Rheol Acta (2009) 48:301–316DOI 10.1007/s00397-008-0318-8

ORIGINAL CONTRIBUTION

Energy elastic effects and the concept of temperaturein flowing polymeric liquids

Markus Hütter · Clarisse Luap ·Hans Christian Öttinger

Received: 1 April 2008 / Accepted: 16 September 2008 / Published online: 30 October 2008© Springer-Verlag 2008

Abstract The incorporation of energy elastic effects inthe modeling of flowing polymeric liquids is discussed.Since conformational energetic effects are determinedby structural features much smaller than the end-to-end vector of the polymer chains, commonly employedsingle conformation tensor models are insufficient todescribe energy elastic effects. The need for a localstructural variable is substantiated by studying a micro-scopic toy model with energetic effects in the settingof a generalized canonical ensemble. In order to ex-amine the dynamics of flowing polymeric liquids withenergy elastic effects, a thermodynamically admissibleset of evolution equations is presented that accountsfor the evolution of the microstructure in terms of aslow tensor, as well as a fast, local scalar variable. It isdemonstrated that the temperature used in the defini-tion of the heat flux is directly related to the Lagrangemultiplier of the microscopic energy in the generalizedcanonical partition function. The temperature equationis discussed with respect to, first, the dependence ofthe heat capacity on the polymer conformation and,second, the possibility to measure experimentally theeffects of the conformational energy.

Keywords Energy elastic effect ·Local structural variable · Viscoelastic fluid ·Nonequilibrium thermodynamics ·Generalized canonical ensemble

M. Hütter (B) · C. Luap · H. C. ÖttingerDepartment of Materials, Polymer Physics,ETH Zürich, Wolfgang-Pauli-Str. 10,8093 Zürich, Switzerlande-mail: markus.huetter@mat.ethz.ch

Introduction

Entropy elasticity of rubbers (Treloar 1975) serves as astarting point for most of the current models to describethe flow of polymeric liquids, while consideration of en-ergetic effects is scarce. Such exclusive subscription toentropy elasticity derives from the following contrast.The slowest degrees of freedom, which are of primaryimportance for the rheology under slow and mod-erate deformation rates, describe large-scale featuresin the polymeric liquid, whereas the energetic effectsdepend on local aspects of the chain conformationsthat have much faster relaxation processes. Further-more, in particular, the variables describing the largescales lead to significant contributions to the entropydue to the large number of corresponding microstates.Nevertheless, it has recently been demonstrated usingconfigurationally biased Monte Carlo simulations thatenergetic effects of the polymer conformation indeedbecome relevant, particularly at large deformations,which supports the significance of fast local degreesof freedom (Mavrantzas and Theodorou 1998; Ionescuet al. 2008b). For these reasons, it is interesting todiscuss the possible ramifications of energy elasticityon the flow behavior of polymeric liquids. In order toarrive at a model suitable for describing conformationalenergetic effects, the following questions must be ad-dressed: (1) What is a suitable choice of variables tocapture energy elastic effects? (2) How can energeticand entropic contributions be distinguished properly inthe thermodynamic potentials?

The proper choice of variables is of fundamentalimportance in modeling complex fluids, as it is oftenstated and also well known. The key aspect of theword “proper” means that the set of variables must be

302 Rheol Acta (2009) 48:301–316

sufficiently detailed to capture the physics one is in-terested in describing. Since polymer chains departsignificantly from equilibrium upon the application offlow, structural variables are chosen to capture thatdeviation (Bird et al. 1977; Larson 1988), including thedistribution function p(R) of the end-to-end vector Rbetween chain ends or the second moment c = 〈RR〉,the distribution function p(n, s) of the tangent vectorn and label s along the primitive path in the reptationmodel, or the local chain stretch λ (Öttinger 1999).Although these choices differ in their applicability topolymeric solutions and melts, respectively, a featurecommon to them is that they are usually employedto describe entropy elasticity only, while energetic ef-fects are neglected, a preference that may be tracedback to the works of Astarita (1974), Astarita andSarti (1976), and Sarti and Esposito (1977). Exceptionsin this regard are some applications of conformationtensor models, in which the entropic spring for R isreplaced by a spring with mixed entropic and ener-getic character (Braun and Friedrich 1990; Peters andBaaijens 1997; Wapperom and Hulsen 1998; Dressleret al. 1999). Such an extension is certainly admissibleif c is to remain the only relevant structural variablein the model. However, by virtue of the physical originof energetic effects, e.g., as described by the rotationalisomeric state (RIS) model (Flory 1988; Mattice andSuter 1994) or the local packing of chain segments, itseems that structural features on scales much finer thanthe end-to-end vector must be resolved to representthe energetics appropriately. Ignoring the existenceof an independent local variable and instead linkingit statically to the conformation tensor seems inade-quate and would be a severe restriction of the physics,in particular with regards to the dynamics, since themore local the structural feature is, the more rapid isits associated relaxation processes (Bird et al. 1977;Larson 1988). Local structural variables have been in-troduced previously for the modeling of flowing poly-meric liquids, e.g., in some versions of the reptationmodel (Ianniruberto and Marrucci 2001; Marrucci andIanniruberto 2003) and in the pompon model for meltsof multiply branched polymers (McLeish and Larson1998; Öttinger 2001; Soulages et al. 2006). While, inboth of these examples, the scalar structural variablerepresents local chain (segment) stretch that relaxessubstantially faster than the tensorial variable, the stepto incorporate energy elasticity has not been taken infull generality by Öttinger (2002, 2005).

The distinction of energetic and entropic effectsbased on a thermodynamic potential in nonequilibriumsituations is more involved than it may seem at firstsight. To address the energy–entropy separation in

equilibrium thermodynamics, one usually departs fromthe free energy F(T, X) with absolute temperature Tand extensive variables X, based on which, the quan-tity E= F−T ∂ F/∂T|X is identified with the energyof the system. This identification can be substantiatedwith statistical mechanics since, in the canonical en-semble, E is the ensemble average of the microscopicHamiltonian (Reichl 1980; Landau and Lifshitz 1980;Callen 1985). Clearly, this procedure for identifica-tion of energetic and entropic contributions rests onthe clear concept of temperature, which is, how-ever, a cumbersome one in beyond equilibrium situ-ations (Grmela 1998; Casas-Vázquez and Jou 2003).Therefore, one must examine critically the ways toachieve the energy–entropy separation based on out-of-equilibrium Helmholtz free energy expressions, asimplicitly invoked, e.g., in Braun and Friedrich (1990),Peters and Baaijens (1997), Wapperom and Hulsen(1998), and Dressler et al. (1999). In order to clarify themeaning of T as a variable out of equilibrium similarto the scenario in equilibrium, one can envision that,also out-of-equilibrium, a statistical mechanics treat-ment must come to the aid to bring clarification. Toour knowledge, such steps have not yet been taken inliterature.

In order to address these issues, the manuscript isorganized as follows: We start with a microscopicallymotivated toy model that includes conformational en-ergetic effects in terms of different RIS states (Flory1988; Mattice and Suter 1994). For this model, boththe coarse-grained conformational energy and the con-formational entropy are calculated in a generalizedcanonical ensemble and expressed in terms of the con-formation tensor c and a local structural variable λ. Inorder to study the dynamic behavior of such a systemunder flow, the general equation for the nonequilibriumreversible–irreversible coupling (GENERIC) frame-work (Grmela and Öttinger 1997; Öttinger and Grmela1997; Öttinger 2005) is then used to formulate a closedset of evolution equations that includes the dynamics ofc and λ, apart from the usual hydrodynamic variables.The resulting model is discussed with regards to theramifications of conformational energetic effects andthe use of a temperature-like dynamic variable, and isillustrated for the previously introduced toy model.

Toy model

With the aid of a simple toy model, which can be treatedanalytically, it will be shown that energetic effects of thepolymer conformation are described with a structuralvariable that resolves length scales significantly more

Rheol Acta (2009) 48:301–316 303

local than what is characterized by the conformationtensor. To that end, a generalized canonical ensembleis considered to relate the microscopic understandingabout the chain with the level of a thermodynamicstate function, valid also out of equilibrium. For thesake of keeping the model tractable, we consider onlyintrachain energetic effects. Possible extensions are ad-dressed in the “Summary and perspectives” section.

While in the usual canonical ensemble, the averagevalue of the energy H is conveniently controlled by theLagrange multiplier β = 1/(kBT), out-of-equilibriumsituations of polymeric systems involve further (exten-sive) variables {Xi}i=1,...,k, the averages of which canbe controlled by corresponding Lagrange multipliers{αi}i=1,...,k. As an example, one structural variable canbe the dyadic of the end-to-end vector, X1 = RR. Thegeneralized canonical partition function can formallybe written as (Öttinger 2005)

Z (β, {αi}) =∑

PS

e−βH−∑k

i=1 αi Xi , (1)

where the summation (integration) runs over all of theavailable phase space (PS). Based on that partitionfunction, one can derive expressions for the entropy Sand for the averages of H and Xi,

S = kB(

ln Z + β〈H〉 +k∑

i=1αi〈Xi〉

), (2a)

E ≡ 〈H〉 = − ∂ ln Z∂β

∣∣∣∣αi

, (2b)

〈Xi〉 = − ∂ ln Z∂αi

∣∣∣∣β,αk,k�=i

, (2c)

all expressed as functions of β and {αi}i=1,...,k. Withthe access to the Lagrange parameters αi in thepartition function Eq. 1, one has a handle on the av-erages 〈Xi〉, which means that out-of-equilibrium situ-ations can be generated in this way, as used in MonteCarlo simulations of polymer melts (Mavrantzas andTheodorou 1998; Mavrantzas and Öttinger 2002; Baigand Mavrantzas 2007; Ionescu et al. 2008b). Using ageneralized canonical partition function of the formEq. 1, it is straightforward to show that the following,model-independent, relations between the entropy andenergy defined in Eq. 2 hold,

∂(S/kB)∂β

∣∣∣∣〈Xi〉− β ∂ E

∂β

∣∣∣∣〈Xi〉= 0 , (3a)

∂

∂〈Xi〉(

SkB

− βE)∣∣∣∣

β,〈Xk〉,k�=i= αi , (3b)

where it must be noted that the averages 〈X〉 ratherthan the Lagrange multipliers α are held constant upondifferentiation.

As a specific example for illustrating conformationalenergetic effects, we consider a single linear polymerchain composed of N (united) atoms with variable bondlengths, bond angles, and torsion angles. To renderanalytic calculations for this toy model feasible andtransparent, we assume that the conformation of sucha chain can be described by (1) well-defined RIS (Flory1988; Mattice and Suter 1994) with fixed bond length,fixed bond angle, and discrete values for the torsionangle, and (2) local vibrations around each RIS state.While the potentials describing the bond length andbond angle degrees of freedom shall have a single min-imum each, the torsional degree of freedom is assumedto take three states (local minima): a trans state (t) ofzero energy by definition, and two gauche states (g+and g−) with a energy penalty of �, each with respectto the trans state. This model is representative of poly-ethylene, and relates to the potentials for bond length,bond angle, and bond torsion as used in realistic unitedatom simulations (Paul et al. 1995; In ’t Veld et al.2006). The number of gauche states along the chainare denoted by n, and hence, �n is the instantaneousconformational energy of a chain.

The energy and entropy for the toy model definedabove can be calculated as follows. In order to describea single polymer chain in situations out of equilibrium,we introduce α and μ as the Lagrange multipliers tothe structural variables X1 = RR and X2 = n, respec-tively, and calculate the generalized canonical partitionfunction Z (β, α, μ). It can be shown (see Appendix Afor details) that the expressions for the energy andentropy of a single chain (sc) of length N in terms of β,the conformation tensor c = 〈RR〉, and the (average)number of gauche states n assume the form

Esc(β, c, n)=(

Floc(β, n)+β ∂ Floc(β, n)∂β

)+�n , (4a)

Ssc(β, c, n)=kBβ2 ∂ Floc(β, n)∂β

+Sconf(n)

+ kB2

[3−tr (c/c0(n))+ln det (c/c0(n))

],

(4b)

where Floc represents the local Helmholtz free energy(Eq. 34) accounting for the local vibrations aroundthe RIS conformations with n gauche states. As dis-cussed in Appendix A, the n-dependence of Floc canbe neglected if the form of the atomistic potentialsaround a RIS conformation with n gauche states iseffectively independent of n, which, e.g., is reasonably

304 Rheol Acta (2009) 48:301–316

well satisfied by the torsion potential for polyethylene.The conformational entropy is given by

Sconf(n) = kB ln(

2n(

N − 2n

)), (5)

which counts the number of possible RIS conforma-tions with n of the N − 2 torsional degrees of free-dom being in the (twofold degenerate) gauche state.In the entropy Eq. 4b, the quantity c0 is related tothe distribution of the end-to-end vector Q of all RISconformations for given n, 〈 Q Q〉 = c0(n)1. Keepingconstant the bond length l and the bond angle ϑ , andwith torsion angle ϕ, one can write for long chains oflength N (Flory 1988)

c0(n) = (N − 1)l2

3C∞

= (N − 1)l2

3

(1 + cos ϑ1 − cos ϑ

)(1 + 〈cos ϕ〉1 − 〈cos ϕ〉

), (6a)

where it has been assumed that the torsion angles areindependent from each other and that the energy of asingle torsion angle ϕ is symmetric in ϕ around ϕ = 0.As the torsion angle in the RIS model can assume onlythree distinct values, e.g., for polyethylene ϕtrans = 0and ϕgauche � ±2, one has

〈cos ϕ〉 = nN − 2 cos ϕgauche +

(1 − n

N − 2)

cos ϕtrans ,

(6b)

i.e., the dependence of c0 on n and N for long chains isof the form f (n/N)N.

The total energy Esc splits into a local and a con-formational contribution depending on n, while theentropy Ssc consists of a local term and two conforma-tional ones, depending on n and c, respectively. Giventhe weak dependence of Floc on n, the influence ofn on the local energy and local entropy, respectively,becomes negligible. Identifying the different contribu-tions is useful for interpreting the deformation of thepolymeric system. For example, the exchange betweenlocal and conformational entropy explains the heatingof an adiabatically stretched rubber. Similar insightscan be expected from accounting for both forms ofenergy, local and conformational. The absence of a c-dependent contribution to the energy is noteworthyand highlights the effect of using the local structuralvariable n.

Since the parameters α and μ in the generalizedcanonical ensemble have been introduced to drive thesystem out of equilibrium, it is now interesting to ex-press them for the specific toy model in terms of the

variables (β, c, n). Application of the identity Eq. 3b tothe single chain energy and entropy Eq. 4 leads to

α = 12

(c−1 − 1

c0(n)1)

, (7a)

μ = −β� + ∂(Sconf/kB)∂n

− β ∂ Floc∂n

− c : α d ln c0dn

. (7b)

In equilibrium, α=0, one thus finds ceq =c0(n)1, whichhighlights the meaning of the function c0(n). Equi-librium furthermore requires μ = 0, from which oneobtains, by virtue of Eq. 7b, a relation of the formneq(β). This in turn leads to the β-dependent confor-mation tensor ceq = c0(neq(β))1 of the polymer chainat equilibrium (see “Implications of the model” sectionbelow for details). On the other hand, if only the localvariable n is equilibrated (μ = 0) but the large-scalevariable c is not (α �= 0), one finds from Eq. 7b byneglecting ∂ Floc/∂n the implicit expression for n/N inthe long chain limit, N − 2 � N,nN

= 2e−β�−Y

1 + 2e−β�−Y , (8a)

Y ≡ 12

[3 − tr(c/c0(n))

] d ln c0(n)dn

. (8b)

In view of the form Eq. 6 for c0, one observes thatd ln c0(n)/dn scales in the long chain limit with fixedn/N as N−1. Therefore, it is reasonable to assume Y �1,which in turn leads to the probability of gauche states

nN

� 2e−β�

1 + 2e−β�(

1 − Y1 + 2e−β�

), (8c)

i.e., linear in the out-of-equilibrium term 3 − tr(c/c0).In order to arrive at an explicit equation for n/N cor-rect to first order in 3 − tr(c/c0), one can replace in Ythe occurrence of n by its value for c = c01.

After these discussions, we proceed with compar-ing our expressions (Eq. 4) with the ones used in theliterature. The c-dependent terms in the entropy arethe well known result found in literature (Sarti andMarrucci 1973; Dressler et al. 1999). However, in Sartiand Marrucci (1973), the unperturbed size of the chaindepends on the temperature, i.e. c0(T), while we haveexplicit access to the number of gauche states and,hence, c0(n). As discussed above, only in the limitof an equilibrated number of gauche states n (i.e.,for μ = 0) do we observe the effect of β on c0. Thecontributions �n and Sconf in the energy Eq. 4a andentropy Eq. 4b, respectively, are new conformationalcontributions. In contrast, in Dressler et al. (1999), aterm proportional to trc is added to the energy torepresent energetic effects. Specifically, the following

Rheol Acta (2009) 48:301–316 305

conformation-dependent contributions to the energyand entropy are used, here expressed for a single chain,

Esc,Dressler(T, c)∣∣conf =

1

2

(K(T)−T dK(T)

dT

)tr(c) , (9a)

Ssc,Dressler(T, c)∣∣conf =

1

2

[−dK(T)

dTtr (c)+kB ln det (c)

],

(9b)

with the overall chain spring constant K(T) withtemperature T. In their case, energetic effects ariseas the dependence of K on T is nonlinear. However,in the light of the microscopic origin of energeticeffects (e.g., in our toy model), it seems to us moredesirable to capture these local effects with a localstructural variable, as exemplified with our toy model.With regard to the approximate form of gauchestates (Eq. 8c), it is clear that, in our approach, theconformational energy �n can be expressed as a termlinear in trc only under special circumstances, namely,if n is equilibrated and if the effect of trc on n is weak.In particular, with the approximation mentioned afterEq. 8c, the energy term linear in trc assumes the form

Esc = −12

d ln c0dβ

tr (c/c0) + . . . , (10)

where the dots stand for terms independent of c. Thederivative of c0 must be understood as a derivative withrespect to n at Y = 0, multiplied by the derivative of nfrom Eq. 8c at Y = 0 with respect to β. It is comfortingto note that the form Eq. 10 is identical to thecorresponding contribution contained in Eq. 9a whenusing their relation K(T) = kBT/c0(T) (Dressler et al.1999; Ionescu et al. 2008a), under the condition T =(kBβ)−1. It will be shown below in the “Discussion”that this condition is indeed satisfied; see Eq. 22.

Formulation of the dynamics

Strategy

Motivated by the above toy model, the use of a lo-cal structural variable in addition to the conformationtensor is promising for addressing the effect of confor-mational energy on the behavior of polymers. In orderto examine the flow behavior, we aim at formulating athermodynamically admissible, closed set of evolutionequations including a scalar and tensorial variable, λand c, for describing the conformation of the polymerchains. To keep this development from being restrictedto the toy model, we do not imply that λ = n, and

we also use the most general form for the energy andentropy as functions of the structural variables. After-wards, the ramifications of the toy model can be studiedas a special case of the general model.

As guard-rail to formulate the dynamic model, wechoose the GENERIC framework by Grmela andÖttinger (Grmela and Öttinger 1997; Öttinger andGrmela 1997; Öttinger 2005) of beyond-equilibriumthermodynamics. In regard to the topic of this paper,this method is particularly suitable for two reasons.First, it is applicable to nonisothermal situations, andsecond, the specification of generalized thermodynamicpotentials, namely energy and entropy, is a key ingredi-ent that allows for a large flexibility of the phenomenaunder consideration. In the following, we summarizeonly briefly those aspects of GENERIC that are ofrelevance for this paper.

When trying to formulate a model in that framework,the first step is to choose a set of variables, x, thatdescribes the situation of interest to sufficient detail.The time evolution of the variables x of an isolatedsystem are then written in the form

∂x∂t

= L(x) · δEδx

+ M(x) · δSδx

, (11)

where the two generators E and S are the total energyand entropy functionals in terms of the state variablesx, and L and M are certain matrices (operators). Thetwo contributions to the time evolution of x generatedby the total energy E and the entropy S are called thereversible and irreversible contributions, respectively.The operator L describes the kinematics, while the op-erator M accounts for a wealth of irreversible processesand, hence, contains material parameters such as dif-fusion coefficients, viscosities, relaxation times, and re-action constants. The matrix multiplications (symbol ·)imply not only summations over discrete indices butmay also include integration over continuous variables,in particular for non-local field theories, and δ/δx typi-cally implies functional rather than partial derivatives.

The GENERIC structure also imposes certain con-ditions on the building blocks in Eq. 11. First, Eq. 11 issupplemented by the mutual degeneracy requirements

L(x) · δSδx

= 0 , (12a)

M(x) · δEδx

= 0 , (12b)representing entropy and energy conservation uponreversible and irreversible dynamics, respectively. Inaddition, L must be antisymmetric, whereas M needsto be positive-semidefinite and Onsager–Casimir sym-metric. As a result of all these conditions, one may show

306 Rheol Acta (2009) 48:301–316

that the GENERIC Eq. 11 implies both the conserva-tion of total energy, as well as a nonnegative entropyproduction. Finally, the GENERIC structure requiresthat the Poisson bracket associated to the operator Lfulfills the Jacobi identity, which expresses the time-structure invariance of the reversible dynamics. Thecomplementary degeneracy requirements, the symme-try properties, and the Jacobi identity are essential forformulating proper L- and M-matrices when modelingconcrete nonequilibrium problems. Various applica-tions have shown that the two-generator idea and thecriteria on the two matrices have strong implications(e.g., Öttinger and Grmela 1997; Öttinger 2002, 2005;Edwards and Öttinger 1997).

Choice of variables x, generating functionals E and S

The toy model in the previous section has illustratedthat, in addition to a conformation tensor c, a localstructural variable, denoted by λ, is useful for capturingconformational energetic effects. Furthermore, in orderto describe nonisothermal flow, we use the “hydrody-namic” variables momentum density u, mass densityρ, and a yet unspecified thermal variable θ . In viewof the toy model and of capturing the fast vibrationsthat are unresolved by λ and c, the variable θ can beinterpreted as related to the Lagrange parameter β, thelocal energy, or the local entropy. In summary, the fullset of variables is thus given by

x = (u, ρ, θ, c, λ) . (13)The tensor c is interpreted as the average c = 〈RR〉introduced in the “Introduction.” In other words, it isa tensor rather than a tensor density and shows upperconvected Maxwell behavior, both of which will be ofrelevance when formulating the reversible dynamics.Note that, in the above choice of structural variables,it is implicit that there are no intermediate time scalesbetween what is described by c and λ, respectively. Inother words, the following description is supposed tobe reasonable for relatively short chains as far as quali-tative aspects of the energetic effects are concerned. Ifone were interested in quantitative predictions, morevariables would have to be included.

The functionals for the total energy and the totalentropy in terms of the variables x are written in theform

E =∫ (

u2

2ρ+ et(ρ, θ, c, λ)

)d3r, (14a)

S =∫

st(ρ, θ, c, λ)d3r, (14b)

with the only assumption that the total energy is thesum of a kinetic contribution plus an internal energyterm, the latter being independent of the momentumdensity. In both expressions, the symbols et and st standfor both local and conformational effects. With theset of variables Eq. 13 and by keeping the functionset and st general, the evolution equations developedin Öttinger (2002, 2005) are extended as to achievehighest flexibility in modeling conformational energeticeffects.

For the microscopic toy model introduced above,one can choose θ = (kBβ)−1 and λ = n (or λ = n/N).The functions et and st are then given by multiplicationof the single chain expressions in Eq. 4 by the numberdensity of polymer chains, ρ/Mw, where Mw denotesthe molecular weight of each polymer.

Operators L and M

The operators L and M, which are involved in thereversible and irreversible contributions to the evo-lution equations (Eq. 11), are discussed in detail inAppendix B. The main points for their specification arethe following.

For L, it is observed, first, that the reversible contri-butions to the full evolution equations of the variablesx are of purely kinematic origin, and hence, directlyrelated to the velocity field v. Second, the reversibleevolution of the conformation tensor c and of the scalarλ is assumed to be given by

∂tc|rev = −v · ∇c + κ · c + c · κT , (15a)∂tλ|rev = −v · ∇λ + κ : g , (15b)with the transpose velocity gradient κ = (∇v)T . Inother words, c shows upper convected Maxwell be-havior, and the evolution of the scalar λ is coupledto c, where g is a function of λ and c. Since λ is ascalar, such coupling is required to describe, for exam-ple, orientation-dependent local chain stretching uponthe application of flow. It can indeed be shown thata Poisson operator L can be found that satisfies allthe GENERIC conditions, provided that the functiong(c, λ) is of the specific form Eqs. 44–47.

In order to construct M, we include as irreversibleeffects anisotropic heat conduction, and the relaxationof the conformational variables c and λ. For simplicity,anisotropic viscous flow and slip effects on the confor-mation tensor, as occurring in Schowalter derivatives,are not included. Due to the restriction to heat con-duction and structural relaxation, only three thermo-dynamic driving forces are relevant. Correspondingly,there are three thermodynamic fluxes, namely, the heat

Rheol Acta (2009) 48:301–316 307

flux, the rate of relaxation of c, and the rate of relax-ation of λ. Transport coefficients are then introduced torelate the forces to the fluxes. In particular, the follow-ing transport coefficients come into play: a second ranktensor k characterizes anisotropic thermal conductivity(Schieber et al. 2004), and a fourth rank tensor � anda scalar rate R describe the relaxation of the conforma-tion tensor c and scalar variable λ, respectively. Cross-effects in the force-flux relations are also possible. Forexample, the relaxation of c can give rise to a heat flux,as will be quantified below by the transport coefficientZ1 of tensorial rank three. Furthermore, the relaxationof λ could affect the relaxation of c, this couplingbeing described below by the transport coefficient Z2of tensorial rank two. Any ansatz for the quantitiesk, �, R, Z1, and Z2 must comply with the positivesemidefiniteness of the matrix of transport coefficients,

D =⎛

⎝T2k T Z1 0T ZT1 � Z2

0 ZT2 R

⎞

⎠ ≥ 0 , (16)

in order to satisfy all conditions of the thermodynamicframework (see Appendix B for details). The quantityT stands for the generalized temperature defined inEq. 18e below.

Time-evolution equations

In view of the functional derivatives Eq. 42, the Poissonoperator Eq. 43, and the friction matrix Eq. 48, andafter some mathematical rearrangements, the full set oftime-evolution equations assumes the form

∂tu=−∇ · (vu) + ∇ · σ , (17a)∂tρ =−∇ · (ρv), (17b)∂tθ =−v · ∇θ+ 1et,θ

[σ s : (∇v) − ∇· jq − et,c : (∂tc|irr)

− et,λ ( ∂tλ|irr)], (17c)

∂tc =−v · ∇c + κ · c + c · κT + T ZT1 · ∇1

T

+� :(

st,c − 1T et,c)

+ Z2(

st,λ − 1T et,λ)

, (17d)

∂tλ=−v · ∇λ + κ : g

+ZT2 :(

st,c − 1T et,c)

+ R(

st,λ − 1T et,λ)

, (17e)

with the velocity field v = u/ρ, the transposed velocitygradient κ = (∇v)T , and where we have defined the

stress tensor σ and heat flux jq, and introduced theabbreviation T as

σ = σ e + σ s , (18a)σ e = (et − ρet,ρ

)1 + 2c · et,c + get,λ , (18b)

σ s = −T (st − ρst,ρ)

1 − 2Tc · st,c − gTst,λ , (18c)

jq = −k · (∇T) + T Z1 :(

st,c − 1T et,c)

, (18d)

T ≡ et,θ /st,θ . (18e)

In Eqs. 17 and 18, as well as in the remainder of themanuscript, we use the notation y(x),xi ≡ (∂y/∂xi)|xk �=ifor partial derivatives. In the expressions for σ ec andσ sc, we have exploited that et,c and st,c commute with c,which is appropriate since c is the only tensor variable.The contributions ∂tc|irr and ∂tλ|irr in Eq. 17c representthe irreversible contributions to the evolution of theconformation tensor c (Eq. 17d) and local structuralvariable λ (Eq. 17e), respectively, i.e., the terms pro-portional to the transport coefficients �, R, Z1, and Z2.

Due to the appearance of σ in the momentum bal-ance Eq. 17a, its interpretation as the physical stresstensor is evident. Similarly, the interpretation of jq

as the heat flux becomes clear when considering theevolution equation for the total entropy density st, thatcan be derived from Eq. 17 by way of the chain rule,

∂tst = −∇ · (stv) − ∇ ·(

1

Tjq)

+⎛

⎜⎝

(∇ 1T)

st,c − et,cTst,λ − et,λT

⎞

⎟⎠ · D ·⎛

⎜⎝

(∇ 1T)

st,c − et,cTst,λ − et,λT

⎞

⎟⎠ . (19)

The evolution equation for the total energy assumeswith

∂tet = −∇ · (etv) + (∇v) : σ − ∇ · jq (20)

the usual form, representing the conservation of thekinetic plus internal energy, E.

It is important to note that the driving forces forstructural relaxation in Eqs. 17d and 17e and theanisotropic contributions to the stress tensor Eqs. 18a–18c can only be expressed as the derivative of theHelmholtz free energy density ft ≡ et − Tst, if the tem-perature T defined in Eq. 18e does not depend on cand λ, respectively. Whether that condition is fulfilleddepends on the choice for the thermal variable θ , asdiscussed below. In the general case, not the Helmholtzfree energy but, rather, the two generators energy andentropy are relevant.

308 Rheol Acta (2009) 48:301–316

Discussion

First, we compare the set of evolution equations(Eq. 17) with the constitutive equations (Eq. 18) withknown expressions from literature. To that end, wenote that the evolution equation for θ (Eq. 17c) canbe replaced by the equivalent evolution of either st(Eq. 19) or et (Eq. 20). In the absence of cross-couplings (Z1 =0 and Z2 =0), the evolution equationfor c (Eq. 17d) and the expression for the stress tensorspecified in Eq. 18 agree with the results in Beris andEdwards (1994), if viscous stresses and nonaffine slipare neglected. The full set of Eqs. 17 and 18 with Z1 =0and Z2 =0 is identical to Eqs. 46 and 47 in Dressler et al.(1999), with the major difference that we have includedthe local scalar variable λ for quantification of theenergetic effects, in contrast to their procedure with thetemperature-dependent overall chain spring constant(see also Eq. 9 above). Furthermore, in contrast toDressler et al. (1999), we have neglected for simplicityviscous stresses, and external effects on energy andmomentum are also not considered. In comparison toÖttinger (2002, 2005), there is agreement about thechoice of a scalar and tensorial structural variable,and about the corresponding evolution equations ingeneral. However, while our treatment is less broadwith respect to the irreversible effects accounted for,we make no restriction on the functional forms of etand st. Particularly, we permit that the conformationalcontributions to these two functions may depend on thethermal variable θ .

Second, we discuss the meaning of the function T de-fined in Eq. 18e. For the choice θ =et, that definition isformally equivalent to the one for the temperature usedin Dressler et al. (1999). On the other hand, if the con-formational energy and the conformational entropy areboth independent of the thermal variable θ , and with θbeing the local energy density, the definition Eq. 18ereduces to the “local” temperature used by Öttinger(2002, 2005). In both cases, the corresponding temper-ature enters in the usual way into Fourier’s law of heatconduction. While we consider a more general case forthe definition of T here, T is still directly related to thedriving force for the heat flux Eq. 18d. Furthermore,T also enters into the expression for the stress tensorEq. 18a–18c. Several terms in the evolution equations(Eq. 17) can be written in a condensed, probably morefamiliar, form if Eq. 18e allows one to make a variabletransformation from θ to T. In this case, one obtainsfrom Eq. 18e the “canonical” consistency relation

T = et,T(ρ, T, c, λ)st,T(ρ, T, c, λ)

, (21)

and et,T in Eq. 17c assumes the meaning of aconformation-dependent heat capacity. Furthermore,with θ =T and defining the Helmholtz free energy den-sity as ft =et−Tst, one can simplify the driving forcesfor structural relaxation in Eqs. 17d and 17e to − ft,c/Tand − ft,λ/T, respectively, and also the stress tensorexpression (Eqs. 18a–18c) assumes a more commonform.

What is the relation between the temperature Tand the parameter β used in the generalized canonicalensemble? Since the GENERIC equations developedabove relate to the choice X1 = RR and X2 =n in thegeneralized canonical ensemble Eq. 1, one concludesfrom Eq. 3a that, for θ =T, the parameter β−1 can beidentified with kBT,

kBT = β−1 . (22)It is pointed out that this identification holds irrespec-tive of a specific model (e.g., the RIS model as used inthe toy example earlier), and it is not restricted to singlechain partition functions. While T was related above tothe driving force for the heat flux and also entered thestress tensor expression, i.e., it can be interpreted as a“transport” temperature, it becomes clear with Eq. 22that this temperature also stands for the Lagrange mul-tiplier for the microscopic energy in the setting of ageneralized canonical ensemble. Therefore, only now,after establishing the relation between the nonequi-librium thermodynamics treatment and the general-ized canonical ensemble, it seems confirmed that usingthe temperature T as a dynamic variable, as done inBraun and Friedrich (1990), Peters and Baaijens (1997),Wapperom and Hulsen (1998), and Dressler et al.(1999), is indeed admissible. Furthermore, according toEq. 3b, the driving forces for structural relaxation inEqs. 17d and 17e are given by the Lagrange parametersα and μ, respectively, and the anisotropic contributionsto the stress tensor expression (Eq. 18a–18c) can alsobe rewritten in the form −kBT(2c · α + gμ).

After clarifying the meaning of T as a temperatureand establishing its relation to the (microscopic) para-meter β, we are now in the position to discuss the evolu-tion equation of the temperature for flowing polymericsystems. In literature, one usually starts by rewritingEq. 20 with the aid of the mass balance Eq. 17b into anequation for the internal energy per unit mass êt =et/ρ,ρDtêt = (∇v) : σ − ∇ · jq , (23a)with the material derivative Dt = ∂t + v · ∇ (Braun andFriedrich 1990; Peters and Baaijens 1997; Wapperomand Hulsen 1998; Dressler et al. 1999; Schieber et al.2004; Ionescu et al. 2008a). Subsequently, as êt is taken

Rheol Acta (2009) 48:301–316 309

as a function of the temperature T, as well as of thepolymer conformation, and the mass density (or thehydrostatic pressure), application of the chain rule toDtêt leads to an equation for the temperature of theform

ρêt,T DtT = (∇v) : σ − ∇ · jq − ρêt,c : Dtc − ρêt,ρ Dtρ .(23b)

While the derived results in all of these references arethus equivalent, our equation for θ = T is most easilycompared to the temperature equations in Braun andFriedrich (1990), Peters and Baaijens (1997), Ionescuet al. (2008a), to which Eq. 17c reduces upon neglectingthe local variable λ. A direct consequence of applyingthe chain rule to Dtêt is the occurrence of the “heatcapacity” êt,T , similar to et,θ in Eq. 17c. It is currentlydebated whether this quantity depends on the confor-mation of the polymer chains (e.g., Schieber et al. 2004;Ionescu et al. 2008a). We will come back to this issuein the next section when discussing the temperatureequation in relation to our microscopic toy model.

The evolution equation Eq. 17c can also be usedfor other purposes than for deriving a temperatureequation. For example, if one is interested in the ex-change between local and conformational entropy, itis convenient to choose the local entropy density asthermal variable, θ = sl, and compare the evolutionequations Eqs. 17c and 19. To simplify the discussion,we start with the special case that the conformationalentropy is independent of sl, i.e., et,θ = T. In this case,the evolution equations for the local entropy density sland conformational entropy density sc = st − sl, basedon Eq. 17c with θ = sl and Eq. 19, can be written in theform

∂tsl = −∇ · (slv) + �l + (s)ex − ∇(

jq

T

), (24a)

∂tsc = −∇ · (scv) + �t − �l − (s)ex , (24b)with rate of entropy exchange defined by

(s)ex =1

T

[ (σ s + Tsl1

) : (∇v) − et,c : (∂tc|irr)

− et,λ (∂tλ|irr)], (24c)

the local entropy production �l = (∇T−1) · jq, and to-tal entropy production �t given by the third contri-bution on the right-hand side of Eq. 19. Note that,for Z1 = 0 and Z2 = 0, the quantity �t − �l expressesthe positive semidefinite entropy production due toconformational relaxation. The first term in Eq. 24cis reversible in nature and is interpreted as follows.

The flow field distorts the chain conformation, whichleads to a decreased conformational entropy. Since thetotal entropy density is not altered by the reversibledynamics, that loss in conformational entropy must becompensated by the local entropy. This effect is wellknown in standard classroom experiments on rubberelasticity. The second term in Eq. 24c is irreversible innature and more subtle since an irreversible exchangealso brings about a corresponding entropy production.Due to this joint occurrence of “current” and entropyproduction, the second term in Eq. 24c must be inter-preted as containing both of these contributions, i.e.,also including production terms. Assuming et,θ = T forthe discussion of entropy transfer is a useful technicaland conceptual simplification. When relaxing that as-sumption, the concept of exchange between local andconformational entropy becomes more subtle, and onemay use st,sl − 1 as an expansion parameter by whichthe scenario for et,θ = T can be extended. Similarly,Z1 �= 0 and Z2 �= 0 introduce coupling between thelocal and conformational entropies. However, we shallnot enter into these extensions to not distract from themain points of the manuscript.

If, on the other hand, one is interested in the ex-change between local and conformational energy, onechooses the local energy density as thermal variable,θ = el. Given that the conformational energy is inde-pendent of el, i.e., et,θ = 1, one can proceed in a similarway as in Eq. 24 to identify the rate of exchange be-tween the local energy density el and conformationalenergy density ec = et − el. Based on Eqs. 17c and 20,one obtains

∂tel = −∇ · (elv) + σ s : (∇v) + (e)ex − ∇ · jq , (25a)∂tec = −∇ · (ecv) + σ e : (∇v) − (e)ex , (25b)with the exchange rate between the local and confor-mational energy densities

(e)ex = el(∇ · v) − et,c : (∂tc|irr) − et,λ (∂tλ|irr) , (25c)that contains an isotropic reversible and an irreversiblecontribution. Note that the contributions proportionalto (∇v) in the evolution equations (Eqs. 25a and 25b)express exchange terms between the kinetic energy onthe one hand and the local and conformational ener-gies, on the other hand.

Implications of the model

We now seek to analyze the full set of evolution equa-tions (Eq. 17) with the constitutive equations (Eq. 18)specifically with regard to identifying the effects of theconformational energy. One should note that, for the

310 Rheol Acta (2009) 48:301–316

choice θ = T, conformational contributions to the en-ergy density become readily identifiable in et. Namely,according to Eqs. 2b and 31, the total energy densityet becomes independent of the polymer conformationexactly if the microscopic conformational energy isset to zero, i.e., for � = 0. In contrast, if one choseθ = et, any explicit dependence of et on the confor-mation would be gone, although that dependence isstill present physically in the system. In that spirit, theseparation of the stress tensor (Eq. 18a) into energeticand entropic contributions (Eqs. 18b and 18c) is par-ticularly meaningful for θ = T. For all these reasons,we choose θ = T = (kBβ)−1 for the remainder of thepaper. Nevertheless, the examination of energetic ef-fects is still complicated since the ramifications of theenergy function et(ρ, T, c, λ) in Eqs. 17 and 18 aremultiple, namely, et enters in (1) the driving forces forstructural relaxation, (2) the stress tensor expression,and (3) several places in the temperature equation.

One possible simplification to the full set of evolu-tion equations consists in looking for spatially homoge-neous solutions in a homogeneous time-dependent andincompressible flow, i.e., v(r, t) = κ(t) · r with trκ = 0.Another possibility is to examine the behavior of thematerial in the absence of flow. In this latter case, andupon neglecting cross-couplings (Z1 = 0 and Z2 = 0),the evolution equations (Eq. 17) become

0 = ∇ · σ , (26a)∂tρ = 0 , (26b)et,T∂tT = −∇ · jq − et,c : ∂tc − et,λ∂tλ , (26c)

∂tc = � :(

st,c − 1T et,c)

, (26d)

∂tλ = R(

st,λ − 1T et,λ)

. (26e)

As a special case of these reduced equations, the sta-tionary solutions can be used to study energetic ef-fects. Particularly, with regard to the microscopic toymodel introduced earlier, the complete relaxation of λis equivalent to μ = 0, according to Eq. 3b. Followingthe discussion after Eq. 7b, the size of the polymercoil in the relaxed state, as described by c0(n), becomestemperature-dependent if conformational energetic ef-fects are involved, since the equilibrium value of ndepends on temperature through Eq. 8. If the polymercoil is fully relaxed also on the large scale, Eqs. 3b and7a lead to α = 0, i.e., c = c0(n)1. Therefore, one can setY = 0 in Eq. 8c to arrive at the temperature depen-dence of the number of gauche states n, and hence ofthe coil size, c0(n(T)). To proceed, we define the local

structural variable to be the fraction of occupied gauchestates,

λ = n/N , (27)where it has again been assumed that the degreeof polymerization N is high, N − 2 � N. Using � =2095 J mol−1 for polyethylene (Hoeve 1961; Paul et al.1995), our toy model predicts a change in relative pop-ulation of gauche states with temperature at T = 420 K(i.e., in the melt state) in equilibrium as

dλeq/dT � 3.5 × 10−4 K−1 , (28)which in turn affects the distribution of the torsionangle ϕ. The average 〈cos ϕ〉 (Eq. 6b) with ϕtrans =0 andϕgauche � ±2 changes with temperature as d〈cos ϕ〉/dT �−5 × 10−4 K−1. Using the relation Eq. 6a, our toymodel thus predicts

d ln C∞dT

� −10−3 K−1 . (29)The good agreement of this value with experimen-tal data, d ln C∞/dT � −1.1 × 10−3 K−1 (Hoeve 1961;Boothroyd et al. 1991; Krishnamoorti et al. 2002) over-estimates the accuracy of the toy model, given thefact that correlations between torsion angles have beenneglected altogether (see Hoeve (1961) for furtherdetails).

The reduced equations (Eq. 26) can also be usedfor studying the material behavior immediately afterstopping the flow (see also Schieber et al. 2004). Inhomogeneous situations and with et,c = 0 as in our mi-croscopic toy model, the temperature equation assumesthe form

et,T∂tT = −et,λ∂tλ , (30a)with, again, λ being interpreted as the relative occu-pation of gauche states defined in Eq. 27. Since themicroscopic toy model was intended to highlight theform of the conformational contributions to the en-ergy and entropy and since the single chain approxi-mation is a drastic abstraction, we base our estimatefor et,T on experimental values rather than on Eq. 4.For polyethylene, the following value is estimated foret,T = ρCV at equilibrium, where CV is the heat ca-pacity at constant volume per unit mass. Using therelation ρ(Cp − CV) = Tα2/κ with the heat capacity atconstant pressure per unit mass Cp = 2202 J kg−1 K−1,the mass density ρ = 780 kg m−3, the thermal expansioncoefficient α = 7.16 × 10−4 K−1, and the isothermalcompressibility κ = 10.5 × 10−10 Pa−1, all at T = 420 K(Orwoll 1996), one obtains et,T = 1.5 × 106 J m−3 K−1.On the other hand, for the microscopic toy model withet = (ρ/Mw)Esc and Esc defined in Eq. 4a, one finds

Rheol Acta (2009) 48:301–316 311

et,λ = �ρ/(Mw/N) � 1.2 × 108 J m−3, where we haveused Mw/N = mCH2 = 0.014 kg mol−1. 1 Therefore, therates of change in the temperature and the relativegauche population are related by

∂tλ

∂tT� −0.013 K−1 . (30b)

For example, if, over some small time interval, λ re-laxes by an amount λ = 0.013, a temperature change

T � −1K occurs as a result of the effects of confor-mational energy. The ability of the local variable λ torelax by a certain amount depends on how stronglythe polymer chain was previously deformed in flow,where the applied flow rate competes with the relax-ation effect. In general, note that the relaxation ofλ is much faster than the relaxation of c. It is thusto be expected that, for experimentally measuring theeffects of conformational energy by way of Eq. 30,one must be able to capture relatively fast changes inthe temperature. Due to the fast relaxation of λ, therestriction in Eq. 30 to homogeneous situations thatpreclude thermal conduction to/from the walls of thesample, i.e., to adiabatic conditions, is reasonable ifthe sample was approximately homogeneous prior tostopping the flow. While detailed flow calculations arerequired to understand how strongly λ can change as aresult of certain flow situations, there is an alternativeroute to at least get a feeling for what λ = 0.013 standsfor. From the relation Eq. 28, one infers that, underequilibrium conditions, a temperature change of theorder of T = 37K is required to achieve λ = 0.013.

Finally, we discuss the dependence of the heat capac-ity et,T on the conformation. Dressler et al. (1999), and,subsequently, Ionescu et al. (2008a), have used Eq. 9a,which leads to a heat capacity with a conformationdependent term proportional to the second derivativeof the overall chain spring constant, K′′(T)trc. In ourtoy model, in contrast, et is completely independentof c, and the term that contains the local structurethrough λ is athermal. As a result, in our case, the heatcapacity et,T is not affected by the chain conformation,neither through c, nor through λ. Due to the absence ofsuch couplings, the heat capacity is rather insensitive toflow. If this is found to be in conflict with experimental

1Note that experimental values for the heat capacity include,in addition to the change in the vibrational behavior describedby the derivative et,T at constant λ, also the change of thetrans/gauche population with respect to temperature. With theabove numbers, one obtains for that contribution ρCV − et,T =et,λdλeq/dT � 4.2 × 104 J m−3 K−1. As this is less than 3% of thetotal value, this correction has been neglected in the estimationof ρCV .

findings, one concludes that the weak dependence ofthe local Helmholtz free energy Floc on n discussedin Eq. 40 is too drastic a simplification, and must,hence, be relaxed. Nevertheless, a formal connectionto Dressler et al. (1999) and Ionescu et al. (2008a)can be made by completely relaxing the fast local vari-able n. Doing so leads to the c dependence of theenergy Eq. 10, which in turn amounts to a heat capacitythat depends on the conformation tensor. However, aspointed out previously, we emphasize again the needto explicitly use the local structural variable as an in-dependent quantity, since the degrees of freedom thatcarry conformational energy are local and have differ-ent dynamics compared to the conformation tensor.

Summary and perspectives

A suitable and manageable treatment of energy elas-tic effects in flowing polymeric systems requires athoughtful discussion of the set of independent vari-ables, namely due to the significant departure fromequilibrium conditions. In this paper, we advocate theuse of a local structural variable for capturing the en-ergetic contributions in a direct manner, in addition toa tensorial variable that accounts for the slow, largescale features. By calculating the generalized canonicalpartition function for a simple microscopic toy model,the microscopic understanding of the energetics wastranslated into two out-of-equilibrium thermodynamicfunctions, the energy density et and the entropy densityst. The latter then enter into the formulation of a setof thermodynamically admissible evolution equations.In the most general case, the driving force for struc-tural relaxation and the conformational contributionsto the stress tensor have been found not to be propor-tional to the derivatives of the Helmholtz free energydensity; rather, combinations of the derivatives of theenergy and entropy densities are the more fundamen-tal quantities. Furthermore, it has been demonstratedthat the common use of the temperature as a dynamicvariable is indeed possible for these systems. In par-ticular, it has been shown that the temperature usedin the definition of the heat flux is directly related tothe Lagrange multiplier of the microscopic energy inthe generalized canonical partition function. Specifi-cally, with regard to the microscopic toy model, thefollowing points are important: First, upon completerelaxation of the local structural variable, the energyof the toy model has been shown to reduce to knownexpressions in literature where only a conformationtensor is used. Second, the prediction about the changein the size of the polymer coil with temperature, which

312 Rheol Acta (2009) 48:301–316

is due to energetic effects, for polyethylene is in agree-ment with experimental data. Third, the heat capacityoccurring naturally in the temperature equation doesnot depend on the polymer conformation. Fourth, it ispredicted for polyethylene that, after stopping the flow,the rate of relaxation λ̇ of the relative gauche popula-tion, λ = n/N, leads to a temperature change Ṫ causedby the conformational energetic effects of the orderṪ � −λ̇/0.013 K−1, under adiabatic conditions. In or-der to discuss the absolute magnitude of the expectedtemperature changes during relaxation, further studiesare needed to understand how strongly λ can departfrom its equilibrium value as a result of certain flowsituations. Finally, based on the general evolution equa-tions, the exchange between local and conformationalenergy (and entropy, respectively) upon deformationhas also been addressed.

The range of applicability of the GENERIC modeldeveloped above goes far beyond the microscopic toymodel in several respects. The latter was specific in thesense that the local structural variable was chosen asthe number of gauche states λ = n and the tensorialvariable as c = 〈RR〉 with R the end-to-end vector.However, there already exist several models in litera-ture that use a scalar and a tensorial variable to describepolymeric liquids. For example, the pompon modelfor melts of multiply branched polymers (McLeish andLarson 1998; Öttinger 2001) is included in the aboveGENERIC model (Öttinger 2002, 2005; Soulageset al. 2006), and also some versions of the reptationmodel (Ianniruberto and Marrucci 2001; Marrucci andIanniruberto 2003) fall into the same class. In both ofthese examples, the scalar structural variable representslocal chain (segment) stretch that relaxes substantiallyfaster than the tensorial variable. Therefore, the abovegeneral set of evolution equations push the door wideopen for studying energy elastic effects also in branchedand entangled polymer melts. The relevance of thelocal chain stretch for describing energetic effects isparticularly clear in the case of polyethylene, where thenumber of gauche states is intimately related to thepersistence length. In this respect, it is helpful to ex-tend Monte Carlo simulations used by Mavrantzas andTheodorou (1998), Mavrantzas and Öttinger (2002),and Ionescu et al. (2008b) as to configurationally biasrather local structural variables, in contrast or in ad-dition to the conformation tensor. Significant steps inthis direction have recently been made by the so-calledsemigrand canonical Monte Carlo method of Bernardinand Rutledge (2007a, b).

In the toy model, conformational energetic effectsentered only through the intrachain torsional degreesof freedom. Certainly, there are other conformational

energies to be considered for realistic polymers, e.g.,nonbonded intrachain and interchain Lennard-Jonesinteractions, as discussed in detail by Gao and Weiner(1992, 1994) and Ionescu et al. (2008b). As far as theirrepresentation in the energy and entropy densities etand st is concerned, it seems clear that density (or pres-sure) effects are important. As for the conformationof the polymer chains, it is to be expected that localstructural features are more relevant than long-rangeones. As an extreme example, crystallizable polymershint in this direction since local orientation and packingeffects are much more relevant than the large-scaleconformation tensor for the formation of nuclei andcrystallization (Ko et al. 2004; Van Meerveld et al.2004). Therefore, again, one is led to use a local struc-tural variable, λ, for an adequate description of suchnonbonded energetic effects. In this regard, it is evidentthat, with the GENERIC model presented here, a largeand rich class of models can now be considered ina setting that is appropriate for beyond equilibriumsituations.

Appendix

A Toy model: calculation of the energyand the entropy for a single chain

For describing the polymer chain in situations out ofequilibrium, we introduce α and μ as the Lagrange mul-tipliers to the structural variables X1 = RR and X2 =n, respectively, and calculate the generalized canonicalpartition function Z (β, α, μ). If R and Q denote theend-to-end vectors of a specific RIS conformation jwith and without vibrations r, respectively, i.e., R =Q + r, and z the Cartesian microscopic positions of theN atoms relative to the RIS conformation, the partitionfunction Eq. 1 can be written in the form

Z (β, α, μ) =N−2∑

n=0e−μn−βHn

×∑

j∈Jn

∫

C je−β Ĥ(z;RIS)−α:( Q+r)( Q+r)dz , (31)

with Jn the set of RIS states that consists of n gauchestates, and C j is the domain in configuration space inthe vicinity of the RIS state j. Note that vibrationsare better characterized with z interpreted as Cartesiandeviations from the RIS state, since even small vari-ations in bond angles and torsion angles would leadto (unphysically) large Cartesian displacements of theatoms for long chains. The right-hand side in Eq. 31

Rheol Acta (2009) 48:301–316 313

illustrates the decomposition of phase space into (1) asum over the number of gauche states, which, in turn, isrefined to (2) a summation over all RIS conformationswith n gauche states, followed by (3) the local vibra-tions around each RIS state [see also Öttinger (2007)for a similar splitting in the context of glasses]. Theenergy H is decomposed into the energy associatedwith the number of gauche states, Hn = �n, and a termthat accounts for the local vibrations around the RISconformation, Ĥ with Ĥ(z = 0) = 0. For rather stiffinteraction potentials, the integration domain C j can beextended to R3N without changing the integral signifi-cantly. Therefore, one can interchange the summation∑

j∈Jn and the integration∫

. . . dz.To proceed, we make the plausible assumption that

Ĥ(z; RIS) depends on local aspects of the RIS con-formation, and not on the corresponding end-to-endvector that is a large-scale quantity. Therefore, the firstand second terms in the second exponential in Eq. 31feel different aspects of the RIS conformation. Forlarge N, Q is Gaussian distributed with zero mean anda variance that depends not only on N but also on thenumber of gauche states n, i.e., 〈 Q〉 = 0 and 〈 Q Q〉 =c0(n; N)1. With the assumption that the energy Ĥ andthe α-term are mutually uncorrelated and that the localfluctuations z are primarily relevant for the energy Ĥbut not for R, i.e., R → Q by virtue of stiff interactionpotentials, one can treat the β and α terms in theintegral in Eq. 31 separately. In particular, e−α: Q Q forgiven j ∈ Jn, can be replaced by its average with respectto the Gaussian distribution function for Q, i.e., by√

det (c/c0), where we have defined the n-dependentquantity

c−1 = 2α + 1c0(n; N)1 . (32)

The partition function Eq. 31 can be rewritten furtherby introducing the definitions for the n contribution tothe conformational entropy and the local Helmholtzfree energy,

Sconf(n) = kB lnN (n) , (33)

Floc(n, β) = −β−1 ln⎧⎨

⎩1

N (n)∑

j∈JnZj

⎫⎬

⎭ , (34)

with N the number of possible RIS conformations forgiven n, and Zj the partition function accounting for thelocal vibrations around the RIS state j,

N (n) = 2n(

N − 2n

), (35)

Zj =∫

e−β Ĥ(z;RIS)dz . (36)

Since the summand in the partition function Eq. 31is strongly peaked with respect to n, a saddle-pointapproximation can be used, leading to the condition forn = n(β, α, μ),

μ=−β�+ ∂(Sconf/kB)∂n

− β ∂ Floc∂n

− (c/c0) : α ∂c0∂n

. (37)

The partition function then assumes the form

Z (β, α, μ) = e−μn−βHn+Sconf

kB−βFloc− 12 ln det(c0c−1) , (38)

with Hn = n�, and Sconf(n) and Floc(n, β) given byEqs. 33 and 34, respectively.

According to Eq. 2 and using the condition Eq. 37,one arrives at the main result of this toy model, namelythe expressions for the energy and entropy, and theaverages of the conformational variables,

E=(

Floc − T ∂ Floc∂T

∣∣∣∣n

)+ �n , (39a)

S=− ∂ Floc∂T

∣∣∣∣n+Sconf+ kB

2

[(3− tr(c)

c0

)−ln det (c0c−1

)],

(39b)

〈RR〉= c , (39c)〈n〉=n . (39d)While the expressions on the right-hand side of Eq. 39are expressed in terms of the variables (μ, β, α), onecan eliminate (μ, α) in favor of (n, c) and express E andS in terms of the variables (n, β, c).

In order to obtain fully explicit expressions for theenergy and entropy, the local Helmholtz free energyFloc defined by Eqs. 34–36 must be specified. Since weconcentrate on the effects of conformation, we shall notdelve into this point. We just mention that the mostpromising route in this respect seems to be the use ofMonte Carlo simulations of (united) atoms in an en-semble with the control parameters β and n, for a singlechain. The influence of the number n of gauche stateson Floc can be examined by looking at the derivative∂ Floc(β, n)/∂n, which can be approximated by

β∂ Floc(β, n)

∂n�∑

j∈Jn∫

e−β Ĥ(z;RIS)(

e−βĤ −1)

dz∑

j∈Jn∫

e−β Ĥ(z;RIS)dz, (40)

where Ĥ stands for the energy difference arising in Ĥas a result of flipping one bond along the chain from thetrans state into a gauche state. If that energy differenceis significantly smaller than the thermal energy kBT forall values of z that contribute to the integral in Eq. 40,then the local energy and entropy can be considered

314 Rheol Acta (2009) 48:301–316

effectively independent of n. That condition is fulfilledif the form of the atomistic potentials around z = 0 isinsensitive to n.

In order to discuss whether the ensemble chosenabove at constant (β, α, μ) is equivalent to other en-sembles that control H, RR, and n, respectively, onemust examine if the fluctuations in the above ensembleare small,

∣∣∣∣∂2 ln Z∂γi∂γk

∣∣∣∣ �∣∣∣∣∂ ln Z∂γi

∣∣∣∣

∣∣∣∣∂ ln Z∂γk

∣∣∣∣ , (41)

where γi and γk are any of the Lagrange parameters(β, α, μ). Except if both γi and γk represent elements ofα, the conditions Eq. 41 can indeed be shown to hold forlarge N, i.e., fluctuations are small. For the conforma-tion tensor c, however, the fluctuations are not small;these fluctuations are given by the Gaussian nature ofthe distribution of R. A generalized microcanonicaldistribution for the conformation, for example, woulddestroy long fluctuations expressing Gaussian behavior.The generalized canonical distribution based on thedyadic RR, on the other hand, nicely represents theunderlying Gaussian statistics.

B Formulation of the evolution equationsin the GENERIC framework

Functional derivatives

The functional derivatives of Eq. 14, that drive the re-versible and irreversible contributions in the evolutionequation (Eq. 11), are given by

δEδx

=

⎛

⎜⎜⎜⎜⎝

v

− 12v2 + et,ρet,θet,cet,λ

⎞

⎟⎟⎟⎟⎠,

δSδx

=

⎛

⎜⎜⎜⎜⎝

0st,ρst,θst,cst,λ

⎞

⎟⎟⎟⎟⎠, (42)

with the velocity field defined by v = u/ρ. In Eq. 42, aswell as in the remainder of the manuscript, we use thenotation y(x),xi ≡ (∂y/∂xi)|xk �=i for partial derivatives.

Reversible dynamics

The reversible contributions to the full evolution equa-tions of the variables x are of purely kinematic origin,and, hence, directly related to the velocity field. For ourspecific choice of variables and in view of the functionalderivatives Eq. 42, one arrives at the following expres-

sion for the Poisson operator (see also Dressler et al.1999; Öttinger 2002)

L = −

⎛

⎜⎜⎜⎜⎜⎝

∇γ uα + uγ ∇α ρ∇α L(uθ)α L(uc)αγ ε L(uλ)α∇γ ρ 0 0 0 0L(θu)γ 0 0 0 0

L(cu)αβγ 0 0 0 0L(λu)γ 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎠, (43a)

with

L(uc)αγ ε ≡ −(∇αcγ ε) − ∇μcμγ δαε − ∇μcμεδαγ , (43b)L(cu)αβγ ≡ (∇γ cαβ) − cαμ∇μδβγ − cβμ∇μδαγ , (43c)L(uλ)α = −(∇αλ) − ∇μgαμ , (43d)L(λu)γ = (∇γ λ) − gγμ∇μ , (43e)

L(uθ)α = −ρ∇αst,ρst,θ

+ st∇α 1st,θ−L(uc)αγ ε

st,cγ εst,θ

− L(uλ)αst,λst,θ

, (43f)

L(θu)γ = −st,ρst,θ

∇γ ρ + 1st,θ ∇γ st

− st,cαβst,θ

L(cu)αβγ −st,λst,θ

L(λu)γ . (43g)

Subscripts (α, β) and (γ, ε) imply contraction with ten-sors multiplied from the left and from the right, re-spectively. All derivative operators act on everything totheir right, also on functions multiplied to the right of L,except when placed inside of parentheses (. . .). The el-ement L(cu) dictates the convection of the tensor c, i.e.,upper convected Maxwell behavior. The convection ofλ, determined by L(λu), is coupled to c through g, thatis a function of λ and c. Since λ is a scalar, such cou-pling is required to describe, for example, orientation-dependent local chain stretching upon the applicationof flow. The complicated expression for L(uθ) is givenby the degeneracy requirement Eq. 12a, which in turnfixes L(θu) by the antisymmetry condition. In the middleterm of L(uθ), the prefactor st is chosen for dimensionalreasons and in view of the evolution equation for thetotal entropy density, st, see Eq. 19.

The Poisson operator Eq. 43 is antisymmetric andsatisfies the degeneracy requirement by construction. Itcan also be shown that the Jacobi identity is satisfiedunder the following condition (Öttinger 2002). Usingthe representation theorem of tensor functions, themost general form of g is given by

g = g1c + g21 + g3c−1 , (44)

Rheol Acta (2009) 48:301–316 315

with the scalar coefficient functions

gj = gj(λ, I1, I2, I3) (45)

depending on λ and the three independent invariants

I1 = trc , (46a)I2 = ln det c , (46b)I3 = −tr

(c−1

). (46c)

It can be shown that the Jacobi identity is satisfiedunder the conditions

gi∂gk∂λ

− gk ∂gi∂λ

= 2(

∂gi∂ Ik

− ∂gk∂ Ii

), (47)

for all pairs (i, k). Applying L to δE/δx leads to thereversible contributions to the evolution equations ofx, which are summarized in Eq. 17.

Irreversible dynamics

We discuss the irreversible effects arising as a resultof anisotropic heat conduction, and relaxation of theconformational variables. For simplicity, anisotropicviscous flow and slip effects on the conformation tensor,as occurring in Schowalter derivatives, are not included.In view of the GENERIC formalism (Eq. 11), all theseirreversible effects come into play through the metricmatrix M. In order to present the procedure in a trans-parent form, we split the metric matrix according toEdwards (1998)

M = C · D · CT . (48)

The special construction Eq. 48 ensures that M issymmetric and positive semidefinite if D has theseproperties. The splitting Eq. 48 is done in such a waythat the degeneracy requirement Eq. 12b is satis-fied by the condition CT · δE/δx = 0, and the quantityCT · δS/δx can be interpreted as the driving force forthe irreversible processes. Since both viscous flow andslip effects are neglected, we expect the thermodynamicdriving forces to be independent of the velocity field. Inorder to achieve that, we choose the u- and ρ-columnsof CT to vanish, since otherwise, the degeneracy condi-tion would bring about spurious velocity terms into thedriving forces. Due to the restriction to heat conductionand structural relaxation, only three thermodynamicdriving forces are relevant. For all these reasons, we

propose the following form of the operator CT , andhence, of C,

CT =⎛

⎜⎝0 0 ∇i 1et,θ 0 00 0 − et,cαβet,θ δαγ δβε 00 0 − et,λet,θ 0 1

⎞

⎟⎠ , (49a)

C =

⎛

⎜⎜⎜⎜⎝

0 0 00 0 0

− 1et,θ ∇k −et,cγ εet,θ

− et,λet,θ0 δαγ δβε 00 0 1

⎞

⎟⎟⎟⎟⎠. (49b)

The three rows in Eq. 49a are unique in that theylead to sensible driving forces for heat conduction andconformational relaxation, that are given by

CT · δSδx

=⎛

⎝∇ 1T

st,c − 1T et,cst,λ − 1T et,λ

⎞

⎠ , (50)

where we have used the abbreviation T = et,θ /st,θ . Forthe matrix of transport coefficients, we write

D =⎛

⎝T2k T Z1 0T ZT1 � Z2

0 ZT2 R

⎞

⎠ ≥ 0 , (51)

where the second rank tensor k takes the meaningof the anisotropic thermal conductivity and the fourthrank tensor � and R describe the relaxation of the con-formation tensor c and scalar variable λ, respectively.The off-diagonal elements are inserted to account forpossible cross-effects in the force-flux relations. Dueto the disparities in length scales, it is assumed herethat the local structural features captured by λ arenot affected by temperature gradients, and hence, thecorresponding off-diagonal element is zero. Any ansatzfor the quantities k, �, R, Z1, and Z2 must complywith the positive semidefiniteness of D. Applying Mto δS/δx leads to the irreversible contributions to theevolution equations of x.

References

Astarita G (1974) Thermodynamics of dissipative materials withentropic elasticity. Polym Eng Sci 14:730–733

Astarita G, Sarti GC (1976) The dissipative mechanism in flow-ing polymers: theory and experiments. J Non-Newton FluidMech 1:39–50

Baig C, Mavrantzas VG (2007) Thermodynamically guided non-equilibrium Monte Carlo method for generating realisticshear flows in polymeric systems. Phys Rev Lett 99:257801

Beris AN, Edwards BJ (1994) Thermodynamics of flowing sys-tems with internal microstructure. Oxford University Press,Oxford

316 Rheol Acta (2009) 48:301–316

Bernardin III FE, Rutledge GC (2007a) Semi-grand canonicalMonte Carlo (SGMC) simulations to interpret experimentaldata on processed polymer melts and glasses. Macromole-cules 40:4691–4702

Bernardin III FE, Rutledge GC (2007b) Simulation of me-chanical properties of oriented glassy polystyrene. Polymer48:7211–7220

Bird RB, Armstrong RC, Hassager O, Curtiss CF (1977) Dy-namics of polymeric liquids. Kinetic theory, vol 2. Wiley,New York

Boothroyd AT, Rennie AR, Boothroyd CB (1991) Direct mea-surement of the temperature dependence of the unperturbeddimensions of a polymer. Europhys Lett 15:715–719

Braun H, Friedrich C (1990) Dissipative behavior of viscoelas-tic fluids derived from rheological constitutive equations.J Non-Newton Fluid Mech 38:81–91

Callen HB (1985) Thermodynamics and an introduction to ther-mostatistics. Wiley, New York

Casas-Vázquez J, Jou D (2003) Temperature in non-equilibriumstates: a review of open problems and current proposals. RepProg Phys 66:1937–2023

Dressler M, Edwards BJ, Öttinger HC (1999) Macroscopicthermodynamics of flowing polymeric liquids. Rheol Acta38:117–136

Edwards BJ (1998) An analysis of single and double gener-ator thermodynamic formalisms for the macroscopic de-scription of complex fluids. J Non-equilib Thermodyn 23:301–333

Edwards BJ, Öttinger HC (1997) Time-structure invariance cri-teria for closure approximations. Phys Rev E 56:4097–4103

Flory PJ (1988) Statistical mechanics of chain molecules. Hanser,Munich

Gao J, Weiner JH (1992) Stress relaxation in a polymer melt offreely-rotating chains. J Chem Phys 97:8698–8704

Gao J, Weiner JH (1994) Simulation of viscoelasticity in polymermelts: effect of torsional potential. Model Simul Mater SciEng 2:755–766

Grmela M (1998) Letter to the editor: comment on “thermo-dynamics of viscoelastic fluids: the temperature equation[J. Rheol. 42, 999 (1998)]”. J Rheol 42:1565–1567

Grmela M, Öttinger HC (1997) Dynamics and thermodynamicsof complex fluids. I. Development of a general formalism.Phys Rev E 56:6620–6632

Hoeve CAJ (1961) Unperturbed mean-square end-to-end dis-tance of polyethylene. J Chem Phys 35:1266–1267

Ianniruberto G, Marrucci G (2001) A simple constitutive equa-tion for entangled polymers with chain stretch. J Rheol45:1305–1318

Ionescu TC, Edwards BJ, Keffer DJ (2008a) Energetic and en-tropic elasticity of nonisothermal flowing polymers: experi-ment, theory, and simulation. J Rheol 52:105–140

Ionescu TC, Mavrantzas VG, Keffer DJ, Edwards BJ (2008b)Atomistic simulation of energetic and entropic elasticity inshort-chain polyethylenes. J Rheol 52:567–589

In ’t Veld PJ, Hütter M, Rutledge GC (2006) Temperature-dependent thermal and elastic properties of the interlamellarphase of semicrystalline polyethylene by molecular simula-tion. Macromolecules 39:439–447

Ko MJ, Waheed N, Lavine MS, Rutledge GC (2004) Charac-terization of polyethylene crystallization from an orientedmelt by molecular dynamics simulation. J Chem Phys 121:2823–2832

Krishnamoorti R, Graessley WW, Zirkel A, Richter D,Hadjichristidis N, Fetters LJ, Lohse DJ (2002) Melt-statepolymer chain dimensions as a function of temperature.J Polym Sci Polym Phys 40:1768–1776

Landau LD, Lifshitz EM (1980) Course of theoretical physics(vols 5 & 9) (3rd ed). Pergamon, Oxford

Larson RG (1988) Constitutive equations for polymer melts andsolutions. Butterworth, Stoneham

Marrucci G, Ianniruberto G (2003) Flow-induced orientation andstretching of entangled polymers. Philos Trans R Soc A361:677–688

Mattice WL, Suter UW (1994) Conformational theory of largemolecules: the rotational isomeric state model in macro-molecular systems. New York, Wiley

Mavrantzas VG, Öttinger HC (2002) Atomistic Monte Carlosimulations of polymer melt elasticity: their nonequilibriumthermodynamics GENERIC formulation in a generalizedcanonical ensemble. Macromolecules 35:960–975

Mavrantzas VG, Theodorou DN (1998) Atomistic simulation ofpolymer melt elasticity: calculation of the free energy of anoriented polymer melt. Macromolecules 31:6310–6332

McLeish TCB, Larson RG (1998) Molecular constitutive equa-tions for a class of branched polymers: the pom-pom poly-mer. J Rheol 42:81–110

Orwoll RA (1996) Densities, coefficients of thermal expansion,and compressibilities of amorphous polymers. In: Mark JE(ed) Physical properties of polymers handbook, chap. 7.American Institute of Physics, Woodbury

Öttinger HC (1999) A thermodynamically admissible reptationmodel for fast flows of entangled polymers. J Rheol 43:1461–1493

Öttinger HC (2001) Thermodynamic admissibility of the pomponmodel for branched polymers. Rheol Acta 40:317–321

Öttinger HC (2002) Modeling complex fluids with a tensor anda scalar as structural variables. Rev Mex Fis 48(Suppl 1):220–229

Öttinger HC (2005) Beyond equilibrium thermodynamics. Wiley,Hobroken

Öttinger HC (2007) Nonequilibrium thermodynamics of glasses.Phys Rev E 74:011113(1–25)

Öttinger HC, Grmela M (1997) Dynamics and thermodynamicsof complex fluids. II. Illustrations of a general formalism.Phys Rev E 56:6633–6655

Paul W, Yoon DY, Smith GD (1995) An optimized united atommodel for simulations of polymethylene melts. J Chem Phys103:1702–1709

Peters GWM, Baaijens FPT (1997) Modelling of non-isothermalviscoelastic flows. J Non-Newton Fluid Mech 68:205–224

Reichl LE (1980) A modern course in statistical physics.University of Texas Press, Austin

Sarti GC, Marrucci G (1973) Thermomechanics of dilute polymer-solutions—multiple bead-spring model. Chem Eng Sci 28:1053–1059

Sarti GC, Esposito N (1977) Testing thermodynamic constitu-tive equations for polymers by adiabatic deformation exper-iments. J Non-Newton Fluid Mech 3:65–76

Schieber JD, Venerus DC, Bush K, Balasubramanian V,Smoukov S (2004) Measurement of anisotropic energy trans-port in flowing polymers by using a holographic technique.Proc Natl Acad Sci U S A 101:13142–13146

Soulages J, Hütter M, Öttinger HC (2006) Thermodynamic ad-missibility of the extended Pom-Pom model for branchedpolymers. J Non-Newton Fluid Mech 139:209–213

Treloar LRG (1975) The physics of rubber elasticity (3rd ed).Clarendon, Oxford

Van Meerveld, Peters GWM, Hütter M (2004) Towards a rhe-ological classification of flow induced crystallization experi-ments of polymer melts. Rheol Acta 44:119–134

Wapperom P, Hulsen MA (1998) Thermodynamics of viscoelas-tic fluids: the temperature equation. J Rheol 42:999–1019

Energy elastic effects and the concept of temperature in flowing polymeric liquidsAbstractIntroductionToy modelFormulation of the dynamicsStrategyChoice of variables x, generating functionals E and SOperators L and MTime-evolution equationsDiscussionImplications of the model

Summary and perspectivesAppendixA Toy model: calculation of the energy and the entropy for a single chainB Formulation of the evolution equations in the GENERIC frameworkFunctional derivativesReversible dynamicsIrreversible dynamics

References

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /Warning /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 150 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 600 /MonoImageMinResolutionPolicy /Warning /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/Description >>> setdistillerparams> setpagedevice