338
Langmuir 1986, 2, 338-341
Mechanical Testing of Monolayers. 2. Substrate Decoupling in the Long-Wave Approximation Kevin C. O’Brien,t J. Adin Mann, Jr., and Jerome B. Lando* Department of Macromolecular Science and Department of Chemical Engineering, Case Western Reserve University, Cleveland, Ohio 44106 Received August 20, 1985. I n Final Form: January 27, 1986 The long-wave approximation was used to analyze the coupling between the water subphase and monolayer film during the course of slow dynamic-mechanical deformation. The motion of tracer particles was used to examine experimentally particle velocity gradients over the length and width of the Tim. For sufficiently slow strain, experimental data and the theory support the contention that the gradient of the particle velocity is negligible over the film. These results imply that the coupling between the monolayer and the water subphase is negligible for the strain frequencies and amplitudes studied. The monolayer behaves mechanically as an autonomous viscoelastic sheet under these conditions. A condition for this behavior is that h / & B while x B- implies that x goes to B with x C B. If is defined as the appropriate Gibbs surface, then [ux,]~computes the differences of a,, when approaching Z from above and approaching Z from below. The linearized momentum balance at the interface can be written as
where uX8and axXs are the velocity component and stress tensor component in the surface. uxxais computed from the constitutive equation for the monolayer and ,a, (=ay,) is the fi.'i.e^, component of the stress tensor for each bulk phase. The surface density is I'o and is assumed constant, small fluctuations being ignored. Equation 1.1 can be written as
-
in the long-wave limit. The result that &,/ax + d u x / & 0 will be shown to be independent of the specific constitutive equation used for uxx8.We comment briefly on constitutive equations before completing the demonstration. The uXx0term can be specified by a constitutive equation11J2for monolayers: = Yo + keu,, + kVUXX
(13) where yois the surface tension of the film in the reference state for which u,, = 0, k, (= G + K,the shear and dilational elastic coefficients) and k , (= ij + t, the shear and dilational coefficients) are the elastic and viscosity coefficients for the film, and u,, and u,, are the strain and time-rate-of-strain tensor components. The major assumptions for this relation to hold are (1)the surface is isotropic or a two-dimensional crystal of a class that admits uxxs
only two coefficients (e.g., hexagonal), (2) the strain and time rate of strain are small so that k, and k, are constants at a given temperature and surface density, and (3) the surface is flat. For the plane long-wave geometry, the viscosity term can be represented for an isotropic fluid as Where f is the dilational viscosity coefficient and ij is the shear viscosity coefficient. If the monolayer is a crystal, only k, is relevant, but it may depend on frequency. If the monolayer is an isotropic, compressible liquid, G 0 since by definition a liquidcannot sustain shear. Regardless of the form of the surface constitutive function, eq 12 becomes
-
where N,, is the appropriate capillary number, N,, = ~ w o X o / y o= 211pco/yo,and is fixed. In the limit h/Xo 0,
(16) which completes the demonstration that in the long-wave limit, the substrate motion can be ignored. Moreover, the terms dux,/dx and du,,/dx in da,,/dx must approach zero since the velocity component, u,* is independent of position along the x axis. We conclude that the effect of the coupling of the substrate to film is negligible during the course of the dynamic-mechanical experiment described herein and in the companion papers.lV2 We further conclude that the deformation is homogeneous (affine);the strain tensor is independent of position on the surface. As a result, the response of the monolayer can be analyzed by the appropriate constitutive equation, e.g., eq 13, directly as was done in ref 1 and 2. 3.4. Experimental Demonstration of Homogeneous Deformation for the Long-Wave Case. Talcum powder particles were used as tracers to determine if the film was being deformed in a homogeneous manner by direct observation during the course of the dynamic-mechanical experiment. The displacement of the particles as a function of time and position along the x axis is directly related to the deformation of the surface as a function of x and t. Consider the geometry of the test as defined in Figure 1. The x axis was taken along the length of the trough. The maximum displacement of the barrier from its rest position was observable and called aB. The maximum displacement of the barrier occurred at times, t , where t = nxT, n = 1 , 3 , 5... and T = l / f where f is the frequency of the periodic deformation of the monolayer. The following terms where defined: ap(T)= maximum displacementof barrier in the x direction at frequency 1/T. M ( T ) = displacement of tracer particle in the x direction at frequency 1/T. Where L and Ware the length and width, respectively, of the monolayer, the dimensionless lengths can be defined as X* = X / L , Z* = Z/W, and M* = M ( T ) / a B ( T ) . Two sets of experiments are reported. In the first set, tracer particles were applied to a clean water surface, Iz, = kv 0. The surface area was then deformed periodically by moving the barrier. Any displacements of the particles were monitored as a function of position along the x and
Langmuir, Vol. 2, No. 3, 1986 341
Mechanical Testing of Monolayers
T 100 -
-0
7 -
mm-mow-ao-w-"-0-0
M" 0 0 0
I 10
100
-
090-
070
-
0 60
060
-
00
02
06
04
00
10
*
-
- -Q*
Mt 0 0 0 -
0 70
0 50
"Om-
0 50
- 0 - -C&A-
*
-
# #
I
I
I
I
0
2"
Figure 2. Plot of M*vs. z* showing affine behavior perpendicular 2, (X) 2.5, (0) 3.0 mm. to the strain axis. (0) z axes. The frequency of deformation was varied by
changing the velocity of the barrier and the amplitude of oscillation. Over a wide range of barrier displacements (2 I aB I 6.5 mm) and frequencies (0.4-0.01 Hz) tracer particles located at distances greater than 3 mm from the moving barrier exhibited only Brownian motion. This control experiment demonstrates that the displacement of tracer particles further than 3 mm from the moving barrier will not be influenced by the motion of the water subphase, induced by the barrier motion. This result confirmed our prediction. In second set of experiments a monolayer of vinyl stearate was spread at the gas-water interface. Tracer particles were then placed on the film in order to monitor film deformation in various locations in the monolayer. Figure 2 is a plot of the characteristic displacement of tracer particles located in the film as a function of the characteristic width of the film. The amplitude of barrier oscillation, cwg(T),was varied from 2 to 6.5 mm. We observed that the characteristic displacement, M*, was independent of the position of the particle along the z axis. This implies that the film is being deformed uniformly along the z axis, perpendicular to the direction of compression, 8,. Figure 3 is a plot of the characteristic displacement as a function of characteristic length of the film. The characteristic displacement, M*, was found to be independent of position along the length of the film for aBless than 6.5 mm. This study of the motion of tracer particles verifies that, neglecting small regions near the transducer and in the vicinity of the moving barrier, the film is deformed in a homogeneous fashion for the barrier amplitudes (2-3 mm) and frequencies (0.44.01 Hz) employed in this study.
Figure 3. Plot of M* vs. x* showing that when the rate of oscillation of the barrier is sufficiently slow and the amplitudes are sufficiently small the strain is uniform and therefore affine. (0) 2, (X) 2.5, (A) 3.0, (0) 4.5, (*) 6.5 mm. (Response is no longer affine; h/X,
