Structure-property relationships via recovery rheology in viscoelastic materials

TL;DR

This study links recoverable strain to microstructure evolution in viscoelastic materials using SANS and rheology, revealing detailed structure-property relationships and stress behaviors under large amplitude oscillatory shear.

Contribution

It introduces a new understanding of how recoverable strain correlates with microstructural changes and stress evolution in viscoelastic materials during LAOS.

Findings

Recoverable strain correlates with microstructure evolution.

Stress responses are dictated by recoverable strain.

LAOS responses contain information from both steady-state and linear regimes.

Abstract

The recoverable strain is shown to correlate to the temporal evolution of microstructure via time-resolved small-angle neutron scattering (SANS) and dynamic shear rheology. Investigating two distinct polymeric materials of wormlike micelles and fibrin network, we demonstrate that, in addition to the nonlinear structure-property relationships, the shear and normal stress evolution is dictated by the recoverable strain. A distinct sequence of physical processes under large amplitude oscillatory shear (LAOS) is identified that clearly contains information regarding both the steady-state flow curve and the linear-regime frequency sweep, contrary to most interpretations that LAOS responses are either distinct from, or somehow intermediate between the two cases. This work provides a physically-motivated and straightforward path to further explore the structure-property relationships of viscoelastic materials under dynamic flow conditions.

Full text

A long-standing challenge in understanding the out-of-equilibrium behavior of soft matter is to link microstructural rearrangements and the macroscopic flow properties. We show, via time-resolved rheo-small-angle neutron scattering (rheo-SANS), that such a link can be made, forming nonlinear rheological structure-property relations, by considering the evolution of the recoverable component of the strain.

The simultaneous collection of macroscopic rheological information and particle- or molecular-level data has been used understand the interplay between material constituents that cover a wide range of length scales. Recent examples include the applications of scattering techniques in polymeric materials Förster et al. (2005); Lopez-Barron et al. (2012); Calabrese and Wagner (2018); Wang et al. (2017); Gurnon et al. (2014); Rogers et al. (2012); Leheny et al. (2015), confocal microscopy in colloidal systems Cheng et al. (2011); Ghosh et al. (2017); Amann et al. (2012); Mutch et al. (2013); Cheng et al. (2012); Sentjabrskaja et al. (2015); Vasisht et al. (2018); Besseling et al. (2007); Struth et al. (2015); Mutch et al. (2016); Smith et al. (2007) and simulation methods Radhakrishnan and Fielding (2016); Hattemer and Arya (2015); Hou et al. (2010); Leishangthem et al. (2017); Xu et al. (2018); Koumakis et al. (2013). Despite these efforts, rheological structure-property relations remain incompletely understood. Oscillatory shearing provides a reproducible way to probe viscoelastic behaviors and has been used to study a wide array of soft materials Cordier et al. (2008); Koos and Willenbacher (2011); Mangal et al. (2015); Kokkinis et al. (2015); Zhong et al. (2016); Boland et al. (2016); Gu et al. (2018). Many processing conditions and practical applications of soft materials can be approximated by large amplitude oscillatory shear (LAOS) because it offers independent control of the length and time scales of structural rearrangements. As such, LAOS has been widely adopted as a model transient flow protocol capable of eliciting nonlinear responses Leblanc (2008); Lee et al. (2010); Lettinga et al. (2012); Lopez-Barron et al. (2012); Koumakis et al. (2013); Münster et al. (2013); Hall et al. (2016); Radhakrishnan and Fielding (2016); Jaspers et al. (2017); Perazzo et al. (2017); Tapadia et al. (2006); Rogers et al. (2018).

Typical mathematically-based descriptions of experimental responses to LAOS are based on Fourier transformation (FT), which represents the complex sequence of processes exhibited by soft materials in the time domain as a sum of harmonic contributions in the frequency domain Wilhelm (2002), performing an averaging of sorts. Physical processes that take place sequentially over intervals of time shorter than a period of oscillation are therefore not easily discerned via such analysis methods. The lack of a generic understanding of the higher harmonics Hyun et al. (2011) has limited the widespread adoption of these methods in time-resolved molecular-level studies Min Kim et al. (2014); Lettinga et al. (2012); Rogers et al. (2012); Lopez-Barron et al. (2012); Eberle and Porcar (2012); Gurnon et al. (2014).

In this letter, we show that the recoverable strain, an often overlooked rheological metric that was first proposed by Weissenberg Weissenberg (1947) and Reiner Reiner (1958), provides an ideal basis for understanding the complex evolution of the microstructure and the shear and normal stresses of an industrially-relevant entangled solution of wormlike micelles (WLMs) and a biopolymer network of fibrin (see SI for material preparation). In discussing the experimental reality of recovery tests, Lodge noted that typical experiments are constrained in one dimension. Constrained recovery was incorporated into his transient network theory of polymers, which makes a prediction about the relationship between recoverable strain and the first normal stress difference Lodge (1958). Several studies Mooney (1951); Lodge (1958); Benbow and Howells (1961); Wagner and Laun (1978); Laun (1986); P Rehbinder (1954); Muenstedt et al. (2008) have adopted this concept to study polymeric and nanocomposite systems, mostly under steady shear or creep flow. The central variable in these studies Weissenberg (1947); Lodge (1958); Mooney (1951); Benbow and Howells (1961); Wagner and Laun (1978); Laun (1986); Reiner (1958) is the amount of deformation recovered after the removal of shear stress. Reiner Reiner (1958) elevated the importance of the recoverable deformation, calling it the strain, a definition that has not stuck. Despite significant early efforts, measurements of constrained recovery remain limited, and have not, until now, been applied to the study of structure-property relationships under oscillatory shearing.

The two polymeric materials we study have very different behaviors, which allows us to more clearly illustrate the benefits of our new approach. Worm-like micelles undergo many breakage and reformation events on the time scale of reptation, leading to a Poisson distribution of length scales at equilibrium, and a single relaxation time Cates and Candau (1990). They have therefore been used to study nonlinear flow behaviors of entangled linear and branched polymers Spenley et al. (1993); Inoue et al. (2005); Boukany and Wang (2008); Förster et al. (2005); In et al. (1999); Cardiel et al. (2013). Fibrin networks are known to possess flow properties that are distinct from common synthetic polymers Mackintosh et al. (1995); Storm et al. (2005); Onck et al. (2005); Münster et al. (2013) because they are primarily elastic and stiffen when strained, protecting tissue from large deformations. In contrast, synthetic polymer solutions often soften and shear-thin when strained.

We use in-situ rheo-SANS to simultaneously monitor the alignment of micellar segments as well as the recovery rheology. Measurements are made in the steady alternating state when all initial transience has decayed. An iterative constrained recovery procedure is employed at 200 distinct evenly-spaced instants along an oscillation (see Supplemental Material for detailed experimental protocol, which includes Ref. Baravian and Quemada (1998)). The unrecovered part of the strain, $\gamma_{un}$, is the strain the system ultimately recovers to, and the recoverable strain, $\gamma_{rec}$, is the part of the total strain that is elastically recovered. All rheo-SANS measurements are made with an Anton-Paar MCR-501 rheometer with a concentric-cylinder Couette geometry at the NCNR on beamline NGB-30. Normal stress differences and additional constrained recovery tests are measured using a DHR-3 rheometer (TA Instruments) with a 4-degree cone and plate geometry.

The alignment of micellar segments is measured in the velocity-gradient (1-2) and velocity-vorticity (1-3) planes using time-resolved rheo-SANS techniques Porcar et al. (2004); Eberle and Porcar (2012). A $q$-range of 0.006 1/$\AA$ to 0.03 1/$\AA$ is probed, corresponding to the rod-like scattering of the micelle segments (see SI). SANS data are reduced according to NCNR guidelines Kline (2006). A temporal deconvolution protocol Calabrese et al. (2016) is used to enhance the resolution and accuracy of the measurement of the underlying structural evolution.

Oscillatory shearing from WLMs with amplitudes ranging from the linear (SAOS) to the nonlinear (LAOS) regimes are performed at Deborah numbers ($De=\omega\lambda$, where $\omega$ is the angular frequency and $\lambda$ is the longest relaxation time) of 0.0625 ($\omega=$0.25rad/s) and 0.25 ($\omega=$1 rad/s), where the inertial effects of the stress-controlled procedure are negligible. The traditional elastic and viscous Lissajous curves of the WLMs are displayed in Fig. 1 (a and e) and (c and g). The distorted elastic Lissajous curve and the secondary loops of the viscous Lissajous curve indicate that the system undergoes a complex sequence of processes over the course of an oscillation. FT-based analysis schemes quantify the departure from linearity in an average sense, taking the response from an entire period of oscillation as the object to be analyzed. The resulting harmonics therefore have no clear physical relation to the structure at any particular instant. We contrast this with the underlying principle of time-resolved rheo-scattering techniques Eberle and Porcar (2012); Calabrese et al. (2016), which is that the structure and rheology at any one instant are causally related. A framework that correlates the rheology with real-time structural evolution is needed.

Strain can be decomposed into recoverable and unrecoverable components Weissenberg (1947); Reiner (1958). Recoverable strain is elastic, while viscous properties are dictated by the rate at which strain is acquired unrecoverably. Elastic and viscous Lissajous curves that reflect this view are therefore presentations of stress vs. recoverable strain ($\sigma-\gamma_{rec}$) and stress vs. unrecoverable strain rate ($\sigma-\dot{\gamma}_{un}$). The slopes of the $\sigma-\gamma_{rec}$ and $\sigma-\dot{\gamma}_{un}$ curves for Hookean solids and Newtonian fluids retain their clear meaning of being the modulus and viscosity, respectively. Plots of stress vs. recoverable strain for a Newtonian fluid will have undefined slopes because purely viscous fluids acquire no strain recoverably. Our proposal therefore removes the current possibility of defining a modulus for a purely Newtonian fluid or a viscosity for a purely Hookean solid.

The proposed elastic Lissajous curves, $\sigma-\gamma_{rec}$, from WLMs under LAOS are shown in Fig. 1(b and d). The new presentations show significant differences from the traditional plots that use the total strain. At the instants close to the total strain extremes (stars in Fig. 1 a and c), the recoverable strain is nearly zero, indicating that the material is closest to its equilibrium configuration and is therefore undeformed. Similar behavior has been observed in polymeric and colloidal systems Rogers et al. (2011); Kim et al. (2014); Lee and Rogers (2017); Rogers (2017); Ramya et al. (2019), where a constant linear-regime elasticity exists near the strain extrema under LAOS. At both investigated frequencies, the $\sigma-\gamma_{rec}$ curves show straight lines of equal slope at $\mid\gamma_{rec}\mid<\sim$0.5, which indicates an elastic modulus of 180 Pa. This value corresponds to the plateau modulus (indicated by lines in Fig. 1 b and d), $G_{0}=180$ Pa, which is typically determined at much higher frequencies ($De>>1$), indicating that the elastic modulus of the transient micellar network is constant across a wider range of lower frequencies than previously thought [Fig. S2(a)]. We contrast these observations with the storage modulus, which is frequently taken as a measure of elastic modulus, which has values of $G^{\prime}(De=0.0625)$= 0.62 Pa and $G^{\prime}(De=0.25)$= 10.3 Pa [Fig. S7].

The new viscous Lissajous curves from WLMs under LAOS, $\sigma-\dot{\gamma}_{un}$, are shown in Fig. 1 (f and h). Similar to the $\sigma-\gamma_{rec}$ curves, straight lines with constant slopes are obtained in the small and intermediate amplitudes. This slope is independent of frequency and equal to the zero-shear viscosity, $\eta_{0}$, determined from steady shearing (straight lines in Fig. 1). The viscosities determined during LAOS and steady shearing are favorably compared in Fig. 1 (i and j). Over the course of an oscillation at $De=0.0625$, the two protocols yield similar data. Even at the higher frequency ($De=0.25$), the viscosity determined from the LVE and thinning portions in LAOS is consistent with the flow viscosity on the upward sweep.

The determination of a constant elastic modulus and a constant viscosity in the linear regime across a broad range of frequencies requires reconciliation with the frequency dependence of the dynamic moduli, $G^{\prime}$ and $G^{\prime\prime}$. The storage and loss moduli are known to represent the average amount of energy stored and dissipated per unit volume over a cycle of oscillation Tschoegl (1989), while the elastic modulus presented here is the amount by which the stress increases with an increase in the recoverable strain, or the modulus in a force-extension perspective. Having determined the elastic modulus and the viscosity, and the amount of deformation that is recoverable and unrecoverable at each instant, the instantaneous energy storage and dissipation rate can be quantified at any time: $W_{s}(t)=\frac{1}{2}G(t)\gamma_{rec}^{2}(t)$ and $\dot{W}_{d}(t)=\eta(t)\dot{\gamma}_{un}^{2}(t)$. Averaging the instantaneous energy storage and dissipation over an oscillation results in metrics that are related to the dynamic moduli: $W_{s,avg}=(\gamma_{0}^{2}/4)G^{\prime}$ and $\dot{W}_{d,avg}=(\gamma_{0}^{2}/2)\omega G^{\prime\prime}$. We show that following an energetic analysis, one can transform from the elastic modulus and viscosity to the energetic parameters, $G^{\prime}$ and $G^{\prime\prime}$ (Figs. S7 and S8). The linear-regime dynamic moduli can therefore be obtained within a LAOS response if one focuses exclusively on the response to small recoverable strains (Fig. S7). We suggest on this basis that LAOS tests, rather than being treated as being fundamentally different from other tests Pipkin (1986); Ewoldt et al. (2008); Ewoldt and McKinley (2017), sequentially present information regarding the linear-regime relaxation spectrum and the steady-state flow curve, and how the material transitions between the two states.

At instants when the total strain is large (stars in Fig. 1), the recoverable strain and the unrecoverable strain rate are minimal, and the WLMs exhibit linear viscoelasticity characterized by $G_{0}$ and $\eta_{0}$. Measuring the evolution of the recoverable and unrecoverable strains establishes a clear sequence of processes during a period of oscillation: LVE behavior followed by softening/thinning and recoil, taking place twice per oscillation.

Unlike the significant differences between the total and recoverable strains in WLMs, the fibrin network shows nearly complete strain recovery (Fig. S3). The oscillatory shearing response of fibrin at an imposed frequency of 1 rad/s is presented in Fig. 2 (a). Across all deformation amplitudes, when $\mid\gamma_{rec}\mid<\sim$0.05, we observe straight lines with the same slope, indicating a single modulus that is approximately equal to $G^{\prime}$. This equivalence is because $\gamma_{rec}\approx\gamma$, which is not true for elastic liquids, such as the WLMs.

With increasing recoverable strain, the fibrin stiffens Münster et al. (2013). Immediately after the recoverable strain reversal, the fibrin recoils to its LVE configuration. Our sequencing of the fibrin responses under oscillatory shearing is in resonance with other studies that account for the unique mechanical properties as a sequence of structural hierarchy with respect to the (recoverable) strain Onck et al. (2005); Piechocka et al. (2010); Kang et al. (2009).

While many studies have focused exclusively on the shear stress response to LAOS, normal stress differences have received little attention Nam et al. (2010); Janmey et al. (2006); Seth et al. (2011). A full exposition of the extra stress tensor necessarily involves both normal and shear components. Janmey et al. Janmey et al. (2006) studied fibrin networks and saw negative normal stresses under shear. A micromechanical model has been proposed and used to study normal stress differences of soft glasses Seth et al. (2011). Despite these works, a clear link between normal stress differences and other components of the extra stress tensor under LAOS remains incompletely understood. In their study of silicone polymers, Benbow and Howells concluded that, “the observation of recoverable elastic strain may be taken as a necessary and sufficient condition for observable normal stress Benbow and Howells (1961).” Their conclusion, however, was drawn from steady, uni-directional shearing, and has not been examined under transient shear, nor linked to any structural measure.

The first normal stress difference, $N_{1}$, from WLMs is shown in Fig. 2 (c) and (e), where the raw response was filtered to remove noise (black lines). When plotted against the total strain, typical bow-tie or butterfly shapes are observed. The normal stress trajectory also depends strongly on the imposed amplitude, as the weakly nonlinear response obtained from medium-amplitude (MAOS) and the fully nonlinear response obtained from large-amplitude cases clearly differ. When the normal stress differences are plotted in the recoverable strain domain, $N_{1}-\gamma_{rec}$, there is collapse onto the same parabolic master curve, showing that the MAOS and LAOS cases undergo similar sequences within a cycle of oscillation. We therefore observe a quadratic dependence of the normal stress difference on the recoverable strain.

To form a more quantitative understanding of the normal stress difference, we apply an expression derived by Lodge, from his transient network theory of polymers Lodge (1958). Lodge showed that the ratio of the first normal stress difference to the shear stress under constrained recovery is $N_{1}/\sigma=2\gamma_{rec}$. We exploit this relationship to provide a description of the first normal stress difference with respect to the recoverable strain, $N_{1}(t)=2G(t)\gamma_{rec}^{2}(t)$, where $G(t)$ is the recoverable strain-dependent elastic modulus determined from the $\sigma-\gamma_{rec}$ plots. The calculated $N_{1}$ values are shown in Fig. 2 (d and f) in comparison with the experimental WLM results at the two frequencies. In both cases there is exceptional agreement. Additionally, we show the theoretical prediction from the linear regime, $N_{1}(t)=2G_{\text{LVE}}\gamma_{rec}^{2}(t)$ as dashed lines, and note the departure from LVE behavior at $\gamma_{rec}\approx$0.5, in agreement with the shear stress rheology.

The fibrin normal stress is shown in Fig. 2(b) as a function of the recoverable strain. The negative normal stress indicates that the fibrin contracts under shearing Janmey et al. (2006). Despite having fundamentally distinct structure and properties from WLMs, the magnitude of the normal stress still increases quadratically with the recoverable strain. While it is known that biopolymers do not follow classical rubbery elasticity Mackintosh et al. (1995), we empirically observe that the normal stress is well described by $N_{1}(t)=-2mG(t)\gamma_{rec}^{2}(t)$, where $m$ is found to be 5. Having $m>$1 agrees with the finding that these biopolymers tend to show larger normal stresses than synthetic polymers Janmey et al. (2006); Kang et al. (2009). With the results from the two distinct systems, we conclude that the recoverable strain dictates not only the shear stress, but also the complex normal stress.

The experimentally-determined physical processes observed in the macroscopic rheological responses are mirrored in the microstructural evolution of the WLMs as shown in Fig. 3 for the $De=0.25$ case for a total strain amplitude of $\gamma_{0}=4$. Shown in Fig. 3 (a) is the $\sigma-\gamma_{rec}$ figure, with a color scale that reflects the degree of alignment observed in the 2-D SANS patterns shown in Fig. 3 (b). We quantify the alignment of the micellar segments by defining an alignment factor ($A_{\text{f}}$) as $A_{\text{f}}=\int_{0}^{2\pi}I_{\text{c}}(q^{*},\phi)cos(2(\phi-\phi_{0}))d\phi/\int_{0}^{2\pi}I_{\text{c}}(q^{*})d\phi$, where $I_{\text{c}}(q^{*},\phi)$ is the azimuthal intensity over $q^{*}$ and $\phi$ is the azimuthal angle with the segmental $q$-range $q^{*}=0.006$ to 0.03 $1/\AA$. $\phi_{0}$ represents an orientation angle. The alignment factors in the 1-2 and 1-3 planes are presented as functions of the recoverable strain in Fig. 3 (c) and (d).

Contrary to the established view that shear-induced alignment is correlated with shear rate at low frequency and strain at high frequency Rogers et al. (2012), we observe that the alignment of micellar segments is linearly proportional to the magnitude of the recoverable strain regardless of the imposed frequency. Further, the same proportionality coefficient is determined in both the 1-2 and 1-3 planes: $A_{f}=0.045\mid{\gamma_{\text{rec}}}\mid$. We therefore build a remarkably consistent physical picture: when the recoverable strain is small, linear viscoelastic responses are elicited, even under LAOS, and the scattering patterns are identical to equilibrium conditions with no alignment of the micellar segments. As the magnitude of the recoverable strain increases, so too do the alignment factor and shear and normal stresses. Even when the modulus begins to drop at large $\mid\gamma_{rec}\mid$ (points ii and iv in Fig. 3 (a)), the alignment is still linearly dependent on $\mid\gamma_{rec}\mid$.

Biopolymers such as fibrin are known to align and stretch with strain (Vader et al., 2009), exhibiting unique force-extension relationships Storm et al. (2005); Janmey et al. (2006). We have shown that almost all strain is acquired recoverably by fibrin networks. Our results from both the shear-thinning linear entangled micelles and the strain-stiffening fibrin networks therefore show that it is the recoverable strain that provides the basis of accurate nonlinear structure-property relations of soft polymeric materials.

Access to NGB 30m-SANS was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No. DMR-1508249. This material is based upon work supported by the National Science Foundation under Grant No. 1727605. We gratefully acknowledge helpful discussions with Michelle Calabrese for the deconvolution protocol, and Connie Hong for preparing the fibrin. Certain commercial instruments or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.