1. Adsorption Modeling of physisorption in porous materials Part 2 European Master Bogdan Kuchta Laboratoire MADIREL Université Aix-Marseille

2. Typical hysteresis of adsorption-desorption cycle Hysteresis loops H 1 and H 2, are characteristic for isotherms of type IV (nanoporous materials). Loop of hysteresis H 1 shows nearly vertical and parallel branches of the loop : it indicates a very narrow distribution of pore sizes. Loop of hysteresis H 2 is observed if there are many interconnections between the pores.

3. Loops of hysteresis H 3 et H 4, appear on isotherms of type II where there is no saturation. They are not always reproducible. Loop of hysteresis H 3, is observed in porous materials formed from agregats, where the capillary condensation happens in a non-rigid framework and porosity not definitly defined. Loop of hysteresis H 4 are often observed in structures built from planes that are not rigidly Typical hysteresis of adsorption-desorption cycle

4. n p/p 0 n

5. n n

6. n

7. Frundlich model Langmuir model BET Theories of adsorption

8. Theory of adsorption by Freundlich : x/m =  c 1/n x – adsorbed mass m – mass of adsorbent c – concentration , n – experimental constants x/m lg(x/m) c lg(c) Conclusion: adsorption is better at higher pressure

9. - 1 one type of » adsorption sites" - No lateral interactions - 1 site of adsorption allows 1 particle to be there: adsorption is limited to one layer N s = number of adsorption sites N a = number of adsorbed molecules  = fraction of the surface covered Langmuir theory

10.  Isotherm of Chemisorption  at low pressure bp << 1, so  Henry’s law  at high pressure, bp >> 1, si  Langmuir theory

11. Langmuir isotherm : influence of the coefficient ‘b’

12. Variations on Langmuir and Henry Henry Freundlich Langmuir Sips (Langmuir-Freundlich) Toth Jensen & Seaton

13. Variations of Langmuir and Henry

14. Methode « BET »

15. Hypothèses – Starting from the second layer E 1  E L energy of molecules in liquid state Théorie de Brunauer Emmett et Teller (BET) E 1 =energy of adsorption of the first layer } - 1 one type of » adsorption sites" - No lateral interactions

16. Basic hypothesis of the BET theory E 1 = Energy of adsorption for the first layer E l = Energy of liquid state Energy of adsorptionRelative pressure p/p° B ELEL E1E1  1

17. surface s o covered with 0 adsorbed layers... s 1... 1............ s i... i Accessible surface A = s o + s 1 + … + s i +... soso s1s1 s2s2 s3s3 A Basic hypothesis of the BET theory

18. Derivation of the BET formula For s 0 Rate of condensation of an empty surface Rate of evaporation from a surface covered with one layer = k i s i-1 p = k -i s i Rate of evaporation from the surface covered with two layers Rate of condensation on the surface covered with one layer = for s 1 Rate of condensation on a surface covered with i layers Rate of evaporation from a surface covered with i+1 layers = General, in the case of s i

19. Total surface of adsorbent Total quantity of adsorbed gas Asuming, that the layer properties are all the same k i s i-1 p = k -i s i C 1 (T)=exp(-E 1 /kT) C i (T)=exp(-E L /kT) Derivation of the BET formula

20. À p° : donc : Derivation of the BET formula

21. Theory of Brunauer Emmett and Teller (BET) Equations – N= number of layers x = p/p 0 = relative equilibrium pressure – if N   – Transformed equation BET

22. Influence of number of layers N on the shape of isotherms of adsorption (BET) p / p o  N = 4 N = 5 N = 6 N = 7 N = 25 à 

23. Influence of the constant ‘C’ on the shape of isotherms of adsorption (BET)

24. Application for calculation of the adsorption surface example : alumin NPL / N 2 / 77 K Pente : Ordonnée :

25. Verifications of BET results example : alumin NPL / N 2 / 77 K

26. Normal interactions Lateral interactions

27. Simulation of adsorption 1.Calculation of energy of adsorption 2.Simulation of isothermes (with different strength of interaction) 3.Analyse the results 1.Simulation Monte Carlo grand canonique (GCMC) 2.Tool: program GCMC (Fortran) Numerical challenge: 1. Simulations of equilibrium between gas and adsorbed phase 2. Modeling of interaction between pore walls and adsorbed particles

28. Problem: Fluid adsorption in cylindrical pores.  VT- constant  (gas) =  (adsorbate)  (gas, ideal) =  0 (gas) + k B T ln(P)  VTPVT - constant Grand Canonical Monte Carlo Working case: MC simulation of adsorption in a pore External ideal gas pressure P

29. P 2 and T fixed R (radius) P 1 and T fixed R (radius) Working case: MC simulation of adsorption in a pore

30. p  0.05234.716.2 0.1362.87.6 0.2385.85.8 0.3401.96.9 0.4421.29.7 0.5448.313.2 0.7558.331.6 0.8691.626.7 0.91259.18.3 T = const Working case: MC simulation of adsorption in a pore

31. Directory Run program (compiled) input files gcmc_H2.dat gcmc_H2_par.dat pos_inp.dat spline* Results files ene.ini- initial molecular energies ene.fin- final molecular energies mc.pos- molecular position after each bin mc_ene.dat - energies after each bin (wall and total) mc_ent.dat - energies pos_inp.res - execute analysis of results OK STOP NO Rename : pos_inp.res  pos_inp.dat

32. Nbin N x y z 1 154 10.886520 -14.887360 9.244424 10.898990 14.983010 21.000650 14.028510 11.913710 2.990251 -14.459990 11.605350.722188 1.908520 18.285780 22.916110 1.256716 -18.238170 4.253842 13.606980 -12.499060 7.607756 -15.536920 -9.600965 16.492660 -15.764630 -9.760927 24.275380 -4.318254 -17.841070 23.939530 15.933630 -9.115001 17.567660 …………. …….. mc.pos Nbin = 1 N = 154 Nbin = 1000 N = 258 Nbin = 2000 N = 615

33. Equilibrium situation Mean values Variation -337.713.5 -1160.215.3 -1497.910.2 234.716.2 p  0.05234.716.2 0.1362.87.6 0.2385.85.8 0.3401.96.9 0.4421.29.7 0.5448.313.2 0.7558.331.6 0.8691.626.7 0.91259.18.3 T = const

34. Experimental results of adsorption Milestones results 1.Isotherms 2.Energy of adsorption 3.Hysteresis properties

35. Approach thermodynamic – energie of adsorption   =  g u  +pv  -Ts  = u g +pv g - Ts g u  -Ts  = u g +RT - Ts g (v  = 0, pv g =RT) s g =s g,0 – R ln(p/p 0 )  ads h = u  - u g - RT  const.  ads s 0 = s  - s g,0  const. Adsorption is a phenomenon exothermic !!!

36. p T1T1 T 2 > T 1 p1p1 p2p2  Isosteric enthalpy Approach thermodynamic – energie of adsorption

37. p/p 0  ads h 1 2 3 5 4 Basic types of adsorption energy curves Curve 5 shows an existence of well defined fomains. Curve 1 is characteristic for heterogeneous surfaces. Curves 3 and 4 correspond delocalized and localized adsorption on a homogeneous surface, with lateral interactions between molecules. Curve 2 appears in homogeneous systems with no lateral interaction.

38. CO & CH 4  Typical for heterogeneous surface   ads h (  2 kJ.mol -1 ) during the capillary condensation Kr   ads h (  5 kJ.mol -1 ) during the capillary condensation  solidification ? Example: mesoporous system: MCM-41 et 77K.. CO Kr CH 4

39. Milestone properties  Capillary condensation is accompnied with histeresis of variable form Ar N2N2 lichrospher CPG

40.  Hysteresis disappears at some high temperature Argon / MCM41 (Morishige et al) Milestone properties

41.  For each temperature, there is a size of pore (and/or equilibrium pressure), that the hysteresis disappears below this value. 2.5 nm 4.0 nm 4.6 nm Nitrogen 2.5 nm 4.0 nm 4.6 nm Argon Llewellyn et al., Micro. Mater. 3 (1994) 345. Milestone properties

42. Adsorption - Desorption Isotherms :  MCM41 à 77K Ar CO N2N2 Llewellyn et al., Surf. Sci., 352 (1996) 468.

43. Nitrogen / black of de carbon (Carbopack) M. Kruk, Z. Li, M. Jaroniec, W. B. Betz, Langmuir 15 (1999) 1435-1441.

44. Adsorption on precipitated silica Isotherms : N 2 & Ar à 77K The conditions of the sample preparation are very important!!!! p / p 0 n a / mmol g -1 300°C 25°C 110°C 200°C P. J. M. Carrott & K. S. W. Sing, Ads. Sci. Tech., 1 (1984) 31.