Adaptive Resolution Simulation of Molecular Liquids

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1 Adaptive Resolution Simulation of Molecular Liquids Matej Praprotnik Max Planck Institute for Polymer Research, Theory Group, Mainz, Germany On leave from the National Institute of Chemistry, Ljubljana, Slovenia Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. /59

2 Collaborators Kurt Kremer, Max Planck Institute for Polymer Research, Mainz, Germany Luigi Delle Site, Max Planck Institute for Polymer Research, Mainz, Germany Christoph Junghans, Max Planck Institute for Polymer Research, Mainz, Germany Rafael Delgado Buscalioni, Universidad Autonoma de Madrid, Madrid, Spain Cecilia Clementi, Rice University, Houston, Texas Silvina Matysiak, Rice University, Houston, Texas Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 2/59

3 Outline Introduction: molecular liquids and soft matter, length- and time-scales, multiscale modeling Hybrid Adaptive Resolution Scheme (AdResS): linking atomic and mesoscopic length-scales Triple-scale AdResS-HybridMD method: coupling particle-based and continuum descriptions of a liquid Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 3/59

4 Molecular systems gases: intermolecular distances are large in comparison with molecular sizes the intermolecular interactions are negligible hard condensed matter: strong intermolecular interactions, long-range orientational and positional order molecular liquids and soft matter: energy-entropy interplay. The free-energy scale: k B T. The relevant properties of the system are determined by the interplay of the various temporal and spatial scales involved. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 4/59

5 Length-scales Wide range of characteristic length-scales: bond length:.nm persistence length of polymers: nm radius of gyration: nm macroscopic domains size: µm Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 5/59

6 Time-scales Wide range of characteristic time-scales: covalent bond vibrations: 5 s angle bending: 2 5 s atom dynamics governed by torsional and van der Waals interactions: 4 5 s atom dynamics governed by long-range Coulomb interactions: 2 s relaxation times of macromolecular segments: 2 9 s maximal macromolecular relaxation times: s Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 6/59

7 Multiscale modeling Universita t Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 7/59

8 Adaptive resolution simulation Motivation: to treat in a simulation only as many degrees of freedom (DOFs) as absolutely necessary for the problem considered. Method: AdResS (Adaptive Resolution Scheme) allows for an dynamical switching between the atomistic and mesoscopic levels of detail = on-the-fly changing of the number of DOFs tailor-made for molecular systems where spatially localized domains with required atomistic resolution exchange particles with the remainder of the system sufficiently described on the mesoscopic scale. Results: accurately reproduces the statistical properties of the reference all-atom system. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 8/59

9 Molecular Dynamics (MD) simulation V F Rauschen Noise Langevin Thermostat t "hard core" Damping Reibung V r F = m. a Electrostatic Elektrostatik Periodic Periodische Boundary Randbedingungen Conditions Geometry r 3 P M Algorithm (Mesh Ewald) hexagonal cell Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 9/59

10 MD simulation All-Atom MD simulation: allows to study processes at the atomic level of detail is often incapable to bridge a gap between a wide range of length and time scales involved in molecular systems Mesoscopic MD simulation: reduces the number of DOFs by retaining only those that are relevant for the property of interest = longer length and time scales can be reached specific chemical details are usually lost in the coarse-graining procedure Combining the best from both approaches: Hybrid Adaptive MD Schemes Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. /59

11 Hybrid atomistic/mesoscopic liquid MP, L. Delle Site, K. Kremer, J. Chem. Phys. 23, 2246, 25. MP, L. Delle Site, K. Kremer, Phys. Rev. E 73, 667, 26. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. /59

Changing number of degrees of freedom A tetrahedral molecule has a defined spatial orientation and 3N = 2 DOFs: 3 translational 3 rotational 3N 6 = 6 vibrational One particle mesoscopic molecule has

12 Changing number of degrees of freedom A tetrahedral molecule has a defined spatial orientation and 3N = 2 DOFs: 3 translational 3 rotational 3N 6 = 6 vibrational One particle mesoscopic molecule has no defined spatial orientation and only 3 translational DOFs. MP, L. Delle Site, K. Kremer, J. Chem. Phys. 23, 2246, 25. MP, L. Delle Site, K. Kremer, Annu. Rev. Phys. Chem. 59, 545, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 2/59

13 Geometrically induced phase transition In thermodynamical equilibrium,boundary conditions analogous to to two-phase coexistence must be satisfied: µ ex = µ cg, p ex = p cg, T ex = T cg. The rotational and vibrational parts of the free energy can be viewed as thelatent heat, which is supplied or taken by the thermostat, at this transition. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 3/59

14 Transition region F(x) A B d +d X Molecules in A and B are physically identical but differently represented. The number of DOFs is n = n(x) with: n A = const A ; n B = const B ; and n = n(x) The system is in equilibrium which implies: lim F A(x) x d = lim F B(x) x x d + = = x lim n A(x) x d = lim n B(x) x x d + = x Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 4/59

15 Weighting Function = Order Parameter w.5..2 r/a The values w = and w = correspond to the atomistic and coarse-grained regions, respectively, while the values < w < correspond to the transition (hyb) regime. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 5/59

16 AdResS: Linking atomic and mesoscopic length-scales AdResS consists of two main steps:. Derive the effective pair potential U cm between coarse-grained molecules on the basis of the reference all-atom system. 2. Couple the atomistic and mesoscopic scales: F αβ = w(x α )w(x β )F atom αβ +[ w(x α )w(x β )]F cm αβ, where F atom αβ = iα,jβ F atom iαjβ is the sum of all pair interactions between explicit atoms of molecules α and β and F atom iαjβ F cm αβ = Uatom r iαjβ, = Ucm R αβ. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 6/59

17 May the Force be with you One must interpolate the forces and not the interaction potentials if the Newton s Third Law is to be satisfied! MP, K. Kremer, L. Delle Site, J. Phys. A: Math. Theor. 4, F28, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 7/59

18 Continuous change of the phase space dimensionality The switching procedure implies that in the transition regime, where < w(x) <, we deal with fractional DOFs, i.e., by switching on/off a DOF we continously change the dimensionality of the phase space. To proper describe this situation we resort here to the fractional calculus. Let us consider a given w(x) = α and use α as a variable parameter. The infinitesimal volume element of the fractional configurational space is defined as dv α = d α xγ(α/2)/2π α/2 Γ(α) = x α dx/γ(α) = dx α /αγ(α) where the positive real parameter α denotes the order of the fractional coordinate differential. MP, K. Kremer, L. Delle Site, Phys. Rev. E 75, 77, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 8/59

19 The extension of equipartition theorem to non-integer DOFs For the fractional quadratic DOF Θ with the weight w = α we can write the partition function as: exp( βf α ) = C exp( βαp 2 Θ /2)dV α = = 2C exp( βαp 2 Θ /2) p Θ α dp Θ Γ(α) = = 2α/2 CΓ(α/2) Γ(α) α α/2 β α/2 β α/2. K α = d(βf α) dβ = α 2β = αk BT 2. In equilibriumt A = T B = T = T and thus: n α α. MP, K. Kremer, L. Delle Site, Phys. Rev. E 75, 77, 27. MP, K. Kremer, L. Delle Site, J. Phys. A: Math. Theor. 4, F28, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 9/59

20 Langevin thermostat Equation of motion for thei-th particle: Fluctuation-dissipation theorem: m i d 2 r i dt 2 = F i m i Γ dr i dt +W i(t). < W i (t) W j (t ) >= δ ij δ(t t )6 m i m j k B TΓ. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 2/59

21 Mapping of structural properties Effective pair potential that reproduces the structure of the all-atom system as closely as possible is determined using the RDF cm of the reference all-atom system via the potential of mean force (PMF ) as U cm (r) PMF(r) = k B T logg cm ex (r), g cm ex (r) is the all-atom RDF cm and U cm (r) is the effective potential. Effective potential is: in general temperature and density dependent softer than the interatomic potentials Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 2/59

22 Iterative Boltzmann inversion In the limit ρ = : U eff (r) = F(r) = k B T lng(r). For systems with ρ > the above relation is used as an initial approximation in the iteration scheme: U eff i+ (r) = Ueff i (r)+k B T ln g i (r) g target (r) Ramp pressure correction: U lin = A( r r cut ). D. Reith, M. Pütz, F. Müller-Plathe, J. Comput. Chem., 24, 624, 23. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 22/59

23 Tetrahedral liquid: effective potential Utab cm U cm ex cg U RDF r r (c) Tabulated effective potential. (d) RDF cm. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 23/59

24 Equation of state 6 5 ex cg ex ex cg 4.5 p 3 2 RDF ρ r (e) EOS forρ =.75 andt =. (f) RDF cm. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 24/59

25 Density profile profile.4.3 ex cg.4.3 ex cg ρ/ρ ρ/ρ r r 2 4 (g) Radius of the explicit regime: (h) Radius of the explicit regime: r = 6.. r =.. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 25/59

26 Artifacts of the method p * RDF ex ex-cg:w=const=/2 U * PMF ex PMF ex-cg(w=const=/2) r * w (i) Pressure r * (j) RDF cm. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 26/59

27 Interface pressure correction (ic) F cm αβ = s[w(r α )w(r β )]F cm αβ o + ( s[w(r α )w(r β )] ) F cm αβ ic. s [,] is defined as s[x] = 4( x 2 )2, where s[] =, s[] = and s[/4] = Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 27/59

28 System of hybrid molecules with w = /2 RDF U Uic cm U cm.6.8 r RDF U Utab cm ic Utab cm r.5 2 ex ex cg(w = /2) ex cg(w = /2) ic r ex ex cg(w = /2) ex cg(w = /2) ic r (k) ρ =.. (l) ρ =.75. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 28/59

29 Density profile with ic correction.4.3 ex cg ic.4.3 ex cg ic ρ/ρ ρ/ρ r r 2 4 (m)ρ =.. (n)ρ =.75. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 29/59

30 Macromolecule in solvent MP, L. Delle Site, K. Kremer, J. Chem. Phys. 26, 3492, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 3/59

31 Maximal monomer distance r = 7. r =. r = 2. rmax t Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 3/59

Static properties of the solvent 2.5 2 ex N= ex N=2 ex N=3 2.5 2 ex N=3 ex cg N=3 ex cg ic N=3 RDFcm.

32 Static properties of the solvent ex N= ex N=2 ex N= ex N=3 ex cg N=3 ex cg ic N=3 RDFcm.5 RDFcm r r Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 32/59

33 Static properties of the polymer.4.2 ex N=3 ex cg N=3 ex cg ic N=3 q 2 S(q) ρ/ρ q ex N=3 ex cg N=3 ex cg icn= r S(q) = N exp(iq (r i r j )) ij S(q) q /ν q 2 S(q) q 2 /ν,ν.57 Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 33/59

34 Dynamic properties of the polymer MSD MSD.. < r 2 > < r 2 exn= > < r 2 exn=2 > < R 2 exn=3 > < R 2 exn= > < R 2 exn=2 > exn=3 t.. < r 2 > < r 2 ex cg/icn= > < r 2 ex cg/icn=2 > < R 2 ex cg/icn=3 > < R 2 ex cg/icn= > < R 2 ex cg/icn=2 > ex cg/icn=3 t D = 6 lim t R i (t) R i () 2 t = 6 lim t R 2 t r 2 = (r i (t) r i ()) 2 t 2/z,2/z.67(N = 3) Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 34/59

35 Diffusion constant of the chain N 2 3 D(ex) D(ex-cg) D(ex-cg ic ) D bulkex =.36,D bulkcg =.57 D polymer D solvent Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 35/59

36 Liquid water The simulation speed-up is 7 2 compared to atomistic simulations. MP, S. Matysiak, L. Delle Site, K. Kremer, C. Clementi, J. Phys.: Condens. Matter, 9, 2922, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 36/59

37 Electrostatics: Reaction field method The electrostatic forces interactions are described using the Reaction field (RF) method: F atom C i αj β (r iα j β ) = e i α e jβ 4πǫ [ r 3 i α j β R 3 c 2(ǫ RF ) +2ǫ RF ] r iα j β. The RF is suitable to be used with AdResS because: it is pairwise like AdResS it must also be applied with a thermostat Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 37/59

38 Weighting function for flat geometry w x/a Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 38/59

39 Static properties q = j=k=j+ ( cosψ jk + ) 2 3 Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 39/59

40 Interface effect of the cg water transition all-atom transition The transition regime neutralizes the interface effect of the cg water = the structure of water in the explicit regime is the same as in the bulk. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 4/59

41 Diffusion across the transition regime I. coarse-grain transition all-atom Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 4/59

42 Diffusion coefficient across the simulation box S. Matysiak, C. Clementi, MP, K. Kremer, L. Delle Site, J. Chem. Phys 28, 2453, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 42/59

43 Position dependent Langevin thermostat The Langevin equation with a position dependent coefficient Γ(x) can be written as: m i dv i /dt = F i m i Γ(x)v i +R i (x,t) () where R i (x,t) is: R i (x,t) =, (2) R i (x,t )R j (x,t 2 ) = 2Γ(x)m i ktδ(t t 2 )δ ij (3) Γ(x) = Γ cg if x.6 αx+β if.6 < x. (4) This choice provides a simple interpolation between the two limit values of Γ(.6) = Γ() = Γ cg = 5ps and Γ() = Γ all atom = 5ps. The parameters α and β are 25ps and 3ps, respectively. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 43/59

44 Diffusion coefficient in the hybrid system Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 44/59

45 Diffusion coefficient across the simulation box II. (w) Position dependent thermostat (x) Regular thermostat Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 45/59

46 Transverse DPD Thermostat The variation of the dissipative particle dynamics (DPD) thermostat includes the damping of the perpendicular components of the relative velocity, yet keeping the advantages of conserving Galilei invariance and within our error bar also hydrodynamics. It allows for controlling transport properties of molecular fluids. C. Junghans, MP, K. Kremer, Soft Matter 4, 56, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 46/59

47 AdResS: Conclusions Adaptive Resolution MD simulation: Changing resolution is formally equivalent to a phase transition latent heat. For a smooth variation of the resolution we introduce a transition regime. The temperature in the transition region can be obtained by extending the equipartition theorem to non-integer dimensions. Hybrid method AdResS: Allows for a dynamical switching of the spatial resolution. We treat only as many DOFs as absolutely necessary for the problem considered. AdResS was so far applied to MD simulations of a simple tetrahedral liquid, a macromolecule in an explicit solvent, and liquid water at standard conditions. The simulation speed-up for liquid water is 7 2 compared to atomistic simulations. Review: MP, L. Delle Site, K. Kremer, Annu. Rev. Phys. Chem. 59, 545, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 47/59

48 AdResS: Future work Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 48/59

49 Concurrent triple-scale simulation Motivation: to simplify the model to the largest extend possible while keeping all the necessary details where this is required Method: Triple-scale AdResS-HybridMD scheme is a combination of two dual-scale models: a particle-based Adaptive Resolution Scheme (AdResS), which couples the atomic and mesoscopic scales, and a hybrid continuum-molecular dynamics scheme (HybridMD) covers the length-scales ranging from the micro- to macro-scale successfully sorts out the problem of large molecule insertion in the hybrid particle-continuum simulations of molecular liquids opens up the possibility to perform efficient grand-canonical molecular dynamics simulations of truly open molecular liquid systems Results: the structural and dynamical properties of the liquid are accurately captured Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 49/59

50 Triple-scale model R. Delgado Buscalioni, K. Kremer, MP, J. Chem. Phys. 28, 4, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 5/59

51 HybridMD: Coupling particle-based and continuum descriptions The hybrid particle-continuum scheme (HybridMD) is designed to connect the dynamics of a molecular domain with that obtained from a continuum description of the surrounding fluid flow. The method is based on flux-exchange. The system is divided in (at least) two domains, described via classical molecular dynamics (MD) and continuum fluid dynamics (CFD), i.e., solving the Navier-Stokes equations. The MD and CFD domains share one unique hybrid interface, H: Flux balance implies the conservation of mass and momentum across H. G. De Fabritiis, R. Delgado Buscalioni, P. Coveney, Phys. Rev. Lett 97, 345, 26. R. Delgado Buscalioni, G. De Fabritiis, Phys. Rev. E 76, 3679, 27. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 5/59

52 AdResS-HybridMD: Combined scheme HybridMD setup AdResS setup 3σ C cg 2.5σ 2.5σ hyb ex hyb cg P P C wall e H B H H x B e H 6 wall Domain decomposition of the combined scheme. The top part of the figure shows the location of the fluid model layers (cg, hyb and ex) within the HybridMD setup. The bottom part of the figure shows the set of control cells used in the HybridMD setup. R. Delgado Buscalioni, K. Kremer, MP, J. Chem. Phys. 28, 4, 28. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 52/59

53 Weighting function w x/l Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 53/59

54 Molecular density profile molecular density molecular density B cg cg hyb ex hyb cg x (σ) ex H hyb cg H H cg B cg B buffer (a) (b) x(σ) (a) ρ m =.σ 3. (b) ρ m =.75σ 3. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 54/59

55 Radial distribution functions 2.5 ex+hyb (HybridMD) ex+hyb+cg (HybridMD) ex (PBC) RDF r (σ) RDF cm s of the liquid in the atomistic and transition domains (ex+hyb) and in the total molecular region (ex+hyb+cg) of the triple-scale model together with the reference RDF cm of the all-atom system (ex(pbc)) at ρ =.75/σ 3. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 55/59

56 Couette flow.2 cg cg hyb ex hyb cg cg velocity (σ/τ) B H H x(σ) B Velocity profile at the particle region of an hybrid simulation of a Couette flow. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 56/59

57 Stokes flow velocity (σ/τ) velocity (σ/τ) cell #5 cell # time (τ) cell #8 cell # time (τ) Velocity in the y-direction at some selected cells in a hybrid simulation of a Stokes flow. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 57/59

58 AdResS-HybridMD: Conclusions AdResS-HybridMD Scheme: We performed a triple-scale simulation of a molecular liquid. Length scales from the micro- to macro-scale are concurrently coupled. The method allows us to perform efficient grand-canonical molecular dynamics simulations of molecular liquids. Future work: Application to realistic systems, e.g., liquid water. Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 58/59

Triple-scale liquid water URL: http://www.mpip-mainz.mpg.

59 Triple-scale liquid water URL: praprot/ Universität Stuttgart - SFB 76, Pforzheim-Hohenwart, September 8-, 28 p. 59/59

Coupling Atomistic and Continuum Hydrodynamics

Coupling Atomistic and Continuum Hydrodynamics Matej Praprotnik praprot@cmm.ki.si Laboratory for Molecular Modeling National Institute of Chemistry Ljubljana, Slovenia & Department of Physics Faculty of