Thermo-Rheological, Piezo-Rheological, and TVγ-Rheological

Macromolecules , 2014, 47 (22), pp 8056–8063. DOI: 10.1021/ma501843u. Publication Date (Web): November 10, 2014. Copyright © 2014 American Chemical...
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Thermo-Rheological, Piezo-Rheological, and TVγ‑Rheological Complexities of Viscoelastic Mechanisms in Polymers K. L. Ngai*,†,‡ and D. J. Plazek§ †

Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, I-56127, Pisa, Italy CNR-IPCF, Largo B. Pontecorvo 3, I-56127, Pisa, Italy § Department of Mechanical Engineering and Material Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261United States ‡

ABSTRACT: Experimental evidence are collected together to show in general the viscoelastic mechanisms of low and high molecular weight polymers have different dependences on temperature T, pressure P, and the same scaling product variable, TVγ, where V is the specific volume and γ is a material constant. The viscoelastic mechanisms include the Johari−Goldstein β-relaxation, the segmental α-relaxation, the sub-Rouse modes, the Rouse modes, and the terminal relaxation if the polymer chains are entangled. Appropriately called the thermo-, piezo-, and TVγ-rheological complexity of polymers, these viscoelastic anomalies are fundamental and deserve attention from the polymer physics community. Although we show that the coupling model can rationalize the origin of the complexity and explain some of the data quantitatively, the objective of this paper is to stimulate others to construct a theory that can do even better.

1. INTRODUCTION Temperature is the variable traditionally and primarily used to study viscoelastic properties of amorphous polymers, particularly in the problem of the glass transition. In recent years, pressure P has been introduced as another variable together with temperature in investigations by photon correlelation spectroscopy,1−8 quasielastic neutron scattering,9 dielectric relaxation spectroscopy,10−39 and molecular dynamics simulations.40 The studies are mostly on the segmental α-relaxation of polymers and the structural α-relaxation of small molecular glass-formers, and relation to the secondary relaxations. Polymers with repeat units having dipole moments are amenable to the study of the segmental α-relaxation and secondary relaxations by dielectric relaxation spectroscopy (DRS). However, global chain dynamics of unentangled and entangled polymers can only be studied by DRS in a handful of polymers with dipole moment having a component parallel the chain backbone, classified as type A polymers by Stockmayer.41−43 The dipole moment of the chain end-to-end vector makes it possible to observe the global chain dynamics by DRS at ambient and applied pressure. Examples include polyoxybutylene (POB),24,25,44 polypropylene glycol (PPG),23,45−49 and polyisoprene (PI).14−16,23,25,33,39,50−55 Besides DRS, there are also photon correlation spectroscopy studies of both the segmental α-relaxation and the global chain relaxation modes in a few polymers at elevated pressure. These include poly(methylphenylsiloxane) (PMPS)1,2,4,7 and poly(methyl-paratolylsiloxane).8 Despite the limited number of polymers for which both segmental α-relaxation and global chain dynamics (often referred to as the normal modes) can be studied together with or without applied elevated pressures, the results are independent of the chemical structure of the polymer. Thus, © XXXX American Chemical Society

the results reviewed in the next section can be taken as general properties of polymers. Studies of the segmental relaxation and secondary relaxation, but without the global chain modes, at elevated pressures by DRS and photon correlation spectroscopy (PCS) are interesting to consider because they reveal also general properties and that will be discussed. The polymers studied include poly(ethyl acrylate),56 poly(methyl acrylate),57 poly(methyl methacrylate),37 polystyrene,3 poly(propylene oxide),5 and poly(vinylacetate).58 The structure of the present paper is organized as follows. In the next section, the pressure and temperature dependence of all viscoelastic mechanisms in unentangled and entangled polymers are brought out from experiments and simulations. These general properties are summarized into three categories worthy of paying attention to in the polymer viscoelasticity and glass transition research communities. Namely, the three are (1) thermorheological complexity, (2) piezorheological complexity, and (3) TVγ-rheological complexity; i.e., all viscoelastic mechanisms are functions of the product variable, TVγ, with the same γ but the functions are different, which has to be the case in order to be consistent with points 1 and 2. Category 1 has a long history. It was mainly discovered by Plazek and coworkers,59 and repeatedly confirmed by others, that different viscoelastic mechanisms have different shift factors on changing temperature. The accomplishment of Plazek was recognized by the Society of Rheology in awarding him the Bingham Medal in 1995.59 Category 2 originates from the studies by DRS and PCS of polymers at elevated pressures in recent years. The Received: September 5, 2014 Revised: November 4, 2014

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creep and dynamic mechanical relaxation measurements73,74 and photon correlation spectroscopy75 in polyisobutylene (PIB). By now, sub-Rouse modes have been found in other polymers including PS, PVAc, PEMA, and PMMA by precision mechanical spectroscopy76−82 and polyisoprene (PIP) by dielectric spectroscopy.83 The resolved sub-Rouse modes have relaxation times, τsR, with temperature dependence weaker than the segmental α-relaxation time, τα. This can be seen from the plot of τsR and τα versus reciprocal temperature for PIB in ref 74, PS in ref 79, PMMA in refs 80 and 81, and PVAc in ref 80. Polymers with molecular weight low enough will have recoverable compliance, Jr(t) = J(t) − t/η, below J(0) = 1.3 × 10−7 Pa−1 in eq 1, and hence, the observed compliance is contributed to by the sub-Rouse modes and the segmental αrelaxation. This situation is realized in the creep compliance measurements of a low MW polystyrene of 3400 g/mol by Plazek and O’Rourke71 and Gray et al.,84 and also in the monodisperse poly(methylphenylsiloxane) (PMPS) with low molecular weight of 5000 g/mol,4 as well as in PPG400076 and Se.46 This can be seen for PMPS in Figure 1 of ref 4, and for PS

acronym, piezorheological complexity, is created to stand for the different dependences on pressure of different viscoelastic mechanisms. Category 3 takes categories 1 and 2 further by combining the pressure and temperature dependences into one product variable TVγ, which is remarkable. Even more remarkable is that all viscoelastic mechanisms in a polymer have the same γ. Theoretical explanations are given in the penultimate section.

2. THREE SALIENT PROPERTIES IN VISCOELASTICITY AND GLASS TRANSITION OF POLYMERS (A). Thermorheological Complexity. In high molecular weight (MW) entangled polymers, viscoelastic mechanisms inside the glass−rubber transition zone (or softening dispersion) shift with temperature differently than the terminal dispersion or viscosity. This property was found in 1965 in PS59−61 and later in PVAc,62 and atactic polypropylene (aPP)63 by Plazek. Some of the data can be found in Ferry’s book.64 The discovery of this kind of thermorheological complexity was verified by others. Here we give as an example of the subsequent verifications the case of aPP by PCS and NMR.65,66 Another example is the high molecular weight poly(cyclohexylemethacrylate) plasticized by dioctyl phthalate.67 More examples can be found in the review article.68 An explanation of the property was given by the coupling model.55,69 The glass−rubber transition zone of high MW PS ranges in creep compliance from the glassy compliance Jg = 0.93 × 10−9Pa−1 to the entanglement plateau compliance Jplateau ≈ 10−5 Pa−1. Over this range of 4 orders of magnitude, one can expect the presence of more than one viscoelastic mechanism. The segmental α-relaxation responsible for glass transition and the generalized Rouse modes for undiluted polymers64 are two mechanisms universally believed to be present. However, the compliance ranges covered by the two mechanisms remain to be determined. It was recognized by M.L. Williams70 that the short time limit of the Rouse modes contribution to the modulus is given by G(0) = vNkT =vρRT/M, where N is the number of molecules per cm3. The number of monomers in a submolecule, z, is given by P/v where P is the number of monomers in a polymer chain. For a polymer of molecular weight 150 000 and a density of 1.5 g/cm3, and assuming that the smallest submolecule that can still be Gaussian consists of five monomer units (i.e., z = 5), Williams found that G(0)=7.5 × 106 Pa or J(0)=1.3 × 10−7 Pa−1. Thus, the Rouse modes contribution, JR̂ (t), only can account for the compliance in the range (1)

Figure 1. Pressure dependences of the segmental α-relaxation time (squares) and the longest normal mode relaxation time (circles) for the entangled PIP with Mn = 26000 g/mol at 320 K. The inset shows the shift factors of the two processes, and the line is the prediction from the CM (see text for explanation and source of the experimental data).

On the other hand, other experimental evidence71,72 enabled determination of the contribution to the compliance by the segmental α-relaxation, Jα̂ (t), to lie within the range bounded by the glassy compliance Jg = 0.93 × 10−9Pa−1 and Jeα ≈ 4 × 10−9Pa−1 (see also ref 4). Therefore, the intermediate compliance range, approximately from 4 × 10−9 to 1.3 × 10−7 Pa−1, remains to be accounted for by some other viscoelastic mechanisms. The missing molecular mechanisms have length-scales smaller than the Gaussian submolecule of the Rouse model but larger than the local segmental motion, which naturally is called the “sub-Rouse modes”. The sub-Rouse modes were first found in 1995 by resolving it from the Rouse modes and the segmental α-relaxation by

from Figure 7 of ref 71, where the final increase of Jr(t) is due to the presence of a higher molecular weight tail in the sample. In all cases, the stronger temperature dependence of τα than τsR is the cause of the strong decrease of the plateau compliance on lowering temperature toward Tg.85,86 The stronger temperature dependence of τα than τsR is directly observed by DRS in poly(propylene glycol) (PPG) with MWs from 1000 to 4000 g/mol,45 and in polyisoprene (PIP) with MWs from 1000 to 8000 g/mol.52 Cochrane et al.76 also found by creep compliance measurements of PPG4000 that the chain normal mode retardation time has a weaker T-dependence than the segmental α-retardation time.

J(0) ≤ JR̂ (t ) ≤ Jplateau

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the α′ relaxation in some earlier studies, were even more clearly resolved from the segmental α-relaxation in poly(methyl-paratolylsiloxane) (PMpTS) of MW = 4500 and 42 000 g/mol by PCS experiments and shear modulus measurements of Patkowski and co-workers.8 From the properties observed, Patkowski et al. concluded that “... both α and α′ processes are due to the relaxation of a similar molecular subunit but with a different cooperativity”, which is consistent with our current understanding of the nature of the sub-Rouse modes and the relation to the segmental α-relaxation. The pressure dependence of the sub-Rouse modes in PMPS is confirmed in detail by a more recent study of the dynamics of PMPS with MW = 10 500 g/mol and PDI = 1.04 by PCS under applied pressure from 1 to 900 bar and temperature in the range from 247 to 278 K by Kriegs et al.7 The α′ mode in this reference is comprised of the sub-Rouse modes. These authors found both the relaxation times of the segmental α-relaxation and the sub-Rouse modes are temperature and pressure dependent, and well described by two different Vogel−Fulcher laws, and the two different forms of pressure dependence ⟨τ⟩ = τP exp[DPP/(P0 − P)]. They found from the pressure dependences that the activation volumes for both processes are very similar, and both relaxation times, τsR and τα, are functions of the combined variable, TVγ, with the same γ where V is the specific volume, like that found in other polymers,23−25 which will be discussed in detail in the next subsection (C). This experiment shows beyond any doubt that the sub-Rouse modes are coupled to density like the segmental α-relaxation, and hence they have thermodynamic properties similar to the segmental α-relaxation including increase of their relaxation times on physical aging. A comprehensive study of the pressure dependences of the viscoelastic mechanisms is the work using DRS by Floudas and co-workers14,16 on cis-polyisoprene (PIP) with number-average MW, Mn =1200, 2500, 3500, 10600, and 26000 g/mol, and polydispersity less than 1.1. The entanglement molecular weight Me of PIP is 5400 g/mol. Since PIP has dipole moment perpendicular as well as parallel to the chain backbone, the chain normal modes are observed together with the segmental α-relaxation. The term, normal modes, is used in the literature for modes of all chain lengths without specifying their nature. Here we distinguish the viscoelastic mechanisms contributed by the normal modes depending on the chain length. For PIP with sufficiently low molecular weight such that the observed compliances are all less than J(0)=1.3 × 10−7 Pa−1 in eq 1, the normal modes are the sub-Rouse modes. Higher molecular weight PIP with observed compliance falling within the range given by eq 1, the normal modes are the Rouse modes. In entangled PIP of high MW, the normal modes are the entangled terminal chain modes. Thus, in the 1200, 2500, and 3500 g/mol samples with Mn< Me, observed are the segmental α-relaxation, the sub-Rouse modes and the Rouse modes. While in the 10600 and 26000 g/mol samples with Mn > Me, the observed normal modes form the terminal relaxation spectrum of the entangled PIP. Nevertheless, in all PIP samples with molecular weight ranging from 1200 to 26000, the relaxation times of the normal modes, τn, have weaker pressure dependence than τα of the segmental α-relaxation. To demonstrate this, we have replotted the pressure dependences of τn and τα of the samples with the highest Mn = 26 000 g/mol and the lowest Mn = 1200 g/mol (degree of polymerization = 18) in Figure 1 and Figure 2, respectively. The insets will be discussed in Section 3 where the

Stonger temperature dependence of τsR than the relaxation times of the Rouse modes, τR, can be deduced from the failure of time−temperature superposition of creep data of high MW PS at the compliance level larger than J(0) contributed by the Rouse modes (see Figure 9 in ref 87. The breakdown of thermorheological simplicity in this region was confirmed by Cavaille et al.88 and Guo et al.89 in PS and by Palade et al.90 in high MW polybutadiene. The summary given above calls attention to the presence of three distinctly different viscoelastic mechanisms in the glass− rubber transition zone of high MW entangled polymers. They are the segmental α-relaxation, the sub-Rouse modes, and the Rouse modes within the glass−rubber transition zone, and the terminal relaxation at times beyond the entanglement plateau. Their relaxation times all have different temperature dependences,68 causing breakdown of time−temperature superposition of data or thermorheological complexity. In the next subsection (B), we consider the pressure dependences of these viscoelastic mechanisms. (B). Piezorheological Complexity. Actually photon correlation spectroscopy (PCS) measurements have found the sub-Rouse modes and their temperature dependence in low MW poly(methylphenylsiloxane) (PMPS) ten years before the discovery of these modes in PIB by the creep compliance measurements73 and PCS experiment75 on PIB, but it was not called the sub-Rouse modes at that time. In 1984, Fytas et al.1 studied by PCS the dynamics of PMPS with low MW = 2500 g/mol at different pressures from 1 to 1750 bar and temperatures between 269 and 308 K. The time correlation function measured shows a faster segmental α-relaxation and a slower q2-dependent diffusional process due to local concentration fluctuations. The latter is actually the sub-Rouse modes in this low MW PMPS, but not recognized as such in this early study, and also in another study of PMPS with MW = 5000 g/ mol by combination of creep compliance and PCS done in 1994 ten years later.4 Also observed in this 1984 paper is that the mean characteristic times of the two processes approach each other at low temperatures on varying temperature and at high pressures on varying pressure, which indicates the temperature and pressure dependences of the sub-Rouse modes and the segmental α-relaxation are different. The explanation of this piezorheological and thermorheological complex behavior was given 2 years later in 1986 by the coupling model2 based on the coupling parameter of the subRouse modes, nsR, is smaller than nα of the segmental αrelaxation, consistent with that given in papers on various polymers published in the next decade.4,73,75,79,85,86,91 In particular, the 1994 paper4 confirms the observed slower viscoelastic mechanism comes from the sub-Rouse modes by the combination of PCS and compliance measurements on another low MW PMPS. This 1994 paper also helps to determine the range of compliance contributed by the segmental α-relaxation in PMPS is given by Jg = 5.39 × 10−10 Pa−1 ≤ Jα̂ (t ) ≤ Jeα ≈ 2.82 × 10−9 Pa−1 (2)

The ratio, Jeα/Jg ≈ 5.2, is comparable to that of PS given before,72 in support that the size of the contribution to compliance from the segmental α-relaxation is about the same in general. By the way, the contribution to Jeα/Jg from the αrelaxation in nonpolymeric glass-formers such as trinaphthal benzene (TNB) is about 2.5.92 The sub-Rouse modes, called C

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the same γ is already challenging to be explained. Moreover, the different dependences of τα and τn on TVγ makes the explanation even more difficult to come by from conventional theory. Notwithstanding, the results have been accounted for quantitatively by the coupling model,25 which will be further discussed in section 3. Mention should also be made here of molecular dynamics simulations of flexible Lennard-Jones chains with rigid bonds by Veldhorst et al.94 in which they show the same TVγ-scaling for the full time dependence of the dynamics, including chain specific dynamics such as the end-toend vector autocorrelation function and the relaxation of the Rouse modes. There are other general properties of the segmental αrelaxation and secondary relaxation found on varying both temperature and pressure in many glass-formers and polymers27−32,34−36 including PIP.33 For the structural αrelaxation in nonpolymeric glass formers and the segmental αrelaxation in polymers, it was found that the frequency dispersion, or the Kohlrausch−Williams−Watts (KWW) stretched exponential correlation function used to fit it, ⎡ ⎛ ⎞1 − nα ⎤ t ϕ(t ) = exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ τα ⎠ ⎥⎦

Figure 2. Pressure dependences of the segmental α-relaxation time (circles) and the longest normal mode relaxation time (triangles) for the unentangled PIP with Mn = 1200 g/mol at 320 K. The inset shows the shift factors of the two processes (see text for explanation and source of the experimental data).

(3)

is invariant to various combinations of P and T while τα is kept constant.27−32,36 This property can be restated as coinvariance of τα and (1 − nα) to variation of P and T. The KWW function in eq 3 is usually written in form, φ(t)=exp[−(t/τ)βKWW]. In eq 3, βKWW is rewritten as (1 − nα), where nα is the segmental αrelaxation coupling parameter of the coupling model to be introduced in Section 3. The references27−32,36 cited above are all from dielectric relaxation measurements. There are viscoleastic bulk modulus measurements of a symmetric three-arm star polystyrene by Guo et al.89 and polystyrene by Meng and Simon,95 which have shown the same relaxation dynamics for various combinations of T and P when the Tg (or relaxation time) is kept constant. Light scattering data of polymers also found invariance of the segmental α-relaxation at constant τα to variations of P and T.28 In some cases where both the α-relaxation and the secondary relaxation belonging to a special class called the Johari− Goldstein (JG) β-relaxation96,97 with relaxation time τβ are observed together, the ratio, τ β /τ α , is also constant.29,30,32,34,36,40 Putting these together, we have coinvariance of τα, τβ, and nα to variation of P and T. For polymers, this property was well elucidated by the molecular dynamics simulations of Bedrov and Smith.40 In practice, all three quantities, τα, τβ, and nα, are determined from experimental data by some procedure involving fits of the spectra with assumption, and some uncertainties are associated with the values obtained. Also, theoretically from the coupling model (CM) the coinvariance of τα and τβ can only be approximate. This is because in the CM, the experimentally obtained value τβ can only be approximately the same as the primitive relaxation time τ0, albeit the model itself has exact coinvariance of τα, τ0, and nα.27,28,34−36,38,98 Notwithstanding, it is important to observe that if the coinvariance of τα and τβ to variations of P and T holds experimentally, and if τα is a function of TVγ then necessarily that τβ also is a function of TVγ with the same γ. Since coinvariance of τα and τβ is only approximate, τβ is not expected to scale exactly as a function of TVγ with the same γ used to scale τα, but should be quite close. This was demonstrated for several nonpolymeric glass-formers in ref

coupling model is used to explain the difference in the pressure dependences of the segmental α-relaxation and the chain normal modes quantitatively. Also using DRS, Schönhals52 found τn has the weaker temperature dependence than τα in PIP with Mn = 1230 and up to 8360 g/mol. This parallel difference in pressure and temperature dependences of the two modes is not an accident but originates from the same physics to be discussed in section 3 later. (C). TVγ-Rheological Complexity. Dielectric relaxation measurements of the segmental α-relaxation and the normal modes were made over some ranges of temperatures and pressures on polypropylene glycol (PPG), polyisoprene (PIP)23,25 and poly(oxybutylene) (POB).24,25 The PPG has a molecular weight equal to 4000 g/mol, and is slightly entangled by transient coupling via hydrogen-bonding of the chain ends.93 For the PI, Mw = 11.1 kg/mol, which is about a factor of 2 larger than Me. Hence the normal modes of PPG and PIP can be considered as the slightly entangled chain modes. The POB sample has Mn = 4800 g/mol and PDI = 1.10, corresponding to 67 repeat units per chain, the chains are probably not entangled and the normal modes are the Rouse modes. Since the dielectric segmental α-relaxation times τα and the normal mode relaxation times τn were obtained over a wide range of frequencies, temperatures, and pressures for the three polymers, for which the equation of state parameters, V(T,P), were also determined, the authors of refs 23 and 24 were able to express τα and τn as functions of the product variable TVγ. The remarkable finding is that the value of γ is the same for the two processes, but the dependences of τα and τn on TVγ are different, with τα stronger than τn. Hence, in analogy to thermorheological and piezorheological complexity, the experimental data show also TVγ-rheological complexity. The data of τα and τn obtained by measurements carried out at constant temperature at varying pressures when plotted as functions of the specific volume V, also show τα varies more rapidly with V than τn. The fact that both τα and τn are functions of TVγ with D

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34. Another study of dielectric relaxation35 of a low molecular weight poly(methyl methacrylate) (PMMA) at various combination of P and T found τα can be scaled to show that it is a function of TVγ over a range of 7 orders of magnitude change in τα. On the other hand, the accompanying data of τβ when scaled in the same way cannot be described exactly by a function of TVγ with the same γ over the same range as for τα. The values of τα and τβ were deduced by fitting the dielectric loss spectra with the assumption that the α- and the βrelaxations are independent and additive, which is at odds with the known connections between them.29,30,32,96,97 Also error estimates or uncertainties in the values of τα and τβ deduced from the fits were not given, making it impossible to make unequivocal assessment. Notwithstanding, the scatters of the τβ data from exactly a function of TVγ are small, and all are within ±0.5 decade over the entire range of the scaling variable TVγ, where τα changes by 7 orders of magnitude (see Figure 4 of ref 35). Such small scatters of τβ from a function of TVγ actually verify the CM expectation that τβ is to a good approximation a function of TVγ with the same γ as τα. However, the functions of the segmental α-relaxation and the JG β-relaxation are different, and therefore the two processes are TVγ-rheologically complex. Summarizing, the experimental findings indicate the JG βrelaxation, the α-relaxation, and the normal modes are all functions of TVγ with the same γ, but the functions are all different, i,e. all these viscoelastic mechanisms are TVγrheologically complex.

tc,m, the many-body effects changes the one-body relaxation (or the primitive relaxation) to become heterogeneous and the correlation is modified from eq 4 to the Kohlrausch stretched exponential function of any mode m, ⎡ ⎛ ⎞1 − nm ⎤ t ⎥ ϕ(t ) = exp⎢ −⎜ ⎟ ⎢⎣ ⎝ τm ⎠ ⎥⎦

For segmental α-relaxation, this is given before by eq 3. Taking advantage of the fact that one-body relaxation holds before the onset of chaos at time tc,m which changes it to many-body relaxation after tc,m, a useful relation between the primitive relaxation time τm0 and the many-body relaxation time τm is obtained and given by τm = [(tc , m)−nm τm0]1/(1 − nm)

(6)

Derived from classical chaos, eqs 5 and 6 are applicable to relaxation and diffusion via various modes in many-body interacting systems,38,98−102 and in particular applicable to all viscoelastic mechanisms of polymers. The reason for affixing the subscript m in eqs 5 and 6 is because different modes do not have the same coupling parameter nm, crossover time tc,m, and primitive τm0. This is the case for the terminal relaxation of entangled polymers compared with the segmental α-relaxation and the sub-Rouse modes. Entanglement interaction occurs at much longer lengthscale than the other processes. Hence the interaction strength is weaker, and correspondingly the temperature insensitive tc,m is of the order of 1 ns,38 longer than 1 to 2 ps for the segmental αrelaxation.38,104 Also nm and τm0 are different because the interaction and the primitive relaxation correspond to processes of different length-scales. The primitive relaxation for the terminal relaxation of entangled chains is the Rouse modes,38,69,105 while that of the segmental α-relaxation, τ0α, is the local relaxation time, comparable in magnitude but not necessarily exactly equal to the experimentally deduced Johari− Goldstein β-relaxation time, τJG, i.e., τ0α ≈ τJG.34,97,102 For segmental α-relaxation, the relations

3. COUPLING MODEL EXPLANATION The main purpose of the present paper is to call attention of the three general and fundamental viscoelastic properties of polymers, namely the thermo-, piezo-, and TVγ-rheological complexities. It is worthwhile because these are fundamental problems deserving attention from the research community. Here we present an explanation of these complexities from the coupling model (CM), with the disclaimer that this is neither the only nor the most satisfactory way to solve the problem. Rather the purpose is to encourage others to provide their own solution to this important problem. In materials in which the basic units are interacting with each other, the relaxation or diffusion of the units is necessarily a many-body problem. In the absence of interunit interactions, any process m is simply a one-body independent relaxation with exponential time correlation function, ⎛ t ⎞ ϕ(t ) = exp⎜ − ⎟ ⎝ τm0 ⎠

(5)

τα = [(tc , α)−nα τα 0]1/(1 − nα)

(7)

and τ0α ≈ τJG immediately imply that if τα is a function of the product variable TVγ (as found experimentally), then also τα0 must be exactly, and τJG approximately, a function of the same variable TVγ. Time being a natural variable, the fact that τα0 and τJG precedes τα implies that the dependence of molecular mobility on TVγ originates from τα0.34 Following the traditional practice of polymer viscoelasticity,64 the dependence on T and P, or T and V of τα0(TVγ) comes the primitive monomeric friction coefficient ζ0(T,P) or ζ0(TVγ). In the CM, one and the same primitive monomeric friction coefficient governs all the primitive relaxation modes m, including that of the segmental α-relaxation, the sub-Rouse modes, and the chain normal modes n, whether unentangled or entangled, i.e.

(4)

proceeding simultaneously and homogeneously for all units. The interunit interaction and the associated many-body effects slow down the relaxation, modify the exponential correlation function to to become the Kohlrausch−Williams−Watts function given by eq 3, and change the process to become dynamically heterogeneous. The CM was proposed from the outset38,98−102 to take into account these many-body effects, and is a general theoretical framework applicable to diverse systems, not just molecular glass-formers and amorphous polymers.38,103 The potential of the interunit interactions in most systems of interest are usually anharmonic, and it follows from classical mechanics that the dynamics in phase space is chaotic and inhomogeneous. Based on classical chaos theory38 as well as simplified models,98−101 after the onset of chaos at

τα 0(TV γ ) ≈ τJG(TV γ ), τsR 0(TV γ ), τn0(TV γ ) ∝ ζ0(TV γ ) (8)

and similar relations τα 0(T , P) ≈ τJG(T , P), τsR 0(T , P), τn0(T , P) ∝ ζ0(T , P) (9) E

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also in PPG23 and POB24 are also consistent with the CM predictions as shown before in ref 25. A rigorous solution of the many-chain cooperative dynamics of entangled polymers from the Langevin equation with intra- and intermolecular interactions104−106 has been given by Guenza.107−110 Her model, where specific intermolecular interactions are responsible for the observed cooperative dynamics, is in the same spirit as the CM when applied to unentangled and entangled polymers, although pressure and temperature dependences have not yet been considered by the model at the present time. Notwithstanding this, we find excellent agreement of the prediction with experiment in the case of Figure 1, we point out that all parameters in eqs 10−12 have been deduced by fitting the spectra, and involves errors and uncertainties. Therefore, in testing the prediction, one should be mindful of the errors and uncertainties of the parameters used but usually not given, which can contribute to deviations from perfect agreement with prediction. This caution must be considered when testing the difference in the TVγ-dependences of τn and τα, as well as the TVγ-dependences of τα andτJG mentioned before in the previous section. Finally we consider the different pressure dependences of τn and τα shown in Figure 2 of the unentangled PIP with Mn = 1200 g/mol at 320 K taken from Floudas and co-workers.16 The temperature dependences of τn and τα are also different, which can be deduced from the data of PIP with Mn = 1230 g/ mol obtained at ambient pressure by Schönhals.52 The inset of Figure 1 shows the pressure shift factors of τn and τα. The line represents the right-hand-side of eq 14 for the prefactor, (1 − nn)/(1 − nα), equal to 1.35. From this, and using once more (1 − nα) = 0.47, we find the value of (1 − nn) is 0.64 for the chain normal mode. The result of nn = 0.36 being smaller than nα = 0.53 is consistent with the explanation of the origin of the thermo- and piezo-rheological complexity of unentangled polymers is due to larger intermolecular coupling of the segmental α-relaxation than the sub-Rouse modes and the Rouse modes as done before.2,4,25,45,46,55,58,74,79,86 Also the nonzero value of the normal mode is consistent with the subdiffusive behavior at short times of the mean-square displacement of the center-of-mass of unentangled and barely entangled polymeric chains before crossing over to the longtime diffusion. This anomalous subdiffusion behavior was observed by simulations,111−115 and verified by neutron spin echo measurements by Zamponi and co-workers116 over an extended time range. The polyethylene samples investigated cover a range of molecular weights from the low degree of polymerization regime of unentangled polymers to the slightly entangled regime. All of the samples displayed anomalous subdiffusive dynamics at short time and crossover to normal diffusion at long times. The Rouse theory of unentangled polymer chain dynamics predicts diffusive motion at short and long times, in disagreement with simulations and experiments. On the other hand, the results from the theory Guenza presented also in ref 116 together with the neutron spin echo measurements are in good agreement with the experimental data. Furthermore, the subdiffusion found by experiment and theory is consistent with the Kohlrausch correlation function of the unentangled chain normal modes with exponent (1 − nn) that is a fraction of unity in the context of the coupling model, such as the value of 0.64 for the unentangled PIP with Mn = 1200 g/mol shown in Figure 2.

for dependences on T and P. Specializing eq 6 to the various viscoelastic mechanisms m each having its own characteristic coupling parameter, nα, nsR, and nn, we have τα(TV γ ) ∝ [ζ0(TV γ )]1/(1 − nα)

or

1/(1 − nα)

τα(T , P) ∝ [ζ0(T , P)]

τsR(TV γ ) ∝ [ζ0(TV γ )]1/(1 − nsR )

(10)

or

1/(1 − nsR )

τsR(T , P) ∝ [ζ0(T , P)]

τn(TV γ ) ∝ [ζ0(TV γ )]1/(1 − nn)

(11)

or

1/(1 − nn)

τn(T , P) ∝ [ζ0(T , P)]

(12)

Because in general the coupling parameters, nα, nsR, and nn, have different values, it follows from the proportionality relations above that the segmental α-relaxation, the sub-Rouse modes, and the chain normal modes n have different dependences on T, P, and TVγ (with the same γ). Thus, conceptually we can explain the thermo-, piezo- and TVγrheological complexities found by experiments as described in the previous section. The CM predictions can be put to the quantitative test against experimental data if the coupling parameters are known. This is the case of the fully entangled 26 000 g/mol PIP data of Floudas and co-workers16 shown in Figure 1 where the pressure dependences of the terminal relaxation time τn and τα at constant T = 320 K are compared. The molecular weight dependence of the terminal relaxation time τn of highly entangled linear polymers without the influence of chain ends has the M3.4-dependence.64 In the CM, the primitive mode of entangled terminal relaxation is the Rouse modes with relaxation time τR ∝M2,64 and another application of eq 6 is38,55,105

τn(M ) ∝ (M2)1/(1 − nn)

(13)

Hence by equating 3.4 and 2/(1 − nn), the value of (1 − nn) = 0.59 is determined. On the other hand, nα has been determined before by fitting the dielectric loss peak by the Fourier transform of the Kohlrausch function to give βK ≡ (1 − nα), the value of 0.47 in ref 106 and 0.46 in ref 33. With (1 − nn) = 0.59 and (1 − nα) = 0.47 known, the CM relations 10 and 12 for pressure dependence at constant T = 320 K can be combined and rewritten as the relations between the pressure shift factors of the two processes ⎤ ⎡ (1 − nn) ⎤ ⎡ τα(320 K, P) log⎢ ⎥ ⎥=⎢ ⎣ τα(320 K, P = 1 bar) ⎦ ⎣ 1 − nα ⎦ ⎡ ⎤ τn(320 K, P) log⎢ ⎥ ⎣ τn(320 K, P = 1 bar) ⎦

(14)

Shown by symbols in the inset of Figure 1 are the two pressure shift factors. The blue line is the right-hand-side of eq 14 calculated with (1 − nn) = 0.59 and (1 − nα) = 0.47. The good agreement of the calculation with the left-hand-side as shown verifies the CM prediction for entangled polymers quantitatively, and as also found before.38,55,65,67,69,74,105 The same values of (1 − nn) and (1 − nα) had been used in conjunction with eqs 10 and 12 to account successfully for the different TVγ-dependence of τn and τα but the same γ of entangled PIP with Mn = 10600 g/mol (see Figure 5 in ref 25). Different TVγ-dependences of τn and τα with the same γ found F

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4. CONCLUSIONS Already 5 decades have passed since the first discovery of the thermo-rheological complexity in unentangled and entangled polymers, i.e., different viscoelastic mechanisms do not have the same temperature dependence. Progress made in the intervening years has identified the thermo-rheologically complex viscoelastic mechanisms, which are not only the more familiar segmental α-relaxation, the Rouse modes, and the terminal relaxation of entangled chains, but also the sub-Rouse modes and the Johari−Goldstein β-relaxation. Advances in experimental techniques to make measurements at elevated pressures have shown these viscoelastic mechanisms are also piezo-rheologically complex. By measuring the variation of the specific volume, V, with temperature and pressure to obtain the equation of state, another recent advance is that segmental αrelaxation times, τα, obtained under various conditions of T and P collapse onto a single master curve, when superposed as a function of the product variable, TVγ, where γ is a materialspecific constant. More recently, this TVγ-scaling was shown to apply to the chain normal mode relaxation times, τn, and also the Johari−Goldstein β-relaxation times, τJG, with the same γ and within the errors and uncertainties accompanying the determination of the relaxation times. However, the dependences of τn, τα, and τJG on TVγ are all different, and thus we have TVγ-rheological complexity in polymers. By collecting thermo-, piezo-, and TVγ-rheological complexities together in this paper and emphasizing their generality and fundamental importance, the objective is to heighten the awareness of this basic unsolved problem in polymer physics. Although the coupling model has identified the origins of these complexities and can explain some of the data quantitatively, it is not the emphasis of this paper. Perhaps the many-body relaxation and diffusion aspect of the model may lead others to construct a more rigorous theory that can explain the complexities as well or even better.

■

AUTHOR INFORMATION

Corresponding Author

*(K.L.N.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS We thank George Floudas for providing experimental data, and Caroline Plazek for a careful reading of the manuscript. REFERENCES

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