CHEN 633: Thermodynamics and Kinetics of Confined Fluids

Adsorption thermodynamics-2Fluid-substrate interactions; structure of fluids inside pores1Why we need microscopic theories/experiments2

Question: where are the goats?2Do you see the goats?3

3Do you see them better?4

4And now?5

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6Microscopic theoriesIntegral equation theories

Classical density functional theory

They have in common the introduction of intermolecular potential functions to describe the system77Capillary action8If a glass tube with a bore as small as the width of a hair is dipped into water then the liquid rises in the tube to a height greater than that at which stands outside. Rise is 3cm in a tube with a bore of 1mm.

Molecular theory of capillarity, J. S. Rowlinson and B. Widom, Oxford, 19828Atomistic structure of matter9LFLiquid filmForce needed to balance the tension in the 2-sided film is proportional to LF = 2 s LNote that the work: F dx = s dAA tension per unit length has the same units that a surface energy per unit areaTension/lengthSurface energy/area9Atomistic structure of matter10plpgLaplaces equation:

pl pg = 2 s /Rinternal pressure (based on cohesive energy)first indication of molecular forcesqssg =sls + slg cos qYoungs equation10wetting coefficient11k = cos q = (ssg sls)/slg-1 < k < 1qk= -1 sls large and >0; liq. does not wet solidsolidliquidweak liq-solid forces or strong liq-liq or sol-sol11wetting coefficient12if -1 < k < 0sls not too large; Hg on glass q = ~140 ostrong forces in the liquid if 0 < k > water/water12inhomogeneity at an interface13lzone phase aboveand one phase below the plane (line)

Helmholtz free energy13Why/when a fluid is heterogeneousAn external potential

Another fluid molecule (like or unlike)

A solid wall

Electric, magnetic fields

1414evolution of the conceptclassical mechanics (Laplace, Young)

classical thermodynamics (Gibbs)

statistical mechanics

1515Statistical Mechanics of FluidsA classical, isotropic, one-component, monoatomic fluid.A closed system, for which N, V and T are constant (the Canonical Ensemble). Each particle i has a potential energy Ui.The probability of locating particle 1 at dr1, etc. is

The probability that 1 is at dr1 and n is at drn irrespective of the configuration of the other particles is

The probability that any particle is at dr1 and n is at drn irrespective of the configuration of the other particles is

16N-particle distribution functionIf the distances between n particles increase the correlation between the particles decreases.

In the limit of |ri-rj| the n-particle probability density can be factorized into the product of single-particle probability densities.

Measures the extent to which the fluid structure deviates from randomness17Radial Distribution FunctionIn particular g(2)(r1,r2) is important since it can be measured via neutron or X-ray diffraction

g(2)(r1,r2) = g(r12) = g(r)

18Radial Distribution Functiong(r12) = g(r) is known as the radial distribution functionit is the factor which multiplies the bulk density to give the local density around a particleIf the medium is isotropic then 4pr2rg(r)dr is the number of particles between r and r+dr around the central particle

19Correlation FunctionsPair Correlation Function, h(r12), is a measure of the total influence particle 1 has on particle 2 h(r12) = g(r12) - 1Direct Correlation Function, c(r12), arises from the direct interactions between particle 1 and particle 2

20Ornstein-Zernike (OZ) EquationIn 1914 Ornstein and Zernike proposed a division of h(r12) into a direct and indirect part.The former is c(r12), direct two-body interactions.The latter arises from interactions between particle 1 and a third particle which then interacts with particle 2 directly or indirectly via collisions with other particles. Averaged over all the positions of particle 3 and weighted by the density.

21Thermodynamic Functions from g(r)If you assume that the particles are acting through central pair forces (the total potential energy of the system is pairwise additive), , then you can calculate pressure, chemical potential, energy, etc. of the system.For an isotropic fluid

22Classical density functional theoryThe thermodynamic grand potential W = W (m, T, V) is a functional of the one particle distribution function, r (r)

So we have W = W [r (r)]

The equilibrium density profile is obtained by minimizing this functional23Classical DFT

Intrinsic Helmholtz free energyChemical potentialExternal potential r(r) is the fluid number density at position r

free energy functional inhomogeneous hard-sphere fluid24Classical DFT25

r (2)(r) is the pair distribution function

mean field approximation for attractive forces25Classical DFT26Repulsive part = ideal gas +excess part (Tarazona, Mol. Phys. 1984)

ideal gas part is directly a functional of r(r)

excess part is a functional of a smoothed density:

weighting function26Summary27

Intrinsic Helmholtz free energyChemical potentialExternal potential r(r) is the fluid number density at position r27Intermolecular potentials (an example)28

28Fluid-fluid interactions (an example)29

29Solid-fluid intermolecular forces30

Tan and Gubbins, JPC, 199230Isosteric heat of adsorption31enthalpy and volume differences betweenadsorbed phase and vapor phasemeasures the strength of the forces betweenadsorbate and substrate

QqQheat of vaporization31experimental heats of adsorption32

32Atomistic simulation/theory methodsMolecular simulations (Gibbs ensemble)Useful to determine phase equilibria, adsorption isotherms, heats of adsorption

Molecular simulations (Grand canonical ensemble)Useful to determine adsorption isotherms, heats of adsorption, density profiles

Classical density functional theoryUseful to determine density profiles, pore size distribution, adsorption isotherms, heats of adsorption3333Phase equilibrium problem34

T1 = T2

P1 = P2

m1 = m2Gibbswhich is the condition for phase coexistence in a one-component system.Given two phases 1 and 2, they will be at equilibrium in all points where:34Gibbs ensemble (Panagiotopoulos, 1987)35gasliquidachieve equilibriumby coupling them3536

Overall system: NVT ensembleN = N1 + N2 V = V1 + V2T1 = T2 distribute N1 particles change the volume V1 displace the particles3637

Particle displacementVolume change equal PParticle exchange equal 3 different kinds of trial moves:3738

38Adsorption isotherm39Panagiotopoulos, Mol. Phys., 1987

39Density profiles for fluids in pores40Panagiotopoulos, Mol. Phys., 1987

40Density profiles at different pore diameters41

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