Iteration numerics

A.3 Iteration numerics

The iteration Eq. (2.59) yields the long-time limit of Φ_q(t), viz. f_q, in the absence of strain. Also, the MCT-EQM (2.57) must be solved iteratively. Two iteration algorithms have been frequently used in preceding works. The first is the momentum algorithm (also callediterated mapping algorithm), which has been (theoretically) elaborated neat to reduce discretization errors. The second one is theBrader-Voigtmann(BV)algorithm, a straight forward discretization of the MCT-EQM (2.57). However, the BV algorithm converges for smaller time-grid resolutions N_t to an asymptote (in N_t). This might be due to numerical subtleties in the decimation of the momentum algorithm, which have not been ‘fixed’ in the version of Ref. [86], which is presented below. Both algorithms are summarized and compared in the following. In consequence, the BV algorithm has been chosen rigorously throughout this work.

A.3.1 Time-grid discretization

The timetis discretized in a 2d time-grid with index parametersiandb, with06i6N_t and 06b6b_max via

timet→t^i,b≡i·h^b, with h^b ≡∆h·2^b. (A.5) The smallest time increment is given as ∆h and b labels blocks of the time-grid. Hence Φ_q(t)→Φ^i,bq , which applies in the same manner for all quantities, which are functions of time.

A.3.2 Iteration schemes Momentum algorithm

A detailed explanation of the momentum algorithm (for the F₁₂^{( ˙γ)}-EQM (5.1)) can be found in Chap. 1 of Ref. [86], whose notation is adopted and adapted in the following.

It is based on the splitting-up of the convolution integral over m_q(t^′₎(τ) in the MCT-EQM (2.57) by using partial integration such that the appearing time derivatives are always slowly varying. This is the reason why it cannot be used for a two-times MCT, because the two-times EQM (2.53) does not have the convolution. In this section, the momentum algorithm is just used for a comparison with the quiescent case (γ˙ = 0) of the F₁₂^{( ˙γ)}model. Thus it is not accounted for the wavevector advection of the index vectorq(t^′) of m_q(t^′₎(τ). AlsoΓ_q is not time dependent, Eq. (2.51). The discrete iteration equation of the MCT-EQM (2.57) reads in the momentum algorithm forb >0andNt/26i < Nt

Φ^i,b_q

n≡1/A^b_q

B_q^dm^i,b_q Φ^i,b_q

n−1

−C_q^i,b

, (A.6)

with iteration counter n. For i < N_t/2, one approximates the b block with the b−1 block via

Φ^i,b_q ≈Φ^2i,b_q ⁻¹, (A.7)

A.3 Iteration numerics

which is called time-grid decimation. The appearing quantities are

A^b_q ≡1 +dM_q^1,b+D_q^b, B_q^b ≡Φ^0,b_q −dΦ^1,b_q , (A.8) Equation (A.6) is initialized with

Φ^i,bq The iteration Eq. (A.6) is aborted when

reaches a lower bound, e.g. 10⁻⁷ ∀q, or after a given amount of maximum iteration cycles, denotedIt.

Brader-Voigtmann algorithm

The Brader-Voigtmann algorithm (BV) [103,104] is a simple straight forward discretiza-tion of the MCT-EQM (2.57).

The index b is dropped, because all quantities live on the same b block. Only in the decimation step, cf. Eq. (A.7), the b blocks intermix. The BV iteration equation for N_t/26i < N_t reads

A.3 Iteration numerics

where on the right hand side Φⁱ_q = Φⁱ_q

n−1 and mⁱ_q⁻k^k = mⁱ_q⁻k^k

h Φ^i−k

n−1

i ∀ i, k, according to Eq. (2.58). It is necessary to set Φ⁻_q¹ ≡ 1; see the discussion below. An advected wavevector with strainγt˙ ^k is denotedq^k=q(t^k).

Fori < N_t/2 the decimation scheme reads Φ^i,b≈ 1

2

Φ^2i,b−1+ Φ^2i−1,b−1

and m^i,b ≈ 1 2

m^2i,b−1+m^2i−1,b−1

. (A.18) For at leasti= 0, the decimation of the BV algorithm must differ from the simpler one of the momentum algorithm, Eq. (A.7), to account for the derivation ∂_t^′Φ_q(t^′)|_t^′₌₀in the MCT-EQM (2.57), i.e. Φ^0,b6= Φ^{! 0,b−1}. This problem has been avoided in the momentum algorithm by partial integration of the memory integral and introduction of the moments dΦ and dM. The term Φ⁻¹ −Φ^0,b assigns for b > 1 ‘some decreased’ slope between 1 and the average (Φ^1,bq ⁻¹+ Φ^0,bq ⁻¹)/2 to this derivation ∂_t^′Φq(t^′)|_t^′₌₀. This is a heuristic choice, which has the same role as B_q^dm^i,bq in Eq. (A.6). The difference is that dΦ^1,bq

differs from the BV algorithm’s Φ^0,bq , see Eq. (A.13) and (A.18).

When choosing the decimation Eq. (A.7) of momentum also for BV algorithm fori >0, Φq(t)gets shifted slightly on the time axis to larger values, but small kinks in the curves at thebblock borders get bigger (seen in Fig.A.5for momentum algorithm). These kinks emerge as the integration weights should be handled explicitly at the b block borders, which is hard due to the convolution in the MCT-EQM (2.57). However, it seems that the BV algorithm implicitly handles them better, which could be the reasons for its faster convergence (inN_t), even though the momentum algorithm is elaborated more neat [86].

Comparison of BV with momentum algorithm

The actual BV algorithm iteratesΦq(t) with the same precision as the momentum algo-rithm, but for smaller N_t, i.e. faster. This suggests to use the BV algorithm, although the momentum algorithm takes more care about aspects like the derivative∂tΦq(t)|_t=0. Figure A.4 shows a comparison of correlators Φq(t), which have been calculated in the quiescent state with both, momentum and BV algorithm. The conclusion is that a maximum of iteration cyclesIt= 150in Eq. (A.6) and (A.17) is sufficient. This holds also under shear, cf. Fig. A.3. Figure A.4shows that the momentum algorithm approaches the BV curves with increasing N_t. The BV curves are close to an asymptote, because doubling their N_t does not change the outcome and the curves approach well the non-ergodicity parametersf_q^c. On the other hand, the momentum algorithm’s correlator has not even reached this asymptote for double the size of BV’s N_t.

Figure A.5 shows a comparison of the shear induced decay of the schematic density correlator Φ(t) of the F₁₂^{( ˙γ)}model, Eq. (5.1). It is evident that, also for schematic MCT, the BV algorithm works much better for much smallerN_tthan the momentum algorithm.

When using the BV algorithm, Φ(t) hardly changes even for an extremely coarse time-grid discretization ofN_t= 8. Figure A.5also illustrates that the momentum algorithm’s Φ(t)approaches that of the BV’s algorithm as asymptote (inN_t). However, a quite large N_t= 1024must be chosen for the momentum algorithm.

A.3 Iteration numerics

Fig. A.4:Quiescent decay of the correlatorΦq(t)for wavevector moduliq as given in the axes labels. Left panel: BV solutions achieved with Eq. (A.17), denoted (BV).Right panel: Both, BV solutions and momentum algorithm solutions of Eq. (A.6), denoted (mom). The angularq-grid resolution isπ/∆ϑq = 8and2π/∆ϕq = 16, the packing fraction is the accordingφc of Tab. 3.1.

In the legends, according parametersItandNtfor the iteration equations are given. Doubling the time-grid discretizationNt for BV algorithm above 64 does not anymore improve the iteration accuracy (see left panel), as well as doublingIt= 150for both algorithms (see both panels). The right panel implies that the momentum algorithm approximates BV’s precision only for much higherNt. BV algorithm curves approach the desired long-time plateausf_q^c from Eq. (2.59).

0 Γ = 100, andγc= 1. The legend shows the time-grid discretizationsNtand ifΦ(t)was calculated with momentum (mom) or BV algorithm (BV). BV’s asymptote is reached already forNt= 8.

This asymptote is approached by the momentum algorithm, but not reached unlessNt= 1024.

A.4 Formulary

A.4 Formulary

A.4.1 Percus-Yevick structure factor

The direct correlation functioncq of a monodisperse hard-sphere fluid can be calculated in three dimensions analytically via the Percus-Yevick closure of the Ornstein-Zernike equation. It is derived in real space in Ref. [60] and inq space in Ref. [25] for a polydis-perse mixture. The outcome for a monodispolydis-perse hard-sphere fluid is

c_q=−4π The structure factor is given by the Ornstein-Zernike equationS_q= 1/(1−nc_q).

A.4.2 Real spherical harmonics

A complete, orthonormal set of real spherical harmonics Y_l,m can be defined in spherical coordinatesϑand ϕ, cf. Fig.3.2, via

Y_l,m≡ MCT-EQMs (2.57) and (2.58). In consequence,Φ_q(t)can have only symmetries in terms of spherical harmonics with l and m being even numbers. The relevant even numbered associated Legendre polynomials P^|_l^m| withl64 read

P⁰₀ = 1 P⁰₂∝3 cos²ϑ_q−1 P⁰₄∝35 cos⁴ϑ_q−30 cos²ϑ_q+ 3 P²₂∝sin²ϑ_q P²₄∝ 7 cos²ϑ_q−1

sin²ϑ_q P⁴₄∝sin⁴ϑ_q,

(A.21)

where the normalization can be shifted towards Nl,|m|, because for the discussion in Sec.3.4 only the functional dependence on angles is important.

A.4 Formulary

A.4.3 Deformation under simple shear and uniaxial compression The rate of strain tensors for simple shear, κs and uniaxial compression κu read

κ_s≡





0 γ˙_s 0 0 0 0 0 0 0



 and κ_u ≡





−γ˙_u 0 0 0 γ˙_u/2 0 0 0 γ˙u/2



. (A.22)

From Eq. (2.38) follows for the deformation tensorsE_s,u= exp(κ_s,ut)and hence for the Finger tensors, Eq. (2.37),

B_s≡





1 + ( ˙γ_st)² γ˙_st 0

˙

γ_st 1 0

0 0 1



 and B_u ≡





e^{−2 ˙γut} 0 0 0 e^γ˙ut 0 0 0 e^γ˙ut



. (A.23)

The invariants I₁ =TrB and I₂ = TrB⁻¹, Eq. (5.4), evaluate as I₁ = I₂ = 3 + ( ˙γ_st)² for simple shear. For uniaxial compression it holds I₁ = exp(−2 ˙γ_ut) + 2 exp( ˙γ_ut) and I₂ = exp(2 ˙γ_ut) + 2 exp(−γ˙_ut) and thusI_1,2≈3 + 3( ˙γ_ut)²+O ( ˙γt)³

. The last equation gives rise to introduce γ˙ = ˙γ_s = √

3 ˙γ_u as common control parameter of shear and compressional flow in Chap.5. As consequence, strain parameters of theF₁₂^{( ˙γ)}model are comparable in different strain geometries, when compared by the value ofγt, which enters˙ the schematic model as isotropic strain contribution in I(t), Eq. (5.3).

B Ad β -Analysis

Im Dokument Time dependent flows in arrested states (Seite 119-125)