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Modeling of densification process for particle reinforced composites
Journal of Alloys and Compounds 487 (2009) 511–516
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Modeling of densification process for particle reinforced composites ´ O. Dimˇcic, ´ B. Dimˇcic´ ∗ , M. Vilotijevic, ´ S. Riznic-Dimitrijevi ´ D. Boˇzic, c´ Institute of Nuclear Sciences, “Vinˇca”, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 30 October 2008 Received in revised form 29 July 2009 Accepted 2 August 2009 Available online 8 August 2009 Keywords: Composite material Hot pressing of two phase systems Densification mechanisms modeling Radial distribution function Partitioning factor
a b s t r a c t It is a well known fact that behavior of materials during consolidation at high temperatures is a very complex issue. Each mechanism that promotes densification, depends on a large number of parameters in many different ways, making the development of the densification process very difficult to plan. For that reason, any kind of model which could encompass large number of the influencing parameters would be a great contribution for easier planning and handling of the densification process. Modeling of densification process based on the papers of well-known authors, including a new appropriate modifications is presented in this work. Importance of this model is that it can be applicable to composite materials. This new model was tested on a real Al–SiC composite systems (Al–10 vol.% SiC, Al–30 vol.% SiC and Al–50 vol.% SiC). Predicted behavior of composites obtained by model calculations and the one defined through the experiments concur in the 7–30% range. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Due to their low specific weight, aluminum based materials present widely used structural material. Recently, with the development of composites, these materials found even wider application as the matrix for different kind of composite materials. The most used secondary phase in these composites is SiC. Aluminum based composite materials hardened by the addition of SiC particles are characterized by the lower density, better corrosion resistance, improved strength, elastic moduli and wear resistance compared to the monolithic alloys. Powder metallurgy techniques, hot isostatic pressing and hot pressing were found to be a very promising for the production of this type of materials [1,2]. Controlling a large number of parameters and understanding certain mechanisms of above mentioned processes is important for the production of composites. By presenting the densification process through densification maps [3], all mentioned important issues are much easier for comprehension. Practical use of densification maps is to link the influence of the crystal structures and atomic bonds on the plastic flow of materials and enabling easier analyzing of certain mechanisms which are necessary for rational designing of the experiments. This densification maps are also very useful in qualitative selection of the material for engineering purposes, and prediction of mechanisms relevant for the deformation under applied load or hardening mechanisms.
∗ Corresponding author. Tel.: +381 11 2439 454; fax: +381 11 2439 454. ´ E-mail address: [email protected] (B. Dimˇcic). 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.08.003
The aim of this work was modeling of densification process for particle reinforced composite materials, using the known papers which considered monolithic materials [4,5,6,7,8], and the experimental confirmation of the results on the real systems: Al–10 vol.% SiC, Al–30 vol.% SiC and Al–50 vol.% SiC.
2. Basic assumptions for the development of densification model The most used densification models are based on the works of Artz and Ashby et al. [6,9]. It should be noted, that although the densification process in this work was observed during simple hot pressing process, the assumptions for the hot isostatic pressing models were used. Explanation for these assumptions can be found in literature [10], from which can be concluded that HIP models are quite acceptable in the case of simple hot pressing. Model presented in this work considers the processes in the initial stage of consolidation from the works of Artz et al. [6]. Basic assumption is that the particles are spherical and that there is a very small number of the small surfaces contacts. Beside that, in order to simplify the calculations, the technique of “particle expanding” was used. This technique is based on the retaining the fixed distance between particles, while allowing the increase of the particle diameter from value R to R . In other words, a particle “expands” and “overlaps” other particles. The number and size of the new contacts is explained by the function of the radial distribution function (RDF). It is, also, assumed that the material distributes evenly from the overlapping area over the remaining free surface forming the “squeezed” particle with the diameter R .
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D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
For better understanding of the model it is necessary to explain the radial distribution function, geometric model, and to further analyze the selection of the equations that are applicable for the possible densification mechanisms of the composite powders during hot pressing process. 2.1. Radial distribution function (RDF) It is a well known fact that the size and the number of interparticle contacts have a great influence on the densification of powders. For the case where all the particles are of the same size, number of all contacts is described by only one function [6]. Stacking of these kinds of powders has been widely investigated leading to the development of large number of approximations for the radial distribution function for these types of powders. The most important parameter in modeling is the number of the initial contacts, N0 , although the number of new contacts has also an important influence at the rate and mechanism of densification process. When the relative density of the initial powder mixture is 0.64, the change of particle size by 16% during hot pressing process, compared to the initial powder particle size, leads to the full densification of compacts. That is the reason why only a part of the RDF diagram from (R1 + R2 ) to 1.16x (R1 + R2 ) is important to us. Considering the complexity of known analytical procedures, number of initial contacts, N0 , is estimated using the simple geometry based on the equations of Suzuki and Oshima [11], for the non-porous powders. These equations propose that the N0 is the function of the relation between the size and the volume fraction of the particles. In the model of the authors mentioned above, dependence of N0 on the stacking density does not exist, but the proportional high stacking density of particles is supposed. More accurate determination of RDF model demands an estimation of the rate of increase of Nij (number of j particles that are in contact with i particles), with the increase of the distance, r, from the center of the particle i, i.e. dNij /dr. A fact that for the large values of r, dNij /dr is determined by the total volume fraction of j particles is a great advantage in developing of model. There are some assumptions that this equation (Eq. (1)) can be used even for the small values of r (for all values of r greater than Ri + Rj ). dNij dr
=
3D0 fj r 2 Rj 3
(1)
where D0 is the initial relative density and fj is a volume fraction of j particles in the mixture. The chosen semi-empirical equation for the calculation of the cumulative RDF curve shows a very good match with the known analytical calculations and experimental results [12,13,14]. That means that accuracy, simplicity and the fact that the important part of the function is in the narrow range (from 1 to 1.16) are the basic reasons why RDF is chosen for the modeling in the case of this work. 2.2. Description of the geometrical model The first step in modeling acquires the determination of the volume value which the “expanded particle” (R1 ) reaches. R2 can be calculated from the condition that R1 /R1 = R2 /R2 . Since the distance between the centers in this stage is constant, it is obvious that:
D = D0
R1 R1
3
Fig. 1. Schematic view of the contact between two particles.
their expansion is presented (Fig. 1). In the further text this particles will be denoted as a i and j particles. Calculation of the overlapped volume is done in a specific manner (Eq. (2)). In the reality this volume is distributed over the surfaces of both particles and therefore causes an appearance of the new contacts. Radii of the “squeezed” particles are denoted as R . In the starting approximation it has been taken that R1 = R1 and R2 = R2 . Overlapped volume that belongs to the particle i, Vi c is: Vi c =
(3)
where the height of the overlapping area, hi , is: hi = Ri −
R 2i + r 2 − R 2j
(4)
2r
where r is the distance between the centers of the particles. All of these equations apply for the overlapped volume that belongs to j particle as well. Therefore, total overlapped volume, VijT is: VijT = Vi c + Vj c
(5)
For the qualitative evaluation of distribution of the total overlapped volume dislocated from the contact (the sum of the volumes of the overlapped particle calottes), a partitioning factor, P, concept is applied [15]. This factor is very important because it determines the influence of the relative deformation of the particles in the mixture. Determination of a specific equation for the P factor between the particles that are not of the same size is still impossible. Generally, it can be assumed that P = 1/2 for the case when the volumes of translated material from every particle are equal [12,15], but for the case of the composite mixtures, this assumption is not applicable. In this work, i.e. for modeling of the densification process of composite powders which are made of the very hard secondary phase (SiC) distributed in the soft matrix (Al), P = 1 [15] for the softer phase, which means that the deformation induced by the hard–soft contact is placed always in the soft particle. If the Piji is the part of the volume of material distributed on the particle i (which is causing the deformation of the particle i), then j Pij can be calculated as (1 − Piji ). Now, a total distributed volume over the particle i is: R +R
(2)
where D0 is the related density before pressing, and D is a current value of density. For the easier comprehension of the model development, a schematic view of the contact between two particles as a result of
1 2 3R1 − hi h 3 i
ViT =
i
j
j=1,2r=R +R i j
Piji VijT
dNij dr
dr
(6)
where dNij /dr is a radial distribution function for j particle calculated from the center of the i particle. This volume has been
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
calculated from the increase of the particle i diameter from Ri to Ri . Knowing that the volume of the particle i must remain constant before and after the “squeezing”, given equations have to be fulfilled: 4 4 3 3 R i − ViT = R i 3 3
(7b)
2.3. Selection of the appropriate equations for the explanation of densification mechanisms during hot pressing
y˙ = A · · cn = A · ·
F 2
n
(8)
where A is a steady-state power-law creep constant, is a radius of the contact circle between two particles, c is a creep stress, and F is the intensity of the force in the contact area. By rearranging the previous equation we can express F as:
A
(2−1/n)
(9)
This equation can be modified for all types of contacts where the power-law creep mechanism is dominant. The rate at which particle centers approach each other is 2y˙ ij , because y˙ ij = y˙ ji . This statement is based under assumption that identical limitations are applied to the both particles. In the mathematical model this rate is correlated with the i and j particle expansion through the following relation: 2y˙ = Ri + Rj
(10)
y˙ 12 = y˙ 21 = 1 +
R2 R1 R1
(11)
2
Intensities of the forces on all types of contacts can be expressed by the unknown variable R˙ 1 . Since y˙ ij is the same for all contacts with the same values i and j, a mean value of the force on the contact, F¯ ij is:
F¯ ij =
y˙ ij
1/n ij∗
A
(12)
where ij∗ is mean value of a radius of circle for all contacts between two particles. R +R
i
ij∗
j
· ij (2−1/n) ·
=
dNij dr
dr/Nij
ij =
√
R 2i
+ r2 2r
− R 2j
dr
(15)
r=Ri +Rj
It can be seen in literature data [12] that: N11 F¯ 11 f1 4R 21
+
N12 F¯ 12 f1 4R 21
+
N21 F¯ 21 f2 4R 22
+
(13)
+ ···
(16)
Observing the contacts, when the particles are squeezed in one another, acquired force for the plastic flow (yield) is: (17)
Combining Eq. (17) with Eq. (9) a critical radius of the circle of the contact between two particles for which the value of the force needed for the plastic flow is equal to the force needed for the power-law creep to occur can be calculated (Eq. (18)). cij =
y˙ ij
A 3y
n
(18)
where y is a yield stress. For the contacts where ij > cij densification will be completed by the power-law creep, while in the case where ij > cij , intensity of the force for the power-law creep becomes so great that it starts to induce plastic flow on the surface of the contact. Not one contact can endure not to deform when the force greater than the force needed for the plastic flow is applied. Therefore, calculation of the mean value of the force that acts on the contacts is needed when the mechanism of plastic flow is assumed (Eq. (19)). p
y F¯ ij =
p
y
y
Nij · F¯ ij + Nij · F¯ ij
(19)
Nij
where p and y exponents present the given values for the appropriate operative mechanisms (p: power-law creep, y: yielding). The total number of ij contacts, Nij , remains unchanged and it can be calculated from the Eq. (15). p y Nij and Nij can also be calculated from the Eq. (15) by changing y
p
the superior limit of the integral (for Nij ) with rc . Here is Nij = Nij + y y N . Similarly, F¯ can be calculated from the Eq. (20). ij
ij
R +R 1 2
y F¯ ij = rcij
ij2 Nij2
dNij dr
dr
(20)
Switching the calculated values of Fij into Eq. (16), we can calculate new, slightly higher values of R1 . This can be expected, because the “additional” load, induced by contact stress of 3 y , is distributed over the rest of the remaining contacts, increasing the creep rate. 2.5. Diffusion The change in the volume of the doped material on the surface of the formed neck of one particle in one second, by the grain boundary diffusion, can be calculated from the Eq. (21): V˙ =
(14)
4R 22
2.4. Plastic flow
2
N22 F¯ 22 f2
where ap is applied stress, and values of F¯ ij and Ri are different compared to the original equation.
r=Ri +Rj
dNij
F y = 3y 2
2.3.1. Power-law creep If it is assumed that the deformation rate is applicable for the entire deformation zone which depth is equal to the radius of the contact area (overlap), a rate at which contact area approach the centre of the particle, y˙ is:
j
dr
ap =
If these equations are not fulfilled, calculation procedure has to be repeated. A new values of and Ri must be calculated and the Rj procedure given in the previous text has to be followed (similar to the procedures from the references [6,12]).
y˙ 1/n
i
Nij =
(7a)
4 4 3 3 R j − VjT = R j 3 3
F =
R +R
513
4 ıDb + Dv kT
˝
dNij dr
pef
(21)
where ˝ is atomic volume, b is Burgess vector, Dv is volume diffusion coefficient, Db is grain boundary diffusion coefficient, ı is
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thickness of the grain boundary, is a radius of circle of the contact between two particle, k is Boltzman constant, and pef is effective pressure on the contact between two particles. In the case of the composite materials, as it is a case in this work, diffusion mechanisms are occurring only between particles of the soft phase. Therefore, in the Eq. (21), dNij /dr must be replaced with dN22 /dr, while pef is calculated from the Eq. (16).
3. Densification mechanisms maps of the particle reinforced composites Based on the mentioned assumptions, selection of the geometry and the distribution of the particles as well as the equations for the certain densification mechanisms, aim of this work was to model the densification process of particle reinforced composite materi-
Fig. 2. Deformation mechanism maps (Xt D0 is a time curve, where t is the time of densification and D0 is the starting density of the powder, and ED0 n is the group of experimental values; red line is the boundary between the power-law creep and diffusion zones. (a) For pure Al powder; (b) for the Al–10 vol.% SiC composite; (c) for the Al–30 vol.% SiC composite; (d) for the Al–50 vol.% SiC composite. Al particle radius in all composites was 2R = 125 m, and SiC particle radius was 2R = 33 m. Applied pressure was P = 35 MPa (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article).
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
515
Fig. 2. (Continued ).
als (Al–SiC) during hot pressing. The results are presented through the adequate densification maps where the relation between the relative density and the homologue temperature is presented (Fig. 2a–d). Results of densification, presented in Fig. 2, are determined only for the certain constant pressure (P = 35 MPa) and for the certain constant size of the matrix particles (2R = 125 m). A model that is made is universal and it allows the comparison of the densification mechanisms fractions for different pressures, and for different values of the matrix particle size. Other parameters of the densification process can easily be changed by changing the boundaries between zones of dominating mechanisms. Due to the modular structure of the model, complete modulus can be corrected or changed. This is practically very useful fact which enables potential corrections of the model caused by further experimental verification of the results. Model considers analysis of different mechanisms of densification during the hot pressing process, such as plastic flow, power-law creep and diffusion. From the chosen examples it can be noted that the process of densification is slowing down as the fraction of the hardening phase increase (slope of the dashed lines), which is in the agreement with the present experimental studies. A change of densification mechanisms during hot pressing induced by the change of the fraction of the hardening phase can also be observed from the maps (different size of the zones for certain densification mechanism).
4. Experimental evaluation of the model Experimental evaluation of the model has been performed on the pure Al powder and three different composite powders (Al–10 vol.% SiC, Al–30 vol.% SiC, Al–50 vol.% SiC) during densification. Mean value of Al powder particles was 125 m and it was produced by the gas atomization process. This material particles are generally rounded in shape, although some plate like and spherical particles can, also be noticed (Fig. 3a). SiC powder particles, with the mean size value of 33 m (Fig. 3b) are mostly polygonal. Metallic and ceramic powders are mixed in the cylindrical mixing dish for 1 h, with the velocity of 70 rpm. The filling rate of the blender was 30%. These parameters are, according to literature data [3], optimal for this process. Al powder and appropriate composite mixtures of powders are hot pressed in argon protective atmosphere at temperatures of 573 K, 723 K and 873 K for 0.5, 1.5 and 4 h.
Fig. 3. SEM. Size and shape of powders: (a) pure Al; (b) SiC.
The applied pressure was 35 MPa. Density of compacted samples was measured by the conventional Archimedes method, while the apparent density was determined using the Hall’s device (Table 1). Comparing the experimental results, and results suggested by the model, for the certain conditions of the hot pressing process, exhibits deviation of 7–30%. All the experimentally determined valTable 1 Values of the apparent densities for Al and Al/SiC composite. Material
Apparent density, %Dtheoretical
Al Al–10 vol.%SiC Al–30 vol.%SiC Al–50 vol.%SiC
0.42 0.48 0.52 0.55
516
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
ues of the density are greater than those that the model predicts. The deviations are greater for the samples pressed at the low temperatures (T = 573 K) (Al, Al–10 vol.% SiC, Al–30 vol.% SiC, Al–50 vol.% SiC). The reason for this lies in fact that applied pressure of 35 MPa is not sufficient for the higher densification rate at low temperature. With increase in temperature, differences between model and experimental results are smaller due to the more efficient transport of mass between the Al particles in the composite. As it can be observed from the constructed densification maps, diffusion mechanism shows the greatest and dominant influence on the transport of mass which speeds up the densification process. Also, one of the reasons for the incomplete matching of theoretical assumptions and experimental values is that the irregular shape of the powder particles is approximated with the ideal spheres. Agreement of the model and experiment increases with the increase of SiC fraction in the mixture, which is only a logical result of the greater homogeneity of the mixtures.
Agreement of the experimental results and the results obtained by using the model is quite acceptable, and becomes greater at high temperatures (for the constant pressure) and with the increase of SiC particle fraction in the mixture. Simplicity of the model which allows changing of the process parameters responsible for the movement of the different densification mechanism zones boundaries is the greatest advantage because it presents a quite rapid way of predicting the behavior of the chosen system.
5. Conclusions
´ Z. Vujovic, ´ Powder Metallurgy, BMG, Belgrade, 1998 (in [1] M. Mitkov, D. Boˇzic, Serbian). ´ ´ Ceramic Technology Basics, TMF, Belgrade, 1990 [2] Marija Tecilazic-Stevanovi c, (in Serbian). [3] R.M. German, Sintering Theory and Practice, John Wiley&Sons Inc., New Jersey, 1996. [4] M.F. Ashby, Proceedings of the International Conference on Hot Isostatic Pressing, Lulea, June, 1987, pp. 29–40. [5] D.S. Wilkinson, M.F. Ashby, Science of Sintering 10 (1978) 67–76. [6] E. Artz, M.F. Ashby, K.E. Easterling, Metallurgical Transactions A 14 (1983) 211–221. [7] M.F. Ashby, Acta Metallurgica 20 (1972) 887–897. [8] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, first ed., Pergamon Press, Oxford, New York, 1982. [9] A.S. Helle, K.E. Easterling, M.F. Ashby, Acta Metallurgica 33 (1985) 2163–2174. [10] F.F. Lange, L.L. Atteraas, F. Zok, J.R. Porter, Acta Metallurgica and Materialia 39 (1991) 209–219. [11] M. Suzuki, T. Oshima, Powder Technology 35 (1983) 159–166. [12] S.V. Nair, J.K. Tien, Metallurgical Transactions A 18 (1987) 97–107. [13] S.V. Nair, B.C. Hendrix, J.K. Tien, Acta Metallurgica 34 (1986) 1599–1605. [14] E.K. Li, P.D. Funkenbusch, Metallurgical Transactions A 24 (1993) 1345–1354. [15] E.K.H. Li, P.D. Funkenbusch, Acta Metallurgica 37 (1989) 1645–1655.
This work presents analytical model of the hot pressing process of particle reinforced composite with the soft matrix and hard particles. Basic assumptions for the hot isostatic pressing are successfully applied on the exhibited case of hot pressing. For the determination of the radial distribution function, a semiempirical equation, suggested by Suzuki and Oshima [11] is used. Although the function is rather simple it exhibits the same results as the precise analytical models. Partitioning factor, P, which determinates the relative deformation of the particles in the mixture has also been used in this model. This factor enables the use of this suggested model for all particle reinforced composites, without considering the difference between deformation abilities of different phases. Model considers following densification mechanisms: plastic flow, power-law creep and diffusion.
Acknowledgement Authors are indebt to the Ministry of Science of Republic of Serbia for their financial support through the Project No. 142027. References
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Modeling of densification process for particle reinforced composites ´ O. Dimˇcic, ´ B. Dimˇcic´ ∗ , M. Vilotijevic, ´ S. Riznic-Dimitrijevi ´ D. Boˇzic, c´ Institute of Nuclear Sciences, “Vinˇca”, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 30 October 2008 Received in revised form 29 July 2009 Accepted 2 August 2009 Available online 8 August 2009 Keywords: Composite material Hot pressing of two phase systems Densification mechanisms modeling Radial distribution function Partitioning factor
a b s t r a c t It is a well known fact that behavior of materials during consolidation at high temperatures is a very complex issue. Each mechanism that promotes densification, depends on a large number of parameters in many different ways, making the development of the densification process very difficult to plan. For that reason, any kind of model which could encompass large number of the influencing parameters would be a great contribution for easier planning and handling of the densification process. Modeling of densification process based on the papers of well-known authors, including a new appropriate modifications is presented in this work. Importance of this model is that it can be applicable to composite materials. This new model was tested on a real Al–SiC composite systems (Al–10 vol.% SiC, Al–30 vol.% SiC and Al–50 vol.% SiC). Predicted behavior of composites obtained by model calculations and the one defined through the experiments concur in the 7–30% range. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Due to their low specific weight, aluminum based materials present widely used structural material. Recently, with the development of composites, these materials found even wider application as the matrix for different kind of composite materials. The most used secondary phase in these composites is SiC. Aluminum based composite materials hardened by the addition of SiC particles are characterized by the lower density, better corrosion resistance, improved strength, elastic moduli and wear resistance compared to the monolithic alloys. Powder metallurgy techniques, hot isostatic pressing and hot pressing were found to be a very promising for the production of this type of materials [1,2]. Controlling a large number of parameters and understanding certain mechanisms of above mentioned processes is important for the production of composites. By presenting the densification process through densification maps [3], all mentioned important issues are much easier for comprehension. Practical use of densification maps is to link the influence of the crystal structures and atomic bonds on the plastic flow of materials and enabling easier analyzing of certain mechanisms which are necessary for rational designing of the experiments. This densification maps are also very useful in qualitative selection of the material for engineering purposes, and prediction of mechanisms relevant for the deformation under applied load or hardening mechanisms.
∗ Corresponding author. Tel.: +381 11 2439 454; fax: +381 11 2439 454. ´ E-mail address: [email protected] (B. Dimˇcic). 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.08.003
The aim of this work was modeling of densification process for particle reinforced composite materials, using the known papers which considered monolithic materials [4,5,6,7,8], and the experimental confirmation of the results on the real systems: Al–10 vol.% SiC, Al–30 vol.% SiC and Al–50 vol.% SiC.
2. Basic assumptions for the development of densification model The most used densification models are based on the works of Artz and Ashby et al. [6,9]. It should be noted, that although the densification process in this work was observed during simple hot pressing process, the assumptions for the hot isostatic pressing models were used. Explanation for these assumptions can be found in literature [10], from which can be concluded that HIP models are quite acceptable in the case of simple hot pressing. Model presented in this work considers the processes in the initial stage of consolidation from the works of Artz et al. [6]. Basic assumption is that the particles are spherical and that there is a very small number of the small surfaces contacts. Beside that, in order to simplify the calculations, the technique of “particle expanding” was used. This technique is based on the retaining the fixed distance between particles, while allowing the increase of the particle diameter from value R to R . In other words, a particle “expands” and “overlaps” other particles. The number and size of the new contacts is explained by the function of the radial distribution function (RDF). It is, also, assumed that the material distributes evenly from the overlapping area over the remaining free surface forming the “squeezed” particle with the diameter R .
512
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
For better understanding of the model it is necessary to explain the radial distribution function, geometric model, and to further analyze the selection of the equations that are applicable for the possible densification mechanisms of the composite powders during hot pressing process. 2.1. Radial distribution function (RDF) It is a well known fact that the size and the number of interparticle contacts have a great influence on the densification of powders. For the case where all the particles are of the same size, number of all contacts is described by only one function [6]. Stacking of these kinds of powders has been widely investigated leading to the development of large number of approximations for the radial distribution function for these types of powders. The most important parameter in modeling is the number of the initial contacts, N0 , although the number of new contacts has also an important influence at the rate and mechanism of densification process. When the relative density of the initial powder mixture is 0.64, the change of particle size by 16% during hot pressing process, compared to the initial powder particle size, leads to the full densification of compacts. That is the reason why only a part of the RDF diagram from (R1 + R2 ) to 1.16x (R1 + R2 ) is important to us. Considering the complexity of known analytical procedures, number of initial contacts, N0 , is estimated using the simple geometry based on the equations of Suzuki and Oshima [11], for the non-porous powders. These equations propose that the N0 is the function of the relation between the size and the volume fraction of the particles. In the model of the authors mentioned above, dependence of N0 on the stacking density does not exist, but the proportional high stacking density of particles is supposed. More accurate determination of RDF model demands an estimation of the rate of increase of Nij (number of j particles that are in contact with i particles), with the increase of the distance, r, from the center of the particle i, i.e. dNij /dr. A fact that for the large values of r, dNij /dr is determined by the total volume fraction of j particles is a great advantage in developing of model. There are some assumptions that this equation (Eq. (1)) can be used even for the small values of r (for all values of r greater than Ri + Rj ). dNij dr
=
3D0 fj r 2 Rj 3
(1)
where D0 is the initial relative density and fj is a volume fraction of j particles in the mixture. The chosen semi-empirical equation for the calculation of the cumulative RDF curve shows a very good match with the known analytical calculations and experimental results [12,13,14]. That means that accuracy, simplicity and the fact that the important part of the function is in the narrow range (from 1 to 1.16) are the basic reasons why RDF is chosen for the modeling in the case of this work. 2.2. Description of the geometrical model The first step in modeling acquires the determination of the volume value which the “expanded particle” (R1 ) reaches. R2 can be calculated from the condition that R1 /R1 = R2 /R2 . Since the distance between the centers in this stage is constant, it is obvious that:
D = D0
R1 R1
3
Fig. 1. Schematic view of the contact between two particles.
their expansion is presented (Fig. 1). In the further text this particles will be denoted as a i and j particles. Calculation of the overlapped volume is done in a specific manner (Eq. (2)). In the reality this volume is distributed over the surfaces of both particles and therefore causes an appearance of the new contacts. Radii of the “squeezed” particles are denoted as R . In the starting approximation it has been taken that R1 = R1 and R2 = R2 . Overlapped volume that belongs to the particle i, Vi c is: Vi c =
(3)
where the height of the overlapping area, hi , is: hi = Ri −
R 2i + r 2 − R 2j
(4)
2r
where r is the distance between the centers of the particles. All of these equations apply for the overlapped volume that belongs to j particle as well. Therefore, total overlapped volume, VijT is: VijT = Vi c + Vj c
(5)
For the qualitative evaluation of distribution of the total overlapped volume dislocated from the contact (the sum of the volumes of the overlapped particle calottes), a partitioning factor, P, concept is applied [15]. This factor is very important because it determines the influence of the relative deformation of the particles in the mixture. Determination of a specific equation for the P factor between the particles that are not of the same size is still impossible. Generally, it can be assumed that P = 1/2 for the case when the volumes of translated material from every particle are equal [12,15], but for the case of the composite mixtures, this assumption is not applicable. In this work, i.e. for modeling of the densification process of composite powders which are made of the very hard secondary phase (SiC) distributed in the soft matrix (Al), P = 1 [15] for the softer phase, which means that the deformation induced by the hard–soft contact is placed always in the soft particle. If the Piji is the part of the volume of material distributed on the particle i (which is causing the deformation of the particle i), then j Pij can be calculated as (1 − Piji ). Now, a total distributed volume over the particle i is: R +R
(2)
where D0 is the related density before pressing, and D is a current value of density. For the easier comprehension of the model development, a schematic view of the contact between two particles as a result of
1 2 3R1 − hi h 3 i
ViT =
i
j
j=1,2r=R +R i j
Piji VijT
dNij dr
dr
(6)
where dNij /dr is a radial distribution function for j particle calculated from the center of the i particle. This volume has been
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
calculated from the increase of the particle i diameter from Ri to Ri . Knowing that the volume of the particle i must remain constant before and after the “squeezing”, given equations have to be fulfilled: 4 4 3 3 R i − ViT = R i 3 3
(7b)
2.3. Selection of the appropriate equations for the explanation of densification mechanisms during hot pressing
y˙ = A · · cn = A · ·
F 2
n
(8)
where A is a steady-state power-law creep constant, is a radius of the contact circle between two particles, c is a creep stress, and F is the intensity of the force in the contact area. By rearranging the previous equation we can express F as:
A
(2−1/n)
(9)
This equation can be modified for all types of contacts where the power-law creep mechanism is dominant. The rate at which particle centers approach each other is 2y˙ ij , because y˙ ij = y˙ ji . This statement is based under assumption that identical limitations are applied to the both particles. In the mathematical model this rate is correlated with the i and j particle expansion through the following relation: 2y˙ = Ri + Rj
(10)
y˙ 12 = y˙ 21 = 1 +
R2 R1 R1
(11)
2
Intensities of the forces on all types of contacts can be expressed by the unknown variable R˙ 1 . Since y˙ ij is the same for all contacts with the same values i and j, a mean value of the force on the contact, F¯ ij is:
F¯ ij =
y˙ ij
1/n ij∗
A
(12)
where ij∗ is mean value of a radius of circle for all contacts between two particles. R +R
i
ij∗
j
· ij (2−1/n) ·
=
dNij dr
dr/Nij
ij =
√
R 2i
+ r2 2r
− R 2j
dr
(15)
r=Ri +Rj
It can be seen in literature data [12] that: N11 F¯ 11 f1 4R 21
+
N12 F¯ 12 f1 4R 21
+
N21 F¯ 21 f2 4R 22
+
(13)
+ ···
(16)
Observing the contacts, when the particles are squeezed in one another, acquired force for the plastic flow (yield) is: (17)
Combining Eq. (17) with Eq. (9) a critical radius of the circle of the contact between two particles for which the value of the force needed for the plastic flow is equal to the force needed for the power-law creep to occur can be calculated (Eq. (18)). cij =
y˙ ij
A 3y
n
(18)
where y is a yield stress. For the contacts where ij > cij densification will be completed by the power-law creep, while in the case where ij > cij , intensity of the force for the power-law creep becomes so great that it starts to induce plastic flow on the surface of the contact. Not one contact can endure not to deform when the force greater than the force needed for the plastic flow is applied. Therefore, calculation of the mean value of the force that acts on the contacts is needed when the mechanism of plastic flow is assumed (Eq. (19)). p
y F¯ ij =
p
y
y
Nij · F¯ ij + Nij · F¯ ij
(19)
Nij
where p and y exponents present the given values for the appropriate operative mechanisms (p: power-law creep, y: yielding). The total number of ij contacts, Nij , remains unchanged and it can be calculated from the Eq. (15). p y Nij and Nij can also be calculated from the Eq. (15) by changing y
p
the superior limit of the integral (for Nij ) with rc . Here is Nij = Nij + y y N . Similarly, F¯ can be calculated from the Eq. (20). ij
ij
R +R 1 2
y F¯ ij = rcij
ij2 Nij2
dNij dr
dr
(20)
Switching the calculated values of Fij into Eq. (16), we can calculate new, slightly higher values of R1 . This can be expected, because the “additional” load, induced by contact stress of 3 y , is distributed over the rest of the remaining contacts, increasing the creep rate. 2.5. Diffusion The change in the volume of the doped material on the surface of the formed neck of one particle in one second, by the grain boundary diffusion, can be calculated from the Eq. (21): V˙ =
(14)
4R 22
2.4. Plastic flow
2
N22 F¯ 22 f2
where ap is applied stress, and values of F¯ ij and Ri are different compared to the original equation.
r=Ri +Rj
dNij
F y = 3y 2
2.3.1. Power-law creep If it is assumed that the deformation rate is applicable for the entire deformation zone which depth is equal to the radius of the contact area (overlap), a rate at which contact area approach the centre of the particle, y˙ is:
j
dr
ap =
If these equations are not fulfilled, calculation procedure has to be repeated. A new values of and Ri must be calculated and the Rj procedure given in the previous text has to be followed (similar to the procedures from the references [6,12]).
y˙ 1/n
i
Nij =
(7a)
4 4 3 3 R j − VjT = R j 3 3
F =
R +R
513
4 ıDb + Dv kT
˝
dNij dr
pef
(21)
where ˝ is atomic volume, b is Burgess vector, Dv is volume diffusion coefficient, Db is grain boundary diffusion coefficient, ı is
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thickness of the grain boundary, is a radius of circle of the contact between two particle, k is Boltzman constant, and pef is effective pressure on the contact between two particles. In the case of the composite materials, as it is a case in this work, diffusion mechanisms are occurring only between particles of the soft phase. Therefore, in the Eq. (21), dNij /dr must be replaced with dN22 /dr, while pef is calculated from the Eq. (16).
3. Densification mechanisms maps of the particle reinforced composites Based on the mentioned assumptions, selection of the geometry and the distribution of the particles as well as the equations for the certain densification mechanisms, aim of this work was to model the densification process of particle reinforced composite materi-
Fig. 2. Deformation mechanism maps (Xt D0 is a time curve, where t is the time of densification and D0 is the starting density of the powder, and ED0 n is the group of experimental values; red line is the boundary between the power-law creep and diffusion zones. (a) For pure Al powder; (b) for the Al–10 vol.% SiC composite; (c) for the Al–30 vol.% SiC composite; (d) for the Al–50 vol.% SiC composite. Al particle radius in all composites was 2R = 125 m, and SiC particle radius was 2R = 33 m. Applied pressure was P = 35 MPa (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article).
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
515
Fig. 2. (Continued ).
als (Al–SiC) during hot pressing. The results are presented through the adequate densification maps where the relation between the relative density and the homologue temperature is presented (Fig. 2a–d). Results of densification, presented in Fig. 2, are determined only for the certain constant pressure (P = 35 MPa) and for the certain constant size of the matrix particles (2R = 125 m). A model that is made is universal and it allows the comparison of the densification mechanisms fractions for different pressures, and for different values of the matrix particle size. Other parameters of the densification process can easily be changed by changing the boundaries between zones of dominating mechanisms. Due to the modular structure of the model, complete modulus can be corrected or changed. This is practically very useful fact which enables potential corrections of the model caused by further experimental verification of the results. Model considers analysis of different mechanisms of densification during the hot pressing process, such as plastic flow, power-law creep and diffusion. From the chosen examples it can be noted that the process of densification is slowing down as the fraction of the hardening phase increase (slope of the dashed lines), which is in the agreement with the present experimental studies. A change of densification mechanisms during hot pressing induced by the change of the fraction of the hardening phase can also be observed from the maps (different size of the zones for certain densification mechanism).
4. Experimental evaluation of the model Experimental evaluation of the model has been performed on the pure Al powder and three different composite powders (Al–10 vol.% SiC, Al–30 vol.% SiC, Al–50 vol.% SiC) during densification. Mean value of Al powder particles was 125 m and it was produced by the gas atomization process. This material particles are generally rounded in shape, although some plate like and spherical particles can, also be noticed (Fig. 3a). SiC powder particles, with the mean size value of 33 m (Fig. 3b) are mostly polygonal. Metallic and ceramic powders are mixed in the cylindrical mixing dish for 1 h, with the velocity of 70 rpm. The filling rate of the blender was 30%. These parameters are, according to literature data [3], optimal for this process. Al powder and appropriate composite mixtures of powders are hot pressed in argon protective atmosphere at temperatures of 573 K, 723 K and 873 K for 0.5, 1.5 and 4 h.
Fig. 3. SEM. Size and shape of powders: (a) pure Al; (b) SiC.
The applied pressure was 35 MPa. Density of compacted samples was measured by the conventional Archimedes method, while the apparent density was determined using the Hall’s device (Table 1). Comparing the experimental results, and results suggested by the model, for the certain conditions of the hot pressing process, exhibits deviation of 7–30%. All the experimentally determined valTable 1 Values of the apparent densities for Al and Al/SiC composite. Material
Apparent density, %Dtheoretical
Al Al–10 vol.%SiC Al–30 vol.%SiC Al–50 vol.%SiC
0.42 0.48 0.52 0.55
516
D. Boˇzi´c et al. / Journal of Alloys and Compounds 487 (2009) 511–516
ues of the density are greater than those that the model predicts. The deviations are greater for the samples pressed at the low temperatures (T = 573 K) (Al, Al–10 vol.% SiC, Al–30 vol.% SiC, Al–50 vol.% SiC). The reason for this lies in fact that applied pressure of 35 MPa is not sufficient for the higher densification rate at low temperature. With increase in temperature, differences between model and experimental results are smaller due to the more efficient transport of mass between the Al particles in the composite. As it can be observed from the constructed densification maps, diffusion mechanism shows the greatest and dominant influence on the transport of mass which speeds up the densification process. Also, one of the reasons for the incomplete matching of theoretical assumptions and experimental values is that the irregular shape of the powder particles is approximated with the ideal spheres. Agreement of the model and experiment increases with the increase of SiC fraction in the mixture, which is only a logical result of the greater homogeneity of the mixtures.
Agreement of the experimental results and the results obtained by using the model is quite acceptable, and becomes greater at high temperatures (for the constant pressure) and with the increase of SiC particle fraction in the mixture. Simplicity of the model which allows changing of the process parameters responsible for the movement of the different densification mechanism zones boundaries is the greatest advantage because it presents a quite rapid way of predicting the behavior of the chosen system.
5. Conclusions
´ Z. Vujovic, ´ Powder Metallurgy, BMG, Belgrade, 1998 (in [1] M. Mitkov, D. Boˇzic, Serbian). ´ ´ Ceramic Technology Basics, TMF, Belgrade, 1990 [2] Marija Tecilazic-Stevanovi c, (in Serbian). [3] R.M. German, Sintering Theory and Practice, John Wiley&Sons Inc., New Jersey, 1996. [4] M.F. Ashby, Proceedings of the International Conference on Hot Isostatic Pressing, Lulea, June, 1987, pp. 29–40. [5] D.S. Wilkinson, M.F. Ashby, Science of Sintering 10 (1978) 67–76. [6] E. Artz, M.F. Ashby, K.E. Easterling, Metallurgical Transactions A 14 (1983) 211–221. [7] M.F. Ashby, Acta Metallurgica 20 (1972) 887–897. [8] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, first ed., Pergamon Press, Oxford, New York, 1982. [9] A.S. Helle, K.E. Easterling, M.F. Ashby, Acta Metallurgica 33 (1985) 2163–2174. [10] F.F. Lange, L.L. Atteraas, F. Zok, J.R. Porter, Acta Metallurgica and Materialia 39 (1991) 209–219. [11] M. Suzuki, T. Oshima, Powder Technology 35 (1983) 159–166. [12] S.V. Nair, J.K. Tien, Metallurgical Transactions A 18 (1987) 97–107. [13] S.V. Nair, B.C. Hendrix, J.K. Tien, Acta Metallurgica 34 (1986) 1599–1605. [14] E.K. Li, P.D. Funkenbusch, Metallurgical Transactions A 24 (1993) 1345–1354. [15] E.K.H. Li, P.D. Funkenbusch, Acta Metallurgica 37 (1989) 1645–1655.
This work presents analytical model of the hot pressing process of particle reinforced composite with the soft matrix and hard particles. Basic assumptions for the hot isostatic pressing are successfully applied on the exhibited case of hot pressing. For the determination of the radial distribution function, a semiempirical equation, suggested by Suzuki and Oshima [11] is used. Although the function is rather simple it exhibits the same results as the precise analytical models. Partitioning factor, P, which determinates the relative deformation of the particles in the mixture has also been used in this model. This factor enables the use of this suggested model for all particle reinforced composites, without considering the difference between deformation abilities of different phases. Model considers following densification mechanisms: plastic flow, power-law creep and diffusion.
Acknowledgement Authors are indebt to the Ministry of Science of Republic of Serbia for their financial support through the Project No. 142027. References