Mesoscopic single and multi-mode rheological models for ...€¦ · The simplest of them basic rheologi-cal model [15,16], allows modeling steady viscomet-ric ﬂow of solutions and

Mesoscopic single and multi-mode rheological models for polymericmelts viscometric flows description

GRIGORY PYSHNOGRAIAltai State Pedagogical University

Department of Mathematical Analysisand Applyed Mathematics

Molodeznaya 55, 656031 BarnaulRUSSIA

[email protected]

DARINA MERZLIKINAAltai State Technical University

Department of MathematicsLinin prospekt 46, 656039 Barnaul

[email protected]

PETR FILIPInstitute of Hydrodynamics AS CRPod Patankou 5, 166 12 Prague 6

CZECH [email protected]

RADEK PIVOKONSKYInstitute of Hydrodynamics AS CRPod Patankou 5, 166 12 Prague 6

CZECH [email protected]

Abstract: The Vinogradov and Pokrovskii rheological model was extended for a description of rheological be-haviour of branched polymer melts. Since stationary elongational viscosity is a nonmonotonic function of theelongational rate, it required a generalization of the law of internal friction for beads of macromolecule. To achievehigh prediction accuracy was proposed multi-mode approximation. The contribution of each independent modeto a stress tensor corresponds to the individual polymer fractions differing in relaxation time and viscosity. Thetheoretical predictions of the generalized model provide a good agreement with the measured steady and transientrheological characteristics of two samples branched low density polyethylenes.

Key–Words: Constitutive equations, Mesoscopic approach, Polymer dynamics, Polymer melts, Viscometric flows,Viscoelasticity

1 IntroductionThe widespread introduction of polymers and prod-ucts based on them in daily practice leads to increasedattention to their production and processing. The de-mand of polymeric materials explains easy recyclingas well as their unique properties, such as elasticity,strength, flexibility, good electrical insulation proper-ties. Moreover, these properties cannot be explainedonly by the chemical composition of raw materials;the determining factor is chain structure of polymermolecules. It is well known that one of the importantproblems facing businesses is the issue of cost reduc-tion in the production and processing. Effective solu-tion to this problem is possible for optimizing tech-nological processes, which it is impossible withoutbuilding a mathematical model of behavior of poly-meric media in different conditions of deformation.The work with such models is significantly compli-cated by the need to take into account non-linear ef-fects and the effects of ”memory” when consideringthe flow of polymer melts and fluids.

Thus a formulation of rheological constitutive re-

lation, which establishes a connection between thekinematic characteristics of the flow and the internalthermodynamic parameters plays an important rolewhen describing flows of solutions and melts of lin-ear polymers in various modes of process equipment.

Experimental studies of various polymer liquidsexhibit their nonlinear viscoelastic behavior. To de-scribe these effects a large number of models that de-scribe the rheological behavior of polymeric liquids,both qualitatively and quantitatively were proposed.It may be noted that there are two fundamentally dif-ferent classes of models: in one case, a phenomeno-logical approach is used, in the second takes micro-scopic approach. In the phenomenological approach,the dynamics of macroscopic bodies is built on thebasis of general laws, which are found from expe-rience. The phenomenological model of Maxwell,Oldroyd [1], Prokunin-Leonov [2] and others belongto this class of models. Another class of models isbased on the mesoscopic approach [3-14]. In suchmodels the dynamics of macromolecules should bedescribed on the basis of model concepts and there-

WSEAS TRANSACTIONS on HEAT and MASS TRANSFERGrigory Pyshnograi, Darina Merzlikina,

Petr Filip, Radek Pivokonsky

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fore takes into account a certain approximation, asthe structure of the polymer molecules and processesof intermolecular interaction. Often in these modelsuse single-molecule approach, in which instead of theentire set of macromolecules is considered one se-lected macromolecule, which moves in a ”effective”environment. In some models of this class of macro-molecule is often presented as a sequence of beads andconnecting them springs (spring force); for example,one of the first of these models is Cargin-Slonimsky-Rouse model [3,4]. Thus, the mesoscopic approach,in contrast to the phenomenological, allows us to tracethe connection between micro and macro propertiesof polymer systems. This helps to explain variousphenomena in polymer melts, such as diffusion, vis-coelasticity and others. However, introduction addi-tional mesoscopic parameters that should be speci-fied empirically is typical for such approach. Thesemodels include the Doi and Edwards[5] model, deGennes[6], Curtiss and Bird[7]. In these models, eachmacromolecule is considered as a flexible chain, mov-ing in a certain tube formed by other macromolecules;while for small molecule observation times can onlymake movements along the inside of the tube. How-ever, the original model of Doi-Edwards does not takeinto account the elongation of each segment of thepolymer chain. Subsequently model was modifiedto account for such extension. Generalization of the”reptation” is in [8], in which the dynamics of macro-molecular chains examined on two dimensional levels- actually macromolecule and the level at which dis-cusses the tube model. In [9] the one-chain problemfor reptation and contour-length fluctuations whichdoes not have any adjustable parameters was solved.

Another model [10], derived from the theory of”reptation”, was developed to describe the dynamicsof branched polymers, and a rheological model in dif-ferential form. It is based on the dynamics are not se-quentially linked beads, but the beads are at the chainends are connected to several. This is so-called pom-pom model. The disadvantage of this model is thezero value of the second normal stress difference insimple shear. Later, it was modified [11,12] and calledthe extended pom-pom model. These models consistof four differential equations containing an additionalparameter that allows you to get a non-zero secondnormal stress difference with the shear flow. Thesemodels are system of differential equations containingan additional parameter that allows you to get a non-zero second normal stress difference with the shearflow. At present, various versions of the model ”rep-tation” is most often used for the interpretation of ex-perimental data in the study of the viscoelastic proper-ties of solutions and melts. Nevertheless, the theoreti-cal predictions are not always consistent with experi-

ments to quantify, although in general and give correctpredictions. In some works, attempts to improve theagreement between theory experiments in quantitativeterms, which leads to the theory of ”double reptation”,when taken into account local entanglements. Also,the presence of long chain macromolecular branchingleads to additional difficulties when applying ”repta-tion” model since in this case the ”reptation” becomeimpossible. In this case, the model should be im-proved; some studies specifically investigated macro-molecular chains with long lateral branches. For ex-ample, in [8] has been shown that the presence of longchain branches per macromolecule leads to the ap-pearance features of the viscoelastic behavior is espe-cially important in processes involving polymer meltsstretching.

In the case of the model Cargin-Slonimsky-Rouseor ”beads-springs” generalization constructed to in-clude consideration of ”internal viscosity”. It made itpossible to describe the independence of the plateaumodulus of the molecular weight of the polymer[13,14]. If thus consider only the slowest relaxationprocess in the polymer chain, it’s possible to go to themodel Vinogradov and Pokrovsky, and consideringthe induced anisotropy in the modified model Vino-gradov and Pokrovsky [15-17]. It describes such ef-fects observed in practice as the first and second nor-mal stress difference, extensional viscosity increase,its output value on the other stationary and others.Furthermore, good correlation between theoreticaland experimental curves of a wide range of deforma-tion rates was observed.

The differential-vector-RHL model is also basedon microstructure approach. In this model the dynam-ics of the macromolecule is described by the vectorconnecting the start and end of the polymer chain.This vector can be prolonged by increasing the lengthof the macromolecule, and also change the orientationby an external flow. RHL model is simple enough, butit was observed a good agreement this model with ex-perimental data. In fact, this model makes it possibleto consider the orientation of the chain and its exten-sion separately with nonlinear elastic parameters ofthe coil macromolecule because it is based on the as-sumption that the orientation relaxation time longerrelaxation time extension. At present, the develop-ment of new models still pays great attention to, forexample, in [20], the authors try to solve the prob-lem of describing the linear viscoelastic long-chainbranched polymers. Besides, in [18, 19] the authorspropose an advanced version of the MSF model allow-ing modeling of the transient and steady-state elon-gational viscosity data of monodisperse polystyrenemelts without using any nonlinear parameter, i.e.,solely based on the linear viscoelastic characteriza-



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tion of the melts and then extend the same approachto model experimental data in shear flow.

Despite the large number of papers in the fieldof polymer media describe the dynamics of the mostpopular of these are: Prokunin and Leonov model [2],pom-pom model [10,11] and its modifications [12].

2 The rheological constitutive rela-tion

To obtain rheological constitutive relation it is con-venient to use the microstructural approach, whichallows state a connection between the macro- andmicro-characteristics of the polymer system [3-24].Thus in the theory of viscoelasticity of the polymerthe most productive is monomolecular approximation,when one selected macromolecule moving in an effec-tive medium formed by the solvent and other macro-molecules is considered instead of the totality of themacromolecules in the volume [13,14]. To investi-gate the relatively slow movements it is possible touse Kargin, Slonimskii, Rouse model [3,4] when dy-namic equations of selected macromolecule are

mdrαidt

= −ζij(uαj − νjkrαk )−

−2µTAαγrγi + φαi (t), (1)

α = 0, 1, 2, . . . , N,

wherem is the mass of a Brownian particle associatedwith a piece of the macromolecule of length M/N,rα and uα = ṙα are the co-ordinates and velocity ofthe Brownian particle, ζij - the tensor coefficient offriction and 2Tµ is the coefficient of elasticity of ’aspring’ between adjacent particles, T is temperature inenergy units. The matrix Aαγ depicts the connectionof Brownian particles in the entire chain and has theform

A =

∥∥∥∥∥∥∥∥∥∥∥

1 −1 0 . . . 0−1 2 −1 . . . 0. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

0 0 0 . . . 1

∥∥∥∥∥∥∥∥∥∥∥.

The dissipative force in equations (1) are introducedby ζij(uαj − νjkrαk ), presents the effective forces fromthe neighbouring macromolecules.

For the considered, linear in velocities, case, thecorrelation functions of the stochastic forces in thesystem of equations (1) can be easily determined fromthe requirement that, at equilibrium, the set of equa-tions must lead to well-known results (the fluctuation-dissipation theorem). It is readily seen that, according

to the general rule [8,14],

〈φαi (t)φγk(t′)〉 = Tζik exp

(− t− t

′

τ

). (2)

It should be mentioned that such an approachleads to rheological defining relations of varying de-grees of complexity, a review of papers in this di-rection and comparison with other approaches can befound in [14]. The simplest of them basic rheologi-cal model [15,16], allows modeling steady viscomet-ric flow of solutions and melts of linear polymers, notonly qualitatively but also quantitatively. Thereforeit is necessary to consider this model in more detail.In this case it is possible to model the dynamics ofmacromolecules on elastic dumbbell (N = 2), whichcorresponds to the consideration of the only one (theslowest), relaxation process of the polymer chain. Inthe inertialess case in the laboratory system the equa-tion of dynamics of macromolecules (1) will take thefollowing form

ṙ2i = νijr2j − 2Tµζ−1ij (r

2j − r1j ) + ζ−1ij φ

2j , (3)

ṙ1i = νijr1j − 2Tµζ−1ij (r

1j − r2j ) + ζ−1ij φ

1j .

If the anisotropy of the considered polymer sys-tem is determined by a symmetric second-rank tensoraij , then for the friction tensor ζij the following ex-pressions should be used [25,26]

ζij = Bζ

(δij + 3β

(aij −

ajj3δij

)+ κajjδij

)−1,

(4)

where ζ - the coefficient of friction of the beads inthe ”monomer” liquid (for spherical particles ζ =6πRηs); B - measure of the friction coefficient gain;and β, κ - scalar coefficients of anisotropy taking intoaccount the isotropic and anisotropic contributions tothe dependence respectively ζij from aij . The expres-sion (4) was obtained previously as a self-consistentgeneralization to the case of large deformations of ex-pression [15]:

ζij = Bζ

(δij − 3β

(aij −

ajj3δij

)− κajjδij

).

The above dynamic equations were designed [23]to describe effects in entangled systems, that is inthe systems consisting of macromolecules of lengthM > 2Me, where Me is the length of the macro-molecule between adjacent entanglements [21]. Inthe case, when ζik = ζδik, the considered equations



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describe the Rouse dynamics of macromolecule in aviscous liquid [3]. For entangled systems in [13-16]are introduced measures of intensities of the externaland internal extra dissipative forces in Eq. 1, con-nected with the neighbouring macromolecules. Thedependence of the quantity B on the length of macro-molecules can be estimated [13] by using simple pic-ture of overlapping coils [24] or the constraint-releasemechanism [22].

The meaning of this expression corresponds withthe notion that for a small deformation of the polymersystem it is more convenient to change its shape in thedirection determined by the tensor aij , which leads toreducing the friction. It is clear that for large defor-mations this relation will become incorrect (the valueof the friction tensor can be negative). Therefore, ex-pression (4) was proposed. However, this generaliza-tion does not include all the features of the processesat large deformations. It is a more realistic situationwhen parameters β and κ themselves depend on theanisotropy tensor and its invariants. The preliminaryinvestigation showed that β and κ have to be slowlyincreasing functions of the first invariant of the tensoranisotrophy aij . Further consider the situation whereas such dependence expressions of the form are used

β(I) =β0 + pI

1 + pI, κ(I) =

κ0 + pI

1 + pI, I = ajj . (5)

Here β0 and κ0- initial values of anisotropy pa-rameters, p - parameter takes account of the effect ofthe first invariant I on β(I) and κ(I).

It is now possible to obtain the rheological consti-tutive relation. When deducing it’s convenient to go tonew coordinates by means of

ρi =1√2

(r1i − r2i ), ρ0i =1√2

(r1i + r2i ). (6)

Coordinate ρ0i describes movement of the gravitycenter of the dumbbell, and ρi - relative movement ofthe beads. Then equations (3) in coordinates (6) takethe form

ρ̇i = νijρj − 4Tµζ−1ij +1√2

(φ1i − φ2i ),

ρ̇0i = νijρ0j +

1√2

(φ1i + φ2i ).

Now receive the equations for the correlation mo-ment mik(t) = 〈ρi(t)ρk(t)〉 , where the averagingprocedure is performed over all possible realizationsof random forces. Differentiating mik(t) on time andusing the recently ratio, we obtain

d

dtmik = νijmjk + νkjmji −

−4Tµζ−1ij mjk − 4Tµζ−1kj mji + (7)

+1√2〈(φ1i + φ

2i

)ρk〉

1√2〈(φ1k + φ

2k

)ρi〉.

Correlation moments 〈(φ1i + φ

2i

)ρk〉 which are

unknown in (7) can be found from the correspond-ing fluctuation-dissipation relation (2), but its possi-ble to do otherwise. Note that the equilibrium (whenνik = 0 ) value of correlation moment mik was de-fined previously [12,13,23] (m0ik = δik/4µ). Besidesequations (7) are linear with respect to mik, so theinfluence of moment 〈

(φ1i + φ

2i

)ρk〉 means that it is

necessary to replace mik on mik − m0ik everywherewhanmik has no factor of velocity gradient tensor νik.Then instead of (7) we obtain

d

dtmik = νijmjk + νkjmji− (8)

−4Tµζ−1ij(mjk −

1

4µδjk

)−

−4Tµζ−1kj(mji −

1

4µδji

).

Now note that tensor mik describes the shapeand size of the macromolecular coil. And since theanisotropic properties of polymer media are deter-mined by the size and shape of the macromolecularcoils, the anisotropy tensor, which describes the devi-ation of the statistically nonequilibrium system fromequilibrium can be defined as

aik =〈ρiρk〉〈ρ2〉eq

− 13δik =

4µ

3mik −

1

3δik. (9)

Then using (9), equations (8) can be rewritten as

d

dtaik − νijajk − νkjaji+

+1 + (κ− β)I

τ0aik =

2

3γik −

3β

τ0aijajk, (10)

where τ0 = ζ/(8Tµ) - initial relaxation time; γik =(νik + νki)/2 - symmetrized velocity gradient tensor.

For comparison (10) with experimental data theexpression for the stress tensor of the polymer systemis required. It can be obtained in a standard way [14-16] and in generalized coordinates (5) has the form



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σik = −2nTδik + 4nTµ〈ρiρk〉.Where n - number of macromolecules in init vol-

ume. Using notation (9), the last expression can bewritten as

σik = −p̂δik + 3η0τ0aik, (11)

where p̂ = nT - hydrostatic pressure; η0 = nTτ0 -initial value of the shear viscosity.

Thus, (4), (10), (11) form a rheological constitu-tive relation of the nonlinear anisotropic viscoelasticfluid with parameters: η0, τ0, κ and β, which in turnhave to depend on the molecular weight of polymerMand its concentration c . For example, the dimensionalparameters can be estimated by formulas [24-26]:

η0 = η0(c∗,M∗)

(c

c∗

)4,5·(M

M∗

)3,4, τ0 =

η0nT

,

whereM∗, c∗ - some fixed values of molecular weightand concentration at which the initial values of viscos-ity and relaxation time are known.

As for the parameters of the inducedanisotropyκ0, β0 and p, there is a reason to as-sume that for linear polymers, they do not dependon the molecular weight or the concentration of[26,27,28].

3 Non-linear effects in simple shearand uniaxial elongation

The rheological constitutive relation (10), (11) ob-tained above needs to be checked for compliancewith the flow of real polymeric liquids. Flows real-ized in experiments with solutions and melts of lin-ear polymers, are very diverse: stationary and non-stationary, linear and non-linear, homogeneous andheterogeneous. It is clear that any check on the ad-equacy of the rheological model should start with thesimplest flows. It should also be noted that the cal-culation of the most complex flows - heterogeneousstationary and non-stationary (when it is necessary todetermine the velocity field of polymer liquid and avelocity gradient tensor) - are very time-consuming,even for fairly simple models and require using nu-merical methods for solving partial differential equa-tions. Therefore such calculations are quite a complexmathematical problem [26-29].

At the beginning consider viscosimetric flows ofpolymer media. Such flows are implemented in vis-cometers of different designs, and they are character-ized by the fact that the velocity gradient tensor is a

known function of time. Therefore, modeling suchflows is reduced to the integration of ordinary differ-ential equations for the internal thermodynamic pa-rameters, which is an easy task.

Simple shear and uniaxial elongation are the mostinvestigated viscometric flows. In the case of a sim-ple shear flow, where only one component of velocitygradient tensor ν12(t) is different from zero, rheolog-ical behavior of the polymer system is characterizedby the following viscometric functions: shear stress -σ12 ; shear viscosity - η ; first normal stress difference- N1 and second normal stress difference - N2, whichare determined as:

η =σ12ν12

, N1 = σ11 − σ22, N2 = σ22 − σ33, (12)

and in general they are functions of the shear rate γand time t.

If the function of shear rate on the time given asν12(t) = γE(t) or ν12(t) = γE(−t), where

E(t) - Heaviside unit function, than the first case de-scribes establishment of shear stress from the steadystate, and second - the stress relaxation after shear de-formation. Then viscosimetric functions (13) are de-noted η+(γ, t), N+1 (γ, t) and η

−(γ, t), N−1 (γ, t) re-spectively.

Also of interest are the stationary values ofη+(γ, t), N+1 (γ, t) :

η(γ) = limt→∞

η+(γ, t), N1(γ) = limt→∞

N+1 (γ, t). (13)

Sometimes it is convenient to consider variouscombinations of variables introduced into considera-tion, for example

Ψ1(γ) =N1γ2, Ψ2(γ) =

N2γ2. (14)

Here Ψ1 and Ψ2 - coefficients of the first and sec-ond normal stress difference.

Numerous experimental data on the shear defor-mation of solutions and melts of linear polymers showthat shear viscosity - is a decreasing function of shearrate; shear stress η(γ) is an increasing function ofshear rate; Ψ1 and Ψ2 - coefficients of the first andsecond normal stress difference are decreasing func-tions of shear rate; ratio of the second normal stressdifference to the first is a small quantity, which de-pendence on the shear rate is not enough investigatedin experiments [30].

In the case of uniaxial tension, when only the di-agonal terms of the tensor of velocity gradients are



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different from zero ν11 = −ν22 = −|nu33 = γ , rhe-ological behavior of the polymer fluid has a viscosityin uniaxial elongation - λ , defined as

λ(γ) =σ11 − σ22

γ. (15)

Viscosity in tension can be both increasing anddecreasing function of strain rate. More detailed in-formation about the behavior of viscometric functionsare given below when comparing detected theoreticaldependencies with the experiments described in liter-ature. The resulting equations require a numerical so-lution. When considering non-linear effects in steadyshear flow system (10), (11) takes the following form

a11 = 2τNsa12 − 3β(I)τN (a211 + a212), (16)a22 = −3β(I)τN (a222 + a212),

a12 =τN3

+ τNsa22 − 3β(I)τNa12(a11 + a12).

Here τN = 1/(1 +κ(I)−β(I))I , I = a11 +a12,s = τ0γ.

The calculation results of viscometric functions(13), (14) on the basis of the system of equations (17)are shown in Figures 1-6. Figures 1,2 shows the de-pendence steady shear viscosity η of dimensionlessshear rate s. Stationary shear viscosity is a decreas-ing function of shear rate. Thus with increasing β0 thedeviation of the initial value of steady shear viscosityincreases, too.

Figures 5,6 shows the ratio of the second normalstress difference to the first defined by (14). This ratiois less than zero and small in absolute value. The pa-rameter β0 affects on the ratio significantly, whereasκ0 - insignificantly. Also influences the coefficient ofthe first normal stress difference Ψ1 , shown in Fig-ures 3,4. All these calculations are in good agreementwith the experimental data.

In the case of uniaxial tension stationary systemof equations (10) and (11) takes the following form

3β(I)τNsb211 + (1− 2τNs)b11 =

2

3τNs, (17)

3β(I)τNsb222 + (1 + τNs)b22 =

1

3τN ,

where I = (b11 + 2B22)s, s = τ0ν11 and bii = aii/s.In this case, the behavior of the polymer is char-

acterized by a stationary elongation viscosity (15),which dependencies of the dimensionless elongationrate for various parameter values κ0, β0 and areshown in Figure 7. At high tension rates λ(s) be-comes constant (λ∞ ∼ 1/β0), if p = 0. When p is

Figure 1: Steady shear viscosity coefficient as a func-tion of dimensionless shear rate s for p = 0 and dif-ferent parameter values κ0, β0.

Figure 2: Steady shear viscosity coefficient as a func-tion of dimensionless shear rate s for p = 0.001 anddifferent parameter values κ0, β0.



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Figure 3: Coefficient of the first normal stress differ-ence as a function of dimensionless shear rate s forp = 0 and different parameter values κ0, β0.

Figure 4: Coefficient of the first normal stress dif-ference as a function of dimensionless shear rate sforp = 0 and different parameter values κ0, β0.

Figure 5: The ratio of the second normal stress differ-ence to the first as a function of dimensionless shearrate s for p = 0 and different parameter values κ0, β0.

Figure 6: The ratio of the second normal stress differ-ence to the first as a function of dimensionless shearrate s for p = 0 and different parameter values κ0, β0.



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Figure 7: Dependence of the steady state elongationviscosity coefficient λ of dimensionless elongationrate s for different parameter values κ0, β0 and p.

different from zero dependence of viscosity in tensiongoes through a maximum. This corresponds to calcu-lations based on similar rheological models, such asProkunin-Leonov model [2] or pom-pom model [10-12].

Obtained non-linear systems of algebraic equa-tions (16) and (17) were solved with the method ofsequential approximations and a modified Newton’smethod, and then the solutions are compared witheach other.

Shown in Figures 1-7 dependencies demonstratequalitative accordance of the model (10), (11) to theactual behavior of solutions and melts of linear poly-mers in the flow. Most often stady state shear viscositycoefficient η and first normal stress difference N1 areinvestigated in experiments. By choosing parametersη0, τ0, κ0, β0 and p it’s possible to describe depen-dences η(ν12) and N1(ν12) for some values of molec-ular weight M and concentration c of polymer quiteaccurately. Therefore, the greatest interest is the prob-lem of describing a series of experiments on steadysimple shear for the given polymer at a different val-ues M and c. The solution of this problem can befound in [26,27] for the case (p = 0), which showsthe applicability of a rheological model (10), (11) todescribe the steady shear flow solutions and melts oflinear polymers in a wide range of shear rates. Also,it is established that in the description of the dynam-ics of linear polymer model parameters κ0 and β0 areweakly dependent on the molecular weight and con-centration, which can be the basis for consideration in

Figure 8: Establishment of simple shear viscosity.

this model, the effects associated with polydispersityof the polymer material.

Now let us consider unsteady effects on the basisof the rheological model (10), (11). To do this we con-sider the problem of establishment stresses in simpleshear, after instant application of shear deformation ata constant shear rate ν12 . It is convenient so because itis possible to compare the results obtained, both withthe experimental data [30] and with calculations basedon other models [2,10-12].

The calculation of viscometric functions (13-15)for the system (10), (11) is carry out. In this case, theequation for the non-zero components of anisotropytensor takes the following form

d

dta11−2γ12+

1 + (κ− β)Iτ0

a11 = −3

τ0β(a211+a

212),

d

dta12 − γ22 +

1 + (κ− β)Iτ0

a12 =

=1

3− 3τ0βa12I, (18)

d

dta22 +

1 + (κ− β)Iτ0

a22 = −3

τ0β(a222 + a

212),

Since the deformation starts from the steady state,the system of differential equations (18) should besolved with the initial conditions aij(0) = 0. Numer-ical calculations of the system (18) were produced byRunge-Kutt method with fourth-order accuracy.

We can construct establishment viscosity in shearand uniaxial tension following similar actions. Thecalculated dependences are given in Figures 8 and



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Figure 9: Establishment stresses in uniaxial elonga-tion.

9. In the case of a simple shear we are seeing non-monotonic output of viscosity on steady-state value(Figure 8). And in the case of a uniaxial elongation- monotonic output of viscosity on steady-state value(Figure 9).

Thus, the model (10), (11) can be chosen as theinitial approximation in the description of nonlinearand viscoelastic properties of solutions and melts oflinear and branched polymers. The obtained resultsonly qualitatively coincide with experimental datapresented in [27,28] and shown in Figure 10. The cal-culation results, despite the simplicity of the system(10), (11) or (18), predicts non-monotonic stress set-ting σ12(γt) acceptable values for in the middle rangeof values γ and time t.

Analyzing the above we should not expect thatthe system of equations (10) and (11) will be fully ad-equate to the real data because only single relaxationprocess was taken into account. Also, the system ofequations (10), (11) does not allow to describe the de-pendence of the dynamic shear modulus on frequency[24]. Therefore the further study of the rheologicalmodel that takes into account the multiplicity of re-laxation times is of great interest.

4 Multi-mode rheological model

The multi-mode character of dynamics of polymericfluids or multiplicity of relaxation processes is appar-ent already in the case of analysis for monodispersepolymer solutions with flexible chains [13, 24]. Thisis first of all connected with topology of branchedpolymer molecules. Obtained relaxation times independence on a mode number exhibit vanishing

Figure 10: Comparison of experimental [30] and the-oretical dependences to establish shear viscosity andelongation viscosity for single-mode model.

behaviour inversely proportional to the mode num-ber, in other words so-called Rouse character. In-crease in concentration of polymer in a system re-sults in macromolecular entanglements, the dynam-ics becomes more complex, long-scale interrelationsoriginate between the macromolecular parts. This isreflected in the additional components in the corre-sponding stress tensor of a polymeric system or in re-spect to new relaxation processes with relatively lowtime relaxations [16]. The rheological relations ob-tained in these papers sufficiently describe the rela-tions of linear viscoelasticity, dependence of storageand loss moduli on frequency. A description of non-linear behaviour represents another important prob-lem in the dynamics of polymeric fluids.

It is apparent that in the case of materials exhibit-ing polydispersity or branched topology, multiplicityof relaxation processes plays still a more importantrole. In this case the behaviour of the characteris-tics of relaxation process (its relaxation time and itsparticipation in stress tensor) has no longer the Rousecharacter and generation is necessary.

Regardless of this, for a description of dynamicsof concentrated polymeric systems, a perspective rhe-ological model should be proposed based on a meso-scopic approach. This approach taking into account adescription of comparatively slow heat macromolec-ular movement uses such model representations ase.g. a dashpot-spring model. Apart from Rouse pa-rameters (friction coefficient of a dashpot, equilib-rium dimension of macromolecular coil, etc.), the ad-ditional parameters are necessary to be included inthe description of concentrated systems correspondthe nature of entanglement in the concentrated poly-



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meric systems. In this sense we will use the modifiedVinogradov-Pokrovskii model [25-30]. An advantageof this model is respecting a tensor character of fric-tion coefficient of the dashpots related with supposedanisotropy of shear flow. Such anisotropy is deter-mined by the dimensions and form of a macromolec-ular coil which reflects additional parameters in theequations of motion. The generalisation of this modelto a multi-mode case proposed in [28,29] is of the fol-lowing form

σik = −p̂δik + 3n∑

α=1

ηαταaαik, (19)

d

dtaαik − νijaαjk − νkjaαji +

1 + (κα − βα)aαjjτα

aαik =

=2

3γik −

3β

ταaαija

αjk,

where p̂ is hydrostatic pressure, σik is the stress tensor,νij is the velosity gradient tensor, γik is the symmet-rical part of velosity gradient tensor, α is the modenumber, n is a number of relaxation modes or pro-cesses, aαik is the dimensionless extra stress tensor cor-responding to α mode, ηα is the coefficient of shearviscosity of α mode, τα is the relaxation time of αmode, κα = κα(aαjj) and βα = βα(a

αjj) are the

parameters of the suggested anisotropy given by thefunctions

βα(aαjj) =

fα + pαaαjj

1 + pαaαjj,

κα(aαjj) = 1, 2 · βα(aαjj). (20)

The equations presented in [23,24] are first for-mulated in a multi-mode approximation and conse-quently simplified. While deriving the relations (20)it was supposed that the parameters of the suggestedanisotropy do not depend or depend negligibly on anumber of modes. However, the comparison withthe experimental data proved this assumption wrong.Hence, it is necessary to return to the multi-mode ap-proximation. It is expected that an application of morerelaxation modes will improve the description of thematerials used and agreement with the experimentaldata.

Influence of the parameters of the suggestedanisotropy is different for each mode. Therefore, itis useful to investigate each mode separately as givenby the formulation of the expressions (20). For settingthere was used the relation κα(aαjj) = 1, 2 · βα(aαjj)was used as proposed in [25].

Hence, in the set of equations (19,20) parametersηα,τα, pα and fα are only unknowns and should be de-termined by means of the experimental data. A num-ber of these parameters are not low, especially with anincreasing number of the modes. Therefore, it is nec-essary to present their determination more carefully.

5 Determination of the parametersfor multi-mode rheological model

It is obvious that the accuracy between a theoreti-cal prediction and the experimental data representsa crucial factor for choosing an adequate rheologi-cal model. It is feasible to use such different ma-terials with the values of some parameters close andothers are different. Such comparison for two differ-ent polymer melts is presented in [29,30].The deter-mining relations were obtained using the experimentaldata in [30] presented the rheological characteristicsof two highly-branched polyethylenes of low density,namely LDPE Bralen RB0323 (Slovnaft, Slovakia)and LDPE Escorene LD165BW1 (Exxon, USA). Theexperiments were carried out at the temperature of200oC. In the paper [30] storage and loss moduli weremeasured by different procedures, a determination ofstresses, transient and steady behaviour of shear vis-cosity and the coefficient of the first normal stress dif-ference. Capillary rheometer RH7 (Rosand Precision)was used for a determination of rheological character-istics under high shear rates. The coefficient of thefirst normal stress difference was determined using aslit geometry. A uniaxial elongational viscosity wasmeasured using the SER Universal Testing Platform(Xpansion Instruments, USA) housed in the ARES2000 rheometer (TA Instruments, Germany). The ex-perimental details are presented in [30].

First, it is necessary to determine parameters andwith the dimensions of viscosity and time, respec-tively. If we solve the second equation of the set (19)with the accuracy up to the components of the firstorder with respect to the velocity gradient, we obtain

d

dtaαjk +

1

ταaαjk =

2

3γik

This equation coincides with Maxwell equationand its usage provides the relations of linear viscoelas-ticity. From here and from (19) we get the relations fordetermination of storage moduli G′(ω) and loss mod-uli G′′(ω) which can be obtained easily [1,2,22]

G′(ω) =n∑

α=1

ω2ταηα1 + (ωτα)2

,



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Figure 11: Comparison of the shear and loss modulivs. frequency dependence with the experimental datafor Bralen [30].

G′′(ω) =n∑

α=1

ωηα1 + (ωτα)2

. (21)

These relations coincide with the results pre-sented in other rheological models [29,30,33,34]. Theprocedure of determination of ηα and τα is based onleast-square minimization.

The values of the moduli calculated according to(21) can be compared with the experimental data. Thecalculations were carried out for 10 modes and thevalues of the parameters obtained by an optimizationprocedure between the calculations and the experi-ments given in Table 1. Relaxation spectra were cal-culated in [30] from the loss and storage moduli by thehelp of the generalized Maxwell model. The values ofviscosity parameterηα were determined with the helpof relation: ηα = gατα. The fitting curves of the gen-eralized Maxwell model for both described materialsare shown in Figures 11,12.

To determine the values of the parameters ofanisotropy in the paper [30] contributions of eachmode and the values of fα and pα were determined ateach part of steady viscosity dependence under uniax-ial elongation. From here it followed that both fα andpα are non-monotonous functions of mode number α.At the beginning the values of these parameters areincreasing with increasing α, then they are decreas-ing. They attain their maxima for α = α0. Hence, inthe following we will approximate these parametersby the expressions

fα =B

1 + (α− α0)2, pα =

P

1 + (α− α0)2. (22)

Figure 12: Comparison of the shear and loss modulivs. frequency dependence with the experimental datafor Escorene [30].

Let us analyses these relations in more detail. Amode denoted by α0 corresponds to a mean value ofmolecular weight Mn. It is possible to expect that ashape and size of a macromolecular coil correspond-ing toMn will exhibit the highest influence on the dy-namics of polymer liquid. The fractions which molec-ular weight apparently differs from Mn have negligi-ble impact on nonlinear viscoelastic properties of sucha system. It implies that for α → ∞ the parame-ters fα and pα will attain infinitely low values. Natu-rally, in reality this limited passage is not possible toput into effect, however, when the difference betweenmode numbers attain 5 then their influence attenuates25 times.

6 Viscometric functions for simpleshear flow and uniaxial elongationflow

Now we analyses the influence of parameters B andP (22) on the relation steady shear viscosity vs. shearrate. The calculations of steady and transient viscosi-metric functions for the cases of simple shear and uni-axial elongation according to the model (19,22) followthose introduced in paragraph 3 for a one-mode case.Therefore, only the final results will be presented.

Figures 13 and 14 show that the curves describinga course of steady uniaxial elongational viscosity ex-hibit non-monotonous character with one or more in-flexion points. The maximal value decreases with theincreasing parameter B. The non-monotonous char-acter converts to the monotonous one forP = 0. Itshows that by the suitable choice of parameters B and



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Figure 13: Relation of steady uniaxial elongation vis-cosity vs. elongation rate (constant P = 0.001) de-pending on parameter B.

Figure 14: Relation of steady uniaxial elongation vis-cosity vs. elongation rate (constant B = 0.2) depend-ing on parameter P .

Figure 15: Comparison of the model predictions forfor steady shear and elongational viscosity with theexperimental data for Escorene [30].

P it is possible to describe a course of steady uniax-ial elongational viscosity in dependence on elongationrate with sufficient accuracy. The question is whetherwith such determined B and P it is possible to de-scribe steady and transient effects for simple shear anduniaxial elongation. Figures15,16 depicts the coursesof steady shear and uniaxial elongation viscosity. Itfollows that the model (19) describes both viscositieswith sufficient accuracy. It is obvious that Cox-Merzrule is fulfilled for the model [30].

Figures 17-20 show the courses of transient shearviscosity and the coefficient of the first normal stressdifference in the case of simple shear. It follows thatfor low values of shear rates the passages to the lim-ited stationary values are monotonous while for highervalues the passages attain their maxima.

Figures 21,22 presents the courses of transientuniaxial elongation viscosity for various values ofelongation rates. The calculation shows monotonousbehaviour of elongation viscosity up to stabilized val-ues. These values are attained more rapidly for highervalues of extension rates. The model (19) depicts ac-curately the initial parts of viscosity curves. Similarly,the experimental data for higher extension rates is ac-companied by non-smoothness. As this factor is nottaken into account in the model (19), the predictionand experimental data for higher elongation rates dif-fers. We should mention that this shortcoming is alsopresent in other rheological models [33,34].

The calculations were carried out with the follow-ing choice of the parameters:B = 0.5, P = 0.002,α0 = 6.5 for Escorene and B = 0.65, P = 0.002,α0 = 5 for Bralen.



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Figure 16: Comparison of the model predictions forfor steady shear and elongational viscosity with theexperimental data for Bralen [30].

Figure 17: Comparison of the model predictions fortransient shear viscosity with the experimental datafor Escorene [30].

Figure 18: Comparison of the model predictions fortransient shear viscosity with the experimental datafor Bralen [30].

Figure 19: Comparison of the model predictions forthe coefficient of the first normal stress difference withthe experimental data for Escorene [30].



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Figure 20: Comparison of the model predictions forthe coefficient of the first normal stress difference withthe experimental data for Bralen [30].

Figure 21: Comparison of the model predictions fortransient uniaxial elongation viscosity with the exper-imental data for Escorene [30].

Figure 22: Comparison of the model predictions fortransient uniaxial elongation viscosity with the exper-imental data for Bralen [30].

7 Conclusion

The papers [33,34] presents a comparison of the givenexperimental data with other rheological models suchas Prokunin and Leonov model [2], the pom-pommodel the extended Pom-Pom model[ 10-13]. Thecomparison of the experimental data with the pre-dictions of other models shows that the introducedmodel (20) provides equivalent accuracy to the ex-tended pom-pom model and better than the modelsin [2,10], and of course much better results whenonly one-mode approximation is used [24,27]. Basedon the preceding comparisons, it is possible to con-clude that the multi-mode model (19) describes steadyand transient rheological characteristics of polymermelts with a sufficient accuracy. It is possible to ex-pect that the model (19) is also applicable for high-concentrated polymer suspensions and linear polymermelts. This model is also possible to apply for morecomplex flows of polymer liquids, for example three-dimensional flow in the converning channel [31], orfor investigate the stability of inhomogeneous flows[32].

Acknowledgements: The authors are grateful to theanonymous referees of the previous versions of thepaper for detailed analysis and helpful commentsand appreciate the fruitful discussions with Prof.Pokrovskii. Authors (GVP and MDA) acknowledgesfinancial support of the Russian Foundation for Ba-sic Research (RFBR) under grant 15-41-04003. Au-thors (PF and RP) acknowledges financial support ofthe GA CR, grant No. 103/09/2066.



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References:

[1] J.G. Oldroyd, On the formulation of rheologi-cal equations of state, Proc R Soc. A 200 (1950)523-541

[2] Leonov A.I., Prokunin A.N., Nonlinear Phe-nomena in Flows of Viscoelastic Polymer Flu-ids, Chapman and Hall, New York.(1994).

[3] Rouse P.E., A theory of the linear viscoelasticproperties of dilute solutions of coiling poly-mers. J. Chem. Phys. 21 (1953) 1271-1280

[4] Kargin V.A., Slonimskii G.A., On the deforma-tion of amorphous liquid linear polymers. Dok-lady Russian Academy of Sciences. 62, (1948),vol. No. 2, 239-242 (in Russian)

[5] Doi M, Edwards SF. The theory of polymer dy-namics. Clarendon, Oxford (1986)

[6] Gennes De, Scaling concepts in polymerphysics, Cornell University Press, Ithaca (1979)

[7] Bird RB, Curtiss CF, Armstrong RC, HassagerO. Dynamics of polymeric fluids. Wiley, NewYork. 2 (1987)

[8] Ottinger H.C. A thermodinamically admissiblereptation model for fast of flows entangle poly-mer. J. Rheol. 43 (1999) p. 1461-1493.

[9] Likhtman A, McLeish T. Quantitative theory forlinear dynamics of linear entangled polymers.Macromolecules. 35 (2002) p. 6332-6343

[10] BishkoG., McLeish T.C.B.,Harlen O.G., Lar-son R.G., Theoretical molecular rheology ofbranched polymers in simple and complex flows:The Pom-Pom Model. Phys. Rev. Letters. 79(1997), 2352-2355.

[11] McLeish T.C.B., Larson R.G. Molecular consti-tutive equations for a class of branched poly-mers: the pom-pom polymer. J. Rheol. 42 (1998)v. p. 81-110

[12] Verbeeten W.M.H., Peters G.W.M. Differentialconstitutive equations for polymer melt: the eX-tended Pom-Pom model. J. Rheol. 45 (2001) p.821-841

[13] Pokrovskii V.N., Dynamics of weakly-coupledlinear macromolecules. Physics-Uspekhi (Adv.Phys. Sci.). 35 (1992) No. 5, 384-399.

[14] Pokrovskii V.N., The mesoscopic theory of poly-mer dynamics, 2nd Ed., Springer, Dordrecht-Heidelberg-London-New York. (2010)

[15] Pokrovskii V.N., Pyshnograi G.V., Simple formsof determining equation of concentrated poly-mer solutions and melts as a consequence ofmolecular viscoelasticity theory. Fluid Dynam-ics. 26 (1991) 58-64.

[16] Pokrovskii V.N., Volkov V.S., The calculation ofrelaxation time and dynamical modulus of linearpolymers on one-molecular approximation withself-consistency (A new approach to the theoryof viscoelasticity of linear polymers). PolymerScience USSR 20 (1978), 3029-3037

[17] Remmelgas J., Harrison G., Leal L.G. A differ-ential constitutive equation for entangled poly-mer solutions. J. Non-Newton. Fluid. 80 (1999)N. 2-3. p. 115-134.

[18] Wagner M., Kheirandish S., Hassager O. Quan-titative prediction of transient and steady-stateelongational viscosity of nearly monodispersepolystyrene melts. Journal of Rheology. 49(2005) p.1317-1327

[19] Wagner M., Roln-Garrido V. The interchainpressure effect in shear rheology. RheologicaActa. 49 (2005) p.459-471

[20] Zatloukal M. Differential viscoelastic constitu-tive equations for polymer melts in steady shearand elongational flow. J. Non-Newtonian FluidMech. 209 (2003) p. 11-27.

[21] Graessley W.W., The entanglement concept inpolymer rheology. Adv. Polym. Sci. 16 (1974),1-179.

[22] W.W. Graessley, Entangled linear, branchedand network polymersystemsmolecular theories,Adv. Polym. Sci. 47 (1982) 67-117.

[23] Pyshnograi G.V., Pokrovskii V.N., YanovskiiYu.G., Karnet Yu.N., Obrazcov I.F. , Equationof state for nonlinear viscoelastic (polymer) con-tinua in zero-approximations by molecular the-ory parameters and sequentals for shearing andelongational flows. Doklady Russian Academyof Sciences 335 (1994) No. 9, pp. 612-615 (inRussian).

[24] Pyshnograi G.V., Gusev A.S., Pokrovskii V.N.,Constitutive equations for weakly entangled lin-ear polymers. J. Non-Newt. Fluid Mech. 163(2009), 17-28.

[25] Golovicheva I.E., Zinovich S.A., PyshnograiG.V., Effect of the molecular mass on the shearand longitudinal viscosity of linear polymers, J.



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Appl. Mech. Techn. Phys. 41 (2000), No. 2, 347-352.

[26] Gusev A.S., Makarova M.A., Pyshnograi G.V.,Mesoscopic equation of state of polymer sys-tems and description on the dynamic character-istics based on it. J. Eng. Phys. Thermophys. 78(2005), No.5, 892-898.

[27] Joda H.N.A. Al , Afonin G.L., Merzlikina D.A.,Filip P., Pivokonsky R., Pyshnograi,G.V. Mod-ification of the internal friction law in meso-scopic theory flowable polymer media. J. Comp.Mech. Design (Mekh. Komp. Mater. Konstr.).19(2013), No. 1, 128-140 (in Russian).

[28] Merzlikina D.A. , Filip P., Pivokonsky R.,Pyshnograi G.V. , Multimode rheological modeland findings for simple shear and elongation.J. Comp. Mech. Design (Mekh. Komp. Mater.Konstr.) 19 (2013), No. 2, 254-261 (in Russian)

[29] Pivokonsky R., Zatloukal M., Filip P., On thepredictive/fitting capabilities of the advanceddifferential constitutive equations for branchedLDPE melts. J. Non-Newt. Fluid Mech. 135(2006), 58-67.

[30] V.H. Roln-Garrido, R. Pivokonsky, P. Filip,M. Zatloukal, M.H. Wagner, Modelling elonga-tional and shear rheology of two LDPE melts.Rheol. Acta 48 (2009) 691-697.

[31] Koshelev K.B., Pyshnograi G.V., Tolstych M.Y.,Simulation of three-dimensional flow of thepolymer melt in a converging channel witha rectangular cross section, Fluid Dynamics(2015) issue 3

[32] N.V. Bambaeva and A.M. Blokhin, The t-hyporboliscity of a nonstationary system gov-erning flows of polymeric media, Journal ofMathematical Science. 188 (2013) 333-343

[33] Pivokonsky R., Filip P., Predictive/fitting ca-pabilities of differential constitutive models forpolymer melts-reduction of nonlinear parame-ters in the eXtended Pom-Pom model// Colloidand Polymer Science 292 (2014) 2753-2763

[34] Pivokonsky R., Filip P., Zelenkova J., The roleof the GordonSchowalter derivative term in theconstitutive modelsimproved flexibility of themodified XPP model// Colloid Polymer Science293 (2015) 1227-1236



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Table 1: Parameters gα, τα and ηα for polymeric ma-terials.

α gα, [Pa] τα, [s] ηα, [Pa · s] gα, [Pa] τα, [s] ηα, [Pa · s]LDPE Bralen RB0323 LDPE Escorene LD165BW1

1 121440 0.00134 162.7 109430 0.00154 168,52 35290 0.0052 183.5 37350 0,00634 236,43 33443 0.0202 673.9 32409 0,026 843,34 19480 0.078 1520 15250 0,107 16305 11923 0.302 3604 11080 0,439 147106 5764 1.171 6749 4836 1,802 87047 2575 4.54 11680 1986,65 7,404 147108 800.9 17.57 14070 494,7 30,4 150469 213.4 68.07 14530 110,16 125 1376010 34.69 263.7 9146 33,38 513 17126



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