2. Theoretical background and model construction
2.3 Materials
2.3.1 Inks
The inks used in this theoretical part of the study are generic inks, which approximate the content and mixing ratio of the actual conductive inks used in experiments, manufactured by GEM. For example, in the case of D58 ink, the modelled ink volumetrically contains ~13% silver micro particles, ~51% Ethanol as solvent, and ~36% Epoxy Resin as bonding agent (the mixing ratio is from that of D58 ink). The combination of constituents is chosen by the manufacturer for best performance. The choices of resin and solvent are often uncommon substances protected by patents. However, close approximation is made to give generic and indicative model simulation results [217]. Table 2.1 shows a few common Gem Inks and properties.
2. Theoretical background and model construction
Name Description
D1 Platinised Carbon Ink; an economic alternative for low temperature applications
D58 Silver ink designed to be used for screen printing working electrodes.
D6 A screen printable Platinum ink designed for printing working electrodes in electrochemical sensor applications.
2.3.2 Substrates
There are a few common substrates used for LADW. The popular types include alumina (Al2O3), polyethylene terephthalate (PET), carbon fibre composite (CFC), polyaramid and
paper, as listed in Table 2.2. This work focuses on the two most commonly used materials, namely alumina and PET. Alumina is used as a well known dielectric material, which is chemically inert and thermally resistant; it also has excellent durability in harsh environments due to its hardness. Alumina is light weight and reasonably cheap. PET is chosen for its light weight, physical flexibility, elasticity against stress, and economical advantage. PET has relatively low thermal conductivity; while Al2O3 is considerably more thermally conductive.
Therefore the LADW curing process on PET is very different from that on Alumina.
Substrate Thermal Conductivity (W/mK)
Advantage Disadvantage Density (kg/m3)
Specific Heat Capacity (J/kgK)
Alumina 18 hard, thermally resistant, rigid
brittle 961 880
CFC 0.059 light, strong brittle 1850 950
PET 0.23 flexible cheap, light easily thermally damaged 1380 1000
Polyaramid 0.04 flexible cheap, light easily thermally damaged, variable surface roughness
1440 1400
Burn Paper 0.01 cheap light, flexible poor strength, poor resistance to humidity
192 1336
Table 2.2: A table of substrate properties
2. Theoretical background and model construction
2.4 Laser intensity distributions
2.4.1 Gaussian
A Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or "TEM00 mode" of the laser optical resonator.
A Gaussian beam mode is described by the following equation
exp(
2
2)
2 0w
r
I
I
=
-
(2.1) where I0 is the peak intensity at the centre of the beam, ω is the beam waist; r is the distanceaway from the centre of the beam, illustrated in Fig 2.3.
The Gaussian beam mode is good for operations such as cutting, drilling and welding. In these processes the beam quality of the laser often determines the quality of the process. However this mode is not ideal for surface heat treatment, due to the uneven energy intensity between the centre and the edge of the laser spot.
2.4.2 Top-hat
A top-hat beam profile can be described by the following
ITH = I1 if r ≤ ω1 (2.2) = 0 if r > ω1
2. Theoretical background and model construction
The circular top-hat intensity distribution is one of the most popular shaped profiles. It homogenises laser intensity and evenly distributes energy across the focus spot. Like the Gaussian beam, the circular shape enables top-hat beams to be used in any direction. It is mostly applied in selectively micro-machining thin films in the electronics, displays and solar cells industry. Also this beam profile is highly desired in surface heat treatment of metal parts compared to Gaussian beams, in the sense that it can reduce the overlapping needed, hence speeding up the process while reducing undesired tempering effect [18].
2.4.3 Annular filled rings
2
I
I
R=
if ω3<r ≤ ω2 (2.3)= I3 if r < ω3
=
0
if r >ω2Fig 2.4 The top-hat intensity distribution.
2. Theoretical background and model construction
Wellburn [18] studied the advantage of the annular ring profile extensively. He suggested that the filled ring has advantages compared to both Gaussian and the top-hat intensity distributions. In his work simulations were used to successfully support this argument. However, the simulation done in his work was based on a slab of metal, which is linear and homogeneous. A direct translation of argument to this study would be invalid since this work investigates the application of a shaped beam in curing ink tracks, that is, surface treating a thin, non-linear material.
2.4.4 Power integration over the beam focal spot
The total power delivered by a laser beam can be calculated by integrating the intensity over the area irradiated:
dA
I
P
=
òò
i (2.4) where P is the power of laser, and dA is the area increment of the processed surface.2.5 Curing mechanism
In this section, a novel theory for the curing mechanism has been proposed and formulated in detail. For the first time, a comprehensive account of the thermal dynamic process for LADW is demonstrated, allowing a parametric study over a range of processes with similar mechanisms and set-ups.
Fig 2.6 illustrates a single pass curing process. As discussed previously, the ink is a composite of silver particles, resin and solvent, and hence, a nonlinear overall curing mechanism takes place. The temperature of the ink rises in a series of steps as the heat source gradually couples energy into the ink. In stage 1, heat from laser brings the irradiated region to an elevated temperature of vaporisation Tv, which is the temperature where the latent heat
of evaporation of solvent becomes dominant, represented by stage 2. After the solvent evaporates, the volume has contracted, resulting in the silver particles becoming more closely packed.
The overall reduced heat capacity results a larger curve gradient in stage 3. Further heating will elevate the temperature of the material untill Tcur is achieved. Tcur is the temperature
2. Theoretical background and model construction
where cross-linking takes place, in stage 4. Once this stage is finished, the ink can be considered cured andextra energy will only go into heating the ink track, stage 5. If an “over- curing” temperature Tov is exceeded, one risks damaging the bonding between the silver track
and the substrate, and therefore this needs to be avoided. For the generic ink of the study illustrated in Fig 2.6,Tv is set to 80 oC (353 K), Tcur is 160 oC (433K) and Tov is 330 oC (603
K). This mechanism approaches the problem from a purely thermal point of view and assumes the material to be homogeneous at during the curing process.
2.5.1 Coupled laser energy
Based on this analysis, we have
(2.5)
where ΔQ is the total energy required to bring unit volume of ink from room temperature to 5 4 3 2 1
Q
Q
Q
Q
Q
Q
Q=
D
i=D
+D
+D
+D
+D
D
å
2. Theoretical background and model construction
Tov; ΔQ1, ΔQ2, ΔQ3, ΔQ4 ΔQ5 are respectively the energy required for the individual stages
described in Fig 2.6.
If we denote the temperature change for each stage with ΔTi, denote the Latent heat of
solvent with Ls, the energy for cross-linking with Lcr, and use Cp for the specific heat capacity
with, ρ for density, and % for volumetric percentage, then we have
(2.6)
where V is unit volume.
The suffixes Ag, r and s respectively represent silver micro particles, resin and solvent.
We assume the density and specific heat capacity of epoxy resin do not change over the curing process. The specific heat capacity and Latent heat of Ethanol is 2.3 KJ/kg and 846 KJ/kg. The cross-linking energy of the resin is approximately 890 J/kg, as estimated from Sato’s result [131]
2.5.2 Effective Specific Heat Capacity
As a consequence of the nonlinearity of the ink system, one needs to define an Effective Specific Heat Capacity Cp-eff for the ink, which is a function of temperature. Use ρink-i for the
corresponding density of the ink for each stage of curing, we have:
and 1 ) ( ) ( ) ( 1