Preprint for submission to Journal of Applied Physics
Thermal and Fluid Processes of a Thin Melt Zone during
Femtosecond Laser Ablation of Glass: The Formation of Rims by
Single Laser Pulses
Adela Ben-Yakar,1, ∗ Anthony Harkin,2 Jacqueline
Ashmore,2 Robert L. Byer,1 and Howard A. Stone2
1
Applied Physics Department, Ginzton Lab,
Stanford University CA, USA 94305
2
Division of Engineering and Applied Sciences,
Harvard University, Cambridge, MA, USA 02138
(Dated: September 29, 2003)
Abstract
We study the formation mechanism of rims created around femtosecond laser-ablated craters on
glass. Experimental studies of the surface morphology reveal that a thin rim is formed around the
smooth craters and is raised above the undamaged surface by about 50-100 nm. To investigate the
mechanism of rim formation following a single ultrafast laser pulse, we perform a one-dimensional
theoretical analysis of the thermal and fluid processes involved in the ablation process. The results
indicate the existence of a very thin melted zone below the surface and suggest that the rim is
formed by the high pressure plasma producing a pressure-driven fluid motion of the molten material
outwards from the center of the crater. The heat transfer calculations show a melt thickness of
the order of 1 µm and a melt lifetime of the order of 1 µs. The numerical solution of fluid motion
of the thin melt, on the other hand, demonstrate that the time for the melt to flow to the crater
edge and to form a rim is about 1 µs, which is of the same order of magnitude as the melt lifetime.
The possibility of controlling or suppressing the rim formation is also discussed.
∗
email: [email protected]; tel: (650) 723-0161, fax: (650) 723-2666
1
I.
INTRODUCTION
Laser micromachining has been widely studied for many materials and has recently been
extended to ultrashort laser pulses that can machine any material to very high precision at
the micron scale [1]. The key benefits of an ultrashort pulse include its ability to produce
a very high peak intensity (> 1016 W/cm2 ) and rapid deposition of energy into the material. While high peak intensities allow energy delivery even into transparent high band-gap
materials such as glass, the rapid absorption of energy allows material removal, or ablation,
before significant heating of the substrate occurs.
Ultrashort pulses interact with the material by a nonlinear mechanism that is very different from that of conventional longer pulse lasers. The high energy density during the
ultrashort laser pulses can initiate nonlinear processes such as multiphoton absorption [2, 3],
which allows machining of any material independent of the laser wavelength. For example,
it is possible to micromachine glass with very high precision using near infrared wavelengths,
around 800 nm, at which the linear absorption properties of glass are very weak.
While the nonlinear effects and short time scales associated with ultrashort laser ablation
are believed to provide a non-thermal micromachining of glass materials, in some cases
complicated surface morphologies result [4, 5]. Detailed experimental studies have shown
that the laser ablation process might actually be thermal in nature, even in the femtosecond
operating range [5]. Our preliminary estimates show that a very small amount of energy, less
than 3% of the incident energy, might be deposited in the undamaged part of the borosilicate
glass as thermal energy and so create a transient shallow molten zone below the ablated area
(where there is the expanding plasma) [5]. The molten material rearranges (flows) due to
forces acting at the surface of the melt. Another recent study also measured that about 8%
of the incoming energy was thermalized and transmitted to the undamaged part of a quartz
material when irradiated with an ultrafast laser pulse [6].
In order to improve the quality and precision of ultrafast laser micromachining, it is
important to understand, model and quantify the thermal nature of the process and any
material rearrangement that occurs. In this paper, we provide scaling arguments for the
depth of the melted layer and model its motion during the lifetime of the melt. If the melt
lifetime is long enough, forces acting on the liquid will be able to drive the molten material
from the center to the edges of the crater, which creates an elevated rim around the ablated
2
crater. Multiple laser pulses show that this rim formation causes a surface roughness and
therefore reduces the precision of an ultrafast laser micromachining process.
Two main forces might affect the flow of a molten layer below the expanding plasma:
1) thermocapillary forces (Marangoni flow) [7, 8] and 2) forces exerted by the pressure of
the plasma above the surface [9, 10]. Thermocapillary flow is induced by the temperature
gradient on the surface which is expected to follow the Gaussian beam intensity profile of
the laser. In studies of laser texturing of silicon surfaces in the absence of ablation, the rim
formation was attributed to the thermocapillary flow in thin films created by nanosecond
laser pulse heating [7]. The temperature gradient on the surface creates surface tension
gradients that drives material from the hot center to the cold periphery. This response is
expected in most materials where the surface tension, γ, decreases as the fluid gets hotter
(dγ/dT < 0). However, in the case of glass dγ/dT is positive [11]. Consequently, such a
thermocapillary flow in laser irradiated glass surfaces would actually drive fluid from the cold
periphery to the hot center of the melt in contrast to what was observed in our experiments.
In addition, the effect of thermocapillary flow in glass is expected to be negligible because
of its high viscosity, which leads to a flow time scale much longer than the typical melt time
scale, as detailed further below.
On the other hand, a hydrodynamic force due to the pressure gradients exerted by the
plasma onto the molten material might play an important role in the rim formation. A
gradient of ablation pressure on the molten surface can induce a lateral melt flow to the
periphery. The pressure gradients are expected to be particularly large at the edges of
the ablated crater, which should be close the plasma/air interface. Because of these large
pressure gradients, we might expect a melt flow to the periphery and rise of a thin rim at
the edges of the melted surface much like a splash of a fluid. Although some of aspects
of this mechanism have been discussed qualitatively in the literature for nanosecond laser
irradiation of metals [9], in this paper we provide a detailed model of the flow processes
associated with the first pulse and so give the time evolution for the profile of the melt
surface. The mathematical model of the physical processes gives rise to a rim topography,
though has one significant unknown which is the pressure gradient along the surface that
results from the plasma. This uncertainty primarily affects time scales and not the physics.
We first present typical experimental results of the surface morphology of craters ablated
using single and multiple femtosecond laser pulses with an emphasis on the rim formation
3
around the craters (Section II). We then discuss the level of the material heating and
estimate the variation of melt depth with time for a single laser pulse (Section III A). Finally,
we present a thin film model to estimate the characteristic time scales of various mechanisms
causing the melt to flow outwards to create the rim and show representative numerical
simulations of fluid flow in a thin molten layer (Section III B). We conclude with some ideas
for suppressing formation of the rim (Section IV).
II.
EXPERIMENTAL RESULTS
Our experimental studies focus on the surface morphology of borosilicate glass
(BorofloatTM ) ablated using single femtosecond laser pulses. We irradiated the glass samples
with 800-nm 100-fs pulses from a regeneratively amplified Ti:sapphire laser. The laser was
focused with a 250-mm focal-length lens to a spot size of about 30 µm. The surface of the
sample was positioned to be normal to the direction of the incident beam. We performed
these experiments in air at atmospheric pressure. Following irradiation, the samples were
analyzed with a scanning electron microscope (SEM). The basic results were reported recently [5] and we summarize them here with some additional features as they form the basis
for the theoretical and modeling considerations given in Section III.
Figure 1 presents three SEM images of crater rims produced by an average laser fluence of
34 J/cm2 . The first image (Fig. 1a) shows a thin circular rim around a nearly smooth crater
following a single pulse of the laser on a flat glass substrate. It is the resemblance of this
rim to a “resolidified splash” of a molten layer that originally motivated our investigation
of the dynamical processes that result in the formation of the rim.
The second image (Fig. 1b) shows that when a second pulse irradiates a previously formed
rim, a new rim is formed inside the original rim. The distance between the two rims is
approximately equal to the wavelength-λ of the light, which suggests that diffraction of
light plays an important role [12]. Upon close inspection of the area between the pulses,
there is a wave pattern apparently due to the diffraction-induced modulation of the second
laser pulse.
When a third pulse irradiates two previously overlapping craters (Fig. 1c), micrometerscale organized features appear along the rim. The smaller scale ripple-like features that are
evident circumferentially along the rim, which is basically semicircular in cross-section, are
4
presumably a manifestation of a Rayleigh-capillary instability familiar from the disintegration of fluid filaments. This aspect of the surface evolution has not been studied. As shown
in Fig. 2, the interplay between rim formation and diffraction results in a perceptible surface roughness when attempting to micromachine channels in a glass substrate by scanning
a pulsed laser across the surface.
A detailed AFM study of a single crater is shown in Fig. 3. The rim is raised by about
50 − 100 nm above the surface and the maximum depth of the ablated crater is about 400600 nm depending on the laser fluence [13].
In order to understand and control the micromachining process using ultrafast lasers, it is
necessary to investigate the formation mechanism of the surface microfeatures. To address
these issues, we first have to examine the rim formation of a single laser pulse, which is the
main focus of this paper.
III.
THEORETICAL MODELLING AND DISCUSSION
A theoretical model is described to characterize the thermal and flow processes associated
with the first laser pulse. The goal is to identify the nature of the rim formation around the
laser ablated craters and suggest ways (such as modification of the laser profile) to eliminate
it. The laser deposits energy to the substrate which melts the glass. To estimate the melt
thickness, and its variation in time, the optical light penetration and heat conduction into
the material are studied (Section III A). A thin film model of the molten surface layer is
then introduced to explore the temporal evolution of the melt surface (Section III B).
A.
Heat Transfer Calculations
In this section, we estimate the thickness and the lifetime of the molten layer and present
a model for the time dependent variations of the melt thickness. To achieve this, we first
calculate the fraction of the laser energy deposited in the bulk of the material as heat by
analyzing 1) the fraction of the incoming energy that is absorbed by the material and 2) how
this energy dissipates from the illuminated regime (i.e., partition sequence of the absorbed
laser energy).
In dielectric materials, the incident laser beam is absorbed by electrons through nonlinear
5
processes (multiphoton and avalanche ionization) [2, 3] and then transferred to the lattice
on the time scale of few picoseconds. At this stage, a high pressure plasma is formed above
the surface; the plasma expands and removes the ablated material away from the surface.
Dissipation of the energy deposited in the substrate begins only after the laser pulse
is gone and when the energy is transferred from the high energy electrons to the lattice.
Numerical simulations of the fluid and thermal dynamics of a laser-induced plasma on an
aluminum target irradiated with 100 picosecond laser pulses show that a few microseconds
after irradiation, ∼ 70 % of the absorbed energy is used by the expanding plasma to move
the ambient gas [14]. About 20 % of the absorbed energy is lost in radiation and only 10 %
of it remains in the target as thermal energy.
In the case of femtosecond laser ablation of glass, a rather complicated and time consuming numerical solution is required to estimate the partition of the absorbed laser energy. In
this paper, we propose to use the optical light penetration depth to estimate the fraction of
the incoming laser energy deposited in the glass as heat (thermal energy) [13].
1.
Initial Energy Deposition Depth
We assume that the absorbed laser energy is deposited uniformly in a layer defined by the
penetration of light as illustrated in Fig. 4. The light intensity is attenuated exponentially
with depth z according to the Beer-Lambert law, which provides a convenient means to
quantify the penetration depth of the absorbed laser energy. Then the absorbed laser fluence
as a function of depth, Fa (z), is given by
z
Fa (z) = AF0 exp − −1
αeff
,
(1)
−1
where A is the surface absorptivity, αeff
is the effective optical penetration depth, and F0 is
the average laser fluence (energy per unit area). Let us next summarize how we determine
−1
from experimental measurements.
the values of A and αeff
The absorptivity of glass depends on the intensity of the laser irradiation and varies
with time during the duration of the laser pulse. When glass is exposed to high intensity
ultrashort pulses, its reflectivity rises with time as the plasma density increases [2]. Once
the critical surface plasma density is formed, any further incident laser energy is reflected
back from the surface due to an induced skin effect. Perry et al. [2] showed that when
6
the incident laser fluence is high above threshold (F > 5 − 10Fth where Fth ≈ 2 J/cm2 for
fused silica), a plasma with a critical density can be achieved early in the pulse and a large
portion of the energy is reflected back from the sample. At a laser irradiance of 1014 W/cm2
they calculate the reflectance of fused silica to be around 70%. We used a similar laser
irradiance, and therefore we assume that on average 30% of the incident energy is absorbed
by the borosilicate glass, namely A = 0.3.
For the effective optical penetration depth, we use experimental data from our previous
−1
measurements [13]. We determined αeff
= 224 nm by calculating the slope of the linear
relationship between the ablation depth, ha , and the logarithm of the incident laser fluence,
F0 , expressed by
ha =
−1
αeff
ln
F0
Fth
.
(2)
Here, Fth = 1.7 J/cm2 is the measured threshold fluence for borosilicate glass when 200-fs
and 780-nm laser pulses are used. Since at laser fluences below Fth no ablation occurs, the
material is ablated to a depth, z = ha , where the laser fluence drops to Fth . Therefore, by
substituting Fa (z = ha ) = AFth in Eq. (1) we can obtain the expression given in Eq. (2).
The distance, where the laser fluence decreases to 1/e of its value, is interpreted as the
−1
“effective optical penetration depth” (αeff
) in accordance with the Beer-Lambert absorption
law. Using Eqs. (1) and (2), we can write the variation of the absorbed laser fluence with
depth, Fa (z), in terms of Fth and ha ,
z − ha
Fa (z) = AFth exp − −1
αeff
,
(3)
where z = 0 corresponds to the location of the flat surface away from the irradiated zone.
We assume that the absorbed laser energy that penetrates beyond the ablation depth,
ha , remains in the material as thermal energy. Since the laser fluence at the ablation
depth is equal to AFth , the thermalized energy remaining in the material as heat is Eheat =
AFth (πw02 ), where w0 is the 1/e2 laser beam radius at the surface. Thus, we find that a fixed
amount of laser fluence, approximately Fheat = AFth = 0.5 J/cm2 , heats the undamaged part
of the material independent of the incident laser fluence when it is well above the threshold.
This means that, in our experiments, presented in Fig. 1, less than 2% of the total incident
laser pulse energy goes into heating of the material (the value Fheat = 0.5 J/cm2 is about
1.5 % of the incident laser fluence of Fheat = 34 J/cm2 ).
7
2.
Initial Temperature Distribution
The initial energy deposition described by Eq. (3) produces an initial temperature distribution in the glass, T (z, 0), with an exponential profile
T (z, 0) − T∞
Fa
Fheat
z − ha
=
−1 =
−1 exp −
−1
ρCp αeff
ρCp αeff
αeff
(4)
where Fheat = AFth is the portion of the absorbed laser fluence that goes into heating, T∞
is the initial temperature, ρ is the density of the substrate, and Cp is the heat capacity.
Melting occurs when the local temperature exceeds the temperature, Tm ≈ 1500 K [15]. The
value Tm used here is the working point temperature of glass, defined as the temperature at
which the glass can readily be formed, which corresponds to a viscosity of 103 Pa.s. Since
the glass materials do not have a latent heat of melting, all of the absorbed energy goes
into melting. We can therefore calculate the initial melting depth, hm (t = 0) = zm − ha ,
−1
−1
from hm = αeff
ln(Fheat /(Tm − T∞ )ρCp αeff
) where Cp = 1250 J/kgK (see the thermophysical
properties of glass in Table I); we have chosen to evaluate all physical parameters at the
mean temperature of 900 K. This calculation yields an initial molten layer thickness of
hm (t = 0) ≈ 424 nm.
In the above discussion of energy deposition, we assume that the incident laser beam
and the resulting initial temperature distribution inside the glass material attenuate with
depth in accordance with the Beer-Lambert law (see the illustration in Fig. 4). When the
material is illuminated with an ultrashort laser pulse, a high-temperature and high-pressure
plasma is formed above the surface, presumably down to the ablation depth. The ablated
material is removed by the expansion of the plasma. Below the plasma there is a thin zone
of a molten glass with an initial thickness of hm = z(Tm ) − ha that can be calculated using
Eq. (4). The molten region grows, then decays according to conduction process and the
shape of the molten layer changes. Both these effects are described in more detail below.
3.
Heat Conduction and Variation of Melt Thickness
Following the cessation of ultrafast energy input, the melting process continues as the
heat flows out of the region where the initial energy is deposited. Diffusion of the thermal
energy determines the movement of the melting front and therefore the variation of the melt
thickness.
8
We calculate the heat flow out of the region where the initial energy is deposited by solving a one-dimensional (1-D) heat conduction equation with an initial temperature profile
described by Eq. (4). The cooling at the top of the melt zone is assumed to be negligible because of the presence of the high temperature plasma. During the expansion, the
plasma cools in tens of microseconds from some very high initial temperature to the ambient temperature. So all of the heat loss is assumed to take place through the solid, as the
rearrangement of the molten layer takes place on shorter time scales.
The one-dimensional heat conduction model can then be described by
∂T
∂
∂T
=
(D
), z > ha
∂t
∂z
∂z
∂T
= 0 at z = ha
∂z
T = T∞
(5)
(6)
as z → ∞
T (z, 0) = T∞ +
(7)
Fheat −(z−ha )/α−1
eff
−1 e
ρCp αeff
(8)
where ha is the ablation depth or, in other words, the depth where the molten region begins.
As a first approximation, we assume that the heat capacity, Cp , the thermal conductivity, k,
the density, ρ, and therefore the thermal diffusivity, D = k/ρCp , are constants at an average
temperature of 900 K.
To find an analytical solution for this heat conduction problem, nondimensionalize variables as:
T̄ =
−1
ρCp αeff
(T − T∞ ),
Fheat
z̄ =
z − ha
−1 ,
αeff
t̄ =
D
−1 2 t.
)
(αeff
(9)
The nondimensional problem statement is then
∂ T̄
∂ 2 T̄
=
, z̄ > 0
∂ t̄
∂ z̄ 2
∂ T̄
= 0 at z̄ = 0
∂ z̄
T̄ = 0 as z̄ → ∞
T̄ (z̄, 0) = f (z̄) = e−z̄ ,
(10)
(11)
(12)
(13)
which has the solution
T̄ (z̄, t̄) = √
1
4π t̄
Z
∞
0
h
i
2
2
f (ζ) e−(z̄+ζ) /4t̄ + e−(z̄−ζ) /4t̄ dζ.
9
(14)
The solution for f (ζ) = e−ζ can be expressed in terms of the complementary error function,
erfc(s), as:
et̄ z̄
−z̄ + 2t̄
z̄ + 2t̄
−z̄
√
T̄ (z̄, t̄) =
+ e erfc
.
e erfc √
2
4t̄
4t̄
(15)
We can now calculate the temperature distribution in the material that is not ablated (i.e.
below the plasma) using the analytical solution given in Eq. (15). The variation of the melt
−1
depth as a function of time follows from hm (t) = zm − ha = αeff
z̄ as where T (zm , t) = Tm . A
plot of the melt depth as a function of time is shown in Fig. 5 for two different laser fluences
thermalized in the material as heat; 1) Fheat = AFth = 0.5 J/cm2 and 2) Fheat = 1.0 J/cm2
since it is possible that radiation from the high temperature plasma also contributes to
the melting of the material. The numerical results show that for the material parameters
representative of the experiment, the melting process continues for 0.25-0.75 µs and then
solidification begins. These calculations provide us estimates for the order of magnitude of
two important characteristic scales of the melt zone:
• An average melt depth, hhm i, of the order of 1 µm.
• An average melt lifetime, tm , of the order of 1 µs.
With these estimates in hand, we will discuss in Section B which hydrodynamic forces acting
on the melt have enough time to rearrange the liquid to form a rim within the melt lifetime.
B.
1.
Fluid Dynamics Calculations
Thin Film Model
In order to examine hydrodynamic conditions under which a crater partially filled with
molten glass can form rims reaching the heights measured in experiments, we develop a
one-dimensional model of the fluid motion as illustrated in Fig. 6. It is natural to work with
cylindrical coordinates. The free surface of the molten glass is described by z = h(r, t) and
the boundary between the liquid and the solid substrate is given by the time-independent
profile z = b(r). I don’t know quite how to state the justification for this. - Tony
The shape of the thin liquid film evolves owing to variations in the surface tension γ along
the free surface and because of the very high pressure ppl (r) of the plasma above the free
surface. Because the melted region has a typical height much less than the typical width of
10
the laser ablated craters, the lubrication approximation (see Appendix A) can be used to
obtain an evolution equation for the time-dependent profile of the free surface h(r, t):


3
3
2
(h − b) ∂ γ ∂
∂h
∂h 1 ∂  (h − b) dγ (h − b) dppl
 = 0,
−
+
r
(16)
+
r
∂t
r ∂r
2µ dr
3µ
dr
3µ ∂r r ∂r
∂r
{z
} |
{z
} |
|
{z
}
Marangoni
Pressure
Curvature
where µ is the liquid viscosity and the different flux contributions have been labelled. The
first term in brackets accounts for motion caused by the surface tension gradients due to
the uneven heating of the surface (thermocapillary or Marangoni-driven flow). The second
term accounts for the motion due to the pressure gradients exerted by the plasma onto the
molten material, and the third term in (16) represents surface tension effects, which enter
as the product of surface tension and surface curvature (the latter has been linearized).
Also, because of the very complicated dynamics of the plasma, for which the details are
unknown, we assume in our model that the plasma pressure field, ppl , at the surface of the
molten material is independent of time and therefore we use a time-averaged plasma pressure
distribution, i.e. ppl (r) only.
There are three assumptions needed to justify the use of Eq. (16): (i) The film must
be thin, i.e. hhm i/L 1, where hhm i is the average melt depth and L is typical of the
radial dimensions of the molten region. (ii) The Reynolds number Re for the film flow (see
Appendix A) must be small, Re = ρhhm i3 hppl i/(µ2 L) 1. (iii) The flow must be quasi-
steady which requires the time for “viscous diffusion” across the thin layer, ρhhm i2 /µ, to be
small compared to the time scale for film evolution. We verify these assumptions below.
When a surface temperature distribution, Ts (r), is imposed, then the surface tension
varies according to
dγ dTs
dγ
=
.
dr
dTs dr
(17)
We assume γT = dγ/dTs is constant and neglect the effect of temperature variations on
the viscosity. Using Eq. (16) we can estimate characteristic time scales associated with the
Marangoni flow (τM ), and the pressure-driven flow (τp ),
Marangoni flow:
Pressure-driven flow:
11
µL2
γT Tm hhm i
µL2
τp ≈
hppl ihhm i2
τM ≈
(18)
(19)
where hhm i is an average melt depth, L is a typical radial dimension, and hppl i is an average
plasma pressure. The thermophysical properties of borosilicate glass are summarized in
Table I. For an average melt depth of hhm i ≈ 1 µm, as estimated in the heat transfer
calculation in Section A, a typical crater radius of L = 5µm, and an average plasma pressure
of hppl i ≈ 1000 atm (the typical plasma pressure drops from millions of atmospheres to about
10 atm during the first microsecond of its expansion [16]), we obtain τp ∼ 10−4 sec and
τM ∼ 10−1 sec. It should be noted that using L = 5µm gives timescales for τp and τM that
are too slow by one or two orders of magnitude in cases where the plasma pressure gradient,
ppl /L, is very large only near the edge of the crater. In such cases, the characteristic length
scale L would better be described by the much shorter distance over which the pressure
gradient changes. In any event, we have
τM
=O
τp
hppl ihhm i
γ T Tm
≈ 103
(20)
and hence the characteristic time scale for Marangoni flow is two to three orders of magnitude
longer than that of pressure-driven flow, and τp τM even if the peak pressure is lowered
more than a factor of ten. It is clear from this estimate that the large plasma pressure above
the free surface acts to move the fluid much more quickly than do surface tension gradients.
This result suggests that once an expanding plasma is formed above the glass surface during
laser ablation [? ], the spatially varying plasma pressure will control the evolution of the
free surface at the interface.
This mechanism for rim formation contrasts with that in laser texturing of silicon surfaces
in the absence of ablation, which is attributed to the Marangoni flow in thin films created by
nanosecond laser pulse heating [7]. The idea that thermocapillary effects do not contribute to
the observed formation of a rim at the edge of the melt zone in our experiments is supported
by the observation that the surface tension coefficient of borosilicate glass is positive [11] in
contrast to the usual negative values of most pure liquids. Therefore, as discussed earlier,
thermocapillary (Marangoni) flow in laser irradiated glass surfaces would be expected to
drive fluid from the cold periphery to the hot center of the melt, which is not what is
observed in the experiments shown in Fig. 1.
We close by verifying the three assumptions used to obtain Eq. (16). First, the molten
region must be thin hhm i/L 1. Since hhm i ≈ 1 µm and L ≈ 5µm, this condition is
reasonably well satisfied. In fact, errors in using Eq. (16) are O ((hhm i/L)2 ), which further
12
justifies the use of the lubrication approximation for this thin film flow. Second, we require
that the effective Reynolds number, Re , for the film flow must be small. Using typical
parameter values and viscosity at T = 2000 K we obtain
ρhhm i3 hppl i
Re = O
≈ 10−4 .
2
µL
(21)
Note that even if the pressure gradient that drives the flow was increased three orders of
magnitude, the low-Reynolds-number assumption, equation (21), is still satisfied. Finally,
the third assumption requires that the viscous effects must act quickly to establish the
velocity profile across the thin region. This time is O(ρhhm i2 /µ) ≈ 10−11 s, which is much
faster than the time scale, τp , of fluid motion. Hence, the three principle assumptions in the
fluid dynamics calculation are verified.
2.
Numerical Results and Discussion for Pressure-Driven Flow
To perform numerical simulations of the evolution of the molten glass we first nondimensionalize the time, length, height, and plasma pressure by characteristic values
S = t/τp ,
R = r/L,
H = h/hhm i,
B = b/hhm i,
Ppl = ppl /hppl i.
Since the key estimate of the previous section, Eq. (20), indicates plasma pressure gradients
primarily drive the fluid flow, then we take γ = constant (the value is given in Table I),
neglect the Marangoni term from Eq. (16), and arrive at the following nondimensional
evolution equation for the free surface height, H(R, S),
∂H
∂H
1 ∂
(H − B)3 ∂
1 ∂
(H − B)3 dPpl
R
= 0.
−
R
+Λ
∂S
R ∂R
3
dR
3
∂R R ∂R
∂R
(22)
The nondimensional parameter Λ is given by
Λ=
γhhm i
≈ 2.8 × 10−5 .
hppl iL2
Even though Λ is small, we will keep the curvature term in the numerical simulations. We
also assume for simplicity that the melt-solid boundary is time independent, i.e., B(R) only.
In Fig. 7 and Fig. 8 we numerically explore the behavior of solutions of Eq. (22). The
rim height is the feature of the solutions in which we are most interested. More specifically,
we investigate how changing the plasma pressure profile, Ppl (R), or modifying the substrate
13
shape, B(R), affect the rim height. Since the plasma pressure and bottom substrate are not
well characterized experimentally, then it is important to at least understand how changes
in these quantities affect the crater and rim formation.
Figure 7 shows the numerical solutions of pressure-driven melt flow for two different
plasma pressure profiles. It is not well understood how the plasma pressure distribution
depends upon the laser beam profile, and because of this uncertainty, we studied the effect
of two different pressure profiles, one with a Gaussian distribution and another with a steeper
gradient at the edges. The solid curve in the top plot describes a pressure profile of Ppl (R) =
exp(−R10 ) and the dashed curve is the gentler distribution Ppl (R) = exp(−R2 ). For the
free surface and bottom substrate profiles used in the numerical simulations, we require that
the initial thickness of molten glass in the center of the crater reflects a reasonable value
from the melt depth plot of Fig. 5. Therefore, the free surface shape of the molten glass
in the numerical simulations is initially described as H(R, 0) = −0.5 exp(−15R 10 ), and the
bottom substrate is given by B(R) = −0.01 − 0.9 exp(−R 10 ). At the center of the crater,
R = 0, the melt depth is H(0, 0) − B(0) ≈ 0.4, or roughly 0.4µm, which based on Fig. 5
is a reasonable melt depth. Solving Eq. (22) numerically for each of the plasma pressure
profiles to a dimensionless time of S = 0.08 gives the two solutions shown in the bottom
plot of Fig. 7 (solid and dashed curves). The rim height of the solid curve solution is larger
than that of the dashed curve solution. These rim heights are consistent with the plasma
pressure gradient of the solid curve in the top plot being larger than that of the dashed
pressure curve. Converting the rim heights to dimensional values gives heights of roughly
210nm for the solid rim and 65nm for the dashed rim after a time of roughly 2µs. There
is consistency between this range of rim heights, the time-scale of their formations, and
rim heights observed experimentally, though it should be noted that the plasma pressure
is not known with a large degree of certainty. We believe that the order of magnitude is
reasonable and so the qualitative and quantitative trends offer insight into the details of the
film rearrangement.
In Fig. 8 we explore how changing the shape of the bottom substrate, B(R), while
keeping all other parameters the same, affects the rim height. A plasma pressure profile of
Ppl (R) = exp(−R10 ) was chosen for the two simulations to be performed. The initial shape
of the free surface is again given by H(R, 0) = −0.5 exp(−15R 10 ). For the first simulation
(dashed curve), the bottom shape was chosen to be B(R) = −0.01 − 0.9 exp(−R 10 ), which
14
reproduces the solid curve solution in Fig. 7. The bottom shape for the second simulation
(solid curve) is given by B(R) = −0.01 − 0.9 exp(−0.3R 20 ), which provides more fluid at
the edges of the domain than in the first simulation. The solutions are shown at S = 0.04,
or roughly 1µs. The solution corresponding to the solid curve has a taller rim height than
that of the dashed curve. The reason for this is that the fluid flux is proportional to
(H − B)3 , as seen in Eq. (22), and there was more fluid available to be transported in the
second simulation. So, although we do not know exactly how the melt thickness changes
in the experiments, these simulations show that a rim of a reasonable height may still be
formed within the order of magnitude of the melt lifetime and that the typical results are
independent of the precise forms of the plasma and bottom profiles.
As a final remark, we note that the formation of transient rim-like structures occur in
other fluid dynamics problems, such as the splash of drop on a solid surface or on a liquid
layer (e.g. the famous Edgerton photograph of a splashing drop of milk). In particular,
in the case of drop splash, numerical calculations for thin-film dynamics in the absence of
viscous effects, which is very different from the dynamical situation we discuss, are reported
by, among others, Yarin and Weiss [17], and experiments with drops splashing following
their impact with thin layers are reported in Shin and McMahon [18].
IV.
HOW TO SUPPRESS THE RIM FORMATION?
We can now answer the question of “how to suppress the rim formation for clean laser
processing” or in other words seek “how to achieve clean borders of the irradiated spot”.
If the pressure-driven flow time, τp , is long enough, the pressure induced forces acting
on the fluid will not be able to drive the molten material from the center to the edges of
the crater during the melt lifetime. From the parametric dependence shown in Eq. (19), τp
depends on the viscosity, µ, the average thickness of the molten layer, hhm i, and the average
pressure above the surface, hppl i. Smaller melt thickness or smaller average pressures above
−1 2
) /D and therefore
the surface lead to longer times for melt flow. The melt lifetime tm ∝ (αeff
−1
(e.g.
a thinner melt thickness can be achieved by reducing the optical penetration depth, αeff
using shorter laser wavelengths or/and shorter pulse durations). Both the melt thickness
and lifetime will be further reduced by the decreased amount of the absorbed thermal energy
(Fheat ∝ Fth ) because the threshold fluence decreases with both laser wavelength and pulse
15
duration [19, 20]. In addition, materials with higher thermal diffusivities can reduce the
melt lifetime by more rapidly removing the absorbed thermal energy.
Another way of suppressing the rim formation may be modifying the plasma pressure
profile. As shown in Fig. 7, with a less steep Gaussian pressure profile the formation of a
rim takes longer time. It may be possible that for even a less steep pressure gradient near
the edge, a rim cannot be formed during the lifetime of the molten material. Therefore, if
one can modify the pressure profile, by modifying for example the laser spatial beam profile,
it may be possible to achieve a cleaner border of the irradiated spot. Although it is easy to
control the intensity distribution of the laser beam profile, it is not clear, however, how this
may impact the pressure distribution. This remains a subject for further investigation.
V.
CONCLUSIONS
The morphology of the single-shot ablated areas revealed a smooth and shallow crater
surrounded by an elevated rim. From these experimental observations, conclusions could be
drawn about the ablation mechanism of borosilicate glass. We argued that a very thin melt
zone existed during the ablation process and calculated the thermal and flow properties of
this thin melt zone. In these calculations, several characteristic time constants associated
with ablation, melting, and flow processes were determined. The comparative values revealed
that a flow of fluid driven by a plasma-induced pressure gradient localized near the radius
of the laser pulse, with reasonable values of the plasma pressure, would have enough time
to move melted material towards the edge and so deposit a thin rim around the ablated
area. Physically based estimates of the melt lifetime and the pressure-driven flow process
then suggest ways to suppress this rim formation.
Acknowledgments
The work has been supported by the TRW research fund and the Harvard NSEC. The
authors gratefully acknowledge the contributions of Prof. Eric Mazur, Dr. Mengyan Shen,
and Dr. Catherine Crouch to this investigation.
16
VI.
APPENDIX
APPENDIX A: DERIVATION OF THE EQUATION OF MOTION FOR THE
THIN FILM MODEL
In this appendix we present a short derivation of the partial differential equation describing viscous fluid flow in a thin film [21–23]. Suppose that the position of the free surface of
the fluid is denoted z = h(x, y, t), and the shape of the time-independent bottom substrate
is denoted z = b(x, y). We assume that the flow is incompressible and described by the
Navier-Stokes equations. Then, the fluid velocity u and pressure p satisfy
1
µ
∂u
+ u · ∇u = − ∇p + ∇2 u,
∂t
ρ
ρ
∇ · u = 0.
(A1)
(A2)
For some of the estimates below, it is convenient to denote the mean film thickness hm =
O(h−b) and the average, or typical, fluid pressure by hpi. Note that in (A1) we are neglecting
the gravitational body force since for the small length scales characteristic of the rims in the
experiments, ρghm /hpi 1. The boundary conditions to be satisfied are: no slip on the
solid substrate, the normal and tangential stress balances across the free surface, and the
kinematic boundary condition on the free surface.
For pressure-driven flows on the scale L typical of the flow direction, we expect a typical
velocity along the film to have magnitude u = O(h2m hpi/(µL), in which case we define
the Reynolds number for the thin-film flow as Re = ρh3m hpi/(µ2 L). Let us suppose first
that the velocity field is u(x, y, z, t) = (u, v, w). Then, under the thin-film (lubrication)
approximation we assume hm /L 1 and inertial effects are negligible, which is equivalent
to the requirement that the Reynolds number is small, Re 1, so that the Navier-Stokes
and continuity equations reduce to
∇2 p = µ
∂ 2 u2
∂z 2
∂p
=0
∂z
∇2 · u 2 +
(A3)
(A4)
∂w
=0
∂z
(A5)
where ∇2 = (∂x , ∂y ) and u2 = (u, v). Within the lubrication approximation, the boundary
conditions are
17
p = ppl − γκ on z = h
(normal stress)
(A6)
on z = h
(tangential stress)
(A7)
u=v=w=0
on z = b
(no slip)
(A8)
∂h
+ u 2 · ∇2 h − w = 0
∂t
on z = h
(kinematic condition)
(A9)
µ
∂u
= ∇2 γ
∂z
where ppl is the plasma pressure above the free surface, γ is the surface tension of the
interface, and κ is twice the mean curvature of the interface. We have assumed ppl to be
time-independent in the numerical simulations reported in this paper.
Equation (A4) and the normal stress balance give the local pressure in the liquid to be
p = ppl − γκ,
b(x, y) ≤ z ≤ h(x, y, t)
(A10)
and we will use a linearized expression for the mean curvature term, κ = ∇22 h. Substituting
the pressure into Eq. (A3), integrating the result twice with respect to z, and using the
no-slip and tangential stress boundary conditions yields
1
1 z 2 b2
− + bh − zh ∇2 ppl − ∇2 (γ∇22 h) + (z − b)∇2 γ.
u2 =
µ 2
2
µ
(A11)
Integrating the continuity equation shows that
∂h
+ ∇2 · q = 0 with q =
∂t
Z
z=h
u2 dz,
(A12)
z=b
where q is the flux vector. Substituting (A11) for u2 to calculate the depth-averaged flux,
q, we then obtain the evolution equation for the height, h, of the thin film:
∂h
(h − b)2
(h − b)3
2
+ ∇2 ·
∇2 γ −
∇2 ppl − ∇2 (γ∇2 h) = 0.
∂t
2µ
3µ
(A13)
In cylindrical coordinates and assuming an axisymmetric shape, h(r, t), Eq. (A13) becomes
(h − b)2 dγ (h − b)3 dppl (h − b)3 ∂ γ ∂
∂h
∂h 1 ∂
+
r
−
+
r
= 0.
(A14)
∂t
r ∂r
2µ dr
3µ
dr
3µ ∂r r ∂r
∂r
18
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and M. Sabsabi, Spectrochimica Acta Part B 56, 973 (2001).
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[18] J. Shin and T. A. McMahon, Phys. Fluids 2, 1312 (1990).
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19
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20
APPENDIX B: LIST OF TABLES
Table I Thermophysical properties of borosilicate glass (BorofloatTM ). The chemical composition includes 81% SiO2 , 13% B2 O3 , 2% Al2 O3 , 4% Na2 O.
21
APPENDIX C: LIST OF FIGURES
FIG. 1 SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser
pulses, and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence
was F0 = 34 J/cm2 . The numbers correspond to the order of the incident laser pulses.
FIG. 2 An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of
F0 = 23 J/cm2 focused to a spot size of about 12 µm. The image shows the microscale
surface roughness created by the rims of overlapping laser pulses when scanning the
laser across the surface.
FIG. 3 An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of
F0 = 12.6 J/cm2 . (a) The AFM image. b) The ablation profile at the center-line of
the crater. The ablation depth, ha , is normalized by an average melt depth of hhm i =
1 µm and the radial distance, r, is normalized by a characteristic radial dimension of
L = 5 µm.
FIG. 4 Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of
the resulting initial temperature distribution inside the material is shown along with
definitions of several variables used in the analysis.
FIG. 5 Melt depth variation with time for two different laser fluences using the approximation
that 1.5-3 % of the incident laser energy remains in the glass as thermal energy. The
calculations are performed assuming constant thermophysical properties at an average
temperature of 900 K.
FIG. 6 Description of parameters used in the thin film model to calculate the evolution of the
surface of the melt.
FIG. 7 Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two different pressure profiles. The plasma pressures are shown in the top plot where the solid
curve is given by Ppl (R) = exp(−R10 ), and the dashed curve is Ppl (R) = exp(−R2 ).
The bottom plot shows the evolution of the free surface for each of the pressure profiles.
FIG. 8 Numerical solutions of pressure-driven melt flow for two different bottom substrate
shapes. The bottom plot shows the evolution of the free surface for each of the
22
substrate profiles (solid and dashed curves).
23
TABLE I: Thermophysical properties of borosilicate glass (Borofloat TM ). The chemical composition includes 81% SiO2 , 13% B2 O3 , 2% Al2 O3 , 4% Na2 O.
Property
Density [15]
Melting
Symbol
Units
Values
ρ
kg/m3
2.23 × 103
Tm
K
1500
µ
Pa s
≈ 103 (at 1500 K)
temperature [15]a
Viscosity [15]
≈ 102 (at 2000 K)
Surface tension [24]
γ
N/m
0.28 (at 300 K)
Temperature coefficient
γT
N/m K
3.4 × 10−5
k
W/m K
1.25 (at 300 K)
of surface tension [11]
Thermal
conductivity [24]b
1.6 (at 600 K)
4.5 (at 900 K)
42.0 (at 1500 K)
Specific heat [25]b
Cp
J/kg K
746 (at 300 K)
1000 (at 600 K)
1250 (at 900 K)
1320 (at 1500 K)
Thermal
m2/s
D
diffusivityb
0.75 × 10−6 (300 K)
0.72 × 10−6 (600 K)
1.60 × 10−6 (900 K)
14.3 × 10−6 (1500 K)
Ben-Yakar et al.
a
Tm is the working point temperature of glass, defined as the temperature at which the glass can readily
be formed and has a viscosity of approximately 103 Pa.s.
b
In the heat conduction calculations, we used data at an average temperature of 900 K, which is an average
between T∞ = 300 K and Tm = 1500 K.
24
(a)
10 mm
x 3,000
(b)
1
2
10 mm
x 4,000
(c)
4
2
1
3
3 pulses
overlapping
2 pulses
overlapping
10 mm
x 4,000
FIG. 1: SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser pulses,
and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence was F 0 = 34 J/cm2 .
The numbers correspond to the order of the incident laser pulses.
Ben-Yakar et al.
25
5 mm
x 5,000
FIG. 2: An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of F 0 =
23 J/cm2 focused to a spot size of about 12 µm. The image shows the microscale surface roughness
created by the rims of overlapping laser pulses when scanning the laser across the surface.
Ben-Yakar et al.
26
(a)
(b)
0.2
Ablation profile, ha/<hm>
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Radial position, x/L
FIG. 3: An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of F 0 =
12.6 J/cm2 . (a) The AFM image. b) The ablation profile at the center-line of the crater. The
ablation depth, ha , is normalized by an average melt depth of hh m i = 1 µm and the radial distance,
r, is normalized by a characteristic radial dimension of L = 5 µm.
Ben-Yakar et al.
27
Incident beam, F0
AFth
AF0
Fa(z)
plasma
ha
melt
hm
solid
z
T(z,0)
ablation
depth
Tth
melt front
Tm
z
ha
hm
FIG. 4: Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of the
resulting initial temperature distribution inside the material is shown along with definitions of
several variables used in the analysis.
Ben-Yakar et al.
28
1.5
2
Fheat = 1 J/cm
hm (µm)
1
2
Fheat = 0.5 J/cm
0.5
0
0
0.5
1
1.5
2
time (µs)
FIG. 5: Melt depth variation with time for two different laser fluences using the approximation
that 1.5-3 % of the incident laser energy remains in the glass as thermal energy. The calculations
are performed assuming constant thermophysical properties at an average temperature of 900 K.
Ben-Yakar et al.
29
p(x) - plasma pressure
x
z=h(x,t) - interface height
plasma
liquid
h(x,0)=<hm>
x
L
Thin melted layer
FIG. 6: Description of parameters used in the thin film model to calculate the evolution of the
surface of the melt.
Ben-Yakar et al.
30
1
Ppl
0.5
0
−1
−0.5
0
0.5
1
0.5
1
0.2
S = 0.08
0
H
plasma
−0.5
molten glass
−1
−1
−0.5
0
R
FIG. 7: Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two different
pressure profiles. The plasma pressures are shown in the top plot where the solid curve is given
by Ppl (R) = exp(−R10 ), and the dashed curve is Ppl (R) = exp(−R2 ). The bottom plot shows the
evolution of the free surface for each of the pressure profiles.
Ben-Yakar et al.
31
1
P
pl
0.5
P = exp(−R10)
pl
0
−1
−0.5
0
0.5
1
0.5
1
0.4
0.2
S = 0.04
0
H
plasma
−0.5
molten glass
−1
−1
−0.5
0
R
FIG. 8: Numerical solutions of pressure-driven melt flow for two different bottom substrate shapes.
The bottom plot shows the evolution of the free surface for each of the substrate profiles (solid and
dashed curves).
Ben-Yakar et al.
32