Microscopic viscosity measurement using orbital rotation in dual-beam fiber-optic trap with transverse offset

Abstract. We presented an optical system that could measure the viscosity coefficient of liquid in a micro-area. The orbital rotation of a polystyrene microsphere was realized by a dual-beam fiber-optic trap with a transverse offset. The rotation rate increased with the viscosity coefficient of the environmental medium. On this basis, the viscosity coefficients of ethanol solutions with different concentrations were measured successfully. The volume of solution samples was less than 1  μL. This provides a basis for the viscosity measurement of rare liquid or enchylema, which is of great significance for biological applications such as cell characteristics and reaction dynamics.


Introduction
The viscosity coefficient of liquid, also known as the dynamic viscosity, is a significant physical parameter for characterizing the resistance of fluid to shear motion. It has been widely used in the industrial, biological, and medical fields. However, for the viscosity measurement of rare liquids or bio-fluids, it is quite important to reduce the required volume of the sample. 1,2 On a microliter scale, within liquid media, the Reynolds number is small, meaning that the forces arising from the viscosity of the fluid are dominant over its inertia. Viscosity is therefore the dominant force in all processes relying on transport, mixing, or diffusion, and hence measurement of the viscosity is paramount in understanding such systems. 3 Researchers have conducted long-term and indepth research on viscosity measuring methods. Among them, the commonly used methods are falling ball, 4 capillary tube method, 5 and vibration measurement method. 6 However, these methods are all only suitable for the measurement of the viscosity coefficient of masses of liquid, not micro-liquid or liquid in micro-areas.
Optical tweezers make measuring the viscosity in micro-areas possible. In 2005, Pesce et al. 7 described a method that combined an unbiased position detector calibration procedure and frequency analysis of the Brownian motion of optically confined polystyrene microspheres. The measurement of the viscosity coefficient of pure water was realized by this method. 8 However, this method could only be used to measure low or medium viscous materials, such as distilled water, bio-fluids, sucrose, glycerol solutions, and silicone oils. 9,10 In 2007, Parkin et al. 11 investigated the performance and accuracy of a micro-viscometer based on rotating optical tweezers and measured the viscosity coefficient of tears. In 2013, Cooper et al. 12 measured the viscosity of a fetal bovine serum using the falling ball method based on optical tweezers. However, these methods are not considered to be widely applicable because they are limited by a highly controlled beam profile or samples with special optical properties. 13,14 *Address all correspondence to Xinlin Chen, xlchencs@163.com; Guangzong Xiao, xiaoguangzong@nudt.edu.cn In 2016, we realized the orbital rotation of the microspheres by a dual-beam fiber-optic trap with a transverse offset. 15 In this paper, we present that the period of orbital rotation of the microspheres varied with the viscosity coefficient of liquid. Based on this, the viscosity coefficient of liquid in a micro-area could be measured. The method we propose not only realizes miniaturization of the optical trap system but also has a simple operation that is reliable. Our method will have a broad application in biological and chemical fields.

Fundamentals
The ray-optics can be used to calculate the trapping force when the microsphere size is much larger than the wavelength of light. 16 The surface of the trapped object is divided into a finite number of small surface fractions to calculate the optical force exerted by a Gaussian beam. The direction and intensity of the ray that shines on each surface fraction can be determined according to the Gaussian beam profile. 16 When a single ray hits the microsphere, the trapping force generated by this ray are divided into the scattering force component dF s and the gradient force component dF g , which are given by 17 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 5 3 6 (1) where the refractive index of water n 1 ¼ 1.333, dP is the light power of the ray, s and g denote the unit vectors parallel and perpendicular to the ray, respectively, and q s and q g are the efficiency factors of the optical force, which are given by 18 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 4 3 6 where α i and α r are the angles of incidence and refraction, respectively, and R and T are the Fresnel reflectance and transmittance at the surface of the microsphere, respectively. For an unpolarized incident beam, the reflection and transmission coefficients are calculated by the average of s and p polarizations. We thus have E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 3 1 5 The total optical force F tot of the dual-beam fiber-optic trap is obtained as the sum of the force applied by two fibers. 19 No torques act on the microsphere. 17 Further, the O-xyz coordinate system is established as shown in Fig. 1. When moving in the water, the microsphere is also affected by the viscosity resistance: 20 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 7 2 3 where η is the viscosity of the surrounding medium, r 0 is the radius of the microsphere, and v represents the velocity of the microsphere, which is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 6 ; 6 6 8 vðt; rÞ ¼ F tot ðrÞ 6πr 0 η : The program designed to simulate the trajectory of the microsphere is based on calculating its position after a small increment of time Δt. The optical forces are set as constants over the time interval Δt. The next position of the microsphere after Δt is located by the Runge-Kutta method: 21 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 5 7 5 The magnitude and direction of the optical forces on the microsphere are recalculated for the new position, and the displacement process is repeated. The process is repeated until the desired dynamic properties are obtained.

Numerical Results
In this section, we carried out the trajectory simulation and motion frequency of the microsphere trapped in the dual-beam fiber-optical trap with a transverse offset. The fibers were separated by a width of 300 μm with an offset distance of d ¼ 9.7 μm. The wavelength of the trapping laser was 1064 nm. The radius of the microspheres was 5 μm with the refractive index of 1.59. We chose deionized water (refractive index n 1 ¼ 1.333, viscosity coefficient η ¼ 1.005 × 10 −3 Pa · s) as the surrounding medium. Figure 2 shows the simulated trajectory of the microsphere. Due to the joint action of the optical trapping force and the viscous resistance, the microsphere rotates along an approximate elliptic orbit in the optical trap. 22 In Fig. 3(a), we plotted the vibration waveform of the microsphere in the z direction. The vibration of the microsphere was periodic, and the amplitude was stable. Figure 3(b) shows the power spectrum, which was calculated by the fast Fourier transformation. The orbital rotation frequency of the microspheres was 1.062 Hz. Figure 4 shows the simulated orbital rotation frequency versus the viscosity coefficient of surrounding medium. According to Eqs. (2) and (3), the velocity of the microsphere decreases with the increase of the viscosity coefficient, resulting in the decrease of the orbital rotation frequency. Finally, the viscosity of the unknown mixture could be estimated based on the orbital rotation frequency of the captured microspheres.

Experiment
The optical layout for the experiment is shown in Fig. 5. In this system, two sets of 1064-nm wavelength lasers, respectively, output two trap lights. For each optical path, a fast variable optical attenuator is used to control the laser power of individual fiber. Then, 10% of the beam  is directed toward a photodetector, and 90% of the beam is directed to the micro-chip for the optical trap. Each fiber is attached to a translation stage to adjust the offset distance. A lightemitting diode is placed under the micro-chip for the dual-beam fiber-optic trap as the light source. The trapped microspheres in the chip are observed by a 10× microscopic objective lens and a CCD camera. A filter is placed in front of the CCD to filter out the capturing laser.
The concept diagram of the micro-chip is shown in Fig. 6. Two fibers are fixed by fiber arrays (FA). The spacing of the waveguide end is 300 μm, and the transverse offset is 9.7 μm. The size of chip is 4 mm × 8 mm × 1 mm. During the experiment, the liquid sample containing microspheres is imported into the microgroove between the FA. The height and width of the microgroove are, respectively, 1 and 2 mm. The volume of the liquid sample required in the experiment is less than 1 μL. The micro-chip has a good sealing property, which enables it to avoid the impact of jerks in flow.
In the experiment, we used ethanol solution at the concentrations of 10% to 50% as the liquid sample. The captured microspheres are polystyrene microspheres with a radius of 5 μm. The experimental phenomenon is shown in Fig. 7, in which the microsphere is rotated along an ellipse in the optical trap.
The measured rotation frequencies of the microspheres with different concentrations of alcohol are shown in Fig. 8. The orbital rotation frequency of the microsphere decreases with the increase of alcohol concentration.
The viscosity coefficient of the alcohol samples with different concentrations are given in Figs. 4 and 8. Figure 9 shows the comparison of experimental and theoretical values of the viscosity coefficient. The dots represent the viscosity coefficient measured in experiment;     the error bar on each point represents the standard deviation of five measurements. The solid curves represent theoretical values, and the squares denote the experimental results. The theoretical value of the viscosity coefficient used in Fig. 9 was measured by the falling ball method. 23 Compared with it, the error of the viscosity coefficient measured by our method is less than 5%, which can achieve accurate measurement of the viscosity coefficient of the liquid.

Conclusion
In conclusion, we realized the orbit rotation of polystyrene microspheres in a dual-beam fiberoptic trap with a transverse offset. The frequency of microspheres orbital rotation increases with the viscosity coefficient of the liquid. Then the viscosity coefficients of ethanol solutions with different concentrations were obtained by measuring the rotation frequency. The measuring error is less than 5%.
Compared with other existing viscosimeter, this system results in a lower fabrication cost, requires a smaller volume of the sample, and has much higher integration. In this experiment, the required volume of the sample was less than 1 μL, which is conducive to the measurement of the viscosity coefficient of rare liquid in micro-areas. The volume of this viscosimeter was only 32 mm 3 .
The method is also expected to achieve greater development in the measurement of the viscosity coefficient of non-uniform liquid in chemistry, biology, microrheology, and other microscopic fields, such as corrosive liquids and cell SAP. In the future, an optical tweezers chip is expected to replace the viscometer as a new tool for measuring the viscosity coefficient.