Jackson M.J. Micro and Nanomanufacturing

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422 Micro- and Nanomanufacturing

2.5 1

3 1.5

I n

0.5

I I I I I I

0 2 4 6 8 10 12

Non-dimensional distance from nozzle exit (X/De)

0.3mm M.L.N.

•0.1mm M.L.N.

0.2mm M.L.N.

-0.05mm M.L.N.

Fig. 8.17. Mach number distribution about the centreline of a minimum length mi-

cro-nozzle

Laser Micro- and Nanofabrication 423

When the highly under expansion condition is reached, the pres-

ence of a Mach shock disc exerts a large penalty on the centerline

pressure distribution. The loss increases with increasing exit Mach

number. The downstream exit Mach number distribution is not sig-

nificantly increased by raising inlet pressure once the Mach shock

disc forms. Sonic nozzles reach this condition very quickly with

inlet pressures as low as 2.6 bar pressure Therefore, the location of

stand-off distance becomes extremely critical. Conversely, the con-

vergent-divergent nozzle is under-expanded slightly in comparison

with the sonic nozzle, but the Mach number fluctuation is still sig-

nificant for laser micro-machining applications and applications

where surface roughness is the governing factor. M.L.N.s provide

the best possible results with uniform potential cores and very small

fluctuations in Mach number distribution (Figure 8.17).

Validation of IVI.L.N. Contours

Predictions of supersonic nozzle flow structures using the M.O.C.

inviscid contours were solved using a C.F.D. linear finite volume

method. The turbulence model used was the re-normalized group-

based (R.N.G.) k-6 model, derived from the instantaneous Navier-

Stokes equations. The near-wall treatment was in non-equilibrium.

The geometry and grid were drawn in the pre-processor using non-

equilibrium mesh seeds to analyze boundary formation. Figure 8.18

shows the Mach number distribution inside the nozzles along the

axi-symmetric line when a viscous R.N.G. first-order diffusive

model is computed through an inviscid geometry predicted by the

M.O.C. theory.

As predicted by the M.O.C. theory, the exit Mach number is re-

duced owing to boundary layer formation. This effect is enhanced as

the throat diameter of the nozzle is reduced. If a smaller diameter

nozzle is used then the boundary layer is proportionally larger. Fig-

ure 8.19 shows Mach number distributions for corrected boundary

layers in M.O.C. inviscid contours.

424 Micro- and Nanomanufacturing

0.2 0.4 0.6 0.8

Non-dimensioned axisymmetric nozzle length

-0.3mm MOC -

•0.1mm CFD -

0.3mm

CFD

•0.05mm CFD

0.2mm

CFD

Fig. 8.18. C.F.D. generated viscid flow through a non-corrected boundary layer

minimum length nozzle inviscid contour. The results are presented as Mach num-

ber (U/A) as function of non-dimensioned axisymmetric nozzle length (x/Xe)

Laser Micro- and Nanofabrication

425

I I I

0.2 0.4 0.6 0.8

Non-dimensioned axisymmetric nozzle length

•0.3mm

MOC -

"0.1mm

CFD -

0.3mm

CFD

-0.05mm

CFD

0.2mm

CFD

Fig. 8.19. C.F.D. generated viscid flow through

corrected boundary layer mini-

mum length nozzle inviscid contour.

The

results

are

presented

Mach number

(U/A)

function

non-dimensioned axisymmetric nozzle length (x/Xe)

426 Micro- and Nanomanufacturing

When the inviscid contour is displaced by the displacement

thickness, all contours from the M.O.C. are representative of C.F.D.

flow propagation predictions. If the flow is supersonic, especially at

high Mach numbers, a second order diffusion model must be used to

highlight areas of interest, i.e., shock capturing, boundary layer in-

formation, etc. The M.O.C. has provided a very accurate method of

predicting supersonic nozzle contours for both de Laval and

M.L.N.s. M.L.N.s offer many advantages over de Laval, not only

due to its small scale in contrast to de Laval, but because boundary

layer formation in M.L.N.s is at a minimum level. This is advanta-

geous when designing supersonic micro-nozzles allowing nozzle di-

ameters to be as small as 50 |Lim.

C.F.D. models using a M.L.N, with a throat diameter of 300 |Lim

and exit diameter of 429 |Lim were generated for drilled holes and

kerf geometries in 1 mm thickness materials. In these circumstances

the inlet pressure was set at 8 bar, and the nozzle stand-off distance

was 1 mm. Figure 8.20 depicts two holes drilled to a depth of 250

|um, one with a diameter equivalent to the nozzle diameter, the sec-

ond hole being eight times smaller.

Although the results represent an ideal case, in reality the hole

geometry would be dependent on the pulse shape, ablation mecha-

nism, and the dynamic characteristics of the plasma. The finding

suggests that a blunt-body normal shock is present on the surface.

This shock wave induces higher compressibility in the jet, producing

oblique shocks upstream to the flow. Bow-like structures above the

entrance to the hole represent a normal shock wave producing a

diamond-shaped pattern above the normal shock in the oblique

shock region. As flow properties across the shock wave change, the

pressure and density increase while velocity decreases significantly.

This phenomenon is further represented in Figure 8.21 as velocity

vector diagrams that indicate the direction and magnitude of flow in-

side the processing region. The flow structures show signs of flow

separation, stagnation regions, and vortex formation. The primary

mechanism for removing molten and evaporated material either in

the drilling or cutting process is initiated by the pressure gradient be-

tween the interaction zone and the surrounding atmosphere. Use of

supersonic assisted gas jet to remove molten material from the hole

may have an adverse effect on the process mechanism.

Laser Micro- and Nanofabrication 427

Fig. 8.20. C.F.D. generated density distribution in the jet stream that impinges

onto the surface of the laser-machined surface, (a) Hole diameter, dn = 429

and length 1^= 250 // m; (b) hole diameter, dH= 53.6 jUm, and length LH= 250

m. Stand-off distance,

X^s,

is 1 mm. The density scale shown to the left of the

density distributions is in kg m"^

428 Micro- and Nanomanufacturing

3.00e+02

2.70e+02

2.40e+02

Z10e+02

:|l.80e+02

1.50e+02

1.20e+02

9,00e+01

6.00e+01

3.00e+01

4.00e-02

M i

/ ^ ^^ ^,'/

-Ml

\ ' %'•

r,r^'s

*;7f/

(a)

V^"A%'

' li

(b)

Fig. 8.21. C.F.D. generated velocity profile inside a laser-drilled hole, (a) Hole

diameter, dn = 429 // m, and length L^ = 250

m; (b) hole diameter, dn =

53.6 jUm, and length LH = 250 // m. The velocity scale shown to the left of the

velocity profiles is in ms"^

Laser Micro- and Nanofabrication 429

As shown in Figures 8.20 and 8.21, the supersonic jet estabHshes

a normal shock wave at the entrance to the hole, raising the pressure

just above the hole that results in a reduction of the overall pressure

gradient between atmosphere and the interaction zone. Laser cutting

and drilling processes are pressure-driven systems.

The difference in the pressure from the leading edge to the exit

of the kerf creates viscous drag, or shear stress, on the molten film

created. Ideally the pressure gradient along the kerf should exponen-

tially decay, but in reality this may not be the case. If flow is super-

sonic a sudden change in the pressure gradient creates an internal

shock formation. If the pressure gradient changes from positive to

negative then flow separation is imminent, generating vortices, or

eddies, inside the processing zone thus creating effects that are det-

rimental to the roughness of

the

processed surface.

A supersonic jet is a pressure driven system, which creates shear

stresses on a molten film. The magnitude of the shear stress applied

depends on the jet velocity and the surface tension of the gas-liquid-

solid interface. For simplicity, walls in C.F.D. models are treated as

a solid surface with no external heat source. Shear stress distribu-

tions along these walls are represented by Figures 8.22 and 8.23 for

hole diameters, dn = 429 |Lxm and 53.6 |Lim, respectively. It can be

seen that the shear stress distribution inside a drilled hole depends on

its aspect ratio with respect to the nozzle diameter. As the aspect ra-

tio is increased the shear stress is reduced. As long as the shear

stress is positive there is no separation of flow; negative shear

stresses indicate a change in velocity direction of 180°. Shear stress

fluctuations indicate that there are variations in mass flow along the

surface of the wall.

Very large fluctuations of shear stress are observed in the hole

with the smallest aspect ratio, which leads to a turbulent flow re-

gime. If the shock width is approximately the same size as the di-

ameter of the hole then shock interaction with the stagnation pres-

sure creates uneven fluid properties inside the processing zone, as

shown in Figure

8.21a.

At higher aspect ratios the stagnation pres-

sure build-up is greater within the processing zone. This higher pres-

sure decreases the velocity along the wall, hence reducing the shear

stress along the wall.

430 Micro- and Nanomanufacturing

7000

6000

5000

4000

3000-

2000-

1000-

-1000 H

-2000

-3000

Non-dimensioned hole length

^—— Hole length/Hole diameter

= 0.58

Hole length/Hole diameter

= 1.16

—

Hole length/Hole diameter

= 1.74

Fig.

8.22.

Shear stress distribution within

the

melt pool inside

the

laser drilled

hole that

has a

diameter

429

The

results

are

presented

function

the

ratio

hole length-to-diameter,

LH/DH-

Note that the unit

shear stress

is N m'^

Laser Micro- and Nanofabrication 431

1200-1

1000 H

800 H

600 H

400 H

200 H

-200 i

-400-"

Non-dimensioned hole length

Hole

length/Hole diameter = 4.66 — - Hole length/Hole diameter = 9.32

Hole length/Hole diameter = 14

Fig. 8.23. Shear stress distribution within the melt pool inside the laser drilled

hole that has a diameter of 53.6 // m. The results are presented as function of the

ratio of hole length-to-diameter, I///D//. Note that the unit of shear stress is N m'^