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

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.

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

n p/p 0 n

n n

n

Frundlich model Langmuir model BET Theories of adsorption

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

- 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

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

Langmuir isotherm : influence of the coefficient ‘b’

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

Variations of Langmuir and Henry

Methode « BET »

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

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

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

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

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

À p° : donc : Derivation of the BET formula

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

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 

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

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

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

Normal interactions Lateral interactions

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

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

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

p  T = const Working case: MC simulation of adsorption in a pore

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

Nbin N x y z …………. …….. mc.pos Nbin = 1 N = 154 Nbin = 1000 N = 258 Nbin = 2000 N = 615

Equilibrium situation Mean values Variation p  T = const

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

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 !!!

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

p/p 0  ads h 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.

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

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

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

 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

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

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

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 °C 25°C 110°C 200°C P. J. M. Carrott & K. S. W. Sing, Ads. Sci. Tech., 1 (1984) 31.