3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE I LECTURE 14: TE ELECTRICAL DOUBLE LAYER (EDL) Outline : REVIEW LECTURE #11 : INTRODUCTION TO TE ELECTRICAL DOUBLE LAYER... SURFACE CARGE : WERE DOES IT COME FROM?... 3 TE ELECTRICAL DOUBLE LAYER...47 General Definition...4 General Mathematical Formulation... 5 PossionBoltzmann(PB) Formulation... 6 PB Formulation for a Monovalent Electrolyte... 7 Potential Profile for Two Interacting EDLs... 7 Objectives: To understand the qualitative origins and mathematical foundations for the electrical double layer Readings: Course Reader Documents 4 and 5 Multimedia : Boundary Lubrication Podcast, Briscoe, et al. Nature 006 444, (191 194) 1
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE REVIEW LECTURE #11 : INTRODUCTION TO TE ELECTRICAL DOUBLE LAYER W(D)= W(D) + W(D) VDW +W(D) + W(D) W(D) STERIC STRUCTURAL ELECTROSTATIC DEPLETION "Electrostatic Double Layer Repulsion" : for charged particles, this force arises from a diffuse, highly mobile surface layer of counterions; an exponential repulsion exists on compression since which is entropic in origin since the counterions want to retain their translational mobility W(D)= W (D) + W(D) VDW ELECTROSTATIC "DLVO Theory"Derjaguin, Landau, Verwey and Overbeek Dispersed Repulsive interactions dominate Potential energy ard sphere h Soft sphere h 1. Colloidal /Nanoparticle Stability Dispersion. Biocompatibility 3. Tissue Nanomechanics: Cartilage Secondary minimum h Weakly flocculated Attractive interactions dominate Primary minimum h Strongly flocculated Figure by MIT OCW. After Lewis. J Am Ceram Soc 83, no. 10 (000): 34159.
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE SURFACE CARGE : WERE DOES IT COME FROM? Net Surface Charge (Fixed Charge Groups on Surface) : 1. Direct ionization or dissociation of surface chemical groups, e.g. weak acid : COO COO + + (dependent on p) strong acid : sulfate SO 4 (independent of p) COO C O O O O SO 3 O β O O NCOC 3 cartilage glycoaminoglycan (chondroitin6sulfate). Adsorption (binding of ions) via van der Waals, hydrophobic, or ionic interactions, e.g. lipids, polyelectrolytes (charged polymers) neutral hydrophobic + charged 3
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE TE ELECTRICAL DOUBLE LAYER (EDL): GENERAL DEFINITION D 1 Stern or elmholtz layer Diffuse Layer electrolyte (aqueous solution containing free ions) bulk ionic strength/ salt concentration D 1 >D negatively charged surface Ion concentration gradient z solvated coion solvated counterion bound ion Electroneutrality maintained; # of counterions= # of surface charge groups + # of bulk coions Diffuse Layer : atmosphere of mobile counterions in rapid motion, attractive ionic forces pulling them to surface (electrical migration force) causes concentration gradient, gain translational /rotational entropy by moving away from surface (diffusion down the concentration gradient) these effects are balanced so their is no net flux of any ionic species Stern or elmholtz Layer : bound, usually transiently, thickness is a few Å, reflects the size of the charged surface groups and bound counterions do not completely neutralize the surface charges When two similarly charged electrical double layers are compressed together and overlap (D 1 <D ), repulsive force entropic/osmotic, deviating the system from its minimum 4
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE energy equilibrium configuration TE ELECTRICAL DOUBLE LAYER : GENERAL MATEMATICAL FORMULATION Concentration gradient ρ= volume density of ions (C/m 3 ) D 1 σ σ ρ s counterions(ρ z +) ρ = ρ o ε ρ s ρ s (ρ z +) ρ s ρ o z=d/ coions(ρ z ) z=0 z=+d/ z=d/ z=0 z=+d/ z Generally; W(D) = C e ELECTROSTATIC ES κd C ES = electrostatic prefactor analogous to the amaker constant for VDW interactions κ 1 = Electrical Debye Length (characteristic decay length of the interaction) will be defined more rigorously later on rule of thumb : range of the electrostatic interaction ~ 5κ 1 5
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE TE ELECTRICAL DOUBLE LAYER : POISSONBOLTZMANN (PB) FORMULATION Assumptions; ions are point charges (don't take up any volume, continuum approximation), they do not interact with each other, uniform dielectric; permittivity independent of electrical field, electroquasistatics (time varying magnetic fields are negligibly small) Start with Poisson's Law, relation between electrical potential,ψ (Volts), at any point within a diffuse ρ (C/m 3 ), is ; space charge region of volume charge density, ρ ψ ψ ψ ψ ; ε = permittivity, ψ = = + + ε x y z 1D Case, z to charged surface: n d ψ (z) = 1 zifψ(z) zfc exp ε i i io dz RT Boltzmann Ion Distribution; thermodynamic equilibrium, describes variation of ion concentration in electric field 1D General P B Equation,describes EDL, defines potential as a function of z in a spatial charge distribution R = Universal Gas Constant = 8.314 J/K mole T = Temperature (K) F = Faraday Constant (96,500 Coulombs/mole of electronic charge) c io 3 3 (moles/cm or mole/l = [M]; 1 ml = 1cm ) = electrolyte ionic strength (IS) = th bulk concentration of the i species, ideally for x, but practically just far enough away from surface charge region, several Debye lengths away z i th = electrolyte valence of i ion 6
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE TE ELECTRICAL DOUBLE LAYER : POISSONBOLTZMANN (PB) FORMULATION FOR A MONOVALENT ELECTROLYTE + For Na, Cl (monovalent 1:1 electrolyte solution) x x e e d ψ (z) Fco Fψ(z) sinh(x)= ; = sinh dz ε RT nd order nonlinear differential eq.; Fψ(z) Linearize when << 1 or ψ <~ 60 mv RT "Debye uckel Approximation" d ψ (z) Fcψ(z) o = κ ψ(z) (*) linear differential equation dz ε RT 1 εrt where : κ = z F c o 1 κ (nm) = Debye Length = is more specifically defined as the distance over which the electric field and potential decay to (1/ e) of their value at x = 0 κ z Solution to (*): ψ( z) = ψ(0) e + K ψ ( z) = electrical potential ψ (0) = ψ s = surface potential (x = 0) = constant K = integration constant, Boundary Conditions; ψ ( ) = 0, K = 0 rule of thumb : range of the electrostatic interaction ~ 5κ 1 σ ψ ο =ψ s ψ ο /e z=0 POTENTIAL PROFILE Ψ(z) c 1 c c 3 1/κ 1 1/κ 1/κ 3 IS [M] k 1 (nm) 10 7 (pure water) 950 [ + ]=[O ] 0.0001[NaCl] 30 0.001 [NaCl] 9.5 0.01 [NaCl] 3.0 0.15 * [NaCl] 0.8 1 [NaCl] 0.3 (not physical!) *physiological conditions z 7
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE COMPARISON OF NONLINEAR AND LINEAR PB SOLUTIONS 8
3.05 Nanomechanics of Materials and Biomaterials Thursday 04/05/07 Prof. C. Ortiz, MITDMSE ELECTRICAL POTENTIAL PROFILE FOR TWO INTERACTING EDLs 9