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Experimental MechanicsAn International Journal ISSN 0014-4851Volume 54Number 4 Exp Mech (2014) 54:593-603DOI 10.1007/s11340-013-9804-8

Dual-Configuration Fiber Bragg GratingSensor Technique to Measure Coefficientsof Thermal Expansion and HygroscopicSwelling

Y. Sun, Y. Wang, Y. Kim & B. Han

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Dual-Configuration Fiber Bragg Grating SensorTechnique to Measure Coefficients of Thermal Expansionand Hygroscopic Swelling

Y. Sun & Y. Wang & Y. Kim & B. Han

Received: 21 February 2013 /Accepted: 3 September 2013 /Published online: 20 November 2013# Society for Experimental Mechanics 2013

Abstract Wepropose amethod based on the dual-configurationfiber Bragg grating (FBG) sensor to measure the coefficientsof thermal expansion (CTE) and hygroscopic swelling (CHS)of polymeric materials. The Bragg wavelength shifts are doc-umented in “two” small but different polymer-FBG assem-blies while they are subjected to environmental loading con-ditions (temperature or moisture). The behavior of the infinitepolymer/FBG assembly is reconstructed numerically from thedata obtained from the two configurations. The coefficients,then, can be determined from the simple governing equationderived for the infinite assembly. The proposed method isimplemented for an underfill material. The validity of mea-surements is corroborated by a commercially available tool,and the repeatability of measurements is verified by an exper-iment with a different configuration.

Keywords Dual-configuration . Coefficient of hygroscopicswelling . Coefficient of thermal expansion . FBG sensor .

Polymer . CHS . CTE

Introduction

Semiconductor packages are often exposed to high tempera-ture and humidity conditions during manufacturing and oper-ation. The coefficient of thermal expansion mismatch amongvarious materials produces thermal stress in the packages[1–3], and directly affects device reliability [4]. Similarly,

the different capacity to absorbmoisture leads to the mismatchof the hygroscopic swelling [5–7]. It has been reported that thedeformation caused by hygroscopic swelling can be as signif-icant as the thermal deformation [8–11]. Consequently, accu-rate determination of the coefficient of thermal expansion(CTE) and the coefficient of hygroscopic swelling (CHS) iscritically required for reliability assessment.

Thermo-mechanical Analyzer (TMA) has been practicedroutinely for CTE measurement [12–14]. In the TMA instru-ment, a probe that rests initially on the surface of a test specimenis pushed up or down as the specimen expands or shrinks duringheating or cooling, and the resultant probe displacement isdocumented. The thermal strain is then determined from theprobe displacement. The CTE can be determined from thethermal strain and temperature change.

The TMA was also used to measure the CHS of variousmold compounds [11]. The CHSmeasurement using the TMAinvolved a procedure to take the specimen out of the environ-mental chamber and to place it in the TMA chamber that waspreset at a constant temperature. During this procedure, thetemperature stabilization process could alter the location of theprobe, which could affect the measurement accuracy.

To cope with the problem, an optical technique called moiréinterferometry was employed to measure the CHS [8, 9]. Areference specimen that was subjected only to the temperaturechange was used to tune a moiré setup during measurements,which allowed to document only moisture-induced deforma-tions. Moiré interferometry also has been used very effectivelytomeasure theCTE of various electronic packaging componentsand printed circuit board (PCB) [15, 16]. However, its practicehas been limited due to the stringent requirement of surfacepreparation including replication of a high frequency diffractiongrating [17].

The Fiber Bragg Grating (FBG) sensor has been used forstrain measurement for various materials, most notably compos-ite materials [18–27]. The FBG was embedded in composite

Y. Sun :Y. Wang :Y. Kim : B. Han (*, SEM Fellow)Department of Mechanical Engineering, University of Maryland,College Park, MD 20742, USAe-mail: bthan@umd.edu

B. HanDepartment of Mechanical Engineering, Sungkyunkwan University,Suwon 440-746, Korea

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material structures to measure the strain caused by thermalexpansion [19, 20, 25], moisture absorption [22, 23, 28] andcuring [24, 26, 27]. A cylindrical configuration was adoptedmost widely due to its mathematical simplicity [18, 22, 24,26–28]. The schematic diagram of the FBG sensor embeddedin a cylindrical configuration is shown in Fig. 1, where theradius of the fiber is rf (typically 125 μm) and the outer radiusof the substrate material is rs.

When the cylindrical configuration is subjected to an ex-ternal strain, a part of the strain is transferred to the embeddedFBG. The amount of strain measured by the FBG is stronglydependent on the ratio of the volume stiffness between thematerial used to fabricate the cylindrical configuration and thefiber. From the definition of the volume stiffness (defined as“volume multiplied by modulus”), the volume stiffness ratio,RVS, can be defined as:

RVS ¼Es r2s−r2f� �E f ⋅r2f

ð1Þ

where Es and Ef are the modulus of the material and the fiber,respectively.

The measured strain normalized by the applied strain isshown as a function of the volume stiffness ratio in Fig. 2. Asthe volume stiffness ratio increases, the normalized measuredstrain converges to “unity” (indicated by the red line in theplot); i.e., for the configuration with the ratio greater than 200(will be referred to as “infinite configuration”), the measuredstrain becomes virtually the same as the applied strain. Underthis condition, the CTE or CHS of the material can be deter-mined by using this “single” infinite configuration. To achievethis condition, some authors used a very large substrate diam-

eter rsr f >> 1� �

[18, 24], while others took advantage of the

large modulus of a substrate material EsE f >> 1� �

[26].

The modulus of the polymer decreases rapidly with tem-perature increase, especially above the glass transition tem-perature. Consequently, a very large substrate diameter will

have to be used when the properties of polymers are to bedetermined over a wide range of temperature. It is important tonote that the applicability of this large configuration is limitedin practice only to polymers that produce very small heatgeneration; otherwise, an excessive amount of heat generatedduring curing would produce a large temperature gradientwithin the specimen, leading to non-uniformmaterial propertyor undesired residual stresses.

More recently, a dual-configuration FBG sensor methodwas proposed to simultaneously measure the chemical shrink-age and modulus evolution of a polymer during curing [29]. Inthis approach, the BW shifts were documented from two smallpolymer-fiber assemblies with different diameters while theycured. Then, the two unknowns (chemical shrinkage and mod-ulus evolution) were calculated from a complex mathematicalformula that defined the relationship between the BW shiftsand the two properties.

In this paper, the concept of the dual-configuration FBGsensor method is adopted and advanced to measure the CTEand CHS of polymeric materials under various environmentalconditions. The behavior of the infinite configuration isreconstructed numerically from the two small finite configu-rations that are subjected to environmental conditions. Then,the coefficients can be determined from the simple governingequation derived for the infinite configuration. The smallconfigurations negate the effect of heat generation duringcuring, which makes the method applicable for a wide rangeof polymeric materials.

The FBG grating sensor method is reviewed in “Background:FBG sensor method” section. The proposed method to recon-struct the infinite configuration is described in “Proposed meth-od: reconstruction of infinite configuration” section. The appli-cation of the proposed method using an underfill material ispresented in “Implementation” section. Finally, the validity ofthe proposed method is confirmed in “Discussion” section.

Fig. 1 Schematic diagram of an FBG sensor embedded in a cylindricalconfiguration

1 10 1000.0

0.2

0.4

0.6

0.8

1.0

niartsdeilpp

A/niartsderusae

M

Volume stiffness ratio

Fig. 2 Schematic of the ratio of the measured strain to the applied strain

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Background: FBG Sensor Method

Governing Equation

Let us consider a case where only the polymer in the assemblyis subjected to a uniform deformation; i.e., the piecewiseloading condition of

εloading ¼ 0; 0≤r < r fΔε; r f < r≤rs�

ð2Þ

where Δε is the strain of the polymer caused by the thermalexpansion or hygroscopic swelling.

The generalized plane strain solution of stress componentsof the fiber can be expressed as [29]:

σ frr ¼ σ fθθ ¼E f

1þ ν f ⋅C1 f

1−2ν fð3Þ

σ fzz ¼2ν f E f C1 f

1þ ν f� �

1−2ν f� � þ

rsr f

� �2−1

� �Es1−νs

Δε−2νsEsC1s

1þ νsð Þ 1−2νsð Þ

−2ν f E f C1 f

1þ ν f� �

1−2ν f� �

1þ EsE f

rsr f

� �2−1

! ð4Þ

where σzzf , σrr

f and σθθf are the axial, radial and hoop stress

components of the fiber, respectively; Ef and ν f are themodulus and Poisson’s ratio of the fiber material; and Es, ν sare the modulus and Poisson’s ratio of the polymer. Thedetailed descriptions of the coefficients C1f and C1s can befound in Ref. [29].

The axial strain of the fiber, ε fσ, can be calculated from the

stress–strain relationship as

εσf ¼1

E fσ fzz−υ f σ

frr−υ f σ

fθθ

� �ð5Þ

By substituting equations (3) and (4) into (5), the axialstrain can be expressed as [29]

εσf ¼ F Es;βð Þ⋅Δε ð6Þ

where β ¼ rsr f (will be referred to as “configuration”); andF(Es, β ) is a nonlinear function that can be expressed explic-itly. The function F (Es, β ) is the ratio of the axial strain of thefiber to the applied polymer strain; therefore, it depends on themodulus of the polymer and the configuration, β . The valuesof F(Es, β ) are plotted as a function of the configuration, β , inFig. 3 for a wide range of polymer moduli to illustrate thisdependency.

Temperature Effect

The BW shifts as the temperature or the strain state of the fiberBragg grating (FBG) changes. The total BW shift can beexpressed as

Δλ ¼ Δλi þΔλd ð7Þ

whereΔλ is the total BW shift;Δλ i is the intrinsic BW shift;and Δλd the stress-induced BW shift.

The intrinsic BW is not associated with any stress induceddeformation and can be defined as [30]

Δλi ¼ λ α f þ 1neffdn

dT

� �ΔT þ 1

neff

dn

dT⋅α fΔT 2

ð8Þ

where λ is the initial BW, α f is the CTE of the fiber corematerial, neff is the effective refractive index, dndT is the thermo-optic constant, andΔT is the temperature change with respectto an initial condition.

Only the stress-induced BW should be used when materialproperties are to be determined. The intrinsic BW shift can bemeasured from a simple calibration test using a bare FBG.Then, the stress-induced BW shift can be determined fromequation (7).

1 10 1000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

F(E

s,)

10 GPa 5 GPa 1 GPa 0.5 GPa

Fig. 3 Values of F as a function of the configuration, β for variouspolymers

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Special Case: Infinite Configuration

The strain of the fiber can be obtained directly from themeasured Bragg wavelength (BW) shift using the followingrelationship [18]:

εσf ¼Δλd

λ 1−Pkð Þ ð9Þ

The effective strain-optic constant, Pk, is defined as

Pk ¼ − n22 P12−ν f P11 þ P12ð Þ� �

, where Pij are strain opticconstants and ν f is the Poisson’s ration of the fiber [19, 31], λis the initial Bragg wavelength.

As mentioned earlier, when the volume stiffness of thepolymer is much larger than that of the fiber, the behavior ofthe assembly is essentially governed by the polymer (“infiniteconfiguration”). The infinite configuration reduces the math-ematical complexity substantially. With the infinite configu-ration, the function F(Es, β ) becomes “unity” regardless ofthe polymer modulus (i.e., the effect of the polymer modulusdoes not have to be considered in the governing equation). Asa result, the strain of the fiber becomes the same as the strainapplied to the polymer; i.e., ε f

σ=Δε ; i.e., with the infiniteconfiguration, the strain of the polymer caused by thermalexpansion or hygroscopic swelling can be determined usingequation (9) even when the polymer modulus is not known.

Proposed Method: Reconstruction of InfiniteConfiguration

The behavior of an infinite polymer/FBG assembly can benumerically reconstructed from the behavior of the specimenswith two finite configurations. The numerical procedure isdescribed below.

From equations (6), and (9), the governing equation can berewritten as

Δλd ¼ F Es;βð Þ⋅λ 1−Pkð Þ⋅Δε ð10ÞThe Bragg wavelength shift ratio, κ , of the infinite config-

uration to a finite configuration β can be defined as:

κ ¼ Δλdjβ¼∞Δλd jβ

ð11Þ

By substituting equation (10) into equation (11), the ratio,κ , can be expressed as:

κ E;βð Þ ¼ F Es;∞ð ÞF Es;βð Þ ð12Þ

The coefficient κ depends only on the polymer modulusand the configuration; i.e., it remains the same regardless of thedeformation state of the polymer. For a given configuration β ,

the coefficient κ becomes a polymer modulus dependentparameter.

The polymer modulus dependent κ values for four mostpractical configurations, β =20 (2.5 mm for the 125 μm fiber),30, 40 and 50, are shown in Fig. 4. The polymer modulusranges from 100 MPa to 10 GPa, which covers the mostengineering polymers [29]. The typical material properties ofthe fiber were used for the calculation; the modulus of 73 GPaand Poisson’s ratio of 0.17.

The value of κ can be determined from the plot in Fig. 4only when the polymer modulus is known. In the actualapplication, it is not always possible to determine the valueof κ from the plot since the polymer modulus can changeduring the environmental loading.

In order to cope with the problem, another parameter isdefined. Let us consider a smaller configuration β2 (the firstone is now called β1 and β1>β2). If the two configurationsare subjected to the same deformations, the second parameter,which is the ratio of the BW shifts between the two configu-rations, can be defined as:

η ¼ Δλdjβ¼β1Δλdjβ¼β2

¼ F Es;β1ð ÞF Es;β2ð Þ

ð13Þ

It is worth noting that this ratio is also independent of thedeformation state of the polymer, but, unlike the coefficient κ ,it is determined directly from the experiment.

For a given set of configurations, β1 and β2, η becomes apolymer modulus dependent parameter. The modulus depen-dent behavior of η is plotted in Fig. 5, where various cases ofβ1/β2 ratios are shown for β2 of (a) 10, (b) 20 and (c) 30.

As mentioned earlier, the coefficients κ and η are depen-dent only on the polymer modulus for a given set of config-urations, and they are related to each other in the followingway:

100001000100

1.0

1.5

2.0

2.5

3.0

β = 20β = 30β = 40β = 50

κ

E (MPa)

Fig. 4 Modulus dependent κ with different configurations

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η ¼ F Es;β1ð ÞF Es;β2ð Þ

¼ F Es;β1ð ÞF Es;β∞� �⋅F Es;β∞ð Þ

F Es;β2ð Þ¼ κjβ¼β2

κjβ¼β1ð14Þ

An analytical relation can be defined to relate these twoparameters as

κ ¼ g ηð Þ ð15Þ

The relationship between κ and η are shown in Fig. 6,where various cases of β1/β2 ratios are shown for β2 of (a) 10,(b) 20, and (c) 30.

In the actual experiment, the value of η is first determinedfrom the experimental data using equation (12). The correspond-ingκ can be determined using equation (14) (or the plot in Fig. 6Relationships between κ and η for (a) β2=10, (b) β2=20, and(c) β2=30). After calculating the BW shifts in the infiniteconfiguration (i.e., Δλd|β=∞) using equation (11), the strain ofthe polymer can be finally determined from equation (10).

Implementation

The proposed method was implemented with an underfill ma-terial of medium filler ratio (40% filler ratio). The fiber diameterwas 125 μm and the Bragg grating was 5 mm long. Twoconfigurators, β =40 and 20, were used in the experiments,which were proven to be optimum configurations by the previ-ous study [29]; the actual diameters were 5 mm and 2.5 mm,respectively. To ensure the generalized plane strain conditionassumed in the governing equations, the length of the specimenwas set to be 10 times as long as the Bragg grating (50mm). TheBragg wavelength was measured by a fiber Bragg gratinginterrogation system (sm125-500: Micron Optics), which hada resolution of 1 pm and repeatability of 0.2 pm.

CTE Measurement

The setup used for CTE measurement is illustrated in Fig. 7.Two configurations were placed inside a convection oven

100 1000 10000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

E (MPa)

100 1000 10000

1.0

1.5

2.0

2.5

E (MPa)

(a) 2 = 10 (b) 2 = 20

100 1000 100000.8

1.0

1.2

1.4

1.6

1.8

E (MPa)

(c) 2 = 30

/

/

/

/

/

Fig. 5 Modulus dependent η with different configurations: (a) β2=10, (b) β2=20 and (c) β2=30

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(EC1A: Sun Systems) and the fibers are connected to theinterrogator located outside the oven. The interrogator wasconnected to a computer for data acquisition.

A block diagram of the fiber Bragg grating interrogationsystem (FBG-IS) layout is also shown in Fig. 7. An LEDilluminates the FBG, which reflects the light at the Braggwavelength. The fiber Fabry-Perot tunable filter (FFP-TF) scansthe reflected light from the FBG as well as the picoWavereference alternatively. The detected signals are then convertedto wavelengths. The picoWave is the multi-wavelength refer-ence of the FBG interrogation system that consists of a FiberFabry-Perot Interferometer, a wavelength marker of a fiberBragg Grating, and a built-in thermo electric controller forthermal stability. It enables real-time wavelength calibration topicometer accuracy.

Before being embedded into the polymer substrate, twoFBGs were calibrated to determine the intrinsic BW shift,Δλ i. The bare FBGs were placed in the environmental cham-ber subjected to a temperature excursion and the BW shift was

1.0 1.5 2.0 2.5 3.0 3.5 4.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

3.5

4.0

1.29

1.11

(a) 2 = 10 (b) 2 = 20

1.0 1.5 2.0 2.5 3.0 3.5 4.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(c) 2 = 30Fig. 6 Relationships between κ and η for (a) β2=10, (b) β2=20, and (c) β2=30

Fig. 7 Experimental setup for CTE measurement

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documented as a function of temperature. The results of thetwo fibers are shown in Fig. 8. The intrinsic BW shift has aquadratic relationship with ΔT [30] and the following rela-tionships were obtained by fitting the results with the quadrat-ic function using the reference temperature of 175 °C:

Δλijβ1 ¼ 0:0103⋅ T−175ð Þ þ 6:78� 10−6 T−175ð Þ2 ð16Þ

Δλijβ2 ¼ 0:0097⋅ T−175ð Þ þ 8:51� 10−6 T−175ð Þ2 ð17Þ

After the calibration, two fiber/polymer assemblies werefabricated using silicone rubber tubes with two different innerdiameters that match to β1=40 and β2=20; the actual diame-ters were 5 mm and 2.5 mm, respectively. The fibers were firstlocated at the center of the tubes using an alignment fixture.Then, the polymer was injected into the tubes and cured at acuring temperature. The silicone rubber tubes were removedafter the polymer cured.

The two assemblies were subjected to the same temperatureexcursion. The temperature of the specimen was measured bya thermocouple embedded in the specimen. The temperatureprofiles are shown in Fig. 9. They show the identical profiles,which confirms that the specimens were at the equilibriumtemperatures.

The total BW shift measured by the FBG is shown in Fig. 10.The stress induced BW shifts were obtained by subtracting theintrinsic BW shift (Fig. 8) from the total BW shift. The stressinduced BW shifts of the two configurations are shown inFig. 11.

The BW shifts of the two configurations are superimposednearly on top of each other at the initial stage of the temperatureramping (below 60 °C). For this region, the parameters η and κshould be “unity” since the modulus of the material is large atlow temperatures. As the modulus gradually decreases, themeasured BW shift of configuration β1 starts to deviate from

that of configuration β2. The slope of BW increase is larger inconfiguration β1 because of the larger volume stiffness.

The values of η were calculated from the stress induced BWshifts using a constant interval of ΔT =5 °C. An example tocalculate the value of η at 140 °C is illustrated in Fig. 11. TheBW shift of β1 caused by increasing the temperature from140 °C to 145 °C is Δλd jβ1 ¼ 0:557 nm. The correspondingBW shift of β2 isΔλdjβ2 ¼ 0:43 nm. Then, the ratio producesthe value of η as

η ¼ Δλdjβ1Δλdjβ2

¼ 1:29

The value of κ for β1 and β2 is determined from Fig. 6(b);it is 1.11 (see the value in Fig. 6(b)). The complete temperaturedependent parameters, η and κ , are plotted in Fig. 12. Usingthe value of κ in Fig. 12 and equation (11), the BW shifts in

20 40 60 80 100 120 140 160 180 200-2.0

-1.5

-1.0

-0.5

0.0

0.5

Fiber used in

Fiber used in

Quadratic fitting of

Quadratic fitting of

(nm

)

Temperature ( C)

Fig. 8 Intrinsic BW shifts of FBGs used in two configurations

0 20 40 60 80 100 120

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(o C

)

Time (min)

Temperature excursion Tmeperature of

Tmeperature of

Fig. 9 Temperatures of two configurations as a function of time

20 40 60 80 100 120 140 160 180 2001526

1528

1530

1532

1534

1536

1538

1540

1542

2

(nm

)

Temperature (oC)

Fig. 10 Total BW shifts of two specimens

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the infinite configuration (i.e.,Δλd|β=∞) were calculated. Thefinal result is shown in Fig. 13.

The polymer strain caused by thermal expansion can beexpressed as

Δεs Tð Þ ¼ αs−α f� �

ΔT ð18Þwhere α s is the CTE of the polymer. Then, from equations (9)to (18), the CTE of polymer can be calculated as:

αs Tð Þ ¼ Δεs Tð ÞΔT þ α f ¼Δλd Tð Þ

λ 1−Pkð ÞΔT þ α f ð19Þ

The slopes before and after the glass transition were deter-mined from Fig. 13; they were 0.0512 nm/°C and 0.128 nm/°C, respectively. Using equation (19), the CTE are determinedto be:

α1 ¼ 0:05121540⋅ 1−0:216ð Þ þ 0:55⋅10−6 ¼ 42:5⋅10−6=o C

α2 ¼ 0:1281540⋅ 1−0:216ð Þ þ 0:55⋅10−6 ¼ 106⋅10−6=o C

where the intrinsic properties of the fiber are λ =1,540 nm andPk=0.216 [18]. The glass transition temperature can be de-fined at the inflection point of the BW shift curve and it wasdetermined to be 120 °C.

20 40 60 80 100 120 140 160 180

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Val

ues

of

and

Temperature ( C)

Fig. 12 Values of η determined from the experimental results; and thecorresponding κ values determined from Fig. 6(b)

20 40 60 80 100 120 140 160 180 200

0.0

1.5

3.0

4.5

6.0

7.5

9.0

10.5

12.0

13.5

15.0

d (n

m)

Temperature (oC)

Tg=120 oC

Fig. 13 BW shift reconstructed for the infinite configuration

Fig. 14 Experimental setup for CHS measurement

20 40 60 80 100 120 140 160 180 2000.0

1.5

3.0

4.5

6.0

7.5

9.0

10.5

12.0

d|

2

d (n

m)

d|

Temperature (oC)

Fig. 11 Deformation induced BW shifts

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CHS Measurement

The experimental setup for CHS measurement is shown inFig. 14. An environment chamber (SH-241: ESPEC) providesthe required stable moisture condition, and the moistureweight gain is measured by a high precision balance (PI-225D: Denver Instrument) that offers a resolution of 0.01 mg.

TheCHSwasmeasured only at room temperature since it hasbeen known that the CHS of polymer is virtually constant belowglass transition temperature [23]. It is important to rememberthat the modulus of the underfill is very large (6 GPa) at roomtemperature. As can be seen clearly from Fig. 11, both config-urations mimic the behavior of the infinite configuration below60 °C; i.e., κ ≈ 1. Consequently, only β1 configuration was usedfor CHS measurement.

In the experiment, an extra specimen (the same configura-tion without the fiber; referred to as a reference specimen) was

prepared for moisture weight gain measurement to avoiddamaging the fiber during frequent weight measurements.The full spectrum was measured before baking as show inFig. 15. Both specimens were then baked at 125 °C for 48 huntil the weight change in the reference specimen was notnoticeable. Then, the dry weight was measured (mdry=308 mg). The BW at the dry condition was also measured atroom temperature (λd

dry=1538.719 nm). The full BW spectraof the specimen obtained before and after baking are shown inFig. 15. The BW change during the baking process wasΔBWbaking=1.36 nm, which was significant and should notbe ignored.

The reference specimen and the test specimen were placedin the environment chamber (Fig. 14) and they were subjectedto 85 °C/85%RH. The weight gain of the reference specimen,

0 40 80 120 1600.0

0.4

0.8

1.2

Moi

stur

e C

onte

nt (

%)

Time (Hours)

1.07

Fig. 16 Moisture absorption history of the reference specimen

20 40 60 80 100 120 140 160 180 2002.935

2.940

2.945

2.950

2.955

2.960

2.965

2.970

TMA Raw data Linear fit below T

g

Linear fit above Tg

Tg=124oC

2= 109 ppm/ oC

1= 41.2 ppm/ oC

Tip

pos

ition

(m

m)

Temperature (oC)

Fig. 17 Measurement of CTE by TMA

1540 1541 1542 1543 1544 15450.0

0.2

0.4

0.6

0.8

1.0

1.2

Nor

mal

ized

Pow

er

BW (nm)

After baking

After saturation

BWsaturation

Fig. 18 Full BW spectrum of the smaller configuration (β =25.4)obtained during the moisture test

1538 1539 1540 1541 15420.0

0.2

0.4

0.6

0.8

1.0

1.2

BWbaking

Before baking

Nor

mal

ized

pow

er

BW (nm)

After baking

BWsaturation

After saturation

Fig. 15 Full BW spectrum of the β1 configuration obtained during themoisture test

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mwet, was measured periodically. Then, the moisture content(%) was calculated from

C ¼ mwet−mdrymdry

� 100% ð20Þ

The complete moisture weight gain history of the referencespecimen is shown in Fig. 16. The specimen was saturated ataround 120 h, and the saturated moisture weight gain wasCsat=1.07 %.

The BW was measured after moisture saturation. The fullspectrum is also shown in Fig. 15; λd

sat=1541.607 nm. Thenet BW change caused by the hygroscopic swelling strain isexpressed as

Δλd ¼ λsatd −λdryd ¼ λ 1−Pkð ÞΔεhs ð21Þwhere λd

sat is the BW shift at the moisture saturation andΔεsh

is the hygroscopic strain of the polymer. Then, the CHS, χ s,can be determined as

χs ¼ΔεhsC

¼ Δλdλ 1−Pkð Þ⋅C ð22Þ

The initial BWafter baking is 1538.719 nm. The saturatedBW was 1541.607 nm, which yielded the BW increase of2.888 nm. Using the well-known linear relationship betweenthe hygroscopic swelling strain and the moisture concentra-tion [8, 9, 17, 22, 23], the CHS was determined from equation(21) as

χs ¼1541:607−1538:719

1540⋅ 1−0:216ð Þ⋅1:071% ¼ 0:223 %εh=%C

Discussion

The CTE of the tested polymer was also measured by a TMA(TMA-7: Perkin Elmer) for verification. The raw data obtainedfrom TMA for testing is shown in Fig. 17. The CTE valueswere 41.2 ppm/°C and 109 ppm/°C for below and above glasstransition temperature, respectively. The glass transition tem-perature was also determined to be 124 °C. By comparing withthe results measured by FBG, the differences (less than 3 %)fall in the typical measurement uncertainty range including theinherent material variability.

In the CHS measurement, the saturated weight gain and theBW change are required for calculation. In order to verify theexperimental procedure, the same procedure was repeated with adifferent configuration, β=25.4. The experimental raw data areshown in Fig. 18. The total BW shift was 2.807 nm, whichresulted in χs=0.22 %εh/%C . The difference was within 2 %and the results confirmed good repeatability of the measurement.

The CHS value reported in this paper was obtained from thelinear relationship between the hygroscopic swelling strain and

themoisture concentration. For some epoxymaterials, this linearrelationship may not be valid [28]. The proposed techniqueshould be practiced with caution when it is utilized to measurethe CHS of those materials; an additional moisture weight gainhistory is needed to determine the moisture concentration-dependent CHS [28].

Conclusion

An experimental technique utilizing the concept of the dual-configuration FBG sensor method was proposed to measurethe CTE and CHS of polymeric materials, and it was im-plemented with a medium filler ratio underfill material. Thebehavior of the infinite configuration was reconstructed nu-merically from the two small finite configurations that weresubjected to environmental conditions. Then, the CTE andCHS were determined from the simple governing equationthat provided a direct relationship between the measured strainsand the two coefficients. The small configurations negated theeffect of heat generation during curing, whichmade the methodapplicable for a wide range of polymeric materials. The mea-surement accuracy and repeatability were confirmed by a com-mercial tool and an experiment with a different configuration.

The sample preparation is simple and the implementationcan be readily automated. Besides the advantage that the meth-od can measure both CTE and CHS from the same specimen,the method can be used effectively to quantify the effect ofdegradation caused by accelerated environmental tests such asthermal cycling, high temperature and relative humidity agingtests, etc..

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Dual-Configuration Fiber Bragg Grating Sensor Technique to Measure Coefficients of Thermal Expansion and Hygroscopic SwellingAbstractIntroductionBackground: FBG Sensor MethodGoverning EquationTemperature EffectSpecial Case: Infinite Configuration

Proposed Method: Reconstruction of Infinite ConfigurationImplementationCTE MeasurementCHS Measurement

DiscussionConclusionReferences