doi: 10.1098/rsta.2011.0020, 2592-2599369 2011 Phil. Trans. R. Soc. A

 G. Gonnella, A. Lamura and A. Tiribocchi of lamellar fluidsThermal and hydrodynamic effects in the ordering  

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Phil. Trans. R. Soc. A (2011) 369, 2592–2599doi:10.1098/rsta.2011.0020

Thermal and hydrodynamic effects in theordering of lamellar fluids

BY G. GONNELLA1, A. LAMURA2,* AND A. TIRIBOCCHI1

1Dipartimento di Fisica, Università di Bari, and INFN, Sezione di Bari,Via Amendola 173, 70126 Bari, Italy

2Istituto Applicazioni Calcolo, CNR, Via Amendola 122/D, 70126 Bari, Italy

Phase separation in a complex fluid with lamellar order has been studied in thecase of cold thermal fronts propagating diffusively from external walls. The velocityhydrodynamic modes are taken into account by coupling the convection–diffusionequation for the order parameter to a generalized Navier–Stokes equation. The dynamicalequations are simulated by implementing a hybrid method based on a lattice Boltzmannalgorithm coupled to finite difference schemes. Simulations show that the ordering processoccurs with morphologies depending on the speed of the thermal fronts or, equivalently,on the value of the thermal conductivity x. At large values of x, as in instantaneousquenching, the system is frozen in entangled configurations at high viscosity while itconsists of grains with well-ordered lamellae at low viscosity. By decreasing the value ofx, a regime with very ordered lamellae parallel to the thermal fronts is found. At verylow values of x the preferred orientation is perpendicular to the walls in d = 2, whileperpendicular order is lost moving far from the walls in d = 3.

Keywords: thermal effects; hydrodynamics; lattice Boltzmann method; lamellar order

1. Introduction

Ordering in complex fluids can be a slow process. Thermodynamic equilibriumis often characterized by the presence of organized structures at mesoscopicscales [1]. However, the dynamical formation of topological defects can make itdifficult to reach the true equilibrium state [2,3]. As there are physically relevantproperties related to the degree of order in the system, it can be crucial to controlthe ordering process by proper driving.

In this paper, we will consider fluids with lamellar order, such as symmetricblock copolymer melts [4]. For recent applications of these systems in micro- andnano-technologies one can, for example, see the work by Man et al. [5], wherethe relevance of having long-range order is discussed. A transient macroscopicsample of a copolymer melt usually consists of different oriented lamellardomains (grains) with defects like dislocations and disclinations. The interplaybetween the mesoscopic structures and the local velocity field is essential forthe evolution of defects [6]. Hydrodynamic modes favour the disentanglement

*Author for correspondence (a.lamura@ba.iac.cnr.it).

One contribution of 25 to a Theme Issue ‘Discrete simulation of fluid dynamics: applications’.

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Thermohydrodynamics of lamellar fluids 2593

of intertwined patterns but simulations have shown how extended defectsbetween different grains remain at late times slowing down the dynamics ofordering [7,8].

Mechanical driving like shear can favour ordering as interfaces generally preferto be aligned with the flow [1]. In lamellar systems under shear, however, differentstable orientations are possible [9], and in the ordering process [10,11] the shearby itself can cause the creation of defects owing to the dilatation or compressionof the layers [12,13]. More complex phenomena like shear banding have also beenobserved [14].

A different controlling procedure could be thermal driving. Recently, in simplebinary mixtures, it has been shown that in thermally controlled quenches withcold fronts propagating into the system at finite speed, interfaces have a preferredorientation with respect to the fronts [15–17]. As a result, ordered patterns areobtained. Applications to polymer systems have been considered, for example, byVoit et al. [18]. In this work, we want to investigate how a thermally controlledquench could be used to produce well-ordered lamellar systems.

We consider a system described in the continuum by the Navier–Stokes andconvection–diffusion equations with the thermodynamics of the lamellar phaseencoded in a free-energy functional. The temperature evolves by following astandard diffusion equation with the system between two walls at a temperaturebelow the critical value. In order to solve the dynamical equations, we apply ahybrid method with a lattice Boltzmann algorithm for solving the Navier–Stokesequations and a finite difference scheme for the convection–diffusion equation.The model and the method will be described in §2 while the numerical resultswill be presented in §3. A few conclusions will complete the paper.

2. The model

We consider a binary mixture having total density n and density difference 4,the order parameter, between the two components. The equilibrium propertiesare described by the free energy [19]

F =∫ [

nT + nT ln(nT ) + a2

42 + b4

44 + k

2(V4)2 + c

2(V24)2

]dr. (2.1)

The first two terms depending on n and on the temperature T (the Boltzmannconstant is assumed to be one) give the ideal gas pressure, which does not affectthe phase behaviour. We consider b, c > 0 to ensure stability. The parameter acan be thought of as a reduced temperature

a = T − Tc

Tc, (2.2)

and is the only one depending on temperature of the terms depending on4. For temperatures above the critical value Tc the fluid is disordered. ForT < Tc and k > 0 there is the coexistence of two homogeneous phases withequilibrium values 4 = √−a/b. Negative values of k favour the formation ofinterfaces, while a reduction of k can induce a transition into a lamellar phase.By using a single-mode approximation with profiles like A sin k0x in the direction

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2594 G. Gonnella et al.

normal to lamellae, one finds a transition, when |a| = b, at a � −1.11k2/c withk0 = √−k/2c and A2 = 4(1 + k2/4cb)/3 [8]. The parameters of this model havebeen mapped on those of copolymer systems by Binder [20].

The evolution of the system is described by a set of coupled partial differentialequations:

vtn + va(nua) = 0, (2.3)

n(vtua + ubvbua) = −vbPab + vb

[h(vaub + vbua) +

(z − 2h

d

)dabvgug

](2.4)

and vt4 + va(4ua) = GV2m, (2.5)

where the Greek subscripts denote the Cartesian components, u is the localvelocity of the fluid, h the shear viscosity, z the bulk viscosity, d the spacedimension and G the mobility coefficient. We model the quench process by takingthe reduced temperature a (equation (2.2)) to be the solution of a diffusionequation vta = xV2a, where x is the thermal conductivity. The pressure tensor Pab

and the chemical potential m can be computed from the free-energy functional(2.1) and have the following forms [6]:

Pab ={nT + a

242 + 3b

444 − k

[4(V24) + 1

2(V4)2

]+ c

[4(V2)24 + 1

2(V24)2

+ vg4vg(V24)]}

dab + kva4vb4 − c[va4vb(V24) + vb4va(V24)] (2.6)

andm = a4 + b43 − kV24 + c(V2)24. (2.7)

Equations (2.3)–(2.5) are solved numerically, by using a mixed approach. Alattice Boltzmann scheme is used for the continuity and Navier–Stokes equations(2.3) and (2.4), and a finite-difference approach for equation (2.5) and forthe temperature diffusion equation. The full study is limited to bi-dimensionalsystems while the equations for 4 not coupled to hydrodynamics will be solvedalso in d = 3. In the case of a simple fluid [21–23], the lattice Boltzmann method(LBM) is defined in terms of a set of distribution functions, fi(r, t), located in eachlattice site r at each time t, and of a set of nine velocity vectors ei , defined on asquare lattice, having moduli |ei| = 0, c,

√2c with c = Dx/Dt, Dx and Dt being the

lattice and the time step, respectively. The distribution functions evolve accordingto a single relaxation time Boltzmann equation [24]

fi(r + eiDt, t + Dt) − fi(r, t) = −Dtt

[fi(r, t) − f eqi (r, t)] + DtFi(r, t), (2.8)

where t is the relaxation time, f eqi are the equilibrium distribution functions and

Fi are the forcing terms to be properly fixed.The physical quantities, that is, the total density n and the fluid momentum

nu, are determined by the relations

n =∑

i

fi and nu =∑

i

fiei + 12FDt, (2.9)

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Thermohydrodynamics of lamellar fluids 2595

with F the force density acting on the fluid. The equilibrium distribution functionsf eqi are given by the standard second-order expansion in the fluid velocity u ofthe Maxwell–Boltzmann distribution function [25].

The forcing term in equation (2.8) is expressed as a second-order expansion inthe lattice vector velocities [26]

Fi =(

1 − Dt2t

)ui

[ei − u

c2s

+ ei · uc4s

ei

]· F, (2.10)

where cs = c/√

3 is the speed of sound and ui = 4/9, 1/9, 1/36 in correspondenceof the lattice directions with |ei| = 0, c,

√2c, respectively. The force F has to have

the following expression:

Fb = va(nc2s dab − Pab) (2.11)

in order to recover equation (2.4), where it results

z = h = nc2s Dt

(t

Dt− 1

2

). (2.12)

The first term on the right-hand side of equation (2.11) allows one to cancelout the athermal ideal gas pressure nc2

s of the LBM. We verified that the Machnumber Ma = |u|max/cs, where |u|max is the maximum value of the fluid velocityduring evolution, stays always smaller than 0.1 so that the fluid is practicallyincompressible with n � 1.

The convection–diffusion equation (2.5) and the temperature equations aresolved by using a finite-difference scheme, which is described with details in thepaper by Tiribocchi et al. [27]. The function 4(r, t) is defined on the same sites ofthe LBM with the same space and time steps. We update 4 using an explicit Eulerscheme [28]. The spatial differential operators are calculated by a second-orderfinite-difference scheme.

Periodic boundary conditions are used along the x-direction (and y-direction ind = 3) and flat walls are placed at the lower and upper rows of the lattice along they-direction (z-direction in d = 3), where the temperature is kept at fixed value Tw.Moreover, neutral wetting condition for the order parameter 4 is implemented atwalls imposing that a · V4|wall = 0, a being an inward normal unit vector to theboundaries. To guarantee exact conservation of the order parameter 4 we alsorequire that a · V(V24)|wall = a · V[(V2)24]|wall = 0. The bounce-back rule [29,30]for the LBM is adopted at walls supplemented by the prescription for densityexact conservation introduced by Lamura & Gonnella [31].

3. Results and discussion

In this section, we will present the results of our simulations of equations (2.3)–(2.5). We used the set of parameters b = 0.1, k = −c = −0.03, Tc = 0.005, andG = 0.1 corresponding to the lamellar phase in the phase diagram. Lattices hadsizes 256 × 256 in d = 2 and 32 × 32 × 256 in d = 3. The system, initially, isin disordered symmetric states with 4 fluctuating around zero in the interval[−0.01, 0.01]. The initial temperature is above Tc with the walls at Tw = 0.9Tc.

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2596 G. Gonnella et al.

(a) (b) (c)

Figure 1. Configurations of the concentration field 4 without hydrodynamics with thermalconductivity x = (a) 10−2, (b) 10−1 and (c) +∞ at times t = 55.7 × 103, 11.7 × 103 and 1.95 × 103,respectively.

We first show in figure 1 typical configurations of systems with very highviscosity. In this case, we neglected the coupling with the velocity field by solvingequation (2.5) without the convective term. Lamellar domains start to formclose to the walls, where the temperature has become lower than the criticalvalue, with interfaces perpendicular to the walls in agreement with the imposedneutral wetting conditions. Then, following the temperature fronts, the regionswith phase separated domains will extend towards the middle of the system. Thepattern morphology, however, depends on the value of the thermal conductivityor, equivalently, on the speed of the temperature fronts. At very large x, thesystem remains frozen in intertwined noodle-like configurations without longrange order, like that in figure 1c, as observed in other simulations (e.g. [6]).However, by decreasing the value of x, in the interval [0.1, 1], in spite of theadopted boundary conditions, the interfaces tend to follow the temperature fronts,parallel to the walls, and exhibit only few defects. The two fronts separating theregions with ordered lamellae from the central disordered region will approacheach other joining at the later times of the simulation. By further decreasing x,interfaces show the tendency to keep the perpendicular orientation with respectto the walls. This phenomenology has to be compared with that observed insimple binary mixtures where at large x usual isotropic phase separation occurs(of course, in binary mixtures, there is no frustration and the average size ofdomains grows by a power law, e.g. [32]), while by lowering the values of thethermal conductivity one observes domains first parallel and then perpendicularto the cold walls [17].

Going to a lower viscosity (t = 5, h = 1.5; figure 2), at very large x, one findsthat hydrodynamics favours the disentanglement of the network pattern [6], butone still observes extended defects between grains of different oriented lamellae(e.g. the grain with vertical orientation on the top-left region of the systemopposed to the large horizontal underlying domain). One could see in systemslarger than those here considered that these extended defects would slow downthe ordering process at very late times [8]. At lower thermal conductivities, in therange [5 × 10−4, 5 × 10−3], one again finds that lamellae are well aligned with thethermal fronts. The velocity helps this ordering and, indeed, very ordered defect-free lamellae can be observed (compare the snapshots of figures 1b and 2b) in theregions close to the walls. In the central part, even if structures can be observed,

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Thermohydrodynamics of lamellar fluids 2597

(a) (b) (c)

Figure 2. Configurations of the concentration field 4 at low viscosity (h = 1.5) with thermalconductivity x = (a) 10−4, (b) 10−3 and (c) +∞ at times t = 144 × 104, 35.6 × 104 and 2.1 × 104,respectively.

(a)

(b)

(c)

Figure 3. Interface configurations in lamellar ordering without hydrodynamics with thermalconductivity x = (a) 10−2, (b) 1 and (c) 102 at times t = 78 × 102, 15 × 102 and 1.1 × 102,respectively. (Online version in colour.)

the local values of the field 4 are not in equilibrium, and further evolution willshow a completely ordered lamellar state. Finally, by further decreasing the valueof x, one sees in the snapshot of figure 2a that hydrodynamics clearly helps thetendency of the system to prefer the perpendicular orientation.

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2598 G. Gonnella et al.

We also studied three-dimensional lamellar systems without coupling to thevelocity field. At large values of x, the system evolves towards metastableconfigurations ordered only on short scales, similar to what occurs in two-dimensional systems, as can be seen in the snapshot of figure 3c. At lower valuesof x (0.1 ≤ x ≤ 10) one can see stacks of lamellae parallel to the thermal frontswith very few defects (snapshot of figure 3b), while, again similarly to the two-dimensional case of figure 1, perpendicular order is lost far from the walls at verylow values of x.

4. Conclusions

In this work, we have studied the ordering of a lamellar phase where thetemperature cold fronts diffusively move from external walls. Phase separationtherefore starts close to the walls but develops in the middle of the systems with aphenomenology depending on the speed of the thermal fronts. In an intermediaterange of values of thermal conductivity x, lamellae appear ordered and parallel tothe thermal fronts while a perpendicular orientation is preferred at very low valuesof x. Hydrodynamics favours these preferred orderings and one can conclude thatquenching with moving temperature fronts is effective in producing lamellar stateswith very few defects. A full analysis of the three-dimensional case will completethis study.

We acknowledge support from CASPUR for HPC grant 2010.

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