surface properties of nanoparticle - ovgu · electrochemical potential electrical neutrality of...

BaSO4

Surface properties of nanoparticle

• the surface of suspended nanoparticles is electrically charged (in many cases) • counter ions are adsorbed onto the surface, more or less to compensate the

electrical charges • the layer of surface charges + the layer of counter ions

Origin of surface charges:

• lattice defects by substituted atoms • adsorption of ions onto surface of the solid particle • adsorption of molecules with functional groups

which have electrical charges and/or are dissociable • chemical (e.g. acid / basic) reactions on the surface

of the solid particles (e.g. by dissociation)

Ba2+

Ba2+

Ba2+

Ba2+

BaSO4

model of the electrochemical double layer on the particle surface

O O O O

Si Si Al

O O O O

solid particle

solid particle

Stabilisation of nano - sized titanium dioxide

Zeta - potential of TiO2 ranging from + 20 mV to + 40 mV (pH < 3.0)

TiO--

O-

O-

O-

base

+ OH-

TiOH

OH

+H+

OH2+

OH2+

OH2+

OH2+

OH

Tiacid

OH

Mechanism of redispersion (peptization) and stabilisation

Acid / basic reactions on surface of solid particles by dissociation

of metal oxide nanoparticles (titanium dioxide)

Inner and outer Helmholtz layer

e.g. negatively charged solid nanoparticle Nernst - potential

• after particle formation / suspending an adsorption of negatively / positively

charged counter ions onto the particle surface

• while adsorbing anions lost their hydrate envelope, van der Waals forces

dominates inner Helmholtz - layer

• positively charged cat ions are bound onto the negative mono layer of ani-

ons, electrostatic and van der Waals forces dominate

outer Helmholtz - layer

• inner and outer Helmholtz - layer

Stern - layer Stern - potential

Gouy - Chapman layer

e.g. negatively charged solid nanoparticle

• bound counter ions in the Stern - layer compensate the electrical charge of the solid particle only partially

• for a whole charge compensation of particle surface there is the need of

more counter ions (condition of electrical charge neutrality)

• counter ions form a diffusive cloud around the particle, counter ions can move independently, concentration of counter ions grows in direction to the particle surfaces

• Gouy - Chapman layer begins on the firm bound Stern - layer and ends in

the infinite (totally compensation of particle surface charge)

Helmholtz model Gouy-Chapman model Stern model

(1) negatively charged particle surface (2) inner Helmholtz layer (3) outer Helmholtz layer

Stern layer Stern - layer Gouy - Chapman layer

(1)

(2)

(3)

(1) 1879 Helmholtz (2) 1910, 1917 Gouy, 1913 Chapman (3) 1924 Stern

Electrochemical double layer model

negatively charged particles

Stern layer shear plane ψ 0

ψ S

Nernst potential

Stern potential

δ diffuse layer

distance

Z P

ψ 0 /e Zeta potential

+

+

+ +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

_ +

+

+

+

+

+

+

+ +

+

_

_ _

_

_ _

_

_ _

_

_ _ _

_ _

_

Gouy - Chapman layer

φ0

φS

φS/e

ζ

distance from particle surface

φS

φi

φ0 φ0

Electrochemical potential • a potential is the work needed for transporting a charge from the infinite to a

defined place divided by charge

• a surface potential of a suspended particle is a potential on the surface in

comparison to the infinite

• in a suspension there is the condition of electrical charge neutrality, that means

the absolute potential can be determined

φ0 Nernst potential

φi inner Helmholtz potential

increased by adsorbed anions

φS outer Helmholtz potential

lowered by adsorbed cations

Stern potential

Electrochemical potential

electrical neutrality of charges in infinite distance from particle for practical calculations :

the thickness of the electrochemical double layer is the Debye length δк the decrease of the potential to 1/e of the surface potential

∑εε

=δ⎥⎦

⎤⎢⎣

⎡δ

−=ϕϕ

κκ

2iiA

20r

S zcNeTk

mitrexp)r(

with

φS Stern potential ε0 absolute dielectric constant

φ potential at distance r εr relative dielectric constant

r distance r from the particle surface k Boltzmann constant

δк Debye length e elementary charge

NA Avogadro constant c concentration of ions

T temperature z valence of ions

potential

φi

φS

φ0

φS/e

electrolyte concentration


C in mol / L3Debye length δκ of different types of electrolytes in nm

(1,1) (1,2) (2,2) (1,3) 10-1 0.96 0.55 0.48 0.39 10-2 3.04 1.76 1.52 1.24 10-3 9.60 5.55 4.81 3.93 10-4 30.40 17.60 15.20 12.40

How to reduce the electrochemical potential

• increasing electrolyte concentration reducing Debye length

• increasing ion valence reducing Debye length

• destabilisation of suspension e.g. with addition of Fe3+ or Al3+ ions

compression of diffuse double layer

van der Waals attraction higher than

electrostatic repulsion

∑εε

=δκ 220

iiA

r

zcNe

Tk

Gouy-Chapman layer radii in nm for different salt types in water at 298 K

ζ - p

oten

tial

potential

φi

φS

φ0

φS/e


shear zones with different shear velocities

Determination of the surface potential of nano particles

measuring of electrical potential :

• suspended particles seems to be neutral, charges of the diffuse double layer are compensated

• by a displacement of a part of the charge cloud

arises a measurable potential difference

• shear forces caused by diffusion only are too low, there is the need of a higher shear velocity

• a removing of the complete double layer is

impossible, only approximately the Stern potential measurable

measurable potentials potential at the shear plane = zeta potential ≈ stern potential

0Ev

ε⋅εη

⋅rr

Determination of the zeta - potential for nanoparticle characterisation

Zeta potential: electrical potential of a charged particle at the shearing plane

A charged particle in motion caused by an electrical field or by diffusion loses a portion of

its counter ions of the electrical double player

It is assumed that the zeta - potential corresponds to the Stern - layer potential

Measurement of the zeta – potential: determination of the electrophoretical mobility

Helmholtz – Smoluchowski equation:

ζ =

ζ zeta - potential

E electrical intensity

v particle velocity

η viscosity

ε·ε0 dielectric constant

Sterical stabilisation of nanoparticles

on the surface there are polymers with hydrophilic groups, polymers form short “hairs” towering into the dispersant stabilisation entropic effects, numbers of possible configurations would be lowered

by coagulation energetic effects, polymers have in the dispersant a lower energy con-

tent than being in contact each other

Electrostatic stabilisation of nanoparticles

DLVO theory: developed by Derjaguin, Landau, Overbeek and Verwey

combines van der Waals attraction and electrostatic repulsion forces

Boris Derjaguin Lew Landau Evert J.W. Verwey J.T.G. (Theo) Overbeek

B. Derjaguin, L.Landau (1941): Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes, Acta Phys. Chem. USSR 14, 633 E.J.W. Verwey, J.T.G. Overbeek (1948): Theory of the stability of lyophilic colloids. The interaction of sol parti-cles having an electric double layer, Elsevier Publishing Company

δ+ δ-

δ-δ+δ-δ+

dipole nonpolar molecule

dipole induced dipole

Stabilisation of disperse systems

a suspension of nanoparticles is then stable, if primary particles stay isolated – the stability is influ-

enced by the interaction of attractive and repulsive forces

attractive van der Waals forces

dispersion forces inductive forces dipole - dipole forces

e-

e-

e-

e-

e-

e-

e-

e-

fast electron motions lead

to charge fluctuations

δ+ δ- δ- δ+

dipole dipole

electrostatic attraction be-tween partial charges with

different algebraic sign

MN

N A⋅=

ρ


binary system of interacting particles: van-der-Waals interaction theories:

a) London-van-der-Waals attraction - London (1930), Hamaker (1937) classical microscopic approach London: atom - atom (a: center-to-center distance) Hamaker: particle - particle

mit

b) Lifschitz (1956) macroscopic approach for A - electromagnetic properties of media εi static dielectric constant εi(iν) εi at imaginary frequency

LondonKeesomDebye

4h3

kT32

2)4(

1Cmit

aC

)a(E24

22

06attr ⎟⎟

⎠

⎞⎜⎜⎝

⎛++=−=

ανμμαεπ

3

26

2

1attr

cmatomsN

vdaCN

vd)a(E

−⋅

−= ∫∫

CNA 22 ⋅⋅= πA Hamaker ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+++++

++++

+++

−=jiji

2ji

2

jiji2

ji

ji2

jiSLSattr rr4dr2dr2d

dr2dr2dln

rr4dr2dr2d

rr2

dr2dr2d

rr2

6A

)d(E

ννενενενε

πεεεε

εεεε

ν

d)i()i()i()i(

4h3

kT43

A1 L1S

L1S

L2S

L2S

L1S

L1S2LS1S ∫

∞

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

≈

contribution of the fluctuat-ing, induced dipoles (Keesom, Debye)

contribution of dispersion (quantum mechanics)

L2SL1SL2S1S2LS1S AAAAA −−+= 221112 AAA ⋅≈ ( )( )L2S2SL1S1S12 AAAAA −−≈

ri rj

d

Stabilisation of disperse systems binary system of interacting particles: van-der-Waals interaction energy

sphere - sphere sphere - plate plate - plate sphere-sphere:

for a close approach (d << ri): for r = ri + rj : sphere-plate:

plate-plate: for a finite thickness δP: for a infinite thickness:

ri

d

AS

d

AS

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++++

++++

+++

−=jiji

2ji

2

jiji2

ji

ji2

jiSLSattr rr4dr2dr2d

dr2dr2dln

rr4dr2dr2d

rr2

dr2dr2d

rr2

6A

)d(E

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+−=ii

iiSLSattr r2d

dln

r2dr

dr

6A

)d(E

2SLS

attr d12A

)d(Eπ

−=

)rr(d6

rrA)d(E

ji

jiSLSattr +

−=d12rA

)d(E SLSattr −=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

+−= 2P

2P

2SLS

attr )2d(2

)2d(1

d1

12A

)d(Eδδπ

d12rA

)d(E SLSattr −=

Xylole can stabilize suspensions (dispersing agent)

Hamaker constant depends only on the materials, but not from geometry!

Hamaker metals 10⋅10-20 - 30 ⋅10-20 J constants: oxides 1⋅10-20 - 3⋅10-20 J halogenides 1⋅10-20 - 3⋅10-20 J organics 0.3⋅10-20 - 1⋅10-20 J toluole 5.4 ⋅10-20 J exp. 1.72 ⋅10-20 J cal.


Hamaker constant AS1LS2 depends on material: solid S1- liquid L - solid S2

L2SL1SL2S1S2LS1S AAAAA −−+=

221112 AAA ⋅≈

( )( )L2S2SL1S1S2LS1S AAAAA −−≈

( )2L1S1S1LS1S AAA −≈ attractionno0AfollowsAAif 1LS1SL1S1S ==

AL

AS1 AS1

AL

AS1 AS1

( ) attractioncaseeveryin0AA2

L1S1S >−

!!!theoryonly

isthatrepulsiveisforceWaalsdervan

!negativebecouldAAAAif 1LS1S2SL1S

−−−>>


repulsive Born force caused by adsorption of water (hydrate cloud around the particle) or other liquids

energy is needed to remove the covering liquids

influence of Born repulsion on the stabilisation of nanoparticles is low

repulsive electrostatic force

influenced by the changing of the Debye length

φS Stern potential

φ potential at distance r

r distance r from the particle surface

δк Debye length

NA Avogadro constant

k Boltzmann constant

T temperature

ε0 absolute dielectric constant

εr relative dielectric constant

e elementary charge

z valence of ions

c concentration of ions

∑=⎥

⎦

⎤⎢⎣

⎡−=

2iiA

2

0r

S zcNe

Tkmit

rexp

εεδ

δϕϕ

κκ

Overview: Approach and Solution of the Poisson-Boltzmann Equation Poisson equation Boltzmann distribution Poisson-Boltzmann equation

Linearised Poisson-Boltzmann equation

Debye radius:

( )r0

rr

²rr²r

1²εε

ρ−=⎟

⎠⎞

⎜⎝⎛

∂ϕ∂

∂∂

=ϕ∇ ( ) ( )⎟⎠⎞

⎜⎝⎛ ϕ

=kT

rezexpNrN i0,ii

( ) ²zNkTr²e

r²r

r²r1

ii

0,ir0∑εε

ϕ−=⎟

⎠⎞

⎜⎝⎛

∂ϕ∂

∂∂

( )⎟⎠⎞

⎜⎝⎛ ϕ−

εε−=⎟

⎠⎞

⎜⎝⎛

∂ϕ∂

∂∂ ∑ kT

rezexpezN1r

²rr²r

1 i

ii0,i

r0

( ) [ ]( )a1r4

)ar(expezrr0

i

⋅κ+επε−κ−

=ϕ

Linearisation:

!1x1)x(exp −≈−

kTezi <<ϕ

0ezN ii

0,i =∑

I²eN2kT

A

r0εε=δκ

²zc21I ii∑=

( ) ⎥⎦⎤

⎢⎣⎡ −

⋅+−−⋅

−⋅

⋅= 1

a1)ar((exp

r4ze

r4ze

rr0

i

r0

i

κκ

επεεπεϕ

Contribution of the central ion and the ion cloud

κ-1 is the Debye length

for nanoparticles with r ≥ a, r is center-to-center distance

van-der-Waals attraction


⎟⎟⎠

⎞⎜⎜⎝

⎛=

+⎟⎠

⎞⎜⎝

⎛

−⎟⎠

⎞⎜⎝

⎛

=

⎟⎠⎞

⎜⎝⎛=

+⎟⎠⎞

⎜⎝⎛

−⎟⎠⎞

⎜⎝⎛

== ∑

kT4

eztanh

1kT2

ezexp

1kT2

ezexp

kT4ez

tanh1

kT2ez

exp

1kT2ez

expand

Tk

zcNe

Sj

Sj

Sj

j

Si

Si

Si

ir0

2iiA

2

ϕϕ

ϕ

Γ

ϕϕ

ϕ

Γεε

κ

Stabilisation of disperse systems DLVO - theory

Interference of attracting and repulsing interactions: for a system of two spheres of the radii ri and rj

the interaction energy ET(a) = Eattr (a) + Erep (a) with a being the surface-to-surface distance

Derjaguin (1934):

Linear superposition approximation LSA - interaction energy Erep (a) between two closely spaced spheres

for the Eattr(a) attractive van-der-Waals interaction energy and

Erep(a) repulsive electrostatic potential energy of the double layer

a2

ji

jiA,iji

ji

jiSLST e

)rr(

TkNcrr128

)rr(a6

rrA)a(E κ

κΓΓπ −∞

⋅++

+−=

iirepji

jirep

areprep

ji

jirep raand5rwithv

rr

rr2)a(Fareaunitperenergypotentialvmitdxv

rr

rr2)a(E <<>⋅

+=

+= ∫

∞

κππ

distance from the particle surface

van‐der‐Waals attraction


Born repulsion addition of acting potentials

addition of acting potentials including Born repulsion

potential

potential barrier


Interaction energy – distance profiles from DLVO theory

a) strong repulsion of surfaces nanoparticle suspension is stable

b) surfaces come into a stable equilibrium near the second minimum, if deep enough suspension is kinetically stable

c) surfaces come into the sec-ond minimum, slow coagula-tion of nanoparticles

d) critical coagulation concen-tration ccc: surfaces stay in the second minimum, or co-agulate, fast coagulation of nanoparticles

e) fast coagulation of nanopar-ticles

a2

ji

jiA,iji

ji

jiSLS e)rr(

TkNcrr128

)rr(a6

rrA κ

κΓΓπ −∞

⋅+=

+ a

ji

jiA,iji

ji

jiSLS e)rr(

TkNcrr128

)rr(a6

rrA κ

κΓΓπ −∞

⋅+=

+


DLVO - theory and Schulze - Hardy rule critical coagulation concentration (ccc) is reciprocal proportional to 6th power of ion valence z

at the ccc for the interaction energy is valid: ET(a) = 0 and

(1) (2)

or

( ) 2

j2i2

SLS66

A

53r04

AezNkT)(

1085.9ccc ΓΓεε⋅=

0a)a(ET =

∂∂

0a

)a(E)a(E

a)a(E att

repT =−−=∂

∂ κ 1

0repatt afollows)a(E)a(E −=−= κ cccTkzeN2

r0

22A

εεκ =

ionapproximatHückelDebyez

ccckT4ez

1kT4ez

z1

ccc11kT4ez

2

4iSi

iSi

6iSi

−

∝≈<<

∝≈>>

ΓϕΓϕ

Γϕ

22

66

31

:21

:1

005.0:016.0:131

:21

:1

Electrochemical double layer

Poisson-Boltzmann equation - Basic approach starting from Poisson’s and Boltzmann’s equation

Maxwell: electrostatic potential = (x,y,z)

),,(),,(),,(),,(

/108541.8),,(

),(

arg),,(

),,(

120

0

zyxzyxgraddivzyxzyxgraddiv

mFzyxgradE

mediumvacuumtypermittivifieldelectricEED

densityechfreezyxfieldntdisplacemeelectricD

operatordivergencedivzyxDdiv

r

Poisson’s equation:

)z,y,x(

zyx)z,y,x()z,y,x(graddiv 2

2

2

2

2

2

electrochemical equilibrium between NP and all ions

)z,y,x(dFzClndRT)P,T(d~d

)z,y,x(FzClnRT)P,T(~with~d~d

mol/C3.485,96FNeFttanconsFaradaydnFzQddW(QddndPVdTSGd

ii0ii

ii0ii

IIi

Ii

A

iiieliii

Boltzmann’s equation:

)z,y,x(Cto)ionconcentratbulk(CfromCand)0)((0z,y,xtofrom)z,y,x(ofegrationint

)z,y,x(dFzClndRTfollows0)P,T(and0~d

i0i

ii0ii

RT)z,y,x(FzexpC)z,y,x(C i

0i

i

iii valenceionzCFz)z,y,x(

Poisson-Boltzmann’s equation:

i

iii RT

zyxFzCFzzyx ),,(exp1),,( 0

possibilities to solve PB equation:

is a non-linear partial differential equation - no analyticasolution, but approximations (e.g. Debye)

possibleionlinearisat0CFzneutralityeargch

x1xexp1RT

)z,y,x(Fzfor

!3x

!2x

!1x1xexpsmallis

RT)z,y,x(Fz

0ii

i

32i


ez1

rez

41

rq

41)r(:potentialCoulomb

rrforDrC)r(ansatz0

rr

rr1

r1r4rrexpez

)r(r14

rexpezA

thus,ezr1rexpA4

r1rexpA4

drrexprA4

drrexprTk

CzFeA4

drr4rTk

rexpACFez

ez

rdr4RT

)r(Fz1CFzrdr4)r(

scounterionofeargchelectricNPofeargchelectric

j

r0r0

j2

2

jr0

jj

jr0

jj

jjjr0

r2

2r0

r

2r0

rr0

i

0i

2ir0

2

r i

0i

2

j

2

r i

i0ii

r

2

j

j

j

j

i

jj

Poisson-Boltzmann equation - Debye-Hückel’s approximation

i

i0ii

i

0ii

ii

i

i0ii

RT)z,y,x(Fz1CFz1)z,y,x(

0CFzandRT

)z,y,x(Fz1RT

)z,y,x(Fzexp

RT)z,y,x(FzexpCFz1)z,y,x(

Linearized PB equation:

Tk

CzFewith

)z,y,x()z,y,x(Tk

CFez)z,y,x(

r0

i

0i

2i

2

i

0i

2i

central NP (radius rj at the origin, electric field has spherical symmetry, thus linear PB equation reads:

)r(r

rrr

1 222

r1r

rexpAEthus,r1r

rexpAr

0Bfollows0r

and0)r(as

rrforr

rexpBr

rexpA)r(ansatz

22

j

inside the central NP (r < rj ) Laplace’s equation is valid


Poisson-Boltzmann equation - Gouy - Chapman’s approximation

jr0

j

r0

j

2j

2jr0

j

rr

jjjjr0

jj

j

r14ez

Dand4

ezC

rC

r4ez

r

rrforDrC

r1r4ez

)r(

:rrfor

j

jr0

j

r0

j

r14ez

r4ez

)r(

Solution of PB equation for charged planes:

)x(exp(x)obviously),x(x S

22

2

2i

i2

2

r0

2

2

r0i

ii

ii

iiiii

ii0ii

IIi

Ii

)x(d21)x(d)x()x(d

x)x()x(d

x)x()x(d)x(

CdRT)x(dx

x)x(CdRT)x(dCFz)x(d)x(

CdRT)x(dCFzand0ClnRT)x(Fz:thus)x(FzClnRT)P,T(~with~d~d:startwe

1RT

)x(FzexpCRT

CCRTx2

1

givesCdRT)x(d21

egrationintCdRT)x(d21

i

i

0i

i

0ii

2

r0

C

C ii

)x(

)x(

2r0

ii

2r0

i

0i

1

RT)x(FzexpCRT2

xi

i

0i

r0

2

RT2)x(Fzexp

RT2)x(Fzexp

RT2)x(Fzexp

1RT

)x(Fzexp

iii

i

for a symmetric z:z electrolyte

follows2

)x(exp)x(exp)x(sinhwith

RT2)x(Fzexp

RT2)x(FzexpCRT4

1RT

)x(Fzexp1RT

)x(FzexpCRT4x

2

r0

0

r0

02

RT2)x(FzsinhCRT8

x r0

0

Poisson-Boltzmann equation - Gouy - Chapman’s approximation - Grahame equation

C4

ztanhlnC))u(tanh(lnx

toleadsegrationint)u2(sinh

du2dxfollows4

zuwith

2zsinh

2zd

1

2zsinh

d2zxd

dRTFdthus,

RT)x(Fngsubstituti

CFz2RT1gsinu,d

RT2)x(Fzsinh

1RT2

Fz

d

RT2)x(Fzsinh

1CRT8

xd

RT2)x(FzsinhCRT8

xegrationint

reargchsurface

022r0

2

0r0

r0

0

rrr0surf

j

x1x1ln

21)x(harctan

potentialzeta)0x(,RT4

)0x(Fztanh

:0xsurfacetheat,xexp4

ztanh

))xtanh(ln)x(tanh)x(coshthus),x(f(ln

)x(f)x(f

dx)x(cosh)x(tanh

121dx

)x(cosh)x(sinh1

21dx

x2sinh1

)x(cosh)x(sinh2)x2(sinh:inthalMathematic

0

0

2

2

Interaction energy between two nanoparticles Question: Why do we have a repulsion force between electrically equal charged nanoparticles? Please remem-ber, the opposite electrically charged counter-ions are screening the charge of the central nanoparticle. Answer: When the ionic double layers start to overlap, there is an excess of ions in the overlap region respect to the electrochemical potential. To compensate this, an os-motic pressure develops between the bulk solution and the overlap region.

xexp1xexp1ln

FzRT2)x(

0

0

Grahame equation

RT2)x(FzsinhCRT8 r0

0surf

Interaction energy between nanoparticles - osmosis and osmotic pressure

ionsions1

120l2

0l

P

P

P

P

0l

0l

120l2

0l

2

220l11

0l

ll

21

XX1lnXln:ionapproximat

XlnRTnV)P()P(

thus,PdnVd:egrationint

.)constT(PdnVdPVdTS

n1

nGdd

XlnRT)P()P(

:follows)solventpure(1Xwith,XlnRT)P(XlnRT)P(

)2phase()1phase(:readsPPPpressureosmotic

forstateinitialtheatlsolventtheofpotentialchemical

2

2

2

2

overlapping potential between two nanoparticles

Osmosis: diffusion of solvents through a semipermeable membrane into a region of higher solute concentration

C1 = 2 C2

H2O transportation

initial state

H2O transportation

C1 = C2

final state

Osmotic pressure:

H2O transportation

C1 = 2 C2

initial state C1 = 2 C2

H2O transportation

ΔP

final state i1

l

CRTXlnVRT:pressureosmotic

overlapping potential surface potential surf

electrostatic potential (x) of single overlapping nano-particles (as flat planes)

x = d x = 0 x = 2d = D

Interaction energy between nanoparticles - repulsion pressure due to osmotic pressure

RT2)d(FzsinhCRT4)d(

2)x(exp)x(exp)x(sinhwith

RT2)d(Fzexp

RT2)d(FzexpCRT2)d(

RT2)x(Fzexp

RT2)x(Fzexp

RT2)x(Fzexp

1RT

)x(Fzexpwith

1RT

)d(Fzexp1RT

)d(FzexpCRT)d(

)planeflatfor)x(ncalculatiosee(:eelectrolytz:zlsymmetricaafor

1RT

)d(FzexpCRT)D(P

pressurebulktocompared)dx(planemiddletheinpressureosmotic

C)d(CRT)d()bulk(P)d(P)D(P:Dcetandisparticleaat)D(Ppressurerepulsive

20

2

0

iii

i

ii0i

i

i

0i

i

0ii

D 20

0

20

0

20

242

22

22

0

53

0

0

0

DexpCRT64dD)D(P)D(W

:areasurfaceperenergyeractionint

NPsbetweencetandissurfaceDd2DDexpCRT64)d(

RT)d(FzCRT)d(

follows1xfor!2

x1!4

x!2

x1)xcosh(

and)x(cosh)x(sinh)x2(cosh),x(cosh1)x(sinhwith

dexpFzRT42)d(

,1xforxx152

3xx)x(tanh:compare

expFzRT4)x(follows1

RT4)x(Fzfor

)theoryChapmanGouysee(


)0x(Fztanh

,xexpRT4

)x(Fztanhisthere

calculation of osmotic pressure ΔP(D):

“weak overlap approximation”: at x = d, overlapping (d) is the sum of each (d) of single NPs!

Interaction energy between spherical nanoparticles - Derjaguin approximation

interaction energy between spherical NPs with radii ri and rj

ji

2

ji

ji22j

22i

22j

22i

ji

22jjj

22iii

2i

22ii

2j

22jj

jjii

ji

r1

r1

2rD)r(zandrdr

r1

r1)r(zd

followsrrandrrfor,rdrr

rrr

r)r(zd

rrrr

rrD)r(z

rrxr

rrxr

,thusrrxr

rrxr

:trianglegreenontheoremnPythagorea

xrxrrrD)r(z

DexpCRT64)D(E

)D(Errrr

2zdPrrrr

2zdPrrrr

2)D(F

zdrrrr

rdrwithrdr2PAdP)D(F

20

0

D ji

jiD

ji

ji

ji

ji

ji

ji

0

TkzcNe

andkT4ez

tanh1

kT2ezexp

1kT2ezexp

kT4eztanh

1kT2ezexp

1kT2ezexp

with

e)rr(

CRTrr128)rr(a6

rrA)a(E

:spheresthebetweencetandistheisa(randrradiithewithspherestwobetween)a(Eenergyeractioninttotal

e)rr(CRTrr128

)a(E


)0x(Fztanh


)0x(Fztanh

:potentialszetadifferentwith

e)rr(

CRTrr128Dd)D(F)a(E

r,rradiiwithspherestwobetween)a(Eenergyrepulsion

planesflatofareasurfaceperenergyrepulsion)D(E

r0

2i

0iA

2j

i

i

j

i

i

i

i

a2

ji

ji0ji

ji

jiSLST

ji

T

a2

ji

ji0jirep

jj

j

ii

i

a2

ji

20

0ji

D

rep

jirep

repulsive force between spherical NPs:

ri

rj

D

z

y

dr annuli

r

rj -xj xj

0 ri

surface properties of nanoparticle - ovgu · electrochemical potential electrical neutrality of...

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