A simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains

Aqueous foams and a wide range of related systems are believed to coarsen by gas diffusion between neighboring domains into a statistically self-similar scaling state, after the decay of initial transients, such that dimensionless size and shape distributions become time independent and the average grows as a power law. Partial integrodifferential equations for the time evolution of the size distribution for such phase separating systems can be formulated for arbitrary initial conditions, but these are cumbersome for analyzing data on non-scaling state preparations. Here we show that essential features of the approach to the scaling state can be captured by an exactly-solvable ordinary differential equation for the evolution of the average bubble size. The key ingredient is to characterize the the bubble size distribution approximately, using the average size of all bubbles and the average size of the critical bubbles, which instantaneously neither grow nor shrink. The difference between these two averages serves as a proxy for the width of the size distribution. To test our model, we compare with data for quasi-two dimensional dry foams created with three different initial amounts of polydispersity. This allows us to readily identify the critical radius from the average area of six-sided bubbles, whose growth rate is zero by the von~Neumann law. The growth of the average and critical radii agree quite well with exact solution, though the most monodisperse sample crosses over to the scaling state faster than expected. A simpler approximate solution of our model performs equally well. Our approach is applicable to 3d foams, which we demonstrate by re-analyzing prior data, as well as to froths of dilute droplets and to phase separation kinetics for more general systems such as emulsions, binary mixtures, and alloys

Average Evolution and Size-Topology Relations for Coarsening 2d Dry Foams

Two-dimensional dry foams coarsen according to the von Neumann law as $dA/dt \propto (n-6)$ where $n$ is the number of sides of a bubble with area $A$. Such foams reach a self-similar scaling state where area and side-number distributions are stationary. Combining self-similarity with the von Neumann law, we derive time derivatives of moments of the bubble area distribution and a relation connecting area moments with averages of the side-number distribution that are weighted by powers of bubble area. To test these predictions, we collect and analyze high precision image data for a large number of bubbles squashed between parallel acrylic plates and allowed to coarsen into the self-similar scaling state. We find good agreement for moments ranging from two to twenty.Comment: 8 pages, 7 figure

Bellybutton: Accessible and Customizable Deep-Learning Image Segmentation

The conversion of raw images into quantifiable data can be a major hurdle in experimental research, and typically involves identifying region(s) of interest, a process known as segmentation. Machine learning tools for image segmentation are often specific to a set of tasks, such as tracking cells, or require substantial compute or coding knowledge to train and use. Here we introduce an easy-to-use (no coding required), image segmentation method, using a 15-layer convolutional neural network that can be trained on a laptop: Bellybutton. The algorithm trains on user-provided segmentation of example images, but, as we show, just one or even a portion of one training image can be sufficient in some cases. We detail the machine learning method and give three use cases where Bellybutton correctly segments images despite substantial lighting, shape, size, focus, and/or structure variation across the regions(s) of interest. Instructions for easy download and use, with further details and the datasets used in this paper are available at pypi.org/project/Bellybuttonseg.Comment: 6 Pages 3 Figure

Aqueous foams in microgravity, measuring bubble sizes

The paper describes a study of wet foams in microgravity whose bubble size distribution evolves due to diffusive gas exchange. We focus on the comparison between the size of bubbles determined from images of the foam surface and the size of bubbles in the bulk foam, determined from Diffuse Transmission Spectroscopy (DTS). Extracting the bubble size distribution from images of a foam surface is difficult so we have used three different procedures : manual analysis, automatic analysis with a customized Python script and machine learning analysis. Once various pitfalls were identified and taken into account, all the three procedures yield identical results within error bars. DTS only allows the determination of an average bubble radius which is proportional to the photon transport mean free path $\ell^*$. The relation between the measured diffuse transmitted light intensity and {$\ell^*$} previously derived for slab-shaped samples of infinite lateral extent does not apply to the cuboid geometry of the cells used in the microgravity experiment. A new more general expression of the diffuse intensity transmitted with specific optical boundary conditions has been derived and applied to determine the average bubble radius. The temporal evolution of the average bubble radii deduced from DTS and of the same average radii of the bubbles measured at the sample surface are in very good agreement throughout the coarsening. Finally, ground experiments were performed to compare bubble size distributions in a bulk wet foam and at its surface at times so short that diffusive gas exchange is insignificant. They were found to be similar, confirming that bubbles seen at the surface are representative of the bulk foam bubbles

Quantifying Structure in Quasi 2D-Foams at All Lengths by Uncovering Long-Range Hidden Order with Hyperuniformity Disorder Length Spectroscopy and Relating Local Bubble Shape to Dynamics

Amorphous materials have constituent particles without translational order and appear to have no long range structure; one way to search for hidden order in these materials is to measure their long range density fluctuations and if the fluctuations are suppressed to the same extent as in crystals then the system is called hyperuniform. Using the tools developed to measure hyperuniformity we introduce a new length scale to quanitfy disorder in amorphous systems at all distances. This quantity is called the hyperuniformity disorder length h and it is a distance that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size; only the particles that land within a distance h from the boundary of each window determine fluctuations. For cubic windows where the window volume scales like Ld, we propose to quantify disorder in particle configurations by the real-space spectrum of h(L) versus L. We call this approach Hyperuniformity Disorder Length Spectroscopy (HUDLS). Completely random patterns have long-ranged density fluctuations and h = L/2. Hyperuniform patterns have h = he is constant at large L. We investigate the short, medium, and long-range structure of soft disk configurations for a wide range of area fractions and simulation protocols by using HUDLS with square windows of growing side length L. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that, surprisingly, approach the totally-random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau, h(L) = he, for L larger than particle size and smaller than a cutoff Lc beyond which there are long-range fluctuations. The value of he is independent of protocol and characterizes the putative hyperuniform limit. Additionally, we use HUDLS to study the structure of foams by analyzing bubble centroids that are either unweighted or given a weight equal to the area of the bubble they occupy. The h\left( L \right) data for the unweighted foam centroids collapse for different ages of a coarsening foam and have random fluctuations at long distances. The area weighted patterns have h\left(L\right)= he for large L and its value shows the foams are most ordered when compared to other cellular structures. Finally, we quantify bubble shape with a quantity called “circularity” that is based on the curvature of the bubble edges and show that it significantly affects individual bubble coarsening. For wet foams we find 6-sided bubbles whose areas change in time; this is in direct violation of the usual von Neumann law but in agreement with a generalized coarsening equation that we describe. The data for these bubbles is in good agreement with predictions made by our generalized equation and the coarsening we observe is entirely due to the bubble shape as defined by its circularity

Quantifying Structure in Quasi 2D-Foams at All Lengths by Uncovering Long-Range Hidden Order with Hyperuniformity Disorder Length Spectroscopy and Relating Local Bubble Shape to Dynamics

Amorphous materials have constituent particles without translational order and appear to have no long range structure; one way to search for hidden order in these materials is to measure their long range density fluctuations and if the fluctuations are suppressed to the same extent as in crystals then the system is called hyperuniform. Using the tools developed to measure hyperuniformity we introduce a new length scale to quanitfy disorder in amorphous systems at all distances. This quantity is called the hyperuniformity disorder length h and it is a distance that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size; only the particles that land within a distance h from the boundary of each window determine fluctuations. For cubic windows where the window volume scales like Ld, we propose to quantify disorder in particle configurations by the real-space spectrum of h(L) versus L. We call this approach Hyperuniformity Disorder Length Spectroscopy (HUDLS). Completely random patterns have long-ranged density fluctuations and h = L/2. Hyperuniform patterns have h = he is constant at large L. We investigate the short, medium, and long-range structure of soft disk configurations for a wide range of area fractions and simulation protocols by using HUDLS with square windows of growing side length L. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that, surprisingly, approach the totally-random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau, h(L) = he, for L larger than particle size and smaller than a cutoff Lc beyond which there are long-range fluctuations. The value of he is independent of protocol and characterizes the putative hyperuniform limit. Additionally, we use HUDLS to study the structure of foams by analyzing bubble centroids that are either unweighted or given a weight equal to the area of the bubble they occupy. The h\left( L \right) data for the unweighted foam centroids collapse for different ages of a coarsening foam and have random fluctuations at long distances. The area weighted patterns have h\left(L\right)= he for large L and its value shows the foams are most ordered when compared to other cellular structures. Finally, we quantify bubble shape with a quantity called “circularity” that is based on the curvature of the bubble edges and show that it significantly affects individual bubble coarsening. For wet foams we find 6-sided bubbles whose areas change in time; this is in direct violation of the usual von Neumann law but in agreement with a generalized coarsening equation that we describe. The data for these bubbles is in good agreement with predictions made by our generalized equation and the coarsening we observe is entirely due to the bubble shape as defined by its circularity