Small-angle Hv light scattering from deformed spherulites with orientational fluctuation of optical axes

Abstract

Mathematical evaluation was done for small-angle light scattering from disordered spherulites under Hv polarization conditions. The calculation was carried out for a two-dimensional deformed spherulite whose major optical axes are oriented at 0 or 45° with respect to the radial direction. The calculated results were compared with the scattering patterns observed for polypropylene (PP) spherulites, whose optical axes are oriented parallel to the radial direction, and poly(butylene terephthalate) (PBT) spherulites, whose optical axes are oriented at 45° with respect to the radial direction. The degree of disorder for PBT was much larger than that for PP. By selecting a parameter associated with the degree of disorder of the optical axes with respect to the radial direction, the patterns calculated as a function of draw ratios were in good agreement with the observed patterns, which changed from four leaves to streaks extended in the horizontal direction. Through a series of observed and calculated patterns, it turns out that an increase in the disorder under the deformation process occurs drastically even for perfect spherulites in an undeformed state.

Appendix

The coefficients A and B in Eq. 16 are given by

$$A = \lambda _{3} \cos \alpha _{1} \sin \theta \cos \mu + \lambda _{2} \sin \alpha _{1} \sin \theta \sin \mu $$

(18)

$${B = \lambda _{3} \cos \alpha _{2} \sin \theta \cos \mu + \lambda _{2} \sin \alpha _{2} \sin \theta \sin \mu }$$

(19)

Substituting ω ₁=ω ₀+Δ₁, ω ₂=ω ₀+Δ₂, and Δ₁₂=Δ₂−Δ₁=ω ₂−ω, Eq. 14 reduces to

$${\left( {{\user2{M}} \bullet {\user2{O}}} \right)}_{1} {\left( {{\user2{M}} \bullet {\user2{O}}} \right)}_{2} = \frac{{\lambda ^{2}_{2} \lambda ^{2}_{3} {\left\{ {\cos 2{\left( {\alpha _{1} - \alpha _{2} } \right)} - \cos 2{\left( {\alpha _{1} + \alpha _{2} + 2\omega _{0} } \right)}\cos 2\Delta _{1} } \right\}}f{\left( {r_{{12}} } \right)}}} {{8{\left\{ {{\left( {\lambda ^{4}_{2} + {A}'\lambda ^{2}_{2} + \frac{{{A}'^{2} }} {4}} \right)} + {\left( {\frac{{{A}'}} {2}\lambda ^{2}_{2} + \frac{{{A}'^{2} }} {4}} \right)}D + \frac{{{A}'^{2} }} {4}F} \right\}}}}$$

(20)

where

$${A}' = \lambda ^{2}_{3} - \lambda ^{2}_{2} $$

(21-1)

$${D = \cos {\left\{ {2{\left( {\alpha _{1} + \omega _{0} } \right)}} \right\}}\cos 2\Delta _{1} + \cos {\left\{ {2{\left( {\alpha _{2} + \omega _{0} + \Delta _{1} } \right)}} \right\}}f{\left( {r_{{12}} } \right)}}$$

(21-2)

$${F = \cos 2{\left( {\alpha _{1} + \omega _{0} } \right)}\cos 2\Delta _{1} \cos 2{\left( {\alpha _{2} + \omega _{0} + \Delta _{1} } \right)}f{\left( {r_{{12}} } \right)}}$$

(21-3)

By using Eq. 6, Eq. 20 can be rewritten as follows:

$$\begin{array}{*{20}l} {{{\left( {{\user2{M}} \bullet {\user2{O}}} \right)}_{1} {\left( {{\user2{M}} \bullet {\user2{O}}} \right)}_{2} } \hfill} & { = \hfill} & {{\frac{{{C}''\exp {\left( { - \frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{8{\left\{ {{A}'' + {B}''\exp {\left( { - \frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}} \right\}}}}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{{C}''}} {8}\left\{ {\frac{{\exp {\left( { - \frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{A}''}} - \frac{{{B}''\exp {\left( { - 2\frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{A}''^{2} }} + \frac{{{B}''^{2} \exp {\left( { - 3\frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{A}''^{3} }} - \frac{{{B}''^{3} \exp {\left( { - 4\frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{A}''^{4} }} + \frac{{{B}''^{4} \exp {\left( { - 5\frac{{{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{A}''^{5} }} - \;.\;.} \right.} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{{C}''}} {8}{\sum\limits_{m = 1}^\infty {{\left( { - 1} \right)}^{{m - 1}} \frac{{{\left( {{B}''} \right)}^{{m - 1}} \exp {\left( { - \frac{{m{\left| {r_{{12}} } \right|}}} {a}} \right)}}} {{{\left( {{A}''} \right)}^{m} }}} }} \hfill} \\ \end{array} \;.$$

(22)

where

$${A}'' = G + H\cos (2\alpha _{1} )\cos (2\omega _{0} ) - H\sin (2\alpha _{1} )\sin (2\omega _{0} )$$

(23-1)

$$\begin{array}{*{20}l} {{{B}''} \hfill} & { = \hfill} & {{\frac{J} {2}\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)} + \frac{J} {2}\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)}\cos {\left( {4\Delta _{1} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + \frac{J} {2}\sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)} - \frac{J} {2}\sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)}\cos {\left( {4\Delta _{1} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + H\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\cos {\left( {2\omega _{0} } \right)} - J\sin {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\sin {\left( {2\omega _{0} } \right)}\cos {\left( {4\Delta _{1} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ - J\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {4\Delta _{1} } \right)}\sin {\left( {2\omega _{0} } \right)}\sin {\left( {2\alpha _{2} } \right)}\cos {\left( {2\omega _{0} } \right)} + \frac{J} {2}\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ - \frac{J} {2}\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {4\Delta _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)} + \frac{J} {2}\cos {\left( {2\alpha _{1} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)}\sin {\left( {2\alpha _{2} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + \frac{J} {2}\sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\alpha _{2} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)}\cos {\left( {4\Delta _{1} } \right)} - H\cos {\left( {\Delta _{1} } \right)}\sin {\left( {2\omega _{0} } \right)}\sin {\left( {2\alpha _{2} } \right)}} \hfill} \\ \end{array} $$

(23-2)

$$\begin{array}{*{20}l} {{{C}''} \hfill} & { = \hfill} & {{ - \cos {\left( {2\alpha _{1} } \right)}\cos {\left( {4\Delta _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)} + \cos {\left( {4\Delta _{1} } \right)}\sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\alpha _{2} } \right)}\cos ^{2} {\left( {2\omega _{0} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + 2\cos {\left( {4\Delta _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)}\sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\omega _{0} } \right)}\cos {\left( {2\omega _{0} } \right)} + 2\cos {\left( {2\alpha _{1} } \right)}\cos {\left( {4\Delta _{1} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)}\sin {\left( {2\alpha _{2} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + \cos {\left( {2\alpha _{1} } \right)}\cos {\left( {2\alpha _{2} } \right)} - \cos {\left( {4\Delta _{1} } \right)}\sin {\left( {2\alpha _{1} } \right)}\sin ^{2} {\left( {2\omega _{0} } \right)} + \sin {\left( {2\alpha _{1} } \right)}\sin {\left( {2\alpha _{2} } \right)}} \hfill} \\ \end{array} $$

(23-3)

where

$${G = \lambda ^{4}_{2} + {\left( {\lambda ^{2}_{3} - \lambda ^{2}_{2} } \right)}\lambda ^{2}_{2} + {{\lambda ^{2}_{3} - \lambda ^{2}_{2} } \over 4}}$$

(24-1)

$${H = {{{\left( {\lambda ^{2}_{3} - \lambda ^{2}_{2} } \right)}\lambda ^{2}_{2} } \over 2} + {{\lambda ^{2}_{3} - \lambda ^{2}_{2} } \over 4}}$$

(24-2)

$${J = {{{\left( {\lambda ^{2}_{3} - \lambda ^{2}_{2} } \right)}^{2} } \over 4}}$$

(24-3)

Substituting Eqs. 23-1, 23-2, 23-3 and 24-1, 24-2, 24-3 into Eq. 16, we have

$$\begin{array}{*{20}l} {{I{\left( {\alpha _{1} ,\alpha _{2} } \right)}} \hfill} & { = \hfill} & {{{\int\limits_0^R {{\int\limits_{ - r_{1} }^{R - r_{1} } {\frac{1} {{16}}\lambda ^{2}_{2} \lambda ^{2}_{3} {C}''{\sum\limits_{m = 1}^\infty {{\left( { - 1} \right)}^{{m - 1}} \frac{{{B}''}} {{{A}''}}\exp {\left( {m\frac{{ - {\left| {r_{{12}} } \right|}}} {a}} \right)}} }} }} }} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ \times {\left[ {\cos k{\left\{ {{\left( {A - B} \right)}r_{1} + Br_{{12}} } \right\}} + \cos k{\left\{ {{\left( {A - B} \right)}r_{1} - Br_{{12}} } \right\}}} \right]}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ \times r_{1} {\left( {r_{1} + r_{{12}} } \right)}dr_{{12}} dr_{1} } \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{\lambda ^{2}_{2} \lambda ^{3}_{2} {\left( {{B}''} \right)}^{{m - 1}} {C}''{\left( { - 1} \right)}^{m} - 1}} {{16{\left( {{A}''} \right)}^{m} }}{\sum\limits_{m = 1}^\infty {\left\{ {H_{1} {\left( m \right)} + H_{2} {\left( m \right)} - H_{3} {\left( m \right)}} \right.} }} \hfill} \\ {{} \hfill} & {{} \hfill} & {{\left. { + H_{4} {\left( m \right)} - H_{5} {\left( m \right)} + H_{6} {\left( m \right)}} \right\}} \hfill} \\ \end{array} $$

(25)

where

$${H_{1} {\left( m \right)} = {{2{\left\{ {{a \over m}k^{2} R^{2} \sin {\left\{ {k{\left( {A + B} \right)}R} \right\}} + 2{a \over m}kR{\left( {A + B} \right)}\cos {\left\{ {{\left( {A + B} \right)}R} \right\}} - 2a\;\sin {\left\{ {k{\left( {A + B} \right)}R} \right\}}} \right\}}} \over {{\left\{ {k{\left( {A + B} \right)}} \right\}}^{3} {\left\{ {{\left( {{a \over m}} \right)}^{2} {\left( {kB} \right)}^{2} + 1} \right\}}}}}$$

(26-1)

$${H_{2} {\left( m \right)} = {{4{\left\{ {{\left( {{a \over m}} \right)}^{3} k^{2} {\left( {A + B} \right)}BR\cos {\left\{ {k{\left( {A + B} \right)}R} \right\}} - a^{3} \;kB\sin {\left\{ {k{\left( {A + B} \right)}R} \right\}}} \right\}}} \over {{\left( {kB} \right)}^{2} {\left\{ {{\left( {{a \over m}} \right)}^{2} {\left( {kB} \right)}^{2} + 1} \right\}}^{2} }}}$$

(26-2)

$$H_{3} {\left( m \right)} = \frac{{4{\left\{ {{\left( {\frac{a} {m}} \right)}^{3} k^{2} B{\left( {A - B} \right)}R\cos {\left\{ {k{\left( {A - B} \right)}R} \right\}} - {\left( {\frac{a} {m}} \right)}^{3} \;kB\sin {\left\{ {k{\left( {A - B} \right)}R} \right\}}} \right\}}}} {{{\left\{ {{\left( {\frac{a} {m}} \right)}^{2} {\left( {kB} \right)}^{2} + 1} \right\}}^{2} {\left\{ {k{\left( {A - B} \right)}} \right\}}^{2} }}$$

(26-3)

$$ \begin{array}{*{20}l} {{H_{4} {\left( m \right)} = } \hfill} & {{\frac{{2{\left( {\frac{a} {m}} \right)}^{2} {\left\{ {{\left( {\frac{a} {m}} \right)}^{2} B^{2} - 1} \right\}}{\left( {\frac{a} {m}} \right)}e^{{ - mR \mathord{\left/ {\vphantom {R a}} \right. \kern-\nulldelimiterspace} a}} }} {{{\left\{ {{\left( {\frac{a} {m}} \right)}^{2} B^{2} + 1} \right\}}^{2} {\left\{ {{\left( {\frac{a} {m}} \right)}^{2} A^{2} + 1} \right\}}^{2} }}\left[ {\cos AR{\left\{ {A^{2} {\left( {\frac{a} {m}} \right)}^{3} - A^{2} R{\left( {\frac{a} {m}} \right)}^{2} - {\left( {\frac{a} {m}} \right)} - R} \right\}}} \right.} \hfill} \\ {{} \hfill} & {{\left. { + {\left( {\frac{a} {m}} \right)}A\sin AR{\left\{ {{\left( {\frac{a} {m}} \right)}^{2} RA^{2} + 2{\left( {\frac{a} {m}} \right)} + R} \right\}}} \right]} \hfill} \\ \end{array} $$

(26-4)

$$H_{5} {\left( m \right)} = \frac{{{\left( {\frac{a} {m}} \right)}^{2} {\left\{ {{\left( {\frac{a} {m}} \right)}^{2} A^{2} - 1} \right\}}}} {{{\left\{ {{\left( {\frac{a} {m}} \right)}^{2} A^{2} + 1} \right\}}^{2} }}$$

(26-5)

$$ \begin{array}{*{20}l} {{H_{6} {\left( m \right)} = } \hfill} & {{\frac{1} {{{\left\{ {{\left( {\frac{a} {m}} \right)}^{2} A^{2} + 1} \right\}}^{2} {\left\{ {{\left( {\frac{a} {m}} \right)}^{2} B^{2} + 1} \right\}}^{2} }}\left[ {2{\left( {\frac{a} {m}} \right)}^{2} \left\{ { - A^{2} {\left( {\frac{a} {m}} \right)}^{3} e^{{ - mR \mathord{\left/ {\vphantom {R a}} \right. \kern-\nulldelimiterspace} a}} + A^{2} {\left( {\frac{a} {m}} \right)}^{3} \cos {\left( {AR} \right)}} \right.} \right.} \hfill} \\ {{} \hfill} & {{ + A^{3} {\left( {\frac{a} {m}} \right)}^{3} R\sin {\left( {AR} \right)} + A^{2} {\left( {\frac{a} {m}} \right)}^{2} R\cos {\left( {AR} \right)} - 2A{\left( {\frac{a} {m}} \right)}^{2} \sin {\left( {AR} \right)} - {\left( {\frac{a} {m}} \right)}\cos {\left( {AR} \right)}} \hfill} \\ {{} \hfill} & {{\left. { + A{\left( {\frac{a} {m}} \right)}R\sin {\left( {AR} \right)} + {\left( {\frac{a} {m}} \right)}R - Rm/a + R\cos {\left( {AR} \right)}} \right\}\left\{ {B^{2} {\left( {\frac{a} {m}} \right)}^{3} \cos {\left( {BR} \right)}} \right.} \hfill} \\ {{} \hfill} & {{ + B^{3} R{\left( {\frac{a} {m}} \right)}^{3} \sin {\left( {BR} \right)} - B^{2} {\left( {\frac{a} {m}} \right)}^{2} r\cos {\left( {BR} \right)} + 2{\left( {\frac{a} {m}} \right)}B\sin {\left( {BR} \right)} - {\left( {\frac{a} {m}} \right)}\cos {\left( {BR} \right)}} \hfill} \\ {{} \hfill} & {{\left. {\left. { + BR{\left( {\frac{a} {m}} \right)}\sin {\left( {BR} \right)} - R\cos {\left( {BR} \right)}} \right\}} \right]} \hfill} \\ \end{array} $$

(26-6)

In numerical calculation, the intensity is normalized by R ⁴ and H ₁(m), for example, is rewritten as follows:

$$H_{1} {\left( m \right)} = \frac{{2{\left\{ {\frac{a} {{mR}}{\left( {\frac{{2\pi R}} {{{\lambda }'}}} \right)}^{2} \sin {\left\{ {\frac{{2\pi R}} {{{\lambda }'}}{\left( {A + B} \right)}} \right\}} + \frac{{2a}} {{mR}}{\left( {\frac{{2\pi R}} {{{\lambda }'}}} \right)}{\left( {A + B} \right)}\cos {\left\{ {\frac{{2\pi R}} {{{\lambda }'}}{\left( {A + B} \right)}} \right\}} - \frac{{2a}} {R}\sin {\left\{ {\frac{{2\pi R}} {{{\lambda }'}}{\left( {A + B} \right)}} \right\}}} \right\}}}} {{{\left\{ {\frac{{2\pi R}} {{{\lambda }'}}{\left( {A + B} \right)}} \right\}}^{3} {\left\{ {{\left( {\frac{a} {{mR}}} \right)}^{2} {\left( {\frac{{2\pi R}} {{{\lambda }'}}B} \right)}^{2} + 1} \right\}}}}$$

(27)

Similar treatment can also be applied to H ₂(m)–H ₆(m).