myprojects-hg-reborn

Fork

#+TITLE:    Generation and characterization of fractal aggregates.
#+AUTHOR:    Lorenzo Isella
#+EMAIL:     lorenzo.isella@gmail.com
#+DATE:      2011-05-02 Mon

#+DESCRIPTION: presentation given at nowhere land
#+KEYWORDS: 
#+LANGUAGE:  en
#+OPTIONS:   H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:t pri:nil tags:not-in-toc
#+INFOJS_OPT: view:nil toc:nil ltoc:t mouse:underline buttons:0 path:http://orgmode.org/org-info.js
#+EXPORT_SELECT_TAGS: export
#+EXPORT_EXCLUDE_TAGS: noexport
#+LINK_UP:   
#+LINK_HOME: 
#+XSLT: 


* Overview
We assume we are in a situation in which we are able to generate
(correctly!) fractal aggregates with a prescribed fractal dimension
$d_f$ and prefactor $k_f$.
** Generation of fractal aggregates
- the algorithm by Filippov & Zurita is slightly more general than the
  one by Thouen, as it allows one to combine clusters of different size.
  However, as far as cluster-cluster aggregation is concerned,
  but there is not much difference in the
  implementation.  When it comes to cluster-cluster aggregation,
  Filippov simply allows  the two clusters not to
  have the same size and calculates the
  distance between the two CMs accordingly.
- If needed this can be easily implemented by Tasos in his already existing
  algorithm (but is it needed at this stage?).
- I should do exactly what Thouen recommends: choose
  two monomers, $m_1$ in cluster A and $m_2$ in cluster B (both on the
  surface of the respective clusters), move clusters B in such a way
  that $m_1$ is in touch with $m_2$ and then rotate cluster B in such a
  way that the distance between the CM of cluster A and cluster B is the 
  desired distance $\Gamma$. By construction, I have at least a contact
  point ($m_1--m_2$) between the two clusters and if there is no
  compenetration between the two clusters, then I am done. If not, I can
  always rotate cluster B along a circle that leaves the distance
  $\Gamma$ between the two CMs unchanged and check again the
  non-compenetration condition.
- What I do in my scripts is not as "clean" as what described above; instead
  I went for something quick and
  dirty to simulate cluster-cluster aggregation. I place cluster A and B
  in such positions that the distance between their CMs is $\Gamma$.
  Then I rotate cluster B by some (small) angle  $\theta$  around e.g. the
  $x$ axis and check the two conditions:
-- no monomer compenetration and

-- at least a contact point.
  
-  However,  by random rotations I will never get a perfect contact
   between any two monomers belonging to aggregate A and B respectively,
   so I consider two monomers
   to be in contact if their distance is $\le d+\epsilon$, where $d$ is
   the monomer diameter and $\epsilon\ll d$ is an accepted tolerance (in
   my simulations, $\epsilon/d=5\cdot10^{-3}$).
- If the steps above fail, I will now rotate cluster B by $2\theta$
  around the $x$ axis and once again check for non-compenetration and at
  the existence of at
  least one contact point and so on and so forth. 
- Prior to that, I build small clusters by adding repeatedly a single
  monomer to an already existing seed.  I know the distance
  (prescribed once again by Filippov) between the seed's CM and the
  monomer I want to add. I place randomly the monomer on a sperical
  shell centered about the seed's CM and test the contact and
  non-compenetration conditions as outlined above.

** Tests to be performed on the aggregates
- The first, most obvious test is to ensure that there is the correct
  scaling of the radius of gyration with the number of monomers in the
  aggregate i.e.
  \begin{equation}\label{scaling}
  N=k_fR_g ^{\;d_f},
  \end{equation}
  where we are measuring the radius of gyration $R_g$ in units of
  monomer radius $a_0$.
- Unfortunately, this test can be passed even by aggregates that do not
  exhibit the right correlation function which is supposed to decay as a
  power law of the distance with exponent $d_f-3$.

  
  


** How to characterize the aggregates?
*** Fractal dimension $d_f$
The fractal dimension is probably the most important single parameter to
classify an aggregate, but taken on its own suffers from some
limitations. Most of the authors are happy to provide only the fractal
dimension with the (more or less) implicit assumption
- $d_f$ small $\mapsto$ elongated aggregate
- $d_f$ large $\mapsto$ compact aggregate

We know from the time of the Langevin simulations the limitations of
this approach. Actually, the problem is really what other quantities of
interest one should consider to fully characterize (and what we mean by
that??) an aggregate.
*** Coordination number
The coordination number is the mean number of (first) neighbors of a
monomer inside the aggregate. I think it should be consider as a measure
of local compactness of the aggregate. In the end of the day, the
fractal dimension is a global property telling us how the radius of
gyration (again, a global property of the aggregate) grows with its size
(i.e. the number of monomers $N$). One can go beyond the mean value and
consider the probability distribution $P(k)$, i.e. the probability that
a randomly chosen monomer within my aggregate has $k$ neighbors.
*** Prefactor $k_f$
The prefactor $k_f$ has to play a role in the global structure of the
aggregate if we think that, according to eq. \ref{scaling}
\begin{equation}
k_f=\frac{N}{R_g^{\; df}}
\end{equation}
i.e. if I fix $d_f$ and $N$, when I increase the prefactor I have to
decrease the radius of gyration of the aggregate and viceversa.
So, it seems to me that $k_f$ has again a global influence on the
aggregate shape, as it indirectly controls (while keeping the fractal
dimension fixed) the radius of gyration of aggregates of the same size
$N$. It is clear however that since the prefactor $k_f$ is always $O(1)$, the
most important role is played by the fractal dimension $d_f$ (does it
make sense???).
 
*** Principle radii
Another way of characterizing the aggregate, and in particular its
anisotropy, is via the calculation of the so-called principle radii
which one obtains via diagonalizing the aggregate anysotropy tensor.
They are related to the radius of gyration via 
\begin{equation}
R_g^2=\frac{1}{2}(R_1^2+R_1^2+R_3^2),
\end{equation}
where $R_1>R_1>R_3$  and one can then define the anysotropy coefficient as
$A_{13}=R_1/R_3$.
The numerical simulations by Sorensen and by others (and my simulations) suggest
that, for a given $\{k_f,d_f\}$ I still have a relatively broad
distribution of a anysotropy coefficients $A_{13}$.
** Conclusions (so to say)
We can calculate a handful of statistics for an aggregate,
i.e. $\{d_f,k_f, P(k), A_{13}, R_g \}$. How are these statistical
quantities related? No clear idea, but
- Sorensen (in one of his papers, aerosol science and technology)
  generates his clusters via Monte Carlo simulations, hence
  he does not control $d_f$ and $k_f$ a priori.
  He gets a lot of aggregates which are described by a single fractal
  dimension. However, for the same $d_f$, he finds a distribution of
  anisotropy coefficients. They correspond usually to clusters having 
  also an overall different $R_g$. In other words, he also allows the
  prefactor $k_f$ to
  vary and he finds a distribution of anisotropies for the same
  $d_f$. The conclusion is that you cannot characterize a cluster by its
  $d_f$ only as two clusters with the same $d_f$ may have very different 
  anisotropy.
- However things are not even this simple. Even if you fix $\{k_f,
  d_f\}$, you can have a distribution of anisotropy coefficient
  relatively broad (see paper by Pranamy). His results are consistent
  with my findings, though I miss some outliers of his $n(A_{13})$
  distribution, but this may be an artifact of the different algorithms
  that we deploy.
- What is the physical meaning of having two different anysotropies for
  a given $\{k_f,d_f\}$? Look at the attached figures for a low fractal 
  dimension ($d_f=1.3$) and the same $k_f$. Regardless of the anisotropy, clusters
  with low $d_f$ tend to look like linear chains (which, as we know, are
  not straight by definition).
  However, if the anisotropy is high, the cluster does look like a
  straight chain, i.e. most of the mass displacement occurs along a
  single direction. This does not matter for the $R_g$ calculation, but
  of course enhances $A_{13}$. Instead, clusters with a low $A_{13}$ and
  low fractal dimensions look more like a spring, i.e. a line that tends
  to wind up around an axis. This way, I measure a mass displacement
  roughly comparable along the three principle directions. 
  To be fair, what does this tell us (above all in the cases of higher
  $d_f$, where this intuitive explanation fails)? I would say mainly
  that even when fixing $d_f$ and $k_f$, we are not (terribly)
  constraining the anisotropy of the cluster, as the radius of gyration
  evaluates the variance of mass distribution in 3D, but does not
  (directly) tell us anything about how it is distributed along each
  principle axis.
- Finally, the big question (perhaps worth a paper by itself, or at
  least tell me if you found the answer). Let us say that I provide you
  with 2 clusters consisting of $N=100$ monomers (or bigger if you
  want).
  Other than resorting to sophisticated statistical analysis (decay of
  the correlation function, structure factors, not to mention the direct
  verification of the scaling law \ref{scaling} that requires a set of
  clusters of varying size), is there a \emph{simple} statistical
  quantities to tell apart their fractal dimensions? Or at least a
  combination of the quantities of interest I have outlined such that I
  can easily conclude that the two clusters have/do not have a similar
  $d_f$, regardless of the prefactor?