Role of Complex Surface Tension in the Dispersion Relation of a Capillary Wave

Abstract

We discuss theoretically the effect of complex surface tension on the dispersion relation of a capillary wave, and we deduce an analytical expression for the attenuation of a capillary wave due to the imaginary part of the complex surface tension. We make use of our recently developed apparent energy dissipation spectroscopy to measure the attenuation of a propagating capillary wave at an air− water interface via a microrheometer based on an atomic force microscope. We show that the revised theoretical expression for the attenuation of a capillary wave can satisfactorily explain the abnormal attenuation of such a wave, which stems from both the apparent energy dissipation spectrum and optical scattering, and we obtain the same value for the imaginary part of the surface tension of water. ■ INTRODUCTIONThe term "capillary wave" usually refers to the mechanical wave at an air−liquid or a liquid−liquid interface 1−3 and is an important way to reveal the properties of the interface, such as the viscoelasticity of a Langmuir−Blodgett film 4,5 or another absorbed molecular layer. 6,7 The attenuation of a capillary wave is described by the expression 2,8where ν is the dynamic viscosity of the liquid and q is the surface wave vector. The attenuation of a capillary wave comes entirely from the viscosity of the fluid. The viscosity of the air− liquid interface is unclear. Goodrich 9 suggested that the anisotropic momentum transport, which is involved in the molecularly diffuse interfacial region, will lead to surface viscosity effects. Earnshaw 10 developed this theory and showed that the surface viscosity causes the imaginary part of surface tension, i.e., the energy dissipation. However, careful experiments by optical techniques show that the surface viscosity is zero or very small for the surfaces of many air−liquid interfaces. 11−13 We noticed that in physics, the surface viscosity and the imaginary part of the surface tension are two different concepts. Both of these concepts can cause damping of the capillary wave. The difference is that whereas the surface viscosity is related to the velocity, the surface tension is related to the displacement. In other words, the surface viscosity is frequency-dependent, and the surface tension is frequency-independent. To date, finding experimental evidence of the imaginary part of the surface tension is still an open problem.The atomic force microscope (AFM) was invented for detection on the submicrometer to subnanometer scale and is an appropriate instrument for probing capillary waves. Recently, a microrheometer developed by us, which is based on the atomic force microscope, has been proven to be an instrument useful for exploring air−liquid interfaces. 14 By using this microrheometer and its corresponding analysis method of apparent dissipation factor frequency spectroscopy (ADFFS), 15 we can obtain those vibrational modes of a complex system that are difficult to detect by means such as X-ray scattering 7 and optical techniques. 2 Compared to these techniques, the advantage...

New Information on the Hydrophobic Interaction Revealed by Frequency Modulation AFM

“…Finally, recent measurements of the dissipation associated with periodic modulation of air–water surface area disclosed enhanced frequency dependent dissipation, which was too large to be accounted for by bulk water properties . The authors of ref suggested that the excess dissipation stems from a complex surface tension coefficient which may explain the large dissipation seen here following the cavitation transition and the formation of a new water–vapor interface. Within the cavitation scenario, the tip oscillation should lead to an oscillatory variation in the capillary surface area and hence, assuming a complex surface tension coefficient, to dissipation.…”

Surface Tension as a Complex Parameter: A Critique of Experimental Evidence from Energy Dissipation Studies on Fibers Partially Immersed at an Air/Water Interface

“…This term is not defined. The contact angle on the fiber is shown as an unlikely 90° at a flat surface in ref and as swinging between ∼45 and ∼135° at the terminus of the standing wave at the fiber in ref . Such a “swing” would constitute a cycle of large positive and negative contributions to the vertical force on the fiber registered by the cantilever, which is not observed, but the geometries and associated displacements at the fiber surface may not be ignored.…”

“…An instructive experiment would be to compare the force profile and power dissipation when withdrawing the fiber through the water surface, for comparison with the reported experiments, which appear to always be in the advancing mode. Varying the tapping amplitude would be similarly instructive Chen et al 4 cite the early light-scattering results of Hammarlund et al 19 on aqueous solutions of carboxymethylcellulose in support of their conclusions from the AFM-fiber measurements on water. From a reanalysis of the light scattering data (presumably for the source water) as interpreted with their model, they conclude that wave damping from the imaginary part of the surface tension is larger than the contribution from the bulk solution viscosity, leading to an estimate of the imaginary part of the surface tension in agreement with that estimated from their AFM-fiber experiments.…”

“…It is reported that when the fiber touches the water surface there is a large but unspecified increase in the observed vertical force. This fact is used to define the zero for immersion, which may be in error by a few microns for the more sensitive cantilevers described in ref due to the displacement of the cantilever when contact with the surface is made. The contact angle on the fiber is not considered as the probable source of the force increase.…”

“…Whether dissipative terms are relevant to periodic surface stresses at the pure water/air interface is addressed by valuable new measurements on glass fibers partially immersed in water with the upper end attached to an ATM-style cantilever operated in the tapping mode and measuring the power dissipation as a function of the depth of immersion of the fiber. − The purpose of this note is to focus on these important new experiments and to indicate additional experiments at the air/water interface and at the lines of contact of the liquid surface with the fiber and the container walls and edges, which can potentially show that the observed energy dissipation in the water surface may be assigned to factors other than or additional to a complex surface tension. Extended references to relevant literature are given in refs − .…”