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The continuum description of conformational entropy effects in viscoelastic flow
77
Journal of Non-Newtonian Fluid Mechanists, 3 (1977/1978) 77-86 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
THE CONTINUUM DESCRIPTION OF CONFORMATIONAL EFFECTS IN VISCOELASTIC FLOW
ENTROPY
G. ASTARITA and G.C. SARTI Istituto
di Principi di Ingegneria Chimica,
Universitci di Napoli, Naples (Italy)
(Received March 2, 1977)
Summary The general continuum thermodynamic theory of viscoelasticity is reviewed, and it is shown that confi~ration~ entropy effects can always be described within its framework. Furthermore, entropy rather than temperature can be taken as the independent variable within the same framework. Recent work on molecular modeling of the thermomechanics of polymer flow is reviewed in this paper.
Introduction The recent literature on viscoelastic flow has shown a renewed interest in its thermodynamic interpretation, and the general area of thermodynamic approach to the development of constitutive equations has attracted attention. A very general framework, within which most work in the area is couched, is the general continuum theory of thermodynamics of materials with fading memory (hereafter referred to as GCT). After Coleman and Noll’s paper in 1963 [ 111 made available the method by which restrictions on constitutive equations can be derived from the second law of thermodyn~ics formulated as the Clausius-Duhem inequality, the GCT was developed in 1964 by Coleman [8,9] by applying the method to the class of simple materials with fading memory. Later works on the GCT are concerned with weakening of the basic postulates [‘7,10), and illustration of its contents in a more general context [21], but not in any way with confutation of any of its results. Although, in the literature, it has never been claimed that the Clausius-Duhem inequality is the definitive complete mathematical formula-
tion of the whole possible conceptual content of the second law, attempts at developing a thermodynamic theory for thermomechanical phenomena based on alternative formulations, when developed with satisfactory rigour [see, for example 161 have not produced results at odds with the GCT. Finally, it is worth stating, that, although the original literature on the GCT is possibly difficult to follow because of heavy mathematical formalism, illustrations of its contents aimed primarily at engineering scientists have been published recently [ 1 and Chapter 4 of 31. Since there is no reason to believe that the theory of simple materials with fading memory is not general enough to encompass the description of the behaviour of real viscoelastic materials, the GCT (unless proved wrong) should be regarded as an acceptable general framework; in fact, so general as to allow an enormous variety of special subcases or classes of subcases to be considered, which by being more specific may allow more detailed predictions than the GCT, though, of course, only for a more restricted class of materials. Indeed, a number of special cases have been discussed in the literature. A phenomenological continuum theory based on the assumption that elastic energy can be accumulated only by a decrease of conformational entropy has been developed both for the incompressible [2] and the compressible [6] case; the same theory (which, by being based on a restrictive assumption, is of course liable to experimental confutation) has been subjected to experimental test [4,19] and shown to be valid for polymer melts provided the temperature is sufficiently high. The theory of viscoelastic materials with entropic elasticity is an obvious extension of classical rubber elasticity theory, and has the same physical justification based on molecular arguments. More specific thermodynamic constitutive equations have been deduced from simple molecular models which do incorporate, albeit roughly, a description of conformational changes due to flow [ 17,201, and the general problem of the connection between such “molecular modeling” approaches and continuum descriptions has been considered in some detail [5,18]. A different kind of restriction to a special subcase of the GCT has been considered by Crochet and Naghdi [ 13-151: namely, the one emerging from the experimentally well-known fact that the time-temperature superposition principle is obeyed by polymeric materials. To recapitulate, a number of works have recently been published which, after introduction of some specific assumption suggested by either molecular considerations or experimental evidence, have produced proper special subcases of the GCT. Of course, none of the results obtained are at odds with the GCT; in particular, any specific argument concerning changes of conformational entropy due to flow can be accommodated in the framework of the GCT, which allows entropy to be an unspecified functional of the deformation history, subject only to such restrictions as are imposed by general principles (determinism, local action) and by the second law. In contrast with the works quoted above is a recent paper by Christensen
79
[ 121. It uses methods which are reminiscent of those typical of the GCT, and does not introduce any specific restrictive assumption; yet it claims to produce results which are not those of the GCT. One is thus left in a position where only three possibilities arise: (i) that the GCT is wrong; (ii) that Christensen’s results simply duplicate the GCT; and (iii) that Christensen’s argument is wrong. Furthermore, Christensen seems to imply that allowance for conformational entropy changes due to flow produces a theory which, in some sense, is more general than the GCT, while, as said before, this is not the case Our aim in this note is twofold. On the one hand, we wish to place in a proper perspective the problem of how conformational entropy effects, and their modeling, can be placed in the framework of the GCT. On the other hand, we wish to analyze Christensen’s work in order to decide which of its results, if any, are either generalizations or particularizations of the GCT. The description
of conformational
entropy
effects
in the GCT
The physical significance of the arguments to be developed in this note is comparatively straightfonvard; unfortunately, the mathematics involved is rather complex, so that the essential simplicity of the argument risks to be overshadowed by mathematical detail. That is why, in what follows, we choose to be rather cavalier in dealing with the mathematical formalism, referring the reader to the original literature for all details. When dealing with any formulation of thermodynamic theory, the concept of the “state” of the body considered is a primitive one. In order to have a non-trivial theory, one needs to allow the state to be identified, at the very least, by one kinematic and one thermal variable; say, in the very simplest example of an ideal gas, the state would be postulated to be the ordered pair V, T, with V the specific volume and T the temperature. When dealing with the thermomechanics of viscoelastic materials, two different issues arise. On the one hand, one needs to take into account the geometrical complexity of strain; this issue is resolved by substituting for V some appropriate tensor-valued measure of strain such as the Cauchy tensor C. On the other hand, the well-known ability of polymeric materials to exhibit memory for past deformations and temperature - or, in other words, the fact that relaxation phenomena take place on a time scale comparable to that of actual experiments - needs somehow to be allowed for. From a physical viewpoint, one may choose t,o argue that, in addition to C and T, some set of parameters $I describing the instantaneous microstructure of the materials determines the state; the microstructure 4 is, in turn, determined by the past history of deformation and temperature. This viewpoint is perfectly well accommodated by the phenomenological approach of the GCT, which chooses to identify the state u directly with C, T, and their past histories. The description of the memory of the materials for past deformation and temperature is handled by the GCT (and, apart from the incorrectnesses dis-
80
cussed in the next section, by Christensen as well) as follows. Let 7 be current time, and consider any quantity which is a function time: w = 1;(T).
of (I)
(Here, and everywhere in what follows, we use a different symbol for a function and for its value. In particular, a superimposed A identifies a function of time.) The history at time r = t of w is defined as the following function of the time lapse s: J(S)
= &t--s),
(2)
restricted to non-negative values of s, i.e. to times r not later than t. The physically intuitive concept that the state of the material at any one time is to be identified with its instantaneous microstructure, and that the latter is determined by the histories of both deformation and temperature, is formalized in the development of the GCT by postulating that the state u is the ordered pair of histories: 0 = P(s),
P(s);
SE [O, -).
(3)
One may, if one wishes, regard the state as being the instantaneous microstructure, and then proceed to describe how the latter is determined by the histories of deformation and temperature; it has been shown [6] that this approach is equivalent to the GCT. Of course, the entire past history of deformation and temperature can never be known, and this difficulty is resolved by introducing the concept of fading memory. Therefore, the space of possible states is assigned an appropriate topology which embodies the concept of fading memory; for technical details, see [lo]. One of the principles which is involed in the development of the GCT is the “principle of equipresence”, which requires the state u to be the same for all quantities which are postulated to be functions of state; since one of the latter is the entropy S, a constitutive functional for it is laid down as follows: S=a(a)
= SE:
.-
,{C’(s), Tf(s))
(4)
(Here, and in what follows, we use script symbols such as h for the functionals mapping the state u as given in eqn. (3) into the constitutive quantities identified by the corresponding roman symbol such as S.) (In the GCT, the temperature gradient VT is also considered as an independent variable; since, in the arguments to be developed below, we do not need to describe heat conduction, this additional complication is omitted from the discussion.) We wish here to emphasize that, in the GCT, the functional h ( *) is only restricted insofar as the second law of thermodynamics is required not to be violated in any process. Therefore, flow-induced changes of conformational entropy, even for spatially and temporally isothermal flows, can be described
81
within the framework of the GCT. Indeed, such descriptions have been presented in the literature, both for specific molecular models [ 1’7,201 and in rather general terms [ 5,181; in particular, a complete theory is available for the case where such confo~ation~ entropy effects are the only source of elasticity [ 2,4,6,19]. The physically intuitive idea that the state of the material is determined by its instantaneous structure, and that the latter determines not only stress but also conformational entropy, is given a formal mathematical description in the GCT by the inclusion of the deformation history C?(s) in the argument of the functional ~3( * }, This inclusion is formally required by the principle of equipresence: we point this out to show that a principle which has often been considered a rather abstract one in fact formalizes a concrete physical description of the microstructure. We now turn our attention to two additional points concerning the GCT which need to be emphasized before the discussion of the next section. First of all, in the structural viewpoint, one may wish to regard the instantaneous structure as described by its conformations entropy, rather than regarding the latter as to be calculated from a geometrical description. This corresponds to considering entropy, rather than temperature, as the independent variable; correspondingly, one would need to consider the internal energy U rather than the Helmholtz free energy A as the basic thermodynamic potential. This type of description is included in the formulation of the GCT; indeed, it is the description chosen in the pioneering work of Coleman and No11 [ 111. A physically suggestive brief outline of this development in the GCT is given below. The domain CD of the constitutive functional cl{ - } in eqn. (4), as well as of all other constitutive functionals, is a Banach space the topology of which embodies an appropriate formalization of the concept of fading memory [lo] ; this domain is represented in suggestive geometrical form in Fig. 1 as the shaded area. Suppose some particular deformation history CL(s) has been fixed, so that the horizontal line labelled “1” in Fig. 1 is the subdomain considered. Equation (4) implies that, along this line, S is a unique functional of T’(s), say: S = S(t) =
n UZ(s), sElO,=)
Tf(s)] = $&
*-)
V%)>.
(5)
The value of 5’ at any time T < t is thus given by: $7) =
X {T(s)}, sE[O,mf
(6)
which shows that, once T’(s) has been assigned, not only the instantaneous value S(t) of entropy is determined, but also its past history. Thus, along the line “l”, a functional transformation T’(s) --f S’(s) is established. If the assumption is made that such a transformation is invertible for all possible choices of C&(s), one may write: T = sE;$
3-j
ICf(s), S”(s)).
(7)
c
T*(s)
Fig. 1. Temperature as independent ature-entropy functional inversion. Fig. 2. Entropy
as independent
St(s)
TtW variable. Line “1”: to be considered Line “2”: isothermal case.
variable.
Line “lr’:
isothermal
for temper-
case.
When eqn. (7) is substituted in all the constitutive functionals, ordinary chain rules of differentiation of functionals yield the GCT formulation for the case where entropy is regarded as the independent variable [Section 11 of ref. 91, say when the state 6 is regarded as: 6 = P(s), sf(s),
s E [O, -).
(8)
(A superimposed - distinguishes the constitutive functionals relative to the “entropy as an independent variable” case.) With reference to Fig. 2, the domain % of the constitutive functional t { *} (and of all other “tilde” functionals) has, in view of the assumed invertibility, a one-to-one correspondence with the domain (5 in Fig. 1. For future reference, we record here one of the results of the GCT relative to the “tilde” functionals, namely:
l--;-l-as(t)
c
sG[O,-)
W(s), fml,
where ii{ * } is the functional which maps a into the internal energy U, and its derivative is taken with respect to the instantaneous entropy. (Such instantaneous derivatives are crucial to the development of the theory; their existence is to be understood throughout, and this is why we have always indicated s E [O,m); for technical details, see [8] .) Equation (9) is the generalization of the classical result of the elementary thermodynamic theory which states that temperature is the partial derivative of internal energy with respect to entropy at constant volume. Equation (9) clearly shows that, although in addition to the kinematic variable C one may choose either temperature or entropy as an independent variable, one cannot, of course,
a3
choose both - as should be obvious also from eqns. (4) and (7). The second point to be stressed concerns the case where one wishes to restrict attention to isothermal flows. This is a very common restriction in fluid mechanics, and it is, of course, easily accommodated by the GCT; again a brief outline is given below. Define the function T* (s) as having the constant value T for all non-negative S: T*(s) = T
‘o”sE [O, -)*
(10)
With reference to Fig. 1, restriction to the isothermal case corresponds to fixing, among all possible temperature histories, a particular one, namely T’(s) = T* (s); the domain CDis restricied to the vertical line labeled “2” in Fig. 1. Correspondingly, the domain Q in Fig. 2 is restricted to some curve such as the one labeled “%“, this can also be seen from inspection of eqn. (7), which shows that fixing one particular value of T assigns an implicit relationship between C’(s) and Sf (s). The “isothermal” form of the GCT is simply read-off from the general results by imposing the condition of constant temperature. Of course, although one may wish to restrict attention to isothermal phenomena, the constitutive functionals are still defined over the whole domain Cz, in Fig. 1 (and the “tilde” functionals over the whole domain 5 ); thus, in particular, eqn. (9) still holds true, although, of course, its right-hand side is not even defined unless one allows for displacements (albeit differential ones) from the Curve 2 in Fig. 2. What is being said is, very simply, that the isothermal theory is no more and no less than a special case of the nonisothermal theory. The approach
of Christensen
In Christensen’s paper, heat conduction is excluded from consideration, and the basic constitutive equation is written in the following form [eqn. (2) of ref. 123 : u = sE[;*_, W’(s),
S’(sH;
(111
i.e. within the framework of the present analy$s, Christensen considers the states ?i as given by eqn. (8), and the domain ‘D in Fig. 2. Christensen then restricts attention to isothermal flow conditions, and therefore correctly expresses entropy as uniquely determined by the deformation history [eqn. (4) of ref. 121: s = sf;,_)
{C*(s), T*(s)) = SE~,~)ICYs)) *
(12)
(For the functional of - ) , the time lapse s needs to be considered only over (0, -), rather than [0, -), because Christensen only considers fluids and thus
84
uses the relative strain in the deformation history, which is zero by definition at s = 0.) Of course, the functional cp{ * } describes the dependence of S on C’(s) along line “2” in Fig. 1; along the same line, also the internal energy is uniquely determined by the history of deformation Cf(s). This is also evident from consideration of the domain a in Fig. 2; again, the restriction to isothermal flows implies the restriction to Curve “2”, along which internal energy could be regarded as being a unique functional of the history of deformation Cf (s) or (under sufficient assumption of smoothness) of the history of entropy S’(s), but not independently on both, unless the restriction to isothermal conditions is removed. This point should be discussed in more detail. Even setting aside completely the GCT, there seems to be an internal inconsistency in Christensen’s approach. In fact, from his eqn. (4) (which is eqn. (12) above) one gets: (13) which shows that, once Cf(s) has been assigned, not only the instantaneous entropy S(t) but also its values S(r) at all previous times r < t are assigned (from a physical viewpoint, once the kinematics are assigned up to time t, the conformational entropy at all times up to t can, in principle, be calculated). Hence, if these values are substituted in eqn. (2) of Christensen (which is eqn. (11) above), the internal energy U is seen to reduce to a functional of only the deformation history Ct(s). Indeed, in isothermal flow it is hard to understand how internal energy could depend on anything else! Yet Christensen chooses not to reduce the functional L?{ * } and proceeds to express its two Frechet differentials corresponding to a variation of S’(s) at constant C’(s), and to a variation of C’(s) at constant S’(s) [eqns. (7) and (8) of ref. 121. A glance at Fig. 2 shows immediately that such variations are impossible ones, unless the restriction to line “2” is removed. Indeed, in isothermal flow, after stating explicitly that entropy is a unique functional of the deformation history, how can one consider a variation of deformation history at constant entropy, or vice versa? If eqns (7) and (8) of Christensen are to make any sense, one must admit that the functional E{ * } is still being considered as defined over the whole domain 6?, and not restricted to line “2”, i.e. temperature variations (albeit possibly only differential ones) are being allowed. This is a perfectly sensible thing to do, but if ones does one recovers the GCT in the “tilde” form, and thus in particular eqn. (9). In very simple physical terms, temperature may be constant in both space and time, but it still equals the partial derivative of internal energy with respect to entropy. We fail to see anything in Christensen’s paper that comes even close to hinting that eqn. (9) is wrong; he simply chooses to ignore it. This has very serious consequences. In fact, the main result of Christensen is the appearance in the constitutive equation of stress of a “coupling term”;
85
his eqn. (20) reads, in the notation
of this note:
(14) where h{ * } and t { . } are two functionals, the exact form of which needs not to be discussed here. The main point is that, in view of eqn. (9), the term in square brackets in eqn. (14) is identically zero, and therefore the “coupling term” (a u/as)E is simply cancelled out by the “entropy term” Tt. Indeed, when eqn. (9) is taken into account; all the equations given by Christensen in the “General derivation” section reduce to those of the GCT for the special isothermal case. The balance of Christensen’s paper consists in a second-order integral expansion of the constitutive equation for stress, followed by some arbitrary simplifications of the same. It is then claimed that “the second-order theory models the shear thinning behaviour. It is believed that this is the first formulation of a second-order theory to provide a realistic behaviour of this type”. The claim is incorrect, and the implication that the result obtained is due to the “coupling term” is also incorrect. Indeed, even Christensen’s equations show the fallacy of the implication: if one sets B, = 0 in eqns. (26) (i.e. if one cancels out the “coupling term”), the shear-thinning terms do not go to zero. While it is true that a second-order differential expansion fails to model shear thinning, second-order in tegrul expansions do predict shear thinning, with no need for spurious “coupling terms”. Incidentally, the coupling term should be cancelled anyway, because eqn. (9) implies that the first square bracket on the right-hand side of eqn. (24) of Christensen’s paper is zero. Conclusions A general continuum theory of the thermomechanics of viscoelastic bodies is available. While, of course, no theory can at any time be asserted as definitive and not liable to improvement and generalization, for this particular case there is no known experimental confutation of any of its predictions, and therefore there seems to be no pragmatic need for generalization or improvement. Quite the contrary, what is needed is specification of the theory, i.e. the formulation of special subcases which, at the cost of describing the behaviour of a more restricted class of materials, allow solution of a wider class of boundary-value problems. In the formulation of such special subcases, molecular arguments may be most helpful; the general nonlinear theory of elasticity has had its most successful specification in the theory of rubber elasticity, which is based on molecular arguments. In fact, the availability of a general theory is helpful in the formulation of special subcases, since it provides a framework for tests of internal consistency. This program of research can only be hindered by vague formulations based on
86
molecular arguments which either explicitly or implicitly contradict the general theory. Since the general theory is a purely mathematical one, it can only be contradicted either by negating some of its axioms (and substituting alternative ones), or by showing it to be wrong somewhere. If neither is done, the results of the theory need to be accepted. References 1 G. Astarita, An Introduction to Non-Linear Continuum Thermodynamics, S.P.A. Editrice di Chimica, Milan, 1975. 2 G. Astarita, Thermodynamics of dissipative materials with entropic elasticity, Polym. Eng. Sci., 14 (1974) 730. 3 G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGrawHill, Maidenhead, 1974. 4 G. Astarita, and G.C. Sarti, The dissipative mechanism in flowing polymers. Theory and experiments, J. Non-Newtonian Fluid Mech., 1 (1976) 39. 5 G. Astarita and G.C. Sarti, An approach to thermodynamics of polymer flow based on internal state variables, Polym. Eng. Sci., 16 (1976) 490. 6 G. Astarita and G.C. Sarti, Thermodynamics of compressible materials with entropic elasticity. In J.F. Hutton, J.R.A. Pearson and R. Walters (Eds.), Theoretical Rheology, Applied Science Publishers, Barking, 1971. 7 B.D. Coleman, Some recent results in the theory of fading memory, Pure Appl. Chem., 22 (1970) 321. 8 B.D. Coleman, On thermodynamics, strain impulses, and viscoelasticity, Arch. Ration. Mech. Anal., 17 (1964) 230. 9 B.D. Coleman, Thermodynamics of materials with memory, Arch. Ration. Mech. Anal., 17 (1964) 1. 10 B.D. Coleman and V.J. Mizel, A general theory of dissipation in materials with memory, Arch. Ration. Mech. Anal., 27 (1967) 255. 11 B.D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963) 167. 12 R.M. Christensen, Entropy effects in viscoelastic fluid theory, J. Non-Newtonian Fluid Mech., 1 (1976) 371. 13 M.J. Crochet and P.M. Naghdi, A class of simple fluids with fading memory, Int. J. 14 15 16 I7 18 19 20 21 22
Eng. Sci., 7 (1969) 1173. M.J. Crochet and P.M. Naghdi,
A class of non-isothermal
viscoelastic
fluids,
Int. J.
Eng. Sci., 10 (1972) 775. M.J. Chrochet and P.M. Naghdi, On a restricted non-isothermal theory of simple materials, J. Met., 13 (1974) 97.‘ W.A. Day, The thermodynamics of simple materials with fading memory, SpringerVerlag, Berlin, 1972. G. Marrucci, The free energy constitutive equation for polymer solutions from the dumbbell model, Trans. Sot. Rheol., 16 (1977) 321. G.C. Sarti and G. Astarita, A thermomechanical theory for structured materials, Trans. Sot. Rheol., 19 (1975) 215. G.C. Sarti and N. Esposito, Testing polymer thermodynamic constitutive equations by adiabatic deformation experiments, J. Non-Newtonian Fluid Mech., 3 (1977) 65. G.C. Sarti and G. Marrucci, Thermomechanics of dilute polymer solutions. Multiple bead-spring model, Chem. Eng. Sci., 28 (1973) 1053. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. C. Truesdell and W. Noll, The non-linear field theories of mechanics. In the Encyclopedia of Physics, Vol. 111/3, Springer-Verlag, Berlin, 1965.
Journal of Non-Newtonian Fluid Mechanists, 3 (1977/1978) 77-86 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
THE CONTINUUM DESCRIPTION OF CONFORMATIONAL EFFECTS IN VISCOELASTIC FLOW
ENTROPY
G. ASTARITA and G.C. SARTI Istituto
di Principi di Ingegneria Chimica,
Universitci di Napoli, Naples (Italy)
(Received March 2, 1977)
Summary The general continuum thermodynamic theory of viscoelasticity is reviewed, and it is shown that confi~ration~ entropy effects can always be described within its framework. Furthermore, entropy rather than temperature can be taken as the independent variable within the same framework. Recent work on molecular modeling of the thermomechanics of polymer flow is reviewed in this paper.
Introduction The recent literature on viscoelastic flow has shown a renewed interest in its thermodynamic interpretation, and the general area of thermodynamic approach to the development of constitutive equations has attracted attention. A very general framework, within which most work in the area is couched, is the general continuum theory of thermodynamics of materials with fading memory (hereafter referred to as GCT). After Coleman and Noll’s paper in 1963 [ 111 made available the method by which restrictions on constitutive equations can be derived from the second law of thermodyn~ics formulated as the Clausius-Duhem inequality, the GCT was developed in 1964 by Coleman [8,9] by applying the method to the class of simple materials with fading memory. Later works on the GCT are concerned with weakening of the basic postulates [‘7,10), and illustration of its contents in a more general context [21], but not in any way with confutation of any of its results. Although, in the literature, it has never been claimed that the Clausius-Duhem inequality is the definitive complete mathematical formula-
tion of the whole possible conceptual content of the second law, attempts at developing a thermodynamic theory for thermomechanical phenomena based on alternative formulations, when developed with satisfactory rigour [see, for example 161 have not produced results at odds with the GCT. Finally, it is worth stating, that, although the original literature on the GCT is possibly difficult to follow because of heavy mathematical formalism, illustrations of its contents aimed primarily at engineering scientists have been published recently [ 1 and Chapter 4 of 31. Since there is no reason to believe that the theory of simple materials with fading memory is not general enough to encompass the description of the behaviour of real viscoelastic materials, the GCT (unless proved wrong) should be regarded as an acceptable general framework; in fact, so general as to allow an enormous variety of special subcases or classes of subcases to be considered, which by being more specific may allow more detailed predictions than the GCT, though, of course, only for a more restricted class of materials. Indeed, a number of special cases have been discussed in the literature. A phenomenological continuum theory based on the assumption that elastic energy can be accumulated only by a decrease of conformational entropy has been developed both for the incompressible [2] and the compressible [6] case; the same theory (which, by being based on a restrictive assumption, is of course liable to experimental confutation) has been subjected to experimental test [4,19] and shown to be valid for polymer melts provided the temperature is sufficiently high. The theory of viscoelastic materials with entropic elasticity is an obvious extension of classical rubber elasticity theory, and has the same physical justification based on molecular arguments. More specific thermodynamic constitutive equations have been deduced from simple molecular models which do incorporate, albeit roughly, a description of conformational changes due to flow [ 17,201, and the general problem of the connection between such “molecular modeling” approaches and continuum descriptions has been considered in some detail [5,18]. A different kind of restriction to a special subcase of the GCT has been considered by Crochet and Naghdi [ 13-151: namely, the one emerging from the experimentally well-known fact that the time-temperature superposition principle is obeyed by polymeric materials. To recapitulate, a number of works have recently been published which, after introduction of some specific assumption suggested by either molecular considerations or experimental evidence, have produced proper special subcases of the GCT. Of course, none of the results obtained are at odds with the GCT; in particular, any specific argument concerning changes of conformational entropy due to flow can be accommodated in the framework of the GCT, which allows entropy to be an unspecified functional of the deformation history, subject only to such restrictions as are imposed by general principles (determinism, local action) and by the second law. In contrast with the works quoted above is a recent paper by Christensen
79
[ 121. It uses methods which are reminiscent of those typical of the GCT, and does not introduce any specific restrictive assumption; yet it claims to produce results which are not those of the GCT. One is thus left in a position where only three possibilities arise: (i) that the GCT is wrong; (ii) that Christensen’s results simply duplicate the GCT; and (iii) that Christensen’s argument is wrong. Furthermore, Christensen seems to imply that allowance for conformational entropy changes due to flow produces a theory which, in some sense, is more general than the GCT, while, as said before, this is not the case Our aim in this note is twofold. On the one hand, we wish to place in a proper perspective the problem of how conformational entropy effects, and their modeling, can be placed in the framework of the GCT. On the other hand, we wish to analyze Christensen’s work in order to decide which of its results, if any, are either generalizations or particularizations of the GCT. The description
of conformational
entropy
effects
in the GCT
The physical significance of the arguments to be developed in this note is comparatively straightfonvard; unfortunately, the mathematics involved is rather complex, so that the essential simplicity of the argument risks to be overshadowed by mathematical detail. That is why, in what follows, we choose to be rather cavalier in dealing with the mathematical formalism, referring the reader to the original literature for all details. When dealing with any formulation of thermodynamic theory, the concept of the “state” of the body considered is a primitive one. In order to have a non-trivial theory, one needs to allow the state to be identified, at the very least, by one kinematic and one thermal variable; say, in the very simplest example of an ideal gas, the state would be postulated to be the ordered pair V, T, with V the specific volume and T the temperature. When dealing with the thermomechanics of viscoelastic materials, two different issues arise. On the one hand, one needs to take into account the geometrical complexity of strain; this issue is resolved by substituting for V some appropriate tensor-valued measure of strain such as the Cauchy tensor C. On the other hand, the well-known ability of polymeric materials to exhibit memory for past deformations and temperature - or, in other words, the fact that relaxation phenomena take place on a time scale comparable to that of actual experiments - needs somehow to be allowed for. From a physical viewpoint, one may choose t,o argue that, in addition to C and T, some set of parameters $I describing the instantaneous microstructure of the materials determines the state; the microstructure 4 is, in turn, determined by the past history of deformation and temperature. This viewpoint is perfectly well accommodated by the phenomenological approach of the GCT, which chooses to identify the state u directly with C, T, and their past histories. The description of the memory of the materials for past deformation and temperature is handled by the GCT (and, apart from the incorrectnesses dis-
80
cussed in the next section, by Christensen as well) as follows. Let 7 be current time, and consider any quantity which is a function time: w = 1;(T).
of (I)
(Here, and everywhere in what follows, we use a different symbol for a function and for its value. In particular, a superimposed A identifies a function of time.) The history at time r = t of w is defined as the following function of the time lapse s: J(S)
= &t--s),
(2)
restricted to non-negative values of s, i.e. to times r not later than t. The physically intuitive concept that the state of the material at any one time is to be identified with its instantaneous microstructure, and that the latter is determined by the histories of both deformation and temperature, is formalized in the development of the GCT by postulating that the state u is the ordered pair of histories: 0 = P(s),
P(s);
SE [O, -).
(3)
One may, if one wishes, regard the state as being the instantaneous microstructure, and then proceed to describe how the latter is determined by the histories of deformation and temperature; it has been shown [6] that this approach is equivalent to the GCT. Of course, the entire past history of deformation and temperature can never be known, and this difficulty is resolved by introducing the concept of fading memory. Therefore, the space of possible states is assigned an appropriate topology which embodies the concept of fading memory; for technical details, see [lo]. One of the principles which is involed in the development of the GCT is the “principle of equipresence”, which requires the state u to be the same for all quantities which are postulated to be functions of state; since one of the latter is the entropy S, a constitutive functional for it is laid down as follows: S=a(a)
= SE:
.-
,{C’(s), Tf(s))
(4)
(Here, and in what follows, we use script symbols such as h for the functionals mapping the state u as given in eqn. (3) into the constitutive quantities identified by the corresponding roman symbol such as S.) (In the GCT, the temperature gradient VT is also considered as an independent variable; since, in the arguments to be developed below, we do not need to describe heat conduction, this additional complication is omitted from the discussion.) We wish here to emphasize that, in the GCT, the functional h ( *) is only restricted insofar as the second law of thermodynamics is required not to be violated in any process. Therefore, flow-induced changes of conformational entropy, even for spatially and temporally isothermal flows, can be described
81
within the framework of the GCT. Indeed, such descriptions have been presented in the literature, both for specific molecular models [ 1’7,201 and in rather general terms [ 5,181; in particular, a complete theory is available for the case where such confo~ation~ entropy effects are the only source of elasticity [ 2,4,6,19]. The physically intuitive idea that the state of the material is determined by its instantaneous structure, and that the latter determines not only stress but also conformational entropy, is given a formal mathematical description in the GCT by the inclusion of the deformation history C?(s) in the argument of the functional ~3( * }, This inclusion is formally required by the principle of equipresence: we point this out to show that a principle which has often been considered a rather abstract one in fact formalizes a concrete physical description of the microstructure. We now turn our attention to two additional points concerning the GCT which need to be emphasized before the discussion of the next section. First of all, in the structural viewpoint, one may wish to regard the instantaneous structure as described by its conformations entropy, rather than regarding the latter as to be calculated from a geometrical description. This corresponds to considering entropy, rather than temperature, as the independent variable; correspondingly, one would need to consider the internal energy U rather than the Helmholtz free energy A as the basic thermodynamic potential. This type of description is included in the formulation of the GCT; indeed, it is the description chosen in the pioneering work of Coleman and No11 [ 111. A physically suggestive brief outline of this development in the GCT is given below. The domain CD of the constitutive functional cl{ - } in eqn. (4), as well as of all other constitutive functionals, is a Banach space the topology of which embodies an appropriate formalization of the concept of fading memory [lo] ; this domain is represented in suggestive geometrical form in Fig. 1 as the shaded area. Suppose some particular deformation history CL(s) has been fixed, so that the horizontal line labelled “1” in Fig. 1 is the subdomain considered. Equation (4) implies that, along this line, S is a unique functional of T’(s), say: S = S(t) =
n UZ(s), sElO,=)
Tf(s)] = $&
*-)
V%)>.
(5)
The value of 5’ at any time T < t is thus given by: $7) =
X {T(s)}, sE[O,mf
(6)
which shows that, once T’(s) has been assigned, not only the instantaneous value S(t) of entropy is determined, but also its past history. Thus, along the line “l”, a functional transformation T’(s) --f S’(s) is established. If the assumption is made that such a transformation is invertible for all possible choices of C&(s), one may write: T = sE;$
3-j
ICf(s), S”(s)).
(7)
c
T*(s)
Fig. 1. Temperature as independent ature-entropy functional inversion. Fig. 2. Entropy
as independent
St(s)
TtW variable. Line “1”: to be considered Line “2”: isothermal case.
variable.
Line “lr’:
isothermal
for temper-
case.
When eqn. (7) is substituted in all the constitutive functionals, ordinary chain rules of differentiation of functionals yield the GCT formulation for the case where entropy is regarded as the independent variable [Section 11 of ref. 91, say when the state 6 is regarded as: 6 = P(s), sf(s),
s E [O, -).
(8)
(A superimposed - distinguishes the constitutive functionals relative to the “entropy as an independent variable” case.) With reference to Fig. 2, the domain % of the constitutive functional t { *} (and of all other “tilde” functionals) has, in view of the assumed invertibility, a one-to-one correspondence with the domain (5 in Fig. 1. For future reference, we record here one of the results of the GCT relative to the “tilde” functionals, namely:
l--;-l-as(t)
c
sG[O,-)
W(s), fml,
where ii{ * } is the functional which maps a into the internal energy U, and its derivative is taken with respect to the instantaneous entropy. (Such instantaneous derivatives are crucial to the development of the theory; their existence is to be understood throughout, and this is why we have always indicated s E [O,m); for technical details, see [8] .) Equation (9) is the generalization of the classical result of the elementary thermodynamic theory which states that temperature is the partial derivative of internal energy with respect to entropy at constant volume. Equation (9) clearly shows that, although in addition to the kinematic variable C one may choose either temperature or entropy as an independent variable, one cannot, of course,
a3
choose both - as should be obvious also from eqns. (4) and (7). The second point to be stressed concerns the case where one wishes to restrict attention to isothermal flows. This is a very common restriction in fluid mechanics, and it is, of course, easily accommodated by the GCT; again a brief outline is given below. Define the function T* (s) as having the constant value T for all non-negative S: T*(s) = T
‘o”sE [O, -)*
(10)
With reference to Fig. 1, restriction to the isothermal case corresponds to fixing, among all possible temperature histories, a particular one, namely T’(s) = T* (s); the domain CDis restricied to the vertical line labeled “2” in Fig. 1. Correspondingly, the domain Q in Fig. 2 is restricted to some curve such as the one labeled “%“, this can also be seen from inspection of eqn. (7), which shows that fixing one particular value of T assigns an implicit relationship between C’(s) and Sf (s). The “isothermal” form of the GCT is simply read-off from the general results by imposing the condition of constant temperature. Of course, although one may wish to restrict attention to isothermal phenomena, the constitutive functionals are still defined over the whole domain Cz, in Fig. 1 (and the “tilde” functionals over the whole domain 5 ); thus, in particular, eqn. (9) still holds true, although, of course, its right-hand side is not even defined unless one allows for displacements (albeit differential ones) from the Curve 2 in Fig. 2. What is being said is, very simply, that the isothermal theory is no more and no less than a special case of the nonisothermal theory. The approach
of Christensen
In Christensen’s paper, heat conduction is excluded from consideration, and the basic constitutive equation is written in the following form [eqn. (2) of ref. 123 : u = sE[;*_, W’(s),
S’(sH;
(111
i.e. within the framework of the present analy$s, Christensen considers the states ?i as given by eqn. (8), and the domain ‘D in Fig. 2. Christensen then restricts attention to isothermal flow conditions, and therefore correctly expresses entropy as uniquely determined by the deformation history [eqn. (4) of ref. 121: s = sf;,_)
{C*(s), T*(s)) = SE~,~)ICYs)) *
(12)
(For the functional of - ) , the time lapse s needs to be considered only over (0, -), rather than [0, -), because Christensen only considers fluids and thus
84
uses the relative strain in the deformation history, which is zero by definition at s = 0.) Of course, the functional cp{ * } describes the dependence of S on C’(s) along line “2” in Fig. 1; along the same line, also the internal energy is uniquely determined by the history of deformation Cf(s). This is also evident from consideration of the domain a in Fig. 2; again, the restriction to isothermal flows implies the restriction to Curve “2”, along which internal energy could be regarded as being a unique functional of the history of deformation Cf (s) or (under sufficient assumption of smoothness) of the history of entropy S’(s), but not independently on both, unless the restriction to isothermal conditions is removed. This point should be discussed in more detail. Even setting aside completely the GCT, there seems to be an internal inconsistency in Christensen’s approach. In fact, from his eqn. (4) (which is eqn. (12) above) one gets: (13) which shows that, once Cf(s) has been assigned, not only the instantaneous entropy S(t) but also its values S(r) at all previous times r < t are assigned (from a physical viewpoint, once the kinematics are assigned up to time t, the conformational entropy at all times up to t can, in principle, be calculated). Hence, if these values are substituted in eqn. (2) of Christensen (which is eqn. (11) above), the internal energy U is seen to reduce to a functional of only the deformation history Ct(s). Indeed, in isothermal flow it is hard to understand how internal energy could depend on anything else! Yet Christensen chooses not to reduce the functional L?{ * } and proceeds to express its two Frechet differentials corresponding to a variation of S’(s) at constant C’(s), and to a variation of C’(s) at constant S’(s) [eqns. (7) and (8) of ref. 121. A glance at Fig. 2 shows immediately that such variations are impossible ones, unless the restriction to line “2” is removed. Indeed, in isothermal flow, after stating explicitly that entropy is a unique functional of the deformation history, how can one consider a variation of deformation history at constant entropy, or vice versa? If eqns (7) and (8) of Christensen are to make any sense, one must admit that the functional E{ * } is still being considered as defined over the whole domain 6?, and not restricted to line “2”, i.e. temperature variations (albeit possibly only differential ones) are being allowed. This is a perfectly sensible thing to do, but if ones does one recovers the GCT in the “tilde” form, and thus in particular eqn. (9). In very simple physical terms, temperature may be constant in both space and time, but it still equals the partial derivative of internal energy with respect to entropy. We fail to see anything in Christensen’s paper that comes even close to hinting that eqn. (9) is wrong; he simply chooses to ignore it. This has very serious consequences. In fact, the main result of Christensen is the appearance in the constitutive equation of stress of a “coupling term”;
85
his eqn. (20) reads, in the notation
of this note:
(14) where h{ * } and t { . } are two functionals, the exact form of which needs not to be discussed here. The main point is that, in view of eqn. (9), the term in square brackets in eqn. (14) is identically zero, and therefore the “coupling term” (a u/as)E is simply cancelled out by the “entropy term” Tt. Indeed, when eqn. (9) is taken into account; all the equations given by Christensen in the “General derivation” section reduce to those of the GCT for the special isothermal case. The balance of Christensen’s paper consists in a second-order integral expansion of the constitutive equation for stress, followed by some arbitrary simplifications of the same. It is then claimed that “the second-order theory models the shear thinning behaviour. It is believed that this is the first formulation of a second-order theory to provide a realistic behaviour of this type”. The claim is incorrect, and the implication that the result obtained is due to the “coupling term” is also incorrect. Indeed, even Christensen’s equations show the fallacy of the implication: if one sets B, = 0 in eqns. (26) (i.e. if one cancels out the “coupling term”), the shear-thinning terms do not go to zero. While it is true that a second-order differential expansion fails to model shear thinning, second-order in tegrul expansions do predict shear thinning, with no need for spurious “coupling terms”. Incidentally, the coupling term should be cancelled anyway, because eqn. (9) implies that the first square bracket on the right-hand side of eqn. (24) of Christensen’s paper is zero. Conclusions A general continuum theory of the thermomechanics of viscoelastic bodies is available. While, of course, no theory can at any time be asserted as definitive and not liable to improvement and generalization, for this particular case there is no known experimental confutation of any of its predictions, and therefore there seems to be no pragmatic need for generalization or improvement. Quite the contrary, what is needed is specification of the theory, i.e. the formulation of special subcases which, at the cost of describing the behaviour of a more restricted class of materials, allow solution of a wider class of boundary-value problems. In the formulation of such special subcases, molecular arguments may be most helpful; the general nonlinear theory of elasticity has had its most successful specification in the theory of rubber elasticity, which is based on molecular arguments. In fact, the availability of a general theory is helpful in the formulation of special subcases, since it provides a framework for tests of internal consistency. This program of research can only be hindered by vague formulations based on
86
molecular arguments which either explicitly or implicitly contradict the general theory. Since the general theory is a purely mathematical one, it can only be contradicted either by negating some of its axioms (and substituting alternative ones), or by showing it to be wrong somewhere. If neither is done, the results of the theory need to be accepted. References 1 G. Astarita, An Introduction to Non-Linear Continuum Thermodynamics, S.P.A. Editrice di Chimica, Milan, 1975. 2 G. Astarita, Thermodynamics of dissipative materials with entropic elasticity, Polym. Eng. Sci., 14 (1974) 730. 3 G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGrawHill, Maidenhead, 1974. 4 G. Astarita, and G.C. Sarti, The dissipative mechanism in flowing polymers. Theory and experiments, J. Non-Newtonian Fluid Mech., 1 (1976) 39. 5 G. Astarita and G.C. Sarti, An approach to thermodynamics of polymer flow based on internal state variables, Polym. Eng. Sci., 16 (1976) 490. 6 G. Astarita and G.C. Sarti, Thermodynamics of compressible materials with entropic elasticity. In J.F. Hutton, J.R.A. Pearson and R. Walters (Eds.), Theoretical Rheology, Applied Science Publishers, Barking, 1971. 7 B.D. Coleman, Some recent results in the theory of fading memory, Pure Appl. Chem., 22 (1970) 321. 8 B.D. Coleman, On thermodynamics, strain impulses, and viscoelasticity, Arch. Ration. Mech. Anal., 17 (1964) 230. 9 B.D. Coleman, Thermodynamics of materials with memory, Arch. Ration. Mech. Anal., 17 (1964) 1. 10 B.D. Coleman and V.J. Mizel, A general theory of dissipation in materials with memory, Arch. Ration. Mech. Anal., 27 (1967) 255. 11 B.D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963) 167. 12 R.M. Christensen, Entropy effects in viscoelastic fluid theory, J. Non-Newtonian Fluid Mech., 1 (1976) 371. 13 M.J. Crochet and P.M. Naghdi, A class of simple fluids with fading memory, Int. J. 14 15 16 I7 18 19 20 21 22
Eng. Sci., 7 (1969) 1173. M.J. Crochet and P.M. Naghdi,
A class of non-isothermal
viscoelastic
fluids,
Int. J.
Eng. Sci., 10 (1972) 775. M.J. Chrochet and P.M. Naghdi, On a restricted non-isothermal theory of simple materials, J. Met., 13 (1974) 97.‘ W.A. Day, The thermodynamics of simple materials with fading memory, SpringerVerlag, Berlin, 1972. G. Marrucci, The free energy constitutive equation for polymer solutions from the dumbbell model, Trans. Sot. Rheol., 16 (1977) 321. G.C. Sarti and G. Astarita, A thermomechanical theory for structured materials, Trans. Sot. Rheol., 19 (1975) 215. G.C. Sarti and N. Esposito, Testing polymer thermodynamic constitutive equations by adiabatic deformation experiments, J. Non-Newtonian Fluid Mech., 3 (1977) 65. G.C. Sarti and G. Marrucci, Thermomechanics of dilute polymer solutions. Multiple bead-spring model, Chem. Eng. Sci., 28 (1973) 1053. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. C. Truesdell and W. Noll, The non-linear field theories of mechanics. In the Encyclopedia of Physics, Vol. 111/3, Springer-Verlag, Berlin, 1965.