Sauro Succi
Center for Life Nanoscience at la Sapienza, Istituto Italiano di Tecnologia, viale Regina Elena 295, 00161, Rome, Italy
sauro.succi@iit.it
Andrea Montessori
Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy
Marco Lauricella
Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy
Adriano Tiribocchi
Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy
Fabio Bonaccorso
Center for Life Nanoscience at la Sapienza, Istituto Italiano di Tecnologia, viale Regina Elena 295, 00161, Rome, Italy
1 Introduction
In the last decades kinetic theory has developed into a very elegant and effective framework to handle a broad spectrum of problems involving complex states of flowing matter, far beyond the original realm of rarefied gas dynamics.
The kinetic theory of gases in its original form as devised by Ludwig Boltzmann was restricted to binary collisions to pointlike particles hence formally limiting its application to dilute gases. Subsequent attempts to extend it to dense gases and liquids were notoriously plagued by several problems, mostly connected with infinities in the treatment of higher order collisions. Several strategies have been developed over the years to cope with such problems, but the kinetic theory of dense, heterogeneous fluids remains a difficult subject to this day. A similar statement applies to complex flows with interfaces often encountered in science, engineering, soft matter and biology. A particularly interesting framework to deal with such complex flows is provided by density functional theory (DFT). Essentially the idea is that much of the physics of the complex many-body problem associated with dense fluids can be explored by investigating the dynamics of the fluid density, namely a single one-body scalar field. Of course, such dynamics is subject to self-consistent closures, typically in the form of well-educated guesses on the generating functional from which the effective one-body equation for the density can be derived via standard functional minimization of the suitable free-energy-functional (FEF). Density functional theory has met with spectacular success for the case of quantum many body problem, leading to the development of powerful theorems and attending computational methods that still form the basis for modern Nobel-prize winning computational quantum chemistry. [7].
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milestone framework to describe an simulate complex flows with interfaces, which we now proceed to illustrate in some more detail [6].
2 Density Functional Kinetic Theory
In the framework of kinetic theory, the main ingredient is the non-ideal force associated with the forced-streaming term ˆS = Fa[ρ]∂vaf. , where Fa is a density-dependent mean field force and latin indices run over spatial dimensions.
The natural question is: why would the kinetic formulation be computationally advantageous over the hydrodynamic one?
The main point is that in the kinetic framework the term S can be brought to the right hand side and treated as a soft-collision term. Note that the partial derivative in velocity space is handled by integrated by parts, This permits to move the distribution function along unperturbed, force-free (straight) characteristics dxv = vdt and include the effect of soft forces as a local correction/perturbation to this free-streaming motion.
In equations:
f (x + vdt, v, t + dt) − f (x, v, t) = (C − S)dt (1) where ˆC stands for standard short-range collisions, and vector indices have been omitted for simplicity. A very popular choice is the single-time relaxation Bhatnagar-Gross-Krook model
C = (feq− f )/τ
where feq is the local equilibrium and τ the relaxation time. The soft-collision term is conveniently turned into an algebraic source term
S = F [ρ] X
k
skHk(v),
by integrating by parts in velocity space and exploiting recurrence relations of tensor Hermite polynomials [2, 3].
The advantage of the above formulation is that the streaming step at the left hand side proceeds along constant characteristics, hence it is exact, i.e. zero round-off error in the numerical treatment. This stands in contrast with the hydrodynamic formulation in which information moves along spacetime dependent material lines defined by the fluid velocity itself, dxu = u(x, t)dt.
This simple but key advantage lies at the heart of the success of lattice kinetic techniques and most notably Lattice Boltzmann method, in which the characteristics are restricted to a suitable set of discrete velocities {vi}, i = 0, Nv, showing sufficient symmetry to recover the correct large scale hydrodynamic limit.
Differently restated, the highly complex physics of moving interfaces is entirely absorbed within the local source S.
Clearly, such perturbative treatment is limited to sufficiently weak forces, as gauged by the so-called cell-Froude number
F r = adt v
where a is the acceleration due to the non-ideal forces. In order to preserve the stability of the numerical scheme, the time-step must be chosen such that F r 10−3, a condition which may eventually go broken in the presence of strong density gradients. This problem
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can be mitigated by improving the time-marching scheme, typically via locally implicit formulations, but it must be watched carefully case by case.
Lattice DFKT as discussed above is currently being used over an amazingly broad spectrum of soft-fluid problems, definitely beyond the original realm of rarefied gas dynamics.
3 Lattice Formulations
The first lattice transcription of the free-energy functional has been proposed back in the mid 90’s [1]. The main idea is to write the collision operator in single-relaxation form, i.e. C = −(f − feq)/τ, where feq is the Maxwell-Boltzmann equilibrium, and incorporate the effect of the soft term S within a generalized non-local equilibrium, reflecting teh non-locality of the Korteweg tensor. Ever since its inception, it has generated a wide body of interesting results, especially in multiphase microfluidics.
An alternative and possibly more straightforward route is to connect with is to write the non-ideal pressor directly in the form of a two-body convolution:
Pab(x) = Z
raψ(x − r
2)G(x, r)ψ(x + r
2)rbdr (2)
where G(x, r) is the two-body density correlator and ψ(ρ) is a local functional of the density.
At variance with the free-energy approach, the correlator is designed top-down, i.e.
by reverse-engineering the expression of G(x, r) so as to obtain the desired physical phenomena, namely i) Non-ideal EoS, ii) Tunable surface tension, iii) Positive disjoining pressure.
The earliest and still very popular such top-down formulation is due to Shan-Chen [5]
and makes use of just a single parameter correlator, taking the value G0 within the first Brillouin lattice cell and zero elsewhere.
The Shan-Chen model has been subsequently extended in many directions, including the formulation of multi-range models designed to simulate dry foams and moderatley dense emulsions.
When it comes to dense emulsions, spurious effects have been reported on the disjoining pressure, due to lack of sufficient lattice symmetry.
To this regard, a very fruitful avenue turned out to be offered by a entirely rule-driven approach, based on the so called Color Gradient technique (”color” is just a mnemonics for different chemical species or different phases of the same species, in analogy with quantum chromodynamics). Essentially the idea is to add an explicit anti-diffusive flux sending particles of each species uphill along their density gradient instead of against it.
By construction, such anti-diffusive flux helps interface formation against the coalescing effect of surface tension. here too, the parameters can be adjusted to recover the properies i)-iii) above independently.
In the lattice chromodynamics models for multicomponent flows, two sets of dis-tributions functions(let’s say red and blue) are introduced to code for two different fluids.
fik(x + ci, t + 1) = fik(x, t) + Ωki(x, t) (3)
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Ωki = (Ωki)3[(Ωki)1+ (Ωki)2] (4) where (Ωki)1 is the BGK collisional, (Ωki)2 is a two-phase collision operator, generating an interfacial tension between the two immiscible components and (Ωki)3 is an anti-diffusive operator which favours phase segregation and keeps interfaces sharp.
To note that, the stress-jump condition across the fluid interface, induced by the perturbation operator, can be augmented with a suitable repulsive term aiming at modelling the effect of short-range, repulsive near-contact forces induced by the presence of surfactants and colloids adsorbed at the fluid-fluid interface. The additional repulsive term can be added efficiently in the LB framework via a forcing term localized at the interface:
Frep = Ah[h(x)]nδΣ (5)
the interested reader is referred to [4] and related literature.
4 Soft matter applications
As a an application, we show the capability of the multicomponent model with near-contact interactions to reproduce the formation of ordered droplets clusters in microfluidic channels. In figure 4(a) we reported the formation of multilayer hexagonal droplet
(a)
(b)
increasing fI
Figure 1: (a) Multi layer hexagonal droplets clusters in a microfluidic channel. Dashed lines represents Voronoi tessellation while solid lines Delaunay triangulation (b) Droplet self-assemblies within a microfluidic channel with a divergent opening angle α = 45◦ for two different inlet channel Capillary numbers (Ca = 0.04 left, Ca = 0.16 right).
clusters in a channel formed by a thin inlet and an outlet chamber. The droplets are continuously injected within the main channel by employing a recently developed internal periodic boundary condition. To note that the sponstaneous ordering of the droplets into hexagonal clusters is drive by a non trivial competition between local, short-range, repulsive interactions (i.e., the near-contact forces) and the surface tension.
In particular, small disturbances introduced by the short-range repulsive action of the near-contact interaction forces trigger the rupture of the initial, single-file crystal symmetry, driving the droplets towards a new spatial arrangement.
It is interesting to note that the process described above is somehow similar to the instability observed in densely packed granular materials subjected to force unbalances.
In panel (b) of figure 4 we reported the formation of dense emulsions in microfluidic devices formed by a divergent inlet channel connected to a downstream channel. Even in
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this case the droplets are continuously injected within the system and let free to assemble within the outlet channel. By simply tuning the inlet capillary number it is possible to observe a spontaneous transition from a high-ordered emulsion, formed by hexagonal clusters flowing within the system (i.e. a wet emulsion) to a foam-like, dry-state, structure which results in a neat distortion of the Delauney triangulation (blue solid lines connecting thecenters of neighboring droplets).
In the simulations, both the dispersed and continuous phases’ discharges are kept constant and the Ca is changed via the surface tension. The observed transition is likely to by due to (i) the breakup processes downstream the injection channel, promoting the formation of liquid films and (ii) the increased deformability of the droplets interface, due to the lower values of surface tension employed.
5 Conclusions
Summarising, lattice formulations of density functional kinetic theory provide a powerful theoretical and computational framework great for the numerical studies of complex flowing soft matter systems.
References
[1] Swift, M. R., Orlandini, E., Osborn, W. R., Yeomans, J. M. (1996). Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Physical Review E, 54(5), 5041.
[2] Succi, S. (2001). The lattice Boltzmann equation: for fluid dynamics and beyond.
Oxford university press.
[3] Succi, S. (2018). The lattice Boltzmann equation: for complex states of flowing matter. Oxford University Press.
[4] Montessori, A., Lauricella, M., Tirelli, N., Succi, S. (2019). Mesoscale modelling of near-contact interactions for complex flowing interfaces. Journal of Fluid Mechanics, 872, 327-347.
[5] Shan, X., Chen, H. (1993). Lattice Boltzmann model for simulating flows with multiple phases and components. Physical review E, 47(3), 1815.
[6] Bray, A. J. (2002). Theory of phase-ordering kinetics. Advances in Physics, 51(2), 481-587.
[7] Hohenberg, P. C., Halperin, B. I. (1977). Theory of dynamic critical phenomena.
Reviews of Modern Physics, 49(3), 435.