The characterisation of irregularlyshaped particles
A reconsideration of finitesized, “porous” and “fractal” grains
Institut d’Astrophysique Spatiale (IAS), Bâtiment 121, Université ParisSud
11 and CNRS,
91405
Orsay,
France
email: Anthony.Jones@ias.upsud.fr
Received:
21
October
2010
Accepted:
15
January
2011
Context. A porous and/or fractal description can generally be applied where particles have undergone coagulation into aggregates.
Aims. To characterise finitesized, “porous” and “fractal” particles and to understand the possible limitations of these descriptions.
Methods. We use simple structure, lattice and network considerations to determine the structural properties of irregular particles.
Results. We find that, for finitesized aggregates, the terms porosity and fractal dimension may be of limited usefulness and show with some critical and limiting assumptions, that “highlyporous” aggregates (porosity ≳80%) may not be “constructable”. We also investigate their effective crosssections using a simple “cubic” model.
Conclusions. In place of the terms porosity and fractal dimension, for finitesized aggregates, we propose the readilydeterminable quantities of inflation, I (a measure of the solid filling factor and size), and dimensionality, D (a measure of the shape). These terms can be applied to characterise any form of particle, be it an irregular, homogeneous solid or a highlyextended aggregate.
Key words: dust, extinction
© ESO, 2011
1. Introduction
Interstellar, interplanetary or cometary dust particles are generally modelled as spheres, or collections of mono or multidisperse spheres. Such approaches are clearly rather idealised starting points from which to model and to perform analogue experiments aimed at understanding the obviously much more complex and diverse forms of dust in astrophysical media. For example, many of the collected interplanetary dust particles, or IDPs, show complex, extended and open structures comprised of aggregates of irregular subgrains with radii typically of the order of hundreds of nanometres (e.g., Brownlee 1978; Rietmeijer 1998). Such types of structures are to be expected in coagulated grains in interstellar and protoplanetary disc particles (e.g., Ossenkopf 1993; Ormel et al. 2007, 2009) and are seen in soot particle experiments and simulations (e.g., Dobbins & Megaridis 1991; Köylü et al. 1995; Mackowski 2006, and references therein).
Highlyporous grains, of apparently arbitrarilylarge porosity, can be constructed mathematically but can they exist physically, i.e., can we build a physical 3D model of them and do such particles bear any resemblance to “real” interstellar grains? Such particles also pose the question as to how one should define porosity in these cases.
In this paper we discuss the issues of aggregate/irregular grain porosity and fractal structure and attempt to define characteristic properties for aggregate grains, composed of singlesized spherical constituent grains (also known as subgrains or primary particles) that can be determined using simple measures of the aggregate geometry. We show that this approach can be extended to any irregularlyshaped particle.
2. Porosity matters
The porosity, P, of a particle is usually defined as, (1)where V_{v} and V_{s} are the volumes of vacuum and of the solid matter making up the particle, and V_{t} is the total volume of the particle within some defined surface. Note that this definition of porosity is independent of the form of the particles, which can be of any arbitrary shape, and is only defined for 0 ≤ P < 1. The problem with this definition of porosity is that it is often difficult to define a physicallyreasonable surface that delimits the particle in an unequivocal way. Clearly, a measure of a particle’s porosity requires an unambiguous definition of the reference volume, which is in many, if not most, cases not possible. Thus, the porosity of very extended particles can be hard to define and it may not make much sense to talk of “highly porous” particles. It therefore seems that a better measure or a more openended definition of the nature of coagulated grains is needed. Ormel et al. (2007, 2009) have considered the “porosity” of coagulated grains in terms of an enlargement parameter and a geometrical filling factor approach to determine the particle crosssections, respectively. We return to a discussion of these ideas later.
3. Fractal matters
We recall that a fractal structure has the property of selfsimilarity and that the constituent parts of its structure can be considered a reduced copy of the whole, i.e., the details of the structure are essentially scale or reference pointinvariant. The characteristic of a fractal structure is its Rényi, Hausdorff or packing dimension (i.e., the fractal dimension). In a rather general way the fractal dimension, D_{f}, of a system can be defined as follows: (2)where N(r) and M(r) are the number and mass, respectively, of particles within a defined radius, r, from a given reference point. This definition of the fractal dimension, or what might perhaps be called the “fractality” of the system, is determined by the spatial distribution of the structural units existing within a defined spherical volume of a much larger entity. For an infinitesimallyreducible structure, such as the Mandelbrot set, this clearly poses no problem. However, for a finite sized, porous particle, which does not exist beyond this assumed spherical volume and does not reproduce itself at ever smaller sizescales, the idea of a scaleinvariant fractal dimension no longer holds. Interstellar grains therefore cannot be fractal in the true sense of the definition of fractal entities because their structures are not scaleinvariant, i.e., a shift in the reference point will not always yield the same fractal dimension. For example, a finite piece of “chicken wire” (essentially a 2D hexagonal grid or mesh) might be considered to have a fractal dimension of two globally but one locally when one descends to the level of the wire and no longer sees the larger mesh. Thus, the use of the term fractal, when referring to finitesized particles, can be rather misleading. It would seem that a better description for irregular interstellar grains ought to be possible and should be rather general.
As we will show low D_{f} particles cannot be “porous” because this leads to a loss of continuity in the structure. Thus, the “fractal” nature of a structure is coupled to its “porosity” and the two properties therefore cannot be independently varied, as already discussed in some detail by Mandelbrot (1982).
4. Sparse lattices
By sparse lattice we here mean stacked constituents on a regular lattice (e.g., cubic, hexagonal or facecentred cubic) in which not every lattice site is necessarily occupied.
4.1. Simple cubic lattices
A simple cubic lattice, i.e., a lattice composed of equidimensional regular cubic constituents, can be packed together with 100% efficiency leaving no intervening, unfilled space. This lattice is clearly not of relevance to the structure of interstellar grains but it does, nevertheless, provide some useful intuitive insight into the nature of porous grain structures.
In a simple cubic lattice each site shares 6 faces, 12 edges and 8 corners with neighbouring lattice sites. If we now randomly remove lattice sites we can look at how the number of neighbouring sites depends on the porosity. For illustrative purposes we consider constituent cubes of edge length l stacked into a cubic particle of edge length nl (i.e., total number of lattice sites = n^{3}). A cubic particle of porosity P then contains (1 − P)n^{3} occupied sites and the average number of occupied adjacent sites (ignoring the macroscopic particle outer edges and surfaces), which we call the coordination number for a given porosity, C_{P}, is simply (1 − P)C, where C is the maximum site coordination number under consideration. For a cubic lattice, C may take the values 6, 18 or 26, depending on the hierarchy of the coordination that we allow, i.e., sites sharing only faces (C = 6), sites sharing faces and edges (C = 6 + 12 = 18) sites sharing faces, edges and corners (C = 6 + 12 + 8 = 26). We then have that (3)We are now interested in the limit where the structure is just continuous and, therefore, can be physically constructed as a single entity. Two limits are of interest here:

1.
the dimer limit where each site has only one occupiedneighbouring site and the system therefore consists only ofdimers. Such a “structure” is clearly not macroscopic, does not“hold up” and is not “constructable”;
 2.
the continuity limit, which is somewhat akin to the percolation threshold for a lattice, where the structure is just bound into a single, “onedimensional” unit by intersite contacts or bonds.
The first case is trivial and C_{P} = 1. The second case is of interest in determining whether a given structure is “constructable” and in this case C_{P} is just equal to 2, which ensures that the network can, minimally, completely interlink. In this case each site has two filled neighbouring sites and the network can, in theory at least, propagate over the whole considered volume as a 1D “chain”. Although, in the latter case the structure would resemble something like a ball of wire, i.e., a onedimensional construct warped into a three dimensional space, which is not a fractal structure.
For the simple cubic lattice, or in fact for any lattice, the relevant critical porosities P_{crit} for a given case are given by (4)For the dimer limit, i.e., C_{P} = 1, we then have that , respectively, for the C = 6, 18 and 26 maximum coordination cases. For the more interesting continuity limit case, where C_{P} = 2, we have , respectively, for the C = 6, 18 and 26 maximum coordination cases. In the latter case, i.e. C = 26, allowing a structure to coordinate only through the corner points is perhaps a rather extreme situation. The equivalent porosities for the three cases are ≈66, 89 and 92%. For comparison, percolation theory, see for example Stauffer & Aharony (1994), predicts site and bond percolation thresholds of 0.3116 and 0.2488, respectively, which would be equivalent to “critical porosities” of ≈69 and 75%. Our simple model therefore indicates values roughly consistent with percolation theory.
4.2. Closepacked lattices
We now turn our attention to the perhaps more astrophysicallyrelevant, but still idealised, case of closepacked lattices of spherical entities. In this case we note that the lattice positions are fixed and that the constituent “monomers” cannot take arbitrary positions. Nevertheless, this is a useful starting point.
By simple closepacked lattices we refer to lattices composed of regular, equalsized spheres packed as efficiently as possible, i.e., facecentred cubic (fcc) and hexagonal close packing (hcp), which are identical for our purposes. However, for the simplicity of the presentation, we will only consider the fcc case for simple solid spheres. For identical size and composition spheres, the fcc lattice unit cell can be considered as a cube where each face of the cube shares a sphere with one adjacent unit cell and each corner of the cube shares a sphere with seven other neighbouring unit cells. It then follows that each fcc unit cell contains spheres within a cube of side length where d is the diameter of the sphere. The volume of the cube is then (5)and the volume occupied by the solid matter of the spheres is (6)The porosity of such a vacancy or defectriddled fcc structure is then given by (7)i.e., a porosity of 26%, which is equivalent a solid filling factor of , which appears to be the most efficient lattice packing possible for equalsized spheres. The same value holds for the hcp lattice. For the simple cubic packing of spheres it can similarly be shown that the filling factor is . These results can be found in standard textbooks on crystallography and the solid state.
We now consider a spherical volume of radius A and volume V_{g}, perhaps equivalent to an aggregate interstellar grain, composed of spherical subgrains, of volume V_{sg}, packed on a fcc lattice. The maximum number of subgrains, N_{max}, within the total grain volume is then (8)We now use the above results to study the effects of introducing porosity into the structure by removing some of the subgrains from the lattice. If N is the total number of subgrains within the particle or aggregate (N ≤ N_{max}) then, following from the definition above (with ), the porosity of the particle is given by (9)or, equivalently, the number of subgrains for a given porosity, N_{P}, is given by (10)where we recover the value N_{max} when P = 0.
The average subgrain coordination number for our porous lattice, which can be expressed in the same way as for the simple cubic lattice discussed above, is simply the maximum coordination number, C (12 for fcc and hcp), multiplied by the fraction of filled sites (1 − P), i.e., C_{P} = (1 − P)C. In this case the dimer limit and continuity limit critical porosities, P_{crit}, are and , i.e., 92% and 83%, respectively. Thus, it appears that, for this idealised closepacked lattice (fcc or hcp) model, there is an upper limit to the porosity of a particle and that this limit is ≈80%. For the cubic lattice the equivalent limiting porosity is ≈70%. Note that these limits are given by (1 − [2/C]) and are therefore independent of the monomer size.
However, remember that this limiting porosity strongly depends upon the above, and widely used, definition of porosity and the fixed lattice positions for our particle subgrains. It also depends on the fact that we have assumed that the subgrains are “evenly” distributed throughout the volume . We now look at the nature and definition of porosity more closely.
5. Sparse networks
By sparse network we here mean contiguous, irregularlyconnected, spherical constituents that do not lie on a regular lattice.
5.1. General packing of spherical grains
We consider a monodisperse collection of connected spherical subgrains in an arbitrarilyshaped, “porous” particle that can be defined by its three semimajor axes (a,b and c), as per an ellipsoid, with an “enclosing” volume, . The maximum number of subgrains of diameter d within this assumed volume is given by (11)similar to that defined for a spherical particle above but where we have now replaced the maximum fcc lattice packing efficiency for monodisperse, spherical particles, , by a moregeneralised maximum packing efficiency φ. The porosity of the particle is given by (12)and the number of subgrains for a given porosity, N_{P}, is (13)We can now consider a more generalised definition of the semimajor axes, a,b and c, of our porous particle that is independent of the filling factor. We normalise the three principal axes of the particle (i.e., 2a, 2b and 2c) by the monomer size, d. If we assume that these axes are parallel to the Cartesian axes x,y and z, and that N_{x} = 2a/d, N_{y} = 2b/d and N_{z} = 2c/d, then the volume within the encompassing surface is (14)We can simplify the above expressions for N_{max}, P and N_{P} to: (15)(16)(17)The key parameters in defining our arbitrary particle are now the number of constituent monomers, N, and its normalised dimensions, N_{x},N_{y} and N_{z} or more generally N_{i} where i = x,y or z. The nature of the particle can be characterised, independent of porosity or fractal measures, through the sum, Σ^{i}, and the product, Π^{i}, of its dimensions, i.e., Table 1 shows the values of Σ^{i} and Π^{i} for the most compact and the most extended particles possible. For highly porous “disclike” and “spherelike” distributions this corresponds to linear structures located along 2 and 3 orthogonal axes, respectively. Note that a onedimensional or rodlike assemblage of spheres can never be porous since any “vacancy” cuts the structure into smaller but separated particles. Thus, the maximum possible particle dimension is Nd and the minimum particle dimension is that of the most compact spherical particle possible, i.e., φN^{1/3}d. Figure 1 shows the relationship between Σ^{i} and Π^{i} as a function of the particle dimensionality, which is defined below.
Fig. 1 Axis sum, Σ^{i}, vs. axis product, Π^{i}, plot for a particle containing 100 identical subparticles as a function of the dimensionality (as indicated): D = 1 (diamond), 1.5 (plus signs), 2 (triangles), 2.5 (crosses) and 3 (squares). Compact particles are to the lower left and extended particles to the upper right. Note that no porosity is allowed for D = 1, rodlike particles. 

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Σ^{i} and Π^{i} for compact and highly extended particles.
We can see from Table 1 that for the most compact particles Σ^{i} ≤ N and Π^{i} = N and for the most extended particles Σ^{i} = N and Π^{i} ≥ N. If we wish we can consider the porosity, P, of these particles and rewrite N_{max}, P and N_{P} as: (20)(21)(22)However, the advantage of using the Σ^{i} and Π^{i} characterisation is that we have a system that can generically describe an arbitrarilyextended particle without the need to define its porosity or its “fractality”.
5.2. Dimensionality and inflation
With a view to generalising the notion of grain “porosity”, or better characterising extended particles, we consider a more openended definition than simply the particle porosity, as did Ormel et al. (2007, 2009). Ormel et al. (2007) considered grains in terms of an enlargement parameter, the ratio of the extended volume to the most compact volume. Whereas Ormel et al. (2009) used a geometrical filling factor approach to determine the particle crosssections. We now consider the idea of particle “inflation” beyond the most compact forms possible. In this case we define the inflation, I, for a given particle as: (23)Inflation is thus, equivalently, the ratio of the volume of the enclosing ellipsoid (, Eq. (14)) to the (minimum) volume of the solid matter in the aggregate (), where φ is the maximum packing efficiency of the subgrains in the minimum volume. For cubic particles (on a cubic grid, see Sect. 4.1), where φ = 1, we have . Note that with this definition an inflation factor I = 10 could be considered akin to a porosity of 90%.
The inflation, I, for any particle has a minimum value of unity and a maximum value of ≈N^{2}/27 (for unit packing efficiency, i.e., φ = 1), which is determined by the mostextended “spherical” particle possible. This hypothetical, and rather physically unrealistic, structure has an equal number of subparticles (N/3) arranged along the three orthogonal, cartesian axes. This corresponds to the extreme upper right part of Fig. 1. For a particle with a dimensionality of 2 the maximum value of I is ≈N/4. We can see that with this definition the increase in the particle size by a factor of f translates to an inflation factor of f^{3}, provided that the condition (N_{x} + N_{y} + N_{z}) ≤ N is fulfilled. Note that the largest values of I are to be found for the most “spherical” extended particles where the product, Π^{i}, is a maximum for a given sum, Σ^{i}.
We now define the dimensionality, D, of any given extended particle in terms of its maximumdimensionnormalised sum Σ^{i}: (24)where N_{L + } = max { N_{x},N_{y},N_{z} } and indicates that we take the longest dimension of the particle, as the normalisation length.
It is clearly possible to extend this definition of D to the general case of any arbitraryshaped particle, independent of whether it is an aggregate or a highly irregular single “lump”, i.e., (25)where L_{i} is the particle size along its ith axis (x, y or z). As an illustrative example, in Table 2 we show the dimensionality and inflation factors for several regular solids, where the value L_{x} (L_{z}) is, generally, the maximum (minimum) dimension of the regular solid. Note that in the case of the cube the longest dimension, L_{x}, is the cube diagonal, and not an edge, and that the other two dimensions are then taken orthogonal to this direction.
The dimensionality, D, and inflation, I, for regular geometrical solids. a and l are the relevant radius and side/edge lengths, respectively.
For our general case we can define a minimum and a maximum particle dimension, N_{L − } and N_{L + }, respectively, defined by the minimum volume (spherical) closepacked particle, of radius a, and the maximum linear chain length. These extreme particle dimensions are given, for the fcc closepacked and the more general case, by (26)and (27)Thus, all possible particles must have sizes, L (expressed in terms of the subunit dimension d), within the range (28)where the lower limit can be equated with a fractal dimension D_{f} ≈ 3 and the upper limit with D_{f} ≈ 1. However, the particles in question are not necessarily “truly” fractal in nature. In fact, this definition of the quantity dimensionality (1 ≤ D ≤ 3) can indeed be applied to any particle, whether solid and homogeneous (I = 1), particles with concavities and protrusions or porous aggregates (I > 1). Additionally, the concept of inflation can be applied to any particle because it is just the volume of the enclosing ellipsoid, that can directly determined from the particle dimensions, divided by the minimum volume of the solid matter (V_{s}/φ, where φ is set to the maximum packing efficiency and is set to 1 for a solid particle).
Clearly any given, finitesized particle can only have dimensionalities in the range 1 to 3. The dimensionality, D, when coupled to I, can perhaps be considered as somewhat analogous to the fractal dimension of a structure and as a representation of the spatial arrangement.
The concepts of porosity and fractal structure (“fractality”) for any given extended and “constructable” particle can now reposed in terms of the related and estimable quantities of inflation (I) and dimensionality (D). In Fig. 2 we show a plot of D vs. I for ≈15 000 randomlygenerated particles containing 50 identical subgrains. For illustrative purposes we have assumed maximum packing efficiency (i.e., φ = 1). These aggregates were generated by imposing the conditions: 1 ≤ N_{i} ≤ N (where i = x, y or z), Σ^{i} ≤ N and Π^{i} ≥ N. From this figure it appears that particles created in this way are dominated by dimensionalities of the order of ≈1.2–2.3 and inflation factors of ≈5–50. The lower cutoff to the data points in this figure is simply due to the relationship between the sum of the particle dimensions, which determines the maximum dimension and therefore D, and the product of the particle dimensions, which determines the volume and hence I.
In Fig. 2 all particles are assumed to be equally “constructable”. However, given that the number of possible structural isomers for the given number of monomers (that we could perhaps call “structomers”) must increase with the enclosing volume V_{e}, the more extended structures are given undue weight in Fig. 2. This is indicated by the higher density of the data points for larger values of I in this linearlog plot. Hence, and for illustrative purposes, in Fig. 3 we plot another simulation in which the size of the data points are volumeweighted by dividing by I and shapeweighted by multiplying by (1 − [D_{m} − 2]^{2}). For the shapeweighting we adopt an illustrative mean value of the dimensionality D_{m} = 2, i.e., close to the typical fractal dimension (i.e., D_{f} = 1.8) seen in soot particle formation experiments. What Fig. 3 shows is that, assuming D_{m} = 2, aggregate particles will probably have inflation factors I < 10. However, this will obviously require a more detailed investigation than presented here because the “structomer” distribution in the D vs. I parameter space will depend upon the adopted particle construction protocol.
Thus, what emerges from the above is that it is perhaps not the fractal dimension or even the porosity that counts for finitesized, extended particles but the “dimensionality” of the structure coupled to its “inflation”. In Table 1 we can see that the “dimensionality” is represented in the exponents of Σ^{i} and Π^{i}.
Fig. 2 Dimensionality, D, versus inflation, I, plot for particles containing 50 equalsized constituents for ≈15 000 randomlygenerated structures (with φ = 1). The inset box shows the limits on D (1 ≤ D ≤ 3) and I (1 ≤ I ≤ N^{2}/27), and the middle vertical line shows the limit for an idealised 2D particle, i.e., I = N/4. 

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Fig. 3 Same as for Fig. 2 except that we have now volume and shapeweighted the data point sizes by a factor (1 − [D − 2]^{2})/I (see text). 

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In Appendix A we consider the implications of the above characterisation of irregular particles for the calculation of their crosssections.
6. Implications for astronomical dust studies
Perhaps in no small part motivated by the then rather recent publication of the Bohren & Huffmann (1983) book, early studies of “porous” and composite interstellar grains, which must implicitly be irregular in form, treated “porosity” as vacuum filling a given fraction of the particle “volume” but did not take into account effects arising from their irregular structure (e.g., Jones 1988; Mathis & Whiffen 1989). These early investigations of grain porosity, and those subsequent to them, adopted a variety of approaches to the problem, including: effective medium theories (EMT, e.g., Jones 1988; Mathis & Whiffen 1989), hollow spheres (HS, e.g., Jones 1988; Min et al. 2005), multilayered sphere models (MLS, e.g., Voshchinnikov et al. 2005) and discrete dipole approximations such as DDA (e.g., Bazell & Dwek 1990; Perrin & Sivan 1990, 1991; Henning & Stognienko 1993; Wolff et al. 1994). These methods can be put into three general classes: 1) implicit orientationaveraged (HS and MLS – spherical shell representations of an infinite number of randomlyorientated structures), 2) optical property averaged methods (EMT – solid “dilution” by the addition of vacuum, e.g., see Appendix B), and 3) fixedshape methods where orientation and structureaveraging is required (DDA). Comparison studies (e.g., Jones 1988; Wolff et al. 1994, 1998; Voshchinnikov et al. 2005; Shen et al. 2008, 2009) reveal that the results of these very different methods actually compare rather well. However, the “averaged” approach EMT, HS and MLS methods, as applied to spheroidal porous grains, cannot account for the details of the scattering properties of porous/irregular particles. In this sense DDAgenerated particle studies do a much better job in determining the scattering properties of porous particles (e.g., Shen et al. 2008, 2009) because they take into account shape effects. Recently (Skorov et al. 2010, and their earlier work cited therein) studied very large, fluffy aggregates (N ≲ 2000) and conclude that their optical properties are sensitive to the aggregate size parameter, that characterisation by only N is insufficient (i.e., more detailed descriptions are needed) and that polarisation data can provide a measure of the aggregate fluffiness.
The concepts of inflation, I, and dimensionality, D, as defined here, can be applied to any arbitraryshaped particle that is “constructable” with the DDA approach. They can also be applied to aggregates of spherical primary particles or monomers that can be analysed with the Tmatrix method (TMM, e.g., Mackowski & Mishchenko 1996) and the generalised multiparticle Mie theory method (GMM, e.g., Xu 1995). All such particles are characterisable in terms of I and D. The “porosity” of very open structures (i.e., “highly porous” particles) is not unambiguously determineable, for the reasons discussed in the earlier sections, and the “fractal dimension” description for aggregates consisting of a rather limited number of monomers or primary particles does not appear tenable (see earlier sections and also Köhler et al. 2011).
The D and I descriptors may be particularly useful for characterising highly elongated structures and more meaningful than is possible with a porous particle description. Elongated structures actually correspond to low values of D and, perhaps rather surprisingly, also to low values of I. The low values of I arise in this case because the enclosing ellipsoid surface lies close to most of the constituent subparticles because of its “cigar” shape.
In a porous aggregate the projected crosssection can be significantly less than that adopted in the “averaged” and “dilution” methods (e.g., EMT, HS and MLS) where the particle is assumed to be spherical and defined by a porosityweighted, effective radius (e.g., see Appendix A.1). In the determination of the optical properties these methods actually do a surprisingly good job. In Appendices B and C we present some aspects of the Bruggemann EMT methods and dust crosssections at long wavelengths that could be useful in studies of inhomogeneous particle studies.
6.1. Irregular particle mean projected crosssections
We now use the methods developed earlier, to explore the relationship between D and I, to estimate the projected crosssections of randomlyconstructed aggregates within an ellipsoidal volume. For this exploratory model we use 81 cubic primary particles (i.e., N = 81) on a cubic grid (see Sect. 4.1), which have unit packing efficiency (φ = 1) and a “spherical” compact form with cube centres that lie within a circumscribed sphere of diameter 5 times the length of the side of a cube. As before, the values of D and I for the particles are randomly generated. The centre of each face of the N_{x} × N_{y} × N_{x} “box” that encompasses the ellipsoid is assigned a primary particle, and this used as a seed to link the faces with randomlygenerated but sequentiallycoordinated primary cubes. This ensures that the values of D and I sogenerated are fulfilled for the particle in question. Extra cubes are added, randomly but also sequentiallycoordinated to existing particle cubes, until the aggregate contains the required 81 primary particles. The coordination number, m_{i}, for each cubic element i in the generated aggregate is calculated by summing the number of occupied nearest neighbour cubes. The mean coordination number for each generated aggregate is then . The minimum condition for a constructable, single particle is the continuity limit for the cubic lattice, i.e., (see Sect. 4.1) and aggregates with are therefore rejected. In constructing an aggregate we only allow neighbourcoordination by cube faces and edges (see Sect. 4.1), which corresponds to the condition . However, coordination numbers up to 26 are, in principle, possible because of the random filling method used. For the aggregates is always greater than 2, and generally less than 4 for these sparse lattices.
In Fig. 4 we show a D vs. I plot for a limited range of I (≤10). The symbols (filled circles) are sizescaled by the mean particle coordination number , which lies in the range . The crosses indicate “particles” with that are not constructable (see Sect. 4.1) and therefore eliminated from the crosssection analysis. Note that most of the rejected particles “envelope” the valid aggregates at higher values of I. The values of are not systematically distributed in the D vs. I space but appear to be somewhat random. Also note that the majority of the constructable aggregates have I < 5, which correspond to porosities ≲80%. The curved series of data points for the valid aggregates, especially visible at the lower values of I are due to the discreteness, in the limited range of compact aggregates for N = 81, and the coupling between D and I (see below).
Fig. 4 The same as Fig. 3 but for ≈500 aggregates (consisting of 81 cubic primary particles) and a more limited range of I (≤10). The symbols (filled circles) are sizescaled by . The crosses indicate “particles” which are not constructable (see Sect. 4.1). 

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Fig. 5 The effective crosssection, normalised to the number of primary particles, N, for ≈500 aggregates constructed from 81 cubic primary particles on a cubic grid. The symbols (black filled circles and coloured plus signs) are sizescaled by the particle dimensionality, D. The plus signs colourcode (see key at bottom right of the figure) the binned dimensionality (red D = 1–1.5; yellow D = 1.5–2; green D = 2–2.5; blue D = 2.5–2). The orange lines shows some representative analytical fits (see text). 

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The projected crosssection for a given aggregate is calculated by viewing it along each of its axes, x,y or z, and summing the number of occupied “pixels” in the plane projected on the other two axes, i.e., σ_{x}(y,z), σ_{y}(x,z) and σ_{z}(x,y). The effective crosssection for the aggregate is then taken to be (29)Figure 5 shows the relationship between the aggregate projected crosssections ⟨σ_{eff}⟩, normalised by the number of primary particles N, and their inflation. The data points (filled circles) are scaled by the value of D for the aggregate and are also colourcoded and binned by D (plus signs): 1–1.5 (red), 1.5–2 (yellow), 2–2.5 (green) and 2.5–3 (blue). The solid orange line in Fig. 5 shows an analytical fit, σ_{fit} = 0.52 N (1.1 + 0.5I^{0.7}), to the principle data points in this ⟨σ_{eff}⟩ vs. I plot. The upper [lower] dashed orange lines are for σ_{fit} = 0.59 N (1.1 + 0.5I^{0.5}) [σ_{fit} = (0.47 N (1.1 + 0.5I^{0.9})]. Perhaps the first thing to note in this figure is that, with our adopted methodology, we do not recover the full crosssection of the 81 constituent primary particles but, at most, only about half of that number. Note that the asymptotic limit for the analytical fit to the data (solid orange line) is ⟨σ_{eff}⟩ = 1.1(0.52 N) l^{2} or ≈57% of the crosssection of the infinitelyseparated cubic primary particles, i.e., 81l^{2}. This arises because we impose the condition that the aggregates must be constructable and contiguous entities, which limits the typical ranges of D (≈1.2−2.5) and I (≲5) and results in structures with significant shadowing (≳50%). As shown in Appendix A.1 even highly extended aggregates with D_{f} = 1 still shadow some 13% of the total primary particle crosssection. Note that for such an aggregate our method predicts and therefore 33% shadowing, rather than the expected 13%. In fact only in a system of isolated particles (and therefore by definition not an aggregate) is it possible to recover the crosssection of the infinitelyseparated primary particles. This only occurs when there can be no shadowing and the crosssection per primary particle, σ_{pp}, is equal to that of the total number of particles (i.e., σ_{pp} ≈ N l^{2} = πr^{2}) and the volume per primary particle is then , which is equivalent to the value of I where all primary particles are “visible” in the crosssection. With our methodology we fitted simulations of the crosssection of a “cloud” of 81 randomlyplaced and isolated primary particles and find σ = N l^{2} (1−1.9e^{−I0.25}), which yields σ = N l^{2} at an inflation value of ~1700.
It is clear from Fig. 5 that ⟨σ_{eff}⟩ increases with increasing I. This is due to what might be considered an “optical depth effect” because the more inflated particles expose more of their constituent primary particles and therefore add to the projected area. It also appears that the effects of I on ⟨σ_{eff}⟩ are stronger than those due to D, which perhaps appears at odds with previous studies of soot particles (e.g., Köylü et al. 1995, see Appendix A.1 for the details). However, such studies generally only characterise the particles in terms of their fractal dimension D_{f}, whereas here we use a dual characterisation with D and I, where D is not equivalent to D_{f}. Hence, a direct comparison of the respective analyses is not really possible.
The data in Fig. 5 show a “yellow spur” in the upper left region at ⟨σ_{eff}⟩/N = 0.44–0.51 and I ≲ 3. An analysis of the aggregate shape distribution shows that the yellow spur particles corresponds to “pancakelike” particles with holes in which one of the dimensions N_{x}, N_{y} or N_{z} is equal to that of a single primary particle. Such particles are not numerous in this simulation, and are not particularly physical, and we therefore ignore them here.
6.2. Irregular particle mean surfacetovolume ratios
The formation of H_{2} is a critical step, and a key starting point for, interstellar chemistry. Given that its formation occurs principally by H atom (re)combination on grain surfaces an aggregate surfacetovolume ratio, S_{V}, is of importance to interstellar chemistry and we have therefore calculated this ratio for our cubic aggregates. S_{V} depends upon the mean facet coordination number, (), which is the mean number of cube faces that coordinate to other primary particle faces and reduce the available surface. For the most compact “spherical” form with 81 primary particle cubes, and a projected crosssection of 21l^{2} along each axis, we have S_{V} = (6 × 21l^{2}/81l^{3}) = (126/81l) = 1.56/l. The maximum possible value, where the cubes are connected only by their edges and all surfaces are therefore exposed, is S_{V} = (81 × 6l^{2}/81l^{3}) = 6/l (we recall that cube apexonly connections are not allowed here). For the constructed aggregates we then have (30)which gives the maximum value when and the minimum value when . For each cube the maximum possible coordination number is the sum of the edge (12) and face (6) connections, i.e., 18. Face or facet connections therefore make up one third of the possible coordination sites and we have , which yields (31)and normalising to the minimum surfacetovolume ratio for the most compact structure gives (32)In Fig. 6 we show this normalised surfacetovolume ratio for the same aggregates as for Fig. 4 but also include the nonconstructable aggregates (crosses). The horizontal dotted line indicates the clear cutoff between constructable () and nonconstructable () aggregates, i.e., . For the constructable aggregates the mean coordination numbers range between 2 and 4, corresponding to a rather narrow range in S_{V}, i.e., 3.00–3.43. Thus, these aggregates do not exhibit large variations in S_{V}, which will also be the case for aggregates of spherical primary particles. The points enclosed by squares are the rather unlikely pancakelike, “yellow spur” aggregates in Fig. 4 and we again ignore them here. Also shown is an illustrative empirical “fit” to the data .
Fig. 6 The surfacetovolume ratio for ≈500 valid aggregates (filled symbols) with 81 cubic primary particles, on a cubic grid, normalised to that of the most compact “spherical” structure. The filled symbols are size and colourcoded as per Fig. 5. The “bare” crosses indicate nonconstructable aggregates, as per Fig. 4. 

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6.3. Applicability to “real” aggregates?
As interesting as the above results, for the effective crosssections of aggregates of cubic grains on a cubic lattice, may appear we really need to consider their relevance to “real” aggregates consisting of a size distribution of “spheroidal” primary particles. Firstly, for spheres on the same cubic grid the effective crosssection would be lower by a factor when we take a mean value based upon views exactly along the cartesian axes of the aggregates, which is a special case that maximises the view of the “pores” between the primary particles. If we imagine taking the mean values offset by only a small angle then we quickly “lose sight of” the pores and approach the value of ⟨σ_{eff}⟩ for the array of cubic primary particles. For spheres on an fcc lattice our derived ⟨σ_{eff}⟩ is also probably a good approximation because, in the fcc lattice where φ = 0.74 (cf. φ = 1 for cubic primary particles) and the maximum coordination number is 12, there is intrinsically significant shadowing and in projection the “pore dilution” of ⟨σ_{eff}⟩ will be negligible. Similarly, if we consider pore filling by a mantling “glue” and/or abundant smaller primary particles, then our “simple cubic crosssection” model probably does a rather good job of estimating the projected crosssection for an aggregate as a function of D and I.
This study is not intended to be exhaustive but is meant to be rather indicative and to give some general guidelines for studies of irregular aggregates. Clearly the exact forms of the particles in a study, and therefore their distributions in the D and I space, will depend on the aggregate construction protocol. Nevertheless, the “simple cubic crosssection” model is probably a very good approximation and ought to be widely applicable.
6.4. Implications for gasgrain coupling
In dynamical studies of porous/irregular grains, where gasgrain coupling plays a key role, it is important to have a good measure of the geometrical, or projected, crosssection of the particles. This is particularly so for porous/aggregate dustgas coupling leading to dust drag/acceleration in circumstellar and disc winds, and in turbulent or shocked gas. Simply “inflating” the grains, using a lowered effective density to model porosity (e.g., Jones et al. 1994) is probably not sufficient because the inflated crosssection is not filled and the surfacetomass ratio will lower than that predicted by the simple grain“inflating” approach. In gasgrain coupling studies the projected crosssection for irregular (and “porous”) grains, and the appropriate orientationaveraging, therefore needs to be carefully considered as it will depend upon the adopted aggregate construction protocol.
7. Concluding remarks
We underline the fact that care needs to exercised in the use of the terms porosity and fractal dimension when considering finitesized interstellar, interplanetary or cometary particles. In detail these grains are not fractal in the true sense of the definition of the term fractal because their structures are not scaleinvariant, i.e., a shift in the reference point will not always yield the same fractal dimension.
Often, in coagulation simulations and experiments, the structures appears as rather psuedocentrosymmetric, in the sense that they appear to have a centre from which extended structures radiate. In contrast, IDP grains are rather “blocky” and do not appear to exhibit “radial”type structures extending away from a centre.
A new means of characterising the averaged “porosity” and “fractality” of irregular particles is proposed. Given that these two properties of an aggregate are not independent of one another, it seems that may be we should rather think in terms of defining our aggregate, “porous” or “fractal” particles in terms of dimensionality and inflation. A particle’s inflation, I, is a measure of its size compared to the most compact form possible and its dimensionality, D, is a measure of the 3D disposition of the constituent matter in the particle. The parameters I and D can be derived from direct measurements of the particle dimensions (normalised to that of the constituent subgrains in the case of monodisperse aggregates). Their usefulness is, in particular, underlined by the fact that I and D can be applied to particles of any arbitrary shape or structure be they solid and homogeneous, have significant concavities and protrusions or consist of porous aggregates of subgrains or primary particles.
We find that the crosssections of irregular aggregates can be rather well approximated by simply modelling them as sparse lattices of cubic primary particles on a simple cubic grid and then averaging along the orthogonal x, y and z axes to obtain the mean crosssection.
This study has focussed on aggregates composed of singlesized subgrains in an attempt to explore their nature and to propose new measures to allow their characterisation. Interstellar grains will in any case be less porous than the rather idealised aggregates considered here because of the effects of ice mantle accretion and the fact that the coagulated grains are formed from a size distribution of primary particles with a clear excess of small over big grains. These factors will tend to lead to pore filling and to aggregates that are more compact and much more strongly bound together due to the extensive surfacetosurface contacts.
A definition of I and D for aggregate particles derived from a size distribution of primary particles will be rather more complex than the simple derivations presented but could, in principle, be derived in an analogous way. However, this may not be necessary because the quantities dimensionality, D, and inflation, I, can be derived from direct measurements of the particle dimensions (L_{i}, where i = x,y or z) and the volume (or mass) of the constituent solid matter, V_{s}, i.e., (33)and (34)
Acknowledgments
I would like to thank Melanie Köhler and Vincent Guillet for a careful reading of the manuscript and for stimulating discussions on this subject. I would also like to thank the referee for an encouraging review and for useful suggestions. This research was made possible through the key financial support of the Agence National de la Recherche (ANR) through the program Cold Dust (ANR07BLAN036401).
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Appendix A: Extended particle properties
With the interpretation of the interstellar dust emission observations from the Herschel and Planck missions very much in mind, we are obviously interested in the long wavelength emission from dust and dust aggregates, and therefore in the absorption crosssection, which is equal to the emission crosssection, C_{abs} = C_{em}. In the following we explore the crosssectional properties of aggregates of subgrains.
A.1. Orientationaveraged crosssections
The orientationaveraged crosssection factor, ⟨S⟩, (the effective crosssection normalised to the maximum crosssection) for a particle is obviously ⟨S⟩ = 1 for a sphere (all particle axes equal). For a thin disc Micelotta et al. (2010) find . For a randomlytumbling cylinder Brown et al. (2005) show that the effective crosssection is , where r is the cylinder radius and l is its length, which corresponds to: (A.1)If we take the particle enclosing volume, V_{e}, normalised to the subparticle size d, to be V_{e}/d^{3} = π (N_{x} N_{y} N_{z})/6 (see Eq. (14)), then the subparticle sizenormalised, orientationaveraged, “enclosing” particle crosssection is given by: (A.2)which is not filled, except for compact particles. However, we need to determine the actual geometrical crosssection of the extended, porous particle (the fraction of the crosssection σ_{e} that is filled), i.e., the total projected crosssection, or the geometrical filling factor of Ormel et al. (2009). Unfortunately, Ormel et al. (2009) and (2007) give no indication as to how the geometrical filling factors and enlargement parameters were actually calculated for their particles.
This problem was studied by Köylü et al. (1995), based on earlier work cited therein, within the context of soot particle aggregate structures. They found that, for agglomerated soot particles with a fractal dimension D_{f} ~ 1.8, the number of primary particles, N, within the aggregate can be expressed as: (A.3)where k_{a} is constant close to unity, A_{a} is projected area of the aggregate and A_{p} is the crosssection of a primary particle, π(d/2)^{2}. Köylü et al. (1995) considered aggregates of primary particles or subgrains. A powerlaw correlation is found to give an excellent fit to their data with k_{a} = 1.15 ± 0.02 and an index α = 1.10 ± 0.004. The experimental uncertainty in α is apparently very small (see Fig. A.1) and we will therefore subsequently ignore it. Thus, the projected area, or effective crosssection, ⟨σ_{eff}⟩, of their fractal particles is given by (A.4)As noted above this appears to be valid for a limited range of fractal dimension applicable to soot aggregate particles (D_{f} ≈ 1.7−1.9). The above expression can be more generally expressed as: (A.5)For an extended, rodlike aggregate (D_{f} = 1), where most, if not all, of the primary particles are “visible”, we have ⟨σ_{eff}⟩ = (N πd^{2})/4 (i.e., c = 1 and β = 1), and for the case of the most compact spherical aggregate (D_{f} = 3) of primary particles it can be shown that (i.e., c = 1 and ). Figure A.1 shows these three determined values for α as a function of D_{f} along with an analytical fit to these data (). Here, and in the absence of evidence to the contrary, we make the (probably incorrect) assumption that k_{a} is indepdent of D_{f} over the entire range of D_{f}. The use of Eq. (A.4), and the dependence of α on D_{f}, now enables us to calculate the projected crosssection for any collection of primary particles of a given fractal dimension or fractality, i.e., (A.6)This expression is plotted in Fig. A.2, as the normalised function ⟨σ_{eff}⟩/((Nπd^{2})/4), for aggregates with 10–1000 primary particles. What this figure shows is that for highly fractal particles (D_{f} = 1) the effective crosssection, ⟨σ_{eff}⟩, scales directly with the number of the primary particles, N, and their crosssection, π(d/2)^{2}. This shows that for D_{f} = 1 the angleaveraged crosssection is only 87% of the total crosssection of all the constituent subgrains, i.e., Nπ(d/2)^{2}, which implies that some of the primary particles are “shadowed” in the orientationaveraged projection as clearly must be the case. For a randomly tumbling cylinder (D_{f} ~ 1) setting ⟨S⟩ = 0.87 in Eq. (A.1) corresponds to a cylinder with l ≈ 10r.
As the particle fractality (and hence the “porosity”) increase Fig. A.2 shows that the normalised effective crosssection decreases rapidly with N as the number of primary particles that are “hidden” within the structure increases. This is simply the effect of decreasing surfacetovolume ratio as the size, or number of primary particles, increases. For typical soot, clustercluster aggregates with D_{f} ~ 1.7−1.9 it can be seen that the normalised crosssection is ≈40–80% of the total primary particle crosssection (Nπ(d/2)^{2}).
Fig. A.1 Aggregate projected area exponent, α, versus fractal dimension, D_{f}, or equivalently the dimensionality, D. The data point at D_{f} = 1.82 (with the uncertainties in D_{f} and α indicated by the horizontal and vertical bars) is the bestfit value of α for soot aggregates, which have 1.7 ≤ D_{f} ≤ 1.9 (shaded range), taken from Köylü et al. (1995). 

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Fig. A.2 The normalised projected crosssection, ⟨σ_{eff}⟩/((Nπd^{2})/4), for aggregates consisting of 10–1000 primary particles of given fractal dimension or fractality (indicated by the number at right). The experimental uncertainties are of about the same order as the thickness of the D_{f} = 3 line. 

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A.2. Extended particle radii of gyration
As pointed out by Mandelbrot (1982); Dobbins & Megaridis (1991); Ossenkopf (1993), Köylü et al. (1995) and Mackowski (2006), in aggregated particle studies a quantity of prime importance for the optical and collisional properties of the particles is the radius of gyration, R_{g}, which in these works is defined as: (A.7)where r_{i} is the distance of the individual primary particle from the centre of mass of the aggregate. The soot particle and simulation studies by Köylü et al. (1995) further indicate that (A.8)where, from experiment, D_{f} = 1.82 ± 0.08 and k_{f} = 8.5 ± 0.5.
Mackowski (2006) in a similar study, but based only on simulations (references therein), find that (A.9)and that the variation of the factor k_{0} with D_{f} can be expressed as k_{0} = (1.27−2.2[D_{f} − 1.82]). We can see that the two k factors above are simply related by k_{0} = (1/2)^{Df}k_{f}. Upon substitution it can be shown that there is an apparent discrepancy between the experimental and simulation fits, in that k_{f}/k_{0} = 1.9 at D_{f} = 1.82. Nevertheless, we can use the above expressions for k_{0}(D_{f}) to derive an equivalent fractal dependence for k_{f}(D_{f}), i.e., k_{f} = 8.5(2.82 − D_{f}), which obviously reduces to the above value of k_{f} = 8.5 when D_{f} = 1.82.
In these studies another characteristic, or rather characterising, parameter is the longest particle dimension, the maximum projected length L, which is simply related to the aboveused quantity, N_{L + } (see Sect. 5.2), by L = N_{L + } × d. Köylü et al. (1995) give the following useful relationship: (A.10)which, when the constant k_{fL} is known, can be used to derive the fractal dimension of a particle from measurements of its number of constituent primary particles, its maximum projected length and the size of the primary particles, i.e., D_{f} = (log N − log k_{fL})/log N_{L + }.
Using the above equations that express N as a function of A_{a} (Eq. (A.3)) and of R_{g} (Eq. (A.8)), and substituting, we obtain the following relationshipfor R_{g} as a function A_{a}, (A.11)This is useful because it allows us to calculate the radius of gyration, R_{g}, of an aggregate from known quantities and the calculated projected crosssection of the particle (Eq. (A.3)). However, its utility does depend on knowing, or assuming, values for the parameters k_{a} and k_{f}.
A.3. Extended particle optical properties
The optical properties of soot aggregates (with D_{f} ~ 1.8) have been studied by Dobbins & Megaridis (1991) who derive the angle and distributionaveraged absorption and scattering properties. Dobbins & Megaridis (1991) find that the absorption crosssection for polydisperse, fractallike aggregates with D_{f} = 1.7−1.9 is: (A.12)where, according to their definitions, is the average number of primary particles per aggregate, N (the first moment of their aggregate size probability function, p(N)), x_{p} = (πd)/λ ≪ 1, E(m) = −Im[(m^{2} − 1)/(m^{2} + 2)] and k = (2π)/λ. With the, partial, substitution for the primary particle radius d = 2a and some rearrangement it can be shown that, with x = (2πa)/λ, the above absorption crosssection is simply, in the more usual nomenclature, (A.13)which is just the total geometrical crosssection of the primary particles, Nπ(d/2)^{2}, multiplied by the absorption crosssection for a single primary particle in the Rayleigh limit. The angleaveraged scattering crosssection derived by Dobbins & Megaridis (1991) is more complex, involving the second moment of their aggregate size probability function, p(N), but for the purposes of the longwavelength emission from aggregates does not concern us here. This approximation, via the RayleighGans (RG) theory, is often used in the context of soot particle studies.
As pointed out by Mackowski (2006), the RG approximation, in the Rayleighlimit, generally under predicts the absorption crosssections of aggregates of spheres that have refractive indices typical of carbonaceous soots. Mackowski (2006) undertook a detailed theoretical/simulation study of the effects of aggregation on the optical properties of soot particles and arrived an elegant and simple model for predicting their absorption in relation to the properties predicted by the RG model. In essence, Mackowski finds that, for large aggregates (N ≥ 100), the RG model results underpredict the relevant crosssections by as much as a factor of two but that this discrepancy strongly depends on the value of the complex refractive index at the wavelength of interest. For soot aggregates the RG model underestimate is found to be of the order of 10% at visible wavelengths and a factor of two in the midinfrared. The reader is recommended to take a close look at the work by Mackowski (2006) for further details of the proposed simple model methodology.
The optical properties of irregular particles have also been dealt with using a Gaussian random particle approach (Muinonen 1996; Muinonen et al. 1996) and a distribution of form factors (DFF) based on the gaussian random particle approach (Min et al. 2006a,b; Mutschke et al. 2009). With these models it is possible to calculate the absorption and scattering of highly irregular particles.
Köhler et al. (2011) have studied the effects of refractive index, particleparticle contact and grain mantling on the emissivity of coagulated particles within the astrophysical context. They find that the graingrain contact surfaces introduced in the discrete dipole approximation (DDA) model for coagulated grains can, depending on the material properties, explain the observed enhancement in the particle emissivity at farinfrared and submm wavelengths.
Appendix B: Multicomponent effective media
Fig. B.1 A comparison of the threecomponent Bruggemann EMTgenerated n (upper curves) and k (lower curves) data for a mix of aSil (0.4), aC:H (0.2) and vacuum (0.4) by volume (orange dashed line), with pairwise mixing using the Bruggemann EMT. The pairwise mixing order [(1+2)_{i}+3_{m}] is: [(aSil+aC:H)+vac] and [(aC:H+aSil)+vac] – green lines, [(aSil+vac)+aC:H] and [(vac+aSil)+aC:H] – upper blue lines, [(aC:H+vac)+aSil] and [(vac+aC:H)+aSil] – lower blue lines. 

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In the Bruggemann effective medium theory (EMT) the effective dielectric function, ϵ_{av}, of a fractional volume of inclusions, f_{i}, of dielectric function ϵ_{i} in a matrix of of dielectric function ϵ_{m} is found from the equation (Bruggemann 1935) (B.1)For a binary mix this involves solving a quadratic equation, for three components a cubic equation etc. In Fig. B.1 we compare the mixing of the Draine & Lee (1984) astronomical silicate (aSil), Rouleau & Martin (1991) BE amorphous carbon (aC:H) and vacuum components, obtained using the Bruggemann rule for three components, with a pairwise mixing of the same three components in the same volume fractions. In this pairwise mixing scheme components 1_{i} (inclusion) and 2_{m} (matrix) are first mixed and then the “effective” result of this pair mixed with a third component, i.e., [(1 + 2)_{i} + 3_{m}]. We note that in the pairwise mixing case the results are not commutative, except for the case of switching inclusions and matrix in the first pairwise effective medium calculation. We find that where the mixing is performed in the order of most refractive to least refractive, i.e., silicate → carbon → ice → vacuum, the pairwise method gives an excellent match to n and k in comparison with the exact threecomponent Bruggemann calculation, except for the lowest values of k at midinfrared wavelengths (see Fig. B.1).
In the absence of having solved the quartic, and higher equations, for four and more component mixes we postulate that this pairwise calculation of an effective medium using the Bruggemann mixing rule can be used to calculate the effective medium of any number of mixed media, provided that the materials are mixed in the order of most to least refractory. Thus, this method (the pairwise Bruggemann EMT or PWBEMT) can be used to limit the complexity of the calculation of multimaterial effective media to the solution of quadratic equations. For example, this method could facilitate studies of inhomogeneous, irregular and “porous” aggregates consisting of amorphous silicates and amorphous carbons with ice mantles, i.e., a fourcomponent mix, without having to solve the quartic version of Eq. (B.1).
Appendix C: C_{abs} at long wavelengths for n constant
At long wavelengths the absorption (and emission) crosssection of a particle of radius a is given by (C.1)Substituting x = 2πa_{eff}/λ, where a_{eff} is the effective radius of the solid matter in the aggregate/particle, and m = n − ik for the complex index of refraction, and solving for the imaginary part of the refractive index term we have (C.2)For astronomical silicate grains (Draine & Lee 1984) at wavelengths longer than 100 μm n is constant and n > k (n ≃ 3.4, k ≲ 0.5, see Fig. C.1, i.e., n^{2} ≃ 11.6, k^{2} ≲ 0.25 and n^{2}k^{2} ≲ 2.9). Neglecting the k^{2} terms the above yields a simple expression for the absorption crosssection for solid and “porous” Draine & Lee (1984) astronomical silicate grains, i.e.,
(C.3)where k_{solid} is the imaginary part of the complex index of refraction of the solid material at wavelength λ and f_{v} is the volume fraction of vacuum in the particle. Note that a is the outer radius of the “inflated” porous particle and a_{eff} is the effective radius for the solid matter within the particle. For nonporous particles a = a_{eff}. The above approximation to C_{abs} is equivalent to a simple dilution of the absorptive index, k_{solid}, of the solid material by a factor (a/a_{eff})^{3}.
In Fig. C.2 we show this fit, together with that calculated using the Mie theory (and the Bruggemann EMT where there is a vacuum component). We find that this approximation can reproduce C_{abs} to within ± 6% for porosities ≲50%. This analytical approximation is only valid at long wavelengths (e.g., the farinfrared and longward) and for materials where n is constant. For amorphous carbons in general n is not constant at long wavelengths and this very simple approximation will not work for these materials.
Fig. C.1 n (upper) and k (lower) for astronomical silicate particles with 0, 10, 20, 30, 40 an 50% vaccuum (line thickness decreases with increasing vacuum fraction) calculated using the Bruggemann EMT. 

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Fig. C.2 The absorption crosssection, multiplied by the wavelength for clarity, for astronomical silicate particles with 0, 10, 20, 30, 40 an 50% vaccuum (line thickness decreases with increasing vacuum fraction), and effective radii 100, 104, 108, 113, 119 and 126 nm, respectively, calculated using Mie theory and the Bruggemann EMT (blue lines). The red lines show the simple analytical fits (see text). 

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This surprisinglysimple approach shows that we need not resort to the Mie theory nor an EMTaveraging to calculate the the dust crosssections for solid or porous grains at long wavelengths where n is constant. However, its usefulness needs to be tested against laboratory data for real interstellar dust analogue materials at long wavelengths.
All Tables
The dimensionality, D, and inflation, I, for regular geometrical solids. a and l are the relevant radius and side/edge lengths, respectively.
All Figures
Fig. 1 Axis sum, Σ^{i}, vs. axis product, Π^{i}, plot for a particle containing 100 identical subparticles as a function of the dimensionality (as indicated): D = 1 (diamond), 1.5 (plus signs), 2 (triangles), 2.5 (crosses) and 3 (squares). Compact particles are to the lower left and extended particles to the upper right. Note that no porosity is allowed for D = 1, rodlike particles. 

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In the text 
Fig. 2 Dimensionality, D, versus inflation, I, plot for particles containing 50 equalsized constituents for ≈15 000 randomlygenerated structures (with φ = 1). The inset box shows the limits on D (1 ≤ D ≤ 3) and I (1 ≤ I ≤ N^{2}/27), and the middle vertical line shows the limit for an idealised 2D particle, i.e., I = N/4. 

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In the text 
Fig. 3 Same as for Fig. 2 except that we have now volume and shapeweighted the data point sizes by a factor (1 − [D − 2]^{2})/I (see text). 

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In the text 
Fig. 4 The same as Fig. 3 but for ≈500 aggregates (consisting of 81 cubic primary particles) and a more limited range of I (≤10). The symbols (filled circles) are sizescaled by . The crosses indicate “particles” which are not constructable (see Sect. 4.1). 

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In the text 
Fig. 5 The effective crosssection, normalised to the number of primary particles, N, for ≈500 aggregates constructed from 81 cubic primary particles on a cubic grid. The symbols (black filled circles and coloured plus signs) are sizescaled by the particle dimensionality, D. The plus signs colourcode (see key at bottom right of the figure) the binned dimensionality (red D = 1–1.5; yellow D = 1.5–2; green D = 2–2.5; blue D = 2.5–2). The orange lines shows some representative analytical fits (see text). 

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In the text 
Fig. 6 The surfacetovolume ratio for ≈500 valid aggregates (filled symbols) with 81 cubic primary particles, on a cubic grid, normalised to that of the most compact “spherical” structure. The filled symbols are size and colourcoded as per Fig. 5. The “bare” crosses indicate nonconstructable aggregates, as per Fig. 4. 

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In the text 
Fig. A.1 Aggregate projected area exponent, α, versus fractal dimension, D_{f}, or equivalently the dimensionality, D. The data point at D_{f} = 1.82 (with the uncertainties in D_{f} and α indicated by the horizontal and vertical bars) is the bestfit value of α for soot aggregates, which have 1.7 ≤ D_{f} ≤ 1.9 (shaded range), taken from Köylü et al. (1995). 

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In the text 
Fig. A.2 The normalised projected crosssection, ⟨σ_{eff}⟩/((Nπd^{2})/4), for aggregates consisting of 10–1000 primary particles of given fractal dimension or fractality (indicated by the number at right). The experimental uncertainties are of about the same order as the thickness of the D_{f} = 3 line. 

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In the text 
Fig. B.1 A comparison of the threecomponent Bruggemann EMTgenerated n (upper curves) and k (lower curves) data for a mix of aSil (0.4), aC:H (0.2) and vacuum (0.4) by volume (orange dashed line), with pairwise mixing using the Bruggemann EMT. The pairwise mixing order [(1+2)_{i}+3_{m}] is: [(aSil+aC:H)+vac] and [(aC:H+aSil)+vac] – green lines, [(aSil+vac)+aC:H] and [(vac+aSil)+aC:H] – upper blue lines, [(aC:H+vac)+aSil] and [(vac+aC:H)+aSil] – lower blue lines. 

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In the text 
Fig. C.1 n (upper) and k (lower) for astronomical silicate particles with 0, 10, 20, 30, 40 an 50% vaccuum (line thickness decreases with increasing vacuum fraction) calculated using the Bruggemann EMT. 

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In the text 
Fig. C.2 The absorption crosssection, multiplied by the wavelength for clarity, for astronomical silicate particles with 0, 10, 20, 30, 40 an 50% vaccuum (line thickness decreases with increasing vacuum fraction), and effective radii 100, 104, 108, 113, 119 and 126 nm, respectively, calculated using Mie theory and the Bruggemann EMT (blue lines). The red lines show the simple analytical fits (see text). 

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In the text 