Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential            Simulations and Post-Processing       Results and Disc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential        Simulations and Post-Processing   Results and Discussion  ...
Introduction   Physical Scales and Interaction Potential        Simulations and Post-Processing   Results and Discussion  ...
Introduction   Physical Scales and Interaction Potential           Simulations and Post-Processing   Results and Discussio...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing     Results and Discussion   Co...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential                  Simulations and Post-Processing   Results and Di...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
Introduction   Physical Scales and Interaction Potential          Simulations and Post-Processing                         ...
Introduction   Physical Scales and Interaction Potential                                                                  ...
Introduction   Physical Scales and Interaction Potential   Simulations and Post-Processing   Results and Discussion   Conc...
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Talk given at the Particle Technology Lab, Zurich, Switzerland, November 2008.

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The process of nanoparticle agglomeration as a function of the monomer-monomer interaction potential is simulated numerically by solving Langevin equations for a set of interacting monomers in three dimensions. The simulation results are 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 the Langevin equations is discussed and its effect on cluster translational and rotational properties is quantified.

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Talk given at the Particle Technology Lab, Zurich, Switzerland, November 2008.

  1. 1. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Nanoparticle collisional dynamics by Langevin simulations Lorenzo Isella and Yannis Drossinos Joint Research Centre, Ispra, Italy ETH, November 2008 EC DG JRC – TFEIP - November 2006
  2. 2. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Problem Formulation Motivation and Goals Simulation of soot particle agglomeration via Langevin equations. Aggregate static properties: radius of gyration, hydrodynamic radius, fractal dimension, coordination number And dynamic properties: transport (diffusion coefficient), response time, thermalization. Agglomeration dynamics and numerical evaluation of the collisional kernel matrix elements, comparison with Smoluchowski kernel. EC DG JRC – TFEIP - November 2006
  3. 3. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Model for Monomer Dynamics Langevin Equation for Mesoscopic Systems 1/2 3D system of interacting monomers, each obeying m1¨i = Fi − β1 m1 ri + Wi (t). ˙ r Force acting on i-th monomer from pairwise monomer-monomer interaction potential   1 Fi = − ri Ui = − ri  u(rij ) . 2 j=i White noise acting on each monomer Wij (t) = 0 Wij (t)Wij (t ) = Γδii δjj δ(t − t ), and noise strength Γ = 2β1 m1 kB T fixed by fluctuation- dissipation theorem. EC DG JRC – TFEIP - November 2006
  4. 4. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Model for Monomer Dynamics Langevin Equation for Mesoscopic Systems 2/2 MD ⇒ microscopic description of the system. Langevin thermostat (among many) to model the coupling of the system with a thermal bath and define temperature for the system. Nanoparticles ⇒ mesoscopic description of the system. Langevin equation as coarse-grained description of the nanoparticle dynamics. Noise term accounting for the effect of fluid-molecule-to-nanoparticle collisions giving rise to nanoparticle diffusion. Simulations performed with a MD package (ESPResSo) but results interpreted for nanoparticles. EC DG JRC – TFEIP - November 2006
  5. 5. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Model for Monomer Dynamics General considerations Langevin equation for interacting monomers ⇒ cluster properties and dynamics fixed by the monomer-monomer interaction potential only. A cluster of monomers is not a primitive concept; only monomer properties are specified in the model hence Cluster fractal dimension, coordination number, friction coefficient, collisional kernel etc. . . are a model output. Langevin equation does not include monomer screening in a cluster ⇒ effects on cluster mobility. EC DG JRC – TFEIP - November 2006
  6. 6. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Dimensionless Formalism Specification of the Units Natural (but not unique!) choice for time, distance and mass units t ≡ τ1˜ r ≡ σ˜, ˜ m1 ≡ m1 m1 . r t, Temperature unit T ∗ is a derived quantity. For a 20nm soot particle (ρp 1.3g/cm3 ) in air at room temperature 182 πµ2 σ m1 σ 2 T∗ = f 650K = 2 6kB ρp kB τmon ˜ ⇒ T ≡ T /T ∗ = 0.5 for exhaust nanoparticles at room temperature. Dimensionless quantities used in the following, unless EC DG JRC – TFEIP -stated. 2006 otherwise November
  7. 7. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Interaction Potential Features of Monomer-Monomer Interaction Potential Repulsion at short separations r ≤ σ (hard-core repulsion) and attraction for separations above σ (sticking upon collision). Simulations performed with two radial interaction potentials: integrated Lennard-Jones potential (model for the attractive part of Van der Waals interaction between two spheres, ∼ r −6 for r σ) and with a short-ranged model potential. Model Potential Van der Waals 50 0 () u(r) −50 −100 1.0 1.1 1.2 1.3 1.4 EC DG JRC – TFEIP - November 2006 r
  8. 8. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Interaction Potential Non-coalescent monomers Hard-core repulsion: monomers do not compenetrate but retain their identity after colliding ⇒ no coalescence. In ESPResSo, a “softer” monomer-monomer interaction potential leads to overlapping but not coalescing monomers. No primitive concept of monomer radius, only of monomer mass and monomer-monomer interaction potential. m1 m1 2m1 2m1 m1 m1 EC DG JRC – TFEIP - November 2006
  9. 9. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Numerical Implementation Overview of the Numerical Simulations 5000 monomers placed randomly in a box with periodic boundary conditions and density ρ = 0.01. Mitigate the role of initial conditions ⇒ results averaged over 10 simulations. At the end of the simulation, the aggregate concentration is almost two orders of magnitude smaller than initially. MD ESPResSo package to solve the 3D Langevin equations (Verlet algorithm and Euler scheme for evaluating stochastic force on monomers). Unlike early studies (Meakin, Mountain), not necessary to look for agglomeration events while evolving the system. Each monomer can be addressed individually at all times (one can “label” it). Only monomer positions and velocities are returned ⇒ how to identify the clusters? EC DG JRC – TFEIP - November 2006
  10. 10. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Post-Processing Distances and Graphs Distance between i-th and j-th monomer along e.g. x-axis in a periodic box of size L xi − xj (x) = xi − xj − L · nint ≤ L/2 Dij L Total distance between i-th and j-th monomer (x) 2 (y ) 2 (z) 2 Dij = Dij + Dij + Dij Fix a distance dthr and calculate symmetric adjacency matrix 1, if Dij ≤ dthr Aij = 0, otherwise In graph theory, a symmetric Aij identifies completely a non-directed graph. What does it have to do with clusters? EC DG JRC – TFEIP - November 2006
  11. 11. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Post-Processing Cluster Detection: Clusters as Graphs C A B 1 2 4 6 1 3 5 5 3 6 4 2 i-th and j-th monomer bound together (adjacent) if their distance Dij ≤ dthr , dthr close to req 1.02σ. Any monomer in a cluster can be reached from any other monomer in the same cluster by strides of length dthr . For a fixed dthr , cluster determination from monomer positions ⇔ determination of connected components of a non-directed graph. Each monomer configuration (hence each cluster) has a unique representation as a graph via the adjacency matrix Aij i,j 2 4 5 C 2 1 1 0 Aij : 2 4 4 1 1 1 5 EC DG JRC – TFEIP - November 2006 5 0 1 1
  12. 12. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Post-Processing Collision Statistic: Clusters as Sets of Monomers t + δt t 9 1 1 2 2 4 6 9 4 7 5 6 7 10 3 8 3 10 5 8 Clusters as not ordered collections of monomers (no monomer label is repeated) ⇒ mathematical definition of a set. If two independent clusters at time t become a proper subset of the same cluster at time t + δt ⇔ collision. Fragmentation as the reverse process of a sticky collision. Kernel elements βij from i and j-mer concentrations and from number of collisions per unit volume in δt Nij /δt = (2 − δij )βij ni nj . EC DG JRC – TFEIP - November 2006
  13. 13. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Determination of the Fractal Dimension EC DG JRC – TFEIP - November 2006
  14. 14. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Determination of the Fractal Dimension Distribution of Aggregate Morphologies EC DG JRC – TFEIP - November 2006
  15. 15. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Determination of the Fractal Dimension Time-averaged Fractal Dimension (ri − rCM )2 2 2 Radius of gyration Rg = i + rp . k Time-independent df from average Rg = αk 1/df from k = 5 (power-law breaks down for smaller k ). Two slopes ⇒ DLA (df 2.4 − 2.5) and cluster-cluster (df 1.7 − 1.8) aggregation. 10.0 Small clusters k ≤ 15 small Rg~k1 df , for 4<k ≤ 15 Large clusters k >15 large Rg~k1 df 5.0 for large clusters Rg~k1 df for k ≥ 5 large df =1.56 2.0 df=1.62 Rg dsmall=2.25 1.0 f 0.5 1 2 5 10 20 50 100 200 500 EC DG JRC – TFEIP - November 2006 k
  16. 16. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Determination of the Fractal Dimension Time-dependent Fractal Dimension t From ensemble data at time t, fit Rg = αk 1/df . Evolution of dft determined by kinetics of large and small cluster populations. large Fractal dimension decreases and tends to df . 2.2 2.0 dtf 1.8 1.6 0 500 1000 1500 2000 2500 3000 Time 2 10 60 500 5000 All clusters Small clusters k ≤ 15 Large clusters k >15 N∞Vbox 0 500 1000 1500 2000 2500 3000 EC DG JRC – TFEIP - November 2006 Time
  17. 17. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Cluster structure Cluster coordination number Coordination number calculated from cluster Aij . 6 5 Mean coordination number 4 3 2 1 0 0 500 1000 1500 2000 2500 3000 Time High coordination number ⇒ low df from elongated shape of large aggregates, not from cluster cavities. df by itself insufficient to characterise cluster morphology. High coordination number as intrinsic feature of Langevin EC DG JRC – TFEIP - November 2006Araki, Phys. Rev. Lett., 85). simulations (Tanaka &
  18. 18. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Cluster structure Cluster restructuring Radial monomer-monomer interaction potential ⇒ locking of relative distance between neighbouring monomers but bonds can move on the monomer surface. 2.8 2.6 2.4 2.2 2.8 Rg 2.0 Rg 2.1 1.8 1.4 1.6 550 575 600 Time 1.4 500 600 700 800 900 1000 Time Restructuring on a time-scale ∼ τ1 after collision. Aggregate restructuring from Langevin equation and radial interaction only. EC DG JRC – TFEIP - November 2006
  19. 19. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Translational diffusion coefficient 1/2 25 0.5 −1 10 20 −5 2 × 10 15 <δr2 > δ CM 0.1 5 15 1 10 Simulation Linear fit Power−law fit 5 0 0 100 200 300 400 Time Ensemble average on 800 trajectories (k = 50 above) 2 δrCM (t) kB T t→∞ Dk − − −→ Cs (Kn). = 6t k m1 βk Consistent with Cs (Kn) = 1 and βk = β1 ⇒ continuum regime and cluster relaxation time identical to τ1 . 2 t γ , γ 3 at early times ⇒ Brownian particle. δrCM (t) EC DG JRC – TFEIP - November 2006
  20. 20. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Translational diffusion coefficient 2/2 k -mer diffusion coefficient Dk ∝ 1/k hence Dk does not “see” cluster structure, only its size k Cluster mobility radius Rm (radius of a sphere with the same diffusion coefficient) defined via kB T 1 ∝. Dk = 6πµf Rm k Rg = αk 1/df , whereas Rm = kr1 ⇒ Rm Rg for large clusters. Effect of introduction of a shielding factor for the i-th monomer in Langevin equation 0 (perfect shielding) ≤ ηi ≤ 1 (no shielding). EC DG JRC – TFEIP - November 2006
  21. 21. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Numerical experiment: adjusting friction and noise strength 1/2 Langevin equation with monomer shielding factor ηi m1¨i = Fi − ηi β1 m1 ri + 2 ηi β1 m1 kB T δt N(0, 1). ˙ r Heuristic expression ηi = 1 − neighbours(i−th monomer) /12 . Fluctuation-dissipation theorem (FDT) enforced for each monomer, not for the whole aggregate. 12 η=1 η= heuristic expression 10 8 <δr2 > δ CM 6 4 2 0 0 20 40 60 80 100 EC DG JRC – TFEIP - November 2006 Time
  22. 22. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Numerical experiment: adjusting friction and noise strength 2/2 The {ηi }’s enhance cluster translational diffusion ⇒ they tend to close the gap between Rm and Rg . Understandable enhancement: from FDT a Brownian particle with friction ηβ1 has diffusion coefficient (η) (η=1) (η=1) D1 = kB T /(m1 ηβ1 ) = D1 . /η≥D1 Assuming FDT, cluster friction is obtained from diffusion {η } simulations: βk = kB T /(km1 Dk i ). Does FDT hold for a cluster? Numerical investigation of time-decay of cluster initial velocity v? How to choose the {ηi }’s? Monomer exposed surface? EC DG JRC – TFEIP - November 2006
  23. 23. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Cluster rotation 1/2 Rotations in 3D identified by 3 Euler angles representing the spatial orientation of any frame as a composition of rotations from a reference frame Cluster as a rigid body: motion fixed by the coordinates of three non-collinear monomers rA (t), rB (t) and rC (t). 3x3 rotation matrix A from X = [rA (0), rb (0), rc (0)] and X = [rA (t), rb (t), rc (t)] −1 T XXT A=XX . EC DG JRC – TFEIP - November 2006
  24. 24. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Transport Properties and Response Time Cluster rotation 2/2 For random√rotation matrices, α and γ uniform in √ δα2 = δγ 2 = π/ 3 1.81, whereas [−π, π] ⇒ δβ 2 β = arccos(1 − 2U(0, 1)) − π/2 ⇒ 0.68. 2.0 Angle standard deviation (rad) 1.5 < δα2 > 10 < δβ10 > 2 < δγ2 > 10 1.0 < δβ50 > 2 0.5 0.0 0 100 200 300 400 Time No preferential cluster rotation angle/orientation; an initially ordered ensemble of large clusters takes longer to reach random orientation. EC DG JRC – TFEIP - November 2006
  25. 25. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Agglomeration Dynamics Analytical Expressions for the Collisional Kernel Smoluchowski kernel 2kB T 1/df i −1/df + j −1/df . Sm + j 1/df i βij = 3µf Diffusion coefficient from Langevin simulations and non-continuum effects in aggregate collisions (Ri ≡ Rg,i ) 4πkB T −1 i + j −1 Ri + Rj βF (Ri , Rj ). LD βij = m1 β1   data from simulations i < 5 where  small αsmall i 1/df , if 5 ≤ i ≤ 15 Ri =  α i 1/dflarge , if i > 15  large Kernel homogeneity exponent λ: βγi,γj = γ λ βi,j ⇒ N∞ ∼ t −1/(1−λ) . βij : λ = 0 ⇒ N∞ ∼ t −1 ; Sm large ) − 1 ⇒ N∞ ∼ t −0.72 . LD βij : λ = (1/df EC DG JRC – TFEIP - November 2006
  26. 26. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Agglomeration Dynamics Simulation Results and Agglomeration Equation 5000 q q q q q qq qqqqq qq qq qqq qqq qq qq qq qq qq q qq qq q qq q qq q q 1300 q q q qq q qq q q qq q q qq q q q qq q q qq q q q qq q q q q qq q q q qq q q q q qq q qq q q qq q q qq q q qq q q q qq q q qq q q q qq q q q q q N∞Vbox q q q q qq q q q q qq 330 q q qq q q q qq q q q qq q q q q qq q q q q qq q q q qq q q q qq q q q q qq q q q qq N ~t−0.77 q q q q qq q q q qq q q q ∞ q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q q N ~t−0.78 q q q q q qq q q q q q 85 q q ∞ q q q qq q q q q qq q q q q qq q q q q qq q q q q qq q q q q Simulations (Van der Waals) N∞~t−0.79 Simulations (Model Potential) q Continuum kernel and dtot N∞~t−1 f large Fuchs correction, Dk ~ 1 k, dsmall and df f 20 3 10 30 100 500 3000 1 Time Late-time decay of N∞ evaluated for 2500 ≤ t ≤ 3000. Smoluchowski kernel: poor agreement at early time and LD different decay at late times; βij reproduces simulation results at early times and similar decay at late times. Van der Waals interaction enhances agglomeration rate. EC DG JRC – TFEIP - November 2006
  27. 27. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Agglomeration Dynamics Evaluation of the kernel elements βij 1.2 × 10−52.4 × 10−5 q Simulations q q q β q 2βijninj q q q q q N13/δt q δ q q q q q q q q q q qq qq qq qqq qqqq q qqqqqq qqqqqqqqqqqq q 0 0 20 40 60 80 100 Time 0.5 1j (numerical) βij/(8kBT/3µf) 1j (analytical) µ 2j (numerical) 0.35 2j (analytical) 0.2 3 1 2 4 j βij numerically determined by fitting Nij /δt from simulations to (2 − δij )βij ni nj at all times. Calculations for i, j = 1 . . . 4 but not enough data for the other kernel elements. EC DG JRC – TFEIP - November 2006
  28. 28. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions Final Remarks Use MD techniques to investigate aggregate collisional dynamics as a function of the monomer-monomer interaction potential. Cluster identification by considering them as connected components of a graph. Time-dependent dft linked to the kinetics of two cluster populations. Diffusion coefficient of a k -mer scaling like k −1 ⇒ aggregates are “transparent” to the fluid and with the same response time of a monomer. Investigation of cluster rotational properties. Numerical calculation of the kernel elements βij for low indexes. EC DG JRC – TFEIP - November 2006

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