U NIVERSITA DEGLI S TUDI DI P ADOVA Corso CFMA. LS-SIMat1 Chimica Fisica dei Materiali Avanzati Part...

Corso CFMA. LS-SIMat 1

UUNIVERSITA’ DEGLI NIVERSITA’ DEGLI SSTUDI DI TUDI DI PPADOVAADOVAUUNIVERSITA’ DEGLI NIVERSITA’ DEGLI SSTUDI DI TUDI DI PPADOVAADOVA

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



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



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



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



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



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



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



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



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



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



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



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



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



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



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.



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.



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



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



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



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




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




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



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.

U NIVERSITA DEGLI S TUDI DI P ADOVA Corso CFMA. LS-SIMat1 Chimica Fisica dei Materiali Avanzati Part...

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