Kinetics of Crystallization

• Majority of studies on 'crystallization' are really thecombined processes of nucleation and growth.

• Considerable commercial interest in the rate oftransformation from the melt to the semi-crystallinestate. -Processing.

• What are the effects of various additives oncrystallization rate, nucleation aids, fillers,lubricants, antioxidants, anti-statics, colors, etc.

• Follow kinetics of crystallization using any methodto 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) andmake certain assumptions regarding thecrystallization process.

• The Avrami approach tries to calculate the volumeof material that crystallizes as a function of time;allows for impingement.

The Avrami equation

• Convert from one phase (polymer melt) to a secondphase (semi-crystalline polymer).

Assumptions

• Random nucleation in space; no preferentialnucleation on the walls of the crystallizing vessel.

• Time dependence of nucleation is either:-Zero order; all nuclei form instantaneouslyFirst 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.

Assumptions

• Rate of crystal growth is first order with time in theprimary growth direction. r = G. tr = the radius of a sphere or plate; or rod lengthG = growth constant and t = time

• Density of the second phase (semi-crystal) doesn'tchange with time; it's independent of how muchmaterial has crystallized.

• DefineWl = Wt. of polymer yet to transform (crystallize)Wo = Total Wt. of crystallizable polymer.

Wl/Wo = exp(-Z.tn)Z = Rate Constantt = timen = 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

Experimental Approach

• Polymer sample starts in bathabove Tm°.

• Place "J" tube in bath tocrystallize at Tc.

• Follow crystallization bychange in level of mercury.

• Plot mercury height vs. time.

Time

Height

t=0

ho

ht

h∞

t

Induction

period

• Use dilatometer heights as a measure of amount ofpolymer transforming (crystallizing), then:-

Wl/Wo = (ht - h∞) / (ho - h∞) = exp( -Z.t )n

• 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, soplot:- ln(-ln(ht - h∞)/(ho - h∞)) vs. ln t

Slope = n and Intercept = ln Z

• Using narrow molecular weight fractions, in thiscase poly(ethylene adipate), generally haveexcellent agreement with Avrami approach.

19°C

35°C

≈40°C

44°C

47°C

0 1 2 3 log time, min

Log(-log(Wl/Wo)

0

-1

-2

• Data fits straight line (n = 4) over a wide range oftemperatures and conversions.

• When unfractionated polymer is used, data hasgreater deviation from Avrami type approach.

16°C

40°C

35°C

25°C

1 2 log time, min

Log (-log(Wl/Wo)

0

-1

-2

• Data may fit a straight line only over a smalldistance. This corresponds to fitting the Avramiapproach only over a small amount ofcrystallization, in this case ≈ 50% conversion.

• In special cases the data may fit a straight line usingthe Avrami type plot, however, the straight line hasa none integer value.

n = 3

n = 4

Log (-log(Wl/Wo)

0

-1

Log time

99.9

50

5

% Cryst

• Poly decamethylene terphthalate crystallized at≈123°C fits a straight line from 0 to 99.9%conversion,

BUT n = 3.587

Deviations from Avrami relationship

• a) Data fits portions of an Avrami plot with integervalues, but deviates at high conversion.

b) Avrami type plots show a linear fit over allconversions but have non-integer values of slope

• Type 'a' deviations can be explained by having atleast two different kinds of crystal growth.

1) different nucleation mechanisms2) different kinds of spherulite3) rod to disk to spherulite conversion

• Type 'b' deviations are inconsistent with theory.

In support of Avrami

• 'n' is never greater than ≈4

• Difficult to obtain information about the nature ofcrystal growth in polymers. Deduce from n.

• Need a practical measure of the effect oncrystallization rate of various additives.

Use 't1/2' or 'induction time'

Fracn.Transformed

0

0.2

0.4

0.6

0.8 120°C 125°C

128°C

129°C

0 500 Time, mins

10 10,000 log time, mins

0

0.2

0.4

0.6

0.8

Fracn.

Transformed

120°C 125°C

128°C

129°C

• Note the influence of temperature on rate ofcrystallization for this sample of linear PE.

Secondary crystallization

• A number of polymers, including PE seem to havebig deviations from Avrami behavior.

• Instead of crystallizing to some constant level h∞,the polymer crystallizes but with a significantchange in slope. See the solid line below.

Time

Height

t=0

ho

ht

h∞

t

Induction period

• Treat by proposing a value of h∞ (guess) and seehow much of the data fits an Avrami plot.

• Adjust h∞ to force most of the data to fit Avrami.

0.01 1 100Time, hr

0

0.2

0.4

0.6

0.8

Fracn.Transformed

Log(-Log(Wl/Wo)

0

-1

-2

-3

0.01 1 100Time, hr

0.9

0.5

0.1

0.01

Ws/Wo

• Ln (-ln) plots can be misleading; it appears thatmore material fits Avrami. In the above only 50%.

Why does crystallinity increaseafter the first Avrami type response?

• a) Crystallization of a more difficult to crystallizecomponent of the polymer; or

b) Initially formed imperfect crystals and later thesecrystals improved their perfection.

• Explain type 'a' by crystallization of rejectedimpurities, more branched material, chainscontaining more comonomer units.

• Expect that crystallization of such 'defective' chainsshould lead to defective crystals. Such crystalsshould have lowered melting points!!

• Experimentally, observe an increase in meltingpoint with crystallization. Favors crystalsreorganizing to a more perfect, and thereforehigher melting state.

Dependence of crystallization rate ontemperature

-60 -40 -20 0 20

Temperature (°C)

Cryst. Rate, hr -1

0.4

0.2

0

Dependence of crystallization rate ontemperature

• General form is a bell-shaped curve; anchored atboth ends by Tg and Tm° for the polymer.

• Rate initially INcreases as temperature is loweredfrom Tm° because thermodynamic driving force forcrystallization increases.

• As temperature is lowered further, chains find itincreasingly more difficult to move about and formcrystals; melt viscosity increases. At Tg there's nochain translation –> no crystallization.

Dependence of crystallization rate onmolecular weight

• From polysiloxane data; higher molecular weightslead to reduction in crystallization rate.

• Also note the characteristic bell shaped curve ofrate vs. temperature for any molecular weight.

≈10 4

≈5 x 10 4

≈10 6

0 50 100 Temp (°C)

Cryst. Rate

1,000

500

0

130°C

128°C

125°C

119°C

115°C

4 5 6 7 log Mv

τ , 0.01

3

2

1

0

-1

Dependence of crystallization rate onmolecular weight

• Plotted for PE samples is the time to reach 1%conversion (1/rate) as a function of temperature andmolecular weight.

• Starting with low mol. wt. :-rate initially ↑ (time ↓) asmol. wt. increases;

further increases in mol. wt.lead to rate ↓. (polysiloxane)

• Extent of change depends oncryst. temperature.

• Avrami crystallization curve is even more complex.

Different molecular weight ranges fit differentAvrami exponents. Within a range, increaseddeviation from fit as molecular weight increases.

n = 4 n = 3 n = 2

≈5K

≈11K

≈20K

≈284K

≈660K

≈1,200K ≈5x10 6

≈8x10 Tc = 130°C 6

Log time

1.0

0.5

0

Fraction converted

Dependence of crystallization rate onorientation/draw

• Shown is crystallization of natural rubber at 0°C forvarious extensions.

0%

≈100%

≈700%

0 320 560 Time, hr

Density Change %

3.0

2.0

1.0

0

• As extension ratio ;rate of crystallization .

• Extension forces coiledchains to be moreelongated. If chains aremore ‘linear’ it’s easier tocrystallize.

Dependence of crystallization rate onbranching or composition

• Overall shape of crystallization curve fits Avramitype relationship.

• However, compare temperatures at which branchedPE crystallizes (102 - 108°C) vs. temperaturesreported earlier for linear PE (120 - 130°C).

Fracn.

Transformed

0

0.4

0.8

≈108°C

≈106°C ≈102°C

Branched PE

10 100 1,000 10,000

Time, mins

• Under the same conditionsbranched PE crystallizesmore slowly than linear PE.

• Copolymer crystallizesslower than homopolymer.

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

Isothermal at Tc

a

T 1

t = 0

Total Area = A

Time

Induction Time

Time taken to Equilibrate to Crystallization Temp. (Tc)

Isothermal at (above Tm°) T 1

Fraction cryst. at time 't' = Xt = a/A = 1- exp(-Ztn)

Problems

• Defining t=0. If crystallization rate is high at someparticular Tc; the thermogram may not reachequilibrium temperature before crystallizationstarts.

• Baseline construction. If crystallization rate is verylow it's difficult to observe a peak spread over alarge distance along the time axis.

Modifications of Rate Methods

• Like to quickly determine effect of additives andother 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 polymermelt to transform to semicrystalline material.

Non Isothermal• Use DSC or DTA to directly determine changes in

crystallization peak temperature, on cooling atsome fixed rate from the melt.