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Electrophoretic Relaxation Dynamics of Domains in Langmuir Monolayers S. Wurlitzer,† H. Schmiedel,‡ and Th. M. Fischer*,† Fakulta¨ t fu¨ r Physik, Universita¨ t Leipzig, Linne´ str. 5, 04103 Leipzig, Germany, and Max Planck Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Am Mu¨ hlenberg 1, 14476 Golm, Germany Received December 19, 2001. In Final Form: March 4, 2002 Equilibrium patterns of a Langmuir monolayer of methyl octadecanoate are perturbed on a micrometer length scale using optical tweezers and laser heating. The consecutive electrophoretic motion of circular domains in liquid expanded or gaseous surroundings is investigated. The experimentally observed domain relaxation is described by a balance of the dissipative hydrodynamic force on the domain and an electrostatic dipole force from a neighboring domain. Drag forces derived from the experiments are in agreement with theoretical predictions (Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. Fluid Mech. 1981, 110, 349.) for the viscous drag on solid domains moving in monolayer surroundings of negligible surface shear viscosity. The dipole interactions are characterized by the surface potential differences between the coexisting phases. On pure water, the relaxation experiments reveal that the surface potential difference between the liquid condensed and the liquid expanded phase depends on the area fractions and the coexistence cannot be described by an ideal first-order phase transition with constant dipole densities of the phases.
1. Introduction Rheological properties of interfaces can be considerably changed by the adsorption of surfactant molecules. They play an essential role in technological problems appearing in the manufacture of synthetic dispersions and coatings, enhanced oil recovery, biotechnology, and separation processes. An overview of interfacial rheology and the experimental techniques can be found in Edwards1 and Miller.2 A real two-dimensional incompressible liquid resists the translational motion of objects. The only stationary solution at vanishing Reynolds numbers is the translation of the complete liquid. The object is immobile (Stokes paradox3,4). In real systems, the Stokes paradox is not observed due to compressibility of the monolayer, finite size, instationary motions, and the coupling force to a three-dimensional liquid. For the motion of circular domains in Langmuir monolayers of low viscosity, the coupling to the three-dimensional subphase is the dominant effect preventing the Stokes paradox. The ratio of the surface shear viscosity ηs [N s/m] and the subphase viscosity η [N s/m2] defines a characteristic length scale called “Boussinesq radius” rB. Above and below rB, different flow profiles are observed. On larger length scales, r > rB, the flow is mainly determined by the viscosity of the subphase. On smaller length scales, r < rB, the flow depends on the surface shear viscosity of the monolayer. For measurements of rheological properties of Langmuir monolayers, different techniques are developed. The rotating knife edge rheometer5 works on a millimeter * To whom correspondence should be addressed. E-mail:
[email protected]. † Max Planck Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. ‡ Universita ¨ t Leipzig. (1) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial transport processes and rheology; Butterworth-Heinemann: Boston, 1991. (2) Miller, R., et al. Colloids Surf., A 1996, 111, 75. (3) Lamb, H. Trans. ix., Papers, iii., 1850, 65. (4) Lamb, H. Hydrodynamics; Cambridge University Press: Cambridge, 1924. (5) Goodrich, F. C.; Allen, L. H.; Poskanzer, A. J. Colloid Interface Sci. 1975, 52, 201.
length scale. It is sensitive to surface shear viscosities [ηs . 1 nN s/m]. Shear deformations6 allow the observation of surface shear viscosities in the same order of magnitude. Heckl, Miller, and Mo¨hwald7 created an electrophoretic motion with an inhomogeneous electric field in the vicinity of a charged needle. With the canal flow experiment,8,9 one can investigate interface flow profiles driven by surface pressure differences. In the present work, local perturbations of the monolayer are achieved by a mechanical manipulation with optical tweezers or by local heating with a focused laser. The device and its calibration are described in detail elsewhere.10 Because of its micrometer working scale,11 the technique is able to detect small surface shear viscosities [ηs > 1 nN s/m]. A single domain is placed in a nonequilibrium position. We investigate the consecutive relaxation dynamics driven by electrostatic dipole forces, calculated in section 2.1, and balanced by the hydrodynamic drag force, discussed in section 2.2. Section 2.1 encloses the surface potential measurements for the characterization of the dipole interactions of the domains. The calculated domain kinetics (section 3.1) is fitted to the experiments (section 3.2), and the friction coefficients of the domains are determined (section 3.3). Finally, discussion, conclusions, and the summary of the results follow. 2. Forces on Circular Domains in Langmuir Monolayers 2.1. Electrostatic Interaction Forces. Patterns of coexisting phases in Langmuir monolayers are the result of electrostatic interactions between asymmetrically aligned electric dipoles of the surfactants at the interface (6) Benvegnu, D. J.; McConnell, H. J. Phys. Chem. 1992, 96, 6820. (7) Heckl, W. M.; Miller, A.; Mo¨hwald, H. Thin Solid Films 1988, 159, 125. (8) Stone, H. Phys. Fluids 1995, 7, 2931. (9) Schwartz, D. K.; Knobler, C. M.; Bruinsma, R. Phys. Rev. Lett. 1994, 73, 2841. (10) Wurlitzer, S.; Lautz, C.; Liley, M.; Duschl, C.; Fischer, Th. M. J. Phys. Chem. 2001, B105, 182. (11) Steffen, P.; Heinig, P.; Wurlitzer, S.; Khattari, Z.; Fischer, Th. M. J. Chem. Phys. 2001, 115, 994.
10.1021/la015747x CCC: $22.00 © 2002 American Chemical Society Published on Web 04/26/2002
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and the line tension between the coexisting phases.12 The interaction energy Wd of two neighboring monolayer domains 1 and 2, embedded in a matrix phase M, can be described by
∫A ∫A
Wd ) µ2
1
da1da2
2
3
|r b1 - b r 2|
) -µ2 I∂A1 I∂A2
ds b1 ‚ ds b2 |r b1 - b r 2|
(1)
where A1 and A2 (da1 and da2) denote the area (elements) of domains 1 and 2 and b r1 and b r2 denote the position vectors of these area elements. Equation 1 considers only dipole components perpendicularly oriented to the interface. Wd is proportional to the material parameter µ2 (unit N), which can be expressed in terms of surface potential differences:13
µ2 )
0 aL (V - VM)(V2 - VM) 2π (a + L) 1
(2)
Here V1, V2, and VM are the surface potentials of phase 1, phase 2, and the matrix phase; 0 is the vacuum permittivity, and a (L) is the relative permittivity of air (liquid). In Langmuir monolayers, the domains usually exhibit nearly circular shapes with radii in the order of several micrometers. The interaction energy Wd of a domain pair with the radii R1 and R2 is calculated for the disk/halfplane configuration (R1 . R2 ) R) and for the disk/disk configuration with equal radii (R1 ) R2 ) R) (Appendix 1). The electrostatic body force Fd onto both domains is obtained from Wd [eq 1] by taking the derivative with respect to the domain separation D:
∂Wd ∂D
Fd ) -
{
(3)
For R1 . R2 ) R, one obtains
[
Fd
) 4π
µ2
(1 + )
x2 + 2
]
x8
π
-1 f
for , 1
2π 2π ≈ 2 for . 1 2 (1 + ) (4)
with ) D/R. At small distances ( , 1), the force diverges with Fd ∝ -1/2, and for large distances it reduces to the force of a point dipole in front of a half-plane (Fd ∝ -2). In the case of two interacting disks with equal radii (R1 ) R2 ) R), one obtains
Fd
d ) 4π {Y()} f d µ2
with
Y() )
1 4π
{
2π x
for , 1
3π2 3π2 ≈ for . 1 (2 + )4 4
(5)
2 2 1/2 where E(k) ) ∫π/2 denotes the complete 0 dϑ(1 - k sin ϑ) elliptic integral of the 2nd kind. In Figure 1, we show a plot of Fd/µ2 versus for two circular domains of equal radius [eq 5]. The general relation (solid line) for 10-4 e e 10 is calculated numerically. The asymptotic expressions are added as dotted lines. They do not describe the relation in the experimentally relevant interval 0.1 e e 10 completely, and hence the numerical result must be used. The dipolar interaction force of a disk in front of a dipolar half-plane [eq 4, no depiction] behaves similarly. A quantitative analysis of the dipole interaction requires the knowledge of the surface potential differences ∆V between the domain and the surrounding phase [eq 2]. In the present work, these measurements are performed by combining the Kelvin probe technique14 with fluorescence microscopy measurements of the area fractions of the coexisting phases (Φ). In an ideal first-order phase LC transition, the molecular areas A LE Mol and A Mol of the coexisting liquid expanded (LE) and liquid condensed (LC) phases are fixed during the transition. The average molecular area of the monolayer AMol and the area fraction ΦLC ) ALC/A of the LC phase (ALC is the area of the LC phase, A is the total area of the monolayer) are then related via
AMol )
LE A LC Mol A Mol LC LE A LC Mol - ΦLC(A Mol - A Mol)
[2x - sin(2x)] sin(2x)
[x
1-
/2 - cos x (6) ( 11 ++ /2 + cos x) ] 2
(12) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. (13) Rivie`re, S., et al. Phys. Rev. Lett. 1995, 75, 2506.
(7)
The surface potential V ) V(ΦLC) averaged over the LC and LE phases fulfills the lever rule:15,16
V(ΦLC) ) ΦLC(VLC - VLE) + VLE
(8)
where VLE and VLC denote the constant (ΦLC-independent) surface potentials of the LE and LC phases. If the lever rule [eq 8] holds, the material parameter µ2 [eq 2] can be determined by using
(VLC - VLE)2 ) [V(ΦLC)1) - V(ΦLC)0)]2
∫0π/2 dx (1 + /2 + cos x)(1 + /2 - cos x)2 × E
Figure 1. Dipolar interaction force of two domains with radius R ) R1 ) R2 as a function of their separation ) D/R, calculated numerically using eqs 5 and 6. The asymptotic expressions are added as dashed lines. For experimentally relevant separations, 0.1 < < 10, none of the asymptotic expressions describes the force completely.
(9)
With the Kelvin probe, the dependency V ) V(AMol) is measured, and from the fluorescence microscopy images one obtains AMol ) AMol(ΦLC). Fitting the latter with eq 7 and combining the result with the Kelvin probe experi(14) Adamson, A. W. Physical chemistry of surfaces; Wiley: New York, 1990. (15) Miller, A.; Helm, C.; Mo¨hwald, H. J. Phys. 1987, 48, 693. (16) Miller A. Dissertation, T. U. Mu¨nchen, Mu¨nchen, Germany, 1986.
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Table 1. Surface Potential Differences ∆VLE/LC, LC Molecular Areas A LE Mol and A Mol, Composition of the Glycerol/Water Subphases [cGly ) mGly/(mH2O + mGly)], and Temperature T of Methyloctadecanoate Monolayers in the LE/LC Phase Coexistence Region cGly
ϑ [°C]
2 A LE Mol [Å ]
2 A LC Mol [Å ]
0
35 ( 1
34 ( 1
19 ( 1
0.6 0.8
25 ( 1 25 ( 1
+20 160-40 180 ( 30
30 ( 5 25 ( 5
a
∆VLE/LC [V] (0.38 ( 0.04)a (0.18 ( 0.05)b (0.27 ( 0.03)a (0.21 ( 0.04)a
Kelvin probe measurements. b Domain electrophoresis.
ments, we obtain V(ΦLC). We measured V(ΦLC) according to this procedure for methyloctadecanoate on various glycerol/water subphases. For the glycerol concentrations cGly ) 0.4, 0.6, and 0.8 [cGly ) mGly/(mH2O + mGly)], we find good agreement with eq 717 and with the lever rule eq 8. On pure water (cGly ) 0), the experimental data deviate from eq 8. The use of eq 9 despite this fact results in ∆VLE/LC ) 0.38 V. Due to the nonideality of the phase transition, the surface potential difference ∆VLE/LC(ΦLC) depends on ΦLC and values at ΦLC * 0 and ΦLC * 1 can differ from those determined via eq 9. Table 1 shows the LE/LC surface potential differences of all subphase compositions of interest obtained by the procedure described above. The surface potentials of the three-phase coexistence region G (gas)/LE/LC of methyloctadecanoate on pure water are taken from refs 17 and 18. 2.2. Hydrodynamic Drag Force. The drag force Fη of a solid circular domain of radius R, moved stationarily in an incompressible two-dimensional monolayer with a surface shear viscosity ηs, is calculated by Hughes19 taking into account the coupling of the monolayer to a threedimensional underlying liquid of shear viscosity η. Analytical expressions for the drag force are obtained for negligible [B , 1, B ) ηs/(Rη)] and dominating (B > 2) surface shear viscosity:
Fη ) - fRηv
(10)
fliquid(B) )
π
∫0
z2[1 + Bz]
{
(solid domain) (11) Here v denotes the domain velocity. The dimensionless friction coefficient f depends on the Boussinesq number B. In our experiments, moving liquid condensed, liquid expanded, and gaseous domains in liquid expanded or gaseous surroundings are investigated. The compression of the LE or the G phase is negligible.20 De Koker has calculated the effect of back flow inside a liquid disk moved in liquid surroundings of the same surface shear viscosity. He obtained (17) Wurlitzer, S. Dissertation, Universita¨t Leipzig, Leipzig, Germany, 2001. (18) Heinig, P.; Wurlitzer, S.; Steffen, P.; Kremer, F.; Fischer, Th. M. Langmuir 2000, 16, 10254. (19) Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. Fluid Mech. 1981, 110, 349. (20) Dimova, R., et al. J. Colloid Interface Sci. 2000, 226, 35.
(12)
dz
where J1 denotes the Bessel function of the first kind. For domains where the internal surface shear viscosity is larger than that of its surroundings, the value of the friction coefficient f lies between the values obtained from eq 11 and eq 12. The relative difference in f ) f(B) does not exceed 10% and is not detectable in the experiments. Therefore, eq 11 is a good approximation for liquid domains. The hydrodynamic interaction of two domains (squeeze flow), moved in incompressible surroundings, can be estimated in the limit of a pure two-dimensional system for R1, R2 . D:21
[
]
3/2 2R1R2 3 Fs ) πηsv 2 D(R1 + R2) (hydrodynamic interaction) (13)
For large surface shear viscosities and small distances, the squeeze force Fs exceeds the drag force Fη. In our experiments, surface shear viscosities of the surrounding phases are small and squeeze forces can be neglected, as we will show in the next chapter. 3. Relaxation Dynamics 3.1. Theory. The balance of the dipole interaction force [eq 4 and eq 5] and the drag force [eq 10] on domains reads
Fd + Fη ) 0
{
(14)
In both geometries, Fd is of the general form Fd ∝ ξ() with
ξ() )
(1 + )
x2 + 2
-1
{ }
dY 2 d
for R1 . R2 ) R (15) for R1 ) R2 ) R
Equation 14 transforms into a differential equation τ˘ ) ξ() for (t) with the solution
with
4πB 4 1 ln(2B) for B > 2 f ) ln(2B) - 0.577 + πB 2B2 8 for B , 1
(liquid domain)
J12(z)
∞
d ) ∫0(t) ξ()
t - t0 τ
where τ )
fηR2 4πµ2
(16)
∆t ) t - t0 is the time from/to the contact of the domains. The parameters t0 and τ are used to fit the experimental data. With eq 15 and eq 16, the time dependence of the distance of the two domains is calculated. One obtains for the disk/half-plane geometry (R1 . R2 ) R; Appendix, section A.2)
∆t 1 3 ) [ + 32 + x(2 + 2)3] f τ 3 x8 3/2 3
{
for , 1 (17)
2 2 (1 + )3 ≈ 3 for . 1 3 3
For the geometry of two disks with equal radii (R1 ) R2 ) R), the motion proceeds along the connecting line of the (21) Fischer, Th. M. Der Langmuir Monolayer, ein quasi zweidimensionales System im thermodynamischen Gleichgewicht und Nichtgleichgewicht; Habilitationschrift; Universita¨t Leipzig: Leipzig, Germany, 1999.
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Figure 2. Relaxation dynamics of two domains with radius R as calculated numerically using eqs 6 and 15 and eq 16 (solid line). The asymptotic relations (dashed lines) cannot be used for experimentally relevant separations (0.1 < < 10).
centers in opposite directions with a velocity v ) 1/2(dD/ dt). The dynamics of the asymptotic cases . 1 and , 1 (dotted lines in Figure 2) can be calculated analytically:
∆t ) τ
∫0(t) d (2
)
dY() -1 f d 2 3/2 3
{
for , 1
2 2 5 (2 + )5 ≈ for . 1 15π 15π
(18)
The relaxation in the range 10-4 e e 10 is calculated numerically using eq 6 and eq 18 (solid lines in Figure 2). One observes that on experimentally relevant length scales none of the asymptotic approximations describes the dynamics completely. It can be seen in Figure 2 that the relaxation time ∆t of the domain is of the order of τ, if the separation is of the order of the radius of the domain. For separations 1 order of magnitude larger than that, ∆t and τ differ by order of magnitudes, such that the time scale τ can be as much as four orders of magnitude lower than the experimental time scale ∆t. The disk/half-plane relaxation [eq 17] shows a similar behavior and is therefore not depicted. 3.2. Experiments. Relaxation experiments are performed with domains of different phases (G, LE, and LC) in G or LE surroundings on altered subphases (different viscosities). The domains were displaced or created in a nonequilibrium position using optical tweezers22 or local heating of the monolayer. Their relaxation kinetics is observed with fluorescence microscopy.23,24 Examples for relaxation experiments are presented in Figures 3-6. Figure 3 shows a monolayer of methyloctadecanoate (Aldrich) on pure water (Millipore) in the LE (bright area)/LC (dark domains) phase coexistence at ϑ ) 35 °C. For the mechanical manipulation, silica beads [Polyscience, L (3-5) µm] are spread into the monolayer from a chloroform suspension. Before being spread, they have been rinsed several times with chloroform. One of the beads (not visible) is trapped in the optical tweezers (white cross). The monolayer is subject to a small drift25 of velocity v ≈ 9 µm/s, directed from the right top to the left bottom. At t ) -3.3 s, the silica bead hits the rear of LC domain no. 1 (R1 ) 12 µm), adheres at the domain (22) Ashkins, A. Phys. Rev. Lett. 1970, 24, 156. (23) Lo¨sche, M.; Sackmann, E.; Mo¨hwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 848. (24) McConnell, H. M.; Tann, L. K.; Weiss, R. M. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 3249.
Figure 3. Relaxation dynamics of two LC domains in LE surroundings. The monolayer is subject to a small drift from the upper right to the lower left (v ≈ 9 µm/s). Domain no. 1 (R1 ) 12 µm) is fixed at the rear with a silica bead (not visible) in the optical tweezers (white cross). Domain no. 2 approaches no. 1 (t ) -3.3 s to t ≈ 0 s). At small domain separation, the tweezers are switched off and the repulsive dipolar separation dynamics follows (t ≈ 0 s to t ) 4.1 s).
Figure 4. Dipolar repulsion of a LC domain (R ) 7 µm) in front of a LC half-plane in LE surroundings.
Figure 5. Dipole attraction of a G bubble (R ) 2 µm) in front of a LC half-plane in LE surroundings. After hitting the LC phase (t ) 0 s), the bubble partially wets the phase boundary with an equilibrium contact angle of 110°.
boundary, and fixes its position. Due to the drift, LC domain no. 2 (R2 ) 10 µm) approaches domain no. 1. At a very small separation, the tweezers are switched off (t ≈ 0 s) and the repulsive relaxation dynamics of both domains can be observed (t ) 1.3 s, t ) 4.1 s). In all pictures, (25) An hydrodynamic drift does not shear the monolayer on the length scale of the fluorescence microscope image (300 µm) and can be easily distinguished from the translational motion of the domains due to electrostatic dipole interactions as the former does not change the separation of the domains. In all experiments, the relative separation of two domains is measured, such that the drift velocity drops from the equations.
Electrophoretic Relaxation Dynamics of Domains
Figure 6. Dipolar repulsion of a LE domain (R ) 6 µm) from a LC half-plane in G surroundings. A large G bubble nucleates due to local heating. Afterward, a LE domain dewets the LC/G boundary (t ) -0.6 s to t ) 0.3 s), its shape relaxes to a disk, and the domain is repelled from the LC half-plane (t ) 0 s to 4.8 s).
the time t denotes the extrapolated time from/to the contact of the domains. For local heating experiments (Figures 4-6) with laser powers P > 200 mW, no silica beads are needed. In the LE/LC phase coexistence region (Figure 4), this technique is used for local melting of a phase boundary, transferring LC into LE. After switching off the heating, a new LC domain (R ) 7 µm) nucleates close to the melted one (LC half-plane), and a repulsive dipolar relaxation dynamics can be observed. The domain/half-plane distance increases with time (t ) 1.9 s to t ) 14.1 s). At temperatures ϑ e 27.5 °C, methyloctadecanoate shows a coexistence region of the G (dark), LE (bright), and LC (gray) phases (Figure 5 and Figure 6). Here local heating causes the nucleation of G bubbles of different sizes, depending on the power of the heating. After switching off the heating, the bubbles shrink a little but remain stable. At low laser powers (P ≈ 200 mW), small G bubbles with radii of several micrometers nucleate in the LE phase. An example is shown in Figure 5. The G bubble is attracted [negative sign of µ2, eq 2] by the LC half-plane at the bottom of the image (t ) -3.2 s to t ) 0 s). After hitting the LC phase, the G bubble partially wets the boundary with an equilibrium contact angle of 110°. At laser powers P > 500 mW, dilatational flow at the interface creates a larger G bubble (R g 100 µm) in LC surroundings (Figure 6). After the laser is switched off, now a reconfiguration of areas with different phases (dipole densities) can be observed. Initially, the G/LC phase boundary is partially wetted by the LE phase. These LE domains dewet the G/LC boundary (t ) -0.6 s to 0.3 s), relax to a circular shape (t ) 0.7 s to t ) 3.3 s), and are pushed into the interior of the G bubble due to their dipole interactions with the LC half-plane. The nature of dissipation in the relaxation dynamics can be investigated by changing the Boussinesq number B ) ηs/(Rη) and determining the dependence of the friction coefficient f on B. To change B, melting experiments such as those presented in Figure 4 have been carried out at different subphase compositions (water/glycerol) and radii of the domains. 3.3. Analysis and Interpretation. The data of all domain experiments are fitted with the relaxation times τ and the offset time t0, using eq 17 (disk/half-plane geometry) and eq 18 (disk/disk geometry). When the normalized time t/τ is plotted versus the normalized distance , all experimental data fall onto two master curves (eq 17 or eq 18, respectively) as shown in Figure
Langmuir, Vol. 18, No. 11, 2002 4397
Figure 7. Normalized time ∆t/τ as a function of the normalized domain separation ) D/R. The experimental data fall on top of the two master curves for the disk/disk or disk/half-plane geometries. (- - -) Theoretical disk/disk kinetics (numerical with eq 6 and eq 18, top -axis); (s) theoretical disk/half-plane kinetics (eq 17, bottom -axis); (/) LC disk/LC half-plane in LE (Figure 4, bottom -axis); (b) LC disk/LC disk in LE (Figure 3, top -axis); (9) LC disk/LC half-plane in G (bottom -axis); (4) LE disk/LC half-plane in G (Figure 6, bottom -axis); (1) G disk/LC half-plane in LE (bottom -axis); (O) LC disk/LC half-plane in LE (η ) 8.8 mN s/m2, bottom -axis); (0) LC disk/LC half-plane in LE (η ) 66 mN s/m2, bottom -axis).
7. The friction coefficients are obtained using eq 2 with eq 16 assuming L . 1:
f)
20τ ηR2
∆V1∆V2
(19)
The values of ∆V (∆V1/2 ) V1/2 - VM), η, R, τ, and f are summarized in Table 2. The experiments on pure water, where deviations of the surface potential dependence ∆V ) ∆V(ΦLC) from the lever rule eq 8 have been observed (see section 2.1), are marked with an asterisk (*). Using the lever rule despite its invalidity results in friction coefficients larger than the theoretical minimum f ) 8 in these cases. For all other experiments, the surface potential fulfills the lever rule and the measured friction coefficients are in the range of 6 e f e 11. The friction coefficient does not depend on the radius of the domain (one order of magnitude changed), the subphase shear viscosity (two orders of magnitude changed), and the state of aggregation (LE, LC, G) of the moving domain or its surroundings. On the micrometer length scale, the relaxation can be described by a balance of dipole and drag forces only. The different elasticity of the LE and the G phase of the droplet phase or the monolayer surroundings has no detectable effect on f. 4. Discussion and Conclusions With the exception of the experiments on pure water, the friction coefficients f are in the order of f ) 8. Changing the subphase viscosities and the radii does not affect f [f * f(B)]. These findings are in agreement with the theoretical prediction for a flow dominated by subphase dissipation and characterized by low Boussinesq numbers B [eq 10 and eq 11 with B , 1]. For B . 1 (two-dimensional flow), one would expect a strong dependence of the friction coefficent on the subphase viscosity and on the radius of the domain. These results are in agreement with those of Klingler,26,27 Heckl,28 and Steffen.11 Experiments in the LE/LC phase coexistence region of methyloctadecanoate on pure water, however, deviate from these general findings. Using a method which does not (26) Klingler, J. F.; McConnell, H. M. J. Phys. Chem. 1993, 97, 6096. (27) Klingler, J. F.; McConnell, H. M. J. Phys. Chem. 1993, 97, 2962. (28) Heckl, W. M.; Miller, A.; Mo¨hwald, H. Thin Solid Films 1988, 159, 125.
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Table 2. Calculated Friction Coefficient f, Fit Parameter τ, and Other Relevant Experimental Data of the Relaxation Experiments kind of relaxation (D., disk; H., half-plane; [Figure])
ϑ [°C]
LC-D./LC-D. in LE, [3]* LE-D./LC-H. in LE, [4]* LE-D./LC-H. in G, [6]
35 ( 1 35 ( 1 25 ( 1
0.8 ( 0.1 0.7 ( 0.1 0.4 ( 0.1
LC-D./LC-H. in G G-D./LC-H. in LE, [5]
25 ( 1 25 ( 1
(5.5 ( 1.5) × 10-3 (-48 ( 10) × 10-3
LC-D./LC-H. in LE LC-D./LC-H. in LE
25 ( 1 25 ( 1
7.5 ( 2 1.9 ( 0.3
a
τ [s]
η [mN s/m2]
∆V1, ∆V2 [V] 0.38 ( 0.04 0.38 ( 0.04 0.27 ( 0.c 0.135 ( 0.03c 0.41 ( 0.06c 0.27 ( 0.05c 0.135 ( 0.03c 0.27 ( 0.03 0.21 ( 0.04
R [µm]
f
0.725 (H2O)a 0.725 (H2O)a 0.893 (H2O)a
11 ( 0.5b 7 ( 0.5 6 ( 0.5
23 ( 6* 50 ( 30* 8(7
0.893 (H2O)a 0.893 (H2O)a
1.3 ( 0.5 2 ( 0.5
11 ( 8 9(8
8.8 (cGly ) 0.6)a 66 (cGly ) 0.8)a
11 ( 0.5 2 ( 0.5
9(6 6(5
Landolt Bo¨ rnstein IV/1, 6th ed.; Springer: Berlin, 1955; pp 600, 613. b Average radius of both domains. c References 17 and 18.
rely on the electrostatics of the monolayer, Steffen et al.11 showed that the surface shear viscosity of this system is indeed negligible. This contradiction can be explained by the failure of the lever rule [eq 8, eq 9] to describe the surface potential data V ) V(ΦLC). Presumably, the value of ∆VLE/LC ) 0.38 V, determined by applying the lever rule, is overestimated. Using Steffen’s11 result (f ) 8) together with eq 19, we obtain a surface potential difference ∆VLC/LE ) (0.18 ( 0.05) V. Hence, if the rheological properties of a configuration of the monolayer are known, the relaxation experiments offer a method for measuring local surface potential differences acting on the micrometer length scale. The relaxation experiments (including those on pure water) fall on top of the theoretical master curves. If the monolayer viscosity were large, hydrodynamic interactions between the domains would lead to deviations of the experimental data at small distances. But this is not observed. We have an a posteriori argument that the squeeze force [eq 13] can be neglected. The relaxations give an experimental proof that the surface shear viscosity of the LE and G phase is negligible on micrometer length scales and that the interaction between domains is of purely electrostatic origin. Experiments of the sort presented here could be an interesting way of probing the rheological properties of viscous monolayers with nonvanishing Boussinesq numbers, because the shear rate ˘ would vary from relatively low values at large separations > 1 to high values at small separations < 1 during the relaxation. Deviations of the relaxation from the master curve in Figure 2 would in this way in principle allow one to get a hand on nonNewtonian constitutive dynamic surface excess equations. Note, however, that the relations 11 and 12 hold only for Newtonian monolayers and have to be replaced by new relations between the drag coefficient and the Boussinesq number derived from the proper non-Newtonian constitutive equation. 5. Summary Equilibrium patterns within monolayers can be perturbed using local mechanical and thermal manipulation techniques. The analysis of the relaxation following the perturbation reveals that in a large range of domain radii and subphase viscosities, and independent of the state of aggregation within the domains or the surroundings, the origin of dissipation is the friction of the subphase. The friction coefficients are in accordance with theoretical predictions of Hughes et al.19 in the limit of low Boussinesq numbers. The surface potential data of methyl octadecanoate on gylycerol/water mixtures show an ideal behavior, but the electrostatic behavior of the coexistence region liquid expanded/liquid condensed on pure water cannot be described as an ideal first-order phase transition
with constant surface potential differences ∆V or dipole densities. In this system, ∆V depends on the area fraction of the phases. Acknowledgment. We thank Professor H. Mo¨hwald for generous support and stimulating discussion. T. Fischer thanks the Deutsche Forschungsgemeinschaft for providing a Heisenberg fellowship. We thank P. Steffen for useful discussion. Appendix A.1. Dipole Forces. A.1.1. Two Circular Domains. The dipole energy eq 1 for two circular domains of radius R with their centers separated by the vector B b reads
Wd ) µ2
∫d2br ∫d2br ′ Θ(R - |br - B2b|) |rb -1 br ′|3 ×
( | B2b|) ∫d br ′ Θ(R - |br ′ + b2r |) × b r r ′ - |) Θ(R - |b 2 r′ + ΘR- b
) µ2
∫d2br |rb -1 Bb|3
) µ2
∫d2br |rb -1 Bb|3 p(rb)
2
(A1)
with
p(r b) )
∫d2br ′ Θ(R - |br ′ - b2r |) Θ(R - |br ′ + b2r |)
(A2)
Θ(x) denotes Heaviside’s function. From the first to the second line in (A1), we have made use of an affine transformation of the integration variables, namely, (r, r′) f (ra ) r′ - r + b, r′a ) r/2 + r′/2). Setting the x′ axis along b r and assuming r E 2R, equation A2 reads
p(r b) )
r
∫-xxRR --(r(r/4)/4) dy′ ∫r -- 2x+Rx-y′R -y′ 2
2
2
2
2
2
2
dx′ )
2
2
∫
xR2 - (r2/4)
-
dy′(2xR2 - y′2 - r)
xR2 - (r2/4)
)|
y′ ) y′xR2 - y′2 + R2 arcsin - ry′ R
(
(
x
) R2 2 arcsin
1-
x
r2 r 4R2 R
[R2-(r2/4)]1/2
-[R2-(r2/4)]1/2
1-
)
r2 4R2
(A3)
Electrophoretic Relaxation Dynamics of Domains
Langmuir, Vol. 18, No. 11, 2002 4399
For r > 2R, we have p(r b) ) 0. Therefore,
Wd ) µ2
(
∫d2br |rb -1 Bb|3 ×
x
R2 2 arcsin
1-
x
r2 r 2 R 4R
1-
Wd )
)
Wd )
2πRµ2 4π
∫02π dφ∫0π/2 dψ × sin ψ cos ψ(2ψ - 2 sin ψ cos ψ)
(
)
b2 b + cos2 ψ - cos ψ cos φ 2 R 4R
3/2
(A5)
∫0π/2 dψ ×
E
(x
1-
∫0
Iν(φ) )
For ν ) 1/2, we find
I1/2(φ) )
(1 + /2 + cos ψ)
2
) 4πRµ2Y() (A11)
2ψ
∫
c+
∫0π/2
4 1/2 3π
1/2
dy
y4 + o() ) 1 + y4
[
]
1 y 3 4 1/2 y+ - arctan y 3π 2 1 + y2 2
π/21/2
+ o() )
0
c′ - 1/2 + o() (A12) while in the limit f ∞ we obtain
) ∫02π xp -dφ q cos φ π/2 ∫ 0 xp + |q|
)
(1 + /2 - cos ψ)2
(2ψ)3 6 1 π/2 dψ E(1) + c + o() ) Y(f0) ) 4π 0 2(/2 + ψ2/2)2 c+
1 ∂ dφ Iν-1(φ) (A6) ) (p - q cos φ)ν ν - 1 ∂p
×
In the limit f 0, the leading contributions to eq A11 occur near ψ ) 0. We may therefore expand the integrand around ψ ) 0 to obtain
The integral over φ in (A5) is of the form 2π
sin 2ψ(2ψ - sin 2ψ)
(1 + /2 + cos ψ)(1 + /2 - cos ψ)2
r2 Θ(2R - r) 4R2 (A4)
Denoting the angle between B b and b r with φ and substituting r ) 2R cos ψ, we obtain
4πRµ2 4π
4
Y(f∞) ) dχ
x
) 2|q| sin2 χ p + |q| 4 2|q| K (A7) p + |q| xp + |q|
1-
(x
)
where K(k) denotes the complete elliptic integral of the first kind. Using the recurrence relation A6 and the relation29
1 4π
sin 2ψ(2ψ - sin 2ψ)
∫0π/2 d ψ
(A8)
π 43 (A13)
E(0) )
Using eq 3 and eqs A11-A13, we obtain eq 5. A.1.2. Circular Domain in Front of a Half-Plane. The dipole energy of a circular domain of radius R separated from a half-plane (x′ < 0) by the distance D reads
∫DD+2R dx ∫-xxRR-(R+D-x) -(R+D-x) 2
W d ) µ2
E(k) K(k) dK(k) ) 2 dk k k(1 - k )
(/2)3
2
2
dy
2
∫-∞0 dx′ ∫-∞∞ dy′ ×
1 (A14) ((x - x′)2 + (y - y′)2)3/2
we find
Evaluating the first three integrals yields
∂ I3/2(φ) ) -2 I1/2(φ) ) ∂p 4
xp + |q|(p - |q|)
E
(x
)
|q| (A9) p + |q|
∫0
π/2
sin 2ψ(2ψ - sin 2ψ) dψ × 2 b b + cos ψ - cos ψ 2R 2R
(
)(
R2 - (R + D - x)2 (A15) x
For the last integral, one finds
and therefore
4πRµ2 Wd ) 4π
∫DD+2R dx x
Wd ) 4µ2
)
( )
2b cos ψ x R E b + cos ψ 2R
(A10)
[x
Wd ) 4µ2
R2 - (R + D - x)2 -
D+R-x (D + R) arcsin R (R + D)(x - D) - DR xD(D + 2R) arcsin xR
]
) 4πµ2[D + R - xD(D + 2R)]
x)D
(A16)
Hence, the body force onto the disk is
Finally, with b ) 2R(1 + /2) we obtain (29) Gradstein, I. S.; Ryshik, I. M. Tables of series, products, and integrals; Harri Deutsch: Thun Frankfurt/M; Vol. 2, Number 8.123 2.
x)D+2R
Fd ) -
[
(1 + ) dW ) 4πµ2 -1 dD x2 + 2
]
(A17)
4400
Langmuir, Vol. 18, No. 11, 2002
Wurlitzer et al.
A.2. Relaxation Dynamics of a Circular Domain in Front of a Half-Plane. The relaxation kinetics of a circular disk in front of a half-plane (R1 . R2) can be calculated, if one neglects the hydrodynamic domain/halfplane interaction, by inserting eq 15 into eq 16:
∫0(t)
d 1+
x2 + 2
) -1
t - t0 ∆t ) τ τ
With the substitution z ) 1 + , eq A18 reads
(A18)
∆t ) τ
∫1z(t)
dz
x
)
2
z -1 z2 - 1
(
∫1z(t) (z2 - 1 + zxz2 - 1) dz )
-z +
)
z3 1 2 + (z - 1)3/2 3 3
z(t) 1
(A19)
and the relaxation kinetics is described by
∆t 1 3 [ + 32 + x(2 + 2)3] ) 3 τ LA015747X
(A20)
