w, mN m-1

1. Introduction

Many physical and chemical phenomena occur at the surface of contacting phases, which is called ‘interfacial surface’ or the ‘phases border’. This

interfacial border, or rather layer, is of 0.5 to several nanometers thickness.

The phenomena taking place in this region are called ‘interfacial phenomena’.

The subject of physical chemistry of interfaces deals with physico- chemical properties of the phenomena and their kinetics, structure of the interfaces, and generally, the processes taking place in this region.

The interfacial phenomena occur commonly and they are of different nature. They are important in many scientific disciplines, like physics, chemistry, biology, pharmacy, medicine, agriculture, ecology, as well as cosmetic and food industry. Many physical, chemical and biological

processes, like the light dispersion, adsorption, photoelectric effect,

heterogenic chemical catalysis, as well the processes in living organisms take

place just at the interfaces.

Moreover, many industrial processes and those occurring in the nature are the interfacial ones. They may involve one or more interfaces simultaneously.

Dr. Kash L. Mittal, who is Doctor Honoris Causa of Maria Curie-Skłodowska University at Lublin, has alphabetically listed the examples of processes

and/or products where the interfacial phenomena occur and play an important role. They are shown in Table 1.1

Interfacial processes or products

Polish translation

A: abhesion, adhesion, adsorption, aerosols,

agriculture, anesthesiology, audiotapes, autophobicity

zapobieganie sklejaniu, adhezja, adsorpcja, aerozoele, rolnictwo, anastezjologia, taśmy grające, autofobowość

B: bioadhesion, bifouling, biomaterials, biomembranes, biotechnology, blood clotting

bioadhezja, biomateriały, biomembrany, biotechnologia, krzepnięcie krwi

Table 1.

C: capillarity, catalysis, chromatography, clearing, coagulation, coalescence,

coatings, colloids, composites, corrosion, cosmetics

włoskowatość, kataliza, chromatografia, czyszczenie, koagulacja, koalescencja, pokrywanie (powlekanie), koloidy,

kompozyty, korozja, kosmetyki D: decontamination, detergency,

dewetting, dispersions, drug delivery

usuwanie zanieczyszczeń, detergencja, suszenie, dyspersje, podawanie leków w organizmie

E: electrocapillarity, electric compo- nents, elecrophoresis, electrowetting, emulsions

elektrokapilarność, części elektroniczne, elekroforeza, elektrozwilŜanie, emulsje

F: flotation, fluidics, foams, foods, fouling, friction

flotacja, przepływy cieczy, piany, Ŝywność, zanieczyszczenia powierzchni, tarcie

G: gels, glass fiber composites, glazing, glueing, grinding

Ŝele, składniki szklanych włókien, pokrycia transparentne, klejenie, mielenie

H: haemodialysis, hard disks, heterocoagulation, hydroplaning, hydrosols, higienie products

hemodializa, twarde dyski,

heterokoagulacja, warstewka wody pod kołami, hydrozole, środki do higieny I: information storage devices, ink jet,

integrated circuits fabrication

urządzenia magazynujące informacje,

drukarki atramentowe, produkcja układów zintegrowanych

J: jelly, jet breakup, joggling, joining,

junctions (semiconductors) galarety, dezintegracja strumienia, wytrząsanie, spawanie, połączenia (półprzewodniki)

K: karyology, kinesiology badania powstawania komórek, badania mechanizmu i anatomii w odniesieniu do poruszania się ludzi

L: LB films, liposomes, liquid crystals, lithography, lubrication

filmy Langmuira-Blodgett, liposomy, ciekłe kryształy, litografia, smarowanie M: Marangoni effect, marine products,

membranes, molecular beam epitaxy

efekt Marangoni’ego, produkty morskie, membrany, molekularne wiązki

epitaksacyjne (krystalografia) N: nanotechnology, nanodevices,

nanomaterials, nanacomposites, NEMS (nanoelectromechanical systems)

nanotechnologia, nanourządzenia, nanomateriały, nanokompozyty,

nanoelektromechaniczne układy (NEMS) O: oil recovery, oil spills, olfaction,

ostwald ripening wydobywanie olejów, rozlewanie olejów, sensory powonienia, starzenie Ostwalda P: painting, paper sizing, paving,

pharmaceuticals, photography, photplitogtaphy, polymer blends, printing, prothetics

malowanie, wykończanie papieru,

brukowanie, przemysł farmaceutyczny, fotografia, fotolitografia, mieszanki polimerów, drukowanie, protetyka

Q: quarrying, quenching, quicksand wydobywanie (kopalnie odkrywkowe), gaszenie, kurzawka

PHYSICOCHEMICAL PHENOMENA AT PHYSICOCHEMICAL PHENOMENA AT

INTERFACES INTERFACES

Lectures for students of Material Chemistry at Faculty of Chemistry

UMCS, Lublin

Emil Chibowski and Lucyna Hołysz

Department of Physical Chemistry, Faculty of Chemistry,

Maria Curie-Skłodowska University, Lublin.

R: reinforced composites, rusting wzmacniane kompozyty, rdzewienie S: self-assemble monolayers,

semiconductors devices, sintering, soap bubbles, sol-gels, sols

samoorganizujące się warstewki,

urządzenia półprzewodnikowe, spiekanie, bańki mydlane, zol-zel, zole

T: tarnishing, tertiary oil recovery, thin films devices, tribology

wytwarzanie warstewki na powierzchni metalu, ulepszone uzyskiwanie olejów (ropy), cienkie filmy, nauka o względnym ruchu dwu powierzchni

U: ultrasonic welding, urology ultradźwiękowe spawanie, urologia V: vehicle skidding, veneer laminatem,

videography, videotapes

pojazdy ślizgowe, laminaty fornirowe, wideografia, taśmy wideo

W: washing, waterproofing, wear,

weathering, welding, wettability, wicking, wine tears, wound healing

pranie, impregnowanie (od wody), ubranie, wietrzenie skał, spawanie, zwilŜanie, łzy wina, gojenie ran

X: xerography kserografia

Y: yeast, yellowing, yogurt droŜdze, proces Ŝółknięcia, jogurt

Z: zygology nauka o łączeniu i przyczepianiu

A few examples of interfacial phenomena important for practical application and in human life.

– Adsorption, i.e. accumulation of molecules at the solid/gas or liquid/gas interface, is important for separation of gaseous or liquid mixtures, their purification or liberation.

– Enrichment of mineral ores, containing only a few percent of useful mineral, by the flotation method. It is based on selective hydrophobization of the mineral grains surface, which then easily attach to gas bubbles and are carried out to the flotation pulp surface, where they are collected. For this purpose suitable chemical compounds are used, which are called collectors, as well as frothers, that form mineralized froth on the top of the pulp.

– Washing process consists in removing oily and mineral dusts from the cloth fibers surface. This is a wetting process and to enhance it there are used surface active

substances, which are here called detergents. They decrease the surface tension of water, which reduces adhesion of the dust particles to the fibers and facilitates their removal to the aqueous phase.

– Stability of various suspensions, e.g. pharmaceuticals and some food products,

depends upon interactions at the solid/liquid and liquid/liquid interfaces. Natural tendency of such systems is their aggregation and sedimentation of the particles or oily droplets.

Between the moieties there are dispersion and electrostatic interactions, which can be regulated by the addition of surface active substances, changes in pH and ionic strength.

2. Capillary phenomena

In this chapter the phenomena occurring on a condensed phase surface will be discussed.

In fact, we always deal with an interfacial surface, i.e. the surface dividing two contacting phases, and therefore properties of this surface (or rather thin interfacial layer) depends on the properties of both phases being in contact.

Depending on the kind of two contacting phases the following interfacial systems can be distinguished:

−liquid - gas (commonly termed as liquid surface)

−liquid - liquid

−liquid - solid

−solid – gas (commonly termed as solid surface)

−solid – solid.

The condition for an interface to exist is increase in Gibbs free energy (free enthalpy ∆∆∆G)∆ during the interface formation. If during formation of such interface the energy would decrease, then in accordance with general phenomenological thermodynamics rules, after a suitable time period the two phases could mutually disperse up to their molecular sizes (sometimes ions), which just takes place in the case of two gases or miscible

liquids.

In the case of interfaces which are mobile and can immediately change their shape and reach equilibrium surface, then the phenomena occurring at such interfaces are called capillary phenomena.

First of all they involve liquids, which at the solid/liquid and liquid/gas interfaces form meniscus or a droplet, as well as thin layers (films) of immiscible liquid formed on the other liquid.

This general term capillary phenomena involves the phenomena considered in the

macroscopic scale occurring at the interfaces and less deals with their molecular level, like for example structure.

3. Molecular description of liquid surface

Intermolecular interactions

In all phases between molecules or atoms (if a phase consists of atoms) there interact intermolecular forces, generally called Wan der Waals forces, which involve London dispersion forces, Keesom orientation forces and Debye induction forces. Moreover,

between some atoms there can appear hydrogen bonding, or more generally the electron- donor and electron-acceptor forces, which are also called Lewis acid-base interactions. All these forces acting between the same molecules are called cohesive forces.

London dispersion forces

(sometimes known as quantum induced instantaneous polarization forces) – named after the German-American physicist Fritz London, they are weak intermolecular

forces that are formed between temporary multipoles in molecules without permanent multipole moments. They are of induced dipole–dipole nature. London dispersion forces are formed between all kinds of molecules, thus also nonpolar ones because electron density fluctuates around a molecule probabilistically (see quantum

mechanical theory of dispersion forces). In consequence, the electron density will not be evenly distributed throughout a nonpolar molecule, which results in a temporary multipole. This multipole will interact with other near multipoles and induce similar temporary polarity in near molecules. Fig.3.1 illustrates schematically the interaction of multipoles. In the case of polar molecules London forces are also present, but they constitute only a small fraction of the total interaction force.

Fig.3.1. Schematic representation of two multipoles interaction.

Fritz London (1900-1954)

• Dispersion Forces are also known as

London forces and van der Waals forces.

• They were named the London forces in

honor of the German physicist; Fritz

London who studied

these forces!!

• Unlike the name London

forces, van der Waals forces refers to the collective

grouping of the weakest

attractions between molecules.

• This includes dispersion forces as well as dipole interactions.

• The name van der Waals

forces is a credit to the Dutch chemist and physicist

Johannes van der Waals who extensively studied

intermolecular forces.

Johannes van Der Waals

(1837-1923)

Although the life time of the multipoles is very short, it is enough to orient themselves because of repeating orientation. As a result between the atoms there exist attractive forces which are commonly called dispersion forces, or London forces. They are inversely proportional to 6th power of the distance and they act up to 0.5 nm. Owing to these forces the noble gases can be liquefied at sufficiently low temperatures.

Keesom interactions

(after Wilhelm Hendrik Keesom) are those of permanent dipole – dipole interactions of two molecules. The permanent dipole forms in a covalently bonded molecule in which the atoms possess significantly different electronegativity. More electronegative atom attracts the electrons toward itself thus becoming slightly negative and thus the other one becomes slightly positive. This creates an electrostatic force between the opposite charges and an alignment of the molecules (to increase the attraction) and thus reduces their potential energy. The dipole-dipole interaction hardly ever appears between two atoms, because no dipole moment forms. The dipole-dipole interactions depend on the temperature, because thermal movement of the molecules affects their ordering.

Hydrogen chloride is an example of dipole-dipole interaction:

δδδ

δ+ δδδδ– δδδ+ δδ δδδ–

H  Cl ---H   Cl 

Fig.3.2. Scheme of dipole – dipole

(interactionshttp://www.docbrown.info/page07/equilibria8a.htm)

Note, this is not a hydrogen bond because the electronegativity of Cl atom is not sufficient to create it. It may also happen that there are dipole moments within a molecule but it does not have any net dipole moment. This occurs in the case of a symmetric molecule where the moments cancel out each other, for example in the tetrachloro-methane molecule.

Debye induction forces

appear when the permanently polar bond in one molecule can induce a dipole in a neighbouring molecule, no matter whether the other molecule is polar or non-polar.

attractions

Fig.3.3. Illustration of Debye induction forces.

http://www.docbrown.info/page07/equilib ria8a.htm

The partial charge of the polar molecule causes distortion of the electron distribution of the other molecule. Hence this molecule acquires the regions of partial positive and

negative charge and thus it becomes polar. Such induced dipole interacts with the polar molecule that originally induced it and the two molecules cohere.

—O—H----O— —N—H----N—

Hydrogen bond

The attractive interaction between the hydrogen atom and the electronegative atom like that of nitrogen N, oxygen O or fluorine F is called hydrogen bond and this interaction should not be confused with the covalent bond. The hydrogen atom that creates hydrogen bond must be covalently bonded to another electronegative atom. The hydrogen atom is a hydrogen bond donor. The electronegative atom: fluorine, oxygen, or nitrogen is a

hydrogen bond acceptor, regardless whether it is bonded to a hydrogen atom or not.

The hydrogen bonds may appear even within different parts of the same molecule (intramolecular – proteins, nucleic acids, some polymers) or between molecules (intermolecules –water, hence high boiling point 100^oC.

Their strength lies between 5 and 30 kJ/mole and they are stronger than van der Waals interactions but weaker than covalent or ionic bonds.

The length of hydrogen bonds depends on bond strength, temperature, and pressure.

As hydrogen bonds are partially of covalent nature the distances between X −−−− H···Y indicate which atom is the 'donor' and which ‘acceptor’, and to which molecule or atom the hydrogen nucleus belongs. The distance X −−−− H is ~110 pm (10^-12 m) and H^...Y

distance is ~160 to 200 pm.

Examples of hydrogen bond donating (donors) and hydrogen bond accepting groups (acceptors)

http://en.wikipedia.org/wiki/Hydrogen_bond#Bonding

www.lbl.gov/MicroWorlds/Kevlar/KevlarClue4.html

Fig.3.4. Models of hydrogen bonds between molecules of water

Tethraedric arrangement of hydrogen atoms around oxygen atom in water.

Because of the hydrogen bonds, water forms associates, where each oxygen atom is surrounded by 4 hydrogen atoms (see figure above), and around each water molecule creates the tetrahedral system of water molecules. Hydrogen bonds are formed both in liquid water and ice, also the bonds are responsible for extraordinary properties of water, as for example increasing specific volume of water during its freezing.

Relative strength of forces

Bond type Dissociation

energy (kJ/mol)

Covalent 1675

Hydrogen bonds 50 -67

Dipole-dipole 8.4 – 2.1

London (Van der Waals) Forces < 4

http://en.wikipedia.org/wiki/Intermolecular_force

4. Interactions at the liquid/gas interface

In bulk of a liquid each molecule is surrounded by alike molecules on all its sides.

However, this is not the case at the liquid surface. The forces acting on the molecules being in the surface that originate from the molecules in the bulk are not compensated from the gas phase. However, between the molecules in the surface the same cohesive forces interact and they are responsible for the phenomenon termed 'surface tension'.

The surface molecules are more strongly attracted to their neighbours. The resulted net force is directed normally to the surface towards the bulk phase. Therefore a small

volume of a liquid tends to spherical shape to minimize its surface. This is shown schematically in figure 4.1.

Fig.4.1. Scheme of unbalanced forces at a liquid surface From [J. Ościk, Adsorpcja, PWN, Warszawa, p. 18]

In fact, the state of molecules on the liquid surface should be considered from kinetics point of view, which is a very turbulent one. The surface molecules both move inwards the bulk phase and evaporate to the gas phase, and reverse. The average life time of a

molecule on the surface is about 10^-6 s, thus on 1 cm² of water surface there take place about 1.2 ××××10²² 'arrivals and departures’ of the molecules, which go down to about 100 Å (10 nm) into the bulk phase. Therefore in the surface region their density changes from that characteristic of the liquid to gas phases.

Whereas in the case of a newly created liquid surface, the equilibrium establishes

within milliseconds, for a solid it is not the case. Here the process of formation of a new surface can be considered as occurring in two stages:

- splitting of a piece of solid into two parts

-reorientation and re-location of the surface atoms (molecules) toward the equilibrium

positions, and this stage lasts for a long time and real surfaces mostly are not in equilibrium.

5. Surface tension of liquid

Surface tension is defined as the force acting along a line of unit length parallel to the surface but perpendicular to the line needed to enlarge the liquid surface by a unit. The force is denoted by symbol γγγγ or sometimes by σσσσ. It can be depicted schematically as follows.

liquid surface Shift of the barrier

Fig.5.1. Illustration of the surface tension appearance.

It pictures a flat soap film bound on one side by a taut thread of length, l. The thread is pulled toward the interior of the film by a force equal to γ⋅γ⋅γ⋅lγ⋅ on each side of the film. Hence the surface tension is

measured as forces perunit length. Its unit is newton per metre, N/m.

However, enlarging the film surface by dx, the work W per unit area is done:

dx l

W = γ ⋅ ⋅

The relation between the surface tension and surface free energy can be visualized as on the picture in Fig.5.2, as a mechanical analogy to the surface tension.

Fig.5.2. Illustration of the relation between surface

tension and surface free energy F ( γγγγ )

W

Thermodynamically surface free energy is defined as follows:

[ J/m² ]

 ≡ γ



 





∂

∂

n , V ,

A

F

A visual sign of the surface tension presence is the fact that a free droplet of liquid naturally assumes a spherical shape because such mechanical systems tend to reach a state of minimum potential energy, that is the minimum surface area for a given

volume – which is the case for a sphere.

Soap bubble

The soap bubble is another example of surface tension presence, and this case is better to consider as the surface free energy.

Fig.5.3. The soap bubble of radius r (from A.W.

Adamson, Physical Chemistry of Surfaces).

For the reasons given earlier its shape is spherical.

Its total surface free energy equals:

γ π ²

4 r

Shrinking of the bubble decreases its surface energy.

It must be equilibrated by the pressure difference inside and outside the bubble ∆∆∆∆p.

Work done to counteract the pressure difference equals:

dr r 4 P

W = ∆ π

(3)

And it is balanced by decrease in the surface free energy of the bubble:

dr r 8

dr P r

4 π

∆ = π γ

Hence: (6)

r P 2 γ

=

∆

It follows from Eq.6 that the pressure inside the bubble increases with decreasing its radius.

Equation 6 is a special case of more general equation of Young-Laplace, which was derived independently by Young in 1805 and Laplace in1806, for any curved surface. It will be derived later. The Young-Laplace equation can be proved by simple experiment schematically shown in Fig.5.4.

R₁ < R₂

Fig.5.3. Illustration of Young-Laplace equation [A.W. Adamson, p. 6]

p₁ > p₂

A smaller bubble shrinks until the radii of both bubbles equal, it means that mechanical

equilibrium occurs.

6. Two fundamental equations in physical chemistry of interfaces

Capillary pressure – Young-Laplace equation

Let us consider a liquid/vapor system denoted as phases αααα and ββββ, respectively.

There is no sharp border between these two phases. It is smooth transition from phase to phase with changing the density.

This is shown schematically in Fig.6.1.

distance from the surface in normal direction arbitrary chousen dividing surface

phase phase

Fig.6.1. Variation in the density changes with the distance normal to the surface from phases αααα to ββββ.

For the thermodynamic description of the system it is helpful to introduce

hypothetical dividing surface in a proper place (see figure). Then real magnitudes of the functions and parameters for the whole system will differ from their sum for the phases α and β by an excess or deficiency amount. Therefore, to describe the system one has to introduce the magnitudes of U, S, n denoted with the symbol σσσσ, dealing with the surface quantities:

The system volume: V = V^α + V^β

Internal energy: U = U^α + U^β + U^σ Entropy: S = S^α + S^β + S^σ Total amount of the moles: n_i = n_i^α + n_i^β + n_i^σ

For any curved surface the complete differential of the internal energy U is written:

∑ ^{µ − − + γ + +}

+

=

i

2 2 1 1

i

i

dn P dV P dV dA C dc C dc TdS

dU

Where C₁ and C₂ are the constants, and c₁ and c₂ are the reciprocal of the two radii of curvature, i.e. c₁ = 1/r₁; c₂ = 1/r₂, µµµµ_i is the chemical potential and the other symbols have their common meaning.

The last two terms can be expressed in a different way, thus leading to Eq.2

( _{1 2} ) ( _{1 2} ) ( _{1 2} ) ( _{1 2} )

i

i i

c c

d C 2 C

c 1 c

d C 2 C

1

dA dV

P dV

P dn

TdS dU

−

− +

+ +

+ γ

+

−

− µ

+

= ∑ ^{α α β β}

(2)

These two terms and plus γγγγdA term describe the effect of curvature and area variation of the surface. However, this effect has to be independent of the dividing surface

location (Fig.6.1), and therefore we can choose such a place that C₁ + C₂ = 0, which is called the surface of tension.

This position of the dividing surface means that a number of moles in the phases αααα and β

β β

β, (n^αααα + n^ββββ), if calculated on the assumption that these two phases continue up to the dividing surface, will be different from the actual number of moles present in the

system. It means that there is an excess (positive or negative) of the moles, even in the case of liquid/vapor system. Such position is convenient from a mathematical point of view, but somewhat difficult to imagine physically. Other possibility is to place the surface in such a place that there is no excess.

There are two special cases where d(c₁ – c₂) = 0, i.e. where the curvature is small compared to the surface region thickness:

1) sphere case - the two radii of curvature are the same

2) plane surface – the both radii of curvature are infinitely large, r₁ = r₂

∞

∑ ^{µ − + + γ}

+

=

i

i

i

dn P dV P V dA

TdS

dU

Because: G = H – TS 4)

H = U + PV (5)

G = U –TS + PV (6)

Where: G is the free enthalpy (Gibbs free energy), H is enthalpy, hence:

(7) And complete differential of G amounts:

G = U –TS + P^ααααV^αααα+ P^ββββV^ββββ

β β

α α

β β

α

α + + +

+

−

−

= dU TdS SdT P dV P dV V dP V dP dG

Inserting dU from Eq. 3 one obtains:

∑ ^{µ + + + γ}

+

−

= ^{α α β β}

i

i

i dn V dP V dP dA

SdT dG

(8)

(9)

At equilibrium the energy of the system must reach a minimum at given S and n_i

Hence: (10)

And because: (11)

0 dA dV

P dV

P − + γ =

− ^{α α β β}

β α +

=

= 0 dV dV

dV

And P > 0 (cannot be zero), therefore dV = 0, and it means that:

dA dV

P dV

P czyli dV

dV ^β = − ^α − ^{α α} + ^{β β} = − γ

( ^{P − P} ) ^{dV = − γ dA}

− ^{α β α}

( ^{P α − P β} ) ^{dV α = γ dA}

Thus: (13)

(14) (12)

Equation 4.16 can be interpreted as describing the case of two bulk phases (here ααα and α β

β

ββ) separated by a membrane exposing tension γγγγ.

Displacing the surface of tension by a distance dt then the change in the surface area amounts:

( ^{c c} ) ^Adt

dA = ₁ + ₂

β

α dV

dV

Adt = = −

but

Therefore:

( ^{P α} − ^{P β} ) ^Adt = γ ^{( c} 1 + ^c 2 ^{) Adt}

(15) (16)

(17)

( ^c ₁ ^c ₂ )

P = γ +

∆

Or:

 



 

 +

γ

=

∆

2

1

R

1 R

P 1

Equation (19) is the Young–Laplace equation and it is one of the two basic equations in physical chemistry of interfaces.

For any surface there can be also derived the other relationship:

( ) ( )

∫ ∫

−

− +

−

=

0

0 a

a

dx p P

dx p

P

γ

Where: x is the distance normal to the surface along, -a and a are the points lying in the phases α and β, respectively. In the case of the plane surface P^α = P^β and therefore:

( )

∫

−

−

=

a

a

dx p

γ P

Where: P is the pressure in bulk phases and p is the local pressure which varies across the interface. Equation (21) describes the change of the liquid surface tension as a

function of local pressure occurring across the interface.

7. Surface tension and vapor pressure above the curved surface – Kelvin equation

Kelvin equation can be derived from the Young – Laplace equation. It desribes vapour pressure above the curved surface of the liquid.

At T = const. molar free enthalpy (Gibbs surface free energy) can be describe by change in mechanical pressure:

∫

=

∆ G V dP

Assuming constant molar volume (because T = constant):

P V G = ∆

∆

V P ∆ G

=

∆

(1)

(2)







 +

γ

=

∆

2

1 R

1 R

P 1 Then using the Young-Laplace equation this results:











 + + + + γγγγ

=

=

=

∆ =

∆

∆ ∆

2 1

1 1

R R

V G

(3)





 



 + + + + γγγγ

= =

= =

∆ ∆

∆ ∆

2 1

1 1

R V R

G

Assuming that the vapor behaves as an ideal gas, Gibbs free energy (free enthalpy) is:

P ln RT G

G =

+

∆

P ln d RT dG =

And differential:

(5) (6)

Then:

^{∆ =} ∫

P₀

P ln d RT

G

P

ln P RT G =

(8) ∆ And using Eq.25 one obtains:











 + + + + γγγγ

=

=

=

=

2 1

1 1

R V R

P ln P RT

o

(9)

Where: P

is the vapour pressure above the flat surface, P is the pressure

above the curved surface.

If the surface is spherical, R

= R

= R then:

R V P

ln P

RT

γγγγ

=

= =

= 2

0

(10) The Kelvin equation is the second fundamental equation in surface chemistry.

For a liquid sphere (droplet) the radius R is positive and it means that at a given temperature the vapour pressure is larger than that over a flat surface.

Kelvin equation probably holds up to R ≈ 30Å.

For water:

R

P / P

10

cm 1.001

10

cm 1.011

10

cm = 100Å = 10 nm

≈33 molecules

1.114

(11.4 %)

8. Theoretical calculation of surface free energy

Statistical mechanical approach allows calculation of probability of finding a molecule at a distance r from a given one applying the radial distribution function.

∫

∞

ε π ρ

= γ

0

4 2

8 g ( r )

( r ) r dr

Where: ρ is an average number of the molecules in a unit volume, and ε′(r) = dε(r)/dr is described with the help of 6-12 Lennard –Jones potential.

 





 



 



 



−  σ

 

 

 ε  σ

= ε

−

12 6

4 r r

) r

(

here: εo – the potential energy in the minimum, and σ – the effective diameter of the molecule^.

Fig.8.1. Interaction energy between two molecules as a function of separation

The results of molecular dynamic computation and Monte Carlo simulation of density profile and pressure difference (P – p) across the interface for argonlike/vapour (Walton, 1983) are shown in Figs.8.2 and 8.3, respectively.

z (Å) z (Å)

Fig.8.2. Density profile Fig.8.3. Pressure difference

across the interface across the interface

s

m m AU

U

U = ⋅ +

s

m

U

m U A

m

U = +

9. Surface energy and interface tension

Summing up hiterto knowledge about the surface region it can be said:

1. Atoms or molecules in the surface region are in different energetic state from those in the bulk phase – this excess energy is called surface energy.

This causes many different specific properties of this region from those of the bulk phase.

2. The total internal energy of a phase can be represented by a sum of two terms:

- internal energy of a unit mass U

- energy of a unit surface U

(1) Where: A – the total surface area, m is the total mass

Then: (2)

where: A/m is the surface of unit mass i.e. specific surface.

If the specific surface is small the second term in the above equation can be neglected.

However, for the system with expanded surface (powdered porous materials, adsorbents), the specific surface is large and the surface (interfacial) energy is significant and affects the properties of the system.

Example: 1 g of a solid of cubic shape having 1 cm edge and 1 g/cm³ density.

1 cm³ – 1 g – 6 cm² surface.

Volume : surface = 6 : 1

However, if it is grounded down and its structure is porous the specific surface can reach a value over 1000 m²/g.

10. Thermodynamic characterization of liquid surface (one-component system)

Let us consider hypothetical system shown in Fig.9.1.

Fig.10.1. Hyphothetical thermodynamic experiment

The liquid fills the box with cover that can be easily slid. It is made of such a material that there are no interfacial interaction, that is the interfacial tension liquid/cover is zero. If the cover is slid in the direction shown by the arrow, the surface dA is formed and it needs work γγγγdA. If the process is conducted at T, p = const. in reversible way, the Gibbs free energy change (increase) amounts:

dG = γγγγdA (1)

This is because the number of moles has not changed and the only work done is reversible mechanical one. Therefore for the whole newly created surface area the Gibbs free energy change (free entalphy) equals:

∆∆∆G∆ ^s = γγγγ⋅⋅⋅⋅A (2) And the surface tension γγγγ can be expressed as follows:

n , p , T s

A

g G



 





∂

= ∂

=

γ (3)

As this process is reversible (done quasi-statically) the heat associated with this surface creation gives the surface entropy S^s

^el

dS

T

Q =

Q

= TdS

= Ts

dA

Where s^s is the surface entropy of a unit areas (e.g. cm²) Because at p = const

T S G

n , p

−

 =



 





∂

∂

(6)

therefore

p s

p s

s T T

g 



 





∂ γ

= ∂

−

 =













∂

∂ (7)

And for the total surface area the entropy of its formation amounts:

p s

A T

S 



 





∂ γ

− ∂

=

From the dependences between the thermodynamic functions it is known that:

TS

G

H = +

Hence

H^s =G^s +TS^s ₍₁₀₎

And for a unit surface area

p s

T T

h 



 





∂

− ∂

= γ

γ

Or for the total one 









 



 





∂

− ∂

=

p s

T T A

H γ

γ ₍₁₂₎

The enthalpy change accompanying formation of a new surface of ∆∆∆∆A area at p,T = const amounts:

 





 



 



 





∂

− ∂

=

p s

T T A

H γ

γ

∆

∆

The change of enthalpy ∆H^s is usually greater than that of Gibbs free energy (free enthalpy) because of the entropy term (Eq.13).

( )

s

s

U pV

H = +

s

s

U

H ≈

(14) (15)

0

G^s +TS^s =F^s +TS^s (16)

s s

G

F ≈

Therefore numerically free enthalpy is practically equal to free energy, and this latter term is used to describe the energy changes.

The units of the specific free energy and surface tension are:

A A G

G

s

s====γγγγ ⇒⇒⇒⇒γγγγ==== ₍₁₈₎

G^s →→→→ [J/m²] ; [mJ/m²]

γγγγ →→→→ [N/m¹] ; [mN/m¹] and J/m² = N·m/m² →→→→ N/m.

As mentioned earlier, in the case of pure liquid surface free energy and surface tension are equivalent, but this is not the case for a solid surface. This is because of small mobility of the atoms or ions in the crystal lattice.

Shutterworth depicted the relationship between the surface free energy and the surface tension for a solid surface.

dA₂ dA₁

γγγγ₁

γγγγ₂ Fig.10.2. Scheme of the

surface enlargement for a solid

( )

dA dA

Ag

d = γ + γ

( )

dA A dg dA g

Ag

d

+

=

= γ

In case of anisotropic solids the surface tension depends on the direction of the surface increase.

(Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies homogeneity in all directions. It can be defined as a difference, when measured along different axes)

The total increase in the Gibbs surface free energy is the sum of the reversible work done against force of surface tension different in the two directions γγγγ1 and γγγγ2.

(19)

Where g^s is the Gibbs free energy of the unit area.

For isotropic solids: γγγγ1 = γγγγ2 = γγγγ and then dA1 + dA₂ = dA

(20)

In the case of liquid the second term on the right hand side equals zero and hence γγγγ = g^s. The same holds true if dA change occurs very slowly (quasi-statically) in such a manner that

the surface atoms (molecules) occupy their equilibrium positions. If this not the case g^s and γγγγ differ between themselves and differ from their equilibrium values.

0 20 40 60 80 100 55

60 65 70 75 80

γw, mN m-1

Temperature, ^oC

11. Influence of temperature on the surface tension

Surface tension of most liquids in a range of temperature decreases almost linearly with the temperature increase. With the temperature increase thermal energy of the molecules

increases and more molecules move from the bulk phase to the surface thus causing increase in the surface area and in consequence, decrease in the liquid surface tension.

Fig.10.1 presents the changes in the surface tension of water as a function of its temperature.

Fig.11.1. Changes in the surface tension of water as a function of temperature.

From γγγγ = f(T) it can be easily determined (∂γγγγ/∂T)p and next entalphy ∆∆∆H∆ ^s of the surface formation.

Fig.11.2. Changes of the surface tension and surface enthalpy for CCl4

p s

T T

h 



 





∂ γ

− ∂ γ

=

S s

A h H = ∆

∆

∆

0 T

and const

T

2 2 p

 =







 





∂ γ

= ∂

 

 





∂ γ

∂

p 2 2

p s

T T T

h













∂ γ

− ∂

 =













∂

∂

const h

and T 0

h

p s

=

 =







 





∂

∂

(∆∆∆H = ∆∆ ∆∆∆G - T∆∆∆S) ∆ For a unit surface area it amounts:

(1)

Because (∂γ/∂T)p < 0, hence ∆h^s > γγγγ, and as the surface tension γγγγ decreases almost linearly with the temperature increase, therefore:

(2)

and (3)

(4)

Hence

which is illustrated in Fig. 11.2.

In fact the empiric equation given by Eötvös is better relationship describing γγγγ = f (T) is.

( ^{T T} ) ^{or V k} ( ^{T T} )

M k

3 c / m2 c

3 / 2

−

= γ

−

 =



 



 γ ρ

Enthalpy is a constant quantity independent of temperature as ∆U^S is.

In fact the empiric equation given by Eötvös is better relationship describing γγγγ = f (T) is.

(5)

Where: M is molar mass, d is the density, Tc is critical temperature at which at which a phase boundary ceases to exist, i.e. the meniscus between a liquid and its vapor

disappears, V_m is the liquid molar volume, k is a constant valid for all liquids.

The Eötvös constant has a value of 2.1×10⁻⁷ J/(K·mol^−2/3). For the associated liquids the coefficient is smaller and depends on the temperature. For water at 298 K it amounts 1.03⋅⋅⋅⋅10^–7 J K^–1 and increases to 1.18 ⋅⋅⋅⋅10^–7 J K^–1 with the temperature increase to 373 K.

More precise values can be gained when considering that the line normally passes the temperature axis 6 K before the critical point. Thus Ramsay and Shields modified Eötvös

equation to: (6)

Then Baczyński and McLeod found the dependence of a given liquid on its density.

(7)

Where: c is the constant, ρ_c is the liquid density, ρ_p is its vapour density.

( ^{T T 6} )

k

V

=

− − γ

(

)

c ρ − ρ

=

γ

4 / m 1 c

4 /

1 M V

P = γ

γ ρ

=

Based on this relationship Garner and Lugden introduced the notion "parachor".

(

)

M

P = γ

ρ − ρ

Neglecting the vapour density because it is small in comparison to the liquid one there is obtained the equation:

(9)

Parachor can be interpreted as the molar volume corrected relative to the surface forces.

For a given liquid parachor is practically independent of the temperature and additive. It means that this is the sum of parachors of atoms and functional groups.

^{P =} ∑ ^nP

⁺ ∑ ^nP

⁺ ∑ ^nP

Where: Pa, Pw and Pp indicate parachors of atoms, bonds and rings in the molecules, respectively, and n their number.

This property can be used for determination of the organic molecules structure calculating the parachor from Eq.10 and comparing with the value calculated from Eq.9.

For example, from the chemical formula: C₂H₆O two compounds can be derived: CH₃CH₂OH and CH₃OCH₃. Because the parachors of PCH₃,PCH₂, POH, andPOdiffer, thus comparing the experimentally determined value with that calculated it is possible to decide which compound is the investigated and find its true structure. The parachors can be found in textbooks.

Tabela 11.1. Parachors of atoms and bonds, [P] –10^–4 _{1 2}

3 4 1

/ /

s kmol

m kg

⋅

⋅

Atom P Atom P Atom P Bond P

C 8.5 P 67.0 Se 111.

1 single 0 H 30.4 F 45.7 Si 44.5 double 41.3 H in

OH 20.1 Cl 96.5 As 89.1 triple 82.9 O 35.6 Br 120.

9 Sb 117.

3

ring 3-atoms

29.7 N 22.2 I 161.

8 Sn 102.

9 4-atoms 20.6 S 85.7 B 29.2 Hg 122.

2 5-atoms 15.1 6-atoms 10.8 O₂ in esters

and acids

106.

7

12. Methods of surface tension measurements

There are several methods of surface tension measurements:

1. Capillary rise method

2. Stallagmometer method – drop weight method 3. Wilhelmy plate or ring method

4. Maximum bulk pressure method.

5. Methods analyzing shape of the hanging liquid drop or gas bubble.

6. Dynamic methods.

1. Capillary rise method.

This is the oldest method used for surface tension determination. A consequence of the surface tension appearance at the liquid/gas interface is moving up of the liquid into a thin tube, that is capillary, which is usually made of glass. This phenomenon was applied for determination of the liquid surface tension. For this purpose, the thin circular capillary is dipped into the tested liquid. If the interaction forces of the liquid with the capillary walls (adhesion) are stronger than those between the liquid

molecules (cohesion) the liquid wets the walls and rises into the capillary to a defined level and the meniscus is hemispherically concave.

In the opposite situation the forces cause decrease of the liquid level in the capillary below that in the chamber and the meniscus is semispherically convex. Both cases are illustrated in Fig. 11.1

Fig. 12.1. Schematic representation of capillary rise method.

If the cross-section area of the capillary is circular and its radius is sufficiently small, then the meniscus will be semispherical. Along the perimeter of the meniscus acts the force due to the surface tension presence.

θ γ

π r cos f ₁ = 2

Where: r – the capillary radius, γγγγ – the liquid surface tension, θθθθ – the wetting contact angle.

(1)

The force f₁ in Eq.(1) is equilibrated by the mass of the liquid raised in the capillary to the height h, that is the gravity force f₂. In the case of non-wetting liquid – it is lowered to a distance –h.

(2)

where: d – the liquid density (g/cm³) (actually the difference between the liquid and gas densities), g – acceleration of gravity.

g d h r f

= π

In equilibrium (the liquid does not moves in the capillary) f₁ = f₂ , and hence

g d h r cos

r

2 π γ θ = π

γ θ

cos g d h r

= 2

(3) or

(4) If the liquid completely wets the capillary walls the contact angle θθθθ = 0^o, and cosθθθθ = 1.

In such a case the surface tension can be determined from Eq. (5).

2 g d h

= r

γ

If the liquid does not wet the walls (e.g. mercury in a glass capillary), then it can be

assumed that θθθθ = 180^o, and cosθθθθ = -1. Because the meniscus is lowered by the distance - h, Eq.(5) gives correct result.

Eq.(5) can be also derived using Young-Laplace equation, ,from which it results that the pressure difference exists across a curved surface, which is called capillary

pressure and is illustrated in Fig.12.2. On the concave side of the meniscus the pressure is p₁. The mechanical equilibrium is attained when the pressure values are the same in the capillary and over the flat surface. In the case of wetting liquid the pressure in the capillary is lower than outside it, (p₂ < p₁). Therefore the meniscus is moved up to a height h when the pressure difference ∆∆∆∆p = p₂ - p₁ is balanced by hydrostatic pressure caused by the liquid raised in the capillary.

r P 2γ

=

∆

Fig. 12.2. The balanced pressures on both sides of the meniscus.

h g d P

P

P =

−

= ∆

∆

(6)

(6)