7. Tools III_ Charged Surfaces

8/12/2019 7. Tools III_ Charged Surfaces

1/72

8. Fundamentals of Charged Surfaces


2/72

Moving the reagents

Quickly and with

Little energy

Diffusion

electric fields


3/72

Yo

Charged

Surface

+

+

+

+

X=0

N

No

G

kT

*

exp

1. Cations distributed thermally

with respect to potential

2. Cations shield surface and

reduce the effective surface

potential

Yo


4/72

Yo

Charged

Surface

+

+

+

+

X=0

N

No

G

kT

*

exp

Yo +

+

+

dx dx

Yo

* ** dx

+

+

***

Yo


5/72

n

n eo

zF

RT

x

o i i

i

z FRTd

dx z FC e

i x2

2

*

Surface Potentials

Poisson-Boltzman equation

Charge near electrode dependsupon potential and is integrated

over distance from surface - affects

the effectivesurface potential

Cation distribution has

to account for all species,i

Dielectric constant of solution

Permitivity of free space

Simeon-Denis Poisson

1781-1840


6/72

ze

kT

o 1 o mV50

x o xe

o i ii

z F

RTd

dx z FC 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


7/72

d

dx

z FC

e

z FCz F

RT

z F

RT

i ii

o

z F

RTi i

i

o

i i i i

i i2

2

2

11

2

* *......

Largest term

d

dx

F z C

RT

i ii

o

x

2

2

2

*

Let

2

2 2

1

x

F z C

RTa

i ii

o

*

Then:

d

dx x

x

a

2

2 2


8/72

General Solution of:

Y xx

x

x

xAe Bea a

d

dx x

x

a

2

2 2

Because Ygoes to zero as x goes to infinity

B must be zero

Y xx

x xAe Aea

Because Ygoes to Y0as x goes to zero (e0

=1)A must be Y0

thus

x o

xe


9/72

Potential decays from the surface potential exponentially with distance

x o oe 1 0367( . )

When =1/x or x=1/then

The DEBYE LENGTH x=1/


10/72

Yo

Charged

Surface

Y=0.36 Yo

+

+

+

+

+

X=0 X=1/

+

+

+

+

What is

Petrus Josephus

Wilhelmus Debye1844-1966


11/72

2 2 2

1

2n z e

kTo

*

z x C( . )( )* /329 107 1 2

Debye Length

Units are 1/cm

26 02 10 1

10

100 160218 10

7849885419 10

100

138065 10298

23

3 3

22

19 2

25

12 2

2

23

1

2moles

L

x

mole

L

cm

cm

m ch e

x C

ch e

unitlessx C

N m

m

cm

Nm

J

x J

K Ko C

.arg

.

arg

.. .

2 160218 10

7849 138065 10 298 6 022 10

2 19

2

25

23

1

2

23

#.

. . .

cm x

x

mole

x ionso C

2 2 21

2C N z e

kT

onc A

o

Does not belong

=1/cm


12/72

zFn z e

kTo

2 2 21

2*

z x C( . )( )* /329 107 1 2

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

C(M) 1/ ()

1 3

0.1 9.6

0.01 30.4

0.001 96.2

0.0001 304

Debye Length

Units are 1/cm


13/72

In the event we can not use a series approximation to solve the

Poisson-Boltzman equation we get the following:

exp

exp exp

exp exp

x

ze

kT

ze

kT

zekT

zekT

2 2

2 2

1 1

1 1

0

0

Ludwig Boltzman1844-1904

Simeon-Denis Poisson1781-1840

Check as

Compared to tanh

By Bard


14/72

Set up excel sheet ot have them calc effect

Of kappa on the decay


15/72

Example Problem

A 10 mV perturbation is applied to an electrode surface bathed in

0.01 M NaCl. What potential does the outer edge of a Ru(bpy)33+

molecule feel?

Debye length, x

z x C

XA

x A

( . )( )

/( . )( . )

.

* /

/

329 10

110

1 329 10 0 01 304

7 1 2

8

7 1 2

Since the potential applied (10 mV) is less than 50 can use

the simplified equation.

Units are 1/cm


16/72

x ox

o

x

x

e e ez

10 7 43

9

30 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 event

from the electrode to the molecule.

Radius of Ru


17/72

Surface Charge Density

The surface charge distance is the integration over all the charge

lined up at the surface of the electrode

oa a adx

d

dx dx

d

dx

0

2

2 0

The full solution to this equation is:

o oo

o

o o

kT n zekT

C z

(8 ) sinh( )

. ( *) sinh( . )

12

1

2

2

117 19 5

C is in mol/L


18/72

Yo

Charg

edSurface

Y=0.36 Yo+

+

+

+

+

X=0 X=1/

+

+

+

+

Can be modeled as a capacitor:

C

d

ddifferential


19/72

For the full equation

Cz e n

kT

ze

kT

o

o

2

2

2 2

0

1

2 cosh

C z C z o228 19 51

2*

cosh . At 25oC, water

d

d

Differential capacitance

Ends with units of uF/cm2

Conc. Is in mol/L


20/72

0

2000

4000

6000

8000

10000

12000

-15 -10 -5 0 5 10 15

y xcosh


21/72

o o o

Can be simplified if (o~ 25 mV),

Specific Capacitanceis the differential

space chargeper unit area/potential

C

A

dq

Ad

d

d

specific

C

A o

Specific Capacitance

Independent of potential

For small potentials

1


22/72

o

Flat in this region

Gouy-Chapman Model

Cz e n

kT

ze

kT

o

o

2

2

2 2

02

cosh

0

20

40

60

80

100

120

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

E-Ezeta

Capacitance


23/72

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

Physical Chemistry Chemical Physics

DOI: 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. FermnandHubert H. Girault*

Laboratoire d'Electrochimie, Dpartement de Chimie, Ecole Polytechnique Fdrale de Lausanne, CH-1015,

Switzerland

Received 16th February 2001 , Accepted 3rd Apr i l 2001

Publ ished on the Web 17th May 2001
http://pubs.rsc.org/ej/CP/2001/B101512P/http://pubs.rsc.org/ej/CP/2001/B101512P/


24/72

Yo

Charged

Surface

+

+

+

+

+

X=0

+

+

+

+

Linear drop

in potentialfirst in the

Helmholtz or

Stern specifically

adsorbed layer

Exponential

in the thermally

equilibrated or

diffuse layer

Cdiffuse

CHelmholtz or Stern

x2

Hermann Ludwig

Ferdinand von Helmholtz1821-1894

O. SternNoble prize 1943


25/72

Capacitors in series

C

z e n

kT

ze

kTDiffuse

o

o

2

2

2 2

0

1

2

cosh

C

AHelmholtz or Stern

o

C

C C C

series

N

1

1 1 1

1 2

......

1 1 1 1

1 2C C

C C Cseriesseries

N

......

Wrong should be x distance of stern layer


26/72

For large applied potentials and/or for large salt concentrations

1. ions become compressed near the electrode surface to

create a Helmholtz layer.

2. Need to consider the diffuse layer as beginning at theHelmholtz edge

1 1

2

2

2

0 2 2

0

12C

x

z e n

kT

ze

kT

o

o

cosh

Capacitance

Due to Helmholtz

layer Capacitance due to diffuse

layer


27/72

Deviation

Is 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

Capa

citance


28/72

Create an excel problemAnd ask students to determine the smallest

Amount of effect of an adsorbed layer


29/72

Experimental data does not

Correspond that well to theDiffuse double layer double capacitor

model

(Bard and Faulkner 2ndEd)


30/72

acitancepotential curve for the Au(111)/25 mM KI in DMSO interface

Physical Chemistry Chemical

Physics

DOI: 10.1039/b101279g

PaperComplex formation between halogens andsulfoxides on metal surfaces

Siv K. SiandAndrew 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 Apri l 2001

Published on the Web 1st June 2001

Model needs to be altered to account

For the drop with large potentials


31/72

This curve is pretty similar to predictions except where specific

Adsorption effects are noted


32/72


33/72

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


34/72

Introducing the Zeta Potential

Yo

ChargedSurface

+

+

+

+

+

+

+

+

+

Imagine a flowing solution

along this charged surface.

Some of the charge will be carriedaway with the flowing solution.


35/72

Introducing the Zeta Potential, given the symbo l

Yo

ChargedSurface

+

+

+

+

+

+

+

+

+

Shear Plane

Flowing solution

Yzeta

Sometimes

assumedzeta

corresponds

to Debye

Length, butNot

necessarily

true


36/72

C C

1

2

1

2

expThe zeta potential is dependent upon how the electrolyte

concentration compresses the double layer. are constants

and sigma is the surface charge density.

Shear Plane can be talked about in


37/72

Shear Plane can be talked about in

two contexts

Yo

ChargedSurface

+

+

+

+

+

+

+

+

+

Shear Plane

+

+

+

++

+

+

+

++

++

Shear

Plane

Particle in motion

In either case if we push the solution along

a plane we end up with charge separation which

leads to potential

St i P t ti l


38/72

Streaming Potentials

From the picture on preceding slide, if we shove the solution

Away from the charged surface a charge separation develops

= potential

Y

P

o

solution resis ce m

zeta potential

vis ity kgm s

tan

cos


39/72

Sample problem here

R i t i t ti l


40/72

Reiger- streaming potential

apparatus.

Can also make measurements on blood capillaries


41/72

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


42/72

Yo

ChargedS

urface

+

+

+

+

+

X=0

+

+

+

+

Cathode

Anode

Vapp+

Jo Jm

Jm

t e sa e way, we ca app y a pote t a a d ove o s a d

solution

Movement of a charged ion in an electric field


43/72

Movement of a charged ion in an electric field

Electrophoretic mobility

applied electric field

f frictional drag rv electrophoretic velocity

6

The frictional drag comes

about because the migrating

ions atmosphere is movingin the opposite direction, dragging

solvent with it, the drag is related to the ion atmosphere

f v z ei i i

The force from friction is equal to the electric driving force


44/72

Electric ForceDrag Force

Direction of Movement

Ion accelerates in electric field until the electric force

is equal and opposite to the drag force = terminal velocity

f z eelectrical i

f r

vis ity

r ionic radius

ion velocity

frictional

6

cos


45/72

f f

r z e

frictional electric

i

6

At terminal velocity

z e

ri

6

The mobility is the velocity normalized for the electric field:

uz e

rii

6

St k Ei t i


46/72

v z e

f

z e

r u

i i i

ep

6

Typical values of the electrophoretic mobility are

small ions 5x10-8m2V-1s-1

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

Frictional drag r6(Stokes Law)

r = hydrodynamic

radius

Stokes-Einstein

equation

Reiger p. 97Sir George GabrielStokes 1819-1903


47/72

Insert a sample calculation


48/72

uepo 2

3

When particles are smaller than the Debye length you get

The following limit:

Remember: velocity is mobility x electric field

Reiger p. 98

What controls the hydrodynamic radius?


49/72

What controls the hydrodynamic radius?

- the shear plane and ions around it

Compare the two equations for electrophoretic mobility

uf

ep

o o 2

3

u

z e

repi

6

f z e

r

o i

6

rz e

f

i

o

6Where f is a shape term which is 2/3 for spherical

particles

Relation of electrophoretic mobility to diffusion


50/72

DkT

f

kT

r

6

Thermal force

Frictional drag r6

DkT

f

uz e

rii

6

DkT

f

kT

ze

uelectrophoretic migration


51/72

Measuring Mobilities (and therefore Diffusion)

from Conductance Cells

- +

+

+

++

++

+

-

-

-

- -

To make measurement need to worry about all the processes

Which lead to current measured

- +


52/72

Ac Voltage- +

OR-

+

+

++

+

Charging

ElectronTransfer

SolutionCharge

Motion = resistance

-

-

-

--

- ++

R-O

Zf1 Zf2Rs

CtC

t


53/72

Z R

C

f ct

s

Cs

11

2

2

1

2

12

1

2

Electron transfer at electrode surface can be modeled as the

Faradaic impedance, Z2

diffusion

Related to ket

An aside


54/72

Zf1 Zf2Rs

Ct Ct

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

1

1 1

1

1 1

( ) ( )

Applying a high frequency, w, drops out capacitance and Faradaic

Impedance so that RT=Rs

What frequency would you have to use


55/72

To measure the solution resistance between

Two 0.5 cm2 in 0.1 M NaCl?

C

A

d

d

d

d

specific o

o

( )

z x C xm

( . )( ) .*329 10 104 10 17 1/2 7

C C A Aspecific o CheckCalculation

To show that

It is cm converted to


56/72

C C A Aspecific o

C A xm

x cm x m

cmx

C

J mo

104 10 1

2 05100

7854 8854 107 22

122

. . . .

C A xm

x cm x mcm

x CJ m

o

104 10 1 2 05100

7854 8854 107 22

12 2. . . .

C x CJ

x CCV

x CV

x F 7 2 10 7 2 10 7 2 10 7 2 1072

72

7 7.. .. .. ..

The predicted capacitance of both electrodes in 0.1 M NaCl would

Be 0.72 microfarads


57/72

For the capacitive term to drop out of the electrical circuit

We need:

1 1

1 1

7 2 10 14 1076

C

C x x

t

t

. .

The frequency will have to be very large.

Solution Resistance Depends upon


58/72

p p

Cell configuration

RA

length

A

Resistivity of soln.


59/72

Sample calculation in a thin layer cell

Resistance also depends upon the shape


60/72

Resistance also depends upon the shape

Of an electrode

Disk Electrode Spherical electrode Hemispherical

electrode

Ra

4

a is the radius

Ra

4R

a

2

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


61/72

From Baranski, U. Saskatchewan

Thioglycole at Hg electrode

Conductivity is the inverse of Resistance


62/72

kRA

1

Conductivity is the inverse of Resistance

Resistivity and conductivity both depend upon

Concentration. To get rid of conc. Term divide

kC C RCA

1

A plot of the molar conductivity vs Concentration has a slope

Related to the measurement device, and an intercept related to

The molar conductivity at infinite dilution

molar conductivity

o d d l d


63/72

o s dard molar conductivity tan

This standard molar conductivity depends upon the solution

Resistance imparted by the motion of both anions and cationsMoving in the measurement cell.

t

t

o

o

Where t is a transference number which accounts for the

Proportion of charge moving


64/72

Transference

Numbers can be

Measured by capturingThe number of ions

Moving.

Once last number needs

To be introduced:

The number of moles of ion

Per mole of salt

o v v

C t th i t f di k l t d


65/72

Compute the resistance of a disk electrode

Of 0.2 cm radius in a 0.1 M CaCl2 solution

o v v

o Ca Cl mmol mmol mmol 2 1 2 000763 1 00119 0027162 2 2

. . .

0 02716 1 1

01 10

100

2

3 3

3.

.

m

mol C mol

L

L

cm

cm

m

1

0 02716 01

10

1000368

2

3 3

3

. .

.m

mol

mol

L

L

cm

cm

m

m

The resistance is computed from


66/72

The resistance is computed from

Ra

m

cmx m

cm

40368

4 0 2 01

4 6.

. .

.

b i bili


67/72

Rememberwe were trying to get to mobility

From a conductance measurement!!!!

uz F

i

oi

i

Also remember that mobility and diffusion coefficients arerelated

D kT

ze

u kT

ze zF

kT

z eF

x

z

J mol

C

io

io

io

2

7

2 2

2 66 10.

D xz

J mol

C

io

2 66 10 7

2 2.

We can use this expression to calculate


68/72

We can use this expression to calculate

Diffusion coefficients

D xz

J molC

io

2 66 10 72 2

.

D xx

m

mol J mol

Cx

m J

C3

7

42

2 2

10 2

2266 10

302 7 10

3892 10

..

( ).

m J

C Vs

C

VC

J

m

s

2

2

2


69/72

D x

x m

mol J mol

C x

m

s47

42

2 2

102

266 10

442 10

4 734 10

. ( ) .

Fe(CN)63-

diffusion coefficient is 9.92x10-10

m2

/s

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

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

How does this effect the rate of electron transfer?


70/72

How does this effect the rate of electron transfer?

k Zet el

G

kT

exp

Probability factor Collisional factor

Z kT

m~

2

12

Where m is the reduced mass.

Z is typically, at room temperature,

104cm/s

Activation energy

Free energy change


71/72

G

Go

2

4

work required to change bonds

And bring molecules together

in out

out

o D A DA op s

e

a a r

2

4

1

2

1

2

1 1 1

a donor radii

a acceptor radii

optical dielectric cons t

regular dielectric cons t

e electron ch e

D

A

op

s

tan

tan

arg

Formal potential


72/72

G e E w wo o p r ( )

( )w w Uz z e e

a

e

aep r r

a pa

D

a

A

rD A

DA

2

04 1 1

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:

p r

The larger kappa the smaller the activation energy, the closer

Ions can approach each other without work

Documents

7. Tools III_ Charged Surfaces