8. Fundamentals of Charged Surfaces

8. Fundamentals of Charged Surfaces

Moving the reagentsQuickly and with Little energy

Diffusionelectric fields

o

Cha

rged

S

urfa

ce

+

+

+

+

X=0

N

N o

G

kT

*

ex p

1. Cations distributed thermallywith respect to potential2. Cations shield surface and reduce the effective surfacepotential

o

o

Cha

rged

S

urfa

ce

+

+

+

+

X=0

N

N o

G

kT

*

ex p

o

+

+

+

dx dx

o

* ** dx

+

+

***

o

n

ne

o

zF

R Tx

o i ii

z F

R Td

dxz F C e

i x2

2*

Surface Potentials

Poisson-Boltzman equation

Charge near electrode dependsupon potential and is integratedover distance from surface - affects the effective surface potential

Cation distribution hasto account for all species,i

Dielectric constant of solution

Permitivity of free space

Simeon-Denis Poisson1781-1840

ze

kTo 1 o m V 5 0

x o

xe

o i ii

z F

R Td

dxz F C e

i x2

2*

Solution to the Poisson-Boltzman equation can be simple if the initial surface potential is small:

Potential decays from the surface potential exponentially with distance

d

dx

z F C

e

z F Cz F

R T

z F

R T

i ii

o

z F

R Ti i

i

o

i i i i

i i2

2

2

11

2

* *

. . . . . .

Largest term

d

dx

F z C

R T

i ii

ox

2

2

2

*

Let

2

2 2

1

x

F z C

R Ta

i ii

o

*

Then:d

dx xx

a

2

2 2

General Solution of:

x

x

x

x

xA e B ea a

d

dx xx

a

2

2 2

Because goes to zero as x goes to infinityB must be zero

x

x

x xA e A ea

Because goes to as x goes to zero (e0 =1)A must be

thus x o

xe

Potential decays from the surface potential exponentially with distance

x o oe 1 0 3 6 7( . )

When =1/x or x=1/ then

The DEBYE LENGTH x=1/

o

Cha

rged

S

urfa

ce

=0.36 o+

+

+

+

+

X=0 X=1/

+

+

+

+

What is

Petrus Josephus Wilhelmus Debye1844-1966

2 2 21

2n z e

kTo

*

z x C( . )( )* /3 2 9 1 0 7 1 2

Debye Length

Units are 1/cm

26 0 2 1 0 1

1 0

1 0 0 1 6 0 2 1 8 1 0

7 8 4 98 8 5 4 1 9 1 0

1 0 0

1 3 8 0 6 5 1 02 9 8

2 3

3 3

22

1 9 2

2 5

1 2 2

2

2 3

1

2m oles

L

x

m ole

L

cm

cm

mch e

x C

ch e

un itlessx C

N m

m

cm

N m

J

x J

KKo C

.a rg

.

arg

.. .

2 1 6 0 2 1 8 1 0

7 8 4 9 1 3 8 0 6 5 1 0 2 9 8 6 0 2 2 1 0

21 9 2

2 5

2 3

1

2

2 3

#.

. . .cm

x

x

m ole

x ionso C

2 2 21

2C N z e

kTonc A

o

Does not belong

=1/cm

zF

n z e

kTo

2 2 2

1

2*

z x C( . )( )* /3 2 9 1 0 7 1 2

Table 2: Extent of the Debye length as a function of electrolyte

C(M) 1/κ ( )

1 3

0.1 9.6

0.01 30.4

0.001 96.2

0.0001 304

Debye Length

Units are 1/cm

In the event we can not use a series approximation to solve the Poisson-Boltzman equation we get the following:

ex p

ex p ex p

ex p ex p

x

ze

kT

ze

kT

ze

kT

ze

kT

2 2

2 2

1 1

1 1

0

0

Ludwig Boltzman1844-1904

Simeon-Denis Poisson1781-1840

Check as Compared to tanhBy Bard

Set up excel sheet ot have them calc effectOf kappa on the decay

Example Problem

A 10 mV perturbation is applied to an electrode surface bathed in0.01 M NaCl. What potential does the outer edge of a Ru(bpy)3

3+

molecule feel?

Debye length, x

z x C

XA

xA

( . )( )

/( . )( . )

.

* /

/

3 2 9 1 0

11 0

1 3 2 9 1 0 0 0 13 0 4

7 1 2

8

7 1 2

Since the potential applied (10 mV) is less than 50 can usethe simplified equation.

Units are 1/cm

x o

xo

x

xe e ez 1 0 7 4 3

9

3 0 4. .

The potential the Ru(bpy)33+ compound experiences

is less than the 10 mV applied.

This will affect the rate of the electron transfer eventfrom the electrode to the molecule.

Radius of Ru

Surface Charge Density

The surface charge distance is the integration over all the charge lined up at the surface of the electrode

oa a a

dxd

dxdx

d

dx

0

2

2 0

The full solution to this equation is:

o oo o

o o

kT nze

kT

C z

(8 ) s in h ( )

. ( * ) s in h ( . )

1

2

1

2

2

11 7 1 9 5

C is in mol/L

o

Cha

rged

S

urfa

ce

=0.36 o+

+

+

+

+

X=0 X=1/

+

+

+

+

Can be modeled as a capacitor:C

d

ddifferential

For the full equation

Cz e n

kT

ze

kT

oo

2

2

2 20

1

2 co sh

C z C z o 2 2 8 1 9 51

2* co sh . At 25oC, water

d

d

Differential capacitanceEnds with units of uF/cm2

Conc. Is in mol/L

0

2000

4000

6000

8000

10000

12000

-15 -10 -5 0 5 10 15

y x co sh

o o o

Can be simplified if (o ~ 25 mV),

Specific Capacitance is the differential space charge per unit area/potential

C

A

dq

A d

d

dspecific

C

A o Specific CapacitanceIndependent of potentialFor small potentials

o

Flat in this regionGouy-Chapman Model

Cz e n

kT

ze

kT

oo

2

2

2 20

1

2 co sh

0

20

40

60

80

100

120

-500 -400 -300 -200 -100 0 100 200 300 400 500

E-Ezeta

Capacitance

Real differential capacitance plots appear to roll off instead ofSteadily increasing with increased potential

Physical Chemistry Chemical PhysicsDOI: 10.1039/b101512p

               Paper

Photoinduced electron transfer at liquid/liquid interfaces. Part V. Organisation of water-soluble chlorophyll at the water/1,2-dichloroethane interface�

Henrik Jensen , David J. Fermn and Hubert H. Girault*

Laboratoire d'Electrochimie, D partement de Chimie, Ecole Polytechnique F d rale de Lausanne, CH-1015, � � �Switzerland

Received 16th February 2001 , Accepted 3rd April 2001 Published on the Web 17th May 2001

o

Cha

rged

S

urfa

ce

+

+

+

+

+

X=0

+

+

+

+

Linear dropin potentialfirst in theHelmholtz orStern specificallyadsorbed layer

Exponentialin the thermallyequilibrated ordiffuse layer

CdiffuseCHelmholtz or Stern

x2

Hermann Ludwig Ferdinand von Helmholtz1821-1894

O. SternNoble prize 1943

Capacitors in series

Cz e n

kT

ze

kTD iffuse

oo

2

2

2 20

1

2 co sh

C

A H elm ho ltz or S terno

C

C C C

series

N

11 1 1

1 2

. . . . . .

1 1 1 1

1 2CC

C C Cseriesseries

N

. . . . . .

Wrong should be x distance of stern layer

For large applied potentials and/or for large salt concentrations1. ions become compressed near the electrode surface to

create a “Helmholtz” layer.2. Need to consider the diffuse layer as beginning at the

Helmholtz edge

1 1

2

2

2

0 2 20

1

2C

x

z e n

kT

ze

kT

oo

co sh

CapacitanceDue to Helmholtzlayer Capacitance due to diffuse

layer

DeviationIs dependent uponThe salt conc.

The larger the “dip”For the lower The salt conc.

0.63

0.64

0.65

0.66

0.67

0.68

0.69

0.7

0.71

-500 -400 -300 -200 -100 0 100 200 300 400 500

E-Ezeta

Capacitance

Create an excel problemAnd ask students to determine the smallestAmount of effect of an adsorbed layer

Experimental data does notCorrespond that well to the Diffuse double layer double capacitormodel

(Bard and Faulkner 2nd Ed)

Fig. 5 Capacitance potential curve for the Au(111)/25 mM KI in DMSO interface with time. �

Physical Chemistry Chemical PhysicsDOI: 10.1039/b101279g               

PaperComplex formation between halogens and sulfoxides on metal surfaces

Siv K. Si and Andrew A. Gewirth*

Department of Chemistry, and Frederick Seitz Materials Research Laboratory, Uni ersity of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

Received 8th February 2001 , Accepted 20th April 2001 Published on the Web 1st June 2001

Model needs to be altered to accountFor the drop with large potentials

This curve is pretty similar to predictions except where specificAdsorption effects are noted

Graphs of these types were (and are) strong evidence of the Adsorption of ions at the surface of electrodes.

Get a refernce or two of deLevie here

Introducing the Zeta Potential

oC

harg

ed

Sur

face

+

+

+

+

+

+

+

+

+

Imagine a flowing solutionalong this charged surface.Some of the charge will be carriedaway with the flowing solution.

Introducing the Zeta Potential, given the symbo l

oC

harg

ed

Sur

face

+

+

+

+

+

+

+

+

+

Shear Plane

Flowing solution

zeta

Sometimesassumedzeta correspondsto DebyeLength, butNot necessarily true

C C1

21

2ex p

The zeta potential is dependent upon how the electrolyteconcentration compresses the double layer. are constantsand sigma is the surface charge density.

Shear Plane can be talked about in two contexts

o

Cha

rged

S

urfa

ce

+

+

+

+

+

+

+

+

+

Shear Plane

+

+

+

+ +

+

+

+

++

++

ShearPlane

Particle in motion

In either case if we “push” the solution alonga plane we end up with charge separation whichleads to potential

Streaming Potentials

From the picture on preceding slide, if we shove the solutionAway from the charged surface a charge separation develops= potential

P

o

so lu tion resis ce m

zeta po ten tia l

v is itykg

m s

tan

co s

Sample problem here

Reiger- streaming potentialapparatus.

Can also make measurements on blood capillaries

o

Cha

rged

S

urfa

ce

+

+

+

+

+

X=0

+

+

+

+

Cathode

Anode

Vappapp

+

Jo Jm

Jm

In the same way, we can apply a potential and move ions and solution

Movement of a charged ion in an electric field

Electrophoretic mobility

app lied electric fie ld

f frictiona l drag r

v electropho retic velocity

6

The frictional drag comesabout because the migratingion’s atmosphere is movingin the opposite direction, draggingsolvent with it, the drag is related to the ion atmosphere

f v z eii

i

The force from friction is equal to the electric driving force

Electric ForceDrag Force

Direction of Movement

Ion accelerates in electric field until the electric forceis equal and opposite to the drag force = terminal velocity

f z eelectrica l i

f r

vis ity

r ion ic rad ius

ion velocity

fr ic tiona l

6

co s

f f

r z e

fr ic tiona l electric

i

6

At terminal velocity

z e

ri

6

The mobility is the velocity normalized for the electric field:

uz e

ri

i 6

v z e

f

z e

ru

i i iep

6

Typical values of the electrophoretic mobility aresmall ions 5x10-8 m2V-1s-1

proteins 0.1-1x10-8 m2V-1s-1

F rictiona l drag r 6(Stokes Law)

r = hydrodynamicradius

Stokes-Einsteinequation

Reiger p. 97Sir George Gabriel Stokes 1819-1903

Insert a sample calculation

u epo

2

3

When particles are smaller than the Debye length you getThe following limit:

Remember: velocity is mobility x electric field

Reiger p. 98

What controls the hydrodynamic radius?- the shear plane and ions around it

Compare the two equations for electrophoretic mobility

uf

epo o

2

3

uz e

rep

i 6

f z e

ro i

6

rz e

fi

o

6

Where f is a shape term which is 2/3 for sphericalparticles

Relation of electrophoretic mobility to diffusion

DkT

f

kT

r

6

Thermal “force”

F rictiona l drag r 6

DkT

f

uz e

ri

i 6

DkT

f

kT

zeu electropho retic m igra tion

Measuring Mobilities (and therefore Diffusion)from Conductance Cells

- +

+

+

++

++

+

-

-

-

- -

To make measurement need to worry about all the processesWhich lead to current measured

Ac Voltage- +

O R-+

+

++

+

Charging

ElectronTransfer

Solution Charge Motion = resistance

--

-

--

- ++

R-O

Zf1 Zf2Rs

CtCt

Z R

C

f c t

sC s

11

2

2

1

2

1

2

1

2

Electron transfer at electrode surface can be modeled as the Faradaic impedance, Z2

diffusion

Related to ket

An aside

Zf1 Zf2Rs

CtCt

Solving this circuit leads to

RZ

Z

C

RZ

Z

C

R

Z C

R

Z C

Tf

f

t

sf

f

t

T

f t

s

f t

1

1

2

2

1 2

1 1

11 1

11 1

( ) ( )

Applying a high frequency, w, drops out capacitance and FaradaicImpedance so that RT=Rs

What frequency would you have to useTo measure the solution resistance betweenTwo 0.5 cm2 in 0.1 M NaCl?

C

A

d

d

d

dspecific o

o

( )

z x C xm

( . )( ) .*3 2 9 1 0 1 0 4 1 017 1 / 2 7

C C A Aspecific o CheckCalculationTo show thatIt is cm converted to m

C C A Aspecific o

C A xm

x cm xm

cmx

C

J mo

1 0 4 1 01

2 0 51 0 0

7 8 5 4 8 8 5 4 1 07 22

1 22

. . . .

C A xm

x cm xm

cmx

C

J mo

1 0 4 1 01

2 0 51 0 0

7 8 5 4 8 8 5 4 1 07 22

1 22

. . . .

C xC

Jx

C

C Vx

C

Vx F 7 2 1 0 7 2 1 0 7 2 1 0 7 2 1 07

27

27 7. . . . . . . .

The predicted capacitance of both electrodes in 0.1 M NaCl wouldBe 0.72 microfarads

For the capacitive term to drop out of the electrical circuit We need:

11

1 1

7 2 1 01 4 1 0

76

C

C xx

t

t

.

.

The frequency will have to be very large.

Solution Resistance Depends uponCell configuration

RA

length

A

Resistivity of soln.

Sample calculation in a thin layer cell

Resistance also depends upon the shapeOf an electrode

Disk Electrode Spherical electrode Hemisphericalelectrode

Ra

4a is the radius

Ra

4

Ra

2

From Baranski, U. Saskatchewan

Scan rate 1000 V/s at two different size electrodes for Thioglycole at Hg electrode

kR A

1

Conductivity is the inverse of Resistance

Resistivity and conductivity both depend uponConcentration. To get rid of conc. Term divide

kC C R C A

1

A plot of the molar conductivity vs Concentration has a slopeRelated to the measurement device, and an intercept related toThe molar conductivity at infinite dilution

m olar conductiv ity

o s dard m olar conductiv ity tan

This standard molar conductivity depends upon the solutionResistance imparted by the motion of both anions and cations Moving in the measurement cell.

t

t

o

o

Where t is a transference number which accounts for the Proportion of charge moving

TransferenceNumbers can beMeasured by capturingThe number of ionsMoving.

Once last number needsTo be introduced:The number of moles of ionPer mole of salt

o v v

Compute the resistance of a disk electrodeOf 0.2 cm radius in a 0.1 M CaCl2 solution

o v v

oC a C l

mm ol

mm ol

mm ol

2 1 2 0 0 0 7 6 3 1 0 0 11 9 0 0 2 7 1 62 2 2

. . .

0 0 2 7 1 61 1

0 11 0

1 0 0

2

3 3

3.

.

m

m ol C m ol

L

L

cm

cm

m

1

0 0 2 7 1 6 0 11 0

1 0 00 3 6 8

2

3 3

3

. .

.m

m ol

m ol

L

L

cm

cm

m

m

The resistance is computed from

Ra

m

cm xm

cm

4

0 3 6 8

4 0 20 1

4 6.

..

.

Remember – we were trying to get to mobilityFrom a conductance measurement!!!!

uz F

i

oi

i

Also remember that mobility and diffusion coefficients are related

DkT

zeu

kT

ze zF

kT

z eFx

z

J m ol

Cio

io

io

27

2 22 6 6 1 0.

D xz

J m ol

Cio

2 6 6 1 0 7

2 2.

We can use this expression to calculateDiffusion coefficients

D xz

J m ol

Cio

2 6 6 1 0 7

2 2.

D xx

m

m ol J m ol

Cx

m J

C3

7

42

2 21 0

2

22 6 6 1 0

3 0 2 7 1 0

38 9 2 1 0

.

.

( ).

m J

C V s

C

V C

J

m

s

2

2

2

D xx

m

m ol J m ol

Cx

m

s4

7

42

2 21 0

2

2 6 6 1 04 4 2 1 0

47 3 4 1 0

.

( ).

Fe(CN)63- diffusion coefficient is 9.92x10-10 m2/s

Fe(CN)64- diffusion coefficient is 7.34x10-10 m2/s

The more highly charged ion has more solution solutes aroundIt which slows it down.

How does this effect the rate of electron transfer?

k Zet e l

G

kT

ex p

Probability factor Collisional factor

ZkT

m~

2

1

2

Where m is the reduced mass.

Z is typically, at room temperature,104 cm/s

Activation energy

G

G o

2

4

Free energy change

work required to change bondsAnd bring molecules together

in ou t

ou t

o D A D A op s

e

a a r

2

4

1

2

1

2

1 1 1

a donor rad ii

a accep tor rad ii

op tica l d ie lectric cons t

regu lar d ie lectric cons t

e electron ch e

D

A

op

s

tan

tan

arg

G e E w wo o p r ( )

( )w w Uz z e e

a

e

aep r

ra p

a

D

a

A

rD AD A

2

04 1 1

Formal potential

Work of bringing ions together

When one ion is very large with respect to other (like an electrode)Then the work term can be simplified to:

( )w w U zep rr

The larger kappa the smaller the activation energy, the closerIons can approach each other without work