Optical characterization method for birefringent fluids using a polarization camera

Various techniques exist for characterizing a birefringent ﬂuid by means of extinction angle and birefringence. We summarize these techniques and present a new procedure. The approach derives the ﬁrst three Stokes parameters from images of a polarization camera and uses them to calculate the corresponding streaming birefringence by applying the Mueller matrix calculus. The required theory and a suitable experimental set-up are described. We apply the new measurement procedure to two Xanthan gum solutions and characterize their optical properties. The results agree with published data. Advantages of the new measurement technique are ease of handling, high robustness against optical imperfections and the possibility to measure a continuous response to shear in a single test series.


Introduction
If a fluid is composed of optically anisotropic particles or molecules, shear induced birefringence may occur during flow. At rest, Brownian motion randomly orientates the constituents. Therefore, on a larger scale, the fluid appears isotropic. Shear and viscous forces, on the other hand, cause local alignment leading to a preferential orientation that induces birefringence as a form of optically anisotropic response [1][2][3] . Physically the alignment is based on either the stretching and orientation of long polymer-like chains or the deformation or orientation of suspended macromolecules or colloidal particles [ 2 , 4 ]. Birefringent fluids exhibit shear-dependent polarization behavior, an effect called photoelasticity. Maxwell [5] used this effect to visualize fluid motion and documented the phenomenon in 1873, and it has continuously attracted attention ever since. Reviews are given by Cerf and Scheraga [1] , Pih [2] and Peterlin [6] , and a summary of these reviews can be found in the study conducted by Hu et al. [7] . Known birefringent fluids are for example Canada balsam [5] , vanadium pentoxide [ 8 , 9 ], cetyltrimethylammonium bromide (CTAB) [10] , bentonite [11] , tobacco mosaic virus [ 7 , 11 , 12 ], milling yellow [13] and Xanthan [14][15][16][17][18] solutions. We focus on Xanthan solutions in this study.
Birefringent fluids can be used in flow measurement and visualization techniques. They have a major advantage in that they allow noninvasive measurement and therefore offer the possibility of studying shear rates and shear stresses within the bulk fluid [19] . This is particularly interesting for the study of biomedical flows where flow-induced stresses can evoke physiological responses. Currently no single ideal The second is the extinction angle . This is the angle between the flow direction and the refractive index axis. The relationship between flow direction, refractive index axis and the resulting extinction angle is illustrated in Fig. 1 . Peterlin and Stuart [ 27 , 28 ] investigated and summarized the relationship between a flow field u in Cartesian coordinates: with the velocity gradient and shear rate ̇= , the particle orientation of rigid submicroscopic optically anisotropic ellipsoids and the position of the resulting main refractive index axes 1 , 2 . Following their findings, it is notable that birefringence is related to size, shape and optical properties of the particles and the solute, whereas the extinction angle only depends on the size and shape of the particles.
These relationships summarize the optical response of a material under shear. Theoretical approaches relating birefringence to strain rates have been published by Wayland [ 29 , 30 ] and Aben [ 31 , 32 ]. Wayland developed a 2D orientation theory that relates birefringence and refractive index axes directions to the principal strain rate. He assumed ellipsoidal particles and took into account their optical anisotropy, their rotary-diffusion coefficient and axial ratio as well as their mean index of refraction and the volume concentration of the solution. Aben presented a photoelastic tomography approach for 3D flow birefringence studies by assuming weak birefringence and interpreting integrated photoelasticity as tensor-field tomography. With the help of the Radon transform, Aben derived 3D field equations which link the optical response of the flow to its velocity field.
For the experimental characterization of birefringent fluids, various test devices and measurement procedures have been developed to determine Eqs. (3) and (4) . Cerf and Scheraga [1] as well as Chow and Fuller [15] summarize the basic concepts. Most test procedures are of the Taylor-Couette type in which the fluid is placed in the gap between two concentric cylinders. This apparatus is considered best practice [3] . Rotating the inner or the outer cylinder causes the fluid to be sheared. If the gap is small compared to the cylinder radius, the velocity gradient within the gap is close to constant. The resulting birefringence in the gap is most easily studied by placing two crossed linear polarizers at the top and the bottom of the cylinders, viewing the gap from one end of the longitudinal axis and illuminating from the other end. Light passing through the first linear polarizer will be linearly polarized with half the intensity ∕2 . When the fluid is at rest the gap appears dark. When rotating one of the two cylinders, the fluid is subject to shear and the shear induces birefringence. As a consequence, the resulting light emerging from the gap after passing through the fluid is in general elliptically polarized. Depending on the amount of birefringence, the gap will appear bright with a dark cross. This cross is referred to as the isoclinic cross. The isoclinic cross occurs when one of the refractive index axes is in line with the linear polarization. The perceivable light intensity 90 varies with [2] :

Fig. 2.
Basic test set-up for birefringence studies, as described by Chow and Fuller [15] .

Fig. 3.
Isoclinic Cross and associated extinction angle of a birefringent fluid in a Taylor-Couette type setting [13] .
Here is the path length, the wavelength of the light, and the angular coordinate. The measurable intensity pattern Eq. (5) depends on the optical parameters Δ and . The subscript indicates the angle of the polarizers relative to each other. The basic test set-up, as stated by Chow and Fuller [15] , is shown in Fig. 2 . Fig. 3 shows part of the resulting isoclinic cross and defines the parameter as the angle between refractive index axis and plane of vibration of the incident light, in our case corresponding to the horizontal polarizer [13] : Generally, the refractive index axis 1 referred to by could be the fast axis 1 , with 1 < 2 , or the slow axis. The definition in Fig. 3 does not distinguish between fast and slow axis.
Chow and Fuller describe different methods for measuring Δ and . The standard approach requires multiple measurements. First, the position of the isoclinic cross and the extinction angle have to be determined.
The isoclinic cross occurs when the first sine term in Eq. (5) becomes zero and thus the light intensity becomes zero. This is equal to ( + ) being zero or multiples of /2. Rotating + /-45°from these locations gives the maximum intensity and birefringence can then be derived from the second sine term in Eq. (5) : This procedure has to be repeated for each desired velocity gradient and is therefore laborious. Osaki et al. [33] use a similar approach, measuring at two different points separated by 45°from each other. From these two measurements the two parameters Δ and are derived. Although being faster, this approach still requires two consecutive measurements per velocity gradient.
Due to these drawbacks, Chow and Fuller [ 15 , 34 ] designed a technique called "two color flow birefringence " which uses two lasers of different wavelengths. With this method two separate measurements can be taken simultaneously and so only one measurement setting is required. Extinction angle and birefringence are measured simultaneously. The disadvantages of this method are the complexity of setting up the lasers and the sensitivity of the lasers to disturbances such as air pockets [35] . Fundamentally, this approach implies that the material parameter Δ is wavelength independent. Although the difference in wavelength might be small, the authors indicate that each system must be checked to ensure that any dispersion of Δ at the chosen wavelengths is negligible. This technique has however been successfully used by other researchers [36] .
Another well-established approach is to use a single monochromatic laser and combine the so-called null method with the classical method of Senarmont [37][38][39] [ 37 39 . In the null method, the position of the isoclinic cross ( ) is determined by rotating a crossed polarizer/analyzer pair. The method of Sénarmont, also referred to as Sénarmont compensation, uses a rotatable analyzer together with a polarizer and a quarterwave plate to determine small retardations (birefringence) with an accuracy on the order of magnitude of the wavelength of the light. Again, the required manual operation may be considered a drawback of the technique.
A precise method to measure birefringence has also been presented by Matsuura et al [40] . They used an advanced setup transmitting circularly polarized light through a birefringent medium and applied a Babinet-Soleil plate and a rotating analyzer to convert the elliptically polarized wave back into a circularly polarized wave.
Considering Eq. (5) it becomes clear that the appearance of the isoclinic cross is due to the first sine term being zero linked with the material parameter . The most common and easiest way to measure is therefore to use two crossed polarizers and to measure the angle between the polarizers and the cross as indicated in Fig. 3 . If the second sine term in Eq. (5) becomes zero the whole gap appears dark. This is referred to as an isochromatic fringe. In this case, the following relationship holds: with N being the fringe order ( = 0, 1, 2, 3, …) and hence: Birefringence Δ can be therefore measured by counting the isochromatic fringes N as the speed of the rotating cylinder is increased. The order N = 0 corresponds to zero birefringence when the liquid is at rest. This simple but effective procedure can be found in McHugh et al. [41] and is often used to determine the birefringence of milling yellow [ 13 , 42 ]. The method can only measure defined points (isochromatic fringes) and it requires the fluid to show strong birefringence within the range of shear rates investigated, as Eq. (9) has to show multiple fringe orders.
In order to use optical fluids for quantitative measurements, reliable data for Δ and are necessary. However, variance in the published data of Δ for similar solutions suggests that the optical properties vary  [14] , Xanthan: 0.05-0.4 wt%, Solvent: Water; 2. Chow and Fuller [15] , Xanthan: 0.03 wt%, Solvent: 90% Glycerin, 10% Water; 3. Kaap [16] , Xanthan: 0.25, 0.5 wt%, Solvent: Water; 4. Smyth et al. [17] , Xanthan: 0.05 wt%, Solvent: Water with 75 wt% fructose; 5. Yevlampieva et al. [18] , Xanthan: 0.011-0.054 −3 , Solvent: Water depending on how the fluids are produced and the general material quality. Fig. 4 shows published results for different Xanthan based birefringent fluids. In practice researchers are therefore advised to calibrate their birefringent fluid. For calibration a simple and convenient characterization method is desirable. It should not require an advanced set-up or a laborious procedure and should be suitable for different types of birefringent fluids. The intention of this study is to present such an approach. Our method allows simultaneous measurement of Δ and even in transient flows whilst using only one light source. It can also be fully automated, and no manual adjustments have to be made. As the light source fully illuminates the flow chamber, single disturbances from air pockets do not affect the accuracy of measurements. By using Stokes parameters to determine Δ any effects from optical elements and surfaces of the test configuration are eliminated.

Materials and Methods
In this section we outline the theoretical approach and experimental evaluation of the measurement procedure. With the help of a polarization camera we can measure three of the four Stokes parameters that characterize the polarization state of an electromagnetic light wave. With this measured data for the polarization state, we can derive the material characteristics Δ and inducing birefringence.

Theory
A polarization camera has small polarizers on each pixel. The polarizers vary in direction depending on the pixel position. A schematic representation of a polarized pixel array is given in Fig. 5 .
A polarization camera is therefore able to simultaneously measure the intensities passing the horizontal (0°), vertical (90°) and two diagonal (45°and 135°) polarizers. The intensities are referred to as: 0 , 45 , 90 , 135 . With these intensities, the first three Stokes parameters can be determined [43] : Ideally, the first Stokes parameter can be computed as 0 = 0 + 90 = 45 + 135 . The fourth Stokes parameter requires information  about the rotation direction of the light ( , ) and therefore cannot be directly measured with a polarization camera. Stokes parameters describe the state of polarization. Changes in this state resulting from optical components can be modelled with the help of Mueller matrices [43] . The basic elements of the optical set-up used in this study are outlined in Fig. 6 . A light source emits unpolarized light of intensity . The corresponding Stokes vector representation is: The Mueller matrix of the linear polarizer 0 with axis orientation 0°is: 12) and the respective matrix of a birefringent medium , , represented as a linear retarder with being the linear phase retardance and the orientation angle of the fast axis 1 ( 1 < 2 ): 0 0 0 0 cos 2 2 + sin 2 2 cos cos 2 sin 2 ( 1 − cos ) − sin 2 sin 0 cos 2 sin 2 ( 1 − cos ) sin 2 2 + cos 2 2 cos cos 2 sin 0 sin 2 sin − cos 2 sin cos The orientation angle lies in the range of [0, ]. The resulting Stokes vector of the light, emitted from the light source and travelling through the polarizer and the birefringent medium can be calculated as: The resulting output parameters are: The relation between phase retardation and the difference of the refractive indexes Δ is given as [44] : For a known path length and wavelength , and Δ are linearly related. We learn from Eqs. (17) that the Stokes parameters 1 and 2 are periodic with period ∕2 in and with 2 in . The first periodicity means that this approach cannot distinguish between fast and slow vibration axis ( 1 and 2 ). The second periodicity leads to Eq. (9) .
In theory, knowing 0 , 1 , 2 at one point is sufficient to calculate and at this point, as Eq. (15) gives: In practice however, measurements with a polarization camera showed that it is not reliable to derive and correctly from a single point measurement. This is particularly the case in the area of isoclinics, where is close to 0 or ⋅ ∕2 and the orientation of the linear polarization is aligned with one of the main refractive indices 1 or 2 . The polarization then remains unchanged, and hence one cannot derive . In this study we measure the distributions of 1 and 2 along an extended circular arc segment and obtain and by fitting Eqs. (17) to the measured distributions. Isoclinics therefore present no longer a problem, as they are part of the distributions.
Eqs. (19) however help us to understand the measurement ranges of and . The inverse of the cosine function is only defined for values between [0, ], but results for could also lie between [-, 0]. In order to determine the sign, knowledge about the direction of rotation and therefore the fourth Stokes parameter 3 would be required. Without 3 , any resulting is either between [0, ] or between [-, 0]. Clearly, the range [-, 0] is equal to [ , 2 ] due to periodicity. The measurable range of is thus [0, ] and that of is [-∕4 , ∕4 ], which corresponds to [0, ∕2 ] due to the periodicity. It is noted once again that this approach cannot distinguish between fast and slow vibration axes, hence the measurement range. Onuma and Otani [45] used a similar set-up to the one outlined in Fig. 6 but placed a quarter-wave plate at 45°behind the linear polarizer 0 to create circular polarized light. The authors employed this approach for two-dimensional birefringence distribution measurements and quantified and by measuring 0 , 45 , 90 , 135 at every point. The use of circular polarized light seems to be advantageous compared to the linear arrangement if and are to be determined from single point measurements of 0 , 45 , 90 , 135 , as no isoclinics appear. However, the (dynamic) measurement ranges for and are different. In the circular case, the range for the phase difference is [0, /2] and that of is [-/2, /2] [45] . As a consequence, the (dynamic) measurable range for the retardation is halved, whereas it is doubled for . When using circular polarized light, the location of the fast axis can be determined. The approach is therefore advantageous when determining two-dimensional birefringence with phase differences up to values of ∕2 , as it can distinguish fast and slow axes of the birefringent sample. If phase differences are larger than /2 however, the use of a linear configuration appears advantageous, with the drawback that information about the position of the fast axis is lost. When characterizing birefringent fluids, the extinction angle only describes the relative position of the refractive index axes. Moreover, phase differences are likely to exceed values of /2. For these reasons, linearly polarized light is recommended for the characterization of birefringent fluids and is used in this study.

Material
For this study two Xanthan solutions of 0.2 and 0.3 wt% were prepared by dissolving Xanthan powder in demineralized water. The pow- der was stirred into the solution until fully dissolved. Being a food additive, Xanthan is safe to handle and cheap. It shows good birefringence behaviour that has been studied in previous literature [14][15][16][17][18] [ 14 -18 ]. The Xanthan gum powder used in this study was purchased from Sigma Aldrich. The mixtures were both prepared a week before testing and the error in concentration is about Δ = +∕− 0.05 wt%.

Experimental Procedure
The experimental set-up based on the concept introduced in section 1 is shown in Fig. 8 . A small DC motor rotates the inner cylinder at a given rotation speed. The height of the inner cylinder, equaling the path length , is 25mm. Two glasses at the top and bottom allow optical access. The outer radius of the inner cylinder and the inner radius of the outer cylinder are 48mm and 49mm respectively, leading to a gap width of 1mm. Due to the gap being small in comparison with the diameter, a constant shear rate within the gap is assumed and modeled as: Here Ω is the angular velocity and , the radii of the respective cylinders. In each test sequence, 49 single measurements were conducted by evenly increasing the angular velocity from 0-0.9 1/s, leading to shear rates between 0-43 1/s. All experiments were therefore well below the critical Taylor number [46] , ensuring laminar flow without instabilities. At every measurement point the angular velocity was kept steady for 5 seconds before the measurements to eliminate any time-dependent birefringence effects [15] . Light from a halogen incandescent lamp was filtered using a long-and a short pass filter, principally limiting transmission to a spectrum of 650-690nm. This red light was then linearly polarized with an extinction ratio of 10000:1, collimated and redirected through the Taylor-Couette flow. A monochrome polarization camera (Lucid Vision Labs Phoenix PHX050S-P, Schneider Xenon 50mm/0.95 lens) mounted above the set-up captured the optical response created by the birefringent fluid. Preliminary investigations showed that camera-related errors did not influence the results significantly. For each concentration, 15 test series were conducted. For the last 5 series the testing conditions were slightly modified to study the impact of several external parameters: First, a variation of the light source intensity led to the same measurement results. This is not surprising, as the procedure is based on normalized Stokes parameters. Second, the linear polarizer was rotated + /-1°from its original position. A change of angle of the incident linear polarized light changes the extinction angle in the same manner with respect to a global coordinate system. Taking this into account, the measured results for remain the same. Third, unpolarized ambient light was employed. Considering the following two equations, Eq. (22) and Eq. (23) , the results were not noticeably affected.
With the test set up being at rest, the polarization camera was adjusted in such a way that maximum light intensity is measured at 0 , minimum intensity at 90 and equal intensities for 45 and 135 . This setting is closest to the desired set-up outlined in Fig. 6 , with a 0°linear polarizer. However, the image of 90 still detects some light passing through the gap. These residual intensities arise from stray light reflections. Therefore, the measured Stokes vector will not be completely polarized. This is quantified by the degree of polarization ( ): A partially polarized Stokes vector ( < 1) can be considered as a superposition of a fully polarized Stokes vector ⃗ and an unpolarized Stokes vector ⃗ [43] : We assume that the remains constant during one test series (during the increase of shear rate ̇) but will vary as a function of due to non-uniform lighting, irregularities induced by the optical path and reflections from the black cylinder surfaces which are manufactured from Polyoxymethylene (POM). Fig. 9 indicates the automated image analysis. The intensities ( ) were measured at select angular coordinates, with being the angle as defined in Fig. 3 .
At rest, the fluid is not birefringent. The Stokes vector will therefore be composed of a purely linear polarized part (linear polarizer) and an unpolarized part (stray contribution). If only linear polarized components are present, 3 is zero by definition. The can then be calculated as: At every shear rate, the four images of the polarization camera were used to obtain the first three Stokes parameters: Together with the DOP ( ), the polarized part of the measured Stokes vector is given as: Here, as 3 ( ) cannot be measured, it is simply left out. With this approach we measure the distribution of ⃗ ( ) along the gap. In the following, we will drop subscript P but all Stokes parameter refer to the fully polarized part ⃗ ( ) .

Data Fitting
In order to compare the measurements with the theoretical distributions Eqs. (17) , a coordinate transformation is applied for each measurement point at every shear rate, following the geometry relationships outlined in Fig. 3 . It is important to note that the extinction angle is a function of the shear rate and thus the relation between and , Eq. (6) , is a function of the shear rate as well. Eqs. (17) can be written as: Using a numerical optimization algorithm, Eqs. (26) can be fitted to the measured distributions. We would like to note that for the numerical fitting of the phase difference only the magnitude of the two amplitudes 1 2 ( 1 − cos ) were considered. This was done in order to avoid distortions due to centerline offset errors. Fig. 10 shows an example of the normalized Stokes parameters 1 and 1 resulting from the polarized part of the measured Stokes parameters ( ) Eq. (25) . The corresponding numerical fits are also plotted in the figure. Each fit gives an associated extinction angle and phase difference for the corresponding shear rate ̇. This characterises the optical response of the fluid to shear: Plotting the uncertainty propagations ( Fig. 11 ) shows that the extinction angle cannot be measured precisely when is close to 0 (or multiples of 2 , i.e. the definition of isochromatics). Propagation of uncertainty for the phase difference is large when ( + ) is 0 or multiples of ∕2 . This means that the amount of birefringence cannot be measured along an isoclinic. It also underlines the statements made in section 2.1. regarding Eqs. (19) , that single point measurement of the Stokes parameter are not reliable for determining and . Our proposed measurement procedure evaluates the Stokes parameters along a distribution of that inevitably includes ( + ) = ∕4 (and/or multiples of it), so we assume that the parameter ( + ) does not affect the quality of our results. However, measurement uncertainties of remain large at = ⋅ (with = 0 , 1 , 2 , …), as shown in Fig. 11 (a). Plotting distributions of 1 and 2 for various shear rates in the -̇space -implying the knowledge of -leads to 3-dimensional plots that show strong similarities to the theoretical results described in section 2.1.. Such resulting data plots are shown in Fig. 12 (a) and (b) for the 0.2 wt% solution and in Fig. 13 (a) and (b) for the 0.3 wt% solution. Fig. 12 (c), (d) and Fig. 13 (c), (d) exemplify the corresponding superimpositions of the experimental results onto the theoretical distributions in the -space.

Results and Discussion
The results of the experiments described in section 2 can be compared with the relevant literature. Advantages and drawbacks of the applied optical characterization method can then be identified. The data considered are the mean values of the experimental series. The error bars indicate full range between minimum and maximum values within which all experiments fall.

Measurement of extinction angle
The evaluated extinction angles are shown in Fig. 14 for Xanthan solutions of 0.2 and 0.3 wt%, respectively. Within measurement errors, the curve of 0.3 wt% is slightly lower than the 0.2 wt% curve. The shapes of both curves follow the usual result of decreasing extinction angle with increasing shear rate [15] . Both figures also show that our results are in good agreement with the corresponding data published by Meyer et al. [14] , and thus we conclude that our measurements are consistent. The variances are bigger for the 0.2 wt% Xanthan solutions. The reason for this may be the parameter sensitivity of the experiments on the δ-axis, as discussed in section 2.4. and shown in Fig. 11 (b). This parameter selection is defined by the experimental design parameters: path length and wavelength (compare Eq. (18) ). For the 0.2 wt% case, most data fits are half way up the slope and not close to the peak, which is defined to be at δ = π and thus at Δ = 1 2 λ = 1 . 32 ⋅ 10 −5 in our case. Therefore, the amplitudes of the corresponding Stokes parameter distributions are lower for the 0.2 wt% compared to the 0.3 wt% solution, which are gathered closer to the peak. The fits are more consistent throughout the test series of 0.3 wt%, leading to smaller deviations of the resulting extinction angles and suggesting that measurement accuracy is higher in the region around δ = π and thus Δ = 1 2 λ = 1 . 32 ⋅ 10 −5 (or multiples thereof).

Measurement of birefringence
Birefringence results are plotted in Fig. 15 together with the data published by Meyer et al. The measurements are in the correct order of magnitude and appear consistent. It can be seen that variances are higher in the area around Δ = 1 2 λ = 1 . 32 ⋅ 10 −5 (equal to δ = π). This is in line with the considerations discussed in section 2.4. and Fig. 11 (a). As outlined in section 2.1., results for Δ are not unique within one period, as any resulting Δ can be either between [0, 2 ] or between [ 2 , ]. Therefore, the fitting algorithm shows decreased accuracy for birefringence values close to λ 2 , as Δ could be allocated below or above the value of λ 2 . (This would also hold for values close to λ ). For all other values the assignment is unambiguous due to the assumption of monotonically increasing birefringence. Fig. 15 shows that a higher concentration in Xanthan leads to higher birefringence. The shapes of both curves are similar to the shape of the curve plotted by Chow and Fuller [15] in Fig. 4 . At low shear rates, the gradient of the curve is highest and with increasing shear rates the gradient decreases. Meyer et al.      [14] . [14] examined two different Xanthan samples from two different suppliers. Despite using similar concentrations, both samples showed different birefringence (data points plotted with and without filling effect in Fig. 15 ). The discrepancy between our results and the data published by Meyer et al. could therefore be related to material differences not taken into account in this study. Another source of error may come from the experimental set-up. Many of the established birefringent measurement techniques described in section 1 apply a laser with a narrowly defined wavelength . Meyer et al. use a helium-neon laser with 632.8nm wavelength to obtain the data given in the plots. We use a halogen lamp and two filters to narrow the transmitted wavelengths down to the range between 650 -690nm (see section 2.3.) and substitute a wavelength of 670nm to obtain our results. Therefore, an influence of within this range cannot be excluded and the propagated error in birefringence is estimated as

Conclusions
The new measurement procedure to characterize birefringent fluids represents an attractive alternative to previous methods. The advantages are a continuous analysis with increasing shear rates, high resilience towards optical imperfections in the beam path and the possibility to determine extinction angle and birefringence simultaneously. Compared to established methods, the experimental set-up appears easier to handle. It was successfully applied to measure extinction angle and birefringence of two Xanthan Gum solutions. The results are in agreement with the literature. It is expected that the accuracy of the method can be improved further by employing a narrowband light source (e.g. LED or laser) instead of the present finite bandwidth source.