Issue 
A&A
Volume 576, April 2015



Article Number  A60  
Number of page(s)  19  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201424278  
Published online  27 March 2015 
Online material
Appendix A: Binary granular mixtures
The strong dichotomy between sizes of chondrules and matrix particles suggests the treatment of the chondritic granular mixture as a binary mixture of two granular components with significantly different sizes. The two components each are idealized as monosized. This case has been studied in powder technology and some general results are available (see e.g. Fiske et al. 1994,and references therein). Here we give a simplified treatment appropriate for chondritic material.
We denote the diameters of the small and large components (matrix and chondrules) as d_{ma} and d_{ch}, respectively, and define the size parameter (A.1)For all known meteoritic classes this size parameter is well above the value of α ≳ 7 above which the small particles easily fit into the interstitials between the large particles. In fact, the value of α is at least 10^{2} and probably above 10^{3} at the early stages of planetesimal evolution before Ostwald ripening of the matrix particulates. Then also “wall effects” are completely negligible where the packing of matrix particles immediately adjacent to a large particle is somewhat less dense than where small particles are only surrounded by small particles. This allows us to treat the chondritic granular mixture as one of two different cases:

Chondruledominated material. The chondrules form a closest packing and the matrix material partially or completely fills the pore space between the closely packed chondrules.

Matrixdominated material. The matrix forms a close packing and rare chondrules are interspersed into the matrix material.
First consider the chondruledominated material. A volume V may be filled with chondrules. The fraction of V filled by chondrules is denoted as (A.2)where V_{ch} is the total volume of all chondrules in V. We assume that the chondrules form a closest packing such that the chondrules are immobile in this packing. We denote the corresponding value of D as D_{cp}, which would typically be equal to ≈0.64 for a random closest packing of spheres. The volume of interstitials (pores) between the chondrules is (A.3)where φ = 1 − D is the porosity.
Now we assume that we have given volumes V_{ch} and V_{ma} of chondrule and matrix material, where the matrix material at most fills the voids between the chondrules. These volumes refer to the true volume filled by the corresponding materials and does not include the voids between the particles. The total volume filled by the granular material is determined by the volume of the chondrules and the void space between them (which, however, is not empty in our case but is partially filled with matrix). This volume is (A.4)The volume filled by matrix and chondrule material is V_{ma} + V_{ch} such that the packing fraction of the mixture is (A.5)We define by (A.6)the volume fraction of matrix to total material. With this we can write (A.7)This is the effective filling factor of the chondrulematrix mixture in the chondruledominated case.
Next we consider the matrixdominated case. The volume filled by the matrix inclusive the void space between matrix particles is V_{ma}/D_{cp} where it is assumed that the filling factor D_{cp} of the porous matrix material is the same as just before. The total volume filled by matrix and chondrules is (A.8)The filling factor of the matrixchondrule mixture is (A.9)and with definition (A.6) of the matrix volume fraction we obtain(A.10)This is the effective filling factor of the chondrulematrix mixture in the matrixdominated case.
At the limit between the two cases the results for the filling factors corresponding to the two cases have to equal each other. From this one finds a matrix volume fraction at the transition of (A.11)The filling factor takes at the transition its maximum which equals (A.12)In terms of the porosity (which is minimum if D is maximum) this takes the obvious form (A.13)If it is assumed that the chondrules are equal sized spheres, the filling factor for the random closest packing is D_{cp} ≈ 0.64 (e.g. Onoda & Liniger 1990; Jaeger & Nagel 1992). The random loosest packing of spheres where the particles just resist to small external forces has a filling factor of D_{cp} ≈ 0.56 (e.g. Onoda & Liniger 1990; Jaeger & Nagel 1992; Güttler et al. 2009). Figure A.1 shows the variation of the effective porosity of the chondrulematrix mixture with matrix fraction for these two cases and Table A.1 gives some numerical values. In principle the initial structure of the chondritic material could be somewhere in between these two cases. Because during the growth of planetesimals by collisions with other planetesimals the surface material is permanently gardened, we assume that the filling factor corresponds to the closest random packing as in experiments this kind of packing is the outcome after vigorous stirring and shaking of a granular material.
Fig. A.1
Variation of effective porosity of a binary granular mixture of matrix and chondrules with volume fraction (with respect to matterfilled volume) of matrix. Full line: the case of random closest packing of spheres, corresponding to D_{cp} = 0.64. Dashed line: the case of loosest close packing of spheres, corresponding to D_{cp} = 0.56. Dotted line: the case of random closest packing of the matrix material where the interspersed chondrules are more densely packed than in the loose close packing. 

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Some characteristic numbers for a chondrulematrix mixture.
The experimental results shown in Fiske et al. (1994) confirm that the equations for the effective packing fraction given above are good approximations for the properties of real binary granular media.
In the case of matrixdominated material one has also to observe the filling factor of the chondrules interspersed in the matrix ground mass. This filling factor is defined as (A.14)where D_{ma} is the filling factor of the matrix ground mass. Eliminating V_{ma} by means of Eq. (A.6) results in (A.15)This filling factor equals the loose random closest packing D_{l} ≈ 0.56 where the packing of chondrules becomes immobile for a value of the volume fraction of matrix material (A.16)In the incompacted state the filling factor of the porous matrix component should equal the closest random packing D_{cp} ≈ 0.64. The value of f_{ram} for this case is shown in Table A.1. For lower filling factors the effective filling factor of the matrixchondrule mixture is (A.17)This is also shown in Fig. A.1. For f below the limit value of f_{tran} corresponding to D_{cp} = D_{l} the packing fraction coincides with the packing fraction of the loose close packing. This type of packing in principle is unstable because the porous matrix does not completely fill the voids between the chondrules. By shaking it would make a transition to the case of random closest packing of both the chondrules and the matrix.
Appendix B: Hot isostatic pressing of a binary granular mixture
For hot isostatic pressing of the chondritic binary mixture one has to observe that sintering of the pure matrix material because of the smallness of the particles occurs at lower temperature by surface diffusion than sintering of the pure chondrule material by dislocation creep. This has to be observed in the modelling of the sintering process.
First we consider the case of a chondruledominated material. Since the matrix does not completely fill the voids between the chondrules if the matrix fraction is less than the value given by Eq. (A.11), the pressure loading rests on the contacts between the chondrules. The matrix is essentially only subject to the low gas pressure and sintering of the matrix is mainly driven by surface tension. Though the filling factor of the initially porous matrix material increases up to unity during sintering of the matrix material, the effective filling factor of the matrixchondrule mixture given by Eq. (A.7) does not change. The only thing one has to do is to use this effective packing fraction and not D_{cp} as the initial value if we solve the differential equation for the time evolution of the filling factor of the chondrules.
In the case of matrixdominated material we have to discriminate between two different cases that are related to the filling factor of the chondrules interspersed in the matrix ground mass. During sintering of the matrix the filling factor D_{ma} increases from the initial value D_{cp} for the closest packing of the nonsintered material to a maximum value of unity. At the same time D^{ch}, given by Eq. (A.15), also increases. This filling factor then could approach the value corresponding to the loose random closest packing D_{l} ≈ 0.56 where the packing of chondrules becomes immobile. If we assume that the chondrule material is more rigid than the matrix material, the sintering of matrix material beyond this point cannot increase the effective packing fraction of the matrixchondrule mixture. The further sintering of the matrix occurs under zero pressure conditions because the still porous matrix material then incompletely fills the space between the chondrules and therefore partially detaches from them. The condition that D^{ch}<D_{l} at complete sintering of the matrix (D_{ma} = 1) is (B.1)The value of this limit is higher than the limit f_{ram}, defined by Eq. (A.16), where the filling factor of the chondrules already equals D_{l} before compaction starts.
If f_{ma}>f_{stop}, reducing of the distance between chondrules during sintering of the matrix ground mass does not increase the packing fraction of the chondrules to the limit D_{l}. Sintering has to be calculated in this case by solving the differential equation for the time evolution of the filling factor of the matrix, D_{ma}, with the initial value D_{cp}. The pressure loading rests on the contacts between the matrix particles. The effective porosity for the matrixchondrule mixture follows from the analogue of Eq. (A.10) (B.2)In the case f_{trans} ≤ f_{ma} ≤ f_{stop} one also has to solve the differential equation for the time evolution of the filling factor of the matrix, D_{ma}, with the initial value D_{cp}. The effective porosity for the matrixchondrule mixture is given by Eq. (B.2) as long as D^{ch} calculated from Eq. (A.15) remains less than D_{l}. The effective filling factor of the mixture at this point follows from Eq. (A.15) by letting D^{ch} = D_{l}; its value is (B.3)If D_{ma} exceeds this value the pressure load is taken over by the chondrules and the calculation of the matrix filling factor has to be continued with zero pressure load for the matrix
until D_{ma} = 1. The effective filling factor of the matrixchondrule mixture is D_{ef} and remains constant until D_{ma} = 1. From this point on the differential equation for the time evolution of the filling factor of the chondrules has to be solved with the initial value D_{ef}.
We can thus discriminate between three different sintering modes of the binary granular mixture of chondritic material, depending on the relative abundance, f_{ma}, of matrix material:

1.
The chondruledominated case f_{ma}<f_{trans} where the shrinking of the material is determined by sintering of the chondrule component.

2.
The matrixdominated case f_{stop}<f_{ma} where the shrinking of the material is determined by sintering of the matrix component.

3.
The twostep case f_{ma} ≤ f_{trans} ≤ f_{stop} where the shrinking of the material occurs in two steps, first by sintering of the matrix component and then by sintering of the chondrule component.
© ESO, 2015
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