Direct numerical simulation of flexible molecules and data-driven molecular conformation

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Amine AMMAR, Francisco CHINESTA - Direct numerical simulation of flexible molecules and

datadriven molecular conformation Comptes Rendus Mecanique Vol. 347, n°11, p.743753

-2019

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Direct

numerical

simulation

of

flexible

molecules

and

data-driven

molecular

conformation

Amine

Ammar

a

,

∗

,

Francisco

Chinesta

b

a_{LAMPA@ArtsetMétiersParisTech,2,boulevardduRonceray,BP93525,49035Angerscedex01,France}

b_{ESIGroupChair@PIMM,ArtsetMétiersInstituteofTechnology,CNRS,CNAM,HESAMUniversity,151boulevarddel’Hôpital,75013Paris,} France

a

b

s

t

r

a

c

t

Keywords:

Flexiblemolecules Directnumericalsimulation Moleculardynamics Suspension Deep-learning

The present work aims at performing a molecular dynamics modeling of suspensions composedofflexiblelinearmolecules.Moleculesarerepresentedbyaseriesofconnected beads,whosedynamicsisgovernedbythreepotentials:theextensionpotentialaffecting theelongationofsegmentsconnectingconsecutivebeads,theonegoverningthemolecule bendingandfinallytheLennard-Jonesdescribingtheinteractionofnon-consecutivebeads. Apopulationofnon-interactingmoleculesissimulatedinelongationandshearflowsfor different flow and molecule parameters. The flow-induced conformation is analyzed in the different considered situations. Finally a model for predicting the evolution ofthe populationconformationwillbeobtainedbyusingdeep-learning.

1. Introduction

Polymersandviscoelasticfluidscanbedescribed atdifferentscales,deeplydiscussedinmanybooks,amongthem [1] [2]. The molecularscale (see [3] [4] and the referencestherein) precedesthe one ofkinetic theory wherethemolecules individualityisreplacedbyascalardistributionfunction–DF–involvingaseriesofconformationalcoordinates[2].Most ofkinetic theory modelsrepresentmolecules asmulti-bead-rod ormulti-bead-springs idealizations,usually coarsenedby using theend-to-end vector to reduce thedimensionality ofthe conformational space[5]. The resulting modelswere in generalsolvednumericallybyusingBrownianandstochasticnumericaltechniquesforcircumventingtheso-calledcurseof dimensionalitythatthehigh-dimensionalityoftheconformationspaceentails[6] [7].

In[8] and[9] authorsconsiderGaussianchainscombinedwiththeLangevinequationforbetteraccountforthemolecular conformation.Worksattainingtheatomicscalearequitescare,beingtheresultingmodelsdifficulttoextendtolargerscales [10–14].

The presentworkaims atperforming a moleculardynamics modelingofsuspensions composedof flexiblemolecules. Suchadescriptionscalehasasmainadvantagethefinerepresentationofthemolecularconfigurations,finerthantheones attainablewhenusingcoarserdescriptions(e.g.kinetictheorydescriptions).Themaindrawbackwhenusingtheseextremely finedescriptions concerns thecomputational cost.However, tohaveaccessto therealbehavior andevaluate thevalidity of coarserdescriptions, high fidelitysimulations atthe finestscale seems compulsory. Moreover, assoon as an accurate approximationofthemicroscopicphysicsisattainedandavailable,itcouldbethenintroducedintocoarserdescriptions.

*

Correspondingauthor.

Fig. 1. Molecule schema.

When describingfinely moleculesasa seriesofconnectedbeads,thewholemolecular conformationstronglydepends on thechoiceofthepotentialsusedforderiving theforcestobe introducedintothemotionequation atthebeadslevel. Different potentials are in general considered: (i) the one describing the extension of the inter-beads segment; (ii) the one describing themolecule bending mechanism,that is expressedateach beadfromits two neighbors;andfinally (iii) the weak interaction (Lennard-Jones) between non-neighborbeads. The evaluation of the population conformation with respecttothedifferentavailablechoiceofpotentialsandinparticulartheirintensityisofmajorinteresttobetterdescribe rheological features. Inthepresentwork andwithoutloss ofgeneralityof theproposed methodologywe considerdilute suspensions,thatallowsneglectingtheinteractionofbeadsbelongingtodifferentmolecules.

Our modelassumes linearmolecules composed ofbeads atwhich, other than the potentialsjustdiscussed, also acts Brownian events representingthe action ofthe moleculesof solventin which themolecule is assumedbe immersed. A population ofnon-interacting molecules is simulated in elongation and shear flows for different flow and molecule pa-rameters. The flow-induced conformation is analyzed in the different considered situations and some statistical outputs extracted.

Finally a modelfor predicting theevolution of the populationconformation willbe obtainedby using deep-learning, deeplyusedinavarietyofapplications,assystemcontrol[15,16] anddynamicalsystems[17] forcitingfew.

2. Fine-scalediscretesimulation

Thefinescalesimulationisperformedbyrepresentingthemoleculeasaseriesofconnectedbeadswhereforcesapply, thelastderivedfromdifferentpotentialsaswellasthebombardmentofsolventmolecules.

Aspreviously discussedweareconsideringthreedifferentpotentialsresponsibleofthreedifferentdeformation mecha-nisms:theLennard-Jonespotential VLJ _{isusedtodescribetheinter-beadsinteractions (ofnon-neighborbeads), andother}

two potentials, VE _{and V}B_{,are used to describe the molecule segments elongationsand chainbending respectively, see}

Fig.1.

Thus,thetotalpotentialreads

V

=

M



j=1 N−1



i=1 V_iE_,i+1;j

+

M



j=1 N−1



i=2 V_iB_−1,i,i+1;j

+

M



j=1 N



i=1 N



k=1 V_iLJ_,k;j

χ

ik;j (1)

where

M

isthenumberofmoleculesinthepopulation, j referstothemoleculechaininthatpopulation,andi referstothe consideredbeadcomposingit,i

=

1

,

. . . ,

N

(with

N

thenumberofbeadscomposingthemolecularchain).Finallywedefine acharacteristicfunctionfordescribingthebeaddirectneighborhood:

χ

ik

=

1 if

|

i

−

k

|

>

1 and

χ

ik

=

0 if

|

i

−

k

|

≤

1

TheLennard-Jonespotentialreads

V_iLJ_,k

=

4

ε



σ

dik



12

−



σ

dik



6



(2)

withdik

= ||

xk

−

xi

||

thedistancebetweenbeads

B

i and

B

k,located atpositions xi andxk respectively, and



and

σ

the twousualparametersinvolvedintheLennard-Jonespotential.

Theelongationpotentialisgivenby

V_iE_,i+1

=

KE 2



1

−

di j deq



2 (3)

wheredeq istheequilibriumdistancebetweentwosuccessivebeads,distanceatwhichthepotentialreachesits minimum

(andconsequentlytheassociatedforcevanishes).Inthepreviousexpression KE reflectsthepotentialintensity.

Thebendingpotentialbetweenthreesuccessivebeads

V_iB_−1,i,i+1

=

KB 2



θ

i−1,i,i+1

− θ

eq



2 (4)

where

θ

i−1,i,i+1 istheangledefinedbyvectorsjoiningbeads i

−

1

,

i andi

,

i

+

1 with

θ

eq theangleatequilibrium. Inour

studyweconsider

θ

eq

=

70

.

5◦ (thatcorrespondstotheanglebetweentwoconsecutivecarbone-carbonebonds).Again,the

potentialintensityisdescribedfromKB.

The force applyingat each bead related to thesethree molecule deformation mechanisms results from the potential derivativeaccordingto

FI_i

= −

∂

V

∂

xi

(5)

Therearetwootherforcesactingonthebead.First,theonerelatedtothefluiddragFD_i.Letv

(

x

)

bethefluidvelocityat positionx,assumedunperturbedbytherodspresence,thedragforcereads

F_iD

= ξ(

v

(

xi

)

− ˙

xi

)

(6)

wherexi

˙

denotesthevelocityofbead

B

i.

FinallyaBrownianforceisalsoconsideredemulatingthesolventbombardment,andisexpressedfrom

FBrow_i

=

√

2D dt Wi (7)

whereD denotesthediffusioncoefficient,dt thecalculationtimestepandWiarandomvariablewithzeromeanandunit standarddeviation.

Thelinearmomentumbalanceinvolvingbead

B

ireads

Fi

=

FDi

+

FiI

+

FBrowi

=

m ai (8)

withtheparticleaccelerationai

=

ddtx˙i

=

d2_x

i

dt2.

Asecond-ordertimeintegrationschemeisconsidered,consistingof

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

an_i+1

=

Fni m xn_i+1

=

xn_i

+ ˙

xn_i

t

+

1₂ai

(

t

)(

t

)

2

˙

xn_i+1

= ˙

xn_i

+

1₂

an_i

+

an_i+1



t (9)

wherethesuperscriptn referstothetimestep,tn

=

n

t.

Priortoproceedwiththetimeintegration,equationsarerewrittenindimensionlessform,byconsideringascharacteristic variables – length:

σ

– energy:

ε

– mass:m – velocity:

√

ε

/

m – acceleration:

ε

/(

m

σ

)

– time:

σ

√

m

/

ε

– diffusion:

σ

−3



5 m

whichallowswritingthedimensionlessmodelinwhich,forthesakeofclarity,wedonotchangethenotationfor designat-ingthedimensionlessvariables.

Thus,weobtain

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

V_iLJ_,j

=

4



_d1 i j



12

−

1 di j



6



V_iE_,i+1

=

KˆE 2

1

−

di j deq



2 VB i−1,i,i+1

=

ˆ KB 2

(θ

i−1,i,i+1

− θ

eq

)

2 F_iI

= −

∂_∂xV i F_iD

= ˆξ(

v

(

xi

)

− ˙

xi

)

FBrow_i

=



2D

ˆ

dt W



i Fi

=

ai (10) whereK

ˆ

E

=

K_{ε ,}E K

ˆ

B

=

K_{ε ,}B

ˆξ = ξ√

ε

m

/

σ

,D

ˆ

=

D

σ

3



m 5 anddt



=

dt σ



 m.

Fig. 2. Initial conformation and its evolution when subjected to elongation. 3. Results

Inthe numericalresultsreportedinthepresentsection weassumedasequilibriumdistancebetweentwoconsecutive beadstheoneassociatedwiththeLennard-Jones(dimensionless)potential,i.e.deq

=

21/6.Inordertoavoidlargefluctuations

ofthemoleculelengthweassumeaquitelargeelongationstiffness K

ˆ

E

=

10000 andabendingpotential K

ˆ

B

=

1000.

Inordertoderiverheologicalproperties(basedonthemolecularconformation),alargeenoughpopulationofmolecules must be considered, allowing one to compute accurate statistics. In the cases reported in the present paper

M

=

1000 chainsinvolving

N

beads,

N

=

100,andconsequently99 inter-beadsegments areconsidered.Inallcasesthedimensionless diffusionanddragcoefficientswereassumedtakingunitvalues.

Twosimplerheologicalflowswereaddressed,elongationalandsimpleshearflows,bothexpressedfromthe dimension-lesseffectivestrainrate



˙

,leadingtothefollowingdimensionlessvelocityfields:

– elongationalflow, v

=

⎛

⎝

vvxy vz

⎞

⎠ =

⎛

⎜

⎝

˙

x

−

1 2



˙

y

−

1 2

z

˙

⎞

⎟

⎠

(11)

– simpleshearflow,

v

=

⎛

⎝

vvxy vz

⎞

⎠ =

⎛

⎝

z

˙

0 0

⎞

⎠

(12)

wherethreevaluesof



˙

willbeevaluated:(i)



˙

=

5;(ii)



˙

=

1,and(iii)



˙

=

0

.

2.

Theinitialstateconsistsofthe

M

=

1000moleculesgeneratedrandomly.Forthatpurposethefirstbeadofeachchainis placeatpointx1.Thenthesecondbeadisrandomlyplacedatpointx2 byensuring



x2

−

x1



=

deq.The thirdbeadisplace

atpointx3ensuringnowthetwoconditions:(i)



x3

−

x2



=

deqand(ii)L



12L23

= θ

eq,withLi,i+1

=

xi+1

−

xi.Theprocedure continuesuntilplacingthefinalbead.

Byproceedinglikethis,thereistheriskthatbeadsinthesamemoleculeapproachtoomuch.Thus,beforeapplyingthe flow,themoleculespopulationissubjectedtotheLennard-Jonespotentialtoensurethefulfillmentoftheexcludedvolume. Fig.2depictstheinitialmoleculeconformationandthen,theonethatresultsfromitselongation.

3.1. Casestudies

Inwhatfollowsdifferentanalysesareperformedbyvaryingtheflowanditsintensity(elongationandsimpleshear,with

˙



=

5 and



˙

=

0

.

2).Eachsimulationleadstoafigurecomposedof8subfigures,each onerepresenting(toptobottom,left toright):

– representation ofthe molecules population wherethe molecule barycentre is placed at theorigin ofthe coordinate system;

– averageofthedifferentmoleculesegmentsjoiningconsecutivebeads(from1to

N

−

1); – theorientationtensorcalculatedbydefining

u_ij_,i+1

=

L j i,i+1



L_ij_,i+1



(13)

where j referstheconsideredmoleculeinthepopulation, j

=

1

,

. . . ,

M,

andi theconsideredbead,i

=

1

,

. . . ,

N

−

1,from whichtheorientationtensorwrites

S

=

1

M

M



j=1 N−1



i=1 u_ij_,i+1

⊗

u_ij_,i+1 (14)

– theconformationtensorC computedfromtheend-to-endvectorL₁j_,N accordingto

C

=

1

M

M



j=1 L₁j_,N

⊗

L₁j_,N (15)

– the distributionofu_ij_,i+1 ontheunit sphere(notedasDFinthefigure),eachvector beingassociatedwithaGaussian distributionforsmoothingtherepresentation;

– asimilarrepresentationbutnowconcerningonlytheend-to-endvectorsu₁j_,N; – averageofthesegmentsofallmoleculeswiththeassociatedstandarddeviation;

– averageofsegmentslocatedatthemoleculeendsandatthecenterwiththeirassociatedstandarddeviations.

3.2. Discussion

Figs.3and4concernelongation.Itcanbenoticedthatcentralsegmentsareveryelongated,withafinallengththatat highelongationdoublestheoneofsegmentslocatedatthemoleculesextremities.Whendecreasingtheelongationintensity central segments become lessextended,andthe end-to-end conformationremains alignedwiththe elongation, however the segments orientation distribution exhibits a particular annular distribution whose radius depends on the elongation intensity.

Figs.5and6concernshearflow.Segmentshaveapproximatelythesamelengthatthealmoststeadystate.Anovershoot is noticed at the central segments during the transient regime. At highshear rates the distribution concentrate at two regions(4bysymmetry)thatattenuateswhendecreasingtheshearrate.

3.3. Derivingamodeloftheconformationevolution

Asdiscussedintheintroductionsection directnumericalsimulations areveryexpensivefromthecomputationalpoint ofview,andinthatcaseequationsdescribingtheconformationevolutiondepending ontheconformationandtheapplied flowkinematicsseemspreferable.

Asthiskindofmodelisquitedifficulttoobtain,hereweproposeusingdeep-learningtechniquesforthatpurpose.The conformationevolutionisassumedevolvingaccordingto

dC

dt

=

f

(

C

,

∇

v

)

(16)

withthevelocitygradientgivenby

∇

v

=

⎛

⎜

⎝

∇

v11

∇

v12 0 0

−

∇v11 2 0 0 0

−

∇v11 2

⎞

⎟

⎠

(17)

Fig. 3. Elongation at˙=5.

Weelaboratedaneuralnetwork–NN–involvingtwohiddenlayersof20neurons,beingtheinputsthe6components ofthe(symmetric)conformationtensorandthetwocomponentsofthevelocitygradient

∇

v11and

∇

v12,beingtheoutputs

thesixcomponentsoftimederivativeoftheconformationtensor.

Werestricttomoleculescomposedof20beads,i.e.

N

=

20,withthelowerbendingstiffness K

ˆ

R

=

100.600flowinduced

conformations aresimulatedbychoosingrandomlyboth gradientofvelocitycomponents,accordingwiththe 101.4(ω−0.5), where

ω

isarandomvariableuniformlydistributedintheinterval

[

0

,

1

]

.Thischoiceallowsensuringvaluesoftheeffective strainrateintheinterval

(

0

.

2

,

5

)

,coveredalmostuniformlyinthelogarithmicscale.

Fig. 4. Elongation at˙=0.2.

When theneural network extractthe internal weights,the solutionis computedagainfordifferentvalues ofvelocity gradient,randomly chosen,inorder tocomparethe directnumericalsolutionwiththeneural-network predictions.Fig.7 comparesboth conformationpredictions.The conformationalpredictions arequalitativelyinagreementwiththeones de-rivedfromdirectnumericalsimulations,howeverquantitativelysomegapsarenoticed.Forreducingthem,differentroutes exist: consideralternative networks, increasing thenumber ofhidden layers orthe numberof nodes(neurons) inthose layers.However,thosechoicesaredetrimentalwithrespecttothetrainingefficiency.AdeepercomprehensionofNNseems necessaryforbetterrepresentingthesubjacentphysics,tobetterperformpredictionswhilekeepingasreducedaspossible

Fig. 5. Shear at˙=5.

the trainingneeds (it isimportanttonote thatdirect numericalsimulationsthat servefortraining theNNareextremely expensivefromthecomputationalviewpoint).

4. Conclusion

In this paper, we proposed a direct numerical simulation of a suspension of non-interacting flexible molecules, and evaluated theflowinduced conformation.Simulationsprovethattheconformationcanexhibitunexpectedfeatures,asthe annulardistributionofthemolecularsegmentsorientation.

Fig. 6. Shear at˙=0.2.

Thesecondcontributionofthepresentworkconcernsthederivationofanevolutionequationfortheso-called conforma-tiontensor,that isexpecteddescribingtherheologicalfindings,byusingartificialintelligencetechniques,andinparticular deep-learning.Ithasbeenprovedthatthelearnedmodelallowsderivingveryaccuratepredictionsforflowsdifferingfrom theonesconsideredinthelearningstage(neuralnetworkconstruction).

The obtentionof richerexpressions of theconformation evolutionin generaltransient andcomplexflows, as well as the conformation / flow kinematics coupling through an adequate constitutive equation, constitute the main works in progress.

Fig. 7. Comparing direct numerical solutions (left) and neural-network based prediction (right). Acknowledgements

TheauthorsgratefullyacknowledgetheANR(“Agencenationaledelarecherche”,France)foritssupportthroughproject AAPG2018DataBEST.

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