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Evaluation of paracrystallinity in blown polyethylene films by generalized Clausius-Mosotti equation.


In some material media there are zones of mixed crystalline and amorphous structures. Furthermore, there are intermediate zones with a degree of imperfect crystallographic order, known as "paracrystalline" or "quasi-crystalline." The Theory of Paracrystallinity, introduced in 1950 (1, 2) has been applied successfully to the study of structures of many substances including polymers, biological materials, colloids, catalysts, glasses, metals and melts. One way to characterize these structures is through the mean percentage of lattice paracrystalline distortion, which ranges from 1% in catalysts, 3% in polymers, 6% in graphite, and 12% in glasses to 15% in molten metals, containing microparacrystals (3, 4). Thus, we assume that the degree of paracrystallinity in the direction corresponding to the extraordinary index, [Z.sub.i], is nearly equal to the mean paracrystalline distortion parameter, [Mathematical Expression Omitted], for low-density blown polyethylene films.

We assume this near-equality, because the concept of the paracrystallinity is a distortion of the second kind (5). This paper describes an efficient process for calculating _the mean degree of paracrystallinity (expressed as [Mathematical Expression Omitted]). by using a simple model for the origin of this paracrystalline boundary structure. The calculations use the ordinary and extraordinary bulk refractive indices for samples of low-density blown polyethylene films, which were previously measured beforehand (6). This can be done by applying a generalization of the Clausius-Mosotti equation (7) in each case, and the application of the Bruggeman's equation to finally get the aforementioned total average value, [Mathematical Expression Omitted].


According to Aspnes (8), where [f.sub.a] and [f.sub.b], the volume fractions of phases a and b, respectively, are comparable, [[Epsilon].sub.a] and [[Epsilon].sub.b] are the dielectric functions of phases a and b in their pure forms. If it is not possible to determine whether a or b is the host medium, an alternative approach is to use Wienner's self-consistent limit, denoted by [Epsilon].

Thus, [f.sub.a] [[Epsilon].sub.a] - [Epsilon] / [[Epsilon].sub.a] - 2 [Epsilon] + [f.sub.b] [[Epsilon].sub.b] - [Epsilon] / [[Epsilon].sub.a] - 2[Epsilon] = 0. (1)

This is Bruggeman's expression, commonly known as the effective-medium approximation (EMA) (9). Equation 1 represents the aggregate or random-mixture microstructure, where media a and b are inserted into the effective medium itself.

In generalizing Eq 1, phase a is considered to be amorphous and b crystalline; the 2 in the denominator is transformed to [g.sub.c] for the crystalline phase; [Epsilon] is rewriten as x, and [f.sub.a] as [x.sub.c] (the degree of crystallinity of the five samples studied), and [f.sub.b] as 1 - [x.sub.c]. We thus obtain a special Bruggeman's equation:

[Mathematical Expression Omitted]. (2)

This results in a biquadratic equation, that may be transformed into a simple quadratic one by making the following changes: [x.sup.2] = y, [Mathematical Expression Omitted], [Mathematical Expression Omitted], [g.sub.c] = g, and [x.sub.c] = f, Thus, we are left with

(q - y)(p + gy)(1 - f) + (p - y)(q + 2y)f = 0, (3)

which in turn, results in

[g(1 - f) + 2f][y.sup.2] +

[p(3f- 1) - q(1 + g)f - g]y + pq = 0. (4)

Equation 4, or its equivalent Eq 2, should be applied in the two cases concerning us here: with the ordinary and extraordinary hulk indices, where [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted]. Initially., we take f = 0.5 for both cases, this being the hypothesis considered in our model. The crystalline and the amorphous phases coexist at 50% in the paracrystalline zone. If we solve Eq 4, we find 2 solutions:

1) Bruggeman's mean ordinary index to be:

[x.sub.1] = [-square root of [y.sub.1]] = 1.50313 [+ or -] 0.00001

2) Bruggeman's mean extraordinary index to be:

[x.sub.2] = [-square root of [y.sub.2]] = 1.52018 [+ or -] 0.00001

This is now applied to the degree of crystallinity values in the five standard samples (5). The results are shown in Table 1. Except for the first one, they all have a value near 50%.

These values are consistent with the ones obtained previously (7), and the values shown in the second column of Table 1 are consistent with those in literature (10) for almost fully crystalline polyethylene. This tends to confirm our hypothesis that in the neutral line direction defined by [n.sup.e], the crystalline component is dominant, while in the direction defined by [n.sup.o], the amorphous component is dominant. Therefore, we have just defined the aspects relating to the uniform refractive index, by calculating a mean index (which will later serve in making a valid approximation in the paracrystalline zones). Thus, in the neutral line direction, where the ordinary and the extraordinary indices were measured, the paracrystalline zones behave as homogeneous amorpho-crystalline zones. However, they have been completely averaged out.

The values represented in Table 1 are substituted in the Clausius-Mosotti equation (7). Then, in the case of the ordinary index, we obtain a number of discrete values [Y.sub.B], plotted in function of the degree of crystallinity [x.sub.c]. From these values we carry out linear fitting [f.sub.B], the representative parameters being [m.sub.1] = 1.29 x [10.sup.6], [c.sub.2] = 0.2956 and correlation coefficient [r.sub.1] = 0.9992. [TABULAR DATA FOR TABLE 1 OMITTED] The entire previous explanation is shown in Fig 1. We may proceed similarly in the case of the extraordinary index. In Fig. 2, [Z.sub.B] is shown with its linear fitting [g.sub.B], whose representative parameters are: [m.sub.2] = 4.69 . [10.sup.-6], [c.sub.2] = 0.3039, and [r.sub.2] = 0.9999.


On the basis of previous studies carried out on samples of low-density blown polyethylene (6, 7), in the latter article, we succeeded in generalizing the Clausius-Mosotti equation for this type of polymer, through the directions of influence of the ordinary and extraordinary bulk refractive indices, by means of the general expression (7):

[Mathematical Expression Omitted] (5)

where: a = amorphous, c = crystalline, o = ordinary and e = extraordinary, and

[Mathematical Expression Omitted], (6)

[Mathematical Expression Omitted], (7)

[Mathematical Expression Omitted], (8)

where [N.sub.j] is the number of molecules in phase j and [Mathematical Expression Omitted] is the polarizability of phase j in direction i. Also considered is the previous result of the special Bruggeman's equation, which we assume locally applies to this special zone, there being 50% crystalline phase and 50% amorphous phase.

In principle, our work falls into three parts:

1) Ordinary case: Application to the direction of influence of the ordinary bulk refractive index.

2) Extraordinary case: Application to the direction of influence of the extraordinary bulk refractive index.

3) General case: a summary of the others. Where total averaged results in relation to complete material are obtained.


In this first case, we start from the values obtained for the amorphous and crystalline zones, which are shown in two tables of (7): [Mathematical Expression Omitted] = 1.503259 and [Mathematical Expression Omitted] 1.502998, and also [Mathematical Expression Omitted]. If we further consider that there to be 50% of crystallinity in this very special zone [ILLUSTRATION FOR FIGURE 3 OMITTED], the overall effect exerted by both phases can be calculated by simply applying Eq 5 and Eq 8 together

[Mathematical Expression Omitted], (9)

Now, after solving Eq 9, the value obtained can be applied to the generalized Clausius-Mosotti equation (7), with a mean index found by the Bruggeman's equation for [x.sub.c] = 50%; as indicated previously: [Mathematical Expression Omitted] = 1.50313, this particular area being completely averaged out:

[Mathematical Expression Omitted]. (10)

From here, we obtain [Mathematical Expression Omitted], and from Eq 7: [Mathematical Expression Omitted]. The values of both structural parameters are constant, the degree of crystallinity, therefore, being independent since this is an intermediate material existing in all samples, supposedly having the same characteristics. If we now take the values of [Mathematical Expression Omitted] and [([g.sub.o]).sub.i] given in (6, 7), we can then calculate the overall effect produced by that index, by applying the generalized Clausius-Mosotti equation to the ordinary bulk index direction of influence, designated as [Z.sub.i]

[Mathematical Expression Omitted]. (11)

Having compiled all these data, we then attempt to calculate the percentage of paracrystallinity, [G.sub.i], that is to say the percentage of paracrystallinity portion whose partial effects appear in this preferential direction. In fact it affects the entire sample, but is observed exclusively from this direction, by means of the equation

[Mathematical Expression Omitted]. (12)

In this expression, the degrees of crystallinity of the crystalline and amorphous zones have been modified. This will be explained later and can be seen in Fig 3 and Fig. 4 based on the following reasoning, We assume that area of paracrystallinity is locally composed [ILLUSTRATION FOR FIGURE 3 OMITTED] of 50% the crystalline zone, and 50% the amorphous zone. Hence, in the ordinary case, if [G.sub.i] is the total percentage of this area, will have lost [G.sub.i]/2, and their degree of amorphousness, 1 - [x.sub.c], will have gained in the same amount. Thus, in the foregoing Eq 12 we have ([x.sub.c] - [G.sub.i] / 2)and 1 - ([x.sub.c] - [G.sub.i] / 2) = 1 - [x.sub.c] + [G.sub.] / 2 and for the crystalline and and amorphous percentages respectively [ILLUSTRATION FOR FIGURE 4 OMITTED]. If we look at Fig. 3, the original situation can be seen, where the crystalline and amorphous zones are synthesised graphically in two blocks (abstracted from the more complex reality), which are totally independent, with no intermediate zone or zone of interaction. In Fig. 4 there is such a zone, with a width [G.sub.i], where it is assumed that the original effect started from the medium line. This gives a degree of growth in the amorphous phase; in the ordinary case, in the direction of the ordinary bulk index, with value + [G.sub.i] / 2 having an outwards direction can be seen along with an equal degree of reduction for the crystalline phase, the inward value being - [G.sub.i]/2. In the former case, the overall effect in the direction of [n.sup.o] is positive:

+ [G.sub.i] / 2 - (-[G.sub.i] / 2) = + [G.sub.i].

In the latter case, in the direction of the extraordinary bulk index, the overall effect in the direction of [n.sup.e] is negative: [K.sub.i] / 2 - (+[K.sub.i] / 2) = -[K.sub.i], being [K.sub.i] [equivalent to] [G.sub.i] as designated previously. These are the two values that go with the last term of Eq 12, which we have found for the extraordinary case.

If [Alpha] denotes the first parenthesis in Eq 12, and its value is [Alpha] = 0.2957. [Beta] denotes the second parenthesis in Eq 12, and its value is [Beta] = 0.2956. Thus

[Z.sub.i] = ([x.sub.c] - [G.sub.i] / 2)[Alpha] + (1 - [x.sub.c] + [G.sub.i] / 2)[Beta] + [G.sub.i]F. (13)

Hence by finding [Mathematical Expression Omitted] (in percentage), we obtain the values of the degree of paracrystallinity whose partial effects appear in this preferential direction, shown in the Table 2. One possible explanation [ILLUSTRATION FOR FIGURE 4 OMITTED] for these extremely small percentages is that in this preferential direction ([n.sup.o]), the active crystalline part is very small, and hence there is little contribution from the paracrystalline zone. The block diagram in Fig. 5 clearly shows the system of calculation in this case.

Finally, the values of [G.sub.i] are plotted as function of degree of crystallinity of five samples [x.sub.c] [ILLUSTRATION FOR FIGURE 6 OMITTED] and its linear fitting ([g.sub.1]) with representative values, [m.sub.3] = 0.0018, [c.sub.3] = - 0.0410 and [r.sub.3] = 0.9899.


The procedure in this case is very similar to that of the previous section. We take the values of bulk refractive indices, calculated in (7) for the amorphous and crystalline zones: [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted]. With a 50/o degree of crystallinity, the effect produced by both phases can be calculated, by simply applying the generalized Clausius-Mosotti equation

[Mathematical Expression Omitted], (14)


When Eq 14 has been calculated, the resulting value of can be applied to the generalized Clausius-Mosotti equation, using the previously provided Bruggeman's mean index [Mathematical Expression Omitted] = 1.52018.

[Mathematical Expression Omitted], (15)

therefore [Mathematical Expression Omitted] and can be found by applying Eq 7, [Mathematical Expression Omitted].

The consequence is immediate and identical to the ordinary case. Despite being perpendicular in direction to the other neutral line of the first case, the structural values are constant. Thus, they are independent of the degree of crystallinity, because it is an intermediate zone found in all samples and presumably have the same characteristics.

If we now take the values of [Mathematical Expression Omitted] and [([g.sup.e]).sub.i] found in (6, 7), the overall effect produced by these values can calculated, using the generalized Clausius-Mosotti equation, which is designated as [Y.sub.i]

[Mathematical Expression Omitted]. (16)

Once all these data have been found, an attempt can be made to calculate the degree of paracrystallinity, in this case, designated as [K.sub.i], by means of the equation

[Mathematical Expression Omitted]. (17)

In Eq 17, if [Epsilon] is taken as the first bracket value and the previous values are applied, then [Epsilon] = 0.30430; if is [Delta] taken as the second bracket value, then [Delta] = 0.30387. Thereafter

[Y.sub.i] = ([x.sub.c] - [K.sub.i] / 2)[Epsilon] + (1 - [x.sub.c] + [K.sub.i] / 2) [Delta] + [K.sub.i]J (18)


Hence, by finding [Mathematical Expression Omitted] the paracrystalline degree values are found whose partial effects appear in the direction of this neutral line (bulk extraordinary index), being equivalent their mean value, [Mathematical Expression Omitted], to the mean paracrystalline distortion parameter, [Mathematical Expression Omitted] (in percentage), as shown in Table 3.

A possible explanation for these higher percentages (excepting the first special one) is that in this preferential direction ([n.sup.e]), it can be assumed that almost all the active crystalline phase is along this direction. This greater paracrystalline zone would diminish its effect in this direction [ILLUSTRATION FOR FIGURE 4 OMITTED]. Therefore, in calculating the mean value [Mathematical Expression Omitted], since the mean dispersion paracrystalline parameter is [Mathematical Expression Omitted] for polymers, the proportionality constant is found to be p = 1.0022, being [Mathematical Expression Omitted].

Figure 7 shows the values of [K.sub.i] plotted as function of degree of crystallinity [x.sub.c] and its linear fitting ([g.sub.2]) with representative values, [m.sub.4] = 0.2215, [c.sub.4] = -6.6371 and [r.sub.4] = 0.99788. For an easier understanding of the calculation in this case, see also the block diagram in Fig. 5.


Despite the importance of the two previous cases, each sample is in fact one, and so in each sample we need to have mean percentage values representing their general averaged degree of paracrystallinity.

Then, following the order established, first calculated is a local mean effect, composed of the crystalline plus the amorphous phases, [Mathematical Expression Omitted]

[Mathematical Expression Omitted]. (19)

Once this is calculated, the mean value of [Mathematical Expression Omitted] can be computed. This is done, as before, using a coupling coefficient equal to one (11), since this is actually a very close cooperative interaction inside the material. Therefore

[Mathematical Expression Omitted], (20)

gives us [Mathematical Expression Omitted]. Likewise, we could apply [Mathematical Expression Omitted] to the Clausius-Mosotti equation and find [Mathematical Expression Omitted] and then (Eq 7) [Mathematical Expression Omitted]. However, using previously calculated values, we have found that the result is roughly the same following the procedure of coupling coefficients (equal to one)(11):

[Mathematical Expression Omitted], (21)

The same holds for the value of (as stated previously). Once again, giving us constant values, [Mathematical Expression Omitted] and, by Eq 7: [Mathematical Expression Omitted].

Once the values of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are known, they can be applied to the generalized Clausius-Mosotti equation (designated by [Mathematical Expression Omitted]). As in the previous cases, this represents the overall or general mean effect

[Mathematical Expression Omitted]. (22)

Following the same order as before, we are now in a position to calculate the overall mean effect as a mutual interaction, with coupling coefficient equal to one (11), between the effect of the ordinary index and that of the extraordinary index. We have designated thus by [Mathematical Expression Omitted] and it can be expressed as

[Mathematical Expression Omitted]. (23)

However, before proceeding to the final calculation, the value of

[Mathematical Expression Omitted], (24)

must be found, giving us [Mathematical Expression Omitted] = 1.51180. The value of

[Mathematical Expression Omitted], (25)

must be found, giving us [Mathematical Expression Omitted] = 1.51146. We must proceed as before to calculate

[Mathematical Expression Omitted], (26)

to get [Mathematical Expression Omitted]. Once all these data are known. the degree of paracrystalline percentage, that affects each sample as an overall mean must be calculated. It is designated as [P.sub.i] and is calculated as follows

[Mathematical Expression Omitted], (27)

Here the same form as the extraordinary case has been used, since the final effect is much greater than in the ordinary ease.

In Eq 27 is now taken [Chi] to be the value of the first bracket and previously found values are applied, then [Chi] = 0,29996; taking [Phi] as the value of the second bracket, the result is [Phi] = 0.29980, Thus,

[Mathematical Expression Omitted]. (28)

If we solve this equation for [P.sub.i], the final average values of the degree of paracrystallinity will be found: [Mathematical Expression Omitted] (percentages), are shown in Table 4.

In order to check whether these results are consistent with the two preceding cases, the arithmetic mean is calculated for the results of the ordinary case [G.sub.i], and extraordinary case [K.sub.i] and is designated as [Mathematical Expression Omitted] (shown in Table 5). Another possible way of confirmating this tesis, is to replace [Mathematical Expression Omitted] in Eq 28, which was the effect of the mean refractive index of the paracrystalline zone (Eq 22), with the local mean effect originating in the crystalline and amorphous zones [Mathematical Expression Omitted]. This provides other, similar values [Mathematical Expression Omitted], which are also shown in Table 5. As can be seen, these are practically the same as the values of [Mathematical Expression Omitted].



If we now plot this value of [Mathematical Expression Omitted] as a function of crystallinity degree [x.sub.c] and calculate its linear fitting (f), the representative values are: m = 0.1115, c = -3.3286 and correlation coefficient r = 0.9977 [ILLUSTRATION FOR FIGURE 8 OMITTED]. If we do the same for the values of [Mathematical Expression Omitted] instead of with [Mathematical Expression Omitted] as before, we obtain the same graph (no visible differences) with similar representative values, but with a correlation coefficient r[prime] = 0.9980. The same occurs for values of [Mathematical Expression Omitted].


In this paper, we have demonstrated the hypothesis that the mean paracrystalline distortion parameter is practically equal to the percentage of paracrystallinity for low density blown polyethylene films, in the particular zone oriented by the direction of the neutral line for the extraordinary bulk refractive index. By using the generalized Clausius-Mosotti equation for this material along with the values of the average refractive indices provided by the special Bruggeman's equation, we obtain results similar to those indicated in literature for the mean distortion paracrystalline parameter (in the specific zone) of this polymers, fully justifying our initial assumption.


1. R. Hosemann. Z. Phys., 128, 1 (1950); and 128, 465 (1950).

2. R. Hosemann and N. Bagchi, Direct Analysis of Diffraction by Matter, Ch. 9, North - Holland, Amsterdam (1962).

3. R. Hosemann, M. P. Hentschel, U. Schmeisser, and R. Bruckner. J. Non-Cryst. Solids, 83, 223 (1968).

4. R. Hosemann, Colloid Polym Sci., 260, 864 (1982).

5. F. J. Balta-Calleja and C. G. Vonk, X-Ray Scattering of Synthetic PoLymers, p. 135, Polymers Science Library 8, Elsevier, New York, (1989).

6. E. Bernabeu, J. M. Boix, A. Larena, and G. Pinto. J. Mater. Sci., 28. 5826 (1993).

7. E. Bernabeu and J. M. Boix, Polym. Eng. Sci., 36, 1203 (1996).

8. D. E. Aspnes, Thin Solid Films, 89, 249 (1982).

9. D. A. G. Bruggeman, Ann. Phys. (Leipzig), 24, 636 (1935).

10. J. Brandrup and E. H. Immergut, Editors., Polymer Handbook 3rd Ed., Chap. V, p. 15-26, R. P. Quirk and M. A. A. Alsamarraie: "Physical Constants of Polyethylene," Young-Wiley Ed., New York (1989).

11. F. A. Firestone, The Mobility and Classical Impedance Analogies, American Institute of Physics Handbook, D. E. Grey, ed, McGraw-Hill Book Co., New York (1957).
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Author:Bernabeu, E.; Boix, J.M.
Publication:Polymer Engineering and Science
Date:Apr 1, 1999
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