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3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE I LECTURE 15: THE ELECTRICAL DOUBLE LAYER (EDL) 2 Outline : THE ELECTRICAL DOUBLE LAYER IN BOUNDARY LUBRICATION PODCAST .................................. 2 REVIEW LECTURE #14 : THE ELECTRICAL DOUBLE LAYER (EDL) 1 ................................................ 3 THE 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 surfaces Readings: Course Reader Documents 24 and 25 Multimedia : Cartilage Podcast, Dean, et al. J. Biomech. 2006 39, 14 2555 Acknowledgement : These lecture notes were prepared with the assistance of Prof. Delphine Dean (Clemson University) 1 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE THE ELECTRICAL DOUBLE LAYER (EDL) IN BOUNDARY LUBRICATION PODCAST 2 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE REVIEW 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 gradient D General Mathematical Form : W(D)ELECTROSTATIC CES 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 today Poisson-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-) Stern or Helmholtz layer Diffuse Layer D1 electrolyte (aqueous solution containing free ions) bulk ionic strength/ salt concentration d 2 (z) 2 Fco Fψ(z) sinh 2 dz RT z solvated coion solvated counterion D1 >D2 bound ion negatively charged surface ( z ) electrical potential R=Universal Gas Constant = 8.314 J/mole K T= 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 3 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE SOLUTION TO 1D LINEARIZED P-B EQUATION FOR A 1:1 MONOVALENT ELECTROLYTE POTENTIAL PROFILE For Na+ , Cl- (monovalent 1:1 electrolyte solution) d 2 (z) 2 Fco Fψ(z) sinh 2 dz RT 2nd order nonlinear differential eq. solve numerically Fψ(z) 1 or ψ <~ 60 m V RT "Debye - Huckel Approximation" - Linearized P - B Equation Linearize when d 2 (z) 2 F 2 coψ(z) 2ψ(z) (*) linear differential equation 2 dz RT where : 1 εRT 2F 2 co Solution to (*) : ( z ) (0)e z K ( z ) electrical potential, K = integration constant Boundary Conditions (1Layer or D > 5 -1 ); (z ) 0, K = 0 1) Constant Surface Potential (0) o s = surface potential (z = 0) = constant -1 (nm) = Debye Length = the distance over which the potential decays to (1/e ) of its value 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 -1 s /e (z) c1 c2 c3 z z=0 c3<c2<c1→"salt screening" IS [M] -1 (nm) 10-7 (pure water) 950 + [H ]=[OH ] 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 4 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE ELECTRICAL POTENTIAL PROFILE FOR TWO INTERACTING EDLs 2) Constant Surface Charge d dz z=0 surface charge density (C/m ), volume charge density of entire electrolyte phase (C/m3 ) 2 0 0 comes from electroneutrality : dz = - d 2 d dz = dz 2 dz o z=0 Poisson's Law d Boundary Conditions for Two Interacting Plane Parallel Layers : 0, m for z = D/2 dz Surface potential constant with D Surface potential changes with D 160 160 (d/dx at surface changes with D) o0 140 140 120 (x)(mV) (mV) (z) 120 100 80 (x) Constant potential solution (mV) as 2 planes are approached together 100 80 60 60 40 40 20 20 0 0 o0 1 2 3 4 0 0 z Constant Charge solution when 2 planes are approached 5 2 4 6 5 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE EDL : 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: P(z = z j ) - P( ) + + + + + z = zj + (x) + + (x) + m + z = -D/2 z =0 + + + + + + + + + + control box z Reference position in the bulk where: (z) = 0 E = d/d z = 0 ci(z) = cio (z→∞) (z∞) z = D/2 Now at any point the pressure is going to be the sum of two terms: P(z = z j ) - P( ) = "electrical" + "osmotic" The “electrical” contribution (a.k.a. Maxwell stress) is due to the electric field: P(z = z j ) - P( ) = (E 2 (z = zi ) - E 2 ( ))+"osmotic" 2 The “osmotic” contribution is due to the ion concentrations (van't Hoff's Law, chemical equation of state): P(z = z j ) - P( ) = (E 2 (z = zi ) - E 2 ( ))+ RT (ci (z = zi ) - ci ( )) 2 all ions Now some things simplify because at z ∞, E 0 and ci(∞) cio 6 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE EDL : FORCE CALCULATION (2-CONT'D) -If we can simplify things if we pick zj to be the point between the surfaces where E = 0 (i.e. if the surface charges are the same then this is the mid point between the two surfaces, z = 0), only have an osmotic component: P(z = 0) - P( ) = RT (ci (0) - cio ) all ions Note: we can pick any point zj because the sum of the “osmotic” and “electrical” terms is a constant. We chose z = 0 because it’s less messy since we don’t have to calculate E. If it’s a monovalent salt solution we can plug in the formula for the concentration for Boltzmann Statistics: m F m - FRT P = RT c0 e + c0 e RT where : m F m - 1 - RTco = 2RTc0 cosh RT is the potential at z=0 and P is the net force/area on the surface. If m<< RT/F ~60mV, then we can actually solve this equation further: 2 2c0 F 2 m 2 F m P 2RTC0 (1 - = εκ 2 m2 ) - 1 = RT RT and we can solve for you get: m if we wanted using the linearized PB equation (it’s a little messy). If you plug in the value for m P 4 e-κD (constant surface potential) 4 -κD P e (constant surface charge) Exponential in this case so you can think of the Debye length, -1, as being the length scale over which electrostatic forces occur. 7 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE APPENDIX : MATHEMATICAL POTENTIALS FOR ELECTRICAL DOUBLE LAYER FOR DIFFERENT GEOMETRIES I (From Leckband, Israelachvili, Quarterly Reviews of Biophysics, 34, 2, 2001) 8 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE MATHEMATICAL POTENTIALS FOR ELECTRICAL DOUBLE LAYER FOR DIFFERENT GEOMETRIES II Z = 9.38 10 -11 tanh 2 ( /107) J m-1 (monovalent electrolyte) σ = 0.116sinh( /53.4) co Cm-2 (monovalent electrolyte) 9 3.052 Nanomechanics of Materials and Biomaterials Tuesday 04/10/07 Prof. C. Ortiz, MIT-DMSE 10