1. ChE 553 Lecture 9 Statistical Mechanics Of Adsorption 1

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 2

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 3

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

5. Typical Surface phases 5 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 6

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 7

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

9. Example:When will Each Of The Following Phases Form? 9 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 o K –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 10

11. Energy Expression -Ei=H 1 +n n h nn +n 2nn h 2nn Ei=Energy molecule on site n n = Number of nearest neighbors occupied n 2nn = = Number of second newest neighbors occupied 11

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 12 (4.202)

13. Random Absorption On A C(2x2) Layer: 13 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) 14

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

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

17. When Does The (1x1) Have The Lowest Energy? 17 E (1x1) < E (2x1) E (1x1) < E C(2x2) E (1x1) < E P(2x2) Solving h nn +h 2nn >O h nn >O Strong attractions and insufficiently weak second nearest neighbors to prevent 1x1

18. When Does The P(2x2) Have The Lowest Energy? 18 P (2x2) < E (2x1) P (2x2) < E C(2x2) P (2x2) < E P(2x2) Solving h 2nn <0 h nn <0 Occurs when First and second neighbors are repulsive

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

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

21. Actual Phase Behavior More Complex 21

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

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

24. Calculate Coverage Analytically 24 (4.115) Fig 4.17 A 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 25 Figure 4.18 A replot of the data from Figure 4.17 versus dimensionless pressure. Fig 4.17 A 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.

26. Key Features Characteristic S like behavior No first order phase transitions 26 Fig 4.17 A 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 27 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. 28