - Req) (Ac/Aaggr - 1) ln (A4) π dt A

x

c

rearranging the terms results in the rate eq 9 with the relaxation time given by eq 10. Supporting Information Available: Video clip 1 shows the entire time sequence of the local collapse including the recovery of the monolayer after turning off the IR-laser. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Schwarztschild, K. S. Deut. Akad. Wiss. Berlin, Kl Math.-Phys. Technol. 1919, 189. (2) Lu, W. X.; Knobler, C. M.; Bruinsma, R. F.; Twardos, M.; Dennin, M. Phys. ReV. Lett. 2002, 89, 146107.

(3) Adamson, A. W.; Gast, A. P. The Physical Chemistry of Surfaces, 6th ed.; John Wiley & Sons 1997. (4) Schurch, S.; Bachhofen, H.; Goerke, J.; Possmayer, F. J. Appl. Physiol. 1989, 67, 2389. (5) Oosterlakendijksterhiius, M. A.; Haagsman, H. P.; VanGolde, L. M. G.; Demel, R. A. Biochem. 1991, 30, 10965. (6) Longo, M. L.; Bisagno, A. M.; Zasadzinski, J. A. N.; Bruni, R, Waring, A. J. Science. 1993, 261, 453. (7) Oosterlakendijksterhiius, M. A.; Haagsman, H. P.; VanGolde, L. M. G.; Demel, R. A Biochem. 1991, 30, 8276. (8) Lipp, M. M.; Lee, K. Y. C.; Zasadzinski, J. A.; Waring, A. J. Science 1996, 273, 1196. (9) Perezgil, J.; Nag, K.; Taneva, S.; Keough, K. M. W. Biophys. J. 1992, 63, 197. (10) Angelova, A.; Vollhardt, D.; Ionov, R. J. Phys. Chem. 1996, 100, 10710. (11) Schief, W. R.; Touryan, L.; Hall, S. B.; Vogel, V. J. Phys. Chem. B 2000, 104, 7388. (12) Schief, W. R.; Hall, S. B.; Vogel, V. Phys. ReV. E 2000, 62, 6831. (13) Ybert, C.; Lu, W. X.; Mo¨ller, G.; Knobler, C. M. J. Phys. Chem. B 2002, 106, 2004. (14) Hatta, E.; Suzuki, D.; Nagao, J. Eur. Phys. J. 1999, B11, 609. (15) Hatta, E.; Fischer, T. M. J. Phys. Chem. 2002, 106, 589. (16) Gopal, A.; Lee, K. Y. C. J. Phys. Chem. B 2001, 105, 10348. (17) Lipp, M. M.; Lee, K. Y. C.; Takamoto, D. Y.; Zasadzinski, J. A.; Waring, A. J. Phys. ReV. Lett. 1998, 81, 1650. Diamant, H.; Witten, T. A.; Gopal, A.; Lee, K. Y. C. Europhys. Lett. 2000, 52, 171. (18) Ries, H. E., Jr. Nature 1979, 281, 287. (19) Wurlitzer, S.; Lautz, C.; Liley, M.; Duschl, C.; Fischer, Th. M. J. Phys. Chem. B. 2001, 105, 182. (20) Khattari, Z.; Hatta, E.; G. Kurth, D.; Fischer, Th. M. J. Chem. Phys. 2001, 115, 9923. (21) Khattari, Z.; Steffen, P.; Fischer Th. M.; Bruinsma, R. Phys. ReV. E. 2002, 65, 041603. (22) Khattari, Z.; Fischer, Th. M. J. Phys. Chem. B 2004, 108, 13696. (23) Denser in the terms of a 2D surface density. The volume density of the monolayer and the aggregate may be similar. (24) Koker R. D.; McConnell, H. M. J. Phys. Chem. B 1998, 102, 6927. (25) The flow profile at the surface is a 2D-incompressible flow without shear, such that neither the surface shear nor the surface dilatational viscosity lead to a damping of the flow. (26) The surface density times the velocity on both sides of the aggregate edge must be the same in the coordinate system co-moving with the aggregate edge.