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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
distance from particle surface
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
distance from particle surface
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
Stabilisation of disperse systems
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.
Stabilisation of disperse systems
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
−−−>>
Stabilisation of disperse systems
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
electrostatic repulsion
⎟⎟⎠
⎞⎜⎜⎝
⎛=
+⎟⎠
⎞⎜⎝
⎛
−⎟⎠
⎞⎜⎝
⎛
=
⎟⎠⎞
⎜⎝⎛=
+⎟⎠⎞
⎜⎝⎛
−⎟⎠⎞
⎜⎝⎛
== ∑
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
electrostatic repulsion
Born repulsion addition of acting potentials
addition of acting potentials including Born repulsion
potential
potential barrier
Stabilisation of disperse systems
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 −∞
⋅+=
+
Stabilisation of disperse systems
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
Electrochemical double layer
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
Electrochemical double layer
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(
potentialzeta)0x(,RT4
)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
potentialzeta)0x(,RT4
)0x(Fztanh
potentialzeta)0x(,RT4
)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