南京市 Nanjing City

1. 南京市 Nanjing City

2. 河海大学 -Hohai University

5. College of Civil Engineering in Hohai University Basic facts of the College (largest in our university): • Close to 200 staffs • 4 departments (“civil engineering”, “survey & mapping”, “earth sciences & engineering”, “engineering mechanics”) • 1 department-scale institute (geotechnic institute) • Around 3000 undergraduate students + 1000 graduate students

6. Modeling of “Anomalous” Behaviors of Soft Matter Wen Chen （陈文） Institute of Soft Matter Mechanics Hohai University, Nanjing, China 3 September 2007

7. Soft matter？ • Soft matters, also known as complex fluids, behave unlike ideal solids and fluids. • Mesoscopic macromolecule rather than microscopic elementary particles play a more important role.

8. Typical soft matters • Granular materials • Colloids, liquid crystals, emulsions, foams, • Polymers, textiles, rubber, glass • Rock layers, sediments, oil, soil, DNA • Multiphase fluids • Biopolymers and biological materials highly deformable, porous, thermal fluctuations play major role, highly unstable

9. Lattice in ideal solids

11. Polymer macromolecules: fractal mesostructures

19. Fractured microstructures

22. Soft Matter Physics Pierre-Gilles de Gennes proposed the term in his Nobel acceptance speech in 1991. widely viewed as the beginning of the soft matter science. _ P. G. De Gennes

23. Why soft matter? • Universal in nature, living beings, daily life, industries. • Research is emerging and growing fast, and some journals focusing on soft matter, and reports in Nature & Science.

24. Engineering applications • Acoustic wave propagation in soft matter, anti-seismic damper in building，geophysics，vibration and noise in express train； • Biomechanics，heat and diffusion in textiles, mechanics of colloids, emulsions, foams, polymers, glass, etc； • Energy absorption of soft matter in structural safety involving explosion and impact； • Constitutive relationships of soil, layered rocks, etc.

25. Challenges • Mostly phenomenological and empirical models, inexplicit physical mechanisms, often many parameters without clear physical significance; • Computationally very expensive; • Few cross-disciplinary research, less emphasis on common framework and problems.

26. Characteristic behaviors of soft matter • “Gradient laws” cease to work, e.g., elastic Hookean law, Fickian diffusion, Fourier heat conduction, Newtonian viscoustiy, Ohlm law; • Power law phenomena, entropy effect; • Non-Gaussian non-white noise, non-Markovian process; • In essence, history- and path-dependency, long-range correlation.

27. More features (courtesy to N. Pan) • Very slow internal dynamics • Highly unstable system equilibrium • Nonlinearity and friction • Entropy significant a jammed colloid system, a pile of sand, a polymer gel, or a folding protein.

28. Major modeling approaches • Fractal (multifractal), fractional calculus, Hausdorff derivative, (nonlinear model?); • Levy statistics, stretched Gaussian, fractional Brownian motion, Continuous time random walk; • Nonextensive Tsallis entropy, Tsallis distribution.

29. Typical “anomalous” (complex) behaviors • “Anomalous” diffusion（heat conduction, seepage, electron transport, diffusion, etc.） • Frequency-dependent dissipation of vibration, acoustics, electromagnetic wave propagation.

30. Mechanics of Soft Matter • Basic postulates of mechanics. conservation of mass, momentum and energy • Basic concepts of mechanics stress, strain, energy and entropic elasticity • Constitutive relations and initial–boundary-value problems.

31. Outline: • Part I: Progresses and problems – a personal view • Part II: Our works in recent five years

32. What in Part I? • Field and experimental observations • Statistical descriptions • Mathematical physics modelings

33. Field and experimental observations

34. Non-dissipation Anomalous electronic transport Normal dissipation Anormalous dissipation

35. The absorption of many materials and tissues obeys a frequency-dependent power law Courtesy of Prof. Thomas Szabo

36. Statistical descriptions

37. Anomalous diffusion • ＝1, Normal (Brownian) diffusion • 1, Anomalous ( >1 superdiffusion， <1 subdiffusion)

38. Random walks Left: Brownian motion; Right: Levy flight with the same number (7000) of steps.

39. Levy self-similar random walks A characteristic Levy walk

40. Stretched Gaussian Distribution

41. Measured probability density of changes of the wind speed over 4 sec

42. Stretched Gaussian diffusion: Gaussian diffusion:

43. Levy stable distribution

44. Two cases of Levy distributions Gaussian (=2) Cauchy distribution (=1)

46. Tsallis distribution (nonequlibrium system) Tsallis non-extensive entropy Max s Boltzmann-Gibbs entropy Tsallis distribution

47. Tsallis distribution cases

48. A comparison of diverse distributions A. Komnik, J. Harting, H.J. Herrmann

49. Progresses in statistical descriptions • Continuous time random walk, fractional Brownian motion, Levy walk, Levy flight; • Levy distribution, stretched Gaussian, Tsallis distribution.

50. Problems in statistical descriptions • Relationship and difference between Levy distribution, stretched Gaussian, and Tsallis distribution? • Calculus corresponding to stretched Gaussian and Tsallis distrbiution? • Infinite moment of Levy distribution?