Why the Cox–Merz rule and Gleissle mirror relation work: A quantitative analysis using the Wagner integral framework with a fractional Maxwell kernel

There are a number of important empirical rules, such as the Cox–Merz rule¹ as well as a second less well-known “forgotten Cox–Merz rule,”^2,3 and the Gleissle mirror relationship,⁴ that are often used in linear and non-linear rheology to supplement the available data. These rules have been applied to a wide variety of polymer melts,^2,5–8 polymer solutions,^5,6,9,10 and hydrogels.¹¹ Similar principles have been used to develop an extended Cox–Merz rule (or Rutgers–Delaware rule) for suspensions as well as highly filled, yielding, and thixotropic materials.^12–14 

These different empirical rules involve relationships between various measurements of the viscous response in a complex fluid such as the steady shear viscosity, the dynamic viscosity, and the shear stress growth coefficient measured following the inception of steady shear flow. For many rheologists, including the present authors, the first exposure to these curious empirical relationships is in Chapter 3 of the pioneering text developed by R. Byron Bird with coauthors and former graduate students Armstrong and Hassager.¹⁵ They note that “the Cox–Merz rule has proven very useful in predicting $η(γ̇)$ when only linear viscoelastic data are available” and show data for a linear a polystyrene (PS) melt in which the agreement between $|η*(ω)|$ and $η(γ̇)$ is “within experimental error” at equivalent values of the shear rate and angular frequency. This similarity between the complex viscosity and the steady shear viscosity was first noted by W. P. Cox and E. H. Merz in their 1958 paper in which they show that experimental measurements of the steady shear viscosity $η(γ̇)$ are a smooth continuation of the complex viscosity $|η*(ω)|$ for polystyrene melts.¹ More recently, Snijkers and Vlassopoulos¹⁶ have carefully reviewed the levels of agreement (and disagreement) that can be achieved using the Cox–Merz rule for a range of different complex fluids.

The Cox–Merz rule establishes an approximate equality between the shear rate-dependent steady shear viscosity $η(γ̇)$ of a complex fluid and the dynamic viscosity at equivalent values of the shear rate and angular frequency,¹ i.e.,

This equality between a non-linear quantity (such as the steady shear viscosity) and a linear viscoelastic property (such as the dynamic viscosity) corresponds to equivalence between two quite distinct material functions when they are evaluated at equal values of the Deborah number (De = $ωτc$⁠) and Weissenberg number (Wi = $γ̇τc$⁠),^17,18 where τ_c is some characteristic (or mean) relaxation time of the complex fluid being studied. When viewed from the perspective of a Pipkin diagram^19,20 (in which the flow unsteadiness is represented by the magnitude of the Deborah number along the abscissa and in which a suitable non-linear measure such as the Weissenberg number reports the strength of the flow along the ordinate axis), this relationship is especially surprising, as it represents equivalence between the relevant material functions measured along two completely orthogonal axes!

Several researchers have reported different cases in which constitutive models for complex fluids obey the Cox–Merz rule.^2,3,11 In an early analysis, Booij arrives at a sufficient condition for a viscoelastic material to follow the Cox–Merz rule that is independent of the linear relaxation spectrum.² The condition states that the non-linear strain measure (denoted S₁₂ in his work) must satisfy

where J₀ is the zeroth Bessel function and $γ̇u$ is the accumulated strain in an elapsed time u. However, he notes that this strain measure oscillates with increasing time and is therefore unphysical; the conditions required to satisfy the Cox–Merz relationship in real fluids thus necessarily involve details of the viscoelastic relaxation spectrum.

It is shown in advanced textbooks (see, for example, Ref. 21) that for models characterized by a single relaxation time such as a simple Maxwell model, when the steady shear viscosity is computed using the Wagner integral constitutive model [which is a special limiting case of the time-strain separable (TSS) Kaye–Bernstein Kearsley Zapas (K-BKZ) constitutive equation]^22–27 with an exponential damping function, the exponent characterizing the rate-dependence of the steady shear viscosity $η(γ̇)$ is different from the exponent characterizing the frequency dependence of the dynamic viscosity $|η*(ω)|$⁠. It is thus evident that one needs to consider more complex relaxation functions than a single mode Maxwell model.

In one early study, Larson²⁸ considers materials with a relaxation modulus of power-law form that follows $G(t) ≃ S t−β$⁠. Again, utilizing the Wagner integral formulation he derives an analytic condition required to satisfy the Cox–Merz rule for two different forms of the damping function [an exponential damping function and the Doi–Edwards (DE) damping function].²⁸ Complex materials exhibiting a power-law stress relaxation response of this functional form are known as critical gels.²⁹ However, the majority of real complex fluids such as polymer solutions and melts are not critical gels and, also, may not follow the exponential or Doi–Edwards strain damping form. Thus, it is not straightforward to apply the Cox–Merz criterion developed by Larson to our understanding of numerous experimental reports in the literature of materials that closely satisfy the Cox–Merz rule.^{1,2,5–7,9–11}

In an early extensive study on polystyrene solutions and melts, Yasuda and coworkers¹⁰ examine the agreement (and disagreement) of their data with the Cox-Merz rule. Recently, Snijkers and Vlassopoulos¹⁶ reviewed data for a large variety of different, well-defined polymers (linear monodisperse and polydisperse polymers, star polymers, model branched polymers with more than one branch point, as well as blends of linear polymers of the same chemistry) and found the Cox–Merz rule to work remarkably well for a large variety of molecular structures of flexible polymers with only minor deviations at high shear rates.

Using general arguments and a general Maxwell–Wiechert model (consisting of multiple discrete relaxation time scales), Renardy argues that for viscoelastic materials with a sufficiently broad relaxation spectrum, the Cox–Merz rule will be followed to within a prefactor.³⁰ It is thus desirable to consider extensions to the analysis of Larson²⁸ to other related functional forms of the relaxation spectrum and other types of strain damping function. In the present work, we do this by using an integral formulation of the fractional Maxwell model and considering a generalized form of the strain damping function.

The empirical relations proposed by Cox and Merz relate dynamic measurements of linear viscoelasticity (at low strains) to steady state measurements of the shear viscosity at large (or infinite) strains. Bird et al. also describe another “interesting alternative” to the Cox–Merz rule for predicting the steady shear viscosity from dynamic linear viscoelastic data, which they refer to as the Gleissle Mirror Relation crediting observations from Gleissle⁴ for a high polymer silicone oil. Gleissle⁴ related the bounding linear viscoelastic envelope of the shear stress growth coefficient (following inception of steady shear) to the steady shear viscosity as follows:

Henceforth in this paper, for compactness, we refer to the limiting curve given by the shear stress growth coefficient during start-up of steady shear at Wi $≪1$ [i.e., the left-hand side of Eq. (3)] as the transient viscosity$η+(t)≡σ+(t)/γ̇0$⁠. This expression is known as the mirror relation since a single measurement of the transient start-up viscosity can be reflected using the mapping $γ̇=1/t$ to estimate the functional form of the steady shear viscosity over a wide range of shear rates. When viewed in the framework of a Pipkin diagram,¹⁹ this relationship is just as puzzling as Eq. (1) as it again relates measurements made as a function of the elapsed time t in the linear viscoelastic regime (i.e., in the limit $τcγ̇0≪1$⁠) to measurements of the steady shear viscosity at all shear rates, even well into the non-linear shear-thinning regime. This relation is discussed by Leblans et al.³¹ for polymer melts and by Jaishankar for Xanthan gum.¹¹ 

Although these empirical rules have been shown to be obeyed for a wide range of complex fluids, there is limited physical understanding or mathematical derivation of these expressions from general constitutive relationships, such as the K-BKZ model, that interrelate the stress and strain tensors for a wide range of complex fluids.^24,32 In this paper, we discuss in detail the conditions required for a general complex fluid to obey the Cox–Merz rule and the Gleissle mirror relationship. To do this, we use the Wagner integral framework, which is a special limiting case of the time-strain separable K-BKZ constitutive equation describing the tensorial stress–strain relationship in a general frame-invariant way. This constitutive equation has been tested extensively and has been demonstrated to describe accurately the steady and transient shear responses of a wide range of complex fluids.^{11,15,22,23,33}

To incorporate the non-linearities that arise at large strain amplitudes within the Wagner integral constitutive equation, we use a generalized form of the damping function often known as the Soskey–Winter damping expression.³⁴ This functional form has been shown to capture the damping or strain-softening of many complex fluids such as polymer melts, polymers, suspensions, magnetorheological fluids as well as polymer blends at large strains.^{22,23,28,35–39} To evaluate the Cox–Merz rule and Gleissle mirror relationships embodied in Eqs. (1) and (3), we also need to specify the form of the viscoelastic relaxation kernel. We first illustrate our approach using the simple Maxwell model that describes materials such as wormlike micellar fluids with a single dominant relaxation time scale, and show analytically using the Wagner integral formulation why such materials do not obey the Cox–Merz relation. Then, in order to represent more realistic complex fluids with a broad relaxation spectrum, we use the fractional Maxwell liquid model (FML) to describe the linear relaxation kernel. The FML model has been shown to compactly describe the linear viscoelastic properties of a wide range of complex fluids that have a finite zero shear viscosity such as polymer melts and polymer solutions.^11,40 In the special limit $η0→∞$⁠, the FML model can also be used to describe critical gels and thus reduces to the same relaxation spectrum considered by Larson.²⁸ 

In a recent study, we have used this constitutive formulation to derive analytical expressions for the rate-dependent viscosity in various weak and strong damping limits.⁴⁰ In the present contribution, we use these analytic results, in conjunction with the linear viscoelastic response of the fractional Maxwell liquid, to show that complex fluids undergoing sufficiently strong strain-softening at large strain amplitudes will obey both the Cox–Merz and Gleissle mirror relations to within a numerical constant. By contrast, other highly elastic materials (which are characterized by very weak damping responses) will deviate from both the empirical Cox–Merz and Gliessele mirror rules. Finally, we illustrate these mathematical expressions using experimental measurements with an 8% alginate solution which is well-described by the FML/Wagner integral framework and indeed satisfies the Cox–Merz and mirror relationships very closely.

The Wagner integral model, which is a special limiting case of the time-strain--separable K-BKZ constitutive equation,⁴¹ gives the relation between the stress and Finger strain tensor as follows:

where $M(t−t′)≡∂G(t−t′)/∂t′$ is known as the memory function and G(t) is the linear viscoelastic relaxation modulus. The damping function h describes the survival probability of elastically active network elements after the application of a sudden large deformation.^23,41 Here, I₁ and I₂ are the first and second invariants of the Finger strain tensor (⁠$C−1$⁠) and I is the identity matrix. For shear flows in which I₁ = I₂, we can simplify Eq. (4) to arrive at the following integral expression for the shear stress $σ(t)$ at time t:

where the minus sign is present in front of the integral since we follow the dynamics of polymeric liquids (DPL)¹⁵ definition for the strain $γ(t,t′)=γ(t′)−γ(t)$ as the accumulated strain between an arbitrary time $t′$ in the past and the present time t. The damping function $h(γ)$ is given by^11,23,33

where $G(t,γ0)$ is the strain-dependent relaxation shear modulus and G(t) is the linear relaxation modulus (which is independent of strain at small strain amplitudes).

For a steady shearing flow at constant shear rate $γ̇$⁠, the accumulated strain can be written as $γ(t,t′)=γ̇×(t′−t)=−γ̇u$⁠. The steady shear viscosity $η(γ̇)$ for a constant shear rate $γ̇$ can then be derived from Eq. (5) and written using the change of variables $u=t−t′$ as

First, we derive the shear rate dependent viscosity for the canonical case of a simple Maxwell relaxation kernel and an exponential damping function as proposed by Rolon-Garrido and Wagner.²³ In differential form, the simple Maxwell element can be pictured as a series arrangement of a single linear (Newtonian) dashpot and a linear Hookean spring as shown in Fig. 1(a). The corresponding memory function for a Maxwell element can be written as¹⁵ 

where η₀ is the zero shear viscosity and τ is the (single) relaxation time scale captured by the model. Thus, the steady shear rate-dependent viscosity for a linear Maxwell element combined with an exponential damping function (which is characterized by a critical strain $γ*$⁠) and written in the form $h(γ)=exp (−γ/γ*)≡ exp (−γ̇u/γ*)$ becomes

This integral expression for the rate-dependent viscosity can be evaluated analytically and simplified to the form

The dimensionless product $γ̇τ$ can be identified as the Weissenberg number, $Wi$ measuring the strength of the non-linear shear flow.¹⁷ Thus, Eq. (10) can be written equivalently in dimensionless form as

The magnitude of the complex viscosity $|η*(ω)|$ in small amplitude oscillatory shear (SAOS) for the linear Maxwell model is given by^15,40

The product $ωτ$ corresponds physically to the ratio of two time scales (τ and the period of oscillation $T∼ω−1$⁠) and thus defines the Deborah number (De) for this flow. Therefore, the linear complex viscosity can be written as

It is clear from Eqs. (11) and (13) that only at low values of both Wi and De is the Cox–Merz rule exactly satisfied for this single relaxation mode model. This corresponds to approaching the origin of a Pipkin diagram¹⁹ and gives the expected simple fluid result

At larger values of the Weissenberg and Deborah numbers, the steady shear viscosity and the complex viscosity deviate systematically from each other. This is illustrated graphically in Fig. 1. At high values of the imposed oscillatory frequency or shear rate (corresponding to De $≫1$ and Wi $≫1$⁠, respectively), the complex viscosity scales as $|η*(De≫1)|∼De−1$ and the steady shear viscosity scales as $η(Wi/γ*≫1)∼Wi−2$⁠.

Next, we consider a more genetalized form of the damping function also known as the Soskey–Winter damping function³⁴ 

where $γ*$ again corresponds to the critical strain and m is the damping exponent. The damping function in the Doi–Edwards constitutive equation with independent alignment assumption (DE-IAA) can be approximated by a special case of this general Soskey–Winter damping form²³ with a damping exponent m = 2 and a critical strain $γ*=15/2$⁠.

The critical strain parameter $γ*$ is connected physically to the idea of a maximum in the effective strain imparted to a transient network during a rapid step deformation {i.e., $max[h(γ)γ]$}. Beyond this critical strain level, the effective strain imparted to the network starts decreasing due to strain-softening effects such as chain disentanglement or non-affine deformation. For the exponential damping function used in Eq. (9), as well as the approximate form of the Doi–Edwards damping function (m = 2) this occurs exactly at $γ*$⁠. For a general Soskey–Winter damping with m > 1 (but $m≠2$⁠), the maximum effective strain is still controlled by the critical strain parameter, but is not identically equal to $γ*$⁠. For weaker damping with $m≤1$⁠, the effective strain increases without bound irrespective of the critical strain parameter.

Using this form of the damping function and combining Eqs. (7) and (14), we obtain

This expression can readily be evaluated numerically for any value of the shear rate, but we can also obtain asymptotic expressions for the rate-dependent viscosity at low and high shear rates, respectively. To do this, we approximate the integral Eq. (15) using the limiting forms of the damping function $h(γ)$ at small strains (⁠$Wi/γ*≪1$⁠) and at large strains (⁠$Wi/γ*≫1$⁠) to obtain

where the argument $v=u/τ$ is a dimensionless time. At low shear rates, this expression is dominated by the first integral⁴⁰ and approaches the zero shear-rate viscosity $η(Wi/γ*≪1)=η0$⁠. Similarly, the asymptote for the rate-dependent viscosity at high shear rates (which is dominated by the second integral) can be found to be⁴⁰ 

where $Γu(s,x)=∫x∞ts−1e−tdt$ is the upper incomplete Gamma function. From Eq. (17), it can be shown⁴² that for values of the damping exponent m < 2, we have $η(Wi/γ*≫1)/η0≈Γ(2−m)(Wi/γ*)−m$⁠, and for the damping exponents $m≥2$⁠, the rate-dependent viscosity scales as $η(Wi/γ*≫1)/η0∼(Wi/γ*)−2$⁠.

In the linear viscoelastic limit of small strains, strain damping is unimportant and the complex viscosity is again given by Eq. (13). In the limit of high frequencies, it thus always scales as $|η*(De≫1)|∼De−1$ regardless of the form of the damping function. This deviation between the rate-dependent viscosity and complex viscosity explains why polymeric materials with very narrow relaxation spectra or single dominant relaxation modes generally do not satisfy the Cox–Merz relationship. For completeness, the asymptotic limits of the complex viscosity and steady shear rate-dependent viscosity are tabulated in Table I.⁴³ 

We now extend the above analysis to consider more realistic complex fluids that are typically characterized by broad relaxation spectra. We use the fractional Maxwell liquid model (FML) to describe the linear viscoelastic properties of such materials in a compact way. The constitutive relation for the evolution of the shear stress in the FML model can be written in differential form as

It is necessary for the exponent β to satisfy $0≤β≤1$ for thermodynamic consistency,⁴⁴ and the parameter η₀ in Eq. (18) corresponds to the zero shear viscosity of the fluid.¹¹ 

It can be shown that this model is also a compact approximation of the linear viscoelastic response predicted by the Rouse and the Zimm models when $β=0.5$ and $β=0.66$⁠, respectively.^45,46 The parameter $G$ in the model is best described as a quasiproperty⁴⁷ with units of Pa s^β.

The critical gel model studied by Winter and coworkers is also embodied within this constitutive framework. In the limit of infinite zero-shear viscosity (⁠$η0→∞$⁠), the mean relaxation time in the FML model diverges to infinity $τc ∼ (η0/G)1/(1−β)→∞$⁠, and the effective modulus goes to zero, $Ge∼(η0−βG)1/(1−β)→0$⁠. After taking this limit in the differential equation (18) and integration [by a fractional order of $(1−β)$], we obtain the familiar limit of a critical gel,²⁹ which corresponds to a “Scott Blair element,”⁴⁷ i.e., a single spring–pot element with power-law exponent β and a gel strength S that is related to the quasi-property $G$ by $S=G/Γ(1−β)$⁠.

Considering a step strain response, we can calculate the relaxation modulus of the FML model from Eq. (18) as⁴⁸ 

where $Ea,b(x)$ is the Mittag–Leffler function.⁴⁹ The characteristic time scale for the three-parameter FML model is given by: $τc=(η0/G)1/(1−β)$ and this allows the argument of the Mittag–Leffler function to be compactly written in terms of a dimensionless time $t/τc$⁠. From here, the memory function can be found from the expression $M(t)≡−∂G(t)/∂t$⁠. Thus,

where the negative sign appears from the differentiation (in time) of the (monotonically decreasing) relaxation modulus.

The fractional differential operator in Eq. (18) is nonlocal in time and captures a broadly distributed response to an imposed deformation. The viscoelastic response described by a fractional differential constitutive equation can also be equivalently represented in terms of a continuous relaxation spectrum, $H(τ)$⁠, which quantifies the contribution of different relaxation time scales τ to the viscoelastic memory exhibited by a hereditary material.^50,51 To see this, we note that the relaxation modulus can be written in terms of the relaxation spectrum as¹⁵ 

Combining this result and the relaxation modulus given in Eq. (19), a Stieltjes transform⁵² can be used to obtain

This is a special limit of the more general result that can be derived for the full fractional Maxwell model^53,54 when α = 1. The evolution in the shape of the relaxation spectrum for the FML model for different values of β is shown in Fig. 2 and illustrates the compactness of using a fractional model to describe broad relaxation spectra. The relaxation spectra shown in Fig. 2 for the values of $β=0.5$ and $β=0.66$ closely approximate the Rouse and Zimm relaxation spectra, respectively.⁴⁵ As the value of β approaches zero, the relaxation spectrum becomes narrower and eventually approaches a singular (delta) function for β = 0 which corresponds to the simple Maxwell model limit described in Sec. III.

The dynamic modulus $G*(ω)$ of the FML model can be obtained by taking a Fourier transform of the linear differential equation in time represented by Eq. (18) and is given by

The real and imaginary parts of the complex modulus can be found from Eq. (23) and hence the expressions for the storage and loss modulus of the FML model can be written as follows:

where $Gc=η0τc−1=Gτc−β$ is the characteristic elastic modulus of the FML model, and the Deborah number can again be identified as De = $τcω$⁠.

The magnitude of the complex viscosity, defined as $|η*(ω)|=G′(ω)2+G″(ω)2/ω$⁠, can then be evaluated from Eqs. (24) and (25) and expressed in compact dimensionless form as

The low and high Deborah number asymptotes for the complex viscosity can be calculated from Eq. (26) to be

and

In all of the above expressions, the simple linear Maxwell model corresponding to a single spring and dashpot can be retrieved by substituting β = 0 in Eqs. (18)–(27). This would lead us to a single dominant relaxation time scale as illustrated in Fig. 2 characterized by a single dominating microstructural length scale. As shown in Sec. III, this would lead to failure of the Cox–Merz rule and Gleissle mirror relation irrespective of the damping function.

The rate-dependent viscosity defined in Eq. (7) evaluated with a linear FML kernel specified by Eq. (20) can then be written quite generally as

In our earlier work,⁴⁰ we derived an approximate analytical expression for the above integral using the exponential damping function $h(γ)= exp (−γ/γ*)$⁠, which can be written in terms of a two-part summation as follows:

where $h$ is the Heaviside step function. In Ref. 40, we denoted exponential damping as a strong damping function since the exponent describing the rate-dependence of the viscosity at high shear rates is set by the FML linear kernel parameter [i.e., $η∼γ̇−(1−β)$] and not by the parameters of the damping function itself.

The low Weissenberg number asymptote for the steady shear viscosity can be obtained from the first term in Eq. (30) above and reduces to

whereas the high Wi asymptote is given by

By comparing Eqs. (27) and (31), it is easy to note that the steady and complex viscosity converge to the same value, corresponding to the zero shear rate viscosity η₀ at low Wi or De, as expected for a “simple fluid.”⁵⁵ By comparing Eqs. (28) and (32), we observe that the power-law exponents describing the dependence on De and Wi are also equal and therefore the Cox–Merz relation is always satisfied (to within a constant numerical prefactor) for the integral constitutive equation given by Eq. (5) with an FML kernel and an exponential strain-dependent function.

We can also arrive at a condition to identically satisfy the Cox–Merz relation, i.e., $η(γ̇)=|η*(ω))|$ at high Wi or De by equating Eqs. (28) and (32). We thus obtain the constraint

This expression provides a relationship between the critical strain parameter $γ*$ and the FML exponent β needed to identically satisfy the Cox–Merz relation. If we consider the Rouse model in which the high frequency linear viscoelastic response is set by $G′∼ω0.5$⁠, then we find $γ(R)*=(0.5)1/(0.5−1)=4$⁠. Similarly, if we consider the Zimm model with $β=0.66$ we obtain $γ(Z)*=(0.66)1/(0.66−1)=3.4$⁠. It is typical for entangled polymer solutions and melts to have $γ*=1≤γ*≤4$ (Ref. 15) (see also the  Appendix). It is apparent that the Cox–Merz relationship will therefore always be very closely approximated [especially given the form of Eq. (33) and the fact that we graphically represent viscometric measurements on double logarithmic axes].

We can also consider the rate-dependent viscosity for the more general damping function of $h(γ)=[1+(γ/γ*)m]−1$ by evaluating Eq. (29) and then obtain⁴⁰ 

where $p=β+k(1−β), q=1−k(1−β)$⁠. The asymptotes of this expression are dominated by the k = 1 terms. The low Weissenberg number asymptote can be calculated from the first term in Eq. (34) to once again be

However, care must be taken in evaluating the viscosity at high Weissenberg numbers as we have shown⁴⁰ that the high Wi limit of Eq. (34) is valid only when $m>1−β$⁠. The two different limiting cases are shown in Figs. 3(a) and 3(b), respectively.

Provided that we satisfy the constraint $m>1−β$⁠, the asymptotic expression for the rate-dependent viscosity given by Eq. (34) is

The exponent of the rate-dependent viscosity at high Wi is thus $η(γ̇)∼γ̇−(1−β)$⁠. In Ref. 40, we derived a uniformly valid expression for the steady shear viscosity for the case $m>1−β$ that accurately approximates the rate-dependent viscosity for all Wi and that can be compactly written in the form

where the constant B is given by

This general functional form is similar to several common empirical expressions, such as the Carreau–Yasuda model and Cross model,^10,15,56,57 which are used in the rheology literature for fitting experimentally measured flow curves of $η(γ̇)$⁠.

However, for the case $m≤1−β$⁠, Eq. (34) is invalid and in our previous paper⁴⁰ we have derived an alternative expression for the rate-dependent viscosity in the high Wi limit:

where

For the case $m≤1−β$⁠, the exponent describing the rate-dependence of the viscosity is therefore $η(γ̇)∼γ̇−m$ as shown in Fig. 3(b). We refer to damping functional forms in which the slope of the rate-dependent viscosity at high Wi is set by the linear viscoelastic parameter (⁠$1−β$⁠) as the strong damping limit. For the general Soskey–Winter functional form of Eq. (14), the strong damping condition is therefore $m>1−β$⁠. Cases in which the high Wi slope is set by the damping parameter m itself correspond to weak damping and in the Soskey–Winter framework correspond to $m≤(1−β$⁠). As we noted above, the damping function in the Doi–Edwards constitutive equation with the independent alignment assumption (DE-IAA) is of the form given by Eq. (14) with a damping exponent m = 2 and a critical strain $γ*=15/2=1.94$⁠.²³ Since the damping exponent m = 2 is always greater than the linear FML parameter $1−β$ (for $0≤β<1$⁠), the damping function from the DE model with independent alignment is thus always of strong damping character.

By comparing Eq. (27) with Eq. (35), we notice that for viscoelastic liquids the Cox–Merz relation is always satisfied at low shear rates and frequencies, or more formally, as the dimensionless values of $Wi/γ*$ and De approach zero. By comparing Eq. (28) with Eqs. (36) and (39), we notice that the Cox–Merz rule is satisfied to within a multiplicative constant factor for strong damping functions, but will not hold at high shear rates or frequencies for functional forms corresponding to weak damping. Again for completeness, the asymptotes for the complex and rate-dependent viscosity for the different damping functional forms are tabulated in Table II.

For the general Soskey–Winter damping expression, we can also determine when the Cox–Merz relationship is identically satisfied by equating the asymptotic expression for the rate-dependent viscosity at high Weissenberg numbers with the frequency-dependent complex viscosity at high Deborah numbers. Setting Eqs. (28) and (36) to be equal, we obtain the dimensionless constraint

This constraint represents a condition interrelating three of the constitutive parameters of a fractional Maxwell/Wagner liquid: the critical strain $γ*$⁠, the damping exponent m, and the linear exponent β (characterizing the short time power-law decay of the relaxation modulus and the power-law dependence of the complex modulus at high frequencies).

Numerical calculations show that even when Eq. (41) is satisfied, there can still be some deviation from the Cox–Merz relation when Wi $∼1$ and De ∼ 1. To illustrate this, we highlight in Fig. 4(a) the deviations between the complex viscosity and the steady shear viscosity. This is reported in terms of the logarithm of the ratio $|η*(ω)|/η(γ̇)$ [i.e., we plot contours of $log (|η*(ω)|/η(γ̇))$] for a range of different damping exponents m and for equal values of De and Wi. Exact correspondence with the Cox–Merz rule corresponds to a contour value of $log(1)=0$ and yellow-colored regions of the contour plot. At low deformation rates, the error is very small for all values of m. By contrast, at high De and Wi the errors are only small in the strong damping limit above the solid black line. The error is minimal and approaches zero for all deformation rates when m is in the vicinity of the value given by Eq. (41) as shown by the broken gray line. When the ratio $|η*(ω)|/η(γ̇)$ is greater than unity, the logarithm is positive and the complex viscosity is therefore greater than the steady shear viscosity. At high Wi or De, we can observe this in Fig. 4(a) for values of $m≥1$⁠, and this corresponds to most experimental observations.¹⁶ However, it is also possible for the ratio to be smaller than unity and the logarithm of the ratio is then negative at high Wi as is visible in Fig. 4(a) for smaller values of the damping exponent $0.8≤m≤1$⁠. This case has also been observed in the literature for highly branched polymer systems.¹⁶ In Fig. 4(b), we show a comparison between the complex viscosity and the steady shear viscosity when the damping exponent m, the FML parameter β, and the critical strain $γ*$ are exactly interrelated by the constraint of Eq. (41). The deviation observed at De $≃$ Wi $≃1$ is only of maximum magnitude 0.17 (corresponding to a ratio of viscosities of approximately 1.5). This close agreement of the Cox–Merz relationship at high and low De or Wi (but with small deviations at intermediate shear rates) is consistent with measurements for worm-like micelles and the non-linear differential viscoelastic model proposed by Manero and coworkers.⁵⁸ 

A similar error contour plot showing $log (|η*(ω)|/η(γ̇))$ is illustrated for the exponential damping function with various values of the FML parameter β and at equal values of Deborah and Weissenberg numbers in Fig. 4(c). At high Wi or De, once again the ratio of the dynamic viscosity to the steady shear viscosity may be bigger or smaller than unity. From Fig. 4(c), we observe the steady shear viscosity to be less than the dynamic viscosity for smaller values of β and vice versa at larger values of β at high Wi or De. For values of $β≥0.5$⁠, the difference between the two material functions is very small and is minimized when the critical strain $γ*$ follows Eq. (33). A direct comparison of the complex viscosity and the steady shear viscosity when a complex liquid exhibits exponential damping with the critical strain $γ*$ and the FML parameter β interrelated by Eq. (33) is shown in Fig. 4(d). It is clear that the Cox–Merz rule is satisfied almost exactly at all deformation rates.

To help illustrate the broad applicability of this analysis and the Wagner model with a fractional Maxwell kernel, we consider experimental data obtained with an 8% alginate solution in Fig. 5. This semiflexible biopolymer can be used to produce biocompatible gel-like materials and the rheology can be varied systematically by changing the alginate concentration and the concentration of divalent crosslinks.^59,60 In order to evaluate the strain-dependent damping function $h(γ)$ for this fluid, we first measure the strain-dependent relaxation modulus $G(t,γ0)$ as shown in Fig. 5(a) for step strains in the range of $0.3≤γ0≤10$⁠. A broad power-law decay in the relaxation modulus with time is observed for all imposed strain amplitudes. At small strains (⁠$γ0≤0.7$⁠), the measured relaxation curves superpose—indicating the linear viscoelastic limit. At higher applied strain amplitudes, the measurements of the relaxation modulus are systematically lower. We then shift the strain-dependent relaxation modulus on to the linear relaxation modulus [here taken to be $G(t,γ0=0.3)$] to determine the damping values $h(γ)=G(t,γ0)/G(t)$⁠. The collapsed master curve of the relaxation modulus and the corresponding values of the damping function are plotted in Fig. 5(b). The generalized damping function given by Eq. (14) is regressed to the damping values and the resulting best fit parameters (shown by the solid line) are found to correspond to a critical strain $γ*=2.5$ and a damping exponent m = 1.4 (which is significantly smaller than the Doi–Edwards limit, $mDE=2$⁠). The storage and loss modulus are obtained by performing SAOS tests for a range of frequencies at a strain amplitude $γ0=0.02$ and are shown in Fig. 5(c). The linear viscoelastic properties of this alginate solution are very well described by the three-parameter fractional Maxwell liquid model as shown in Figs. 5(b) and 5(c), respectively. By regressing the data to Eq. (19), (24), and (25), the best-fit FML parameters are evaluated to be $G=23$ Pa $sβ, β=0.56$⁠, and $η0=7.0$ Pa s. The characteristic relaxation time is thus $τc=(7/23)1/(1−0.56)=0.067$ s and this value can be detected visually by the change in the slope of the storage and loss modulus curves at frequencies around $ωc=1/τc≃15$ rad/s.

The material properties for this alginate solution thus correspond to the strong damping limit with $m>(1−β)$ and we should expect the Cox–Merz relationship to be valid. The resulting predictions for the complex viscosity $|η*(ω)|$ and the steady shear viscosity $η(γ̇)$ are illustrated in Fig. 5(d). The FML predictions from Eq. (26) and the prediction for the steady shear viscosity computed from Eq. (29) (with no adjustable parameters) are also compared with the experimental data in Fig. 5(d). The agreement between the measured steady shear data and the Cox–Merz predictions is excellent. The consistency between the experimental data and our chosen time-strain separable (TSS) integral K-BKZ formulation provides validation to the fractional constitutive framework through which we have derived the conditions for the Cox–Merz rule to be satisfied.

The Gleissle mirror relation relates the time-dependent growth of the transient viscosity at a very low and constant shear rate (i.e., in the linear viscoelastic limit $γ̇0τc≪1$⁠) to the rate-dependent steady shear viscosity^4,61 and can be written in the form

Using the results above, we can determine the conditions required for the Wagner integral constitutive equation with the FML kernel and a generalized damping function to satisfy this empirical relationship. In order for the Gleissle mirror relation to hold true exactly, we find that the following implicit condition must be satisfied between the two parameters characterizing the general damping function (⁠$γ*$ and m) and the parameter β in the FML relaxation kernel:

Derivation of Eq.(43): The transient viscosity at very low shear rates (in the linear viscoelastic limit) can be found from the integral expression

Substituting the form of the memory function given in Eq. (20) for the FML model into Eq. (44), we obtain

At early times, during start-up of a steady shear flow, i.e., when $u/τc≤t/τc≪1$⁠, the Mittag–Leffler function can be approximated⁶² and the above integral reduces to