Initial Electrical Conductivity Methods

vbr/vbrCore/functions/DK2014.m

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+function [ VBR ] = DK2014( VBR )
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  %
+  % [ VBR ] = DK2014( VBR )
+  %
+  % experimental results on the electrical conductivity in 
+  % hydrated olivine single crystals measured under a broader temperature range
+  %
+  % Parameters:
+  % ----------
+  % VBR    the VBR structure
+  %
+  % Output:
+  % ------
+  % VBR    the VBR structure, with VBR.out.electric.DK2014_ol structure
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+  % read in eletric parameters
+  ele = VBR.in.electric.DK2014_ol;
+  T = VBR.in.SV.T; % K (Temperature)
+  Ch2o = VBR.in.SV.Ch2o; % ppm (water content)
+  P = VBR.in.SV.P; % GPa (Pressure)
+
+  % Low Temperature Conduction
+    S1 = ele.S_1; % S/m    
+    Va_1 = ele.Va_1; % cc/mol (activation volume)
+    H1 = ele.H_1 + Va_1.*P; % kJ/mol
+    R = ele.R_1; % kJ/(mol*K)
+
+  % High Temperature Conduction
+    S2 = ele.S_2; % S/m
+    Va_2 = ele.Va_2; % cc/mol (activation volume)
+    H2 = ele.H_2 + Va_2.*P; % kJ/mol
+
+    ch2o_o = ele.ch2o_o; % ppm, experimental reference water content
+    r = ele.r; % unitless
+
+  % calculate arrhenius relation for each conduction mechanism
+  esig_1 = arrh_dry(S1,H1,R,T);
+  esig_2 = arrh_dry(S2,H2,R,T);
+
+  % summation of conduction mechanisms
+  esig = esig_1 + esig_2; % S/m
+  esig = ((Ch2o./ch2o_o).^r).*esig;
+
+  % store in VBR structure
+  DK2014_ol.esig_i = esig_1;
+  DK2014_ol.esig_h = esig_2;
+  DK2014_ol.esig = esig;
+  VBR.out.electric.DK2014_ol = DK2014_ol;
+end
+
+function sig = arrh_dry(S,H,k,T)
+    exponent = -(H)./(k.*T);
+    sig = (10^S).*exp(exponent);
+end

vbr/vbrCore/functions/SEO3.m

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+function [ VBR ] = SEO3( VBR )
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  %
+  % [ VBR ] = SEO3( VBR )
+  %
+  % Standard Electrical Olivine 3 model based on point defects in a dunite 
+  % from temperature for anhydrous olivine
+  %
+  % Parameters:
+  % ----------
+  % VBR    the VBR structure
+  %
+  % Output:
+  % ------
+  % VBR    the VBR structure, with VBR.out.electric.yosh2009_ol structure
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+  % read in eletric parameters
+  ele = VBR.in.electric.SEO3_ol;
+  T = VBR.in.SV.T; % K (Temperature)
+
+  % calculate oxygen fugacity from SV.T
+    fO2 = OxF(T); % Pa
+
+  % calculation of conductivity 
+  sig = SEO3_ne(T, fO2);
+
+  % store in VBR structure
+  SEO3_ol.sig = sig;
+  VBR.out.electric.yosh2009_ol = SEO3_ol;
+end
+
+function fO2 = OxF(T)
+qfm = -24441.9./(T) + 13.296; %revised QFM-fO2 from Jones et al 2009
+fO2 = 10.^qfm;
+end
+
+function sT = SEO3_ne(T, fO2)
+    e = 1.602e-19;
+    k = 8.617e-5;
+    kT = k*(T);
+    bfe = (5.06e24)*exp((-0.357)./kT);
+    bmg = (4.58e26)*exp((-0.752)./kT);
+    ufe = (12.2e-6)*exp((-1.05)./kT);
+    umg = (2.72e-6)*exp((-1.09)./kT);
+    concFe = bfe + (3.33e24)*exp((-0.02)./kT).*fO2.^(1/6); 
+    concMg = bmg + (6.21e30)*exp((-1.83)./kT).*fO2.^(1/6); 
+    sT = concFe.*ufe*e + 2*concMg.*umg*e;
+    return 
+end

vbr/vbrCore/functions/UHO2014.m

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+function [ VBR ] = UHO2014( VBR )
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  %
+  % [ VBR ] = UHO2014( VBR )
+  %
+  % Review of experimental hyrous conductivity of Olivine evaluated with
+  % water concentration correction (Withers, 2012)
+  %
+  % Parameters:
+  % ----------
+  % VBR    the VBR structure
+  %
+  % Output:
+  % ------
+  % VBR    the VBR structure, with VBR.out.electric.UHO2014_ol structure
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+  % read in eletric parameters
+  ele = VBR.in.electric.UHO2014_ol;
+  T = VBR.in.SV.T; % K (Temperature)
+  Ch2o = VBR.in.SV.Ch2o; % ppm (water content)
+  P = VBR.in.SV.P; % GPa (Pressure)
+
+  % Vacancy Conduction
+    Sv = ele.S_v; % S/m
+    Va_v = ele.Va_v; % cc/mol
+    Hv = ele.H_v + Va_v.*P; % kJ
+
+  % Polaron Conduction
+    Sp = ele.S_p; % S/m
+    Va_p = ele.Va_p; % cc/mol
+    Hp = ele.H_p + Va_p.*P; % kJ
+
+  % Hydrous Conduction
+    Sh = ele.S_h; % S/m
+    Va_h = ele.Va_h; % cc/mol
+    Hh = ele.H_h + Va_h.*P; % kJ
+    R = ele.R_h; % kJ/(mol*K)
+    a = ele.a_h; % unitless
+    r = ele.r_h; % unitless
+
+  % calculate arrhenius relation for each conduction mechanism
+  esig_v = arrh_dry(Sv,Hv,R,T);
+  esig_p = arrh_dry(Sp,Hp,R,T);
+  esig_h = arrh_wet(Sh,Hh,R,T,Ch2o,a,r);
+
+  % summation of conduction mechanisms
+  esig = esig_v + esig_p + esig_h; % S/m
+
+  % store in VBR structure
+  UHO2014_ol.esig_i = esig_v;
+  UHO2014_ol.esig_h = esig_p;
+  UHO2014_ol.esig_p = esig_h;
+  UHO2014_ol.esig = esig;
+  VBR.out.electric.UHO2014_ol = UHO2014_ol;
+end
+
+function sig = arrh_dry(S,H,k,T)
+    exponent = -(H)./(k.*T);
+    sig = (10^S).*exp(exponent);
+end
+
+function sig = arrh_wet(S,H,k,T,w,a,r)
+ exponent = -(H-a.*(w.^(1/3)))./(k.*T);
+    sig = (10^S).*(w.^r).*exp(exponent);
+end

vbr/vbrCore/functions/poe2010.m

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+function [ VBR ] = poe2010( VBR )
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  %
+  % [ VBR ] = poe2010( VBR )
+  %
+  % calculates the electrical conductivity in single crystal San Carlos 
+  % olivine (Fo90 ) at 8 GPa were determined by complex impedance spectroscopy.
+  %
+  % Parameters:
+  % ----------
+  % VBR    the VBR structure
+  %
+  % Output:
+  % ------
+  % VBR    the VBR structure, with VBR.out.electric.poe2010_ol structure
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+  % read in eletric parameters
+  ele = VBR.in.electric.poe2010_ol;
+  T = VBR.in.SV.T; % K (Temmperature)
+  Ch2o = VBR.in.SV.Ch2o; % ppm (water content)
+  P = VBR.in.SV.P; % GPa (Pressure)
+
+  % hydrous 100 axis
+    S_H100 = ele.S_H100; % S/m   
+    Va_H100 = ele.Va_H100; % cc/mol
+    H_H100 = ele.H_H100 + Va_H100.*P; % eV
+    a_H100 = ele.a_H100; % unitless
+    r = ele.r_H100; % unitless
+    k = ele.k_H100; % eV/(mol*K)
+
+    % hydrous 010 axis
+    S_H010 = ele.S_H010; % S/m 
+    Va_H010 = ele.Va_H010; % cc/mol
+    H_H010 = ele.H_H010 + Va_H010.*P; % eV
+    a_H010 = ele.a_H010; % unitless
+
+    % hydrous 001 axis
+    S_H001 = ele.S_H001; % S/m 
+    Va_H001 = ele.Va_H001; % cc/mol
+    H_H001 = ele.H_H001 + Va_H001.*P; % eV
+    a_H001 = ele.a_H001; % unitless
+
+%   Anhydrous params 
+    S_A100 = ele.S_A100; % S/m 
+    Va_A100 = ele.Va_A100; % cc/mol
+    H_A100 = ele.H_A100 + Va_A100.*P; % eV
+
+    S_A010 = ele.S_A010; % S/m
+    Va_A010 = ele.Va_A010; % cc/mol
+    H_A010 = ele.H_A010 + Va_A010.*P; % eV
+
+    S_A001 = ele.S_A001; % S/m
+    Va_A001 = ele.Va_A001; % cc/mol
+    H_A001 = ele.H_A001 + Va_A001.*P; % eV
+
+  % calculate hydrous arrhenius relation for each crystal axis
+     esig_H100 = arrh_wet(S_H100,H_H100,k,T,Ch2o,a_H100,r);
+     esig_H010 = arrh_wet(S_H010,H_H010,k,T,Ch2o,a_H010,r);
+     esig_H001 = arrh_wet(S_H001,H_H001,k,T,Ch2o,a_H001,r);
+     esig_H = geomean(esig_H001,esig_H010,esig_H100);
+
+  % calculate anhydrous arrhenius relation for each crystal axis
+     esig_A100 = arrh_dry(S_A100,H_A100,k,T);
+     esig_A010 = arrh_dry(S_A010,H_A010,k,T);
+     esig_A001 = arrh_dry(S_A001,H_A001,k,T);
+     esig_A = geomean(esig_A001,esig_A010,esig_A100);
+
+
+  % summation of conduction mechanisms
+  esig = esig_H + esig_A; % S/m
+
+  % store in VBR structure
+  poe2010_ol.esig_H = esig_H;
+  poe2010_ol.esig_A = esig_A;
+  poe2010_ol.esig = esig;
+  VBR.out.electric.poe2010_ol = poe2010_ol;
+end
+
+function sig = arrh_dry(S,H,k,T)
+    exponent = -(H)./(k.*T);
+    sig = (10^S).*exp(exponent);
+end
+
+function sig = arrh_wet(S,H,k,T,w,a,r)
+ exponent = -(H-a.*(w.^(1/3)))./(k.*T);
+    sig = (10^S).*(w.^r).*exp(exponent);
+end

vbr/vbrCore/functions/sun2019.m

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+function [ VBR ] = sun2019( VBR )
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  %
+  % [ VBR ] = sun2019( VBR )
+  %
+  % Hydrogen-Deuterium Interdiffusion on single crystal San Carlos Olivine
+  %
+  % Parameters:
+  % ----------
+  % VBR    the VBR structure
+  %
+  % Output:
+  % ------
+  % VBR    the VBR structure, with VBR.out.electric.sun2019_ol structure
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+  % read in eletric parameters
+  ele = VBR.in.electric.sun2019_ol;
+  T = VBR.in.SV.T; % K (Temperature)
+  Ch2o = VBR.in.SV.Ch2o; % ppm (water content)
+  P = VBR.in.SV.P; % Pa (Pressure)
+
+  k = ele.k_i; % eV/(mol*K)
+    % Ionic Conduction
+    Si = ele.S_i; % S/m
+    Va_i = ele.Va_i; % cc/mol
+    Hi = ele.H_i + Va_i.*P; % eV
+
+    % Hopping Conduction
+    Sh = ele.S_h; % S/m
+    Va_h = ele.Va_h; % cc/mol
+    Hh = ele.H_h + Va_h.*P; % eV
+
+    % Hydrogen Diffusion
+    Sd = ele.S; % (m^2)/s  
+    Va = ele.Va; % cc/mol
+    Hd = ele.H + Va.*P; % kJ/mol    
+    R = ele.R; % kJ/(mol*K) 
+    a = ele.a; % unitless
+    r = ele.r; % unitless
+
+    k_B = ele.k_B; % J/K (Nernst-Eistien constant)
+    q = ele.q; % C (Elementary charge)
+
+  % calculate arrhenius relation for each conduction mechanism
+  esig_i = arrh_dry(Si,Hi,k,T);
+  esig_h = arrh_dry(Sh,Hh,k,T);
+  D = arrh_wet(Sd,Hd,R,T,Ch2o,a,r);
+  esig_p = (D.*Ch2o.*(q^2))./(k_B*T);
+
+  % summation of conduction mechanisms
+  esig = esig_i + esig_h + esig_p; % S/m
+
+  % store in VBR structure
+  sun2019_ol.esig_i = esig_i;
+  sun2019_ol.esig_h = esig_h;
+  sun2019_ol.esig_p = esig_p;
+  sun2019_ol.esig = esig;
+  VBR.out.electric.sun2019_ol = sun2019_ol;
+end
+
+function sig = arrh_dry(S,H,k,T)
+    exponent = -(H)./(k.*T);
+    sig = (10^S).*exp(exponent);
+end
+
+function sig = arrh_wet(S,H,k,T,w,a,r)
+ exponent = -(H-a.*(w.^(1/3)))./(k.*T);
+    sig = (10^S).*(w.^r).*exp(exponent);
+end