1. THERMODYNAMIC OF INTERFACES

2. 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

3. 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

4. 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

5. 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 is
V = Vα + Vβ

6. U = Uα + Uβ + Uσ Ni = Nαi + Nβi + Nσi S = Sα + Sβ + Sσ
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 + Nβi + 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

7. 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 cαi and cβi, the additional quantity that is present in the system due to the interface is:
Nσi = Ni - cαiVα – cβiVβ
With equation above, it is possible to define something like a surface concentration, the so called interfacial excess:

8. A is the interfacial area
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

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

10. For two or multi-component liquids such as a solvent with dissolved substances substituting Vα = V – Vβ we can write
Nσ1 = N1 – cα1V + (cα1 - cβ1)Vβ
For the first component which is taken to be solvent. For all other components we get similar equations
Nσi = Ni – cαiV + (cαi - cβi)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
(cαi - cβi)/(cα1 – cβ1) and subtracting 3.8 from 3.9

11. 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
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

12. 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.

13. Concentration profile of a solute (2) dissolved in a liquid (1)
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

14. 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

15. 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, Nβi and Nσi are not independent because:
Ni = Nαi + Nβi + Nσi is constant
Only two at a time as an example Nαi and Nβi can be varied independently. Nσi is then determined by the other two amounts because
dNσi = -dNαi - dNβi

16. 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:

17. 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

18. 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

19. 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

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

21. 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.

22. Increasing the surface area size by tilting a test tube

23. 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

24. 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

25. 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

26. 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

27. 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)

28. 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

29. 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

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

31. Isoterm adsorpsi Gibbs adalah hubungan antara surface tension dan konsentrasi interfacial excess
dUσ = TdSσ + SσdT + μidNσi + Nσidμi + 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

32. Sistem 2 Komponen
Aplikasi paling sederhana dari isoterm adsorpsi Gibbs adalah sistem 2 komponen mis. Solvent 1 dan solut 2
d = -1d1 - 2d2
Interface ideal umumnya 1 = 0 sehingga
d = 2(1)d2
Potensial kimia solut diuraikan oleh pers
a adalah aktifitas dan a0 adalah aktifitas standar (1 mol/L)

33. Jika persamaan kita turunkan terhadap a/a0 pada T konstan
Persamaan ini menyatakan bahwa saat solut terkumpul di surface (2(1) > 0) surface tension turun jika konsentrasi larutan ditingkatkan

34. 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

35. 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?

36. Marangoni Effect : Tears of Wine