Fluid particle models for the simulation of microfluids

Fluid particle models for the simulation of microfluids Marco Ellero1,2∗ 1

2

Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨ at M¨ unchen, Boltzmannstr. 15, 85748 Garching, Germany. Departamento de F´ısica Fundamental, UNED, Apartado 60141, 28080 Madrid, Spain.

Abstract. In the present work, some of the mesoscopic particle methods generally used for the simulation of microstructured fluids are reviewed. In particular, the Dissipative Particle Dynamics (DPD) method, designed by Hoogerbrugge and Koelman in 1992, offers a good compromise of performance and flexibility. Some aspects of the method are discussed as well as the main conceptual shortcomings which limit its current applicability to some micro-flow conditions. Refined models of DPD are therefore presented, i.e. Smoothed Dissipative Particle Dynamics (SDPD) [Espa˜ nol, Revenga. Phys. Rev. E 67, 026705 (2003)]. The method is a thermodynamically consistent version of DPD and, at the same time represents a direct discretization of the continuous Navier-Stokes equations on a Lagrangian framework. This feature is common to another macroscopic particle method, i.e. Smoothed Particle Hydrodynamics (SPH) [R. A. Gingold and J. J. Monaghan, Mon. Not. R. Astron. Soc, 181, 375 (1977)]. SDPD can be therefore understood as a mesoscopic version of SPH with thermal fluctuations consistently included and provides the unifying multiscale framework linking DPD to SPH. Finally, applications of the model to microfluids are discussed. In particular, results for polymer molecules and colloidal particles suspended in Newtonian solvent are presented.

1

Introduction

The increasing technological interest in the design of micro-scale flow devices characterized by components of size smaller than 1 millimeter (microfluidics) or 1 micron (nanofluidics) is providing strong stimuli in the understanding of the hydrodynamic processes occurring at the micro-scales [1]. Typical target areas in microfluidics include control flow devices for cooling of electronic systems, multiphase flows in lab-on-a-chip, combustors, micro-mixing devices as well as fabrication processes involving individual organic and/or inorganic constituents [2]. Among the latter ones, particular focus has been recently given to the development of sensors for the detection of biological cells, manipulation of single DNA molecules or other micron-sized components. It is therefore evident how the ability to improve the numerical models at this spatio-temporal level and for these class of complex microstructured fluids ∗

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would be of great benefit. It is worth to notice that the Navier-Stokes equations describing the dynamics of a Newtonian liquid at the macroscopic level still remain valid at the microfluidics scales, therefore providing a natural framework based on the continuum description. On the other hand, it is also clear that, whenever the physical dimensions of the considered objects (i.e. polymer macromolecules, colloidal particles etc.) are in the sub-micrometer range, the surrounding fluid starts to feel the presence of its underlying molecular structure and hydrodynamic variables will be influenced by thermodynamics fluctuations according to the Landau and Lifshitz theory [3]. Standard macroscopic approaches, based for example on finite volumes or finite elements methods, are not suitable for this type of simulations. They neglect thermal fluctuations which, as mentioned above, are the most crucial ingredient of the mesoscopic dynamics. On the other hand, direct microscopic approaches, such as molecular dynamics, are able to resolve the smallest details of the molecular structures but they are computationally very expensive and are limited by the available computer resources. Nowadays, these approaches are restricted to computational domains of length of the order of nanometers which represent only the smallest scales (dimensions of one nanostructure) in the range covered by the considered system. Dissipative Particle Dynamics (DPD) as originally invented by Hoogerbrugge and Koelman, is perhaps the most popular particle model for the simulation of Newtonian fluids at mesoscopic scales [1,2]. In DPD, a Newtonian fluid is represented by a collection of points with prescribed stochastic interactions that conserve momentum and produce hydrodynamic behaviour at a coarse-graining level. Moreover, DPD includes thermal fluctuations in a thermodynamically consistent way [2] and it is thus applicable to mesoscopic scales where diffusive processes are important. Since its development, the method has been applied to a wide class of problems and is now emerging as a powerful numerical technique for simulations in the area of micro/nano science. Despite its recent success, DPD suffers from a number of conceptual shortcomings which can limit its applicability and physical understanding. In particular, they are related to the following issues: (i) the resulting equation of state turns out to be quadratic in density; (ii) no direct connection between the model parameters and the transport coefficients of the simulated fluid (kinetic theory or preliminary runs are necessary to measure the transport coefficients); (iii) unclear definition of the particle size. The latter point represents a big drawback preventing an ”a priori” control of the spatio-temporal scales simulated. Indeed, due to the lack of a specific physical size associated to the particles, DPD is unable to characterize in a unambiguous way the external lengths of the problem under study. This is crucial, for instance, in the case of suspended colloidal particles or in mi-

crofluidics applications where the physical dimensions of the external objects determine whether thermal fluctuations come into play. In this work, we discuss in detail a modified DPD formalism, i.e. Smoothed Dissipative Particle Dynamics (SDPD), recently proposed by Espanol [9]. The new method is able to solve the problems mentioned above and, in addition, it helps to bridge the gap with another particle approach operating at the macroscopic level: Smoothed Particle Hydrodynamics (SPH) which represents a Lagrangian mesh-less discretisation of partial differential equations [8]. This article is composed as follows. Section 2 introduces the DPD governing equations; we review the refined particle SDPD model in Section 3. Section 4 is devoted to the modelling of complex microstructured fluids, in particular the SDPD modelling of a colloidal particle and a polymer molecules suspended in a Newtonian fluid will be discussed. Finally, in Section 5 we validate the model by performing simulations in both, Brownian and nonBrownian environments.

2

Dissipative Particle Dynamics

Before discussing the problems of the methodology, let us resume briefly the original equations of motions for the DPD particles [4,5]. If we assume to have N particles of mass m distributed over a physical domain V , each of them follows the Newton’s equations of motion r˙ i = vi and mv˙ i = Fi for i = 1, .., N . Here ri and vi represent respectively position and velocity of particle i. Fi represents the net force acting on particle i and it is evaluated as the sum of conservative, dissipative and stochastic interparticle components P −1/2 D R , where ∆t is the time step and as follows: Fi = j FC ij + Fij + Fij ∆t C FC ij = F (rij )eij

(1)

D FD ij = −γω (rij )(vij · eij )eij

(2)

R FR ij = σω (rij )ξij eij

(3)

and being rij = |ri − rj | the relative distance, vij = vi − vj the relative velocity and eij = rij /rij the unit vector joining particles i and j. As usual, ξij represent symmetric Gaussian random variables with zero mean and unit variance. In Eq. (2), γ is a friction parameter which is connected to the kinematic viscosity of the coarse-grained fluid. Validity of the fluctuation-dissipation theorem requires σ and γ to be linked by the relation σ 2 = 2γkB T , being kB the Boltzmann factor and T the system temperature [5]. In addition, thermodynamic consistency requires also that ω D (rij ) = [ω R (rij )]2 . Finally,

for the conservative part of the force a Mexican function is usually adopted F C (rij ) = aij max{1−(rij /rc ), 0}, where aij is a particle interaction constant and rc is a cutoff radius. One of the main problems of DPD is not only related to the absence of a model parameter which determines a physical measure of the particle size but, more importantly, to the lack of connection of this particle size with the magnitude of thermal fluctuations in Eq. (3), i.e. the specification of σ. Indeed, although the choice of γ remains uniquely specified through the fluctuation-dissipation relation, σ remains arbitrary and it is usually specified in terms of the algorithm’s performance [6]. This represents a serious drawback of the technique because the level of fluctuations which affects a fluid particle on a given scale is dictated by numerical, rather than physical motivations. It would be therefore highly desirable to develop a formalism where the particle’s thermal energy is uniquely linked to physical parameters, i.e. particle size. In order to correct these drawbacks, a new formulation of DPD (denoted as Smoothed Dissipative Particle Dynamics: SDPD) has been recently introduced [9]. In the following section it will be discussed and its advantages over the classical DPD highlighted.

3

Smoothed Dissipative Particle Dynamics

The new mesoscopic model makes use of an additional extra variable for every fluid particle, c.f. a thermodynamic volume, which enters directly in the definition of the noise, characterizing the physical particle length and, at the same time, prescribing their thermal energy uniquely. Furthermore, SDPD deepens the connection with another particle method known as Smoothed Particle Hydrodynamics (SPH) in the sense that interparticle forces will be now considered as numerical discretizations of suitable sets of continuum equations with prescribed transport coefficients. SPH is a macroscopic particle method designed in the late seventies to study astrophysical flow problems [7,8]. The basic idea of SPH is to use an interpolant function to evaluate spatial derivatives of the field at a given particle location. Therefore hydrodynamic equations, written in the form of an arbitrary set of partial differential equations, can be solved to the prescribed order of accuracy. For example, the gradients of Pa function f (r) defined over a domain V are computed in SPH as ∇f (r) ≃ j φj fj ∇W (|r − rj |, h) where W (r, h) is a bell-shaped interpolant (i.e Lucy function) with finite support h and normalized to unity, while φj represents the volume associated to the P −1 particle j and is defined as φj = ( k W (|rj − rk |, h)) . Note that in SPH, there is a direct definition of the particle size given in terms of its volume,

that is l = φ1/3 . In a previous work [9] it has been shown how, by using the so-called GENERIC framework [10], it is possible to cast the original SPH model, in a form which encodes automatically the First and Second Laws of Thermodynamics. In particular, it allows to introduce thermal fluctuations in a systematic way, which by construction satisfies the fluctuation-dissipation theorem and where the fluctuations are governed by the Einstein equilibrium distribution. According to [9], the new equations of motion for the fluid particles are given below. Let us consider first the deterministic part; they read r˙ i = vi # "  X X Pi ωij Pj 5η + ω r − vij − ζ mv˙ i = ij ij 2 2 di dj 3 di dj j j  η  X ωij −5 ζ+ eij eij ·vij 3 j di dj

(4)

where Pi is a pressure variable which can be related to the mass density via an arbitrary equations of state. In this work P (ρi ) = (c2s /2ρ0 )ρ2 is considered where cs is the input speed of sound. In Eq.(4) η and ζ are respectively the fluid shear and bulk viscosity, and κ the thermal conductivity, all input parameters, while the geometrical factor ωij is given by ωij = −W ′ (rij )/rij . Concerning the random terms, they can be introduced in the equations by postulating a tensorial generalization of the stochastic Wiener process which enters the momentum and internal energy equation in the following way: md˜ vi =

X j

1 Ti dS˜i = − 2

c ij ·eij Aij dW X j

c ij : eij vij + Aij dW

X

Cij dVij

(5)

j

  c ij = (1/2) dWij + dWT is the symmetric part of dWij . In this where dW ij expression we have introduced, for each pair i, j of particles, a matrix of independent increments of the Wiener process dWij . In Eqn. (5) we have also introduced an independent increment of the Wiener process for each pair of particles, dVij . For the amplitudes Aij , Cij of the noises we select the very specific form 

Ti Tj ωij 40η kB Aij = 3 Ti + Tj di dj

1/2



ωij Cij = κkB Ti Tj di dj

1/2

(6)

By postulating the noise terms in Eq. (5) with the amplitudes (6), the amount of energy produced is exactly the same as that dissipated in the deterministic part through viscous forces in Eq. (4). Some remarks are here in order: • Being the conservative forces between fluid particles dependent on a additional pressure variable, arbitrary equations of state can be adopted which are not restricted to the quadratic form usually assumed in standard DPD. • Thanks to the analogy with SPH, the set of equations (4) represents a second-order accurate discretization of the Navier-Stokes (NS) equations in a Lagrangian framework. Analogously to standard mesh-based discretization schemes, ”second-order” here means that the discrete equations (4) are consistent with the exact NS up to terms of order O(h2 ), being h the cutoff radius of the kernel interpolant. This solves the problems with the derivation of the transport coefficients which represent now input parameters of the simulation and remain constant, independently of the resolution considered. • The definition of a particle volume φ provides a typical length scale for the particle which is l = φ1/3 . Notice that in a macroscopic particle method this does not represent any physical quantities. For instance, in SPH particles are simple Lagrangian moving nodes on which the NS equations are discretized. By increasing the resolution, that is reducing the particle volumes, convergent results must be recovered if the method is consistent. However, due to the fact that di enters directly in the definition of the stochastic noise, particle volumes acquire in SDPD a physical meaning that is: the dynamics of the solvent is not longer scale-invariant. Indeed, it can p be shown that the averaged particle velocity fluctuations scale as h∆vi i = (kB T /ρ)φ−1/2 . The size of thermal fluctuation is given by the typical length size of the fluid particle scaled as the the inverse of its square root, in accordance with usual concepts of equilibrium statistical mechanics. Therefore, for large enough fluid particles, the thermal fluctuations in the momentum and energy equation can be neglected. On the other hand, if we consider a submicron structure we will need to resolve the surrounding solvent liquid with fluid particles of one order or more smaller than the typical size of the object, which will produce non-vanishing stochastic terms giving rise to its ultimate microscopic diffusional dynamics. Note also that, the numerical sizes of the fluid particle are completely specified, being determined by the external lengths of the problem.

Fig. 1. Light-grey particles represent solvent particles, dark-grey particles represent either the boundary particles forming the colloid (left) or the monomers describing the polymer molecules (right).

4 4.1

Microstructured fluids modelling Colloidal particle

In order to model a colloidal particle, we select a certain number of SDPD solvent particles which lie within a spherical region and denote them as ’boundary particles’. In the computation of the hydrodynamic forces, when a solvent particle interact with a boundary particle an artificial velocity is assigned to the boundary particle in order to have zero perpendicular and tangential velocity on the nominal boundary surface. This guarantees impermeability and no-slip boundary conditions for the fluid at liquid-solid interface [11,12]. In order to take into account the translational motion of the colloidal particle, the following procedure is considered: at every time step, the total force exerted by the fluid on the solid particle is evaluated as X F coll = Fk (7) j∈Ω

where Ω is the domain represented by the colloidal particle (filled with SDPD boundary particles) and Fk represents the total force acting on the boundary particle k dueP to the interactions with all its neighbouring solvent particles, that is: Fk = l∈V Fkl where V is the volume occupied by the fluid. Once the total force on the colloid F coll is evaluated, we simply update its center-of-mass velocity and position according to a predictor-corrector

scheme: this defines the nominal surface at the next time step which is needed to evaluate the artificial boundary velocities. Accordingly, all the boundary particles are translated as a rigid body. Rotational motion of the colloidal particle can be modelled in the same way, by evaluating a total torque and defining an additional angular velocity for the colloid. A sketch of the model is shown in fig. 1 (left). 4.2

Polymer molecule

A polymer molecule can be modelled as a linear chain composed of N monomers interacting by finitely extendable nonlinear elastic springs (FENE potential)   r2 HR02 FENE (8) ln 1 − 2 . U (r) = − 2 R0 p where H is the spring constant, r = tr(rij rij ) is the monomer-monomer distance and R0 represents the maximum extensibility. Monomers belonging to the polymer chain interact, besides the FENE forces (only adjacent ones), also by the usual hydrodynamic SDPD forces with the neighbouring fluid particles and monomers. A typical configuration is sketched in Fig. 1 (right). The physical basis of the model has been discussed in detail in [13].

5 5.1

Simulations Colloidal particle in suspension

In order to test the colloidal particle model, we first consider the 2D case of a colloidal disk of radius R suspended in a deterministic Newtonian solvent (no thermal fluctuations) defined on a square domain of size L with periodic boundary conditions applied to every directions. The fluid and the colloid are initially at rest. At time t = 0, the colloidal particle is perturbed by assigning it a constant velocity U (0) = U0 in the x-direction. We monitor the velocity of the colloidal particle V (t) as a function of time in the reference frame of the center of mass for the total system (colloid+fluid), that is: V (t) = (U (t) − UCM )/(U0 − UCM ) where UCM is the center of mass velocity of the total system. The box domain is L = 2, while the colloidal radius is R = 0.1 giving a sufficiently small concentration ratio φ = (πR2 )/L2 ≈ 0.0078: this enable us to neglect hydrodynamic interactions between images of the colloidal particle produced by the replicas of the main box. Speed of sound is cs = 1, initial particle velocity U0 = 0.01 and solvent kinematic viscosity is ν = 0.02. The dimensionless numbers characterizing this flow problem are the Reynolds number Re = RU0 /ν = 0.05 and the Mach number M a = U0 /cs = 0.01.

Fig. 2. SDPD simulation (no thermal fluctuations) of the flow field around a moving disk. The double vortex structures behind the disk are clearly visible.

Fig. 2 shows a snapshot of the flow field around the colloidal particle suddenly after flow start-up in the laboratory frame. The double vortex structures around the moving particle are well reproduced by the method. It has been shown that, due to the presence of hydrodynamic fluid-particle interaction, the velocity decay is not exponential but, at sufficiently long times a typical algebraical decay ∝ t−1 should be observed. Fig. 3 shows the decay of the velocity in the center-of-mass frame vs. time. It can be seen that for t > 2, the decay is algebraical with exponent α = −1 (see enlarged box). This shows that HI between particle and surrounding fluid are properly taken into account in our model. As a further test, we show here the results of the diffusional motion of a colloidal particle suspended in a Brownian solvent. The solvent is modelled with SDPD particles whose dynamics is governed by the equations (4) with the random terms given in (5). No external perturbations are considered here. Temperature of the solvent is T0 = 1. According to the Einstein’s distribution function, a colloidal particle of mass M suspended in a Brownian solvent should be characterized by a Gaussian velocity probability distribution func-

Fig. 3. Decay of the normalized colloidal particle’s velocity. In the enlarged box, the typical algebraical decay at long times can be appreciated.

tion (PDF) with variance given by hV2 i = D

kB T . M

(9)

We have performed a simulation and computed explicitely the equilibrium distribution function of the velocity of the colloidal particle. Figure 4 shows the histogram of the x-component of its velocity. There is a very good matching between the normalized histogram and the theory. It should be noticed that the result does not depend on the particle resolution used to simulate the problem. Indeed, SDPD posses a correct scaling of the thermal fluctuations [14]. 5.2

Polymer molecule in suspension

In this section, the conformational properties of the polymer molecule are investigated in a 2D case. We apply the SDPD method to the study of a polymer molecule in an infinite Brownian solvent medium under zero flow condition. Under these conditions, the flow is isotropic and theoretically predicted universal scaling laws for several polymer properties can be tested numerically. Conformational properties of a polymer chain, in particular deformation and

Fig. 4. Velocity probability distribution function of the colloidal particle compared with the analytical solution.

orientation, can be analyzed by monitoring the evolution of several tensorial P quantities as, for instance the gyration tensor: G ≡ (1/2N 2 ) i,j hrij rij i or the end-to-end tensor defined as R ≡ h(rN − r1 )(rN − r1 )i here, rij = rj − ri with ri being the position of the i−th monomer in the chain. The indices i, j run from 1 to N , the√total number of beads. Related quantities √ are the radius of gyration RG = trG and the end-to-end radius RE = trR. The effect of the number of monomers N on RG and RE is known to follow the analytical expressions RE = aE (N − 1)ν

RG = aG [(N 2 − 1)/N ]ν

(10)

where ν is called static factor exponent and, according to the Flory’s formula, it assumes the value ν ≈ 0.75 in two dimensions with aE and aG being suitable constants [15]. In order to extract the exponent ν, SDPD simulations have been carried out with five different chain lengths characterized by N = 20, 40, 60, 80, 100 beads. In all cases the time-averaged values of the gyration radius RG has been evaluated from several independent steady-state polymer configurations. Fig. 5 shows a log-log plot of the time-averaged RG versus N . Error bars are within point dimensions. The results can be fitted (dotted line in the figure) by a power-law with exponent ν = 0.76 ± 0.012 which is in good

Fig. 5. Scaling of the radius of gyration RG for several chain lengths corresponding to N = 20, 30, 40, 50, 60, 80, 100 beads. The dotted line represents the best fit consistent with the theory (RG ∝ N ν ) and gives a static exponent ν = 0.76±0.012.

agreement with theoretical results. It should be noticed that this way to evaluate ν is quite time consuming since simulations at large N are necessary in order to fit accurately the data in Fig. 5. An alternative way to extract ν is, instead of using the scaling law (10), by employing the static structure factor defined as: S(k) ≡

1 X hexp (−ik · rij )i. N i,j

(11)

In the limit of small wave vector |k|RG ≪ 1, the structure factor can be approximated by S(k) ≈ N (1 − k2 RG /3), while for |k|RG ≫ 1 holds S(k) ≈ 2N/k2 RG . The intermediate regime |k|RG ∼ 1 contains information about the intramolecular spatial correlations. In absence of external perturbation and close to equilibrium, S(k) is isotropic and therefore depends only on the magnitude of the wave vector k = |k|. S(k) probes therefore different length scales even for a single polymer and in the intermediate regime is shown to behave like S(k) ∝ k −1/ν

(12)

˜ Fig. 6. Normalized equilibrium static structure factor S(k) = S(k)/S(0) versus RG k corresponding to several chain lengths. All the curves collapse on a master ˜ line for 2 < RG k < 8 (scaling regime). In this region S(k) ∝ k−1/ν with ν = 0.75 (dotted line).

Fig.6 shows a log-log plot of S(k) vs. RG k. From this figure it is possible to see how curves evaluated from simulations with different chain lengths (N ) collapse on a single curve for 2 < RG k < 8, the slope of the linear region being −1/ν. The dotted line in the figure represents the theory with ν = 0.75 and shows very good agreement with the SDPD results.

6

Conclusions

In this work, a refined particle model for the description of mesoscopic complex flows has been discussed. The method is known as Smoothed Dissipative Particle Dynamics and represents a generalization of Smoothed Particle Hydrodynamics for micro-flows. Several advantages of the method over standard mesoscopic DPD techniques have been highlighted which include: (1) flexibility in the use of an arbitrary equation of state; (2) control of the transport coefficients and (3) physical determination of particle size. Furthermore, thermal fluctuations in SDPD depend on the particle volumes in such a way that

they can be neglected for sufficiently coarse fluids while they are automatically present whenever the physical dimension of the problem under study become small (i.e microfluidics conditions). This feature of the scheme makes SDPD a good candidate for the simulations of multiscale/multi-resolution phenomena. In order to validate the numerical method, applications in the area of microfluids have been considered. In particular, modelling of colloidal particle and macromolecule suspended in a Newtonian liquid have been discussed. Finally, simulations of these systems in Brownian and non-Brownian environments showed good agreement with the theory and/or previous numerical results.

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