STOCHASTIC EULERIAN-LAGRANGIAN METHODS FORFLUID-STRUCTURE INTERACTIONS WITH THERMAL

FLUCTUATIONS AND SHEAR BOUNDARY CONDITIONS(PREPRINT ROUGH DRAFT: August 3, 2009)

PAUL J. ATZBERGER ∗

Abstract. Motivated by the study of rheological properties of complex fluids and soft materialsa general modeling and simulation framework is introduced. The approach allows for a consistenttreatment of the elastic mechanics of microstructures, hydrodynamic coupling, thermal fluctuations,and externally driven shear flows. In the modeling framework microstructures are represented in aLagrangian reference frame while momentum is accounted for in an Eulerian reference frame. Inorder to couple these disparate descriptions in a consistent manner general coupling conditions arederived. Thermal fluctuations are introduced into the formalism by deriving appropriate stochasticdriving fields consistent with principles from statistical mechanics. To allow for the study of thematerial shear response, shear stresses are introduced at the domain boundaries along with gener-alized periodic boundary conditions. Efficient stochastic numerical methods are formulated for theresulting system of stochastic partial differential equations. To demonstrate the applicability of themethodology, simulation results are presented for the shear response of a polymeric fluid, lipid vesiclefluid, and a gel-like material.

Key words. Statistical Mechanics, Complex Fluids, Soft Materials, Stochastic Eulerian La-grangian Methods, Stochastic Immersed Boundary Methods, Hydrodynamic Coupling, FENE, Vesi-cles, Gels.

PREPRINT NOTE:. This is a rough draft which is being actively checked andrevised. As such, the materials presented here may contain errors. Please feel free tosend any comments, errors, or suggestions to

atzberg@math.ucsb.edu

1. Introduction. Soft materials and complex fluids are comprised of microstruc-tures which have mechanics and interactions characterized by energy scales com-parable to thermal energy. This feature results in interesting bulk material prop-erties and phenomena which often depend sensitively on temperature and appliedstresses (4; 5; 13). Example materials include colloidal suspensions, foams, poly-meric fluids, surfactant solutions, lipid vesicles, and gels (11; 13; 21; 22; 27; 32; 33).Microstructures of such materials include flexible filaments, bubbles, colloidal parti-cles, lipid chains, and polymers. These microstructures are typically surrounded bya solvating fluid which further mediates interactions through solvation shells (25; 28)and hydrodynamic coupling (5; 8; 13). In addition, given the energy scales of themicrostructure mechanics and interactions, thermal fluctuations often play an impor-tant role both in microstructure organization and kinetics (8; 13; 32). A fundamentalchallenge concerning soft materials is to relate microstructure mechanics, interactions,and kinetics to bulk material properties.

For the study of soft materials we introduce a modeling and simulation approachwhich consistently accounts for microstructure elastic mechanics, hydrodynamic cou-pling, and thermal fluctuations. The modeling approach is based on a mixed Eulerianand Lagrangian description. The microstructure configurations are modeled in a La-grangian reference frame, while an Eulerian reference frame is used to account formomentum of the system. When coupling these disparate descriptions an important

∗University of California, Department of Mathematics , Santa Barbara, CA 93106; e-mail:atzberg@math.ucsb.edu; phone: 805-893-3239; Work supported by NSF DMS-0635535.

1

2 P.J. ATZBERGER

issue is to formulate methods which do not introduce artificial dissipation of energyor loss of momentum. These properties are especially important when introducingstochastic driving fields to account for thermal fluctuations. We shall discuss a gen-eral approach for developing such coupling schemes, focusing primarily on one suchrealization the Stochastic Immersed Boundary Method (3).

To facilitate studies of the rheological properties of soft materials we shall intro-duce methods to account for externally driven shear flows. To account for shearingof the material, we shall generalize the usual periodic boundary conditions so thatperiodic images are shifted relative to the unit cell. Our approach is based on theboundary conditions introduced for Molecular Dynamics simulations in (26), referredto as Lees-Edwards boundary conditions. These conditions present a number of chal-lenges in the context of numerically solving the hydrodynamic equations. We shalldevelop numerical methods which utilize jump conditions in the velocity field at do-main boundaries and utilize a change of variable to facilitate handling of the shiftedboundaries. Further issues arise when accounting for the thermal fluctuations. Forthe introduced discretizations we shall develop stochastic driving fields which yieldstochastic numerical methods which are consistent with principles from statisticalmechanics.

We shall consider primarily two physical regimes (i) the relaxation dynamics of thefluid modes are explicitly resolved, (ii) the fluid modes are treated as being in a quasi-steady-state ensemble. In the first regime we shall develop efficient computationalmethods for the generation of the corresponding fluctuating fields. In the second,to account for thermal fluctuations in the microstructure dynamics we shall developefficient stochastic numerical methods for the required correlations in the stochasticdriving terms.

As a demonstration of the proposed stochastic numerical methods simulations areperformed for specific applications, including (i) a polymeric fluid, (ii) a vesicle fluid,and (iii) a gel-like material. To relate microstructure interactions and kinetics to bulkmaterial properties we develop estimators for an associated macroscopic stress tensor.The estimators take into account n-body interactions in the microstructure mechanicsand the generalized boundary conditions. For the polymeric fluid this notion of stressis used to investigate the dependence of the shear viscosity and normal stresses onthe rate of shear. The vesicle fluid is subject to oscillating shear and simulationsare preformed to characterize the frequency response in terms of the elastic storagemodulus and viscous loss modulus over a wide range of frequencies. As a furtherdemonstration of the methods, the time dependent shear viscosity of a gel-like materialis studied through simulations.

The ability to simulate explicitly the microstructure mechanics, hydrodynamiccoupling, and thermal fluctuations provides an important link between small length-scale phenomena and bulk material properties. The presented framework and relatedstochastic numerical methods are expected to be applicable in the modeling and sim-ulation of a wide variety of soft materials and complex fluids.

2. Stochastic Eulerian-Lagrangian Modeling Approach. We shall use amixed Eulerian-Lagrangian description. The momentum of the entire system includ-ing the fluid and microstructures will be accounted for in an Eulerian reference frame.While microstructure deformations will be accounted for in a Lagrangian referenceframe. (see Figure (ref)). More precisely, we shall use

Dp(x, t)

Dt= ∇ · σ(x, t) + Λ(x, t) + g(x, t)(2.1)

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 3

∂X(q, t)

∂t= Γ(q, t) + Z(q, t)(2.2)

where p accounts for the momentum of the material occupying location x, Dp

Dt denotesthe material derivative, X(q, t) denotes the configuration of the material at time t asparameterized by q. Thermal fluctuations will be taken into account through thestochastic fields g and Z. We shall consider systems where the total energy is givenby

E[p,X] =

∫

1

2ρ−1(x)|p(x)|2dx + Φ[X],(2.3)

where ρ(x) = ρ[X](x) is the mass density of material occupying location x, and Φ isthe potential energy for a given configuration. This energy has the associated forceF = −δΦ/δX. In this notation, the local material stress is denoted by σ = σ[p,X,F]with the convention that σ accounts only for the dissipative stress contributions inthe system. The operators Λ, Γ couple the Eulerian and Lagrangian descriptions ofstate of the material. The operator Λ = Λ[X,F] accounts, as the material deforms, foradditional momentum introduced or lost through non-dissipative stresses and bodyforces. The operator Γ = Γ[p,X] determines from the momentum of the system therate of deformation of the material. The equations 2.2 and 2.1 may also be furthersubjected to kinematic constraints, such as incompressibility, as we shall discuss.

In order for the operators which couple the Eulerian and Lagrangian descriptionsto be physically consistent, the following should hold: (i) they should not introduceany loss or gain of energy, (ii) momentum should only change through forces actingwithin the system, and (iii) for thermally fluctuating systems all configurations Xshould be equally probable at statistical steady-state when Φ[X] = 0. More precisely,

∫

F(q) · Γ(q)dq =

∫

[

ρ−1(x)p(x)]

· Λ[F](x)dx(2.4)

∫

Ω

Λ[F](x)dx =

∫

F(q)dq(2.5)

∫

δΓ

δXdq = 0(2.6)

must hold for any realization of p and F. To simplify the discussion, it has beenassumed that the stress contributions denoted by σ are entirely dissipative and thatthere is no net in-flux of momentum from boundary stresses

∫

∂Ω σ(x) · ndx = 0.The condition 2.4 ensures the coupling operators conserve energy. The condition 2.5ensures that the total momentum change of the system is equal to the total force actingon the system. The condition 2.6 ensures that the uniform probability distribution isinvariant under the stochastic dynamics of X when the potential energy is constant,i.e. Φ = 0. The significance of this last condition is that the uniform distributioncorresponds to the marginal Boltzmann distribution in this case.

We now discuss how to account for thermal fluctuations using the stochastic fieldsg and Z. To simplify the discuss we shall assume that the dissipative processes canbe accounted for by a negative definite self-adjoint linear operator L in p, so that∇ · σ = Lp, and that conditions 2.4 - 2.6 are satisfied. With these assumptions,the thermal fluctuations can be accounted for using for g and Z Gaussian stochasticfields which are mean zero and δ-correlated in time (19; 29). The main issue thenbecomes to determine an appropriate spatial covariance structure for these stochastic

4 P.J. ATZBERGER

fields. By requiring that the Boltzmann distribution be invariant under the stochasticdynamics of equations 2.1 - 2.2, it is required that Z = 0, and that

G(x,y) = 〈g(x)g(y)〉 = −2kBTLρ(x)δ(x − y),(2.7)

see Appendix A. As we shall discuss, this operator is subject to further modificationwhen introducing kinematic constraints, such as incompressibility.

This gives a formulation of the basic Stochastic Eulerian-Lagrangian Method(SELM). We shall discuss in this paper practical approaches to discretizing the stochas-tic equations, efficient generation of the associated stochastic fields, and related com-putational methods to perform simulations. Similar formulations as equations 2.1 -2.2, with g = 0, Z = 0, are the starting point for the derivation of a wide varietyof computational approaches used for systems in which fluids interact with immersedbodies. These include Immersed Boundary Methods (3; 30), Immersed Finite Ele-ment Methods (38; 40), Arbitrary Lagrangian-Eulerian Methods (14; 15), SpectralElement Fluid Structure Methods (7), and Phase-Field Methods (31)

3. Semi-Discretization of the Momentum Equations, Microstructures,and the Eulerian-Lagrangian Coupling. We shall now consider semi-discretizationsof the SELM equations. The momentum equations will be spatially discretized on auniform mesh. The pm will denote the momentum at the mesh site indexed bym = (m1, m2, m3) and the composite vector of such values will be denoted by p. Thedeformation state which describes the microstructure configurations will be discretizedusing a finite number of degrees of freedom denoted by X[j] indexed by j = 0, 1, . . . , Mand the composite vector denoted by X. As an energy for this discretized system weshall use

E[p,X] =∑

m

1

2ρ−1m |pm|2∆xd + Φ(X)(3.1)

where ∆x is the mesh spacing and d is the number of dimensions. The semi-discretizedmomentum and deformation equations can be expressed as

Dp

Dt= Lp + Λ + g(3.2)

∂X[j](t)

∂t= Γ[j](3.3)

where D/Dt and L denote respectively the spatially discretized approximation of thematerial derivative and L. The p, Λ, g denote the composite vector of values on themesh and X[j], Γ[j] denote values associated with the jth configurational degree offreedom. We shall assume the discrete dissipative operator is symmetric L = LT andnegative semi-definite. For the coupling operators of the discretized equations thecorresponding consistency conditions 2.4-2.6 are

Γ[p]TF =(

ρ−1 : p)T

Λ[F]∆xd(3.4)∑

m

Λ[F]m∆xd =∑

j

F[j](3.5)

∑

j

∇X[j] · Γ = 0(3.6)

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 5

where[

ρ−1 : p]

m= ρ−1pm, and the superscript T denotes the matrix transpose.

The first condition ensures in the discretized system that the coupling preserves theenergy, the second that changes in momentum only occur from forces acting withinthe system, and the third that the uniform distribution for the configurations X isinvariant under the stochastic dynamics of equation 3.2-3.3 when the potential energyis constant, i.e. Φ = 0. We remark that when Λ and Γ are linear operators and ρ = ρ0

is constant, the energy condition 2.4 amounts to the coupling operators being adjoints,

Γ = ΛT ρ0∆xd.(3.7)

Provided conditions 3.4-3.6 are satisfied, the thermal fluctuations can be ac-counted for using a Gaussian stochastic field on the lattice. The appropriate co-variance structure can be determined by requiring invariance of the Boltzmann dis-tribution under the stochastic dynamics of equation 3.2-3.3. This yields for the semi-discrete system

G = 〈ggT 〉 = −2LC(3.8)

with Cm,n = ρmkBTδm,n/∆xd, where δm,n is the Kronecker δ-function, see Ap-pendix A. As we shall discuss, this covariance structure is subject to further modifi-cation when introducing kinematic constraints, such as incompressibility.

The Stochastic Immersed Boundary Method developed in (3) is one such realiza-tion of this SELM approach. In this case the coupling operators are given by

ΛIBF =

M∑

j=1

F[j](X(t))δa(xm − X[j](t))(3.9)

[ΓIBu][j]

=∑

m

δa(xm − X[j](t))um(t)∆xd.(3.10)

where ρ = ρ0, u = ρ−10 p, and δa is a special kernel function approximating the Dirac

δ-function. The conditions 3.4 and 3.5 can be readily verified to hold exactly for thesemi-discretized SIB method of (3). However, condition 3.6 only approximately holds.This follows since as ∆x → 0 we have

∑

j

∇X[j] · Γ =∑

m

−∇δa(xm − X[j])um∆xd →∫

−∇δa(y − X[j])u(y)dy(3.11)

=

∫

δa(y − X[j])∇ · u(y)dy = 0

where we used that the fluid is incompressible ∇ · u = 0. In practice, this hasthe tolerable consequence that the exhibited fluctuations deviate from Boltzmannstatistics up to discretization error (2; 3).

Other coupling approaches have been introduced in (cite). The Immersed FiniteElement yields another way to realize coupling operators for the proposed SELMapproach. In this case,... This allows for...(PJA: details future draft). The SpectralFluid Structure Methods also yield another way to realize coupling operators for theproposed SELM approach. In this case,...This allows for...(PJA: details future draft).

4. Soft Materials and Complex Fluids Subject to Shear. We now discusshow the SELM approach can be used for the study of rheological properties of softmaterials and complex fluids. We shall then discuss specific stochastic numerical

6 P.J. ATZBERGER

methods for performing simulations in practice. For the type of materials we shallconsider, it will be assumed that the solvent hydrodynamics is described well bythe constitutive laws of Newtonian fluids in the physical regime where the Reynoldsnumber is small. We shall also assume that the explicitly represented microstructuresoccupy only a relatively small volume fraction and are effectively density matchedwith the solvent fluid. In this regime, the momentum of the system will be accountedfor using the time-dependent Stokes equations

ρ∂u

∂t= µ∆u−∇p + Λ + g(4.1)

∇ · u = 0(4.2)

where u(x, t) is the local velocity of the fluid body at x in the Eulerian reference frame,ρ is the fluid density, µ is the dynamic viscosity, p is the pressure. This correspondsto σ = −pI + µ

(

∇u + ∇uT)

and p = ρu in equation 2.1.To introduce shear we shall generalize the usual periodic boundary conditions.

Our basic approach is motivated by the molecular dynamics methods introduced byLees-Edwards (16; 17; 26). In this work, molecules in the base unit cell have modifiedinteractions with molecules in periodic images. To simulate a bulk material undergo-ing a shear deformation at a given rate, the periodic images are treated as shifting intime relative to the unit cell, see Figure (ref). This has the effect of modifying boththe location of periodic images of molecules and their assigned velocities. This hassome advantages over other approaches, where an affine-like deformation is imposedon the entire material body (18; 23; 37). In contrast the Lees-Edwards approach theshear deformation is only imposed at the boundaries allowing within the unit cell forthe molecular interactions to determine the form of the shear deformation.

Motivated by this molecular dynamics approach we shall develop a correspondingmethodology for the SELM approach. For momentum accounted for by the time-dependent Stokes equations we introduce the following generalized periodic boundaryconditions

u(x, y, L, t) = u(x − vt, y, 0, t) + vex.(4.3)

For concreteness we shall consider the case where a shear is imposed in the z-directiongiving rise to velocities in the x-direction. The L is the side length of the periodic cellin the z-direction, v = Lγ is the velocity of the top face of the unit cell relative tothe bottom face, γ denotes the rate of shear deformation, and ej is the standard unitvector in the jth direction. The interactions between microstructures of the systemcan be readily handled in the same manner as in the molecular dynamics simulation.This is done by shifting the location of any microstructure of a periodic image involvedin an interaction.

While relatively conceptually straight-forward, these boundary conditions presentsignificant challenges in practice for the numerical discretization of the momentumequations. The conditions introduce both a jump discontinuity at periodic boundariesand a shift which potentially leads to misalignment of discretization nodes at thedomain boundaries. For commonly employed approaches such as spectral Fouriermethods the jump discontinuity results in a degradation of accuracy through theresulting Gibbs’ phenomena (cite). For uniform finite difference methods on the unitcell the mesh misalignment requires modified stencils or interpolations at the domainboundary. When considering the stochastic driving fields these issues are furthercompounded.

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 7

Fig. 4.1. Discretization Mesh

To address these issues, we shall develop discretization methods which utilize amoving coordinate frame which deforms with the unit cell, see Figure (ref). Let thevelocity field in this frame be denoted by w(q, t) := u(φ(q, t), t), where q = (q1, q2, q3)parameterizes the deformed unit cell. Let φ(q, t) = (q1 + q3γt, q2, q3) denote the mapfrom the moving coordinate frame to the fixed Eulerian coordinate frame x = φ(q).The time-dependent Stokes equations in the deforming coordinate frame become

dw(d)

dt= ρ−1µ [ed − δd,3γtex]T ∇2w(d) [ed − δd,3γtex] −∇p + F + J(4.4)

∇ ·w − eTz ∇w exγt = K(4.5)

where q = (q1, q2, q3) parameterizes the deformed unit cell, γ denotes the rate of theshear deformation, ei the standard basis vector in the i direction with i ∈ x, y, z.In the notation the parenthesized superscript denotes a vector component and δk,ℓ

denotes the Kronecker δ-function. We also use the notational convention

[

∇2w(d)]

i,j=

∂2w(d)

∂qi∂qj(4.6)

[∇w]d,j =∂w(d)

∂qj.(4.7)

In the equations, the terms J,K are introduced to account for the jump introducedby the boundary conditions 4.3. This allows in the new coordinate frame for use ofthe usual periodic boundary conditions

w(q1, q2, L, t) = w(q1, q2, 0, t).(4.8)

We now discuss a discretization for equations 4.4 and 4.5 and the correspondingsource terms J,K. The following central finite difference approximations will be used

∂w(d)

∂qi→ w(d)(q + ei) − w(d)(q − ei)

2∆x(4.9)

8 P.J. ATZBERGER

∂2w(d)

∂qi∂qj→ w(d)(q + ei + ej) − w(d)(q − ei + ej)

4∆x2(4.10)

− w(d)(q + ei − ej) − w(d)(q − ei − ej)

4∆x2, i 6= j

∂2w(d)

∂q2i

→ w(d)(q + ei) − 2w(d)(q) + w(d)(q − ei)

∆x2.(4.11)

These approximations are substituted into equations 4.6–4.7 to approximate the op-erators in equation 4.4–4.5.

An important issue when using such deforming reference frames is that the dis-cretization stencils may become excessively distorted. We shall avoid this issue byexploiting the periodic symmetry of the system in the x and y directions. This peri-odicity has the consequence that for any coordinate frame with s = Lγt > L there isanother coordinate frame with s < L which has aligned mesh sites, see Figure (ref).By adopting the convention that the coordinate frame with s < L is always used whenevaluating stencils the distortion is controlled.

To obtain approximations for the source terms J,K the discretization stencilsare applied at the shear boundaries of the unit cell. For any stencil weights involv-ing values at mesh sites which cross the boundary the modified image value is usedwm ± γLt. The contributions of the stencil weights multiplied by ±γLt are collectedover all boundary mesh sites to obtain the source terms J, K. This allows for theusual finite difference stencils to be used on the unit cell with regular periodic bound-ary conditions, which when including the source terms gives the equivalent result ofimposing the jump boundary condition 4.3. As we shall discuss, this formulation hasa number of advantages when numerically solving the discretized equations and whenintroducing thermal fluctuations.

The Stokes equations 4.4– 4.5 discretized in this manner on a uniform periodicmesh can be expressed as

∂w

∂t= L(t)w + F + J(4.12)

D(t)w = K(4.13)

where L(t) denotes the finite difference operator approximating the Laplacian in themoving coordinate frame, and D(t) the approximation of the Divergence operator inthe moving frame. The discretization L(t) can be shown to be symmetric and negativesemi-definite for each t.

An important property of L(t), D(t) is that for each time t the correspondingstencils are translation invariant with respect to lattice shifts of the mesh. This hasthe important consequence that the matrix representations are circulant and thereforediagonalizable by Fast Fourier Transforms (36). As a result, the incompressibilityconstraint can be handled using FFTs to obtain an exact projection method (10).This allows for the Stokes equations to be expressed as

∂w

∂t= ℘ [L(t)w + F + J](4.14)

where ℘ is the operator which projects to the null space of D(t). The incompressibilitycondition is then satisfied for all time provided D(t) · w(0) = K.

We shall discuss stochastic numerical methods for two particular physical regimes:(i) relaxation of the hydrodynamic modes of the system are explicitly resolved, (ii)

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 9

given the current configuration X the hydrodynamic modes are treated as havingrelaxed to statistical steady-state. We remark that the case of resolving the hydrody-namic relaxation of the system is amenable to stochastic numerical methods similarto those introduced in (3). We shall discuss this case briefly and then focus primarilyon new stochastic numerical methods for handling the second case.

5. Regime I : Resolution of Hydrodynamic Relaxation. We now discusshow the stochastic fields may be generated in practice. For this purpose we shallexpress the equation (ref) in differential form as

dw = ℘(t) [L(t)w + F + J] dt + QdBt(5.1)

where QdBt denotes the stochastic driving field accounting for thermal fluctuationscorresponding to g, Bt ∈ R

3N denotes the composite vector of a standard Brownianmotion process at each of the mesh sites. Throughout our discussion the stochasticdifferential equations will given the Ito interpretation (19).

Using 〈QdBtdBTt QT 〉 = QQT dt = Gdt, we see that Q denotes a matrix square-

root of the covariance of the stochastic driving field G = QQT . Given the discretiza-tions introduced in Section (ref) the dissipative operator L(t) depends on time, whichfrom equation (ref) has the consequence that G = −2LC may be time dependent,where L = ℘L.

In the discretized system the numerical stencils dependent on time, However, sincethe shear deformation is volume preserving the discretized sum introduced to modelthe kinetic energy of the discrete system is not required to depend on time. Sincewe made this choice, we have the important consequence that the Boltzmann equi-librium fluctuations of the velocity field w associated with this energy are stationary(independent of time).

To obtain an explicit form for G we need to compute C taking into account theincompressibility constraint (ref). The equilibrium covariance under these constraintsis given by

C =2

3

kBT

ρ∆xdI.(5.2)

The factor 2/3 arises from application in Fourier space of the projection operatorwhich equivalently enforces the incompressibility. This yields that the time dependentcovariance structure is of the form G(t) = −2L(t)C.

An important issue is whether this will indeed yield a consistent treatment ofthe thermal fluctuations so that the resulting stochastic dynamical system has therequired equilibrium fluctuations. We establish a Fluctuation-Dissipation principle forsuch time dependent systems in Appendix B, which shows the required consistency.In order for these relations to be useful in practice we must have efficient methods bywhich to generate the stochastic driving fields. We now discuss one such approach.

6. Generating the Stochastic Driving Field I. We now discuss one com-putational approach for generating the required stochastic fields. The equilibriumfluctuations can be expressed as C = αI so that condition (ref) can be expressed as

G(t) = −2α℘(t)L(t).(6.1)

To obtain an efficient method we shall utilize the specific properties of the op-erators C, L(t), and ℘(t) arising from the discretization introduced in Section (ref).

10 P.J. ATZBERGER

The first is that each of the operators corresponds to use of numerical stencils whichare translation invariant on the mesh. This has the important consequence that allof these operators are diagonalizable in the Fourier basis. This has the further im-portant consequence that all of these operators commute. We shall also use that ℘is a projection operator, so that ℘2 = ℘ and ℘ = ℘T . Finally, we have that thediscrete approximation of the Laplacian is symmetric negative semi-definite so it canbe factored as L(t) = −U(t)UT (t).

By using these important properties of the operators we can express the covarianceof the stochastic driving field as

G(t) =(√

2α℘U(t))(√

2α℘U(t))T

.(6.2)

In this form the required matrix square-root is readily obtained as Q(t) =√

2α℘U(t).Since the operators U(t) and ℘ are diagonalizable in Fourier space, the matrix actionof this operator on any vector can be computed in O(N log(N)) time using FFTs.In practice, this allows for the stochastic driving fields to be efficiently computed. Incontrast, a direct Cholesky factorization of G would have cost O(N3). The resultingfactors Q are not guaranteed to be sparse, so in general computing the matrix actionwould cost O(N2).

7. Regime II : Under-resolution of Hydrodynamic Relaxation (Quasi-Steady-State Limit). For many problems the equations of motion can be simplifiedby exploiting a separation of time-scales between the time-scale on which the hydrody-namic modes relax to a statistical steady-state and the time-scale associated with themotion of the microstructures. In this case the fluid equations can be approximatedby

w = −℘L(t)−1 [Λ + J] + a.(7.1)

where we have introduced a to account for the thermal fluctuations in this regime. Weshall refer to this as the Quasi-Steady-State Stokes approximation. Using this in 3.2 weobtain the following closed system of equations for the motion of the microstructures

dX(t)

dt= HSELM(t) [F] + J + A(7.2)

where

HSELM(t) = −Γ℘L(t)−1Λ(7.3)

J = −Γ℘L(t)−1J.(7.4)

and A accounts for the thermal fluctuations. We shall refer to HSELM as the hydro-dynamic coupling tensor.

In this regime the thermal fluctuations arise from the hydrodynamic modes whichare relaxed to statistical steady-state. A key challenge is to determine the appropriatestatistics of A which accounts for these time integrated thermal fluctuations in themicrostructure equations. For this purpose we shall rewrite equation 7.2 in differentialform as

dX(t) = HSELM(t) [F] dt + R(t)dBt(7.5)

neglecting for the moment J, and representing the contributions of A by R(t)dBt. Weshall derive the covariance structure S(t) = R(t)R(t)T by requiring consistency with

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 11

the principle of Detailed-Balance of statistical mechanics (34). The Fokker-Planckequation associated with equation 7.5 is

∂Ψ(X, t)

∂t= −∇ · J(7.6)

J = HSELM(t)FΨ − 1

2S(t)∇XΨ.(7.7)

The equilibrium fluctuations of the system are required to have the Boltzmann dis-tribution

ΨBD(X) =1

Zexp (−Φ(X)/kBT )(7.8)

where Z is a normalization constant which ensure the distribution integrates toone (34). Substituting this above gives

J =

(

HSELM(t) − 1

2kBTS(t)

)

FΨBD.(7.9)

where F = −∇XΦ. The principle of Detailed-Balance requires at thermodynamicequilibrium that J = 0, which must hold for all X and F. This requires that

S(t) = 2kBTHSELM(t).(7.10)

In order for S(t) to correspond to a real-valued stochastic driving term this requiresthat the hydrodynamic coupling tensor HSELM(t) must be symmetric and positivesemi-definite. In this case that Λ and Γ are linear operators this is ensured by condi-tion 3.4, which from expression 7.3 gives

qT HSELM(t)q = −vT(

℘L(t)−1)

v∆xd ≥ 0,(7.11)

where v = ΓTq. We have also used that L(t) is symmetric negative semi-definite and℘ is a projection operator. We now discuss one approach to generating the stochasticdriving terms with the required covariance structure S(t) in practice.

7.1. Generating the Stochastic Driving Field II. A significant challengein practice for such methods is to generate the Gaussian stochastic driving termswith the required covariance structure. A commonly used approach is to generate avariate with uncorrelated standard Gaussian components ξ and set A = R(t)ξ. Theresulting variate A then has covariance 〈AAT 〉 = R(t)〈ξξT 〉R(t)T = R(t)R(t)T =S(t). However, to carry this out in practice encounters two challenges (i) given S(t) thefactor R(t) must be determined, (ii) the matrix-vector multiplication R(t)ξ must becarried out. For (i) the Cholesky algorithm is typically used which has a computationalcost of O(M3), where M is the number of microstructure degrees of freedom. For (ii)the resulting factors R(t) are generally not sparse incurring a computational cost ofO(M2).

We shall discuss an alternative approach for SELM methods when the couplingoperators Λ and Γ are linear. In this case

HSELM(t) = −Γ℘L(t)−1ΓT ∆xd(7.12)

by condition 3.4. Using properties of the operators discussed in Section 4 we canexpress the hydrodynamic coupling tensor as

HSELM(t) =(

Γ(t)V (t)∆xd/2)(

Γ(t)V (t)∆xd/2)T

.(7.13)

12 P.J. ATZBERGER

We have used that the operators L(t) and ℘ commute and since ℘ is an exact pro-jection that ℘ = ℘T , ℘ = ℘2. Since L(t) is symmetric we can factor −℘L−1(t) =V (t)V (t)T , with V (t) readily obtained since L(t) and ℘ are diagonalizable in theFourier basis. From equations 7.13 and 7.10 we have the required factor of the covari-ance S(t) = R(t)R(t)T with

R(t) =(

2kBT∆xd)1/2

Γ(t)V (t).(7.14)

This expression for the factor can be used to compute the required Gaussian stochasticdriving term A with a computational cost of O(N log(N) + M), where N is the totalnumber of mesh sites in the momentum discretization and we assumed that the actionof Γ can be computed with a cost of O(M) and that M < N .

In this case the underlying discretization mesh of the momentum equations isutilized. By using FFTs the action of R(t)ξ can be computed with a computationalcost of O(N log(N)). This is accomplished by generating Gaussian random variatescorresponding to the Fourier coefficients of uncorrelated standard Gaussian randomvariates ξ on the mesh. Since V (t) is diagonal in the Fourier basis computing theaction V (t)ξ has only computational cost of O(N log(N)). Assuming the operatorΓ(t) can be computed with computational cost of O(M). The scalar multiplicationalso incurs a cost computational cost of O(M). When there is a sufficient number ofmicrostructure degrees of freedom this allows for a significantly more efficient gener-ation of the stochastic driving terms compared to the traditional approach based onCholesky factorization and matrix-vector multiplication.

7.1.1. Effective Hydrodynamic Coupling Tensor : HSELM. We now discussan approach to analyzing the effective hydrodynamic coupling tensors HSELM whichmay appear in the SELM approach. From equation 7.3 many types of hydrodynamiccoupling tensors are possible depending on the choice of the coupling operators Λ andΓ. For concreteness we shall discuss the specific case corresponding to the StochasticImmersed Boundary Method (SIB) (3; 30). In the case of the SIB method the specificcoupling operators Λ and Γ are given by 3.9 and 3.10. From equation 7.3 the effectivehydrodynamic coupling tensor is given by

(7.15)

[HIB(t) [F]][j]

= −∑

m

δa(xm − X[j](t))

℘L(t)−1

M∑

j=1

F[j]δa(xm − X[j](t))

m

∆xd.

In the notation, the superscript [·][j] denotes the components of the composite vectorassociated with the jth microstructure degree of freedom. The [·]m denotes the vectorcomponents associated with the mth mesh site. An analysis of variants of this tensorfor point particles and slender bodies was carried-out in (1; 9).

Since HIB is linear in the microstructure forces without loss of generality we canconsider the case of only two microstructure degrees of freedom. We denote these asX[1], X[2] and the displacement vector by z = X[2] − X[1]. In making comparisonswith other hydrodynamic coupling tensors we shall find it helpful to make use ofapproximate symmetries satisfied by HIB. From equation 7.15, HIB depends on z up toa shift of X[1] relative to the nearest mesh site, and is similarly rotationally symmetryabout the axis of z. This allows the tensor components for all configurations to berelated to a canonical configuration with z = (z1, 0, 0). For any configuration this

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 13

Fig. 7.1. Comparison of the Immersed Boundary Method Effective Hydrodynamic CouplingTensor HIB with Brownian-Stokesian Dynamics HBSD and Rotne-Prager-Yamakawa HRPY. Thehydrodynamic coupling tensor for displacement r = |z| are shown with the the parallel directioncomponents (circles) and perpendicular direction components (squares). For two particles subject toforce the velocity field corresponding to the Rotne-Prager-Yamakawa Hydrodynamic Tensor is shownon the right. The lighter shaded regions indicate a larger magnitude in the velocity.

is accomplished by introducing the rotation matrix U so that Uz = (z1, 0, 0) andconsidering H = UHIBUT . In our comparisons we shall consider HIB = 〈H〉, wherethe average is taken over all rotations and shifts with respect to the nearest mesh site.

In practice, to numerically compute HIB we shall sample random configurationsof X[1] and X[2]. A useful expression for the tensor components is Hij = eT

i Hej =eT

i v. In this notation, ek are the standard basis vectors in direction k and v is themicrostructure velocity. For a computational implementation of the SELM method,this can be used by applying the force ej to the microstructure degrees of freedomand measuring the components of the realized microstructure velocities v.

When using such SELM approaches the hydrodynamic coupling tensor has fea-tures which depend on the the discretization of the momentum equations and mi-crostructures and a dependence on the specific choice of coupling operators. Forthe specific choice of the IB coupling operators we now discuss how the effective hy-drodynamic coupling tensor compares with other widely used approaches. We shallconsider two specific approaches, the Brownian-Stokesian Hydrodynamics of (8) whichhas tensor

HBSD(z) =2

6πηa

[

I − 3

4

a

r

(

I +zzT

r2

)]

.

and Rotne-Prager-Yamakawa Hydrodynamics of (35; 39) which has tensor

HRPY(z) =2

6πηa

I − 3

4

a

r

(

1 + 2a2

3r2

)

I +(

1 − 2a2

r2

)

zzT

r2 , for r ≥ 2a

r2a

[

(

83 − 3r

4a

)

I + r4a

zzT

r2

]

, for r < 2a

.

14 P.J. ATZBERGER

10−3

10−2

10−1

100

101

10−5

10−4

10−3

10−2

\lambda_H \dot\gamma

oute

rPro

dQQ

/ K

_B T

\lam

bda_

H \d

ot\

gam

ma

OuterProd QQ compIndex = 3

10−3

10−2

10−1

100

101

102

10−5

10−4

\lambda_H \dot\gamma<|

Q|2 >

/ KB T

λH

<|Q|2>

Fig. 8.1. The second moments of the extension vector are shown as the shear rate is varied.The second moment matrix is M = 〈zz〉. On the left is shown the off diagonal entries M1,3. Fordimers with a harmonic linear force extension law this component is related closely to the shearviscosity. On the right is shown the variance of the extension vector Trace[M ].

where η denotes the dynamic fluid viscosity, and a denotes the effective particle sizein terms of the radius of a sphere.

In Figure 7.1 these tensors are compared with HIB. It is found that the effectivehydrodynamic coupling tensor of the Immersed Boundary Method agrees well withboth of the tensors in the far-field r ≫ a. An important finding is that in the near-fieldHIB shows very close agreement to HRPY in the near-field limit.

8. Applications. We now discuss applications of the proposed stochastic nu-merical methods. These include (i) rheological study of the shear rate dependence ofthe shear viscosity in a FENE polymeric fluid, (ii) frequency response of the elasticstorage and viscous loss moduli of a lipid vesicle fluid, (iii) rheological response overtime of a gel-like material subject to shear.

8.1. Application I: Complex Fluid of Finite Extensible Non-linear Elas-tic (FENE) Dimers. As a demonstration of the proposed computational method-ology we shall consider a fluid with microstructures consisting of elastic polymers Weshall model the polymers as elastic dimers which have the potential energy

V (r) =1

2Hr2

0 log

(

1 −(

r

r0

)2)

(8.1)

where r denotes the length of extension of the dimer and r0 denotes the maximumpermitted extension length (6). The configuration of each dimer will be representedusing two degrees of freedom X(1), X(2).

The effective macroscopic stress tensor contributions of the fluid arising from thepolymeric microstructures will be estimated using the Irving-Kirkwood formula

σp =⟨

FzT⟩

(8.2)

where 〈·〉 denotes the shear induced non-equilibrium steady-state ensemble average,F is the particle force acting on X[1], and z the dimer extension vector.

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 15

10−3

10−2

10−1

100

101

100

101

102

\lambda_H \dot\gamma

η p /

KB T

λH

Shear Viscosity

10−3

10−2

10−1

100

101

100

101

102

103

104

\lambda_H \dot\gamma

Ψ1 /

KB T

λH2

First Normal Stress Coeff.

Fig. 8.2. Rheological properties of the FENE polymeric fluid are shown as the rate of shear isvaried. The shear viscosity is shown on the left and the first normal stress difference is shown onthe right.

Parameter DescriptionN Number of mesh points in each direction.∆x Mesh spacing.L Domain size in each direction.T Temperature.kB Boltzmann’s constant.µ Dynamic viscosity of the solvent fluid.ρ Mass density of the solvent fluid.H Bond stiffness.r0 Maximum permissible bond extension.γs Stokesian drag of a particle.λH Relaxation time-scale of FENE dimer, λH = γs/4H .γ0 Shear rate amplitude.γ0 Strain rate amplitude.a Effective radius of particle estimated via Stokes drag.

Table 8.1

Description of the parameters used in simulations of the FENE polymeric fluid.

With this notion of macroscopic stress the shear viscosity ηp and first normalstress coefficient Ψ1 are given by

ηp =σ(s,v)

γ(8.3)

Ψ1 =σ(s,s) − σ(v,v)

γ2(8.4)

where s is the direction of shear, v is the direction of the fluid velocity, and thesuperscript indicates the tensor component (5; 6).

Simulations are performed using the SELM method in the regime where the hy-drodynamic modes are relaxed to statistical steady-state. For Λ and Γ the couplingtensor 3.9 and 3.10 are used. In Figure 8.1 the moments of the extension vector z are

16 P.J. ATZBERGER

shown as the shear rate is increased. As the shear rate increases the dimer approachesthe maximal extension permitted by equation8.1, which results in a non-linear increasein the force of the dimer. This has the consequence that the rheological propertiesare shear rate dependent. For the FENE polymeric fluid the SELM simulations showclear shear thinning phenomena, as is expected in such fluids, see Figure 8.2.

8.2. Application II: Polymerized Lipid Vesicle Fluid. As a further demon-stration of the applicability of the proposed methodology we show how the stochasticnumerical methods can be used to investigate the material properties of a complexfluid with polymerized vesicle microstructures. In particular, we discuss show howthe methods can be used to compute the response of the complex fluids over a widerange of frequencies.

We now discuss the model used for the polymerized vesicles. The vesicle surfaceis approximated by a triangulated mesh of control points subject to the followingpotential energy

E[X] = E1[X] + E2[X](8.5)

E1[X] =K1

2

∑

(i,j)∈N1

(rij − ℓij)2(8.6)

E2[X] =K2

2

∑

(i,j,k)∈N2

|τ ij − τ jk|2 .(8.7)

where the vesicle configuration is represented by X, the composite vector of controlpoint locations. We shall use the convention that X[j] denotes the location of the jth

control point.The first energy term E1 accounts for stretching of the membrane surface and is

computed by summing over all local two body interactions N1 defined by the topol-ogy of the triangular lattice. The energy penalizes the deviation of the distancerij = |X[i] − X[j]| between points with index i and j with a preferred distance ℓij .For example, the preferred distances ℓij can be defined by the geometry of a spher-ical reference configuration for the vesicle. To ensure the two body interactions arerepresented by a unique index in N1 we adopt the convention that we always havei < j.

The second energy term E2 accounts for curvature of the membrane surface andis computed by summing over all local three body interactions N2 defined by thetopology of the triangular lattice. The energy penalizes the deviations of the tangentvectors τ ij = (X[i] −X[j])/rij and τ jk = (X[j] −X[k])/rjk from aligning in the samedirection. In the set of indices in N2 the tacit assumption is made that the point withindex j is always adjacent to both i and k. To ensure the three body interactions arerepresented by a unique index in N1 we adopt the convention that we always havei < k.

To obtain a triangular mesh to capture the shape of vesicles having a sphericalgeometry we start with an icosahedral which circumscribes a sphere of a given radius.Initially we consider the faces of the icosahedral which defines a triangular mesh. Toobtain a mesh which better approximates the sphere we bisect the three edges ofeach triangular face to obtain four sub-triangles. The newly introduced vertices areprojected radially outward to the surface of the sphere. The process is then repeatedrecursively to obtain further refinements of the mesh. This yields a high quality meshfor spherical geometries, see Figure 8.3 which shows vesicles represented by meshesobtained using two levels of recursive refinement.

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 17

Fig. 8.3. Simulations results showing the vesicle response when subject to an oscillating shearflow. At low frequency the vesicle shape distortion is small and is masked by thermal fluctuations. Atlow frequency the vesicle membrane stresses equilibrate to a good approximation with the bulk shearstresses, as illustrated in the plot of σxz . At high frequency the vesicle shape is visibly distorted andthe membrane stresses do not have time to equilibrate with the bulk shear stresses, as illustrated bythe configurations for phase θ = 1.6, 0.4 and the plot of σxz. For the vesicle configurations shown,the low frequency response corresponds to ω = 3.9294 × 10−3ns−1, γ = 1.9647 × 10−3ns−1, σ0 =3.7114×108amu ·nm−1 ·ns−2, and the high frequency response corresponds to ω = 1.2426×102ns−1,γ = 6.2129 × 101ns−1, σ0 = 4.6314 × 1010amu · nm−1 · ns−2. The phase θ = ωt is reported in therange [0, 2π). For additional parameters used in the simulations see Table 8.2 and 8.3.

We discuss how the methodology can be used to investigate the frequency re-sponse of the complex vesicle fluid. We shall consider the dilute regime and in ourmodel the case of a single vesicle subject to an oscillating shear flow. The shear flowwill be induced using the Lees-Edwards boundary conditions of the domain with atime varying shear rate given by γ = γ0 cos(ωt). We shall compute an estimate of the

effective stress tensor of the complex fluid by using σℓ,z = σ(2)ℓ,z + σ

(3)ℓ,z , where the first

and second term account for the contributions of the two-body and three-body inter-actions respectively. The stress contributions of the n-body interactions are computed

18 P.J. ATZBERGER

10−4

10−2

100

102

104

105

106

107

108

109

ωG

’’(ω

)

Viscous Loss Modulus

10−4

10−2

100

102

104

105

106

107

108

109

ω

G’(

ω)

Elastic Storage Modulus

Fig. 8.4. Frequency response of the dynamic complex modulus of the vesicle fluid subject toan oscillating shear flow. Throughout the simulations the total strain was held fixed to be half thedomain length, γ0 = 1

2L. For a description of the parameters and values used in the simulations,

see Table 8.2 and 8.3.

using

σ(n)ℓ,z =

1

L

⟨

∫ b

a

Λ(n)ℓ,z (ζ)dζ

⟩

(8.8)

where L = b − a is the length of the domain in the z-direction, 〈·〉 denotes averaging

over the ensemble, and Λ(n)ℓ,z denotes the microscopic stress arising from the n-body

interactions. The microscopic stress is defined as

Λ(n)ℓ,z (ζ) =

1

A

∑

q∈Qn

n−1∑

k=1

k∑

j=1

f(ℓ)q,j

k∏

j=1

H(ζ − x(z)qj

)n∏

j=k+1

H(x(z)qj

− ζ)(8.9)

where H(ξ) is the Heaviside function which is one if ξ > 0 and zero otherwise, Qn

is the set of n-tuple indices q = (q1, . . . , qn) which are permissible for the n-bodyinteractions, fq,j denotes the force acting on the jth particle of the interaction, andxqj

denotes the jth particle involved in the interaction. The cross-sectional area ofthe domain is denoted by A. This expression corresponds to a sum over all the forcesexerted by particles of the material above the cross-section at ζ = z on the particlesof the material below. For a derivation of this expression and for details of howwe compute the stress efficiently in practice in the case of two-body and three-bodyinteractions, see Appendix C.

For many materials the response of the stress component τxz(t) is linear to agood approximation in the bulk stress and strain over a wide range of frequen-cies provided the stress and strain amplitudes are sufficiently small (32). As ameasure of the material response we shall consider the dynamic complex modulusG(ω) = G′(ω) + iG′′(ω), whose components are defined from experimental data asthe best least-squares fit of the periodic stress component τxz(t) by the functiong(t) = G′(ω)γ0 cos(ωt) + G′′(ω)γ0 sin(ωt). This offers one characterization of theresponse of the material to oscillating bulk shear stresses and strains at differentfrequencies ω.

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 19

Parameter DescriptionN Number of mesh points in each direction.∆x Mesh spacing.L Domain size in each direction.T Temperature.kB Boltzmann’s constant.µ Dynamic viscosity of the solvent fluid.ρ Mass density of the solvent fluid.K1 Vesicle bond stiffness.K2 Vesicle bending stiffness.D Vesicle diameter.ω Frequency of oscillating shearing motion.θ Phase of the oscillatory motion, θ = ωt.γ Shear rate.γ0 Shear rate amplitude.γ Strain rate.γ0 Strain rate amplitude.

Table 8.2

Description of the parameters used in simulations of the vesicle fluid.

To estimate the dynamic complex modulus in practice the least squares fit isperformed for τxz(t) over the entire stochastic trajectory of the simulations (after sometransient period). Throughout our discuss we shall refer to θ = ωt as the phase ofthe periodic response. In our simulations the maximum strain each period was chosento always be half the periodic unit cell domain in the x-direction, corresponding tostrain amplitude γ0 = 1

2 . This was achieved by adjusting the shear rate amplitudefor each frequency using γ0 = γ0ω.

Simulations were performed with the SELM approach in the regime where thehydrodynamic modes were treated as relaxed to statistical steady-state. The specificcoupling operators Λ and Γ from 3.9 and 3.10 were used. The simulation resultsof the complex modulus response of the vesicle when subject to a wide range offrequencies is shown in Figure 8.4 and Figure 8.3. It was found that at low frequencythe vesicle shape distortion is small and masked by thermal fluctuations. At lowfrequency the vesicle membrane stresses equilibrate to a good approximation with thebulk shear stresses, as illustrated in the plot of σxz . It was found at high frequencythe vesicle shape is visibly distorted and the membrane stresses do not have timeto equilibrate with the bulk shear stresses, as illustrated by the configurations forphase θ = 1.6, 0.4 and the plot of σxz . For the vesicle configurations shown, the lowfrequency response corresponds to ω = 3.9294 × 10−3ns−1, γ = 1.9647 × 10−3ns−1,σ0 = 3.7114× 108amu · nm−1 · ns−2, and the high frequency response corresponds toω = 1.2426 × 102ns−1, γ = 6.2129 × 101ns−1, σ0 = 4.6314 × 1010amu · nm−1 · ns−2.The phase θ = ωt is reported in the range [0, 2π). A description of the parametersand specific values used in the simulations can be found in Table 8.2 and 8.3.

20 P.J. ATZBERGER

Parameter ValueN 27∆x 7.5 nmL 2.025× 102 nmT 300 KkB 8.3145× 103 nm2 · amu · ns−2 · K−1

µ 6.0221× 105 amu · cm−1 · ns−1

ρ 6.0221× 102 amu · nm−3

K1 2.2449× 107 amu · ns−2

K2 8.9796× 107

D 50 nmTable 8.3

Fixed values of the parameters used in simulations of the vesicle fluid.

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 21

8.3. Application III: Rheology of a Gel-like Material.

22 P.J. ATZBERGER

Fig. 8.5. Study of the thixotropy of a gel-like material. Shows the configuration of the mi-crostructures at three times (i) No shear yet applied, (ii) intermediate time corresponding to peakshear stress, (iii) long times corresponding to the relaxed shear stress. For the specific physicalparameters and times used in these simulation see Table (ref).

0 500 1000 15000

1

2

3

4

5

6x 10

9

time (ns)

shea

r st

ress

Thixotropy Study

Fig. 8.6. Study of the thixotropy of a gel-like material. At time zero the material has weak bondsbetween short polymeric strands. Under the shear stress the bonds are strained ultimately breakingand the polymers stretch and align with the flow field. This microstructure reordering is reflected inthe shear viscosity of the material as a function of time. For the specific physical parameters usedin these simulation see Table (ref).

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 23

9. Conclusions.

10. Acknowledgements. The author P.J.A. acknowledges support from re-search grant NSF DMS-0635535. We would especially like to thank Aleksandar Donevand Tony Ladd for stimulating conversations about this work. This paper is dedi-cated in memorial to Tom Bringley, who had a passion for life and whose academicpublications continue to inspire.

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Appendix A. Invariance of the Boltzmann Distribution under SELMStochastic Dynamics.

The probability distribution of the stochastic equations 2.1-2.2 are governed bythe Fokker-Planck equation

∂Ψ

∂t= −∇ · J(A.1)

with the probability flux given by

J =

[

LpΨ + ΛΨ − 12G∇pΨ

ΓΨ − 12W∇XΨ

]

(A.2)

where G, W are the covariance operators associated with g and Z. We remarkthat for the present purposes our discuss will only be formal since the SPDEs areinfinite dimensional and full rigor would require a more involved technical treatment,see (12; 19; 29). For the systems we shall consider, the Boltzmann distribution hasthe form ΨBD = 1

Z exp [−E[p,X]/kBT ], where Z is a normalization constant so thatΨBD integrates to one (34). The requirement that this distribution is invariant underthe stochastic dynamics of 2.1-2.2 is equivalent to ∇ · J = 0. This requires

∇ · J = A1 + A2 + ∇ ·A3 = 0(A.3)

A1 = (Λ · ∇pE + Γ · ∇XE) (−kBT )−1Ψ(A.4)

A2 = (∇p · Λ + ∇X · Γ)Ψ(A.5)

A3 =

(

Lp +G∇pE + W∇XE

2kBT

)

Ψ.(A.6)

For the energy given by equation 2.3 we have

∇pE = ρ−1p(A.7)

∇XE = ∇XΦ = −F(A.8)

where the force for the configuration is denoted by F.Now we can derive conditions for the coupling operators by requiring that A1 =

A2 = 0 for all possible values of p and F. The requirement that A1 = 0 corresponds tothe energy being conserved under the dynamics of equations 2.1-2.2 when g = Z = 0and σ = 0. For these dynamics the energy satisfies dE/dt = Λ · ∇pE + Γ · ∇XE = 0.By using the functions representing the variational derivatives (20) given in A.7-A.8we have Λ · ∇pE =

∫

Λρ−1pdx and Γ · ∇XE =∫

−ΓFdq. By substituting theseexpressions into A.4, we obtain from A1 = 0 the condition 2.4.

The requirement that A2 = 0 requires that ∇X · Γ = 0, since we have assumedthat Λ = Λ[X,F]. This condition ensures that the dynamical flow in configurationspace defined by Γ is volume preserving. For the dynamics when g = Z = 0, σ = 0,and E = 0 this condition is equivalent to requiring that the uniform distributionis invariant under the dynamics. The condition 2.6 follows by using the function

26 P.J. ATZBERGER

representing the variational derivatives (20) appearing in the divergence operationwhich gives ∇X · Γ =

∫

δΓ/δXdq.The requirement that A3 = 0 requires from equation A.7-A.8 that

Lp +[

(Gρ−1p − WF)/2kBT]

= 0 for any p and F. This requirement correspondsto the condition of Detailed-Balance of statistical mechanics (34). Since p and F arearbitrary, this requires that W = 0 so that Z = 0. This also requires that G = −2LCwith C = kBTρ−1I, where I is the identity operator. This yields condition 2.7 Fromthe form of the energy in 2.3 and the Boltzmann distribution we see the equilibriumfluctuations of p are Gaussian with covariance C. This condition relates the equi-librium fluctuations to the dissipative operator of the system and is a variant of theFluctuation-Dissipation Principle of statistical mechanics (34). This shows that pro-vided the coupling operators and stochastic fields satisfy conditions 2.4, 2.6, and 2.7,the Boltzmann distribution is invariant under the SELM stochastic dynamics.

For the discretized equations, we now derive conditions 3.4, 3.6, and 3.8. Thecalculations follow similarly to the case above so we only state the basic features ofthe derivation. For the discretized equations the probability flux is given by

J =

[

LpΨ + ΛΨ − 12G∇pΨ

ΓΨ

]

(A.9)

where p and X are now finite dimensional vectors. The Boltzmann distribution nowuses the energy of the discrete system

E[p,X] =∑

m

1

2ρ−1m |pm|2∆xd + Φ(X)(A.10)

with

∇pE = ρ−1p∆xd(A.11)

∇XE = ∇XΦ = −F.(A.12)

Substituting these expressions in A.3 - A.6 and reasoning as above yields the condi-tions 3.4, 3.6, and 3.8.

Appendix B. A Fluctuation-Dissipation Principle for Time-DependentOperators.

Appendix C. Derivation of Effective Macroscopic Stress Tensor Esti-mator for Periodic and Lees-Edwards Domains.

In order to estimate from the microscopic model the contributions of the mi-crostructures to the overall effective macroscopic stress of the material we follow anapproach similar to the Irving-Kirkwood-Kramer estimators (13; 24). In these ap-proaches, the central idea for a specified normal direction is consider the correspond-ing plane which cuts through the sample and to define the microscopic stress as theforce transmitted across this plane per unit area. To estimate the effective macro-scopic stress this microscopic stress is averaged over the ensemble of microstructurestates and all planes having the specified normal, see Figure (ref).

In the case of a material modeled microscopically by a sample having periodic orLees-Edwards boundary conditions it is important to carefully take into considerationhow interactions are handled which cross the boundaries of the finite domain. Addi-tional issues arise when forces arise from more than two body interactions. We shall

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 27

discuss here these issues and how such interactions augment the formulas traditionallyused to estimate the effective macroscopic stress tensor.

For concreteness we shall consider the estimate for the stress components σℓ,z

which corresponds to the stress-plane having normal in the z-direction, ℓ ∈ x, y, z.The σℓ,x and σℓ,y components can be estimated similarly. With the models developedin this paper in mind we shall only consider the special case when for forces arise fromeither two body and three body interactions.

In the case of two-body interactions the stress component is estimated by themicroscopic stress averaged over the ensemble of micro-states and volume by

σ(2)ℓ,z =

1

L

⟨

∫ b

a

Λ(2)ℓ,z(ζ)dζ

⟩

(C.1)

where L = b−a is the length of the domain in the z-direction, < · > denotes averaging

over the ensemble, and Λ(2)ℓ,z denotes the microscopic stress defined as

Λ(2)ℓ,z(ζ) =

1

A

∑

m,n

f (ℓ)m,nH(ζ − x(z)

m )H(x(z)n − ζ).(C.2)

H(ζ) =

0 ζ ≤ 01 ζ ≥ 0

(C.3)

where A is the cross-sectional area of the periodic domain for z = ζ (which we shall

assume is constant independent of ζ) and f(ℓ)m,n is the ℓth component of the force

exerted by the particle with index n on the particle with index m (which we shallassume always occur in equal and opposite pairs fm,n = −fn,m). This corresponds tothe interpretation of the stress as the total force per unit area that the microstructuresof the material above the cross-section exerts on the material below the cross-section.

To cope with the effects of the periodic and Lees-Edwards boundaries we shallfind it convenient to consider the system periodically repeated in space. In the caseof Lees-Edwards boundaries we shall consider as the unit cell the rectangular domainfor which the boundary conditions correspond to shifted image cells, see Figure (ref).We shall consider the following three domains (i) B for the region corresponding tothe explicitly represented unit cell (ii) B−z for the periodic image below the unit celland (iii) B+z for the periodic image above the unit cell, see Figure (ref).

With these conventions we can express the estimator for the stress as

σ(2)ℓ,z =

1

AL〈I1 + I2 + I3〉(C.4)

I1 =∑

m∈B,n∈B

f (ℓ)m,n

∫ b

a

H(ζ − x(z)m )H(x(z)

n − ζ)dζ(C.5)

I2 =∑

m∈B−z,n∈B

f (ℓ)m,n

∫ b

a

H(ζ − x(z)m )H(x(z)

n − ζ)dζ(C.6)

I3 =∑

m∈B,n∈B+z

f (ℓ)m,n

∫ b

a

H(ζ − x(z)m )H(x(z)

n − ζ)dζ.(C.7)

We now show that each of these contributions can be expressed in terms of the cu-mulative forces acting on a particle of the domain by particles which are also within

28 P.J. ATZBERGER

the domain or outside the domain. The cumulative force contributions have threedifferent sources which we denote by

fm =∑

n∈B

fm,n, f−zm =

∑

n∈B−z

fm,n, f+zm =

∑

n∈B+z

fm,n.(C.8)

Now by noting that all particles contributing to equation C.5 are within the domainwe can perform the integral and by noting that the indices m, n can be interchangedwithout changing the value of the sum we have

I1 =1

2

∑

m∈B,n∈B

f (ℓ)m,n(x(z)

n − x(z)m )H(x(z)

n − x(z)m )(C.9)

+1

2

∑

m∈B,n∈B

f (ℓ)n,m(x(z)

m − x(z)n )H(x(z)

m − x(z)n )

=1

2

∑

m∈B,n∈B

f (ℓ)m,n(x(z)

n − x(z)m )(H(x(z)

n − x(z)m ) + H(x(z)

m − x(z)n ))

=1

2

∑

m∈B,n∈B

f (ℓ)m,n(x(z)

n − x(z)m )

= −∑

m∈B

f (ℓ)m x(z)

m .

We have also used that fm,n = −fn,m and the definition of the cumulative force givenin equation C.8.

We now derive expressions for I2, I3. We use that the domain extends in thez-direction from a to b to perform the integrals and equation C.8 to obtain

I2 =∑

m∈B−,n∈B

f (ℓ)m,n(x(z)

n − a) = −∑

m∈B

f−z,(ℓ)m (x(z)

m − a)(C.10)

I3 =∑

m∈B,n∈B+

f (ℓ)m,n(b − x(z)

m ) =∑

m∈B

f+z,(ℓ)m (b − x(z)

m ).(C.11)

We also use that fm,n = −fn,m.

This gives for the periodic domain the following estimator for the effective macro-scopic stress in the case of two body interactions

σ(2)ℓ,z = − 1

AL

∑

m∈B

⟨

f (ℓ)m x(z)

m + f−z,(ℓ)m (x(z)

m − a) − f+z,(ℓ)m (b − x(z)

m )⟩

(C.12)

The estimators for the components in the x and y directions can be derivedsimilarly.

We now discuss an estimator for the contributions to the stress arising fromgeneral n-body interactions. For example, three body interactions can arise if bondangle terms are included in the potential energy. To handle this general case we shalluse an estimator of the form

σ(n)ℓ,z =

1

L

⟨

∫ b

a

Λ(n)ℓ,z (ζ)dζ

⟩

(C.13)

SELM FOR SOFT MATERIALS AND COMPLEX FLUIDS : ROUGH DRAFT 29

where L = b−a is the length of the domain in the z-direction, < · > denotes averaging

over the ensemble, and Λ(n)ℓ,z denotes the microscopic stress arising from the n-body

interactions and is defined as

Λ(n)ℓ,z (ζ) =

1

A

∑

q∈Qn

n−1∑

k=1

k∑

j=1

f(ℓ)q,j

k∏

j=1

H(ζ − x(z)qj

)

n∏

j=k+1

H(x(z)qj

− ζ)(C.14)

where Qn is the set of n-tuple indices q = (q1, . . . , qn) describing the n-body interac-tions of the system, fq,j denotes the force acting on the jth particle of the interaction,and xqj

denotes the jth particle involved in the interaction. As a matter of convention

in the indexing q we take that i ≤ j implies x(z)qi ≤ x

(z)qj . This expression corresponds

to a sum over all the forces exerted by particles of the material above the cross-sectionat ζ = z on the particles of the material below. Each term of the summation overk = 1, . . . , n−1 corresponds to a specific number of particles of the n-body interactionlying below the cross-section at ζ = z.

When integrating the microscopic stress a useful relation is that

∫ b

a

Πkj=1H(ζ − x(z)

qj) · Πn

j=k+1H(x(z)qj

− ζ)dζ = x∗,(z)qk+1

− x∗,(z)qk

(C.15)

where

x∗,(z)qj

=

b, if x(z)qj ≥ b

x(z)qj , if a ≤ x

(z)qj ≤ b

a, if x(z)qj ≤ a.

(C.16)

Substituting (ref) into (ref) we obtain

Λ(n)(ℓ),z(ζ) =

1

A

∑

q∈Qn

n−1∑

k=1

k∑

j=1

f(ℓ)q,j ·

(

x∗,(z)qk+1

− x∗,(z)qk

)

.(C.17)

This can be further simplified by switching the order of summation of j and k andusing the telescoping property of the summation over k. From (ref) this yields thefollowing estimator for the stress contributions of the n-body interactions

σ(n)ℓ,z =

1

AL

∑

q∈Qn

n−1∑

j=1

⟨

f(ℓ)q,j ·

(

x∗,(z)qn

− x∗,(z)qj

)⟩

.(C.18)

Appendix D. Table.

30 P.J. ATZBERGER

Parameter DescriptionNA Avogadro’s number.amu Atomic mass unit.nm Nanometer.ns Nanosecond.kB Boltzmann’s Constant.T Temperature.η Dynamic viscosity of water.γ = 6πηR Stokes’ drag of a spherical particle.R Effective particle size.D = (kBT/γ) Einstein’s diffusivity constant.τD = R2/D Diffusive time scale to move distance R.τF = Rγ/F Time scale for forced particle to move distance R.

Parameter ValueNA 6.02214199× 1023.amu 1/103NA kg.nm 10−9 m.ns 10−9 s.kB 8.31447× 103 amu nm2/ns2 K.T 300K.η 6.02214199 amu/cm ns.R 0.1 nm.γ 11351500 amu/ns.D 0.219736743473623 nm2/ns.τD 0.045509002463215 ns.τE ≈ 10−3 ns (empirical).