Commit: 6fdd8eefbaa9dee63335bdb4959b2598b1c0ef8d - myprojects-hg-reborn (hg)

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• Revisión: 6fdd8eefbaa9dee63335bdb4959b2598b1c0ef8d (tree)
• Tiempo: 2008-11-05 03:51:01
• Autor: iselllo
• Commiter: iselllo

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New revision of the paper before Yannis's mission to Amsterdam. No modification the the figures, only to the text.

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• modified: latex-documents/coagu_draft.tex

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diff -r 99f0ffe97048 -r 6fdd8eefbaa9 latex-documents/coagu_draft.tex
--- a/latex-documents/coagu_draft.tex	Tue Nov 04 18:47:58 2008 +0000
+++ b/latex-documents/coagu_draft.tex	Tue Nov 04 18:51:01 2008 +0000
@@ -14,6 +14,12 @@
 % or use the epsfig package if you prefer to use the old commands
 % \usepackage{epsfig}
 
+
+ %\usepackage[pdftex]{thumbpdf}
+%\usepackage[pdftex]{hyperref}
+
+
+
 \usepackage{graphicx}
 \usepackage{url}
 \usepackage{natbib}
@@ -168,12 +174,23 @@
 
 \begin{abstract}
 % Text of abstract
-We study the Langevin dynamics of diesel exhaust nanoparticles.
+The process of nanoparticle agglomeration as a function of the
+monomer-monomer interaction potential
+ is simulated by solving via Langevin
+equations for a set of interacting monomers in three dimensions.   The simulation output  is used to investigate the
+structure of the generated clusters and the collision
+frequency between small clusters. Cluster restructuring is also
+observed and discussed.
+We identify a time-dependent fractal dimension whose evolution is linked
+to the kinetics of two cluster populations.
+The absence of screening  in Langevin equations is discussed and its
+effect on 
+cluster translational and rotational properties is quantified.   
 \end{abstract}
 
 \begin{keyword}
 % keywords here, in the form: keyword \sep keyword
-Langevin, fractal, aggregate, coagulation
+Langevin, fractal, aggregate, agglomeration.
 % PACS codes here, in the form: \PACS code \sep code
 \PACS 
 \end{keyword}
@@ -182,9 +199,73 @@
 % main text
 \section{Introduction}
 \label{introduction}
-Add references to give a general background on the field.
+Nanoparticle aggregates are of paramount importance in 
+technological and industrial processes such as combustion, filtration,
+ gas-phase-particle synthesis and many more.
+ The fractal nature of these aggregates 
+ has profound implications on their
+transport  \citep{filippov_drag, ybarra_drag, moskal} and thermal
+\citep{filippov_thermal} properties.
 
-\section{Model Formulation}
+
+Fractal aggregates arise from the agglomeration of smaller spherules,
+hereafter called monomers, which do not coalesce, but rather retain
+their identity in the resulting aggregate.
+Individual monomers in a quiescent fluid are Brownian particles whose
+dynamics is described by Langevin equation \citep{risken_book}.
+Langevin simulations have been employed in aerosol science to investigate aggregate
+agglomeration \citep{mountain}, aggregate collisional properties
+\citep{pratsinis_kernel}, the limits of validity of Smoluchowski
+equation \citep{pratsinis_ld} and aggregate films
+\citep{friedlander_deposition, deposition_interaction}.
+
+In this study we investigate nanoparticle dynamics relying solely on
+Langevin equations for a set of interacting monomers in three
+dimensions.
+Unlike the  works mentioned above, no assumptions are made about the
+structure or mobility of the aggregates generated by the dynamics.
+This is reminiscent of other applications of Langevin simulations in
+the field of dilute colloidal suspensions to study agglomeration and the
+structures it gives rise to  \citep{videcoq, hutter_langevin}. 
+The main limitation of this approach is the lack of screening of the
+inner monomers in an aggregate. 
+We  quantify the effect of this
+approximation  on the aggregate diffusional properties, while we argue
+that it has a limited effect on the structure of the generated aggregates.
+
+Since we solve numerically
+Langevin equations for interacting monomers rather than aggregates,
+the raw output of the simulations does not contain direct information
+on aggregate formation. We infer this datum, together with the
+detailed structure of each aggregate, the record of the collisions
+it underwent and its eventual restructuring, by using  techniques borrowed from graph theory \citep{book_algorithms}.
+
+The paper is organized as follows: Section \ref{model}  provides the
+theoretical framework for  Langevin nanoparticle simulations. 
+Emphasis is given to the description and justification of the
+monomer-monomer interaction potentials employed in the numerical
+experiments.
+Section \ref{simulations} offers an overview of the numerical work and  introduces
+ the quantities of the interest monitored to investigate
+the system dynamics.
+Section \ref{results} contains the results and discussion  of
+Langevin simulations, whereas the final remarks in Section \ref{conclusions} conclude
+the paper.
+
+ 
+% For instance, the calculation of the drag force felt by an aggregate
+% in a Stokes flow typically requires the solution of Stokes equation
+% for the flow surrounding the aggregate to be matched with the solution
+% of Brinkman equation for the creeping flow inside the aggregate
+% \citep{filippov_drag, ybarra_drag}, which can be a numerically
+% demanding task for which analytical  treatments are available only in
+% the case of spherically-symmetric fractal aggregates.
+
+
+
+
+
+\section{Model Formulation}\label{model}
 
 \subsection{Langevin Equation for Mesoscopic systems}
 \label{sec:lang-equat-mesosc}
@@ -201,7 +282,7 @@
 solving the  equations for a system of interacting clusters
 in three dimensions.
 We use the word cluster with the same meaning as the term aggregate in
-Ref. \cite{konstandopoulos}, i.e. a set of physically bound spherules, here called monomers.
+Ref. \cite{konstandopoulos}, i.e. a set of physically bound spherules (monomers).
   The $i$-th monomer  obeys the Langevin equation
 \begin{equation}
   \label{eq:Langevin}
@@ -378,19 +459,22 @@
  $\sigma$, the monomers 
   feel a very strong repulsive force much larger than the other energy
   scale in the system, namely the thermal energy $k_BT$, where  $T$ is
-  the system temperature. Although the potential does not diverge at
+  the system temperature. Although neither potential  diverges at
   separations $r<\sigma$ and one should call $\sigma$ the soft-core
   monomer diameter, monomer separations  below $\sigma$ are energetically
-  unfavourable and extremely unlikely occur in the system dynamics. This
+  unfavorable and extremely unlikely occur in the system dynamics. This
   justifies the identification of $\sigma$ with the monomer hard-core diameter.
 
  
 The model potential also exhibits a deep and  narrow
-  attractive part, responsible for the sticking of monomers when they
-  undergo a collision. In our numerical simulations,  the potential has
-  a cut-off length $r_{\rm cut}$ such that $r_{\rm cut}-\sigma\ll\sigma$,
-  i.e. it is attractive part on a length much smaller than the monomer diameter.   
-On the contrary, Van der Waals potential is a long-ranged, though
+  attractive part, responsible for the sticking of monomers upon
+  collision, while smoothly  going to zero at $r= r_{\rm cut}$        , where $r_{\rm
+    cut}$ is a cut-off  distance such that $r_{\rm
+    cut}-\sigma\ll\sigma$. This avoids the introduction of the so-called
+ impulsive forces in the system \citep{md_book}.
+ The model potential is attractive on a length scale much smaller than
+ the monomer diameter, whereas   
+ Van der Waals potential is a long-ranged, though
 quickly decaying, interaction.
 % A simple potential satisfying the above requirement is a very narrow
 % and deep potential well. However, the crude implementation of a potential well would lead
@@ -400,10 +484,9 @@
 %  monomer-monomer interaction potential in the following 
  
 In the following, we give the analytical expression of the
-monomer-monomer model interaction potential $u(r)$ used in
-the numerical experiments. The model interaction potential goes smoothly to zero  
- at $r=r_{cut}$ in order to avoid the introduction of the so-called
- impulsive forces in the system \citep{md_book}
+monomer-monomer model interaction potential $u_M(r)$ used in
+the numerical experiments. 
+
 %  and has a constant gradient for
 % $r\le \sigma$ to model a strong (constant) repulsive force when the physical
 % distance between two monomers falls below $\sigma$.
@@ -519,21 +602,22 @@
 Eqs. \ref{eq:potential_well}  is \ref{eq:van_der_waals} is
 shown in Fig. \ref{plot_potential}.  
 
-\section{Simulations and Post-Processing}
+\section{Simulations and Post-Processing}\label{simulations}
 
 
 \subsection{Preparation of the initial state}
 % \label{sec:prep-init-state}
 
-The initial state is created by placing, with a uniform random
-distribution, $N_{\infty}=5000$ monomers in a cubic box of size $L$
+The initial state is created by placing randomly in a cubic box of size $L$
 (measured, like all distances, in units of monomer diameter
-$\sigma$).
+$\sigma$),  $N_{\infty}(0)V_{\rm box}=5000$ monomers, where $V_{\rm
+  box}=L^3$ is the box volume and $N_{\infty}(0)$ is the initial
+monomer concentration. 
  The box size $L$ is chosen to ensure a given initial monomer
-density $\rho=0.01$ according to:
+density $N_{\infty}(0)=0.01$ according to:
 \begin{equation}
   \label{eq:box_size}
-  L=\lro\f{N_{\infty}}{\rho}\rro ^{1/3},
+  L=\lro\f{5000}{N_{\infty}}\rro ^{1/3},
 \end{equation}
 which leads to $L\simeq 80$.
 The initial random displacement of the monomers in the box  could give rise 
@@ -542,7 +626,7 @@
 strong repulsive force.
 In order to avoid these problems, we use a well-known technique of
 MD simulations and we ramp up the repulsive force
-monomers feel when $r<1$ (in units of $\sigma$) to its constant final value  during a time $\tau_{\rm
+monomers feel when $r<1$  to its constant final value  during a time $\tau_{\rm
   ramp}=1$.
 The system is then evolved until a final dimensionless time
 $3000$, when the cluster concentration is almost two orders of
@@ -553,10 +637,9 @@
 % is the kinetic energy scale for the monomers.
 The results we show in this study were obtained by averaging the
 output of $10$ simulations,  in
-order to ensure to have enough data for the statistical analysis. It
+order to ensure that we have enough data for the statistical analysis. It
 is worth pointing out that  the results in Ref. \cite{mountain} were
-based on a system containing one tenth of the monomers whose dynamics
-is investigated in this study.
+based on a system containing one tenth of the monomers in this study.
 
 
 \subsection{Cluster Detection}
@@ -564,21 +647,20 @@
 The determination of the clusters is one of the most time-consuming
 tasks of the post-processing of the simulation results.
 The simulations we ran with \esp return the individual monomer
-positions and velocity obtained by solving Eq. \ref{eq:Langevin} for a
-3D system of interacting monomers.  Unlike the previously mentioned works
+positions and velocity obtained by solving Eq. \ref{eq:Langevin}.
+  Unlike the previously mentioned works
 \citep{meakin_cluster_models, mountain},
 we do not have to  look for agglomeration events while
 evolving the system.  However, we are left with the problem of
 determining a posteriori which clusters formed during the system evolution. 
+
+
 In this study we resort to an approach based on graph theory.
 A perfectly rigid cluster is a
 set of connected monomers such that all  monomer-monomer distances
 are constant in time. 
-% For such a cluster, therefore, the relative distance between any two
-% first neighbour monomers is fixed, rather than the distance between any
-% two monomers in the cluster. 
 For the specific case of our simulations, since both the model
-potential Eq. \ref{eq:potential_well} and Van der Waals potential Eq. \ref{eq:van_der_waals} is
+potential Eq. \ref{eq:potential_well} and Van der Waals potential Eq. \ref{eq:van_der_waals} are
 not infinitely narrow, monomers are allowed tiny oscillations around
 the equilibrium point.
 Despite of that, the bottom of the potential well is so deep that when
@@ -600,7 +682,7 @@
 For a periodic system, the distance between the $i$-th and $j$-th
 monomer  is the distance between the $i$-th monomer and the nearest
 image of the $j$-th monomer \citep{md_book}. For instance, the
-distance between two monomers along the $x$ axis  is given by:
+distance between two monomers along the $x$ axis  is given by
 \begin{equation}
   \label{eq:distance_calc}
   D_{ij}^{(x)}= x_i-x_j-L\cdot{\rm nint}\lro\f{x_i-x_j}{L} \rro,
@@ -641,9 +723,9 @@
 The adjacency matrix is usually introduced in the context of graph
 theory \citep{book_algorithms}, as a convenient way of uniquely representing a
 graph.
-Monomers in a cluster can be formally regarded as the vertices of a graph and the
+Monomers in a cluster can be formally regarded as the graph  vertices   and the
 physical bounds due to the interaction potential are, in this analogy,
-the edges of the graph.  
+the  graph  edges.  
 The problem of determining the clusters, given the distance matrix
 ${\bf D}$, is then re-formulated equivalently as the determination of
 the connected components in a non-directed graph expressed by the
@@ -658,7 +740,7 @@
 the dependence of the number of monomers in a cluster on its radius of
 gyration can be used to determine the fractal dimension of the system.
 The radius of gyration is defined as the root-mean-square distance
-mass displacement around the centre of mass of the aggregate:
+mass displacement around the aggregate   centre-of-mass 
 
 
 \begin{equation}
@@ -671,6 +753,8 @@
  We stress that the cluster radius of gyration is a static property
  since it does not depend on the diffusional properties of the
  cluster, but only on its mass distribution.
+
+
 In order to evaluate $R_g$ according to Eq. \ref{eq:r_gyr_definition}
 we need again to take into account the periodicity of the box.
 We adopt a reference frame centred on the position of the first
@@ -679,21 +763,21 @@
  Since all the
 monomer-monomer distances along each axis are known from
 Eq. \ref{eq:distance_calc}, one can calculate the position of
-all the other monomer in a cluster with respect to the first monomer;
+all the other monomers in a cluster with respect to the first monomer;
 for instance the $j$-th monomer position along $x$ will be given by:
 \begin{equation}
   \label{eq:reconstruct_cluster}
   x_j=x_1+D_{1j}^{(x)}=D_{1j}^{(x)}
 \end{equation}
 and similarly along the $y$ and $z$ axis.
-Once all the monomer coordinates have been re-calculated, one can
-safely calculate the coordinates of the centre of mass of the
-aggregate (e.g. $x_{CM}=\s_ix_i/k$) and apply straightforwardly
+Once all the new monomer coordinates have been obtained from Eq \ref{eq:reconstruct_cluster}, one can
+safely calculate the coordinates of the 
+aggregate centre-of-mass  (e.g. $x_{CM}=\s_ix_i/k$) and apply straightforwardly
 Eq. \ref{eq:r_gyr_definition}.
 We point out that this procedure is independent on the specific
-monomer we choose to locate in the origin of the reference frame
-(since the radius of gyration of cluster is, of course, independent on
-the spatial position of the cluster itself) nor does it depend on the
+monomer whose coordinates are taken as the origin  of the new reference frame
+(the radius of gyration  is independent on
+the cluster spatial position) nor does it depend on the
 sign convention chosen for $D_{1j}^{(x)}$.
 
 Equation \ref{eq:distance_3D} allows one  to also calculate the
@@ -714,9 +798,9 @@
 mathematical definition of a set.
 Viewing clusters as sets of monomers provides a computational method
 to investigate the process of cluster collisions.
-First of all, we should note that both the model and Van der Waals
-potential used in the simulations are much deeper than $k_BT$ and this
-means that we can safely assume a sticking probability upon collision
+First of all, we should remind ourselves that both the model and Van der Waals
+potential used in the simulations are much deeper than $k_BT$ hence we
+can safely assume a sticking probability upon collision
 equal to one.
 To fix the ideas, let us consider two individual clusters  at time
 $t$.
@@ -725,7 +809,7 @@
 An equivalent description of the collision process is that  both
 sets of monomers at time $t$ (the two initial clusters) end
 up being proper subsets of the same set of monomers (the cluster
-formed by their collision) at time
+formed by their collision) detectable  at time
 $t+\delta t$. One can then record the number of collisions taking
 place in $(t, t+\delta t]$ and the clusters involved in the collisions
 simply by comparing different sets.
@@ -756,7 +840,7 @@
 
 
 
-\section{Results and discussion}
+\section{Results and discussion}\label{results}
 
 
 \subsection{Cluster diffusion coefficient and cluster thermalization}
@@ -767,8 +851,7 @@
 investigation of its rotational properties to the next section.
 
 Equation \ref{eq:Langevin} determines the dynamics of each monomer in
-the system and indirectly also the diffusional properties of the
-clusters which arise due to agglomeration.
+the system and indirectly also the cluster diffusional properties.
 The numerical determination of the diffusion coefficient of a $k$-mer
 (i.e. a cluster consisting of
  $k$ monomers)  and its scaling with $R_g$ and the number of monomers $k$ can provide valuable
@@ -783,6 +866,8 @@
 Langevin equation \ref{eq:Langevin} characterises the fluid only via
 its viscosity which is modelled by the damping and noise terms and no
 information about the fluid molecule mean-free path is available.
+
+
 We can indirectly extract  information about the Knudsen number
 from the numerical evaluation of the diffusion coefficient.
 From Einstein relation, the diffusion coefficient of a single (spherical)
@@ -791,10 +876,10 @@
   \label{eq:monomer-diffusion-coefficient}
   D_1=\f{k_BT}{m_1\beta_1}.
 \end{equation}
-Equation \ref{eq:monomer-diffusion-coefficient} can be extended to the
-case of an aggregate containing $k$ monomers by introducing Cunningham
-slip factor $C_s(Kn)$ and the cluster inverse of the relaxation time
-$\beta_k$:
+Equation \ref{eq:monomer-diffusion-coefficient} can be extended for a
+$k$-mer  by introducing Cunningham
+slip factor $C_s(Kn)$ and the inverse of the $k$-mer relaxation time
+$\beta_k$
 \begin{equation}
   \label{eq:cluster-diffusion-coefficient}
   D_k=\f{k_BT}{km_1\beta_k}C_s(Kn).
@@ -819,6 +904,8 @@
 cluster
  mean square displacement (variance of the  cluster center-of-mass
  positions at a given time).
+
+
 We followed the motion of the cluster centre-of-mass along $800$ trajectories up
 to time $t=400$ for clusters with $k=4,10,50,98$.
 Aggregate transport is investigated by placing the cluster  in the
@@ -838,7 +925,8 @@
 We notice that $D_k\propto 1/k$ for $k$ spanning almost two orders of
 magnitude, thus confirming that we are in the continuum regime.
 Furthermore, the estimated value of $\beta_k$ is equal to the inverse
-of the monomer relaxation time  $\beta_1$ within a few percents and, 
+of the monomer relaxation time  $\beta_1$ within a few percents, the
+non-perfect agreement being   
 due to the relatively limited number of simulated stochastic trajectories.
 
 
@@ -872,7 +960,7 @@
 \end{equation}
 
 
-The fluid viscosity can be expressed in terms of the monomer properties
+The fluid viscosity can in its turn be expressed in terms of the monomer properties
 \begin{equation}
   \label{eq:fluid_viscosity}
   \mu_f=\f{m_1\beta_1}{3\pi\sigma}.
@@ -910,11 +998,12 @@
  $k=4$ to $k=98$, but that they both severely underestimate the
  mobility radius. 
 The major approximation we are making when studying cluster mobility
-is that we are applying Eq. \ref{eq:Langevin} the same way both to an
-isolated monomer and to a monomer in a cluster, which is very often at
+is that we use  Eq. \ref{eq:Langevin} to describe the dynamics of both an
+isolated monomer and  a monomer in a cluster, which is very often at
 least partially shielded by the other monomers from the surrounding fluid.  
 The recent study by \cite{moskal}   suggests that the approximation may
-be extremely poor, but there is not universal agreement about this.
+be extremely poor, but this approximation is often deemed reasonable
+for clusters having $d_f\le 2$ (see for instance \cite{friedlander_deposition}).
 
 
 Finally, we also investigated the velocity fluctuations in order
@@ -927,17 +1016,17 @@
 \end{equation}
 with $k=1$.
 Having determined that $\beta_k=\beta_1$ and that $C_s(kn)= 1$,
-we have already tentatively  trivially generalised Eq. \ref{eq:delta_v_squared} for
-a cluster of mass $km_1$.
+we have already tentatively  trivially generalised the well-known
+equation for a monomer (see \cite{risken_book}) for
+a  $k$-kmer.
 In Figure \ref{diffusion_plot} (right) we plot  $\langle \delta v^2
 \rangle$ as a function of time again for the same cluster considered
 in the diagram on the left. We notice that $\langle \delta v^2 \rangle$   
 clearly fluctuates about $0.03$, the value predicted by
-Eq. \ref{eq:delta_v_squared} for $k_bT=0.5$ and $k=50$ and $m_1=1$.
+Eq. \ref{eq:delta_v_squared} for $k_bT=0.5$, $k=50$ and $m_1=1$.
 
 The main result of this section is that  from Eq.
-\ref{eq:Langevin}, i.e. neglecting the screening of monomers in a
-cluster, a $k$-mer  is
+\ref{eq:Langevin}, i.e. neglecting monomer screening, a $k$-mer  is
 found to have the same relaxation time of a monomer
 and the same diffusional and thermal properties of a sphere  with a
 mass $k$ times the one of a single monomer.
@@ -989,8 +1078,7 @@
 Each cluster can be treated as a rigid body
  and any rotation of a rigid body in 3D can be
 described resorting to three angles $\theta$, $\phi$ and $\psi$, usually
-called Euler angles
-\emph{Goldstein}.
+called Euler angles \citep{goldstein}.
 In the post-processing, we eliminate the contribution of the translational motion by rigidly
   translating the aggregate centre-of-mass to the origin of the
   computational box coordinate system at all times.   
@@ -1234,7 +1322,10 @@
 This quantity can provide valuable information about the openness of
 the aggregates and the presence of cavities in their structure. 
 The knowledge of the mean coordination number complements
-the information from the fractal dimension about the aggregate structure.
+the information from the fractal dimension about the aggregate
+structure.
+
+
 As one can see from Fig. \ref{evolution_coord_number}
   the mean coordination number goes well above seven.
 This relatively high value provides new information about the cluster
@@ -1250,7 +1341,14 @@
 cavities in their structure.
 This conclusion (valid in a statistical sense) cannot be reached on the basis of the fractal
 dimension alone, hence we advocate the use of the fractal dimension
-and the coordination number for a better characterisation of aggregate morphology.
+and the coordination number for a better characterisation of aggregate
+morphology.
+The generation of clusters with a high coordination number (as most
+likely those in Ref. \cite{videcoq}) is an intrinsecal property of
+Langevin simulation  due to their
+neglecting the interparticle hydrodynamic interaction, as demonstrated
+in Ref.  \cite{prl_hydro}.
+
    
 
 % \subsection{Distribution of cluster morphologies}
@@ -1269,7 +1367,6 @@
 Cluster restructuring was investigated in the framework of 2 and 3D
 lattice models in Refs.
 \cite{meakin_reorganization_2D,meakin_reorganization_3D}.
-
 The novelty of the approach that we propose here is that cluster
 restructuring does not have to be implemented as an additional feature
 of the cluster dynamics. The monomers interact via a deep interaction  \emph{radial}
@@ -1328,18 +1425,18 @@
 
 \subsection{Evolution of the number of clusters}
 
-The total number of cluster in the system, $N_{\infty}$, is one of
+The total  cluster concentration in the system, $N_{\infty}$, is one of
 the most important quantities of interest since it shows the
 progress of agglomeration and can be used to compare the results of
 the simulations with the numerical solution of the agglomeration
 equation \citep{friedlander book}
 \begin{equation}
   \label{eq:agglomeration_equation}
-  \f{dn_k}{dt}=\f{1}{2}\s_{i+j=k}\mathcal{K}_{ij}n_in_j-n_k\sum_i\mathcal{K}_{ik}n_i,
+  \f{dn_k}{dt}=\f{1}{2}\s_{i+j=k}\beta_{ij}n_in_j-n_k\sum_i\beta_{ik}n_i,
 \end{equation}
 where $n_k$ is the concentration of clusters consisting of $k$
-monomers and $\mathcal{K}_{ij}$ is the collisional kernel between
-clusters with $i$ and $j$ monomers, respectively.
+monomers and $\beta_{ij}$ is the collisional kernel between
+ $i$ and $j$-mers, respectively.
 
 % Fig. \ref{cluster_decay}  shows the decay of the number of clusters as
 % time progresses on a double logarithmic scale. One notices that, after an
@@ -1410,6 +1507,9 @@
 % take into account the fact that the approximation $R_m\simeq R_g$ is
 % not justified for the clusters generated in our simulations,
 % especially when $k$ is large (cfr. Table \ref{table_diffusion}). 
+
+
+
 On the other hand, we can use the information obtained from the numerical simulations to
 derive a kernel which describes more accurately the aggregate
 collisional dynamics starting from Eq. \ref{eq:cont_kernel_general}.
@@ -1432,10 +1532,8 @@
 
 
 
-Furthermore, although we are in the continuum regime, as far as the
-cluster diffusion is concerned, 
-non-continuum effects can play a role anyhow in cluster collisions and
-they are usually accounted  by expressing  introducing Fuchs
+Furthermore, non-continuum effects can play a role in cluster collisions and
+they are usually accounted for  by   introducing Fuchs
 correction \citep{fuchs book} $\beta_f$ in the kernel.
 % Furthermore, the presence of two different fractal dimensions in the
 % system should be mirrored in the kernel by determining the average
@@ -1475,14 +1573,14 @@
 
 The results of the simulations (with Van der Waals and model potential) and the numerical
 solution of the agglomeration equation \ref{eq:agglomeration_equation}
-with $\beta^{Sm}$ and $\beta^{LD}$ in Fig. \ref{Smoluchowsky
+with $\beta^{Sm}$ and $\beta^{LD}$ are shown in Fig. \ref{Smoluchowsky
   comparison}. Langevin kernel in 
 Eq. \ref{eq:kernel_complete} leads to an enhanced agreement at
 short times between the simulations and the numerical solution of the
 agglomeration equation \ref{eq:agglomeration_equation}.
 We point out that the asymptotic limit in the agglomeration equation
 is reached at times about one order of magnitude longer than the
-duration of the simulation, mainly due to Fuchs factor.
+duration of the simulations, mainly due to Fuchs factor.
 
  Nevertheless, we fit the  time-decay of the total cluster
  concentration to a power-law, $N_\infty\sim t^{-\xi}$, at late
@@ -1634,15 +1732,46 @@
 Figure \ref{calculated_beta_ij} (top) shows an example of the fitting
 procedure  we use to determine $\beta_{13}$ from the quantities $N_{13}/\delta t$ and
 $n_in_j$ evaluated directly from the simulations.
-We calculate numerically a few $\beta_{ij}$  we compare to the
+We calculate numerically a few $\beta_{ij}$ which  we compare to the
 analytical estimates $\beta_{ij}^{LD}$, both expressed in units of the
-diagonal of the continuum kernel $\beta^{LD}_{ii}=8k_BT/3\mu_f$.
+diagonal of the continuum kernel $\beta^{Sm}_{ii}=8k_BT/3\mu_f$.
 We notice an excellent agreement between numerical and analytical
 estimates for the $\{(1,1), (1,2), (1,3), (2,2)\}$
 kernel matrix elements, which play an important role in the evolution of $N_\infty(t)$  above all at
 early times and are responsible for the agreement between numerical
 simulations and agglomeration equation at early times in Fig. \ref{Smoluchowsky comparison}. 
  
+\section{Concluding remarks}\label{conclusions}
+In this study we employed MD dynamics techniques (and in particular a
+MD research code) to investigate aggregate collisional dynamics as a
+function of the monomer-monomer interaction potential.
+The problem of cluster identification is tackled in the
+post-processing of the simulation results   by considering each cluster
+as the connected components of a graph. 
+The time evolution of the system fractal dimension is linked to the
+kinetics of two cluster populations, namely  small and large
+clusters.
+
+
+
+The diffusion coefficient of a $k$-mer is found to scale as
+$D_k\propto k^{-1}$. In other words, 
+aggregates diffuse like massive monomers as a consequence of the
+absence of any screening effect in the Langevin equation for
+interacting monomers. This is an inevitable drawback of the chosen
+method, unless a shielding coefficient is explicitly introduced in
+Langevin equation \ref{eq:Langevin}. 
+
+We also successfully calculate numerically and compare to the
+analytical prediction the kernel elements $\beta_{ij}$ for low
+indexes. An extensive numerical investigation of the collisional  is
+beyond the purpose of the present study, but it can be performed using
+the methodology described here. 
+
+
+
+
+
 
 % The Appendices part is started with the command \appendix;
 % appendix sections are then done as normal sections
@@ -1721,11 +1850,12 @@
 \includegraphics[width=0.5\columnwidth]{cluster_thermalization.pdf}
 %\includegraphics[width=0.5\columnwidth, height=6cm]{figure8_b.pdf}
 \caption{Left: time-dependence of the ensemble-averaged mean-square
-  displacement (crosses), linear fit (continuous line). Inset:
-  early-time behaviour of $\langle \delta r^2  \rangle$, linear fit
+  displacement (crosses) and linear fit (solid line). Inset:
+  early-time behaviour of $\langle \delta r^2  \rangle$ (crosses),
+  linear fit (solid line)
   and power-law fit (dashed line). Right: velocity fluctuations as a
   function of time (crosses) and analytical expression in
-  Eq. \ref{eq:delta_v_squared} (continuous line). }
+  Eq. \ref{eq:delta_v_squared} (solid line). }
 \label{diffusion_plot}
 \end{figure}
 
@@ -1742,7 +1872,7 @@
 angles $\l\delta\theta_{10}\r$ (solid line), $\l\delta\phi_{10}\r$
 (long-dashed line), $\l\delta\psi_{10}\r$ (short-dashed line) for the
 same clusters as in the left diagram and $\l\delta\theta_{50}\r$ for a
-simulation of clusters with $k=50$. The horizontal lines are the theoretical values of $\l\delta\psi(\phi)\r$
+simulation of clusters with $k=50$. The horizontal lines are the theoretical values of $\l\delta\psi(\delta\phi)\r$
 and $\l\delta\theta\r$ for random rotation matrices.}
 \label{cluster_rotation}
 \end{figure}
@@ -1773,9 +1903,10 @@
 \includegraphics[width=\columnwidth]{figure_two_slopes_extended_complete.pdf}
 % \includegraphics[width=0.5\columnwidth]{evolution_df.pdf}
 %\includegraphics[width=0.5\columnwidth, height=6cm]{figure8_b.pdf}
-\caption{Time-averaged fractal dimensions for small and large
-  clusters, together with a single fit. Clusters down to $k=5$
-  considered for the statistics.}
+\caption{Time-independent mean radius of gyration versus cluster
+  size. Linear fits performed on a double-logarithmic scale 
+  for $k>5$ (long-dashed line), $5\le k\le 15 $ (short-dashed line)
+  and $k>15$ (solid line).}
 \label{two_df}
 \end{figure}
 
@@ -1813,7 +1944,8 @@
 %\includegraphics[width=0.5\columnwidth, height=6cm]{figure8_b.pdf}
 \caption{Evolution of the total number of clusters calculated with Van
 der Waals potential (diamonds), model potential (filled circles),
-solution of the agglomeration equation with Smoluchowski kernel ()}
+solution of the agglomeration equation with Smoluchowski kernel
+(long-dashed line) and with Langevin kernel (short-dashed line).}
 \label{Smoluchowsky comparison}
 \end{figure}
 
@@ -2074,7 +2206,9 @@
 
 
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@@ -2096,10 +2230,14 @@
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+Farias, T.L., Carvalho, M.G., Koylu, U.O. (2001). Numerical characterization of the morphology
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 \end{thebibliography}