Difference between revisions of "Thin-Film Fluid Flows over Microdecorated Surfaces: Observation of Polygonal Hydraulic Jumps"

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== Summary ==
 
== Summary ==
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 authors investigate these 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.  
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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 ==
 
== Experimental Setup ==
  
Small numbers of polystyrene (PS) microspheres were placed in cylindrical microwells filled with polyNIPAM nanoparticles (Fig 1A, 1D). The microwells have depth and diameter 30 <math>\mu</math>m, and they are chemically functionalized so that the particles cannot stick to the surfaces. The microspheres have diameter 1.0 <math>\mu</math>m. The nanoparticles have diameter 80 nm and induce a depletion attraction between 2 microspheres (sticky spheres, see Fig 1B). This depletion attraction is very short-ranged (< 1/10 PS sphere diameter) which means that the interactions are pairwise additive (see Fig 1C). Therefore the total potential energy U of a given structure is well approximated by <math>U = CU_m</math>, where <math>C</math> is the number of contacts or depletion bonds and <math>U_m</math> is the depth of the pair potential.  
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Water is pumped into a nozzle of radius 1.2 mm at a flow rate Q 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.
  
The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.
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[[Image:dressaire1.jpg|400px|thumb|center|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>). ]]
  
[[Image:dressaire1.jpg|400px|thumb|center|Fig. 1. A. Experimental system: microwells filled with polyNIPAM microgel particles and PS microbeads. B. Close-up view of two sticky microspheres and surrounding nanoparticles. C. Pair potential. Note that the depletion attraction is very short-range, so the interaction is strictly pairwise additive. D. Micrograph of several microwells, each with different individual clusters. E. Optical micrographs of colloidal clusters in microwells with N = 2, 3, 4, 5 particles.]]
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Intriguingly, patterning of asperities on the substrate can break the symmetry of the typically circular hydraulic jump. In Fig. 1c, the jump adopts an eight-corner star shape. The remainder of the paper examines the role of the lattice geometry and the fluid properties. The paper concentrates on effects on the thin film and assumes that the flow in the outer layer are constant.  
  
 
== Results ==
 
== Results ==
  
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.  
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The authors begin by varying the symmetry of the lattice (Fig. 2). They look at jumps of the same mean radius but formed over different micropatterned surfaces, with hexagonal (b) and square (c) shapes. These have an effect on the jump symmetries.  
  
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.  
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[[Image:dressaire2.jpg|400px|thumb|center|Fig. 2. Influence of pattern structure. Impact of water jets on (a) a smooth surface, (b) a square lattice, and (c) a square lattice. Lattice parameters: <math> D = 200 \mu m, R = 50 \mu m, H = 50 \mu m </math>.]]
  
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.  
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This shape selection is robust in that it is independent of the viscosity and surface tension of the water that is being flowed. This tells us that the symmetry breaking is not caused by viscocapillarity or inertiocapillary instability.  
  
[[Image:dressaire2.jpg|400px|thumb|center|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.]]
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The authors vary systematically the parameters of a square lattice, both the lattice distance D and the height of the posts H (see Fig. 3). For each set (D, H), they look at the average jump radius as a function of the flow rate Q:
  
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:
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- For all (D, H), the mean jump radius increases nonlinearly with the flow rate, and the corners of the polygonal structures become sharper with the flow rate.
  
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).
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- For a given flow rate Q, the radius decreases for smaller D and larger H, and the polygon corners become sharper.  
  
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).  
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[[Image:dressaire3.jpg|400px|thumb|center|Fig. 3. Influence of pattern geometry on the jump. (a) Average radius versus flow rate for different square lattices with post radius <math>R = 50 \mu m</math> and different lattice spacing and posts heights. Inset, the data are compared to predictions. (b) Images of the polygonal jumps over different substrates for Q = 2.5 L/min.]]
  
[[Image:dressaire3.jpg|400px|thumb|center|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.]]
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== Discussion/Modeling ==
  
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.
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The authors attribute the nonlinearity of the data in Fig. 3 as indicative of the reduction of the average flow rate in the thin film above the posts: some of the liquid flows through the microtexture. Indeed, when comparing experimental data to classical predictions, only data corresponding to nearly circular jumps collapse onto the curve of an existing model for smooth substrates (Fig. 3b, inset).  
  
The full theoretical free-energy landscape is shown in Fig 4.  
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One can think of the flow rate of the jet as divided between a 'leakage' flow rate through the roughness and a thin-film flow above the posts. Above the posts, the flow has an intermediate Reynolds number. On the other hand, the fluid velocity through the posts decreases rapidly and the 'leakage' flow has low Reynolds number. The authors therefore use a boundary-layer approach to consider the two flows separately and thus find an analytic solution. They use a slip boundary condition at the interface (top of the posts), where the radial velocity u must satisfy <math>u = \lambda \partial u / \partial z</math>.  
  
[[Image:dressaire4.jpg|400px|thumb|center|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.]]
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The model depends on two parameters describing the substrate surface: the roughness porosity <math>\epsilon</math>, equal to <math>1 - \pi R^2/D^2</math>, and the aspect ratio of the posts <math>\kappa = H/R</math>. <math>\epsilon</math> and <math>\kappa</math> affect the model in two ways. First, the leakage rate is modeled as being proportional to the total flow rate Q: <math>q_{leak} = \alpha Q </math>, with <math>\alpha</math> a function of <math>\epsilon</math> and <math>\kappa</math>. Second, <math>\lambda</math> is proportional to the magnitude of the slip <math>\xi</math>, which is also a function of <math>\epsilon</math> and <math>\kappa</math>.  
  
== Discussion ==
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[[Image:dressaire4.jpg|400px|thumb|center|Fig. 4. Results of the modeling. (a) View from above of a unit cell of a square lattice. Black and white arrows indicate, respectively, the flow direction and the main axis of the lattice. (b) Comparison between the results of the model (solid line) and the experimental data. Inset: Maximum deformation of the shape as predicted by the model. (c) Shapes predicted square lattices with different lattice parameters.]]
  
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.
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Using this approach, the authors obtain an expression of the jump radius as a function of angle. Then, for each lattice structure, one can determine the parameters <math>\alpha</math> and <math>\xi</math> by fitting to experimental data. Given this model, the authors can plot the evolution of the mean radius of the jump with the flow rate of the jet, and they find that the experimental data collapses to the analytic prediction (Fig. 4b). The authors also find that the prediction for the jump deformation conforms to experimental results (Fig. 4b, inset).
 
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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.
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== Relation to Soft Matter ==
 
== 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.
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In class, we talked about calculating the contact angle for liquids on rough surfaces using the Wenzel model (in spite of all of its problems). Here the authors are addressing a similar problem, that of the shape of the hydraulic jump for surfaces with spatially patterned roughness. The paper shows that regular roughness can have a large effect on the properties of the hydraulic jump, with some fairly surprising results.
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A note about the paper: the paper is written quite well, and it is fairly straightforward to understand. But the figures, especially the plots in Figures 3 and 4, are quite difficult to interpret. The different marker shapes are quite distracting, and they seem to have been picked pretty arbitrarily. Fig 3 talks about the flow rate for 4 different lattices, but it shows plots for 6 flows in part (a), and shows 5 patterns in part (b). The marker '+' in Fig 4a is never explained, and it does not reappear anywhere in the paper.

Latest revision as of 23:52, 4 October 2010

Entry by Leon Furchtgott, APP 225 Fall 2010.

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

Summary

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 Q 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>).

Intriguingly, patterning of asperities on the substrate can break the symmetry of the typically circular hydraulic jump. In Fig. 1c, the jump adopts an eight-corner star shape. The remainder of the paper examines the role of the lattice geometry and the fluid properties. The paper concentrates on effects on the thin film and assumes that the flow in the outer layer are constant.

Results

The authors begin by varying the symmetry of the lattice (Fig. 2). They look at jumps of the same mean radius but formed over different micropatterned surfaces, with hexagonal (b) and square (c) shapes. These have an effect on the jump symmetries.

Fig. 2. Influence of pattern structure. Impact of water jets on (a) a smooth surface, (b) a square lattice, and (c) a square lattice. Lattice parameters: <math> D = 200 \mu m, R = 50 \mu m, H = 50 \mu m </math>.

This shape selection is robust in that it is independent of the viscosity and surface tension of the water that is being flowed. This tells us that the symmetry breaking is not caused by viscocapillarity or inertiocapillary instability.

The authors vary systematically the parameters of a square lattice, both the lattice distance D and the height of the posts H (see Fig. 3). For each set (D, H), they look at the average jump radius as a function of the flow rate Q:

- For all (D, H), the mean jump radius increases nonlinearly with the flow rate, and the corners of the polygonal structures become sharper with the flow rate.

- For a given flow rate Q, the radius decreases for smaller D and larger H, and the polygon corners become sharper.

Fig. 3. Influence of pattern geometry on the jump. (a) Average radius versus flow rate for different square lattices with post radius <math>R = 50 \mu m</math> and different lattice spacing and posts heights. Inset, the data are compared to predictions. (b) Images of the polygonal jumps over different substrates for Q = 2.5 L/min.

Discussion/Modeling

The authors attribute the nonlinearity of the data in Fig. 3 as indicative of the reduction of the average flow rate in the thin film above the posts: some of the liquid flows through the microtexture. Indeed, when comparing experimental data to classical predictions, only data corresponding to nearly circular jumps collapse onto the curve of an existing model for smooth substrates (Fig. 3b, inset).

One can think of the flow rate of the jet as divided between a 'leakage' flow rate through the roughness and a thin-film flow above the posts. Above the posts, the flow has an intermediate Reynolds number. On the other hand, the fluid velocity through the posts decreases rapidly and the 'leakage' flow has low Reynolds number. The authors therefore use a boundary-layer approach to consider the two flows separately and thus find an analytic solution. They use a slip boundary condition at the interface (top of the posts), where the radial velocity u must satisfy <math>u = \lambda \partial u / \partial z</math>.

The model depends on two parameters describing the substrate surface: the roughness porosity <math>\epsilon</math>, equal to <math>1 - \pi R^2/D^2</math>, and the aspect ratio of the posts <math>\kappa = H/R</math>. <math>\epsilon</math> and <math>\kappa</math> affect the model in two ways. First, the leakage rate is modeled as being proportional to the total flow rate Q: <math>q_{leak} = \alpha Q </math>, with <math>\alpha</math> a function of <math>\epsilon</math> and <math>\kappa</math>. Second, <math>\lambda</math> is proportional to the magnitude of the slip <math>\xi</math>, which is also a function of <math>\epsilon</math> and <math>\kappa</math>.

Fig. 4. Results of the modeling. (a) View from above of a unit cell of a square lattice. Black and white arrows indicate, respectively, the flow direction and the main axis of the lattice. (b) Comparison between the results of the model (solid line) and the experimental data. Inset: Maximum deformation of the shape as predicted by the model. (c) Shapes predicted square lattices with different lattice parameters.

Using this approach, the authors obtain an expression of the jump radius as a function of angle. Then, for each lattice structure, one can determine the parameters <math>\alpha</math> and <math>\xi</math> by fitting to experimental data. Given this model, the authors can plot the evolution of the mean radius of the jump with the flow rate of the jet, and they find that the experimental data collapses to the analytic prediction (Fig. 4b). The authors also find that the prediction for the jump deformation conforms to experimental results (Fig. 4b, inset).

Relation to Soft Matter

In class, we talked about calculating the contact angle for liquids on rough surfaces using the Wenzel model (in spite of all of its problems). Here the authors are addressing a similar problem, that of the shape of the hydraulic jump for surfaces with spatially patterned roughness. The paper shows that regular roughness can have a large effect on the properties of the hydraulic jump, with some fairly surprising results.

A note about the paper: the paper is written quite well, and it is fairly straightforward to understand. But the figures, especially the plots in Figures 3 and 4, are quite difficult to interpret. The different marker shapes are quite distracting, and they seem to have been picked pretty arbitrarily. Fig 3 talks about the flow rate for 4 different lattices, but it shows plots for 6 flows in part (a), and shows 5 patterns in part (b). The marker '+' in Fig 4a is never explained, and it does not reappear anywhere in the paper.