THE ELECTRICAL DOUBLE LAYER (EDL) IN BOUNDARY LUBRICATION PODCASTREVIEW LECTURE #14 : THE ELECTRICAL DOUBLE LAYER (EDL) 1ELECTRICAL POTENTIAL PROFILE FOR TWO INTERACTING EDLsEDL : FORCE CALCULATION (1)EDL : FORCE CALCULATION (2-CONT'D) APPENDIX : MATHEMATICAL POTENTIALS FOR ELECTRICAL DOUBLE LAYER FOR DIFFERENT GEOMETRIES I (From Leckband, Israelachvili, Quarterly Reviews of Biophysics, 34, 2, 2001)MATHEMATICAL POTENTIALS FOR ELECTRICAL DOUBLE LAYER FOR DIFFERENT GEOMETRIES II3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSEILECTURE 15: THE ELECTRICAL DOUBLE LAYER (EDL) 2Outline :THE ELECTRICAL DOUBLE LAYER IN BOUNDARY LUBRICATION PODCAST ....................................2REVIEW LECTURE #14 : THE ELECTRICAL DOUBLE LAYER (EDL) 1..................................................3THE ELECTRICAL DOUBLE LAYER .....................................................................................................4-7 Solution to 1D Linearized P-B Equation For a 1:1 Monovalent Electrolyte............................4 Electrical Potential Profile for Two Interacting EDLs..............................................................5 EDL Force Calculation (1).....................................................................................................6 EDL Force Calculation (2)......................................................................................................7 Objectives: To understand the mathematical formulation for the repulsive EDL Interaction between two charged surfacesReadings: Course Reader Documents 24 and 25 Multimedia : Cartilage Podcast, Dean, et al. J. Biomech. 2006 39, 14 2555Acknowledgement : These lecture notes were prepared with the assistance of Prof. Delphine Dean (Clemson University)13.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSETHE ELECTRICAL DOUBLE LAYER (EDL) IN BOUNDARY LUBRICATION PODCAST23.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSEREVIEW LECTURE #14 : THE ELECTRICAL DOUBLE LAYER (EDL) 1● Applications, Origins of Surface Charge, Definitions : EDL, Stern layer→ balance between attractive ionic forces (electrical migration force) driving counterions to surface and entropy/diffusion/osmotic down the concentration gradientGeneral Mathematical Form : D-=kELECTROSTATIC ESW(D) C e CES= electrostatic prefactor analogous to the Hamaker constant for VDW interactions-1= Electrical Debye Length (characteristic decay length of the interaction)- will be defined more rigorously todayPoisson-Boltmann (P-B) Formulation : 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)Derived 1D P-B Equation for a 1:1 monovalent electrolyte (e.g. Na+, Cl-) 22=yeo2Fcd (z) Fψ(z)sinhdz RT( ) =y electrical potentialzR=Universal Gas Constant = 8.314 J/mole KT= Temperature (K)F= Faraday Constant (96,500 Coulombs/mole electronic charge)co (moles/cm3 or mole/L=[M], 1 ml=1 cm3)=electrolyte ionic strength (IS) = bulk salt concentration, ideally for z→∞, but practically just far enough away from surface charge region, several Debye lengths away(C2J-1m-1)=permittivity 33.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSESOLUTION TO 1D LINEARIZED P-B EQUATION FOR A 1:1 MONOVALENT ELECTROLYTE221 or=�<<ye+ -For Na , Cl (monovalent 1:1 electrolyte solution)2nd order nonlinear differential eq. solve numericallyLinearize when <~ 60 "Debye -Huckel Approximati mn" -Vo Lo2Fcd (z) Fψ(z)sinhdz RTFψ(z)ψRT2212(*) ( ) (0)( )--� === +=kykeky yylinear differential equationSolinearized P -B EquationBoundary Cution to (*) : Kelectrical potential, K = integration constoant2o22ozF cψ(z)d (z)ψ(z) dz RTεRTwhere : 2F cz ezo s(z ) 0,(0)=� == = yy ykky--11 ; K = 0= surface potential (z = 0) = constant (nm) = Debye Length = the distannditionsce over (1 Layer or D > 5 )1) Constant Surface which the potential decays to (1/Pot) enoftial its ek-1value at z = 0, depends solely on the properties of the solution and not on the surface charge or potential-rule of thumb : range of the electrostatic interaction ~ 5 POTENTIAL PROFILEc3<c2<c1→"salt screening"IS [M]-1k(nm)10-7 (pure water)[H+]=[OH-]9500.0001[NaCl] 300.001[NaCl] 9.50.01[NaCl] 3.00.15* [NaCl] 0.81[NaCl] 0.3 (notphysical!)*physiological conditions 43.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSEELECTRICAL POTENTIAL PROFILE FOR TWO INTERACTING EDLs0 0),� �== ==- -� �142 43 y ses rys r e2 3Poisson's Lawsurface charge density (C/m volume charge density of entire electrolyte phase (C/m )comes from electroneutra2) Constant Surface Charlity ge :z=022ddzddz = -dzom0,== =ye y ekyy yBoundary Conditions for Two Interacting Plane Parallel Laye :rs z=0ddz =dzdfor z = D/2dz53.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSEEDL : FORCE CALCULATION (1)-So now that we can solve the P-B equation and get (z) and (z) in space, how do we calculate a force?-We can calculate a pressure (P = force per unit area) on these infinite charged surfaces if we know the potential. Using a control box, the pressure on the surface at x = D is the pressure calculated at any point between the surfaces relative to a ground state: �jP(z = z ) - P( )Now at any point the pressure is going to be the sum of two terms:�jP(z = z ) - P( ) = "electrical" + "osmotic"The “electrical” contribution (a.k.a. Maxwell stress) is due to the electric field:e� �2 2j iP(z = z ) - P( ) = (E (z = z ) - E ( ))+ "osmotic"2The “osmotic” contribution is due to the ion concentrations (van't Hoff's Law, chemical equation of state):e� � ��2 2j i i i iall ionsP(z = z ) - P( ) = (E (z = z ) - E ( ))+ RT (c (z = z ) - c ( ))2Now some things simplify because at z  ∞, E  0 and ci(∞)  cio63.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSEEDL : FORCE CALCULATION (2-CONT'D)-If we can simplify things if we pick zj to be the