Analytical Description of the Pore Structure of Porous Powder Materials

Abstract

In this chapter, we propose a method for the analytical description of the porous powder materials’ (PPMs) pore distribution based on the pore structure data obtained by mercury porosimetry. The mercury porosimetry method is mostly informative and reliable when speaking about the recurrence of results as compared with other methods of pore distribution investigation. In this chapter, we present a calculation method of correcting experimental data of mercury porosimetry, based on the presentation of a porous body by a statistical model of a serial type.

1. Introduction

Consistent with the multiple functions performed by porous powder materials (PPMs) in various technical devices, a variety of computational methods were developed to assess the effectiveness of the PPMs’ varying pore structure. The relevant calculations use characteristics of the pore structure of the PPMs determined experimentally. The pore volume distribution on size and the average hydraulic pore size are considered as main, most common, characteristics of the pore structure. In this chapter, we propose a method for the analytical description of the PPMs’ pore distribution based on the pore structure data obtained by mercury porosimetry. The mercury porosimetry method is mostly informative and reliable when speaking about recurrence of results as compared with other methods of pore distribution investigation. However, a pore distribution function provided by this method has a distorted character. It increases the volume of small pores that is provided by the narrowing and widening of pore channels on the way of mercury travel. In this chapter, we present a calculation method of correcting experimental data of mercury porosimetry, based on the presentation of a porous body by a statistical model of a serial type.

2. Analytical description of the pore structure

Mercury porosimetry is the most accurate and informative method of studying the pore volume distribution on size. The essence of this method consists of measuring the quantity of mercury pressed in the pre-evacuated porous material, depending on the applied external pressure [1].

Mercury porosimeter operates as follows. The test sample is placed in a sealed cell which is evacuated; simultaneously the sample is degassed. Then, mercury is introduced into the cell so that mercury completely closes the sample. The mercury is automatically subjected to a predetermined pressure, which is left for a certain time so that the mercury fills all the pores that have the size larger than the critical value. At each table pressure value, the volume of mercury, which went down in the pores of the sample, is measured with a permittance method. According to the experimental data, the integral

$$F\left(d\right)=\frac{V\left(d\right)}{V_0}$$ E1

and the differential

$$f\left(d\right)=-\frac{1}{V_0}\frac{\mathit{dV}\left(d\right)}{\mathit{dd}}$$ E2

functions of pore volume distribution on size are calculated. Here V(d)—the volume of mercury—went down into the sample at the pressure corresponding to the critical pore size d; V₀—the total amount of mercury—went down into the sample at the maximal pressure.

For processing the experimental data, the following technique was developed. Because usually the minimum and maximum pore sizes of the PPMs differ by 1–2 orders of magnitude; the logarithmically uniform pressure table is pre-assigned that corresponds to the logarithmically uniform sequence of pore size values d₀^T, d₁^T, …, d_N^T:

$$\frac{{d_i}^T}{{d_{i-1}}^T}=\mathit{\text{const}},i=1,\dots ,N.$$ E3

However, because the automatically applied pressure is not exactly equal to the table value, and may differ from it by 1.5%, the real critical pore sizes d_i coincide with the table values with the same deviation:

$$d_i={d_i}^T\pm 0.015{d_i}^T,i=0,\dots ,N.$$ E4

According to the obtained values of the volume of mercury which went down into the sample V₀, V₁, …, V_N, integral function of pore distribution is approximated by a finite Fourier series [2]. As a first approximation, the volume V_i is deemed as related to the values of pore size distributed logarithmically evenly between d₀ and d_N and equal to d₀(d_N/d₀)^i/N, i = 0, …, N. Assuming

$$V_{2N-i}=V_i,i=1,\dots ,N,$$ E5

the volume values are calculated in the points d_i by the approximating function:

$${V_i}^{\left(j\right)}=\frac{{a_0}^{\left(j\right)}}{2}+\sum _{k=1}^{N-1}{a_k}^{\left(j\right)}\text{cos}\frac{\mathit{\pi k}\text{ln}\frac{d_i}{d_0}}{\text{ln}\frac{d_N}{d_0}}+{\left(-1\right)}^N\frac{{a_N}^{\left(j\right)}}{2},i=0,\dots ,N,$$ E6

where

$${a_k}^{\left(j\right)}=\frac{1}{N}\sum _{m=0}^{2N-1}{y_m}^{\left(j\right)}\text{cos}\frac{\mathit{\pi mk}}{N},k=0,\dots ,N;j=1,\dots ,J.$$ E7

In the last expression in the first approximation, as it was said,

$${y_m}^{\left(1\right)}=V_m,m=0,\dots ,2N-1,$$ E8

and successive approximation of the volume values in the points d_i is given by the approximating function to the experimental values provided by the next iteration:

$${y_m}^{\left(j\right)}={y_m}^{\left(j-1\right)}+V_m-{V_m}^{\left(j-1\right)},m=0,\dots ,2N-1;j=2,\dots ,J.$$ E9

A satisfactory accuracy of the approximation of the experimental results (deviation less than 1%) is usually achieved when the number of iterations is J = 5. Obtained values of the expansion coefficients a_k^(J) allow one to calculate the approximating function of pore volume distribution on size for any values of the pore size d₀ ≤ d ≤ d_N by the expression:

$$F\left(d\right)=\frac{1}{V_0}\left(\frac{{a_0}^{\left(j\right)}}{2}\right)+\sum _{k=1}^{N-1}{a_k}^{\left(j\right)}\text{cos}\frac{\mathit{\pi k}\text{ln}\frac{d}{d_0}}{\text{ln}\frac{d_N}{d_0}}+{\left(-1\right)}^N\frac{{a_N}^{\left(j\right)}}{2}.$$ E10

It is easy to obtain the expression for the approximating differential function by differentiating the last expression:

$$f\left(d\right)=\frac{1}{V_0}\frac{\pi }{d\text{ln}\frac{d_N}{d_0}}\sum _{k=1}^{N-1}{{\mathit{ka}}_k}^{\left(j\right)}\text{sin}\frac{\mathit{\pi k}\text{ln}\frac{d}{d_0}}{\text{ln}\frac{d_N}{d_0}}.$$ E11

Figures 1 and 2 show the processed, accordingly described, technique data on the experimental study of pore volume distribution on size of the PPMs obtained by sintering a freely poured copper powder PMS-N with a particle size from −315 to 200 μm.

Figure 1.

The results of the experimental investigation of volume pore distribution on size of sintered copper PMS-N. Particle size (−315 to +200) mm, sample weight 1.7 g.

Figure 2.

The approximating differential function of pore volume distribution on size of sintered copper.

The average hydraulic pore size characterizes the transport and evaporative capacity of the PPMs at full saturation of its pore space with working fluid. Experimental determination of the average hydraulic pore size is based on the use of Laplace’s law. The test sample in the form of a tablet is placed in the sleeve so that the rubber gasket is tightly compressed on the side surface of the sample (Figure 3). At the bottom of the sleeve is a socket, connected to a hose of sufficient length, filled with a liquid which completely wets the sample. The lower end of the hose is placed in a vessel containing the same liquid. A slow rise of the sample is produced. At the moment of separation of the liquid in the hose from the sample, the height of the sample over the liquid level in the vessel h is recorded. The average hydraulic pore size d is calculated by the expression:

$$d=\frac{4\sigma }{\mathit{\rho gh}},$$ E12

Figure 3.

Experimental determination of the average hydraulic pore size.

where $\sigma$ is the surface tension, $\rho$ is the density of the liquid, and $\mathrm{g}$ is the acceleration of free fall.

In case of partial draining of the pore space (e.g., with intense evaporation of the liquid inside the PPMs, the action of the mass forces, etc.), the pore size distribution becomes significant. The question arises about the relationship between the function of pore distribution and average pore size of PPMs. Special experiments and subsequent calculations showed that for the PPMs, fabricated with the same technology from different fractions of one powder, such a relationship exists. If the integral function of pore distribution of PPMs with the average hydraulic pore size of d₁ is F₁(d), then the integral function of pore distribution of PPMs with the average hydraulic pore size of d₂ can be calculated from the function F₁(d) by the coordinate transformation d → d′ = d d₁/d₂:

$$F_2\left(d\right)=F_1\left(d\frac{d_1}{d_2}\right);$$ E13

respectively, for the differential pore distribution function:

$$f_2\left(d\right)=-\frac{{\mathit{dF}}_2\left(d\right)}{\mathit{dd}}=-\frac{{\mathit{dF}}_1\left(d^{\prime }\right)}{d{d}^{\prime }}\frac{d{d}^{\prime }}{\mathit{dd}}=f_1\left(d\frac{d_1}{d_2}\right)\frac{d_1}{d_2}.$$ E14

Express provision is illustrated in Figure 4, where the experimental data for a porous bronze material BrOF10-1 depicts in the conventional coordinates (f, d) (a) and in the coordinates normalized by the average hydraulic pore size (f_n, $\frac{d}{d_i}$ ) (b). As shown, in the normalized coordinates the experimental points lie almost on the same curve.

Figure 4.

Experimental data on pore distribution of the porous bronze in conventional coordinates (a) and the normalized coordinates (b): 1, particle size <63 μm, d₁ = 8.1 μm; 2, 63–100 μm, d₂ = 15.2 μm; 3, 160–100 μm, d₃ = 24.2 μm; 4, 200–160 μm, d₄ = 33 μm.

3. Method to correct the data of mercury porosimetry

It is known [1] that pore distribution function, derived from the method of mercury porosimetry, is of a distorted character. It raises the volume of small pores that is caused by narrowing and widening porous channels on the way of mercury travel. Therefore, to use the data of mercury porosimetry in calculations of operational properties of porous materials, a correction of this data is necessary.

The developed method of correction of mercury porosimetry data is based on using a statistical model of a porous body of a serial type [3, 4]. In this model, a porous body is presented as a block of parallel capillaries, each of which consists of a number of successively disposed cylindrical elements. The diameter ζ and the length ξ of each element are random values, which do not depend on adjacent elements and are distributed with the probability density Ψ(ζ,ξ). Such a model is very similar to a real PPM structure and discloses the corrugateness of channels, as well as the accidental character of narrowing and widening.

Let us consider a process of mercury pressing into a model porous body. Let the mercury be on the left from the plane x = 0; on the right there is a porous body as a layer with the thickness l₀. If mercury pressure is p, it penetrates in the elements with the diameter, exceeding the critical one ζ_p = 4γ cosΘ/p, where γ is a coefficient of the mercury surface tension and Θ is an angle of moistening with the mercury of the porous body material. Thereby, mercury will enter the ν first

$$z_{\nu }=\sum _{i=1}^{\nu }{\xi }_i.$$ E15

elements, if ζ₁, …, ζ_ν > ζ_p, and ζ_ν+1 < ζ_p. Then the length of the mercury part of the given capillary is

Let the function of the mercury capillary part length be P(z). If suppressing in k-element, a corresponding distribution function is P_k(z), and a probability density is Ψ_k(z):

$$P_k\left(z\right)=\underset{0}{\overset{z}{\int }}{\mathrm{\Psi }}_k\left(x\right)\mathit{dx},$$ E16

then

$$P\left(z\right)=\sum _{k=1}^{\infty }{\omega }_k{P}_k\left(z\right),$$ E17

where ω_k is the probability of mercury suppression in the k-element, distributed as per a geometrical law:

$${\omega }_k={\mu }^{k-1}\left(1-\mu \right),$$ E18

μ is the relative number of elements with the diameter exceeding a critical one.

$$\mu =\underset{{\varsigma }_p}{\overset{\infty }{\int }}\mathit{d\varsigma }\underset{0}{\overset{\infty }{\int }}\mathrm{\Psi }\left(\varsigma ,\xi \right)\mathit{d\xi }.$$ E19

Under pressure p all the ξ_i are distributed in the same may with the probability density:

$${\mathrm{\Phi }}_p\left(\xi \right)=\underset{d_p}{\overset{\infty }{\int }}\mathrm{\Psi }\left(\varsigma ,\xi \right)\mathit{d\varsigma }/\underset{0}{\overset{\infty }{\int }}\mathit{d\xi }\underset{d_p}{\overset{\infty }{\int }}\mathrm{\Psi }\left(\varsigma ,\xi \right)\mathit{d\varsigma }.$$ E20

Ψ_k(x) is calculated via Ψ_{k − 1}(x) as follows:

$${\mathrm{\Psi }}_k\left(x\right)=\underset{0}{\overset{x}{\int }}{\mathrm{\Psi }}_{k-1}\left(x^{\prime }\right){\mathrm{\Phi }}_p\left(x-x^{\prime }\right)d{x}^{\prime }.$$ E21

Substituting Eq. (21) in Eq. (16), we shall get:

$$P_k\left(z\right)=\underset{0}{\overset{z}{\int }}{\mathrm{\Phi }}_p\left(x\right)P_{k-1}\left(z-x\right)\mathit{dx}.$$ E22

Next, substituting Eq. (22) in Eq. (17):

$$P\left(z\right)=\left(1-\mu \right)P_1\left(z\right)+\sum _{k=2}^{\infty }{\mu }^{k-1}\left(1-\mu \right)\underset{0}{\overset{z}{\int }}{\mathrm{\Phi }}_p\left(x\right)P_{k-1}\left(z-x\right)\mathit{dx}=\underset{0}{\overset{z}{\int }}{\mathrm{\Phi }}_p\left(x\right)\left(1-\mu +\mathit{\mu P}\left(z-x\right)\right)\mathit{dx}.$$ E23

Thereby, we derived an equation to find a function of the length z distribution of the mercury capillary part under the pressure in the mercury p, which is an integral Volterra equation of the second gender:

$$P\left(z\right)=\underset{0}{\overset{z}{\int }}{\mathrm{\Phi }}_p\left(x\right)\left(1-\mu +\mathit{\mu P}\left(z-x\right)\right)\mathit{dx}.$$ E24

Thereafter we shall consider a model, in which all the elements have the same length ξ₀, that is, the density of the distribution probability ξ_i is equal to the δ-function under any pressure p:

$${\mathrm{\Phi }}_p\left(\xi \right)=\delta \left(\xi -{\xi }_0\right).$$ E25

Justifying such a simplification is based on a smooth-changing a pore diameter. When the value of ξ₀ is rather small, a transversal size of the pore part, the length of which is ξ₀, may be considered as constant. Substituting Eq. (25) in (24), we derive an equation for P(z):

$$p\left(z\right)=1-\mu +\mu p(z-{\xi }_0),$$ E26

giving a step-by-step solution:

E27

where n = l₀/ξ₀ is the number of elements in one capillary.

The derived solution for P(z) must be connected with an experimental value of the entered mercury volume. If in the given capillary the length of the mercury part is z, the mercury volume in it is

$$v_{1p}\left(z\right)=v_{1}z\underset{d_p}{\overset{\infty }{\int }}f\left(\varsigma \right)\mathit{d\varsigma },$$ E28

where f(ζ) is the true density of pore volume distribution on sizes and v₁ is a medium volume of capillary length unit. A total volume of mercury, entered under the pressure p, is

$$v\left(p\right)=\underset{0}{\overset{l_0}{\int }}{N}_0\frac{\mathit{dP}}{\mathit{dz}}{v}_{1p}\left(z\right)\mathit{dz}+v_1{l}_0\underset{l_0}{\int }{N}_0\frac{\mathit{dP}}{\mathit{dz}}\mathit{dz}\underset{{\varsigma }_p}{\overset{\infty }{\int }}f\left(\varsigma \right)\mathit{d\varsigma },$$ E29

where N₀ represents a total quantity of the capillaries. In the expression (29), the first component considers the volume of partially filled capillaries, and the second component considers the volume of fully filled ones. Convert the expression (29), considering V₀ = v₁N₀l₀ (V₀ is the total volume of the porous area):

$$v\left(p\right)=V_0\left(1-\frac{1}{l_0}\underset{0}{\overset{l_0}{\int }}P\left(z\right)\mathit{dz}\right)\underset{d_p}{\overset{\infty }{\int }}f\left(\varsigma \right)\mathit{d\varsigma }.$$ E30

Using the solution (27), we may make a calculation:

$$\frac{1}{l_0}\underset{0}{\overset{l_0}{\int }}P\left(z\right)\mathit{dz}=1-\frac{\mu }{n}\frac{1-{\mu }^n}{1-\mu }.$$ E31

Substituting Eq. (31) into Eq. (30):

$$v\left(p\right)=V_0\frac{\mu }{n}\frac{1-{\mu }^n}{1-\mu }\underset{d_p}{\overset{\infty }{\int }}f\left(\varsigma \right)\mathit{d\varsigma }.$$ E32

A dependence exists between the functions μ(ζ) and f(ζ):

$$\frac{\mathit{d\mu }}{\mathit{d\varsigma }}=\frac{f\left(\varsigma \right)}{s{\varsigma }^2},$$ E33

where we use the designation, $s={\int }_0^{\infty }\frac{f\left(\varsigma \right)}{{\varsigma }^2}\mathit{d\varsigma }.$

Using Eq. (33), let us convert an integral in Eq. (32) (later on for convenience of writing let us consider v = v(d)):

$$\underset{\varsigma }{\overset{\infty }{\int }}f\left(\varsigma \right)\mathit{d\varsigma }=-s\underset{\varsigma }{\overset{\infty }{\int }}\frac{\mathit{d\mu }}{\mathit{d\varsigma }}{\varsigma }^2\mathit{d\varsigma }=s\left({\mathit{\mu \varsigma }}^2+2\underset{\varsigma }{\overset{\infty }{\int }}\mathit{\text{μςdς}}\right),$$ E34

Substituting Eq. (34) in Eq. (32), we get:

$$\frac{\mathit{bv}}{\mu }\frac{1-\mu }{1-{\mu }^n}-{\mathit{\mu \varsigma }}^2=2\underset{\varsigma }{\overset{\infty }{\int }}\mathit{\text{μςdς}},$$ E35

where there is marked b=n/(sV₀). Differentiating the expression (35) on d, we get the following equation after converting:

$$\frac{\mathit{d\mu }}{\mathit{d\varsigma }}=\frac{b\frac{\mathit{dv}}{\mathit{d\varsigma }}\mu \left(1-\mu \right)\left(1-{\mu }^n\right)}{{\mu }^2{\varsigma }^2{\left(1-{\mu }^n\right)}^2-\mathit{bv}\left(\left(1-\mu \right)n{\mu }^n-1+{\mu }^n\right)},$$ E36

which forms the Cauchy problem together with a boundary condition

$$\mu \left({\varsigma }_{\min }\right)=1$$ E37

to determine a true function of pore quantity distribution on sizes μ(ζ). In Eq. (21), the values b and n are indefinite and are connected with the desired function μ(ζ). Therefore, to solve the Cauchy problem there is an iterative method as follows.

Zero approximation μ₀(ζ) is obtained, considering:

$$f_0\left(\varsigma \right)=-\frac{1}{V_0}\frac{\mathit{dv}}{\mathit{d\varsigma }};s_0=\underset{0}{\overset{\infty }{\int }}\frac{f_0\left(\varsigma \right)}{{\varsigma }^2}\mathit{d\varsigma };{\xi }_{00}=\underset{0}{\overset{\infty }{\int }}\varsigma f_0\left(\varsigma \right)\mathit{d\varsigma };n_0=\frac{l_0}{{\xi }_{00}};b_0=\frac{n_0}{s_0{V}_0},$$ E38

where l₀ is the size of the porous material sample being investigated in the direction of mercury travel. The following approximations μ_i(ζ) are obtained using the following calculations:

$$f_i\left(\varsigma \right)=\frac{\frac{d{\mu }_{i-1}}{\mathit{d\varsigma }}}{2\underset{0}{\overset{\infty }{\int }}{\mu }_{i-1}\mathit{\varsigma d\varsigma }};s_i=\sqrt{s_{i-1}\frac{1}{2\underset{0}{\overset{\infty }{\int }}{\mu }_{i-1}\mathit{\varsigma d\varsigma }}};{\xi }_{0i}=\frac{3\underset{0}{\overset{\infty }{\int }}{\mu }_i{\varsigma }^2\mathit{d\varsigma }}{2\underset{0}{\overset{\infty }{\int }}{\mu }_{i-1}\mathit{\varsigma d\varsigma }};n_i=\frac{l_0}{{\xi }_{0i}};b_i=\frac{n_i}{s_i{V}_0}.$$ E39

Calculation of μ_i(ζ) on each step of the iteration is made by the Runge-Kutta method [5]; therewith, the initial value of the calculated function is μ(ζ_min) = 1, and the derivative is calculated using the equality

$$\underset{\mu \to 1}{\text{lim}}\frac{1-\mu }{1-{\mu }^n}=\frac{1}{n},$$ E40

with which it is possible to obtain:

$$\underset{\mu \to 1}{\text{lim}}\frac{\mathit{d\mu }}{\mathit{d\varsigma }}=\frac{b\frac{\mathit{dv}}{\mathit{d\varsigma }}}{n{\varsigma }^2-\frac{n-1}{2}\mathit{bv}}.$$ E41

In Figure 5 the results of calculating the functions of pore volume on size distribution from Eq. (36) and—for comparison—directly from the experimental data are given. It is seen that as a result of data correction of mercury porosimetry, the curves of pore distribution displace considerably in the direction of large pores.

Figure 5.

Function of pore volume distribution on sizes, calculated by the developed methodic (1) and directly from experimental data (2).

4. Conclusions

The relationship between the function of pore distribution and average hydraulic pore size, eliminating the need for a time-consuming set of experiments to determine the function of pore distribution of porous powder material, allowing to calculate the pore distribution function of porous powder material with any hydraulic average pore size from the known pore distribution function of the reference porous powder material with the fixed average hydraulic pore size, is explained.

The true function of pore distribution, obtained as a result of correcting mercury porosimetry data, enables to improve considerably the accuracy of calculations of processes and facilities parameters, where porous powder materials are used.