polymers: calculate degree of crystallinity?

in [1[2,3]] an equation for quiscent crystallisation kinetics, calculating "material derivative" or relative degree of crystallisation is given:

(1*) (D/Dt)theta = n *** K(T) *** (1-theta) * (-ln(1-theta))^((n-1)/n) (D/Dt) is "material derivative" - what is its meaning? theta is the relative degree of crystallisation (theta=theta_abs/theta_max)

is my interpretation right, that (D/Dt)theta is the speed of change in crystallinity degree? my problem is, that if initially theta=0 then (D/Dt)theta=0 everytime, because ln(1-(theta=0))=ln(1)=0? there is obviously something wrong

is this rearrangement of (1***) allowed? (2) (D)theta = (Dt) *** n *** K(T) *** (1-theta) * (-ln(1-theta))^((n-1)/n)

to calculate theta(t)? (3) theta(t+Dt) = theta(t) + (D)theta(t)

in [1[4,5,6,7]] this equations are given for K(T), Delta_T, f and T_inf: (4*) K(T) = (ln2)^(1/n) * (1/t_12) * exp(-U_/(R***(T-T_inf))) *** exp(-K_k/(T*** Delta_T ***f))

Delta_T = T_m_0 - T f = 2*T / (T + T_m_0) T_inf = T_g - 30

R = 8.314 [J/mol K]

for profax PP-6723 (IPP) [1] states: n = 3 avrami-index (1/t_12) = 3.858E8 [1/s] factor: influcences crystallisation independent of T K_k = 3.930e5 [1/K] nucleation exponent U_ = 6284 [J/mol] activation energy for semental jump

i took missing data from [8]: Tg = -3 [K] glass-temperature T_m_0 = 187 [K] equilibrium melting point xi_inf = 0.460 maximum crystallisation degree

substituting with those values leads to huge K(T): T = 250

(ln2)^(1/n) ~ 1 (1/t_12) ~ 1E8 exp(-U_/(R*(T-T_inf))) ~ exp(-6284/8.314*288) ~ 1E-3 exp(-K_k/(T*** Delta_T ***f)) ~ exp(3.9E5/(250***63***1)) ~ 1E10

K(T) ~ 1E15 ? using a simple simulation K(T) is increasing to infinity (for float numers in c).

where is my thinking-error?

I would be glad to get some hints.

Thank you,

Bernd Schmitt

[1] x. guo, a.i. isayev* and i. guo, "crystallinity and microstructure ..." polymer engineering and science, oct 1999, vol. 39, no. 10, p2096ffhttp://130.83.61.160/~robert/temp/ISA99a_CrystallinityAndMicroStructurePartII.pdf [2] k. nakamura, t. watanabe, k. katayama and t. amano, j. appl. polym. sci, 16, 1077 (1972)

[3] k. nakamura, k. katayama and t. amano, j. appl. polym. sci., 17, 1031 (1973) [4] t.w. chan, a.i. isayev***, polym. eng. sci., 34, 461 (1994) [5] t.w. chan, g.d. shyu, a.i. isayev***, polym. eng. sci., 35, 733 (1995) [6] b. wunderlich, "crystal nucleation, ...", academic press, new york (1976) [7] b. wunderlich, macromolecular phzsics, vol. 2, academic press, london (1976) [8] m. moneke, "die kristallisation von verstaerkten thermoplasten ...", dissertation, tu-darmstadt (2001)