Introduction

The nature of green strength is still to be understood. In particular, the contribution of different components to strength yet remains to be determined [1]. Bonds between individual particles are considered to result from adhesion or self-adhesion interaction [2, 3]. Pietsch [2] mainly considered forces associated with van der Waals, electrostatic, and magnetic interaction. Zimon and Andrianov [3] calculated the self-adhesion force (F1) as a sum of the following components:

$$ {F_1} = {F_{\text{m}}} + {F_{\text{c}}} + {F_{\text{e}}} + {F_{\text{mech}}} - {F_{{\text{e}}{\text{.a}}}}, $$
(1)

where F m , F c , F e , F mech, and F e.a are van der Waals, cohesion,Footnote 1 electric, mechanical, and elastic aftereffect forces. The green strength can be divided into the molecular or van der Waals (σVW), mechanical (σmech), electrostatic (σel), chemical (σchem) [4], magnetic (σmag), and elastic aftereffect (σe.a) components. The van der Waals component is represented as:

$$ {\sigma_{\text{VW}}} = {\sigma_{\text{o}}} + {\sigma_{\text{p}}} + {\sigma_{\text{d}}} + {\sigma_{{\text{d}}{\text{.e}}{\text{.l}}}} - {\sigma_{\text{r}}}, $$

where σo, σp, σd is the orientation, polarization, and dispersion components, respectively [5]; σd.e.l is due to the formation of a double electric layer [6]; and σr is the component allowing for repulsion between molecules [7]. The mechanical component

$$ {\sigma_{\text{mech}}} = {\sigma_{\text{i}}} + {\sigma_{\text{s}}} + {\sigma_{\text{en}}} $$

is represented by contributions from interlocking σi, seizure σs, and entanglement σen of particles [8].

Elastic aftereffect σe.a of a compact differs in vertical (σe.a.vert) and horizontal or lateral (σe.a.lat) directions:

$$ {\sigma_{{\text{e}}{\text{.a}}{\text{.lat}}}} = \xi {\sigma_{{\text{e}}{\text{.a}}{\text{.vert}}}},{\sigma_{{\text{e}}{\text{.a}}{\text{.vert}}}} = {{{ - {P_{\text{c}}}}} \left/ {{{S_{\text{com}}}}} \right.}, $$

where P c, σe.a.vert, and σe.a.lat are the compacting pressure and vertical and lateral stresses, respectively; ξ is the lateral pressure coefficient (from 0.2 to 0.8 [9]); and S com is the area of the compact contacting with the punch.

Note that this paper does not address components σel and σmag resulting from magnetization or charging of powders. Component σchem occurs only under special deformation processing of powder blanks [4], and can be neglected in ordinary conditions of warm and cold pressing [1]. Hence, the green strength in our case is determined by molecular and mechanical components.

There are two methods to evaluate the molecular interaction between macroscopic bodies such as powders: microscopic and macroscopic [10, 11]. With the first method, the energy of interaction is determined by integrating pairwise molecular interactions, additivity of dispersion forces being taken into account. However, calculations based on adding together the energy of pair interactions have no adequate theoretical justification and can apply only to systems consisting of isolated particles, i.e., to the ideal case. With the other method, the interacting bodies are regarded as a continuum. Their interaction is due to a fluctuating electromagnetic field present inside every matter and going beyond its boundaries. This approach can apply to any bodies, regardless of their molecular nature and the distance between particles. The calculations use the Maxwell equation for the distance between particles longer than electromagnetic wavelengths of the material. The equation also accounts for delay effects. The experimental results agree well with the theory developed by Lifshitz [12].

The papers [13–16] attempt to evaluate the effect of van der Waals forces in pressing of different powders, but no calculations were performed because particle material constants were missing. The only exception was pressing of fine ZrO2 powders, when an approximate value of the Hamaker constant was used [15].

Easterling and Thölén [13] provided only an expression for the interaction between two spherical particles:

$$ {f_{\text{LW}}} = \frac{{\hbar \omega d}}{{32z_0^2}}, $$
(2)

where ħ ω is the Lifshitz–van der Waals constant (ħ is Planck’s constant, ω is characteristic frequency of the absorption spectrum of particle material); d is the diameter of a particle; and z 0 is the shortest possible distance between two particles.

Grechka [14], with reference to Rumpf and Orr, gives only a formula (without experimental data) to calculate green tensile strength ensured by van der Waals forces:

$$ {\sigma_{{\text{VW}}\left( {\text{Gr}} \right)}} = \frac{{3A{\rho_{\text{comp}}}K}}{{64{z^2}\pi d}}, $$
(3)

where A is the Hamaker constant depending on particle material and varying from 10–13 to 10–12 erg (10–20 to 10–19 N∙m); z is the distance between two particles (<100 nm); ρcomp is the compact relative density; and K is the coordination number.

To determine the strength of a compact made of ceramic equigranular powder in the absence of binder, Bortzmeyer [15] proposed the following equation:

$$ {\sigma_{{\text{VW}}\left( {\text{H}} \right)}} = \frac{{A{\rho_{\text{comp}}}}}{{24z_0^2\left( {1 - {\rho_{\text{comp}}}} \right)d}}, $$
(4)

where the force contributing to this strength was calculated as

$$ {f_{\text{H}}} = \frac{{Ad}}{{24z_0^2}}. $$
(5)

The strength due to van der Waals forces (~0.03 MPa) was much lower than the values obtained for compacts of fine ZrO2 powder. Since there is no van der Waals force calculated for metal powders, we use data for pharmaceutical powders whose particles are close to metal particles in size. The strength of different pharmaceutical pills is directly proportional to the measured force of molecular interaction, which varied from 1.2 to 66 nN [16].

Van der Waals forces have not calculated for pressing of metal and nonmetal medium-size and coarse powders. In this regard, the contribution of van der Waals forces to the strength of such compacts is of particular interest.

Our goal is to determine and calculate the effect of van der Waals forces on the strength of metal and nonmetal medium-size and coarse powders and to compare the results with the tensile strength of compacts produced at different compacting pressures.

Materials and methods

Table 1 summarizes the properties of some powders of five metals (Al, Zn, Cu, Ni, and Mo) and one nonmetal (FeSi) [8]. The following parameters are determined for the powders: relative apparent density (RAD) as per standard DSTU 2495–94; average particle size, dry screening method (as per GOST 23402–78); and oxygen content, hydrogen reduction method (GOST 18897–98). There are also characteristic temperature (t*), onset recrystallization temperature (t or ) [17]; atomic spacing (b) [18], and strength of particle material [19].

Table 1 Properties of Starting Powders and Particle Material
Full size table

Compacts 11.3 mm in diameter and 11–14 mm in height were produced by double-ended pressing using the standard procedure to determine densification at room temperature (as per GOST 25280–90). The compacting pressure was 200, 400, 600, and 800 MPa for most powders. The green relative apparent density was determined geometrically (compacts were measured with a micrometer and weighed at a VLR-200 analytical balance). The green tensile strength was determined with an indirect method [20]. For the metals, the Lifshitz–van der Waals constants (ħ ω), taken from different sources [21, 23, 24], ranged from 2.08∙10–19 to 8.5∙10–19 N∙m.

Experimental

The following simplifications were used for approximate calculations.

  • To determine the maximum van der Waals contribution to strength, atomic spacing in crystallites (b) [18] was used as the distance between particles; i.e., it was considered that there was no air space between particles and their surface was not covered with layers of any other molecules or chemical compounds.

  • The green relative density without applied pressure corresponded to the relative apparent density of the powder and, in other cases, to the relative density of the compact after pressing.

  • All powders were regarded as equigranular, with average size of particles. All particles were spherical.

  • For a reference calculation, we used the Hamaker constants for our metals in combination with platinum [22] and the value for ceramic materials [15] (1∙10–19 N∙m) for FeSi, which is acceptable (Hamaker constant for Si and SiO2 is 2.3∙10–19 and 0.853∙10–19 N∙m, respectively [21]).

  • To evaluate the van der Waals contribution to green strength, we took into account only the order of magnitude since, given the low accuracy of measured van der Waals forces, the orders of values match well [25].

  • In analysis of green strength, the Brazilian test is not appropriate for comparison of tensile strength (break) of materials with different yield stresses because of different stress states that occur during tests, but it can be used to qualitatively compare the tendency of materials to breaking. These simplifications should not significantly affect the results.

Table 2 shows the tensile strength of powder compacts determined indirectly (σtl.av) [1], the van der Waals component calculated with the Hamaker constant (σVW(H)max and σVW(Gr) calculated by formulas (3) and (4), respectively) and with the Lifshitz–van der Waals constant (σVW(LW)max by formula (3), where (2) was used instead of the force calculated by (5)), and their ratios. Note that the van der Waals component increased with compacting pressure from 0.0002 MPa for compacts from electrolytic copper powder at zero pressure to 0.36 MPa for compacts from gas-atomized copper powder at a pressure of 800 MPa.

Table 2 Tensile Strength Calculated for Powders by Formulas (3)–(4)
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The strength changes from ~0 MPa for atomized copper powder compacts to 50 MPa for carbonyl nickel powder compacts (compacting pressure). The σtl.av/σVW ratio at compacting pressure from 200 to 800 MPa varies from 41 for Al to 870 for Nielectrolytic. We calculated the coordination number (K) at relative density (ρ) in the range 0.177–0.7 as [26]

$$ K = \frac{1}{{1 - \sqrt[3]{\rho }}}. $$
(5)

For ρ > 0.7, K = 12.

Discussion

Compare the van der Waals forces that occur between two aluminum powder particles of the same size at the minimum possible distance between them (b = 0.286 nm) calculated by formulas (2) and (5) using the Lifshitz–van der Waals [13] and Hamaker [15] constants. Figure 1 shows the van der Waals forces versus the diameter of interacting particles, the distance between them being constant. Although formulas (2) and (5) to calculate the van der Waals force were derived using fundamentally different assumptions, the calculated value of f LW is greater than f H by a factor of 1.3 for all particle diameters. For interacting particles 1 μm in diameter, f LW = 0.245 and f H = = 0.199 μN; for particles 500 μm in diameter, f LW = 122 and f H = 97 μN. The force calculated for particles 1 μm in size is close to the forces measured experimentally for pharmaceuticals (0.066–0.0012 μN) [16].

Fig. 1
figure 1

The Van der Waals force calculated using the Hamaker and Lifshitz–van der Waals constants versus the diameter of aluminum powder particles

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The adhesion strength calculated for aluminum powder by formulas (1)–(3) increases differently depending on compacting pressure (Fig. 2): σVW(LW)max increases most intensively, σVW(H)max increases slower, and σVW(Gr)max hardly changes in the pressure range 200–800 MPa.

Fig. 2
figure 2

The strength of aluminum powder compacts provided by van der Waals forces and calculated by formulas (3) and (4) using the Hamaker and Lifshitz–van der Waals constants versus compacting pressure

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According to the notion of characteristic temperature [17], pressing of the powders in question is classified as cold (FeSi nonmetal and Mo bcc metal powders) and warm (Al, Cu, Ni fcc and ZnFootnote 2 hcp metal powders). By their properties (in particular, RAD and particle shape), the powders can also be divided into two groups: (i) spherical or near-spherical shape and high RAD (atomized Cu, Al, and Zn powders) and (ii) irregular (irregular, branched) shape and average or low RAD (carbonyl Ni, electrolytic Cu and Ni).

In the initial state and at low compacting pressures, the bonds between powder particles or the strength of compacts are mainly due to van der Waals forces, and the green strength and stress to be applied to overcome the van der Waals force practically match. However, when compacting pressure increases and the compact becomes noticeably strong (units or tens of megapascals), σtl.av is higher than σVW by one to three orders of magnitude (Table 1). The van der Waals contribution to strength is greater than the green strength only for the atomized copper powder. To correctly evaluate the effect of molecular interaction on the green strength, the former should be compared with adhesion strength, which corresponds to tensile strength in the direction opposite to pressing (σt). For compact brittle materials, σtl/σt = 0.9–1.1 [27] (tensile strength σtl is determined indirectly), but it is not the case for green compacts made of plastic powders. For example, for the iron powder (compacts 12 mm in diameter and 8 mm in height) in the RAD range 0.7–0.93, the σtl/σt ratio changes from 10 to 17, i.e., by one order of magnitude at least [28]. The dependence of the σtl/σt ratio on the parameters of plastic metal powders has hardly been studied, so we assume that σt ≅ 0.1 σtl for an approximate evaluation. On this basis, adhesion strength (σt) of zinc powder compacts, provided that there is no elastic aftereffect, can be provided only by the van der Waals component (Fig. 3). This is quite possible for zinc because the compacting temperature is close to the onset recrystallization temperature (t or ~75°C), t or decreasing with higher strain; moreover, local heating may occur in deformation areas.

Fig. 3
figure 3

The tensile strength (σt) and contribution of the maximum possible van der Waals force to the green strength (σVWmax) calculated by formula (4) using the Lifshitz–van der Waals constant versus compacting pressure for spherical powder PTs-1

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For the copper powder with spherical particles, the onset recrystallization temperature is much higher than the compacting temperature (t or  ≈ 405°C), and the elastic aftereffect in pressing is greater than the van der Waals contribution to strength (σmech = 0 and σe.a > > σm (1)). The aluminum powder, similar to the copper powder, has σtl.av twice as high as σVW, which can be attributed to the mechanical component originating from the penetration of solid oxide films into adjacent particles. This assumption can be confirmed by fractography of compacts [29]. Powders with branched particles show σtl.av that is two to three orders higher than the van der Waals strength, and σt, considering [28], is greater by one to two orders (for warm pressing and branched particles). The maximum van der Waals strength constitutes 0.1–0.01 of the green tensile strength in the direction opposite to pressing.

Using homological compacting temperature (Θc) compared with ductile–brittle temperature (Θ*) and onset recrystallization temperature (Θ or ) of particle material, we analyzed how temperature at which the powders were formed influenced the σtl.av/σVW ratio. For the metals, we accepted Θ* ≅ 0.2 (taking into account that Θ* < < 0.2 for fcc metals) and Θ or  = 0.5. Note that the homological compacting temperature induces ((Θc < Θ or ) and (Θc < Θ*)) or does not induce ((Θc > Θ or ) and (Θc < Θ*)) elastic aftereffect and plasticity in particle material. For the metal powders with spherical particles, Θc = 0.216 for Cuatomized (fcc)Footnote 3, i.e., σtl.av < σVW; Θc = 0.432 for Zn (hcp)Footnote 4, i.e., Θtl.av ≈ σVW; and Θc = 0.314 for Al (fcc)3, i.e., σtl.av > σVW .

For the metal powders with nonspherical particles, Θc = 0.216 for Cuelectrolytic (fcc)3, i.e., σtl.av > > σVW; Θc = 0.170 for Nicarbonyl and Nielectrolytic (fcc)3, i.e., σtl.av > > σVW; Θc = 0.101 for Mo (bcc)Footnote 5, i.e., σtl.av > σVW.

For the nonmetal FeSi4 powder with nonspherical particles, Θc = 0.195 and Θ* ≅ 0.6–0.8, i.e., σtl.av > σVW. The absolute value of σtl.av for the compact is two orders lower than for the metal powders with branched particles.

The σp/σVW ratio for the Mo powder should be explained: it is practically identical with that for the Ni powder, though Θc < Θ* for molybdenum. This is because characteristic temperature is always higher than coldbrittleness temperature, which can be determined using the Ioffe–Orowan scheme [30]. However, its value for chemically pure metals still has not been introduced into academic use since it depends on the purity of metals and on testing conditions and other factors. Given the nature of mechanical component, σtl is influenced by increased strength of Mo particle material (784–1177 MPa) compared with Ni powder (343–667 MPa).

We have calculated the maximum green strength governed by the van der Waals force σm (1) (it is much lower than elastic aftereffect (–σe.a)). In addition, elastic aftereffect (stresses that occur in the compact are equal to stresses induced during pressing) may increase the distance between particles (decreasing molecular forces) and destroy single contacts [31]. It is clear that only the mechanical component, mechanical interlocking, can resist such great stresses; otherwise (in case of atomized copper powder), the compact just falls apart.

Conclusions

The van der Waals contribution to the strength of compacts made of medium-size and coarse metal and nonmetal powders produced at different temperatures has been calculated for the first time. This allowed us to evaluate its contribution to actual strength.

The ratio of the van der Waals force calculated using the Lifshitz–van der Waals constant to the force calculated using the Hamaker constant for two identical spherical aluminum particles (1–500 μm in diameter) is 1.3.

The van der Waals component of the green strength calculated by three different methods for aluminum powders differs from the experimentally determined green strength by three times (corresponding to compacting pressure 200 MPa).

The comparison of the σtl.av/σVW ratios has shown that σtl.av and σVW are of one order of magnitude for the zinc powder; σVW > σtl.av for the atomized copper powder, though σtl.av is greater than σVW by two to three orders for most powders; and σtl.av is greater than σVW by one to two orders for plastic metals according to [28]. This great difference between σtl.av and σVW can be attributed only to the effect of particle shape and mechanical interlocking [16] or seizure of particles when compacted. In fact, the powders of Cuatomized (σtl.av < σVW) and Zn (σtl.av = σVW) have spherical particles, Al (σtl.av > σVW) near-spherical particles, and other powders (σtl.av > > σVW) irregular particles.

To predict the green strength, both the shape of particles (or relative apparent density of the powder) and temperature of the particle material (homological temperature in pressing, homological characteristic temperature, and recrystallization temperature) need to be taken into account.