Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

Aditya Jaishankar, Gareth H. McKinley

Abstract

Consumer products, such as foods, contain numerous polymeric and particulate additives that play critical roles in maintaining their stability, quality and function. The resulting materials exhibit complex bulk and interfacial rheological responses, and often display a distinctive power-law response under standard rheometric deformations. These power laws are not conveniently described using conventional rheological models, without the introduction of a large number of relaxation modes. We present a constitutive framework using fractional derivatives to model the power-law responses often observed experimentally. We first revisit the concept of quasi-properties and their connection to the fractional Maxwell model (FMM). Using Scott-Blair's original data, we demonstrate the ability of the FMM to capture the power-law response of ‘highly anomalous’ materials. We extend the FMM to describe the viscoelastic interfaces formed by bovine serum albumin and solutions of a common food stabilizer, Acacia gum. Fractional calculus allows us to model and compactly describe the measured frequency response of these interfaces in terms of their quasi-properties. Finally, we demonstrate the predictive ability of the FMM to quantitatively capture the behaviour of complex viscoelastic interfaces by combining the measured quasi-properties with the equation of motion for a complex fluid interface to describe the damped inertio-elastic oscillations that are observed experimentally.

1. Introduction

A multitude of consumer products, especially foods, owe their structure, stability and function to the presence of interfaces. Common examples include foams and emulsions such as milk, soups, salad dressings, mayonnaise, ice cream and butter (see McClements [1] and references therein). Although many of these foams and emulsions are thermodynamically unstable, the kinetics of phase separation can be controlled with the addition of various proteins, surfactants, gums and other stabilizing agents, which have very important implications for the shelf life of foods [2]. However, the presence of these additives often leads to complex rheological properties and gives rise to distinctive power laws in the creep response (i.e. the strain varies as γ(t)∼tα) and also in the corresponding frequency response (i.e. the elastic modulus varies with frequency as G′(ω)∼ωα). Such power-law responses are not well described by canonical rheological models such as the Maxwell or Kelvin–Voigt models [3]. Moreover, the sensory perception of foods in terms of textural parameters plays an important role in the assessment of food quality, and is strongly related to the viscoelastic properties of the interfacial layers present [4]. New rheological tools such as the double wall ring (DWR) interfacial rheometer [5,6] enable us to experimentally quantify such responses with unprecedented accuracy over a wide range of frequency and time scales. We now seek a framework for modelling these power-law responses in a simple yet robust constitutive theory that can then be used to predict the material response in other, more complex, flows.

Constitutive modelling in rheology often involves constructing models that can be viewed conceptually as an arrangement of elastic Hookean springs and viscous Newtonian dashpots. Tschoegl [3] compiles many such arrangements along with their response to various applied deformations. The canonical example is the linear viscoelastic Maxwell model, which consists of a spring and a dashpot in series. When a step strain deformation is imposed, the stress in the material responds exponentially, and this fundamental mode of response is commonly referred to as the Maxwell–Debye response [7]. Some model complex fluids (for example, entangled worm-like micellar solutions in the fast-breaking chain limit) are well described by this simple model [8]. However, there are many classes of materials in which stress relaxation following a step strain is not close to exponential, but is in fact best represented as a power law in time, i.e. G(t)∼tβ. Examples of such materials include physically cross-linked polymers [9], microgel dispersions [10], foams [11], colloidal hard sphere suspensions [12], soft glassy materials [13] and hydrogels [14]. Non-exponential stress relaxation in the time domain also implies power-law behaviour in the viscoelastic storage modulus, G′(ω), and loss modulus, G′′(ω), measured in the frequency domain using small-amplitude oscillatory shear (SAOS) deformations. This broad spectral response is indicative of the wide range of distinct relaxation processes available to the microstructural elements that compose the material, and there is no single characteristic relaxation time [7]. The irregular nature of relaxation events in complex fluids such as foods and consumer products is also often manifested in micro-rheological experiments as anomalous sub-diffusion or sticky diffusion, in which the mean square displacement of Brownian tracer particles is found to scale as 〈x2〉∼tα, 0<α<1 [15,16].

To describe these so-called power-law materials, one may add progressively more mechanical elements in series or parallel to the initial Maxwell element or Voigt element [3], and in the process, provide additional modes of relaxation. We thus obtain a broad spectrum of discrete relaxation times that characterize the material response. Most real systems can thus be described in an ad hoc way using a sum of exponentials [17]. However for power-law materials to be modelled accurately, it is often found that a very large number of corresponding mechanical elements are required. For many complex fluids, this approach is frequently impractical from a modelling point of view. Moreover, the values of the fitted parameters in any model with a finite array of relaxation modes depend on the time scale of the experiment over which the fit is performed. Consequently, the model parameters that are obtained lack physical meaning [18].

Scott-Blair [19] pioneered a framework that enabled the power-law equation proposed by Nutting [20] to be made more general through the use of fractional calculus. With analogy to the classical ideas of (i) the Hookean spring, in which the stress in the spring is proportional to the zeroth derivative of the strain and (ii) the Newtonian dashpot, in which the stress in the dashpot is proportional to the first derivative of the strain, he proposed a constitutive equation in terms of a fractional derivative Embedded Image1.1where 0<α<1, effectively creating an element that interpolates between the constitutive responses of a spring and a dashpot. Here, the material property V is a quasi-property, and dα/dtα is the fractional derivative operator [21], both of which are discussed in further detail below. Scott-Blair and co-workers used equation (1.1) as a constitutive equation in itself; Koeller [22] later equated this canonical modal response to a mechanical element called the spring-pot (sometimes known as the Scott-Blair element [23]) and identified it as the fundamental building block from which more complex constitutive models could be constructed.

One of the consequences of Scott-Blair and co-workers' detailed study into these so-called fractional models is the emergence of the concept of material quasi-properties, denoted in equation (1.1) by the quantity V (with SI units of (Pa sα)). Quasi-properties differ from material to material in the dimensions of mass M, length L and time T, depending on the power α. It may thus be argued that they are not true material properties because they contain non-integer powers of the fundamental dimensions of space and time. However, such quasi-properties appear to compactly describe textural parameters such as the ‘firmness’ of a material [24]. They are numerical measures of a dynamical process such as creep in a material rather than of an equilibrium state. In the present paper, we show how we can compactly represent the wide range of microstructural relaxation processes in a power-law material in terms of these so-called quasi-properties and the associated fractional derivatives with only a few parameters.

Bagley & Torvik [25] have demonstrated that, for long chain molecules with many submolecules per chain, the Rouse molecular theory [26] is equivalent to a fractional constitutive equation, and compactly represents the polymer contribution to the total stress in terms of the fractional half-derivative of the strain. The fractional Maxwell model (FMM) and other fractional constitutive models have been considered in detail in the literature [22,2730].

We demonstrate in this paper that fractional stress–strain relationships are also applicable to viscoelastic interfaces, and result in simple constitutive models that may be used to quantitatively describe the power-law rheological behaviour exhibited by such interfaces. We first briefly outline the basic definitions of fractional calculus in a form most useful for applications in rheology. We then connect the framework to the studies of Scott-Blair and co-workers [24,31], and show, using Scott-Blair et al.'s [31] original data on ‘highly anomalous butyl rubber’, how the use of the FMM to extract the quasi-properties of this material is superior to the use of conventional spring–dashpot models that characterize creep and stress relaxation. Next, we emphasize the utility of fractional constitutive models, and highlight the shortcomings of linear constitutive models for describing complex fluid interfaces using interfacial rheology data obtained from highly viscoelastic bovine serum albumin (BSA) and Acacia gum interfaces. Finally, we present a discriminating comparison of linear and fractional viscoelastic constitutive models using the phenomenon of creep ringing that arises from the coupling between surface elasticity and instrument inertia. We show that combining fractional constitutive models with the concept of material quasi-properties enables the quantitative description of complex time-dependent interfacial phenomena.

2. Mathematical preliminaries

(a) Definitions

In this paper, we use the Caputo derivative for fractional differentiation, which is defined as [3234] Embedded Image2.1where n−1<αn, n is an integer, and f(n)(t) indicates an integer order differentiation of the function f(t) to order n. The Caputo derivative operator Embedded Image is itself a linear operator, so that Embedded Image where b is any scalar, and f and g are appropriate functions [21]. In what follows, we choose the lower limit of integration a=0, which enables us to reformulate the Caputo definition as a Laplace convolution [29]. In essence, we restrict our attention to the domain t>0 because f(t)=0 for t≤0. Consequently, we henceforth use the more compact notation for the Caputo derivative, Embedded Image2.2The Laplace transform of the Caputo derivative is given by Podlubny [32], Embedded Image2.3where Embedded Image. The Fourier transform of the Caputo derivative is given by Schiessel et al. [29], Embedded Image2.4where Embedded Image. Using these definitions, we can now formulate the FMM for a complex fluid, which is the simplest general rheological model involving spring-pots, and contains only four constitutive parameters.

(b) The fractional Maxwell model

The spring-pot, whose constitutive equation is given in equation (1.1), bridges the gap between a purely viscous and a purely elastic material response by interpolating between a spring and a dashpot. For dimensional consistency, the constant V must have the units (Pa sα) where 0≤α≤1, and can be equated to Scott-Blair's concept of a quasi-property [31]. The formulation of fractional constitutive equations in terms of quasi-properties has fallen out of use in the recent rheological literature. It is often preferred to write the constitutive equation of a spring-pot (equation (1.1)) as Embedded Image, where the modulus G0 has units of (Pa) and λ0 has units of (s) [35]. While this initially seems simply to be a matter of notational convenience, the latter formulation draws attention away from the fact that the fundamental material property that characterizes the behaviour of power-law-like materials is the unique quasi-property Embedded Image, which characterizes the magnitude of the material response in terms of a single material parameter. In fact, it can be shown that it is not possible from simple rheological tests to isolate the individual components (G0,λ0,α), but only the product Embedded Image.

The two parameters V and α are the only parameters required to characterize a spring-pot. It is evident that the spring-pot—schematically shown in figure 1a—reduces to a spring when α=0 and a dashpot when α=1. The quasi-property V also reduces, respectively, to the limits of a modulus G (units: (Pa)) or a viscosity η (units: (Pa s)) in these limiting cases. The preferred term proposed by Scott-Blair et al. [31] for a constitutive law exhibiting spring-pot-like behaviour was ‘the principle of intermediacy’. We present a vectorial graphical representation of the spring-pot in the electronic supplementary material to aid in understanding this intermediate nature of the spring-pot.

Figure 1.

(a) Schematic of a spring-pot as an element that interpolates between a spring (α=0) and a dashpot (α=1). (b) The fractional Maxwell model (FMM).

We may now use these spring-pot elements to construct more complex constitutive models. This approach has been discussed in some detail in the literature, notably by Koeller [22], Nonnenmacher [27], Schiessel & Blumen [28], Bagley & Torvik [36], Torvik & Bagley [37], Friedrich [38] and Heymans & Bauwens [39]; we therefore summarize the primary result without derivation. The FMM consists of two spring-pots in series characterized by the parameters (V,α) and (G,β), respectively (figure 1b). The constitutive equation for the FMM can be obtained from assuming equality of the stress (σ=σ1=σ2) in the spring-pots, and additivity of the strains (γ=γ1+γ2) to give Embedded Image2.5where we take α>β without loss of generality [29]. The ratio (V/G)1/(αβ) with units of (s) represents the fractional generalization of a characteristic relaxation time for the model.

Friedrich [40] has shown that this model results in a non-negative internal work and a non-negative rate of energy dissipation, and is hence consistent with the laws of thermodynamics. Lion [41] has argued more generally that a constitutive model containing fractional elements is thermodynamically admissible only if the resulting constitutive equation represents some physically realizable combination of springs, dashpots and spring-pots. In other words, models that do not have mechanical analogues are thermodynamically inadmissible.

In a stress relaxation experiment, a step strain of the form γ=γ0H(t) is imposed (where H(t) is the Heaviside step function), and the resulting stress is measured as a function of time. The solution of equation (2.5) following the imposition of such a step strain can be solved analytically, and the relaxation modulus is expressed as [29] Embedded Image2.6where Ea,b(z) is the two-parameter Mittag–Leffler function defined as [32] Embedded Image2.7Note that in equation (2.6), we have written the expression for G(t) in terms of the quasi-properties G and V of the two spring-pot elements, and the power-law exponents α,β.

We may also analytically solve for the compliance J(t) in a creep experiment wherein a step stress of the form σ(t)=σ0H(t) is imposed, with σ0 denoting the step in stress. By substituting this equation into the fractional Maxwell constitutive equation (equation (2.5)), we can now solve for the evolution of the strain in the Laplace domain. We note that the necessary initial conditions are determined from the fact that the sample is initially at rest, and we have Embedded Image. Moreover, the state of zero strain may be fixed arbitrarily, and we set the material strain to be γ(0)=0 at t=0. Hence, upon inverting the Laplace transformed strain Embedded Image we arrive at Embedded Image2.8where Γ(z) is the Gamma function. It is evident that setting α=1 and β=0 retrieves the creep response of the linear viscoelastic Maxwell model given by J(t)=(t/η+1/G).

In this paper, we are interested in the properties of complex interfaces. Because interfacial stresses correspond to a line force, they have units of (Pa m) or (N/m). This additional length dimension also influences the units of the corresponding interfacial quasi-properties Vs, Gs that characterize complex interfaces, and they now have units of (Pa m sα) and (Pa m sβ), respectively, which remain quasi-properties in time. To construct a constitutive equation for complex fluid interfaces, we write the interfacial counterpart of equation (2.5) as Embedded Image2.9in which σs(t) is the interfacial stress (any symbol in this paper with the subscript ‘s’ is to be interpreted as an interfacial quantity unless otherwise specified). This model is confronted with experimental data in §4 of the present paper.

3. Techniques and materials

To demonstrate the ability of the fractional models discussed earlier to describe viscoelastic interfaces, we performed interfacial rheological experiments on BSA and Acacia gum solutions. Interfacial rheological experiments were performed with a TA Instruments ARG2 stress-controlled rheometer using the DWR fixture. The construction and operation of the DWR has been described in detail by Vandebril et al. [5]. The test fixture consists of a platinum–iridium ring that is placed at the air–liquid or liquid–liquid interface of interest. The ring has a square cross-sectional angled at 45° to pin the location of the interface and minimize the effects of meniscus curvature. Steady or oscillatory shear deformations may be applied to the fluid interface using the DWR, and the resulting torque is measured.

In any interfacial shear rheological experiment, torque contributions are present from the flow that is induced in the subphase in addition to the interfacial torque contribution. For accurate measurements, it is important to identify these subphase contributions and ensure that they do not dominate over the interfacial torque measurements. The selective sensitivity of a specific test geometry to interfacial effects, in comparison with the induced subphase flow, is characterized by the Boussinesq number Bos [42]. For Bos≫1, interfacial effects dominate, and the characteristic length scale ls=AB/Ps of the geometry is thus crucial in determining the relative sensitivity of a particular fixture to interfacial viscous effects when compared with subphase effects from the bulk. In all our experiments, we ensured that the requirement Bos≫1 is satisfied. The relatively small value of the length scale ls for the DWR fixture when compared with other available rheometer fixtures leads to a higher value of the Boussinesq number. Additional details regarding the importance of the Boussinesq number in interfacial measurements are discussed elsewhere [5,42,43].

BSA, extracted by agarose gel electrophoresis, was obtained from Sigma-Aldrich Corp (St Louis, MO, USA) in the form of a lypophilized powder. 0.01 M phosphate buffered saline (PBS) solution (NaCl 0.138 M; KCl 0.0027 M; pH 7.4, at 25°C.) was prepared by dissolving dry PBS powder obtained from Sigma-Aldrich Corp. A precisely weighed quantity of BSA was dissolved in the PBS, and the solution was brought up to the required volume in a volumetric flask to finally obtain solutions with a BSA concentration of 50 mg ml−1. The uncertainty in composition from solution preparation was determined to be only 0.002 per cent. The prepared solutions were stored under refrigeration at 4°C and were allowed to slowly warm up to room temperature before being used for experiments. All BSA solutions used in this study had a concentration of 50 mg ml−1, unless otherwise specified.

Acacia gum in powdered form was also obtained from Sigma-Aldrich Corp (SKU:G9752). A known quantity of Acacia gum was dissolved in deionized water by slow stirring for approximately 6 h to make solutions at a concentration of 3 wt%. The solutions were then double-filtered using Whatman filter paper grade #595 (pore size: 4−7 μm) to remove any residual insoluble material. Prior to rheological testing, all solutions were stored at 4°C for 24 h to ensure complete biopolymer hydration [44].

4. Results

(a) Stress relaxation and creep without inertia

We first consider the stress relaxation in a complex material after the imposition of a step strain. The broad spectrum of relaxation times exhibited by power-law materials often present challenges in modelling such experiments [45]. It has already been noted that the inclusion of additional relaxation modes, which is equivalent to including additional Maxwell or Voigt units in parallel, gives improved fits to experimental data. The resulting expression for linear viscoelastic stress relaxation is a Prony series [14,46], Embedded Image4.1where ηk and λk are fitting constants. The number of modes Nm required to fit experimental data varies depending on the time scale over which the relaxation modulus is measured and the degree to which the experimental data deviates from the exponential Maxwell–Debye response. Although describing data in this manner is a well-posed exercise, it is often cumbersome because of the large number of fitting parameters required. Tschoegl [3] remarks presciently ‘If the number of Maxwell of Voigt units is increased to the minimum number required for a series-parallel model to represent such a (power-law) distribution at all adequately, the simplicity of the standard models is lost and, in addition, arbitrary decisions must be made in assigning suitable values to the model elements’.

Another empirical approach often used to describe experimental observations of power-law-like relaxation is a stretched exponential response, known as the Kohlrausch–Williams–Watts expression (KWW) [14], given by σ(t)=0 e−(t/τ)β, where the characteristic relaxation time τ, the exponent β and the modulus scale G are the fitting constants. The KWW expression works well in practice for describing the step strain excitation. However, it is in general not possible using standard procedures to find the underlying form of the constitutive model that could subsequently be used to predict the response of the material to another mode of excitation [3]. Scott-Blair et al. [31] attempted to model measurements of anomalous stress relaxation in a range of materials using a higher-order Nutting equation of the form Embedded Image4.2with AB,C,… However, we show in the electronic supplementary material that this equation is not thermodynamically admissible.

To demonstrate the ability of properly formulated fractional constitutive models and the resulting quasi-properties to compactly describe the complex time-dependent properties of real viscoelastic materials, we revisit Scott-Blair et al.'s [31] original stress relaxation data and fit the measurements with the FMM discussed in §2b. In figure 2, we re-plot representative data reported for the original stress relaxation and creep experiments performed by Scott-Blair et al. [31]. We plot the relaxation modulus G(t) and the corresponding creep compliance J(t) for compactness, instead of the original stress and strain values, respectively. It can be seen that the data collapse onto a rheological master curve, as expected for experiments performed in the limit of linear deformations. We now fit equation (2.6) to the measured G(t) values shown in figure 2a. We set one of the elements in the FMM to be a spring (i.e. β=0); this accounts for the instantaneous elastic response in the stress at the start of the experiment. The FMM fit (solid line) describes the material response extremely well over a wide range of time scales (10 st≤400 s) in terms of just three material parameters α=0.60±0.04, V=2.7±0.7×107 Pa s0.60 and G=2.3±0.2×106 Pa (with β=0). The error bars in the figure and the error estimates of the individual parameters α,V and G correspond to 95% confidence intervals for the nonlinear least-square parameter fits. A satisfactory fit using a sum of relaxation modes (equation (4.1)) is obtained only if three relaxation modes are used, leading to the use of six fitting parameters, instead of the three required in the fractional Maxwell case.

Figure 2.

Rheological data for ‘highly anomalous’ butyl rubber taken from Scott-Blair et al. [31]. (a) Representative data of stress relaxation experiments performed at two different strain amplitudes. The solid line depicts the FMM fit (equation (2.6)), with one of the elements set to be a spring (β=0). For comparison, the fit obtained from a linear Maxwell model is shown as a dashed line. (b) Creep data at two different stresses for the same ‘highly anomalous’ butyl rubber. The solid line represents the prediction of the FMM (equation (2.8)) based on the quasi-properties determined from the stress relaxation fit.

If the values of the quasi-properties found above truly characterize the material, then we should be able to predict the constitutive response of the material to other deformations using the same rheological equation of state. To demonstrate this, we next consider the creep data for the same ‘highly anomalous’ rubber presented by Scott-Blair et al. [31], which has been plotted as the creep compliance J(t) in figure 2b. We can use equation (2.8) to predict the creep response of the ‘highly anomalous’ rubber based on the power-law exponent and quasi-properties found from fits to the relaxation modulus. Substituting these values into equation (2.8) leads to the solid curve shown in figure 2b. It can be seen that the prediction of the model again agrees very well with measured data, indicating that the FMM quantitatively describes the power-law-like behaviour observed by Scott-Blair in these ‘anomalous’ materials.

From this analysis of some previously published data, the superiority of fractional models in compactly describing the broad power-law-like response of real materials is apparent. Similar power-law creep responses are commonly observed in both micro-rheological experiments [47,16] and macroscopic experiments [48,49]. Scott-Blair's concept of quasi-properties is intimately connected to the framework of fractional calculus models and provides a physical material interpretation of the predictive power of these apparently abstract constitutive models.

(b) Interfacial dynamics

Interfacial rheology or ‘two-dimensional rheology’ studies the dynamics and structure of interfacial viscoelastic thin films or skins formed by solutions containing surface active molecules [42]. Understanding the mechanics of viscoelastic interfaces is critical to a number of applications, including the use of food additives and stabilizers [2], medicine, physiology and pharmaceuticals [50,51]. Although static surface tension measurements are sufficient to characterize the interfacial properties of surfactant-free solutions with clean interfaces, accurate descriptions of solutions or dispersions containing surface active molecules with dynamically evolving interfaces necessitate correct accounting of the mass and momentum transport processes occurring at the interface [52]. In this paper, we will only concern ourselves with the interfacial response of surface-active solutions to shearing deformations, although dilatational interfacial phenomena can also be important in other modes of deformation [53].

Two common examples of surface active materials are Acacia gum solutions and BSA solutions, which form the focus of the present study. The surface characteristics of BSA solutions at the air–water interface have been studied extensively using multiple techniques, and it is well established that these solutions form rigid viscoelastic interfaces [43,5355,]. On the other hand, although some interfacial studies have been performed on Acacia gum solutions [49,56,57], there is comparatively less literature available for these solutions. Furthermore, there is significant variability present between Acacia gums extracted from different sources.

For each sample, we first performed interfacial time sweep experiments at a fixed frequency of ω=1 rad s−1 and a fixed strain amplitude of γ0=1% to monitor the time evolution of interfacial viscoelasticity at the interface. We find that the interfacial viscoelastic storage and loss moduli, Gs′(ω) and Gs′′(ω), respectively, reach equilibrium about 2.5 h after sample loading, indicating that the interfacial structure has reached steady state. It is observed that Gs′(ω)>Gs′′(ω), indicating that the interfacial microstructures formed are predominantly elastic. The solid-like nature of the microstructures formed at the interface can also be observed in the strain sweep performed at an angular frequency of ω=1 rad s−1 shown in figure 3a. In the linear regime, we measure Gs′≈0.025 Pa m>Gs′′≈5×10−3 Pa m. The interfacial structure yields at a strain amplitude of about γ0≈3%. In figure 3b, we show the values of the interfacial moduli as a function of excitation frequency for the 3 wt% Acacia gum solution. Throughout the frequency range tested, Gs′(ω)>Gs′′(ω) signifying that viscoelastic solid-like behaviour persists, even at lower frequencies. Testing at frequencies lower than ω=10−2 rad s−1 was avoided to prevent evaporation effects from interfering with the measurements. Erni et al. [49] have reported that the values of Gs′ and Gs′′ measured in a frequency sweep are unchanged upon changing the concentration of Acacia gum in the subphase from 10 wt% to 20 wt%, which has been attributed to the saturation of the interface by Acacia gum molecules.

Figure 3.

Interfacial small-amplitude oscillatory shear data of 3 wt% Acacia gum solutions carried out using the DWR. (a) Strain amplitude sweep performed at ω=1 rad s−1. (b) Frequency sweep performed at a strain amplitude γ0=1%, which lies in the linear regime. The viscoelastic interface shows weak power-law behaviour. Filled squares, Gs′; open squares, Gs′′.

The viscoelastic data obtained from the frequency sweep exhibit a weak power-law behaviour, which is typical of many physical and chemical gels [9] as well as soft glassy materials [13]. Numerous recent reports of bulk rheology in soft solids have shown examples of such power-law behaviour in SAOS deformations [18,48,58]. We have already demonstrated the utility of fractional models in describing bulk creep and stress relaxation experiments in §4a. We next examine the ability of the FMM to describe the power-law responses observed in interfacial oscillatory deformations.

(c) The fractional Maxwell model in small-amplitude oscillatory shear deformations

The complex fluid examples discussed earlier, including the Acacia gum and BSA interfaces tested in this study, exhibit broad power-law responses when subjected to SAOS. Winter & Mours [9] have presented a model for critical gels in which the storage and loss moduli in the bulk are described by the power laws Embedded Image and Embedded Image, respectively, where S is the gel strength parameter (units: (Pa sn)). It may be shown by inverse Fourier transforming the complex modulus G*(ω)=G′(ω)+iG′′(ω) and finding the resulting constitutive equation that this is equivalent to a constitutive model consisting of a single spring-pot, and the gel strength parameter is closely related to the quasi-property of the spring-pot V=(1−n).

One may achieve a more versatile constitutive model for describing foods and other gels and soft glasses that show power-law-like rheology by considering the FMM depicted schematically in figure 1b. For a viscoelastic interface, the corresponding interfacial constitutive equation is equation (2.9). Following the procedure outlined by Friedrich [38] and Schiessel et al. [29], we evaluate the complex modulus of the interface by Fourier transforming equation (2.9) using equation (2.4) to obtain Embedded Image4.3By evaluating the real and imaginary parts of the right-hand side of equation (4.3), we find that the storage and loss moduli are given, respectively, by Embedded Image4.4and Embedded Image4.5The asymptotic behaviours of equations (4.4) and (4.5) in the limit of low and high frequencies are given in table 1. Several different limits can be distinguished in the special cases corresponding to β=0,1 and α=0,1, respectively. These expressions (4.3)–(4.5) reduce correctly to those of the linear Maxwell model when α=1 and β=0. When multiple Maxwell modes are used to generate a satisfactory description of the behaviour of power-law materials, we often require a very large number of discrete relaxation times [3], something that can be readily circumvented with the use of a fractional model such as equation (2.9). The fractional calculus description captures the dynamics of the broad spectrum of relaxation times very succinctly, by collapsing them into a single spring-pot [28].

View this table:
Table 1.

Asymptotic behaviour of Gs′(ω) and Gs′′(ω) in the FMM. Because 0<β< α<1, Gs′ and Gs′′ reduce identically to 0 for the cases β=1 and α=0, respectively, and the result holds for all frequencies.

One limitation of the critical gel model is that the elastic and viscous moduli remain parallel to each other over all frequencies, and the loss tangent Embedded Image is independent of frequency. By contrast, many experiments show broad power-law signatures over some frequency range, but ultimately a crossover at low enough frequencies to a limiting viscous-like material response. The existence of a characteristic relaxation time in the FMM enables such a material response to be described. The crossover frequency ωc at which Gs′(ω)=Gs′′(ω) for the FMM is found by equating equations (4.4) and (4.5), and we then find Embedded Image4.6Equation (4.6) makes it evident that the characteristic relaxation time scale in this model is Embedded Image, provided the argument in square brackets is positive. However, there is no crossover predicted by the model if 0<β<α<0.5 or if 0.5<β<α<1 (the total model response is then predominantly elastic or viscous, respectively, at all frequencies). For such materials, no clear characteristic time scale exists.

In figure 4, we show SAOS measurements of the interfacial viscoelasticity for 3 wt% Acacia gum solutions and 50 mg ml−1 BSA solutions. The black solid lines in figure 4a,c show the fit of the FMM for the elastic interfacial modulus Gs′(ω) (equation (4.4)) for the 3 wt% Acacia gum solutions and 50 mg ml−1 BSA solutions, respectively. The dashed lines show the fitted values of the interfacial loss modulus Gs′′(ω) (equation (4.5)). From these fits, the power-law exponents that characterize the Acacia gum solution are determined to be α=0.8±0.2,β=0.124±0.003, and the corresponding quasi-properties are Vs=3±2 Pa m s0.8,Gs=0.027±0.003 Pa m s0.124. (In the electronic supplementary material, we discuss the reason for the large confidence interval estimates for V). The material parameters of the 50 mg ml−1 BSA solution are α=0.80±0.07,β=0.11±0.02,Vs=0.048±0.008 Pa m s0.80 and Gs=0.017±0.001 Pa m s0.11. When the loss modulus is plotted against the storage modulus in a Cole–Cole representation, we do not observe the simple semicircular response expected from a linear Maxwell material, but instead power-law materials produce Cole–Cole plots with more complicated elliptical shapes [35]. It can be seen from the figures that the FMM captures the frequency dependence of the interfacial material functions accurately. On the other hand, the single-mode linear Maxwell model (indicated by broken lines in figure 4b,d) is unable to capture the power-law behaviour of these viscoelastic interfaces.

Figure 4.

The FMM fitted (lines) to interfacial storage Gs′(ω) and loss Gs′′(ω) moduli data (symbols) obtained from (a,b) 3 wt% Acacia gum solutions and (c,d) 50 mg ml−1 BSA solutions. The FMM fits are given by equations (4.4) and (4.5), respectively. Cole–Cole plot of (b) the same Acacia gum solution and (d) the same BSA solution showing the fractional Maxwell fit as a solid line with a linear Maxwell fit shown for comparison by the dashed line.

It is possible to estimate the crossover point and hence the relaxation time of the viscoelastic interface from the FMM fit. Calculating the value of ωc using equation (4.6), we find that for the Acacia gum solution, ωc=7×10−4 rad s−1, corresponding to a characteristic time constant of tc≈1430 s. As we have noted previously, it is challenging to measure linear viscoelastic properties at such low frequencies and at room temperature due to the long times it takes for test completion, which can result in solvent evaporation. In the case of the BSA solutions, the interfacial relaxation time is shorter, and the crossover point can be measured directly using the DWR fixture, giving ωc=0.16  rad s−1 (tc≈6.4 s). This crossover to a viscously dominated response is also captured accurately by the FMM. Acacia gum clearly produces a predominantly elastic interface with a very long relaxation time.

The values of the interfacial quasi-properties of the Acacia gum and BSA solutions we have found here fully characterize the linear viscoelastic interfacial properties of the two solutions, and these parameters may now be used to predict the response of these rheologically complex materials to other modes of excitation. In §4d, we discuss the transient response of the materials in creep experiments when inertial effects in the flow cannot be neglected.

(d) Creep ringing and power-law responses

In stress-controlled bulk rheometry, the effects of inertia can be coupled with material elasticity, which leads to damped periodic oscillations in a step-stress experiment at early times [59,60]. We have shown in a previous study that this inertio-elastic phenomenon can be observed not just in the bulk, but at interfaces as well [6]. These periodic oscillations decay exponentially with time due to viscous dissipation, and this phenomenon is often termed creep ringing. Although the presence of these oscillations is generally regarded as an intrusion, these transients can, in fact, be exploited to extract useful information about the linear viscoelasticity of soft materials [59,60]. In previous work, using BSA solutions exhibiting interfacial viscoelasticity [6], we have shown that this technique of extracting interfacial properties even presents certain advantages over the conventional technique of conducting frequency sweep measurements to high frequencies. In this earlier study, we also noted that solutions of BSA exhibit a power-law creep response at long times, which could not be adequately captured with the linear Maxwell–Jeffreys model that was considered analytically. In the electronic supplementary material, we show a creep experiment performed on 50 mg ml−1 BSA solutions with significant inertial effects, as well as the best fit prediction of the Maxwell–Jeffreys model [3] with an added inertial mass. It is evident from the figure that linear models such as this are incapable of capturing the full viscoelastic response of the material. In the current work, we extend the creep-ringing analysis to fractional viscoelastic constitutive models for the interface; we aim to predict the power-law creep behaviour over the entire time range of the experiment using the power-law exponents and quasi-properties of the materials determined previously in frequency sweep experiments (figure 4).

In figure 5, we show measurements of the interfacial creep compliance Js(t) (units: (Pa−1 m−1)) of 3 wt% Acacia gum solutions for different values of the imposed interfacial stress σ0s. We observe that the interfacial compliance Embedded Image measured at different stresses collapses onto a single curve, indicating the measurements are in the linear viscoelastic regime. The inset plot shows the creep compliance response on logarithmic axes, which exhibits a power-law scaling in time with Js(t)∼t0.13, instead of the slope of unity or zero expected from, respectively, a purely viscous or purely elastic material response.

Figure 5.

Creep compliance for 3 wt% Acacia gum solutions performed at various values of imposed interfacial stress σ0s. All experiments collapse onto a single curve as expected for a linear viscoelastic response. The interfacial viscoelasticity is coupled with instrument inertia, giving rise to creep ringing at early times. The inset plot shows that at long times, the creep compliance exhibits power-law behaviour with Js(t)∼t0.13. (Online version in colour.)

To overcome the poor predictions achieved from single-mode linear viscoelastic models, and without resorting to the ad hoc introduction of a large number of superposed relaxation modes, we instead use the FMM (equation (2.9)) coupled with the inertia of the test fixture to describe both the ringing observed in the creep experiment at short times, as well as the power-law behaviour seen at long times. We begin with the equation of motion for the spindle of the stress-controlled rheometer [59,60], Embedded Image4.7where I is the total moment of inertia of the spindle of the rheometer and the attached test geometry (i.e. the DWR fixture), σs(t) is the retarding interfacial stress applied by the sample on the spindle and γ(t) is the resulting strain. The factor bs=Fγ/Fσ (units: (m2)) is a geometric factor determined by the specific instrument and geometry used. The quantities Embedded Image (dimensionless) and Fσ=σs/T (units: (m−2)) convert the measured quantities of torque T and angular velocity Ω into the rheologically relevant quantities of interfacial stress σs and strain rate Embedded Image, respectively. Equation (4.7) can now be coupled with equation (2.5) to yield the fractional differential equation Embedded Image4.8where we introduce the parameter A=I/bs for compactness. In this equation, we have used the composition rule for fractional derivatives, which states that (dq/dtq)(dpf/dtp)=(dp+qf/dtp+q), provided f(k)(0)=0, where k=0,1,…,m−1; m<p<m+1 [32]. The fractional differential equation (4.8) is of order 2+αβ, and Heymans & Podlubny [61] have shown that a fractional differential equation of arbitrary real order k requires k* initial conditions, where k* is the lowest integer greater than k. Because we have 0≤βα≤1, we find that we need three initial conditions. The spindle is initially at rest and hence Embedded Image. However, the step in stress causes an instantaneous acceleration and the third initial condition is σs(0)=0, which is equivalent to Embedded Image from equation (4.7).

Before we solve equation (4.3) for γ(t), we first seek to determine its asymptotic behaviour in the limits of early times and long times. Evaluating the Laplace transform of equation (4.8) using equation (2.3) and using the three initial conditions given earlier, we find that Embedded Image4.9

It may be shown (see the electronic supplementary material) that at short times, equation (4.9) yields Embedded Image4.10This quadratic response is independent of the fractional orders of the spring-pots α and β, as expected, because the short-time response in the equation of motion (4.7) is dictated solely by the inertial response of the fixture; at very early times, the interface has not had time to build up any stress and hence σs(t)≈0. The solution of equation (4.7) under the condition σs(t)=0 yields the quadratic expression in equation (4.10). Similarly, at long times, we obtain (see the electronic supplementary material for details) Embedded Image4.11which is, to the leading order, the same as the inertia-free creep response derived in equation (2.8). This means that the effects of inertia become unimportant at long times, as observed in the experimental measurements shown in figure 5.

The value of A=I/bs can be calibrated once the rheometer fixture is selected and, in our case, was found to be A=1.72×10−4 kg. Figure 6a shows the asymptotic short-time response (line) given by equation (4.10) plotted against the measured interfacial creep compliance of a 3 wt% Acacia gum solution (filled symbols). It can be seen that the short-time asymptotic response agrees very well with the measured data. The inset plot also shows the value of the long-time asymptote derived in equation (4.11). From the Cole–Cole fits for the FMM shown in figure 4, the fit values that characterize the Acacia gum solutions are found to be α=0.8±0.2,β=0.124±0.003,Vs=3±2 Pa s0.8 and Gs=0.027±0.003 Pa s0.124. Because tβ/Gs≈6(tα/Vs) at t=60 s, we find that the first term in equation (4.11) is smaller than the second. Therefore, to a first approximation, at long times γ(t)≈(σ0s/Gs)(tβ/Γ(1+β)). Calculating the value of the coefficient 1/GsΓ(1+β), we find it equals 39.3 Pa−1 m−1 s−0.124. When we fit a power law of the form γ(t)=atb directly to the measured data, where a and b are fitting constants, we find that the measured data at long times is described by Js(t)≈40.4t0.130 Pa−1 m−1, which is in excellent agreement with the analytically derived asymptotic predictions for long times. This asymptotic power-law creep behaviour, shown as the solid line in the inset plot in figure 6a, cannot be conveniently captured using conventional spring–dashpot models.

Figure 6.

(a) Experimentally measured values of the interfacial compliance response for 3 wt% Acacia gum solutions (symbols), and the short- and long-time asymptote in the FMM coupled with instrument inertia. At early times, we retrieve the expected quadratic response of Embedded Image which is in accordance with the equation of motion at very early times. At long times, the effect of inertia only appears as a higher order correction (equation (4.11)). (b) The predicted interfacial creep compliance from solving equation (4.8) numerically using the exponents and quasi-properties found from the SAOS experiments that characterize the Acacia gum solutions. The prediction made by the model is in excellent agreement with the measured data, and it captures both the creep ringing at early times as well as the power-law behaviour observed at long times. (Online version in colour.)

We now proceed to predict the interfacial creep response of the Acacia gum solutions on the basis of the FMM fit parameters and quasi-properties found in §4c. To this end, we solve equation (4.8) for the strain γ(t) with the values of Vs,Gs,α and β determined from the fits of the FMM to the SAOS data. Equation (4.8) is amenable to an analytical solution and can be found by calculating the inverse Laplace transform of equation (4.9), in terms of the Mittag–Leffler function defined in equation (2.7). However, the resulting expression is cumbersome to evaluate because it contains a double infinite sum. Instead, we circumvent this difficulty by solving equation (4.8) numerically using the procedure outlined by Podlubny et al. [62] and a modified version of a Matlab code freely available from the same group. We refer the reader to the paper by Podlubny et al. [62] for details of the numerical scheme used.

The resulting numerical solution of equation (4.8) obtained using the quasi-properties found from SAOS is plotted in figure 6b as a solid line overlaid onto the experimentally measured compliance data. It is observed that the prediction of Js(t) based on the previously fitted quasi-property values is in very good agreement with the measured temporal response over the entire range of the creep experiment, indicating that the quasi-properties of the FMM characterize the rheological response of the material over a wide range of time scales. This fractional constitutive model can predict the material response to other excitations once the quasi-properties have been found from SAOS fits. This would not be possible using empirical laws such as the KWW expression, or the critical gel equation, even though these laws are able to capture power-law behaviour. It is noteworthy that the FMM contains only two additional parameters (α,β) beyond a simple Kelvin or Maxwell response and yet enables excellent predictions, accounting for the damped inertio-elastic effects at short times as well as the long-time power-law response.

5. Conclusions

We have revisited the concept of quasi-properties for describing the rheology of complex microstructured materials and interfaces, and demonstrated how their inclusion in fractional constitutive models containing spring-pot mechanical elements leads to the natural and quantitative description—using only a few constitutive parameters—of power-law behaviour frequently observed experimentally. Not only is this fractional constitutive approach more compact than the traditional approach of using a multi-mode Prony series, it is also more physical; in the latter approach, the number of fitted parameters, as well as their magnitudes, depend on the time scale of the experiment used for model fitting.

In the spring-pot constitutive equation, the elastic modulus, G′(ω), and the loss modulus, G′′(ω), increase as a function of frequency while maintaining a constant ratio between them. This is reminiscent of the behaviour observed in critical gels and soft glassy materials [13]. In fact, it can be shown that the soft glassy rheology (SGR) model under certain conditions yields exactly the same constitutive relationship as a single spring-pot defined in the Caputo sense, and the ‘effective noise temperature’ x in the SGR model is intimately related to the fractional exponent α (or β). Both these aspects are discussed in the electronic supplementary material.

Not only can fractional models accurately model the complex relaxation behaviour exhibited by bulk materials (as demonstrated here using Scott-Blair's [19] original data on ‘highly anomalous’ butyl rubber), they can also be extended to describe complex viscoelastic interfaces as well. Using SAOS experiments, we measured the power-law linear viscoelastic behaviour exhibited by interfaces formed from adsorbed films of BSA and Acacia gum. By fitting the data to the FMM, we could extract the quasi-properties Vs, Gs and exponents α,β that characterize these rheologically complex interfaces. We then considered the transient flow generated by an interfacial creep experiment in which inertial contributions are significant. We were able to predict a priori the inertio-elastic creep ringing observed at short times as well as the long-time power-law response using the values of the quasi-properties determined previously. There is excellent agreement between the model predictions and the experimental data across a wide range of time scales. These measurements demonstrate that once the quasi-properties of a material have been determined from one particular excitation, they characterize this rheologically complex interface and help determine the material response to other modes of deformation.

Finally, we note that all of the models presented here describe the linear viscoelastic limit and cannot describe nonlinear viscoelastic behaviour (for example, the onset of shear thinning or strain softening) exhibited by many complex fluids and interfaces at large strains. Extending the capability of fractional constitutive models into the nonlinear regime remains an open research problem for future investigation.

Acknowledgements

We are grateful to NASA Microgravity Fluid Sciences (Code UG) for supporting this research under grant no. NNX09AV99G. We also thank Prof. Pamela Cook, Dr Adam Burbridge and Dr Vivek Sharma for useful discussions on fractional calculus and interfacial rheology.

  • Received May 11, 2012.
  • Accepted September 26, 2012.

References

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