Transcript Chimica Fisica dei Materiali Avanzati Parte I

UNIVERSITA’ DEGLI STUDI DI PADOVA
Laurea specialistica in Scienza e Ingegneria dei Materiali
Curriculum Scienza dei Materiali
Chimica Fisica dei Materiali Avanzati
Part 4 – Forces between particles and surfaces
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Combining relations
 The fundamental forces involved in the interaction among
particles and between particles and surfaces are the same as
those already described for atoms and molecules
(electrostatic, VdW, solvation forces, etc.).
 However, they can manifest themselves in quite different ways
 There are also similarities expressed by certain
semiquantitative relations known as combining relations.
 We may express the binding energy of molecules A and B in
contact as
WAA   A2 ;
WBB   B 2
WAB   AB
(for like molecules)
(for unlike molecules)
where A and B are the appropriate molecular properties of the
corresponding molecules (e.g., charge, dipole, polarizability, etc.;
cf. Part 1, Slide 6).
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Combining relations (cont’d)
Difference in energy between
dispersed and associated clusters
W  Wass  Wdisp  9 A  B
2
W  22 A  B 
2
in 3 - D
with 12 nearest neighbors (c.p. structure)
In the general case
W  Wass  Wdisp  n A  B
2
where n is the number of like bonds
that have been formed upon
association.
Since
 A  B 2 is always positive
( Wass  Wdisp ) the associated state of
identical molecules is energetically
favored, i.e., there is always an
effective attraction between like
particles in a binary mixture.
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Surface and interfacial energy
 The association energy can be recast in different forms
W  Wass  Wdisp  n A  B 
2
 nA2  B 2  2 AB   nWAA  WBB  2WAB 
 Consider two flat macroscopic surfaces of A, each of unit area, in a
liquid of B.
 The above eqn. can be read as (negative) free energy change for bringing
the two surfaces into adhesive contact; equivalent to twice the interfacial
energy, i.e. W  2 .
AB
 Let’s assume n bonds per unit area: nW
is the adhesion energy in
AB
vacuum per unit area of the A-B interface

is the (negative) energy change for bringing unit area A into contact
with unit area A in vacuum, known as cohesion energy. The cohesion
energy is related to the (positive) surface energy or surface tension by
nWBB  2 B . Similarly, nWAA  2 A .
nWAA
 Combining these definitions with the above eqn., one gets
 AB   A   B  nWAB
per unit area
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Surface and interfacial energy (cont’d)
 Because of their definition, WAB  WAAWBB . Hence,
 AB   A   B  nWAB   A   B  2  A B    A   B 
2
i.e., the interfacial energy can be estimated solely from the surface
energies or surface tensions of the pure liquids in the absence of any
data on the energy of adhesion.
Particles or surfaces in a third medium
 By similar arguments, in a three-component mixture one gets
W  Wass  Wdisp   A  C B  C 
for the association of two unlike molecules A and B in a solvent
composed of molecules C.
 This result shows that the association energy can be positive or
negative. If positive, the particles effectively repel each other and
remain dispersed.
 This happens when C is intermediate between A and B.
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Summary
 There is always an effective attraction between like molecules or
particles in a multicomponent mixture.
 Unlike particles may attract or repel each other depending on the
properties of the medium.
Proviso
 There are two very important exceptions:
 Coulomb interactions between atomic or molecular ions: the sign
of W is reversed; the dispersed state is favored
 Hydrogen-bonding molecules: the strength of the H-bond between
different molecules cannot be expressed simply in terms of
WAB   AB .
 Example: repulsive forces due to hydration of hydrophilic molecules.
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Long-range forces between macroscopic bodies
 The properties of gases and the cohesive energies of
condensed phases are determined mainly by the interaction
energies of molecules at contact w  (Coulomb forces may be
an exception).
 For macroscopic bodies, when all the pair potentials between
the molecules in each body is summed, we find
 The net interaction energy is proportional to the size of the
particles
 It can be much larger than kBT even at separations of 100 nm or
more
 The energy and force decay much more slowly with the separation
 A variety of different behaviors may arise depending on the
specific form of the long-range distance dependence of the
interaction
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Interaction potentials between macroscopic bodies
Molecule-surface interaction
Assume an additive molecular pair potential:
wr    C r n .
By integration of the interactions between the
isolated molecule and those contained in rings
of volume 2xdxdz , for the total interaction of
a molecule at a distance D from the surface we
get
wD   2C
z 

zD

x 
dz

x 0
z
xdx
2
2C
n  2n  3D n 3
x
2

n

2
2C  dz
n  2 D z n  2
for n  3.
Assuming VdW forces with n  6
wD   
C
6 D3
i.e., an interaction potential with much longer range
than the original pair potential.
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Interaction potentials between macroscopic bodies (cont’d)
Sphere-surface interaction
The volume of a thin circular section of
radius x within the sphere is
 x 2dz   2R  z z dz (use is made of
the chord theorem).
Using the previous result for w D  and
integrating over a number of molecules
2 R  z z dz at a distance  D  z 
from the planar surface, we get
2 2C 2
W D   
n  2n  3
2 R  z zdz
n 3

z  0 D  z 
z 2R
For D  R , only small values of z contribute to the integral
2 2C 2
W D   
n  2n  3
z 2R

z 0
2 Rzdz
4 2C 2 R

n 3
n  2n  3n  4n  5D n 5
D  z 
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Interaction potentials … (cont’d)
For VdW forces with n  6
W D   
 2C 2 R
6D
The interaction energy is proportional to the radius of the sphere and decays
as 1 D .
For D  R we may replace  D  z  in the denominator by D, and obtain
2 2C 2
W D   
n  2n  3

2 R  z zdz   2 C 4R3  3 .
 D n 3
n  2n  3D n 3
z 0
z 2R

Since 4R3  3 is the number of molecules in the sphere, the result is
equivalent to that for the interaction of a molecule with a surface.
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Interaction potentials between macroscopic bodies (cont’d)
Surface-surface interaction
Consider the energy per unit surface area;
starting from a sheet of thickness dz and unit area
at a distance z from an extended surface of larger
area.
The interaction energy of this sheet with the
surface is  2C dz  n  2n  3z n 3. Thus for
the two surfaces
z 
2C 2
dz
2C 2
W D   

n  2n  3 z D z n 3 n  2n  3n  4Dn  4
For n = 6
W D   
C 2
12D2
per unit area
When D is small compared to the lateral dimensions, this result holds to
two unit areas of both surfaces.
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Non-retarded VdW interactions between macroscopic bodies
 Assume a pair potential (in
vacuum, additive and nonretarded) wr    C r n .
 The resulting interaction laws
for some common geometries
are given in terms of the
conventional Hamaker
constant
A   2C12
 Typical values of Hamaker
constants for condensed phases
are in the range of 10-19 J.
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Van der Waals forces in condensed media
 The definition just given for the Hamaker constant ignore the influence of
neighboring atoms on the interaction between any pair of atoms; but,
straightforward additivity breaks down in condensed media.
 Lifshitz theory avoids the problem of additivity: large bodies are treated
as continuous media characterized by bulk properties as the dielectric
constant and refractive index.
 As a result, although the expressions for the interaction energies remain
valid, the Hamaker constant is calculated in a different way.
 Example: for two identical phases 1 interacting across a medium 3 the Hamaker
2
constant is
2
2 2
3      3h e n1  n3 
A  kT  1 3  
4  1   3  16 2 n12  n32 3 2
 The first term accounts for the Keesom and Debye energies, the second for
dispersion (  is an effective electronic excitation frequency)
e
 The Hamaker constant of metals and metal oxides can be an order of magnitude
higher
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Surface Tension
 For any material, it costs energy to make a surface.
To make a small element dA, the work done is
dW  d A  dA  Ad
i.e. if we make new surface, we get the first term, but if we
stretch or change the surface as we create more, the second
term will contribute. There is strain created.
 For a simple one component system:
dW  dA


at constant T , p
For liquids, we measure  using a variety of techniques such as:
drop-weight, the Du Nouy ring method, static drop, capillary rise.
For solids:
crystal cleavage or heat of solution
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Typical Values of Surface Tension*
Material
Surface Energy (erg/cm2)
Temperature (oC)
W
2900
1727
Au (s)
1410
1027
Ag (s)
1140
907
H2O (liq)
72.7
20
Ag (liq)
879
1100
Fe (s)
2150
1400
Fe (liq)
1880
1535
NaCl (s)
227
25
KCl (s)
110
25
MgO (s)
1200
25
Hg (liq)
487
16.5
He (liq)
0.308
-270.5
* Somorjai, Principles of Surface Chemistry, 1972.
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Computed Values of Specific Surface Energies*
Material
Es(erg/cm2)
Es(0)erg/cm2
Difference
 Es
Crystal Face
Ne
19.7
19.76
-0.06
111
Ne
20.34
20.52
-0.18
100
NaCl
158
210.9
-52.7
100
NaCl
354
469.7
-115.6
110
NaF
216
265.9
-49.5
100
Ag
2537
2560
-23
111
* Somorjai, Principles of Surface Chemistry, 1972.
Es (erg/cm2) = Es(0) + Es
Here Es(0) is the specific surface energy of the rigid lattice and Es
is the relaxation energy of opposite sign.
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Curved Interfaces - Laplace Pressure
If we imagine a bubble of radius r, the
work done to expand it is

dWV   pdV   Pin  Pout 4r 2dr

but this is opposed by the creation of
more surface, which would cost
dWS  dA   8rdr
Pin
At equilibrium dWV =  dWS,hence
Pout
Pin  Pout  2 r
With a little more work we can see that
ln Pin Pout   2Vm rRT
The equilibrium pressure inside a curved surface, whether
solid or liquid, rises as the sphere decreases in size.
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Young’s Equation - Contact Angle and Adhesion
Consider the work done in separating a solid
phase and a liquid phase.
By balancing the surface forces, Young showed
(1805)
 lg cos   sl   sg
lg
sg

(Young equation)
Measurement of the equilibrium contact angle
allows one to measure
sl
cos   sg   sl   lg
(You can measure the difference
the absolute values.)
 sg   sl  but not
Consider the work done in separating a solid phase
and a liquid phase
solid
WS   lg   sg   ls
Combining Young’s and Dupre’s equations we get
WS   lg 1  cos  (Young - Dupré eqn.)
In principle, one should work in vacuum for reference
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work.
solid
gas
liquid
liquid
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Nucleation
Consider the simplest case where n moles
of gas M are transferred to the liquid
phase at constant temperature.
The work of isothermal compression is
G  nRT ln  p peq 
M(g) F M(liq)
K = peq
If the pressure of gas is greater than the
equilibrium value, then growth is
favorable. If it is less than peq , the liquid
will evaporate.
However, for a curved droplet, the
growing liquid must also create new
surface as it forms
M
M
M
M
Gtotal  nRT ln  p peq   4r 2
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Nucleation (cont’d)
 If Vm = MW/ is the molar volume of M in the liquid phase, then
n moles will cause a volume increase of, 4r 3 3  nVm hence
Gtotal  4r 3RT 3Vm ln  p peq   4r 2
 There is a balance point when d G  dr  0
Rcrit 
2Vm
RT ln  p peq 
 Also
Gmax
2
4Rcrit


3
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Nucleation (cont’d)
2.0.10-18
Free Energy of Nucleation vs Nucleus Radius (nm)
2
f or Various Supersaturation Ratios assuming g=100mJ/m
S=2
1.5.10-18
S=4
Free Energy (J)
S=5
1.0.10-18
S=6
S=10
S=20
5.0.10-19
0.0.100
-5.0.10-19
-1.0.10-18
0.0
0.4
0.8
1.2
Radius (nm)
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1.6
2.0
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Nucleation (cont’d)
Eff ect of Nucleation Temp on Free Energy of Nucleation
1.10-18
Free Energy (J)
5.10-19
0
-5.10-19
-1.10-18
S=20, T=600
S=20,T=400
-1.10-18
S=20, T=300
S=20, T=200
-2.10-18
0
1
Radius of Nucleus/nm
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Ostwald ripening
The growth of large particles at the expense of smaller ones,
owing to a difference in solubility rates of different size particles
 There is only one critical radius for any given set of S, T, p,.
Hence all crystallites in an ensemble are either above or below
Rcrit. If they are above, they will grow, if they are below, they will
evaporate.
 Ostwald ripening starts once nucleation is complete and near the
end of the growth phase as monomer decreases.
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