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Hydrostatics

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ComiXology Thousands of Digital Comics. East Dane Designer Men's Fashion. The height of the ridge is approximately the elastocapillary length 11 ; for an elastomeric material with a modulus of a few kilopascals, this is a few microns. Therefore, depending on the measurement resolution, deformation due to the normal component of surface tension can be easily measured only for sufficiently compliant materials, with a modulus in the tens of kilopascals or lower, using optical measurement techniques. In this work, we show that liquid drops placed under a compliant film with a modulus of several megapascals induce deflections of the solid film in the tens of microns, which is observable by the naked eye.

For compliant materials, this elastocapillary length can be many microns; thus, its effects are more significant and easily measured. Examples include the foregoing one on contact line deformation 6 , 10 , 19 , mechanical instabilities in a gel 20 , adhesion instabilities 21 , 22 , limitation on shape replication by compliant materials 23 , flattening of a surface due to surface tension 12 , effect on surface creasing 24 , influence on cavitation rheology 25 , and deformation of a compliant cylinder immersed in a liquid By using a thin-film geometry, we circumvent the existing limitations in surface tension measurements and present a method for measuring the surface tension in relatively stiff solids.

We placed a droplet on the underside of the thin film and measured the surface profile of the top surface of the film via optical profilometry. We find that these forces cause significant film deformation Fig. However, for our suspended thin-film geometry, deformation is mainly in the opposite direction, that is, in the direction of the Laplace pressure P Fig.

The action of gravity causes overall deflection opposite to the Laplace pressure; however, our drops are sufficiently small that this deflection only shifts the datum for deflection, but not the shape of the deformed film, as shown below. Extraction of tension by measurement of deformation due to a drop placed under a solid film. A Schematic of experimental setup for surface tension experiments. B Schematic drawing shows the balance of forces near the axis of symmetry and at the triple line.

Laplace pressure is denoted by P. Note that deformation is mainly in the direction of P, whereas all previous measurements of deformation due to a drop show the formation of a cusp in the direction of surface tension, opposed to P. C Comparison of measured and theoretical surface profiles. There is virtually no difference in the calculated shape of the deformed membrane near the axis of symmetry with or without bending and gravity, consistent with the assumption that the membrane essentially supports only in-plane tension. D Profile of the top surface of an Although the PDMS used in this work is three orders of magnitude stiffer than the silicone gel used by Style et al.

Here, and are the total tensions in the film just outside and inside the contact line in newtons per meter. As argued later, the value of tension has contributions due to stretch of the film and the sum of the solid surface tensions at the two film surfaces. In the limit of vanishing film thickness, the values of tensions approach the sum of the surface tensions of the two film surfaces.

Density Measurement

If the liquid—vapor surface tension is measured independently, Eqs. Because the elastic film has finite thickness and the drop has finite weight, a natural question arises about the influence of bending and gravity. The blue line in Fig. The tension, , was estimated independently as described below. Other parameters were measured or are known independently.

Density Measurement Workflow

The red line, overlaid on the same plot, is the theoretical prediction within a vertical offset for the case where bending and gravity are neglected. Note that the shape of the membrane near its axis of symmetry is captured very well by either theoretical model i. We retained data only for cases where, a posteriori,.

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Thus, we can neglect the influence of bending and gravity on the film shape near the axis of symmetry, where the film supports mainly biaxial tension. We also independently confirmed, by direct computation based on measured deflection, that shear forces in the film are small compared with measured tension; SI Text. Let R be the radius of curvature at the axis of symmetry.

Using the geometric relation , Eq. We obtain R by fitting a sphere to a small region near the center of deformation for various values of c SI Text.

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Based on known values of the liquid surface tension of the liquids used in this work 29 , Eq. The radial equilibrium Eq. The effects of gravity and bending result in vertical offsets that affect neither the measurement of the radius of curvature nor the angles used to calculate tension by Eq.

These liquids were chosen because they are the least capable of absorbing into PDMS, essentially eliminating deformations due to swelling of the elastomer Extraction of tension in the film from its measured deformation. A Time evolution during drying of the deformed shape of the bulge for DI. The deflection data have an arbitrary vertical offset that does not affect the measurement of the radius of curvature or the angles.

B Bulge shape for different liquids on the same PDMS film and approximately the same drop radius the deflection data have been shifted to a common datum of zero. C As the drop dries, its radius, c , reduces; the radius of curvature of the bulged film reduces in proportion. Inset Consistent with Eq. D Estimated value of tension in the solid film, , remains approximately constant as the drop dries, consistent with Eqs. We observe that the radius of curvature decreases linearly as a function of droplet contact radius and that the ratio is approximately a constant.

This would not be the case if bending were significant, and it further supports the assumption that the membrane supports mainly in-plane tension. Data in either case are discarded. The finding that measured tension is approximately independent of the contact radius again supports the assumption that the film deforms as a membrane i.


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For a sufficiently thick film, where bending is expected to be important, Fig. S4 shows that the extracted tension is no longer independent of drop radius. For a given film thickness and liquid, we performed three droplet experiments on three separate samples. The reported tension value is the average of the tension over three-drop radii from the three experiments.

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The tension in the solid film depends distinctly on the fluid used and is ranked according to the liquid—vapor surface tensions The experiments and their analysis discussed so far allow us to compute the tensions in the parts of the film inside and outside the liquid droplet. These tensions can arise due to the surface tension of the fluid—solid interfaces, stretching of the elastic film, and, potentially, bulk residual stresses.

To separate out the contribution of tension due to stretching, we measured tension newtons per meter as a function of film thickness for three different liquids: We find that the measured tensions increase significantly with film thickness, presumably due to increasing contributions from film stretching. Extrapolation by straight-line fits to zero thickness yields values that we interpret as the surface tension.

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The values we obtain are shown in Tables 1 — 3. In SI Text , we outline a model confirming that, in the limit of small film thickness, the contribution to tension due to stretch vanishes, supporting the interpretation of the intercept as the surface tension. Extrapolation of film tension to zero thickness yields surface tension. A Film tension in the region in contact with liquid as a function of film thickness.

We find that all three cases exhibit a nonzero intercept, which we interpret as the surface tension. Inset There is a systematic difference on the same sample depending on the type of liquid in contact with the film. The data plotted here therefore represent the difference in film tension simply due to a change of the liquid in contact with the solid. B Film tension in the region immediately outside the contact line as a function of film thickness.

Tension in the film in the limit of zero thickness, interpreted as surface tension. Difference in tension in the film inside the drop extrapolated to zero thickness, interpreted as the difference in surface tension. Tension in the film just outside the drop extrapolated to zero thickness, interpreted as surface tension. Intercept values for absolute tension and for the difference in tension between DI and EG or DMSO both show a significant dependence of tension inside the contact on the liquid used.

That there is a systematic difference in the intercept tension for the same sample PDMS film , depending on the liquid, rules out the possibility of bulk residual stress playing an important role and further supports the assertion that we are measuring surface tension. The intercept values should be compared with approximately the sum of the surface energy of the PDMS surfaces on two sides of the film. Of course, there is no necessity for surface tension and energy to hold the same value, but one might expect their magnitude to be similar.


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With significant confidence, we can accept the hypothesis that intercept values represent nonzero solid—fluid surface tensions. We note that unlike previous experiments with drops on thin films 31 that showed hoop buckling, we observe no such phenomenon. We ascribe this difference to the strong influence of the surface tension and the geometry of our setup, which keeps hoop stresses tensile. The argument and method presented so far represent a direct, model-independent measurement of tension that relies only on force balance. Once it has been established that thickness is sufficiently thin, one ought to be able to dispense with the need for measuring films of different thickness.

However, if one wishes to improve on the accuracy of the method, it may be possible to estimate the contribution to film tension due to film stretching in terms of measured parameters. As outlined in SI Text , this contribution is on the order of. We studied liquid surface tension-induced deformation of films with a thickness of several micrometers spanning relatively large cylindrical holes. The interplay of liquid—vapor surface tension and tensions in the solid film results in a bulged membrane. In the limit of vanishing membrane thickness, film tension may be interpreted as solid—fluid surface tension.