we fabricate nanopores from glass capillaries using a programmable 
 laser puller ( p-2000 , sutter instruments ) .
the pore sizes are characterized 
 by direct imaging using an sem and from conductance measurements by 
 taking a current  voltage ( i  v ) curve .
the three pore diameters used are 15  3 , 
 148  26 , and 1018  30 nm .
the pulled capillaries are assembled 
 into a pdms - based microfluidic sample cell which is sealed onto a 
 glass coverslide .
the capillary connects two reservoirs filled with 
 kcl of varying concentrations buffered by tris - edta ( te ) at ph 8 . 
 for concentrations greater than 100 mm , 1 ( 10 mm )
te is used ; 
 for lower concentrations the buffer is diluted to give a final concentration 
 of 10% ( i.e. , for a 10 mm kcl solution , 1 mm te is used ) .
because 
 it is not possible to change the salt concentration once the sample 
 cell is filled , it is necessary to fill a new capillary for measurements 
 at a different concentration . to take into account the variations 
 between pores , the results presented are an average over several pores 
 at each salt concentration .
the optical tweezer setup is a single - beam 
 gradient trap based on an inverted microscope .
we use a 5 w ytterbium fiber laser ( ylm-5-lp , ipg laser ) operating 
 at a wavelength of 1064 nm which backfills a 60 , na 1.2 olympus 
 uplansapo water immersion objective to create a stable three - dimensional 
 optical trap near the laser focus . real time position tracking with 
 a bandwidth of a few khz
the sample cell is mounted onto a piezoelectric 
 nanopositioning device ( p-517.43 and e-710.3 , physik instrumente ) 
 which allows the relative position of the trap and pore to be adjusted 
 with an accuracy of 100 nm .
spherical 2 m streptavidin  polystyrene 
 beads ( kisker ) are flushed into the reservoir and captured with the 
 trap .
force calibration is achieved for every trapped particle using 
 a power spectral density method ; the resulting trap stiffness is in 
 the range 1060 pn/m , corresponding to applied laser 
 powers of 50300 mw at the sample plane .
voltages 
 in the range + 1 to 1 v are applied using ag / agcl electrodes 
 connected to a commercial electrophysiology amplifier ( axopatch 200b , 
 molecular devices ) , which also allow for simultaneous low - noise ionic 
 current recording .
the entire experimental setup is controlled using 
 custom - written labview software ( labview 2009 , national instruments ) .
it integrates a fast emccd camera ( andor ixon3 865 ) with ionic current 
 measurements and has been described previously .
a heka epc 800 electrophysiology amplifier is 
 used to apply voltages across the nanopore and record ionic currents . 

the reservoir containing the nanopore is seeded with a dilute solution 
 of 540 nm diameter streptavidin coated polystyrene particles that 
 are embedded with the nilered dye ( spherotec ) . commercially available 

solutions ( 0.1% w / v ) are centrifuged for 10 min at 5000 g , the supernatant removed , and the particles resuspended in a  washing 
 buffer  of 100 mm kcl buffered with 1  te at ph 8 .
this 
 is repeated thrice , after which the fluorescent particles are resuspended 
 in the measurement buffer of choice .
the particles can therefore be 
 added to the reservoir surrounding the nanopore without affecting 
 the salt concentration in the reservoir .
a green laser operating 
 at 1 mw ( laser quantum ) is used to illuminate a wide region ( 30 
  30 m ) surrounding the nanopore opening . 

the motion of the fluorescent particles due to the flows is recorded 
 at 500 frames per second .
the emccd chip is cooled to 20 c 
 and operated at an emccd gain of 3 .
individual particles are tracked 
 using custom - written software ( labview 2009 , national instruments ) 
 which allows for the extraction of average particle velocities at 
 each point in a grid surrounding the pore .
the stokeslet limit of the 
 landau  squire solution describes the flow field resulting from 
 a point force applied to a quiescent fluid at a low reynolds number .
the stokes stream function for this solution is given by8where  is the dynamic 
 viscosity , p the magnitude of force required to set 
 up the flow , and r and  are spherical polar 
 coordinates centered at the pore .
the coordinate system is defined 
 with the polar axis coincident with the pore axis . from the stream 

function we can obtain velocity components:910 by moving 
 the bead to different locations in the plane of the pore a force map 
 can be created , which is converted to a velocity map using the equation fi = 6rvi  , where r is the bead radius and fi and vi are the i - th components of the force and velocity .
the self - similar 
 nature of the flow allows the data to be linearized : if we let  
 = cos /r and  = sin /r , plotting ur against  or u against 
  gives straight lines which allow p to be 
 determined.1112 in practice , once the landau 
squire nature of the flow 
 has been verified , p can be determined from just 
 a single point measurement of force , as long as the coordinates r and  ( or equivalently x and y ) are known .
boltzmann equation for 
 the electric potential  , followed by the stokes equation for 
 the fluid velocity . when e/kbt < 1 , the poisson  boltzmann 
 equation can be linearized , which permits analytic solutions .
although 
 at low salt concentrations this condition is not fulfilled , the qualitative 
 features of the analytic model are preserved . the linearized poisson 
. we can solve for the 
 electric potential subject to the boundary condition that  
 does not diverge anywhere , and the gradient of  at the glass 
 surface depends on the surface charge  according to gauss s 
 theorem .
inside an infinite cylinder of radius a the 
 solution is given by14and outside the cylinder , by15 the velocity profiles are determined from the stokes equation 
 in the absence of pressure gradients . inside the cylinder , this is 
 given bywhere we have used relations from 
 the poisson and debye  hckel equations , and finally 
 our solution for the potential to rewrite the result .
outside the 
 cylinder , the equation is given by19 we can directly integrate these equations to obtain the final 
 results for the velocity profiles inside and outside the cylinder:20and21 finite - element 
 simulations were carried out using the comsol multiphysics package , 
 version 4.4 .
planck equations ( eqs 1 and 2 ) are solved in a first 
 step neglecting convection ( the ciu term ) , which outputs fluxes and concentrations ji , c , ci and hence the charge density 
 e(r ) and electric potential (r ) .
(r ) and e(r ) are the inputs for the body force in the stokes 
 equation ( eq 3 ) , which is solved in a second 
 step to produce the velocity and pressure fields , u(r ) and p(r ) . the ratio 
 of diffusion to convection in the nernst 
emax  
 10 v / m and umax  0.1 
 m / s , giving a diffusive to convective ratio of around 100 ; thus the 
 neglect of convection in the first step of the simulations is a reasonable 
 approximation . after each run , quantities such as ionic current and 
 flow rate through the pore were calculated . in order to compare with 
 experiments ,
a quantity p was also extracted by measuring 
 the velocity at a point on the pore axis 1 m from the pore 
 opening .
this simulates placing a bead close to the pore and using 
 it to probe the local velocity ( although in reality the measured force 
 is due to an average velocity over the entire bead surface , which 
 is not taken into account here ) .
 we report observations of a striking 
 reversal in the direction of electroosmotic flow ( eof ) outside a conical 
 glass nanopore as a function of salt concentration . at high ionic 
 strengths 
 ( > 100 mm ) , we observe eof in the expected direction as 
 predicted by classical electrokinetic theory , while at low salt concentrations 
 ( < 1 mm ) the direction of the flow is reversed . 
 the critical crossover 
 salt concentration depends on the pore diameter . 
 finite - element simulations 
 indicate a competition between the eof generated from the inner and 
 outer walls of the pore , which drives flows in opposite directions . 
 we have developed a simple analytical model which reveals that , as 
 the salt concentration is reduced , the flow rates inside the pore 
 are geometrically constrained , whereas there is no such limit for 
 flows outside the pore . 
 this model captures all of the essential physics 
 of the system and explains the observed data , highlighting the key 
 role the external environment plays in determining the overall electroosmotic 
 behavior . 
Materials and Methods