Orthotropic elastic constants for polyimide film

Orthotropic Elastic Constants for Polyimide Film

SEO HYUN CHO, GENE KIM, THOMAS J . McCARTHY, and RICHARD J. FARRIS*

Polymer Science and Engineering Department University of Massachusetts Amherst

Amherst, MA 01 003

The orthotropic constants of polyimide film have been characterized using the theory of elasticity of an anisotropic material. Experimental techniques coupled with the mechanics of orthotropic materials are used to determine all 9 independ- ent orthotropic elastic constants (3 tensile moduli, 3 shear moduli, and 3 Poisson’s ratios) and 3 coefficients of thermal expansion. Vibrational holographic interferom- etry is used to determine the orthotropic axes of symmetry. For this polyimide film, the two principal axes coincided with the machine and transverse directions. It is also used to evaluate the 2 in-plane Poisson’s ratios by measuring residual stresses in 2-D and 1-D square membranes. Using other instruments such as a high pres- sure gas dilatometry apparatus, a tensile tester, a pressure-volume-temperature apparatus, a thermomechanical analyzer, and a torsion pendulum, the 7 other orthotropic constants and the 3 coefficients of thermal expansion are determined.

INTRODUCTION orthotropic material, there exists 3 mutually perpendic-

n recent years, polyimides have emerged as a new I class of insulating materials owing to the increased demand on the dimensional and thermal requirements of electronic packaging materials. Polyimides are often used in electronic applications as interlevel insulators, adhesives, or flexible circuit board substrates. One reason for the popularity of these materials is their ex- cellent thermal and chemical stability under very harsh environments. During processing, residual stresses can cause failures like delamination and cracking (1). Residual stresses can arise due to thermal shrinkage, mechanical force, and other factors. Hence, a complete understanding of the material properties is necessary to predict the reliability of the material.

Orthotropic linear elasticity theory has been used to characterized polyimide films. Most commercial films such as polyimide, polyethylene terephthalate, and ply- propylene are anisotropic in nature. They have differ- ent physical and material properties in the machine and transverse direction as well as in the out-of-plane direction. For some films, the orthotropic axes may not coincide with the machine and transverse directions. The orthotropic axes should be determined before the material properties are measured. The orthotropic axes can be easily determined using vibrational holographic interferometry (2). If the orthotropic axes for an aniso- tropic film are identified, the polymeric film may be described by the generalized Hooke’s Law for a linearly elastic orthotropic material in matrix form (3). For an

‘Corresponding author.

ular planes of symmetry (4). and due to the symmetry in the compliance matrix the number of independent material constants can be reduced from 21 to 9 inde- pendent constants. This means that only 9 independ- ent material constants are required to described the be- havior of orthotropic materials. These are the 3 normal compliances, the 3 shear compliances, and the 3 com- pliances associated with Poisson’s ratios. A more complete characterization should also include the two in-plane CTEs (Coefficient of Thermal Expansion) and the out-of-plane CTE. Therefore if the 12 independent elastic constants have been determined, the load-de- formation characteristics in any arbitrary direction can be predicted using tensor transformation relations.

This paper discusses how to obtain these elastic constants. All 12 elastic constants were measured for the polyimide film. The techniques used in this paper are generally applicable to thin films made from any type of material.

Material

The polyimide film used in this work is Kapton E@ film which is a DuPont tradename. All polyimide films were supplied by DuPont. The density of the poly- imide film is 1.3668 g/cm3.

Measurements Vibrational Holographic Interferometry

The vibrational holographic interferometry equipment is schematically illustrated in Fig. 1 . The equipment is

EXPERIMENTAL

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Seo Hyun Cho, Gene Kim Thomas J. McCarthy, and Richard J. Farris

U

Camera

El f- Sine wave

generator

Photo plate Sample chamber

Rg. 1. The vibrational holographic interferometry equipment.

a typical two-beam holographic interferometer. The co- herent light source is a 5mW helium-neon laser with wavelength of 6328A. The laser beam is split into two beams by a variable beam splitter, which adjusts the intensity ratios of the two split beams. The reference beam and the object beam are superimposed on a ther- moplastic holograhpic plate. The thermoplastic holo- graphic plate used to record the interference pattern of the two beams as a hologram can be developed in- situ by electronic charging of a holographic camera (Newport Research Corporation, HC301). All experi- ments were performed under vacuum conditions and at a temperature of 22OC-23"C.

The film sample on a constraining square washer is rigidly mounted inside a vacuum chamber. The en- tire chamber is connected to a piezoelectric shaker (Wilcoxon Research) driven by a frequency generator (Wavetek, Model 190). The frequency generator can produce sinusoidal vibrations with various frequencies and adjustable amplitudes. More detailed explana- tions are reported elsewhere (2, 5, 6).

Instron Tesile Tester With an Instron Tester Model 5564, the Young's mod-

uli were measured at a temperature of 22T-23"C. Following ASTM D882-88 guidelines (7). rectangular samples of 50 mm in length by 5 mm in width with a cross-head speed of 5 mm/min. (strain rate = lo%/ min.) were tested. A 1 kN load cell was used.

High Pressure Gas Dilatomety The out-of-plane Poisson's ratios for polyimide film

were evaluated from high pressure gas dilatometry experiments. The change in stress of a thin ribbon sample with applied hydrostatic pressure can be ob- tained using this technique. Thin ribbon samples 5 mm in width and 50 mm in length were cut along both orthotropic 1 and 2 axes. The ribbon sample is then

held between sample clamps inside the sample cham- ber. The sample is uniaxially strained (under 1%) and held iso-strain. The chamber is then pressurized with high-pressure nitrogen gas to pressures up to 7MPa (about 1000 psi). This results in an uniform hydro- static stress on the sample with no shear effects. The change in the stress of the sample with the applied pressure is then recorded.

ThermoMechanical Analyzer (7MAl The in-plane CTEs were measured on thin films

using a TA Instruments 2940 Thermomechanical An- alyzer. A force of 0.01 N was applied to 5 mm wide by 25 mm long sample. The samples were purged using nitrogen gases. An average linear CTE was determined from the second heating run from 30°C to 200°C at a heating rate of 5"C/min.

Pressure-Volume-Temperature (PW, Apparatus The out-of-plane CTE and out-of-plane Young's Mod-

ulus were determined using a PVT apparatus made by Gnomix Inc. This apparatus allows one to measure the volume change of a sample as a function of tem- perature and hydrostatic pressure. In their design, a rigid sample cell having a total volume of approximately 7.45 cm3 is sealed at one end with the other end having a flexible stainless steel bellows.

The sample cell configuration used for all the PVT ex- periments is bellows #5, long tube #24, partition #44, and sample container #64. The cross-sectional area of bellows #5 is 1.145 cm2 (8). If this area and the verti- cal displacement are known, the volume change of the sample can be determhed. Isothermal and isobaric cali- brations are done to account for the volume change of mercury due to pressure and temperature as well as any dimensional changes of the sample cell and the Linear Variable Differential Transducer (LVDT) assem- bly.

302 POLYMER ENGINEERING AND SCIENCE, FEBRUARY 2001, Vol. 41, No. 2

Orthotropic Elastic Constants for Polyimide Film

Typically a sample weight of 0.5-2.0 g is used with mercury as the confining fluid. The polyimide film weight of 1.1 g was tested. A n Enerpac P-2282 high pressure hand pump is used to pressurize from 10 MPa to 90 MPa. A heating jacket around the pressure ves- sel, connected to a temperature controller, can vary temperature between from 30°C to 90°C.

Torsion Pendulum

The shear moduli were determined using a torsion pendulum made by the Virtis Company Inc. The tor- sion pendulum experiment was done on seven ribbon- like samples cut in each orthotropic direction. One end of the sample was held fured [no rotation possible] and a circular disk, with a moment of inertia of 106 gmm2, was attached to the other end. The samples were 3 cm long and ranged from 200 pm to 250 pm wide. The pendulum was displaced by some angle about the long axis of the ribbon, and released. The torsion pendu- lum experiments resulted in a frequency of 0.40-0.42 Hertz and a temperature of 2ZoC-23"C. The period of oscillation is measured by watching the movement of a mark on the pendulum past a mark on the lower platform. Several periods were measured and an aver- age period was used in the calculation.

THEORY

Orthotropic Elasticity Theory

As explained in the Introduction, orthotropic mate- rials have three mutually perpendicular axes of sym- metry. For the orthotropic materials, only 9 independ- ent constants need to be determined to characterize a materials mechanical behavior in any direction. These include 3 tensile moduli, 3 shear moduli, and 3 Pois- son's ratios. The CTEs (coefficients of thermal expan- sion) in all the 3 orthotropic directions also need to be measured. Polyimide film is an anisotropic orthotropic material, so only 12 constants are needed. The consti- tutive behavior of an anisotropic orthotropic materials

u13

u23

,912

where,

E~~ : strain crij : stress ai : CTE

C,i : compliance

Eij : tensile modulus G, : shearmoddulus vij : Poisson's ratio

Membrane Theory of Vibrational Holographic Interferometry

Developed in our laboratory several years ago, vi- brational holographic interferometry is a useful tech- nique for studying residual stresses in polymeric films. The detailed explanation for the membrane theory of vibrational holographic interferometry is reported else- where (2, 5, 6, 9). In this paper, equations for the resid- ual stress are introduced.

For an anisotropic film with known orthotropic axes, the two principal stresses under a 2-dimension constraint can be expressed as Eq 3.

a:? - m2 + cr;? n2 = 4pL2 [fmnl2 (3)

where, u2D 11, u22 2 D : biaxial stresses in the two principal

directions [MPal p : material density (kg/m3) L : membrane length (m)

f,, : resonant frequency for (m,nl mode [Hz)

For the same sample under a 1-D constraint the stress from vibrational holography is given by,

a:? = 4pL2 [ Jy i ] " (4)

where,

cr ;7 : uniaxial stress in the 1 principal direction (MPa)

L : length of the ribbon (m) fi : resonant frequency for ith node (Hz] i : nodeofinflection

The relationship between elongation and biaxial stress can be expressed as Eq 5.

(5) c11 ClZ E 2 D -

[ E l i ] [ c21 c2,1 [ And Eq 5 can be expressed as Eq 6 and 7.

E;? = c11 x US: + cl2 x IT;: =

(l/J%l) x a?? - (VlZ/ELI) x &? (61

If the sample is cut in the 2 direction,

POLYMER ENGINEERING AND SCIENCE, FEBRUARY2001, V d . 41, NO. 2 303

Seo Hyun Cho, Gene Kim, Thomas J. McCarthy, and Richard J. Fani s

Using Eqs 6 to 9, the equations for the two in-plane Poisson's ratio can be expressed as Eq 10.

(of? - o 3 0;:

u12 =

RESULTS AND DISCUSSION

Principal Directions of Polyhide Film

Figure 2 shows a vibration pattern for a polyimide film. Two symmetric axes represent the two principal axes. For this polyimide film, the two principal axes coincide with the machine and transverse directions.

Poisson's Ratio (vl2. v2,)

Using vibrational holographic interferometry, the 2-D stresses and the 1-D stresses in the 1 and 2 prin- cipal directions were measured. Firstly, the 2-D stress was measured and then the same sample was cut in the principal direction and the 1-D stress was meas- ured. Table 1 represents the 2-D stresses and the 1-D stresses in the 1 and 2 principal directions. The two in-plane Poisson's ratios, u12 and uzl. calculated from Eqs 10, are 0.53 and 0.42, respectively.

In-Plane Young's Moduli (El,, E22) and In-Plane Shear Modulus (G,,)

Standard tensile testing experiments were carried out to determine the in-plane Young's moduli (Ell, E,,) and the in-plane shear modulus (G1& The two in-plane tensile moduli, Ell and E,,, are determined from stress-strain curves for these samples.

The in-plane shear modulus, G,,, is obtained by measuring the Young's modulus at some angle, 8, to the principal axes in the plane of the film. The in- plane modulus is related to the Young's modulus (E,) atanangleOasshowninEq11 (10).

Uniaxial samples are cut at a 45" angle to the princi- pal axes and Young's modulus, E,, is determined. Since Ell, E,, and ul, have already been determined, the in- plane shear modulus (G12) is readily calculated. The two in-plane tensile moduli (El 1, q,) and the in-plane shear moduli (G12) of polyimide film are shown in Table 2.

F g 2. One of the vibration patterns for polyimideflm ((2, 1 ) mode).

(12)

The-out-of-plane Poisson's ratios (q3, ~ 2 3 ) are deter-

(ao/aP),,, = 1 - U], - u13

(aa/aP),z, = 1 - U,] - u23

mined using Eq 12; u I 3 is 0.21 and u23 is 0.19.

In-Plane CTEs

In-plane coefficients of thermal expansion (CTEs) were measured using a thermomechanical analyzer (TMA). The in-plane CTEs of polyimide film are shown in Table 2. The polyimide film processing is intended to produce a planar film. The in-plane CTEs, a1 and a, (see Table 2) were equal.

Out-of-Plane CTE (ag)

A Pressure-Volume-Temperature (PVT) apparatus was used to evaluate the out-of-plane CTE (a3). This constant is one of the most difficult to determine for

Table 1. The 2-0 Stresses and the 1-D Stresses in the 1 and 2 Principal Directions.

~

2-D stress (MPa) 1-D stress (MPa)

1 principal direction 2.26 (u:?) 1.30 (a:?) 2 principal direction 1.81 (u$$) -

1 principal direction 2.83 (u:?) - 2 principal direction 2.33 (uzf) 1.14 (a:!)

Table 2. Physical Properties of Polyimide Film.

Out-of-Plane Poisson's Ratios (v13. ~ 2 3 )

A high-pressure gas dilatometer was used to deter- mine the out-of-plane Poisson's ratios (u13, ~ 2 3 ) . FSg- wes 3 shows high pressure gas dilatomem of poly- imide film in the 2 principal directions. The stress in the polyimide film increases linearly as the applied pressure increased. The relationship between the out- of-plane Poisson's ratios and the stress change with the applied pressure is,

~ ~ ~~

Ell 6.31 i- 0.15GPa E,, 5.71 2 0.13GPa E,, 5.94 t 0.13 GPa GI, 1.97 2 0.04 GPa a1 20.0 ppmPC 20.0 ppm/"C aY 163.2 ppm/"C a, 123.2 pprn/"C K 0.124 GPa-I E,, 10.90 2 0.08 GPa

1) E,,, E, : in-plane tensile moduli. 2) E,, : in-plane tensile moduli at 45" angle to the principal axes. 3) G,, : in-plane shear moduli. 4) a,, a2 : in-plane CTEs. 5) al, : volumetric CTE; a3 : out-of-plane CTE. 6) K : bulk compressibility; E,, : out-of-plane Young's modulus.

304 POLYMER ENGINEERING AND SCIENCE, FEBRUARY 2001, Vol. 41, No. 2

Orthotropic Elastic Constants for Polyimide Film

Fig. 3. High pressure gas dilatometry on polyimideflm. *: in the first princi- pal direction (slope: 0.26, R2 = 99.9); A: in the second principal direction (slope: 0.39. R2 = 99.9).

thin films as conventional methods can not be used to evaluate it. Hence, an indirect approach has been developed to determine the constant using a PVT ap- paratus. The sum of the linear CTEs along the 3 or- thotropic axes is defined as the volumetric CTE (a,) and is written as

@V/aT), could be measured from an isobaric PVT run and V, is the initial volume of the sample (0.8048 cm3). With Eq 13, a3 could be evaluated.

Figure 4 shows an isobaric PVT run of polyimide film. The volumetric CTE at the constant pressure could be evaluated from the slope of the data. The vol- umetric CTE at atmosphere pressure was estimated by extrapolation of the 4 data points.

The volumetric CTE (a,) and the out-of-plane CTE (a3) of polyimide film are shown in Table 2. The value of a3, 123.2 ppm/"C, is much higher than that of a1 and a2.

Out-of-Plane Young's Modulus (Es) For thin films, Young's modulus in the out-of-plane

direction (i.e. along the thickness direction) is prac- tically impossible to evaluate using conventional methods. A method has been developed to determine this quantity indirectly by finding the bulk compress- ibility of the film using the PVT apparatus. The bulk compressibility (K) is calculated by carrying out an isothermal test on a sample in the PVT apparatus. For

12 ! I I I

0 2 4 6 8

Pressure (MPa)

an isothermal test (AT = 0). the dilatation of the sam- ple is measured with varying pressure. Under isother- mal conditions,

K = - (l/vo)(av/aP), (15) The relation between out-of-plane Young's modulus

and compressibility (2) is,

VE33 =

- [ l /El l + 1/E22 - 2(u12/E11 + u13/Ell + v23/E22)

(16)

Since the elastic constants (Ell, E22, uI2. ~ 1 3 . and u2) were already determined, the-out-of-plane Young's modulus can be indirectly determined using Eq 16.

Figure 5 shows the isothermal PVT runs of polyimide film. The specific volume changes of the polyimide film with pressure were almost constant regardless of tem- perature. The bulk compressibility and the out-of-plane Young's modulus (E33) of polyimide film are shown in Table 2. The value of E,, , 10.90 GPa, is slightly higher than that of Ell and E22.

Out-of-Plane Shear Moduli (Gsl. Gs2)

A Torsion Pendulum can be used to measure the two out-of-plane shear moduli (G31. G32). The relationship between the period of oscillation and the out-of-plane shear modulus can be expressed as follows (2, 11):

P-~=--- 8a3bG32 1 1 AL m = l 2 m ~ ( l - - t a n h Q , Qm

abm Q m = z G Z G

POLYMER ENGINEERING AND SCIENCE, FEBRUARY2001, Vol. 41, No. 2

Seo Hyun Cho, Gene Kim, Thomas J. McCarthy, and Richard J. Fanis

Det

0.75 I

cll c12 c13

c12 G Z c23 > (23) c13 c3Z G3

0.74 n

P E

E 2 0.73

m

0

W v

0 > 0 c .-

8

/ 0.74 -/ n

P E

E 2 0.73

m

0

W v

0 > 0 c .-

8 a v)

0.72 1 0

0 0

: lOMPa (s = 1.3453e'4, r2= 99.4)

: 30MPa (s = 1. 108e-5, r2 = 99.6)

: SOMPa (s = 1.06e-4, c2= 98.7)

: 70MPa (s = 1.09e-4, r2 = 96.4)

0.71 1 I I I I I I

20 30 40 50 60 70 80 90

Ternperature('C) Fig, 4. Isobaric PVT run for polyirnideflm

where

p : period of oscillation (s) a : width of sample (m) b : thickness of sample (m) L : length of sample (m) I : momentum of inertia of disc (kg-m2)

I t is very difficult to solve Eq 17 directly. The shear moduli are determined by using a program to find the moduli values, which yield a value of p, which is iden- tical to the one measured experimentally.

The shear modulus (G31) of polyimide film is 0.49 1- 0.05 GPa, and the shear modulus (G32) of polyimide film is 0.27 1- 0.05 GPa.

The value of C,, (- 0.0840) should be the same as the value of C,, (- 0.0736) but is slightly different from that of C,, due to experimental error. The average val- ue of C,, and C,, was used as the value of C12 or C,,. The value of C,, (C31) may be different from that of C,, (C,,). For polyimide film, they have almost the same values.

The 9 components of Eq 18 should satisfy Eqs 19, 20, 21, 22, and 23. If not, it violates the energy con- servation rule.

(19)

(20)

(21)

C,, > 0, G2 > 0, andC33 > 0 c,, x c,, - (C12), > 0

c,, x c3, - (C,,), > 0

Orthotropic Elastic Constants of Polyimide Film c22 x c3, - (%,I2 > 0 (22)

All orthotropic elastic constants of polyimide film are determined and expressed as Eq 18. The unit of these constants is 1 /GPa.

0.1585 +- 0.0370 - 0.07882 0.0052 - 0.0333 5 0.0008 0 0 0 -0.07881-0.0052 0.1751Z0.0041 -0.03331-0.0008 0 0 0 - 0.03331-0.0008 - 0.03335 0.0008 0.09 17 -C0.0007 0 0 0

0 0 0 0.507620.0 105 0 0 0 0 0 0 2.04081-0.2319 0 0 0 0 0 0

306 POLYMER ENGINEERING AND SCIENCE, FEBRUARY 2007, Vol. 41, No. 2

0.78

0.76

h

u) m'

v 5 0.74 E 3 0 > 0

0 Q) Q v)

-

E 0.72

0.70

0.68

Orthotropic Elastic Constants for Polyimide Film

0

W

A

: 30°C (slope : -9.45e.') : 70°C (slope : -1.015e-4)

: I 10°C (slope : -1.005e'4) : 150°C (slope : -1.145e-4)

I I I I

0 20 40 60 80 100

Pressure (MPa)

Hg. 5. Isothermal PVT run for polyimidefilrn

The orthotropic constants of this polyimide film sat- is@ all the above equations.

SUMMARY The orthotropic constants of polyimide film have

been characterized using the theory of elasticity of an anisotropic material. These techniques are generally applicable to orthotropic polymeric films.

Vibrational holography interferometry is used to de- termine the orthotropic axes of symmetry. I t is also used to evaluate the 2 in-plane Poisson's ratios by measuring residual stresses in 2-D and 1-D square membranes. Using a commercially available PVT appa- ratus, the bulk compressibility and volumetric thermal expansion of polyimide film were measured. In combi- nation with other techniques such as high pressure gas dilatometry, tensile testing, and thermomechanical analyzer, the out-of-plane Young's modulus and the out-of-plane coefficient of thermal expansion were de- termined. The out-of-plane shear moduli were walu- ated using a torsion pendulum.

ACKNOWLEDGMENT The authors would like to thank Hewlett-Packard

for financial and material support. One of authors (Seo Hyun Cho) thanks the Korea Science and Engi- neering Foundation in Korea for the partial financial support.

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1026 (1988). 2. M. A. Maden, The Determination of Stress and Material

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3. I. M. Ward, Structure and Properties of Oriented Poly- mers, p. 271, Chapman & Hall (1997).

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5. R. M. Jennings, An Investigation of the Eflects of Curing Conditions on the Residual Stress and Dimensional sta- bility in Polyirnide Films. PhD thesis, University of Mass- achusetts, Amherst (1993).

6. Q. K. Tong, A Structure Property Investigation ofa Mufti- Component Potyacrylate Photoresist, PhD thesis, Univer- sity of Massachusetts, Amherst (1993).

7. ASTM D882-88: Standard Test Method for Tensile Prop- erties of Thin Plastic Sheeting, Annual Book of ASTM (1988).

8. P. Zoller, PW Manual Wersion 2.01), Gnomix Inc., Boul- der, Colo. (1987).

9. M. J. Chen, Pressure-Volume-Temperature and Wave Propagation Studies of Polyimide Films. PhD thesis, Uni- versity of Massachusetts, Amherst (1998).

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