© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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Adsorption in mesopores

See an example of
surface area / pore volume analysis report
BJH scheme details and pore filling graphics

NOTE
Adsorption in mesopores according to the classical Kelvin model (below) occurs in vapours. Liquids require a different approach (e.g. DFT).

Adsorption in mesopores (usually pores of 2 - 50m - IUPAC - or sometimes 2 - 100 nm in diameter).

Typical adsorbates are N₂ (nitrogen vapours at the temperature of liquid nitrogen bath at atmospheric pressure, approx. 77-78K - depending on liquid N₂ purity and pressure) and vapours of benzene, C₆H₆ , at room (i.e. 25°C) temperature. Krypton is often used for low pressure isotherms. Other gases like CH₄, CO₂ are also used.

Adsorption in a so called "mesopore range" is a combination of physical adsorption on mesopore walls (described by one of theoretical or experimental isotherm equations like HJ or FHH or other - sometimes with wall curvature corrections) with physical condensation of adsorbate in pores, where meniscus reached a critical radius. It is assumed that in this "mesopore" range all micropores are already filled-up and monolayer is filled, too.

Generally, experimental isotherms measured on mesoporous solids display hysteresis loops (adsorption and desorption go along different paths) in the mesopore region. However, hysteresis is observed only when the menisci in the adsorption and desorption "paths" are different in shape/diameter. Such situation is typical for open-ended cylindrical mesopores, bottle-shaped (ink bottle), or spaces between parallel sheets (slits). If cylindrical pore is closed at one end and does not contain any narrows, the adsorption isotherm will not diplay hysteresis loop. The same is true for conical pores.
For typical adsorbates the opening/closing of hysteresis loop is found at:

• For N₂ at 78K (liquid nitrogen bath at std. pressure): x_hist = p/p_s ≈ 0.40
• For benzene at room temperature (25°C or 298K) : x_hist = p/p_s ≈ 0.175.

As the true pore diameter is a sum of 2 adsorbed layer thicknesses and meniscus diameter, the methods of data analysis use meniscus radii calculated from Kelvin equation (or its modifications including e.g. influence of menicus radius on adsorbate surface tension - e.g. by Dubinin) and statistical layer thickness of adsorbate (often given by Harkins-Jura (HJ) or by Halsey / Frenkel-Halsey-Hill (FHH) equations).
Very often in simple data analysis specific expressions for N₂ adsorption are used: t_HJ and t_Halsey, respectively - see also above). Those simple equations (i.e. HJ and FHH) are also used in a so called t-plot method which is a simple method of determination of total micropore adsorption and external (i.e. mesopore + macropore) surface area. A better method is an α_s-plot which compares adsorption data on porous solid with experimental isotherms measured on a similar in nature/chemical structure - but non-porous (or having much less micro- and mesopores) - adsorbent.

Calculations of mesopore size distribution depend on many factors:

• Mesopore shape:
  • Cylindrical mesopores:
    Evaporation occurs at lower pressures than condensation in the same pore (hysteresis):
    p_evap(r_pore) < p_cond(r_pore)
    • Adsorption on cylindrical walls acc. to Kelvin eq. with:
      r_cond = r_K,cyl = r_x , r_y = +∞
    • For desorption, the meniscus is spherical, i.e.:
      r_cond = r_K,sph = r_x = r_y
      (see below)
  • Spherical mesopores
    Kelvin eq. with:
    r_cond = r_evap = r_K,sph = r_x = r_y
  • Bottle shaped mesopores
    • Condensation (ads. branch) occurs acc. to Kelvin eq. for spherical pore, i.e. with:
      r_cond = r_K,sph = r_x = r_y
      where this radius is the radius of inner pore space
      r_K,sph = r_sph,in
    • Evaporation (des. branch) occurs at the much lower pressure determined by the radius of spherical meniscus formed in a narrow bottleneck:
      r_evap = r_K,sph = r_neck
  • Slit-shaped mesopores
    • Adsorption on "flat" pore walls
    • Desorption occurs on slit openings with cylinrical meniscus (along slit edge), i.e. acc. to Kelvin eq. with:
      d_evap = 2 r_K,cyl = r_x , r_y = +∞
  • Other, e.g. conical pores (no rapid condensation at any fixed pressure, condensation and evaporation along the same path)
• Mesopore filling model / pore distribution calculation scheme:
  • Barret-Joyner-Halenda (BJH)
  • simplified BJH
  • Dubinin (like BJH, but takes into account the dependence of N₂ surface tension on meniscus radius)
  • Density-Functional Theory (DFT)

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