the technology 
 opens new exciting research and technological possibilities in dissipative 
 systems out of equilibrium . a plethora of applications , such as lab - on - a - chip 
 analysis , diagnostics , and particle synthesis , can be accessed , to 
 mention a few examples . one particularly important highlight is the 
 promise of new cheap technologies for health care , which are critical 
 for underdeveloped countries . importantly , 
 microfluidics can also be used as a probe for more fundamental studies . this is true not only to study phenomena where the micrometer - length 
 scales are a relevant part of the phenomenon itself ( e.g. , blood rheology ) but also because many systems can be better 
 controlled on micrometer - length scales where laminar flow regimes 
 dominate , allowing the simultaneous control of experimental parameters 
 that were previously hard to achieve ( e.g. , rate of mixing , shear 
 rate , concentration gradients , confinement , and geometry ) . the advantages of using such a platform for small - angle scattering 
 are also enormous . first , sample consumption is reduced to the microliter scale , allowing 
 experiments with expensive and rare materials . second , the constant 
 flow of material prevents radiation damage ( critical for x - ray synchrotron 
 radiation ) . third , flow - alignment effects can be used to our advantage , 
 allowing a gain in resolution and the possibility to probe material 
 properties and dynamic behavior that are usually not visible in nonoriented 
 samples . given the small length scales , confinement , 
 and far - from - equilibrium 
 conditions , unexpected phenomena can also arise . in some cases , very 
 complex phenomena such as folding instabilities can be simplified 
 by the confinement of one of the dimensions / degrees of freedom to 
 which the system could randomly evolve , allowing systematic studies 
 to be performed . liquid crystals ( lcs ) 
 have an intermediate degree of order between liquids and crystals , and earlier work has shown that under flow , 
 shear effects can induce unexpected out - of - equilibrium structural 
 changes . more recent studies of lcs under flow but now in 
 microfluidic devices have also started to increase considerably . , the phase has only short - range positional order such 
 as that in a regular liquid , but because of some specific molecular 
 features ( e.g. , anisotropy , amphiphilicity ) , molecules exhibit a long - range 
 orientational order , specified by a unit vector n , the nematic director . this long - range 
 orientational order gives rise to a wealth of interesting phenomena , 
 the most famous being the application in liquid - crystal displays ( lcds ) . the orientation state of the nematic director n under 
 flow is determined by the ericksen parodi ( elp ) 
 theory , which considers the effects of the viscous v and elastic el torques on n. according to the elp theory , the elastic torque vanishes at high 
 shear rates ( and hence flow rates ) , and only v is 
 significant . when the ratio  between the shear ( 2 ) and rotational ( 1 ) viscosities (  
 = 2/1 ) meets the condition 
 || > 1 ( the most common situation for nematics , including 
 5cb ) , the stable solution of the equation 
 of motion 
 ( v = 0 ) is then achieved when n aligns at a critical angle s with the flow direction 
 given by1 the presence 
 of an interface may also lead to a preferred orientation . if the orientation imposed by the interface is different from that 
 imposed by flow , then conflicting alignments may give rise to rich 
 flow behavior . these effects can be greatly enhanced in microfluidic 
 devices , where the small geometric length scales can accentuate the 
 effects of conflicting surfaces , leading to , for example , director 
 reorientations over smaller length scales than usual . this has been 
 recently explored by sengupta et al . , who observed the emergence of 
 different flow regimes resulting from the viscoelasticity of a nematic 
 phase ( 5cb ) accentuated at smaller channel gaps . in a related approach , although without microconfinement , 
 the combination between capillary flow alignment and saxs was successfully 
 employed to study the nematic  smectic transition . in this work , we are interested in the competition between 
 flow 
 and boundary conditions with respect to the orientation of nematic 
 liquid crystals . by combining small - angle x - ray scattering ( saxs ) 
 with a microfluidic device , we describe an approach to measure the 
 orientation of a flowing nematic liquid crystal directly under confinement 
 near solid or liquid surfaces . for the solid - wall confinement , , 
 we use the hydrodynamic focusing technique , where a sheet of liquid is narrowed by a surrounding fluid . in the 
 case of immiscible fluids , much of the attention has been placed on 
 droplet formation under different regimes , including nematic droplets . here , however , we aim to create 
 stratified flows and inspect the nematic director orientation , with 
 particular emphasis on the role of viscous forces from flow and surface 
 anchoring forces imposed by the solid and liquid interfaces . the microfluidic chips , made 
 of cyclic olefin copolymer ( coc or topas ) were purchased from microfluidic 
 chipshop ( catalog no . 02 - 0757 - 0166 - 02 ) and used without modification 
 of the hydrophobic surface . the channel consists of four inlets meeting 
 at a cross , with inlets 14 having lengths of 80 , 5 , 5 , and 
 6 mm , respectively ( figure 1a ) . the microchannel 
 cross section is square , with a width of 100 m . in this work , contrary to what is more conventional , the main fluid flows in the 
 reverse mode from what is usually the outlet along the main channel , 
 meeting the cross at the end ( figure 1a ) . this 
 method allows the study of pure 5cb in simple flow ( without hydrodynamic 
 focusing ) along the main channel without perturbations from the cross . more importantly , in the hydrodynamic focusing experiments , allowing 
 the nematic to flow in this direction helps to stabilize the nematic 
 sheet and avoid the displacement of the lc into side channels 2 and 
 3 . ( a ) the flow is along the + y direction ( inlet 
 1 to outlets 24 ) . ( b ) the channel cross section is square 
 with a width of 100 m . the velocity profile can be described 
 with some level of approximation by a poiseuille flow being maximized 
 at the center of the channel and fading to zero at the surfaces as 
 a result of the no - slip boundary condition . ( c ) far from the edges , 
 the nematic director n aligns preferentially along 
 the velocity direction ( y ) , making an angle  
 in the velocity gradient direction ( along x and z ) . is the angle between the velocity gradient and 
 the z direction and defines the nematic director 
 cone around the velocity direction y. yz is the angle between y and n projected onto the y  z plane ( nyz ) . yx is the angle between y ( velocity direction ) and n projected 
 onto the y  x plane ( nyx ) . ( d ) alternative visualization 
 of yz , the angle measured in the 
 2d saxs detector ( y  z plane ) , 
 and yx . in the simple flow experiments , a nemesys syringe pump ( cetoni , 
 korbuen , germany ) was used . in the hydrodynamic focusing experiments , 
 a custom - built step - motor - driven syringe pump system with three independently 
 controlled pumps was used to drive fluid flow in the main channel 
 and both side inlets . in both experiments , tygon tubing ( formulation 
 s-54-hl ) with a 250 m inner diameter was used . the tubing is 
 connected to the syringes via a needle ( gauge 29 ) with a suitable 
 grip and connected to the microfluidic chip using custom - built connectors . for the hydrodynamic focusing experiments , 
 the side fluid is a 
 2 wt % solution of triton x-100 , well above the cmc ( 0.019 wt % ) . for the fluorescence microscopy observations , 
 0.1 mm fluorescein is added . for each flow rate , a stability 
 test is performed to make sure the flow regime is stable . the stable 
 flow rates are also approached from different paths to determine if 
 they are robust or history - dependent ( cf . after the initial flow mapping through 
 microscopy and the identification 
 of stable flow regions , selected flow regimes are probed by saxs . 
 for the transmission saxs measurements , the chip is mounted on a custom - built 
 device holder , with translation capacity in the x , y , and z directions , and rotation 
 of the  ( about y ) and  ( about x ) angles ( figure 2a , bottom ) . before 
 the measurements the  alignment is more complicated , and a maximum tilt angle 
 of 2 can occur . the sample is placed normal to the x - ray 
 beam and scanned in the z and y directions . the 
 simple flow experiments were performed on the csaxs beamline at the 
 swiss light source ( sls ) , paul scherrer institute ( psi ) , villigen , 
 switzerland . the hydrodynamic focusing experiments were performed 
 on saxs / d beamline 4 - 2 at the stanford synchrotron light source ( ssrl ) , 
 menlo park , ca , usa . for the csaxs instrument , the x - ray wavelength 
 (  ) used was 1.11  , and the sample - to - detector 
 distance was 2.16 m. the data was collected on a pilatus 2 m detector . the beamline optics allowed a very small beam size of fwhm  
 6 m in the z direction and 40 m in 
 the y direction . for the saxs / d ( bl 4 - 2 ) instrument , 
 the x - ray wavelength (  ) used was 1.03  , and the sample - to - detector distance was 1.1 m. data was collected 
 on a mx-225 rayonix ccd detector . the x - ray beam had a size of 30 
 m ( fwhm ) in the z direction and 300 m 
 in the y direction . for both instruments , the obtained 
 2d scattering patterns were corrected for background by subtracting 
 the 2d pattern of the chip outside of the microchannel corrected for 
 the different thicknesses . the resulting 2d background - corrected patterns 
 are subsequently analyzed with matlab routines to extract structural 
 information through qh and ql scans or the nematic director orientation through  
 scans ( figure 2a , top ) . ( a ) typical saxs 
 pattern ( top ) and schematic of the microfluidic device ( bottom ) . the 
 device holder ( shown at the bottom ) can be translated in the x , y , and z directions 
 and rotated in  ( about y ) and  ( about x ) . the x - ray beam is along the x direction , 
 and the saxs pattern ( in the qy , qz plane ) is 
 collected in a detector behind the device . in both the chip and detector , 
 the flow is along the + y direction . a typical saxs 
 pattern ( flow rate q = 3 l / min , z = 30 m ) at small q is dominated by the nematic 
 phase correlation peak , which is inversely proportional to the nearest - neighbor 
 distance between the nematogens along their main axis . therefore , 
 the peak maximum q = 2n / d is along the nematic director n projected 
 onto the y  z plane ( nyz ) , and yz is the angle between nyz and the flow direction ( y ) . the 
 direction qh is defined as being parallel 
 to nyz , and ql is perpendicular to nyz . in our convention ,  = 0 is the negative qz axis , and the positive rotation is counterclockwise . ( b ) snapshot of 5cb flowing slowly ( approaching 0 l / min ) before 
 data acquisition at the csaxs beamline . the green box represents 
 the x - ray beam s fwhm ( 6 m along z and 40 m along y ) and hence the 
 area probed by each x - ray measurement . the orange box represents a 
 typical area probed in a vertical scan ( z intervals 
 of 4.85 m ) . ( c ) representative scattering patterns of a flowing 
 nematic phase recorded along the width ( z direction ) 
 of the microchannel at a flow rate of 3 l / min ( left ) and at 
 rest ( right ) . the patterns rotate with roughly mirror symmetry with 
 respect to the center of the channel ( clockwise in the upper part 
 of the channel ( + z ) and counterclockwise in the lower 
 half ( z ) ) . close to the edges , the magnitude 
 of the rotation is larger at rest . at the edges of the microchannel , the scattering intensity is also lower because approximately half 
 of the x - ray beam is outside of the microchannel and may have a contribution 
 from the y  x plane due to 
 a small tilt ( < 2 ) of the device in . figures 1a , b show a simplified 
 picture of the microchannel geometry , and figures 1b  d show the coordinate system used in this work . the 
 channel cross section ( x  z plane ) is square with a width of 100 m ( figure 1b ) . because of the micrometer - sized dimensions in both the z and x directions , the velocity gradient 
 ( or shear rate ) extends in both of these directions ( i.e. , vy/x  0 and vy/z  0 ) , and therefore there 
 is no neutral direction . the velocity profile can be described with 
 some level of approximation by a 2d poiseuille flow , which is maximized 
 at the center of the channel and fades to zero at the surfaces ( no - slip 
 boundary condition assumed , cf . importantly , throughout this work , the flow ( velocity v ) in the microchannel is always along the + y direction . the average orientation of a nematic liquid crystal is given by the 
 director n , and far from the walls , n tends to align along the velocity direction ( y axis ) 
 but with an angle  with v ( figure 1c , d ) . because of the square geometry , the velocity gradient 
 extends over both the x and z directions 
 ( cf . figure s3 ) ; therefore , the director 
 can rotate about a cone determined by a second angle  ( i.e. , 
 rotation about the y axis ) in which not changing 
 the magnitude of  changes its projection in the y  z plane ( containing v and zv ) and y  x plane ( containing v and xv ) ( figure 1c ) . 
  = 0 in the vicinity of the horizontal walls ( z = zwall = 50 m ) , 
 and  = 90 in the vicinity of the vertical walls ( x = xwall = 50 m ) . anywhere in the channel ( away from the walls ) , (x , z ) can be calculated from the velocity gradient 
 profile in the x  z plane , 
 according to  = tan(xv/zv ) . importantly , yz = tan(cos  tan  ) is the projection 
 of  in the y  z plane 
 ( the 2d - detector plane in the saxs regime ) and is equal to  
 when  = 0 . yz is the orientation 
 angle measured in our microfluidic saxs setup ( figures 1c and 2a ) . yx = tan(sin  tan  ) is the 
 projection of  in the y  x plane and is equal to  when  = 90. the 
 flow rate q is varied from 3 to 0 l / min , 
 covering a range of shear rates  = 200 to 0 
 s ( table s1 ) . after 
 we switch to a new flow condition , 1030 min of waiting is 
 allowed before the next measurement starts . below 0.1 l / min , the pump is switched off , and after 10 min , a new scan is started , 
 which we denote as q approaching 0 l / min . 
 throughout the duration of the scan ( 10 min ) , the phase continued 
 flowing slowly , as observed with the video camera . after this measurement , 
 we waited 30 min more before a new scan was started , which we denote 
 as q  0 l / min . in this last scan figure 2 shows the experimental setup used 
 for saxs data collection and an example of a 2d saxs pattern in reciprocal 
 ( qh , ql ) space 
 ( a ) , a snapshot of 5cb in the microfluidic chamber at a low flow rate 
 ( b ) , and a series of 2d saxs patterns with the x - ray beam going through 
 different positions ( z ) across the width of the microfluidic 
 channel ( c ) . each saxs pattern corresponds to a displacement in the z direction ( across the width of the channel ) with a step 
 size of 4.85 m . the saxs pattern collected by a 2d detector 
 is displayed with the same orientation as for the microfluidic device . the saxs pattern is dominated by a broad peak centered at qh = 0.254  ( figure 2a , c ) . this peak results from the short - range positional 
 liquid order between nearest neighbors along the direction parallel 
 to n. hence , the azimuthal ( in plane ) orientation 
 of the peak directly gives the orientation of n projected 
 onto the y  z plane ( i.e. , nyz ) . in other words , the saxs 
 patterns directly give the director angle projection onto the y  z plane : yz . the peak is not perfectly radially symmetric but is somewhat 
 extended in the ql direction . as such , 
 normal azimuthal integration smears the pattern to some extent . for 
 this reason , we choose to integrate the intensity in projections along 
 the ql and qh directions ( figure 2a , top ) . the most striking 
 effect that can be seen in figure 2c is the 
 variation in the peak orientation as we scan along z. the rotation in the 2d saxs patterns exhibits roughly mirror symmetry 
 with respect to the center of the channel ( z = 0 
 m ) i.e. yz ( the angle between 
 the axis connecting both peaks and the y axis , figure 2 ) , is approximately zero in the middle of the channel , 
 and rotates clockwise in the upper half of the channel ( + z ) and counterclockwise in the lower half ( z ) . this effect also seems to be pronounced at rest ( q  0 l / min ) because the tilt in yz seems to be larger when compared to q = 3 l / min . figure 3a shows qh scans through the peak maximum ( qh = 
 0.254  ) of 5cb at q = 3 
 l / min for the 2d saxs patterns at z = 0 m 
 ( figure 2c ) averaged over different ql intervals . 1.4 times the 
 size of a 5cb molecule , in good agreement with the suggestion that 
 5cb molecules are predominantly in a dimeric form , with their benzene 
 rings forming a pair , and the alkyl chains protruding outward . the half - width at half - maximum ( hwhm ) of the peak is 0.057  . this corresponds to a correlation domain with 1/0.057 
  = 17.5  , which is expected for liquids 
 with near - neighbor short - range positional order . one - dimensional scattering 
 profiles of the 5cb nematic confined 
 in the microfluidic device . the coordinates ( qh and ql ) in reciprocal space are 
 defined in figure 2a ( top ) . ( a ) qh scans integrated over different ql intervals in the center of the channel ( z  0 m ) at a flow rate of q = 3 l / min . ( b ) qh scans ( integrated over a ql range from 0.036 to 0.036  ) at q = 3 l / min and at different z positions along the channel ( cf . figure 2c , which shows that the z range goes from 
 50 to +50 m ) . the proximity to the 
 microchannel boundaries does not modify or distort the 1d profiles 
 integrated over ql . ( c ) qh scans ( integrated over the ql range from 0.036 to 0.036  ) at 
 different flow rates . the 1d profiles are not modified or distorted 
 in the investigated flow rates . in both ( b ) and ( c ) , the curves are 
 normalized and slightly offset for ease of visualization . in all panels 
 ( a  c ) , the circles constitute scattering data and the lines 
 constitute lorentzian fits . in figure 3b , qh scans of the central region of the pattern ( ql interval of 0.036 to 0.036  ) are shown for z at the center and close to the 
 edges of the channels . as shown , no noticeable differences can be 
 spotted between the curves , and the peak positions remain the same . this extends also to the case of no shear , in which the peak position 
 and width are essentially the same ( figure 3c ) . this indicates that flow effects are not distorting the short - range 
 positional order of the nematic liquid crystal . instead , these effects 
 result in changes in the orientation of n , which 
 is the most common situation for nematics under flow . the 
 shift in the director orientation  is more clearly seen in azimuthal 
  scans in the ( qh , ql ) plane ( figure 2a , top ) . by fitting 
 the peak positions in we fit the peaks with 
 a double lorentzian equation of the form2where we restrict the position 0 of peak 2 to exactly 180 away from peak 1 because both 
 peaks are related through a two - fold rotational axis . a1 and a2 are proportional 
 to the heights of peaks 1 and 2 , b1 and b2 are the hwhm s of the peaks , and c  0 is a constant related to the background . in figure 4 , normalized  scans and 
 respective fits are shown for q  0 l / min . 
 as can be seen , when going from the top to the bottom of the channel 
 ( bottom to top in figure 4a ) , 0  90 at z = 0 m . 
 because the flow direction is along the y axis and 
 is at an angle of  = 90 , the director angle with the 
 flow direction yz is directly extracted 
 through yz = 0  90.  scans and respective double - lorentzian fittings 
 ( cf . figure 2a , top , for the definition of 
 angle  in the 
 ( qh , ql ) plane ) 
 measuring the orientational width of the nematic correlation peak 
 for 5cb in a microfluidic device at q  0 
 l / min . the quality of the data ( and hence 
 the fitting ) is reduced at the edges of the microfluidic device ( z  53 m ) , as a result of part of the 
 x - ray beam being outside the microchannel and hence not hitting the 
 5cb nematic . in figure 5a , b , we show the dependence of 
 yz on z for flow 
 rates of 0.2 l / min  q  3 l / min 
 and q < 0.1 l / min . note that to facilitate 
 the observation of symmetry ( or lack thereof ) of yz across z with a mirror plane in 
 the middle of the channel ( z = 0 m ) , we display 
 the absolute value |yz| instead . 
 also , for easier association of the yz dependence with the poiseuille profile immediately 
 evident is the fact that for q  0.2 l / min 
 all of the curves overlap very well . likewise , q approaching 
 0 and q  0 l / min also display very 
 similar profiles to each other . more striking is the obvious difference 
 in curve profiles between the q  0.2 and q < 0.1 l / min groups . in all cases , yz is negative ( clockwise rotation ) in the 
 upper half of the microchannel ( z > 0 ) and is 
 positive 
 ( counterclockwise rotation ) in the lower half ( z < 
 0 ) . the same information can be seen in figure 5c for flow rates of q = 3 l / min and q  0 l / min and where the nematic director 
 projection nyz orientation 
 is represented in the form of an orientation field as a function of z and y. figure 5d shows a schematic of the 5cb molecules and nematic director orientation 
 and corresponding 2d saxs patterns . fitting results for the nematic director 
 angle projection yz at different z positions 
 along the microchannel ( cf . results for 
 the ( a ) 0.2 l / min  q  3 l / min 
 and ( b ) q < 0.1 l / min regimes . the z position is rescaled by centering  = 0 at zrescaled = 0 m . note that the abscissa 
 represents |yz| in reverse order 
 for ease of visualization because the resulting z  yz dependence 
 resembles the poiseuille flow profile . solid data points result from 
 x - ray patterns where the beam is mostly inside the microchannel in 
 the y  z plane . open data 
 points result from patterns where a significant fraction of the x - ray 
 beam was hitting outside of the y  z plane , catching part of the y  x plane because of a small rotation of the microfluidic 
 device holder about the y axis parallel to the flow 
 ( i.e. , the  angle shown in figure 1a ) . note that some points close to z = 55 m shift 
 to smaller angles . a clear distinction in the flow profiles can be 
 seen for q  0.2 and q < 
 0.1 l / min . in the first group ( a ) , the director tilt is controlled 
 by flow ( i.e. , the leslie solution discussed in the text , where the 
 nematic director assumes a dominant angle s ) . the red and blue lines represent the leslie solution for s = 11 using the poiseuille and non - newtonian velocity 
 profiles , respectively ( which are very similar ) . in the second group the line represents the 
 escaped director configuration model ( eq 3 ) 
 assuming r = 71 m ( half of the diagonal length 
 of the square channel ) , resulting in r ( the angle of n at the surface ) = 32. 
 ( c ) schematic representation of the orientation of the nematic director 
 as a function of microchannel z position for q = 3 and 0 l / min . each blue bar represents 
 the average orientation of a rectangle with height z  6 m and length y  40 m ( the fwhm of the beam profile ) . the director 
 angle yz is multiplied by 3 to 
 facilitate visualization of the changes across z and 
 for the different q values . ( d ) schematic illustration 
 of the molecular arrangement for selected points along the orientation 
 field in ( b ) and respective x - ray scattering patterns . under pressure - driven flow ( e.g. , poiseuille flow ) , far from the 
 boundaries , and when the ratio  between the rotational ( 1 ) and shear ( 2 ) viscosities (  = 2/1 ) meets the condition || > 1 
 ( which 
 is the case for 5cb and most known nematics ) , 
 the torque  on the nematic director usually vanishes when the 
 director aligns at a critical angle ( eq 1 ) s = cos(/)/2 ( henceforth called the shear angle ) between the flow and gradient 
 directions . if the orientation imposed 
 by the surface is different from the alignment imposed by flow , then 
 a transition layer of length e1  
 ( k/ ) ( where k is related to the nematic - phase elastic 
 constant and  is an average nematic 
 lc viscosity , cf . si ) close to the surface 
 will occur , where the director gradually shifts from the boundary 
 orientation to the flow orientation . in addition , because the velocity 
 field is symmetric about the center of the channel ,  switches 
 signs in the center along a transition region of sizee2  ( he12 ) ( h = 100 m gap size in a microfluidic 
 device ) . it is therefore important to determine if the observed variation 
 in yz with z is 
 an effect of confinement , where the boundary layers could have a greater 
 influence on the orientation field far from the walls or they are 
 simply a consequence of the fact that saxs data measures yz ( the projection of  on the y  z plane ) rather than  directly 
 ( cf . figure 1d ) . as shown in figure 1c , d ,  is the 
 angle between the nematic director n and the velocity 
 ( or flow ) direction y.  is the angle measuring 
 the rotation of n about the y axis . the projection of  in the y  z plane ( saxs detector plane ) is given by yz = tan(cos  tan  ) . to compare our results with leslie s solution , we assume a 
  profile equal to s throughout the whole 
 microchannel except within the boundary layer e2 , where for simplicity  decreases linearly from  
 = s at z and x = e2 to  = 0 at z and x = 0 ( figure s2 in si ) . ( e1 is neglected because 
 our measurements in the vicinity of the channel walls are not accurate . ) 
 to obtain  , we require the 2d velocity profile ( v(x , z ) ) from which the velocity 
 gradients can be calculated ( figures s1 and s2 in si ) . from the velocity gradient components in the x and z directions ,  is easily obtained 
 (  = tan(xv/zv ) ) . yz is then computed as a function 
 of x and z using the respective 
 (x , z ) and (x , z ) and subsequently averaged over the x direction to obtain yz(z ) . a further step in the fitting includes finite 
 beam size effects . in the fitting , we compare two different velocity 
 profiles . we use the normal poiseuille flow , which yields a parabolically 
 shaped velocity profile and is suitable for newtonian fluids under 
 laminar flow . because the nematics are non - newtonian , the actual flow 
 profile deviates slightly from the parabolic shape , and we also use 
 the flow profile determined by jewell et al . for 5cb flowing in a 
 microchannel with a rectangular geometry . a more detailed description of the fitting procedure can be found 
 in the supporting information . as can be 
 seen in figure 5a , both models ( poiseuille , 
 red curve ; non - newtonian , blue curve ) fit the data reasonably well 
 for s = 11 , which validates the use of the 
 poiseuille flow profile as an approximation in this case . more importantly , 
 the goodness of the fits confirms that the leslie solution holds under 
 these conditions ( q > 0.1 l / min ) . the variations 
 in the observed shear angle as a function of z therefore 
 most likely results from the azimuthal angle  , which varies 
 from 0 to 90 ( as a result of the equivalence of walls y  x and y  z in the square cross section ) and decreases the projection 
 of angle  in the saxs detector plane ( y  z ) . using s = cos(/)/2 ( eq 1 ) , one can 
 extract  from the determined s , obtaining 
 || = 1.08 , which is in reasonable agreement with the values 
 of || = 1.12 and s = 13 reported elsewhere . one should also note that even though e2 ( and e1 ) depends 
 on  , our data does not change in the range of 0.2 l / min 
  q  3 l / min ( figure 5a ) . this is mainly a result of  being convolved 
 with . as we see in figure 5b , 
 at a low flow rate 
 below 0.1 l / min the behavior is very different . ( there is a small lack of symmetry 
 that is discussed in the si . ) despite the 
 movement of 5cb in the microchannel , clearly seen in the camera and 
 microscope at q approaching 0 l / min , yz has reoriented across the whole channel , 
 except at z = 0 m where yz is still zero . even though in our measurements 
 we can only probe the projections on the y  z plane , it is very likely that this relaxation also takes 
 place across x. hence , the line parallel to y in the center of the channel at x and z = 0 m is almost a smoothed disclination line , where 
 yz ( and yx ) takes opposite signs on each side of this line . this configuration 
 resembles the escaped radial configuration ( typically observed in 
 tubes of cylindrical and square cross sections at rest , where the nematic director reorients 
 near the central region of the channel along the channel axis in order 
 to accommodate competing orientation demands from walls perpendicular 
 to each other ) . in the cylindrical geometry , at rest or low 
 flow , where boundary 
 conditions and elastic energy terms dominate , the angle between the 
 director and cylinder axis ( approximately equal to our condition with 
 a square cross section ) (  ) can be described by3where r is the radial distance 
 from the center of the cylinder , r is the cylinder 
 radius , and r is the angle of n at the surface ( for strong homeotropic anchoring , r = 90 ) . this equation , convoluted with 
 the (x , z ) profile and beam 
 size , fits our data to some degree ( figure 5b ) , resulting in r = 32 
 if r is taken to be 71 m ( half of the diagonal 
 of our square geometry ) . if r is 
 fixed to r = 90 ( strong homeotropic 
 anchoring ) and we fit r instead , then we obtain r = 250 m . however , one also has to recall 
 that the above equation was derived for cylindrical geometry . in the 
 case of square geometry , the elastic deformation that causes the escaped 
 configuration is already very significant at the corners of the channel , 
 which may force the escaped configuration to change more abruptly 
 than what eq 3 would predict for cylinders . hence , it is not clear on the basis of these measurements if the anchoring 
 is strong or weak . it is our view that the observed behavior is in 
 better agreement with a situation of strong anchoring . note that we 
 started at q = 3 l / min and slowly decreased 
 the flow rate until q  0 l / min , which 
 resulted in the strong preorientation of the director in the flow 
 field . given the fact that stopping the flow leads to a relaxation 
 of the preoriented profile , with the director reorienting all of the 
 remaining  , except the  disclination  line at x and z = 0 , then this should give a hint 
 that the elasticity emanating from the boundary condition is strong . 
 in any case , regardless of strong or weak anchoring , it is evident 
 that at low flow rates ( q < 0.1 l / min ) 
 the elasticity ( resulting from competing preferential homeotropic 
 alignment at the walls ) becomes dominant . to conclude this section , 
 we stress that through our microfluidic 
 saxs measurements we were able to directly probe the nematic director 
 orientation profile of a nematic liquid crystal flowing in a microchannel 
 with micrometer resolution in z and identify two 
 different flow regimes ( one for 0.2 l / min  q  3 l / min and the other for q < 0.1 l / min ) . for the strong flow 
 regime ( high flow rates ) . by using channels with rectangular cross 
 section , the interesting intermediate and weak flow regimes observed 
 in that work are also expected to be suitably determined . one advantage 
 of our method compared to microscopy is the ability to obtain a more 
 quantitative description of the director tilt across the channel . the tilt angle can be accurately determined , with its resolution being 
 only limited by the averaging effects from the beam size and by the 
 shift in the angle projection due to the azimuthal orientation . ( note 
 that the azimuthal shift is caused by the square cross - section . hence , 
 resolution can be significantly improved by using channels with a 
 rectangular cross section of high aspect ratio . ) another advantage 
 of this present method is the ability to map changes in the molecular 
 arrangements of the fluid simultaneously . here we clearly demonstrated 
 that despite reorientation the nematic peak position and width remain 
 the same , indicating that the studied flow conditions do not affect 
 the local arrangements of the fluid simultaneously with the director 
 orientation . this may not always be the case in lyotropic systems 
 that are more compressible and where concentration fluctuations and 
 instabilities induced by flow can occur . in section 3.1 , we described the alignment 
 of 5cb and the interplay between viscous and surface forces in simple 
 flow , where the boundary conditions are due to the walls of the microchannel 
 ( solid interfaces ) . in this section , we describe a similar approach , 
 but where the nematic liquid crystal is bound by flowing liquid from 
 above and below ( liquid interfaces ) and where capillary forces arising 
 from the interfacial tension between the immiscible fluids ( nematic 
 and water with some dissolved surfactant ) also play an important role . 
 for this purpose , figure 6a , b ) as before , with a flow rate of qm , but now a solution of 2 wt % triton x-100 also flows in channels 
 2 and 3 at a flow rate of qs ( figure 6a ) . the ratio of the total flow rate from the side 
 channels ( 2qs ) to the flow rate of the 
 middle fluid ( qm ) is rq = 2qs / qm . the three fluids meet at the cross and flow 
 together into the outlet ( channel 4 ) . for a suitable range of flow 
 rates qm and flow rate ratios rq between 5cb and the triton 
 solutions , stable sheets of 5cb with fluid on both sides can form 
 ( table s2 and figure 6c , d ) . the nematic sheet was found to be stable for a surfactant solution 
 total flow rate of 2qs = 24 l / min , 
 with 5cb flowing in the range of qm = 
 1 to 3 l / min ( table s2 ) . at lower qm < 1 l / min , jetting occurs , with 
 a thin thread of 5cb forming for a limited length ( 200300 
 m ) before rupturing to form droplets . at higher qm > 3 l / min , displacement instabilities occur , leading to an invasion of middle fluid 
 5cb into the side channels . the system is also stable when increasing qm while keeping the same flow rate ratio rq , but the same does not happen 
 when qm is lowered at constant rq . instead , it is 
 surrounded by side fluid along the z direction , in 
 which the y  x interfaces 
 are now the side fluid , but along the x direction , 
 it is still bound by the chip walls ( y  z surfaces , figure 6b ) . ( note in 
 figure 6d that in the nematic region there 
 is no trace of fluorescence . furthermore , the relief pattern indicated 
 by the white arrow suggests a concave shape of the nematic sheet . ) it was found that qm = 3 l / min 
 with rq = 8 is the most 
 stable flow condition . for lower qm ( < 
 1.5 l / min ) , the flow condition becomes history - dependent because 
 the nematic stream is not able to form by itself but remains stable 
 if it is formed at higher qm and then 
 decreased slowly to the final value . summary of the hydrodynamic focusing experiments 
 with the nematic 
 phase of 5cb confined in a microfluidic device by solid and liquid 
 interfaces . 5cb flows 
 along channel 1 and is focused into channel 4 by the side fluids ( 2 
 wt % triton x-100 in water ) coming from channels 2 and 3 . ( b ) schematic 
 representation of the 5cb nematic sheet focused by the triton solution . 
 ( c ) micrograph of a typical hydrodynamic focusing experiment . here the side fluids contain 0.1 mm fluorescein to facilitate the visualization 
 of the nematic sheet ( black ) . 5cb flows at 3 l / min , and each 
 side fluid flows at 12 l / min ( rq = 2qs / qm = 8) . the green 
 box represents the x - ray beam s fwhm ( 30 m along z and 3000 m along y ) and 
 hence the area probed by each x - ray measurement . the orange box represents 
 a typical area probed in a vertical scan ( z intervals 
 of 1020 m ) . ( d ) the same as in ( c ) but with a 5cb flow 
 rate of 1 l / min and rq = 24 . the focused nematic sheet seems to have a concave shape 
 along the x direction ( note the relief pattern indicated 
 by the white arrow compared to the dark arrow for the formation of 
 the sheet ) , with its width being ca . 22 m on the top and ca . 
 8 m in the middle of the channel . ( e ) saxs along the width 
 of the channel for qm = 3 l / min , rq = 8 . as in the simple flow 
 experiments , here also the nematic director rotates from a negative 
 yz on the top to a positive yz on the bottom side of the sheet . have studied 
 the coflow of 5cb and silicon oil . 
 although the goal was the formation of nematic droplets , stable stratified 
 flow regimes were also identified . in addition to capillary and viscous 
 forces , wetting of the device walls by 5cb was identified as an important 
 parameter stabilizing the nematic sheet . the non - newtonian nature 
 of 5cb and defect structures , however , cause instabilities that promote 
 droplet formation . in a more general 
 study , although restricted to newtonian fluids , cubaud and mason have 
 studied different flow regimes between coflowing immiscible liquids 
 with different viscosities and interfacial tension contrasts , with 
 the flow regimes being mapped into a flow map of capillary number ca in middle fluid cam versus 
 side fluid cas . the interfacial tension between 5cb and water has been reported 
 to be approximately 26 mn / m . this leads 
 to a predominance of capillary over viscous forces , which does not 
 allow the formation of a stable nematic sheet of liquid . instead , 
 the system drips , with the immediate formation of droplets ( i.e. , 
 to reduce the surface energy cost ) , in agreement with the flow map . the addition of 2 wt % triton x-100 reduces 
 the interfacial tension , allowing the formation of nematic sheets 
 sandwiched between two aqueous surfactant solutions ( figure 6c , d ) . also important for the stabilization of the 
 sheets against breakup is the poor wettability of the microchannel 
 walls by the side fluid when compared to that by the nematic fluid . ( note that the hydrophobic channel walls should 
 have more affinity for 5cb than for water . ) the nuances of the 
 flow map and the role of capillary and viscous 
 forces coupled with the non - newtonian behavior will be the subject 
 of a more detailed study elsewhere . in this work , we are mainly interested 
 in the effect of the liquid boundaries on the nematic director . note 
 that despite the nematic sheet being bound by liquid interfaces at 
 wm/2 m and solid surfaces 
 at 50 m , 5cb is squeezed along the z direction , reducing the contact with the solid walls . also , as noted previously , in these saxs measurements , what 
 is measured is yz , which is the 
 projection influenced by the liquid boundaries . at the synchrotron , 
 two flow conditions were studied : qm = 
 3 l / min , rq = 
 8 ; and qm = 1 l / min , rq = 24 . in figure 6e , 2d saxs profiles due to scattering from the hydrodynamically 
 focused 5cb at different positions along the z axis 
 are shown . as in the simple flow experiment , the same rotation in 
 the nematic director is observed , with yz being negative in the upper half of 5cb and positive in the 
 lower half . in this case , the beam size is much larger ( fwhm 30 
 m ) , which introduces much stronger averaging effects but is 
 still suitable for an overall determination of the focused nematic 
 stream . the larger scanning step size ( 10 and 20 m ) also needs 
 to be taken into account . figure 7a shows qh scans averaged over different ql intervals at qm = 3 l / min , rq = 8 , and z = 0 m . in figure 7b , qh scans of the central region of the pattern ( ql interval of 0.036 to 0.036  ) are shown across the nematic stream , in contact 
 with the triton solutions , and compared to the pure system . all hydrodynamic 
 focusing patterns show a slight shift in the peak position to lower 
 angles ( q  0.006  ) , which may indicate a slight penetration / contamination of triton 
 or triton / water particles within the 5cb sheet . even though this shift 
 in the peak is very small , this phenomenon is occurring mainly on 
 the interface , which makes the signal weaker compared to that of the 
 remaining nematic sheet . we also note that although the residence time of 5cb 
 in the channel is small ( 0.1 s ) , the surfactant concentration 
 is two orders of magnitude above the cmc ( 0.019 wt % ) , which could lead to some solubilization of 5cb at the 
 interface and lead to some instabilities that could affect the observed 
 results . the coordinates ( qh , ql ) and angle  in reciprocal space are 
 defined in figure 2a . ( a ) qh scans integrated over different ql intervals in the center of the channel ( z  0 m ) for a hydrodynamic focusing experiment . the 
 flow rate of 5cb ( middle channel ) is qm = 3 l / min , and the ratio rq = 2qs / qm of the combined side fluids ( triton x-100 2% solutions ) with the 
 inner fluid is rq = 8 . ( b ) comparison between qh scans ( integrated 
 over ql = 0.036 to 0.036  ) in the system with no hydrodynamic focusing ( black , q = 3 l / min ) and the hydrodynamic focusing system 
 ( q5cb = 3 l / min , rq = 8) . all hydrodynamic focusing patterns 
 show a slight shift in the peak position to lower angles ( q  0.006  ) , which may 
 indicate a slight penetration / contamination of triton or triton / water 
 particles within the 5cb sheet . nonetheless , given the very small 
 shift in the peak , both the contamination and resulting structural 
 changes should be minimal . ( c )  scans and respective double - lorentzian 
 fittings for a hydrodynamic focusing run with qm = 3 l / min and rq = 8 . data is normalized and displaced along the ordinate axis 
 for ease of visualization . in all panels ( a  c ) , circles constitute 
 scattering data and lines are lorentzian fits . as with the simple flow case ( cf . section 3.1 ) , also here it is convenient to extract yz from the  scans . figure 7c shows 
 the resulting fits of scans with the double - lorentzian expression 
 ( eq 2 ) . figure 8 shows 
 a plot of the measured yz at different z positions along the channel . the data extends over a wider z range than the widths of the nematic sheets wm . this results from the fact that the beam size ( fwhm 
  30 m ) is larger than the nematic stream , which enlarges 
 the range in z over which the nematic phase gives 
 a signal . more importantly , the averaging effects due to beam size 
 also modify the observed values of yz . in figure 8a , a comparison of the 
 simple flow system ( i.e. , no hydrodynamic focusing ) with the small 
 ( fwhm  6 m , blue circles ) and large ( fwhm  
 30 m , green squares ) beams can be seen . the curves are slightly 
 different , with the data from the larger beam showing a shift to smaller 
 yz angles . despite this , the same 
 overall trend is still observed , which allows the characterization 
 of the system . the shift in yz to 
 smaller angles is easily understood if one recalls that yz has opposite signs in the upper and lower 
 halves of the microchannel . if the beam is large enough that it partially 
 overlaps both halves , then the opposite signs of yz partially cancel each other , making the observed 
 angle yz smaller . this effect is 
 more significant for narrower 5cb streams because the yz variations are more compressed in space , 
 accentuating the smearing from the larger beam size . in figure 8b , |yz| versus z curves are shown for different rq . even with the finite beam size effects , 
 clear differences in the slopes of the different curves are easily 
 observed . it can be clearly seen that as rq increases from rq = 0 ( simple flow experiments , green squares ) to rq = 8 ( blue circles ) and to rq = 24 ( red diamonds ) , the variation 
 of |yz| with z also increases . results for the nematic director angle projection yz at different z positions 
 along 
 the microchannel in the hydrodynamic focusing experiments . ( a ) comparison 
 of measured yz for the simple flow 
 experiments ( no hydrodynamic focusing ) at q = 3 l / min 
 using the ssrl ( green ) and csaxs ( blue , finer beam size ) beamlines . 
 even with the loss in spatial resolution due to the ca . 5-fold - larger 
 x - ray beam at ssrl , which also shifts yz to lower values , the same overall evolution in director rotation 
 is observed . the z position is rescaled by centering the director angle  
 = 0 at zrescaled = 0 m . compared to the simple 
 flow experiments , fitting the hydrodynamic 
 focusing data is now significantly more complex . now , parameters such 
 as the viscosity mismatch , nematic sheet width , and interfacial curvature , 
 among others , should clearly play a role in the velocity profile across 
 the nematic sheet , which in turn should reflect on the measured yz profile . in addition , the nonideal resolution 
 along z ( combination of large beam size and large 
 step size ) smears the measured yz profiles significantly . as such , rather than fitting the data lockwood et al . have shown that surfactants with branched chains ( as is 
 the case for triton x-100 ) have a tendency to induce planar alignment 
 in nematics . here we clearly observe a tilt in the director , which 
 is also seen to increase as rq is increased . we note that the observed shift in yz to larger angles as rq is increased ( figure 8b ) should be even larger in reality because of the above - mentioned 
 averaging effects due to the large beam size ( that reduce the measured 
 yz to lower values ) . hence , as 
 the nematic stream gets narrower , we can conclude that the average 
 angle yz increases substantially . what is not clear is whether this angle is approximately constant 
 or varies smoothly across the stream width . in either case , these 
 observations seem more consistent with a picture where triton x-100 
 imposes an angle / tilt at the interface ( i.e. , homeotropic anchoring 
 or some angle intermediate between 90 and 0 ) rather than planar 
 alignment . we note that if triton was imposing planar alignment under 
 these ( flow ) conditions , then it should be imposing the same orientation 
 as flow . then , at most ( as the elp theory predicts ) , a shear angle 
 of s could be observed across the nematic 
 sheet as a result of flow , but this angle should not increase when 
 the nematic stream becomes narrower at higher rq , as we observe in the results . this 
 work is the first step toward more detailed explorations and 
 characterizations of the formation of nematic sheets and flowing liquid 
 interface effects . in future work , an improved spatial resolution 
 ( achieved with smaller beam sizes ) will be employed , along with a 
 systematic variation of other surfactants with different anchoring 
 properties ( e.g. , nonbranched alkyl chain surfactants that are known 
 to impose homeotropic alignment and bolaamphiphile surfactants , that 
 are known to impose planar alignment ) , 
 to look for their effects on the nematic sheets . we anticipate that 
 the coupling of flow with the different anchoring properties of different 
 materials ( e.g. , surfactants , lipids , polymers , and proteins ) will give rise to a wealth of interesting phenomena 
 for both fundamental and applied fields . we also stress that this 
 method can be applied to many different materials in a high - throughput 
 fashion , which can be particularly useful for lc displays , where switching 
 orientations are known to induce local flows , and for boundary lubrication , 
 where the orientations of lcs under flow determine the tribological 
 response of this type of material . in this work , we investigated the interplay between flow and boundary 
 condition effects on the orientation field of a thermotropic nematic 
 liquid crystal under flow and confinement by using a combination of 
 microfluidics with in situ saxs . two different experiments were performed . 
 in the first , a nematic liquid crystal flowed through a square - channel 
 cross section at varying flow rates , and the projection of the nematic 
 director angle on the y  z shear plane was measured . at moderate - to - high flow rates , the director 
 was found to be predominantly aligned in the flow direction , but with 
 a small tilt angle of 11 in the velocity gradient 
 direction , in agreement with the leslie solution for poiseuille flow . 
 at low flow rates , a flow profile with a resemblance to the escaped 
 radial director was observed , with the director varying more smoothly 
 from the edges to the center of the channel . in the second experiment , hydrodynamic focusing was employed to 
 confine the nematic phase in a thin sheet sandwiched between two triton 
 x-100 aqueous solutions . the average director orientation was found 
 to tilt by some amount away from the flow direction toward the liquid 
 boundaries , although it remains unclear if one tilt angle is dominant 
 through most of the nematic sheet or if the tilt angle varies smoothly 
 between two extreme values ( 90 and 0 ) . we anticipate 
 that expanding these studies to materials with different anchoring 
 properties will give rise to a wealth of new phenomena under flow , 
 which is interesting for both fundamental and applied sciences . in 
 particular , this method can be used for high - throughput routine measurements 
 to assess the influence of different surfactants or other additives 
 on the orientation of nematic phases , which can lead to further improvements 
 in areas such as boundary lubrication and lcds .