Various light scattering and optical techniques have been investigated as potential candidates for characterization of multiphase polymeric materials. The Mueller matrix formalism is briefly mentioned as an effective procedure of complete optical description of the medium. The effects of multiple and dependent scattering as well as coherent and diffuse light attenuation are discussed in model calculations and corresponding experiments (small-angle light scattering, time-resolved light scattering, transmission or diffuse reflectance measurements and image analysis). The following physical processes has been investigated: kinetics of phase separation and dissolution (demixing) of polymer blends, stress whitening process and photon migration in polymer composites. Light scattering and the diffuse reflectance method were used to characterize heterogeneous structures of polypropylene-based materials before and after solid-state deformation. The application of a hybrid model of Monte Carlo simulation and diffusion theory recently developed to describe propagation of photons into sufficient depth in the turbid media is presented for polypropylene samples with various amounts of ethylene-propylene rubber particles. The value of reduced scattering coefficient determined from experiment is an important structural parameter with direct relationship to size of particles and their volume fraction in the polymer matrix. Since speckles play an important role in many physical phenomena, it is essential to fully understand their properties. Structural aspects of the laser speckle techniques and their application in polymer physics are presented.

Keywords: light scattering and transmission, Mueller matrix, turbidity, diffusion approximation, speckle effect, optical properties of polymers

2.2.1. Incoherent (diffusion) approximation

(a) The validity of the diffusion approximation of the transfer theory is restricted to cases in which albedo a is close to unity^10,11,20

where C_abs, C_sca are absorption and scattering cross-section. For the light transmission calculation reported here, the turbidity is given by

where N is number of spherical particles (having radius R) per unit volume, dispersed in a nonabsorbing medium and C_sca is scattering cross-section calculated according to the L-M theory⁹. For particle sizes 2p R/l >1, eq.(2) has been modified¹⁰ adding a multiplicative factor A(cos q ) on the right-hand side moderating a too strong reduction in turbidity caused by the 1-<cos q > factor for larger particles. The value A(cos q ) = 1.5 has been used¹⁰ for particle radii 5 < 2p R/l < 70. The parameter g = <cos q > (asymmetry factor) is the average value of cosq (q is the scattering angle) with the angular intensity as the weighting factor , i.e.

<cos q > = [i₁(q ) + i₂ (q )]cos q sinq dq /[i₁(q ) + i₂ (q )]sinq dq (3)

where i₁(q ), i₂ (q ) are the L-M angular intensity functions^9,13. The average photon may be regarded as tracing a route the basic steps of which are of length L (mean free path), being scattered through polar angle q at each step. The effect of anisotropy of scattering is to introduce a correlation between its pre- and post-scattering directions. The weight factor which determines a non-zero correlation is the distribution of intensity scattered by a particle as described in eq.(3). It is well known that <cos q > is also related to the radiation pressure C _pr .

We used the following relation between C_ext , C_sca and C _pr

where C_ext is the extinction cross section and C_pr is the radiation pressure cross-section⁹. Reduced scattering cross-section C_sca (1- <cos q > ) signifies the total scattered radiation minus the radiation scattered in the forward direction. We included into calculations of the L-M functions (by means of the Mie3 code²¹) both the effects of refractive index dispersion (the dependence of refractive index on the wavelength if the calculation includes several wavelengths) and the change in refractive index of the matrix due to the presence of the minor phase. The concentration dependence of the refractive index of the medium (effective refractive index of matrix) n_m has a form of the modified mixing rule^11,22,23

where n₁, n₂ are refractive indices of both phases, v₁ is the volume fraction of the minor phase and y is a correcting factor depending on the relative refractive index n₁/n₂ and on the size of the particle relative to the wavelength²³.

Even if the scattering arises principally from the fluctuating part of the index (index mismatch between scatterer and immediate surrounding), the fact that the relative refractive index of any particle must decrease as v₁ increases is generally accepted and very well verified experimentally^9,11.

(b) Thick-layer approximation.

In the thick-layer approximation, when collimated transmitted flux is very small, we obtain for a diffuse reflectance¹¹ R

where S is scattering parameter from the Kubelka-Munk theory^11,13 and Z is layer thickness. The S parameter is then

For the ratio of diffuse reflectances at two different wavelengths, we get relation

At present, several software packages exist^1,2,24,25 enabling computations of C_sca and g and thus determination of S and R(l ₁)/R(l ₂). We used the Mie3 program^21,25 (implementation of the L-M theory) and eq.(2) in our calculation of scattering cross-sections as well as asymmetry factors and corresponding quantities in eq.(6) and (7).

It is well known that the resonances in the Mie coefficients may cause oscillations of the scattering cross-sections for some sizes and relative refractive indices²⁶. To "smooth" the oscillations (an effect observable on the ideal sphere or spherical monodispersions only), the calculated values have been averaged over the diameter polydispersity of the sample which corresponded to the periodicity of resonances d x. Thus for any particular sphere it was assumed that there existed a spread of total width²⁷ of d x.

(c) Semi-infinite medium
For a semi-infinite medium whose C_a << C_sca, a simple and rapid approach to measure the reduced scattering coefficient,

has been derived^28,29. A laser beam with an oblique angle of incidence to the medium a _i causes the centre of the diffuse reflectance, which is several transport-mean-free parts away from the incident point, to shift away from the point of incidence by an amount D x . This amount is used to compute C4_sca by
 

The following subsections describe the experimental testing of eq.(10) and a comparison with the computations of the reduced scattering coefficient in the diffusion approximation using the L-M theory, asymmetry factor and effective refractive index of the medium composed from known particular components³⁰. The calculations of the L-M functions by means of the Mie3 code ²¹ have been performed.

The value of N (N is the number of spherical particles per unit volume having diameter d) has been calculated from the respective volume fraction and the volume of the single particle.

The effective refractive index of the medium reflects the change in the refractive index of the matrix due to the presence of the minor phase. The concentration dependence of the refractive index of the medium (effective refractive index of matrix), n_m has the form of the modified mixing rule (eq.(5)).

2.2.2. Coherent (dependent) scattering and light attenuation

For the interpretation of the turbidity data, the hard-sphere model is available elaborated also for polydisperse adhesive spheres. The theory is based on the Percus-Yevick (P-Y) approximation³¹ in the Ornstein-Zernike equation and has been developed by Baxter³². It has been found that the P-Y approximation provides a satisfactory description of scattering properties of hard spheres up to high volume fractions. If the reduction in transmitted intensity is not caused by absorptions of radiation, but solely by scattering, a simple conservation law then relates scattering and turbidity^9,19

t = 2p I(q)sinq dq (11)

with

Here I(q) is the scattering intensity, q the scattering wave vector, q the scattering angle, n the refractive index of the sample, and l ₀ the wavelength in vacuo.

For a system containing equal-sized spheres of radius R, the scattering intensity can be written in the form³³

where F(qR) is a shape factor accounting for the interference of the light scattered from different parts within one particle and the structure factor S(qR) accounts for the interference of light scattered from different particles. We used the closed form³⁴ for S(qR) in the explicit form given by Hayter³⁵. The L-M theory has been used in calculation of F²(qR).

2.3.1. Stress whitening in polypropylene materials

The origin of turbidity and stress whitening in semicrystalline polymers and blends are still a matter of some controversy^36-39. We studied transmission and reflectance properties of stress-whitened specimens of semicrystalline polypropylene (PP), with or without embedded particles (various fractions of amorphous phase, ethylene-propylene rubber particles (EPDM) or open voids), by model computations, measurements of relative diffuse reflectance in the visible region and by immersion experiments which showed a long-term transparence recovery of the stress-whitened specimen. The theory²⁰ (based on the diffusion approximation of the transfer theory and Kubelka-Munk theory) predicts a decreasing spectral dependence of turbidity with increasing wavelength for a matrix with embedded particles of slightly different refractive indices, but a flat dependence of this quantity for a matrix material with microvoids^27,30. It is demonstrated that the diffuse reflectance displays the same type of wavelength dependence for the thick layer approximation. Indeed, diffuse light reflectance experiments on bulk specimens using an integrating sphere accessory reveal the first type of behaviour for nondeformed neat and rubber-modified polypropylenes. On the other hand, the second type of behaviour was observed with stress-whitened neat and rubber-modified polypropylenes after solid-state drawing.

Figure 1. Comparison of turbidity ratio t(l₁)/ t(l₂) for various polypropylenes (l₁ = 0.4 m m, l₂ = 0.8 m m); 1 PP with EPDM rubber, 2 PP with spherical inclusions of amorphous phase, 3 PP with spherical voids. Volume fractions circles 0.1,   squares 0.3

Model calculations were carried out for three systems: (a) matrix of isotactic polypropylene containing voids, (b) polypropylene containing particles of ethylene-propylene (EPDM) rubber and (c) matrix of isotactic polypropylene containing spherical inclusions with refractive indices corresponding to those of amorphous and crystalline phases. We included into the calculation of turbidity both the effects of refractive index dispersion (the dependence of refractive index on the wavelength) and the change in refractive index of the matrix medium due to the presence of the minor phase^23,27. A reasonable agreement has been found between the experimental and calculated values of the scattering cross-section for various polymer systems up to a volume fraction of 0.2 under the diffusion approximation.

In all three solid systems considered hereinafter, monodisperse spherical particles³⁰ or voids were assumed (with the diameter spread eliminating resonance effects). An overview of relevant refractive indices for voids, rubber particles and PP inclusions in PP matrix can be found elsewhere³⁰.

We introduced only spectral dependences t (0.4)/t (0.8), i.e. the ratios of turbidities at l = 0.4 and 0.8 m m and volume fractions of the minor phase, 0.1 and 0.3, which can be compared with experimental data. The calculations indicated that the spectral dependence of turbidity is negligible for voids in the PP matrix (horizontal plateau of t (0.4)/t (0.8) is close to unity). A much steeper spectral dependence of turbidity can be expected for all types of inclusions as seen in Figure 1, where the turbidity ratios at two wavelengths are compared for different particle sizes and two concentrations for all three systems.

It can be inferred from Figure 1 that the turbidity ratio is virtually insensitive to the concentration of the minor phase. The data points calculated for two different volume fractions (0.1 and 0.3) of the minor phase are located on the same curves. However, the curves distinctly differ for voids, EPDM particles or inclusions with refractive index slightly different from the PP matrix, respectively. The t (0.4)/t (0.8) ratio for larger inclusions (diameters larger than ca. 1 m m) depends on the inclusion size but slightly.

The ratio t (l ₁)/ t (l ₂) for voids displays a very mild dependence in comparison with both PP + EPDM and PP + amorphous inclusions. Due to practical independence of turbidity ratios on the particle diameters (cf. Fig. 1), the size polydispersity effects do not play any significant role for diameters larger than ca 1 m m.

The effect of particle anisometry caused by stress is usually dealt with by means of an assumption of spheroidal shape (with dimensions a, a, c), where a root-mean-square radius of a spheroid can be expressed as r = [(2a² + c²)/3]^1/2. From calculations for spherical and ellipsoidal particles, it appears that such approximation of anisometric scatterer by an equivalent spherical particle is quite reasonable¹⁰.

2.3.1.1. Results and discussion

We measured relative diffuse reflectance (proportional to turbidity, cf.eqs.(2, 6-8) in the spectral region 0.4 0.85 m m with a Perkin-Elmer 340 spectro-photometer using an integrating sphere accessory. The experimental results for typical specimens are given in Figure 2. We used standard dumb-bell test specimens of neat polypropylene Mosten 58412 (Chemopetrol, Czech Republic), cross-section 104 4mm (thickness) and a specimen of the same polymer deformed at drawing temperature T_d = 100 ^oC. The reactive type of rubber-modified polypropylene Kelburon KLB 9569X(DSM Holland) was deformed at T_d = 23 ^oC. As predicted in Figure 2, the stress-whitened samples (both with voids and voids plus EPDM inclusions) show only mild spectral dependences as compared with the semicrystalline undeformed test specimen. Hence, a flat spectral dependence indicates voids as light scatterers, whereas a decreasing spectral dependence is typical of light scattering by embedded particles. The experimental data are normalized at l = 0.85 m m by shifting curves to intersect in this point to allow a comparison of all samples in the same figure. The experimentally observed increasing R (l ) dependence cannot be obtained from the t (l ) theoretical dependences for any reasonable particle diameters (1 -80 m m), whereas the C_sca(0.4)/ C_sca(0.8) ratios may be smaller than unity for very large particles (diameters tens of microns). This might indicate the role of collimated flux and needs a further detailed study.

Figure 2. Comparison of calculated and experimental relative diffuse reflectances for various polypropylenes. (x) non-deformed injection-molded PP (experimental); (full square) deformed injection-molded PP after solid-state drawing at 100 0C (experimental); (full circle) PP with EPDM microspheres after solid state drawing at 100 0C (experimental). PP with spherical voids (empty square) R = 4 m m, (hollow circle) R = 15 m m (calculated); PP with spherical inclusions of amorphous phase (downward triangle) R = 4 m m (calculated); PP with spherical inclusions of EPDM (upward triangle) R = 4 m m (calculated). All curves are shifted to cross at l = 0.85m m. Volume fraction of the minor phase is 0.1

A direct proof of interconnected voids in the studied stress-whitened samples can be obtained by immersion of the sample into a suitable solvent³⁸. We selected n-heptane, known as a preferential solvent of polypropylene amorphous phase⁴¹. The almost total transparency recovery was seen at both sides of the broken sample (where the solvent uptake is supported by the connected open voids system) after two-weeks immersion in n-heptane at room temperature.

The results can be summarized as follows:

(a) The turbidity ratios and diffuse reflectance values at two different wavelengths are markedly larger for polypropylene with rubber particles or polypropylene with inclusions of crystalline phase as compared with those with voids. The difference offers a possibility of diagnosing the origin of stress-whitening in real polymeric materials.

2. Experimental results (relative diffuse reflectance) support the predicted spectral dependence of turbidity for all three systems under consideration.
3. Direct measurements of transparency recovery in stress-whitened specimens in an immersion liquid confirmed the existence of a connected void system

A simple procedure characterizing light attenuation by transmission (based on the diffusion approximation of the transfer theory, asymmetry factor and effective refractive index of the scattering medium) is compared with the coherent aproaches based on (i) published MC simulations, (ii) scattering from a system of closely packed spherical particles with interparticle correlation characterized by the P-Y approximation and (iii) experimental data. The use of the procedure in analysis of phase separation in polymer blends is presented. It has been shown that the results obtained by this procedure are in reasonable accord with theoretical and experimental data so far published for systems with a volume fraction of the minor phase less than ca. 0.2 and without any specific interactions in the system. The more complicated coherent approaches (such as MC simulations and scattering by systems with interparticle correlations) used for calculation of coherent flux attenuation in concentrated scattering systems depend on the pair distribution function of the particle position in which the P-Y approximation is used to describe the correlation of positions of particles of finite sizes. Since all the mentioned theories deal with the same physical problem, there must exist some relationship between them. It should be noted that the measurements require a technique to distinguish incoherent multiple-scattered light from the coherent transmission. Experimental details of measurements of the coherent and incoherent scattered radiation are given in the cited papers.

2.3.2.1. Objectives

A comparison of light attenuation by scattering based on coherent and incoherent (diffusion) approach is presented for: (a) influence of increasing fractional volume of spherical scatterers on the light scattering parameters (Teflon dispersions in water^15-18, silica particles in toluene¹⁹, polystyrene (PS) dispersions in water²⁰, poly(vinyl acetate) (PVA) dispersions in water¹⁵, (b) the time changes in turbidity during the late stages of phase separation (polystyrene/poly(methyl methacrylate))^42,43 (PS/PMMA)

2.3.2.2. Results and discussion

Some polymer systems have been studied during processes accompanied by considerable changes in supermolecular structure and morphology (critical opalescence, phase separation, etc.) The late stage of the phase separation process is characterized by scaling laws relating the growing particle size with time evolution (various exponents of the power law). The value of exponent is characteristic of the type of the phase separation process^42,43. Growing particles (spherical domains) produce scattered light under illumination due to the refractive index unmatched with the surrounding matrix. The transmission of light decreases significantly during the particles growth. The temporal changes in turbidity are related to the size vs. time growth law in the form

where R is radius of the growing spherical domain, R(0) is the starting domain radius and the exponents l = 1/3 and 1/4 have been used^43,44 in our calculations .

To test the extent of the possible use of the diffusion approximation, we compared published experimental data⁴⁵ with calculation for the same system (eq.(2)). The scattering cross-section calculated by the quasi-crystalline approximation (QCA) with the P-Y expression for the pair distribution function of hard spheres is in good accord with the experimental data^15-18. The same quantity calculated using the diffusion approximation gives good results up to volume fractions ca. 0.15- 0.2.

In Fig. 3 we compared the results of controlled experiments and calculations¹⁵ for a dispersion of PVA particles with small absorption (particle refractive index¹⁵ n₂ = 1.425 + i.0.0006). A reasonable accord between both approximations and experiment has been found for v₁ < 0.2. A comparison of turbidity versus volume fraction of silica particles¹⁹ calculated by the P-Y approximation and by the diffusion approximation is shown in Fig. 4.

Fig. 3. Comparison of measured and theoretical turbidities for a dispersion of PVA spheres; l = 0.6328 m m, R = 0.09 m m, n (PVA) = 1.425 + i.0.0006, n (water) = 1.33. Compared are: Independent scattering (full line), (triangles) diffusion approximation with asymmetry factor and mixed indices, QCA (full curve), (circles ) experimental data¹⁵

Fig. 4. Comparison of theoretical turbidities for diffusion approximation (triangle ) and P-Y approximation¹¹ (circle) for silica particles with average particle radius 0.035 m m, refractive indices of particles 1.430 and toluene (solvent) 1.490 

Fig. 5. Turbidity versus volume fractions for diffusion and P-Y approximation for PMMA particles in PS matrix. Particle radii (a) R = 0.1 m m, (b) R = 1.0 m m; n(PS) = 1.5859, n(PMMA) = 1.4868 and l = 0.6328 m m

Again, a reasonable accord is obtained up to volume fractions v₁ < 0.2. So far, scattering from relatively small particles has been considered. Whereas an extension of diffusion theory to the large particles brings no principal problems¹⁵, the S(qR) function (eq.(13)) could be calculated for a limited range of R values only. In Fig. 5, we plotted calculations for PMMA particles with radii R = 0.1 m m and R  = 1.0 m m in a PS matrix, the latter value being the largest that can be computed in our implementation of the P-Y approximation. It appears that the growing size of scatterers does not lead to an increasing discrepancy between coherent and diffusion approximations. From the practical point of view, an addition of the minor phase with volume fraction lower than ca. 0.2 is typical of many technological processes and is sufficient to achieve final products (polymer blends, paintings, composites) with desired properties. One of these processes is the phase separation, where two partly miscible components separate due to an unfavourable Gibbs energy of mixing. The time evolution of transmission during the phase separation process calculated by means of the diffusion theory is shown in Fig. 6, where we compared the PS/PMMA system with two possible exponents of the power law dependence (eq.(14)) and two sample thicknesses. It can be inferred immediately that there exists an induction time for transmission measurements where the time evolution of domain sizes of separating phases has no observable effects on transmission. At later stages of the phase separation and at constant values of volume fraction of the minor phase, a saturation of transmission and eventual increase in transmission is observed.

Fig. 6. Time evolution of transmittance for spherical domains of PS in PMMA matrix. Volume fraction of PS = 0.15. Starting domain size 2R(0) = 0.01 m m. The growth law is of the form 2R = 2R(0) t^l, the exponent l = 1/4 and 1/3, n_m(medium) = 1.50166. Sample thicknesses 0.1 mm (top) and 1 mm (bottom)

2.3.2.3. Summary

(1) The presented diffusion approximation with the asymmetry factor and effective refractive index of the scattering matrix is in relatively good accord with the coherent P-Y approximation and MC simulations and also with the published experimental data for volume fraction up to ca. 0.2. For higher volume fractions of the minor phase, this approximation gives higher values of turbidity than the coherent P-Y approximation and experiments. The same limitations are also valid for slightly absorbing particles (cf. Fig. 3). For systems with higher volume fractions of scatterers, much more complicated calculations taking into account specific correlations in particle positions are necessary. A comparison of coherent and diffusion approximations for larger particle diameters is impossible due to dificulties in calculation of interference function S(qR).

(2) The above mentioned limitations are still valid for many physico-chemical processes or at least for some of their stages (e.g. crystallization⁹, phase separation⁴², phase dissolution⁴³, stress whitening^27,30, painting) and the diffusion approximation can be practically used for structural characterizations.

(3) Absolute value of turbidity can be calculated from the known optical parameters of the scattering system.

(4) The application of the above approach to other scattering systems with known optical parameters is straightforward.

2.3.3. Light scattering and reflectance in multiple scattering regime. Single-side illumination

Real polymeric materials inevitably contain defects and heterogeneities of different nature and origin. They can be intrinsic (semicrystalline polymers), intentionally introduced (polymer blends) or developed under mechanical stress (in stress-whitening process). A great majority of blends are immiscible from the thermodynamic viewpoint and also most blends of commerce are immiscible. Such blends form a multiphase system with a deformable minor phase and, under appropriate conditions, morphological structures such as spheres, ellipsoids, fibres, and plates can be produced. Under certain conditions, co-continuous phases may also be formed^45-47. As a result, development of rapid on- and off-line techniques to measure the size, shape and volume fraction of the minor phase is important. Optical methods are attractive for the on-line characterization of transparent or translucent materials because they are noninvasive, rapid and applicable to high-temperature materials. A limitation of conventional light scattering techniques is their requirement for relatively thin samples in order to avoid multiple scattering effects ^1,9,11. The approach described in this paper is proposed to circumvent this limitation; as a result, a semi-infinite material may be inspected with an access to a single side of the material.

Recently, a hybrid model of the MC simulation and diffusion theory has been described, which combines the accuracy advantage of MC simulation and the speed advantage of diffusion theory being, at the same time, faster than pure MC simulation and more accurate than the pure diffusion theory^28,29. The application of the above mentioned method to polypropylene samples with various amounts of ethylene-propylene rubber particles (EPDM) is presented here. The value of the reduced scattering coefficient determined from experiment is an important structural parameter with direct relationship to the size of particles and their volume fraction in the polymer matrix.

2.3.3.1. Experiment

To test eq.(10) experimentally, we used a video reflectometer^28,29,47 (Fig. 7) to measure the diffuse reflectance from a series of specimens made of polypropylene composites (Table 1).

Light from a He-Ne laser (output 10 mW, wavelength l = 632.8 nm), attenuated by a neutral filter and focused by lens (focal length f = 180 mm) was directed to the medium surface at an angle of incidence a _i = 80^o. A 8-bit video CCD camera Pulnix 765 measured the diffuse reflectance, and the computer collected and then analyzed the image using the LUCIA D system^49,50. The dynamic range of the CCD camera is limited to 255 intensity levels.

 

 
Figure 7. Schematic diagram of the video reflectometer. 1 laser, 2 attenuation filter, 3 objective lens, 4 diaphragma, 5 tilting stage, 6 sample, 7 CCD camera, 8 CCD controller, 9 computer
 

 

Figure 8. A diffuse reflectance pattern of sample PPR1 obtained with a CCD camera (Table 1). Several thresholds are used to visualize the center B of the diffuse reflectance far from the incident laser beam (position A). A shift away D x is indicated. The scale bar is 1 mm
 

Therefore, two images with different intensities of the laser beam were taken to measure the diffuse reflectance on a larger surface area. The laser beam for this experiment has an elliptic shape on the surface of the turbid medium and hence has a mirror symmetry about the y axis (perpendicular to x in Figure 8). Figure 8 illustrates a diffuse reflectance pattern for sample PPR1 with thresholding to visualize better the shift D x = x(B)-x(A) between the centre of the incident laser beam A and the center of the diffuse reflectance B on a larger surface area. Standard dumb-bell test specimens were used with the parameters given in Table 1.

2.3.3.2. Results and discussion

We usually used two measurements, with and without the attenuation of the laser source intensity, to obtain a sufficient dynamic range of the reflectance for determination of D x values. The resolution in the setup was 6.34 10^-3 cm/pixel and the experimental error was ca. 1 pixel. The results are summarized in Table 2.
    The reduced scattering coefficients in Table 2 are calculated using eq.(10), where the refractive index nm is taken from eq.(5) with y = 1 for the last but one column and nm for the last column is determined for EPDM particles with diameter d = 0.6 m m according to Heller4s theory23. As can be seen, the difference in reduced scattering coefficients thus calculated is very small and is within an experimental error.
    The experimentally obtained C4sca and calculated values of reduced scattering coefficients for various diameters of EPDM particles at appropriate volume fractions are plotted in Figure 9.
It can be inferred from Figure 9 that the reduced scattering coefficients of EPDM particles in the polypropylene matrix relatively strongly depend on their diameters and concentrations; for the studied system, the diameter d = 0.6 m m is in the good accord with experimental data. The SEM image of the PPR2 sample presented in Figure 10 indicates that the size value obtained by the C4_sca measurement is in reasonable agreement with the SEM image.

The results can be summarized as follows:

The laser beam diffuse reflectance can be used to measure reduced scattering coefficients C4_sca of polymer composites in the multiple-scattering regime. The experimentally obtained C4_sca values are in good accord with the values calculated by means of the L-M theory in the diffusion approximation. A reasonable agreement has been found between size parameters obtained by the diffuse reflectance method and by direct image analysis of the SEM images.

 

Figure 9. Comparison of theoretical reduced scattering coefficients C4_sca for various diameters d of EPDM particles in polypropylene matrix and experimental C4_sca values for samples PPR1, PPR2 and PPR3. 1 (d = 0.5 m m), 2 (d = 0.6 m m), 3 (experimental data), 4 (d = 1 m m)

 

 
Figure 10. SEM image of PPR2 sample. A histogram of the size distribution (diameters in m m) of EPDM particles obtained by image processing is given in the inset (the scale bar is 1 m m)

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27.J. Holoubek and M. Raab, "Stress-whitening in polypropylene: Light scattering theory and model experiments", Collect. Czech. Chem. Commun. 60, pp. 1875-1887, 1995

28.L. Wang, and S. L. Jacques, "Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium", Appl. Opt. 34, pp. 2362-2370, 1995

29.L. Wang, and S. L. Jacques, "Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media", J. Opt. Soc. Am. A10, pp. 1746-1752, 1993

30.J. Holoubek, J. Kotek, and M. Raab, "Light scattering by optically heterogeneous polymer systems: stress-whitening in polypropylene materials", Polym. Bull. 37, pp. 631-638, 1996

31.J. K. Percus and G. J. Yevick, "Analysis of classical statistical mechanics by means of collective coordinates", Phys. Rev. 110, pp. 1-13, 1957

32.J. R. Baxter, "Percus-Yevick equation for hard spheres with surface adhesion", J. Chem. Phys. 49, pp. 2770-2274, 1968

33.W. K. Bertram, "Correlation effects in small-angle neutron scattering from closely packed spheres", J. Appl. Crystallogr. 29, pp. 682-685, 1996

34.N. W. Ashcroft and J. Lekner, "Structure and resistivity of liquid metals", Phys. Rev. 145, pp. 83-90, 1966

35.J. B. Hayter and J. Penfold, "An analytic structure factor for macroion solutions", Mol. Phys. 42, pp. 109-118, 1981

36.H. Breuer, F. Haaf, and J. Stabenow, "Stress-whitening of rubber-modified thermoplastics", J. Macromol. Sci. Phys. 14, pp. 387-394, 1977

37.Y. Liu, C. H. L. Kennard, R. W. Truss, and N. J. Calos, "Characterization of stress-whitening of tensile-yielded isotactic polypropylene", Polymer 38, pp. 2797-2805, 1997

38.Y. W. Lee and S. H. Kung, "Elimination of stress-whitening in high-molecular-weight polyethylene", J. Appl. Polym. Sci.46, pp. 9-18, 1992

39.R. Rengarajan, S. K. Kesavan, and K. L. Fullerton, "Ethylene-methacrylic acid copolymers as stress-whitening suppressant in polypropylenes", J. Appl. Polym. Sci. 45, pp. 317-324, 1992

40.V. S. Lee and L. Tarassenko, "Absorption and multiple scattering by suspensions of aligned red blood cells", J. Opt. Soc. Am. 8, pp. 1135 -1142, 1991

41.D. R. Jenke, "Solute migration through polypropylene blend films", J. Appl. Polym. Sci. 44, pp. 1223-1234, 1992

42.J. Holoubek, "A note on light attenuation by scattering: Comparison of coherent and incoherent (diffusion) approximations", Opt. Eng. 37, pp. 705-709, 1998

43.J. Holoubek and C. C. Han, "Optical transmission of phase separating polystyrene/poly(methyl methacrylate)blends", Polym. Mater. Sci. Eng., Vol.71, pp. 368-369, 1994

44.J. Holoubek, "Optical transmission and light scattering of phase separation structures in polymer blends", Proc. PARTEC 95, 4th Int. Congress Optical Particle Sizing, Nuernberg, 21-25 March 1995, pp. 387-396, 1995

45.C. Belanger, P. Cielo, B. D. Favis, and W. I. Patterson, "Analysis of polymer blend morphology by transmission and reflection light scattering techniques", Polym. Eng. Sci. 30, pp. 1090-1097, 1990

46.D. R. Paul and S. Newman, Eds, "Polymer Blends", Elsevier Appl. Sci. Publ., London, 1978.

47.L. Utracki and R. A. Weiss, Eds, "Multiphase Polymers: Blends and Ionomers", ACS Symp. Ser. 395, 1989

48.P. Cielo, B. D. Favis, and X. Maldague, "Light scattering characterization of polyblends in the presence of multiple scattering conditions", Polym. Eng. Sci. 27, pp. 1601-1610, 1987

49.J. Holoubek, "Light scattering and reflectance of optically heterogeneous polymers in multiple scattering regime", Polym.Commun, 1998, in press

50.LUCIA D, system for image processing and analysis, Laboratory Imaging Ltd., Prague, Czech Republic.

51.J. Holoubek, "Light scattering, transmission and image analysis in polymer blends and composites", PARTEC 98, 7th Eur. Symp. Particle Characterization, Nuernberg, Germany 10-12 March 1998, Preprints III, pp. 899-908, 1998

52.J. C. Dainty, Ed., "Laser Speckle and Related Phenomena", Springer Verlag, Berlin, 1975

53.R. K. Erf, Ed., "Speckle Metrology", Academic Press, 1978

54.N. Takai, T. Iwai, and T. Asakura, "Correlation distance of dynamics speckles", Appl. Opt. 22, pp. 170-177, 1983

55.J. Holoubek, "Application of speckle techniques in polymer physics", Proc. SPIE, Vol. 473, Symposium Optika '84, pp. 160-163, 1984

56.J. Holoubek, J. Mikes, and B. Sedlacek, "Application of speckle techniques in macromolecular physics", in "Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems", Ed. B. Sedlacek, Walter de Gruyter , Berlin, pp.477-480, 1985

57.H. Krug, J. Holoubek, and E. W. Fischer, "Application of interactive image analysis to light scattering patterns of some polymer systems", Colloid Polym. Sci. 265, pp. 779-785, 1987

58.J. Holoubek and H. Krug, "Laser speckle techniques: Application in polymer physics", Makromol. Chem./Macromol. Symp.18, pp. 113-133, 1988

59.J. Holoubek and H. Krug, "Application of single beam speckle interferometry in macromolecular physics", Proc. SPIE, Vol. 1121, Interferometry '89, pp. 400-410, 1990

60.J. Holoubek, "Light scattering speckle photography: Determination of slow correlation times", Optica Acta 31, pp. 1283-1291, 1984

61.J. Holoubek, "Light scattering speckle photography: Determination of slow correlation times" in "Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems", Ed. B.Sedlacek, Walter de Gruyter , Berlin, pp. 481-484, 1985

62.J. Holoubek, "A strain analysis technique by scattered light speckle photography", Proc. SPIE, Vol. 35, Milestone Series, Selected Papers on Speckle Metrology, Ed. R. S. Sirohi, pp.114-122, 1991

63.J. Holoubek, "Structural aspects of dichromatic laser speckle patterns: Light scattering from polymer films", Proc. SPIE, Vol. 556, Int. Conference on Speckle, pp. 55-62, 1985

64.J. Holoubek, H. Krug, J. Krepelka, J. Perina, and Z. Hradil, "Light scattering speckle patterns and their correlation properties", J. Mod. Opt. 34, pp. 633-642, 1987

65.J. Holoubek and H. Krug, "Correlation properties of light scattering speckle patterns" in "Holography and Speckle Phenomena and Their Industrial Application", Ed. R.S. Sirohi, World Scientific, Singapore, pp. 333-336, 1990

66.J. Holoubek, C. Konak, and P. Stepanek, "Measurement of dynamic light scattering by means of a CCD camera", Jem. Mech. Opt. 41, No. 11-12, pp. 352-354, 1996 (in Czech)

67.J. Holoubek, C. Konak, and P. Stepanek, "Time resolved small-angle light scattering apparatus", 5th Int. Congress on Optical Particle Sizing, Minneapolis, August 10-14, p.81, 1998