Introduction

Flowing materials containing anisotropic microscale structures are omnipresent in chemical, geophysical, and biomedical systems, e.g., chemically synthesized slurries or solutions of liquid crystals1, surfactants2,3,4, and gels5,6, botanically produced crystalline particles such as tobacco mosaic virus7,8 and cellulose nanocrystals9,10,11,12, suspended sediment like mineral flakes13,14, biological macromolecules like synovial fluids15 and red blood cells16,17,18. The microstructures dynamically alter their conformation and orientation in response to macroscopic processes, such as background flows, thereby leading to complex fluid motion. Such materials often exhibit mesmerizing kaleidoscopic patterns for optical anisotropy induced by flows, coined flow birefringence19. Light rays transmitted through the flowing media are double-refracted and exhibit interference patterns (colored fringes) under polarized environments, delivering intuitions of the complexities of microscopic internal flow structures.

Birefringence measurements, also called photoelastic measurements, have been developed successfully to quantify the degree of the anisotropy, securing a firm position in sensing solid materials’ states optically and thus noninvasively20,21,22,23,24. Such a methodological success calls for a great expectation of application of flow birefringence, building on the “stress-optic law,” linking the microscopic and the macroscopic physical processes25. The latter, originally established for solid materials, associates the internal stress (principal stress difference Δσ) with the state of index ellipsoid (IE), characterized as birefringence Δn = ∣ne−no∣, the difference of refractive indices experienced by the extraordinary and ordinary rays, together with its orientation angle χ, i.e., Δn = f(Δσ). It implicitly assumes the prompt elastic deformation (strain γ) of the materials to the stress, and thus it is also coined as the “strain-optic law”26,27, i.e., Δn = f(γ), as there is a unique relationship between stress and strain in solid materials. The simple principle is applied in quantifying the intertwined force chains in granular materials in the context of solid mechanics only via optical visualization28,29,30. With regard to flow birefringence, such a noninvasive nature is a great advantage, making in situ and in vitro measurements possible, as the microstructures in the fluid themselves work as visualizers. Therefore, flow birefringence has garnered multidisciplinary attention, foreseeing its wide application, such as quality control in chemical and food manufacturing31, and blood diagnosis of cardiovascular flows32, as well as studying hemodynamics toward biomedical applications33,34. Earlier studies have identified one-on-one correspondences between Δn and shear stress τ (or shear rate \(\dot{\gamma }\)) under steady forcing conditions35,36,37,38,39,40,41,42, highlighting the analogy to solid birefringence, although their relationships can be nonlinear43,44,45. These efforts imply the potential of birefringence as a promising approach to measure stress fields even in flowing materials, grounded on the establishment of the stress-optic law, i.e., Δn = f(τ) or \(f(\dot{\gamma })\). But, is it as simple as that?

We question such an unverified paradigm relying on a tacit agreement, the stress-optic law in flow birefringence. For fluids, the causation between birefringence and force acting on the bulk is way more complex than that of solids. The above attempts studying “steady flow birefringence” may lead to a misinterpretation as the steady forcing compels the one-on-one correspondence between the control parameter, τ or \(\dot{\gamma }\), and the output value Δn, as a result of equilibration between the microstructures forming the IE and the imposed mechanical forcing at infinite time. In this context, birefringence represents merely the optical property of the microstructures under a given force balance, not the macroscopic information of the flows. The unique correspondence in such extreme scenarios establishes a superficial stress-optic law in flow birefringence, which is in fact a rule of thumb valid in specific conditions rather than a firm law37,39. The crucial difference from solid materials is rooted in the relaxation of the microstructures and the indirect connection of macroscopic force to the structures through the continuous phase. The microscopic state represented by Δn at an arbitrary instance does not necessarily match the local properties of the flows like τ and \(\dot{\gamma }\)46,47,48, unless the spontaneous recovery of the microstructures to their original equilibrium states is sufficiently slow, and their elongation and contraction always follow the bulk deformation. Strictly speaking, the latter is never true even for solid materials because the timescale of the light ray passing through the material is way shorter than that of the material’s deformation. Therefore, adopting the analogy of solid birefringence to explain flow birefringence without accounting for such multi-timescale processes may mislead physical interpretation.

The question remains: What does flow birefringence tell us? Incorporating all the discussion above, the unsteady nature of fluid flows is vital in fending off the superficial establishment of the stress-optic law, which hinders the profound understanding of flow birefringence. In other words, to claim the correct establishment of the law, one must prove it under unsteady forcing conditions. Yet, only a few works have considered transient processes49,50, remaining qualitative or not accounting for spatial distributions; “unsteady flow birefringence” has been largely unexplored. Identifying a robust answer to the question above is pivotal for the practical use of flow birefringence. We delve into the mechanism of unsteady flow birefringence to attain its integrated physical interpretation through a solid but novel laboratory experiment.

Results

Oscillation experiment

We utilized xanthan gum (SatiaxaneTM CX 930, Cargill) to produce test solutions with deionized water at different concentrations, c = 0.1, 0.2, and 0.4 wt.%. Its viscosity-average molecular weight is estimated to be 8.8 × 105 g/mol using the Mark–Houwink–Sakurada equation51. Figure 1a is a photograph of the solution, c = 0.4 wt.%, forced by a rotating impeller in a glass beaker placed in front of a backlight. The left half is sandwiched with orthogonally aligned linear polarization filters. The solution exhibits high transparency, as demonstrated in the right half of the figure, and produces vivid interference coloration under polarized conditions when subjected to external forces. The corresponding video is available from Supplementary Movie 1. Their rheological properties were measured using a standard torque-type rheometer (MCR702e, Anton Paar) with a double-gap measuring system at 20 °C. As previously known in other works52,53, the solutions were characterized to be shear-thinning as shown in Fig. 1b. They also exhibit viscoelasticity under oscillatory shear flows. Figure 1c shows the storage \({G}^{{\prime} }\) and loss G″ moduli at the strain amplitude of 10%. Strain sweep tests were first performed at a fixed oscillation frequency to determine the linear viscoelastic regime, where \({G}^{{\prime} }\) and G″ remain constant. Subsequently, frequency sweep tests were performed as shown in Fig. 1c. In the xanthan gum and many other polymer solutions, a single crossover frequency at which \({G}^{{\prime} }\) and G″ intersect is observed54. A representative relaxation time Tr for each concentration is defined as the reciprocal of the frequency at a cross-point of \({G}^{{\prime} }\) and G″ as shown with the dashed lines in Fig. 1c. The cross points were determined by interpolation for c = 0.1 and 0.2 wt.%, and extrapolation for c = 0.4 wt.% as Tr = 6.2, 116, and 1635 s, respectively. In the linear viscoelastic regime, the classical Maxwell model serves as an essential and the most straightforward framework for decomposing the viscous and elastic contributions to the shear stress. Estimating the representative relaxation time from the cross-point frequency, generally called the crossover method, therefore, does not compromise the generality. In addition, since viscoelasticity is a generic feature of polymer solutions55, the following results and discussion will hold without loss of generality. The rheological properties we characterized here will depend on the polydispersity of the xanthan gum and the temperature54,56,57, but we ensured that those were identical by consistently utilizing the same materials for the rheological measurements and the later unsteady measurements.

Fig. 1: Birefringent fluids, aqueous solutions of xanthan gum.
figure 1

a Photograph of the xanthan gum solution (SatiaxaneTM CX 930, Cargill), concentration of c = 0.4 wt.%, continuously forced by a rotating impeller. Only the left half is sandwiched by orthogonally aligned linear polarization filters. See the corresponding video, Supplementary Movie 1. b Viscosity curves for different concentrations. c Frequency dependence of the storage \(G{\prime}\) and loss G″ moduli at the shear strain amplitude of 10%. The dashed lines indicate the cross points of the storage and loss moduli at each concentration, corresponding to the inverse of the relaxation time Tr.

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To impose unsteady forcing on the test fluids, we employed a cylindrical system utilized earlier58. Its fluid domain is a radius of R = 75 mm and a height of L = 40 mm, as schematized in Fig. 2a. The wall was oscillated azimuthally at a given frequency of fo and azimuthal rotation amplitude of Θ, providing the unsteady (periodic) forcing to the system. The wall velocity is controlled as \({u}_{{{\rm{wall}}}}(t)={U}_{{{\rm{wall}}}}\sin (2\pi {f}_{{{\rm{o}}}}t)\), where Uwall = 2πRfoΘ. We varied fo to be 0.25, 0.5, and 1 Hz (corresponding to the periods of To = 4, 2, and 1 sec) while fixing Θ = π/4. This configuration realizes axisymmetric and azimuthally unidirectional flows in the cylindrical domain due to the azimuthal momentum transfer from the wall to the interior, as shown in Fig. 2b, and is often used to examine local rheological properties of complex fluids59,60,61,62. The detail of the oscillatory cylinder system is described in “Methods” and Supplementary Information. Please note that Uwall was carefully set while assuring the reversible deformation, i.e., we did not observe any signatures which may be observed at very high shear rate, such as the shear-induced structures (SIS), effectively altering the material’s rheological properties locally3,63. We, therefore, consider the material properties like Tr to be unchanged in space and time.

Fig. 2: Overview of the oscillation experiment.
figure 2

a Schematic of the azimuthally oscillating cylinder, producing axisymmetric and unidirectional flows. b Snapshot of velocity distribution measured by particle tracking velocimetry. c The corresponding color image from the flow birefringence measurement. Time variation of d the azimuthal velocity imposed on the wall uwall, e azimuthal velocity uθ, f shear rate \(\dot{\gamma }\), g orientation angle χ, and h birefringence Δn. The data are for c = 0.4 wt.% and fo = 1 Hz. Colors represent different radial positions. Only one period of oscillation To is displayed. Experimental movies corresponding to (b, c), including other conditions, are available from Supplementary Movies 2–4.

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The setup was comprised of transparent materials that can maintain the material’s temperature constant at 20 °C, the same as that for the rheological measurements (Fig. 1), using a thermostatic bath while enabling optical visualization with two different configurations: One is for direct flow field measurements with particle tracking velocimetry (Fig. 2b), consisting of laser sheet that sliced mid-height of the cylindrical domain to illuminate tracer particles; another is for birefringence measurements utilizing light rays from an LED backlight source transmitted vertically through the fluid layer sandwiched by crossed linear polarizing filters recorded as color information (Fig. 2c). These two experiments, performed individually, quantified radial and time variations of azimuthal velocity uθ(r, t), shear rate \(\dot{\gamma }(r,t)\), orientation angle χ(r, t), and birefringence Δn(r, t). Here, \(\dot{\gamma }\) is obtained as a radial derivative of uθ, representing the macroscopic deformation rate. χ and Δn are the state of IE from the perspective of optics, and simultaneously characterize the coarse-grained microscopic polymer’s state, orientation, and elongation, respectively. We thus primarily utilize these quantities as metrics of macroscopic, uθ and \(\dot{\gamma }\), and microscopic, χ and Δn, features of the system. The results were confirmed to be highly reproducible, allowing direct comparison of the physical quantities. The details of the analytical procedures are provided in “Methods” and Supplementary Information, Figs. S1–S4. Experimental videos for all concentrations are available from Supplementary Movies 2–4.

Figure 2d–h showcases an exemplary case of c = 0.4 wt.% and fo = 1 Hz, which exhibits typical behaviors common across all the experimental conditions. Colors represent the radial positions. The azimuthal velocity uθ in Fig. 2e shows the decay of azimuthal momentum in the radial direction, as well as the phase shift of the propagating oscillatory shear flow toward the interior. The sinusoidal forcing imposed at the wall (Fig. 2d) does not maintain its shape, indicating the viscoelastic properties of the fluid under the condition of large amplitude oscillatory shear (LAOS)64. The shear rate \(\dot{\gamma }\) in Fig. 2f shows two characteristic phases, showing quasi-linear change and a plateau. Analogously, the orientation angle χ in Fig. 2g exhibits the two phases of linear change and plateau, but with a phase shift to \(\dot{\gamma }\). The birefringence Δn also shows periodicity like the other quantities; however, the values never return to zero, especially at radially outward positions. We remark that these phase shifts and non-recovery nature were identical in the other concentrations outside the LAOS regime, assuring their generality in unsteady shear flows (see further details in Supplementary Information).

Exploration of the stress-optic law in flow birefringence

To explore the stress-optic law (the shear rate-optic law) in unsteady flow birefringence, we directly compare \(\dot{\gamma }\) with Δn as shown in Fig. 3 for the concentration of c = 0.4 wt.% at different fo, a 0.25, b 0.5, and c 1 Hz. Colors indicate the radial positions. Assuming that the stress-optic law can hold in flow birefringence, a given \(\dot{\gamma }\) must take a specific Δn, i.e., \(\Delta n=f(\dot{\gamma })\), for a given birefringent fluid37,39,45. However, Δn takes multiple values at a given \(\dot{\gamma }\) for all the conditions. Particularly evident for the data at radially outward positions in Fig. 3b, c, the plots exhibit butterfly-shaped loops, suggesting the significant role of hysteresis and phase lag between the two quantities. Moreover, Δn does not necessarily take zero, and its minimum \(\Delta {n}_{\min }\) increases with r. These results prove the failure of the stress-optic law in unsteady flow birefringence, as Δn does not represent \(\dot{\gamma }\) instantaneously.

Fig. 3: Direct comparison of shear rate \(\dot{\gamma }\) and birefringence Δn.
figure 3

The data are for c = 0.4 wt.% at different fo, a 0.25 Hz, b 0.5 Hz, and c 1 Hz. The plots represented with circles and lines are projections of the instantaneous relationship of \(\dot{\gamma }(t)\) and Δn(t), and orange hexagons are the comparison of their effective quantities for a single cycle, \({\dot{\gamma }}_{{{\rm{eff}}}}\) and Δneff. Colors represent radial positions.

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What if we compare their effective values? Orange hexagons in Fig. 3 represent the relationship between the effective quantities, \(\Delta {n}_{{{\rm{eff}}}}=\sqrt{{\langle \Delta {n}^{2}\rangle }_{t}}\) and \({\dot{\gamma }}_{{{\rm{eff}}}}=\sqrt{{\langle {\dot{\gamma }}^{2}\rangle }_{t}}\), where the operator 〈⋅〉t denotes the time-average over one oscillation period, To (see their derivations in Supplementary Information). They show unique correspondences in each fo, however, they vary across fo despite the same concentration, c. The latter highlights the fact that flow birefringence is not unique to a given material property but varies depending on frequencies fo. It also suggests the frequency dependence of the stress-optic coefficient, often denoted as C in unit of Pa−1, to establish the linear stress-optic law Δn = CΔσ. It is thus reasonable to assert that the failure of the stress-optic law (or the shear rate-optic law) persists even when considering effective quantities.

Mechanism of unsteady flow birefringence

The experimental results underscore the failure of the stress-optic law in flow birefringence. Yet, why does it not hold? Here, we delve deep into the mechanism of unsteady flow birefringence based on the results shown in Figs. 2 and 3. Back to the origin of birefringence, it exhibits the retarded light rays through an anisotropic IE originating from the dielectric constant tensor formed by the microstructures, i.e., polymer chains in this study. The polymer chains consist of anisotropic monomers, who has intrinsic birefringence. But under the state of rest, each monomer orients randomly, resulting in zero birefringence on average of the polymer chain, Δn ≈ 0. When the system is forced, the polymer chains deform and rotate, resulting in Δn > 0 and χ ≠ 0. The polymer’s length scale, \({{\mathcal{O}}}(100\,{{\rm{nm}}})\)53, is comparable to the visible light wavelength, and thus let us assume that the IE immediately reflects the microstructure’s state integrated over a small volume for the brevity of the following discussion. That is, we consider Δn and χ as measures of coarse-grained polymer’s states, locally averaged elongation, and orientation.

Inspired by the results shown in Fig. 2d–h, we provide an illustrative interpretation of the mechanism of unsteady flow birefringence observed in the oscillatory experiment in Fig. 4a. The top row illustrates the conformation of the polymer chains in dilute solutions53,55 forming the IE, and the three panels beneath are the schematic time evolution of χ, Δn, and \(\dot{\gamma }\), depicted from Fig. 2g, h, and f, respectively. Note that the trends in these profiles are common for all conditions (see Supplementary Information, Fig. S5). The profiles are aligned to set t = 0 when χ = 0 at which Δn takes its minimum \(\Delta {n}_{\min }\). We identified four phases according to the deformation and the orientation of the IE. First, χ increases monotonically at the beginning as well as Δn. Note that Δn departs from \(\Delta {n}_{\min } > 0\), suggesting that the IE is always elongated due to a certain offset deformation under the oscillatory forcing. The latter indicates that the polymer chains do not recover their isotopic structures at the state of rest. After this first “elongation-rotation phase,” χ shows a plateau while Δn keeps increasing, the “elongation phase.” During these two phases, \(\dot{\gamma }\) is almost constant, meaning the steady shear is imposed effectively. The signals of χ and Δn synchronously change in the first phase can be considered to be a signature of the small amplitude oscillatory shear (SAOS) regime, keeping the polymer’s deformation small and linear64. Conversely, the asynchronous signals of χ and Δn in the second phase appear to reflect the large, nonlinear deformation of the LAOS regime. The elongation phase finishes once \(\dot{\gamma }\) starts decreasing, corresponding to the moment that Δn reaches its peak. With the decrease of \(\dot{\gamma }\), Δn rapidly decreases while χ remains approximately constant (“contraction phase”). The contraction occurs more rapidly than elongation under identical deformation conditions, presumably due to the spontaneous restoring forces of the polymer chains facilitating contraction. In contrast, elongation proceeds against these restoring forces, resulting in a slower response. The contraction phase continues, accompanied by the decrease of χ while \(\dot{\gamma }\) keeps decreasing towards negative and Δn approaches \(\Delta {n}_{\min }\). The latter “contraction-rotation phase” comes back to the first elongation-rotation phase when χ reaches zero, but in the other azimuthal direction. The oscillatory forcing repeats these four phases periodically. This suggests that the Δn represents the total deformation (shear strain) accumulated in the microstructures since t = 0. The above physical interpretations of the polymer’s conformation are aligned with those for the well-known Oldroyd-B65 and FENE-P66 models, derived theoretically by coarse-graining the bead-spring model suspended in viscous fluids67,68, and are now tied with the consequential optical anisotropy that manifests as Δn and χ.

Fig. 4: Mechanism of flow birefringence.
figure 4

a Illustrative interpretation of temporal evolution of χ, Δn, and \(\dot{\gamma }\) over one oscillatory period To shown in Fig. 2. The evolution of the IE's state over half a cycle is schematized at the top. The oscillatory forcing results in four different phases in one cycle, identified by rotation χ and deformation Δn of the IE. Schematic of the relationship between the macroscopic deformation (parallelograms) and the IE (ellipsoids) for b large De and c small De. Once an oscillatory forcing with large De > 102 is applied, the IE does not recover its original isotropic shape because of negligible relaxation during the one oscillation cycle, i.e., \(\Delta {n}_{\min } > 0\), and the IE immediately reflects macroscopic deformation irrespective of the amplitude of the oscillatory forcing. On the other hand, the IE does not represent macroscopic deformation due to significant relaxation for small De ≤ 102. Steady forcing represents the limiting case as De → 0, where the birefringence Δn∞ emerges due to the equilibrium established between macroscopic forcing and microscopic relaxation in the limit of infinite time.

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Deborah number dependency

Division of the mechanistic processes of unsteady flow birefringence into four distinct phases (Fig. 4a) allows deepening the comprehension of the underlying physics of flow birefringence that underscores the importance of the characteristic relaxation time Tr of the microscopic polymer structures. Δn keeps increasing while a constant shear rate is imposed macroscopically and drops when the stress is loosened. Although the polymer’s deformation cannot be directly characterized, it suggests that the microstructures accumulate the elastic deformations by the shear stress and consequently exhibit them as birefringence. However, the microstructures inherently undergo relaxation to restore their original isotropy, as underpinned by the existence of Tr (Fig. 1c), thereby mitigating the accumulation. Its importance differs depending on the characteristic timescale Tc of the process that imposes the external force and is characterized by the Deborah number69 as

$${{\rm{De}}}=\frac{{T}_{{{\rm{r}}}}}{{T}_{{{\rm{c}}}}}.$$
(1)

This quantifies the relative slowness of the relaxation of the deformed microstructures with respect to the characteristic timescale of the externally applied forcing—small De indicates liquid-like (viscous) behaviors, whereas large De dictates solid-like (elastic). In the oscillatory systems, we naturally consider Tc = To. Accordingly, the polymer relaxation is negligibly slow for large De conditions, and the system is ultimately considered solid for De → ∞. Under this scenario, the deformation is promptly accumulated on the microstructures and is maintained as long as the external force is applied (see Fig. 4b). Contrarily, when De is small, the structures relax rapidly even while experiencing forcing (see Fig. 4c), leading to a large phase lag between the forcing and the actual deformation. The De dependency, as conventionally thought67,68, results primarily in such a phase lag, and consequently manifests as that between the forcing and Δn (see Supplementary Information, Fig. S5 for signals of different De conditions achieved with the same fo). On top of the De dependence, the response of IE to macroscopic deformation is also amplitude-dependent: In the SAOS regime64, the birefringence may be scarcely observed due to the relatively small deformation of the microstructures, and can easily recover as illustrated by the smaller closed loops in Fig. 4b, c. The viscoelasticity in the SAOS regime can be described by the Maxwell model, which is theoretically derived by assuming internal structures with linear springs. Therefore, when De is sufficiently greater than unity, a unique relationship exists between the macroscopic deformation and the internal structural response. In the LAOS regime, the deformation is carried out at the polymer or larger scale, and these internal structures deform nonlinearly against the applied external force, leading to the nonlinear evolution of Δn. The latter may explain the manifestation of nonlinear Δn-\(\dot{\gamma }\) relationships in steady but infinitely large shear strain conditions reported elsewhere43,44,45. See a further discussion in Supplementary Information, Fig. S6.

Since the relaxation is negligibly slow for large De conditions, Δn(t) directly reflects the macroscopic deformation (shear strain) of fluids γ(t) at the same moment as the microscopic structures promptly deform in response to the macroscopic flows. For fluids, the deformation is the temporal integration of \(\dot{\gamma }(t)\) from an initial state γ0 at t = t0, that is

$$\gamma (t)={\gamma }_{0}+\int_{{t}_{0}}^{t}\dot{\gamma }(t)\,{{\rm{d}}}t.$$
(2)

This general definition indicates the need to trace the shear from its initial state in a Lagrangian manner, which is usually not feasible. In the oscillatory system, however, we can readily compute the equivalent value to the effective shear over one period as \({\gamma }_{{{\rm{eff}}}}=\sqrt{{\langle {\left[\gamma (t)-{\gamma }_{0}\right]}^{2}\rangle }_{t}}\), even without knowing γ0. This is enabled by our experimental configuration, achieving axisymmetric and unidirectional flows, which offers tracking of a continuously deforming fluid parcel in Lagrangian-equivalent observation (see the derivation in Supplementary Information).

The De dependency is demonstrated as the plots of the effective birefringence Δneff versus γeff shown in Fig. 5. The colors represent different c and fo. We obtain Tr for each concentration with separate frequency sweep measurements with a rheometer as shown in Fig. 1c, resulting in a wide range of De, \({{\rm{De}}}={{\mathcal{O}}}(1{0}^{0}\,{\mbox{-}}\,1{0}^{3})\): De ∈ [1.5, 6.2] for c = 0.1 wt.%, De ∈ [29, 116] for c = 0.2 wt.%, and De ∈ [409, 1635] for c = 0.4 wt.%. Strikingly, the highest concentration, c = 0.4 wt.%, with large \({{\rm{De}}}\ge {{\mathcal{O}}}(1{0}^{2})\) shows a sound collapse across different fo, providing a one-on-one relationship between Δneff and γeff. The collapse becomes less evident with the decrease of De as lower c. In particular, the curves fully spread for c = 0.1 wt.% with \({{\rm{De}}}={{\mathcal{O}}}(1{0}^{0})\). Within each concentration for lower c conditions, the larger fo records, the larger the value of Δneff for a given γeff. Because of the rapid relaxation of the structures for low De, the obtained birefringence does not fully represent the accumulated deformation by the external force, leading to the underestimation of Δneff.

Fig. 5: Effective birefringence Δneff vs effective shear strain γeff.
figure 5

Symbols colored green, blue, and purple correspond to c = 0.1, 0.2, and 0.4 wt.%, respectively, and gradations from light to dark represent variations of fo = 0.25, 0.5, and 1 Hz. The corresponding De numbers are noted in the legend. Both effective quantities are computed over one oscillatory period (see Supplementary Information for their definitions).

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Discussion

What does flow birefringence tell us?

The results allow a conceptual formulation of the evolution equation of flow birefringence Δn(t) as

$${{\Delta}} n(t)=\underbrace{{{\Delta}} n_0}_{{{\rm{initial}}}}+\int_{t_0}^{t}\left[\underbrace{\alpha(\tau,\dot{\gamma})}_{{{\rm{macro}}}}+\underbrace{\beta({{\Delta}} n,\chi,\alpha)}_{{{\rm{micro}}}}\right]\,{{{\rm{d}}}}t,$$
(3)

where Δn0 is the initial birefringence at t = t0. We remark that Eq. (3) does not lose generality by considering the integration along streamlines in a Lagrangian manner, analogously to Eq. (2). Also, we assume that the microstructures do not aggregate or segregate spatially, unlike those in the SIS-present systems3,63, i.e., the material properties are macroscopically invariant in space and time. Here, the rate (speed) of change of birefringence, i.e., ∂Δn/∂t, is explicitly decomposed into the macroscopic α and microscopic β contributions, and their relationship is tightly related to Tr (and De). α primarily contributes to deforming the IE by the macroscopic forcing characterized by τ and \(\dot{\gamma }\), and its sign depends upon the forcing direction with respect to the IE’s state. β represents consequential relaxation and includes the polymer’s state represented by Δn and χ, as well as α, since the relaxation of the structural anisotropy depends not only on the original restoring potential but also on the interaction between the microscopic and macroscopic flow states in a superimposed manner. Although β is ordinarily negative because of relaxation, it may become positive when longitudinal compression is applied due to the slow rotational response of the structures to the flow and is released with a time delay greater than Tc.

The above formulation builds on the experimental results obtained using polymer solutions; however, it is indeed generic for other complex fluids exhibiting flow birefringence, such as suspensions containing crystalline particles with anisotropic microstructures, i.e., rod- and flake-like shapes8,9,11,12,13,14. Birefringence evolves through α originating from the orientation and rotation of the microstructures described by Jeffery orbits70 in response to the macroscopic forcing. In general, the higher the shear rate is, the faster the microstructures spin, increasing effective birefringence; however, their relationship remains largely unexplored71,72,73,74. Concomitantly, since the length scale of the microstructures is sufficiently small to affect the refractive index, \(\lesssim {{\mathcal{O}}}(1{0}^{-6}\,{{\rm{m}}})\), diffusive Brownian motions of fluid media are of fundamental importance in determining the orientation75,76. The latter inherently forces the microstructures to orient randomly, and the birefringence integrated through the optical path cancels out. Therefore, the birefringence is zero-sum without external force13, resulting in β being a negative definite. To sum up, although the mechanisms of the emergence of flow birefringence and those to diminish differ across the polymer solutions and the crystalline suspensions, the above formulation, Eq. (3), holds for both scenarios.

By combining the concept of the Deborah number (Eq. 1) and the formulation of Eq. (3), we here provide a profound understanding of flow birefringence in diverse scenarios. For a long Tr material (illustrated in Fig. 4b as large De conditions \({{\rm{De}}}\ge {{\mathcal{O}}}(1{0}^{2})\)), the change of Δn carried out by the microscopic contribution β is negligibly small, and only α effectively determines the IE’s state as α ≫ β (see the analogy between Eqs. 2 and 3). In this context, we remark that the strain-optic law established for large De (Fig. 5) is essentially the same as that in solid materials, i.e., the law is valid only when the elastic deformation is dominant. For a short Tr material, leading to small De, \({{\rm{De}}}\le {{\mathcal{O}}}(1{0}^{2})\), schematized in Fig. 4c, however, β plays a significant role in determining the consequential IE’s state by mitigating the accumulated deformation.

We also shed light on the importance of the initial state Δn0. This can be intuitively understood by considering a birefringent fluid subjected to uniform flows with initial perturbations, where non-zero Δn arises solely due to the initial conditions, though Δn diminishes because of β. This behavior is analogous to a system undergoing plastic deformation, where residual birefringence persists even after the external forcing is removed77. Steady forcing cases that have been conventionally studied can be interpreted as limiting cases as De → 0 and also with an infinite shear strain. Under this scenario, the birefringence is obtained as the equilibrium state of IE in a balance between the macroscopic and microscopic contributions at infinite time, Δn∞ = Δn(∞) (see Fig. 4c). Integrating Eq. (3) compels a mapping τ→t→∞Δn∞ for a given τ, establishing the superficial stress-optic law. It is valid only along the same integral paths (streamlines) as formulated in Eq. (3), and thus the opposite mapping does not hold in general, i.e., τ ↚ Δn. Considering general unsteady flows consisting of physical processes across multiple timescales like turbulent flows, De and relative importance of α and β will vary both in time and space, complicating the integration in Eq. (3). To characterize stress/strain fields from flow birefringence, it is essential to track macroscopic and microscopic contributions along streamlines to decompose each other, although it persists as a technical challenge. Our conclusion is consistent with the fact that the constitutive equations for representing viscoelasticity, such as the Oldroyd-B and FENE-P models, theoretically derived by coarse-graining the bead-spring model, can predict macroscopic flow behaviors but cannot be used to track back the individual deformation and orientation states of internal structures78,79,80.

How can we harness flow birefringence?

Perceptually attracting the function of flow birefringence in diverse complex fluids has garnered multidisciplinary prospects of its utilization toward noninvasive stress measurements based on the stress-optic law originally established for solid materials. However, one must be cautious when adopting the law as the physical causation between the birefringence and the stress is arduous in flowing materials. Our experiments of unsteady flow birefringence demonstrate the failure of the stress-optic law in flow birefringence under unsteady forcing. This finding urges the undertaking of a radical overhaul of birefringence measurements in flowing materials. Yet, how can we profit from flow birefringence practically?

Flow birefringence itself still conveys microscopic characteristics when subject to given steady forcing conditions. This hints at its potential usage for determining the additives’ material properties, like molecular weight and their polydispersity81. Although our experiment utilized specific polymer solutions, and therefore, a further validation is required, the derived framework (Eq. 3) may not compromise its generality even if it is extended to various conditions of polydispersity in molecular weight distributions by considering α and β as effective values of local integrations. This is because the macroscopic and microscopic contributions to Δn, α and β, are defined for each polymer chain, and each polymer chain has its own mechanical properties—a longer chain has a longer Tr82,83,84,85. The effect of polydispersity will manifest as the integrated time evolution of Δn (see a further discussion in Supplementary Information, Figs. S7 and S8). In other words, the time-dependent signals of Δn in response to an external force can be indicative of polydispersity of molecular weight distributions, warranting its future venue from the perspective of material science. Moreover, although such approaches will be rather ad hoc, in situ and in vitro applications may be plausible to statistically detect defects or outliers, as they may exhibit noticeable differences in anomalies, considering chemical/food industrial and cardiovascular processes essential in engineering and biomedical fields.

Our formulation is general and brings an unprecedented view to the current paradigm on how flow birefringence is interpreted, highlighting the intrinsic and herculean challenge toward in situ and in vitro applications. It warns against viewing it as a practical tool characterizing macroscopic physical quantities due to its complex, nonlinear response to background flows, while promising proper venues for utilization.

Methods

Oscillatory cylinder

The oscillatory cylinder system, introduced earlier58, is a triple-wall construction. The outermost tank is a water jacket to circulate temperature-controlled water supplied from a thermostatic bath to maintain the temperature constant at 20 ± 0.2 °C during measurements. A cylinder filled with the test fluids is placed inside the tank. The innermost cylinder, whose inner radius of R = 75 mm and height of 120 mm, is then placed in it and is motorized to oscillate sinusoidally in the azimuth. A fluid inside the oscillatory cylinder is confirmed to realize azimuthally unidirectional and axisymmetric flow below a certain threshold value of the wall velocity Uwall, above which secondary flows manifest. Therefore, below the threshold, only the azimuthal component of the momentum equation (Cauchy’s equation) remains as

$$\rho \frac{\partial {u}_{\theta }}{\partial t}=\left(\frac{\partial }{\partial r}+\frac{2}{r}\right)\tau .$$

This simple system allows only a single component in the strain rate tensor in the azimuthal direction, \(\dot{\gamma }=(\partial /\partial r-1/r){u}_{\theta }\). A further detail is described in Supplementary Information. The setup is highly reproducible and is capable of implementing different optical configurations explained below to characterize and directly compare the macroscopic and microscopic features of the system.

Velocity field measurements

One of the experimental setups serves to quantify macroscopic features of the system, spatiotemporal distributions of velocity and shear rate. We seeded spherical particles, whose mean diameter of 70 μm and density of 1.02 g/cm3, into the test fluids as tracer particles. Their motions were then visualized at the half-depth of the cylindrical domain by a 500-mW continuous wave blue laser sheet, and recorded by a high-speed camera (FASTCAM MINI AX50, Photron) mounted above. Particle velocities were then quantified with the in-house particle tracking code utilized earlier58. Since the velocity vectors were obtained at the randomly distributed particle positions, we utilized the moving least-squares method to map uθ and \(\dot{\gamma }\) every 0.5 mm in the radial direction. Further detail is described in Supplementary Information, Fig. S1.

Birefringence measurements

Another optical setup is to characterize the microscopic features of the system, spatiotemporal distributions of birefringence, and orientation angle. We placed a white LED backlight source beneath the tank, and the tank was sandwiched by two orthogonally aligned linear polarization filters. The light intensity of a transmitted light ray is denoted as

$$I(\chi,\Delta n;\lambda )={I}_{0}(\lambda ){\sin }^{2}(2\tilde{\chi }){\sin }^{2}\left(2\pi \frac{\Delta nL}{\lambda }\right)$$

for a given light wavelength λ3,13. Here, \(\tilde{\chi }\) is the orientation angle of the IE relative to the crossed polarization filters. In the axisymmetric flows and the present configuration of the cylindrical coordinate, the dark fringe pattern appears every π/2 along θ with the minimum brightness signal. On the extinction angle \({\theta }_{\min }\), the IE coincides with one of the linear polarizers. The orientation angle χ, relative to the r axis, is essentially the same as the extinction angle (isoclinics) \({\theta }_{\min }\) at each r and is readily obtained by identifying the minimum brightness along θ. Also, the brightest angles, \({\theta }_{\max }\) are found at \({\theta }_{\max }={\theta }_{\min }\pm \pi /4\). The light signals at \({\theta }_{\max }\) are \(\tilde{\chi }\)-independent, i.e.,

$$I(\Delta n;\lambda )={I}_{0}(\lambda ){\sin }^{2}\left(2\pi \frac{\Delta nL}{\lambda }\right).$$

A further detail is described in Supplementary Information and illustrated in Fig. 2. We recorded the color images using the camera, same as the velocity measurements (see Fig. 2c and Supplementary Movies 2–4). The relationship between the digitized spectral information recorded as brightness values of red, green, and blue, and Δn can be theoretically computed as the Michel–Lévy interference color chart21,86 (Supplementary Information, Fig. 3). Since the relationship is nonlinear and multivariate, we utilized a deep neural network to solve the nonlinear multiple regression problem. The detailed description is provided in Supplementary Information, Figs. S3 and S4.