DIELECTRIC PROPERTIES OF POROUS ROCKS WITH AN APPLICATION TO SEA ICE BY LARS BACKSTROM

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Diane Holt
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1 DIELECTRIC PROPERTIES OF POROUS ROCKS WITH AN APPLICATION TO SEA ICE BY LARS BACKSTROM PRESENTATION GIVEN AS PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE CLASS GEOS FALL 2006

2 Contents of presentation What is complex dielectric permittivity How is it measured Bulk dielectric properties Percolation threshold and boundary effects Using complex dielectric permittivity to estimate the bulk salinity of sea ice

3 What is complex dielectric permittivity The complex dielectric permittivity is a measurement of how a easily a material can polarize in response to an imposed external electric field. An elastic response. When an electric field is present in a material a current flows. It consists of a conductive part and a reversible dislocation current: J tot =J c +J d =(σ+iωε * )E ε*=ε -iε

4 How dielectric permittivity is The Schering capacitance bridge is first balanced using an electrolyte of known capacitance C s resistivity, R s and dielectric permittivity, k s. It is then balanced with the specimen, with unknown capacitance C x resistivity R x, and dielectric permittivity, k x. The unknown permittivity is given by: measured k x = k s C 2 R 1 C s R 3

5 Dielectric permittivity spectrum The response of a material to an electric field is frequency dependent The polarization is not instantaneous but has a relaxation time, τ Loss is caused when ions orient in a viscous medium or from resonance close to the characteristic absorption frequencies of ion, atoms, or electrons

6 Debye relaxation time for ionic and dipolar relaxation ε = ε' iε"= ε + (ε l ε ) (1 + iωτ) i σ ε 0 ω Valid for ideal dielectric material with only one relaxation time

7 Cole-Cole diagram In real material many processes contribute to dielectric relaxation Plot ε against ε to determine distribution of relaxation times

8 Bulk dielectric permittivity - 1 Maxwell-Wagner model Φ 1 = V 1 V Φ 2 = V 2 V f 2 = 3ε' 1 2ε' 1 +ε' 2 E =Φ 1 E 1 +Φ 2 E 2 E = 1 V 1,2 Edv v E 1,2 = E 1,2 dv f 2 = E 2 E ε'= ε' 1 +(ε' 2 ε' 1 )Φ 2 f 2 ε'= ε' 1 +3ε' 1 ε' 2 ε' 1 2ε 1 + ε 2 Φ 2 f 2 = i= 3 i=1 cos 2 Θ i 1+ A i (ε' 2 ε' 1 ) 1 [ ] The first case is for ellipsoidal Inclusions. The second case is for Ellipsoidal inclusions. A i are the three Depolarization factors (=1/3 in case of Spherical inclusions)

9 Bulk dielectric permittivity - 2 Maxwell-Wagner-Bruggeman-Hanai model Infinitesimal increments in volume Of inclusions. Integrate from ε=ε 1 to ε n =ε, and Φ 2 =0 To Φ n =Φ 2 It is possible to arrive at a formula For complex dielectric permittivity From which the real part can be derived. The problem with the MWBH model is that it does not depend on the actual dimensions of the inclusions. It assumes inert boundary between inclusion and embedding material. It also assumes an even distribution of inclusions within the sample. Δε' n = ε' ε' n = 3ε' n ε * ε 2 * ε 1 * ε 2 * ε'= ε 1 * ε * 1/3 ε' 1 (1 Φ 2 ) 3 =1 Φ 2 ε' 2 ε' n 2ε' n +ε' 2 ΔΦ n

10 Surface effects Ionic displacement at low frequencies can produce very large dielectric permittivity MWBH theory only looks into bulk properties Assumes even distribution of inclusions Only cares about aspect ratio and relative volume of inclusions Assumes inert interface between inclusions and matrix

11 Surface effects Very complicated, many different mechanisms Small amounts of water can increase dielectric permittivity much more than suggested by MWBH theory at low frequencies

12 Percolation theory Percolation transition at Φ c, which is the critical concentration of the inclusions Theory describes conductivity and dielectric permittivity as a function of ΔΦ=Φ 2 -Φ c as ΔΦ 0. Lattice of bonds which have probability p of being occupied, and (1-p) of being empty Clusters of bonds of infinite length start forming at p=p c The other important parameter in percolation theory is L c, which can be defined as the length of clusters below the percolation threshold, Δp=pp c,this factor indicates that there is a dependence on sample size and scale.

13 Dielectric permittivity close to the percolation threshold ε~a Φ 2 -Φ c -s A is a constant and s =1.9 Critical percolation volume is around 0.15=Φ c

14 Dielectric permittivity of some earth materials at 100 khz and up Hematite (25) Obsidian ( ) Rock salt (5.6) Quartz (4.2-5) Gneiss (8.5) Packed sand Dry to moist ( ) Sandstone Dry to moist (4.7-12) Basalt (12) Petroleum ( ) Dry granite ( ) Water at 20ºC (80.36) Sulphur ( ) Diorite (6.0) Biotite ( ) Soil Dry to moist ( )

15 Dielectric permittivity of sea ice Sea ice is a porous material, made up by pure ice, liquid brine, air bubbles, and other impurities Can have complex structure that varies with temperature and age Important for the global climate system, since it changes the ocean-atmosphere interface

16 Interior of first-year undeformed sea Ice grows and traps liquid brine within matrix of pure ice The interior of the ice can be assumed to consist of vertically aligned ellipsoidal brine inclusions sandwiched between layers of pure ice ice

17 Measurements 50 MHz capacitance probe enveloped by naturally growing ice. Additional measurements of temperature and bulk salinity

18 Dielectric mixture model of sea ice Developed by Tinga et al. ε AVE is the dielectric permittivity of the mixture. Sea ice both anisotropic and brine inclusions large in relation to the dimensions of the probe ε AVE ε 1 = V 2 ε 2 ε 1 ε 1 V 1 ( V 2 V 1 )n( ε 2 ε 1 )+ n( ε 2 ε 1 )+ ε 1 [ ] ε' MIX = ε' AVE(a) 2 + ε' AVE(b) 2

19 Results

arxiv:cond-mat/ v2 [cond-mat.mtrl-sci] 27 Nov 2001

arxiv:cond-mat/ v2 [cond-mat.mtrl-sci] 27 Nov 2001 arxiv:cond-mat/0111254v2 [cond-mat.mtrl-sci] 27 Nov 2001 submitted to IEEE Trans. on El. Insul. and Dielec. Dielectric mixtures electrical properties and modeling 1. Introduction En ıs Tuncer, Yuriy V.