Kinetics of Crystallization. Majority of studies on 'crystallization' are really the combined processes of nucleation and growth.

Transcription

1 Kinetics of Crystallization Majority of studies on 'crystallization' are really the combined processes of nucleation and growth. Considerable commercial interest in the rate of transformation from the melt to the semi-crystalline state. -Processing. What are the effects of various additives on crystallization rate, nucleation aids, fillers, lubricants, antioxidants, anti-statics, colors, etc.

2 Follow kinetics of crystallization using any method to measure crystallinity:- Density Chain mobility (NMR) 3-D order (X-ray) Chain conformation (IR) Birefringence (OM) Heat of fusion (DSC) Classic approach is to use density (volume) and make certain assumptions regarding the crystallization process. The Avrami approach tries to calculate the volume of material that crystallizes as a function of time; allows for impingement.

3 The Avrami equation Convert from one phase (polymer melt) to a second phase (semi-crystalline polymer). Assumptions Random nucleation in space; no preferential nucleation on the walls of the crystallizing vessel. Time dependence of nucleation is either:- Zero order; all nuclei form instantaneously First order; number of nuclei formed increases linearly with time. Sporadic. Crystal growth is either 1, 2 or 3 dimensional; we either see rods, platelets or spheres.

4 Assumptions Rate of crystal growth is first order with time in the primary growth direction. r = G. t r = the radius of a sphere or plate; or rod length G = growth constant and t = time Density of the second phase (semi-crystal) doesn't change with time; it's independent of how much material has crystallized.

5 Define Wl = Wt. of polymer yet to transform (crystallize) Wo = Total Wt. of crystallizable polymer. Wl/Wo = exp(-z.tn) Z = Rate Constant t = time n = Avrami Exponent The Avrami exponent (n) consists of two terms:- Nucleation(N) is either 0 or 1. Crystallization(C) is 1, 2 or 3, and n = N + C

6 Experimental Approach Polymer sample starts in bath above Tm. Place "J" tube in bath to crystallize at Tc. Follow crystallization by change in level of mercury. Plot mercury height vs. time.

7 Height Induction period ho ht t=0 t h Time Use dilatometer heights as a measure of amount of polymer transforming (crystallizing), then:- Wl/Wo = (ht - h ) / (ho - h ) = exp( -Z.t n )

8 Take ln of both sides twice and reorganize to :- ln(-ln(wl/wo)) = ln(-ln(ht - h )/(ho - h )) = ln Z + n.ln t Above is a linear equation of the form y = c + mx, so plot:- ln(-ln(ht - h )/(ho - h )) vs. ln t Slope = n and Intercept = ln Z

9 Using narrow molecular weight fractions, in this case poly(ethylene adipate), generally have excellent agreement with Avrami approach. Log(-log(Wl/Wo) C 44 C C 40 C 47 C log time, min Data fits straight line (n = 4) over a wide range of temperatures and conversions.

10 When unfractionated polymer is used, data has greater deviation from Avrami type approach. 0 Log (-log(wl/wo) 16 C C 35 C 40 C 1 2 log time, min Data may fit a straight line only over a small distance. This corresponds to fitting the Avrami approach only over a small amount of crystallization, in this case 50% conversion.

11 In special cases the data may fit a straight line using the Avrami type plot, however, the straight line has a none integer value. Log (-log(wl/wo) % Cryst n = 4 n = Log time Poly decamethylene terphthalate crystallized at 123 C fits a straight line from 0 to 99.9% conversion, BUT n = 3.587

12 Deviations from Avrami relationship a) Data fits portions of an Avrami plot with integer values, but deviates at high conversion. b) Avrami type plots show a linear fit over all conversions but have non-integer values of slope Type 'a' deviations can be explained by having at least two different kinds of crystal growth. 1) different nucleation mechanisms 2) different kinds of spherulite 3) rod to disk to spherulite conversion Type 'b' deviations are inconsistent with theory.

13 In support of Avrami 'n' is never greater than 4 Difficult to obtain information about the nature of crystal growth in polymers. Deduce from n. Need a practical measure of the effect on crystallization rate of various additives. Use 't1/2' or 'induction time'

14 Fracn. Transformed C 129 C C 125 C Time, mins

15 0 Fracn. Transformed C 125 C 128 C 129 C ,000 log time, mins Note the influence of temperature on rate of crystallization for this sample of linear PE.

16 Secondary crystallization A number of polymers, including PE seem to have big deviations from Avrami behavior. Instead of crystallizing to some constant level h, the polymer crystallizes but with a significant change in slope. See the solid line below. Height Induction period ho ht h t=0 t Time Treat by proposing a value of h (guess) and see how much of the data fits an Avrami plot. Adjust h to force most of the data to fit Avrami.

17 Fracn. Transformed Time, hr

18 Log(-Log(Wl/Wo) 0-1 Ws/Wo Time, hr 0.01 Ln (-ln) plots can be misleading; it appears that more material fits Avrami. In the above only 50%.

19 Why does crystallinity increase after the first Avrami type response? a) Crystallization of a more difficult to crystallize component of the polymer; or b) Initially formed imperfect crystals and later these crystals improved their perfection. Explain type 'a' by crystallization of rejected impurities, more branched material, chains containing more comonomer units. Expect that crystallization of such 'defective' chains should lead to defective crystals. Such crystals should have lowered melting points!! Experimentally, observe an increase in melting point with crystallization. Favors crystals reorganizing to a more perfect, and therefore higher melting state.

20 Dependence of crystallization rate on temperature Cryst. Rate, hr Temperature ( C)

21 Dependence of crystallization rate on temperature General form is a bell-shaped curve; anchored at both ends by Tg and Tm for the polymer. Rate initially INcreases as temperature is lowered from Tm because thermodynamic driving force for crystallization increases. As temperature is lowered further, chains find it increasingly more difficult to move about and form crystals; melt viscosity increases. At Tg there's no chain translation > no crystallization.

22 Dependence of crystallization rate on molecular weight From polysiloxane data; higher molecular weights lead to reduction in crystallization rate. Cryst. Rate 1, x Temp ( C) Also note the characteristic bell shaped curve of rate vs. temperature for any molecular weight.

23 Dependence of crystallization rate on molecular weight τ, 0.01 Plotted for PE samples is the time to reach 1% conversion (1/rate) as a function of temperature and molecular weight C 115 C 130 C 128 C 125 C log Mv Starting with low mol. wt. :- rate initially (time ) as mol. wt. increases; further increases in mol. wt. lead to rate. (polysiloxane) Extent of change depends on cryst. temperature.

24 Avrami crystallization curve is even more complex. Different molecular weight ranges fit different Avrami exponents. Within a range, increased deviation from fit as molecular weight increases. Fraction converted 1.0 n = 4 n = 3 n = 2 20K 5K 11K 284K K 1,200K 5x Tc = 130 C 8x10 6 Log time

25 Dependence of crystallization rate on orientation/draw Shown is crystallization of natural rubber at 0 C for various extensions. Density Change % % 100% 0% As extension ratio ; rate of crystallization. Extension forces coiled chains to be more elongated. If chains are more linear it s easier to crystallize Time, hr

26 Fracn. Transformed C Dependence of crystallization rate on branching or composition Overall shape of crystallization curve fits Avrami type relationship. However, compare temperatures at which branched PE crystallizes ( C) vs. temperatures reported earlier for linear PE ( C). Branched PE 108 C 106 C ,000 10,000 Time, mins Under the same conditions branched PE crystallizes more slowly than linear PE. Copolymer crystallizes slower than homopolymer.

27 Thermal methods (DSC) approach to kinetics Follow kinetics using modified DSC experiment. **Sample in DSC at temperature T1 (>Tm ). **Quench DSC to crystallization temperature Tc. **Follow crystallization (exotherm) at Tc vs. time. E T 1 Total Area = A t = 0 a Induction Time Isothermal at Tc Time Isothermal at (above Tm ) T 1 Time taken to Equilibrate to Crystallization Temp. (Tc) Fraction cryst. at time 't' = Xt = a/a = 1- exp(-ztn)

28 Problems Defining t=0. If crystallization rate is high at some particular Tc; the thermogram may not reach equilibrium temperature before crystallization starts. Baseline construction. If crystallization rate is very low it's difficult to observe a peak spread over a large distance along the time axis.

29 Modifications of Rate Methods Like to quickly determine effect of additives and other parameters on crystallization rate. Isothermal Define 'Induction Time' as the time at which a small fraction of the material is converted. e.g. Wl/Wo = 0.95 or only 5% of the material has crystallized. Define 't 1/2' as the time for 50% of the polymer melt to transform to semicrystalline material. Non Isothermal Use DSC or DTA to directly determine changes in crystallization peak temperature, on cooling at some fixed rate from the melt.