RHEOLOGY OF SOLID POLYMERS. Igor Emri

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1 I. Emri, Rheoloy Reviews 5, pp RHEOLOGY OF SOLID POLYMERS Ior Emri Center or Experimental Mechanics, Faculty o Mechanical Enineerin University o Ljubljana, 115 Ljubljana (SLOVENIA) ABSTRACT The mechanical properties o solid polymeric materials quite enerally depend on time, i.e., on whether they are deormed rapidly or slowly. The time dependence is oten remarkably lare. The complete description o the mechanical properties o a polymeric material commonly requires that they be traced throuh 1, 15, or even decades o time. The class o polymeric materials reerred to as thermo-rheoloically and/or piezo-rheoloically simple materials allows use o the superposition o the eects o time and temperature and/or time and pressure in such materials as a convenient means or extendin the experimental time scale. This paper presents a critical review o models proposed to describe the eect o temperature and/or pressure on time-dependent thermo-rheoloically and/or piezorheoloically simple polymeric materials. The emphasis here is on the theoretical aspects, althouh experimental results are used as illustrations wherever appropriate. KEYWORDS: Excess enthalpy, excess entropy, FMT model, ree volume, lass transitions, pressure eects, rate processes, shit unctions, temperature eects, timetemperature-pressure superposition, WLF model 1. INTRODUCTION Polymeric materials exhibit time-dependent mechanical properties that can prooundly aect the perormance o polymer products. The deree o chane in the mechanical properties o polymeric materials over time depends on many actors. These are primarily the temperature, pressure, humidity, and stress conditions to which the material is subjected durin its manuacture and durin its application. Thereore, processin parameters, like pressure and temperature, play an important role in determinin the quality o parts made by injection mouldin, compression mouldin, extrusion, etc. Unsuitable processin conditions may cause parts to warp or crack. These phenomena can occur in manuactured articles even in the absence o any mechanical loadin, in particular, in the presence o hih-modulus illers. Explosions (detonations) represent special cases entailin extremely hih temperatures and pressures. The British Society o Rheoloy, 5 ( 49

2 I. Emri, Rheoloy Reviews 5, pp The mechanical behaviour o polymeric materials is enerally characterized in terms o their time-dependent properties in shear or in simple tension. The time dependence o their bulk properties is almost universally nelected (see, e.., Kralj et al., [1]). The eect o temperature on the shear and tensile properties o polymeric materials has been airly well understood since about the orties o the last century. By contrast, little eort went into the determination o their time-dependent bulk properties and ater initial years o activity research on the eect o pressure lay eectively dormant. This was probably due mainly to the diiculty o makin precise measurements at rather small volume deormations. The materials used in current applications simply did not appear to require a deeper understandin o the time dependence o their bulk properties and o the eect o pressure on their timedependent shear properties to warrant exertin the exactin eort required. This situation has now chaned. The demand or sustainable development 1 requires optimization o the unctional and mechanical properties o new multi-component systems such as composite and hybrid materials, structural elements, and entire structures. Optimization o material use requires a much deeper understandin o the eect that temperature and pressure exert on the time dependence o the bulk as well as the shear properties o the constituents o these materials than is currently available. Our primary ocus in this paper is the eect o pressure. However, we also review the eect o temperature, essentially as necessary backround. In particular, we examine whether and in what manner the eects o time and temperature, respectively, those o time and pressure, may be assumed to superpose. In what ollows we shall thereore pay attention larely to time-temperature and/or time-pressure relations. Measurements o a linear viscoelastic response unction are, however, quite oten reported as a unction o requency. This may be either the cycles/second requency,, or the radian requency, ω = π. Because ln t ln, time and requency are, mutatis mutandis, equivalent in considerations o the eects o temperature and/or pressure. 1.1 Thermo- and piezo-rheoloical simplicity shit unctions. For a special class o polymeric materials, primarily sinle-phase, sinletransition amorphous homopolymers and random copolymers, it is possible to establish temperature and/or pressure shit unctions that collapse, as it were, the two-dimensional contour plots into a simple raph o the chosen response unction as a unction o the loarithmic time, usually shown as lo t, or the loarithmic requency, shown as loω or lo, recorded at a reerence temperature, T, or pressure, P. Such materials are reerred to as thermo-rheoloically and/or piezo-rheoloically simple materials. 1 Sustainable development aims at preservin, or the beneit o uture enerations, the environment and the natural resources culled rom it without lowerin currently excepted standards o livin. 5 The British Society o Rheoloy, 5 (

3 I. Emri, Rheoloy Reviews 5, pp Thermo-rheoloical simplicity requires that all response times (i.e., all relaxation or retardation times, see Tschoel [], p. 86), depend equally on temperature. This is expressed by the temperature shit unction: τ i ( T ) at ( T ) =, τ ( T ) i = 1,,.... (1) i An analoous requirement applies to piezo-rheoloical simplicity. This demands that all that all response times depend equally on pressure and is expressed by the pressure shit unction: τ i ( P) ap ( T ) =, τ ( P ) i = 1,,.... () i Thermo-rheoloically simple materials allow, by deinition, time-temperature superposition, Gross [3, 4], i.e., the shitin o isothermal sements into superposition to enerate a master curve, thereby extendin the time scale beyond the rane that could normally be covered in a sinle experiment. Fiure 1a shows a schematic illustratin this procedure. Fillers and Tschoel [5], and Moonan and Tschoel [6] showed that piezorheoloically simple materials permit analoous, time-pressure superposition, i.e., the shitin o isobaric sements in a similar manner. Because o the equivalence o time and requency, shitin o isothermal (or isobaric) sements alon the loarithmic requency axis applies just as it applies to shitin alon the loarithmic time axis. T 1 (a) (b) lo G ( t ) T a T1 T T lo a T T T 3 a T a T3 T <T 1 <T <T 3 T > T lo t/a T lo t/a T Fiure 1: (a) Shitin isothermal sements into a master curve; (b) two master curves at dierent temperatures. The British Society o Rheoloy, 5 ( 51

4 I. Emri, Rheoloy Reviews 5, pp Thermo- and piezo-rheoloical complexity inappropriate shitin. The requirements embodied in equations (1) and () severely restrict the classes o polymers where superposition can be utilized to simpliy the presentation o their physical properties. In eneral, superposition deinitely does NOT apply to multiphase materials such as block and rat copolymers, hybrid materials, blends, and (semi-) crystalline polymers. All these are examples o thermo-rheoloically and piezo-rheoloically complex materials. For reasons to be stated in section 6. it is doubtul whether in the strict sense o the word thermo and/or piezo-rheoloically simple materials exist at all and, thereore, whether superposition is truly applicable even or the materials just listed. Because the class o thermo-rheoloically simple materials comprises several quite important polymers, and because these dominated in the early days o the emerence o linear viscoelastic theory, it is sometimes assumed that superposability is a eneral eature o the mechanical properties o polymeric materials. However, thermo-rheoloically and piezo-rheoloically complex materials display two or more distinct distributions o response times, each with its own timetemperature (or time-pressure) dependence. Over limited ranes o the experimental data these multi-phase materials nevertheless oten seem to allow shitin o isothermal sements into superposition, eneratin misleadin master curves. Rusch [7] has examined the relaxation behaviour o three typical commercial ABS (acrylonitrile-butadiene-styrene) polymers. Althouh these were almost certainly heteroeneous, the author concluded that time-temperature superposition applied reasonably well. Such conclusions may be reached when the behaviour is viewed throuh an experimental window (the span o time or requency within which the measurements are taken) that is simply not suiciently wide or the lack o superposability to reveal itsel (Plazek [8]; Fesko and Tschoel [9, 1]; Kaplan and Tschoel [11]; Cohen and Tschoel [1]). O course, the master curves resultin rom such inappropriate shitin are in error. Fesko and Tschoel [9] have developed a procedure or the time-temperature superposition o two-phase materials. Another procedure was proposed, more recently, by Brinson and Knauss [13, 14]. However, the mechanical responses o the constituent homopolymers and their temperature unctions must be known and this, in eect, bes the point at issue. The matter has been examined throuh computer simulations by Caruthers and Cohen [15]. 1.3 Glass transition temperature and lass transition pressure. Superposition is intimately linked with the concept o the lass transition phenomenon in polymers. Shen and Eisenber [16] compiled an early, exhaustive review o the lass transition resultin rom a chane in temperature. This transition is characterized by the so-called lass transition temperature, T, which, by deinition, is that temperature below which the micro-brownian thermal motion o the polymer chain sements eectively ceases, i.e., slows down to such an extent that the semental rearranements cannot be ollowed experimentally. It may be deined operationally as shown in iure, taken rom Moonan and Tschoel [17]. 5 The British Society o Rheoloy, 5 (

5 I. Emri, Rheoloy Reviews 5, pp Speciic Volume [m k x1 ] Hypalon 4 Viton B Temperature [ º C].5 ºC/min Fiure : Operational deinition o T The iure shows T obtained rom the intersection o a plot o the speciic volume aainst the temperature, here or two rubbers havin the same T. As determined experimentally, the lass transition temperature is a kinetic phenomenon that, however, shows many o the ormal aspects o a second-order thermodynamic phase transition. Below T the polymer is not in equilibrium, metastable or otherwise. A certain amount o entropy is rozen in as a consequence o the tremendous loss o chain mobility that prevents the detection o semental rearranements on any inite experimental time scale. Thus, Nernst s heat theorem (see, e.., Tschoel [18], p. 47), which requires the entropy to vanish at the absolute zero o temperature, is not obeyed. DiMarzio and Gibbs [18, 1, ] have shown that it is possible to postulate the existence in polymers o a second-order thermodynamic transition temperature, T, that could, however, be reached only by an ininitely slow coolin process. For reasons to be stated in section.7 we shall desinate this temperature as the threshold temperature and assin the symbol T L to it. The lass transition temperature depends on pressure, and this has been noted by several authors (see, e.., Paterson [6], Bianchi [7]). In analoy to the lass transition temperature, T ( P ), there exists also a lass transition pressure, P ( T ), above which the micro-brownian motion o the polymer chain sements ceases. Above P a polymer is aain not in thermodynamic equilibrium or the same reason that is not in equilibrium below T. One may postulate that in polymers there exists also a second-order thermodynamic phase transition pressure, P (or P L ), but nothin seems to be known about it at this time, except its existence. Temperature would presumably aect the lass transition The British Society o Rheoloy, 5 ( 53

6 I. Emri, Rheoloy Reviews 5, pp pressure but data on this phenomenon also appear to be lackin. Bein kinetic rather than thermodynamic phenomena, the lass transitions depend on the time rate at which they are determined. They are actually narrow ranes rather then speciic temperatures or pressures. Superposition requires that the material be in thermal and mechanical equilibrium. In order to reach the equilibrium in a relatively short period o time, material must thereore be above the lass transition temperature, T, and/or below the lass transition pressure, P. 1.4 Purpose and scope o the article. Despite the reservations expressed in the previous sections, superposition has been and is bein used widely. The literature on the eect o temperature on the mechanical properties o polymeric materials is extensive. That on the eect o pressure is less so. This article presents a review o what has been done to date in modellin temperature and/or pressure shit unctions or polymeric materials in equilibrium with respect to temperature and pressure. Two papers concerned solely with the eect o pressure on the viscosity o non-polymeric liquids, e.., Matheson [8], have not been reviewed. Our discussion will be acilitated by reconizin certain reions o the linear viscoelastic response unctions. Those that one records on crosslinked materials (e.., rubbers) exhibit three distinct reions: the lassy, the transition, and the rubbery reion. In uncrosslinked materials there exist our: the lassy, the transition, the plateau, and the low reion. Fiure 3 displays a schematic o these reions. We initially consider the eect o temperature and pressure on the transition reion o amorphous, isotropic, homoeneous materials. This reion is commonly reerred to as lassy lo G ( t ) transition rubbery plateau low Time Fiure 3: Schematic displayin the distinct reions o crosslinked and uncrosslinked materials. 54 The British Society o Rheoloy, 5 (

7 I. Emri, Rheoloy Reviews 5, pp the main transition reion to distinuish it rom possible other transitions (see section 6.. The scope will be widened in sections 6, 7, and 8 to include the other reions o the linear viscoelastic response. We shall consider primarily deormation in shear. Brie reerence will be made to other types o deormation in section 9, and to anisotropic materials in section 1. In sections, 3, and 4 we will introduce various models describin the eect o temperature and pressure on the mechanical properties o thermo- and/or piezo-rheoloically simple polymeric materials. Each section will contain a critique o these models. These critiques will be deerred, however, until the models in each section have all been introduced. We shall urther attempt also to point out to the best o our understandin areas where inormation on the eect o temperature or pressure on the mechanical properties o thermo- and/or piezorheoloically simple materials is currently lackin.. MODELLING THE EFFECT OF TEMPERATURE The eect o temperature on the time-dependent mechanical properties o polymers is relatively well understood althouh a number o problems remain open. Above T it is enerally modeled by the well-known WLF equation, named ater its oriinators, Williams, Landel, and Ferry. We review the derivation and properties o this equation as necessary backround to our discussion o the eect o pressure. The WLF equation is concerned with the relaxation or retardation behaviour arisin rom the micro-brownian (or semental, or backbone) motion o the polymer chains. It has been derived rom consideration o the ractional ree volume. However, equations o the same orm have been proposed also by considerin the relaxation or retardation process as the physical counterpart o a chemical reaction, or by considerin chanes in the excess (coniurational) entropy, or in the excess enthalpy. While all these approaches lead to equations o the same orm, the empirically obtained parameters o the equation are interpreted dierently..1 The Doolittle equation. Many theories or modellin the eect o thermodynamic parameters such as temperature or pressure on the time-dependent behaviour o polymers are based on the ree-volume concept. Doolittle and Doolittle [9] introduced this concept in their work on the viscosity o liquids. They assumed that the chane in viscosity depends on the distribution o molecule-size holes in the luid. The sum o these holes represents the ree-volume, which directly aects the mobility o the liquid molecules. Doolittle and Doolittle [9] expressed this in their semi-empirical expression or the viscosity,η, o liquids: The excess (coniurational) entropy may be deined as the dierence between the (coniurational) entropy o the polymer above the lass transition temperature and that o the lass in a hypothetical reerence state rom which, even in principle, no more o the property can be lost due to relaxational processes, Chan et al. [35]. The British Society o Rheoloy, 5 ( 55

8 I. Emri, Rheoloy Reviews 5, pp Vφ V V 1 η = Aexp B = Aexp B = Aexp B 1 V V.. (3) Here, A and B are empirical material constants, V φ is the occupied volume (i.e., the volume occupied by the molecules o the liquid), and V is the ree volume. This is the unoccupied intermolecular space available or molecular motion. It can also be viewed as the excess volume, i.e., the volume that exceeds the occupied volume. Hence, the total macroscopic volume is V = Vφ + V, and = V / V called the ractional ree volume. Fiure 4 illustrates these concepts schematically. Althouh these concepts cannot be ormulated riorously, they make intuitive sense. Temperature aects the ree volume. In equation (3), η and represent the viscosity and the ractional ree volume at a iven temperature, T. Lettin η, and be the same quantities at a conveniently chosen reerence temperature, T, leads to the viscosity ratio: η 1 1 = exp B.. (4) η The Doolittle equation oriinally reerred to liquids consistin o small molecules. Williams et al. [3] adapted it or use with polymers by modiyin the Rouse theory o the behaviour o polymers in ininitely dilute solution to the behaviour o polymers in bulk, Ferry [31], p. 5. On the basis o this approach they let: where η ( T) τ i ( T ) η = or η ( T ) τ ( T ) η i τ τ i =,. (5) i η is the steady-state viscosity and τ i is any relaxation time at the same Fiure 4: Volume-temperature diaram o an amorphous polymer. 56 The British Society o Rheoloy, 5 (

9 I. Emri, Rheoloy Reviews 5, pp temperature, T. Further, η and τ i are the same quantities at the reerence temperature, T. Makin use o this proportionality, one may deine a temperature shit actor, a T, as an abbreviation or the temperature shit unction, a ( ) T T, to yield: τ 1 1 at = = exp B..(6) τ I another temperature, say, the lass transition temperature, the reerence temperature, we have: T, is chosen as a τ 1 1 T exp B = =,.(7) τ where is the ractional ree volume at T. In loarithmic orm equation (6) becomes: lo a T B 1 1 B 1 1 = =,. (8).33 ( T ) ( T ).33 where lo a T is the distance required to brin (shit) data recorded at the temperature, T, into superposition with data recorded at the reerence temperature, T, alon the loarithmic time axis (c. iure 1). Equation (8) is the startin point or the temperature shit unctions that are based on the ree volume concept. In sections 3 and 4 we will show that equations o the same orm can serve to model the eect o pressure, and that o both pressure and temperature.. The Williams-Landel-Ferry (WLF) model. Williams et al. [3] modelled the eect o temperature on the mechanical properties o amorphous homopolymers or random copolymers by considerin that the ractional ree volume o such polymers increases linearly with temperature. They thus let: = + α ( T T ),. (9) where α is the expansivity (i.e., the isobaric cubic thermal expansion coeicient) o the ractional ree volume (c. iure 4). Substitution o equation (9) into equation (8) leads to: lo a T ( B /.33 )( T T ) =..(1) / α + T T This is usually used in the orm: c ( T T ) lo a T =,. (11) c T T 1 + The British Society o Rheoloy, 5 ( 57

10 I. Emri, Rheoloy Reviews 5, pp where: c = B /.33. (1) 1 and c = α.. (13) / Equation (11) is the WLF equation. The superscript reers the parameters o the equation to the reerence temperature, T. I a dierent reerence temperature, say, T r, is chosen, the constants must be chaned usin the symmetric relations: and c = c + T T,. (14) r r c c = c c.. (15) r r 1 1 The product, c1c, is independent o the reerence temperature and may thereore be written without superscripts. Reerred to the lass transition temperature, the WLF equation becomes: lo a T c1 ( T T ) =,. (16) c + T T where now, however: and c1 = B /.33. (17) c = α,. (18) / with as the ractional ree volume at the lass transition temperature. The constants, c 1 and c, are ound experimentally by shitin isothermal sements o the response unction into superposition alon the loarithmic abscissa to yield a master curve as illustrated in iure 1a. Fiure 1b shows master curves at the reerence temperature, T, and at the experimental temperature, T. The constants are ound rom the slope and the intercept o a straiht line throuh a plot o T lo a T vs. T, Ferry [31], p. 74. The numerical procedure or calculatin the constants is presented in the Appendix A. Williams et al. [3] also proposed a universal orm o the WLF equation havin a sinle adjustable parameter, T S. This orm: lo a T 8.86( T T ) =,. (19) T T 58 The British Society o Rheoloy, 5 (

11 I. Emri, Rheoloy Reviews 5, pp has been ound not to be truly universal even thouh it applies reasonably well to many polymeric materials, see, e.., Schwarzl and Zahradnik [3]. It should thereore be used only as a last resort when the constants, c 1 and c, are not available. It is o interest to obtain the molecular parameters contained in equations (1) and (13). However, there are only two known empirical parameters, the shit constants, c 1 and c, available or the determination o the three unknown parameters, B,, and α. The most common way to deal with this problem has been to assume that the proportionality actor B is unity, Ferry [31], p. 87. In section 4.3 we adduce evidence that this assumption cannot be upheld and show how B can be determined experimentally..3 Bueche s temperature (BT) model. Bueche [33] proposed a hole theory o the ree volume that he considered to be a modiied orm o the WLF equation. He derived: where and lo at = b 1 [ 1 + b ( T T )] 1 1 l,. () 9( v * / v L ) b1 =,. (1).33π b α m P =.. () ρv L In these equations, v * is the critical hole volume or molecular rearranements, v L is the ree volume per molecular sement at T L, the threshold temperature (see sections.5 and.7), α P is the thermal expansion coeicient, m is the molecular weiht o the sement, and ρ is the density. Some o these parameters are not easily determined. On the strenth o the scanty data available to him then, Bueche concluded that his equation provided a better approximation at low temperatures while the WLF equation ave better areement at hiher temperatures. The equation does not seem to have ound many users..4 The Bestul-Chan (BC) model. Bestul and Chan [34], and Chan et al. [35], modelled the eect o temperature by considerin the (excess) coniurational entropy rather than the ree volume. Their loarithmic shit actor takes the orm: The British Society o Rheoloy, 5 ( 59

12 I. Emri, Rheoloy Reviews 5, pp lo a T B 1 1 BC =,. (3).33 Sc ( T) Sc ( T ) [c. equation (8)], where Sc ( T ) and Sc ( T ) are the excess entropy o the polymer above the lass transition temperature, and that o the lass, respectively. Insertin: T Sc ( T ) = Sc ( T ) + CP d ln T = Sc ( T ) + CP ln( T T ),. (4) T where CP is the dierence in the speciic heats o the polymer in the rubbery or plateau reion, and the lassy reion, respectively, yields: lo a T [ BBC.33 Sc ( T )]ln( T T ) =.. (5) S ( T ) C + ln( T T ) c P Since or values near unity ln x x 1, the authors let ln( T T ) = ( T T ) T. This leads to an equation o the orm o the WLF equation, where now, however, BBC c1 =,. (6).33 Sc ( T ) and c T Sc ( T ) =. C P. (7) When the experimental temperature, T, is siniicantly larer than T, the approximation used to obtain an equation o the orm o the WLF equation may not be adequate. In that case, however, the loarithm may be expanded to any deree desired, usin: ln x = ( x 1) ( x 1) + ( x 1), < x as lon as T T. L,. (8).5 The Goldstein (GS) model. Goldstein [36] considered excess enthalpy instead o excess entropy. His model may be derived in analoy to the BC model by lettin the loarithmic shit actor be iven by: lo a T B 1 1 GS =,. (9).33 H ( T ) H ( T ) where H ( T ) and H ( T ) are the excess enthalpy o the polymer above the lass transition temperature, and that o the lass, respectively. This aain leads to a model o the temperature shit actor that is identical in orm with that o the WLF model except that now: 6 The British Society o Rheoloy, 5 (

13 I. Emri, Rheoloy Reviews 5, pp c 1 BGS =,. (3).33 H ( T ) and c H ( T ) =.. (31) C P.6 The Adam-Gibbs (AG) model. Goldstein [36] objected to the use o the entropy alone (as in the BC model) instead o its product with temperature as the principle variable overnin lass ormation. The model o Adam and Gibbs [37] is ree o this objection. They let: where lo a T B 1 1 AG =,. (3).33 TSc ( T ) Tr Sc ( Tr ) S ( T ) = S ( T ) + C lnt T,. (33) c c r P r is the coniurational entropy at the temperature, T, and T r is the reerence temperature [c. equations (4) and (8)]. CP is the dierence in speciic heat between the equilibrium melt and the lass that is considered to be independent o the temperature. Accordin to DiMarzio and Gibbs [18, 1, ], althouh at hiher temperatures the polymer chains are able to assume a lare number o coniurations, at their second-order thermodynamic transition temperature, T, there is only one (or at most a very ew) coniurations available to the backbone. T is thus the threshold temperature below which the relaxation or retardation processes cannot be activated thermally. We thereore assin to it the symbol T L, takin the subscript L rom the Latin limen, a threshold. The coniurational entropy o the lass is thus deemed to vanish at T = T. Settin S ( T ) =, equation (33) yields: L c L S ( T ) = C ln T T..(34) c r P r L Substitutin this into equation (33) ives Sc ( T ) = CP ln T TL and equation (3) thus aain leads to an equation o the orm o the WLF equation, with: and r BAG c1 =,. (35).33 C T ln( T T ) P r r L T ln( T T ) c =.. (36) ln( ) 1 ( ) ln( ) r r r L Tr TL + [ + Tr T Tr ] Tr TL The British Society o Rheoloy, 5 ( 61

14 I. Emri, Rheoloy Reviews 5, pp The second parameter is a weak unction o the temperature, T, involvin the reerence temperature, T r, as well as the threshold temperature, T L. With ood approximation: and thus: [ ] 1 + T ( T T ) ln( T T ) 1,. (37) r r r L T ln( T T ) c Tr TL.. (38) 1 ln( ) r r r L + Tr TL The second o these equations aain ollows upon lettin ln x x 1..7 The rate process (RP) model. A temperature shit actor identical in orm with at o the WLF equation may be derived also rom a modiication o the Arrhenius equation well known rom the theory o chemical reaction rates. Relaxation (or retardation) may be rearded as a physical process in which the material is carried rom state A into state B. The response time (relaxation or retardation time) is the time constant or this process. An increase in temperature shortens the response time because it causes the stress to relax, or the strain to reach its ultimate value, aster than at a lower temperature. Consequently, the response time is the reciprocal o the rate at which the physical chane takes place, and is thus akin to the reciprocal o a chemical reaction rate. Provided that the conventional theory o reaction rates, Kauzmann [38], is applicable to relaxation and retardation phenomena, one would expect the temperature dependence o these processes, in analoy to that o a chemical reaction, to be iven by the equation: τ = Aexp a,. (39) R T where a is the chane in the (molar) ree enthalpy 3 o activation or the process per deree o temperature, and R is the universal as constant. The pre-exponential actor A is considered to be a constant since it is at most a weak unction o temperature. Equation (39) is the physical equivalent o the Arrhenius equation. Accordin to this equation the relaxation time becomes ininite at the absolute temperature T =. However, an ininite relaxation time implies that the material does not relax. For polymers, relaxation or retardation behaviour can become eectively impossible well above T =. The backbone motion, i.e., the main chain semental motion that ives rise to the response time distribution, ceases below the lass transition temperature, 3 The ree enthalpy (c. Tschoel [18], p. 56) is also called the Gibbs ree enery or Gibbs potential. 6 The British Society o Rheoloy, 5 (

15 I. Emri, Rheoloy Reviews 5, pp T. When this is measured at the usual experimental coolin rate o 5 to 15 C/min, it is about 5 C above T. We may thereore set: T T (4) L Subtractin T L rom T modiies equation (39) to allow the response time to become ininite when T = T. We then have: L a τ = Aexp.. (41) R ( T T ) L Model 1 WLF c1 B.33 c c = c = α Bestul-Chan c1 = B.33 S ( T ) c = T S ( T ) C BC c c P Goldstein c1 = B.33 H ( T ) c = H ( T ) C GS Adams-Gibbs c1 = B.33 C T ln( T T ) c = T T AG P L Rate Process c1 =.33R c c = T T a Table 1: Comparison o the parameters o the temperature superposition models. L L P Writin equation (41) or τ, dividin it into equation (4), and takin loarithms leads to: lo a T 1 1 =.33 a R T TL T TL.. (4) This may be rearraned to yield equation (11) where now, however, and 1 = a.33r c c L,. (43) c = T T.. (44) The ormer yields an estimate o the apparent molar ree enthalpy o activation, while the latter yields one o T L. Moreover, a plot o lo a T vs. 1 ( c + T T ) yields a straiht line with constant slope: The British Society o Rheoloy, 5 ( 63

16 I. Emri, Rheoloy Reviews 5, pp =.33R c c,. (45) a 1 as lon as these interpretations o the constants c 1 and c are accepted..8 Critique o the time-temperature superposition models. Except or Bueche s Temperature (BT) model, all other time-temperature superposition models have the orm o the WLF equation or can be reduced to it. They dier only in the interpretation o the parameters o the equation. The Williams- Landel-Ferry (WLF), the Bestul-Chan (BC), the Adam-Gibbs (AG), and the Goldstein (GS) models orm a roup in that all our consider excess quantities. The rate process (RP) model is, strictly speakin, not based on an excess property althouh one miht consider T TL [c. equation (41)] as the excess o the experimental temperature, T, over the hypothetical reerence temperature, T L. The BC and AG models are concerned with chanes in the coniurational entropy. The coincidence between the WLF and BC and AG models becomes understandable i one considers that chanes in the coniurational entropy ariserom cooperatively rearranin reions that require ree volume in which to move. At constant pressure the chane in enthalpy is iven by dh = du + PdV, where U is the internal enery. Thus a chane in volume underlies also the GS model. Excess (ree) volume, excess entropy, and excess enthalpy are undamentally related, Eisenber and Saito [39], Shen and Eisenber [16], and Hutchinson and Kovacs [4]. Theoretically, the excess entropy is an appealin concept. However, volume is much more easily measured experimentally than entropy or enthalpy, and is plausibly related to molecular mobility in a simple manner, Ferry [31], p. 85. We note that the WLF model is the only one that depends on three unknown molecular parameters. Since C P can be determined calorimetrically, all others sport only two. It would clearly be o interest to decide which o the models is the correct one. To do this, one would need to determine separately the molecular parameters in terms o which the models interpret the constants, c 1 and c (see table 1). Unortunately, only some o those parameters are accessible independently. To see, nevertheless, how one miht distinuish between the models based on the ree volume, the excess entropy, and the excess enthalpy concepts, we irst recall that: and that: α = α α α α = α,. (46) e φ q κ = κ κ κ κ = κ,. (47) e φ q where the subscripts, e, q, and reer to the expansivities and the compressibilities o the ree, entire (total), (pseudo-)equilibrium, and lassy volumes, respectively (c. iure 4). In the second o these equations we have substituted the (pseudo)- equilibrium values or those o the entire volume, and the lassy values or those o the occupied volume. For a justiication o the latter, see Moonan and Tschoel [41]. The chane o the ree (excess) volume as a unction o both temperature and 64 The British Society o Rheoloy, 5 (

17 I. Emri, Rheoloy Reviews 5, pp pressure is iven by: V T V P ex ex dvex = dt + dp = V dt V dp α κ.. (48) Assumin, ollowin Goldstein [36], to whom this treatment is due, that in the absence o pressure chanes the ree volume is constant, i.e., dvex = at and below the lass transition temperature, we have: dt dp i V ex determines T. because: and κ =,. (49) α For the excess entropy chane we have: S S C α T P T κ ex ex P dsex = dt + dp = dt,. (5) Sex CP dt = dt,. (51) T T Sex dp = VdP P α But, i we urther assume with Goldstein [36] that P.. (5) dvex = implies dsex =, then: dt TV α =,. (53) dp C i it is S ex that determines T. Goldstein then shows on the basis o a slihtly more involved derivation, that consideration o H ex as the determinin actor o T also leads to equation (53). Thus, measurements o dt dp, the chane o the lass transition temperature with pressure, combined with determinations o α, κ, and CP should allow one to decide whether it is the ree volume on the one hand, or the excess entropy or excess enthalpy on the other hand, that are rozen in at and below the lass transition temperature. So ar, measurements have not been made with suicient accuracy on the same material in the same laboratory to decide the issue unambiuously. Should a valid comparison o equations (49) and (53) have decided aainst the volume as the overnin actor, the question whether it is the excess entropy or the excess enthalpy that determines the lass transition temperature would still appear to be moot. We remark that the hypothetical reerence temperature o an excess property is property T L, not T. The Adam Gibbs (AG) and the Rate Process (RP) models correctly reer to T L. The British Society o Rheoloy, 5 ( 65

18 I. Emri, Rheoloy Reviews 5, pp Measurements as unctions o temperature. Hitherto we have considered isothermal measurements as a unction o time, or o requency. Measurements are, however, requently made isochronically, i.e., at ixed times or at ixed requencies. Determinations with the orced-oscillation torsion pendulum are a point in case. The time required to make the measurements must be short compared with the rate at which the temperature is chaned. In practice it is chaned incrementally and the measurements are taken when thermal and mechanical equilibrium has been reached ater each increment. Fiure 5 shows a schematic plot o lo G( T ) as unction o T. The temperature at the inlection point, i.e., the inlection temperature, T inl, is sometimes considered to be a rouh indication o the lass transition temperature, T. A plot o, or instance, the shear modulus, as a unction o temperature shows qualitatively the same eatures as a plot o the same modulus as a unction o the loarithmic time. Yet measurements as a unction o temperature at a ixed time or requency cannot be brouht into superposition by shitin alon the T-axis. While the time shit actor, a T, and hence also its loarithm, is independent o time or requency [c. equation (11)], the temperature shit actor, T = T T, is evidently a unction o the temperature. A typical set o G( T ) vs. T curves, or NR (natural rubber), at our isochronal times t 1 < t < t 3 < t 4 is shown in iure 6, taken rom Prodan [4]. The shorter the time (or the hiher the requency), the slower is the rate o response to an increase in temperature. The slope o the lo G( T ) vs. T curve thereore depends both on the steepness o G( T ) and the steepness o at at the particular isochronal time, in accordance with: lo G ( T ) t = t T inl Temperature Fiure 5: A schematic plot o lo G(T ) vs. T. 66 The British Society o Rheoloy, 5 (

19 I. Emri, Rheoloy Reviews 5, pp lo G (T ) [MPa] t 1 t =.16 s t t =.4 s t 3 t = 1. s t 4 t =.51 s Temperature [ o C] Fiure 6: G( T ) vs. T or our isochronal times, t i [4]. lo G( T ) lo G( T) lo t at = T t lo t a T T.. (54) t T Thus, when the temperature response at a ixed time, t, or at a ixed requency, ω is iven, and at is known or any reerence temperature T r, the temperature response at any other ixed time t, or ixed requency, ω, can be constructed, but this must be done point by point. To do this, we must ind the temperature, T, to which the measurement at temperature T must be shited or a iven value o t t or ω ω. All this can be understood by the ollowin consideration. Let the parameters o the WLF equation be known or the polymer under investiation. The shit alon the T-axis is then iven by the relations: c lo t t c loω ω T = =,. (55) c lo t t c loω ω 1 1 which are obtained rom equation (11) by lettin at = t t, or ω ω, T T = T, and then solvin or T. A modulus measured at the temperature, T, and at the ixed time, t (or the ixed requency, ω) is thereore equivalent to a modulus measured at the temperature, T + T, and at the ixed time, t (or ixed requency, ω ). Thus G( T; t ω) = G( T + T; t ω). For each value o the modulus there exists, thereore, a translation alon the temperature axis. However, the amount o the translation is not constant but varies with T. Equation (55) allows the construction o master curves o the temperature dependence o a iven viscoelastic response unction when measurements at a iven ixed time or requency do not cover the entire desired rane. The British Society o Rheoloy, 5 ( 67

20 I. Emri, Rheoloy Reviews 5, pp MODELLING THE EFFECT OF PRESSURE Contrary to the case o the eect o temperature, very little has been reported on the eect o pressure. Due to the time-dependent nature o polymers, pressure related material properties, such as the bulk modulus, K, are also unctions o time and are not constant as is oten assumed. Furthermore, while K is the reciprocal o the bulk compliance, B, the time-dependent bulk relaxation modulus, K( t ), is not the reciprocal o the time-dependent bulk creep compliance, B( t ). The two are linked throuh the convolution interal: t K( t u) B( u) d ln u = t,. (56) and are said to be time reciprocal. Research on the mechanical properties o materials under isotropic pressure bean in the early nineteen orties. P. W. Bridman 4 investiated pressure-induced chanes in the compressibility, viscosity, and thermal conductivity o metals and nonmetals [43]. First papers on pressure dependence o the lass transition temperature, T, appeared in the sixties [7, 33, 44, 46, 47 ]. The eect o pressure on the relaxation behavior o piezo-rheoloically simple polymers is analoous to the eect o temperature on that o thermo-rheoloically simple ones, takin into account that hih temperature corresponds to low pressure and vice versa. In contrast to the eect o temperature, data on the eect o pressure on the time-dependent properties o polymeric materials are relatively scant because they are diicult to determine experimentally. To model the eect o pressure one thus establishes a pressure shit unction, ap ( P ), in analoy to the temperature shit unction, at ( T ). Experimentally, ap ( P ) is ound by shitin isobaric sements o a iven response unction into superposition. The easibility o time-pressure superposition is demonstrated by the experimental data shown in iure 7, taken rom Moonan and Tschoel [6]. The iure actually an instant o time-temperature-pressure superposition to be discussed in section 4.3 displays three shear relaxation master curves at dierent pressures. The curves are similar in shape and can be shited into superposition alon the loarithmic time axis. Thus, knowin only one relaxation curve, say, the one at the reerence pressure, e.., atmospheric pressure, P =.1MPa, the other two curves can be predicted i the pressure shit actor, a P, is known. We now turn to the discussion o models or the superposition o the eects o time (or requency) and pressure. With the exception o the BP model (see section 3.) they can be shown to be based on the equation: 4 P. W. Bridman: General survey o certain results in the ield o hih-pressure physics, Nobel Lecture, December 11, (1946) 68 The British Society o Rheoloy, 5 (

21 I. Emri, Rheoloy Reviews 5, pp lo G (t/a ) P [MPa ] ,1 MPa MPa 34 MPa lo t/ a P [s] Fiure 7: The eect o pressure on the shear modulus, G( t ) [6]. lo a P B 1 1 B 1 1 = =,. (57).33 ( P) ( P ).33 the pressure analoue o equation (8). 3.1 The Ferry-Stratton (FS) model. The model proposed by Ferry and Stratton [46] (see also McKinney and Belcher [47], Tribone et al. [48], and Freeman et al. [49]) assumes that the ractional ree volume decreases linearly with pressure. Thus: ( P) = κ ( P P ),. (58) where κ is the isothermal compressibility o the ractional ree volume, and P is the test pressure while P is the reerence pressure at which = ( P ) is known. Introducin equation (58) into equation (57) we obtain the Ferry Stratton equation as: or lo a B P P P =.33 / κ ( P P ),. (59) s ( P P) lo a P =,. (6) s + P P 1 where: The British Society o Rheoloy, 5 ( 69

22 I. Emri, Rheoloy Reviews 5, pp B s1 =,. (61).33 and s =.. (6) κ Equations (6) to (6) are the pressure analoues o the correspondin temperature relations, equations (11) to (13). The equations assume that the temperature is constant and that κ does not depend on either T or P. where: and Equation (6) may be reerred to the lass transition pressure, P by settin: lo a s 1 s P s ( P P) =,. (63) s + P P 1 B =,. (64).33 =.. (65) κ These equations were obtained usin: and s P s P s s + = +,. (66) = s s,. (67) 1 1 the pressure analoues o equations (14) and (15). 3. Bueche s pressure (BP) model. Bueche [33] proposed a time-pressure superposition model also. His model is not based on equation (57). It ives the loarithmic pressure shit actor as: lo a P * P v =,. (68).33kT * where v is aain the critical hole volume [c. section.3], and k is Boltzmann s constant. Bueche considered his model to be no more than a useul approximation. 7 The British Society o Rheoloy, 5 (

23 I. Emri, Rheoloy Reviews 5, pp The O Reilly (OR) model. O Reilly [5] derived his model rom the observation (Ferry [31], p. 93, Tribone et al. [48], Freeman et al. [49]) that the pressure dependence o the shit actor, a P, can be described by the simple exponential relation: a = exp θ ( P P ),. (69) P where θ = 1 Π, and Π is a material-dependent empirical parameter that is considered to be independent o the pressure. The model is also reerred to as the exponential model. Its loarithmic orm becomes: B lo ap = ( P P ),. (7).33Π Equatin this with equation (57) ives: 1 1 P P B =,. (71) Π and solvin or the ractional ree volume,, yields: Π P =,. (7) Π + P upon settin B equal to 1. The model supplies the compressibility o the ractional ree volume as: Π κ = = = P Π + P P Π + P P T ( ).. (73) In the O Reilly model, thereore, the ractional ree volume,, depends inversely on the pressure while its compressibility, κ, depends inversely on the square o the pressure. 3.4 The Kovacs-Tait (KT) model. Kovacs and his associates Ramos et al. [5] (see also Tribone et al. [48], and Freeman et al. [49]) proposed a model derived rom the compressibility o the Tait equation [53] (see also Fillers and Tschoel [5]), κ KT 1 V 1 = =.. (74) V P K + k P * T KT KT In thermodynamic equilibrium the compressibility is the reciprocal o the bulk modulus. The subscript KT on the parameters above identiies them as bein parameters o the Tait equation. The asterisk siniies that: K * KT ( T ) K KT ( T ) P = P =. The British Society o Rheoloy, 5 ( 71

24 I. Emri, Rheoloy Reviews 5, pp * Equation (74) assumes that the pressure dependence o KKT ( T ) can be accounted or adequately by the irst term o a polynomial expansion in the pressure. Interation between the limits V, V, and P, P, where the subscript denotes the reerence state, then leads to: * V V KKT ( T ) + kkt P = ln * V KKT ( T ) + kkt P 1/ k KT The let-hand side o this equation can be expanded to read: V V V V φ + V V = V V + V φ φ.. (75).. (76) where V φ, V are the occupied and ree volumes, respectively [c. section.1], and V φ, V are the same in the reerence state. I we now consider that Vφ Vφ, i.e., that the occupied volume is sensibly the same in the experiment and in the reerence conditions, we ind that: V V V V.. (77) V V V One may now arue that V V V V = and that V V =. The latter appears to be a satisactory assumption. I the ormer is accepted also, equation (75) becomes: * KKT ( T ) + kkt P = ln * KKT ( T ) + kkt P or, equivalently, * KKT ( T ) + kkt P = ln * KKT ( T ) + kkt P 1/ k KT 1/ k KT.. (78).. (79) Equation (78) shows that the ractional ree volume o the KT model, just as its compressibility, κ, depends inversely on the pressure. Combinin equation (57): KT B ln P a =,. (8) and equation (79) leads to: lo a P Alternatively, * B KT 1 + kkt P / KKT ( T ) = ln * kkt P / KKT ( T ) θc 1/ k KT.. (81) P P lo ap = lo 1+,. (8) CKT 7 The British Society o Rheoloy, 5 (

25 I. Emri, Rheoloy Reviews 5, pp * where θ C = BKT kkt, and CKT = KKT ( T ) kkt + P is a material-dependent parameter that is a unction o the ixed experimental temperature, T, but is independent o the pressure. When obtain: P is small, we may expand the loarithm accordin to equation (8) to θ P P C C lo ap = lo 1 + ( P P ).33 CKT.33 and the Kovacs Tait model reduces thus to that o O Reilly in orm. θ,. (83) 3.5 Critique o the time-pressure superposition models. Table lists the assumptions made by the Ferry Stratton (FS), the O Reilly (OR), and the Kovacs Tait (KT) models or the compressibility o the ractional ree volume, κ, and or the ractional ree volume,, itsel. We have shown that, leavin Bueche s Pressure (BP) model aside, the timepressure superposition can all be related to equation (57), the pressure analoue o the loarithmic temperature shit actor, equation (8) and thus to the ree volume concept reardless o the startin point o their derivation. There is, nevertheless, a undamental dierence between time-temperature and time-pressure superposition. While the expansivity o the ractional ree volume, does not depend on the temperature above T, the compressibility o the ractional ree volume, κ, must be considered to depend on the pressure below P. The dierences between the models listed in table thus reside in the assumptions they make about κ. The Ferry Stratton (FS) model does not make any assumption concernin it. It requires κ to be ound rom equations (61) and (6) just as α is to be ound rom equations (1) and (13). In both cases, however, either parameter can be determined correctly only i B is known. Model lo a P κ Ferry- Stratton O Reilly Kovacs- Tait Table : Comparison o models or the time-pressure shit actor, a. P lo a B P P P =.33 / κ ( P P ) B lo ap = P P.33Π ( ) P P lo ap = lo 1+ CKT θ C κ κ = Π + P P 1 κ KT = K + k P * KT KT = κ ( P P ) Π = Π + P P n/a The British Society o Rheoloy, 5 ( 73

26 I. Emri, Rheoloy Reviews 5, pp Superposition models can be applied successully only within their rane o validity. The pressures encountered in the manuacture o polymeric materials easily exceed 1 MPa (Boldizar et al. [54]). A iven time-pressure superposition model should thereore yield valid results or pressures at least up to 1 MPa. The Ferry Stratton (FS) model achieves satisactory superposition only or small pressures, usually up to 1 MPa. The model is not successul at hiher pressures because it does not take into account the pressure dependence o the compressibility. The O Reilly (OR) and the Kovacs Tait (KT) models attempt to take this into account. Both models incorporate an inverse dependence o the compressibility on the pressure, albeit in dierent manner. The OR model predicts a linear dependence o lo a P on P which is not born out by experiment. The assumption that V / V V / V, inherent in the derivation o the KT model, imposes the restriction that V not be very dierent rom V. Thus, the KT model would seem to be valid only at small pressures. We do not know whether the predictions o the model have ever been tested. A dierent approach has been taken by Tschoel and co-workers [c. equation (111)]. Their time-temperature-pressure superposition model, called the FMT model or short, will be the subject o section Measurements as unction o pressure. Measurements can, o course, also be made as unction o pressure at a ixed time or requency, just as they can be made as unctions o temperature (c. iure 6). Aain, care must be taken that the material be in thermal and mechanical equilibrium ater each pressure increment beore the load is applied. A typical set o G( P ) vs. P curves, or NR (natural rubber), at our ixed times t 1 < t < t 3 < t 4 is shown in iure 8 (taken rom Prodan [4]). The curves do not extend much (i at all) beyond the transition reions. It should nevertheless be clear that superposition alon the P-axis would not be possible. A point-by-point shitin procedure analoous to that discussed in section.9 can, however, be devised aain. The shorter the time (or the hiher the requency), the slower is the rate o response to an increase in pressure. The slope o the lo G( P ) vs. P curve thereore depends both on the steepness o G( P ) and the steepness o a P at the particular isochronal time, in accordance with: lo G( P) lo G( P) lo t ap = P t lo t a P P.. (84) t P Thus, when the pressure response at a ixed time, t (or at a ixed requency, ω), is iven, and a P is known or any reerence pressure P r, the pressure response at any other ixed time, t, or ixed requency, ω, can be constructed, but this must be done point by point. To do this, we must ind the pressure, P, to which the measurement at the pressure, P, must be shited or a iven value o t / t or ω / ω. In analoy to what has been said about T in, the inlection temperature, in a plot o G( P ) aainst P the pressure at the inlection point, i.e., the inlection pressure, P, see Paterson [6], may point to the lass transition pressure, P. in l 74 The British Society o Rheoloy, 5 (

27 I. Emri, Rheoloy Reviews 5, pp lo G (P) [MPa] t 1 t t 3 t 4 =.16 s =.4 s = 1. s =.51 s Pressure [MPa] Fiure 8: G( P ) vs. P or our isochronal times, t i [6]. 4. MODELLING THE EFFECT OF TEMPERATURE AND PRESSURE The precedin section was concerned with time-pressure superposition models. We now turn to time-temperature-pressure superposition models. These are concerned with establishin shit actors at, P = at, P ( T, P) that accommodate simultaneously both temperature and pressure chanes, and are thus applicable to materials that are both thermo-rheoloically and piezo-rheoloically simple. 4.1 The Simha-Somcynsky (SS) model. Somcynsky and Simha [55, 56, 57] proposed a hole theory o the liquid state that deines the state in terms o occupied and vacant molecular lattice sites. There are two simultaneous equations to be obeyed. These are: and 1/ 6 1/ 3 1 PV % % / T% = 1 y( yv% ) + y T% yv% yv% ( / )( ) 1,11( ) 1,45 1 y ln(1 y) = ( y / 6 T% )( yv% ) [ ( yv% ) ] + [ ( yv% ) 1/ 3][1 y( yv% ) ] 1/ 6 y 1/3 1/ 6 1/3 1,. (85),. (86) where P % = P / P, V% V / V v / = = v, and T % = T / T are reduced variables deined in terms o the characteristic values labelled with the superscript. In these equations The British Society o Rheoloy, 5 ( 75

28 I. Emri, Rheoloy Reviews 5, pp y denotes the raction o occupied lattice sites. I we identiy the raction o empty lattice sites with the ractional ree volume,, we may set, Curro et al. [58] = 1 y.. (87) Equations (8) and (59) then ive: lo a T, P BSS 1 1 =,. (88).33 1 y 1 y where y is the raction o unoccupied sites under reerence conditions. Equation (88) may be called the Simha Somcynsky (SS) model. Moonan and Tschoel [6] modiied this equation in liht o suestions by Utracki [59, 6]. The reader is reerred to the oriinal papers or details. 4. The Havlìček-Ilvalský-Hrouz (HIH) model. Havlìček et al. [45] extended the Adam Gibbs theory to account or the eect o pressure. The HIH model expresses ln a T, P as: where lo a T, P BHIH 1 1 =,. (89).33 TSc ( T, P) T Sc ( T, P ) Sc ( T, P) = T CP ln( T / TL ) V ( α P ϑp / ),. (9) is the (excess) coniurational entropy at the temperature, T, and the pressure, P. T r and P r are the reerence temperature and pressure, respectively. CP is aain the dierence in speciic heats between the equilibrium melt and the lass at T and P, the latter bein taken at.1mpa MPa. Its temperature dependence is nelected. α is the chane in the coeicient o thermal expansion at atmospheric pressure, and ϑ is a proportionality constant. The pressure dependence o the volume, V, is considered neliible. We could now obtain the prediction o the HIH model or lo a T, P by substitutin equation (9) into equation (89) while lettin Sc ( T, P ) =, since the entropy o the lass is deemed to vanish. The resultin expression must aain undero certain simpliications to reduce it to the orm o the WLF equation. The authors have not done so, probably because the expressions or c 1 and c are rather cumbersome. For P = the equations reduce to those o the Adam and Gibbs (AG) model. The equations or dealin with pressure dependence at constant temperature may be ound in the paper by Moonan and Tschoel [6] who also ound that the authors had used an inappropriate assumption concernin the pressure dependence o the expansivity, and oered a better orm o the temperature dependence. 76 The British Society o Rheoloy, 5 (

29 I. Emri, Rheoloy Reviews 5, pp The Fillers-Moonan-Tschoel (FMT) model. The FMT model (Fillers and Tschoel [5], Moonan and Tschoel [6]) can be viewed as an extension o the WLF equation to account or the eect o pressure in addition to that o temperature. Fillers and Tschoel (see the Appendix or errors in the oriinal publication) take the ractional ree volume to depend on both temperature and pressure, and thereore write it as: = + P ( T ) ( P).. (91) An alternative derivation may be based on: P T = + ( T) T ( P).. (9) Althouh there is no dierence in principle between the two approaches, the ormer is slihtly preerable since it requires knowlede o the pressure dependence o the expansivity while the latter requires knowlede o the temperature dependence o the compressibility, Fillers and Tschoel [5], that is more diicult to determine. Substitution o equation (91) into equation (8) yields: lo a T, P B P ( T) T ( P) =,. (93).33 + P ( T ) T ( P) as the equation or the time-temperature-pressure superposition shit actor, Fillers and Tschoel [5, 6]. Now, ( T) = α ( P)[ T T ],. (94) P because the expansivity must be considered to depend on the pressure but its temperature dependence may be nelected as is, indeed, done in derivin the WLF equation. Introducin equation (94) into equation (93) then yields: lo a B T T θ ( P) T, P =.33 / α ( P) + T T θ ( P),. (95) where θ ( P) = T ( P) / α ( P). It remains to ind an expression or T ( P ). I, in analoy to the temperature independence o α ( P), we assume that the compressibility o the ractional ree volume is independent o pressure, we may set: ( P) = κ ( P P ),. (96) T and, lettin T = T, we reain the Ferry Stratton model, equation (58). I, on the other hand, we assume that T ( P ) is inversely proportional to the pressure, we reain the model o O Reilly (7), by settin: ( P) T Π P =.. (97) Π + P Fillers and Tschoel [5] also considered the compressibility to depend inversely on the pressure, but their approach diers rom that used in the OR and the KT models. The British Society o Rheoloy, 5 ( 77

30 I. Emri, Rheoloy Reviews 5, pp Lettin κ = κ e κφ, where κ e is the compressibility o the entire, and κ φ is that o the occupied volume, they obtained: ( P ) = κ ( T ) dp κ ( T ) dp φ T P P T P P.. (98) For the irst interal they used the compressibility as deined by: 1 V 1 κe = =,. (99) V P K ( T) + k P * T e e where, as in section 3.3 [c. equation (74)]: * Ke ( T) Ke ( T ) P = =, is the bulk modulus at zero pressure, and ke is a proportionality constant deemed independent o either pressure or temperature. In the derivation o the FMT equation its authors urther assumed that the pressure dependence o the compressibility o the occupied volume, κ φ, obeys an analoue o equation (99). The interation: then yields: where P 1 P 1 T ( P) = dp dp K ( T ) + k P K ( T ) + k P lo a,. (1) P * * P e e φ 1 T, P = φ c [ T T θ ( P)],. (11) c ( P) + T T θ ( P) 1+ c4 P 1+ c6 P θ ( P) = c3 ( P)ln c5 ( P)ln,. (1) 1+ c4 P 1+ c6 P and the c s ollow as: c = B /.33,. (13) 1 = / ( ) α,. (14) c P c P 1 keα P 3 ( ) = / ( ),. (15) c = k K,. (16) * 4 e / e c P k P 5 ( ) = 1/ φα ( ),. (17) c = k K.. (18) * 6 φ φ The superscript indicates that the parameter is reerred to the reerence temperature T (irst place) and to the reerence pressure P (second place). A sinle superscript 78 The British Society o Rheoloy, 5 (

31 I. Emri, Rheoloy Reviews 5, pp reers to the reerence temperature only. The * superscript reers to zero (in practice, atmospheric) pressure. Equation (11) is the Fillers Moonan Tschoel or FMT equation. K* φ and k e, and thus c 4, can be determined rom separate volume-pressure measurements throuh a it to the equation: * V 1 Ke ( T ) + kep ln = ln *,. (19) V ke Ke ( T) + kep by a non-linear least-squares procedure. Equation (19) is known in the orm: * Ke ( T ) + kep V = V K * e ( T ) + k e P 1/ k e,. (11) as the equation o Murnahan 5 [61]. The latter is obtained by interatin equation (99) at constant temperature between the limits V, P, and V, P. In conjunction with c 4 the ive equations or c 1, c, c 3, c 4, c 5, and c 6 are clearly suicient to uniquely determine B,, α, K* φ and k φ. This stronly suests that the temperature and/or pressure dependence o polymers be characterized always by carryin out both isothermal and isobaric measurements. The measurements by Fillers and Tschoel [5], Moonan and Tschoel [6], Simha and Wilson [56], and the recent ones by Kralj et al. [1], Prodan [4], and Emri at al. [6] indicate that or most polymers B is not equal to 1, as has been assumed or many years. This leads to the conclusion, that many data on and α available today or several polymers may not represent the best estimates. We note that settin θ ( P) = (i.e., perormin the experiments at the reerence pressure), the FMT equation reduces to the WLF equation. On the other hand, executin the experiments at the reerence temperature, i.e., settin T = T, leads to: lo a P B θ ( P) c1 θ ( P) = =.. (111).33 / α ( P) θ ( P) c ( P) θ ( P) Equation (111) becomes the Ferry Stratton equation (6), upon the substitutions θ ( P) ( P P ), c1 s1, and c ( P) s. Within its domain o validity the FMT model can thereore handle time-temperature, time-pressure, and time-temperaturepressure superposition equally well. 5 The dierence between the Tait and the Murnahan compressibilities, Equations (74) and (99), is rather similar to that between the enineerin or nominal stress, σ = F / A, and the true stress, σ = F / A, where F is the orce, and A is the area upon which the ormer acts. In both relations the subscript denotes the initial, i.e., undeormed quantity. The British Society o Rheoloy, 5 ( 79

32 I. Emri, Rheoloy Reviews 5, pp Critique o the time-temperature-pressure superposition models. Table 3 lists the loarithmic shit actors, lo a T, P, o the Simha Somcynsky (SS), the Havlìček-Ilvalský-Hrouz (HIH), and the Fillers Moonan Tschoel (FMT) models alon with the approach on which each model is based. Model Simha-Somcynsky Havlìček-Ilvalský- Hrouz Fillers-Moonan- Tschoel lo a lo a lo a T, P T, P lo a T, P BSS 1 1 =.33 1 y 1 y B 1 1 HIH =.33 TSc ( T, P) T Sc ( T, P ) B T T θ ( P) T, P =.33 / α ( P) + T T θ ( P) Approach Hole theory Excess entropy Free volume Table 3. Comparison o models or the time-temperature-pressure shit actor, a T, P. 8 6 FMT SS HIH measured lo a To,P 4 Hypalon Pressure [MPa] Fiure 9: Comparison o lo a T, P, obtained rom the HIH, SS, and FMT models. 8 The British Society o Rheoloy, 5 (

33 I. Emri, Rheoloy Reviews 5, pp The three models dier considerably in their approach and in their useulness. Moonan and Tschoel [6] discussed all three in depth. Fiure 9, which reproduces part o iure 1 o the paper by Moonan and Tschoel [6], compares the loarithmic shit actors, lo a T P, or Hypalon 4 rubber, that were obtained usin the three models., Clearly, the HIH model is not in areement with the experimental values while the SS and the FMT models both describe the data about equally well. Moonan and Tschoel [6] stated that the improved orm o the HIH model its the data as well as the FMT equation, but is certainly not more convenient to use. Furthermore, it is perhaps lawed since it requires knowlede o the chane in the heat capacity, CP, and when this is determined rom a it to the shit data, it does not aree with calorimetric data. All this, paired with the capability o the FMT model to describe all three types o superposition, appears to make it the model o choice. 4.5 CEM Pressure apparatus. Recently Kralj et al. [1], and Emri and Prodan [6] reported on the experimental setup or characterization o the eect o pressure and temperature on behavior o solid polymers. The system assembly is shown schematically in iure (1). The pressure is enerated by the pressurizin system usin silicone oil. The pressure vessel is contained within the thermal bath, where another silicone oil circulates rom the circulator, used or precise temperature control. The apparatus utilizes two separate measurin inserts, which can be inserted into the pressure vessel: the dilatometer and the relaxometer. Sinals rom the measurin inserts pass throuh thermal bath pressure vessel electromanet measurin inserts circulator carrier ampliier data acquisition manet and motor charer pressurizin system Fiure 1. Schematic o the CEM Measurin System [1, 6]. The British Society o Rheoloy, 5 ( 81