Fractal dimension of rough surfaces in the solid-on-solid model

Experimental evidence for logarithmic fractal structure of botanical trees

The area-preserving rule for botanical trees by Leonardo da Vinci is discussed in terms of a very specific fractal structure, a logarithmic fractal. We use a method of the numerical Fourier analysis to distinguish the logarithmic fractal properties of the two-dimensional objects and apply it to study the branching system of real trees through its projection on the two-dimensional space, i.e., using their photographs. For different species of trees (birch and oak) we observe the Q^{-2} decay of the spectral intensity characterizing the branching structure that is associated with the logarithmic fractal structure in two-dimensional space. The experiments dealing with the side view of the tree should complement the area preserving Leonardo's rule with one applying to the product of diameter d and length l of the k branches: d_{i}l_{i}=kd_{i+1}l_{i+1}. If both rules are valid, then the branch's length of the next generation is sqrt[k] times shorter than previous one: l_{i}=sqrt[k]l_{i+1}. Moreover, the volume (mass) of all branches of the next generation is a factor of d_{i}/d_{i+1} smaller than previous one. We conclude that a tree as a three-dimensional object is not a logarithmic fractal, although its projection onto a two-dimensional plane is. Consequently, the life of a tree flows according to the laws of conservation of area in two-dimensional space, as if the tree were a two-dimensional object.

Observation of nucleic acid and protein correlation in chromatin of HeLa nuclei using small-angle neutron scattering with D_{2}O-H_{2}O contrast variation

The small-angle neutron scattering (SANS) on HeLa nuclei demonstrates the bifractal nature of the chromatin structural organization. The border line between two fractal structures is detected as a crossover point at Q_{c}≈4×10^{-2}nm^{-1} in the momentum transfer dependence Q^{-D}. The use of contrast variation (D_{2}O-H_{2}O) in SANS measurements reveals clear similarity in the large scale structural organizations of nucleic acids (NA) and proteins. Both NA and protein structures have a mass fractal arrangement with the fractal dimension of D≈2.5 at scales smaller than 150 nm down to 20 nm. Both NA and proteins show a logarithmic fractal behavior with D≈3 at scales larger than 150 nm up to 6000 nm. The combined analysis of the SANS and atomic force microscopy data allows one to conclude that chromatin and its constitutes (DNA and proteins) are characterized as soft, densely packed, logarithmic fractals on the large scale and as rigid, loosely packed, mass fractals on the smaller scale. The comparison of the partial cross sections from NA and proteins with one from chromatin as a whole demonstrates spatial correlation of two chromatin's components in the range up to 900 nm. Thus chromatin in HeLa nuclei is built as the unified structure of the NA and proteins entwined through each other. Correlation between two components is lost upon scale increases toward 6000 nm. The structural features at the large scale, probably, provide nuclei with the flexibility and chromatin-free space to build supercorrelations on the distance of 10^{3} nm resembling cycle cell activity, such as an appearance of nucleoli and a DNA replication.

Contact line dynamics near the pinning threshold: a capillary rise and fall experiment

We used video microscopy to study the pinning dynamics of air/water contact lines in vertical glass capillaries. Stick-slip behavior and avalanches are observed in tubes with rough interior walls and strong pinning forces. In tubes with smooth interior walls, we find that receding contact lines in falling water columns show no evidence of pinning, but advancing contact lines in rising water columns exhibit algebraic slow down. The measured value of the critical exponent beta varies from run to run, but it is always larger than unity. Furthermore, we find that the rise dynamics varies with the waiting time preceding the experiments. These observations led us to conclude that the wetting film on the surface and other microscopic changes in the slipping region near the contact line affect the macroscopic dynamics. We discuss the differences between the real system and the existing theories that might explain the results. We also present a brief review of other studies of contact line dynamics and a numerical study of a one-dimensional model.