Thixotropic properties of multi-functional binder and compaction behaviour of the low alloyed binder-treated powder

Abstract

A new multi-functional binder with adhesive and lubricant effect was designed, and the thixotropic properties of the binder components and the resulting binder, the adhesive effect and compaction behaviour of the binder-treated powder were investigated. The results demonstrate that the thixotropic components in the binder are polyethylene wax and microcrystalline wax. With the increase of polyethylene wax content, the critical solid–liquid shear stress of binders decreases gradually. The main lubrication components in the low-pressure and high-pressure stages during the compaction process are found to be polyethylene wax and microcrystalline wax, respectively. The fine metal or graphite additives are effectively bonded on the surface of iron particles and filled in the pits. The green density reaches 7.25 g cm⁻³ at 600 MPa when the content of polyethylene wax is 30%. From 400 to 750 MPa, the content of polyethylene wax corresponding to the minimum ejection pressure decreases from 20 to 0%.

Keywords

Iron-based powder metallurgy binder-treated powder thixotropic property shear stress green density ejection pressure adhesion effect

1 Introduction

Iron-based powder metallurgy is known as a cost-effective technique to produce intricate and near-net-shape components [1,2]. The metallic additives of Cu, Ni, Mo, C or the other alloying elements are used to improve the mechanical properties of iron-based powder metallurgical products [3,4], while the addition of lubricant to the powder is effective to reduce the friction between mould wall and powder as well as the internal friction between powder particles during the compaction process [5 –8]. However, the resulting powder mixtures are susceptible to segregation during mixing or handling due to the different particle size, morphology, and specific gravity of the powders [9 –12]. The chemistry homogeneity of the powder mixture determines the die cavity filling performance, the morphology or size of the pores, and the resulting mechanical properties of the PM components. In order to improve chemical homogeneity, varied kinds of methods have been proposed in terms of partially diffused method, prealloy method and bonding technology. Among them, the binder treatment technology retains the compressibility of the powder greatly. Compared to the diffusion bonded and pre-alloy method, the annealing step is eliminated during the binder-treated process, which can further reduce the production costs.

The binder-treatment technology is a method that consists of bonding graphite or alloying additives to the coarse iron particles, which is the most important method used in iron-based powder metallurgy [13,14]. The main advantages of binder-treated mixes are better flowability, improved productivity and part consistency and reduced dusting and segregation [15 –19]. Gosselin [20] found the using of a polyvinyl pyrrolidone can effectively reduce dusting and composition segregation of the powder mixture. Kazuhisa [16] proposed a binder containing an epoxy resin and a curing agent and provided a method for producing the mixed powder using the binder. Chawla [1] compared the fatigue behaviour of binder-treated versus diffusion bonded powder metallurgy steels and found PM materials processed using binder treatment have equivalent tensile and fatigue to conventional diffusion bonded materials. Zhang [21] carried out binder treatment experiments on Fe-C mixture by using single binder of epoxy resin, and the bonding mechanism of small particles on large iron particles was explained by the migration of small particles caused by spontaneous capillary flow. Although a large number of patents and papers emerged for binder treatment technology, most of them focused on the binder with simple component, or the compaction behaviour of the mixture by using the binder without the introduction of the detailed composition. Formulation and compositions optimisation of the binder were seldom reported, and the thixotropy properties study of the binder was even less.

In this work, a new type of binder was designed. The thixotropic properties of the four binder components and the resulting binder were characterised. After that, the orthogonal experiment was conducted to optimise the composition of the binder. Through rheological properties analysis of the binder, the relationship between thixotropy of the binder and compaction behaviour of the powder mixture were further explored.

2 Materials and experimental procedure

Conventional Fe–1.75Ni–0.5Mo–1.5Cu–0.5C low alloyed steel powder was prepared by binder-treatment technology. The iron powder used in the experiment was LAP 100.29 water-atomised iron powder (Laiwu Iron&Steel Group Powder Metallurgy Co., LTD) with average particle size of 140 μm. Considering the slow diffusion rate of Mo in Fe in the sintering process, Fe–Mo alloy powder was used instead of pure Mo powder in order to achieve better alloying effect. Elemental powder Cu, Ni, C and Fe–Mo alloy powder with particle size less than 5 μm were supplied by Beijing Xing Rong Yuan Technology Co., Ltd. The binder components mainly included ethylene bis stearamide (EBS), stearic acid, polyethylene wax and microcrystalline wax which account for 95% of the binder. The remaining 5% were 168 antioxidants, isooctanoic acid and lauric acid, denoted by ‘The others’ in tables. The total content of the binder in the powder mixture was 0.3 wt-%. Most of the components in the binder were provided by ShangHai Joule wax Co., LTD. The grade information of metal additives and binder components are detailed in Table 1. A double-cone spray mixer was used as binder treatment equipment, as showed in Figure 1. The alloy powder system was heated by the heat-conducting oil, and the solution in which the binder was dissolved was sprayed and mixed with the alloy powder uniformly under the high pressure of N₂. In the process of bonding, the iron powder and the other alloying elements were first added to the mixing cylinder and premixed for 45 min. In this stage, the ingredient layers were broken-down [22]. At the same time, 800 mL of n-heptane and trichloromethane (3:1) mixed solution was added to the prepared binder. The mixture of the binder and the solvent were heated and dissolved completely in a water bath at 80°C, and sprayed into the premixed powder by five times. Each interval was 20 min to ensure that the solvent was completely volatised. For comparison, the conventional premixed powder was also prepared using the same equipment with the 3-h mixing time.

Table 1

Grade information of metal additives and binder components.

Powder	Cu	Ni	Fe-Mo	C
D50 (μm)	2	3	2	1
Binder composition	EBS	Stearic acid	Polyethylene wax	Microcrystalline wax
Molecular weight	593.02	284.48	3000–3500	600–650

The rheological properties of the binder and its components were analysed with the RS6000 rotary rheometer from HAAKE of Germany. The experimental temperature was 25°C and the shear rate was set to 0.1–100 rad s⁻¹. The YAW-600G pressure tester was used to press premixed powder under the pressure of 400–750 MPa. The mould size was Φ15 mm, and the loading speed was set at 5 KN s⁻¹. Five samples were prepared under different pressures for each binder-treated powder, green density was measured by mass-volume method and the final results was average of the five. Particle size distribution of the premixed powder was analysed by BT-9300H laser particle size analyser. The morphology and distribution of alloying elements of the powder was examined using TESCAN tungsten filament scanning electron microscope (SEM) equipped with EDS system.

3 Results and discussion

3.1 Rheological analysis of binder components

Viscosity and thixotropy are the two most important characterisation of non-Newtonian fluids. Viscosity reveals the internal friction of fluid, which is the resistance of a layer of fluid to the other, while the thixotropy describes the relationship among fluid viscosity, shear force and shear rate [23 –25]. The introduction of the thixotropic component is aimed at reducing the viscosity of the binder under the compaction pressure, thereby enhancing the lubricating effect and further increasing the density of the compact. The relationship between the triggering phase of binder thixotropy during the compaction process and compact density is not clear. Therefore, the thixotropic properties of each component need to be tested, and the triggering phase is analysed to explain the lubrication mechanism of the binder. The thixotropic properties of EBS, stearic acid, polyethylene wax and microcrystalline wax are characterised by using thixotropic-loop, as shown in Figure 2. The shear rate increases continuously from 0 to 100 rad s⁻¹ and then gradually decreases from 100 to 0 rad s⁻¹, while the changes of shear stress with shear rate are measured. In this process, the shear stress shows two curves that do not coincide with each other. The upward curve is located above the downward curve and forms a thixotropic-loop. The larger the area of the thixotropic-loop, the greater the thixotropic properties. It is seen that the upward and downward curves of EBS and stearic acid are basically coincident, and it is considered that they have no thixotropy behaviour. With the increase and decrease of the shear rate, the shear stress of polyethylene wax and microcrystalline wax appears a curve of upward and downward misalignment. Therefore, the thixotropic components in the binder are polyethylene wax and microcrystalline wax. The viscosity of each component is further analysed as a function of shear rate, as shown in Figure 3(a). At room temperature without shear, the viscosity of all the four components is higher than 10,000 Pa s. As shear rate increases, the viscosity decreases gradually and the components change from solid state to liquid state. This phenomenon is known as shear thinning. The sequence of critical solid–liquid shear rates corresponding to the four components is: ω (microcrystalline wax: 17 rad s⁻¹)> ω (polyethylene wax: 5 rad s⁻¹)> ω (EBS: 0.4 rad s⁻¹)> ω (stearic acid: 0.1 rad s⁻¹). From the aspect of average molecular weight (denoted by M), the heavier the molecular weight, the longer the relaxation time of the molecular chain deformation and the molecular chains oriented in the flow cannot easily restored to their original shape, the flow resistance decreases earlier. Moreover, due to the large number of internal entanglement points in the large molecular weight component, the dynamic process of disentanglement and reentanglement is likely to occur at low shear rate [26 –28]. M (polyethylene wax) > M (microcrystalline wax), therefore, the shear thinning of polyethylene wax occurs at a relatively low shear rate, which is in accordance with the rheological test results. Figure 3(b) shows the variation of viscosity with shear stress. The critical solid–liquid shear stress is consistent with the shear rate: τ (microcrystalline wax: 670 Pa)> τ (polyethylene wax: 150 Pa)> τ (EBS: 16 Pa)> τ (stearic acid: 11 Pa).

3.2 Thixotropic properties of binder and compaction behaviour of powder

Orthogonal experiment was designed to optimise the content of binder component, as shown in Table 2. The content range of each component was EBS (15–65%), polyethylene wax (0–20%), microcrystalline wax (0–20%), stearic acid (30–50%). The pre-mixed powder was obtained by the binder treatment under the same process conditions, and the binders with different components and varied contents correspond to 1#–9# samples respectively. Then the binder-treated powder was pressed under the pressure of 400, 500, 600, 700, and 750 MPa. Figure 4 displays a histogram of green density for different binders under five pressures. With the increase of pressing pressure, green density of the nine kinds of alloy powder increases. The continued increase in compaction pressure makes less contribution to the growth of density when the green density reaches 7.2 g cm⁻³ or more. For example, when the compaction pressure increases from 700 to 750 MPa, the green density increases slightly, only 0.05 g cm⁻³. At this time, the plastic deformation of the powder is close to the limit, and small amount of particles undergoes brittle fracture due to the hardening effect. The fine powder produced by the fracture can be filled into the pores between the particles, but it exerts little influence on the density of compact. Under different pressures, the density variation trends of the nine kinds alloy powders are similar and the density decreases from both sides to the centre and reaches the lowest level for the 5#. The highest density is obtained at different alloy powders under a range of compaction pressure. From 400 to 750 MPa, the powder with the highest density is 7#, 7#, 9#, 1#, and 1#, respectively. 7#, 8# and 9# samples perform well under various pressures. The density of 8# powder reaches 7.21 g cm⁻³ at 600 MPa and 7.32 g cm⁻³ at 700 MPa.

Table 2

Composition of the multi-functional binders (wt-%).

No.	EBS	Polyethylene wax	Microcrystalline wax	Stearic acid	PW + MW	The others
1	65%	0%	0%	30%	0%	5%
2	45%	0%	10%	40%	10%	5%
3	25%	0%	20%	50%	20%	5%
4	45%	10%	0%	40%	10%	5%
5	25%	10%	10%	50%	20%	5%
6	35%	10%	20%	30%	30%	5%
7	25%	20%	0%	50%	20%	5%
8	35%	20%	10%	30%	30%	5%
9	15%	20%	20%	40%	40%	5%

Figure 5 shows the thixotropy of binders with different composition. The results indicate that the thixotropic properties differ sharply due to the different contents of binder components. The thixotropic properties of binders 6#–9# are significantly better than the other five binders, and the thixotropic property of the 9# binder is the best. In order to quantitatively characterise thixotropy property of the binder, the thixotropic-loop was integrated to obtain the thixotropy-loop area (denoted by S), as shown in Figure 5(b). Taking Table 2 into consideration, it is found that EBS and stearic acid exert little influence on the thixotropy of binders. The thixotropy of the binder is grouped and compared according to the content of polyethylene wax and microcrystalline wax. It is clearly seen that the area of the thixotropic-loop increases with the increased content of polyethylene wax and microcrystalline wax, such as S9 > S8 > S7, S9 > S6 > S3, indicating that polyethylene wax and microcrystalline wax act as thixotropic components, which is consistent with the previous results. When the content of polyethylene wax is 20% (9#), the thixotropic-loop area reaches 59000 Pa rad s⁻¹, but when its content drops to 10% (8#) and 0% (7#), the thixotropic-loop area decreases to 29,400 and 13,800 Pa rad s⁻¹ respectively. The penultimate column of Table 2 is the content summation of polyethylene wax and microcrystalline wax (PW + MW). The area of thixotropic-loop of the samples with the same content of PW + MW is different, which is associated with the difference in thixotropy of polyethylene wax and microcrystalline wax. The total content of polyethylene wax and microcrystalline wax of 7# and 5# binders is both 20%, but the area of thixotropic-loop of later is only 3500 Pa rad s⁻¹, less than half of that of 7#.

3.3 Lubrication mechanism

On the basis of the 8# binder, two kinds of thixotropy components of polyethylene wax and microcrystalline wax were selected. The influence of binder composition on compacting behaviour of the powder mixture was studied by adjusting the ratio of the two thixotropy components, as shown in Table 3. Figure 6 shows the variation viscosity of the binders with different polyethylene wax content as a function of shear rate and shear stress. With the increase content of polyethylene wax, the curves of viscosity-shear rate and viscosity-shear stress continue to shift to the left, and the critical solid–liquid shear rate and critical shear stress of the binders decrease gradually, which is similar to the curves shown in Figure 3.

Table 3

Composition of the binders (wt-%).

No.	EBS	Polyethylene wax	Microcrystalline wax	Stearic acid	The others
1#	35%	0%	30%	30%	5%
2#	35%	10%	20%	30%	5%
3# (8# before)	35%	20%	10%	30%	5%
4#	35%	30%	0%	30%	5%

Figure 7 shows the variation of green density with polyethylene wax content in the binder. Under the compaction pressure of 600, 700 and 750 MPa, the variation trends of the green density with the polyethylene wax content are basically the same, decreasing from both sides to the centre and reaching the minimum at the polyethylene wax content of 10%. The variation trends are slightly different at the pressure of 400 and 500 MPa. Combining the rheological analysis of binder component and specific slow loading speed (5 KN s⁻¹), the pressing process can be divided into two stages: low-pressure stage and high-pressure stage. The two stages correspond to low shear stress and high shear stress, respectively. Polyethylene wax is first shear thinning under low shear stress and transform from solid state to liquid state. With the increase of compaction pressure, the shear stress reaches the critical value of microcrystalline wax, and the viscosity of microcrystalline wax decreases, thus the lubrication effect is enhanced. Therefore, the major lubricants in the low-pressure and high-pressure stages are polyethylene wax and microcrystalline wax, respectively. The lubricating effect of the binder is mainly manifested in the high-pressure stage for the samples without the addition of polyethylene wax. However, when the polyethylene wax content is 30%, the lubricating effect of the binder is mainly reflected in the low-pressure stage. In the two cases, the lubricating components and phases of the binder are inconsistent, but both microcrystalline wax and polyethylene wax can reduce the friction and increase the green density. When the content of polyethylene wax is 30%, the green density reaches 7.25 g cm⁻³ at 600 MPa and 7.33 g cm⁻³ at 700 MPa.

The ejection pressure reflects the lubrication effect of the binder directly, the lower the strip pressure, the better the lubricating effect. Figure 8 shows the ejection pressure over a range of compaction pressure. From 400 to 750 MPa, the content of polyethylene wax corresponding to the minimum ejection pressure decreases from 20 to 0% gradually. When the polyethylene wax content reaches 20%, the corresponding ejection pressure at 400 and 500 MPa both falls to their lowest point, which is 12.4 and 17.2 MPa respectively. When the compaction pressure increases to 600 MPa, the lowest point of the curve shifts to the left, and the minimum ejection pressure is obtained at polyethylene wax content of 10%, which is 22 MPa. Compaction pressure continues to increase, and the lowest point of the curve continues to move to the left. Finally, when the compaction pressure reaches to 750 MPa, the minimum ejection pressure is achieved when the content of polyethylene wax is 0%, indicating that the enhanced effect of the polyethylene wax on the lubricating of the binder is exhibited under low compaction pressure.

3.4 Binder-treated powder

The purpose of binder treatment is to bond the fine alloying elements particles that easily produce dust or segregation to the larger iron particles. As shown in Figure 9, the atomised iron powder exhibits concave–convex surficial morphology, which will help the fine powder adhere to its surface. In theory, the most ideal and effective bonding method in binder treatment is the binder covered on the surface of iron powder evenly, and the fine powder can be bonded to the surface of the iron particle or its pits under the adherence force of the binder. Figure 10 shows the surface morphology and distribution of alloying elements of the optimal binder-treated powder (30% of polyethylene wax content in binder) and the conventional premixed powder. For the binder-treated powder, most of the additives are bonded to the iron particles and the fine alloying elements (Ni, Mo, Cu) can be filled into the pits, which improves the smoothness of the particles. The proportion of free graphite and metal additives not attached to the iron particle is significantly low. As for the conventional premixed powder, only a few alloying elements are adhered on the surface of iron particles, and most of them are still in free. Figure 11 shows particle size distribution of the binder-treated powder, conventional premixed powder and raw iron powders. It is clearly seen that the average particle size of iron powder is 140 μm. The particle size distribution of binder treatment powder is concentrated, with only one main peak at 125 μm. The corresponding particle size of the peak is close to that of the raw iron powder and the width of the peak is slightly larger than that of the raw iron powder, indicating that the alloying elements are effectively bonded on the surface of the iron particles. In the case of the premixed powder, bimodal size distribution of the powder mixture is observed. The small peak located at 6 μm corresponds to the alloying elements. It is seen that the alloying elements are not bonded on the iron particles.

The excellent adhesion effect of the binder-treated powder mainly depends on the following aspects. The binder solution forms a layer of liquid film on the surface of powder during the binder treatment process. This liquid layer weakens the collision kinetic energy between the particles in the mixing process to a certain extent, reduces the rebound of the powder particles and increases the adhesion probability of the alloy powder. On the other hand, the liquid film provides the adhesion force during mixing: (1) the capillary force caused by the curvature of inner surface of the liquid; (2) the interfacial tension between liquid film and powder particles [29 –31]. During the mixing process, with the heating of heat conducting oil, the solvent was removed from the powder mixture in a short time, forming a uniform solid binder film between the powder particles, which facilitated the effective and stable adhesion between the particle.

4 Conclusions

A new multi-functional binder with adhesive and lubricant effect was designed, and the thixotropic of polyethylene wax and microcrystalline wax is confirmed by relatively large thixotropic-loop. The sequence of critical solid–liquid shear stress of the four components is: τ (microcrystalline wax)> τ (polyethylene wax)> τ (EBS)> τ (stearic acid).

The powder mixture, containing 10% microcrystalline wax and 20% polyethylene wax in binder, performs well under various pressures in the orthogonal experiment, and the green density reaches 7.21 g cm⁻³ at 600 MPa. By adjusting the ratio of the two thixotropy components, the green density reaches 7.25 g cm⁻³ at 600 MPa when the content of polyethylene wax is 30%. With the increase of polyethylene wax content, the critical solid–liquid shear rate and critical shear stress of binders decrease gradually. From 400 to 750 MPa, the content of polyethylene wax corresponding to the minimum ejection pressure decreases from 20 to 0%. The major lubricants in the low-pressure and high-pressure stages during the compaction process are polyethylene wax and microcrystalline wax, respectively.

The binder-treated powder exhibits excellent adhesion and lubricant effect of metal or graphite additives. Almost all the metal and graphite additives are effectively adhered on the surface of the iron particle and filled in the pits, which improves the smoothness and flowability of the powder mixture.

Footnotes

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Yan Chen is a doctoral candidate.

Lin Zhang is a professor. His research interest includes oxide dispersion strengthened alloy and PM superalloys.

Haishen Sun is a doctoral candidate.

Fangfang Chen is a master graduate.

Xuanhui Qu is a professor. His research interest is advanced powder metallurgy technologies and materials.

References

Chawla

, Murphy

, Narasimhan

, Koopman

, Chawla

KK.

Axial fatigue behavior of binder-treated versus diffusion alloyed powder metallurgy steels. Mater Sci Eng A. 2001;308(1):180–188.

Narasimhan

KS.

Sintering of powder mixtures and the growth of ferrous powder metallurgy. Mater Chem Phys. 2001;67(1):56–65.

, Lund

, Gowri

Synergistic effects of silicon and phosphorus on sintering of ferrous metal powder compacts. Can Metall Q. 1989;28(4):345–352.

Haynes

Development of sintered low alloy steels. Powder Metall. 2013;32(2):140–146.

Nor

SSM

, Rahman

, Tarlochan

, . The effect of lubrication in reducing net friction in warm powder compaction process. J Mater Process Technol. 2008;207(1):118–124.

Jiang

, Daehn

, Lannutti

JJ.

Effects of lubrication and aspect ratio on the consolidation of metal matrix composites under cyclic pressure. Acta Mater. 2001;49(8):1471–1477.

Rahman

, Nor

SSM

, Ariffin

AK.

Effect of lubricant content to the properties of Fe-based components formed at above ambient temperature. Procedia Eng. 2013;68(12):425–430.

Hanejko

Single press/single sinter solutions to high density. Powder Metall. 2013;53(2):100–102.

Spitzig

, Smelser

, Richmond

The evolution of damage and fracture in iron compacts with various initial porosities. Acta Metall. 1988;36(5):1201–1211.

10.

Douib

, Mellanby

, Moon JR.

Fatigue of inhomogeneous low alloy PM steels. Powder Metall. 1989;32(3):209–214.

11.

Choi

, Lee

, Song

JI.

Improvement of powder properties and chemical homogeneity of partially alloyed iron powder by a nanopowder process. Mater Trans. 2014;55(8):1356–1362.

12.

German

RM.

Powder metallurgy of iron and steel. Metall Mater Trans A. 1998;198:578–579.

13.

Poszmik

, Narasimhan

KS.

Segregation-Free premixes for increased productivity and improved performance. Mater Sci Forum. 2007;534–536:313–316.

14.

Semel

FJ.

Ancorloy premixes: binder treated analogs of the diffusion alloyed steels. In: Advances in Powder Metallurgy & Particulate Materials. 1999, Vol. 2, p. 7–93, compiled by Charles L. Rose and Martin H. Thibodeau, published by Metal Powder Industries Federation, Princeton, NJ.

15.

St-Laurent

, Gélinas

, Thomas

, . Using binder-treatment technology for high performance steel powder mixes. Powder Metall Indust. 2007;17(1):1–9.

16.

Kazuhisa

, Suzuki

Binder for powder metallurgy, mixed powder for powder metallurgy and method for producing same, P. US, US7074254, 2006.

17.

Ozaki

, Uenosono

, Ogura

Iron-based powder composition for powder metallurgy having higher flowability and higher compactibility and process fir production thereof, P. US, US20010028859. 2001.

18.

Cui

, Li

, Liu

, . Progress in R&D of nonsegregation premixed steel powder. Powder Metall Indust. 2007;1:43–48.

19.

Luk

SH.

Advances in binder-treatment technology statistical data on ancorbond plus. 2000.

20.

Gosselin

Segregation-free metallurgical powder blends using polyvinyl pyrrolidone binder, P. US, US5069714. 1991.

21.

Zhang

, Wang

, . Process analysis of binder treatment on iron-based alloy powder. Powder Metall. 2017;1:1–7.

22.

Mota-Aguilar

, Velázquez

Dynamics of the dry premixing stage of a hydrophobic formulation and potential implications on the wet granulation process. Powder Technol. 2015;286:318–324.

23.

Rivlin

RS.

The hydrodynamics of non-newtonian fluids. Math Proc Cambridge Philos Soc. 1997;45(1):88–91.

24.

Goodeve

CF.

A general theory of thixotropy and viscosity. Trans Faraday Soc. 1939;35(1):342–358.

25.

Cheng

DCH

, Evans

Phenomenological characterization of the rheological behaviour of inelastic reversible thixotropic and antithixotropic fluids. Br J Appl Phys. 1965;16(11):1599–1617.

26.

Robinson

, Kellaway

, Marriott

Proceedings: The effect of molecular weight on the rheological properties of acid and alkaline processed gelatin gels. J Pharm Pharmacol. 1975;27(2):76–82.

27.

Rudd

JF.

The effect of molecular weight distribution on the rheological properties of polystyrene. J Polym Sci A. 1960;44(44):459–474.

28.

Kosinski

, Caruthers

JM.

The effect of molecular weight on the rheological properties of poly(dimethylsiloxane) filled with fumed silica. Rheol Acta. 1986;25(2):153–160.

29.

Princen

HM.

Comments on ‘The effect of capillary liquid on the force of adhesion between spherical solid particles’. J Colloid Interf Sci. 1968;26(2):249–253.

30.

Johnson

, Kendall

, Roberts

AD.

Surface energy and the contact of elastic solids. Proc R Soc Lond A. 1971;324(1558):301–313.

31.

Derjaguin

, Muller

, Toporov

YP.

Effect of contact deformations on the adhesion of particles. J Colloid Interf Sci. 1975;53(2):314–326.