1 Introduction

Arguments hased upon excluded volume considerations have played a key role in numerous areas of both polymer science [1,2,3] and percolation theory [4,5,6]. Estimates for the number of contacts between objects based upon a prescribed definition of when a pair of entities are considered “connected” have typically rested upon the mutually excluded volume and been a key ingredient in theoretical estimates of the percolation threshold [4, 5]. Given that the excluded volume between objects is strongly dependent upon their shape [7, 8], it follows that percolation thresholds for particles of well-defined and regular dimensions show significant variation as a function of their aspect ratio [9,10,11]. Comparatively less, however, is known regarding the percolation behavior of aggregates whose architecture is characterized by their fractal dimension (denoted \({d}_{F}\)) and that are comprised of primary particles (referred to hereafter in this account as “subunits”) which have a given regular structure and aspect ratio [12,13,14]. For such aggregates the fractal dimension is a sensitive function of the experimental conditions under which the assemblies are generated and is particularly dependent upon factors that influence the kinetics of aggregation [15, 16].

An early investigation [13, 14], which to the knowledge of the author is among very few that address this problem, suggested that percolation by such fractal aggregates occurs when the volume fraction is such that the spheres enclosing each aggregate achieve tangency and just start to make contact and interpenetrate. This result leads to the prediction that \({\phi }_{c} \approx {\left(R/\delta \right)}^{{d}_{F}-3}\), where R denotes the radius of an aggregate, \(\delta\) is a lengthscale comparable to the size of a subunit, and \({\phi }_{c}\) denotes the volume fraction at the percolation threshold. A number of experiments have been interpreted on the basis of this finding [17,18,19,20] and have found support for this prediction for systems with fractal dimensions in the range \(1.8 \le {d}_{F} \le 3\). However, the established result [9,10,11] that for randomly distributed, isotropically oriented rod-like particles of length L and diameter d the threshold scales approximately as: \({\phi }_{c}\approx \left(d/L\right)\) suggests that this picture may not be entirely complete. More recent efforts [21] that applied excluded volume considerations to estimate the number of contacts that are formed within a mean-field perspective found a non-monotonic dependence of \({\phi }_{c}\) upon \({d}_{F}\) that provides a possible avenue for resolving this seeming inconsistency. For the case of spherical subunits, the dependence of the percolation threshold upon fractal dimension exhibited a minimum at intermediate values of \({d}_{F}\) which restored consistency with the asymptotic dependence of \({\phi }_{c}\) upon aspect ratio for the case of highly elongated rod-like particles.

The present work builds upon and generalizes that reported in Ref. [21] in two key directions: (i) a simplification is introduced in how the number of contacts is estimated that enables an almost entirely analytical treatment in terms of closed-form results, and: (ii) we address the impact of the aspect ratio of subunits by considering subunits that are cylindrical as opposed to spherical in shape. For situations where percolation is achieved when the spheres enclosing each aggregate start to make contact, the percolation threshold is found to be insensitive to the subunit shape. However, for more tenuous aggregates where greater interpenetration is required for percolation, deviations from unity in the aspect ratio in either direction (that is, subunits that are either longer rods or wider disks) reduces the value of \({\phi }_{c}\). In addition to their fundamental interest, these findings may be not without relevance to the design and understanding of electrodes for lithium ion batteries, where efforts are under way to optimize the conductivity at a minimal volume fraction of carbon-binder domains (“CBD”s) [22, 23].

The remainder of this account is organized as follows. Section 2 presents our model and the method adopted for estimating the percolation threshold. The asymptotic behaviors for \({\phi }_{c}\) anticipated in the limit of large aggregate radii are discussed in Sect. 3, along with the Critical Path Approximation (“CPA”) [24, 25] which we employ to approximately describe the dependence of the conductivity upon fractal dimension for a fixed volume fraction. Results from illustrative calculations based on the formalism developed in Sects. 2 and 3 are presented along with our conclusions in Sect. 4.

2 Model for Percolation in Systems of Fractal Aggregates

2.1 Introduction and Development of Model

Our model envisages each fractal aggregate as being comprised of identical cylindrical primary particles (the “subunits”) with lengths (or thicknesses) and diameters (or widths) denoted \(l\) and δ, respectively. The centers of each of the subunits that comprise an individual aggregate are located within a sphere of radius R, which defines the physical extent of the assembly. The volume of the impenetrable core of each subunit, denoted \({v}_{c}\), satisfies: \({v}_{c}= \pi l{\delta }^{2}/4\) and is held constant as the aspect ratio is varied. The basic unit we shall use for expressing lengths, denoted \({x}_{0}\), is defined in terms of \({v}_{c}\) as follows:

$${x}_{0}= {\left(4 {v}_{c}/\pi \right)}^{1/3},$$
(1)

such that the core volume per subunit equals \({v}_{c}\) for a cylinder for which \(l\) and \(\delta\) are each equal to \({x}_{0}\). The aspect ratio for the subunits, denoted \(\alpha\), is defined as: \(\alpha = l/d\). Thus \({x}_{0}\) represents the length (and diameter) of a subunit with aspect ratio equal to unity and core volume given by \({v}_{c}\). For prescribed values of \({v}_{c}\) and \(\alpha\) the lengths and diameters of the subunits satisfy:

$$l= {\alpha }^{2/3 }{x}_{0} \text{, and: }\delta = {\alpha }^{- 1/3}{x}_{0}.$$
(2)

The larger and smaller of \(l\) and \(\delta\) are denoted \({l}_{1}\) and \({l}_{2}\), respectively, that is, \({l}_{1}= \text{Max }\left[{l}_{1}, {l}_{2}\right]\) and \({l}_{2}= \text{Min }\left[{l}_{1}, {l}_{2}\right]\).

Given our subsequent use of estimates for the excluded volume that assume an isotropic orientational distribution for the subunits, our specification of the volume fraction approximately accounts for possible crossovers between isotropic and ordered phases. As a simple order of magnitude estimate for the volume fraction above which the isotropic phase gives way to orientationally ordered arrangements for non-spherical particles, denoted \({\phi }_{max}\), we assign:

$${\phi }_{max}= \pi {l}_{2}/{l}_{1},$$
(3)

an estimate that approximately captures the dependence of the onset of ordered phases upon aspect ratio for both rodlike and disklike particles [26,27,28]. We denote by \({A}_{max}\) the largest value of subunit aspect ratios that are explored in our calculations, so that: \(1/{A}_{max} \le \alpha \le {A}_{max}\). In terms of \({A}_{max}\) we enforce the following upper bound on the volume fraction occupied by the subunit cores internal to an individual aggregate:

$${\phi }_{max}= \pi /{A}_{max},$$
(4)

to ensure a degree of internal consistency with our assumption of an isotropic orientational distribution for the subunits. In a similar vein we denote by \({l}_{0}\) the minimum value of \({l}_{2}\), that is: \({l}_{0}= \text{Min }\left[\text{Min }\left(l, \delta \right)\right]\) over our range of calculations, such that:

$${l}_{0}= {{A}_{max}}^{- 2/3} {x}_{0}.$$
(5)

The volume fraction occupied by the cores of the subunits belonging to an individual aggregate, denoted \({\phi }_{1}\), is defined in terms of mass fractal dimension (denoted dF ) and the aggregate radius R as follows:

$${\phi }_{1}\approx \epsilon {\phi }_{max }{\left(2 R/{l}_{0}\right)}^{{d}_{F}-3 },$$
(6)

where the dimensionless prefactor \(\epsilon\) lies between zero and unity. The total number of subunits per aggregate, denoted N, follows from Eq. (6) as:

$$N= \frac{2 \epsilon {\phi }_{max}}{3} {\left(\frac{2 R}{{x}_{0}}\right)}^{3} {\left(\frac{2 R}{{l}_{0}}\right)}^{{d}_{F}-3}.$$
(7)

The overall volume fraction occupied by the cores of the subunits of all the aggregates in the system, and the number density of aggregates per unit volume, are denoted \(\phi\) and \(\theta\), respectively. These variables are related to each other and to \({\phi }_{1}\) in the following way:

$$\theta {R}^{3}= 3 \phi /4\pi {\phi }_{1}.$$
(8)

The parametrizations in Eqs. (1, 2), and (5, 6, 7) enable varying the subunit aspect ratio \(\alpha\) at a fixed core-occupied volume fraction and number of subunits per aggregate (for prescribed values of the fractal dimension \({d}_{F}\) and aggregate radius R).

The impenetrable core of each subunit is embedded within a penetrable concentric, coaxial cylindrical shell of length \(l+2 \lambda\) and diameter \(\delta +2 \lambda\) that represents the region within which contacts leading to connectedness can be formed. Each overlap between the penetrable connectedness shells (while observing the excluded volume between the impenetrable cores) of pairs of subunits from different aggregates defines a connectedness contact between the aggregates. A pair of aggregates is considered to be connected to each other if there exists at least one such contact between their subunits, and a schematic illustration of a pair of aggregates that experience one such connectedness contact is provided in Fig. 1. The shell thickness \(\lambda\) is expressed in terms of \({x}_{0}\) through the dimensionless parameter \(\gamma\) defined as: \(\gamma = \lambda /{x}_{0}\). A mapping is introduced between this “real” system of aggregates with impenetrable cores onto an analogous “model” system of aggregates that are completely interpenetrable to provide a mean-field estimate for the number of such contacts as a function of the distance between aggregates centers. The aggregates in the model system are assigned the same radius R as those in the real, physical, system, and the cores and shells of the subunits in the model system have the same dimensions as for the real system with the difference being that the cores of the model subunits are assumed to be mutually interpenetrable. We next delineate our mapping between variables corresponding to the real (subunits with impenetrable cores) and model (fully penetrable subunits) systems.

Fig. 1
figure 1

A schematic image depicting a pair of interpenetrating fractal aggregates, each comprised of rod-like cylindrical primary particles, that experience one connectedness contact between a pair of primary particles of which one belongs to each aggregate. The large dashed circles indicate the radii R of the aggregates, and the cores of the individual primary particles (with lengths and diameters equal to l and δ, respectively) are shown as solid line segments. The cylinders indicated by dotted lines depict the connectedness shells around each primary particle, and the (sole) connectedness contact is indicated by the point marked “X”. In the interests of clarity only a small number (seven) of primary particles for each aggregate are shown. The leftmost (rightmost) seven primary particles each are envisaged as being constituents of the aggregate on the left (right) side of the illustration

Full size image

Given that our approach adopts a mean-field viewpoint, the subunits in the model aggregates are assumed to be uniformly and randomly distributed throughout the interior of the enclosing sphere. The nominal volume fraction occupied by the cores of these subunits inside each aggregate is denoted by \({\xi }_{1}\), where: \({\xi }_{1}= \pi \rho l {\delta }^{2}/4\) and where \(\rho\) denotes the number density of subunits in the model aggregate:

$$\rho = \frac{3 {N}_{1}}{4 \pi {R}^{3}} .$$
(9)

In Eq. (9), \({N}_{1}\) denotes the number of (penetrable) subunits per model aggregate. The quantities \({N}_{1}\) and \({\xi }_{1}\) are chosen to enforce equality between (i) the core-occupied volume fraction \({\phi }_{1}\) for an individual aggregate in the real system of impenetrable subunits, and (ii) (i) the volume fraction internal to the (overlapping) cores of the model subunits, leading to [29]:

$${\xi }_{1}=\text{Ln }\left(\frac{1}{1- {\phi }_{1}}\right)= \frac{3 {N}_{1} {v}_{c}}{4 \pi {R}^{3}}= \rho {v}_{c}.$$
(10)

The nominal volume fraction occupied by the cores of the subunits of the (penetrable) model system and the number density of model aggregates are denoted \(\xi\) and \(\psi\), respectively. These variables are related through:

$$\xi = 4 \pi {R}^{3} \psi {\xi }_{1}/3.$$
(11)

The number density of model aggregates \(\psi\) is determined by enforcing equality between (i) the core-occupied volume fraction \(\phi\) for all the aggregates in the real system with impenetrable subunits, and (ii) the volume fraction internal to the (overlapping) subunit cores in the model system, leading to:

$$\xi = \text{Ln }\left(\frac{1}{1- \phi }\right),$$
(12)

and:

$$\psi {R}^{3}= \frac{3}{4\pi } \left[\frac{\text{Ln }\left(1- \phi \right)}{\text{Ln }\left(1- {\phi }_{1}\right)}\right].$$
(13)

The expected number of contacts between pairs of subunits leading to connectedness between a pair of aggregates is estimated in the next step of our calculation of the percolation threshold.

2.2 Estimate for the Number of Connectedness Contacts Between a Pair of Aggregates

We start by considering the number of connectedness contacts, denoted \(\upsilon \left(x\right)\), between subunits that belong to a pair of model (penetrable) aggregates for which the center-to-center separation is denoted x. The construction of our model implies that \(\upsilon \left(x\right)\) is equal to zero for \(x \ge 2 R+ {l}_{1}+2 \lambda\). In order to form our estimate it will be helpful to define the two following measures of excluded volume:

  1. (i)

    We let \({\upsilon }_{1}\)denote the volume excluded to the center of one subunit by the cylinder representing the core (length \(l\), diameter \(\delta\)) of another subunit, and:

  2. (ii)

    We let \({\upsilon }_{2}\) denote the volume excluded to the center of one subunit by the cylinder representing the core and shell (length \(l+2 \lambda\), diameter \(\delta +2 \lambda\)) of another subunit.

After having been averaged over an isotropic orientational distribution, the values of \({\upsilon }_{1}\) and \({\upsilon }_{2}\) are [7, 8]:

$${\upsilon }_{1}= \frac{\pi }{2} \left({l}^{2}\delta + \frac{\left(\pi +3\right) l {\delta }^{2}}{2}+ \frac{\pi {\delta }^{3}}{4}\right) ,$$
(14)

and:

$${\upsilon }_{2}= \frac{\pi }{8}\left(\delta +2\lambda \right) \left[4 {\left(l+2\lambda \right)}^{2}+2\left(\pi +3\right)\left(l+2\lambda \right)\left(\delta +2\lambda \right)+ \pi {\left(\delta +2\lambda \right)}^{2}\right].$$
(15)

If the center of a subunit enters the region of volume \({\nu }_{1}\) centered upon a subunit of another aggregate, there occurs an overlap between the cores of the subunits. Similarly, if the center of one subunit is in the region of volume \({\nu }_{2}\) centered upon another subunit while remaining outside the region of volume \({\nu }_{1}\), the pair of subunits establish a contact leading to connectedness without an overlap between their respective cores.

The nominal volume fractions internal to (i) the cores of subunits from the model system, and (ii) the cores and connectedness shells for the subunits, denoted \({\xi }^{{\prime }}\) and \({\xi }^{{\prime }{\prime }}\), respectively, are given by: \({\xi }^{{\prime }}= \rho {\upsilon }_{1}\) and \({\xi }^{{\prime }{\prime }}= \rho {\upsilon }_{2}\), respectively. Given our assumption of a uniform, random, distribution of the subunits in the model system, the volume fractions internal to: (i) the overlapping cores, and (ii) the overlapping {core + shell} regions, can then be estimated as: \({\phi }^{{\prime }}=1- {e}^{- {\xi }^{{\prime }}}\) and \({\phi }^{{\prime }{\prime }}=1- {e}^{- {\xi }^{{\prime }{\prime }}}\), respectively [29]. We are thereby led to the following approximation for the volume fraction that is (i) internal to the connectedness shell regions, but also (ii) external to the (overlapping) cores, which we denote \({\phi }_{2}\):

$${\phi }_{2}\left(r<R\right)={\phi }^{{\prime }{\prime }}- {\phi }^{{\prime }}= {e}^{-{\xi }^{{\prime }}}- {e}^{- {\xi }^{{\prime }{\prime }}}= {e}^{-\rho {\upsilon }_{1}}- {e}^{- \rho {\upsilon }_{2}}= {\phi }_{\text{2,0}},$$
(16)

where \({\upsilon }_{1}\) and \({\upsilon }_{2}\) are specified by Eqs. (14 and 15) and \({\phi }_{\text{2,0}}\) denotes the volume fraction of the region that is outside the overlapping cores but inside the connectedness shells. The expression for \({\phi }_{2}\) in Eq. (16) approximates the volume fraction of the space internal to an aggregate where the center of a subunit from another aggregate experiences a connectedness contact while observing the excluded volume condition prohibiting overlap of the cores of subunits belonging to the real system.

When the centers of the pair of model aggregates coincide, that is, in the limit of full penetration of one aggregate by the other (\(x=0\)), our assumption of a uniform, random distribution of the subunits in the model system leads to the following estimate for the number of pairwise contacts between subunits:

$${\upsilon }_{0}= \upsilon \left(x=0\right)= {N}_{1}{\phi }_{\text{2,0}}= \frac{16}{3} \left(\frac{R}{l}\right){\left(\frac{R}{\delta }\right)}^{2}\text{Ln} \left(\frac{1}{1-{\phi }_{1}}\right) {\phi }_{\text{2,0} }.$$
(17)

Given that, additionally: (i) \(\upsilon \left(x\right)\) must be a monotonically decreasing function of \(x\), and: (ii) \(\upsilon \left(x\right)\) vanishes for \(x \ge 2 R+ {l}_{1}+2 \lambda\), we adopt the following simple and approximate representation for \(\upsilon \left(x\right)\):

$$\upsilon \left(x\right)= {\upsilon }_{0} \left(1- \frac{x}{{x}_{max}}\right), \text{for }x \le {x}_{max} ,$$
(18)

and:

$$\upsilon \left(x\right)=0, \text{for } x \ge {x}_{max}$$
(19)

where \({x}_{max}=2 R+ {l}_{1}+2 \lambda\).

As we shall see in the subsequent section, the (admittedly very simple) approximation in Eqs. (18, 19) enables us to express the percolation threshold condition in terms of compact and closed-form expressions. Our results for the case of subunits with unit aspect ratio (\(l=\delta = {x}_{0}\)) will be found to be quite similar to those from a more sophisticated approach applied to aggregates comprised of spherical subunits [21]. Despite the simplicity of Eqs. (18, 19), dependences on the key variables in the problem, namely, R, \(l, \delta , {d}_{F}\) and \(\lambda\) remain incorporated in our picture.

2.3 Calculation of the Percolation Threshold

In this section, we present our calculation of the percolation threshold that combines our approximation for the number of connectedness contacts from Sect. 2.2 together with the approach adopted in Ref. [21] appropriate to the model system of randomly distributed, fully penetrable entities. If we make the assumptions that:

  1. (i)

    The radial distribution function between the centers of aggregates is equal to unity for all separations (which is appropriate for fully penetrable objects), and:

  2. (ii)

    The contact numbers can be described by a Poisson distribution such that the probability that a pair of aggregates for which the separation between centers is equal to x have at least one mutual contact (and are thereby connected to each other) can be estimated as: \(1- {e}^{- \upsilon \left(x\right)}\), and:

  3. (iii)

    The percolation threshold is given by the condition that each aggregate experiences an average of 2.74 connectedness contacts with other aggregates in the system (as is the case for percolation by fully penetrable spheres in three dimensions [30]),

The percolation threshold is determined by the following condition:

$$4\pi {\psi }_{c} \mathop\int\limits_{0}^{2R+ {l}_{1}+2 \lambda } dx {x}^{2} \left[1- {e}^{- \nu \left(x\right)}\right] = 2.74.$$
(20)

Upon transformation of variables back to the volume fractions appropriate to the real system, Eq. (20) can be expressed as:

$$3 \left(\frac{I}{{R}^{3}}\right) \left(\frac{\text{Ln }\left(1- {\phi }_{c}\right)}{\text{Ln }\left(1- {\phi }_{1}\right)}\right)=2.74,$$
(21)

where:

$$I= \mathop\int\limits_{0}^{2R+ {l}_{1}+2\lambda } dx {x}^{2} \left[1- {e}^{-\nu \left(x\right)}\right].$$
(22)

Within the approximate representation for \(\upsilon \left(x\right)\) introduced in Eqs. (18, 19) the integral that appears in Eqs. (21, 22) can be evaluated in closed form, leading to:

$$\frac{I}{{R}^{3}}= {\left(2+ \frac{\left({l}_{1}+2\lambda \right)}{R}\right)}^{3}\left[\frac{1}{3}- \frac{1}{{\upsilon }_{0}}+ \frac{2}{{\upsilon }_{0}^{2}}- \frac{2}{{\upsilon }_{0}^{3}} \left\{1-{e}^{- {\upsilon }_{0}}\right\}\right] .$$
(23)

Equations (17, 22, and 23) provide our estimate for the percolation threshold. The following section presents the asymptotic behaviors anticipated for \({\phi }_{c}\) in the limit of sufficiently large aggregates (sufficiently large value of \(R/{x}_{0}\)) and our use of the Critical Path Approximation (“CPA”) [24, 25] to estimate the conductivity for cases where the subunits are electrically conductive.

3 Volume Fractions at the Percolation Threshold and the Critical Path Approximation (CPA)

We next consider the limiting behaviors expected for \({\phi }_{c}\) as a function of the fractal dimension \({d}_{F}\) for aggregates with sufficiently large values of \(R/{x}_{0}\) for fixed values of the connectedness range \(\lambda\), the parameters \(\epsilon\) and \({A}_{max}\) Eqs. (4 and 6), and subunit dimensions \(l\) and \(\delta\). Under these conditions (\(R \gg {x}_{0}, {l}_{0}\)), both \({\phi }_{1}\) and \({\xi }_{1}\) become much smaller than unity and are approximately equal to each other. Given that \({\upsilon }_{c}\), \({\upsilon }_{1}\), and \({\upsilon }_{2}\) are held fixed, each of \(\rho {\upsilon }_{1}\) and \(\rho {\upsilon }_{2}\) are also then much smaller than unity and \({\phi }_{\text{2,0}}\) can be approximated as: \({\phi }_{\text{2,0}}\approx {\phi }_{1}\left({\upsilon }_{2}-{\upsilon }_{1}\right)/{\upsilon }_{c}\). The number of connectedness contacts between a pair of fully interpenetrating aggregates, \({\upsilon }_{0}\), is then given by:

$${\upsilon }_{0}\approx \frac{16}{3}\left(\frac{R}{l}\right){\left(\frac{R}{\delta }\right)}^{2}\left(\frac{{\upsilon }_{2}- {\upsilon }_{1}}{{\upsilon }_{c}}\right) {\phi }_{1}^{2}= \frac{2{\epsilon }^{2}{\phi }_{1,max}^{2}}{3{A}_{max}^{2}} \left(\frac{{\upsilon }_{2}- {\upsilon }_{1}}{{\upsilon }_{c}}\right) {\left(\frac{2R}{{l}_{0}}\right)}^{2{d}_{F}-3}.$$
(24)

Equation (24) shows that for large enough values of the aggregate radius R, if \({d}_{F}\) is larger (smaller) than 1.5, then \({\upsilon }_{0}\) is far larger (smaller) than unity and a monotonically increasing (decreasing) function of R. Equations (23, 24) lead to the following approximations for the integral that appears in Eqs. (21, 22) when \(R/{l}_{0}\to \infty\) for fixed values of \(\lambda\), \(l\), and \(\delta\):

$$I/{R}^{3} \approx 8/3 \text{for }{d}_{F} >1.5,$$
(25)

and:

$$I/{R}^{3} \approx 2 {\upsilon }_{0}/3 \text{for }{d}_{F} <1.5.$$
(26)

Equations (21) and (24, 25, 26) lead to the following predicted behaviors for \({\phi }_{c}\) under these circumstances:

$${\phi }_{c} \approx \epsilon {\left(2R/{l}_{0}\right)}^{{d}_{F}-3}, \text{for } {d}_{F} >1.5,$$
(27)

and:

$${\phi }_{c} \approx \left(\frac{1}{\epsilon }\right) \left(\frac{{\upsilon }_{c}}{{\upsilon }_{2}- {\upsilon }_{1}}\right) {\left(\frac{2R}{{l}_{0}}\right)}^{-{d}_{F}} , \text{for } {d}_{F} <1.5.$$
(28)

In a similar manner to that observed for the case of spherical subunits [21], the percolation threshold is expected to show a non-monotonic dependence upon \({d}_{F}\). For compact aggregates, Eq. (27) shows that percolation ensues when the spheres enclosing each aggregate are approximately tangent to each other and just start to interpenetrate. In this situation \({\phi }_{c}\) is not anticipated to be sensitive to the shape (aspect ratio) of the subunits. For more tenuous aggregates, however, deeper interpenetration of individual assemblies becomes necessary for percolation Eq. (28). Equations (14, 15) show that for subunits with a fixed core volume \({\upsilon }_{c}\), the quantity \(\left({\upsilon }_{2}- {\upsilon }_{1}\right)\) increases with increasing departures of the subunit aspect ratio from unity (towards either longer and more slender rods or wider and thinner disks), and thus \({\phi }_{c}\) is expected to be lowered by such changes in the shape of the subunits. This reduction of the percolation threshold is predicted to be found in the prefactor Eq. (28) and to not substantially alter the dependence upon \({d}_{F}\). It should be noted that a similar substantial change in the mutual penetrability of fractal aggregates when \({d}_{F}\)is larger or smaller than 1.5 had been found in a previous analysis that also used mean-field arguments [31].

In the complementary scenario where \(\lambda \gg R \gg l, \delta\), arguments similar to those presented above show that, again, both \({\phi }_{1}\) and \({\xi }_{1}\) are each much smaller than unity and approximately equal to each other. However, in this situation \({\phi }_{\text{2,0}} \approx 1\), as can be verified from either: (i) a straightforward but tedious analysis, or: (ii) the physical argument that when \(\lambda\) becomes sufficiently large the overlapping connectedness shells fill the entire interior of the sphere that encloses the centers of the subunits of one aggregate. The counterpart to Eq. (24) then becomes:

$${\upsilon }_{0} \approx \frac{16}{3} \left(\frac{R}{l}\right) {\left(\frac{R}{\delta }\right)}^{2} {\phi }_{1} \approx \frac{\epsilon {\phi }_{1,max}}{{A}_{max}^{2}} {\left(\frac{2R}{{l}_{0}}\right)}^{{d}_{F}},$$
(29)

which is always much larger than unity for all values of \({d}_{F}\) provided R is sufficiently large. In this limit the integral from Eqs. (21, 22) simplifies to:

$$I/{R}^{3} \approx 8{\lambda }^{3}/3{R}^{3},$$
(30)

and the percolation threshold is given by:

$${\phi }_{c} \approx \frac{\epsilon {\phi }_{1,max}}{{\gamma }^{3} {A}_{max}^{2}} {\left(\frac{2R}{{l}_{0}}\right)}^{{d}_{F}}$$
(31)

where \(\gamma \equiv \lambda /{x}_{0}\). Equation (31) may also be written in terms of the critical connectedness length (or value of \({\gamma }_{c}\)) that is required in order to reach the percolation threshold for a prescribed volume fraction \(\phi\):

$${\gamma }_{c} \approx \frac{1}{{A}_{max}^{2/3}} \left(\frac{R}{{l}_{0}}\right) {\left(\frac{{\phi }_{1}}{\phi }\right)}^{1/3},$$
(32)

which is consistent with a picture of randomly located penetrable spheres with the entire “mass” of each sphere concentrated at their centers. The asymptotic results in Eqs. (31, 32) are expected to apply for all values of \({d}_{F}\) when \(\lambda \gg R \gg l, \delta\), and the findings of Eqs. (27, 28) and (31, 32) are consistent with what was previously shown for the case of spherical subunit-based aggregates from a somewhat more cumbersome and detailed approach [21].

If we adopt a point of view in which the connectedness range is varied so that percolation can be achieved at a prescribed value of \(\phi\), we can express the threshold in terms of a critical value for either \(\lambda\) or \(\gamma\), denoted \({\lambda }_{c}\) or \({\gamma }_{c}\). We can then employ the Critical Path Approximation (CPA) to provide a simple estimate [24, 25] for the conductivity (denoted \(\sigma\)) in situations where the subunits that comprise the aggregates are electrically conductive. The CPA estimate for \(\sigma\) is applicable when the conductivity is governed by electron hopping between aggregates and can be expressed as follows:

$$\sigma \approx {\sigma }_{0 }{e}^{- {\lambda }_{c}/\tau },$$
(33)

where \(\tau\) denotes the electron tunneling and is typically in the approximate range \(\tau \approx 1-10\) nm. Previous work [25, 32] within this approximation has shown that the CPA predicts a monotonic but non-power-law rise in the conductivity with increasing volume fraction for composites comprising conductive filler nanoparticles dispersed within an insulating matrix. For the illustrative calculations presented in the following section, we shall assume for simplicity that \(\tau = {x}_{0}\) in all cases and depict the conductivity as calculated from: \(\sigma = {\sigma }_{0} {e}^{- {\lambda }_{c}/{x}_{0}}= {\sigma }_{0} {e}^{- {\gamma }_{c}}\).

4 Results and Conclusions

This section presents results for the percolation threshold calculated from Eqs. (21, 22, 23) as a function of the aggregate radius, connectedness length scale, and subunit aspect ratio (Figs. 2, 3 and 4), and the conductivity estimated from the CPA (Fig. 5). For all of the calculations presented in Figs. 2, 3 and 4 we set \({A}_{max}=50\) and \(\epsilon =0.25\); the range of subunit aspect ratios is therefore restricted to \(0.02 \le \alpha \le 50\).

Fig. 2
figure 2

The volume fraction at the percolation threshold, \({\phi }_{c}\), is shown as a function of the fratal dimension \({d}_{F}\). The solid, dashed, and dotted lines correspond to systems where the subunits have aspect ratios of unity, 25, and 0.04, respectively. For each set of curves, from top to bottom, the aggregate radius \(R/{x}_{0}\) equals 102, 104, 106, and 108. In each case \({A}_{max}\) = 50, \(\epsilon\) = 0.25, and \(\gamma = \lambda /{x}_{0}\) = 0.2

Full size image
Fig. 3
figure 3

The volume fraction at the percolation threshold, \({\phi }_{c}\), is shown as a function of the fratal dimension \({d}_{F}\). The solid, dashed, and dotted lines correspond to systems where the subunits have aspect ratios of unity, 25, and 0.04, respectively. For each set of curves, from top to bottom, the connectedness range \(\gamma = \lambda /{x}_{0}\) = 0.2, unity, and 5, respectively. In all cases the aggregate radius \(R/{x}_{0}\) equals 103, \({A}_{max}\) = 50, and \(\epsilon\) = 0.25

Full size image
Fig. 4
figure 4

The volume fraction at the percolation threshold, \({\phi }_{c}\), is shown as a function of the aspect ratio of the subunits for various aggregate fractal dimensions \({d}_{F}\). In all cases, the connectedness range \(\gamma = \lambda /{x}_{0}\) = 0.2, the aggregate radius \(R/{x}_{0}\) equals 103, \({A}_{max}\) = 50, and \(\epsilon\) = 0.25. The curves numbered from 1 through 6 correspond to \({d}_{F}\) = 3, 2.5, 2.0, 1.5, 1.25, and 1, respectively

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Fig. 5
figure 5

The conductivity of a system of fractal aggregates estimated from the CPA is shown as a function of the fractal dimension. In all cases \(\epsilon =0.25\), \(R/{x}_{0}\) = 102, and we have set \({x}_{0}= \tau\). The solid, dashed, and dotted lines correspond subunits with aspect ratios of unity, 25, and 0.04, respectively. For each pair of solid, dashed, and dotted curves the upper and lower members correspond to \(\phi ={10}^{-3}\) and \(\phi ={10}^{-4}\), respectively

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Figures 2 and 3 show that, as expected from Eqs. (27, 28), \({\phi }_{c}\) exhibits a minimum at an intermediate value of \({d}_{F}\). This minimum deepens and migrates towards smaller values of \({d}_{F}\) as either the radius of the aggregate (Fig. 2) or connectedness shell thickness (Fig. 3) are increased, indicating a larger range of fractal dimensions over which percolation is achieved at tangency of the enclosing spheres. Deviations of the subunit aspect ratio from unity in either direction, towards either longer and narrower rods or wider and thinner disks, in each case lowers the percolation threshold. Increasing the connectedness range \(\lambda\) at fixed values of the other variables has the effect of raising \({\phi }_{\text{2,0}}\) and consequently the likelihood of forming contacts for modest degrees of interpenetration and “percolation at tangency” [13, 14] characterized by the dependence: \({\phi }_{c}\approx {\left(R/{x}_{0}\right)}^{{d}_{F}-3}\). The impact of altering the aspect ratio \(\alpha\) of the subunit cores is also reduced as \(\lambda\) is increased (Fig. 3) as the eccentricity of the entire {core + shell} region within which contacts can be formed is thereby decreased.

The variation of \({\phi }_{c}\) with subunit aspect ratio for various fixed values of \({d}_{F}\) is further examined in Fig. 4. For aggregates that are sufficiently compact (large \({d}_{F}\)) that percolation is achieved when the spheres first start to overlap, the percolation threshold is insensitive to the subunit shape. However, for less compact aggregates (smaller values of \({d}_{F}\)), varying the subunit aspect ratio away from unity in either direction (rods or disks) leads to a reduction in \({\phi }_{c}\) as suggested by the prefactor in Eq. (28). For a given ratio of major to minor dimensions of the subunits, this reduction is somewhat more pronounced for the case of rod-like than disk-like shapes. These observations suggest that for a given overall volume fraction occupied by the subunit cores, the connectedness length for which percolation is achieved should be smaller for aggregates comprised of rod-like than disk-like subunits, and for each of these choices smaller than for subunits with an aspect ratio of unity. The greater effect of rod-like as compared to disk-like subunit shapes in lowering \({\phi }_{c}\) can be seen from the dependence upon subunit aspect ratio of the prefactor \({\upsilon }_{c}/\left({\upsilon }_{2}- {\upsilon }_{1}\right)\) in Eq. (28) for very large (or small) values of \(l/\delta\):

$$\frac{{v}_{c}}{\left({v}_{2}- {\upsilon }_{1}\right)} \approx \frac{1}{4\gamma } {\left(\frac{\delta }{l}\right)}^{4/3} , \text{for } l \gg \delta ,$$
(34)

and:

$$\frac{{v}_{c}}{\left({v}_{2}- {\upsilon }_{1}\right)} \approx \frac{1}{\left(5 \pi +6\right)\gamma } {\left(\frac{l}{\delta }\right)}^{2/3} , \text{for } \delta \gg l.$$
(35)

The effects of subunit aspect ratio and \({d}_{F}\) upon the conductivity at a fixed volume fraction \(\phi\) estimated from the CPA Eq. (33) are depicted in Fig. 5. The conductivity exhibits a non-monotonic dependence upon \({d}_{F}\) with a maximum that is the counterpart to the minima for \({\phi }_{c}\) observed in Figs. 2 and 3. The primary effect of varying the subunit aspect ratio is a slower decrease in \(\sigma\) from its peak value with decreasing values of \({d}_{F}\) on the low-\({d}_{F}\) side of the peak. This attenuation of the decline in \(\sigma\) is modestly more efficacious for the case of rod-like than for disk-like subunit shapes, consistent with our findings for \({\phi }_{c}\) reported in Fig. 4.

In concluding, we have generalized our prior approach [21] to describe aggregates comprised of non-spherical primary particles and explored the impact of primary particle aspect ratio upon the percolation threshold. The treatment of our model has been simplified with an approximation for the number of connectedness contacts that enables an almost entirely analytical study based upon closed-form results. When examined as a function of the fractal dimension, the percolation threshold exhibits a minimum at intermediate values of \({d}_{F}\) that is similar to that found previously for the case of spherical primary particles. Varying the aspect ratio of the subunits away from unity in either sense, that is, either more elongated or flattened, always lowers the percolation threshold (or reduces the connectedness range at which percolation occurs for a fixed volume fraction). An estimate for the conductivity obtained from the CPA suggests that choosing primary particles that are needle-shaped in an aggregate with fractal dimension in the range of \({d}_{F}\approx 2-2.5\) for modest aggregate sizes provides an optimal architecture that maximizes the conductivity for a fixed volume fraction.