5. 3P PIV Measurements

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1 Micro PIV Last Class: 1. Data Validation 2. Vector Field Operator (Differentials & Integrals) 3. Standard Differential Scheme 4. Implementation of Differential & Integral quantities with PIV data 5. 3P PIV Measurements Today s Contents: 1. Introduction of μpiv 2. Considerations of Microscopy in μpiv 3. Depth of Correlation 4. Physics of Particles in Micro PIV 5. Measurement Errors 6. Special Processing Methods 1

Introduction There are many areas in science and engineering where it is important to determine the flow field at the micron scale. In 1998 Santiago et al.

2 Introduction There are many areas in science and engineering where it is important to determine the flow field at the micron scale. In 1998 Santiago et al. demonstrated the first μpiv system a PIV system with a spatial resolution sufficiently small enough to be able to make measurements in microscopic systems. 2

3 High Resolution Velocimetry Techniques 3

Considerations of Microscopy in μpiv (1) http://www.microscopyu.

4 Considerations of Microscopy in μpiv (1) Inverted epi fluorescent microscope: large workspace, long working distance 4

Considerations of Microscopy in μpiv (2)

configured to deliver a significant increase

5 Considerations of Microscopy in μpiv (2) Resolution: In order to overcome the low light levels associated with live cell imaging, modern digital CCD cameras can be configured to deliver a significant increase in sensitivity by implementing a process known as binning 5

Considerations of Microscopy in μpiv (3) The real world size derived from a CCD camera can be expressed as The significant limitations of the microscope optical system

6 Considerations of Microscopy in μpiv (3) The real world size derived from a CCD camera can be expressed as The significant limitations of the microscope optical system (panel (a)) and the detector (digital camera; panel (b)) in performing live cell investigations. 6

7 Considerations of Microscopy in μpiv (4) Long working distance for accommodating large microfluidic chips. 7

Phase Contrast for transparent specimen P wave = S

$Dwave: a deviated or diffracted spherical$ wavefrontthattraversesthespecimen.

8 Phase Contrast for transparent specimen P wave = S wave +D wave Swave: it passes through and around the specimen, but does not interact with it. Dwave: a deviated or diffracted spherical wavefrontthattraversesthespecimen. Pwave: it combines through interference to produce a resultant particle wave normal Phase contrast 8

9 Differential Interference Contrast (DIC) for 3D imaging 9

10 Overview of μpiv Three fundamental problems differentiate μpiv from conventional macroscopic PIV: The particles are small compared to the wavelength of the illuminating light; the illumination source is typically not a light sheet but rather an illuminated volume of the flow; and the particles are small enough that the effects of Brownian motion must be considered. 10

11 3D Diffraction Pattern The dimensionless diffraction variables are defined as: where f is the radius of the spherical wave as it approaches the aperture (which can be approximated as the focal length of the lens), λ is the wavelength of light, and r and z are the in plane radius and the out of plane coordinate, respectively, with the origin located at the point source (left figure). Within the focal plane, the intensity distribution reduces to the expected result which is the Airy function for Fraunhofer diffraction through a circular aperture. 11

12 3D Diffraction Pattern (Cont.) Along the optical axis, the intensity distribution reduces to The focal point is located at the origin, the optical axis is located at v =0,andthe focal plane is located at u =0.Themaximumintensity,I 0, occurs at the focal point. Along the optical axis, the intensity distribution reduces to zero at u = ±4π, ±8π,while a local maximum occurs at u = ±6π. 12

$Depth of Field The depth of field of a standard microscope objective lens is given by Inou e and Spring as: where n is the refractive index of the fluid between the microfluidic device and the$

13 Depth of Field The depth of field of a standard microscope objective lens is given by Inou e and Spring as: where n is the refractive index of the fluid between the microfluidic device and the objective lens, λ 0 is the wavelength of light in a vacuum being imaged by the optical system, NA is the numerical aperture of the objective lens, M is the total magnification of the system, and e is the smallest distance that can be resolved by a detector located in the image plane of themicroscope(forthecaseofaccdsensor,e is the spacing between pixels). The above equation is the summation of the depths of field resulting from diffraction (first term on the right hand side) and geometric effects (second term on the right hand side). Substituting NA = n sin θ = n a/f, and λ 0 = nλ yields the first term on the right hand side of the above equation. 13

14 Depth of Correlation The depth of correlation is defined as twice the distance that a particle can be positioned from the object plane so that the intensity along the optical axis is an arbitrarily specified fraction of its focused intensity, denoted by ε. Beyond this distance, the particle s intensity is sufficiently low that it will not influence the velocity measurement. d p d e 14

15 Depth of Correlation (Cont.) Approximating D a2 /(s o + z) 2 D a2 /s o2 =4[(n/NA) 2 1] 1, combining the both above equations, and solving for z corr yields an expression for the depth of correlation 15

16 Depth of Correlation (Cont.) Because the above DOC has been developed without accounting for the presence of an immersion medium other than air, this model must be slightly modified to extend the case. The theoretical contribution of an unfocused particle to the correlation function is estimated by considering (1) the effect due to diffraction, (2) the effect due to geometric optics, and (3) the finite size of the particle. one can estimate the depth of correlation due to diffraction as: 16

17 Depth of Correlation (Cont.) the thickness of the measurement plane, 2z corr, for various microscope objective lenses and particle sizes based on the previous DOC. Olsen & Adrian used a small angle approximation to derive the depth of correlation as Because it is given in terms of f # instead of NA, it is only applicable for air immersion lenses and not oil or water immersion lenses. 17

18 Particle Visibility The quality of μpiv velocity measurements strongly depends upon the quality of the recorded particle images from which those data are calculated. In macroscopic PIV experiments, it is customary to use a sheet of light to illuminate only those particles that are within the depth of field of the recording lens. The light sheet has two important effects it minimizes the background noise from out of focus particles and ensures that every particle visible to the camera is well focused. However, in μpiv, the microscopic length scales and limited optical access necessitate using volume illumination. where J p is total light flux emitted by a single particle. 18

19 Particle Visibility (Cont.) Making the simplifying assumption that particles located outside a distance z > δ/2 from the object plane as being completely unfocused and contributing uniformly to background intensity, while particles located within a distance z < δ/2 as being completely focused, the total flux of background light J B can be approximated by where C is the number of particles per unit volume of fluid, L is the depth of the device, and A v is the average cross sectional area contained within the field of view. Combining equation both previous equations, correcting for the effect of magnification, and assuming s o >> δ/2, the intensity of the background glow can be expressed as 19

20 Particle Visibility (Cont.) The visibility V of a focused particle can be obtained by combining I(r,z) and d τ, dividing by I B at r =0andz =0, From this expression it is clear that for a given recording optics configuration, particle visibility V can be increased by decreasing particle concentration C or by decreasing test section thickness L. For a fixed particle concentration, the visibility can be increased by decreasing the particle diameter d p or by increasing the numerical aperture NA of the recording lens. Visibility depends only weakly on magnification and object distance so. 20

21 Particle Fluorescence dye Exciting Polystyrene Emission LP filter HP filter Dichroic Mirror λ 21

μpiv Seeding > Scaling Law Since we are dealing

Length =L Area =L 2 Volume=L 3 Surface Forces :

22 μpiv Seeding > Scaling Law Since we are dealing with microscale flow, it is necessary to understand a little physics underlying the microscopic world. Length =L Area =L 2 Volume=L 3 Surface Forces : Surface Tension, Frictional Forces Body Forces : Gravity, Electric Forces, Magnetic Forces 22

23 Knudsen Number (Kn) : : Of the order of Kn 0.1, the fluid can be treated as a continuous medium and described in terms of the macroscopic variables In the range 0.1 Kn 1.0, termed the slip flow regime, it is sometimes possible to obtain useful results by treating the gas as a continuum, but allowing for discontinuities in velocity and temperature at solid boundaries For Kn 10, intermolecular collisions in the region of interest are much less frequent than molecular interactions with solid boundaries, and can be ignored. Flows under such conditions are termed collisionless or free molecular. 23

24 Flow/Particle Dynamics Basedonasimplefirst order inertial response to a constant flow acceleration (assuming Stokes flow for the particle drag), the response time τ p of a particle is: 18 Considering typical μpiv experimental parameters of 300 nm diameter polystyrene latex spheres immersed in water, the particle response time would be 10 9 s. This response time is much smaller than the time scales of any realistic liquid or lowspeed gas flow field. Inthecaseofhigh speed gas flows (10 3 < Kn < 0.1), a correction offered by MELLING suggests the following relation for the particle response time:

25 Velocity Errors A more in depth consideration of the phenomenon of Brownian motion is necessary to completely explain its effects in μpiv. An ideal, non Brownian (i.e., deterministic) particle following a particular streamline for a time period Δt has x and y displacements of: The relative errors, ε x and ε y, incurred as a result of imaging the Brownian particle displacements in a two dimensional measurement of the x and y components of particle velocity, are given as: This error can be reduced by both averaging over several particles in a single interrogation spot and by ensemble averaging over several realizations. The diffusive uncertainty decreases as 1/ N, wheren is the total number of particles in the average 25

26 Particle Position Errors In addition to the flow velocity measurement error associated with particle displacement measurements, the Brownian motion incurred during the exposure time t exp may also be important in determining the particle location, especially for slow flows with long exposures and small tracer particles. This random displacement during image exposure can increase the uncertainty associated with estimating the particle location. This particle location uncertainty is typically negligible for exposure times where the typical Brownian displacement in the image plane is small compared to the particle image diameter or a value of the diffusion time d τ2 /(4DM 2 ) much less than the exposure time. For the experimental parameters mentioned above, d τ2 /(4DM 2 )is 300ms and the exposure time is 5 ns for a typical Nd:YAG laser. Particle image long t exp short t exp 26

27 Measurement Errors Two measurement errors are usually discussed in PIV, namely random error (σ) and biaserror (β). Both constitute a total measuremnt error 1 N 1 N i 1 ( x i x0) The first term (σ) is resulted from various causes such as human operations and some unknown uncertainties. Since it is random and no direction oriented, one can simply eliminate it by averaging with a large number of data. The second term (β) is usually a periodic fluctuation unable to be removed by averaging. In this paper, this value is expressed as 27

28 Special Processing Methods : Overlapping of LID-PIV in many cases, especially in μpiv measurements, the particle image density in the PIV recordings is usually not high enough. These PIV recordings are called low image density (LID) recordings and are usually evaluated with particletracking algorithms. 28

component, for example u, can be approximately determined with the forward

29 Special Processing Methods : CDI Schematic diagram of the four roll mill Overlapped PIV images (inverted) from the four roll mill flow In the experiment the velocity component, for example u, can be approximately determined with the forward difference interrogation (FDI) and central difference interrogation (FDI), respectively, as 29

30 Realization of CDI 2 2 The left figure clearly shows that the bias error of FDI is proportional to the radial position. The bias error of CDI looks independent on the radial position (right figure). 30

31 Special Processing Methods : Image Correction Technique (CDIC) 31

32 Special Processing Methods : Image Correction Technique (CDIC) 32

33 Effects of CDIC Reduction of Peak Locking Effect the image correction does not noticeably change the bias error distribution but effectively reduces the random (precision) evaluation error (here by half). 33

Micrometer and Nanometer Spatial Resolution with µpiv

Micrometer and Nanometer Spatial Resolution with µpiv Steve Wereley Associate Professor of Mechanical Engineering Birck Nanotechnology Center Purdue University (USA) wereley@purdue.edu Experiments in Fluids