1. 1 DIELECTRIC RELAXATION IN POROUS MATERIALS Yuri Feldman Tutorial lecture 5 in Kazan Federal University

2. 2 Initial sodium borosilicate glass of the following composition (% by weight): 62.6% SiO 2, 30.4% B 2 O 3, 7%Na 2 O heat treatment at 650 0 C for 100h heat treatment at 490 0 C for 165h immersion in deionised water 0.5N HCL drying at 200 0 C rinsing in deionized water additional treatment in 0.5N KOH drying at 200 0 C rinsing in deionized water Porous borosilicate glass samples

3. 3 additional treatment in 0.5M KOH drying rinsing in deionized water drying bithermal heat treatment treatment at 650 0 C and at 530 0 C thermal treatment at 530 0 C immersion in deionised water 3M HCL rinsing in deionized water Commercial alkali borosilicate glass DV1 of the following composition (mol.%): 7% Na 2 O, 23% B 2 O 3, 70% SiO 2

4. 4 Structure parameters and water content

5.  Sample C Sample C  Sample C after heating Sample C after heating Dielectric response of the porous glass materials

6. 6 3-D PLOTS OF THE DIELECTRIC LOSSES FOR THE POROUS GLASS MATERIALS Sample C Sample II

7. 7 Low frequency behaviour ~20 Hz High frequency behaviour ~ 100 kHz   C   C

8. 8 1 2  * (  ) = B *  n-1,   >> 1  * (  ) = -i  0 /  0  1) Jonscher Conductivity  * (  ) =  / [1 + ( i   )  ]  +   2) Havriliak-Negami The fitting model

9. 9 A - 50 kJ/mol B - 42 kJ/mol C - 67 kJ/mol D - 19 kJ/mol Ice - 60 kJ/mol I - 64 kJ/mol II - 36 kJ/mol III - 61 kJ/mol Ice - 60 kJ/mol 1 st Process

10. 10 Samples Humidity h, % II 0.63 A 1.2 B 1.4 D 1.6 C 3.2 III 3.39 I 3.6 Dependence of the Cole-Cole parameter  from ln(  )

11. 11 Temperature dependence of the dielectric strength

12. 12 Parallel and anti-parallel orientation B(T) anti-parallel Temperature Orientation of the relaxing dipole units parallel non- correlated system

13. 13 The symmetric broadening of dielectric spectra The Empirical Cole-Cole law (1941 ) (1-  )  / 2 Character of interaction Temperature Structure etc is a phenomenological parameter  is the relaxation time ?   is the dielectric strength ?  13

14. N. Shinyashiki, S. Yagihara, I. Arita, S. Mashimo, JPCB,102 (1998) p. 3249  (  )? What is behind the relationship  (  )?  How can we use experimental knowledge about  and  ? For instance does their temperature or concentration dependencies explain the nature of dipole matrix interactions in complex systems? 14

15. The Traditional Theoretical Models Fractional Cole-Cole equation for relaxation function f(t)‏ Anomalous Diffusion Dipole-Matrix interactions Fractal set  Due to space averaging both space and time fractal properties are incorporated in parameters . Continuous time random walk (CTRW) model. The random Energy Landscape r Levy flights R.Metzler, J. Klafter, Physics Reports, 339 (2000) 1-77 W.T. Coffey, J. Mol. Liq. 114 (2004) 5-25 R.Hilfer, Phisica A, 329 (2003) 35-40 15

16. 16 Dipole-matrix interaction The symmetric broadening of dielectric spectra Ryabov et al J. Chem. Phys. 116 (2002) 8611. Fractal set

17. All dependences for different CS can be described by Universal function N is the average number of relaxation acts in the time interval t=   is the macroscopic relaxation time  0 is the cutoff relaxation time  - fractal dimension of the relaxation acts in time <0<0 >0>0 A is the asymptotic value of fractal dimension not dependent on temperature ,  and N depends on temperature, concentration, etc is a minimum number of relaxation acts If is a monotonic function Scaling relations

18. 18 Sample C A  0.19 is the fractal dimension of the time set of interactions SamplePorous Size, nm Specific porous area, m 2 /g Porosity%H,% C280-4009.880383.2 Rich water content The total number of the relaxation acts during the time  >0>0 t  0 00 During the time of 1 ps, 70 relaxation acts occurs. The density of the relaxation acts on the time interval 

19. Sample D Poor water content <0<0 A=0.495 t 0  00  <   SamplePorous Size, nm Specific porous area, m 2 /g Porosity%H,% D3008.74501.2 t  0 00 19

20. 20 How can we link the numbers of the relaxation acts in time and the molecular structure, in which they occurred ? Additional parameters should be considered : which can be incorporated by using the Kirkwood-Froehlich approach Kirkwood-Froehlich approach Temperature B Orientation of the relaxing dipole units anti-parallel parallel non- correlated system is the average dipole moment of the i -th cell indicate a statistical averaging over all possible configurations. Θ is the angle between the dipole moment of a given cell and neighboring ones, N n is the number of the nearest cell dipoles.

21. 21 For water molecules in porous glasses The effective number of the correlated water molecules is Sample C T m  195 K θ is the angle between the dipole moment of a given cell and neighboring ones, N n is the number of the nearest cell dipoles. reflect the system state with balanced parallel and anti parallel dipole orientations. The corresponding values of parameters are : The maximum conditions:

22. Sample C:l The kinetic and structural properties The CC relaxation process is associated with the anomalous sub-diffusion. R. Metzler and J. Klafter, Phys. Rep., 339,1(2000). R. Hilfer, Applications of Fractional Calculus in Physics, Ed. By R. Hilfer,(World Scientific, Singapore,2000). The time-space scaling relationship Anomalous sub-diffusionArrhenius temperature dependence is a monotonically decreasing function of temperature throughout the temperature range An anti parallel orientation of the cell dipoles, m, is stipulated by the influence of the porous matrix interface Two main scales of cluster in the Ice-like layer on the matrix interface L l L 2 L 2 is the macroscopic scale of the matrix interface area l 2 l 2 is the area of the mesoscopic scale of the Kirkwood-Froehlich elementary unit with an average dipole moment m At T<<T m

23. Sample C: Coupling the kinetic and structural properties The parallel orientation of the cell dipoles, m, is stipulated by the influence of the external layers of the water molecules At T >> T m

24. T m = 195K L R R L The Kirqwood- Froehlich cell - F 1 - F 2 - H

25. 25 2 Second Process

26. 26 L -defect V * is the defect effective volume V f is the mean free volume for one defect N is the number of defects in the volume of system V, where Si O Si O O Si Orientation Defect Orientation DefectD-defect

27. 27  H a is the activation energy of the reorientation  H d is the activation energy of the defect formation  o is the reorientation (libration) time of the restricted water molecule in the hydrated cluster  is the maximum possible defect concentration The fitting results for the second process

28. 28  ( t /  ) ~ e  t / , D f = 3, where D f is a fractal dimension Percolation: Percolation: Transfer of electric excitation through the developed system of open pores Dielectric relaxation in percolation

29. 29 The Fractal Dimension of Percolation Pass

30. 30 w w : size distribution function , , A , , A: empirical parameters   : porosity of two phase solid-pore system V p : volume of the whole empty space V : whole volume of the sample , , : upper and lower limits of self- similarity D D : regular fractal dimension of the system  = /   : scale parameter   [ ,1] Porous medium in terms of regular and random fractals

31. 31 Porosity Determination (A.Puzenko,et al., Phys. Rev. (B), 60, 14348, 1999)

32. 32 O Percolation The transition associated with the formation of a continuous path spanning an arbitrarily large ("infinite") range. The percolation cluster is a self-similar fractal. Static condition of renormalization