# Simultaneous Measurement of Effective Chemical Shrinkage and Modulus Evolutions During Polymerization

## Abstract

A novel method is proposed to *simultaneously* measure the effective chemical shrinkage and modulus evolutions of advanced polymers during polymerization. The method utilizes glass fiber Bragg grating (FBG) sensors. They are embedded in two uncured cylindrical polymer specimens with different configurations and the Bragg wavelength (BW) shifts are continuously documented during the polymerization process. A theoretical relationship is derived between the BW shifts and the evolution properties, and an inverse numerical procedure to determine the properties from the BW shifts is established. Extensive numerical analyses are conducted to provide general guidelines for selecting an optimum combination of the two specimen configurations. The method is implemented for a high-temperature curing thermosetting polymer. Validity of the proposed method is corroborated by two independent verification experiments: a self-consistency test to verify the measurement accuracy of raw data and a warpage measurement test of a bi-material strip to verify the accuracy of evolution properties.

### Keywords

Effective chemical shrinkage Elastic modulus Evolution Fiber Bragg grating Gelation point## Introduction

Thermosetting polymers are widely used in the semiconductor industry. During polymerization, they undergo significant reduction of volumes as molecules move from a van der Waals distance of separation to a covalent distance of separation. This volumetric shrinkage induced by the polymerization is called “*chemical shrinkage*.” The chemical shrinkage produces dimensional changes and residual stresses in packaged devices, which can significantly affect product reliability [1, 2, 3]. In some cases, the chemical shrinkage affects the functionality of special polymers. For example, electrically conductive adhesives with higher curing shrinkage generally exhibit higher conductivity [4]. Measuring the chemical shrinkage of base polymers is critically needed for performance optimization.

Numerous methods have been developed in the past few decades [5, 6, 7, 8, 9, 10, 11] to measure the *intrinsic (or total) chemical shrinkage* (denoted as *ε*_{k}) both during [7, 8, 9, 11] and at the end of the polymerization process [5, 6, 10]. It is important to note that not all of the intrinsic chemical shrinkage contributes to the dimensional stability or the residual stress, since some portion of the chemical shrinkage occurs before the gelation point where polymers start to build mechanical strength [12, 13, 14]. The chemical shrinkage after the gelation point, called the “*effective chemical shrinkage*” (denoted as *ε*^{ch}) [15], should be used for calculation of curing-induced deformations; otherwise, the magnitude of deformations can be significantly overestimated. The existing methods measure the intrinsic chemical shrinkage simply because they cannot detect the gelation point. Additional techniques such as the Rheometrics Dynamic Spectrometer [16] can be used together with the existing methods to measure the effective chemical shrinkage. However, this additional technique greatly increases the complexity of the experimental procedure and does not seem practical for routine practice.

Another important property that is required for accurate determination of the residual deformations is *Young’s modulus*. During the dynamic curing process of polymers, the Young’s modulus evolves non-linearly with the time as polymerization proceeds. To the best of the authors’ knowledge, a rheometer or a Dynamic Mechanical Analyzer (DMA) is the only available technique that can measure the modulus evolution during polymerization [17, 18, 19].

*dε*

_{k}, contributes to the residual deformation during polymerization [18]. In the case of uni-axial loading, the increment of the residual stress,

*dσ*

_{R}, can be expressed as

*σ*

_{R}, can be estimated as

*t*

_{gel}and

*t*

_{final}are the times of gelation point and of complete polymerization, respectively.

The evolution of the effective chemical shrinkage, *ε*^{ch}(*t*), and the evolution of the modulus, *E*(*t*), can be measured by two different techniques. It is important to note, however, that the correct time correlation and identical curing conditions are extremely difficult to achieve in practice when different techniques are employed to measure each quantity separately.

Many researchers have attempted to characterize the evolution of curing-induced residual stresses of bi-material strips [20, 21] by measuring the bending deformation during polymerization. However, the measured results pertain to specimens with the specific dimensions and configurations used in their experiments, and the experimental results cannot be extended to the complex configurations of packaged assemblies.

We propose a novel method to cope with the technical difficulties associated with the existing techniques. It utilizes fiber Bragg grating (FBG) sensors to simultaneously measure the effective chemical shrinkage and modulus evolutions. The basic concept of polymer property measurements using FBG sensors has been reported in Ref. [15]. In this study, the concept is advanced for measurement of the property evolutions during the entire polymerization process.

In our previous study, only the final chemical shrinkage in the complete polymerization state was of interest and it was decoupled from the modulus by employing a large specimen configuration. The large configuration reduced the mathematical complexity substantially and thus simplified the numerical procedure. For the current study of interest, namely evolutions during polymerization, however, the large configuration is applicable only to polymers that have extremely long curing time (or very small heat generation), as otherwise the measurement accuracy can be affected significantly by a temperature overshooting and a temperature gradient within a specimen. In this paper, the issue is addressed and a more general method is developed.

The concept of the proposed method is first described. A numerical procedure to determine the property evolutions from the BW shifts is followed. The rationale to determine a proper set of specimen configurations is described and demonstrated for different polymers. The proposed method is implemented with an underfill material. The evolutions of effective chemical shrinkage and modulus are measured and the results are validated by direct measurement data.

## Mathematical Formulations

### Elastic Solution of an FBG Embedded in Cylindrical Substrate

*ε*

^{ch}occurs in the polymer substrate over the time increment, which has an instant modulus of

*E*

_{s}. The loading condition of the incremental time duration can be written mathematically as

*a*is the radius of the FBG and

*b*is the radius of the polymer substrate. From the theory of elasticity, the generalized plane strain solution of stress components within the fiber can be derived as:

*E*

_{f}and

*v*

_{f}are the modulus and the Poisson’s ratio of the fiber material; and

*E*

_{s}and

*v*

_{s}are the modulus and the Poisson’s ratio of the polymer. The details of the derivation can be found in Ref. [15]. The coefficients,

*C*

_{1f}and

*C*

_{1s}in equations (4) and (5), have the form of

*A*,

*B*,

*C*and

*D*are coefficients and they can be expressed as

Equations (3) to (11) complete the mechanical description of the stress states of the FBG when the polymer matrix undergoes a chemical shrinkage of Δ*ε*^{ch}.

### Bragg Wavelength Shift and Evolution Properties

*intrinsic*” BW shift, which is not associated with any stress-induced deformation. It is defined as

*λ*

_{B}is the initial Bragg wavelength (BW),

*α*

_{f}is the CTE of the fiber core material,

*n*

_{eff}is the effective refractive index, \( \frac{{dn}}{{dT}} \) is the thermo-optic constant, and Δ

*T*is the temperature change with respect to a initial condition.

*deformation*” induced BW shift. It has been shown in equation (4) that the FBG is subjected to a uniform transverse loading condition, i.e., the radial stress equals to the hoop stress while embedded in a cylindrical substrate. Then the “

*deformation*” induced BW shift can be formulated in terms of stress components as [15]

*P*

_{11}and

*P*

_{12}are strain-optic coefficients (Prockel’s coefficients).

*F*, is written as

*G*

_{1},

*G*

_{2},

*H*

_{0},

*H*

_{1}and

*H*

_{2}are functions of

*β*and they can be expressed as

*β*; they will be referred to as C-1 and C-2, respectively, which correspond to

*β*

_{1}and

*β*

_{2}(

*β*

_{1}>

*β*

_{2}).

*d*, of the deformation induced BW shift will hereafter be omitted for convenience. Dividing equation (22) side by side yields

### Calculation of Evolution Properties

*t*

_{eff}) is defined as \( {t_{eff}} = t - {t_{gel}} \).

*m*small constant segments. For the

*j*th (

*j*= 1 to

*m*) time segment \( \left( {{t_{j - 1}} < {t_{eff}} < {t_j}} \right) \), we define

\( \Delta \lambda_{B,j}^{C - 1} \) and \( \Delta \lambda_{B,j}^{C - 2} \): BW shift increments of configuration C-1 and C-2

\( {E_{s,j}} \): Instant equilibrium modulus (averaged over the time segment)

\( \Delta \varepsilon_j^{ch} \): Chemical shrinkage increment

*j*th segment as

*E*

_{s,j}, can be determined using equation (25). Then, by substituting

*E*

_{s,j}back into either one of equation (24), \( \Delta \varepsilon_j^{ch} \) can be calculated. By repeating the above calculations, the shrinkage increment and the instant equilibrium modulus in every segment can be determined. The property evolutions can then be expressed as

At a given instant, *t*_{eff}, the modulus is defined as an instantaneous value while the chemical shrinkage is defined as a value accumulated up to that instant.

To enhance the accuracy in solving equation (25), it is desired to make the BW shift ratios \( \left( {\frac{{\Delta \lambda_B^{C - 1}}}{{\Delta \lambda_B^{C - 2}}}} \right) \) as large as possible, which can be achieved by increasing the ratio of the two configurations \( \left( {M = {\beta_1}/{\beta_2}} \right) \).

## Optimum Specimen Configurations

The ratio can be increased by using either a large value of *β*_{1} or a small value of *β*_{2}. In practice, the lower bound of the smaller configuration C-2 is limited by the resolution of a measurement system, while the upper bound of the larger configuration C-1 is limited by excessive heat produced by the exothermic polymerization process. The following numerical analysis is conducted to provide a guideline for selecting the optimum specimen configurations.

### C-2 Configuration

*n*th order model [17, 22, 23, 24, 25, 26, 27] is considered to illustrate the above criterion quantitatively. The model can be expressed as

*p*is the curing extent and

*k*

_{c}is a temperature-dependent rate coefficient. The rate coefficient is related to the temperature through Arrhenius’s equation

*A*is a material constant,

*E*

_{a}is the activation energy and

*R*is the ideal gas constant (8.314

*J/mol·K*).

*ϕ*(

*p*), can take any mathematical form of

*p*; e.g., a linear function of

*ϕ*(

*p*) has been reported for various polymer systems [28, 29].

*ϕ*(

*p*), can also take any mathematical form of

*p*. The percolation theory [30, 31] is adopted in this study, which can be expressed as

*p*

_{gel}is the curing extent at the gelation point; a typical value of

*p*

_{gel}is around 0.5 [14, 32, 33, 34, 35].

The most representative values of thermosetting polymers were used in the simulation. The curing kinetics parameters were *A* = 29.1/s, *E*_{a} = 35.8 *kJ*/*mol*, *n* = 2 with a curing temperature of 165°C [17]. They yield a polymerization rate of *k*_{c} ≈ 0.002/*s* at the curing temperature [17, 22, 23, 24, 25, 26, 27]. Polymers develop a small modulus at the end of the curing, especially when they are cured at elevated temperatures. Depending on the material, \( {E_\infty } \) can vary from hundreds of MPa to several GPa. A value of 100 MPa was chosen for the calculation, which was the low end of reported moduli [18, 20, 30, 36, 37, 38, 39, 40]. For the total chemical shrinkage, a representative value of 0.9% was chosen based on the literature [5, 6, 7, 8, 11, 41, 42, 43, 44, 45, 46, 47]. Assuming the linear function of *ϕ*(*p*) and *p*_{gel} = 0.5, the corresponding effective chemical shrinkage at the gelation point was 0.45%.

*m*small uniform segments with the time increment of Δ

*t*. The time at the beginning and the end of the

*j*th (

*j*= 1 to

*m*) segment are (

*j*–1)Δ

*t*and

*j*Δ

*t*, respectively. The curing extent at the beginning and the end of the

*j*th (

*j*= 1 to

*m*) segment was determined from equation (28). Then, the chemical shrinkage increment and the average equilibrium modulus of the

*j*th segment were determined from equations (30) and (32). The evolution properties of this material system are plotted in Fig. 3(a). The figure also shows the effective chemical shrinkage and the effective time defined earlier.

For the material system plotted in Fig. 3(a), the simulated BW shift evolution of configuration *β* = 20 is plotted in Fig. 3(b). Typical properties of glass FBGs were used in the simulation: *P*_{11} = 0.121, *P*_{12} = 0.270 [48, 49, 50], *E*_{f} = 73.1 GPa [51], *v*_{f} = 0.17 [49, 52] and *n*_{eff} = 1.457 [53]. The BW evolves non-linearly with the time. A total BW shift of 0.522 nm is observed at the end of the curing process.

*β*) are plotted in Fig. 4, where a dashed line indicates the criterion of equation (27). In Fig. 4, two different cases of the total chemical shrinkage

*ε*

_{k}= 0.45% and 0.90% are plotted. As expected, the BW shift decreases considerably as the chemical shrinkage decreases. The analysis indicates that, for typical polymers, the configuration of

*β*

_{2}larger than 17 will produce a BW signal, which is sufficiently large for accurately determination of properties. It is important to note that the amount of BW shift is function of only the radius ratio (

*b*/

*a*). Therefore, the above analysis will be valid for any FBGs. In the following calculation,

*β*

_{2}of 20 is used as C-2 for simplicity.

### C-1 Configuration

*β*

_{2}= 20) as a function of various C-1 configurations (

*β*

_{1}), where the BW shift ratios are shown for three different elastic moduli (100 MPa, 500 MPa and 1 GPa). It clearly shows that the BW shift ratio and thus the measurement accuracy increases as C-1 configuration becomes larger. The practical upper value of

*β*

_{1}is limited by the effect of heat generated during polymerization (exothermic process) on BG shift measurements. A combined numerical analysis of curing kinetics and heat generation is conducted to provide a guideline to select a proper C-1 configuration.

*k*is the thermal conductivity,

*T*is the temperature,

*t*is the time,

*ρ*is the density,

*c*

_{p}is the specific heat and \( \dot q \) is the heat generation rate, which is directly related to the reaction speed through the equation

*p*is the curing extent and Δ

*H*is the total exotherm of the reaction. An

*n*th order curing kinetics model in equation (28) is used in the following analysis for its mathematic simplicity, but any other complicated explicit models can also be used.

*h*is the convective heat transfer coefficient and

*T*

_{a}is the ambient temperature and

*T*

_{0}is the initial temperature. Equation (35) is highly nonlinear but can be solved using the finite element method.

*N*nodes. The whole polymerization process is divided into

*l*steps with a small time interval of

*Δt*. The temperature distribution of the first time step is calculated through a transient analysis with a uniform heat generation. The resultant non-uniform temperature distribution is used to determine the curing extent of the second time step at each node using equation (36): in the

*i*th (from 2 to

*l*) time step, the curing extent at node

*j*(from 1 to

*N*) can be approximated as

*T*

_{i-1,j}is the temperature at node

*j*in the

*i*-1th (from 1 to

*l*−1) time step. The heat generation rate of each node is then updated according to equation (34). Using the updated curing extent and heat generation rate at each node, the temperature distribution is calculated for the next time step. The procedure is iterated until all time steps are completed. The evolutions of the curing extent and the temperature distribution are obtained by recording the intermediate results during the polymerization process.

The maximum temperature and thus maximum curing extent difference will occur between the center of C-1 and the surface of C-2. This maximum curing extent difference, Δ*p*_{max} can be used as a metric to select a proper C-1 configuration. For the predetermined C-2 configuration (*β*_{2} = 20), Δ*p*_{max} of various C-1 configurations were calculated using the numerical analysis used in the previous section.

*H*from 200 to 400

*J/g*and

*k*

_{c}from 0.8·10

^{-3}to 3.2·10

^{-3}/

*s*. This range covers most of the thermosetting polymers used in semiconductor packaging applications. The simulation results are shown in Fig. 6, where Δ

*p*

_{max}for three discrete total exotherm values of 200, 300 and 400

*J/g*are plotted as a function of C-1 configuration (

*β*

_{1}) in (a), (b) and (c), respectively. Each plot contains Δ

*p*

_{max}distributions for three rate coefficient values of 0.8·10

^{-3}/

*s*, 1.6·10

^{-3}/

*s*and 3.2·10

^{-3}/

*s*.

When Δ*p*_{max} is less than 1%, it is reasonable to assume that polymer in the two configurations cure uniformly (within the configurations) and consistently (between the two configurations). This guideline is also illustrated in Fig. 6.

In practice, the curing kinetics of the material can be first measured by a differential scanning calorimeter (DSC) and a proper C-1 specimen configuration can be determined using Fig. 6. For example, if a polymer has a small exotherm (≈200 *J/g*) [Fig. 6(a)], *β* = 40 can be used for C-1 configuration even when it cures at a high rate of 3.2·10/*s*. For typical polymers used in the previous analysis (*k*_{c} = 2·10^{-3}/*s* and 300 *J/g*), *β* ≈ 50 can be used for C-1 configuration [Fig. 6(b)]. For those polymers with a higher total exotherm, the value of *β* has to be proportionally reduced for C-1 configuration [Fig. 6(c)].

## Implementation

The proposed method was implemented with a high temperature curing underfill material. After the description of a mold to fabricate the specimen is given, a supplementary work using DSC is reported, which guides to determine C-1 configuration (Fig. 6). The evolution property measurements using the selected configurations are reported.

### Experimental Setup

The length of the specimen is also critical because the generalized plane strain condition is assumed in the derivation of governing equations. If the specimen is not sufficiently long, the strain field of the area where the Bragg gratings are located (in the middle of the specimen) can be affected by the free-edge effect. A supplementary numerical analysis was conducted to investigate the effect of the specimen length. The results indicated that the effect was ignorable (less than 0.1%) when the specimen length was longer than 10 times of the gratings length. Since the length of the FBG employed in the experiment was 5 mm, the total length of both specimens was made to be 50 mm. The actual C-1 and C-2 configurations of the underfill material fabricated by the mold assembly are shown in Fig. 7(b).

An FBG data acquisition system consists of an FBG interrogation system (FBG-IS), an environmental chamber and a computer. The Bragg wavelength is measured by a commercial FBG interrogation system (Micron Optics sm125-500), which has a resolution of 1 pm and repeatability of 0.2 pm. The temperature is controlled by a computer-controlled oven (EC1A, Sun Systems).

### Supplementary Experiment Using DSC

*J*. The weight of the specimen was 0.0733

*g*and the total exotherm,

*ΔH*

_{total}, was thus determined to be 330

*J*/g. At arbitrary time

*t*, the partial reaction heat was calculated by the partial integral of the area. The corresponding partial exotherm can be defined as

*ΔH*(

*t*). Then, the curing extent,

*p*(

*t*), at time

*t*can be simply determined as [54, 55, 56, 57, 58, 59]

*nth*order model (equation 28) was used to fit the measured curing extent, and the result is also shown in Fig. 8(b). The curing kinetics are determined to be

*k*

_{c}= 0.0014/

*s*.

Referring to Fig. 6(c), a specimen dimension of *β* = 45 or smaller can be used for the C-1 configuration. Considering the practicality and commercial availability of silicone rubber tubing in the implementation, the inner diameter of the tube is 3/16” in specimen configuration C-1 and 3/32” in configuration C-2, which results in *β*_{1} = 38 and *β*_{2} = 19, respectively. It is to be noted that the curve of *ΔH* = 400 *J*/g *k*_{c} = 0.0016/*s* in Fig. 6(c) is a conservative estimation and the final configurations assure uniform curing.

### Evolution Properties

*T*[60] and the results are:

## Verification

Two tests are conducted to verify the validity of the proposed method: a self-consistency test to confirm the accuracy of the raw data obtained from the two configurations, and a deformation measurement test to verify the validity of the evolution properties inversely calculated from the raw data.

### Self-consistency Test

*β*= 25 (denoted as C-3). The BW shift of the specimen during the curing process was measured experimentally. The experimental results of BW evolution are shown in Fig. 13(a).

The corresponding \( \Delta \lambda_B^d \) after the apparent gelation point is plotted in Fig. 13(b). From the mathematical relationships described in “Mathematical Formulations,” the BW shift of the specimen with a configuration of C-3 was calculated using the evolution properties of Fig. 12. The predicted results are compared with the experimental results Fig. 13(b). The two results agree well with each other, which confirms the accuracy of the raw data obtained from the proposed experimental procedure.

### Warpage Prediction and Verification by using T/G Interferometry

The above self-consistency experiment verified that the results obtained from the proposed two specimen configurations can be used for prediction of the behavior of specimens in other configurations. However, it does not necessarily guarantee that the inversely determined evolution properties from the BW shift accurately represent the intrinsic properties of the material. The polymerization induced warpage of a bi-material strip was studied to further validate the accuracy of measured properties.

Twyman-Green (T/G) interferometry is a classical form of interferometry which involves the interference of coherent light [61]. A coherent light source, usually a laser, is collimated and then passed through a beam splitter where it is split into active and reference paths. The reference and active paths are reflected by the reference flat and the specimen, respectively. The reflected beams combine to interfere and create a fringe pattern seen by the observer, which represents the contour map of out-of-plane displacement (or warpage).

*w*, can be determined by [61]

*n*= 1.5 which is the refractive index of the index matching fluid,

*N*is the order of fringes, and

*λ*is the wavelength of the laser.

*λ*= 514 nm was employed in the experiment, which produced a contour interval of 171.3 nm per fringe. The warpage as a function of the effective time is shown in Fig. 15(b).

*Δε*

^{ch}occurs in the polymer layer that has a thickness of

*t*

_{1}and an instant modulus of

*E*

_{1}, the curing induced warpage of the glass substrate can be calculated by

*t*

_{2}and

*E*

_{2}are the modulus and the thickness of the glass strip, respectively, and

*L*is the half length of the strip.

Since both effective chemical shrinkage and modulus evolve nonlinearly with time, the warpage change during each small time segment can be calculated first and the evolution of curing induced warpage can be then determined by accumulating the warpage increments. The numerically predicted warpage evolution is also plotted in Fig. 15(b) for a comparison. The two results agree very well, which confirms the validity of the property measurement.

## Discussions

The polymer modulus, *E*_{s}, was the only unknown parameter in the numerical procedure using equation (23). As can be seen from the explicit form of the nonlinear function *F* (from eqautions (16) to (21)), however, *F* is also a function of the Poisson’s ratio of the polymer, *ν*_{s}, which can change as the polymer cures. As stated in our previous study [15], the effect of Poisson’s ratio is negligible since the deformation of FBG is mainly induced by the axial stress of the polymer.

*F*were calculated as a function of

*ν*

_{s}for polymers with a wide range of modulus (

*E*

_{s}= 100 MPa, 500 MPa and 1GPa) and specimen configurations (

*β*= 20, 60 and 100). The results are plotted in Fig. 16, which confirms the negligible effect of Poisson’s ratio (less than 1% change in

*F*). In the implementation, a typical value of Poisson’s ratio (say 0.3) can be used to define the function,

*F*.

It should be noted that all the governing equations are based on the elasticity theory. However, polymeric materials in general exhibit some visco-elastic behavior. Stress relaxation can occur during curing progresses. Within the scope of linear visco-elasticity, the rates of relaxation would be identical in the two configurations since the modulus is stress-independent and only a function of the time [62, 63]. Consequently, the real-time modulus can be captured effectively in the two configurations. For the same reason, the effective chemical shrinkage can be overestimated if polymers show significant nonlinear visco-elastic behavior. Some form of compensation through a mathematical modeling may be needed for such polymers.

## Conclusions

A novel measurement method utilizing embedded FBG sensors was proposed and implemented to simultaneously measure the effective chemical shrinkage and modulus evolutions of advanced polymers during the entire polymerization process. A theoretical basis and a numerical procedure were first established to determine inversely the evolution properties from the experimentally documented BW shifts of two specimen configurations. The numerical analyses were extended to provide general guidelines for selecting an optimum combination of the two specimen configurations. The proposed method was applied to a high-temperature curing thermosetting polymer using 20X and 40X configurations. The BW was documented after the effective gelation point, and the effective chemical shrinkage and modulus evolutions of the material during polymerization were determined successfully from the measured BW shifts.

Two independent tests were conducted to verify the validity of the proposed method. A self consistency test was performed using 25X and a deformation measurement test was conducted using a bi-material strip. The predicted results from the measured evolution properties were compared with the results of the verification tests. The two results agreed well with each other for both cases, which corroborated the accuracy and the validity of the proposed method.

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