Thermodynamics of Interfaces

1. Thermodynamics of Interfaces And you thought this was just for the chemists...

2. Terms • Intensive Variables • P: pressure •  Surface tension • T: Temperature (constant) •  Chemical potential • Extensive Variables • S: entropy • U: internal energy • N: number of atoms • V: volume •  Surface area Key Concept: two kinds of variables Intensive: do not depend upon the amount (e.g., density) Extensive: depend on the amount (e.g., mass)

3. Phases in the system • Three phases • liquid; gaseous; taut interface • Subscripts • ‘•’ indicates constant intensive parameter • ‘g’; ‘l’; ‘a’; indicate gas, liquid, and interface Gaseous phase ‘g’ Interface phase ‘a’ Liquid phase ‘l’

4. Chemical Potential • refers to the per molecule energy due to chemical bonds. • Since there is no barrier between phases, the chemical potential is uniform • g = a = l = • [2.21]

5. Fundamental Differential Forms • We have a fundamental differential form (balance of energy) for each phase • TdSg = dUg + PgdVg - •dNg (gas) [2.22] • TdSl = dUl + PldVl - •dNl (liquid) [2.23] • TdSa = dUa - d (interface) [2.24] • The total energy and entropy of system is sum of components • S = Sa + Sg + Sl [2.25] • U = Ua + Ug + Ul [2.26]

6. How many angels on a pin head? • The inter-phase surface is two-dimensional, The number of atoms in surface is zero in comparison to the atoms in the three-dimensional volumes of gas and liquid: • N = Nl + Ng [2.27]

7. FDF for flat interface system • If we take the system to have a flat interface between phases, the pressure will be the same in all phases (ignoring gravity), which we denote P• • The FDF for the system is then the sum of the three FDF’s • TdS = dU + P•dV - •dN - d(system) [2.27]

8. Gibbs-Duhem relationship • For an exact differential, the differentiation may be shifted from the extensive to intensive variables maintaining equality. • TdS = dU + P•dV - •dN - d(system) SadT =  d  [2.29] • or • Equation of state for the surface phase (analogous to Pv = nRT). Relates temperature dependence of surface tension to the magnitude of the entropy of the surface.

9. Laplace’s Equation from Droplet in Space • Now consider the effect of a curved air-water interface. • Pg and Pl are not equal • g = l =  • Fundamental differential form for system TdS = dU + PgdVg + PldVl - (dNg+dNl ) - d [2.31]

10. Curved interface Thermo, cont. • Considering an infinitesimally small spontaneous transfer, dV, between the gas and liquid phases • chemical potential terms equal and opposite • the total change in energy in the system is unchanged (we are doing no work on the system) • the entropy constant TdS = dU + PgdVg + PldVl - (dNg+dNl) - d [2.31] • Holding the total volume of the system constant, [2.31] becomes • (Pl - Pg)dV - d = 0 [2.32]

11. Droplet in space (cont.) • where Pd = Pl - Pg • We can calculate the differential noting that for a sphere V = (4r3/3) and  = 4r2 • [2.34] • which is Laplace's equation for the pressure across a curved interface where the two characteristic radii are equal (see [2.18]).

12. Simple way to obtain La Place’s eq.... • Pressure balance across droplet middle • Surface tension of the water about the center of the droplet must equal the pressure exerted across the area of the droplet by the liquid • The area of the droplet at its midpoint is r2 at pressure Pd, while the length of surface applying this pressure is 2r at tension  Pd r2 = 2r [2.35] • so Pd =2s/r, as expected

13. Vapor Pressure at Curved Interfaces • Curved interface also affects the vapor pressure • Spherical water droplet in a fixed volume • The chemical potential in gas and liquid equal • l = g [2.37] and remain equal through any reversible process • dl = dg [2.38]

14. Fundamental differential forms As before, we have one for each bulk phase • TdSg = dUg + PgdVg - gdNg (gas) [2.39] • TdSl = dUl + PldVl - ldNl (liquid) [2.40] Gibbs-Duhem Relations: • SgdT = VgdPg - Ngdg (gas) [2.41] • SldT = VldPl - Nldl(liquid) [2.42]

15. Some algebra… • SgdT = VgdPg - Ngdg (gas) [2.41] • SldT = VldPl - Nldl(liquid) [2.42] • Dividing by Ng and Nl and assume T constant • vgdPg = dg (gas) [2.43] • vldPl = dl(liquid) [2.44] • v indicates the volume per mole. Use dg = dl [2.38] to find • vgdPg = vldPl [2.45] • which may be written (with some algebra)

16. Using Laplace’s equation... • or • since vl is four orders of magnitude less than vg, so suppose (vg - vl)/vlvg/vl • Ideal gas, Pgvg = RT, [2.49] becomes

17. Continuing... • Integrated from a flat interface (r = ) to that with radius r to obtain • where P is the vapor pressure of water at temperature T. Using the specific gas constant for water (i.e., = R/vl), and left-hand side is just Pd, the liquid pressure:

18. Psychrometric equation • Allows the determination of very negative pressures through measurement of the vapor pressure of water in porous media. • For instance, at a matric potential of -1,500 J kg-1 (15 bars, the permanent wilting point of many plants), Pg/P is 0.99.

19. Measurement of Pg/P • A thermocouple is cooled while its temperature is read with a second thermocouple. • At the dew point vapor, the temperature decline sharply reduces due to the energy of condensation of water. • Knowing the dew point T, it is straightforward to obtain the relative humidity • see Rawlins and Campbell in the Methods of Soil Analysis, Part 1. ASA Monograph #9, 1986

20. Temperature Dependence of  • Often overlooked that all the measurements we take regarding water/media interactions are strongly temperature dependent. • Surface tension decreases at approximately one percent per 4oC!