# A rheo-optical analysis of converging wedge flow for estimation of stress-optical coefficient.

INTRODUCTION

When a polymer melt is subjected to flow deformation, it becomes double-refracting or birefringent. This optical anisotropy is attributable to the orientation of the polymer macromolecules. Flow orientation can cause molecular orientation leading to anisotropy in transport properties [1-3]. Therefore, understanding flow orientation is important both from a fundamental as well as a practical point of view.

For flexible polymer solutions and melts, the net optical anisotropy due to flow can be obtained by measuring the differences in refractive indices in the direction of the principal stresses. When the direction of propagation of a polarized monochromatic light beam coincides with the direction of one of the principal stresses of an optically anisotropic polymeric fluid flowing through a transparent channel, the difference in the refractive index in orthogonal directions is related to the differences in the corresponding principal stresses via a relationship known as the stress-optical law. In fact, the use of birefringence results to infer the components of the stress tensor in a flowing or deformed polymer depends upon the existence and validity of the stress-optical law. This relation gives a proportionality between the components of the refractive index (polarizability) tensor and the stress tensor. In a wide range of conditions involving not too large stresses, the linear stress-optical law is expressed as

[Delta]N = C[Delta][Sigma] (1)

where C is a material constant known as the stress-optical coefficient, [Delta]N is the birefringence, and [Delta][Sigma] is the corresponding difference of principal stresses. The sign and magnitude of the stress-optical coefficient C depends on the chemical structure of the polymer, which is governed by the polarizability of the bonds between the atoms of the polymer molecule and by the direction of the bonds with respect to the polymer backbone. According to rubber-like photoelastic theory [4] for a network of freely jointed Gaussian chain, the stress-optical coefficient is

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the mean refractive index of the material, [[Alpha].sub.1] - [[Alpha].sub.2] is the principal polarizability difference of the chain-link parallel and transverse to the link, k is the Boltzmann constant, and T is the absolute temperature.

Since the deviatoric components of the stress and polarizability tensors are linearly related, the principal stress orientation axes and optical axes will be identical.

[[Chi].sub.optical] = [[Chi].sub.stress] = [Chi] (3)

Extensive experimental work has been done to extend and confirm stress-optical laws for solids to concentrated polymeric solutions [5-7]. Janeschitz-Kriegl [8] has shown experimentally that, for many polymers, the linearity of the stress-optical law is valid up to a stress of [10.sup.4] Pa in shear and up to [10.sup.6] Pa in extension. It should be noted that the proportionality of stress and refractive index does not imply or require proportionality of stress and strain rate. Extensive research has been done indicating that Eq 1 holds good for polymer melts well into the non-Newtonian region [9]. The only requirement is that both stress and refractive-index tensors are governed by the orientation function. This will be true as long as the flow does not create a change in the magnitude of the chain end-to-end vector.

Most of the rheo-optical studies done so far to evaluate the linear stress-optical coefficient of polymer systems have involved mechanical measurements of stress and optical measurements of birefringence [10-13]. In this paper we illustrate the use of non-intrusive optical techniques (laser Doppler anemometry and flow birefringence) to measure stress and birefringence, and evaluate the stress-optical coefficient of a polydimethylsiloxane (PDMS) melt in a converging wedge flow cell. It is an extension of a previous work [14] where we had shown that the orientation angle can be effectively used as a test of constitutive equations for flexible polymer melts and solutions.

EQUATIONS IN A CONVERGING WEDGE FLOW

Velocity Vector, Rate-of-Strain and Stress Tensors

Assuming a two-dimensional flow in the converging wedge flow cell, the components of the velocity vector in circular coordinates [ILLUSTRATION FOR FIGURE 1 OMITTED! are

[V.sub.r] = [V.sub.r](r, [Theta])

[V.sub.[Theta]] = [V.sub.z] = 0 (4)

The rate-of-strain tensor [Mathematical Expression Omitted] is

[Mathematical Expression Omitted]

Since there is no flow in the z-direction (direction of light propagation), the stress tensor [Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Linear Stress-Optical Relations

For a converging wedge flow, Adams et al. [10] give the following forms of the linear stress-optical law:

2C[[Tau].sub.r[Theta]] = [Delta]N(Sin 2[Chi] Cos 2[Theta] - Cos 2[Chi] Sin 2[Theta]) = [Delta]N Sin(2[Chi] - 2[Theta]) (7)

C([[Tau].sub.rr] - [[Tau].sub.[Theta][Theta]]) = [Delta]N(Cos 2[Chi] Cos 2[Theta] + Sin 2[Chi] Sin 2[Theta]) = [Delta]N Cos(2[Chi] - 2[Theta]) (8)

where the birefringence ([Delta]N) is equal to R[Lambda]/d, d is the thickness of the birefringent medium, [Lambda] is the wavelength of the monochromatic light used and R is the relative retardation of the two plane-polarized components of the monochromatic light emerging from the birefringent medium. Here [Chi] is the angle between the principal molecular orientation axis and the cell centerline and [Theta] is the angular coordinate of the measuring point (intersection of the two split light beams) inside the flow cell [ILLUSTRATION FOR FIGURE 1 OMITTED].

EXPERIMENTAL SETUP

Flow Cell Geometry

Figure 1 shows a schematic of the converging (60 [degrees] inward wedge) flow cell used for this study. The width of the cell is 5.00 cm, and it is capable of withstanding pressures up to 1 MPa. Two such cells were used, one for velocity measurements (unmodified) and the other for orientation angle and birefringence measurements (modified) as shown in Figs. 2 and 3. Extra care was needed in designing windows for the modified cell so that their contribution to the measured birefringence is negligible.

PDMS Fluid Properties

Linear poly(dimethyl siloxane) (PDMS) melts are ideally suited for flow birefringence studies for several reasons. PDMS fluids are very stable thermally, physically, and chemically. They have relatively low surface tension (19-21 x [10.sup.-3] N[m.sup.-1]), minimizing the problem of formulation and entrainment of bubbles which can be a tremendous practical difficulty in flow birefringence measurements. PDMS melts exist in the liquid or molten state at room temperature, thus eliminating the need to maintain high temperatures which often hinders optical measurements [15]. PDMS fluids are highly transparent and the polymer chain monomer unit has a relatively small refractive index anisotropy.

For this study, a Dow Corning Fluid 200 PDMS (obtained from Dow Corning Inc., Mississauga, Ontario) with a nominal kinematic viscosity of 3000 St was used. The pertinent material and viscometric properties at ambient temperature are listed below:

Properties of Dow Corning Fluid 200 Polydimethylsiloxane

Velocity Measurement

The velocity measurements were made using a one-color DANTEC 55X laser Doppler anemometry (LDA) optical system in Differential Doppler mode or Fringe mode with forward scattering. A helium-neon laser beam ([Lambda] = 632.8 nm) is split into two beams and made to converge in a small reference volume (located midpoint between the two windows of the unmodified cell) in the flowing polymer. A diffraction pattern is set up in this volume and the motion of polymer molecules large enough to scatter light produces a Doppler frequency proportional to the velocity of the polymer. The LDA optical train is shown in Fig. 4.

Orientation Angle and Birefringence Measurement

Orientation angle ([Chi]) is measured using a set of cross-polarizers (a half-wave plate and an analyzer) and the modified flow cell [ILLUSTRATION FOR FIGURE 5 OMITTED!. The linearly (vertical) polarized light beam traverses the cell in a direction parallel to the z-axis [ILLUSTRATION FOR FIGURE 1 OMITTED! and the transmitted beam is then analysed to obtain the net molecular orientation. Once the orientation angle measurements are completed, the flow birefringence is measured by adding a quartz Soleil-Babinet compensator as shown in Fig. 5.

Subramanian [16] gives the experimental details for measurement of velocity, molecular orientation, and flow birefringence.

RESULTS AND DISCUSSION

Limits of Validity of Linear Stress-Optical Law for PDMS Fluid 200

When a shearing or extensional strain is applied to a polymer fluid, the network chains develop a preferred orientation, and at high strain rates, these can be stretching of the macromolecular chains. As a first approximation, a simple proportionality exists between stresses (entropic) and birefringence (orientational) as given by the linear stress-optical law. In the limits of small macroscopic deformation, the root mean square of the end-to-end distance of a Gaussian coil is rather small compared with the length of the stretched molecule and with unbounded increase in strain, both orientation and birefringence approach saturation. There are two limits to the validity of the linear stress-optical law:

1) An upper bound for stress.

2) Time scale for establishment of the internal thermodynamic equilibrium.

We assumed that internal equilibrium is established quickly. In order to determine the upper bound of stress, it is useful to make measurements in purely elongational flows. For the converging wedge flow cell, measurements in purely elongational flows were done along the cell centerline ([Theta] = 0 [degrees]) at different pressure drops across the flow cell, and it was shown in our previous study [14] that the linearity of the stress-optical law for PDMS Fluid 200 is valid up to a pressure drop of 689 kPa across the flow cell.

Shear and Extension Rates

The shear and extension rates for PDMS flowing through the converging wedge flow cell were obtained by differentiating the local radial velocities ([V.sub.r]) measured using LDA. Figs. 6 and 7 show the shear rate [(1/r)([Delta][V.sub.r]/[Delta][Theta])] profiles for radial positions (r) from 2.5 to 0.9 cm at off-center angles ([Theta]) of 5, 10, 15, and 20 degrees for pressure drops of 276 and 483 kPa across the flow cell. Note that there is no shear along the cell centerline since the flow is purely extensional (i.e., irrotational). Figs. 8 and 9 show the extension rate [[Delta][V.sub.r]/[Delta]r] profiles for radial positions from 2.5 to 0.9 cm at off-center angles of 0, 5, 10, 15, and 20 degrees for pressure drops of 276 and 483 kPa across the flow cell. From these plots it can be seen that at any radial position, as that angular position ([Theta]) increases, the shear rate increases but the extension rate decreases.

Linear Stress-Optical Coefficient of PDMS

In our previous work [14] we have shown that the orientation angle ([Chi]) can be used as a test of constitutive equations. For the PDMS Fluid 200, the two-term version of the Goddard-Miller model [16, 17] with a single relaxation time ([[Lambda].sub.o]) of 0.009 sec and a zero-shear rate viscosity ([[Eta].sub.o]) of 300 Pa.s [18] fitted the experimental orientation angle data quite accurately. We use this information along with experimental measurements of flow birefringence, orientation angle, and local radial velocity and strain rate computations to estimate the linear stress-optical coefficient C of PDMS in a converging wedge flow cell at room temperature.

From Eq 7 one can observe that in order to evaluate the linear stress-optical coefficient for a birefringent polymeric fluid in a converging wedge flow cell, one must plot [Delta]N vs. 2[[Tau].sub.r[Theta]]/sin[2[Chi] - 2[Theta]]. This plot should be straight line and the slope will be equal to the stress-optical coefficient C. Flow birefringence ([Delta]N) and orientation angle ([Chi]) were obtained experimentally at an off-center angle ([Theta]) of 20 [degrees] for pressure drops of 276 and 483 kPa across the flow cell [ILLUSTRATION FOR FIGURES 10 AND 11 OMITTED]. From Fig. 11 it can be seen that, at [Theta] = 20 [degrees], the orientation angles at different radial position are nearly equal (40-44 [degrees]), indicating an approximately constant relation between the effect of extensional and shear strain.

The shear stress ([[Tau].sub.r[Theta]]) was obtained using the two-term Goddard-Miller model, which, for a converging wedge flow, is written as [16]

[Mathematical Expression Omitted]

The shear rate [(1/r)([Delta][V.sub.r]/[Delta][Theta])], extension rate [[Delta][V.sub.r]/[Delta]r], and velocity ([V.sub.r]) terms in Eq 9 were obtained from the LDA experiment, the zero-shear rate viscosity ([[Eta].sub.o]) from viscometric measurements [18], and the relaxation time ([[Lambda].sub.o]) from the orientation angle experiment [14]. Using the above information, the shear stress for flow of PDMS Fluid 200 at an off-center angle of 20 degrees for pressure drops of 276 and 483 kPa across the converging flow cell was evaluated [ILLUSTRATION FOR FIGURE 12 OMITTED!.

For this study, the stress-optical plot ([Delta]N vs. 2[[Theta].sub.r[Theta]]/sin[2[Chi] - 2[Theta]] for PDMS is shown in Fig. 13 and it was found to be linear with a slope of 1.475 x [10.sup.-10] [Pa.sup.-1] which is the linear stress-optical coefficient C at room temperature. This compares well with values given by Wales [11] of 1.35 x [10.sup.-10] [Pa.sup.-1], Liberman et al. [12] of 0.909 x [10.sup.-10] [Pa.sup.-1], and Galante and Frattini [13] of 1.56 x [10.sup.-10] [Pa.sup.-1] for a low molecular weight PDMS and 1.84 x [10.sup.-10] [Pa.sup.-1] for a high molecular weight PDMS. The positive sign of the stress-optical coefficient confirms that the polarizability of the backbone of PDMS chain is indeed highest in the chain direction. Note that Fig. 13 shows a zero offset for birefringence at zero shear stress, which is probably a boundary layer effect due to end walls of the converging wedge flow cell.

CONCLUSIONS

We have shown rheo-optical analysis using flow-induced birefringence and LDA measurements of a birefringent polymeric fluid in a converging wedge flow cell permits an estimation of the linear stress-optical coefficient via rheological constitutive equations. The key to such an analysis is the test of constitutive equations to predict the measured orientation angles. Since stress can be evaluated from the constitutive equations that best describe the rheo-optical behavior of the polymer, there is no need to make mechanical measurement of stress, which has been the case in most previous rheo-optical studies. It was demonstrated by using a two-term version of the Goddard-Miller model to compute the stress and evaluate the linear stress-optical coefficient of a PDMS fluid at room temperature in a converging wedge flow cell.

NOMENCLATURE

C = Linear stress-optical coefficient.

d = Thickness of the birefringent medium or width of the converging wedge flow cell.

k = Boltzmann constant.

[M.sub.w] = Weight averaged molecular weight.

[Mathematical Expression Omitted] = Mean refractive index of the birefringent material.

r = Radial position in the converging wedge flow cell.

R = Relative retardation of the two plane-polarized components of emergent monochromatic light.

T = Absolute temperature.

[V.sub.r], [V.sub.[Theta]], [V.sub.z] = Velocity vector components in circular coordinates.

[[Alpha].sub.1], [[Alpha].sub.2] = Polarizability of the macromolecular chain link parallel and transverse to the link.

[Chi] = Orientation angle or angle between the principal molecular orientation axis and the converging wedge flow cell centerline (reference direction).

1/r [Delta][V.sub.r]/[Delta][Theta] = Shear rate.

[Delta][V.sub.r]/[Delta]r = Extension rate.

[Delta]N = Birefringence of the optically anisotropic material.

[Delta][Sigma] = Difference in principal stresses.

[[Eta].sub.o] = Zero-shear rate viscosity.

[[Lambda].sub.o] = Relaxation time constant.

[Lambda] = Wavelength of the monochromatic light beam.

[Theta] = Off-center angle or angular position of the measuring point inside the converging wedge flow cell.

[[Tau].sub.r[Theta]], [[Tau].sub.rr], [[Tau].sub.[Theta][Theta]] = Stress tensor components in circular coordinates.

REFERENCES

1. A. A. Cocci and J. J. C. Picot, Polym. Eng. Sci., 13, 337 (1973).

2. J. J. C. Picot and F. Debeauvais, Polym. Eng. Sci., 15, 373 (1975).

3. F. H. Gortemaker, PhD thesis, Technische Hogeschool, Delft, The Netherlands (1976).

4. P. J. Flory, Statistical Mechanics of Chain Molecules, Interscience, New York (1969).

5. W. Philippoff, J. Appl. Phys., 27, 984 (1956).

6. W. Philippoff, Trans. Soc. Rheol., 1, 95 (1957).

7. W. Philippoff, Trans. Soc. Rheol., 5, 163 (1961).

8. H. Janeschitz-Kriegl, Adv. Polym. Sci., 6, 170 (1969).

9. H. Janeschitz-Kriegl, Polymer Melt Rheology and Flow Birefringence, Springer-Verlag, New York (1983).

10. E. B. Adams, J. C. Whitehead, and D. C. Bogue, A.I.Ch.E.J., 11, 1026 (1965).

11. J. L. S. Wales, The Application of Flow Birefringence to Rheological Studies of Polymer Melts, Delft University Press, The Netherlands (1976).

12. M. H. Liberman, Y. Abe, and P. J. Flory, Macromolecules, 5, 550 (1972).

13. S. R. Galante and P. L. Frattini, J. Non-Newtonian Fluid Mech., 47, 289 (1993).

14. R. Subramanian and J. J. C. Picot, J. Non-Newtonian Fluid Mech., 53, 113 (1994).

15. J. M. Piau, N. El Kissi, and B. Tremblay, J. Non-Newtonian Fluid Mech., 30, 197 (1988).

16. R. Subramanian, PhD Thesis, University of New Brunswick, Fredericton, New Brunswick, Canada (1994). Microfilm available from National Library of Canada, Canadian Thesis Service, 395 Wellington Street, Ottawa, Ontario K1A ON4, Canada.

17. J. D. Goddard and C. Miller, Rheol. Acta, 5, 177 (1966).

18. P.-A. Lavoie, Ecole Polytechnique, Montreal, Quebec, Canada, personal communication (March 1990).

When a polymer melt is subjected to flow deformation, it becomes double-refracting or birefringent. This optical anisotropy is attributable to the orientation of the polymer macromolecules. Flow orientation can cause molecular orientation leading to anisotropy in transport properties [1-3]. Therefore, understanding flow orientation is important both from a fundamental as well as a practical point of view.

For flexible polymer solutions and melts, the net optical anisotropy due to flow can be obtained by measuring the differences in refractive indices in the direction of the principal stresses. When the direction of propagation of a polarized monochromatic light beam coincides with the direction of one of the principal stresses of an optically anisotropic polymeric fluid flowing through a transparent channel, the difference in the refractive index in orthogonal directions is related to the differences in the corresponding principal stresses via a relationship known as the stress-optical law. In fact, the use of birefringence results to infer the components of the stress tensor in a flowing or deformed polymer depends upon the existence and validity of the stress-optical law. This relation gives a proportionality between the components of the refractive index (polarizability) tensor and the stress tensor. In a wide range of conditions involving not too large stresses, the linear stress-optical law is expressed as

[Delta]N = C[Delta][Sigma] (1)

where C is a material constant known as the stress-optical coefficient, [Delta]N is the birefringence, and [Delta][Sigma] is the corresponding difference of principal stresses. The sign and magnitude of the stress-optical coefficient C depends on the chemical structure of the polymer, which is governed by the polarizability of the bonds between the atoms of the polymer molecule and by the direction of the bonds with respect to the polymer backbone. According to rubber-like photoelastic theory [4] for a network of freely jointed Gaussian chain, the stress-optical coefficient is

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the mean refractive index of the material, [[Alpha].sub.1] - [[Alpha].sub.2] is the principal polarizability difference of the chain-link parallel and transverse to the link, k is the Boltzmann constant, and T is the absolute temperature.

Since the deviatoric components of the stress and polarizability tensors are linearly related, the principal stress orientation axes and optical axes will be identical.

[[Chi].sub.optical] = [[Chi].sub.stress] = [Chi] (3)

Extensive experimental work has been done to extend and confirm stress-optical laws for solids to concentrated polymeric solutions [5-7]. Janeschitz-Kriegl [8] has shown experimentally that, for many polymers, the linearity of the stress-optical law is valid up to a stress of [10.sup.4] Pa in shear and up to [10.sup.6] Pa in extension. It should be noted that the proportionality of stress and refractive index does not imply or require proportionality of stress and strain rate. Extensive research has been done indicating that Eq 1 holds good for polymer melts well into the non-Newtonian region [9]. The only requirement is that both stress and refractive-index tensors are governed by the orientation function. This will be true as long as the flow does not create a change in the magnitude of the chain end-to-end vector.

Most of the rheo-optical studies done so far to evaluate the linear stress-optical coefficient of polymer systems have involved mechanical measurements of stress and optical measurements of birefringence [10-13]. In this paper we illustrate the use of non-intrusive optical techniques (laser Doppler anemometry and flow birefringence) to measure stress and birefringence, and evaluate the stress-optical coefficient of a polydimethylsiloxane (PDMS) melt in a converging wedge flow cell. It is an extension of a previous work [14] where we had shown that the orientation angle can be effectively used as a test of constitutive equations for flexible polymer melts and solutions.

EQUATIONS IN A CONVERGING WEDGE FLOW

Velocity Vector, Rate-of-Strain and Stress Tensors

Assuming a two-dimensional flow in the converging wedge flow cell, the components of the velocity vector in circular coordinates [ILLUSTRATION FOR FIGURE 1 OMITTED! are

[V.sub.r] = [V.sub.r](r, [Theta])

[V.sub.[Theta]] = [V.sub.z] = 0 (4)

The rate-of-strain tensor [Mathematical Expression Omitted] is

[Mathematical Expression Omitted]

Since there is no flow in the z-direction (direction of light propagation), the stress tensor [Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Linear Stress-Optical Relations

For a converging wedge flow, Adams et al. [10] give the following forms of the linear stress-optical law:

2C[[Tau].sub.r[Theta]] = [Delta]N(Sin 2[Chi] Cos 2[Theta] - Cos 2[Chi] Sin 2[Theta]) = [Delta]N Sin(2[Chi] - 2[Theta]) (7)

C([[Tau].sub.rr] - [[Tau].sub.[Theta][Theta]]) = [Delta]N(Cos 2[Chi] Cos 2[Theta] + Sin 2[Chi] Sin 2[Theta]) = [Delta]N Cos(2[Chi] - 2[Theta]) (8)

where the birefringence ([Delta]N) is equal to R[Lambda]/d, d is the thickness of the birefringent medium, [Lambda] is the wavelength of the monochromatic light used and R is the relative retardation of the two plane-polarized components of the monochromatic light emerging from the birefringent medium. Here [Chi] is the angle between the principal molecular orientation axis and the cell centerline and [Theta] is the angular coordinate of the measuring point (intersection of the two split light beams) inside the flow cell [ILLUSTRATION FOR FIGURE 1 OMITTED].

EXPERIMENTAL SETUP

Flow Cell Geometry

Figure 1 shows a schematic of the converging (60 [degrees] inward wedge) flow cell used for this study. The width of the cell is 5.00 cm, and it is capable of withstanding pressures up to 1 MPa. Two such cells were used, one for velocity measurements (unmodified) and the other for orientation angle and birefringence measurements (modified) as shown in Figs. 2 and 3. Extra care was needed in designing windows for the modified cell so that their contribution to the measured birefringence is negligible.

PDMS Fluid Properties

Linear poly(dimethyl siloxane) (PDMS) melts are ideally suited for flow birefringence studies for several reasons. PDMS fluids are very stable thermally, physically, and chemically. They have relatively low surface tension (19-21 x [10.sup.-3] N[m.sup.-1]), minimizing the problem of formulation and entrainment of bubbles which can be a tremendous practical difficulty in flow birefringence measurements. PDMS melts exist in the liquid or molten state at room temperature, thus eliminating the need to maintain high temperatures which often hinders optical measurements [15]. PDMS fluids are highly transparent and the polymer chain monomer unit has a relatively small refractive index anisotropy.

For this study, a Dow Corning Fluid 200 PDMS (obtained from Dow Corning Inc., Mississauga, Ontario) with a nominal kinematic viscosity of 3000 St was used. The pertinent material and viscometric properties at ambient temperature are listed below:

Properties of Dow Corning Fluid 200 Polydimethylsiloxane

Characteristics Value [M.sub.w] 232,700 Polydispersity Ratio 3.43 [[Eta].sub.o] (Pa.s) 300

Velocity Measurement

The velocity measurements were made using a one-color DANTEC 55X laser Doppler anemometry (LDA) optical system in Differential Doppler mode or Fringe mode with forward scattering. A helium-neon laser beam ([Lambda] = 632.8 nm) is split into two beams and made to converge in a small reference volume (located midpoint between the two windows of the unmodified cell) in the flowing polymer. A diffraction pattern is set up in this volume and the motion of polymer molecules large enough to scatter light produces a Doppler frequency proportional to the velocity of the polymer. The LDA optical train is shown in Fig. 4.

Orientation Angle and Birefringence Measurement

Orientation angle ([Chi]) is measured using a set of cross-polarizers (a half-wave plate and an analyzer) and the modified flow cell [ILLUSTRATION FOR FIGURE 5 OMITTED!. The linearly (vertical) polarized light beam traverses the cell in a direction parallel to the z-axis [ILLUSTRATION FOR FIGURE 1 OMITTED! and the transmitted beam is then analysed to obtain the net molecular orientation. Once the orientation angle measurements are completed, the flow birefringence is measured by adding a quartz Soleil-Babinet compensator as shown in Fig. 5.

Subramanian [16] gives the experimental details for measurement of velocity, molecular orientation, and flow birefringence.

RESULTS AND DISCUSSION

Limits of Validity of Linear Stress-Optical Law for PDMS Fluid 200

When a shearing or extensional strain is applied to a polymer fluid, the network chains develop a preferred orientation, and at high strain rates, these can be stretching of the macromolecular chains. As a first approximation, a simple proportionality exists between stresses (entropic) and birefringence (orientational) as given by the linear stress-optical law. In the limits of small macroscopic deformation, the root mean square of the end-to-end distance of a Gaussian coil is rather small compared with the length of the stretched molecule and with unbounded increase in strain, both orientation and birefringence approach saturation. There are two limits to the validity of the linear stress-optical law:

1) An upper bound for stress.

2) Time scale for establishment of the internal thermodynamic equilibrium.

We assumed that internal equilibrium is established quickly. In order to determine the upper bound of stress, it is useful to make measurements in purely elongational flows. For the converging wedge flow cell, measurements in purely elongational flows were done along the cell centerline ([Theta] = 0 [degrees]) at different pressure drops across the flow cell, and it was shown in our previous study [14] that the linearity of the stress-optical law for PDMS Fluid 200 is valid up to a pressure drop of 689 kPa across the flow cell.

Shear and Extension Rates

The shear and extension rates for PDMS flowing through the converging wedge flow cell were obtained by differentiating the local radial velocities ([V.sub.r]) measured using LDA. Figs. 6 and 7 show the shear rate [(1/r)([Delta][V.sub.r]/[Delta][Theta])] profiles for radial positions (r) from 2.5 to 0.9 cm at off-center angles ([Theta]) of 5, 10, 15, and 20 degrees for pressure drops of 276 and 483 kPa across the flow cell. Note that there is no shear along the cell centerline since the flow is purely extensional (i.e., irrotational). Figs. 8 and 9 show the extension rate [[Delta][V.sub.r]/[Delta]r] profiles for radial positions from 2.5 to 0.9 cm at off-center angles of 0, 5, 10, 15, and 20 degrees for pressure drops of 276 and 483 kPa across the flow cell. From these plots it can be seen that at any radial position, as that angular position ([Theta]) increases, the shear rate increases but the extension rate decreases.

Linear Stress-Optical Coefficient of PDMS

In our previous work [14] we have shown that the orientation angle ([Chi]) can be used as a test of constitutive equations. For the PDMS Fluid 200, the two-term version of the Goddard-Miller model [16, 17] with a single relaxation time ([[Lambda].sub.o]) of 0.009 sec and a zero-shear rate viscosity ([[Eta].sub.o]) of 300 Pa.s [18] fitted the experimental orientation angle data quite accurately. We use this information along with experimental measurements of flow birefringence, orientation angle, and local radial velocity and strain rate computations to estimate the linear stress-optical coefficient C of PDMS in a converging wedge flow cell at room temperature.

From Eq 7 one can observe that in order to evaluate the linear stress-optical coefficient for a birefringent polymeric fluid in a converging wedge flow cell, one must plot [Delta]N vs. 2[[Tau].sub.r[Theta]]/sin[2[Chi] - 2[Theta]]. This plot should be straight line and the slope will be equal to the stress-optical coefficient C. Flow birefringence ([Delta]N) and orientation angle ([Chi]) were obtained experimentally at an off-center angle ([Theta]) of 20 [degrees] for pressure drops of 276 and 483 kPa across the flow cell [ILLUSTRATION FOR FIGURES 10 AND 11 OMITTED]. From Fig. 11 it can be seen that, at [Theta] = 20 [degrees], the orientation angles at different radial position are nearly equal (40-44 [degrees]), indicating an approximately constant relation between the effect of extensional and shear strain.

The shear stress ([[Tau].sub.r[Theta]]) was obtained using the two-term Goddard-Miller model, which, for a converging wedge flow, is written as [16]

[Mathematical Expression Omitted]

The shear rate [(1/r)([Delta][V.sub.r]/[Delta][Theta])], extension rate [[Delta][V.sub.r]/[Delta]r], and velocity ([V.sub.r]) terms in Eq 9 were obtained from the LDA experiment, the zero-shear rate viscosity ([[Eta].sub.o]) from viscometric measurements [18], and the relaxation time ([[Lambda].sub.o]) from the orientation angle experiment [14]. Using the above information, the shear stress for flow of PDMS Fluid 200 at an off-center angle of 20 degrees for pressure drops of 276 and 483 kPa across the converging flow cell was evaluated [ILLUSTRATION FOR FIGURE 12 OMITTED!.

For this study, the stress-optical plot ([Delta]N vs. 2[[Theta].sub.r[Theta]]/sin[2[Chi] - 2[Theta]] for PDMS is shown in Fig. 13 and it was found to be linear with a slope of 1.475 x [10.sup.-10] [Pa.sup.-1] which is the linear stress-optical coefficient C at room temperature. This compares well with values given by Wales [11] of 1.35 x [10.sup.-10] [Pa.sup.-1], Liberman et al. [12] of 0.909 x [10.sup.-10] [Pa.sup.-1], and Galante and Frattini [13] of 1.56 x [10.sup.-10] [Pa.sup.-1] for a low molecular weight PDMS and 1.84 x [10.sup.-10] [Pa.sup.-1] for a high molecular weight PDMS. The positive sign of the stress-optical coefficient confirms that the polarizability of the backbone of PDMS chain is indeed highest in the chain direction. Note that Fig. 13 shows a zero offset for birefringence at zero shear stress, which is probably a boundary layer effect due to end walls of the converging wedge flow cell.

CONCLUSIONS

We have shown rheo-optical analysis using flow-induced birefringence and LDA measurements of a birefringent polymeric fluid in a converging wedge flow cell permits an estimation of the linear stress-optical coefficient via rheological constitutive equations. The key to such an analysis is the test of constitutive equations to predict the measured orientation angles. Since stress can be evaluated from the constitutive equations that best describe the rheo-optical behavior of the polymer, there is no need to make mechanical measurement of stress, which has been the case in most previous rheo-optical studies. It was demonstrated by using a two-term version of the Goddard-Miller model to compute the stress and evaluate the linear stress-optical coefficient of a PDMS fluid at room temperature in a converging wedge flow cell.

NOMENCLATURE

C = Linear stress-optical coefficient.

d = Thickness of the birefringent medium or width of the converging wedge flow cell.

k = Boltzmann constant.

[M.sub.w] = Weight averaged molecular weight.

[Mathematical Expression Omitted] = Mean refractive index of the birefringent material.

r = Radial position in the converging wedge flow cell.

R = Relative retardation of the two plane-polarized components of emergent monochromatic light.

T = Absolute temperature.

[V.sub.r], [V.sub.[Theta]], [V.sub.z] = Velocity vector components in circular coordinates.

[[Alpha].sub.1], [[Alpha].sub.2] = Polarizability of the macromolecular chain link parallel and transverse to the link.

[Chi] = Orientation angle or angle between the principal molecular orientation axis and the converging wedge flow cell centerline (reference direction).

1/r [Delta][V.sub.r]/[Delta][Theta] = Shear rate.

[Delta][V.sub.r]/[Delta]r = Extension rate.

[Delta]N = Birefringence of the optically anisotropic material.

[Delta][Sigma] = Difference in principal stresses.

[[Eta].sub.o] = Zero-shear rate viscosity.

[[Lambda].sub.o] = Relaxation time constant.

[Lambda] = Wavelength of the monochromatic light beam.

[Theta] = Off-center angle or angular position of the measuring point inside the converging wedge flow cell.

[[Tau].sub.r[Theta]], [[Tau].sub.rr], [[Tau].sub.[Theta][Theta]] = Stress tensor components in circular coordinates.

REFERENCES

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3. F. H. Gortemaker, PhD thesis, Technische Hogeschool, Delft, The Netherlands (1976).

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12. M. H. Liberman, Y. Abe, and P. J. Flory, Macromolecules, 5, 550 (1972).

13. S. R. Galante and P. L. Frattini, J. Non-Newtonian Fluid Mech., 47, 289 (1993).

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15. J. M. Piau, N. El Kissi, and B. Tremblay, J. Non-Newtonian Fluid Mech., 30, 197 (1988).

16. R. Subramanian, PhD Thesis, University of New Brunswick, Fredericton, New Brunswick, Canada (1994). Microfilm available from National Library of Canada, Canadian Thesis Service, 395 Wellington Street, Ottawa, Ontario K1A ON4, Canada.

17. J. D. Goddard and C. Miller, Rheol. Acta, 5, 177 (1966).

18. P.-A. Lavoie, Ecole Polytechnique, Montreal, Quebec, Canada, personal communication (March 1990).

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Author: | Subramanian, R.; Picot, J.J.C. |
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Publication: | Polymer Engineering and Science |

Date: | May 15, 1996 |

Words: | 2888 |

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