Corso CFMA. LS-SIMat 1
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Chimica Fisica dei Materiali AvanzatiChimica Fisica dei Materiali Avanzati
Part 4 – Forces between particles and surfacesPart 4 – Forces between particles and surfaces
Laurea specialistica in Scienza e Ingegneria dei MaterialiCurriculum Scienza dei Materiali
Corso CFMA. LS-SIMat 2
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Combining relationsCombining 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
where A and B are the appropriate molecular properties of the corresponding molecules (e.g., charge, dipole, polarizability, etc.; cf. Part 1, Slide 6).
molecules) unlike(for
molecules) like(for ; 22
ABW
BWAW
AB
BBAA
Corso CFMA. LS-SIMat 3
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Difference in energy between dispersed and associated clusters
In the general case
where n is the number of like bonds that have been formed upon association.
Since is always positive ( ) the associated state of identical molecules is energetically favored, i.e., there is always an effective attraction between like particles in a binary mixture.
Combining relations (cont’d)Combining relations (cont’d)
29 BAWWW dispass
structure) (c.p. neighborsnearest 12with
D-3in 22 2BAW
2BAnWWW dispass
2BA
dispass WW
Corso CFMA. LS-SIMat 4
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The association energy can be recast in different forms
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. .
Let’s assume n bonds per unit area: is the adhesion energy in 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 . Similarly, .
Combining these definitions with the above eqn., one gets
Surface and interfacial energySurface and interfacial energy
ABBBAA
dispass
WWWnABBAn
BAnWWW
2222
2
ABW 2ABnW
AAnW
AAAnW 2BBBnW 2
areaunit per ABBAAB nW
Corso CFMA. LS-SIMat 5
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Because of their definition, . Hence,
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
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.
Surface and interfacial energy (cont’d)Surface and interfacial energy (cont’d)
BBAAAB WWW
22 BABABAABBAAB nW
CBCAWWW dispass
Corso CFMA. LS-SIMat 6
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SummarySummary 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.
ProvisoProviso
There are two very important exceptions: Coulomb interactions between atomic or molecular ions: the sign
of 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
. Example: repulsive forces due to hydration of hydrophilic molecules.
W
ABWAB
Corso CFMA. LS-SIMat 7
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Long-range forces between macroscopic bodiesLong-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 (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
w
Corso CFMA. LS-SIMat 8
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Interaction potentials between macroscopic Interaction potentials between macroscopic bodiesbodies
Molecule-surface interaction
Assume an additive molecular pair potential:
By integration of the interactions between the isolated molecule and those contained in rings of volume , for the total interaction of a molecule at a distance D from the surface we get
.nrCrw
xdxdz2
.3for 32
2
2
22
3
02
222
nDnn
C
z
dz
n
C
xz
xdxdzCDw
n
z
Dz
x
xD nn
Assuming VdW forces with
i.e., an interaction potential with much longer range
than the original pair potential.
36
6
D
CDw
n
Corso CFMA. LS-SIMat 9
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Interaction potentials between macroscopic bodies Interaction potentials between macroscopic bodies (cont’d)(cont’d)
Sphere-surface interaction
The volume of a thin circular section of radius x within the sphere is
(use is made of the chord theorem).
Using the previous result for and integrating over a number of molecules
at a distance from the planar surface, we get
dzzzRdzx 2 2
Dw
dzzzR 2 zD
Rz
znzD
zdzzR
nn
CDW
2
03
22 2
32
2
For , only small values of z contribute to the integralRD
5
222
03
22
5432
42
32
2
n
Rz
zn Dnnnn
RC
zD
Rzdz
nn
CDW
Corso CFMA. LS-SIMat 10
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Interaction potentials … (cont’d)Interaction potentials … (cont’d)
For VdW forces with
The interaction energy is proportional to the radius of the sphere and decays as .
For we may replace in the denominator by D, and obtain
Since is the number of molecules in the sphere, the result is equivalent to that for the interaction of a molecule with a surface.
6n
D
RCDW
6
22
D1
RD zD
.32
34 22
32
23
32
03
22
n
Rz
zn Dnn
RC
D
zdzzR
nn
CDW
34 3R
Corso CFMA. LS-SIMat 11
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Interaction potentials between macroscopic bodies Interaction potentials between macroscopic bodies (cont’d)(cont’d)
Surface-surface interactionConsider 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 . Thus for the two surfaces
3322 nznndzC
4
2
3
2
432
2
32
2
n
z
Dzn Dnnn
C
z
dz
nn
CDW
For n = 6
When D is small compared to the lateral dimensions, this result holds to two unit areas of both surfaces.
areaunit per 12 2
2
D
CDW
Corso CFMA. LS-SIMat 12
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Non-retarded VdW interactions between macroscopic Non-retarded VdW interactions between macroscopic bodiesbodies Assume a pair potential (in
vacuum, additive and non-retarded)
The resulting interaction laws for some common geometries are given in terms of the conventional Hamaker constant
Typical values of Hamaker constants for condensed phases are in the range of 10-19 J.
.nrCrw
212 CA
Corso CFMA. LS-SIMat 13
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Van der Waals forces in condensed mediaVan 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
constant is
The first term accounts for the Keesom and Debye energies, the second for dispersion ( is an effective electronic excitation frequency)
The Hamaker constant of metals and metal oxides can be an order of magnitude higher
232
321
223
21
2
31
31
216
3
4
3
nn
nnhkTA e
e
Corso CFMA. LS-SIMat 14
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Surface TensionSurface Tension For any material, it costs energy to make a surface.
To make a small element dA, the work done is
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:
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
pTdAdW ,constant at
AddAAddW
Corso CFMA. LS-SIMat 15
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Typical Values of Surface Tension*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.
Corso CFMA. LS-SIMat 16
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Computed Values of Specific Surface EnergiesComputed 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.
Corso CFMA. LS-SIMat 17
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Curved Interfaces - Laplace PressureCurved Interfaces - Laplace Pressure
If we imagine a bubble of radius r, the work done to expand it is
but this is opposed by the creation of more surface, which would cost
At equilibrium dWV = dWS,hence
Pin
Pout
drrPPpdVdW outinV24
rdrdAdWS 8
rPP outin 2
With a little more work we can see that
The equilibrium pressure inside a curved surface, whether solid or liquid, rises as the sphere decreases in size.
rRTVPP moutin 2ln
Corso CFMA. LS-SIMat 18
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Young’s Equation - Contact Angle and AdhesionYoung’s Equation - Contact Angle and AdhesionConsider the work done in separating a solid phase and a liquid phase.By balancing the surface forces, Young showed (1805)
Measurement of the equilibrium contact angle allows one to measure
(You can measure the difference but not the absolute values.)
equation) (Young coslg sgsl
lgcos slsg
slsg
sglg
sl
solid
liquid
solid
liquid
gas
Consider the work done in separating a solid phase and a liquid phase
Combining Young’s and Dupre’s equations we get
In principle, one should work in vacuum for reference work.
lssgSW lg
eqn.) Dupré-(Young cos1lg SW
Corso CFMA. LS-SIMat 19
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NucleationNucleation
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
If the pressure of gas is greater than the equilibrium value, then growth is favorable. If it is less than , the liquid will evaporate.
However, for a curved droplet, the growing liquid must also create new surface as it forms
M(g) M(liq)
K = peq
M
M
MM
eqppnRTG ln
eqp
24ln rppnRTtotalG eq
Corso CFMA. LS-SIMat 20
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Nucleation (cont’d)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, hence
There is a balance point when
Also
mnVr 34 3
23 4ln34 rppVRTrtotalG eqm
0 drGd
eq
mcrit ppRT
VR
ln
2
3
4 2
max
critRG
Corso CFMA. LS-SIMat 21
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Nucleation (cont’d)Nucleation (cont’d)
0.0 0.4 0.8 1.2 1.6 2.0-1.0.10-18
-5.0.10-19
0.0.100
5.0.10-19
1.0.10-18
1.5.10-18
2.0.10-18
Radius (nm)
Fre
e E
nerg
y (J
)Free Energy of Nucleation vs Nucleus Radius (nm)
S=20
S=2
S=4
S=10
S=6
S=5
for Various Supersaturation Ratios assuming g=100mJ/m2
Corso CFMA. LS-SIMat 22
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Nucleation (cont’d)Nucleation (cont’d)
0 1 2-2.10-18
-1.10-18
-1.10-18
-5.10-19
0
5.10-19
1.10-18
Radius of Nucleus/nm
Fre
e E
nerg
y (J
)
Effect of Nucleation Temp on Free Energy of Nucleation
S=20, T=600
S=20,T=400
S=20, T=300
S=20, T=200
Corso CFMA. LS-SIMat 23
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Ostwald ripeningOstwald ripeningThe 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.