In a recent paper [Hassager, J. Rheol. **64**, 545–550 (2020)], Hassager performed an analysis of the start up of stress-controlled oscillatory flow based on the general theory of linear viscoelasticity. The analysis provided a theoretical basis for exploring the establishment of a steady strain offset that is inherent to stress controlled oscillatory rheometric protocols. However, the analysis neglected the impact of instrument inertia on the establishment of the steady periodic response. The inclusion of the inertia term in the framework is important since it (i) gives rise to inertio-elastic ringing and (ii) introduces an additional phase shift in the periodic part of the response. Herein, we modify the expressions to include an appropriate inertial contribution and demonstrate that the presence of the additional terms can have a substantial impact on the time scale required to attain the steady state periodic response. The analysis is then applied to an aqueous solution of wormlike micelles.

## I. INTRODUCTION

Under stress controlled conditions (which is the focus of the present work), the initial strain response to the applied oscillatory stress consists of a periodic response, a transient response, and a strain offset [1]. In a standard stress controlled small amplitude oscillatory shear (SAOS) experiment, the periodic response is isolated by (i) introducing a “conditioning time” during which the transient part of the response decays to a negligible magnitude and which is omitted from data analysis and (ii) removing the strain offset as part of the post-processing algorithms, either by Fourier transform (where it would appear as the DC offset term) or by baseline subtraction where cross correlation based algorithms are used. Motivated by the recent work of Lee *et al.* [2] who demonstrated the utility of the strain offset in probing the zero shear viscosity, Hassager [1] derived expressions for the offset, transient and periodic contributions to the total strain for the generalized Maxwell fluid. However, this analysis neglected the impact of instrument inertia on the strain waveforms. The impact of instrument inertia on stress controlled experiments is perhaps most clearly seen in the context of a creep experiment where the coupling of the material’s elasticity and the rheometer inertia (i.e., the total system inertia inclusive of the geometry inertia, which is often calibrated separately) gives rise to inertio-elastic ringing at short times, i.e., free oscillations that appear superimposed upon the expected creep curve. Useful information can be extracted from such creep-ringing effects [3–7] but, in the context of an oscillatory shear experiment performed on a combined motor transducer (CMT) type rheometer, careful correction of instrument inertia artifacts is required [7,8].

Expressions for the evolution of the strain following the sudden imposition of steady stress (i.e., a creep experiment) in the presence of instrument inertia for several models can be found in the literature [3–6,9,10]. Baravian *et al.* [9] considered the onset of stress controlled flows (both creep and forced oscillation) for the Jeffrey’s fluid (which can be represented as a Voigt element in series with a viscous “dashpot”) in the presence of instrument inertia and found that, for both flows, the inertial term gave rise to a transient contribution to the total strain taking the form of a damped oscillation decaying with $ e \u2212 t / \lambda 2$, where $ \lambda 2$ denotes the retardation time of the Voigt element [9]. The analysis of Baravian *et al.* [9] for forced (stress controlled) oscillations results in an expression for the strain evolution, $\gamma (t)$, which only contains transient and periodic contributions; the offset, which is often observed in experimental data [2], and which appears in Hassager’s analysis of the generalized Maxwell fluid [1], is absent. Lauger and Stettin also considered the impact of both sample and fluid inertia on steady state oscillatory shear in rotational rheometers, demonstrating the dramatic influence these phenomena can have on the experimental data [11].

In the present work, we extend the analysis of Hassager [1] to include the effects of instrument inertia and examine its effect on the evolution of the strain waveform as it approaches steady state conditions in stress controlled oscillatory experiments. Finally, we demonstrate the use of the analysis in a wormlike micellular system of cetylpyridinium chloride (CPyCl) and sodium salicylate (NaSal).

## II. MODELING

### A. Start up of oscillatory shear

*et al.*[7] (and references therein) and Lauger and Stennin (2016) [11] for further information concerning sample inertia. In the present study, we restrict attention to the gap loading condition for which the total stress ( $ \sigma t$) can be expressed as the sum of the sample stress, $ \sigma s$ and the inertial stress as follows [13]:

*t*).

Note that Eqs. (8) and (9) make use of the initial conditions $\gamma (0)=0$ and $ \gamma \u02d9(0)=0$.

The expression for the evolution of $\gamma (t)$ following initiation of the sinusoidal driving stress can hence be determined via the inverse Laplace transform of Eq. (13). The inverse Laplace transform may be obtained by considering the poles of Eq. (13). The expression for $\gamma (t)$ will contain three components, (i) a strain offset, $ \gamma o f f$, associated with the pole at $s=0$; (ii) a periodic response, $ \gamma p(t)$, associated with the pole at $s=i\omega $; and (iii) a transient component, $ \gamma t(t)$, associated with the zeros of the function $g(s)+Is$. We now treat each component in turn.

In Eqs. (20) and (21), we have dropped the notation $ G \u2032(\omega )$ in favor of the simpler $ G \u2032$, however, $ G \u2032(\omega )$ is implied.

### B. Frequency dependency

Figures 2–4 show normalized strain profiles [normalized by the peak periodic strain, $ \sigma 0/ G \u2217(\omega )$] for the start-up of stress controlled oscillatory shear in the presence and absence of inertia for three frequency ranges denoted as low, intermediate, and high, respectively. The designation of a frequency as low, intermediate, or high is made based on the transient part of the strain profile. Frequencies at which the transient is negligible may be considered “low frequency,” frequencies at which there is an observable transient that decays to negligible magnitude within a single period of the applied oscillation to be “mid-frequency,” and frequencies for which the transient response persists beyond a single period to be “high frequency” waveforms. We also compute a viscosity weighted average relaxation time ( $ \tau \xaf=[ \eta 1 \tau 1+ \eta 2 \tau 2]/ \eta 0)$ and use this to define an appropriate Deborah number ( $De= \tau \xaf\omega $). In terms of this Deborah number, the high frequency range corresponds approximately to frequencies for which $De>10$ while the low frequency range corresponds approximately to $De<1$.

It is also interesting to note that the amplitude of the periodic signal decreases in the high frequency range since most of the applied torque is used to accelerate the geometry/instrument; furthermore, a resonant frequency is observed as noted by several authors [9,16] (see Fig. 5). In Figs. 2–4, a spectrum with $ \tau 1=0.01$ s, $ \tau 2=1.0$ s, $ \eta 1=1.0$ Pa s, $ \eta 2=10.0$ Pa s has been employed with $I=0.1 Pa s 2$. For this spectrum, the function $p$ has one real and two complex conjugate roots leading to a damped oscillatory transient response.

### C. Short time expansion and the zero frequency case

*et al.*[4,7],

### D. Generalized Maxwell model

In Eq. (43), $ \rho i , k$ refers to the $k$th root of $a s 3+b s 2+cs+d=0$ for the pair of modes appearing at $[ \tau i, \tau i + 1]$. There are three roots (and hence, $k=1,2,3$) if the discriminant of this polynomial (i.e., $\Delta =18abcd\u22124 b 3d+ b 2 c 2\u22124a c 3\u221227 a 2 d 2$) for the pair of modes is positive (three real roots) or negative (one real and two complex conjugate roots). If the determinant evaluates to 0 the polynomial has a repeated root; a single repeated root if $3ac= b 2$ resulting in $k=1$, otherwise, there are two roots and $k=1,2$. It is interesting to note that if $I=0$ then $a=b=0$ and the conditions for a single repeated root are satisfied. Hence, the roots of the polynomial return to the retardation times of the model as per Hassager’s analysis [1].

### E. Newtonian solvent

### F. Effect of inertia on “conditioning time”

When designing stress controlled oscillatory experiments, it is important to recognize that a conditioning time (the time which is allowed to elapse between the initiation of the perturbation waveform and the collection of “periodic” data) based on an arbitrary criterion such as “the period of a single cycle” may not be sufficient. Indeed, for the inertialess case, Hassager demonstrated that the transient response decays with $exp\u2061 ( \u2212 t / \lambda k )$. A conditioning time of $\u22484 \lambda k$ allows the transient to decay to $\u22482%$ of its initial magnitude. However, in Figs. 3 and 4, the transient term persists for longer in the presence of inertia than in the inertialess case for all frequencies. Plotting the three poles associated with the transient response (see Fig. 7), it can be seen that in the presence of inertia the pair of complex conjugate poles are (i) dominant over the single real pole (i.e., they appear significantly closer to the imaginary axis) and (ii) the real part of the complex conjugate pair lies to the right of the single pole of the inertialess case. The rate of decay of the transient response is predominantly determined by an effective retardation time $ \lambda \u2217=\u22121/Re[ \rho k +]$ (where $ \rho k +$ refers to the dominant pole). Consequently, selection of an appropriate “conditioning time” should involve an assessment of inertio-elastic effects with $ t c\u22484 \lambda \u2217$, allowing the transient to decay to $\u22482%$ of its initial magnitude. For certain relaxation spectra, the presence of inertia may accelerate the decay of the transient component. Experimentally, where the relaxation spectrum is unlikely to be known *a priori*, direct observation/analysis of the transient response may be the quickest way to determine the appropriate conditioning time.

## III. EXPERIMENTS

### A. Materials and methods

#### 1. Sample preparation

Cetylpyridinium chloride (CPyCl) and sodium salicylate (NaSal) (Sigma-Aldrich) were dissolved at a molar ratio of 2:1 in 0.5M sodium chloride (NaCl) solutions prepared using de-ionized water. Appropriate quantities of dry NaCl, NaSal, and CPyCl, in powdered form, were added to de-ionized water to generate a 4.1 wt. % solution of CpyCl in a fume hood. This concentration has previously been shown to display shear thinning (rather than shear banding) characteristics [17,18]. The mixtures were stirred for 24 h at 40 $ \xb0$C (in a sealed beaker atop a heated plate) to completely disperse the powder before measurements were performed. All chemicals were used as received without further purification.

#### 2. Rheometry

Small amplitude oscillatory shear (SAOS) and start-up of stress controlled oscillations with $\psi =0$ (referred to as “transient” experiments herein) were performed using a TA Instruments HR-30 rheometer fitted with a 60.0 mm 2 $ \xb0$ aluminum cone. The sample was loaded to the temperature controlled (Peltier) plate of the rheometer and a thin layer of low viscosity silicone oil was added to prevent evaporation. All experiments were performed at 25 $ \xb0$C. For transient data collected at the two lowest frequencies (0.1 and 1.0 rad $ s \u2212 1$), it was necessary to correct the strain waveforms for strain drift. This was achieved by fitting a straight line ( $y=mx+c$) to the final few peaks of the strain waveform in order to identify the drift rate ( $m$) before subtracting the product $mt$ from the measured strain signal. This correction was unnecessary for the higher frequencies due to the far shorter duration of data acquisition.

The moment of inertia of the instrument and geometry were determined prior to experiments using standard calibration procedures. The instrument moment of inertia constant was $ M i=21.02$ $\mu N ms 2$ and the geometry moment of inertia constant was $ M g=8.80$ $\mu N ms 2$. The total moment of inertia was then converted to $I$ [as it appears in Eq. (4), with units of $ Pa s 2$] using the appropriate geometry factors [4,9] ( $ F \sigma =3/2\pi R 3$ and $ F \gamma =1/tan\u2061\alpha $, where $R$ denotes the cone radius and $\alpha $ the cone angle) such that $I=( M i+ M g)\xd7 F \sigma / F \gamma =0.0185 Pa s 2$.

### B. Results and discussion

Figure 8 shows SAOS data along with a two mode Maxwell model fit (with $ \tau 1=0.251 s$, $ \tau 2=0.013 s$, $ \eta 1=10.37 Pa s$, $ \eta 2=0.08 Pa s$, $[ \tau \xaf=0.25 s]$) which shows excellent agreement over the entire frequency range. The 2 mode model was then used to “predict” the strain profiles occurring in response to the start up of oscillatory stress controlled oscillations (at $\omega =[0.1 rad / s,1.0 rad / s,10.0 rad / s,100.0 rad / s]$) in the presence and absence of inertia using Eq. (33) with $I=0.0185 Pa s 2$ and $I=0 Pa s 2$, respectively. Corresponding experimental data were also obtained at the same angular frequencies using the transient acquisition mode of the HR-30. Figure 9 shows excellent agreement between the predictions (thin red line) and experimental data (thick gray lines) at all frequencies when inertia is included within the model (left hand column). Where inertia is omitted (as per the solution of Hassager [1]), the model captures the dynamics of the strain waveform only at low frequencies, as would be expected. At higher frequencies, the effects of (i) inertial-ringing, (ii) reduced strain amplitude due to resonance effects, and (iii) the inertial phase shift are missing. Consequently, the inertialess model is insufficient at 10 and 100 rad/s. Comparison of the measured and predicted strain waveforms in this manner establishes confidence in the data and in the discrete relaxation spectrum (DRS) fitted to the SAOS data since small errors in the dynamic moduli [ $ G \u2032(\omega )$ and $ G \u2032 \u2032(\omega )$] are well known to generate large deviations in the DRS.

*et al.*[2]), the zero shear viscosity can be determined by measuring the strain offset (determined herein as the mean value of strain in the long time limit, see Fig. 11) through a simple rearrangement of Eq. (15) to

Figure 10 shows values of $ \eta 0$ determined from the strain offset at each frequency as red (filled) circles (for ease of visualization, these are plotted at $ \gamma m a x\omega $ for each frequency, where $ \gamma m a x$ denotes the maximum strain observed in the waveforms). Excellent agreement is observed between values of $ \eta 0$ obtained from (i) the flow curve, (ii) the two mode Maxwell model fit, and (iii) strain offsets.

## IV. CONCLUSIONS

Herein, we have demonstrated how rheometer inertia (including instrument and geometry contributions) influences the establishment of the steady state periodic response to a stress controlled oscillatory perturbation in the context of the general theory of linear viscoelasticity. In contrast to the inertialess case, in which the transient terms decay exponentially with time constants set by the retardation times of the discrete spectrum [1], the presence of inertia modifies this response to include inertio-elastic ringing reminiscent of that seen in creep experiments [3–6,9,10] (which is included in the present analysis as the zero frequency case). Consequently, the timescale for the establishment of the steady state periodic response is dramatically affected by the presence of rheometer inertia. In practice, this highlights the importance of selecting an appropriate “conditioning time” when designing experimental procedures that rely on the steady state response either for direct determination of the dynamic moduli or probing the zero shear viscosity via the strain offset. Direct observation/analysis of the transient response during preliminary experiments may be the most robust way to determine the appropriate conditioning time.

The effect of instrument inertia on experiments involving more complex waveforms, for example, stress-controlled optimally windowed chirps ( $\sigma \u2212OWCh$) [19,20] is anticipated to be more complicated and we leave the analysis of this problem to a later paper.

## ACKNOWLEDGMENTS

The authors acknowledge funding from the Engineering and Physical Sciences Research Council, UK, through Grant Nos. EP/N013506/1 and EP/T026154/1 and the European Regional Development Fund via the Smart Expertise program of the Welsh Government. The authors are also grateful to Dr. Rowan Brown of Swansea University for helpful discussion.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

### APPENDIX: DERIVATION OF ZERO FREQUENCY RESPONSE

## REFERENCES

*Complex Fluids in Biological Systems*, edited by S. Spagnolie (Springer, New York, 2015).

*Advanced Engineering Mathematics*