Adaptive Resolution Molecular Dynamics Simulation (AdResS): Changing the Degrees of Freedom on the Fly

Adaptive Resolution Molecular Dynamics Simulation (AdResS): Changing the Degrees of Freedom on the Fly Luigi Delle Site Max Planck Institute for Polymer Research Mainz Leipzig December 2007

Collaborators Matej Praprotnik, Mainz Kurt Kremer, Mainz Simon Poblete, Mainz Christoph Junghans, Mainz Silvina Matysiak, Rice University, Houston Cecilia Clementi, Rice University, Houston

Motivations many problems in the condensed matter are inherently multiscale interplay between different length scales plays the crucial role in the physics of the system an exhaustive description of the system requires the simultaneous treatment of all the relevant scales implied the aim is to treat in a simulation only as many degrees of freedom as absolutely necessary for the problem considered some regions require a treatment on a higher level of detail than the remainder of the system

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 the complex molecular systems Coarse-Grained MD simulation: reduces the number of degrees of freedom by retaining only those degrees of freedom that are relevant for the particular property of interest = longer length and time scales can be reached specific chemical details are usually lost in the coarse-graining procedure Solution: Hybrid Adaptive MD Schemes

Example I: Solvation of Complex Molecules

Example II: Large Molecules near Inorganic Surfaces

Method and model: general idea Adaptive Resolution MD Scheme allows an on-the-fly interchange between given molecule s atomic and coarse-grained levels of description, i.e., changing the molecular degrees of freedom Hybrid Model with: µ ex = µ cg, p ex = p cg, T ex = T cg

Theoretical basis of the idea: A simple schematic picture Simple schematic picture F(x) A high resolution B low resolution FA > F B x A and B are physically identical but differently represented For B the only assumption (tested numerically) is that for a state point (ρ, T) it is possible to reduce the many body potential of the high resolution description to a dimensionally reduced effective potential.

Theoretical basis Question to address:what are the basic statistical principles governing the changing of DOF? The passage from A to B can be thought as a geometrically induced phase transition, i.e. formally equivalent to a standard phase transition the equivalent of the latent heat is the energy one must give to or take from the molecules depending on the fact that the molecules acquires or looses DOF In a simulation this would means the use of a THERMOSTAT

Theoretical Basis:Transition region F(x) A B d +d X What happens in, i.e. the infinitesimal region at the interface between A and B? The number of DOF 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 + = 0 x If the condition above did not hold, a molecule would see a free energy density gradient along x within the same level of resolution leading to a drift along the x axis

Theoretical Basis:Weighting function lim F A(x) x d x lim n A(x) x d x = lim F B(x) x d + = 0 can be shown to be equivalent to x = lim n B(x) x d + = 0 x This suggests that we can formally describe the switching on/off of a given DOF via a a weighting function w(x) w(x) is such that w(x) = 1; x A and w(x) = 0; x B, with lim w(x) x d + = lim w(x) x x d = 0. x In accordance with the equation above, we require that the weighting function w(x) be continuous up to the first derivative and that goes monotonically from the value one to zero in the region.

Temperature in the region According to the Equipartition Theorem T A = 2 K A n A ; T B = 2 K B n B but in, is T = 2 K n? And what is K? We must know T to control it and to avoid kinetic barriers.

Switching resolution and the fractional dimensions of the phase space The switching procedure implies that in the transition regime, where 0 < w(x) < 1, 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 By this have derived the fractional analog of the equipartition theorem: K α = α 2β = αt 2. where K α is the average kinetic energy per fractional DOF with α degree of fractionality.

Theoretical Basis: Conclusions (1) Changing Resolution formally equivalent to a Phase Transition Latent Heat (2) At the interface it is required the introduction of a switching region,, and a related function, w(x), to allow for a smooth variation of the resolution (3) The temperature in the switching region, T, can be obtained by extending the equipartition theorem to non integer dimensions M.Praprotnik, K.Kremer and L.Delle Site, Phys.Rev.E, 75, 017701 (2007); M.Praprotnik, K.Kremer and L.Delle Site, J.Phys.A:Math.Th.40, F281 (2007). Still to address for a practical MD algorithm: How to express the interactions between molecules.

Adaptive Resolution Scheme The Adaptive resolution scheme scheme is a two-stage procedure: Derive the effective pair potential U cm between coarse-grained molecules on the basis of the reference all-atom system. Couple the atomistic and mesoscopic scales: F αβ = w(x α )w(x β )F atom αβ + [1 w(x α )w(x β )]F cm αβ, (1) where: F atom αβ = X iα,jβ F atom iαjβ (2) is the sum of all pair interactions between explicit atoms of molecules α and β and F atom iαjβ = Uatom r iαjβ, (3) F cm αβ = Ucm R αβ. (4)

Weighting Function 1 w 0.5 0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 x/a w(x) = 8 >< >: sin 2 [ π (x + d)]; 4d d cos 2 [ π 4d (x a 2 + d)]; cos 2 [ π 4d (x + a 2 + d)]; 1; d < x a 2 d 0; a 2 + d x < d x d a 2 d < x a 2 a 2 x < a 2 + d where a is the box length and d the half-width of the interface layer.

Interactions Shifted 12-6 Lennard-Jones potential: 8 < 4εˆ` σ 12 ` ULJ atom r (r iαjβ ) = iαjβ : σ 6 r + 1 iαjβ 4 ; riαjβ 2 1/6 σ 0; r iαjβ > 2 1/6 σ FENE potential: U atom FENE (r iαjα) = 8 < : 1 2 kr2 0 lnˆ1 ` r iαjα R 0 2 ; riαjα R 0 ; r iαjα > R 0

Coupling the atomistic and mesoscopic scales via a POTENTIAL approach One could, in principle, also define the coupling scheme using the respective potentials instead of forces as: U αβ = w(x α )w(x β )U atom αβ + [1 w(x α )w(x β )]U cg αβ A force defined as a gradient of this potential would not obey Newton s Third 1 w(x Law unless, α ) = 1 i.e. w(x w(x α ) x α w(x β ) α ) = w(x β ) = const w(x β ) x β In general, the existence in mathematical terms of a proper (generic) switching function for potentials is a serious problem. L.Delle Site, Phys.Rev.E 76; 047701 (2007).

Results for a medium dense liquid 3 2.5 ex ex-cg ex-cg/ex ex-cg/cg 60 50 ex ex-cg ex-cg/ex ex-cg/cg 2 40 RDF 1.5 n n 30 1 20 0.5 10 0 0 1 2 3 4 5 r * 0 0 1 2 3 4 5 r * (c) RDF cm. (d) Number of neighbors. Global and local structure are properly reproduced

Number of DOFs 5000 4500 4000 45000 40000 n DOF n ex n cg n int 3500 n 35000 n 3000 2500 30000 0 2000 4000 6000 t * 2000 1500 1000 500 0 0 1000 2000 3000 4000 5000 6000 7000 t * n DOS = 3n[4w + (1 w)] Numerical proof of the Zero Flux

Diffusion across the transition regime 70 60 t * =0 t * =10 t * =50 70 60 t * =0 t * =10 t * =50 50 50 40 40 n n 30 30 20 20 10 10 0-15 -10-5 0 5 10 15 x * 0-15 -10-5 0 5 10 15 x * (e) Coarse-grained side. (f) Explicit side. M.Praprotnik, L.Delle Site and K.Kremer, J.Chem.Phys. 123, 224106 (2005).

Dense Liquid Spherical Symmetry: F αβ = w(r α )w(r β )F atom αβ + [1 w(r α )w(r β )]F cm αβ,

Tabulated effective pair potential 8 7 6 Utab cm U cm 2.5 2 ex cg U 5 4 3 RDF 1.5 1 2 1 0.5 0 1.4 1.6 1.8 2 2.2 2.4 r 2.6 2.8 3 3.2 3.4 0 0 1 2 r 3 4 5 (i) Tabulated potential. (j) RDF cm.

Equation of state 6 5 ex cg 2.5 2 ex ex cg 4 1.5 p 3 2 RDF 1 1 0.5 0 0.025 0.05 0.075 0.1 0.125 ρ 0.15 0.175 0.2 0.225 0 0 1 2 r 3 4 5 (k) Equation of state for ρ = 0.175 and T = 1. (l) RDF cm. M.Praprotnik, L.Delle Site and K.Kremer, Phys.Rev.E 73, 066701 (2006).

Conclusions Hybrid method (AdResS) for MD simulation: Allows for a dynamical switching of the spatial resolution. The number of DOFs is changing on-the-fly. We treat only as many DOFs as absolutely necessary for the problem considered. Method successfully reproduces statistical properties of the model liquid.

Further application: Solvating a Polymer A generic Polymer solvated by a Tetrahedron Liquid M.Praprotnik, L.Delle Site and K.Kremer, J.Chem.Phys.126 134902 (2007).

Further application: Liquid Water In collaboration with C.Clementi Group at Rice University. M.Praprotnik et al., J. Phys.: Condens. Matter; 19; 2007; 292201; S.Matysiak et al., J.Chem.Phys. in press; M.Praprotnik, L.Delle Site and K.Kremer, Multiscale Simulation of Soft Matter: From Scale Bridging to Adaptive Resolution, Annu.Rev.Phys.Chem. Vol. 59, (2008).

Projects in progress Development of the algorithm for switching large molecules, e.g. polymers, where part of the molecule remains atomistic and part coarse-grained. Polymer systems, melts and solution, on specific inorganic surfaces. Projects of: Simon Poblete and Christoph Junghans

Contacts The MMG group @ MPIP www.mpip mainz.mpg.de/~dellsite/multiscale Contact: L.Delle Site dellsite@mpip mainz.mpg.de