Concentrated polymeric solutions are used in a variety of industrial applications primarily to increase viscosity, including in foods¹ and other consumer products², enhanced oil recovery³, and soil remediation⁴. During their processing and use, they are necessarily subjected to large deformations over a range of timescales. Under such processes, they demonstrate rich and complex nonlinear rheological behaviors that depend on the flow or deformation conditions¹. Understanding these complex nonlinear rheological behaviors is essential for successfully controlling processes, designing superior products, and maximizing energy efficiency. Aside from the industrial importance, there is a great deal of academic interest in understanding the rheological behaviors of polymeric materials far from equilibrium.

Oscillatory shear tests are a staple component of every thorough rheological characterization because of the orthogonal application of strain and strain rate⁵, and the ability to independently control the length and time scales probed by tuning the amplitude and frequency. The stress response to small amplitude oscillatory shear strains, which are small enough not to disturb a material's internal structure, can be decomposed into components in phase with the strain and in phase with strain rate. The coefficients of the components in phase with the strain and the strain rate are collectively referred to as the dynamic moduli^6,7, and individually as the storage modulus, , and loss modulus, . The dynamic moduli lead to clear elastic and viscous interpretations. However, interpretations based on these dynamic moduli are valid only for small strain amplitudes, where the stress responses to sinusoidal excitations are also sinusoidal. This regime is generally referred to as the small amplitude oscillatory shear (SAOS), or the linear viscoelastic regime. As the imposed deformation becomes larger, changes are induced in the material microstructure, which are reflected in the complexity of the non-sinusoidal transient stress responses⁸. In this rheologically nonlinear regime, which more closely mimics industrial processing and consumer usage conditions, the dynamic moduli act as poor descriptions of the response. Another way to understand how concentrated soft materials behave out of equilibrium is therefore required.

A number of recent studies^{9,10,11,12,13,14,15,16} have shown that materials pass through diverse intra-cycle structural and dynamical changes elicited by larger deformations in the medium amplitude oscillatory shear (MAOS)^15,17 and large amplitude oscillatory shear (LAOS) regimes. The intra-cycle structural and dynamical changes have different manifestations, such as breakage of microstructure, structural anisotropy, local rearrangements, reformation, and changes in diffusivity. These intra-cycle physical changes in the nonlinear regime lead to the complex nonlinear stress responses that cannot be simply interpreted with the dynamic moduli. As an alternative, several approaches have been suggested for the interpretation of the nonlinear stress responses. Common examples of this are Fourier transform rheology (FT rheology)¹⁸, power series expansions¹¹, the Chebyshev description¹⁹, and the sequence of physical processes (SPP)^5,8,13,14,20 analysis. Although all of these techniques have been shown to be mathematically robust, it is still an unanswered question as to whether any of these techniques can provide clear and reasonable physical explanations of nonlinear oscillatory stress responses. It remains an outstanding challenge to provide concise interpretations of rheological data that correlate to structural and dynamical measures.

In a recent study, the nonlinear stress response of the Soft Glassy Rheology (SGR) model⁸ and a soft glass made of colloidal star polymers⁷under oscillatory shear was analyzed through the SPP scheme. Temporal changes in the elastic and viscous properties inherent in nonlinear stress responses were separately quantified by the SPP moduli, and . Furthermore, the rheological transition represented by the transient moduli was accurately correlated to microstructural changes represented by the distribution of mesoscopic elements. In the study of the SGR model⁸, it was clearly shown that rheological interpretation via the SPP scheme accurately reflects the physical changes under all oscillatory shear conditions in the linear and nonlinear regimes for soft glasses. This unique capability to provide accurate physical interpretation of nonlinear responses of soft glasses makes the SPP method an attractive approach for researchers studying out-of-equilibrium dynamics of polymer solutions and other soft materials.

The SPP scheme is built around viewing rheological behaviors as occurring in a three-dimensional space () that consists of the strain (), strain rate (), and stress ()⁵. In a mathematical sense, the stress responses are treated as multivariable functions of the strain and strain rate (). As the rheological behavior is regarded as a trajectory in (or a multivariable function), a tool for discussing the properties of a trajectory is required. In the SPP approach, the transient moduli and play such a role. The transient elastic modulus and viscous modulus are defined as partial derivatives of the stress with respect to the strain () and the strain rate (). Following the physical definition of differential elastic and viscous moduli, the transient moduli quantify the instantaneous influence of strain and strain rate on the stress response respectively, whereas other analysis methods cannot provide any information on elastic and viscous properties separately.

The SPP approach enriches the interpretation of the oscillatory shear tests. With the SPP analysis, the complex nonlinear rheological behaviors of concentrated polymeric solutions in LAOS can be directly related to the linear rheological behaviors in SAOS. We show in this work how the maximum transient elastic modulus (_max) near the strain extrema corresponds to the storage modulus in the linear regime (SAOS). Furthermore, we show how the transient viscous modulus () during a LAOS cycle traces the steady state flow curve. In addition to providing details of the complex sequence of processes that concentrated polymer solutions go through under LAOS, the SPP scheme also provides information regarding the recoverable strain in the material. This information, which is not obtainable through other approaches, is a useful measure of how much a material will recoil once stress is removed. Such behavior has impact on the printability of concentrated solutions for 3D printing applications, as well as screen printing, fiber formation, and flow cessation. A number of recent studies^5,8,13 clearly indicate that the recoverable strain is not necessarily the same as the strain imposed during LAOS experiments. For instance, a study of soft colloidal glasses under LAOS¹³ found that the recoverable strain is only 5% when significantly larger total strain (420%) is imposed. Other studies^{16,21,22,23,24} using the cage modulus²¹ also conclude that linear elasticity can be observed under LAOS at the point close to the strain maxima, implying that the materials experienced relatively small deformation at those instants. The SPP scheme is the only framework for understanding LAOS that accounts for a shift in the strain equilibrium that leads to a difference between the recoverable and the total strains.

This article aims to facilitate understandings and ease of use of the SPP analysis method by providing a detailed protocol for a LAOS analysis freeware, using two concentrated polymer solutions, a 4 wt% xanthan gum (XG) aqueous solution and a 5 wt% PEO in DMSO solution. These systems are chosen because of their broad range of application and rheologically interesting properties. Xanthan gum, a natural high-molecular-weight polysaccharide, is an exceptionally effective stabilizer for aqueous systems and commonly applied as a food additive to provide desired viscosification or in oil drilling to increase viscosity and yield points of drilling muds. PEO has a unique hydrophilic property and is often used in pharmaceutical products and controlled release systems as well as soil remediation activities. These polymeric systems are tested under various oscillatory shear conditions that are intended to approximate processing, transport, and end-use conditions. Although these practical conditions may not necessarily involve flow reversal as in oscillatory shear, the flow field can be easily approximated and tuned with the independent control of applied amplitude and imposed frequency in an oscillatory test. Furthermore, the SPP scheme can be used as described here to understand a broad range of flow types, including those that do not include flow reversals such as the recently-proposed UD-LAOS²⁵, in which large amplitude oscillations are applied in one direction only (leading to the moniker "uni-directional LAOS"). For simplicity, and for illustrative purposes, we restrict the current study to traditional LAOS, which does include periodic flow reversal. The measured rheological responses are analyzed with the SPP approach. We demonstrate how to use the SPP software with simple explanations on salient calculation steps to improve readers' understanding and usage. A legend for interpreting the SPP analysis results is introduced, according to which the type of rheological transition is identified. Representative SPP analysis results of the two polymers under various oscillatory shear conditions are displayed, in which we clearly identify a sequence of physical processes that contains information on the material's linear viscoelastic response as well as the steady-state flow properties of the material.

This protocol provides salient details of how to accurately perform nonlinear rheological experiments, as well as a step-by-step guide to analyzing and understanding rheological responses with the SPP framework, as shown in Figure 1. We begin by providing an introduction to the instrument setup and calibrations, followed by specific commands for making a commercially-available rheometer collect high-quality transient data. Once the rheological data have been obtained, we introduce the SPP analysis freeware, with a detailed manual. Further, we discuss how to understand the time-dependent response of the two concentrated polymer solutions within the SPP scheme, by comparing the results obtained from LAOS with the linear-regime frequency sweep and the steady-state flow curve. These results clearly identify that the polymer solutions transition between distinct rheological states within an oscillation, allowing for a more detailed picture of their nonlinear transient rheology to emerge. These data can be used to optimize processing conditions for product formation, transport, and use. These time-dependent responses further provide potential pathways to clearly form structure-property-processing relationships by coupling the rheology with microstructural information obtained from small-angle scattering of neutrons, X-rays, or light (SANS, SAXS, and SALS, respectively), microscopy, or detailed simulations.