5.4 Aggregation
5.4.2 Properties of the Asymptotic Cluster
5.4.2.5 Inertia Tensor and Spatial Extension
Throughout this section 5.4.2we repeatedly observe a fractal dimension of Df ≈2 for the asymptotic cluster; this raises the question, whether or not the cluster largely consists of planar topologies. To gain quantitative information about the shape of an object, it is useful to compute its moment of inertia tensor[seee.g.Landau
& Lifshitz,1976] and look at its eigenvalues, the principal moments of inertia I1 ≤ I2 ≤ I3. The latter quantify the mass distribution perpendicular to the respectiveprincipal axes e1,e2,e3, as they obey the equation:
I1= Z
%(r) (r22+r32) dr, (5.47) and equivalent equations for I2, I3. Here r = P3
ν=1rνeν is the distance from the center of mass in the coordinate system of the principal axes and %(r) the density at that point.
The principal moments of inertia can easily be related to a more intuitively comprehensible quantity, the spatial extensions ∆rν along the three principal axes:
∆r2ν :=
RR%(r)rν2dr
%(r) dr for ν = 1,2,3. (5.48) While the denominator is simply the mass of the object, the numerator can be expressed in terms of the principal moments; e.g.forν= 1:
−I1+I2+I3
2 =
Z
%(r) −(r22+r23) + (r21+r32) + (r12+r22)
2 dr
= Z
%(r)r12dr. (5.49)
The reason for not calculating Eq. (5.48) directly, but using the detourviathe prin-cipal moments of inertia, is that a priori the principal axes are not known. Note furthermore that from these spatial extensions one can trivially obtain the radius of gyration r2g = P3
ν=1∆r2ν, as one can easily see in the definition (5.48). The choice of the principal axes is always such that ∆r1≥∆r2≥∆r3. For spherical sym-metric objects obviously ∆r1= ∆r2= ∆r3; for plate-like, or so-called oblate objects
∆r1≈∆r2À∆r3, and for rod-like, orprolate objects ∆r1À∆r2≈∆r3.
In order to obtain information about the structure on a wide range of length scales, we analyze partial clusters deduced from the largest cluster, as done in Sec. 5.4.2.1: starting from a random initial particle of the cluster, we define all particles that can be reached byior less neighbor-to-neighbor steps as partial clus-ter N(i) (see Fig. 5.19). The spatial extensions ∆r1,2,3 of N(i) are calculated for different values of iand the data are averaged over several initial particles.12 The result is shown in Fig. 5.31. As one can see, the structures are clearly not
spher-12For simplicity only the particlecenters are considered for the calculation of the inertia tensor.
As we can easily convince ourselves in Eq. (5.49), due to the Huygens-Steiner theorem, accounting for the extension of the individual particles would just lead to a constant shift of ∆r21,2,3→∆r1,2,32 +
1
2Ispherewith the moment of inertia of a single sphereIsphere=101d2.
5.5. CONCLUSIONS 123
0 10 20 30 40 50 60
number of steps i 0
2.5 5 7.5 10 12.5 15 17.5
Dr1,2,3
d
Figure 5.31: Spatial extension of partial clustersN(i) along the principal axes. The blue curve shows the largest component∆r1, the green curve the second largest∆r2, and the red curve the smallest ∆r3. The inset shows data for the same procedure, for which, however, the particle positions are not obtained from the simulation, but randomly placed into the system.
ically symmetric, as the spatial extensions in the principal directions are different for all shown length scales. However, also the assumption that the cluster involves planar structures cannot be supported, since that would imply ∆r1≈∆r2À∆r3 at least on some intermediate length scales. Indeed, we observe the most general case
∆r16≈∆r26≈∆r3 throughout the investigated range.
5.5 Conclusions
We have analyzed a simple model of a wet granulate allowing for large scale event driven simulations. A central feature of wet granulates is the existence of an energy scale ∆E associated with the rupture of a capillary bridge between two grains. This energy scale has important consequences not only for the phase diagram [Fingerle et al.,2008] but also for the free cooling dynamics investigated in this paper. The most important feature is a well defined transition at a time t0, when the kinetic energyT of the particles becomes equal to ∆E.
Fort < t0 the particles are energetic enough to supply the bond breaking energy
∆E, so that very few collisions result in bound pairs and most particles are unbound.
Cooling is very effective in this regime, but drastically different from a dry granulate.
Whereas in dry granulates the dissipated energy is proportional to the energy of the colliding particles, in wet granulates the dissipated energy is ∆E, independent of the energy of the colliding particles so that ˙T ∝√
T. Consequently Haff’s law does
not hold and is replaced by T(t) = T(0)(1−t/t0)2 fort < t0. The simulations are in very good agreement with this cooling law for t < t0.
For t > t0, the kinetic energy of the particles is too small to provide the bond breaking energy, so that larger and larger clusters form. We call this regime the aggregation regime and analyze the properties of the aggregates. For not too long times and sufficiently small volume fractions, we observe flocculation characterized by nonoverlapping, weakly interacting clusters. The fractal dimension of the ag-gregates is Df ≈ 2. The cluster mass distribution follows a simple scaling form, Nm(t)∝m−2f¡
m/m(t)¯ ¢
, which has been applied successfully to different aggrega-tion models before. The increase of the typical cluster size ¯m(t) can be understood by a simple scaling analysis: Assuming that clusters irreversibly stick together when they hit upon each other and that their radiusrgrows with the number of particles m like rDf ∝m, yields a cluster growth ¯m ∝t2Df/(3Df−2D+2). This scaling relation shows good agreement with the simulation for fractal dimension Df = 2.
At larger times, a spanning cluster forms, and a gelation transition is observed for all finite volume fractions. At the gelation transition a spanning cluster coexists with many small ones, whereas at very long times almost all particles are connected to one large cluster. On the largest length scales the final cluster is no longer a fractal but compact, as one would expect for a spanning cluster in the percolating phase.
On smaller length scales, however, we still find fractal structures with Df ≈2. The range where a nontrivial fractal dimension can be observed increases with decreasing density as |logφ|.
Even on the longest time-scales, the temperature continues to decay. In this regime the limiting process is the breaking of a bond. The probability for this process becomes exponentially small Pbb ∝ p
∆E/Te−∆E/T as the temperature goes to zero. Hence the cooling law for high temperatures is replaced by ˙T ∝ e−∆E/T, yielding a slow (logarithmic) temperature decay, which is in very good agreement with the data.
Several extensions of the presented work might be interesting. So far we have completely neglected all inelasticities except for the bond rupture. One expects the collisions at the hard core to be dissipative as they are in dry granular media. In the simplest model these could be described by normal restitution. Furthermore, real wet grains experience frictional forces, coupling translational and rotational motion of the grains [Brilliantovet al.,2007]. However, the author is not aware of any such studies for wet granulates at the present time. In the context of gelation it would be interesting to see, if and how a finite shear modulus emerges in the system, as the percolation transition is passed.
Appendix A
Properties of ⊥- and k-vectors
The properties of⊥- andk-vectors discussed here apply to the calculations in chap-ters2 and3. For any given replicated vector ˆx=¡
x(0), ...,x(n)¢T
, define:
xk:=
Pn
α=0x(α)
√n+1 (A.1)
and
x⊥:=
x(0)
... x(n)
− 1 n+1
Pn
α=0x(α) ... Pn
α=0x(α)
= ˆx− xk⊗εˆ
√n+1 (A.2)
with ˆε= (1, ...,1)T. Then these vectors have the following properties:
The mean of all components ofx⊥ isx(0)⊥ +...+x(n)⊥ =0
Therefore (xk⊗ε)ˆ ·x⊥= 0
xˆ·yˆ=xk·yk+x⊥·y⊥
When integrating over a function f(ˆx) =f(xk, x⊥), we might ask the the question how to change integration variables
Z
Vn
f(ˆx)dˆx→ Z
Vn
f(xk, x⊥)dxkdx⊥, (A.3) whereby
dˆx = dx(0)· · ·dx(n) and
dx⊥ = dx(0)⊥ · · ·dx(n)⊥ δ(x(0)⊥ +...+x(n)⊥ )
= dx(1)⊥ · · ·dx(n)⊥ . (A.4) We already noted that the mean component of x⊥ must be zero, therefore it is sufficient to integrate over only nof the n+1 components, or use the δ-function to satisfy this constraint. For the variable transformation (A.3), we now successively
change the individual variables: In each transformation the determinant of the Jacobian matrix for (e.g.|∂x∂x(n)⊥(n)|) has to be evaluated keeping the variables constant, which are currently not transformed (denoted by | · |x(0),...,(n−1)). With the definition (A.1) and (A.2) one can easily
Hence the transformation can be written as:
Z
127
Appendix B
Calculations for the Randomly Cross-Linked Particle Model
Contents
B.1 Calculation of the Replica Free Energy . . . . 128 B.1.1 Emergence of the Replica Free Energy . . . . 128 B.1.2 Introduction of the Replicated Density Field . . . . 129 B.2 Hubbard-Stratonovich Transformation. . . . 130 B.2.1 Applying the Hubbard-Stratonovich Transformation . . . . 130 B.2.2 Meaning of the Order Parameter Ω(ˆq) . . . . 132 B.3 Expansion of the replica free energy to 3rd Order in Ω. . . 134 B.4 Fluctuations around the Saddle-Point Solution . . . . 135 B.5 Replica Free Energy with Shear Deformations . . . . 136 B.5.1 Preparing the Replica Free Energy for the Ansatz . . . . 136 B.5.2 Expansion of the one-particle partition functionz. . . . 137 B.5.3 Insertion of the Order Parameter inz . . . . 137 B.5.4 Insertion of the Order Parameter infn+1{Ω} . . . . 145 B.5.5 Recomposing the Replica Free Energy . . . . 146 B.5.6 Analysis of the quantity Ξr,a2 . . . . 149 B.6 Auxiliary Calculations for the Results Section . . . . 150 B.6.1 Distribution of the Number of Cross-Links. . . . 150 B.6.2 Shear modulus . . . . 152 B.6.3 Square of the expansion for (1−Q) . . . . 153 B.7 The Order Parameter in the One Replica Sector . . . . 154