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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 mono-sized. 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 10² and probably above 10³ 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:

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

2. Matrix-dominated material. The matrix forms a close packing and rare chondrules are interspersed into the matrix material.

The limit between both cases corresponds to the case where the matrix completely fills the interstitials in a closest packing of chondrules. This is the maximum volume of matrix material in the chondrule-dominated case and the maximum of chondrules that can be interspersed into matrix material in the matrix-dominated material.

First consider the chondrule-dominated 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 chondrule-matrix mixture in the chondrule-dominated case.

Next we consider the matrix-dominated 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 matrix-chondrule 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 chondrule-matrix mixture in the matrix-dominated 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 chondrule-matrix 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.

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 matrix-dominated 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 matrix-chondrule 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 chondrule-dominated 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 matrix-chondrule 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 matrix-dominated 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 non-sintered 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 matrix-chondrule 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 matrix-chondrule 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 matrix-chondrule 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 matrix-chondrule 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 chondrule-dominated case f_ma<f_trans where the shrinking of the material is determined by sintering of the chondrule component.

• 2.

  The matrix-dominated case f_stop<f_ma where the shrinking of the material is determined by sintering of the matrix component.

• 3.

  The two-step 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.

The first mode applies to the ordinary chondrites, the second for the CI, CM, CK, CV, and K chondrites, and the third mode to the CR and CO chondrites.

© ESO, 2015