Dependence of permeability on porosity and pore size. Size distribution (pores, particles) Estimation of pore volume distribution
Used adsorbents:
1) Nitrogen (99.9999%) at liquid nitrogen temperature (77.4 K)
2) If the customer provides reagents, it is possible to carry out measurements using various, incl. liquid adsorbents: water, benzene, hexane, SF 6, methane, ethane, ethylene, propane, propylene, n-butane, pentane, NH 3 , N 2 O, He, Ne, Ar, Xe, Kr, CO, CO 2 ( after agreement with the specialists of the RC).
Operating range of absolute pressures - 3.8 10 -9 - 950 mm Hg. Art.
Instrumental measurement error - 0.12-0.15%
It is possible to measure the adsorption rate at given values of relative pressure. It is also possible to measure the isosteric heat of adsorption (if the user provides liquefied gases other than liquid nitrogen for the low temperature bath).
Required characteristics:
1) it is desirable to have information about the absence / presence of porosity in the sample, in case of presence - the nature of porosity (micro- and meso-), the order of magnitude of the specific surface
2) the purpose of the study: BET surface, pore size distribution and pore volume (hysteresis loop of the isotherm and / or low pressure region) or full adsorption isotherm
3) the maximum allowable sample degassing temperature in vacuum (50-450°С with a resolution of 1°С, recommended for oxide materials 150°С, for microporous materials and zeolites 300°С).
Sample requirements and remarks:
1) Measurements of adsorption isotherms are carried out only for dispersed (powder-like) samples.
2) The minimum required amount of an unknown sample is 1 g (if the specific surface area of the sample is more than 150 m 2 /g, then the minimum amount is 0.5 g, if the specific surface area exceeds 300 m 2 /g, then the minimum amount is 0.1 g). The maximum sample quantity is 3-7 g (depending on the bulk density of the material).
3) Samples must be degassed in a vacuum during heating before measurement. The sample must first be dried in an oven, no toxic substances should be released during degassing, the sample must not react with a glass measuring tube.
4) The minimum specific surface area of the material used for measurement is 15 m 2 /g (may vary depending on the nature of the surface and the composition of the sample).
5) Determination of the specific surface by the BET method due to theoretical limitations is impossible for materials with microporosity.
6) When measuring nitrogen adsorption from the gas phase, the determination of the pore size distribution is possible for pores with a width/diameter of 0.39 - 50 nm (when using the HBD method up to 300 nm, depending on the sample). The construction of the pore size distribution curve is based on various structure models: slit-like, cylindrical or spherical pores; it is not possible to determine the shape of the pores from the adsorption isotherm, this information is provided by the user.
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Usage: in instrumentation to study the filtration and hydraulic properties of filter materials, in particular to determine the pore size distribution. The essence of the invention: the speed and time of free flow of a given mass of gas, located in a sealed chamber under excess pressure, through dry and liquid-impregnated samples are measured at the same pressure drop across them. The pore size distribution is calculated from the ratio F i F =W ci T ci /W at i T at i where F i is the total area of open pores at the i-th pressure drop on the liquid-impregnated sample; F is the total area of through pores of all sizes in the material; W ci , W at i - gas velocity through dry and liquid-impregnated samples at i-th pressure drop on them, T ci , T at i - time of expiration of a given mass of gas through dry and liquid-impregnated samples at i-th pressure drop on them.
The invention relates to control and measuring equipment, namely, to the field of studying the filtration and hydraulic properties of filter materials, and can be used to evaluate their quality indicators. A known method for determining the pore size distribution, the implementation of which is obtained integral dependence of the change in the area of opening pores in a sample impregnated with a liquid from the pressure drop across it. The disadvantage of this method is the low sensitivity of gas flow control, due to the fact that the chain of series-connected elements is not reversible, which reduces the accuracy of determining the pore size distribution. Closest to the claimed technical solution is a method for determining the basic parameters of the structure of porous permeable bodies, which consists in passing gas under pressure through dry and liquid-impregnated samples. However, the known method has disadvantages, namely, that during mathematical processing of gas flow characteristics from pressure, graphical differentiation of experimental dependences is performed, which significantly reduces the accuracy of the method and increases the complexity due to the large amount of calculations. The purpose of the proposed method is to increase the accuracy and reduce the complexity of determining the pore size distribution. This goal is achieved by measuring the speed and time of the free flow of a given mass of gas, located in a sealed chamber under excess pressure, through dry and liquid-impregnated samples at the same pressure drop on them, and the pore size distribution is calculated from the ratio = , where F i - the total area of open pores at the i-th pressure drop on the sample impregnated with liquid: F - the total area of through pores of all sizes in the material; W ci , W at i - gas velocities through dry and liquid-impregnated samples at the i-th pressure drop across them; T ci , T at i - the time of the expiration of a given mass of gas through dry and liquid-impregnated samples at the i-th pressure drop on them. A comparative analysis of the proposed solution with the prototype shows that the proposed method differs from the known one in that the pore size distribution is determined by the ratio of the products of the velocities and times of free flow of a given mass of gas located in a sealed chamber under excessive pressure through dry and liquid-impregnated samples at the same pressure drop across them. Thus, the claimed method meets the criterion of the invention "novelty". A technical solution is known in which the gas contained in the chamber is passed through reference and controlled samples impregnated with liquid. However, the sequence of actions used in it does not allow us to determine the pore size distribution, which is determined in the claimed technical solution. This gives grounds to conclude that the proposed solution meets the criterion of "significant differences". Comparison of the outflow of a given mass of gas under excess pressure in a sealed chamber through dry and liquid-impregnated samples provides, at the same pressure drop across them, the possibility of determining the fraction of the area of open pores in a liquid-impregnated sample in relation to the products of the velocities and times of gas outflow through these samples. In accordance with the Boyle-Mariotte law for a given mass of gas, the process of its outflow from the chamber is characterized by the constancy of the product of pressure and the occupied volume. Therefore, the change in gas pressure from its initial value to the residual pressure value in the chamber characterizes the same amount of gas passing through dry and liquid-impregnated samples, with the same pressure drop across them in the specified range. Since, as the pressure drop decreases, the area of open pores in a sample impregnated with a liquid decreases, while in a dry one it remains constant, the product of the velocity and time of the outflow of identical specific volumes of gas will be inversely proportional to the ratio of the areas of open pores of these samples at the same value of the pressure drop across them. The proposed method for determining the pore size distribution is implemented as follows. An overpressure P is created in a sealed chamber, the value of which should be equal to or somewhat greater than the opening pressure of the smallest pore size, determined by the well-known Kantor dependence for equilibrium capillary pressure. In this case, the given mass of gas will occupy the volume U. By opening the quick-acting valve, the free outflow of gas through the sample impregnated with liquid is ensured. The pressure in the chamber will change from its initial value to some residual value characterizing the size of the maximum pores. For each fixed value of the gas outflow pressure in the specified range of its drop, the speed and time of the outflow are measured by known methods. The rate and time of gas flow through a dry sample is measured at the same fixed values of gas pressure in the specified range of pressure drop. The amount of gas passing through the sample impregnated with liquid is determined by the dependence U1 i = W at i T at i F i , where W at i is the gas flow rate through the sample impregnated with liquid at the i-th pressure drop across it; T at i - the time of the expiration of a given mass of gas through the sample impregnated with liquid at the i-th pressure drop across it; F i is the total area of open pores in the liquid-impregnated sample at the i-th pressure drop across it. Since U1 i =U2 i , and F = const, where U2 i is the amount of gas passing through a dry sample at the i-th pressure drop across it; F is the total area of through pores of all sizes in the material, then = .
Claim
METHOD FOR DETERMINING PORE SIZE DISTRIBUTION, which consists in passing gas under pressure through dry and liquid-impregnated samples located in a sealed chamber, and calculating the desired parameter, characterized in that at the same pressure drop on the samples, the speed and time of free flow of a given mass of gas are measured, and the pore size distribution is calculated from the relation
= ,
where F i is the total area of open pores at the i-th pressure drop on the liquid-impregnated sample;
F is the total area of through pores of all sizes in the material;
W ci , W at i - gas velocity through dry and liquid-impregnated samples at the i-th pressure drop on them;
T ci , T at i - the time of the expiration of a given mass of gas through dry and liquid-impregnated samples at the i-th pressure drop on them.
original document?
LECTURE4
Pore size distribution
The permeability of a porous medium depends mainly on the size of the filtration channels. Therefore, much attention is paid to the study of the structure of the pore space.
The dependence of permeability on the size of filtration channels can be obtained by jointly applying the laws of Darcy and Poiseuille to a porous medium represented by a system of tubes having the same cross section along the entire length. According to Poiseuille's law, the fluid flow rate ( Q) through such a porous medium will be
(1)
where n- the number of pores per unit area of filtration;
R- average radius of filtration channels;
F- filtration area;
DP- pressure drop;
m - dynamic viscosity of the liquid;
L is the length of the porous medium.
The porosity coefficient of the porous medium model is equal to
(2)
Then, substituting (2) into (1), we obtain
(3)
According to Darcy's law, the flow rate of liquid through such a porous medium will be
(4)
Here k- coefficient of permeability.
Solving (3) and (4) with respect to k, we get:
Where
If we measure the permeability in mkm 2 and the radius in mkm, then
(5)
The resulting expression is not very suitable for calculating the dimensions of filtration channels in real porous media, but it gives an idea of the parameters of these media, which have the strongest effect on permeability.
Studies of the reservoirs of the fields of Udmurtia and the Perm region made it possible to obtain correlations between the average radius of the filtration channels and the porosity and reservoir characteristics of the rocks. For terrigenous and carbonate rocks, this dependence is described, respectively, by the equations
Thus, in the entire range of changes in the reservoir characteristics of rocks, the average dimensions of filtration channels in carbonates are 1.2-1.6 times higher than in terrigenous rocks.
Distribution of filtration channels by size
One of the main methods for studying the structure of filtration channels in porous media is capillarometry - obtaining a capillary pressure curve and processing it in order to obtain information of interest on the nature of the size distribution of filtration channels, calculation of the average radius, and characteristics of the inhomogeneity of a porous medium. Capillary pressure curves characterize the dependence of rock water saturation on capillary pressure. They are obtained by indentation of mercury, semi-permeable membrane or centrifugation. The first one is now practically not used due to toxicity and the inability to reuse the studied samples in other studies. The second method is based on the displacement of water from a sample under pressure through a finely porous (semi-permeable) membrane saturated with water. In this case, the pressure in the sample increases stepwise, and after stabilization of the weight of the sample or the volume of the displaced liquid, the water saturation of the porous medium is calculated at the established pressure, which, at the achieved equilibrium, is considered equal to the capillary one. The process is repeated until the residual (or irreducible) water saturation, characteristic of the geological conditions of the region under study, is reached. The maximum pore pressure is established empirically for a particular region based on the results of a comparison of direct and indirect determinations of residual water saturation in the studied rocks.
The third method is based on the same principles, but is implemented by centrifuging samples saturated with water in a non-wetting liquid, such as kerosene. If in the first two methods the pressure in the sample is measured, then during centrifugation it has to be calculated based on the data on the speed and radius of rotation, the length of the sample, and the densities of the saturating liquids. To calculate the pressure created during the rotation of the sample, the formula obtained under the assumption that the porous medium is modeled by a bundle of filtration channels of variable cross section is used.
,
where Pi- average pressure in the section of the filtration channel with a length l i having a constant cross section.
and represent in the form of a probability density distribution curve of filtration channels by size. The average equivalent radius of the filtration channels is defined as
Rav = S(R i cf * W i)/ S W i ,(9)
where R i cf =(R i + R i+1)/2 is the average radius in the range of capillary pressure from R ki to R ki+1.
W i = (K i -K i+1)/(R i -R i+1) - probability density in this range of radii.
Another area of application of capillary pressure curves is related to the assessment of the nature of changes in the water saturation of rocks in the transition zone of the reservoir. To do this, the results of capillarometry are presented in the form of a Leverett function
Depending on the water saturation of the porous medium in the transition zone of the formation, phase permeabilities are determined and hydrodynamic parameters and the ability to produce oil with one or another amount of produced water are evaluated.
Surface wettability
The rock surface is wetted by reservoir fluids to varying degrees, which is reflected in the nature of their filtration. There are several methods for measuring wettability.
First, a method is widely used, which is based on measuring the geometric dimensions of an oil drop placed on a rock section and immersed in water or a chemical solution. Using an optical bench, static and kinetic contact angles can be measured. Static contact angles characterize the general physical and chemical characteristics of oil-bearing rocks and the wetting properties of liquids. Kinetic angles are important to know when studying the selective wetting of rocks in the process of oil displacement by water from porous media and for assessing the sign and magnitude of capillary pressure in filtration channels.
where h is the height of the drop;
d- landing area diameter.
The contact angle refers to a more polar liquid (water), so when calculating the contact angle of an oil drop in water, the measured angle is subtracted from 180°
.
All commonly used methods for measuring the angles of leakage and leakage on inclined plates do not make it possible to reproduce the processes occurring in real porous media.
Some idea of the wetting properties of water and the nature of the surface of the filtration channels can be obtained by measuring the rate of impregnation of a porous medium with a liquid or capillary displacement of this liquid by another.
One of the most simple and informative now is the Amott-Hervey method for assessing the wettability of the surface of filtration channels. It is based on the study of capillary pressure curves obtained when water is absorbed and drained from rock samples. The wetting index is then defined as the logarithm of the ratio of areas under the capillary pressure curves during drainage and imbibition. The wettability index value varies from -1 for absolutely hydrophobic surfaces to +1 for absolutely hydrophilic ones. Rocks with a wetting index ranging from -0.3 to +0.3 are characterized as having intermediate wettability. It is likely that the value of this wettability index is equivalent to Cos Q. At least it changes in the same range and with the same signs. In the reservoirs of the Udmurtia fields, the wetting indices vary from -0.02 to +0.84. That is, there are predominantly hydrophilic rocks and rocks with intermediate wettability. And the latter predominate.
It should be noted that with all the variety of surface properties, the wettability indices represent a certain integral characteristic, since in real porous media there are always channels that have never had oil and therefore have always remained hydrophilic. Therefore, it can be assumed that the main large filtration channels in which hydrocarbons move are much more hydrophobic than we can estimate using integral characteristics.
Specific surface area
The specific surface area is measured in m 2 /m 3 or m 2 /g. The specific surface area depends on the mineral and particle size distribution, grain shape, content and type of cement. ashes.
To assess the specific surface, adsorption, filtration, optical, electron microscopic, granulometric and other methods of laboratory research have been developed.
Adsorption methods can be static and dynamic and are based on: 1) adsorption of vapor and nitrogen, argon, krypton, water, alcohols, hydrocarbons; 2) adsorption of substances from solutions; 3) surface exchange; 4) heat of vapor adsorption and wetting.
Filtration methods are based on the filtration of compressed gases or liquids and rarefied gases in equilibrium and non-equilibrium modes.
Mercury porosimetry is based on the study of capillary phenomena and a method for displacing a non-wetting liquid that wets the pore space of rocks, or vice versa.
One way to evaluate the specific surface of filtration channels (Kozeny-Karman) involves the study of porosity, permeability and electrical conductivity in a rock sample. Then, knowing these parameters, it is possible to calculate the value of the specific surface of the filtration channels
here T d - hydraulic tortuosity;
f is the Kozeny constant;
To pr - permeability, m 2;
m p - porosity, d.
It is generally accepted that , where (here vpk and in - electrical resistivity of water-saturated rock and water). The disadvantage of the method is a very conditional calculation of the tortuosity coefficient and an unknown Kozeny coefficient.
Another method is based on the filtration of helium and argon through a sample of a porous medium. In this case, the value of the specific filtration surface is calculated by the formula
where S UD - specific filtration surface, cm -1 ;
P He, P Ar- pressure in the line of helium and argon, Pa;
m– porosity;
D, L- diameter and length of the sample, cm;
h ef - effective viscosity of the gas mixture, Pa× With;
R- gas constant 8.31× 10 7 ;
T- temperature, about K;
J , J D - total and diffusion flux of He through the sample, mol× with -1 .
where W- volumetric velocity of the gas mixture, cm 3 /s;
FROM- volume concentration of He in the gas mixture,%.
Volume concentration He in the total flow of a mixture of gases is determined by the calibration graph of the katharometer, built in the coordinates U(v)-C(%). The diffusion flux He is found from the dependence J= f(P He 2 -P Ar 2) as a segment cut off on the y-axis, a straight line passing through a number of experimental points.
For the reservoirs of the Udmurtia deposits, the dependences of the specific filtration surface on the porosity and permeability characteristics of the rocks were obtained. For terrigenous reservoirs, this dependence is described by a regression equation with a correlation coefficient of -0.928
with a correlation coefficient of -0.892.
Similar equations were obtained for a number of specific development objects.
There is no direct relationship between rock permeability and porosity. For example, low-porosity fractured limestones have high permeability, while clays, sometimes having high porosity, are practically impermeable to liquids and gases, because clays contain channels of subcapillary size. On average, of course, more permeable rocks are more porous. The permeability of rocks depends mainly on the size of the pore channels. The form of this dependence can be established on the basis of the laws of Darcy and Poiseuille (fluid flow in a cylinder).
We represent porous rocks as a system of straight tubes of the same cross section, length L (the length of the rock volume).
According to Poiseuille's law, the liquid flow rate Q through this porous medium is:
where n is the number of pores (tubes) per unit filtration area, R is the radius of the pore channels (or the average radius of the pores of the medium), F is the filtration area, ΔP is the pressure drop, μ is the dynamic viscosity of the liquid, L is the length of the porous medium.
Since the coefficient of porosity (m) of the medium:
then substituting into (1.15) instead of 
the value of porosity m, we get:
(1.16)
On the other hand, the fluid flow rate Q is determined by Darcy's law:
(1.17)
Equating the right sides of formulas (1.16) and (1.17), we find
(1.18)
(1.19)
(if [k]=µm 2, then [R]=µm).
The R value determines the pore radius of an ideal porous medium with permeability k and porosity m (rock models with straight tubes).
For a real porous medium, the value of R has a conventional meaning, because m takes into account the layered structure and tortuosity of the pores. F.I. Kotyakhov proposed a formula for determining the average pore radius (R) of real porous media:
(1.20)
where λ, φ are dimensionless parameters (φ is the structural coefficient of pores with porosity m≈ 0.28÷0.39, φ≈ 1.7÷2.6), λ= 
- a constant value.
The structural coefficient for granular rocks can be approximately determined by the empirical formula:
(1.21)
Pore size distribution. Curves. Capillary pressure is the saturation of pores with the wetting phase.
The main methods for determining the content of pores of various sizes in porous rock (radius R):
method of indenting mercury into the sample;
method of semi-permeable partitions;
centrifugal method.
Mercury indentation method.
A dry rock sample washed from oil is placed in a chamber filled with mercury (after evacuation). Mercury is pressed into the pores of the sample by a special press with a stepwise increase in pressure. The capillary pressure in the pores, which depends on the radius of the pores and the wetting properties of mercury, prevents the indentation of mercury. The "radius" of the pores into which mercury is pressed is determined by the formula:
(1.22)
where P K is the capillary pressure, δ is the surface tension (for mercury δ=430 mN/m), θ is the contact angle (for mercury θ=140 0 is taken), R is the pore radius.
When the pressure rises from P 1 to P 2 in the chamber, mercury is pressed only into those pores in which the applied pressure has overcome the capillary pressure of the mercury menisci, i.e. mercury enters the pores, the radius of which varies from R 1 = 
before 
. The total volume of these pores with radii (R 2 ≤R≤R 1) is equal to the volume of mercury pressed into the sample when the pressure increases from P 1 to P 2 .
The pressure is successively increased and the volume of mercury pressed in is recorded until the sample no longer accepts it. Thus, the volume of pores of various sizes is determined.
Method of semi-permeable (low-permeable) partitions.
Use the installation (Fig. 9):
1 – sample saturated with liquid (water or kerosene);
2
- camera;
3 - semi-permeable partition (membrane);
4 - manometer;
5 – graduated liquid trap;
6 - supply of gas (nitrogen) under pressure.
Sample and membrane are saturated with liquid.
Membrane pores (ceramic, porcelain and other tiles) must be significantly smaller than the average sample pores.
: Liquid from the sample is displaced by nitrogen, which is pressurized inside chamber 2 and measured by pressure gauge 4.
With an increase in pressure, nitrogen first enters the large pores of the sample and the liquid leaves them through the pores of the membrane 3 into the graduated trap 5. Nitrogen from chamber 2 through the membrane 3 can only break through when the pressure in it exceeds the capillary pressure of the minis in the pores of the membrane () - this the pressure is high due to the small pore sizes in the membrane and limits the upper threshold of the test pressures in the chamber.
By increasing the pressure in chamber 2 in steps and registering the corresponding volumes of liquid displaced from the sample, according to the formula (1.22), the pore volume is determined depending on the intervals of their radii (sizes) (you must first find the values of δ and θ of the liquid).
The results of the analysis, as a rule, are depicted in the form of differential pore size distribution curves (Fig. 10). In this case, the radii of the pore channels are plotted along the abscissa axis in micrometers, and along the ordinate axis - 
- relative change in pore volume per unit change in their radius R. According to the data of experimental studies of reservoirs, the movement of fluid occurs through pores with a radius of 5 - 30 microns.
centrifugal method.
It is based on the rotation of a core saturated with liquid in a centrifuge. As a result, centrifugal forces are developed that help remove liquid from the pores. As the rotation speed increases, the liquid is removed from the pores of a smaller radius.
In the experiment, the volume of liquid flowing out at a given rotation speed is recorded. The rotational speed is used to calculate the centrifugal force and the capillary pressure holding the liquid in the sample. The value of capillary pressure is used to determine the size of the pores from which liquid has flowed out at a given rotation speed, and a differential pore size distribution curve is plotted.
The advantage of the centrifugal method is the speed of research.
According to the data of all the indicated measurement methods, in addition to the differential curve of pore size distribution, it is possible to construct another curve - the dependence of capillary pressure on the water saturation of the pores (Fig. 11).
P 
permeability of rocks:
K3>K2>K1
The method of semi-permeable partitions allows to obtain dependences Рк=f(S В) closest to reservoir conditions, since you can use water and oil as a saturating and displacing medium.
Dependence Рк=f(S В) is widely used in assessing the residual water saturation of the reservoir in the transition zones oil-water, water-gas.
Laboratory methods for determining rock permeability.
Due to the fact that the permeability of rocks depends on many factors (rock pressure, temperature, interaction of fluids with the solid phase, etc.), methods for experimental study of these dependencies are needed. For example, set:
the permeability of rocks for gas is always higher than for liquid (due to partial slippage of gas along the surface of the channels - the Klinkenberg effect and absorption of liquid on the walls of reservoirs, swelling of clays, etc.);
with an increase in temperature and pressure, the gas permeability of rocks decreases (a decrease in the free path of molecules and an increase in friction forces): at a pressure of 10 MPa, in some rocks, gas permeability decreases by 2 times, compared with that at atmospheric pressure (0.1 MPa); with an increase in temperature from 20 0 C to 90 0 C, the permeability of rocks can decrease by 20 - 30%.
Such characteristics can be estimated in several ways from desorption isotherms. Broekhoff and Lineen have provided a fairly detailed review on this subject. In addition to the laborious method of accurately measuring adsorption isotherms, most methods provide for separate calculations for a large number of intervals of the isotherm under consideration. However, with a significantly improved method of measuring and issuing the results obtained, the possibility of processing the data obtained and compiling programs for calculating pore sizes on a computer, such work is greatly simplified,
There are currently two types of commercial instruments available for this kind of measurement. One uses a vacuum system like the original method
BET (Micromeritics instrument), and in the other, a gas flow system (Quantachrome instrument). An isotherm with 10-15 equilibrium points can be measured within a few hours, and surface area values and pore size distributions can be obtained fairly quickly.
Over the past century, various mathematical approximations have been developed to calculate the pore size distribution.
Most methods provide for the construction of a t* curve, since it is necessary to take into account the fact that adsorption occurs on a relatively smooth surface in the absence of pores and the adsorption film turns out to be several molecular layers thick before the vapor pressure reaches the value p/po = 1D corresponding to the formation of a liquid. Obviously, in such a thick film consisting of several layers, the properties of nitrogen will not be the same as for a normal liquid. As already noted, the determination of pore sizes requires not only the use of the Kelvin equation to calculate the size of the pores that are filled with liquid nitrogen, which has the properties of a normal liquid, but also the knowledge of the thickness of the adsorption film on the inner surface of the pores not yet filled with nitrogen.
To obtain experimental data that take into account the film thickness, the silica under study should not contain micropores. Harris and Sing studied a number of such silica samples (with a specific surface less than 12 m2/g) and showed the possibility of drawing an isotherm averaged over their samples in the form of a dependence of vjvm on pipe. Since then, however, numerous studies have been carried out on the corresponding non-porous silicas in order to accurately determine the t-values. Bebris, Kiselev and Nikitin "prepared a very homogeneous wide-pore silica, not containing micropores, by heat treatment of pyrogenic silica (aerosil) in water vapor at 750 ° C, obtaining the indicated silica with a specific surface of about 70-80 m2 / g and pores with a diameter of about 400 A The generally accepted values of film thickness t for different values of p, po using nitrogen are based on data from Lippens, Linsen and de Boer and de Boer, Linsen and Osind.
In table. 5.4 shows typical ^-values as a function of p/p0. The following equation makes it possible to calculate film thickness with most of the published data on average values of t at a relative pressure p/po greater than 0.3:
T_ 4.58 ~ Mg/V/>o)I/3
|
Table 5.4 Partial pressure of nitrogen and film thickness of nitrogen adsorbed on a non-porous surface at a temperature of -195°C (according to ) |
As described by Broekhoff and Linsen, many researchers have contributed to the development of methods for calculating pore size distributions from adsorption isotherms. The original approach and general equation developed by Barrett, Joyner, and Halenda were carried through to completion by Peirce and later by Cranston and Inckley. The subsequent development of this problem has been detailed by Greg and Sing.
Cranston and Inckley method. Cranston and Inckley (39), using the known film thickness t of adsorbed nitrogen on the inner walls of pores, along with the filling of pores with nitrogen according to the mechanism described by the Kelvin equation, developed a method for calculating the volume and size of pores from the desorption or adsorption branches of the isotherm. The calculation is carried out in the area of the isotherm above p/po>0.3, where there is already an adsorbed at least monomolecular layer of nitrogen.
The method is a step-by-step calculation procedure, which, although simple, provides for such calculations at each successive stage. The desorption isotherm consists of a series of experimental points, each of which contains data on the measured volume of adsorbed gas at a certain pressure. Starting from the point p/po = 1.0 with completely filled pores, the pressure is reduced stepwise and at the same time the adsorbed volume is measured at each stage (this applies to the desorption isotherm, but the calculation procedure will be the same when considering the adsorption isotherm). As the pressure decreases from pi/p0 to Pg/Poi, the following statements are true:
1. The volume of liquid nitrogen AVuq evaporates from the pores, thus forming a gas with a volume AVg, which is usually expressed in cubic centimeters under normal conditions per 1 g of adsorbent.
2. The volume AVnq of liquid nitrogen, which was removed from the pores in the size range of their radii between r i and r2, leaves a nitrogen film of thickness t2 on the walls of these pores.
3. In the pores emptied at the previous stages, the thickness of the nitrogen film on the walls decreases from t\ to t2.
The reader unfamiliar with this subject may be helped by the schematic representation of the process shown in Fig. 5.11. The figure shows a cross section of a sample with idealized cylindrical pores that differ in their diameters. It can be seen that when the pressure in the system decreases from pі (position A) to p2 (position B), the thickness of the nitrogen film on the walls of the emptied capillaries decreases from tx to t2, the amount of liquid nitrogen decreases as a result of desorption, and the number of empty pores increases.
In position A (Fig. 5.11) there is one partially filled pore with a diameter of 2r, in which liquid nitrogen is currently in equilibrium with steam at a pressure px. Similarly, in position B we have one pore with a diameter of 2r2, which contains liquid nitrogen, which is in equilibrium at pressure p2. In these pores, the radius is defined as fp = t + rk, where rb is the radius calculated from the Kelvin equation at a given pressure. The calculations are based on the use of the following equations. Let L be the length equal to the total length of all empty pores with radii in the range from r to r2, and rp be the average value of the radius. Then the total volume of evaporated liquid nitrogen Vuq at this stage is equal to
Vuq = 3.14 (rp - t2f L + (t2- tx) Z L
Where A is the surface of the adsorption film remaining in the indicated emptied pores.
The average volume of pores with a radius c is
A V p = nfpL Eliminating the value of L, we obtain
Since rv - t = ru, where Gk is found from the Kelvin equation, then
The volume of released gas, measured at pressure p and temperature TK, corresponds to the volume of liquid
Vid \u003d 2 377 "_
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Rice. 5.11. Scheme of an imaginary adsorbent with a set of cylindrical pores, shown in section, when nitrogen is adsorbed at two pressures and pr - A-pressure pi. All pores with a radius less than n are filled with a liquid adsorbed substance. The adsorption film has a thickness tu and a Kelvin radnus in the pore, Filled under the influence of surface tension, is equal to r, . B - pressure Рg (P2 Born as the pressure dropped from pt to pe (see text). |
The area A of the inner surface of the pores under consideration, assuming that they are cylindrical, turns out to be equal to
A -2 (Vp/rp) ■ 104
Where Vp is expressed in cubic centimeters, and the radius gr is in angstroms.
Using desorption data, calculations start at p/p0 near 1.0, when the pores are actually filled with liquid nitrogen. Cranston and Inckley described step by step calculations of pore volume and pore surface areas. Nevertheless, the details of such a consideration would be useful.
Calculations are performed at each stage at a fixed pressure, starting with filled pores and relative pressure p/po close to 1.0. For each stage, the following values are calculated:
1. Average value? b. of the two Kelvin radii Rk, and Rg at the corresponding pressures p1 and p2, expressed in angstroms. Each value is calculated from the Kelvin equation
4.146 Gk~ lgPo//>
2. Film thicknesses 11 and t2 at pressures px and p2 expressed in angstroms. Each thickness t is taken from tables or determined from the equation
T - 4.583/(lg Po/r)"/3
3. Average pore radius gr in this interval:
Gy = 0.5 [g + g k, + t2)
4. The value t=t\ - t2, expressed in angstroms.
5. The volume of desorbed liquid nitrogen AVnq per unit mass of the adsorbent, AVuq = 1.55-10-3 AVg, cm3/g, where AVg is the volume of released gaseous nitrogen reduced to normal conditions, cm3.
6. The volume of liquid nitrogen lost at this stage due to thinning of the films on the pore walls and equal to (A0 "(Z^)> where 2 A is the surface of the walls of all pores emptied during the desorption process at all previous stages (or AL for the first stage The indicated volume is equal to (At) (E A) 10-4 and has the dimension of cm3, since At is expressed in angstroms, and
in square meters.
7. AA - 2(AVnq) RR 104.
8. The value of £ A is found by summing up all the values of YES from the previous stages.
The specified calculation process is necessary at each stage of such a stepwise method. A series of calculations is performed for each stage in turn as the pressure decreases, and the results obtained are summarized in tables.
The cumulative pore volume Vc from p/po=0.3 up to the highest p/po is simply the sum of the AViiq values obtained in each step. As a rule, a graphical dependence of Vc on lg gr is drawn.
The cumulative surface LS is the total sum of the AL values obtained at each stage. If micropores are absent, then Ac usually amounts to values reaching 85-100% of the surface area determined by the BET method. Since the latter is obtained by measurements in the region of lower p/p o values from 0 to 0.3, such agreement indicates the absence of micropores in the sample.
Cranston and Inckley came to the conclusion that for many silica gels it is expedient to use the considered method in the opposite direction, starting from the value p/p0 = 0.3 and carrying out measurements and calculations at subsequent stages as the adsorption isotherm is obtained.
Hougen presented a further discussion of Cranston and Inckley's method and gave some useful nomograms. However, it turned out to be not so easy to translate the system of equations into the method of practical calculations, which is why the calculation of the stages considered above was shown in such detail.
The pore size distribution can be estimated from the ^-diagram according to the Broekhoff and de Boer data.
Micropores. A special kind of problem arises in the measurement and description of extremely small pores. It is not possible in this book to give an overview of all the vast amount of literature that has emerged over the past decade, but an attempt will be made to describe some aspects of this problem, accompanied by examples.
According to Brunauer, it is generally accepted that "the mechanism of adsorption of molecules in micropores is not well understood". Sing stated in 1976 that "no credible method has been developed to determine the size distribution of micropores." However, it is quite clear that adsorption in micropores differs fundamentally from adsorption on the surface of the walls of wide pores and on open surfaces, and that the molecules in such thin pores are attracted by the surrounding solid and are in a state of strong compression. Dubinin discussed the theory of adsorption under such conditions, which includes the concept of "micropore volume", which describes the process more accurately than the concept of the surface of such pores.
According to Okkers, the specific surface area in microporous materials cannot be determined if the micropore radius is less than 12 A. This author used the term "submicropore", meaning by this concept
the same as other researchers, including Ayler, who used the term "micropore". Okkers generalized the possible application of a number of equations that had been proposed for the smallest pores.
As was clearly demonstrated by Broekhoff and Linsen, micropores can be detected by examining adsorption isotherms plotted as /-curves. If on the graph the line depicting the dependence of Va on / deviates downward towards the /-axis, then this is an indication of the presence of micropores in the sample. Similar graphs obtained by Mikhail are shown in fig. 5.12 for two silica gels. Since the values of the specific surfaces of the samples are close, the lines on the /-diagrams have approximately the same slope. For silica gel A, which is microporous and dense, the /-curve begins to deviate downward towards the /-axis at a relative pressure p/po=0.1. For mesoporous silica B, which has a low density, the /-curve deviates upward at about p/po = 0.5, i.e., when wide pores begin to fill. In such gels having pores of uniform size, it is easy to show the presence of micropores. However, for many
In most silica gels, a large proportion of the surface belongs to meso-pores and only a small part belongs to micropores. In such a case, the deviation from linearity on the /-curve is difficult to determine. Mieville investigated solid materials of mixed structure, which had mesopores and micropores. He applied the /-diagram method and showed that in such a sample with a mixed structure, 10% are micropores.
Using the as-diagram, Synge showed the presence of mesopores by the deviation from linearity with respect to the ao-axis at higher values of as. The presence of micropores is evidenced by the deviation of the curve towards the as-axis at lower values of cc. s. Extrapolation of the linear section to the x-axis allows you to determine the volume of micropores (Fig. 5.13). The authors of the work carried out further research in this direction with a large set of silicas and gave an explanation for the deviations based on the concepts of micropores and mesopores.
Ramsay and Avery obtained data on nitrogen adsorption in dense compacted microporous silicas. They plotted their data using the equation
Pyrogenic silica powder with a particle size of 3-4 nm was pressed to obtain pore volumes of 0.22-0.11 cm3/g (silica packing density was 67-80%), which corresponded to the formation of pores with a diameter of 22-12 A. In the graphs, presented in the coordinates of this equation, one can see a decrease in the slopes of the lines for a series of samples, which indicates the changes occurring in them in the region from complete filling of the pore volume to a monolayer coating (when the adsorbate monolayer fills even the thinnest pores). In this work, the constant C on the BET plot was 73 for the original, uncompressed powder and increased from 184 to over 1000 as time went on. as the pore diameter decreased from 22 to 12 A.
Method of "model pores" (MP). Brunauer, Mikhail, and Bodor developed a method for determining the characteristic pore size distribution, including even a portion of the area occupied by micropores.
According to the Cranston-Inckley method, which also includes the /-curve and the Kelvin equation, the curves characterizing the porous structure of the sample can be calculated for pores with radii from 10 to 150 A. However, the results obtained depend on the assumption made about the cylindrical shape of the pores. Since the pores are not cylindrical, the calculation of the pore size distribution does not reflect the real state of affairs, especially in the presence of small pores.
The model pore method introduces the concept of hydraulic radius rh, defined as rh = V/S, where V is the volume of the porous system and 5 ■ is the surface of the pore walls. The ratio is applicable to pores of any shape. V and S values are calculated from adsorption or desorption isotherms. When desorption occurs and some group of pores is emptied, a monolayer of nitrogen molecules remains on their walls at a pressure p. The emptied space of the pore is called the “core”. This value represents the desorbed volume ■ as the pressure decreased from p0 to p.
This method differs from the method of Cranston and Inckley also in that instead of the Kelvin equation, it uses the Kiselev equation
Y ds = ar da "
Where y is the surface tension; ds is the surface that disappears as the pore is filled; is the change in the chemical potential, da is the number of liquid molecules in the pore. (The Kelvin equation is a special case of the above Kiselev equation when cylindrical pores are considered.) The change in chemical potential is calculated by the equation -Rp = -RT In (p/p0). Integration gives
S = -\ - RT In da
Where ah is the number of adsorbed molecules at the beginning of the hysteresis loop and as is the number of adsorbed molecules at saturation.
The last equation is integrated graphically in steps:
1. During desorption of ai moles of a substance, the relative pressure р/ро decreases from 1.0 to 0.95.
2. The resulting volume of all cores will be equal to the product of a\ and the molar volume of the adsorbate; for the case of nitrogen it is 34.6 a/cm3.
3. Si-surface area of the formed cores is determined by the equation
Integration is carried out graphically.
4. rh is the hydraulic radius equal to the resulting volume of the cores (step 2) divided by the surface area of such cores (step 3).
Then, at the nth stage, when apmoles are desorbed, the following is observed:
1. Decreasing the relative pressure r/ro from rp/ro to pn-l/po-
2. The resulting core volume is 34.6 ap cm3. However, when the substance is desorbed, some volume is added
Adsorbate v„ from the walls of the pores formed in the previous
stages. This volume vn is calculated based on the construction of the /-curve, which makes it possible to determine the value of At, i.e., the decrease in the thickness of the liquid film over the entire total surface of the cores formed up to this point. The volume is thus equal to the product of At and the total surface area of the cores. The introduction of such a correction is the key moment of the calculation.
3. The difference a - vn gives the value of the volume of newly formed cores at the n-th stage.
4. The surface area of the new cores Sn is determined by graphical integration, as in the previous steps.
This explanation is sufficient to show the difference between this "corrected model pore method" and the Cranston-Inckley method. For a more detailed acquaintance with the description of the method and examples of calculations, it is necessary to refer to the original source.
In most cases, the "model pore" method gives a smaller value of the pore radius at the maximum of the distribution curve than that obtained by the Cranston and Inckley method. For example, for samples with pore radii in the range of 5-10 A when using the desorption isotherm by this method the value of the radius at the maximum of the distribution curve was about 6 A, and by the Cranston-Inckley method 10 A. Hanna et al.
For a wide range of different silica gels, good agreement was obtained in pore sizes using nitrogen or oxygen as the adsorbate at two different experimental temperatures. In some cases noted in this work, silica samples contained both micro- and mesopores.
Standard for determining pore sizes. Howard and Wilson
Described the application of the "model pore" method on a sample of mesoporous silica Gasil(I), consisting of spheres with an average radius of 4.38 nm, packed with a coordination number of 4. Such silica is one of the standards
SCI/IUPAC/NPL for specific surface area and can also be used as a standard for pore size determinations and for calibrating BET instruments over the entire pressure range.
The MP method was demonstrated by Mikhail, Brunauer and Bodo. They showed the applicability of this method to the study of micropores, and the "corrected model pore method" to the study of large pores. When this method is applied to silica gel having both micro- and mesopores, the MP method gives an aggregate pore surface value consistent with that found by BET methods. This fact indicates that, despite the objections raised against the use of the BET method for the study of microporous samples, this method, as one might hope, is able to provide reliable data on specific surface areas even in these cases.
The detailed examination of the pore structure for five silica gels by Hagemassy and Brunauer can be considered typical of work of this kind, in which the pore structure was evaluated using the MP method. In this article, water vapor and nitrogen vapor were compared as adsorbates, and the data obtained were in fairly good agreement, giving values of pore diameters at the maxima of the distribution curves equal to 4.1 and 4.6 A, respectively. However, for adsorbents having any hydrophobic surface areas, nitrogen should be used.
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Supermicro - |
The basis for such a proposed classification is that supermicropores and mesopores, but not micropores, can be subjected to a detailed study.
The MP method was criticized, followed by a refutation of the criticisms.
Ultramicropores or submicropores. Such pores have a radius of less than 3 A. The mechanism by which such pores are filled has remained a major topic of discussion. Obviously, if the smallest known gas molecule (helium) is not able to penetrate into the pore, then the pore simply does not exist, since this is confirmed
Experiment. Thus, the lower limit of pore sizes at which these pores can be detected depends on the size of the adsorbate molecule used.
The main issue is to consider the situation when a molecule enters a pore whose diameter is less than twice the size of the molecule. In this case, the van der Waals interaction is very strong, and the heat of adsorption is noticeably higher than on a flat surface. Therefore, such a situation is different from the one when the formation of a single polymolecular? loya or capillary filling of pores.
According to Dollimore and Hill, pores that are probably 7-10 A in diameter, if determined from nitrogen adsorption isotherms, are in fact only 4-5 A in diameter. Submicropores in silica gel prepared from sol particles only ~ 10 A, turn out to be so small that even krypton molecules cannot enter them. Monosilicic acid is known to polymerize rapidly at low pH to form particles of about the same size. Dollimore and Hill prepared such a gel by the method of freeze-drying a 1% solution of monosilicic acid at a temperature below 0°C. Since a large amount of "water" was removed during evaporation and freezing, the pH value of the system was 1-2 during gelation, i.e., exactly the value when the slowest growth of particles is observed. Such silica could be called "porous", since helium molecules (and only these molecules) penetrated into such "pores". Note that helium molecules also penetrate fused quartz. So, in the conventional approach, such silica is considered non-porous.
Isosteric heat of adsorption. The value of the heat of adsorption in micropores turns out to be anomalously high. Sing and Ramakrishna found that through careful selection of adsorbates and the use of the a5 method of investigation, capillary adsorption and adsorption on high-energy surface sites can be distinguished. It is shown that in the p/po range of 0.01-0.2, the isosteric heat of nitrogen adsorption on mesopore-free silica gel remains essentially constant at the level of 2.0 kcal/mol. On silica gel containing mesopores, a drop in heat from 2.3 to 2.0 kcal/mol is observed, and on microporous silica gel, the isosteric heat drops from 2.7 to 2.0. Isosteric heat qst is calculated from adsorption isotherms using the Clausius-Claiperon equation.
Microporosity can be simply characterized by plotting the isosteric heat against p/p0 from nitrogen adsorption isotherms.
Calorimetric studies of microporosity were carried out, in which the heat released during the adsorption of benzene on silica gel was measured. They confirmed that the adsorption energy is highest in micropores and measured the surface that was still available for the adsorption of nitrogen molecules at different stages of benzene adsorption.
Dubischin characterized microporosity using the equation
Where a is the amount of adsorbed substance; T is the absolute temperature; Wo - limiting volume of micropores; v* is the molar volume of the adsorbate; B is a parameter that characterizes the size of micropores.
In the case when there are pores of two sizes in the sample, then a is expressed as the sum of two similar terms that differ in the values of Wo and B.
At a constant temperature, the equation takes the form
Where C to O can be calculated from adsorption isotherms and converted to Wo and B. Dubinin used this method to characterize a silica gel sample containing micropores with diameters in the range of 20-40 A. This method is still being finalized.
Adsorbates differing in molecular size. Such adsorbates can be used in the study by plotting /-curves in order to obtain the micropore size distribution. Mikhail and Sheble used substances such as water, methanol, propanol, benzene, hexane and carbon tetrachloride. Differences in the obtained data were associated with the pore size of the silica sample, as well as with the degree of hydroxylation of its surface. The molecules of most of the listed adsorbates are not suitable for measuring silica surfaces containing fine pores.
Bartell and Bauer had previously performed studies with these vapors at temperatures of 25, 40 and 45°C. Fu and Bartell, using the method of free surface energy, determined the surface area using various vapors as adsorbates. They found that the surface values in this case generally agreed with those determined from nitrogen adsorption.
Water can be used to measure the surface in solid materials containing micropores of such a size that it is difficult for relatively large nitrogen molecules to penetrate into them. The MP method, or “corrected model pore method”, was applied by the authors of the work in the study of hydrated calcium silicate.
Another way to determine microporous characteristics is to make measurements at relative pressures near saturation. Differences in adsorption volumes show that this volume and pore size does not allow large selected adsorbate molecules to penetrate into them, while the smallest molecules used, such as water molecules, show "full" penetration into these pores, determined by the adsorption volume. .
When the micropores are too small for methanol or benzene molecules to enter, then they are still able to absorb water. Vysotsky and Polyakov described a type of silica gel that was prepared from silicic acid and dehydrated at low temperature.
Greg and Langford have developed a new approach, the so-called pre-adsorption method, to determine micropores in coals in the presence of mesopores. First, nonane was adsorbed, which penetrated into the micropores at 77 K, then it was pumped out at ordinary temperature, but the micropores remained filled. After that, the surface of the sample was measured by BET using nitrogen in the usual way, and the results of such a determination were consistent with the geometrically measured surface, which was found A similar pre-adsorption technique for studying micropores can no doubt be used with silica, but in such a case a much more polar adsorbate should probably be used to block the micropores, such as decanol.
X-ray scattering at small angles. Ritter and Erich used this method and compared their results with adsorption measurements. Longman et al. compared the scattering method with the mercury indentation method. Even earlier, the possibilities of this method were described by Porai-Koshitz et al., Poroda and Imelik, Teichner and Carteret.
18 Order No. 250
Mercury injection method. Mercury does not wet the surface of the silica, and high pressure is required to force liquid mercury into small pores. Washburn derived the equation
Where p is the equilibrium pressure; a - surface tension of mercury (480 dynes/cm); 0 - contact angle between mercury and pore wall (140°); r is the pore radius.
It follows from this equation that the product pgr = 70 000 if p is expressed in atmospheres, and rp is in angstroms. Mercury can penetrate into pores with a radius of 100 A at pressures above 700 atm. Consequently, very high pressures must be applied to penetrate the mercury into the micropores.
One problem is that if the silica gel is not very strong, the structure of the sample collapses under the external pressure of the mercury before the mercury can penetrate the fine pores. It is for this reason that the method of measuring nitrogen adsorption isotherms is preferable for research purposes. However, for strong solids like industrial silica catalysts, mercury porosimetry is much faster, not only in terms of performing the experiment itself, but also in terms of data processing in order to plot pore size distribution curves.
Commercial mercury porosimeters are widely available, and improvements to this method are described in . De Wit and Scholten compared the results obtained by mercury porosimetry with those based on nitrogen adsorption. They concluded that the mercury indentation method could hardly be used to study pores smaller than 10 nm in diameter (i.e., radius less than 50 A). In the case of compressed aerosil powder, the pore radius determined from mercury indentation at the maximum of the distribution curve turned out to be about 70 A, while the nitrogen adsorption method gave values of 75 and 90 A when the distribution curve was calculated by different methods. The discrepancy may be due to a curved mercury meniscus with a radius of about 40 A, which has a lower (almost 50%) surface tension than in the case of mercury contact with a flat surface. According to Zweitering, there is excellent agreement between these methods when the pore diameter is about 30 nm. A detailed description of the work on a commercial mercury porosimeter (or penetrometer), the introduction of the necessary corrections, and the actual method for calculating the pore sizes were presented by Frevel and Kressley. The authors also gave theoretical porosimetric curves for different packings of spheres of uniform size.
