TM Opaque Abril03 2014. Nilson

8/10/2019 TM Opaque Abril03 2014. Nilson

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Pulsed Photothermal Mirror Technique: Characterization of

Opaque Materials

O. A. Capeloto1, G. V. B. Lukasievicz

1,2,3, V. S. Zanuto

1,2,3, L. S. Herculano,N. E. Souza

Filho4,A. Novatski

5, L. C. Malacarne

1, S. E. Bialkowski

2, and N. G. C. Astrath

1,2,3,*

1 Departamento de Fsica, Universidade Estadual de Maring, Maring, PR 87020-900, Brazil

2 Department of Chemistry and Biochemistry, Utah State University, Logan, UT 84322-0300, USA

3

CAPES Foundation, Ministry of Education of Brazil, Braslia, DF 70040-020, Brazil4Departamento de Engenharia Acstica, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil

5Departamento de Fsica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil

ABSTRACT:

In this work, the time-resolved Thermal Mirror technique is developed under pulsed laser

excitation for measuring thermal and mechanical properties of opaque materials. The heat

diffusion and thermoelastic equations are solved analytically considering pulsed excitation

assuming surface absorption and instantaneous pulse. The analytical results for the temperature

change in the sample is compared to all numerical solutions using finite element method analysis

taking the pulse width laser into account. Experiments are realized in order to validate theoretical

model and regression fitting is performed to obtain the thermal diffusivity and the linear thermal

expansion coefficient of the samples. The values obtained for these properties are in good

agreement with literature data. The experimental technique combined with the theoretical model

is shown to be useful for quantitative determinations of the physics properties of metals with highthermal diffusivity.

PACS numbers: 61.82.Bg; 65.60.+a; 78.20.nb

*E-mail address: [email protected]


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INTRODUCTION

Photothermal Mirror spectrometry has been developed both experimentally and

theoretically over the past few years [1-10]. The thermoelastic effect has been known and used

for material characterization for decades, and most recently applied as a time-resolved

quantitative method for investigating transparent and opaque materials under continuous [1-8]

and pulsed laser excitation [9,10]. Advances in theoretical treatment of this complex dynamic

thermoelastic problem made possible to predict for effects of heat coupling between sample and

the surrounding fluid [7-9], in addition to account for different time-distribution pulses [10]. Theaccuracy of the theoretical descriptions have been, in most cases, tested against all numerical

modeling by comparing the analytical solutions under various assumptions with finite element

method analysis.

The Photothermal Mirror effect arises from photo-induced heat generation in a sample,

followed by the expansion or contraction of the surface. This deformation acts, in this case, as an

optical element that can be probed by analyzing the wavefront distortion of a second weak laser

reflected off of the surface. The surface deformation then focus or defocus the probe beam,

depending on the expansion coefficient of the sample, and the central portion of the probe beam

intensity is monitored in the far field. This configuration is mostly referred to as Thermal Mirror

(TM).

The TM transient depends direct on thermo-physical properties of the sample. For

instance, the maximum amplitude of the surface expansion or contraction is related to the linear

expansion coefficient and absorption coefficient, and the relaxation time - the characteristic

transient time - brings direct information on the thermal diffusivity of the sample. These

characteristics made this method a powerful tool for quantitative analysis of solid materials. Due

to the high values of the thermal diffusivity of most metals, the TM transient formation time is

much smaller than that observed in glasses for instance. The use of continuous laser excitation in

these cases could induce large errors due to the shutter opening time.

Here, we extend the theoretical and experimental capabilities of the TM technique to

studying opaque materials under pulsed laser excitation. First, it is presented a theoretical model

to describe the temperature, the surface displacement and the TM signal. Then, the model is used

to study a series of opaque metals, providing quantitative information on the thermal diffusivity


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and thermophysical properties of the samples.

THEORY

In the TM experimental apparatus explored in this work, the sample is excited by a pulsed

TEM00 Gaussian laser beam of radius 0ew . The TM signal arises from optical absorption in a

sample. Sample heating occurs due to non-radiative decay processes. The Gaussian profile

produces a radial temperature gradient perpendicular to the propagation of the laser beam, which

in turn results in a surface deformation of the sample. During the pulse, the deformation tends to

follow the temperature rise distribution and then stress and strain govern the deformation. The

curvature of the deformation acts as an optical element to a second beam reflected off of the

surface. A cw TEM00Gaussian beam (probe beam) is assumed to have a radius 0Pw at its waist,

located at a distance 1Z from the sample, and a radius 1Pw at the sample surface. The TM affects

the propagation of the probe beam, resulting in a change in its intensity profile measured in the

far field. The central portion of the reflected probe laser intensity variation gives information on

physical properties of the sample.

The propagation of the excitation and probe beams is along the direction z , and the sample

surface is located at 0z= . A semi-infinite sample is considered here. The radial dimensions of

the sample are assumed to be large compared with the excitation beam radius to avoid edge

effects. The absorption of light for the opaque sample is assumed to be completely superficial and

can be safely represented by a Dirac delta function as ( )zd . The schematic of the mode-

mismatched TM technique is showed in Fig. 1.

FIG. 1. Schematic of the geometric positions of the beams in a TM technique.


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The temperature change at the surface of the sample and within it is described by the

solution of the heat conduction equation [11],

( )( ) ( ) ( ) ( )2 2 20 0

, ,., , = exp 2 e

T r z t T Q z Q t D r z t Q r w

t

D D

- -

(1)

PD k Cr= is the thermal diffusivity of the sample, k , PC and r are the thermal conductivity,

specific heat, and mass density, respectively. ( ) ( )Q t td= , and the Dirac delta function in time

( )td represents the pulsed excitation. The initial condition for this problem is ( ), , 0 0T r zD = , and

the boundary conditions are ( ), , 0T z tD = and ( ) 0, , 0zT r z t z =

D = . The later assumes that

there is no heat conduction from the sample to the surrounding fluid; this is a good approximation

when air is the surrounding fluid [8, 9]. ( ) ( )2 eQ z z Ad= and eA is the optical absorption

coefficient at the excitation beam wavelength. ( ) 20 0 02 1e p eQ E A R C w f r p= - ; 0E is the pulse laser

energy, R is the surface reflectivity, and f is a heat yield parameter accounting for energy loss

due to luminescence. The mathematical procedure for solving Eq. (1) has been explored in other

reports [2,3,10], and consists of applying Laplace, Hankel and Fourier cosine transforms of the t-

time, r - and z -dependent temperature change, respectively, yielding

( )

2 2 2 20 0

0

2exp exp

1 22, , .

1 2

e e

c c

ce e c

r w z w

t t t t QT r z t

t tA w t t p

- - + D =

+ (2)

The characteristic thermal time constant is 20 4c et w D= .

The non-uniform temperature change in the sample causes a non-uniform surface

contraction or expansion that evolves following the Navier-Stokes thermoelastic equation, in the

quasistatic approximation, as [12-14]

( ) ( ) ( ) ( ) ( )u u21 2 , , . , , 2 1 , , .Tr z t r z t T r z t n n a - + = + D (3)

u is the displacement vector, n is the Poisson's ratio, and Ta is the linear thermal expansion

coefficient. The boundary conditions at the free surface are on the normal stress components,

0 0rz zs = = and 0 0zz zs = = . The problem presents axial symmetry due to the radial nature of the


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Gaussian pulsed heat source, and is treated in cylindrical coordinates. The solution has been

proposed by introducing scalar potential, as described in details in Refs. [5,12], and can bewritten in terms of the Hankel transform of the temperature change, yielding at the surface

displacement

( ) ( ) ( ) ( ) ( )2

0 0 2 20 0 00

, 0, 2 1 exp 8 Erfc 2 J .4

ez T e e c

e

Q wu r t w w t t r d

An a a a a a a

= - + - (4)

( )nJ x is the Bessel's function of the first kind and ( )Erfc x is the complementary error function.

Eq. (4) represents the surface deformation induced on an opaque sample by a Gaussian profile

pulsed laser beam.

Assuming an absorption completely superficial using ( )zd in addition to the Dirac delta

time representation, ( )td , simplify considerably the numerical calculations of Eq.(4), although

leading to divergent results for the temperature change at 0t= and discrepant results in

nanosecond time scales because of finite absorption coefficient. These points were investigated

by solving Eq. (1) numerically using finite element method (FEM) analysis, employing Comsol

Multiphysics 4.2a software. This software provides numerical solutions to the heat transfer and

thermoelastic equations with the realistic boundary conditions imposed by the experiment. We

considered the absorbance to follow the Beer-Lambert law as ( ) ( )exp eQ z A z = - in the source

term of the heat diffusion equation and, in addition, we represent the time-dependence of the

pulse as [10] ( ) ( ) ( )2 22 exp 1 Erf Q t t x t t p x t

= - - + , where t is pulse width, x is the

time to the maximum irradiance for the Gaussian pulse, ( )Erf x is the error function. The

temperature change is obtained and compared to the solution given by Eq. (2).

UNS C10100 Copper Alloy was the test sample, and the parameters used in the

simulations are listed in the caption of Fig. 2. We assumed the mechanical and thermal properties

of the sample are independent of temperature. This assumption limits the applicability of the

solution obtained to a small range of temperature change. Fig. 2 (a) shows the calculated

temperature change on the surface at the center of the excitation beam ( 0z= and 0r= ),

obtained with Eq. (2) and simulations using numerical FEM considering different pulse widths.

Fig. 2 (b) shows the dependence of temperature with different optical absorption coefficients. The

figures reveal excellent agreement between both solutions for500

t ns

>, when used pulses


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widths shorter than 50nsand samples with optical absorption bigger than 5 15 10eA m-> .

0

2

4

6

8

10

0 100 200 300 400 5000

1

2

3

4

5

Pulse width: = 15ns

(t)

= 5ns

= 15ns

= 30ns

= 50ns

Temperature(C)

(a) Different pulse width

Superficial Absorption

(b) Different absorption coefficients

Analytical model: (t) and (z)

Superficial Absorption

Ae= 1. 107m

-1

Ae= 2. 106m

-1

Ae= 1. 106m

-1

Ae= 5. 105m

-1

Ae= 2. 105m

-1Temperature(C)

Time (ns)

FIG. 2. Physical properties of copper used for the simulations: 4 2 11.14 10D m s- -= , 1 1391k Wm K - -= ,

1 1385c J kg K - -= , 38940kg mr -= , 6 117 10T Ka - -= , 0.31n= , 0 100ew mm= and

50 1 10eQ A K m

-= .

The surface displacement of the sample act as a concave/convex mirror for the probe

beam, causing a phase shift given by ( )2 2 ( , 0, ) (0, 0, )p z zu r t u t p l= -F [1-3]. By Eq. (4), we get

the TM phase shift as

( ) ( ) ( ) ( )20 2 2

0 00 00exp 8 Erfc 2 J, 1 ,eTM e e

c

ecg t w

w w t t m w dt

gq a a a a a - - -F = (5)


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where 0(1 ) (1 )TTM PE R kq a n f l= - + , ( )2

1 0P em w w= , 1 2( / )Pwg r= , and pl is the probe

beam wavelength. The complex electric field of TEM00 Gaussian probe beam reflected by the

deformed surface of the sample is given by [1, 15]

2 2

1 21 1

( , ) exp ,pp p P

r rU r Z B i

R w

p

l

= - + - F (6)

where 11

12 / exp 2 /P PPB w P i Z p p l- = - . 1pR and pP are the radius of curvature of the probe

beam and the reflected probe beam power at 1Z , respectively. Considering only the center of the

beam reflected at the detector's plane we can use diffraction Fresnel theory to calculate the

complex amplitude of the electrical field of the probe beam. In cylindrical coordinates it is [15]

1 2 0( , ) exp (1 ) ( , ) ,U Z Z t C iV g i g t dg

+ = - + - F (7)

in which, ( ) ( ) ( )1 2 12

1c c cV Z Z Z Z Z Z

= + + , 22 1 2/exp( 2 )P P PC B i i Z Zp l pw l= - , cZ is the probe's

beam confocal distance, and 2Z is the distance from the sample to the detector plane. The

normalized intensity of the electric field at the detector's plane is given [15]

2

0

2

0

exp (1 ) ( , )

( )

exp (1 )

iV g i g t dg

I t

iV g dg

- + - =

- +

F

(8)

The fit of the experimental data with Eq. (8) gives the thermal diffusivity and the

parameter TMq , which is related with the optical, thermal and mechanical properties of the

samples.

EXPERIMENTAL RESULTS AND DISCUSSION

The experimental set-up for the Thermal Mirror system for opaque samples is shown in

Fig. 3. A Q-switched pulsed Nd:YAG laser (Brilliant B, Quantel), operating at the second

harmonic wave with a wavelength of 532nm and pulse width of 15ns, is used as pumping source.


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Part of the laser beam is reflected by a beam sampler (ThorLabs, Model BSF05-A) and then

focused in the sample by a biconvex lens of focal length 0.75f m= . The transmitted beamthrough the beam sampler is used as reference for energy measurement by a Pyroelectric Energy

Sensor (Thorlabs, Model ES120C) plugged in a digital energy meter console (Thorlabs, Model

PM100D). A continuous TEM00Gaussian He-Ne laser at 632.8nm (Newport, Model R-32734)

was arranged almost collinear to the excitation beam and used to probe the TM effect. After

being reflected off of the sample surface, the probe beam propagates to the photomultiplier

detector P1(Hamamatsu, Model R928), which was assembled with a pinhole and a laser line filter

(ThorLabs, Model FL632.8-10). The PMT is biased with a high voltage power supply (Newport,

Model 70706). Only the central part of the probe beam is detected by the photomultiplier tube

and recorded by a digital oscilloscope (Tektronix, Model DPO 4102B). The oscilloscope is

triggered by P2 (Newport, Model 818BB-22). The apparatus is mounted on an optical table

(TMC, size 2.0x1.0m).

FIG. 3. Schematic diagram of the TM experimental setup used: Mi stands for mirrors, Li stands for lenses,

PMT stands for the photomultiplier tube and P2stands for the photodiode.

The geometric parameters of the excitation and probe beams are: 0 123ew mm= ,

1 1467pw mm= , 1.47cZ cm= , 1 41.1Z cm= , 2 387.1Z cm= , 30.93V = , and 142.2m= . The probe

radius along the z-axis is measured with a Beam Profiler (Thorlabs, BP104-UV) to determine the


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confocal distance. The radius of the excitation and probe beam is measured with a beam profile

camera (Coherent, LaserCam-HR) at the sample position.

TM measurements are performed in four opaque samples. The samples are UNS

C10100/C10200 Copper Alloy (Copper) [16], UNS A96351 Aluminum Alloy [16, 17], UNS

C36000 Copper Alloy (Brass) [16], and UNS C90300 Cooper Alloy (Bronze SAE 620) [16]. The

cylindrical samples are 15.0mm thicker with diameter of 25.4mm . All the samples are polished

with polycrystalline diamond compound in order to ensure minimum roughness on the surface.

The values of surface reflectivity are highly dependent of the surface polishing [18].

FIG. 4. Normalized TM signal ( ) / (0)I t I for copper sample. Circles: experimental data; Solid

lines: best fit curves.

Figures 4 and 5 show the normalized TM signal ( ) ( )0I t I for Copper and Aluminum

Alloy samples, respectively. More than 100 transients were averaged to produce each transient.

The repetition rates of excitation beam are 5Hz for copper and 2Hz for aluminum. The

normalization is chosen by the minimum value of the transient in order to discriminate all curves

in function of the energy in the graph. The detector response time is less than 2 sm . The surface

deformation produced by the excitation beam is outward, acting as convex mirror due the positive

linear thermal expansion coefficient of these samples, which decreases instantaneously the signal

at the detector and eventually returning to the initial value as the heat diffuses along the sample.


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FIG. 5. Normalized TM signal ( ) / (0)I t I for aluminum sample. Circles: experimental data; Solid lines:

best fit curves.

The same characteristics in the TM signals are observed for brass and bronze samples as

shown in Figs. 6 and 7, respectively. The excitation beam repetition is 2Hz for both samples.

Note that the time to reach the steady-state differs for each sample, which is directly related to the

thermal diffusivity of the samples.

FIG. 6. Normalized TM signal ( ) / (0)I t I for brass sample. Circles: experimental data; Solid lines: best fit

curves.

The theoretical model presented in this paper is in good agreement with the experimental


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data. Regression using the function "NonLinearModelFit" is used to fit experimental data, which

provides the thermal diffusivities and the parameter TMq . To speed up the regression, the

experimental data is smoothed and about 30 points is taken from the decay part of the dataset.

FIG. 7. Normalized TM signal ( ) / (0)I t I

for bronze sample. Circles: experimental data; Solidlines: best fit curves.

TMq yields information about optical, thermal and mechanical properties of the sample.

Figure 8 shows TMq versus 0E , the slope value of the linear fit gives (1(1 ) )T PkR a n l+- . Here

1f= , i.e., all the energy absorbed by sample is converted into heat. With the additional values of

R, and k, the linear thermal expansion coefficient can be obtained from 0TM Eq . Poisson's ratio

and thermal conductivity are available in Refs. [16, 17]. The surface reflectivity at 532nm were

measured in a spectrophotometer (Perkin Elmer, Model Lambda 1050) assembled with

integrating sphere (Labsphere, Model# 150MM RSA ASSY). Table 1 shows reference properties

of the samples and the average parameters obtained from regression. The obtained properties are

in excellent agreement with those find in the literature.


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0 10 20 30 40 50 60 70

0

2

4

6

8

10

12

Bronze

Brass

Aluminum

Copper

Linear Fit

TM(

s)

E0(J)

FIG. 8. The fittedTMq as a function of excitation energy. Open symbols are experimental data

obtained from TM signals fitted and solid lines are best linear curve fit.

Table 1 - Comparative table with thermal, optical, and mechanical properties obtained from TM

experiment and literature data [16, 17].

Samples D

(Measured)

5 2( 10 )m s-

D

(Literature)

5 2( 10 )m s-

TM/E0

(Measured) 1W-

R@532nm

(Measured)

T

(Measured)

6 1( 10 )K- -

T

(Literature)

6 1( 10 )K- -

Bronze 2.4(0.1) 2.2 0.1415(0.001) 0.719 17.6 18.0

Brass 3.4(0.2) 3.6 0.077(0.002) 0.810 22.5 20.5

Aluminum 7.8(0.3) 7.3 0.0472(0.003) 0.857 27.6 23.4

Copper 12.6(0.8) 11.4 0.029(0.002) 0.619 14.4 17.0


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CONCLUSIONS

In conclusion, we have presented a simple mathematical model to describe the induced

effect when a Gaussian pulsed laser shines on a high absorbing material. The temperature change

in the sample was calculated and compared with FEM simulations for different pulse widths and

optical absorption coefficients. Experiments in four opaque metals were performed to validate the

model and determine the thermal diffusivity of these samples. The results obtained by regression

have yielded good values for the fitted parameters when compared with literature data. The TM

technique using pulsed excitation has been shown be very useful to measure thermal and

mechanical properties of opaque materials with high thermal diffusivity, since, the continuous

excitation presents difficulties to measure such samples.


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TM Opaque Abril03 2014. Nilson