Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA.
Phys Rev E. 2017 Sep;96(3-1):032805. doi: 10.1103/PhysRevE.96.032805. Epub 2017 Sep 27.
For dry foams, the transport of gas from small high-pressure bubbles to large low-pressure bubbles is dominated by diffusion across the thin soap films separating neighboring bubbles. For wetter foams, the film areas become smaller as the Plateau borders and vertices inflate with liquid. So-called "border-blocking" models can explain some features of wet-foam coarsening based on the presumption that the inflated borders totally block the gas flux; however, this approximation dramatically fails in the wet or unjamming limit where the bubbles become close-packed spheres and coarsening proceeds even though there are no films. Here, we account for the ever-present border-crossing flux by a new length scale defined by the average gradient of gas concentration inside the borders. We compute that it is proportional to the geometric average of film and border thicknesses, and we verify this scaling by numerical solution of the diffusion equation. We similarly consider transport across inflated vertices and surface Plateau borders in quasi-two-dimensional foams. And we show how the dA/dt=K_{0}(n-6) von Neumann law is modified by the appearance of terms that depend on bubble size and shape as well as the concentration gradient length scales. Finally, we use the modified von Neumann law to compute the growth rate of the average bubble area, which is not constant.
对于干泡沫，气体从小高压气泡向大低压气泡的传输主要通过穿过分隔相邻气泡的薄皂膜的扩散来实现。对于更湿的泡沫，随着 Plateau 边界和顶点膨胀为液体，膜面积变小。所谓的“边界阻塞”模型可以根据膨胀边界完全阻止气体通量的假设来解释一些湿泡沫粗化的特征；然而，在泡沫变得紧密堆积的球体且即使没有膜，粗化仍在进行的湿或去阻塞极限下，这种近似方法会严重失效。在这里，我们通过由边界内气体浓度的平均梯度定义的新长度尺度来解释始终存在的边界穿越通量。我们计算出它与膜和边界厚度的几何平均值成正比，并通过扩散方程的数值解验证了这种标度。我们类似地考虑了在准二维泡沫中膨胀顶点和表面 Plateau 边界处的传输。我们展示了 von Neumann 定律 dA/dt=K_0(n-6) 如何通过依赖于气泡大小和形状以及浓度梯度长度尺度的项来进行修正。最后，我们使用修正后的 von Neumann 定律来计算平均气泡面积的增长率，该增长率不是恒定的。
