Langmuir 1992,8, 2747-2752
2747
Damping of Cylindrical Propagating Capillary Waves on Monolayer-Covered Surfaces Qiang Jiang and Yee C. Chiew’ Department of Chemical and Biochemical Engineering, Rutgers University, P.O.Box 909, Piscataway, New Jersey 08855-0909
Jose E. Valentini Imaging Systems Department, E. I. du Pont de Nemours & Company, P.O. Box 267, Brevard, North Carolina 28712 Received January 13, 1992. In Final Form: July 22,1992 Cylindricalpropagating capillarywaves are generated by an electrical “point source”and detected with a reflected laser beam. The surface properties of monolayers of soluble (Triton X-100)and insoluble (n-dodecylp-toluenesulfonate)surfactant solutions were investigated through both propagating plane and cylindricalcapillarywaves. We foundthat the values of measured dampingcoefficiente and wavelengths of plane waves are equal to those of cylindricalwaves. Computed values of the surface properties, elastic modulus z and surface pressure II,for both waves are obtained and found to be identical. Our results show that even though plane and cylindrical transverse waves possess different geometries, they can be used to obtain surface dilationalPropertiesaccurately. Also, the dispersionequation for propagatingcylindrical waves is explicitly derived. 1. Introduction
The study of the equilibrium and nonequilibrium propertiesof &/liquid interfacescontainingsurface-active substances (for example, surfactants, biomolecules,lipids, macromolecules) has received considerable attention recently. It is an area of research that is of considerable technologicalrelevance to many interfacial processes such as high-speed coating, wetting, foaming, emulsification, detergency, etc. In this work, we study the dilational properties of monolayer films. The surface dilational modulus e characterizesthe response of the surfacetension to local compression and expansion and is defined by e = duld In A (1) where Q is the surface tension and A represents the area of the surface element. The modulus e, in general, is represented by a complex function, i.e. e
= ed
+ iwqd
(2)
here, Cd and Vd are known as the surfacedilationalelasticity and dilational viscosity, respectively. Capillary (transverse) wave is one of the powerful methods used in the study of surface viscoelasticity. The theory of plane capillary waves has been investigated extensively in recent In this theory, the relationship between the wave characteristics and surface propertiesis establishedvia the dispersionequation;hence, the surfaceproperties may be obtained from experimental measurementsof the wavelength X and damping coefficient @ of the capillary waves. A number of experimental investigationsof surfacedilationalviscoelasticityhave been carried out using standine’as well as propagating plane waves.*ll Because both shear and dilationaldeformations
* To whom correspondence should be addressed.
(I) Levich, V. G. Physico-chemical Hydrodynamics; Prentice Halk Englewocd CWn, NJ, 1962. (2) Hamen, R.; Mann,J. J. Appl. Phys. 1964, 35, 158. (3) Temple, V. D.; Riet, V. D. J. Chem. Phys. 1966,42, 2769. (4) Lucaeeen-Reynders,E.H.; Lucassen, L. Adu. Colloidlnterface Sci. 1969,2, 347. (5) H-n, R.;Ahmnd, J. Prog. Surf. Membr. Sci. 1971,4, 1. (6) Mann, J.; H-n, R. J. Colloid Interface Sci. 1963, 18, 757. (7) Lucaesen, J.; Hansen, R. J. Colloid Interface Sci. 1966,22, 32.
are present in plane capillarywaves,these experimentsdo not provide a clear separation of dilational and shear properties. Furthermore, in an actual plane wave experiment, the wave invariably is excited (through electrocapillarity or mechanical means) by using a generator of finite length; thus, the surface ripple must propagate and shear against two bodies of “quasi-stagnant” liquids that lie on either edge of the generator. This will result in an additional shear deformation (edgeeffect)in experimente. Hence, the dilational elasticity obtained from plane wave experimentswill inevitablyreflect some of these effe~t.a.~~J~ In this work, we generate propagating cylindricalwaves through electrocapillarity and compare their wave prop ertfes (i.e. wavelength X and damping coefficient 8) with those of propagating plane capillary waves. Because cylindrical waves possess a geometry that is distinctly different from plane waves, they will have different combinationsof shear and dilational contributions to the surface stress tensor (see, for example, ref 16). Furthermore, unlike plane waves, cylindrical waves propagate isotropically in the radial direction, the aforementioned edge effect is absent in these ripples. Thus, a direct comparison of the surface dilational elasticity obtained from these two types of waves will provide valuable information for a quantitative assessment of reliability of dilational elasticities determined from both plane and cylindricalwaves. To ensure a careful comparisonof these waves, cylindricalwaves are generated and detected using (8) Garrett, W. D.; Zieman, W. J. Phys. Chem. 1970, 74,1796. (9) Sold, C. H.;Miyano,K.; Ketterson, J. B. Reo. Sci. Instrum. 1978, 49,1464. (10) Stenvot, C.; Langevin, D.Langmuir 1988,4, 1179. (11) Jiang, Q.;Chiew, Y.; Valentini, J. J . Colloid Interface Sei., in preee. (12) Thiewen, D.; Scheludko,A. Kolloid Z . Z . Polym. 1967,218 (2), 139. (13) Bird, R.;Steward, W. E.;Lightfoot, E. N. Transport Phenomena; John Wiley: New York, 1960,Chapter 3. (14) Lucassen, J.; Giles, D. J . Chem. SOC.,Faraday D a n . 1 1972,68, 2129. (15) Lucaesen-Reynders, E. H.Surface Elasticity and Viscosity in CompressionlDibtion in AnionicSurfactants;Marcel Dekker: New York, 1981; Chapter 5. (16) Mann,J. Dynumic Surface Tension and Capilbry Waoecr in Surface and Colloid Science; Plenum Preas: New York l9e4, Vol. 13.
0743-7463/92/2408-2747$03.00/00 1992 American Chemical Society
2748 Langmuir, Vol. 8, No.11,1992
Jiang et al.
the same technique on the same airlwater interface at the same temperature. Cylindrical waves are generated by modifying the plane wave setup used in our laboratory and reported earlier." This modified apparatus permits the generation of both cylindrical and plane propagating waves in the frequency range of 50-1000 Hz. The wavelength X and damping coefficient B are detected with specular reflection of a laser beam. Both wave generation and detection avoid direct contact with the interface. To determine the damping coefficient 0 and surface elasticity €d from wave damping experiments,the solution of the wave amplitudeas a function of propagating distance and the dispersionrelationmust be used. Resultsfor plane waves have been considered by severala ~ t h o r showever, ;~~~ to our knowledge, these equations for cylindrical propagating waves are not available. In the work by Thiessen and Scheludko,12stationary or standingcylindricalwaves (as opposed to propagating cylindrical waves examined here) were considered and generated electromechanically by vibratinga small circular cuvette (of radius R ) vertically on the airlliquid interface. In their experiment, the standing cylindrical waves generated were confined to a bounded circular region (r < R ) enclosed by the cuvette. In contrast, in the propagating wave experiment investigated here, cylindrical ripples propagate from the wave generator (Le. the origin at r = 0) outward isotropically in the radial direction to r = (in an unbounded domain). To facilitate the determination of the damping coefficient B from experimentally measured values of surface wave profile, we obtain solutionsto the hydrodynamic equation and the dispersion relation for cylindrical propagating waves. These equations permit us to obtain damping coefficientB, calculate the surface elastic modulus €d, and carry out a comparison of the results obtained from these two surface waves (which is the main focus of this work). Such a comparisonis carried out for soluble (Triton X-100) surfactants. and insoluble (n-dodecylp-toluenesulfonate) In section 2, the solution of the hydrodynamic equation and derivation of the dispersion relation for cylindrical propagating waves are presented. The experimental approach and apparatus are described in section 3. Resulta of the study are reported in section 4.
2. Theoretical Section 2.1. Solution of Hydrodynamic Equations for Propagating Cylindrical Waves. Following refs 4 and 5, let us consider the linearized Navier-Stokes equation that governs capillary waves in cylindrical coordinates (r,B,z)
pairiat = -vp
+p
~
+ pg 2
~
(3)
Also, the continuity equation for incompressiblefluid may be written as V 4 =0 (4) where 0 is the velocity field which has only r and z components. The quantities p and p represent the bulk viscosity and density,respectively,andg is the gravitational acceleration. It is convenient to resolve 0 into two components 01 and 02 so that ir = 5,
+ ir,
paallat = -vp - pg
(5)
(6)
p a ~ 2 / a=t p ~ 2 0 2 (7) Equation 6 implies that V X 01 = 0, and thus 01 may be written as
5, = -vq (8) where q(r,B,z) is a scalar function._ Similarly, 02 can be described by a vector function $, which has only B component, i.e.
a2=vxJ (9) In cylindrical coordinates, the wave velocities can be written in terms of these functions, i.e. and (11)
Here, u p and uz represent velocitiesin the r and z directions, respectively. It should be noted from eqs 4 and 9 that V*B1 = v-5, = 0
(12) Combining eqs 8 with 12yields the followingequation for the function q(r,z) v2q = 0
From eqs 7 and 9, we see that p a w = pv2$ (14) Consider a liquid filling the half-space z < 0, its surface at rest being the plane z = 0, with a cylindrical wave generated at r = 0 and propagating from r = 0 to r = a. Under these conditions, the solutions to eqs 13and 14 are (ref 17)
= ~ , ( l ) ( k ~ ) ~e-kzi w t
(15)
and
q = BH,")(kr)e-'Wtemz
(16) Here, H#)(kr) represents the zeroth-order Hankel function, k = K + is is the complex wavenumber, w = 27ru is the angular frequency of the wave, K is the wavenumber, and 0 is damping coefficient, where
m2 = k 2 - iwp/p (17) When Ikd >> 1,the zeroth-order Hankel function Ho(l) in eqs 15 and 16 becomes
H,") = (2/*kr)'/2eikr and eqs 15 and 16 reduce to
(18)
= A(2/7rkr)'/2e'k"'WtekP
(19)
and $, = A(2/"'/2e'k"'wtemZ
(20) Upon substituting eqs 19 and 20 into eqs 10 and 11, we obtain the followingequations for the velocity components
u, = eik"'wt(2/7rkr)'/2[-iAkekz- Bmem1
(21)
and u z = eikpiwt ( 2 / ~ k r ) ~ / ' [ - A-k ikBem] e~~ (22) where terms to order of (kr)-3/2have been neglected. Combination of eqs 6, 8, and 19 leads to
(17)Jahnke,E.;Emde,F. Tableoffunetionsurithformuheandcurues,
4th ed.; Dover Publications: New York, 1945.
Langmuir, Vol. 8, No. 11, 1992 2749
Damping of Cylindrical Propagating Capillary Waves
K,=.(a sin a)l,,+b-(r+ Ar)-A9 (a sin a)l;r.A9 - 2(a sin a)lHb12 cos (7d2 AZ
and
0
Figure 1. An element containingthe interface when a cylindrAa wave passes by.
+ Ar)=A9- (a cos a)l;r.AB
2(a COS a)lH,p
I
He-Ne LASER
K, = (a cos a)l,,*(r
COS
( ~ / -2 A9/2).Ar
%
where cos ( ~ / -2 A9/2) = A9/2 and the angle a is related to the displacement 5' as
SURFACE WAVE PROFILER c - - - - 1
F MOTOR
L e -
t-'
I
for sufficiently small values of a. The other forcesacting on the elementare due to viscous stress from the liquid that lies underneath the surface. Applying Newton's secondlaw to the prism shown in Figure 1 gives
Trough
LOCK-IN
FUNCTION
AMPLIFIER
ENERATOR
and 1 pAz6r69r -2 at
Figure2. Schematicdiagram of the apparatusfor the generation and detection of propagating plane and cylindrical waves. A sharp blade (line source) is used to generate plane waves, and a sharp needle (point source) is used to excite cylindrical waves in an experiment.
p = -pgz
- ipwA(2/7rkr)'/2eikpiwtekz
The first terms on the right-hand side of eqs 30 and 31 come from the linearized Navier-Stokes equation (i.e. eq 3) when it is expressed in terms of stress tensor 7,13 i.e. rm= -p[
2
(23)
Let 5: and 5' represent the r and z displacements of the transverse wave, respectively. They can be obtained by integratingthe correspondingvelocitiesover time, to yield
rrz= rZr= -p[
21 -1
av, av, + dr az
(33) (34)
-
In the limit of Az 0, the terms on the left-hand side of eqs 30 and 31 go to zero; these, in turn, lead to the following expressions for the normal boundary condition
and
5'=
1
-(p eikt-iwt
5'O-p
+ rz,)p?r69+ K, = 0
+
where 5'0
=
-kAek" - ikBem2(:k)l/z -iw
Equation 25 implies that the amplitude of a propagating cylindrical wave will decay with distance r according to e-fir/r1/2, and thus, it may be used to determine the wave dampingconstant #? from the measured values of the wave amplitude. 2.2. Surface Stress Boundary Conditions. Figure 1 shows a small element of the interface (shade area) that is tilted over a tiny angle a by a cylindricalwave. The net forces in the normal direction K,and in the r direction Kr due to interfacial stress are respectively given by
(35)
and the tangential boundary condition -rrzr6r69 K,= 0 Substituting eq 34 and eq 37 into eq 35 gives
(36)
(37)
Also, if eq 33 and eq 28 are combined with eq 36, we obtain
The gradient of a in eq 38 can be expressed in tern of the surface dilational modulus 6 defined as AA Ad=€(39) A The area A of the undisturbed element in Figure 1 ia equal
w
-n
s h
> E
n -
v
0 0
.--a $
N
'
s
0
5
5
5!
'
. n N -
f n 1
15
2
25
1
1
1s
2
25
3
ot' rsrse. The boundary at r = r will move to r = r + [ and the boundary at r = r + Ar will move to r = r + Ar + [ + (a[/dr)Ar due to the horizontal displacement [of the cylindrical wave; hence, the area change AA ((a[/ar)rsr + [Ar). If this expression for AA is used in eq 39, we have
-
Substitution of eq 40 into eq 38 leads to the tangential stress balance
Using eqs 22 and 25 to evaluate the velocity derivatives and substituting eq 23 into eq 37 leads to
+
+
pw2A - (pgk &')(A - iB) + 2pkw(ikA mB) = 0 (42) Here, the pressure p (i.e. eq 23) is evaluated at z = f and terms of order (kr)-3I2have been neglected. Similarly, upon substituting eqs 21,22, and 24 into eq 41 yields
- i(m2 + k2)B1+ k2c(ikA + mB) = 0 .uwr2k2A ,
-
(43) .~
As before, we have neglected the terms to order (kr)-'I2 in eq 43. Elimination of A and B from eqs 42 and 43 gives the dispersion equation for propagating cylindricalwaves
+
+
[pw2- pgk - uk3 2iqwk21[iqw(m2 k2) - mk2el = [-i(pgk uk') - 2qwmkI [2qwk2 + ik3e] (44)
+
The dispersion equation is one of the main theoretical results of this paper and is found to be identical to that for propagating plane waves.4~5J8 In the case of small soluble amphiphilic surfactant molecules, it has been found that the aidwater interface behaves like an "elastic insoluble" monolayer film when it is subjected to deformation at high frequencies. This implies that the surface Viscosity qd is negligible at high frequencies; hence, by setting q d = 0 in the dispersion equation (i.e. eq 441,the resulting equation can be used to calculate the surface tension and elastic modulus Cd (e = Cd, since Vd = 0 (see eq 2)) of the surface?*" The detailed computational scheme will not be reproduced here; it can (18) Ito,K.;Sauer,B.B.;Skarlupka,R.J.; Sano, M.; Yu,H.Langmuir
1990,6, 1379.
-10
-8
-6
log concentration (mole/cc)
F-
4. Wavelength at 160 Hz va concentration of Triton
X-LOO: propagating plane waves (0);cylindrical wavea (A).
be found in ref 7. For the soluble nonionic surfactant (Triton X-100) used in our experimenta, the surface viscosity q d is expected to be negligible a t frequenciea higher than approximately 1 H2.12J' 3. Experimental Section The experimentalsetup ia shown in Figure 2. This apparatua waa described in detail earlier." Briefly, electrocapillary wavea are excited by applyinga sinusoidal and dc o f f a t voltage between a metal generator and the water surface. The Propagatingplane and cylindricalwaves are generated by wing a thin metal blade with length of about 8 cm and a sharpmetal needle, respectively. The surface wave cnn be scanned over a distance of 80 m m by a computer-controlledlinear translationmotor. Both generators (Le. the blade and the needle) are aet up in nuch a way that the scanning line goea through the 'point source" (Le. the n e d e tip) for cylindrical waves and ia perpendicular to the blade for plane waves. Both plane and cylindrical wavea can be generated by simply switching the high voltageline between the blade and the needle. The relaxation time required for the surface tension to reach its equilibrium or static value variea from about 1h at low surfactant concentrationto about 16 min near the cmc for Triton X-100 and meaaurementa were carried out after the surface has attained equilibrium. For the insoluble surfactant, we allow the surface to equilibrate for about 30 min before mo(uIutementa were taken.
Langmuir, Vol. 8, No. 11, 1992 2761
Damping of Cylindrical Propagating Capillary Waves
1
I
I
8
-
A44
h
0
I
b
-
v
4
A
A
0
4
0
0.5 0
-10
-8
-10
-6
log concentration (mole/cc)
-6
log c (mole/cc)
Figure 8. Damping coefficient at 150 Hz va concentration of
Triton X-100: propagating plane waves (0);cylindrical waves (A). I
-7
-%
-9
Figure 7. Calculated values of surface spreading pressure 11 plotted as a functionof Triton X-100 concentration: propagating plane waves (0);cylindricalwaves (A);experimentally measured values ( 0 ) .
I
1
0.23
k0.22
W
1 -
x
a
t. . . .
0
log concentration (mole/cc)
Figure 6. Calculated values of surface modulus t plotted as a function of Triton X-100 concentration at 150 Hz: propagating plane waves (0);cylindrical waves (A). The waves are detected by specular reflection of a focused laser beam from the surface to a photodiode. The in-phase and out-of-phase amplitudes of the surface wave, Ai. and Amt, are recorded simultaneouslyas a function of scanned distance using a two-phase lock-in amplifier. A typical signal of a propagating cylindrical wave is shown in Figure 3a. The product of the wave amplitude A and r1I2(i.e. ArW indeed decaysexponentiallywith the scanned distance as predicted by eq 25 (seeFigure 3b). Here, A = (Ab2 A,t2)1/2 and r is the distance from the point source. To ensure that the condition Ikd >> 1is satisfied, the amplitude of the cylindrical wave was scanned for r 2 1.1 cm from the generator. The slope of the curve shown in Figure 3b gives the damping coefficient @.The wavenumber (i.e. 2 d X ) is obtained from the plot of the phase angle (i.e. tan-' (A,JAi.)) plotted as a function of scanned distance (see Figure 3c). With our apparatus, the wavelength and damping coefficient can be measuredwith accuracy to better than 0.1 % and 1% ,respectively. All surfactants wed in the present experimentswere made by Nikkol (Japan) supplied by Mitsui and Co. All surfactants were used as received without further purification. Experiments on adsorbed monolayers of Triton X-100 were carried out at frequencyof 16OHzandtemperatureof24f0.5OC. Experimenta on spread monolayers of n-dodecyl p-toluenesulfonate were carried out at frequency of 200 Hz and at temperature of 20 0.5 OC. Low boiling (20-40OC) petroleum ether was used as the spreading solvent.' Blank petroleum ether spread on the pure water surface does not affect the measurements of pure water. A Teflon-coated trough (15 cm X 80 cm X 1 cm) is used as a
+
1
1
.
.
.
.
I
.
2
.
.
.
'
.
3
'
.
.
'
.
.
.
4
.
I
5
surface concentration (mole/cm2x10") Figure 8. Wavelength at 200 Hz va surface concentration of
n-dodecyl p-toluenesulfonak propagating plane waves (0); cylindrical waves (A).
solution container. A temperature control unit is coupled to the trough to maintain a constant temperature during a experiment. Doubladietilled water was used in all experimenta.
4. Results and Discussion The measured wavelength X and damping coefficient 0 of Triton X-100for both plane and cylindricalpropagating waves, expressed as a function of bulk concentration at an excitation frequency of 150 Hz,are shown in Figures 4 and 5, respectively. In both figures, the open circles and triangles represent data for the plane and cylindricalwaves, respectively. It is seen from these figures that the values of wavelengthsand damping coefficientsof the plane wavea agree with those of cylindrical waves within experimental error. This means that the calculated surface properties from both the plane and cylindrical propagating waves are equal to each other, since, aa discussed in section 2, the dispersion equation (i.e. eq 44)for both waves is the same. The moduli E of Triton X-100 are expected to be real at a frequency of 150 Hz and in the concentration range studied.12 The calculated moduli E and surface pressure Il (i.e. uo- u) from both wave measurements are presented in Figures 6 and 7, respectively. As expected both plane and cylindrical waves yielded identical values
2752 Langmuir, Vol. 8, No. 11, 1992
J i a g et ai. I
1 -
I
I
40
I
0
A A
n n
c
I
5
5 a * E20 -
.
-
2.5-
8
i
0' 0
A
4
2
I
4
-
4
8
0
A A
A A A I
I
I
A
A A I A
I
surface concentration (mole/cm2xl 0")
F W 9. Damping Coefficientat 200 Hz vs surfaceconcentration of n-dodecyl p-toluenesulfonata propagating plane waves (0); cylindrical waves (A).
for the surface properties of e and II. Included in Figure 7 are the measured values of the surface pressure data.It is seen that good agreement is obtained between the measured and calculated values. Spread monolayers of insoluble n-dodecyl p-toluenesulfonate have been studied by Lucassen and Hansen with the standing plane wave technique.' We consider the properties of surface film of this insoluble surfactant through both propagating plane and cylindrical wave techniques. Our measurements of the wavelength X and damping coefficient /3 for both propagating waves are displayed in Figures 8 and 9, respectively. Values of X and /3 obtained from the propagating plane waves agree with those from propagating cylindrical waves within experimental error, i.e., 1 % for damping coefficient and 0.1 % for wavelength. It can be seen from Figure 9 that the anticipated peak in the damping coefficient B occurs at a surface concentration of about 2.1 X 10-lo mol/cm2. Our results of damping coefficient /3 and wavelength X are in good agreement with ref 7. The calculated moduli and surface pressure II expressed as a function of surface concentrationare shown in Figures 10 and 11, respectively. In Figure 11, good agreement between the measured and calculated values of surface pressures is observed. In summary, both propagating plane and cylindrical waves were studied on monolayer-covered aidwater interfaces. The measured wavelength and damping coefficient of both propagating waves are the same for both soluble (Triton X-100) and insoluble (n-dodecyl p-toluenesulfonate),and identical surface properties (i.e. elastic
2ow
7
p
15
V >r v
ea
to-
m
'
AA
.
0
i
i : P
8
)
.
0
5-
j j : : 0
4
0 " " ~ " " ' ~ ~ " ~ " " i " " 2
3
surface Concentration (mole/cm2X10
Figure 11. Calculated values of surface spreading pressure II w surface concentration of n-dodecyl p-tolueneaulfonate: prop agating plane waves (0); cylindrical wavea (A); experimentally measured values ( 0 ) . moduli e and spreading pressure II) can be obtained from both techniques. This suggests that, even though cylindrical and plane surface waves having distinctly different geometries,they yield identical dilational elasticmodulus.
Acknowledgment. Y.C.C.acknowledges the partial support through the trustees' research fellowship,Rutgers University; we thank Professor H. Yu for bringing ref 18 to our attention. Registry No. Triton X-100, 9002-93-1; dodecyl p-toluenesulfonate, 10157-76-3.
