Methods for determining pores of various sizes. Size distribution (pores, particles). See also in other dictionaries

Such characteristics can be estimated in several ways from desorption isotherms. Brockhoff and Lineen provide a fairly detailed review of this issue. In addition to the labor-intensive technique of accurately measuring adsorption isotherms, most methods involve performing separate calculations for a large number of intervals of the isotherm in question. However, with a significantly improved method of measuring and issuing the results obtained, the ability to process the received data and compile programs for calculating pore sizes on a computer, such work is greatly simplified,

There are currently two types of commercial instruments available to perform this type of measurement. One uses a vacuum system, just 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 specific surface area values and pore size distributions can be obtained quite quickly.

Over the past century, various mathematical approximations have been developed to calculate the pore size distribution.

Most methods involve constructing 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 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, determining pore sizes requires not only the use of the Kelvin equation to calculate the sizes of pores that are filled with liquid nitrogen, which has the properties of a normal liquid, but also knowledge of the thickness of the adsorption film on inner surface pores not yet filled with nitrogen.

To obtain experimental data that takes into account film thickness, the silica under study must not contain micropores. Harris and Singh studied a number of such silica samples (with a specific surface area of less than 12 m2/g) and showed the possibility of drawing an isotherm averaged over the samples they examined in the form of a dependence of vjvm on pipe. However, since then, numerous studies have been carried out on corresponding non-porous silicas to accurately determine t-values. Bebris, Kiselev and Nikitin “prepared a very homogeneous wide-pore silica, not containing micropores, by heat treating fumed silica (aerosil) in water vapor at 750 ° C, obtaining the specified silica with a specific surface area of about 70-80 m2 / g and pores with a diameter of about 400 A Generally accepted values of film thickness t for various values of p!rho when using nitrogen are based on data from Lippens, Linsen and de Boer and de Boer, Linsen and Osinda.

In table 5.4 shows typical ^-values depending on p/p0. The following equation allows one to calculate film thickness using most of the published data based on average t values at p/po pressures above 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 data)

As described by Brockhoff 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 completed by Peirce and later by Cranston and Inkley. Subsequent developments of this problem have been described in detail by Greg and Singh.

Cranston and Inkley method. Cranston and Inkley (39), using the known film thickness t of adsorbed nitrogen on the inner walls of the pores along with filling the 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 section of the isotherm above p/po>0.3, where there is already an adsorbed at least monomolecular layer of nitrogen.

The method is a stepwise calculation procedure, which, although simple, provides for such calculations at each successive stage. A 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 each stage the adsorbed volume is measured (this applies to the desorption isotherm, but the calculation procedure will be the same when considering the adsorption isotherm). As the pressure decreases from the value pi/p0 to Pr/Poi, the following provisions are true:

1. A volume of liquid nitrogen AVuq evaporates from the pores, thereby 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 range of their radius sizes between r i and r2, leaves a nitrogen film of thickness t2 on the walls of these pores.

3. In pores emptied at previous stages, the thickness of the nitrogen film on the walls decreases from t\ to t2.

A reader unfamiliar with this issue may benefit from 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 vary in diameter. 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 at the same time 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 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 determined as fp = t + rk, where rz is the radius calculated from the Kelvin equation at a given pressure. Calculations are based on the following equations. Let L be the length equal to the total length of all emptied pores with radii in the range from r to r2, and let r 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 radius g is

A V р = nfpL Eliminating the value L, we get

Since rv - t = ru, where Γk is found from the Kelvin equation, then

The volume of released gas, measured at pressure p and temperature TC, corresponds to the volume of liquid

Vid = 2 377"_

Rice. 5.11. Diagram 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 liquid adsorbed substance. The adsorption film has a thickness tu and radius Kelvin in the pore,

Filled under the influence of surface tension, is equal to g, .

B - pressure Pr (P2

Those born as the pressure dropped from pt to pe (see text).

The area A of the internal surface of the pores under consideration, assuming that they are cylindrical, turns out to be equal to

A -2 (Vp/rr) ■ 104

Where Vp is expressed in cubic centimeters, and the radius gr is expressed in angstroms.

Using desorption data, calculations begin at p/p0 near 1.0, when the pores are essentially filled with liquid nitrogen. Cranston and Inkley described step-by-step calculations of pore volume and pore surface area emptied. Nevertheless, the detail of such consideration will be useful.

Calculations are performed at each stage at a fixed pressure, starting with filled pores and a relative pressure p/po close to 1.0. For each stage the following values are calculated:

1. Average? b. of two Kelvin radii Tk, and Tr at the corresponding pressures pі 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 рх and р2, expressed in angstroms. Each thickness t is taken from the tables or determined from the equation

T - 4.583/(lg Po/r)"/3

3. Average pore radius gr in this interval:

Gr = 0.5 [g + g k, + t2)

4. The value of t=t\ - t2, expressed in angstroms.

5. 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 nitrogen gas, reduced to normal conditions, cm3.

6. The volume of liquid nitrogen lost at this stage due to the thinning of 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) (£ A) 10~4 and has the dimension cm3, since At is expressed in angstroms, and

In square meters.

7. AA - 2(AVnq) Рр 104.

8. The value of £ A is found by summing all the DA values from the previous stages.

The specified calculation process is necessary at each stage of such a stepwise method. A series of calculations are performed for each stage in turn as the pressures decrease, and the results are tabulated.

The total pore volume Vc, starting from p/po = 0.3 and up to the largest value of p/po, is simply the sum of the AViiq values obtained at each stage. As a rule, a graphical dependence of Vc on log gr is drawn.

The total surface Lc is the total sum of AL values obtained at each stage. If there are no micropores, 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 values of p/p o from 0 to 0.3, such agreement indicates the absence of micropores in the sample.

Cranston and Inkley came to the conclusion that for many silica gels it is advisable 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 provided further discussion of the Cranston and Inkley method and provided some useful nomograms. However, it turned out to be not so easy to translate the system of equations into a method of practical calculations, which is why the calculation of the stages discussed above was shown in such detail.

The pore size distribution can be estimated from the ^-diagram according to data from Brockhoff and de Boer.

Micropores. Special problems arise when measuring and characterizing extremely small pores. It is impossible in this book to give an overview of all the vast literature that has appeared 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." Singh stated in 1976 that “no reliable method has been developed for determining the micropore size distribution.” It is clear, however, that adsorption in micropores is fundamentally different from adsorption on the surface of the walls of wide pores and on open surfaces, and that the molecules in such fine pores are subject to the attraction of 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 more accurately describes the process than the concept of the surface of such pores.

According to Okkers, the specific surface area in microporous materials cannot be determined if the radius of the micropore is less than 12 A. This author used the term “submicropore”, meaning by this concept
the same as other researchers, including Eyler, who used the term "micropore". Ockers summarized the possible application of a number of equations that have been proposed for the smallest pore sizes.

As clearly demonstrated by Brockhoff and Linsen, micropores can be detected by studying adsorption isotherms depicted as /-curves. If on the graph the line depicting the dependence of Va on / deviates downwards towards the /-axis, then this is an indication of the presence of micropores in the sample. Similar graphs obtained by Mikhail are presented in Fig. 5.12 for two silica gels. Since the values of the specific surface areas 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 downwards towards the /-axis at a relative pressure p/po = 0.1. For mesoporous silica gel B, which has a low density, the /-curve deviates upward at approximately p/po = 0.5, i.e., when wide pores begin to fill. In such gels, which have pores of uniform size, it is easy to demonstrate the presence of micropores. However, for many
In many silica gels, a large proportion of the surface belongs to mesopores and only a small part belongs to micropores. In this case, the deviation from linearity on the /-curve is difficult to determine. Mieville studied solid materials of mixed structure that 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, Singh showed the presence of meso-pores by deviation from linearity with respect to the a-axis at higher values of as. The presence of micropores is proven by the deviation of the curve towards the as-axis at lower cc values. s. Extrapolation of the linear section to the x-axis allows us to determine the volume of micropores (Fig. 5.13). The authors of the work conducted 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 the adsorption of nitrogen in dense compressed 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 densities were 67-80%), which corresponded to the formation of pores with a diameter of 22-12 A. In the graphs, presented in the coordinates of the specified equation, a decrease in the slopes of the lines for a series of samples is visible, which indicates changes occurring in them in the region from complete filling of the pore volume to a monolayer coating (when a monolayer of adsorbate fills the thinnest pores). In this work, the constant C on the graph plotted in BET coordinates had a value of 73 for the original, unpressed powder and increased from 184 to more than 1000 with time. how the pore diameter decreased from 22 to 12 A.

“Model pore” (MP) method. Brunauer, Mikhail and Bodor developed a method for determining the characteristic pore size distribution, including even part of the area occupied by micropores.

Using the Cranston-Inkley method, which also includes the /-curve and the Kelvin equation, 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 in fact 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.

In the “model pores” method, the concept of hydraulic radius “rh” is introduced, 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 applies to pores of any shape. The 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 empty 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 Cranston and Inkley method in that it uses the Kiselev equation instead of the Kelvin equation

U ds = Ar da "

Where y is surface tension; ds is the surface that disappears as the pore fills; - change in chemical potential, da - the number of liquid molecules located in the pore. (The Kelvin equation is a special case of the above Kiselev equation if cylindrical pores are considered.) The change in chemical potential is calculated by the equation -Ар = = -RT In (р/р0). 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 stages:

1. During the desorption of ai moles of a substance, the relative pressure p/po 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 (stage 2) divided by the surface area of such cores (stage 3).

Then at the nth stage, when an mole is desorbed, the following is observed:

1. Decrease in relative pressure p/po from rp/po to pn-l/po-

2. The resulting volume of the cores is 34.6 ap cm3. However, when the substance is desorbed, some volume is added

Adsorbate v„ from the walls of the pores formed on 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 of the cores. The introduction of such an amendment is a key point in the calculation.

3. The difference a„ - vn gives the value of the volume of newly formed cores at the nth stage.

4. The surface area of the new cores Sn is determined by graphical integration, as in the previous stages.

The above explanation is sufficient to show the difference between this “corrected model pore method” and the Cranston-Inkley method. For a more detailed description of the method and examples of calculations, you must 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 Inkley method. For example, for samples with pore radii in the range of 5-10 A when using the desorption isotherm according to this method the radius value at the maximum of the distribution curve was obtained to be about 6 A, and using the Cranston-Inkley method 10 A. Hannah et al.

For a wide range of different silica gels, good agreement in pore sizes was obtained using nitrogen or oxygen as an 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

We described the use of the “model pores” 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 determination of specific surface area and can also be used as a standard for determination of pore sizes and for calibration of equipment operating on the principle of the BET method over the entire pressure range.

The MP method was demonstrated by Mikhail, Brunauer and Baudot. They showed the applicability of this method to the study of micropores, and the “adjusted model pore method” to the study of pores big size. When this method is applied to silica gel, which has both micro- and mesopores, the MP method gives an aggregate value of the pore surface area that is consistent with the value found by the BET methods. This fact indicates that, despite the objections raised against the use of the BET method for studying microporous samples, this method can hopefully provide reliable data on specific surface areas even in these cases.

The detailed examination of the pore structure of five silica gels by Hagemassy and Brunauer can be considered typical of work of this kind in which the pore structure was assessed using the MP method. This article compared water and nitrogen vapor as adsorbates, and the data obtained were in fairly good agreement, giving pore diameters at the maxima of the distribution curves of 4.1 and 4.6 A, respectively. However, for adsorbents that have any hydrophobic surface areas, nitrogen must be used.

Supermicro -

The basis for this proposed classification is that supermicropores and mesopores, but not micropores, can be subjected to 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 the main topic of discussion. Obviously, if the smallest known gas molecule (helium) is not able to penetrate into a pore, then the pore simply does not exist, since this is confirmed

An 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 where a molecule enters a pore whose diameter is less than twice the size of the molecule. In this case, the van der Waal interaction is very strong, and the heat of adsorption is noticeably higher than on a flat surface. Therefore, such a situation differs from the one when the formation of a single polymolecular? loya or capillary filling of pores.

According to Dollimore and Heale, pores that are probably 7-10 A in diameter when determined from nitrogen adsorption isotherms are actually 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 values to form particles of approximately the same size. Dollimore and Hill prepared such a gel using the freeze-drying method of 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 during the gelation process was 1-2, i.e., exactly the value when the slowest growth of particles is observed. Such silica could be called “porous”, since helium molecules penetrated into such “pores” (and only these molecules). Note that helium molecules also penetrate into fused quartz. So, with the generally accepted approach, such silica is considered non-porous.

Isosteric heat of adsorption. The heat of adsorption in micropores turns out to be abnormally high. Singh and Ramakrishna found that through careful selection of adsorbates and the use of the a5 method of investigation, it was possible to distinguish between capillary adsorption and adsorption at high-energy surface sites. It was shown that in the p/po range of 0.01-0.2, the isosteric heat of nitrogen adsorption on silica gel not containing mesopores remains essentially constant at the level of 2.0 kcal/mol. On silica gel containing mesopores, a drop in heat is observed from 2.3 to 2.0 kcal/mol, and on microporous silica gel the isosteric heat drops from 2.7 to 2.0. Isosteric heat qst under - is read from adsorption isotherms using the Clausius-Cliperon equation.

Microporosity can simply be characterized by plotting the dependence of isosteric heat on p/p0, obtained 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 adsorption energy was highest in micropores and measured the surface area that was still available for the adsorption of nitrogen molecules at different stages benzene adsorption.

Dubischin characterized microporosity using the equation

Where a is the amount of adsorbed substance; T - absolute temperature; Wo is the maximum 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 the sample contains pores of two sizes, then a is expressed as the sum of two similar terms that differ in the values of Wо and B.

At constant temperature the equation takes the form

Where C in O can be calculated from adsorption isotherms and converted into Wо and B values. Dubinin used this method to obtain the characteristics of a silica gel sample containing micropores with diameters in the range of 20-40 A. This method is still being refined.

Adsorbates that vary in molecular size. Such adsorbates can be used in research by constructing /-curves in order to obtain the size distribution of micropores. Mikhail and Shebl used substances such as water, methanol, propanol, benzene, hexane and carbon tetrachloride. The differences in the data obtained were associated with the pore size of the silica sample, as well as the degree of hydroxylation of its surface. The molecules of most of the listed adsorbates are not suitable for measuring the surfaces of silicas containing fine pores.

Bartell and Bauer had previously carried out studies with these vapors at temperatures of 25, 40 and 45°C. Fu and Bartell, using the surface free energy method, determined the surface area using various vapors as adsorbates. They found that the surface values in this case were generally consistent with the values determined from nitrogen adsorption.

Water can be used to measure the surface of solid materials containing micropores of a size that makes it difficult for relatively large nitrogen molecules to penetrate them. The MP method, or “corrected model pore method,” was used by the authors of the work to study hydrated calcium silicate.

Another way to determine microporous characteristics is to take measurements at relative pressures near saturation. The differences in adsorption volumes show that this pore volume and size does not allow large selected adsorbate molecules to penetrate into them, while the smallest molecules used, such as water molecules, show “complete” 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 developed a new approach, the so-called pre-adsorption method, to identify 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 this, the sample surface was measured using the BET nitrogen method in the usual way, and the results of this determination were consistent with the geometrically measured surface that was found By electron microscopy, a similar pre-adsorption method for studying micropores can certainly be used for silica, but in this case, a much more polar adsorbate would probably have to be used to block the micropores, such as decanol.

X-ray scattering at small angles. Ritter and Erich used this method and compared the results obtained 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 Poraj-Kositz et al., Poroda and Imelik, Teichner and Carteret.

18 Order No. 250

Mercury pressing method. Mercury does not wet the surface of silica, and it is necessary to apply high pressure to force liquid mercury to enter 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 the pore wall (140°); gr - pore radius.

From this equation it follows that the product pgr = 70,000 if p is expressed in atmospheres and grp in angstroms. Mercury can penetrate into pores with a radius of 100 A at pressures above 700 atm. Therefore, very high pressures must be applied to penetrate mercury into micropores.

One problem is that unless the silica gel is very strong, the structure of the sample is destroyed by the external pressure of the mercury before the mercury can penetrate into 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 processing the data to construct pore size distribution curves.

Commercial mercury porosimeters are widely available, and improved versions of this method are described in the works. De Wit and Scholten compared the results obtained by mercury porosimetry with the results of methods based on nitrogen adsorption. They concluded that the mercury indentation method is unlikely to be used to study pores whose diameter is less than 10 nm (i.e., a radius less than 50 A). In the case of pressed Aerosil powder, the pore radius, determined by the indentation of mercury, 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 calculating the distribution curve 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 around 30 nm. Detailed description work on a commercial mercury porosimeter (or penetrometer), the introduction of the necessary corrections and the actual method for calculating pore sizes were presented by Frevel and Kressley. The authors also gave theoretical porosimetric curves for cases different packages spheres of uniform size.

Original document?

LECTURE4

Pore size distribution

The permeability of a porous medium depends primarily on the size of the filtration channels. Therefore, much attention is paid to studying the structure of the pore space.

The dependence of permeability on the size of filtration channels can be obtained by jointly applying Darcy's and Poiseuille's laws to a porous medium represented by a system of tubes having the same cross-section along the entire length. According to Poiseuille's law, fluid flow ( Q) through such a porous medium will be

(1)

Where n- number of pores per unit filtration area;

R- average radius of filtration channels;

F- filtration area;

DP- pressure drop;

m - dynamic viscosity of the liquid;

L- length of the porous medium.

The porosity coefficient of the porous medium model is equal to

(2)

Then, substituting (2) into (1), we get

(3)

According to Darcy's law, the fluid flow through such a porous medium will be

(4)

Here k- permeability coefficient.

Solving (3) and (4) for k, we get:

Where

If we measure permeability in mkm 2 and radius in mkm, then

(5)

The resulting expression is of little use for calculating the size of filtration channels in real porous media, but it gives an idea of the parameters of these media that have the strongest effect on permeability.

Studies of reservoirs in fields in Udmurtia and the Perm region made it possible to obtain correlations between the average radius of filtration channels and the filtration-capacitive characteristics of rocks. For terrigenous and carbonate rocks, this dependence is described, respectively, by the equations

Thus, over the entire range of changes in the filtration-capacitive characteristics of rocks, the average sizes of filtration channels in carbonates are 1.2-1.6 times higher than in terrigenous rocks.

Distribution of filtering 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 about the nature of the size distribution of filtration channels, calculating the average radius, and the characteristics of the heterogeneity of the porous medium. Capillary pressure curves characterize the dependence of rock water saturation on capillary pressure. They are obtained by mercury indentation, semi-permeable membrane or centrifugation. The first 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 stabilizing the weight of the sample or the volume of displaced liquid, the water saturation of the porous medium is calculated at a set pressure, which, when equilibrium is achieved, is considered equal to the capillary pressure. The process is repeated until the residual (or irreducible) water saturation characteristic of the geological conditions of the region being studied is achieved. The maximum pore pressure is established empirically for a specific 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, for example, kerosene. If in the first two methods the pressure in the sample is measured, then during centrifugation it must be calculated based on 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 when the sample rotates, a formula is used, obtained under the assumption that the porous medium is modeled by a bunch of filtration channels of variable cross-section.

Where P i- average pressure in a section of the filtration channel length l i, having a constant cross section.

and is presented in the form of a probability density distribution curve of filtering channels by size. The average equivalent radius of filtering channels is defined as

R av = S(R i av * W i)/ S W i ,(9)

where R i av =(R i + R i+1)/2 is the average radius in the range of changes in capillary pressure from P ki to P ki+1.

W i = (K i -K i+1)/(R i -R i+1) - probability density in this interval of radius changes.

Another area of application of capillary pressure curves is associated with assessing the nature of changes in water saturation of rocks in the transition zone of the formation. For this purpose, the results of capillarometry are presented in the form of the 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 a certain amount of associated water are assessed.

Surface wettability

The rock surface is wetted to varying degrees by formation fluids, which is reflected in the nature of their filtration. There are several methods for measuring wettability.

Firstly, a widely used method is based on measuring the geometric dimensions of an oil drop placed on a rock section and immersed in water or solution chemical substance. 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. It is important to know kinetic angles when studying selective wetting of rocks during the process of displacement of oil by water from porous media and for assessing the sign and magnitude of capillary pressure in filtration channels.

Where h– drop height;

d– diameter of the landing area.

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 inflow and outflow angles 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 filtration channels can be obtained by measuring the rate of saturation of a porous medium with a liquid or the capillary displacement of this liquid by another.

One of the simplest and most 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 by absorbing and draining water from samples rocks. The wetting index is defined as the logarithm of the ratio of the areas under the capillary pressure curves during drainage and absorption. The value of the wettability index varies from -1 for absolutely hydrophobic surfaces to +1 for absolutely hydrophilic ones. Rocks with a wettability 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 reservoirs of Udmurtia fields, wetting indices vary from -0.02 to +0.84. That is, predominantly hydrophilic rocks and rocks with intermediate wettability are found. Moreover, the latter predominate.

It should be noted that with all the variety of surface properties, wettability indicators represent a kind of integral characteristic, because in real porous media there are always channels that never contained oil and which therefore 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 is measured in m 2 / m 3 or in m 2 / g. The size of the specific surface depends on the mineral and granulometric composition, the shape of the grains, the content and type of cement. Natural adsorbents have the largest specific surface: clays, tripoli, certain types of bauxite, tuff ashes.

To assess the specific surface area, adsorption, filtration, optical, electron microscopic, granulometric and other laboratory research methods have been developed.

Adsorption methods can be static and dynamic and are based on: 1) adsorption of steam 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 nonequilibrium modes.

Mercury porosimetry and the method of displacing a non-wetting liquid that wets the pore space of rocks, or vice versa, are based on the study of capillary phenomena.

One way to estimate the specific surface area of filtration channels (Kozeny-Karman) involves studying the porosity, permeability and electrical conductivity in a rock sample. Then, knowing these parameters, you can calculate the specific surface area of the filtration channels

Here T g - hydraulic tortuosity;

f- Kozeny constant;

TO pr - permeability, m2;

m n - porosity, units

It is generally accepted that , where (here  vpc and  v are the electrical resistivity of water-saturated rock and water). The disadvantage of the method is the very conditional calculation of the tortuosity coefficient and the 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 using the formula

Where S sp - specific filtration surface, cm -1;

P He, P Ar- pressure in the helium and argon line, 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, o K;

J  , J D - total and diffusion flux of He through the sample, mol× s -1 .

Where W- volumetric velocity of the gas mixture, cm 3 /s;

WITH- volume concentration of He in the gas mixture,%.

Volume concentration He in the total flow of the gas mixture is determined from the calibration graph of the katharometer, plotted in coordinates U(v)-C(%). The magnitude of the He diffusion flux is determined by the dependence J= f(P He 2 -P Ar 2) as a segment cut off on the ordinate axis, a straight line passing through a number of experimental points.

For reservoirs of Udmurtia fields, dependences of the specific filtration surface on the filtration-capacitive characteristics of 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 the permeability of rocks and their 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 type of this dependence can be established on the basis of Darcy’s and Poiseuille’s laws (fluid flow in a cylinder).

Let's imagine porous rocks as a system of straight tubes of the same cross-section with length L (length of the rock volume).

According to Poiseuille's law, the fluid 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, ΔР is the pressure drop, μ is the dynamic viscosity of the liquid, L is the length of the porous medium.

Since the porosity coefficient (m) of the medium:

then substituting in (1.15) instead
porosity value m, we get:

(1.16)

On the other hand, the fluid flow Q is determined by Darcy's law:

(1.17)

Equating the right-hand sides of formulas (1.16) and (1.17), we find

(1.18)

(1.19)

(if [k]=µm 2, then [R]=µm).

The value of R 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 pores. F.I. Kotyakhov proposed a formula for determining the average pore radius (R) of real porous media:

(1.20)

where λ, φ – dimensionless parameters (φ – structural coefficient of pores with porosity m≈ 0.28÷0.39, φ≈ 1.7÷2.6), λ=
- constant value.

The structural coefficient for granular rocks can be approximately determined using the empirical formula:

(1.21)

Pore size distribution. Curves. Capillary pressure is the saturation of pores with a wetting phase.

Basic methods for determining the content of pores of various sizes (radius R) in porous rock:

method of pressing mercury into a sample;

semi-permeable partition method;

centrifugal method.

Mercury pressing 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 using a special press with a stepwise increase in pressure. The indentation of mercury is prevented by its capillary pressure in the pores, which depends on the radius of the pores and the wetting properties of mercury. The “radius” of the pores into which mercury is pressed is determined by the formula:

(1.22)

where P K is capillary pressure, δ is surface tension (for mercury δ=430 mN/m), θ is the contact angle (for mercury it is assumed θ=140 0), R is the pore radius.

When the pressure increases 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. In this way, 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 – pressure gauge;

5 – graduated liquid trap;

6 – supply of gas (nitrogen) under pressure.

The sample and membrane are saturated with liquid.

The pores of the membrane (ceramic, porcelain, etc. tiles) should be significantly smaller than the average pores of the sample.

: The liquid from the sample is displaced by nitrogen, the pressure of which is created inside chamber 2, and is measured by pressure gauge 4.

When the pressure increases, nitrogen first enters the large pores of the sample and the liquid leaves them through the pores of membrane 3 into a graduated trap 5. Nitrogen from chamber 2 through 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 tested pressures in the chamber.

By increasing the pressure in chamber 2 in steps and recording the corresponding volumes of liquid displaced from the sample using formula (1.22), the volume of pores is determined depending on the intervals of their radii (sizes) (it is first necessary to find the values of δ and θ of the liquid).

The results of the analysis are usually depicted in the form of differential pore size distribution curves (Fig. 10). The radii of pore channels in micrometers are plotted along the abscissa axis, and along the ordinate axis -
- relative change in pore volume per unit change in their radius R. According to experimental studies of reservoirs, fluid movement occurs through pores with a radius of 5 - 30 microns.

Centrifugal method.

Based on the rotation of a core saturated with liquid in a centrifuge. As a result, centrifugal forces develop, facilitating the removal of liquid from the pores. As the rotation speed increases, liquid is removed from pores of smaller radius.

The experiment records the volume of liquid flowing out at a given rotation speed. Based on the rotation speed, the centrifugal force and capillary pressure holding the liquid in the sample are calculated. Based on the value of capillary pressure, the size of the pores from which liquid flowed out at a given rotation speed is determined, and a differential pore size distribution curve is constructed.

The advantage of the centrifugal method is the speed of research.

Based on the data from all of the above measurement methods, in addition to the differential pore size distribution curve, it is possible to construct another curve - the dependence of capillary pressure on pore water saturation (Fig. 11).

P rock resistance:

K 3 >K 2 >K 1

The method of semi-permeable partitions makes it possible to obtain dependences Рк=f(S В) that are closest to reservoir conditions, because You can use water and oil as a saturating and displacing medium.

The dependence Рк=f(S В) is widely used when estimating the residual water saturation of a reservoir in the oil-water, water-gas transition zones.

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 experimentally studying these dependencies are needed. For example, installed:

the permeability of rocks for gas is always higher than for liquid (due to partial sliding 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 increasing 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 the 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%.

Adsorbents used:

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 RC specialists).

Working range of absolute pressure - 3.8 10 -9 - 950 mm Hg. Art.

Instrumental measurement error - 0.12-0.15%

It is possible to measure the adsorption rate at specified relative pressures. It is also possible to measure the isosteric heat of adsorption (if the user provides liquefied gases different in temperature from liquid nitrogen for a low-temperature bath).

Required characteristics:

1) it is desirable to have information about the absence/presence of porosity in the sample; if present, the nature of the porosity (micro- and meso-), the order of magnitude of the specific surface area

2) purpose of the study: BET surface, pore size distribution and pore volume (isotherm hysteresis loop and/or low pressure region) or complete adsorption isotherm

3) the maximum permissible temperature of sample degassing in vacuum (50-450°C with 1°C increments, recommended for oxide materials 150°C, for microporous materials and zeolites 300°C).

Sample requirements and notes:

1) Adsorption isotherm measurements are carried out only for dispersed (powdery) 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 amount of sample is 3-7 g (depending on the bulk density of the material).

3) Before measurement, samples must be degassed in a vacuum when heated. The sample must first be dried in an oven; no toxic substances must be released during degassing; the sample must not react with the 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 composition of the sample).

5) Determination of the specific surface area using the BET method, due to theoretical limitations, is impossible for materials with microporosity.

6) When measuring nitrogen adsorption from the gas phase, determining the pore size distribution is possible for pores with a width/diameter of 0.39 – 50 nm (when using the BDC method up to 300 nm, depending on the sample). The construction of a pore size distribution curve is made on the basis of various structural models: slit-like, cylindrical or spherical pores; It is impossible to determine the pore shape from the adsorption isotherm; this information is provided by the user.

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, micropores , monodisperse , morphology of nanostructures , nanopowder , nanopores , nanostructure , nanoparticle Determination of the dependence of the number (volume, mass) of particles or pores on their size in the material under study and the curve (histogram) describing this dependence. Description

The size distribution curve reflects the dispersion of the system. In the case when the curve looks like a sharp peak with a narrow base, i.e. particles or pores have almost the same size, they speak of a monodisperse system. Polydisperse systems are characterized by distribution curves that have broad peaks with no clearly defined maxima. If there are two or more clearly defined peaks, the distribution is considered bimodal and polymodal, respectively

It should be noted that the calculated particle (pore) size distribution depends on the model adopted for interpreting the results and the method for determining the particle (pore) size, therefore the distribution curves constructed according to various methods for determining the particle (pore) size, their volume, specific surfaces, etc. may vary

The main methods for studying the particle size distribution are statistical processing of data from optical, electron and atomic force microscopy, and sedimentation. The study of pore size distribution is usually carried out by analyzing adsorption isotherms using the BJH model. Authors

Links

Manual of Symbols and Terminology // Pure Appl. Chem. - v.46, 1976 - p. 71
Setterfield Ch. Practical course heterogeneous catalysis - M.: Mir, 1984 - 520 p.
Karnaukhov A.P. Adsorption. Texture of dispersed and porous materials - Novosibirsk: Nauka, 1999. - 470 p.

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Methods for determining pores of various sizes. Size distribution (pores, particles). See also in other dictionaries

Pore size distribution. Curves. Capillary pressure is the saturation of pores with a wetting phase.

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Pore size distribution. Curves. Capillary pressure is the saturation of pores with a wetting phase.