## Abstract

We discuss a unified mesoscale framework (*chimaera*) for the simulation of complex states of flowing matter across scales of motion. The chimaera framework can deal with each of the three macro–meso–micro levels through suitable ‘mutations’ of the basic mesoscale formulation. The idea is illustrated through selected simulations of complex micro- and nanoscale flows.

This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

## 1. The concept of chimaera simulation

Multiscale modelling is the natural response to the hierarchical organization of the interactions that shape the complexity of the world around us. Traditional multiscale computing is based on the notion of *overlapping* between the four basic levels of the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, macro, meso, micro and quantum [1,2]. The overlap between these levels is sustained by the continued advances in computer hardware and modelling techniques. Yet the gap remains daunting, easily 10 decades in space (e.g. angstroms to metres) and twice as many in time (femtoseconds to hours). In addition, the practical implementation of the handshaking interfaces across the different levels remain laborious and technically intensive [3–6]. Under such a state of affairs, it is highly desirable to explore different simulation strategies based on a *unified* treatment of the different levels of the hierarchy. In this context, the concept of overlapping is replaced by the idea of *morphing*, i.e. the unified scheme should be able to morph into the desired level, e.g. micro and continuum, wherever and whenever needed, ‘on demand’. This is what we shall refer to as *chimaera* simulation^{1} ; at variance with the original mythological monster, our chimaera is a graceful and flexible creature!

We hasten to clarify that chimaera simulation, as described in this paper, is *not* multiscale in the classical sense of coupling different levels, but rather *transitions* smoothly from one to another, separately. Genuine multiscale modelling based on chimaera methods is of course possible but will not be discussed further in this work.

With the above premises, it is clear that mesoscale representations, and particularly kinetic theory, lie at a vantage point to implement the chimaera strategy, as they offer a natural intermediate description between the micro and macro levels. Indeed, Boltzmann kinetic theory is based on a continuum field, the probability distribution function, propagating along particle-like trajectories. This offers a natural particle–field duality, which can prove quite useful for multiscale purposes. For instance, instead of coupling continuum hydrodynamics to atomistic models, the mesoscale chimaera would morph, so to speak, into both these levels by suitable ‘mutations’, which take it close to Navier–Stokes on the upper end and to Newtonian mechanics on the lower one. In the rest of this paper, we shall provide a cursory description of the mutations that promote this chimaera behaviour and discuss its potential, as well as its limitations.

## 2. Weakly broken universality

Before doing so, let us address a natural question: how far can we push the (unconventional) idea of chimaera simulation? In other words, how close can one take the mesoscale description to the upper macro and lower micro levels, before hitting the ceiling (floor) where handshaking can no longer be postponed?

The ceiling part (hydrodynamics) is firm: hydrodynamics, the kingdom of universality dictated by symmetry and conservation laws, is attained at local thermodynamic equilibrium. Likewise, it is well known that, under conditions of weak departure from local equilibrium, the Boltzmann equation converges to the Navier–Stokes equations. Here, the morphing is guaranteed to be convergent.

The floor side (molecular dynamics, MD) is more shaky. It is indeed obvious that Boltzmann’s kinetic theory cannot be made equal to MD. How close the two can be brought together is controlled by the degree of universality exposed by each given problem. More precisely, by the degree of *broken* universality, meaning by this the departure from universal behaviour, the one associated with the continuum description of moving matter. On the other hand, complex moving matter is characterized, and we might say even *defined*, by the coexistence of universality and molecular individualism, to an extent which changes from problem to problem. It is hereby surmised that the chimaera approach is well suited to capture the fascinating physics stemming from such coexistence. If universality is only weakly broken (WBU), the physics under exploration requires more descriptors, fields and parameters than continuum hydrodynamics, but still no strict knowledge of molecular details [7]. Under such conditions, the chimaera approach can provide major computational returns. Otherwise, conventional handshaking cannot be helped. In the following, we shall substantiate the above programme by means of concrete examples based on a specific mesoscale technique, known as the lattice Boltzmann (LB) method, which we next proceed to briefly illustrate.

## 3. Lattice Boltzmann in a nutshell

The LBE, in single-relaxation form for simplicity [8], reads as follows [9–12]:
3.1where the index *i* labels the set of discrete velocities (figure 1). In the above, denotes a lattice version of the Maxwell–Boltzmann local equilibrium, parametrized by the fluid density and velocity , both functions of space and time.

The parameter *ω*=Δ*t*/*τ*_{c} is the ratio of free-streaming version collision time scales, *S*_{i} is a source term associated with external/internal forces and *γ*=Δ*t*/*τ*_{S} is a measure of the strength of the external versus collisional relaxation. In other words, *ω* measures the strength of collisions versus free propagation, so that connotates the strongly coupled hydrodynamic regime, whereas *ω*→0 denotes the weakly coupled non-hydrodynamic regime leading to free molecular streaming. Likewise, denotes strong interaction with external (or internal) driving sources, while *γ*→0 indicates weak coupling to the source. For the case of sources of momentum, the parameter *γ* is proportional to the Froude number *Fr*≡*Fτ*_{c}/*mv*_{th}, where *v*_{th} is the thermal speed and *F* is the force. Linear stability imposes the constraint 0<*ω*<2, which secures positive kinematic viscosity through the relation . Likewise, the parameter *γ* should be taken well below unity, on pain of ruining the stability of the scheme, as we are going to detail in the following.

### (a) Interacting fluids

One of the main merits of the LB formalism is its flexibility towards the inclusion of complex physics beyond the realm of Navier–Stokes hydrodynamics, through the source term *S*_{i}. The typical case are force fields, representing either coupling with the external environment, or internal forces resulting from intermolecular interactions. A very general and fruitful expression of LB force fields is [13]; 14
3.2where the sum extends to a suitable neighbourhood of the lattice site ** r** and

*G*

_{i}is the coupling strength of the interactions along the

*i*th link. A suitable tuning of the local functional

*ψ*(

**;**

*r**t*)≡

*ψ*[

*ρ*(

**;**

*r**t*)] permits one to recover the main features of interacting fluids, namely non-ideal equation of state, surface tension and various types of dispersion forces.

The key observation is that such forces can be reabsorbed into a shifted local equilibrium of the form
where *f*_{M} denotes the Maxwell–Boltzmann distribution and is the thermal speed.

As a result, the powerful stream–collide paradigm of the force-free formulation is kept intact: the new physics is simply encoded in the force shift. This permits the incorporation of a broad class of (weak) interactions beyond hydrodynamics, without losing the major computational perks of the force-free formalism.

Of course, this is limited to weakly coupled fluids, with Froude numbers well below unity, *Fr*≪1, a situation which covers nonetheless a broad variety of soft matter flows.

In the following, we shall provide a few concrete examples drawn from recent and past experience of this author in the field.

## 4. Chimaera in action: nano- and micro-flows

The LB approach can be taken to the nanoscale, provided fluctuations are properly reinserted through the source term *S*_{i}, this time a stochastic source compliant with the fluctuation–dissipation theorem. On the other hand, for flow quantities resulting from time and ensemble averaging over sufficiently long intervals as compared with the collisional scales, one may stay with deterministic forces expressing just the proper time-averaged interactions. This allows one to simulate a host of interesting nano-fluidic phenomena which are not directly amenable to a continuum description and yet are too demanding for MD.

### (a) Super-hydrophobicity

Super-hydrophobicity is the counterintuitive phenomenon by which fluids may experience less friction by flowing on corrugated surfaces than on smooth ones. The reason for this ‘paradox’ is that, under proper physico-chemical conditions, the development of a vapour film between the solid wall and the flowing liquid is able to lower the free-energy budget required to sustain the three phases (solid–liquid–vapour). Formally,
where *σ*_{AB} denotes the surface tension between phases A and B, inclusive of the associated contact angle, and *S*_{AB} is the area of the corresponding contact surface. Super-hydrophobicity was pointed out by MD simulations of microscale flows over nanometric corrugations [15] (figure 2*a*). In particular, a non-monotonic relation for the pressure deficit between the liquid and vapour phases as a function of the size of the corrugation was computed and shown to exhibit a loop structure, supporting a so-called de-wetting transition: the fluid in contact with the wall transits to the vapour phase, while the bulk remains liquid. If the physics of super-hydrophobicity is controlled only by transport parameters such as surface tension and contact angle, there are good reasons to believe that MD should not be needed to unravel it. On the other hand, it is not immediately obvious that the above nanoscale features can be readily included within a continuum description. Under such a state of affairs, it may be argued that a mesoscale approach, equipped with suitable interfacial interactions, should be able to capture super-hydrophobic behaviour at a tiny fraction of the cost as compared with MD.

This is indeed the case, as first shown in LB simulations [16], which were able to reproduce the pressure–geometry ‘equation of state’ to quantitative accuracy as compared with MD data (see figure 2*b*).

#### (i) What did we gain?

It is of interest to point out that such simulations required a sub-nanometric mesh spacing, Δ*x*_{LB}=0.3 nm, i.e. basically the range of molecular interaction. One might thus wonder what have we gained in the process of using LB instead of MD.

The advantage is arguably threefold. First, even though the LB mesh spacing is the same as the molecular range, the LB time step still is at least an order of magnitude larger: Δ*t*_{LB}∼Δ*x*/*c*_{s}∼*r*_{0}/*c*_{s} versus Δ*t*_{MD}∼0.1(*r*_{0}/*c*_{s}). This is because LB is stable up to Δ*t*_{LB}<2*τ*_{c}, where *τ*_{c}∼*r*_{0}/*c*_{s}, with mass and momentum conserved to machine accuracy, while MD requires a time step much smaller than *τ*_{c} in order to secure accurate energy conservation.

Second, updating a single LB degree of freedom (the discrete population *f*_{i}) is significantly faster than the update of a single MD molecule. This is because the LB interactions are lattice-bound, hence do not require the construction of a dynamic linked list to identify the interacting neighbours. This counts easily another order of magnitude in favour of LB.

Finally, LB requires no statistical average, as, by definition, the distribution function represents a large number of molecules.

Taken all together, this gives a two or three orders of magnitude LB/MD speed-up, for the same spatial size of the problem. In other words, LB performs a substantial coarse graining in time.

Of course, this is **not** MD, since MD would account for molecular fluctuations completely neglected in the present LB formulation. However, for the purpose of investigating the physics of super-hydrophobic transport, such fluctuations are inessential, thus setting a needless tax on the MD simulations.

### (b) Suppression of fast water transport in graphene oxide

The same effect has been highlighted again for the study of fast water transport (FWT) in graphene oxide nano-flows. While pure graphene sustains slip flow, with mass flow rates far exceeding the hydrodynamic value, graphene oxide, exhibiting hydroxyl groups protruding from the walls, is believed to suppress such an FWT regime [17,18].

The debate about whether the nano-flow is hydrodynamic or molecular is typically pursued by either Navier–Stokes with slip boundary conditions or MD.

In order to mimic the friction effect due to the protruding molecules, a mesoscale LB model including a frictional interaction at the interface has been recently developed [19]. The frictional force takes the form
where *γ*_{0} is a typical collision frequency between the water molecule and the hydroxyl groups, *w* the size of the hydroxyl molecules and *u*_{x} the water flow along the mainstream direction *x*, *y* being the crossflow direction (see figures 3*a* and 3*b*).

This model proves capable of predicting the breakdown of the FWT regime and also highlighted a subtle coexistence of collective motion in the bulk and individual molecular motion in the near-wall region. The LB model runs about two orders of magnitude faster than MD simulation.

### (c) The Knudsen paradox

A typical manifestation of non-equilibrium effects beyond the hydrodynamic picture is the so-called Knudsen paradox. According to hydrodynamics, the mass flow rate should scale inversely with the Knudsen number,
where *g*=∇*p*/*ρ* is the pressure drive along the channel of width *h* and the Knudsen number is defined as *Kn*=λ/*h*, with λ being the mean-free path. Since the Knudsen number is proportional to the kinematic viscosity, small *Kn* implies low dissipation, hence high mass flow and vice versa. The vice versa part, however, is deceiving, as the above proportionality only holds if *Kn*≪1.

Indeed, actual experiments show that, beyond a critical value, around *Kn*∼1, the mass flow starts to increase with the Knudsen number, a typical signature of individual molecular behaviour, i.e. free streaming.

Such behaviour cannot be quantitatively reproduced by the Navier–Stokes equations, nor by the standard LB. However, regularized LB with higher-order lattices has proved capable of reproducing these prototypical non-equilibrium phenomena. In a nutshell, the regularization consists of projecting out the higher-order non-equilibrium moments after the streaming step, so as to minimize their effect on the lower-order kinetic moments governing the transport property of the flow [20].

As one can appreciate, a suitable LB variant (R41KB, see caption of figure 4) model comes very close to DSMC, providing a concrete example of LB morphing into Boltzmann. The morphing is not perfect, yet encouraging. Further work is needed to assess whether the morphing reaches up to the details of the spatial structure of the velocity profile within the Knudsen layer, thereby relieving the need for a multiscale LB/DSMC procedure [22].

## 5. Future perspectives

Many fascinating challenges lie ahead of the chimaera approach, all being aimed at enhancing the degree of molecular individualism within the basic mesoscale formalism.

Below, we mention a few instances specific to lattice kinetic theory, together with some speculations on prospective chimaera implementations of other mesoscale techniques.

### (a) Lattice kinetic theory

Various extensions of the LB formalism can be envisaged in order to increase its microscopic fidelity. In the following, we sketch a few tentative ideas along this line.

#### (i) Lattice BBGKY

The idea is to develop a lattice version of the two-body Liouville equation for the joint distribution *f*(*r*_{1},*r*_{2},*v*_{1},*v*_{2};*t*). This would permit heuristic pseudo-potentials to be replaced with *ab initio* atomistic potentials. The computational cost is steep: a very modest 32 space resolution demands of the order of 20×20×32^{6}∼0.4 trillion degrees of freedom! In principle, this can be attempted at least as a proof of concept. The physical issue is how to concoct a realistic two-body local equilibrium in regions with strong inhomogeneities, say sharp interfaces, where the two-body correlation is neither homogeneous nor isotropic.

#### (ii) Lattice Boltzmann with internal degrees of freedom

Here, the idea is to enrich LB with internal degrees of freedom, say descriptors of the molecular structure such as angular or other types of internal variables. This would be particularly useful to study active states of matter, where universality is typically broken by the coupling between internal and external degrees of freedom. Each internal degree of freedom adds an extra dimension to phase space, and, consequently, economic discretizations are key to the computational viability of this approach.

#### (iii) Lattice Boltzmann with strong fluctuations

Inclusion of stochastic fluctuations is the canonical route to take LB closer to MD [23]. However, the strength of the stochastic force, , is subject to severe limitations on account of numerical stability. It would be highly desirable to extend the (fluctuating) LB to regimes of strong fluctuations, where the stochastic force can become comparable with the thermal one, , without compromising stability. A possibility is as follows. Let us define a chimaera distribution, blending the Boltzmann distribution with particles (Klimontovich distribution), . The regular component carries the hydrodynamic content, while the singular one takes care of non-equilibrium fluctuations, so as to absorb strong fluctuations which would violate the positivity of the regular one. This is similar to standard LB/DSMC coupling, except that the weight *w* can be made a function of space and time, so that both regular (*f*) and singular (*δ*) components are evolved concurrently. The advantage is that the particle component can be kept much more economical than stand-alone, since the low-order moments are in charge of the smooth part *f*. Once again, a well-founded and efficient exchange protocol between the two representations is key to the success of this strategy.

### (b) Chimaera simulation with mesoparticles

This paper is heavily centred upon LB as a concrete vehicle to chimaera implementations, but it is clear that other mesoscale approaches, such as dissipative particle dynamics [24] or multi-particle collision (MPC) methods [24,25], may well serve the same purpose. In principle, mesoparticle methods are less suited to cover the broad range scales embraced by LB, because they do not share the same particle–field duality. In other words, the fact of moving smooth fields (the discrete distributions) along straight particle trajectories is a major strength of the LB formalism and, although this entails limitations in reaching down to molecular scales, it provides nonetheless an amazing capability to escalate the large hydrodynamic scales up to fully developed turbulence.

On the other hand, (stochastic) mesoparticle methods naturally incorporate the thermal and statistical fluctuations which are essential to reach down to the microscale physics.

Thus, there appears to be a nice complementarity between the two approaches, one that reflects the two major avenues of kinetic theory: Boltzmann distribution functions versus Langevin stochastic dynamics.

The computational implications of such complementarity appear well worth pursuing in detail for the years to come. The subject is exciting and up for grabs.

## 6. Summary

The LB is broad and flexible; it can navigate across many scales of motion with relatively minor enrichments, all of which fit into the powerful stream–collide paradigm [26]. As a result, LB provides a handy access to complex states of moving matter characterized by the coexistence of universality and molecular individualism, a hallmark of soft matter at the biology–chemistry–physics interface. LB is not perfect; whenever the lattice spacing approaches relevant physical scales, close scrutiny of lattice artefacts is necessary and higher-order lattices become necessary but possibly not sufficient to capture the relevant short-scale physics. Under such conditions, other mesoscale methods, closer in spirit to MD, might well offer an appealing alternative for the implementation of the chimaera strategy portrayed in this paper.

## Competing interests

The author declares no competing interest.

## Funding

This work was partially supported by the Integrated Mesoscale Architectures for Sustainable Catalysis (IMASC) Energy Frontier Research Center (EFRC) of the Department of Energy, Basic Energy Sciences, award no. DE-SC0012573. Funding from the European Research Council under the European Union Seventh Framework Programme (FP/2007-2013)-ERC Grant agreement no. 306357 (NANO-JETS) is also kindly acknowledged.

## Acknowledgements

Enlightening discussions with P. V. Coveney, J. P. Boon and P. Wolynes are kindly acknowledged. The author also thanks the Solvay Foundation for supporting the Solvay Symposium ‘Bridging the gaps at the PCB interface’, which inspired the present contribution.

## Footnotes

One contribution of 17 to a theme issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

↵1 The notion of chimaera in this paper bears no relation to the University of California San Francisco (UCSF) Chimera package for interactive molecular visualization and analysis, see http://www.cgl.ucsf.edu/chimera/about.html.

- Accepted August 3, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.