Introduction

The non-invasive measurement of the internal stress states of fluids in flow is of great importance for flow engineering and biomechanics. For bulk fluid pressure measurements, the method of measuring velocity fields with particle tracking velocimetry1 or particle image velocimetry2 is commonly used. However, they require spatial derivatives of velocity fields to determine the stress, in which the noise amplification problem appears. Other methods have also been proposed to evaluate the information corresponding to the stress based on unique phenomena, such as light scattering and magnetic resonance3,4,5. These methods can convey information relating to the stress driving the fluid intact. Among these approaches, measurement of stress or velocity gradient distributions by the photoelastic method6 has been applied and is attractive due to its high measurement sensitivity and experimental simplicity.

The photoelastic method was developed as a stress-measurement technique for solids, and it has been further developed and widely studied for 50 years7. Among solid materials, various studies considering the measurement of stresses and residual stresses, especially in glass, have been undertaken6,8. Stress-loaded materials change their refractive index with regard to the direction of polarization vibration in response to strain. Therefore, when two orthogonally polarized light beams are transmitted through a strained material, retardation will appear between them. The incident and emitted polarized light is represented by a composite vector of the two linearly polarized light beams, and their trajectories will differ due to the retardation of the composite vector. Depending on the trajectory difference, the result can be classified as linearly, elliptically, or circularly polarized light. When circularly polarized light is incident onto a stress-loaded material, it will be emitted as elliptically polarized light with retardation \(\Delta\) and orientation angle \(\phi\). The values of \(\Delta\) and \(\phi\) (photoelastic parameters) correspond to the principal-stress difference and the principal-stress direction, respectively6. In the photoelastic method, stress can be estimated from \(\Delta\) based on the stress-optic law (SOL)8,9:

$$\begin{aligned} \Delta = \int C_1(\sigma _1 - \sigma _2)~{{{\text {d}}}}h, \end{aligned}$$
(1)

where: \({{{\text {d}}}}h\) is the infinitesimal thickness of the material; \(C_1\) is the stress-optic coefficient; and \(\sigma _1\) and \(\sigma _2\) are the maximum and minimum principal stresses, respectively. However, Eq. (1) holds only for two-dimensional (2D) stress fields with no stress along the optical axis, or, if there is stress along the optical axis, then it must be uniform. This means that \(\Delta\) can be obtained by only considering the secondary principal stress difference10,11. Here, the secondary principal stress difference (\(\sigma _1 - \sigma _2\)) is the principal stress difference projected onto a plane perpendicular to the camera’s optical axis. In the case of 3D stress fields, applying the SOL is more complicated than in the above equation. For stress distributions along the camera’s optical axis, it is necessary to introduce the concepts of the “optically equivalent model”12 and “integrated photoelasticity”8,13. This makes it possible to consider that the 3D stress fields consist of sufficiently thin linear polarizers that can be assumed to be 2D stress fields. Thus, the polarization state transmitting through the 3D stress field can be calculated by multiplying the Mueller matrices of each optically equivalent model13.

McAfee and Pih observed 3D flow in a channel and described the dependence of the “isoclinic” (optical-anisotropy in this study) on shear strain rates perpendicular and parallel (camera’s optical axis) to the light beam3. However, they did not provide a direct relationship between the SOL and the measurements. Some attempts have been made in the past to extend the SOL by including stress components along the camera’s optical axis as a second-order term14,15. However, this has not been experimentally validated and the impact of the stress components along the optical axis continues to be neglected in photoelastic methods. Despite its recognised importance, it has not been incorporated into many studies.

In recent years, research has been conducted on the application of photoelastic methods to fluids, demonstrating that the principle of Eq. (1) can be applied to quasi-2D flows16,17. Despite this, several studies using quasi-2D flow channels have highlighted discrepancies between the birefringence predicted by SOL in the 2D case and the experimental data, particularly in the centre of the channel18,19,20. These reports can be considered experimental evidence that as the flow field becomes more 3D (e.g., circular tubes21 or rectangular channels with an aspect ratio close to 120,22), the stress distribution along the optical axis becomes more critical. These experimental findings have also been corroborated by numerical calculations, which have established that regions, where the impact of 3D effects is dominant, are close to the “plane of symmetry”23.

By introducing the concept of “rheo-optics”, studies of optical-anisotropy responses to the fluid’s deformation have been conducted. This is the simultaneous measurement of rheological and optical-anisotropy and has contributed to the understanding of the internal structures of complex fluids. Starting from the apparatus proposed by Lodge24 and Philipoff25, the filament stretching rheometer26 and the capillary breakup extensional rheometry dripping-onto-substrate (CaBER-DoS) method27,28 were developed to perform birefringence measurements on liquid polymers under uniaxial extension flow. To establish the relationship between the shear rate (shear stress) and optical-anisotropy, which is the aim of the present study, the flow given to fluids should be simple and well-controlled. Consequently, the use of rheometers (concentric cylinder (CC)-type17,29,30,31, parallel plate (PP)-type32,33,34,35, and cone plate (CP)-type36) has been reported, as they are easy to incorporate into optical-anisotropy measurement systems. The optical-anisotropy induced by secondary principal stress difference (principal stress difference acting in a plane perpendicular to the optical axis) can be measured from the shear-vorticity direction, e.g., in uniaxial extension systems and CC-type rheometers. Conversely, measurements from the shear direction provide a direct visualization of the stress along the optical axis rather than showing the principal stress difference. This is due to the existence of a velocity gradient along the optical axis, which is of interest to the present study.

This study aims to understand the 3D effect of birefringence by revisiting the stress-optic law in its original form (including the stress components along the optical axis) and verifying it quantitatively. Our study is novel and important because it is the first quantitative verification of this phenomenon. To simplify the discussion, fluid with properties similar to a Newtonian fluid was used as a first step in the study. This is to minimise the contribution of birefringence due to elastic or normal stress during shear imposition. As a validation of the stress-optic law, the tendency of birefringence induced by the stress along the optical axis was investigated experimentally as well as quantitatively and compared with the results obtained in previous studies. Note that the optical axis in our experiments is along the velocity gradient direction perpendicular to the parallel plates in a PP-type rheometer.

Principles

This section outlines the basic theory of polarization measurements, mainly focusing on flow birefringence and the SOL9.

Flow birefringence of fluids

Birefringent fluids are composed of crystals or polymer chains with large aspect ratios. This is the key to changing the refractive indices \(n_{\perp }\) and \(n_{\parallel }\) perpendicular and parallel to the direction of vibration of the transmitted light, respectively. One of the optical-anisotropies, birefringence \(\delta _n\), is a physical quantity that indicates the magnitude of the anisotropy of the refractive index. Birefringence is defined as the absolute value of the difference between the major and minor diameters of the index ellipsoid:

$$\begin{aligned} \delta _n = |n_{\perp } - n_{\parallel }|. \end{aligned}$$
(2)

As long as there is no stress loading, birefringent fluids show optically isotropic properties (\(n_{\perp } = n_{\parallel }\)) because the particles are randomly oriented by Brownian motion. However, when shear is applied, the crystals or polymers become aligned in the direction corresponding to the stress, resulting in optical-anisotropy (\(n_{\perp } \ne n_{\parallel }\)). When the particles are strongly oriented in a particular direction, the anisotropy of the refractive index becomes stronger and the value of the birefringence increases (see Fig. 1a). Conversely, when the applied shear is reduced, the orientation becomes random again and birefringence is no longer induced (see Fig. 1b). This phenomenon is known as flow birefringence37; as examples of birefringent fluids, aqueous cellulose nanocrystal (CNC) suspensions38, wormlike micelle solutions18, and xanthan gum solutions39 are well known for showing birefringence.

Fig. 1
figure 1

Schematic of nanocrystals in solution (a) when no stress is applied, (b) under stress loading. Although needle-like nanocrystals are shown, the mechanism of onset is the same with macromolecular chains.

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Stress-optic law (SOL)

The retardation \(\Delta\) obtained by the photoelastic method is the summation (integrated value) of the birefringence along the optical axis. The retardation \(\Delta\) caused by flow birefringence \(\delta _n\) is related to the strain rate \(\dot{e}_{ij}\) inside the fluid14:

$$\begin{aligned} \Delta {{{\text {cos}}}}2\phi&=\int \alpha _1({\dot{e}_{{ yy}}}-{ \dot{e}_{{ xx}}})+\alpha _2[(\dot{e}_{yy}+\dot{e}_{xx})(\dot{e}_{yy}-\dot{e}_{xx})+\dot{e}_{zy}^2-\dot{e}_{xz}^2]~{\mathrm d}z, \end{aligned}$$
(3)
$$\begin{aligned} \Delta {{{\text {sin}}}}2\phi&=\int 2\alpha _{1}\dot{e}_{xy}+\alpha _{2}[{2}(\dot{e}_{yy}+\dot{e}_{xx})\dot{e}_{xy}+{2}\dot{e}_{xz}\dot{e}_{yz}]~{\mathrm d}z. \end{aligned}$$
(4)

Here, the optical axis is defined as the z axis of a Cartesian coordinate system, and \(\alpha _1\) and \(\alpha _2\) are functions of the physical properties of the fluid. For Newtonian fluids, the stress is proportional to the strain rate. Therefore, Eqs. (3) and (4) can be expressed using stress20:

$$\begin{aligned} \Delta {{{\text {cos}}}} 2\phi&=\int { C}_1({\sigma _{{ yy}}}-{\sigma _{{ xx}}})+C_2[( \sigma _{yy}+ \sigma _{xx})(\sigma _{yy}- \sigma _{xx})+ \sigma _{zy}^2- \sigma _{xz}^2]~{\mathrm d}z, \end{aligned}$$
(5)
$$\begin{aligned} \Delta {{{\text {sin}}}}2\phi&=\int 2 C_{1} \sigma _{xy}+C_{2}[{2}( \sigma _{yy}+ \sigma _{xx}) \sigma _{xy}+{2} \sigma _{xz} \sigma _{yz}]~{\mathrm d}z. \end{aligned}$$
(6)

In these equations, \(C_1 =\alpha _1/\eta\) and \(C_2 = \alpha _2/\eta ^2\), in which \(\eta\) is the shear viscosity of the fluid. Here, \(C_1\) represents the sensitivity of optical-anisotropy to secondary principal stress difference, whereas \(C_2\) represents the sensitivity to stress along the optical axis. Aben and Puro15 also discussed the optical relationship based on Eqs. (5) and (6) and assumed that the stress components along the optical axis, i.e., \(\sigma _{xz}\), \(\sigma _{zy}\), and \(\sigma _{yz}\), were negligible. In other words, they made the assumption that \(C_2 = 0\), which leads to the proposal of:

$$\begin{aligned} \Delta = \int C_1\sqrt{(\sigma _{xx} - \sigma _{yy})^2 + 4{\sigma _{xy}}^2}~{\mathrm d}z. \end{aligned}$$
(7)

It should be emphasized again that Eq. (7) is a relation that holds only for 2D stress fields. Moreover, Eq. (7) is an often-used expression in the solid-state photoelastic method15,40.

Methodology

In this section, the details of the experiments are presented. Unless otherwise indicated, all experiments were repeated at least three times per case at \(25^\circ\)C.

Experimental setup

Figure 2a shows a schematic of the rheo-optical measurement system. The stress-controlled rheometer (MCR 302, Anton Paar Co., Ltd.) was equipped with a parallel plate (PP43/GL-HT, Anton Paar Co., Ltd., plate radius \(R_o = 21.5\) mm) made of quartz glass and a flat glass plate (PTD200/GL, Anton Paar Co., Ltd.). In this system, shear flow is induced by clockwise rotation of the plate, and the average shear stress and torque are logged. Left-handed circularly polarized light is generated by attaching a polarizer and 1/4-wave plate to an LED light source (SOLIS-525C, Thorlabs Co., Ltd., wavelength \(\lambda = 525\) nm). This polarized light is reflected from the top of the plate by a mirror and emitted as elliptically polarized light with retardation \(\Delta\) and orientation angle \(\phi\). The transmitted elliptically polarized light is reflected by the mirror again and enters the polarization camera (CRYSTA PI-1P, Photron Co., Ltd.), which is equipped with a 524-nm band-pass filter. The polarization camera has a spatial resolution of up to \(512 \times 512\) pixels and a temporal resolution of 1.55 Mfps. In these experiments, all measurements were made with a resolution of \(512\times 512\) pixels (44.5 \(\upmu\)m/pixel) at 1000 fps. Additionally, the gap height between the plates was always fixed at \(H = 100\ \pm 5\) \(\upmu\)m, and the shear rate was set to 1000–10,000 \(\hbox {s}^{-1}\). Note that the shear rate is specified at 2/3 of the plate radius. This is the range in which it is expected that no artefacts will appear in the rheological measurement results (for further details, please see Appendix 1).

Fig. 2
figure 2

(a) Schematic of the rheo-optical setup, in which \(R_i\) and \(R_o\) are the inner and outer radii of the transparent part of the plate, respectively. (b) Representative intensity image for CNC suspension 1.0 wt% at shear rate \(\dot{\gamma }= 1000\) \(\hbox {s}^{-1}\). As shown in inset 1 of panel (b), retardation is obtained from four neighbouring polarizers. (c) Temporal evolution of retardation at no flow.

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Polarization camera

A polarization camera (CRYSTA PI-1P, Photron Co., Ltd.) was used to detect the retardation \(\Delta\), which is the integrated value of birefringence along the optical axis of the light transmitted through the apparatus. Using the phase-shifting method41, this was obtained from the radiance through linear polarizers oriented in four different directions \((0^\circ\), \(45^\circ\), \(90^\circ\), and \(135^\circ\)) in an area of \(2 \times 2\) pixels (as shown in inset 1 of Fig. 2b). Defining the light intensities detected at each of these pixels as \(I_1\), \(I_2\), \(I_3\), and \(I_4\), respectively, the retardation can then be given by:

$$\begin{aligned} \Delta = \int \delta _n\ \textrm{d}z=\frac{\lambda }{2\pi }{{\text {sin}}^{-1}}\frac{2\sqrt{(I_3-I_1)^2+(I_2-I_4)^2}}{I_1+I_2+I_3+I_4}, \end{aligned}$$
(8)

where \(\lambda\) [m] is the wavelength of the light source. As the retardation is obtained from four linear polarizers, the spatial resolution was \(512 \times 512\) pixels, which is 1/4 of the \(1024 \times 1024\)-pixel light-intensity image. As there is no stress distribution along the optical axis, the retardation \(\Delta\) can be calculated as the product of the birefringence \(\delta _n\) and the gap height H. Therefore, in this study, we calculated the birefringence by dividing the retardation measurements by H.

Data acquisition and analysis

An arc-shaped region of interest (ROI-A) located at a distance \(L\approx 3\) mm from the edge of the plate (outlined in yellow in Fig. 2b) was chosen, and the background-subtracted retardation field during flow was provided by the CRYSTA Stress Viewer software package (Photron Co., Ltd.). This background-subtraction process is based on the presence of a certain degree of unevenness on the plate surface or birefringence in the plate or stage itself, which can cause a spatial distribution of retardation even when there is no flow. The position of ROI-A was chosen to avoid possible interference at the edges of the plate by the liquid–air interface or the liquid meniscus42. ROI-A was also chosen to keep the uniformity of the lighting system, as the polarized image sensors use the light intensity itself to calculate the birefringence, as explained in the previous section. Another arc-shaped region, ROI-B (shown in red in inset 2 of Fig. 2b), with a width of 10 pixels (440 \(\upmu\)m), was also chosen for quantitative characterization of the flow birefringence. This was selected to ensure that the shear-rate variation along the radial direction was kept to a minimum, with \(\approx\)0.8% difference. Therefore, within ROI-B, the shear rate applied to the microscopic fluid is \(\dot{\gamma }_{{\text {rep}}} = 0.75R_0{\Omega }/H\), and this was considered as the representative shear rate in the polarization measurements. Here, \(\Omega\) [rad/s] is the angular velocity of the rotating plate.

The parallelism of the plate to the stage was not perfect. Therefore, due to changes in the integral of the birefringence, the measured retardation also changed with time. To exclude this effect, measurements were taken after a sufficient time had passed from the start of the plate’s rotation and averaged over 1000 frames. As the polarization cameras measure the light intensity distribution as an image, minute noise in the intensity values is included in the calculation results as a constant retardation value, even if the fluid is not flowing. Figure 2c shows the time variation of the spatial average retardation in ROI-B measured with no flow. The retardation at a certain level was measured to be 1.46 nm; therefore, the birefringence was calculated on the assumption that this value (\(\delta _{n, {\text {error}}}\approx 1.5\times 10^{-5}\)) was included as a measurement error or uncertainty.

Working fluid: cellulose nanocrystal suspensions

Suspensions of CNCs (Alberta Pacific Co., Ltd.) of two different concentrations (weight percentage, wt%) were studied: 0.5 wt% and 1.0 wt%. These suspensions were prepared by mixing CNCs with ultrapure water using a hot stirrer at \(40^\circ\)C and 650 rpm for more than 24 h. The shear viscosities of these prepared CNC suspensions were then measured using a rheometer equipped with a 50-mm-diameter cone plate (CP50-0.5, Anton Paar Co., Ltd.) to obtain \(\eta\) values. The measurement results are shown in Fig. 3. Both CNC suspensions showed a significant shear-thinning nature at low shear rates (\(\dot{\gamma }< 10~{\text {s}}^{-1}\)) whereas they exhibited relatively weak shear-thinning at high shear rates (\(10^2~{\text {s}}^{-1}<{\dot{\gamma }}\)). Shear thinning of CNC suspensions is commonly observed as typical of lyotropic liquid crystals. The shear-thinning nature exhibited in the low shear rate region is due to the alignment of the chiral nematic liquid crystal domains. In contrast, at high shear rates, it is due to the disruption of the liquid crystal domains and the orientation of the individual CNC rods along the shear flow direction43. Here, the crowding number N is introduced to note the particle-particle interaction of the CNCs. N is expressed as \(N=2c_v(l/d)^2/3\) [–] using the particle volume concentration \(c_v\) (can be estimated from the CNC’s density), particle length l and particle width d44. For \(N<1\), particles are relatively free to move; conversely, collisions between particles occur for \(N>1\). The aspect ratio of the CNCs used in the present study is comparable to those used in previous studies (\(l/d\approx 16\))17,31,43. Referring to morphology and the CNC density of 1500 kg/m338, the N for 0.5 wt% and 1.0 wt% suspension was 0.56 and 1.13, respectively. \(N>1\) may be one of the reasons for the non-Newtonian nature of the 1.0wt% CNC suspension as shown in Fig. 3. The normal stress (Weissenberg effect) was also measured using the rheometer and confirmed to be small enough to be regarded as a measurement error.

Fig. 3
figure 3

Steady shear viscosity \(\eta\) for CNC suspensions at different concentrations. The inset shows an enlarged region at \(\dot{\gamma }= 10^2\)–\(10^4\) \(\hbox {s}^{-1}\).

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Results and discussion

Visualized birefringence fields at different shear rates are shown in Fig. 4a,b. As can be seen, the birefringence increased significantly as the shear rate was increased. This is because the CNCs attain more uniform orientations with increasingly strong shear leading to a stronger optical-anisotropy. The magnitude of birefringence obtained was \(\delta _n \sim O(10^{-5})\), comparable to the values found in previous studies29,31,45,46. In the experiments, we observed an uneven distribution of birefringence outside ROI-A, which was seen to vary unsteadily. This was considered to be an effect of interference at the liquid–air interface due to the plate’s rotation.

We now consider an analytical model to help understand the measurement results. A simple analytical model of shear flow between the plates can be expressed:

$$\begin{aligned} \textbf{u}(x,y,z) = {\Omega }\frac{z}{H}[-y,x,0]^\textrm{T}, \end{aligned}$$
(9)

where H is the gap height. The strain tensor \(\dot{e}\) is:

$$\begin{aligned} \dot{e} =\frac{{\Omega }}{H} \begin{bmatrix} 0 & 0 & -y \\ 0 & 0 & x \\ -y& x & 0 \\ \end{bmatrix}. \end{aligned}$$
(10)

This means that the birefringence measured in the present study was induced by strain (stress) components along the optical axis. Note that the birefringence measured by the present experimental system cannot be explained as far as the SOL used previously (which does not include the stress component along the optical axis, Eq. (1) or (7)) is applied. Substituting each component of the above tensor into Eqs. (3) and (4) yields:

$$\begin{aligned} \delta _n= (C_2 \eta ^2){\Omega }^2\frac{x^2+y^2}{H^2} = (C_2 \eta ^2)~\left( \frac{r\Omega }{H}\right) ^2. \end{aligned}$$
(11)

Thus, the birefringence is particularly dependent on the square of the radius r and the square of the plate’s angular velocity \(\Omega\), as long as \(C_2\) and \(\eta\) are constant.

Fig. 4
figure 4

Visualized birefringence fields within ROI-A for (a) 0.5 wt% and (b) 1.0 wt% CNC suspensions under steady-state conditions. Radial birefringence distributions of the plate in (c) 0.5 wt% and (d) 1.0 wt% CNC suspensions, in which the grey shaded areas correspond to ROI-A. The error bars show the standard deviations and all data were time and circumferential averaged.

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Line profiles of birefringence distributed across the radial direction of ROI-A at varying shear rates are shown in Fig. 4c,d. These show that the shear rate increases outwardly from the centre of the plate, which leads to an increase in the birefringence. When these results are compared with the analytical model, the quadratic dependence of the birefringence on r is evident in the experimental results; however, the birefringence is clearly not proportional to the square of \(\Omega\). This deviation indicates that \(C_2\) is not a constant value, and in particular that it varies with the shear rate. Details are given later in this section.

To further investigate the variation of the stress-optic coefficient \(C_2\), its magnitude was estimated and compared. From Eq. (11), the magnitudes of each parameter are \(\delta _n \sim O(10^{-5})\), \(\eta \sim O(1)\) mPa\(\cdot\)s, \(r\sim O(10)\) mm, \({\Omega } \sim O(10^1\)-\(10^2)\) rad/s, and \(H\sim O(100)~\upmu\)m. By magnitude comparison, the magnitude of \(C_2\) was estimated to be \(C_2\sim O(10^{-7}\text{--}10^{-6})~\rm Pa^{-2}\). To the best of our knowledge, this is the first systematic measurement report to identify the \(C_2\) value of a birefringent fluid. The stress-optic coefficient \(C_1\), generally known as the photoelastic modulus, has been widely investigated in solid polymers, and it is known to have a magnitude of \(O(10^{-12})~\rm Pa^{-1}\)47. In contrast, there have been a few reports on investigations of the \(C_1\) values of fluids. For reference, the \(C_1\) values obtained for fluids are: worm-like micelles \(O(10^{-7})~\rm Pa^{-1}\)18,45,48, aqueous xanthan gum solutions \(3.3\times 10^{-8}~\rm Pa^{-1}\)39, and a 0.5 wt% CNC suspension \(O(10^{-5})~\rm Pa^{-1}\)20. Although simple comparisons are difficult because of the different units, i.e., \(\hbox {Pa}^{-1}\) and \(\hbox {Pa}^{-2}\), the magnitude of \(C_2\) was found to be smaller than that of \(C_1\). Nevertheless, it is inappropriate to assume that \(C_2\) can be ignored in the presence of stress distribution along the optical axis as discussed in the Introduction of this paper.

Next, the trend of birefringence to the shear rate was investigated. For nanofiber suspensions, it has been reported that the no-slip condition in the velocity profile near the rotating plate is no longer applied49. Although the present experiment used a rigid-rod suspension, it could be worthwhile to consider the wall slip. To correct the shear rate applied to the suspension, wall slip was quantified by following the analysis described in the literature50, as only one of the correction methods. Hereafter, the discussion will be based on this corrected shear rate. The detail of the present analysis is described in Appendix 2. In Fig. 5, the vertical axis shows the spatiotemporally averaged birefringence \(\delta _{n,{\text {ave}}}\) in ROI-B, while the horizontal axis shows the representative shear rate \(\dot{\gamma }_{{\text {rep}}}\). It can be seen that the birefringence is increasing exponentially. When modelling the relationship between the flow birefringence and shear rate, the following empirical nonlinear model was proposed by Lane et al.31:

$$\begin{aligned} \delta _n=(A\cdot \dot{\gamma })^{n} \cdot w^{m}, \end{aligned}$$
(12)

where w [–] is the mass fraction of CNC defined as \(w = ({\text {weight of CNC}})/({\text {total weight of suspension}})\), and A [s], n [–], and m [–] are fitting parameters. The experimental results were fitted using Eq. (12), and the black dash-dotted lines in Fig. 5 show the results. The fitting parameters were \(A=4.80\times 10^{-7}\) s, \(n=0.574\) and \(m = 1.28\). It should be emphasized that this model is based on the results of polarization measurements conducted from the vorticity direction to the shear using a CC-type rheometer, which is different from that used in the present study.

As described in the previous section, flow birefringence is induced by the aligned orientation of crystals or polymer chains dispersed in a fluid. For generalization, and to provide a better prospect for the physical interpretation of the present experimental results, we attempt to disentangle the birefringence \(\delta _n\) from the suspension concentration c and the direction of the polarization measurement from the effect of CNC alignment due to the flow. We used tensor invariants that are independent of the coordinate system. Therefore, the discussion henceforth will be based on the second invariant of the deformation-rate tensor, \(\Pi ~[\rm s^{-2}]\), i.e.,

$$\begin{aligned} \delta _{n}=(B\cdot {\Pi })^{\alpha } \cdot w^{\beta }. \end{aligned}$$
(13)

Here, \(B~\hbox {[s}^{2}\)], \(\alpha\) [–], and \(\beta\) [–] are defined as new parameters. In the present system, \(\Pi\) can be calculated as:

$$\begin{aligned} {{\Pi }}(\textbf{E})= \sum \limits _{i=1}^3\sum \limits _{j=1}^3 \textbf{E}_{ij} \textbf{E}_{ji}=\frac{1}{2}\left( \frac{{\Omega }}{H}\right) ^2(x^2+y^2)=\frac{9}{8}{\dot{\gamma }_{{{\text {rep}}}}}^2, \end{aligned}$$
(14)

where \(\textbf{E}\) is the deformation-rate tensor:

$$\begin{aligned} \textbf{E} = \frac{1}{2}\lbrace (\nabla \textbf{u})+(\nabla \textbf{u})^\textrm{T}\rbrace . \end{aligned}$$
(15)

Since \(\Pi\) is proportional to the square of the shear rate \(\dot{\gamma }\) from Eq. (14), \(n = 2\alpha\) holds here. The \(\delta _n\) measurements were fitted using Eq. (13), and the results are shown by the black dashed lines in Fig. 6. The fitting parameters were \(B = 2.10\times 10^{-13}~\rm s^2\), \(\alpha = 0.287\) (\(n = 0.574\)), and \(\beta = 1.28\). Also represented in Fig. 5, it has been well reported that birefringence generally increases nonlinearly as the shear rate is increased29,31,51,52. This trend remains unchanged when organized by invariant, and the increase in birefringence per invariant decreases with decreasing gradient at higher invariants.

We focus our discussion on the exponent which characterizes the tendency of birefringence with the deformation of the fluid (solvent). Calabrese et al.38 studied the birefringence of 0.1 wt% CNC suspensions, and they proposed a proportionality of \(\delta _n\propto {\dot{\gamma }}^{0.9}\) from the results of their experiments. They reported CNC geometries with an average length of 260 nm and an average width of 4.8 nm. Note that the 0.1 wt% CNC suspension was stated to be a dilute region with no particle interactions. Lane et al. also found the relationship \(\delta _n\propto {\dot{\gamma }}^{0.537}\) in part of an investigation into whether CNC suspensions (0.7–1.3 wt%) could be used in studies of flow birefringence31. These suspensions were at concentrations above the dilute region, where particle interactions need to be considered. The exponent of 0.9 differs significantly from the present results, while 0.574 is relatively close. We assume that the difference in n results from different CNC particle-interaction behaviours. In the existing literature, the relationship between CNC particle shear alignment and rotational diffusion is described by the Péclet number \(Pe=|{{\text {E}}}|/D_r\), where \(|{{\text {E}}}|\) is the characteristic deformation rate53. When \(Pe\ge 1\), convective forces are strong enough to align CNCs to the flow direction, eventually inducing birefringence. Here, \(D_r\) is the rotational diffusion coefficient which plays an important role in flow birefringence. The \(D_r\) is determined by the CNC rod length l and the suspension concentration c, giving \(D_r\propto c^{-2}l^{-9}\)53,54. This indicates that the optical properties of CNC suspensions are due to different rod lengths and concentrations, resulting in different exponents. The present experimental results and those of Lane et al.31 indicate that regardless of the direction of polarization measurement with respect to shear, there seems to be a common physical background that leads to the flow birefringence following a power law.

Fig. 5
figure 5

Spatiotemporally averaged birefringence \(\delta _{n,{\text {ave}}}\) and corresponding fit using Eq. (12) with \(A=4.80\times 10^{-7}\), \(n=0.574\) and \(m = 1.28\), in which the error bars represent the standard deviation.

Full size image
Fig. 6
figure 6

Spatiotemporally averaged birefringence \(\delta _{n,{\text {ave}}}\) and corresponding fit using Eq. (13) with \(A=2.10\times 10^{-13}\), \(\alpha =0.287\) and \(\beta = 1.28\), in which the error bars represent the standard deviation.

Full size image

The consistency between the SOL (Eq. (11)) and the empirical relation (Eq. (13)) is now discussed. Since both equations are equally related by \(\delta _n\),

$$\begin{aligned} C_2\eta ^2\dot{\gamma }^2 = (B\cdot {\Pi })^\alpha \cdot w^\beta \rightarrow C_2 = \frac{B^\alpha w^{\beta }}{\eta ^2}{\Pi }^{\alpha -1}. \end{aligned}$$
(16)

It can be seen from Fig. 3, the change the shear viscosity \(\eta\) was only O(0.1) mPa\(\cdot\)s. for changes in shear rate \(\dot{\gamma }\) between \(10^3\)-\(10^4~[\rm s^{-1}]\). To simplify the discussions, if we assume that the CNC suspension in the present study behaves as a Newtonian fluid, i.e., the change in the viscosity coefficient \(\eta\) with the shear rate is sufficiently small,

$$\begin{aligned} C_2\propto B^\alpha w^\beta {\Pi }^{\alpha -1}=(2.30\times 10^{-4}) w^{\beta } {\Pi }^{-0.713}. \end{aligned}$$
(17)

From Eq. (17), the coordinate-independent invariant \({\Pi }\) and the pre-factor B, which reflects the direction of polarization measurement, along with the mass fraction w, disentangle the magnitude of the stress-optic coefficient \(C_2\), which depends on the coordinate. In other words, birefringence (optical-anisotropy) can be universally described by an invariant and certain kinds of biases. The specific values of \(C_2\) in the present study are given in Fig. 7 using the infinite-shear viscosity at \(\dot{\gamma }= 10^4~\rm s^{-1}\) with 1.3 and 1.7 mPa\(\cdot\)s for 0.5 wt% and 1.0 wt% CNC suspensions, respectively.

Fig. 7
figure 7

Determined values of \(C_2\) for each CNC suspension in base-10 logarithmic scale calculated from Eq. (17).

Full size image

These values are consistent with the range of the earlier magnitude-approximation result. Interestingly, as can be seen from Eq. (16) and Fig. 7, \(C_2\) decreased with the invariant. This agrees with the result in Fig. 4c,d, where it was experimentally shown that \(C_2\) is not a constant value. As described previously, \(C_2\) represents the sensitivity of stresses along the optical axis. In the present experiment, the sensitivity depended on the strain rate. Previous studies have reported that \(C_1\) is determined by the internal structure inside the respective fluid, suggesting that the value changes with structural changes. Ito et al. conducted rheo-optical experiments on worm-like micelles and reported that \(C_1\) value changed with shear-induced structure (SIS)45. Muto et al. also performed experiments on worm-like micelles and suggested that \(C_1\) depended only on the total amount of worm-like micelles28. Sridhar et al. reported that the \(C_1\) value of polymer decreased with increasing extension strain55. Therefore, the decrease in \(C_2\) values may reflect the internal structure of the CNC suspensions. Additionally, \(C_2\) was found to increase with increasing mass fraction of CNC suspension. This means that the sensitivity to stress (degree of optical-anisotropy) increased with increasing concentration, which is a well-known trend.

In summary, as in the PP-type rheometer used in the present experiments, \(C_2\) can be calibrated using a flow field with a shear-velocity distribution along the optical axis. The polarization measurement direction-dependent pre-factor B is then incorporated into the SOL as \(C_2\). We suggest that \(C_2\) as obtained in the present method can be replaced by \(C_2=f(B,~\Pi ,~ {w})\) in the SOL to give a good expression for the optical-anisotropy due to the stress distribution along the optical axis. A detailed study of the physical quantities dominating the pre-factor B, which determines \(C_2\), is a subject for future work. We believe that a systematic investigation of B (\(C_2\)) also has the potential to contribute to our understanding of the flow dynamics of complex fluids.

Conclusion

This paper provides the results of an investigation into the effect of the stress distribution along the camera’s optical axis on optical-anisotropy. The original form of the stress-optics law including the stress components along the camera’s optical axis, which has not been quantitatively verified, was revisited. In the experiments, rheo-optical measurements were performed using a PP-type rheometer and a polarization camera. For the birefringent fluid, a dilute aqueous CNC suspension was used, showing properties close to those of a Newtonian fluid. The birefringence was found to be induced by the shear-stress distribution along the camera’s optical axis, rather than by a principal stress difference as considered in conventional photoelastic theory in a 2D stress field. Note that in the present system, the optical axis is along the direction of the velocity gradient perpendicular to the parallel plate of the PP-type rheometer. The birefringence was found to increase nonlinearly and monotonically with the shear rate, following a power function of the second invariant of the deformation-rate tensor. In addition, from the well-known SOL and an empirical equation proposed in a previous study, the contribution of stress to optical-anisotropy can be determined by a pre-factor representing the direction of polarization measurement and the invariant. The degree of optical-anisotropy (birefringence) in the present study corresponds to the coefficient (stress-optic coefficient \(C_2\)) of the nonlinear term of the SOL. Moreover, it was found that the \(C_2\) is not constant to the second invariant (strain rate). This significant finding demonstrates the need for the SOL that accounts for these components. Based on the present experimental results, it can be said that the assumption that \(C_2=0\) is not appropriate, especially when the SOL is applied to 3D fluid stress fields in which the stress is distributed along the camera’s optical axis. By defining \(C_2\) as a non-zero invariant function, the contribution of the stress component along the camera’s optical axis in each experimental system can be quantitatively described. In the future, SOL with \(C_2\) term may provide a deeper understanding of complex fluids showing non-Newtonian properties such as particle-particle interactions and polymer extension and contraction.