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ChE 553 Lecture 9

ChE 553 Lecture 9. Statistical Mechanics Of Adsorption. Objective. Start to discuss thermodynamics of surface phase transitions Landau analysis – what phases do we see Solve one dimensional surface – what is phase behavior like. Background.

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ChE 553 Lecture 9

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  1. ChE 553 Lecture 9 Statistical Mechanics Of Adsorption

  2. Objective • Start to discuss thermodynamics of surface phase transitions • Landau analysis – what phases do we see • Solve one dimensional surface – what is phase behavior like

  3. Background • So far we discussed simple adsorption with no interactions between adsorbed species, complete mobility • Results in Langmuir behavior • At saturation densities like liquid densities • Molecules interact • Diffusion rates not always fast • Leads to defects in adsorbed layer • Predicts no ordering of adsorbed layer

  4. Real Surface systems Are Different • Usually a series of phases that vary with coverage and temperature

  5. Typical Surface phases Figure 4.22 The absorption of molecules in a P(2x2), C(2x2), (2x1) overlayer. The dark circles represent sites, the red circles represent adsorption on the sites.

  6. Why Do These Phases Arise? • Forces Between Adsorbates keep them apart • Direct forces (Van der waals repulsions) • Indirect forces (Adsorbate attracts local electrons, depleting nearby regions for adsorption

  7. Today • Consider the effects of pair-wise interactions on the behavior of adsorbed layer • Qualitative features: Landau analysis • Quantifying results, analytically and with monte carlo • Try to predict surface phase behavior

  8. Landau Analysis • Attempt to predict what phases are produced as gas adsorbs on solid surfaces

  9. Example:When will Each Of The Following Phases Form? Figure 4.22 The absorption of molecules in a P(2x2), C(2x2), (2x1) overlayer. The dark circles represent sites, the red circles represent adsorption on the sites.

  10. Steps in Landau Analysis • Make a list of possible arrangements of the adsorbate molecules to consider • Calculate the energy of each arrangement at 0 oK • Usually use pairwise additive assumption to calculate energy • The structure with the lowest energy fills first, the one with the second lowest energy fills second, …, until the surface fills up

  11. Energy Expression -Ei=H1+nnhnn+n2nnh2nn Ei=Energy molecule on site nn = Number of nearest neighbors occupied n2nn= = Number of second newest neighbors occupied

  12. Solution Random adsorption on a (1x1) layer: For random absorption the probability of any site being filled is . the energy of a atom absorbed at site i, becomes 4 because 4 first and second nearest neighbors (4.202)

  13. Random Absorption On A C(2x2) Layer: With a C(2x2) layer there are no first nearest neighbors, but there are several second nearest neighbors. If it is assumed that the C(2x2) layer is partially filled, then (4.203)

  14. Random Absorption On A P(2x2) Layer: With a P(2x2) layer there are no first or second nearest neighbors (4.204)

  15. Random Absorption On A (2x1) Layer: With a (2x1) layer there are two first nearest neighbors and no second nearest neighbors

  16. Summary Of The Equations (1x1) C(2x2) (p2x2) (2x1)

  17. When Does The (1x1) Have The Lowest Energy? E(1x1)< E(2x1) E(1x1)< EC(2x2) E(1x1)< EP(2x2) Strong attractions and insufficiently weak second nearest neighbors to prevent 1x1 Solving hnn+h2nn>O hnn>O

  18. When Does The P(2x2) Have The Lowest Energy? Occurs when First and second neighbors are repulsive Solving h2nn<0 hnn<0 P(2x2)< E(2x1) P(2x2)< EC(2x2) P(2x2)< EP(2x2)

  19. What Happens If We Squeeze On More Molecules When P(2x2) Is Lowest Energy?

  20. Also Get Incommensurate Adsorption (2x2) Domain Domain wall Domain wall Figure 3.15 The domain wall structure of CO on Pt(100). (Proposed by Persson et al. [1990].)

  21. Actual Phase Behavior More Complex

  22. Need Statistical Mechanics To Solve Real System Define occupancy number by: ξ=1 (occupied site) ξ =-1 (empty site) Partition function definition: (4.67) Combining with expression (4.78)

  23. Analytical Solution Of Equations In One Dimension • Assume a circular chain with So sites • Calculate partition function analytically (see p 270 in Adsorption) Qso=exp(Hso){(1)So+(2)So} (4.103) (4.104)

  24. Calculate Coverage Analytically (4.115) Fig 4.17A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The Langmuir line (βh = 0) is shaded.

  25. Replot vs. Pressure Fig 4.17A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The Langmuir line (βh = 0) is shaded. Figure 4.18A replot of the data from Figure 4.17 versus dimensionless pressure.

  26. Key Features • Characteristic S like behavior • No first order phase transitions Fig 4.17A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The Langmuir line (βh = 0) is shaded.

  27. Real Surfaces Show Phase Transitions Need Monte Carlo Calculation

  28. Summary • Can use statistical mechanics to calculate surface phase behavior • Qualitatively – almost Langmuir behavior if only nearest neighbor interactions in 1 D • More complex in 2 D.

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