Basic principle and good practices of rheology for polymers for teachers and beginners

Introduction

Rheology is the study of the way in which materials respond (deform or flow) to applied forces or stresses. In 550–480 BC, a Greek philosopher and scholar, Heraclitus of Ephesus once said ‘everything flows’. The flow of a material can be defined as continuous deformation over a certain period of time and not only displayed by gases and liquids, but also solids. Rheology has various applications for example in materials science (Markov et al., 2019), engineering (Rathinam et al., 2018), pharmaceutical (Ahmadi et al., 2020), cosmetics (Tafuro et al., 2021), foods (Laguna et al., 2021), etc. Rheological studies and the characterization of viscoelastic properties of materials are of great importance for the stated applications. The flow behaviour of materials is one of the important quality control tools to maintain the quality and consistency of the materials. For example, in food industry, rheology provides the information of the consistency of foods (i.e., how it flows in liquid state), the stability of food storage in a certain period of time, as well as the foods’ texture (experience of biting, chewing and swallowing). Another example of rheology application is in polymeric paints. Rheology is important to provide the information on the application properties such as brushing, spraying and dipping. On application, the paints’ viscosity must be low as low as possible to ensure homogeneous spreading and at the same time fast recovery in avoiding dripping and sagging even on vertical supports.

Numerous books and articles on polymer rheology are available (Estanislao, 2013; Ferry, 1980; Kumaran, 2010; Macosko, 1994; Menard & Menard, 2020). However, they may be too difficult for teachers or course instructors to start with. Consequently, the objective of this article is to serve as an alternative reference for readers which is rather easy to follow and use it for educational purposes.

Hence, in the following discussion, some fundamentals about polymer rheology, the experimental methods using parallel-plate oscillatory rheometer, and step-by-step guides for the estimation of the power law dependence of storage and loss modulus as well as the relaxation time at the crossing frequency of both moduli. The exercises for both estimations are also provided. This polymer rheology text is relevant to courses of chemistry, physics, and others depending on how the topic is being discussed. In chemistry context, the chemical nature (i.e., intermolecular forces) of the polymer is controlling the flow behaviour when subjected to a force. In physics, the discussion is based on interrelationships between stresses, strains, their rates and the explanation on the deformation and flow response of a polymer when subjected to applied forces or stresses. Furthermore, in some extensive subjects, this topic can be also integrated in the industrial application topic where the discussion of correct and suitable processing parameter of polymers while taking into account the economical point of view can be conducted.

Basic principles

Rheology is the study of the flow and deformation of matter (gases, liquid- and solid-like matter) under the influence of a mechanical force (continuum mechanics). Rheological behaviour of materials can be divided into three categories; a purely viscous material, where all energy added is dissipated into heat, an elastic material, where all energy added is stored in the material and a viscoelastic material that exhibits viscous as well as elastic behaviour in response to force, deformation and time. Viscoelastic materials often cannot be described by linear models of hydrodynamics and elasticity, respectively. Some of these deviations are due to the presence of colloidal particles, the influence of surfaces and structural features like temporary aggregations, orientations or certain states of order. Modern rheometer test modes commonly use rotation, shear, torque, extension and compression in continuous or oscillatory (dynamic) mode.

A common method to characterize the viscoelastic properties of soft matter such as putty, viscose oils, gels and soft rubbers, yoghurt, cheese, slurries, etc. The deformation of a matter, for example, can be described using a two-plate model of shear rheometry (c.f. Figure 1). Shear is in this case applied to a sample sandwiched between the two plates where the lower plate act as the stationary plate that will not move around, while the upper plate is the moving plate that will move with the presence of shear force, F. The external force takes the form of a shear stress (σ) which is defined as the force (F) acting over a unit area (A) as shown in Eq. (1).

Figure 1:

An illustration of an object under simple shear stress and give rise to shear strain using a two-plate model.

In response to this force, the upper layer will move to a given distance Δx, while the bottom layer remains constant and parallel with the given height, h, called the gap. This geometry is called parallel plate or plate-plate arrangement. The displacement gradient across the sample, shear strain (γ) can be determined using Eq. (2) as shown below.

For a solid material, the strain will be limited for an applied stress where flow behaviour is impossible. Contrarily, the shear strain will continue to increase for the period of applied stress for a liquid material where the constituent components can move relative to one another and creates the shear rate, γ ˙ which is the rate of change of shear strain with time (c.f. Eq. (3)).

Flow behaviour

Figure 2 shows a plot of shear stress and viscosity against shear rate for various fluid types. Newtonian fluids are ideally viscous that obey the Newton’s law of viscosity, which means the shear stress is linearly related to the shear rate, and hence the viscosity is independent with shear rate or shear stress. Non-Newtonian fluids show non-linear dependency of the viscosity to the applied shear rate or shear stress.

Figure 2:

The various types of fluids with (a) shear stress as a function of shear rate and (b) viscosity versus shear rate.

Shear thinning or pseudoplastic flow shows the most common type of non-Newtonian behaviour, where the fluid viscosity decreases with increasing shear rate. Typical materials showing this behaviour are coatings, shower gel, polymer melts, paints, ketchup and glues. Shear thinning fluids show a constant viscosity value or zero shear viscosity plateau, η° at low shear rates where the state is at rest. However, at a critical shear rate or shear stress, one could observe a significant decrease in viscosity, which implies the beginning of the shear thinning region as illustrated in Figure 2(b). As an example, the observed shear thinning of polymer melts and solutions is caused by disentanglement of polymer chains during flow. Polymers with a sufficiently high molar mass are always entangled and randomly oriented when at rest. When a sufficient shear is applied, they begin to disentangle which causes the viscosity to drop. If the shear thinning is time-dependent, it is called thixotropic.

Meanwhile, shear thickening behaviour occurs where viscosity increases with increasing shear rate or shear stress. This phenomenon is called dilatancy. If the shear thickening is time-dependent, it is called rheopectic. Typically, dispersions or particulate suspensions with high concentration of solid particles exhibit shear thickening such as starch dispersions, ceramic suspensions and wet sand.

Thixotropic and rheopectic fluids are non-Newtonian fluids and these are considered as rare fluids. The key difference between both fluids is that in thixotropic fluids, the viscosity of the fluid decreases with stress over time whereas, in rheopectic fluids, the viscosity of the fluid increases with stress over time.

Some materials require a specific threshold (finite yield stress) of shear stress before they start to flow where a shear rate can only be induced when the yield stress has been exceeded. A Bingham plastic is one that has a yield stress but shows Newtonian behaviour after yielding. This idealized behaviour is rarely seen and most materials with an apparent yield stress show non-Newtonian behaviour after yielding which is generalized as plastic behaviour. Several examples of Bingham plastics are clay suspensions, drilling mud, toothpaste and mustard.

The physical meaning of storage and loss moduli

The viscoelastic properties that result from the microscopic relaxation processes in a material can be described over a range from ideal energy elastic of hard (but not brittle) material like steel, where there is no practically dissipation of energy due to irreversible deformation and heat, over visco-plastic behaviour where an irreversible deformation of a soft specimen occurs (e.g., putty), to the complete loss of shape of a liquid drop and spreading by viscous flow on a surface (spray neglected) and all the potential energy is converted to heat. The viscoelastic response of polymers lies between the extremes of complete recovery of the potential energy and complete conversion of the potential energy to heat. The physical meaning of the storage modulus, G ' and the loss modulus, G″ is visualized in Figures 3 and 4. The specimen deforms reversibly and rebounces so that a significant of energy is recovered (G′), while the other fraction is dissipated as heat (G″) and cannot be used for reversible work, as shown in Figure 4.

Figure 3:

Response of different types of matter on a deformation. Ideal elastic response, irreversible plastic deformation, viscous flow and viscoelastic response.

Figure 4:

Visualization of the meaning of the storage modulus and loss modulus. The loss energy is dissipated as heat and can be measured as a temperature increase of a bouncing rubber ball.

Viscoelastic behaviour

Polymers typically show both, viscous and elastic properties and behave as viscoelastic behaviour. Viscous behaviour can be represented by a dashpot (Newton’s Law) and elastic behaviour by a spring (Hooke’s Law), so that a viscoelastic material can be modelled by an appropriate combination of dashpot(s) and springs. There are two basic combinations of the viscoelastic behaviour: the Maxwell element and the Voigt-Kelvin element, as shown in Figure 5. In this Figure, the downward arrows indicate application of the load (stress applied) while the upward arrows are the removal of the load (stress removed).

Figure 5:

Hooke, Newton, Maxwell and Voigt-Kelvin elements with the corresponding creep behaviour at constant stress. In a Maxwell element, the fluid can completely ‘flow away’, while in a Voigt-Kelvin element this is limited to the maximal extension of the spring.

The deformation for Newtonian fluids is irreversible and time dependent. For a dashpot one obtains for the deformation rate as

where σ and η is shear stress and dynamic viscosity, respectively, and γ ˙ is the shear rate as described previously in Eq. (3). Meanwhile, for a spring, the shear stress can be described as

where | G ∗ | is the complex modulus that relates to the stiffness and γ is the shear strain which is dimensionless. It is inconvenient to associate Hooke’s Law for a spring with the shear modulus, G (modulus of rigidity) and the shear (angle) where this is used for simple shear experiments. A spring, however, correlates the stress, σ with the elongation (engineering strain), ε and the Young’s modulus, E (modulus of elasticity) in a simple stress-strain experiment. The correlation of the moduli is G ≅ 3E if the Poisson ratio is 0.5 (at constant volume on either extension or compression of the material).

In case of a simple extensional deformation the shear modulus G of a simple shear experiment (c.f. Figure 1, which also describes a simple shear experiment for a solid, where the force is applied to the upper surface while the lower surface is fixed) in Eq. (5) is replaced by the Young’s modulus, E and the shear is replaced by the extension strain, ε with ϵ = ( L − L 0 ) / L 0 where L ₀ is the initial length and L the length of the extended sample.

The combination of dashpot and spring in series is Maxwell-element with the relationship as shown in Eq. (6) below.

In a Maxwell element, when external load is applied and being removed, part of the deformation energy is used to get back to the original state (elastic response) and governed by G while the other part has longer times (viscous response) governed by η.

If a stress/load is applied to a Voigt-Kelvin model (a dashpot parallel to a spring), the shear strain, γ takes time to develop since the presence of the dashpot retards the response of the spring and the system behaves like a viscous liquid initially and then elastically over longer time scales, as the spring becomes more stretched. The timescale or rate at which this transition occurs depends on the relaxation time, τ which is given by η G which described as

A simple spring behaves ‘hard elastic’. A dashpot is ‘viscous’ and shows flow. A Maxwell model reacts hard-elastic with flow increasing with time while a Voigt-Kelvin model reacts with a retarded elasticity, and the extend of flow is limited by the extensibility of the spring. Combinations of Maxwell- and Voigt-Kelvin-elements are suited to describe the behaviour of viscoelastic materials, e.g., by the following 4-element model, as described in Figure 6. An experiment where a constant stress is applied to a viscoelastic material and the strain response measured is called creep test, which used to describe the slow steady flow of a material. The test involves applying a constant shear stress over a period of time and measuring the resultant shear strain and the test must be performed in the linear viscoelastic region at constant temperature. A typical creep and recovery profile for a material of 4-element model is shown in Figure 6. After the stress has relaxed (after time, t ₁), there is only a partial recovery that is controlled by the retardation and the corresponding creep.

Figure 6:

4-element model consisting of a Maxwell and a Voigt-Kelvin-element in series with the creep and recovery profile diagram.

The creep-function of the 4-element model in Figure 6 is then given by:

Small amplitude oscillation

Small amplitude oscillatory shear (SAOS) measurement is the most common technique to investigate the viscoelastic behaviour of a material. Again, the two-plate model is used to explain the oscillatory measurement. In this measurement, the sample is sheared (oscillated) between two parallel plates, where the upper plate is oscillated and the lower plate remains stationary. The upper plate oscillated clockwise and counter clockwise at a given stress or strain amplitude and angular frequency, as shown in Figure 7. Oscillatory motion involves circular motion, thus a full oscillation cycle is equivalent to 360° or 2π radian. The oscillation amplitude is the maximum stress (or strain) applied, and the angular frequency is the number of oscillations per second.

Figure 7:

Illustration of a sample placed between the parallel plates with an oscillatory torsional shear applied. The shear rate can be varied in a broad range by varying the angular frequency and the gap width. The shear rate is not constant along the radius.

In the two-plate model, both plates are equipped with sensors. The first sensor detects the strain or deformation, γ of the upper plate, called the angle of torque. The signal is represented by the strain versus time plot in a sinae curve with the strain amplitude γ _A, shown in Figure 8. A sinae curve is presented by its amplitude (maximum deflection) and the oscillation period (c.f. Figure 8). The oscillation frequency is the reciprocal of the oscillation period.

Figure 8:

Illustration of the two-plate model in oscillatory test. The upper plates deflect at the angle of 0° and 90°.

The second sensor detects the torque that acts upon the lower plate. This force is the counter force to keep the lower plate in position and recorded as shear stress, σ. When a sample is strained by a small torsional angle, a sinusoidal curve of the shear stress with the amplitude is obtained. Two sinae curves which corresponds to the preset and the samples’ response oscillate at the same frequency. The sinae curves will present different behaviour depending on the mechanical behaviour of the sample as shown in Figure 9. For a pure elastic material, for example steel, the maximum stress occurs at the maximum strain and both stress and strain are in phase, δ = 0° [c.f. Figure 9(a)]. Whereas for a pure viscous material for example water, the maximum stress occurs when the strain rate is maximum and the stress and strain are said to be out of phase by 90° [c.f. Figure 9(b)]. On the other hand, most samples show viscoelastic behaviour where the phase difference between stress and strain occur between the two extremes, i.e., 0° < δ < 90°, as shown in Figure 9(c). The phase angle shall not exceed 90° as the upper plate would turn in the opposite direction once the rotation angle exceeds 90°.

Figure 9:

The sine curve of shear strain and shear stress for (a) pure elastic, (b) pure viscous and (c) viscoelastic materials.

The ratio of the applied strain (or stress) to the measured stress (or strain) yields the complex modulus, G* which describes the resistance of the sample to deform (viscoelastic behaviour). G* is written as shown in Eq. (9).

The phase difference explained previously which describes the elastic and viscous components is used to determine G*. A vector diagram can be used to determine the two components as shown in Figure 10.

Figure 10:

Illustration of the relationship between complex shear modulus, G*, storage modulus, G′ and loss modulus, iG″ in a Gaussian vector diagram.

Using trigonometry, the elastic and viscous components in G* can be described in G′ and G″ terms, respectively in Eq. (10).

The G′ represents the elastic component of the viscoelastic behaviour, also known as the real part of G*. On the other hand, the G″ represents the viscous component, also known as imaginary part of G*. The description of G′ and G″ is provided in previous section (The physical meaning of storage and loss modulus). For a viscoelastic solid, for example hand cream, the storage modulus is higher than loss modulus (G′ > G″). Conversely, for viscoelastic liquid, for example honey, the loss modulus is higher than the storage modulus (G″ > G′).

Phase angle, δ is also expressed as the loss tangent, defined as tan δ = G ′′ G ′ . For a pure elastic material (δ = 0°), the viscous component is not present, hence tanδ = 0. Conversely, for a pure viscous material (δ = 90°), there is no elastic component, thus tanδ approaches infinity.

Oscillation frequency sweep

In an oscillatory test, the time-dependent behaviour of a sample can be measured by varying the frequency of the applied stress or strain in the non-destructive deformation range. High frequencies correspond to the short time scales, whereas the low frequencies correspond to longer time scales. This frequency sweeps test provides information on the behaviour and inner structure of materials. For example, comparison between the crosslinked and uncrosslinked polymers can be made in the frequency sweeps test. The uncrosslinked polymer exhibits G ′ ′ > G ′ at lower frequencies (liquid-like behaviour predominates), as well as a crossover point at G ′ = G ′ ′ . In contrast, the crosslinked polymer shows G ′ < G ′ ′ over the entire frequency range with no crossover point. The crossover point ( G ′ = G ′ ′ ) quantifies the balance between storage and loss modulus. This point is also called as gel point which represents the transition from liquid-like to solid-like behaviour during gelation process which can be determined at tanδ = 1. The temperature at gel point can be determined from the temperature dependent sweeps measurement where the tanδ is independent to the frequency.

The frequency sweeps measurement of a viscoelastic solid, viscoelastic liquid and gel-like materials is shown in Figure 11. For a viscoelastic solid, G′ is constant and dominates at low frequencies, whereas G″ increases with increasing frequency and dominates at high frequencies. This type of material displays a rigid consistency at rest, however, it is easily deformed when a sufficiently large force is applied. Examples of such materials are toothpaste and lotions. For viscoelastic liquid materials, G″ > G′ is observed at low frequencies (liquid-like behaviour predominates), while G′ < G″ (solid-like behaviour predominates) is displayed at high frequencies. The frequency of G′ and G″ cross, f  cross G ′ − G ′′ is equal to 1 τ , with τ is the relaxation time. This relaxation time is the process when stored elastic stresses are relaxed through rearrangement of the microstructure and converted to viscous stresses. Examples of viscoelastic liquids are shampoo and glues. For a gel-like material, G′ and G″ are parallel throughout the entire frequency range. Gel is defined as a non-fluid colloidal network or polymer network that is expanded throughout its whole volume by a fluid and has finite and usually rather small yield stress (Jones et al., 2009). A gel contain a covalent polymer network formed by crosslinking polymer chains or by non-linear polymerization, polymer network formed through the physical aggregation of polymer chains caused by hydrogen bonds, crystallization, helix formation, complexation, etc., lamellar structures including mesophases, e.g., soap gels, phospholipids and clays (Jones et al., 2009).

Figure 11:

Typical frequency response in an oscillatory at constant temperature for viscoelastic solid, viscoelastic liquid and gel materials.

Viscoelastic materials such as polymers are often discussed based on models (i.e., Maxwell or Voigt-Kelvin) to discuss their stress and strain interactions. The Maxwell model describes that stress decays exponentially with time (relaxation), which is relevant to most polymers. Whereas Voigt-Kelvin model is best in modelling creep in materials. In the following discussion, the relaxation behaviour of the polymer melts is discussed, thus the discussion will be based on the Maxwell model.

Based on Maxwell model, G′ and G″ vary with angular frequency, ω while the temperature is kept constant according to the following expressions;

where ω is angular frequency and τ is relaxation time. A typical Maxwell oscillatory response is shown in Figure 12. At the low frequency range, G″ is higher than G′ where a liquid-like behaviour predominates. As the frequency increases, G′ becomes higher and takes over, thus solid-like behaviour predominates. When G″ reaches a maximum, G′ is equal to G″ which marks the crossover frequency. At sufficiently low frequencies, the G′ and G″ in Eq. (12) are written as

Figure 12:

Modulus of G′ and G″ based on Maxwell model.

This means that, in a double logarithmic plot at the low-frequency region, the slopes of G′ and G″ versus ω curves can be expressed by power law exponents of 2 and 1, respectively (c.f. Figure 12). However, most materials for example polymers, do not display ideal behaviour of the Maxwell behaviour. The power law exponents of these materials may show deviations. Commonly, the power law exponents are estimated using commercial rheometer software, which serves as a “black box” to many users. In order to have better insight on estimation of the exponents, manual graphical approach or commercial graphical software is encouraged here as the first attempt for a beginner of the rheological studies. We note here, with good practices, the estimation of the exponents following manual graphical approach or commercial graphical software versus commercial rheometer software should yield comparable results as shown in Table 1. Further description on the step-by-step guidelines to estimate the power law exponents are provided in S1a, S1b and S1c supplementary files. The guides in S1a and S1b enable the users to understand the basic working principles for estimation of exponents when they rely on the commercial rheometer software in the subsequent attempt.

Normally, it is not practical for an analyst to make each rheology experiment, say, 10 times under identical conditions. If we did so, the “real” errors of the exponents accompanying rheological experiments out of the ensemble of 10 pairs of the exponents can be determined. Where there is experimental time constrain, the statistical errors of linear regression coefficients are estimated based on 2-tailed student t-test distribution with confidence level of 95%. We keep in mind that each regression calculation refers to just one rheological experiment. We suggest having at least 20 data points for each regression calculation.

The estimation and the error of the power law exponents for both G′ and G″ are crucial as it may provide insights on the morphologies of the systems. For example, the data could provide more information on the morphological behaviour in immiscible polymer blends as presented by (Castro et al., 2005). In this analysis, the power law exponents for PEO/poly(methyl methacrylate (PMMA)) blends were calculated and plotted as a function of PEO content. Next, linear regressions were done on both the G′ and G″ power law exponent values. The crossover points of the G′ and G″ linear regressions are suggested to be the critical compositions which corresponds to the limit of the co-continuous morphology. Hence, without the estimation of error in the estimation of the power law exponents, the region of co-continuous morphology could be wrongly interpreted.

Furthermore, the power law exponents of both G′ and G″ at the low-frequency region are often used to discuss the flow behaviour of polymers based on Maxwell behaviour. As discussed earlier, the exponents of both G′ and G″ based on Maxwell are 2 and 1, respectively. The power law exponents for an immiscible blend of PEO/NR-g-PMMA extracted from (Zainal et al., 2020) are listed in Table 2. As shown, the exponents deviate from the ideal Maxwell behaviour. Hence, the discussion should not be based on Maxwell model anymore. Furthermore, the exponents of NR-g-PMMA are far away from PEO which indicates both polymers display a completely different flow behaviour. Under the experimental condition, NR-g-PMMA is at the plateau zone, whereas PEO is at the terminal zone. This will be explained in detail in the next section as well as in the supplementary file S1a.

Viscoelastic spectrum

A typical viscoelastic spectrum of a polymer spanning from low to high frequencies is shown in Figure 13. However, in most cases, it is only possible to observe a fragment of this spectrum using a standard rheometer which depends on the sensitivity of the rheometer as well as the relaxation behaviour of the material. A widely used principle used to extend the relaxation spectrum is the time-temperature superposition (TTS) principle which uses the concept of time and temperature. Further explanation is described in section Time-temperature Superposition (TTS) Principle.

Figure 13:

Sketch of a typical oscillatory response of a viscoelastic material.

The typical viscoelastic spectrum in Figure 13 consists of terminal, plateau, transition and glassy zones. In the terminal zone, G″ > G′ is observed which indicates liquid-like behaviour. In other words, the sample exhibit flowing behaviour. For a Maxwell fluid, the G′ and G″ curves in the lower frequency range shows a slope of 2 and 1, respectively. Furthermore, the time-scale is associated to long-range relaxations in which the relaxation of the polymer chain happens faster than the deformation.

On the other hand, in the plateau zone, G′ > G″ is observed which indicates solid-like behaviour where the sample is not flowable. The time-scale at the plateau zone is associated to short-range relaxation where it involves the local segmental relaxation (Ferry & Myers, 2007). Hence, it can be said that the f  cross G ′ − G ′′ at terminal zone and plateau zone are referring to different relaxation process.

The relaxation time, τ at f  cross G ′ − G ′′ can be estimated using different approaches, i.e., commercial graphical software, manual graphical approach and commercial rheometer software. For example, τ at f  cross G ′ − G ′′ in the terminal zone estimated using these approaches showing comparable results (c.f. Table 3) suggesting one can use any of the approaches, provided that good practices are well employed. Further description on the step-by-step guidelines to estimate the relaxation time at f  cross G ′ − G ′′ are provided in S2a, S2b and S2c supplementary files.

Meanwhile, in the transition to the glassy zone, the response of the polymer chains is less dependence on the frequency applied. The high frequency applied is too high where the configurational rearrangements of the chains do not have enough time to take place. The glassy zone designates a glassy state, where the polymer chains have low mobility and thermodynamically a non-equilibrium state.

Experimental methods (parallel-plate oscillatory rheometer)

Measurement apparatus (plate and sample assembly)

The plate comprises of two concentric and circular parallel plates. The sample is placed between the two rigid, concentric and circular parallel plates. One of the plates is oscillating at a constant angular frequency, while the other plate remains stationary. The plates shall have a maximum arithmetical mean roughness value, R _a = 0.25 μm with no visible imperfections. The R _a value is defined as the absolute average relative to the base length which indicates the roughness of the plates’ surfaces. A diagram of the parallel-plate rheometer geometry is shown in Figure 14. The diameter of the plate, D, is typically in the range of 0.5–3.0 cm. The ratio of the plate diameter to the sample’s thickness is recommended to lie in the range of 10–50 to minimize error during measurement. The plates must be flat and the total variation due to non-parallelism of the plates must be less than ±0.01 mm. If the plates are not parallel, the measurements should be discarded. If the plates are not shiny, it indicates the plates’ surfaces are too rough. To avoid these conditions, proper maintenance of the instrument must be conducted regularly. Furthermore, one should also obey the advice of the instruction manual of the producer of the equipment and the supervisor. It is trivial that experiments should be carried out accurately.

Figure 14:

Parallel-plate rheometer geometry.

A rheometer measures the amplitudes of the torque, the angular displacement, and the phase difference between them of a sample subjected to either a sinusoidal torque or a sinusoidal displacement of constant angular frequency. A torque-measuring rheometer is connected to one of the plates and shall be able to measure the torque within ±2% of the minimum torque used in the measurement. Whereas, an angular-displacement measuring device is fitted to the moving plate and shall be able to measure the angular displacement within ±20 × 10⁻⁶ rad and angular frequency within ±2% of the absolute value.

Linear-viscoelastic (LVE) region

The results of amplitude sweeps are presented in a logarithmic plot of G′ and G″ against strain (or stress), as shown in Figure 15.

Figure 15:

The functions of G′ and G″ display constant values within the LVE region in an amplitude sweep test.

In determining the strain value, the G ' function curve is often preferred. In the LVE region, this function shows a constant value which indicates that the test carried out in this range will not destroy the structure of the sample. The linearity limit of the LVE region defines the LVE region and is calculated in terms of the strain as percentage, abbreviated as γ _L. The maximum value of the strain to be used in the subsequent test has to be less than the lowest value of the strain (or stress) in the LVE region with a tolerance of 5% deviation.

To confirm the measurements have been performed within the LVE region, the resultant output of displacement or torque must also be sinusoidal. If the resultant output is non-sinusoidal, it indicates non-linear behaviour and the analysis of the data are not valid and incorrect.

Temperature-dependent flow behaviour

In a temperature-dependent viscoelastic measurement of a polymer, shearing in oscillatory tests is performed under constant dynamic mechanical conditions. This means that both amplitude and frequency are kept constant. Principally, the tested polymer will go through a number of states depending on the chain length (distribution), molecular weight, degree of crosslinking, degree of cure, degree of crystallinity and thermal stability. The mechanical properties of a material are controlled by the molecular architecture and by the local mobility of the polymer chains and their constituents. Figure 16 shows the molecular origin (motions) of the macroscopic effects observed in the temperature-dependent measurement.

Figure 16:

Temperature-dependent functions of tanδ of a polymer which gives the thermal transitions and the corresponding molecular motions as the temperature increases.

Underlying the mechanical behaviour of amorphous polymers are conformational changes and diffusion processes. In particular the multitude of conformational changes and the corresponding rates can cover a broad temperature (or frequency) range. Short local (i.e., rotation or reorientation) sequences require less time to rearrange (like in liquids) as compared to long sequences, in particular when they occur in cooperative modes involving a number of neighbour segments. Relaxation processes that involve a polymer chain as a whole are in particular strongly depending on the molar mass and they occur over long distances. These processes form the long-time end of the relaxation spectrum.

As shown in Figure 16, the relaxation processes are not homogeneously distributed over the spectrum but accumulate in certain areas and can be traced back to certain processes. The transitions in the glassy state also known as sub-glass transition temperature, T _g are denoted as ε, δ, γ, β transition according to increasing temperature. Sub-T _g transitions are important for toughness effects in polymers because these transitions can dissipate energies that would otherwise cause mechanical failure. These transitions are also called relaxations since they describe the onset of the molecular motion (i.e., rotation and reorientation) and returning to the initial orientation (conformation) after disturbance as indicated in Figure 16. On the other hand, micro-Brownian motion refers to thermally induced motion of segments or parts of a polymer chain. In amorphous polymers the strongest process is called the α-process, which is identified as the glass transition and originates from the cooperative motion of neighbouring polymer chains and is a continuous transition leading from a solid-like, i.e., a glassy state to a liquid-like state, or vice versa. The glass transition is not a thermodynamic transition between two equilibrium states, it is a kinetic process and physical ageing (annealing) and enthalpy/volume relaxation can be observed near T _g. In semi-crystalline polymers the strongest process is frequently the melting of crystalline areas. As a rule of thumb, the T _g is at about 2/3 of the melting temperature, T _m (in K) – Tammann’s rule (Heinrich, 1934).

There is a direct relation of the viscoelastic properties of a polymer with its molecular motions, in particular cooperative motions. This is caused by the fact that each deformation of a polymer chain changes its equilibrium conformation, hence giving rise to an entropy-driven tendency to restore the initial equilibrium state. There are always four parts in the temperature-modulus curve of an amorphous polymer (c.f. Figure 17): the metastable glassy solid (frozen liquid) at low temperatures followed by the leathery-region (or glass-rubber-) transition, the rubber-elastic plateau, and finally the viscous flow (terminal flow range).

Figure 17:

Phases of an amorphous polymer determined from thermomechanical analysis.

Modulus and hence mechanical stability of an amorphous material decreases over decades at the T _g. The polymer becomes increasingly fluid and finally undergoes viscous flow if the polymer is not crosslinked. Around the glass transition (glass-rubber state), the material can show a behaviour that is called ‘leather-like’. It is not too soft, mechanically stable and not brittle – like leather (i.e., skim milk powders become sticky and caked when the temperature is around its glass transition temperature). In the leathery-region, the modulus is reduced by up to three orders of magnitude from the glassy modulus. The rubber plateau has a relatively stable modulus until the viscous flow is developed at higher temperature range.

In the case of semi-crystalline polymer (c.f. Figure 18), the material is still stable between the glass transition and the melting because the crystalline phases of the polymer act as crosslinking sites and keep the polymer in shape but soft, even rubber-like as long as the crosslinking areas are not destroyed by melting, after which the material is fluid and can be melt-processed. Examples of semicrystalline polymers are high-density polyethylene and commercial polypropylene.

Figure 18:

Phases of a semicrystalline polymer determined from thermomechanical analysis.

The thermomechanical materials properties depend strongly on the molecular architecture and can be influenced by physical or chemical measures, see Figure 19. Increasing (chemical) crosslink density increases the mechanical modulus and the glass transition becomes broader and shallow. The brittleness usually increases. Increasing the crystallinity of a sample has a similar effect. It can be controlled by addition of nucleating agents and the thermal history of the sample, the cooling program of the injection moulding process, for example. Increasing the molar mass of a material increases the probability of physical entanglements that act as reversible crosslinks, which can disentangle and creep depending on the chain length and side chains.

Figure 19:

Influence of crosslink density, crystallinity and molar mass on the temperature dependence of the storage modulus, which controls a major mechanical property.

Time-temperature superposition (TTS) principle

The relaxation behaviour can be monitored at a fixed temperature with a frequency scan or at a fixed frequency with a temperature sweep. In the first case resonance is observed when the applied frequency matches a corresponding frequency of the molecular motion at this temperature, in the second case resonance is observed when the energy provided by the applied temperature fits in with a molecular motion that matches with the chosen frequency. This reflects a time-temperature relation, known as time-temperature superposition principle (TTS), as shown in Figure 20.

Figure 20:

Superposition of the individual curves to a form master curve. Each colour represents data at different temperatures.

Despite its popularity, TTS has to be applied and interpreted with caution (Dealy & Wissbrun, 2012). The assumption that all relaxation times are affected by temperature to the same extend does not always hold. Processes like crosslinking, degradation, depolymerization, to name a few, are not necessarily affected in the same way over the whole temperature range under investigation. The same is true for multi-phase polymers. Also, not all mechanisms of change induced by temperature are not necessarily unaffected by a phase transition and there can be a significant effect of the temperature on the rate of processes. In general, superposition shows reliable results provided the spectrum of relaxation times is temperature independent and the thermal activation is unchanged over the whole time-temperature range. There are examples of polymer systems which show deviations from TTS (Bousmina et al., 2002; Chopra et al., 2002; Lin et al., 2020; Yokokoji et al., 2021).

In this TTS method, the data collected at different temperatures can be shifted relative to the loading frequency based on a chosen temperature, T _ref, and can be merged to a single master curve. The frequency shifts are based on a shifting factor, a _T. which described by the Williams-Landel-Ferry (WLF) equation (Williams et al., 1955), written in Eq. (13).

where C ₁ and C ₂ are the material constants which vary with the nature of the polymer (Williams et al., 1955). The so-called ‘universal’ constants are C 1 = 17.44 and C 2 = 51.6 , used for many polymers. T _o is a generalized transition temperature, frequently taken as T _g. Some examples of the material constants were listed by Aklonis and MacKnight (1983) shown in Table 4.

The (semi-empirical) WLF equation can be derived using the free volume theory, and a quantitative description is frequently possible in the melt in a temperature range from T _g to T _g + 100 K. The importance of a relation like the WLF equation becomes clear recalling the fact that the experimental techniques usually only cover a rather narrow time slot, e.g., 10⁰ s … 10⁵ s (corresponding to a frequency range). The TTS principle allows an estimate of the relaxation behaviour and related properties of polymers – such as the melt viscosity – over a wide temperature range (e.g., 10⁻¹⁴ h … 10² h) with the WLF-equation and the shift factor a _T .

In cases where there not all the relaxation processes in a polymer show the same temperature-dependence, the time-temperature superposition does not fit to form one continuous master curve but a “branching-off” is observed.

Conclusions

The basic principle and good practices of the rheology of polymers were presented here specially for teachers and beginners. This article covers the background of the rheology particularly for polymers (i.e., viscoelastic materials), basic consideration of the experimental methods using parallel-plate oscillatory rheometer, step-by-step guidelines to estimate the power law dependence of G′ and G″ on frequency and to estimate the relaxation time at f  cross G ′ − G ′′ at terminal zone (c.f. Figure 13) using commercial graphical software, manual graphical approach and commercial rheometer software. Several examples and exercises are also provided in the supplementary files for better understanding. This initiative can be served as the self-learning resources and suitable for teachers as well as course instructors for educational purposes on how to perform good measurements together with correct data acquisition using various approaches.

Supplementary information

Total of seven supplementary files which divided into two step-by-step guidelines on (S1) – Estimation of power law dependence of G′ and G″ on frequency at terminal zone and (S2) – Estimation of the relaxation time at f  cross G ′ − G ′′ at terminal zone are provided separately for educational, practical, and self-learning purposes. There are three approaches supplement here (i.e., (a) – using commercial graphical software, (b) – using manual graphical approach including the exercises (S3), (c) – using commercial rheometer software.