On Measuring 3D Flow Within Inkjet Droplet Streams

On measuring 3D flow within inkjet droplet streams

S.A.Wormald[*] and J.M. Coupland

Wolfson School of Mechanical and Manufacturing Engineering, LoughboroughUniversity, Loughborough, United Kingdom

Abstract

With a view to measuring the internal and external flow fields in an inkjet droplet stream, this paper discusses the problem of imaging through droplet surfaces and more generally interfaces in two-phase flows. First the propagation of optical fields through interfaces between media of different refractive index is discussed with reference to scalar diffraction theory. Some approximations suitable for droplet imaging are then discussed and the use of a-priori information is then explained. The imaging technique is applied to holograms recorded using a digital holographic microscope and is illustrated in the synthesis of a 3D image that is reconstructed through a cylindrical telecommunication fibre. Subsequently we demonstrate for the first time, high resolution imaging throughout an inkjet droplet.The practical implementation and improvement of these imaging methods for the measurement of two-phase flows is then discussed.

Keywords:digital holographic microscope, two-phase flow analysis, droplet surface reconstruction.

1. Introduction

Central to the design of products such as inkjet printers [1], fuel injectors [2] and asthma inhalers [3], two-phase flows represent one of the most challenging areas of modern fluid dynamics.Flowsdepend strongly on the interactions between phases and are greatly influenced by parameters such as viscosity and surface tension [1]. Nevertheless, certain two-phase flows such as inkjet droplet streams can be made to be some of the most steady and predictable of any flow phenomena [4].

Experimental measurements of two-phase flows have been limited. The shape of fuel injector plumes has been measured using light sheet methods [5] and the droplet size and velocity distributions has been estimated using phase Doppler anemometry [6]. Two-dimensional (2D) measurements of the internal velocity field have been attempted in relatively large (4mm) droplets using particle image velocimetry (PIV) [7]. At smaller scales the 3D flow within a droplet in a channel was recently measured using confocal microscopy [8]. In this case the droplet was in contact with the channel walls and imaging through a planar interface was possible.

In essence, the large refractive index change between different phases causes refraction and reflection at fluid interfacesand this can be attributed to the failure of commercial optical diagnostics.For the most part optical metrology has been developed for use in homogenous fluids or for operation through planar windows butin some cases can be modified for use through windows of known form. For example, corrective optics have been used in PIV measurements made through the glass cylinder of an optical engine [9] and phase conjugation and ray tracing techniques have been used in similar measurements using holographic PIV (HPIV) [10].This work shows that if the optical distortion or aberration is known then it can be compensated; however, a clear requirement for a non-intrusive measurement technique which compensates for the irregular boundaries in two phase flows remains.

Optical tomography is one candidate technology. In essence, optical tomography synthesises an image from measurements of the field scattered under a set of different illumination conditions [11]. At present, however, two problems prohibit its use in two phase flow mapping. First, a significant time is required to record the scattered data sets and this precludes tomographic studies of transient phenomena. Second, the strong scattering caused bylarge changes in the refractive index require sophisticated non-linear inversion methods to compute an image [12]. Although a-priori knowledge can be used to expedite the inversion process [13], it is unlikely that the computational resources to compute 3D images of actual size droplets will be available in the near future.

Holography, provides a single record of the scattered field and a holographic reconstruction can be thought of as a simple tomographic image [14]. Since a hologram is typically a recording of the optical field at a surface, the reconstruction can only be considered as a 3D image if it is interpreted with a-priori knowledge. For example, the identification and location of particles inholographic PIV relies on the fact that the particles are known to be sparsely distributed, point-like objects. For the case of a seeded two-phase flow we also have considerable a-priori knowledge. In addition to knowledge of the seeding particle properties, we often know the refractive indices of the various phases. For the case of droplet imaging, the droplets themselves are frequently well dispersed and we are usually dealing with phases bounded by a finite number of closed surfaces. In addition to knowledge of the incident wave-front this information is sufficient to make detailed 3D measurements of sparse flows.

In the following we consider the problem of imaging within inkjet droplets with radii of curvature of around 50-200 m. The scattered field is recorded using a digital coherent microscope. First we consider the general problem of reconstructing fields in two phase media and discuss some of the simplifications that can be made to reduce the problem. We demonstrate the technique first by imaging through a telecommunications fibre which has particles and debris on its outer surface. It is shown that the strong lensing effect of the fibre can be efficiently compensated by application of a-priori knowledge. The more general problem of imaging through seeded inkjet droplets is then considered.

2. Theory

Let us consider the propagation of light through a water droplet in ambient air as shown in Figure 1. According to scalar diffraction theory, the fields, , and , within the droplet and that in surrounding air respectively, obey the Helmholtz equations,

(1)

(2)

where and are wave-numbers in water and air at wavelengths, and . In addition to these relations the external and internal fields are coupled via the boundary conditions. In this case both the complex amplitude of the field and its normal derivative must be continuous at the boundary, , such that,

(3)

In order to calculate the fields for a particular illumination condition, the field outside the droplet is written as the sum of the light scattered by the object,, and the reference field, , that is defined by the illumination in the absence of the object, such that,

(4)

In principle, for a given illumination condition, equations 1-4 can be solved using the boundary element method to find and [15]. It is then possible to calculate the fields within the regions using the Fresnel-Kirchhoff diffraction formula [16].

A more direct solution, that we have found to be quite useful for calculating the field scattered by smooth objects, is based on the assumption that the internal and scattered fields can be written as surface integrals such that,

(5)

(6)

where and are surfaces just outside of and just inside the boundary as shown in Figure 2, and and are source strengths per unit area. In this way the internal field, , must be a solution to equation 1 since it is produced by sources that are external to the region of interest and similarly, , must be a solution to equation 2.

In a numerical environment it is relatively simple to solve these equations by replacing the integral with a discrete set of sources uniformly spaced on internal and external surfaces. It is found that both the spacing of the internal and external surfaces and the sources should be approximately equal and substantially less than one wavelength. In general, all of the sources contribute to the boundary fields and consequently the solution includes the effects of multiple scattering within the droplet. For the purposes of the work described here, however, we simplify the droplet scattering problem further by assuming, first, that the internal field can be adequately modelled by integrating equation 5 over a finite portion of the droplet surface and second, that the field at the droplet surface is dominated by local sources. In this case,, is the front region of the droplet surface (closest to the recording device) and we assume,

7.

With these assumptions we have neglected the effects of multiple scattering both within the droplet and any external refraction and/or reflection that causes light to re-enter the droplet. We will return to the validity of our assumptions later; for the moment, however, we note that in practice the droplet surface is usually fairly smooth and although present, these effects are of secondary importance.

The reconstruction process is illustrated for the case of imaging through amultimode telecommunications fibre that was sparsely coated with 3.7 m PMMA micro-spheres by dipping in a diluted aqueous suspension of these particles. The fibre was placed under the coherent microscope shown in Figure 3 with its axis approximately perpendicular to the z-direction defined by the optical axis of the instrument. This microscope illuminates the sample from behind with a plane wave from a diode pumped Nd:YLF laser of wavelength = 523 nm. The microscope utilises an objective lens of NA = 0.55 to provide diffraction limited resolution with a cut-off frequency (amplitude) of 1000 cycles/mm when imaging in air. Reconstruction of the 3D scattered field using this microscope has been discussed elsewhere and the reader is referred to reference [17] for details. It is noted, however, that with the comparatively large NA and high resolution imaging offered by this microscope, the Fresnel approximation (that is equivalent to the paraxial approximation [16]) cannot be justified.

3.1 Images of a telecommunications fibre

First a holographic reconstruction is performed with the assumption that the particles are suspended in air; Figure 4 shows the absolute value of the reconstructed holographic image in 3 distinct x-y planes. Figure 4a) shows the image in a plane which is tangential to the front surface (closest to the objective) of the fibre. Here only a small portion of the surface is illuminated since the plane wave illumination is focused by the rear surface of the fibre. In the illuminated region, images of particles on the front surface of the fibre can be seen. Figure 4b) shows a focal plane 70 m further from the objective and approximately half way through the fibre. In this figure the edge of the fibre is brought into focus and particles on the edge can clearly be seen. The diffracted images of the particles that are on the front surface can also be identified. Figure 4c) shows a focal plane 70 m still further from the objective that approximately corresponds to a plane that is tangential to the rear surface of the fibre. No images can be identified in this plane due to the image aberrations introduced by the front of the fibre.

Figure 5 shows an x-z section through the reconstruction. It can be seen that as the illumination passes through it is brought to focus slightly in front of the fibre due to the curvature of both the front and rear surfaces.It is noted that figure 5 does not resemble the fibre section and shows that although holography is closely related to tomography a single hologram does not produce an accurate 3D image [14]. However, there is significant a-priori knowledge with which to improve the image.In this case, it is known that the fibre is a cylinder we may assume that it has a refractive index of approximately n=1.5. From the best focus centre-plane, the edge of the fibre was identified and the fibre diameter was measured (140 m). This is sufficient information to estimate the fibre surface.Accordingly regularly spaced points were defined on a cylindrical surface, +,in the reconstruction space and the complex valued field interpolated at these points. The field approximately 70 m into the fibre (i.e. the central plane) was then calculated using equations 7 and 5. The complex field propagating throughout the fibre volume was then calculated in the usual way.

Figure 6 shows the absolute value of the reconstructed holographic image in 3 distinct x-y planes corresponding approximately to those in Figure 4. Figure 6a) shows the image in a plane which is tangential to the front surface (closest to the objective) of the fibre. Here, the same particles that were identified in Figure 4a) can be seen. Figure 6b) shows the central plane approximately half way through the fibre. Here there are no particles in focus but the diffracted images of the particles that are on the front and rear surfaces surface can be identified. Figure 6c) shows the x-y plane that is tangential to the rear surface of the fibre and the particles on the rear surface are clearly brought into focus. The characteristic spot at the centre of these particles (an image of the source) and comparison with Figure 4a) shows that the reconstruction is good quality and is approximately diffraction limited.

In the following section the same basic approach is used to produce high fidelity images in a two-phase flow.

3.2.Images of an ink jet

To investigate the suitability of this method to the study of droplet break-up the same basic technique was applied to the imaging of a pulsed ink jet. The ink jet was created using filtered de-ionised water that was driven through a 100 m sapphire nozzle at a pressure of 1.3 Bar. The jet was seeded with 3.7 m PMMA micro-spheres at a concentration of approximately 1% by volume. To form a sequence of droplets the jet was controlled by a The Lee Company INKX0514300AA solenoid valve which supplied a square pulse stream at 5Hz with 0.1% duty cycle. With strobed illumination a regular stream of droplets was observed with an exit velocity of approximately 5 m/s.

The ink jet was placed under the coherent microscope with the nozzle axis approximately perpendicular to the optical axis (z-axis) of the instrument. The instrument was focused approximately at the position of the nozzle axis. Figure 7. shows the reconstructed hologram of a droplet close to the nozzle exit approximately 1ms after the solenoid valve opened. At this stage the droplet is in the form of an extruded filament of fluid. Out-of-focus and aberrated images of particles within the fluid stream can be seen. In order to estimate the position of the droplet surface, the plane where edge of the droplet was in best focus was found. In this case, the droplet has a similar geometry to the telecommunications fibre described previously and was assumed to be cylindrical. The diameter of the droplet and the direction of propagation was estimated from the best focus reconstruction. It was found that the diameter was approximately 92m.The field on surface, , corresponding to the front surface of the droplet was then interpolated according to equation 7. and the internal field at the jet axis, calculated using equation 5. The complex field throughout the droplet was then calculated.

Figure 8 shows the field in three planes separated by 45m. Figure 8a) is tangential to the front surface and particles located in this plane are in good focus and are relatively free from aberration. Moving through the planes in Figure8b), particles move into and out of focus as expected. In Figure 8c) closest to the rear of the fibre the best focus images show signs of astigmatism in regions A and B. Here the particle images are elliptical in shape and exchange major and minor axes as they passthrough focus. It is clear that the effect of the front surface, that is similar to a cylindrical lens, has not been completely compensated and further measures, that we will discuss later, are required.

Figure 9 shows a holographic reconstruction of a droplet further from the nozzle exit where a droplet is beginning to form. The reconstruction assumes homogenous air and is focussed on the axial plane. In this case, the droplet is far from cylindrical and for this reason represents a more challenging reconstruction. It is reasonable, however, to assume that the droplet has axial symmetry, at least in the first instance. The surface now is formed by estimating the edge of the droplet and calculating the surface of revolution about an estimate of its central axis. The intersection of a portion of this surface subtending an angle of ±25° with the focused image has been superposed in Figure 10 for illustrative purposes.As before the field on surface, , corresponding to the front surface of the droplet was then interpolated according to equation 7. and the internal field at the jet axis, calculated using equation 5.

Figure 11 shows the reconstruction of the internal field of the droplet. Figure 11(a) is once again tangential to the front surface, like the reconstruction of the filament in Figure 8(a), the particles focus to a well defined spot in this plane. Figures 11(b) and (c) show planes close to the central axis of the droplet. Most of the particles in focus in these planes are suffering from astigmatism, which we attribute to an incorrect model of the droplet surface. Figure 11(d) shows a plane close to where we estimate the back plane of the droplet. Here particles can still be identified, but they are more severely aberrated.

4. Discussion

Our preliminary results are based on certain assumptions and a-priori information and these deserve further discussion. For the general problem of holographic imaging through seeded droplets we typically know the following;

• Refractive index of the droplet
• Size and optical properties of the seeding
• Illumination conditions

In the procedure outlined in this paper we have exploited some, but not all of this knowledge, and have used this with additional assumptions in order to construct a 3D image. The most critical step in the process is estimating the droplet surface. Inherent in this process was the assumption that the illumination was unperturbed by other droplets and the outline of the droplet provided a good estimate of the plan form of the droplet. With the refractive index and the assumption that the droplet was axisymmetric, this is sufficient information to estimate the droplet surface and image within it.