# A new method for estimating the cellular structure of plastic foams based on dielectric anisotropy.

1. INTRODUCTIONThe physical properties of plastic foams are based mainly on the material, expansion ratio and cellular structure. Several investigations (1-3) have been conducted on the cellular structure, which influences strength, modulus of elasticity and dimensional stability. Measurement of the cellular structure is very important from the viewpoint of investigating its physical properties. This cellular structure traditionally has been analyzed using microscopy (4), but this measures too small an area to analyze the whole specimen and measures only the surface of the specimen. Indeed, there are few suitable methods for characterizing morphology except for measuring the compressive strength or the thermal shrinkage after heating at a temperature above the glass transition point. These methods demand much time and labor, so we focused on the dielectric anisotropy based on the shape of cells. It is well known that the shape of cells in the islands-sea structure causes the macroscopic anisotropy of the dielectric constant even if both the islands and the sea are isotropic by themselves. A theory for the permittivity of a double-layer ellipsoid was developed from the viewpoint of electromagnetic theory by Bilboul (5). From this model, equations for the dielectric constants along the major axis and minor axis of ellipsoid in an islands-sea structure such as plastic foams can be derived (6). A microwave cavity resonator method (7) was used to investigate the relation between the dielectric anisotropy based on the shape of cells and the physical properties of plastic foams. This method, which has been developed in our research center, can measure the anisotropy of the dielectric constant as a function of resonance frequency of microwave. In this article, the relationship between the thermal shrinkage and the dielectric anisotropy based on the shape of cells is examined. Second, a better manufacturing condition to minimize the thermal shrinkage for polystyrene foams is described from the viewpoint of extrusion rate, foaming temperature and die gap. Finally, a new approach to the measurement of the shape of cells is performed by using the equations, which show the maximum and minimum dielectric constant as a function of eccentricity of ellipsoid, volume fraction and the dielectric constants of both materials in the islands-sea structure (8).

2. THEORY AND BACKGROUND

2.1 Anisotropy of Dielectric Constant

Figure 1 illustrates the angular dependence of the dielectric constant based on the shape of cells. If the shape of cells is ellipsoidal and each cell is arranged parallel to the extruded direction, the macroscopic dielectric constant has a maximum in the direction of major axis of ellipsoid even if both polymer and cells are dielectrically isotropic by themselves. The maximum dielectric constant [[Epsilon].sub.a] can be expressed by Eq 1, which is derived from the electromagnetic theory (5). The minimum dielectric constant [[Epsilon].sub.b] similarly is given by Eq 2.

[Mathematical Expression Omitted] (1)

[Mathematical Expression Omitted] (2)

Here, [[Epsilon].sub.m] and [[Epsilon].sub.f] are the dielectric constants of the polymer and the cells, respectively. [V.sub.f] is the volume fraction of the cells. [A.sub.a] and [A.sub.b] are parameters shown as Eq 3 and Eq 4, respectively. The parameter of "e" is the eccentricity of ellipsoidal cells shown as Eq 5.

[A.sub.a] = {([e.sup.2] - 1)/2[e.sup.2]) [2 + (1/e) 1n {(1 - e)/(1 + e)}] (3)

[A.sub.b] = (1/2[e.sup.2])[1 + {(1 - [e.sup.2])/2e} 1n {(1 - e)/(1 + e)}] (4)

e = [(1 - [(b/a).sup.2]).sup.1/2] (5)

From above equations, it can be numerically explained that anisotropy of dielectric constant appears in totality, even if both polymer and cells are dielectrically isotropic. Therefore the shape parameter of cells, which is defined as the ratio of major axis to minor axis of ellipsoid, can be computed by using the above equations.

3.2 Principle of Measurement

Figure 2 shows a schematic drawing of the apparatus with a microwave cavity resonator for measuring the anisotropy of dielectric constant of foam materials. This apparatus also can measure the orientation pattern (9) based on the dielectric anisotropy of the sample. The microwaves oscillated by OSC are transmitted through the sample to be detected by DET. The transmitted microwave intensity is transformed into a voltage and is transmitted to a computer after digitization. The orientation pattern shown in Fig. 3 is obtained from the angular dependence of the transmitted microwave intensity while the sample is rotating. The reason why the transmitted microwave intensity changes according to the dielectric constant of the sample in the cavity resonator can be explained by perturbation theory (10), which is shown by Eq 6.

[Mathematical Expression Omitted] (6)

W = 1/2 [integral of] [Epsilon] [[absolute value of [E.sub.a]].sup.2] dv with limit V (7)

Here, [Omega] is the angular eigenfrequency when the sample is inserted in the cavity resonator, [[Omega].sub.a] is the angular eigenfrequency when no sample is inserted in the cavity resonator, [E.sub.a] is the electric field strength of microwave, [H.sub.a] is the magnetic field strength of microwave, W is the total energy stored in the cavity resonator, [Delta]V is the volume of the sample where the microwaves passes, V is the volume of the whole cavity resonator, P is the dielectric polarization of the sample, M is the magnetization and J is the electric current density. This equation means that the resonance frequency shifts according to the dielectric constant in the direction of electric field of the polarized microwave. If the sample has an anisotropy of its dielectric constant, then the resonance curve shifts according to the sample's rotation as shown in Fig. 4. The shift of the resonance curve can be detected as the change of transmitted microwave intensity because the measuring frequency is fixed at "fl" in Fig. 5. "fl" is determined as the frequency at which the transmitted microwave intensity is half of the peak level in the higher side of the resonance curve. By plotting the transmitted microwave intensity on the R-[Theta] coordinates, an orientation pattern shown in Fig. 3 is obtained. The dielectric constant has a maximum in the direction of short axis of this orientation pattern. MOR-c, which is defined as the ratio of the maximum to the minimum of the transmitted microwave intensity as shown in Fig. 6, shows the degree of orientation because the shift of the resonance frequency corresponds to the [Delta][Epsilon][prime] of the sample. The greater the degree of orientation becomes, the greater becomes the MOR-c value. If the sample is entirely isotropic, MOR-c is 1.0. On the other hand, it may be over 500 if the sample has a strong orientation, like a liquid crystalline polymer. Figure 6 shows the relation between the shape of cells and the orientation pattern.

Equation 8 can also be obtained from Eq. 6 by using electromagnetic theory (11).

[Epsilon][prime] = 1 + (A/t) ([[Omega].sub.a] - [Omega]) / [[Omega].sub.a] (8)

Here, [Epsilon][prime] is the dielectric constant of sample, A is a constant based on the cavity resonator, t is the sample thickness, [[Omega].sub.a] is the resonance angular frequency with no sample, and [Omega] is the resonance angular frequency with the sample. The angular dependence of the dielectric constant in the plane of the sample can be obtained from the above equation by using our microwave cavity resonator method because the sample can be rotated in noncontact with the cavity resonator.

3. EXPERIMENTS

Five samples were prepared by slicing the extruded polystyrene foam to examine the orientation for each layer as shown in Fig. 7. Each sample, which is 100 mm long, 100 mm wide and 1.8 mm thick, was measured by Microwave Molecular Orientation Analyzer (MOA-3001A) equipped with the microwave cavity resonator set at 4.0GHz.

Next, fifteen samples of polystyrene foams, which were produced at the various conditions in Table 1, were prepared to examine the relation between the thermal shrinkage and the orientation based on the shape of cells. The extrusion rate was 8.8 to 28.5 m/min, the extruding temperature was 115 [degrees] C to 125 [degrees] C and the die gap was 1.18 to 1.63 mm. Two samples for measurements were sliced out of the center part of an extruded original sample with a thickness of 10 min. One, whose size was 100 mm long, 35 mm wide and 1.8 mm thick, was prepared for the thermal shrinkage measurement. The other, whose size was 35 mm long, 35 mm wide and 1.8 mm thick, was prepared for the orientation measurement. The thermal shrinkages S in the extrusion direction for the fifteen samples were measured after conditioning at a temperature of 70 [degrees] C for 24 h. The thermal shrinkage S is given as follows.

S = ([L.sub.1] - [L.sub.0]) / [L.sub.0] x 100% (9)

Here, [L.sub.0] is the length before thermal treatment and [L.sub.1] is the length after thermal treatment. The orientations of these samples were measured by Microwave Molecular Orientation Analyzer too. Figure 8 shows the results of measurement for Sample A and Sample B. The former is comparatively anisotropic, and the latter is almost isotropic. Figure 9 shows the angular dependence [TABULAR DATA FOR TABLE 1 OMITTED] of the dielectric constants of sample A and B. These data were used to estimate the shape of the cells.

4. RESULTS AND DISCUSSION

4.1 Thermal Shrinkage and Orientation

The MOR-c for each layer of polystyrene foam is plotted against the depth from the surface in Fig. 10. The MOR-c has a minimum value at the middle layer in the thickness range and increases gradually as closing to the surface. This result can be explained by the cells' transformation caused by the shear stress. The closer the cells are to the surface, the bigger the elongation of the cells becomes. This elongation means the ratio of the major axis to the minor axis of the ellipsoidal cell. Table 1 summarizes the results of the measurement of the thermal shrinkage and the orientation degree (= MOR-c) for the fifteen samples which were made under the various production conditions. The thermal shrinkage S is plotted against the MOR-c in Fig. 11. The absolute value of $ increases as the MOR-c increases to -1%, above which it is saturated. There is a strong correlation between them and the correlation coefficient is 0.931. Equation 10 is the linear multiple regression equation, which shows the relation between the MOR-c and the three production conditions. The latter were the extrusion rate, the foaming temperature and the die gap. The MOR-c calculated using Eq 10 is plotted against the measured MOR-c in Fig. 12. The multiple correlation coefficient is 0.941

MOR-c = 0.036 x Speed - 0.016 x Temp. + 0.300 x 1.849 (10)

Here, "Speed" is extrusion rate (m/min), "Temp," is foaming temperature ([degrees]C) and "Die" is die gap (ram). We therefore suggest that the extrusion rate should be small, the foaming temperature should be high and the die gap should be small in order to decrease the MOR-c, in other words, in order to minimize the thermal shrinkage.

4.2 Dielectric Anisotropy based on the Shape of Cells

Even if polymer and cells have no anisotropy of dielectric constant by themselves, total anisotropy appears in the foam materials such as polystyrene foams because of the shape of cells. Uniaxially stretched polystyrene films and PET films were prepared to confirm this assumption. The stretch ratio was up to about 3.0. [Delta][Epsilon](= [[Epsilon].sub.max] - [[Epsilon].sub.min]) of polystyrene films and PET films were measured by MOA-3001A. Each [Delta][Epsilon] is plotted against the stretch ratio in Fig. 13. The polystyrene films remains almost dielectrically isotropic over the whole stretch ratio range 1.0 - 2.5. On the contrary, the [Delta][Epsilon] of the PET films increases up to 0.16 with increasing the stretch ratio up to 3.0. This is due to the negative birefringence (12) caused by the polarization of the side chains and the amorphous state of polystyrene films. Therefore, it can be stated that the MOR-c for the polystyrene foam is based not on the [Delta][Epsilon] of the polymer, but on the shape of cells. Thermal shrinkage doesn't show a zero value at a value of 1.0 of the MOR-c in Fig. 11. The approximated quadratic curve deviates from the expected one. This is attributed to the following. When the shape of cells is transformed to ellipsoidal by the shear stress in the extruder, the membranes of the polystyrene are stretched strongly in the direction of the major axis of cells. In this case, the shape of cells is estimated to be circular at 1.0 of the MOR-c and to be long from side to side in the MOR-c range 0.8 to 1.0, Figure 6 shows the change of shape of cells in the MOR-c range 0.8 to 2.5. It is conjectured that the thermal shrinkage In the extruded direction of polystyrene foams is due to the shrinkage of membranes, which is caused by the stress relaxation at temperatures greater than the glass transition temperature. Therefore, the thermal shrinkage increases with increasing the ratio of major axis to the minor axis of the cell.

4.3 Estimation of the Shape of Cells

If the shape of cells is ellipsoidal, the dielectric constants of foam materials can be calculated according to the Eq 1-5. [[Epsilon].sub.a] and [[Epsilon].sub.b] are plotted against the ratio of major axis to the minor axis of the ellipsoid in Fig. 14. Here, it is assumed that the dielectric constant of polystyrene (= [[Epsilon].sub.m]) is 2.5 and that of the cells (= [[Epsilon].sub.f]) is 1.0. The volume fraction [V.sub.f] is 0.968 because the expansion ratio is 30. [Delta][Epsilon](= [[Epsilon].sub.a] - [[Epsilon].sub.b]) is plotted against the ratio of major axis to minor axis In Fig. 15. As increases up to 0.01 in a sharp curve, above which it saturates gradually. [Delta][Epsilon] of sample A is 0.0075 and that of sample B is 0.0004 according to the results shown in Fig. 9. By using these values, the ratios of major axes to minor axes of cells for both samples can be calculated according to the relation between [Delta][Epsilon] and the ratio of major axis to minor axis shown In Fig. (15). It is found that the ratio of major axis to minor axis of cells for sample A is estimated to be 2.2, and that of B is estimated to be about 1.0. Though these results are the averaged values, the shape of cells for sample B may be almost circular. This result is consistent with the data of the thermal shrinkage mentioned in Section 4.1.

Image analysis was used to confirm this result. A1 and B1 in Fig. 16 show the x 130 enlarged digital photographs taken by a CCD camera with 512 x 475 pixels. A2 and B2 in Fig. 16 show their binary figures for sample A and sample B. Both of them are for the surfaces of the samples, which are 100 mm long, 100 mm wide and 1.8 mm thick. The parts of cells in this binary figure of sample A, which are indicated in white, seem to be ellipsoid and those of sample B seem to be circle on average. It is found that our microwave method based on the dielectric anisotropy is useful for measuring the shape of cells of plastic foams such as polystyrene foams. We will further investigate the numerical confirmation of the accuracy of this method and the applicability for other composite materials that have islands-sea structures.

5. CONCLUSIONS

The following conclusions were derived from the results and discussion.

1. The dimensional stability of plastic foams like polystyrene foam can be estimated from the shape of cells, which is expressed by the macroscopic anisotropy of the dielectric constant.

2. There is a strong correlation between orientation degree (= MOR-c) and thermal shrinkage for polystyrene foams.

3. There is a strong multiple correlation among the three production conditions of polystyrene foams and the orientation degree (= MOR-c). The linear multiple regression equation is shown as follows.

MOR-c = 0.036 x Speed - 0.016 X Temp. + 0.300 x Die + 1.849

4. The shape of cells of polystyrene foams can be characterized by measuring the anisotropy of the dielectric constant.

REFERENCES

1. G. Gioumousis, J. Appl. Polym. Sci., 7, 947 (1963).

2. H. Briscall and C. R. Thomas, British Plastics, July 1968, p.79.

3. G. W. Schael, J. Appl. Polym. Sci., 2131 (1967).

4. R. H. Harding, ASTM Special Technical Publications, 414, 3 (1967).

5. R. R. Bilboul, J. Appl. Phys., Ser. 2, 2, 921 (1969).

6. S. Nagata and K. Koyama, Sen-i Gakkai Preprints, F-112 (1997).

7. S. Nagata and K. Koyama, Journal of JSPP, 9, 11, 897 (1997).

8. Y. Abe, Japanese Patent Application, No. 09-118866 (1997).

9. S. Nagata and K. Koyama, Polym. Proc. So., 13th Annual Meeting. 3-Z2 (1997).

10. F. Okada, Microwave Engineering, Gakkensha, 369 (1993).

11. S. Osaki, J. Appl. Phys., 64, 8, 4181 (1988)

12. Y. Oyagi, Encyclopedia of Plastic Forming, Processing and Recycling, p. 380, Sangyo Chosakai (1997).

Printer friendly Cite/link Email Feedback | |

Author: | Nagata, Shinichi; KOyama, Kiyohito |
---|---|

Publication: | Polymer Engineering and Science |

Date: | May 1, 1999 |

Words: | 2902 |

Previous Article: | Influence of molecular parameters on material processability in extrusion processes. |

Next Article: | In situ formation and processing of ultra high molecular weight polyethylene blends into precursors for high strength and stiffness fiber. |

Topics: |