THERMODYNAMIC OF INTERFACES

Purposes

The purpose of this chapter is to present some important equations, learn to apply them, provide a broader base of understanding and point out some of the difficulties

Surface Excess

The presence of an interface influences generally all thermodynamic parameters of a system

To consider the thermodynamic of a system with an interface, we divide that system into three parts: The two bulk phases with volumes Vα and Vβ and the interface σ

In Gibbs convention the two phases α and β are separated by an ideal interface σ which is infinitely thin: Guggenheim explicitly treated an extended interphase with a volume

Gibbs convention

In Gibbs convention the two phases are thought to be separated by an infinitesimal thin boundary layer, the Gibbs dividing plane (this is of course an idealization)

Gibbs dividing plane also called an ideal interface

In Gibbs model the interface is ideally thin (Vσ = 0) and the total volume isV = Vα + Vβ

All other extensive quantities can be written as a sum of three components : one of bulk phase α, one of bulk phase β and one of the interfacial region σ :

U = Uα + Uβ + Uσ

Ni = Nαi + Nβ

i + Nσi

S = Sα + Sβ + Sσ

The contribution of the two phases and the interface are derived as follows. Let uα and uβ be the internal energies per unit volume of the two phases

The internal energies uα and uβ are determined from the homogeneous bulk regions of the two phases, close to the interface they might be different

The contribution of the volume phases to the total energy of the system is:

uα Vα + uβ Vβ so the internal energy of the interface is:

Uσ = U - uα Vα - uβ Vβ At an interface the molecular constitution changes. The

concentration (number of molecules per unit volume) of the ith material is, in the two phases, respectively cα

i and cβ

i, the additional quantity that is present in the system due to the interface is:

Nσi = Ni - cα

iVα – cβiVβ

With equation above, it is possible to define something like a surface concentration, the so called interfacial excess:

A

N ii

A is the interfacial area. The interfacial excess is given as a number of molecules per unit area (m-2) or in mol/m2.

In the Gibbs model of an ideal interface there is one problem, where precisely do we position the ideal interface?

The density of a liquid-vapor interface decreases continuously from the high density of the bulk liquid to the low density of the bulk vapor

There could be a density maximum in between since it should in principle be possible to have an increased density at the interface

Dependence of the surface excess on the position of the Gibbs dividing plane

For two or multi-component liquids such as a solvent with dissolved substances substituting Vα = V – Vβ we can write

Nσ1 = N1 – cα

1V + (cα1 - cβ

1)Vβ

For the first component which is taken to be solvent. For all other components we get similar equations

Nσi = Ni – cα

iV + (cαi - cβ

i)Vβ

All quantities on the right side of the equations, except Vβ do not depend on the position of the dividing plane and are measurable quantities

Only Vβ depends on the choice of the dividing plane We can eliminate Vβ by multiplying equation by (cα

i - cβi)/(cα

1 – cβ1) and subtracting 3.8 from 3.9

The right side of the equation does not depend on the position of the Gibbs dividing plane and thus, also the left side is invariant. We divide this quantity by the surface area and obtain the invariant quantity

1111

111 )(

cc

ccVcNVcN

cc

ccNN ii

iiii

i

111

)1(

cc

cc iiii

It is called relative adsorption of component i with respect to component 1.

This is an important quantity because it can be determined experimentally and it can be measured by determinig the surface tension of liquid versus the concentration of the solute

Example

To show how our choice of the position of the Gibbs dividing plane influences the surface excess, we consider an equimolar mixture of ethanol and water. If the position of the ideal interface is such that H2O = 0, one finds experimentally that ethanol = 9.5 x 10-7 mol/m2. if the surface is placed 1 nm outward then we obtain ethanol = -130 x 10-7 mol/m2.

Concentration profile of a solute (2) dissolved in a liquid (1). The area of the dotted region correspond to the surface excess Γ(1)

2 of solute

Fundamental Thermodynamic Relation

Let us consider a process in a system with two phases, α and β which are divided by an interface; we could for instance do work on that system

As a consequences the state quantities like the internal energy, the entropy stc. change

How do they change and how can we describe this mathematically?

In contrast to the usual “bulk” thermodynamics we have to take the interface into account

Equilibrium Conditions

In equilibrium, eq.3.16 can be simplified even further because the chemical potentials in the three phases are equal

We assume that there is no exchange material with the outside world (dNi = 0) we have a closed system

Then the three parameters Nαi, Nβ

i and Nσi are not

independent because:

Ni = Nαi + Nβ

i + Nσi is constant

Only two at a time as an example Nαi and Nβ

i can be varied independently. Nσ

i is then determined by the other two amounts because

dNσi = -dNα

i - dNβi

At equilibrium the chemical potentials are the same everywhere in the system

We can further simplify eq.3.17 dF = -(Pβ – Pα)dVβ + γdA + ΣμidNi

This equation allows us to define the surface tension based on thermodynamics:

iNVVTA

F

,,,

The surface tension tells us how the Helmholtz free energy of the system changes when increasing the surface area while keeping the temperature, the total volume, the volume of phase β and the total numbers of all components constant

Is this a useful equation? It is not difficult to control T, V and Ni but Vβ might be difficult to keep constant

For planar surfaces (and those which have small curvatures) the condition that Vβ has to be constant can be dropped

Location of the Interface

At this point we should note that fixing the bending radii, we define the location of the interface

A possible choice for the ideal interface is the one that is defined by the Laplace equation

If the choice for the interface is different, the value for the surface tension must be changed accordingly, otherwise the Laplace equation would no longer be valid

Gibbs Energy and Definition of the Surface Tension In this part we introduce a more useful equation

for the surface tension. This we do in two steps First we seek an equation for the change in the

Gibbs free energy The Gibbs free energy G is usually more

important than F because its natural variables, T and P are constant in most applications

Second we have just learned that for curved surfaces the surface tension is not uniquely defined and depends on where precisely we choose to position the interface

Therefore we concentrate on planar surface

The surface tension is the increase in the Gibbs free energy per increase in surface area at constant T, P and Ni

iNPT

ii

ii

A

G

dAdNVdPSdTdG

PPP

dAdNdPVdPVSdTdG

,,

(planar)flat is interface assuming

Free Surface Energy, Interfacial Enthalpy and Gibbs Surface Energy

Now we consider the interfacial excess quantities starting from the internal interfacial or internal surface energy

The term PdVσ disappear because the ideal interface has no volume

We integrate the expression keeping the intrinsic parameters T, μi and γ constant.

dAdNTdSdU ii

Increasing the surface area size by tilting a test tube

AdNTS ii U

Result

Defining Interfacial Enthalpy

Enthalpy is equal to the internal energy minus the total mechanical work γA – PVσ. Since in Gibbs convention PVσ = 0

Hσ ≡ Uσ – γA This definition is recommended by IUPAC The differential is obtain to be

AddNTdSdH ii

Alternatively we can argue that enthalpy is equal to the internal energy minus the volume work PVσ. Since the volume work is zero in the Gibbs convention we get

Hσ ≡ Uσ

Next is, what is the interfacial excess Gibbs energy? The difference between Uσ and Fσ should be the same as the one between Hσ and Gσ, therefore we definie

AddNdTSdG

dNAFTSHG

ii

ii

is aldifferenti The

Surface Tension Cairan Murni

Untuk cairan murni, kita mulai dari bagaimana surface tension memiliki hubungan dengan kuantitas surface excess? Khususnya energi dalam surface dan entropi surface?

Untuk cairan murni kita pilih Gibbs dividing plane sehingga Γ = 0 sehingga surface tension sama dengan energi bebas surface per unit area

A

Ff

Untuk entropi cairan murni, posisi interface dipilih sedemikian sehingga Nσ = 0. untuk sistem homogen juga diketahui bahwa

sσ ≡ Sσ/A = Sσ/A, sehingga dapat dirumuskan:

Entropi surface per unit area diberikan oleh perubahan surface tension terhadap temperatur. Agar dapat menentukan entropi surface, kita harus mengukur bagaimana surface tension berubah dengan temperatur

APTs

,

Persamaan menyiratkan jika volume interfacial nol tekanan harus dijaga konstan, hal ini terkait dengan perubahan tekanan juga akan mengubah kualitas interface sehingga juga mengubah entropi

Untuk sebagian besar liquid, surface tension akan turun dengan kenaikan temperatur

Entropi di permukaan akan meningkat yang menyiratkan bahwa molekul di surface kurang teratur dibandingkan didalam fasa ruah (bulk phase)

Untuk energi dalam cairan murni kita ketahui : Uσ = TSσ + γA, jika persamaan ini kita bagi dengan A dengan asumsi sistem homogen maka diperoleh

Dimungkinkan menentukan energi dalam dan entropi surface dengan mengukur ketergantungan surface tension terhadap temperatur

APTTu

TsA

Uu

,

.

atau

Isoterm Adsorpsi Gibbs

Telah diketahui bahwa surface tension air akan turun saat ditambahkan detergent. Detergent akan sangat terkonsentrasi di surface sehingga menurunkan surface tension

Perubahan surface tension akibat adsorpsi zat dipermukaan diuraikan oleh isoterm adsorpsi Gibbs

Surface tension, entropi, enthalpi dan energi dalam surface beberapa liquids pada 25oC

Isoterm adsorpsi Gibbs adalah hubungan antara surface tension dan konsentrasi interfacial excess

dUσ = TdSσ + SσdT + μidNσi + Nσ

idμi + γdA + Adγ, jika pers 3.27 dimasukkan

0 = SσdT + Nσidμi + Adγ

Pada temperatur konstan menjadi dγ = - idμi

Pesamaan 3.45 dan 3.46 disebut isoterm adsorpsi Gibbs, secara isoterm adalah plot fungsi keadaa versus tekanan, konsentrasi dll pada temperatur konstan

Sistem 2 Komponen

Aplikasi paling sederhana dari isoterm adsorpsi Gibbs adalah sistem 2 komponen mis. Solvent 1 dan solut 2

d = -1d1 - 2d2

Interface ideal umumnya 1 = 0 sehingga

d = 2(1)d2

Potensial kimia solut diuraikan oleh pers

a adalah aktifitas dan a0 adalah aktifitas standar (1 mol/L)

0

022 ln.

a

aRT

Jika persamaan kita turunkan terhadap a/a0 pada T konstan

TaRT

a

a

daRT

aa

aadRTd

.

kansubstitusi ./

/.

)1(2

0

02

Persamaan ini menyatakan bahwa saat solut terkumpul di surface (2

(1) > 0) surface tension turun jika konsentrasi larutan ditingkatkan

Solut demikian biasanya bersifat surface active dan dinamakan surfactants atau surface active agents

Istilah molekul amphiphilic atau amphiphile juga biasa digunakan

Molekul amphiphilic terdiri dari 2 bagian; pertama oil-solubel (lifofilik atau hidrofobik) dan kedua water-soluble (hidrofilik)

Jika solut tidak terkumpul di surface (2(1)

< 0) surface tension meningkat dengan penambahan solut

Example 3.3

Seorang mahasiswa menambahkan 0,5 mM SDS (sodium dodecyl sulfat, NaSO4(CH2)11CH3) ke dalam air murni pada 25oC. Penambahan menyebabkan penurunan surface tension dari 71,99 mJ/m2 menjadi 69,09 mJ/m2. berapa surface excess dari SDS?

Marangoni Effect : Tears of Wine