Thin-Film Fluid Flows over Microdecorated Surfaces: Observation of Polygonal Hydraulic Jumps

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Entry by Leon Furchtgott, APP 225 Fall 2010.

Thin-Film Fluid Flows over Microdecorated Surfaces: Observation of Polygonal Hydraulic Jumps (2010). PRL, 102, 194503.


The paper is interested in the behavior of liquid films flowing over rough substrates whose asperities are the same sizes as the film thickness. The flow of thin films on substrates is of interest in many different fields, from industrial processes to biology and vulcanology. There has been some research on how individual isolated features of substrates affect flow, but little on the collective influence of patterns of micron-size asperities. In addition, previous studies have examined films thicker or thinner than the characteristic length scales of the substrate, while this study looks at thin liquid films with thickness on the same order of magnitude as the roughness of the substrate. The authors investigate flows through experimental observations of flows on regular arrays of micron-size pores with different sizes and arrangements of pores. The authors then model the flow using a boundary-layer approach and fit experimental results to the model.

Experimental Setup

Water is pumped into a nozzle of radius 1.2 mm at a flow rate between 0.5 and 2.5 L/min (Fig. 1a). The jet impacts the center of a patterned disc of PDMS of radius 2.5 cm, which is embedded in a smooth clear acrylic plate (Fig. 1b). The patterned surface consists of square or hexagonal arrays of cylindrical posts with height H, radius R and lattice distance D of the order of 100 <math>\mu m</math>. The water spreads radially over the rough substrate, then over the smooth surrounding area, before spilling into the collection reservoir. The water sheet experiences a hydraulic jump at a particular radius. The thickness <math>h(r)</math> of the thin film propagating before the jump is of the order of a few hundreds of microns, whereas the depth of the water layer outside the jump is d = 4.3 mm. Measurements show that d does not depend on the flow rate and the substrate topography.

Fig. 1. (a). Schematic of the experimental setup. (b). Micropattern in the heart of the impact plate. The pattern depends on 3 factors: R and H, the radius and height of the posts, and D, the lattice distance. (c). Photograph of a polygonal jump taken from above. The jump is formed by a water jet (Q = 1 L/min) impacting a square lattice (<math> D = 200 \mu m, R = 50 \mu m, H = 50 \mu m </math>).


The cluster classifications can then be compared to experimental predictions, which are found in a previous theoretical PRL paper (Arkus, Manoharan, Brenner. Phys. Rev. Lett. 103, 118303 (2009)). Observed structures agree with experimental predictions. For N = 2, 3, 4, 5, one unique structure (dimer, trimer, tetrahedron, triangular dipyramid), as shown in Fig 1E.

The first interesting case is N = 6. Two structures are observed, an octogon and a polytetrahedron. Both have C = 12 contacts and thus have the same potential energy. What explains the 20-fold preference for the polytetrahedron (3 kT difference) is an entropic difference. The entropy can be divided into two factors: a rotational entropy and a vibrational entropy. The rotational entropy makes the largest contribution to the free-energy difference and is proportional to the moment of inertia or equivalently the permutational degeneracy. This contributes a factor of 12, the remaining factor of 2 coming from the vibrational entropy. In the case of N = 6 as for N = 7 and 8, highly symmetric structures are extremely unfavorable among clusters with the same potential energy.

Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.

Fig. 2. Comparison of experimental and theoretical cluster probabilities P at N = 6, 7, and 8. Structures that are difficult to differentiate experimentally have been binned together at N = 7 and 8 to compare to theory. The calculated probabilities for the individual states are shown as light gray bars, and binned probabilities are dark gray. Orange dots indicate the experimental measurements, with 95% confidence intervals given by the error bars.

The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:

1. nonrigid structures, in which one of the vibrational modes is a large-amplitude, anharmonic shear mode. These have high vibrational entropy. (Fig 3A, 3B).

2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).

Fig. 3. A. Optical micrographs and renderings of nonrigid structures at N = 9 and (B) N = 10 (C) Structures of 3N – 5 = 25 bond packings at N = 10. The anharmonic vibrational modes of the nonrigid structures are shown by red arrows. Experimentally measured probabilities are listed at top. Annotations in micrographs indicate clusters corresponding to subsets of HCP or FCC lattices. Scale bars, 1 µm.

While the average probabilities at N = 9 and N = 10 should be 1% and 0.25%, the single nonrigid structure at N = 9 has probability 10%, and the 3 extra-bond structures at N = 10 occur 10% of the time. This confirms that nonrigid structures and structures with extra bonds are favored.

The full theoretical free-energy landscape is shown in Fig 4.

Fig. 4. Calculated minima of the free-energy landscape for 6 <math>\leq</math> N <math>\leq</math> 10. The x axis is in units of the rotational partition function, where I is the moment of inertia (calculated for a particle mass equal to 1) and <math>\sigma</math> is the rotational symmetry number. Each black symbol represents the free energy of an individual cluster. The number of spokes in each symbol indicates the symmetry number (dot = 1, line segment = 2, and so on). Orange symbols are nonrigid structures, which first appear at N = 9, and violet symbols have extra bonds, first appearing at N = 10. Vertical gray lines indicate the contribution to the free energy due to rotational and vibrational entropy. The reference states are chosen to be the highest free-energy states at each value of N.


The paper shows/confirms that the most stable small clusters of hard spheres with short-ranged attractions can be determined by geometrical rules: (1) rotational entropy favors structures with fewer symmetry elements; (2) vibrational entropy favors nonrigid clusters, which have half-octahedral substructures sharing at least one vertex; and (3) potential energy favors clusters with both octahedral and tetrahedral substructures, allowing them to have extra bonds.

The paper's description of free-energy landscape is still incomplete because of some of the simplifications it makes. In particular, while the interaction energy in the paper is extremely short-ranged, this is not generally true. For longer-range interactions, the effects will no longer be fundamentally entropic, and they will depend on temperature as well.

Relation to Soft Matter

This is a neat paper that fits well with the first few lectures and readings in our course. It makes you think about fundamental ideas in statistical mechanics and how to measure them experimentally. It also shows how complex configurations can quickly come about from simple short-range interactions.