Abstract

This paper proposes and demonstrates a particle free method for flow field visualizations by analyzing liquid crystal polarizations. The proposed concept is implemented by imaging of liquid crystal flow under microfluidic environment using a crossed polarization microscopy configuration. Fringe patterns give good representation of flow characterizations for different nozzle/diffuser microchannel designs. The obtained results demonstrate that the flow field under various conditions can be evaluated. Visualizations of the flow fields are carried out by the liquid crystal polarization induced fringe patterns in nozzle/diffuser microchannels. We achieve good match between the flow field obtained by LC polarization and the simulated one. It is envisaged that the proposed methodology can make a potential impact in flow field visualization studies and related analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Liquid crystals (LCs) are well known for their optical properties, whose orientations can be tuned via various stimuli. In fact, LCs’ orientations are sensitive to electromagnetic fields, chemical presence, surface property and hydrodynamic pressure gradient [1–4]. Recent studies showed that their sensitivity to bio samples allow themselves to be bio sensor [2,4,5]. LCs also respond to a surface treatment, namely surface anchoring condition [6–8]. More importantly, flow field also can influence LCs’ orientation, which makes a perfect condition for optofluidic research [9].

Investigation on behaviors of liquid crystal under external fields has been reported in the past decades, such as electric field [10–13], temperature field [1, 14, 15] and shear flow at normal scale [16]. But the LCs’ flow behavior is still unexplored in modern microfluidics [17,18]. The coupling of flow field with molecular reorientation at micro scale is yet to be studied so as to reveal the relation between each other. The flow confinement of LCs provided by microfluidics is ideal for research upon surface treatment, bio sensing and flow visualization due to the refined control of flow conditions. Microchannel fabricated by modern microfluidic technology has a resolution of the order of μm [19–21]. The ease in geometry design and channel fabrication also provides benefits. Besides, nano-liter per hour flow rates can be achieved by modern syringe pump, which opens the door for exploring at a different scale. Under different flow conditions caused by varying flow rates, channel geometries or surface anchoring conditions, LCs will display different fringe patterns when illuminated by a polarized light source. Thus fringe patterns obtained by LCs’ polarization reveal the flow condition, which achieves the flow visualization. Traditionally adopted dye based flow visualization methods are weak in revealing the detailed flow patterns straightforwardly. Here we propose an alternative approach for flow visualization under microfluidic environment by employing LCs polarization.

We choose 4-Cyano-4′-pentylbiphenyl (5CB) as it remains nematic phase under room temperature which facilitates the flow visualization. 5CB is a type of nematic liquid crystal (NLC) called calamitics and it has a rod-like molecular structure [22]. The rod-like molecular structure become observable on illumination by polarized light when it is in nematic phase. Most of the literatures have been focusing on the dynamic behavior under electric field [10–13,23]. Very few works have been done to study the reorientation under hydrodynamic pressure gradient at micro scale [3]. The nozzle/diffuser microchannel, also called rectifier [18,24–26], is a well reported geometry which is utilized to show the inter connection of the flow field and 5CB molecular reorientations.

In this article, we present dynamic responses of 5CB liquid crystal in the nozzle/diffuser microchannel under various hydrodynamic conditions. We choose the nozzle/diffuser design as it is known for creating key flow pattern, the flow recirculation [18, 25]. When 5CB flows within the geometry, the rod-like molecules are reoriented due to the flow induced patterns including the flow recirculation. The interconnection between the flow condition and molecular reorientation is investigated and analyzed. Three opening angles of the nozzle/diffuser geometry - 15°, 30°, 45° - are designed to study the geometrical effects on the flow field visualized by the LC polarization. The results clearly show the influence of flow condition on reorientation of molecular structure, which demonstrates LC’s potentials in providing novel solutions for flow visualizations at microscale [27,28].

2. Methods

Figure 1 demonstrates the overall setup configured for performing the experiments. The optical setup shown in Fig. 2 consists of a diode pumped solid state laser of wavelength 405 nm, a collimating lens, two objective lenses, one set of polarizer-analyzer combination, a tube lens and a high speed camera interfaced to a PC. The syringe which contains 5CB is driven by the syringe pump and connected with the nozzle-diffuser microchannels via polytetrafluoroethylene (PTFE) tubing. The camera acquiring the images is interfaced to the personal computer (PC). The PC is used for image processing as well as syringe pump control through the controllers as shown in the Fig. Further details of the optical set up and fluidic set up configured for this study are as given below.

 

Fig. 1 Experimental setup for visualizing single phase liquid crystal flow in nozzle-diffuser microchannels.

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2.1. Microchannel

The nozzle/diffuser microchannel is prepared by standard soft lithographic procedure followed by polydimethylsiloxiane (PDMS, Dow Corning Sylgard 184) process and plasma bonding with one slice of microscope glass spin-coated with one thin layer of PDMS to ensure uniform surface property. After exposure to the plasma, the PDMS surface becomes hydrophilic, the surface anchoring changes from homeotropic to degenerate planar [7]. It will gradually turn back to hydrophobic and homeotropic within 1–2 hours, depending on the ambient parameters such as room temperature and humidity. The microchannel was generally stored for more than one day to ensure the homeotropic surface anchoring.

Figure 2(1) shows the overall layout of the microchannel while Fig. 2(2) shows the flow cell of the microchannel. In total 10 flow cells built the microchannel. Both nozzle and diffuser directions are tested in the experiments. Width of the microchannels are W1=150 μm, and W2=25 μm respectively with the depth d=35 μm. The parameter α represents the opening angle of the flow cell. Microchannels with three opening angles - 15°, 30° and 45° - are designed and fabricated for investigations.

 

Fig. 2 Schematic diagrams for (a) flow field, (b) directional field of LC and (c) LC orientations under crossed polarizer-analyzer.

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2.2. Optical setup

The dynamic response of the 5CB liquid crystal in nozzle/diffuser microchannels is observed with the optical setup as shown in Fig. 2. A Diode pumped Solid State laser (MDL 100 mW) at 404 nm wavelength is chosen for illumination. A laser light source is considered to avoid the interferometric patterns from various light wavelengths. A polarizer and analyzer set (from Thorlabs Inc.) is configured at cross polarization state for the purpose of revealing 5CB dynamic response to the flow induced shear. A microscope objective lens (Olympus 10X/0.3NA) is used to focus light into the microchannel and an infinity-corrected objective lens (Olympus 50X/0.5NA) is used for collection of images from the microchannel which is then mapped onto the high speed camera (Phantom m310) through a tube lens (Thorlabs Inc.). The high speed camera can capture 3,200 frames per second at full resolution of 1280 × 800, which has the ability of capturing the 5CB response in the nozzle/diffuser microchannel. All experimental results are recorded at the speed of 500 frames per second.

2.3. Fluidic setup

Centoni neMESYS, high-precision syringe pump and gas tight microliter syringe (Hamilton Gastight 1725) are utilized to create confined flow condition. PTFE micro tubing is chosen to connect syringe and microchannel to avoid chemical corrosion.

2.4. Materials

The liquid crystal adopted is the single component liquid crystal 4-Cyano-4′-pentylbiphenyl (5CB), with chemical formula C18H19N, purchased from Frinton Laboratories (FR-2240). 5CB is prepared without additional treatment and it is in the nematic phase as temperature of the microchannel is controlled at 24 °C. Its transition from nematic phase to isotropic phase occurs at temperature of 33 °C.

3. Results

3.1. Theory of nematic LC dynamics

A brief derivation of theory of nematic LC dynamics is introduced for physical interpretation [29,30]. As indicated in Fig. 2, the dynamics of the nematic LC flow can be described by velocity field v = (vx, vy) and directional field n = (nx, ny). Figure 2 (c) illustrates that light intensity can be tuned via LC molecular orientations. Under crossed polarizer-analyzer setup, when LC molecule is located at position “1”, maximum light intensity is obtained and white fringe appears, while for LC molecule at position “2” the light is blocked rendering the dark fringe.

By applying the general conservation law of mass, linear momentum and angular momentum respectively and assuming one-constant elastic constant, we can get the following equations:

vi,i=0
ρv˙i=ρFi(p+wF),i+g˜jnj,i+Gknk,i+t˜ij,j
Kni,jj+g˜i+Gi=λni
with:
t˜ij=α1nkAkpnpninj+α2Ninj+α3niNj+α4Aij+α5njAiknk+α6niAjknk
g˜i=γ1Niγ2Aipnp
where Fi is the generalized body force per unit, p is the applied pressure, wF is the energy density, K is the elastic constant, Gi represents the external body force and λ is the Lagrange multiplier. ij is called as viscous stress, which can be express in the form of Eq. (4). αi is known as the Lesile viscosity determined by the LC fluid properties. γ1 and γ2 are viscosity constants defined as such: γ1 = α3α2, γ2 = α6α5. The co-rotational time flux N is defined as the curl of relative angular velocity and direction field: N = ω × n, while A is the rate of strain tensor: A=12(v+(v)T).

It’s seen from Eq. (1)Eq. (5), apart from the conventionally velocity field v, the directional field n is also to be solved. Considering pressure driven flow in nozzle/diffuser microchannels where the external body force (including gravity) can be neglected, the hydrodynamic energy is to be balanced by both the viscous dissipation (ij) and varying the direction field of LC (wF,i & jnj,i). However the viscous dissipation ij is also determined by the directional field n. The coupling of the two fields leads to the complexity of the phenomena, yet raises the potential for flow field visualization via imaging of LC molecular orientation.

3.2. Flow field induced fringe patterns in nozzle/diffuser microchannels

Figure 3 presents the characteristics of the fringe pattern obtained from the optical setup shown in Fig. 1 when the LC flows through the nozzle/diffuser microchannel. Fringe patterns can be divided into two major groups based on their locations: (1) at the center region and (2) at the corner region. Fringes at the center determined by the hydrodynamic pressure gradient which visualizes the flow field are of great interest to us.

 

Fig. 3 Fringe pattern characteristics along both diffuser and nozzle directions.

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We obtain the results for 15° opening angle microchannel at different flow rates along both nozzle and diffuser directions. Within the scope of this article, 15° is the smallest opening angle among all microchannels. Small angle leads to less pressure variation along flow direction and thus the optical pattern varies gradually. A gradual and clear shifting of the central fringes towards the expansion on increasing the flow rate from 5 μL/h to 25 μL/h is observed which is highlighted by the dashed line and the arrows in Fig. 4. To validate the concept of this flow visualization method, we conduct a set of numerical simulations on general fluidic flow in the nozzle/diffuser microchannel with viscosity =32 cSt and density = 1008 kg/m3 [7] via finite volume method whose results are well accepted in fluid mechanics. The comparison between the fringe patterns via LC polarization and the simulation is presented in Fig. 4. Noted that one constant velocity scale is set during the result processing for comparison purpose. Interestingly, by comparing the fringe pattern via the LC polarization and the flow contour calculated by simulation we find great similarities in not only the shape of the patterns but the shifting trend of these patterns via increasing the flow rate as shown in Fig. 4. The shifting of the fringe pattern is determined by the variation of the LC molecular orientation which is induced by the flow field. In this case, within the range of flow rates from 5 μL/h to 25 μL/h, the distribution of the fringe pattern gives potentials for general investigations of fluidic flow via introducing LCs.

 

Fig. 4 Fringe patterns of 15° opening angle microchannel along diffuser and nozzle directions.

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We observe similar but more obvious fringe patterns when using the 30° and 45° microchannels as shown in Fig. 5. Larger opening angle generates greater pressure gradient which leads to comparatively more obvious fringe patterns. The fringe patterns also shift towards the expansion portion of the channel with the increasing of flow rate or Reynolds number (Re). The shifting trend of the fringe pattern is also caused by the flow shear induced reorientation of the LC molecules. Noticeably, the thread-like patterns, normally termed as topological defects, appear under higher flow rates (35 μL/h for diffuser, highlighted by the blue circle in Fig. 4). When large molecular reorientation gradient is applied and 5CB’s molecular orientation cannot change continuously, “threads” will occur. When the LC flows in a microchannel with a larger opening angle, the threads appear at a relatively low flow rate making the observation difficult. We will discuss the limitation of this method in the “Discussion” section.

 

Fig. 5 Fringe patterns of 30° and 45° opening angle microchannels along both diffuser and nozzle directions. (a) Fringe patterns of 30° opening angle microchannel, (b) fringe patterns of 45° opening angle microchannel.

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4. Discussion

Figure 6(a) illustrates the dependence of the representative fringe contours on Re. An inverse dependence of fringe number on Re is observed. The characteristics of the fringe contour have great influence on the flow field visualization. The influence of the Re on the fringe number implies the application scope of this visualization method.

 

Fig. 6 Applicable conditions for flow visualizations via 5CB polarization. (a) Representative fringe contours of various Reynolds numbers (15° opening angle), (b) variation in fringe densities as function of Re for various opening angles, (c) critical Er number for different opening angles.

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A direct impact of the fringe patterns induced by flow conditions (Re) and quantified by the term of fringe pattern as shown in Fig. 6(b). Fringe density is defined as:

χ*=χχDi15°,35=N/SχDi15°,35
Where χ* is the dimensionless fringe density, χ is the dimensional fringe density, χDi−15°,35 represents the fringe density of the 15° opening angle channel along the diffuser direction flowing at Re=0.0089 (35 μL/h), N is the total number of fringes and S is the area of one flow cell.

The fringe density in the central region is a combination of Re and channel geometries. A threshold of fringe densities exists, meaning flow field tuned fringe pattern is only sensitive under a certain range of flow rates depending on the geometries. As Re increases, the flow shear applied is strong enough to realign the molecules. When Re < 0.0013 (flow rate <5 μL/h), the fringe cannot be stabilized and a constantly shifting fringe becomes observable. When Re goes beyond the range, the molecular orientation of 5CB is incapable of capturing the flow field resulting in a lower and constant fringe density χ*. Results in Fig. 6(b) show that the opening angle also affects the fringe density χ* as the flow induced shear is determined by both the flow condition and opening angle. The topological defect in the larger opening angles appears at a lower flow rates which reduces the application scope of this method.

Normal flow stability is characterized by Reynolds number Re. However, LCs flow show instability at very low Re (typically Re ≈ 10−3). As such Erikson number should be considered for LCs flow [6]. Erikson number (Er) is defined as:

Er=vμLK
Where v is the flow velocity, μ is the dynamic viscosity, L & K represent the characteristic length and elastic constant respectively. For simplicity, we have μ = 32 cSt, K = 5.5 × 10−12 N [7].

Figure 6(c) shows the critical Er number for different opening angles. Critical Er number indicates the critical flow condition when the threads appear. We can conclude that the critical Er number drops drastically when 5CB flows in larger opening angle while difference between nozzle and diffuser direction is not significant. Large opening angle promotes the flow shear and has a relatively high fringe density and it tends to show the thread even at low Er number. When Er number is high enough, the thread will appear and make it hard for fringe pattern observation. Therefore the critical Er number defines the range of the flow condition, within which the flow visualization via liquid crystal polarization is applicable.

In summary, we have investigated visualization of the flowing through nozzle/diffuser microchannels by analyzing the fringe pattern obtained from liquid crystal polarization. The method is implemented by imaging of the flow of the liquid crystal in microchannels under a polarization based optical configuration. The flow field is visualized and characterized by analyzing the obtained fringe patterns. Flows along both the nozzle and diffuser directions are visualized via analyzing the associated fringe patterns and validated by the simulation results. Our results prove that the optical imaging of LCs flowing provides a novel avenue for flow field visualizations [31,32].

The concept of particle free imaging of flow field achieved by liquid crystal polarization provides a distinctive method for flow field visualization and its related analysis. Although the method is limited by the appearance of the topological defects, various LCs can introduced to expand the applicable flow conditions. Overall it shows great advantages in low Re flow where brownie motion might be a issue for particle based visualization method. Further more, this flow visualization method utilizes the reorientation of the LCs’ molecules whose scalar is smaller than particles and therefore might be able to give clues to the flow dynamics at interfaces. We are to explore the interfacial dynamics at microscale in the near future.

Funding

COLE-EDB, Singapre MOE Tier 1 RG 98/14, RG94/16 and RG 82/15.

Acknowledgments

The in-depth discussion with associate prof. Zhizhao Che from Tianjin Univ., China, is greatly appreciated. The authors would also like to thank prof. Nam-Trung Nguyen and Dr. Say Hwa Tan from Griffith University, Australia for their inspiring suggestions. S.V.K. completed the optical setup when he was a research fellow of School of Mechanical Aerospace Engineering, Nanyang Technological University.

References and links

1. Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012). [CrossRef]   [PubMed]  

2. C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008). [CrossRef]   [PubMed]  

3. A. Sengupta, “Tuning fluidic resistance via liquid crystal microfluidics,” Int. J. Mol. Sci. 14, 22826–22844 (2013). [CrossRef]   [PubMed]  

4. E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006). [CrossRef]   [PubMed]  

5. T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 2, 1–7 (2014).

6. A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012). [CrossRef]  

7. A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011). [CrossRef]  

8. S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006). [CrossRef]   [PubMed]  

9. A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012). [CrossRef]  

10. L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012). [CrossRef]  

11. V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

12. B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010). [CrossRef]  

13. R. Ranjini, M. V. Matham, and N. T. Nguyen, “Conoscopic analysis of electric field driven planar aligned nematic liquid crystal,” Appl. Opt. 53, 2773–2776 (2014). [CrossRef]   [PubMed]  

14. M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999). [CrossRef]  

15. D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012). [CrossRef]  

16. S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983). [CrossRef]  

17. Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017). [CrossRef]  

18. T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005). [CrossRef]  

19. K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998). [CrossRef]  

20. N.-T. Wereley, Fundamentals and applications of microfluidics (Artech House, 2002).

21. K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018). [CrossRef]  

22. S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991). [CrossRef]   [PubMed]  

23. X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011). [CrossRef]  

24. P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010). [CrossRef]  

25. N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008). [CrossRef]   [PubMed]  

26. A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004). [CrossRef]   [PubMed]  

27. Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017). [CrossRef]  

28. L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017). [CrossRef]   [PubMed]  

29. T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013). [CrossRef]  

30. I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, 2004).

31. Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017). [CrossRef]   [PubMed]  

32. H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017). [CrossRef]   [PubMed]  

References

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  1. Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
    [Crossref] [PubMed]
  2. C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
    [Crossref] [PubMed]
  3. A. Sengupta, “Tuning fluidic resistance via liquid crystal microfluidics,” Int. J. Mol. Sci. 14, 22826–22844 (2013).
    [Crossref] [PubMed]
  4. E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
    [Crossref] [PubMed]
  5. T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 02, 1–7 (2014).
  6. A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
    [Crossref]
  7. A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
    [Crossref]
  8. S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006).
    [Crossref] [PubMed]
  9. A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
    [Crossref]
  10. L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
    [Crossref]
  11. V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).
  12. B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
    [Crossref]
  13. R. Ranjini, M. V. Matham, and N. T. Nguyen, “Conoscopic analysis of electric field driven planar aligned nematic liquid crystal,” Appl. Opt. 53, 2773–2776 (2014).
    [Crossref] [PubMed]
  14. M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999).
    [Crossref]
  15. D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
    [Crossref]
  16. S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
    [Crossref]
  17. Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
    [Crossref]
  18. T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005).
    [Crossref]
  19. K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
    [Crossref]
  20. N.-T. Wereley, Fundamentals and applications of microfluidics (Artech House, 2002).
  21. K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
    [Crossref]
  22. S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
    [Crossref] [PubMed]
  23. X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
    [Crossref]
  24. P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
    [Crossref]
  25. N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
    [Crossref] [PubMed]
  26. A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004).
    [Crossref] [PubMed]
  27. Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
    [Crossref]
  28. L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
    [Crossref] [PubMed]
  29. T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
    [Crossref]
  30. I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, 2004).
  31. Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
    [Crossref] [PubMed]
  32. H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
    [Crossref] [PubMed]

2018 (1)

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

2017 (5)

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
[Crossref] [PubMed]

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
[Crossref]

2014 (2)

R. Ranjini, M. V. Matham, and N. T. Nguyen, “Conoscopic analysis of electric field driven planar aligned nematic liquid crystal,” Appl. Opt. 53, 2773–2776 (2014).
[Crossref] [PubMed]

T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 02, 1–7 (2014).

2013 (2)

A. Sengupta, “Tuning fluidic resistance via liquid crystal microfluidics,” Int. J. Mol. Sci. 14, 22826–22844 (2013).
[Crossref] [PubMed]

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

2012 (5)

Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
[Crossref] [PubMed]

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
[Crossref]

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

2011 (2)

A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
[Crossref]

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

2010 (2)

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

2008 (2)

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

2006 (2)

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006).
[Crossref] [PubMed]

2005 (1)

T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005).
[Crossref]

2004 (1)

A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004).
[Crossref] [PubMed]

1999 (1)

M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999).
[Crossref]

1998 (1)

K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
[Crossref]

1992 (1)

V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

1991 (1)

S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
[Crossref] [PubMed]

1983 (1)

S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
[Crossref]

Abbott, N. L.

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Aguirre, L. E.

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

Allender, D.

S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
[Crossref] [PubMed]

Alves, M. A.

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

An, T.

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Anna, S. L.

S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006).
[Crossref] [PubMed]

Anoardo, E.

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

Archer, A. J.

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

Avelino, P. P.

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

Bahr, C.

A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
[Crossref]

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
[Crossref]

Bera, T.

T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 02, 1–7 (2014).

Bondar, V. G.

V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

Brittain, K. S.

K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
[Crossref]

Buka, A.

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

Cadwell, K. D.

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Caggioni, M.

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

Caruso, F.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Chang, H.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Cohen, I.

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

Cui, M.

M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999).
[Crossref]

Cummings, L. J.

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

de Oliveira, B. F.

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

Duan, F.

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

Éber, N.

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

Faetti, S.

S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
[Crossref]

Fang, J.

T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 02, 1–7 (2014).

Feng, H.

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

Fronzoni, L.

S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
[Crossref]

Groisman, A.

A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004).
[Crossref] [PubMed]

Herminghaus, S.

A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
[Crossref]

Ho, S. S.

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

Huang, Y.

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
[Crossref] [PubMed]

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

Johnston, A. P. R.

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Kelly, J. R.

M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999).
[Crossref]

Kim, Y. K.

Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
[Crossref] [PubMed]

Kondic, L.

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

Kralj, S.

S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
[Crossref] [PubMed]

Lam, Y. C.

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

Lavrentovich, D.

V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

Lavrentovich, O. D.

Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
[Crossref] [PubMed]

Lim, W.

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Lin, T.-S.

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

Low, C. L.

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

Lye, S. W.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Matham, M. V.

Miao, J.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Moraes, F.

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

Nguyen, N. T.

R. Ranjini, M. V. Matham, and N. T. Nguyen, “Conoscopic analysis of electric field driven planar aligned nematic liquid crystal,” Appl. Opt. 53, 2773–2776 (2014).
[Crossref] [PubMed]

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

Oliveira, J. C. R. E.

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

Oliveira, M. S. N.

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

Ouskova, E.

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

Pergamenshchik, V. M.

V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

Pinho, F. T.

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

Porter, D.

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

Postma, A.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Priest, C.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Quake, S.

A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004).
[Crossref] [PubMed]

Quake, S. R.

T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005).
[Crossref]

Quinn, A.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Quinn, J. F.

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Ralston, J.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Ranjini, R.

Rolla, P. A.

S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
[Crossref]

Savage, J. R.

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

Schulz, B.

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

Sengupta, A.

A. Sengupta, “Tuning fluidic resistance via liquid crystal microfluidics,” Int. J. Mol. Sci. 14, 22826–22844 (2013).
[Crossref] [PubMed]

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
[Crossref]

A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
[Crossref]

Senyuk, B.

Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
[Crossref] [PubMed]

Shojaei-Zadeh, S.

S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006).
[Crossref] [PubMed]

Sousa, P. C.

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

Spicer, P.

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

Squires, T. M.

T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005).
[Crossref]

Stewart, I.

I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, 2004).

Sun, H.

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Tang, L.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Tao, K.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Thiele, U.

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

Tian, Y.

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

Tjipto, E.

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Tkalec, U.

A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
[Crossref]

Wang, R.

Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
[Crossref]

Wang, Y.

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Wang, Y. L.

Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
[Crossref] [PubMed]

Wen, S.

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

Wereley, N.-T.

N.-T. Wereley, Fundamentals and applications of microfluidics (Artech House, 2002).

Whitesides, G.

K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
[Crossref]

Wong, T.

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Wong, T. N.

Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
[Crossref] [PubMed]

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

Wu, J.

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Xiao, L.

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Xiong, Y.

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

Yang, C.

Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
[Crossref]

Zelikin, A. N.

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

Zhao, X.M.

K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
[Crossref]

Zhao, Y.

Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
[Crossref]

Žumer, S.

S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Sengupta, S. Herminghaus, and C. Bahr, “Opto-fluidic velocimetry using liquid crystal microfluidics,” Appl. Phys. Lett. 101, 164101 (2012).
[Crossref]

Biomicrofluidics (1)

N. T. Nguyen, Y. C. Lam, S. S. Ho, and C. L. Low, “Improvement of rectification effects in diffuser/nozzle structures with viscoelastic fluids,” Biomicrofluidics 2, 34101 (2008).
[Crossref] [PubMed]

Int. J. Heat Mass Transf. (1)

Y. Zhao, R. Wang, and C. Yang, “Interdroplet freezing wave propagation of condensation frosting on micropillar patterned superhydrophobic surfaces of varying pitches,” Int. J. Heat Mass Transf. 108, 1048–1056 (2017).
[Crossref]

Int. J. Mol. Sci. (1)

A. Sengupta, “Tuning fluidic resistance via liquid crystal microfluidics,” Int. J. Mol. Sci. 14, 22826–22844 (2013).
[Crossref] [PubMed]

J. Chem. Phys. (1)

S. Faetti, L. Fronzoni, and P. A. Rolla, “Static and dynamic behavior of the vortex-electrohydrodynamic instability in freely suspended layers of nematic liquid crystals,” J. Chem. Phys. 79, 5054 (1983).
[Crossref]

J. Mater. Sci. Chem. Eng. (1)

T. Bera and J. Fang, “Interaction of surfactants and polyelectrolyte-coated liquid crystal droplets,” J. Mater. Sci. Chem. Eng. 02, 1–7 (2014).

J. Non-Newtonian Fluid Mech. (1)

P. C. Sousa, F. T. Pinho, M. S. N. Oliveira, and M. A. Alves, “Efficient microfluidic rectifiers for viscoelastic fluid flow,” J. Non-Newtonian Fluid Mech. 165, 652–671 (2010).
[Crossref]

Journal of Microelectromechanical Systems (1)

K. Tao, L. Tang, J. Wu, S. W. Lye, H. Chang, and J. Miao, “Investigation of multimodal electret-based mems energy harvester with impact-induced nonlinearity,” Journal of Microelectromechanical Systems PP, 1–13 (2018).
[Crossref]

Lab Chip (1)

Y. Huang, Y. L. Wang, and T. N. Wong, “Ac electric field controlled non-newtonian filament thinning and droplet formation on the microscale,” Lab Chip 17, 2969–2981 (2017).
[Crossref] [PubMed]

Lab. Chip (1)

C. Priest, A. Quinn, A. Postma, A. N. Zelikin, J. Ralston, and F. Caruso, “Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis,” Lab. Chip 8, 2182–2187 (2008).
[Crossref] [PubMed]

Langmuir (1)

S. Shojaei-Zadeh and S. L. Anna, “Role of surface anchoring and geometric confinement on focal conic textures in smetic-a liquid crystal,” Langmuir 22, 9986–9993 (2006).
[Crossref] [PubMed]

Microfluid. Nanofluid. (1)

A. Sengupta, B. Schulz, E. Ouskova, and C. Bahr, “Functionalization of microfluidic devices for investigation of liquid crystal flows,” Microfluid. Nanofluid. 13, 941–955 (2012).
[Crossref]

Molec. Cryst. Liquid Cryst. Sci. Technol. (1)

M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5cb,” Molec. Cryst. Liquid Cryst. Sci. Technol. 331, 49–57 (1999).
[Crossref]

Nano Lett. (1)

E. Tjipto, K. D. Cadwell, J. F. Quinn, A. P. R. Johnston, N. L. Abbott, and F. Caruso, “Tailoring the interfaces between nematic liquid crystal emulsions and aqueous phase via layer-by-layer assembly,” Nano Lett. 6, 2243–2248 (2006).
[Crossref] [PubMed]

Nanoscale (1)

L. Xiao, Y. Wang, Y. Huang, T. Wong, and H. Sun, “Self-trapped exciton emission from carbon dots investigated by polarization anisotropy of photoluminescence and photoexcitation,” Nanoscale 9, 12637–12646 (2017).
[Crossref] [PubMed]

Nat. Commun. (1)

Y. K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion,” Nat. Commun. 3, 1133 (2012).
[Crossref] [PubMed]

Phys. Fluids (1)

T.-S. Lin, L. J. Cummings, A. J. Archer, L. Kondic, and U. Thiele, “Note on the hydrodynamic description of thin nematic films: Strong anchoring model,” Phys. Fluids 25, 082102 (2013).
[Crossref]

Phys. Rev. A (1)

S. Kralj, S. Žumer, and D. Allender, “Nematic-isotropic phase transition in a liquid-crystal droplet,” Phys. Rev. A 43, 2943–2952 (1991).
[Crossref] [PubMed]

Phys. Rev. E (3)

L. E. Aguirre, E. Anoardo, N. Éber, and A. Buka, “Regular structures in 5cb liquid crystals under the joint action of ac and dc voltages,” Phys. Rev. E 85, 041703 (2012).
[Crossref]

D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M. Caggioni, “Temperature dependence of droplet breakup in 8cb and 5cb liquid crystals,” Phys. Rev. E 85, 041701 (2012).
[Crossref]

B. F. de Oliveira, P. P. Avelino, F. Moraes, and J. C. R. E. Oliveira, “Nematic liquid crystal dynamics under applied electric fields,” Phys. Rev. E 82, 041707 (2010).
[Crossref]

Phys. Rev. Lett. (1)

A. Groisman and S. Quake, “A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers,” Phys. Rev. Lett. 92, 094501 (2004).
[Crossref] [PubMed]

Phys. World (1)

K. S. Brittain, X.M. Zhao, and G. Whitesides, “Softlithography and microfabrication,” Phys. World 11, 31–36 (1998).
[Crossref]

Rev. Mod. Phys. (1)

T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoliter scale,” Rev. Mod. Phys. 77, 977–1026 (2005).
[Crossref]

Rheol. Acta (1)

X. Zhang, X. Zhang, Y. Xiong, Y. Tian, and S. Wen, “Anti-electroviscous effect of near-surface 5cb liquid crystal and its boundary lubrication property,” Rheol. Acta 51, 267–277 (2011).
[Crossref]

Small (1)

Y. Huang, L. Xiao, T. An, W. Lim, T. Wong, and H. Sun, “Fast dynamic visualizations in microfluidics enabled by fluorescent carbon nanodots,” Small 13, 1700869 (2017).
[Crossref]

Soft Matter (2)

H. Feng, Y. Huang, T. N. Wong, and F. Duan, “Electrolyte effect in induced charge electroosmosis,” Soft Matter 13, 4864–4870 (2017).
[Crossref] [PubMed]

A. Sengupta, U. Tkalec, and C. Bahr, “Nematic textures in microfluidic environment,” Soft Matter 7, 6542 (2011).
[Crossref]

Sov. Phys. JETP (1)

V. G. Bondar, D. Lavrentovich, and V. M. Pergamenshchik, “Threshold of structural hedgehog-ring transition in drops of a nematic in an alternating electric field,” Sov. Phys. JETP 74, 60–67 (1992).

Other (2)

N.-T. Wereley, Fundamentals and applications of microfluidics (Artech House, 2002).

I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, 2004).

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Figures (6)

Fig. 1
Fig. 1 Experimental setup for visualizing single phase liquid crystal flow in nozzle-diffuser microchannels.
Fig. 2
Fig. 2 Schematic diagrams for (a) flow field, (b) directional field of LC and (c) LC orientations under crossed polarizer-analyzer.
Fig. 3
Fig. 3 Fringe pattern characteristics along both diffuser and nozzle directions.
Fig. 4
Fig. 4 Fringe patterns of 15° opening angle microchannel along diffuser and nozzle directions.
Fig. 5
Fig. 5 Fringe patterns of 30° and 45° opening angle microchannels along both diffuser and nozzle directions. (a) Fringe patterns of 30° opening angle microchannel, (b) fringe patterns of 45° opening angle microchannel.
Fig. 6
Fig. 6 Applicable conditions for flow visualizations via 5CB polarization. (a) Representative fringe contours of various Reynolds numbers (15° opening angle), (b) variation in fringe densities as function of Re for various opening angles, (c) critical Er number for different opening angles.

Equations (7)

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v i , i = 0
ρ v ˙ i = ρ F i ( p + w F ) , i + g ˜ j n j , i + G k n k , i + t ˜ i j , j
K n i , j j + g ˜ i + G i = λ n i
t ˜ i j = α 1 n k A k p n p n i n j + α 2 N i n j + α 3 n i N j + α 4 A i j + α 5 n j A i k n k + α 6 n i A j k n k
g ˜ i = γ 1 N i γ 2 A i p n p
χ * = χ χ Di 15 ° , 35 = N / S χ Di 15 ° , 35
Er = v μ L K

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