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)
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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
Courtesy of Jacob Klein. Used with permission.
Source: Briscoe, W. H., et al. "Boundary Lubrication Under Water." Nature 444 (November 9, 2006): 191-194.
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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
−κ D
General Mathematical Form : W(D)ELECTROSTATIC = CES e
CES= electrostatic prefactor analogous to the Hamaker constant for VDW interactions
κ-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
D1 >D2
negatively
charged surface
d 2ψ (z) 2 Fco
Fψ(z)
=
sinh
2
dz
RT
ε
z
solvated
coion
solvated
counterion
bound
ion
ψ ( 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
ε (C2J-1m-1)=permittivity
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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
For Na+ , Cl- (monovalent 1:1 electrolyte solution)
d 2ψ (z) 2 Fco
Fψ(z)
=
sinh
2
ε
dz
RT
2nd order nonlinear differential eq. → solve numerically
Fψ(z)
<< 1 or ψ <~ 60 mV
Linearize when
RT
"Debye - Huckel Approximation" - Linearized P - B Equation
d 2ψ (z) 2 F 2 coψ(z)
≈
= κ 2ψ(z) (*) linear differential equation
2
ε RT
dz
εRT
where : κ −1 =
2F 2 co
Solution to (*) :ψ ( z ) = ψ (0)e −κ z + K
ψ ( z ) = electrical potential, K = integration constant
Boundary Conditions (1Layer or D > 5κ -1 );ψ (z = ∞) = 0, K = 0
1) Constant Surface Potential
ψ (0) = ψ o = ψ s = surface potential (z = 0) = constant
κ -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 ~ 5κ -1
POTENTIAL PROFILE
σ
ψο=ψs
ψο/e
Ψ(z)
c1
c2
c3
z=0
z
1/κ1 1/κ2 1/κ3
c3<c2<c1→"salt screening"
k-1 (nm)
IS [M]
950
10-7 (pure water)
+
[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
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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
dψ
dz
=
z=0
σ
ε
σ = surface charge density (C/m ), ρ = volume charge density of entire electrolyte phase (C/m3 )
2
∞
∞
d 2ψ
dψ
comes from electroneutrality : σ = − ∫ ρ dz = - ∫ −ε 2 dz = ε
dz
dz
0
0 = ψ oεκ
z=0
Poisson's Law
dψ
Boundary Conditions for Two Interacting Plane Parallel Layers :
= 0,ψ = ψ m for z = D/2
dz
Surface potential constant with D
Surface potential changes with D
1 6 0
160 (dψ/dx at surface changes with D)
ψΦo0
Φ(x)(mV)
(mV)
ψ(z)
140
1 4 0
120
1 2 0
100
1 0 0
80
Constant potential solution
as 2 planes are approached
together
8 0
60
6 0
40
4 0
20
2 0
0
0
1
2
3
ψΦo 0
4
0
Constant Charge solution
when 2 planes are approached
0
z
2
4
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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 ψ(z) and ρ(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( ∞ )
σ
+
+
+
+
+
z = zj
+
ψ(x)
+
+
( )
+
ψm
+
z = -D/2 z =0
+
+
+
+
+
+
+
+
+
+
control box
z
Reference position
in the bulk where:
ψ(z) = 0
E = dψ/d z = 0
ci(z) = cio
(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:
ε
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):
ε
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 Æ ∞, E Æ 0 and ci(∞) Æ cio
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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:
ψm
Fψ m
⎛ - FRT
P = RT ⎜ c0 e
+ c0 e RT
⎝
where : ψ
m
⎞
⎛
⎛ Fψ m ⎞ - 1 ⎞
⎟ ⎟
⎟ - RTco = 2RTc0 ⎜ cosh ⎜
⎝ RT ⎠ ⎠
⎝
⎠
is the potential at z=0 and P is the net force/area on the surface.
If ψm<< RT/F ~60mV, then we can actually solve this equation further:
2
⎛
⎞ ⎛ 2c0 F 2ψ m 2 ⎞
Fψ m ⎞
⎛
P ≈ 2RTC0 ⎜ (1 - ⎜
= εκ 2ψ m2
⎟ ) - 1⎟ = ⎜
⎟
RT
⎝ RT ⎠
⎠
⎝
⎠ ⎝
and we can solve for ψ m if we wanted using the linearized PB equation (it’s a little messy). If you plug in the value for ψm
you get:
P ≈ 4εκ 2ψ ο 2 e -κD (constant surface potential)
4σ 2 -κD
P≈
e (constant surface charge)
ε
Exponential in this case so you can think of the Debye length, κ-1, as being the length scale over which electrostatic forces
occur.
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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 (From Leckband, Israelachvili, Quarterly Reviews of Biophysics, 34, 2, 2001)
Image removed due to copyright restrictions.
Table 2 in Leckband and Israelachvili, Quarterly Reviews of Biophysics, 34, 2, 2001
Z = ( 9.38 × 10 -11 ) tanh 2 (ψ ο /107) J m-1 (monovalent electrolyte)
σ = 0.116 sinh(ψ ο /53.4) co Cm-2 (monovalent electrolyte)
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