kinetics of crystallization • majority of studies on ... of crystallization • majority of...
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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.