Characterizing Porous Materials and Powders
N2 Adsorption Isotherms
Laleh Emdadi
Advisor: Dr. Dongxia Liu
Department of Chemical & Biomolecular Engineering
University of Maryland
College Park, MD, 20742
04/04/2013
Characterizing Porous Materials and Powders
Objective:
To give you a basic understanding of adsorption chemistry
and how it is used to characterize powders and porous
materials. We focus on physisorption.
Good References to Study:
1]
2] K.Y. Foo, B. H. Hameed. Insights into the modeling of adsorption isotherms
systems. Chemical Engineering Journal 156 (2010) 2-10.
3] K. S.W. Sing, R. T. Williams.
Physisorption hysteresis loops and the
characterization of nonporous materials. Adsorption Science and Technology
22(10) (2004) 773-782.
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Characterizing Porous Materials and Powders
- Surface Area
- Multipoint BET Method
- Single Point BET Method
- Multipoint/ Single Point Comparison
- Porosity by Gas Adsorption
- Isotherms
- Total Pore Volume and Average Pore Radius
- Pore Size Distributions (Mesopore): - BJH Method
- DH Method
- Surface Area of Microporous Samples by Langmuir Method
- Micropore Analysis
- V-t Method
- Alpha-s Method
- MP Method
- DR Method
- DA Method
- HK Method
- SF Method
- Density Functional Theory (DFT) and Monte Carlo Simulation Methods
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Characterizing Porous Materials and Powders
Surface Area
The Brunauer-Emmett-Teller (BET) method is the most
widely used procedure for the determination of the surface
area of solid materials and involves the use of BET equation:
Usually N2 is used as
the adsorbate and C for
nitrogen is (50-250)
Intercept (i) Slope (s)
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Characterizing Porous Materials and Powders
Multiple BET Method
The BET equation requires a linear plot of
versus
which for most solids using nitrogen as adsorbate is restricted to a
limited region in the P/P0 range of 0.05 to 0.35. This linear region is
shifted to lower relative pressures for microporous materials.
A Typical BET Plot
The standard multipoint BET procedure requires a minimum of 3
points in this range. Wm can be obtained from slope and intercept of
the BET plot.
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Characterizing Porous Materials and Powders
Multiple BET Method
The second step in the application of the BET method is the
calculation of the surface area. The total surface area of the sample
can be expressed as:
N : Avogadro's number (6.0221415 × 1023 molecules/mol),
Wm: weight of adsorbate constituting a monolayer of surface coverage
Acs: molecular cross section of the adsorbate molecule, (16.2 A°2 for N2 at 77 K)
M: molar mass (molecular weight) of the adsorbate gas
The second parameter can be calculated is specific surface area of the
solid can be calculated as:
St: total surface area,
W: sample weight
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Characterizing Porous Materials and Powders
Single Point BET Method
For routine measurements of surface areas, a simplified procedure
may be applied. In this method, only a single point on the adsorption
isotherm in the linear region of the BET plot is used ( = 0.3).
For microporous materials P/P0 as high
as possible, yet still in the linear region of
the BET plot should be chosen!!!
The assumption is that the intercept in the BET equation is zero:
Ideal gas equation
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Characterizing Porous Materials and Powders
Multipoint/Single Point Comparison
The relative error in single point versus multipoint for determining of
surface area of the sample is a function of BET C constant and the
relative pressure used.
For multipoint BET we have:
For single point BET we have:
As a result the relative error of single point method is:
Usually if you chose the P/P0 properly, this error is very low and so it
is acceptable.
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Characterizing Porous Materials and Powders
Single Point BET Error
This table shows the relative error for various C calculated using P/P0
of 0.3.
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Isotherms (IUPAC classifications)
Microporous (chemisorption) Non-porous or macroporous
porous materials
consisting of welldefined
cylindrical-like pore
channels
or agglomerates of
compacts of
approximately uniform
spheres
disordered materials
and the distribution
of pore size and shape
is not well defined
Not common;
N2 adsorption on
polyethylene
Mesoporous
non-rigid aggregates
of plate-like particles
giving rise to slit-shaped
pores
Mesoporous
Uniform non-porous surface
narrow slit pores,
but now including
pores in the micropore
region
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Characterizing Porous Materials and Powders
Porosity by Gas Adsorption
Pores are characterized according to their sizes:
a) Pores > 50 nm in diameter are called macropores.
b) Pores 2-50 nm in diameter are called mesopores.
c) Pores < 2 nm in diameter are called micropores.
Porosity of powders and other porous solids can be characterized by
gas adsorption studies.
There are 2 common techniques for describing porosity:
1) Determination of total pore volume
2) Pore size distribution
For most solid materials, nitrogen at 77 K is the most suitable
adsorptive.
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Characterizing Porous Materials and Powders
Total Pore Volume and Average Pore Radius
The total pore volume is derived from the amount of vapor adsorbed at a
relative pressure close to unity (P/P0 ≈1), by assuming that the pores are
then filled with liquid adsorbate. The volume of nitrogen adsorbed (Vads)
can be converted to the volume of liquid nitrogen (Vliq) contained in the
pores using this equation:
Pa: ambient pressure, T: ambient temperature
Vm: molar volume of the liquid adsorbate (34.7 cm3/mol for nitrogen)
R: universal gas constant
Since pores which would not be filled below P/P0 ≈1 have a negligible
contribution to the total pore volume, the average pore size can be
estimated from the pore volume.
For example for cylindrical pore:
S: BET surface area
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Characterizing Porous Materials and Powders
Pore Size Distributions (Mesopore)
It means the distribution of pore volume versus pore size. The desorption isotherm generally is
more appropriate than the adsorption isotherm for this purpose. For some special cases like
when a sample exhibits Type H2 hysteresis, the adsorption isotherm is recommended!
Mesopore size calculations based on cylindrical pore geometry assumption are done using
Kelvin equation:
For nitrogen as adsorbate:
rp: actual pore radius
t: thickness of the adsorbed layer; t=3.54 (Vads/Vm) or from de Boer
equation:
Vads: volume of nitrogen adsorbed at a given relative pressure
Vm: volume of nitrogen adsorbed at the completion of a monolayer
for a nonporous solid of the same composition as the porous sample
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Characterizing Porous Materials and Powders
BJH Method (for pore size distribution)
This method proposed by Barrett, Joyner, and Halenda (BJH). This method
is based on the assumption that the initial relative pressure is close to unity,
all pores are filled with liquid.
rp: pore radius
Vp: pore volume
rK: inner capillary radius
∆t: thickness of adsorbed layer of nitrogen
Ac: area exposed by the pore from which
the physically adsorbed gas is desorbed.
Ac can be calculated from Ap which is the area of each pore (see the
AsiQwin software manual for more details).
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Characterizing Porous Materials and Powders
DH Method (for pore size distribution)
This method proposed by Dollimore and Heal (DH).
∑Ap: areas of all the pores emptied of condensate in previous desorption steps.
∑Lp: lengths of all the pores emptied of condensate in previous desorption steps.
Assuming cylindrical pore geometry, the cumulative pore areas and length
can be estimated for each desorption step by:
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Characterizing Porous Materials and Powders
Surface Area of Microporous Samples by Langmuir Method
In the absence of meso and/or macropores, a sample containing micropores will
exhibit a Type I (see slide 10) or Langmuir isotherm. The Langmuir equation is a
limiting case of BET equation for the adsorption of a single molecular layer of
adsorbate:
W: weight of adsorbate at P/P0,
Wm: weight of adsorbate in a monolayer
It can be rewritten in the form of a straight line: (Vm can be found from slope)
This method is not applicable to composite
materials containing micropores and meso-and/or
macropores.
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Characterizing Porous Materials and Powders
Micropore Analysis (V-t Method)
The procedure is same as BET surface area measurement, but it extends the
pressure range to higher pressures to permit calculation of the external surface
area, that is, non-microporous part of the material. A t-plot which is a plot of
volume of gas adsorbed versus t (the statistical thickness of an adsorbed film) is
used.
The t values are calculated as a function of the relative pressure using the de Boer
equation:
the Carbon Black equation:
Or the Halsey equation:
For nitrogen adsorption
at 77 K
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Characterizing Porous Materials and Powders
Micropore Analysis (V-t Method)
Surface area of all the pores can be calculated as:
Vads STP: volume of gas adsorbed, corrected to standard conditions of temperature and pressure
15.47: the conversion of the gas volume to liquid volume
SMP: micropore surface area
St: external surface area
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Characterizing Porous Materials and Powders
Micropore Analysis (Alpha-s Method)
An empirical analogue to the t-method was proposed by Sing. It does
not need statistical thickness, t which depends on BET surface area of
nonporous reference material. Instead the BET area can be replaced by
the amount adsorbed at some arbitrarily chosen P/P0 (usually = 0.4) .
It is used for the determination of micropore volumes by extrapolation
of Alpha-s curves to αs=0, and for the determination of nonporous
surface contributions (from the slopes of linear portions of Alpha-s
curves).
In practice, complications arise using this method because it is found to
depend on the exact nature of the material chosen as nonporous
reference!
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Characterizing Porous Materials and Powders
Micropore Analysis (MP Method)
An extension of de Boer’s t-method for micropore analysis was proposed by Mikhail,
Brunauer, and Bodor.
This method calculate the t independently of the solid by:
t(Aͦ) = 104Vliq/SBET
A V versus t plot is constructed from t versus
P/P0 a shown in figure below, using linear
slopes constructed for t-value intervals from
the origin to 4 Aͦ, 4 to 4.5 Aͦ, 4.5 to 5 Aͦ, etc.
The calculations are continued until no further
decrease in the slope.
The area of pores in the thickness range from
4 to 4.5 A, for example is the difference
between the values calculated from first and
second slopes, and so on.
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Characterizing Porous Materials and Powders
Micropore Analysis (MP Method)
Pore volumes can similarly be calculated, using the relation:
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Characterizing Porous Materials and Powders
Micropore Analysis (Dubinin-Radushkevich (DR) Method)
Dubinin and Radushkevich considered that the fraction of the adsorption
volume V occupied by liquid adsorbate at various adsorption potentials ε can
be expressed as a Gaussian function:
A: free energy of adsorption, A= ε = RT ln (P/P0)
V0: microspore volume
E0: characteristic energy of adsorption
β: affinity coefficient
v: liquid molar volume of a given adsorbate
vC6H6: molar volume of benzene as the reference liquid
Micropore volume (V0) and E0 parameter can be calculated from the linear
fit of the isotherm data plotted as log (V) versus [log(P/P0)]2.
The linear range for these plots is usually found at relative pressures of less
than 10-2.
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Characterizing Porous Materials and Powders
Micropore Analysis (Dubinin-Astakhov (DA) Method)
For the microporous materials with heterogeneous distributions or strongly
activated carbons, the Dubinin-Radushkevich equation fails to linearize the
adsorption data, the Dubinin-Astakhov equation is used:
W: weight adsorbed at P/P0 and T
W0: total weight adsorbed
E: characteristic energy
n: non-integer value (typically between 1 and 3
n and E are calculated re-iteratively by
non-linear curve fitting to the adsorption
isotherm in the low relative pressure,
micropore region. The they are used in this
equation where r is pore radius and K is a
constant.
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Characterizing Porous Materials and Powders
Micropore Analysis (Horvath-Kawazoe (HK) Method)
Many pore size distribution methods are based on capillary condensation
phenomenon which is questionable for small confines of micropores.
Horvath and Kawazoe proposed a method which is independent of this
assumption!
The HK method expresses the adsorption potential function within slit-like
micropores as a function of the effective pore width:
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Characterizing Porous Materials and Powders
Micropore Analysis (Horvath-Kawazoe (HK) Method)
By selecting effective pore widths in the
micropore range, the HK equation can be
used to calculate the corresponding
relative pressures.
From the adsorption isotherms, the
amount of adsorption at each of these
relative pressures is determined.
Differentiation of weight (or volume) of
gas adsorbed relative to the total uptake,
W/W0, with respect to the effective pore
width yields a pore size distribution in
the micropore range.
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Characterizing Porous Materials and Powders
Micropore Analysis (Saito-Foley (SF) Method)
HK method is good for materials with a predominance of slit-like pores (like
activated carbons), but for some solids like zeolites that usually have
cylindrical pore geometry it is better to use the method that Saito and Floey
presented:
Everything (even numerical solution of this equation above) and the other
parameter definitions except the pore shape are analogous to that of HK
method and yields size distributions in the micropore range.
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Characterizing Porous Materials and Powders
Density Functional Theory (DFT) and Monte Carlo Simulation Methods
Classical macroscopic theories that we have introduced until now such as
DR method and BJH method and also semi-empirical treatments such as
HK method and SF method do not give a realistic description of the filling
of micropores and even narrow mesopores which leads to an
underestimation of pore sizes.
Treatments such as the Density Functional Theory (DFT) or methods of
molecular simulation (Monte Carlosimulation (MC) and Molecular
Dynamics (MD)) provide a much more accurate approach for pore size
analysis and bridge the gap between the molecular level and macroscopic
approaches.
DFT is the best suggested method for micro-mesoporous zeolite samples!
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Characterizing Porous Materials and Powders
Density Functional Theory and Monte Carlo Simulation Methods
The Non-Local Density Functional Theory (NLDFT) and the Grand
Canonical Monte Carlo simulation (GCMC) methods correctly describe
the local fluid structure near curved solid walls; adsorption isotherms in
model pores are determined based on the intermolecular potentials of
the fluid-fluid and solid-fluid interactions. The relation between
isotherms determined by these microscopic approaches and the
experimental isotherm on a porous solid can be interpreted in terms of a
Generalized Adsorption Isotherm (GAI) equation :
N(P/P0): experimental adsorption isotherm data
W: microspore volume
N(P/P0,w): isotherm on a single pore of width W
f(W): pore size distribution function
This equation can be solved numerically via a fast non-negative least
square algorithm.
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Characterizing Porous Materials and Powders
Library of DFT and GCMC methods in Quantachrome’s Data Reduction Software
This table is just
part of the large
library we have, for
a complete table
please read the
manual page 382
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Thank you for your attention
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