Non-Isothermal Crystallization Kinetics of Polyether Ether Ketone (PEEK)

Non-isothermal crystallization kinetics of a commercial machine screw made from polyether ether ketone (PEEK) was determined using the Avrami and Ozawa models. Activation energies were calculated using rate constants obtained from the Avrami rate constant as well as an isoconversional method using instantaneous crystallization rate data based on Friedman’s method. Both methods for determining activation energy show good agreement. Finally, the Lauritzen-Hoffman parameters for non-isothermal crystallization were calculated. The results suggest complex nucleation that contains primary and secondary crystallization processes.

Differential scanning calorimetry (DSC) is a practical tool for studying the crystallization kinetics of polymers. The rate of crystallization is proportional to the heat released during a given time interval so the fraction crystallized as a function of time or temperature can be determined by the partial area divided by the total area of the crystallization exotherm. Isothermal kinetics is the commonly used method of obtaining kinetics parameters, but the non-isothermal kinetics experiment offers some practical advantages, such as simplified experimental design. Instead of holding at an isothermal temperature, the cooling rate is varied in a series of DSC runs and crystallization temperatures at some level of conversion are obtained. Cooling rates are chosen that cover a range of at least 10 °C. Samples need to be heated and held sufficiently above the equilibrium melting temperature to destroy the thermal history. This temperature can be determined experimentally [1] after determining the decomposition temperature of the polymer. Care must also be taken to choose cooling rates and sample masses to minimize errors due to thermal inertia.

For the DSC experiment, the crystallization occurring during cooling at a time t and the maximum crystallization possible at infinite time can be expressed as the fractional area of the crystallization exotherm (X(t)) normalized to the entire exotherm area as shown in Equation 1.

Where dH_C/dt is the heat flow measured by the DSC and ΔH_C is the total heat of crystallization. Equation 1 describes the fraction crystallized during the isothermal crystallization experiment.

Many authors have proposed macro kinetic models for determining crystallization kinetics under non-isothermal conditions, including Ozawa, Liu, and Jeziorny; often these models modify the Avrami equation for the non-isothermal experiment. Others have used the Avrami equation to model non-isothermal crystallization [2, 3]. It is important to note that the Avrami parameters generally do not have the same meaning in the non-isothermal experiment as they do in the isothermal experiment but may be useful in empirical comparisons and relatable to other polymer properties.

For non-isothermal data, Equation 2 describes the fraction converted as a function of temperature:

where:
X(T) is volume fraction crystallized at temperature T
ΔH_C = Overall heat of crystallization – area under the crystallization curve of the DSC experiment
dH_C = enthalpy of crystallization released during infinitesimal temperature range dT
T₀ = the temperature at crystallization onset taken from the DSC curve
T = temperatures during the crystallization process

The crystallization temperatures at some degree of conversion (typically X(T) = 0.5) are obtained from the DSC data.

Transforming the data from the temperature domain to the time domain is done by using Equation 3.

where:
t = time in minutes or seconds
T₀ = temperature at crystallization onset (arbitrarily set at X(t) = 0.01 for this data set)
T = Temperature during crystallization
β = cooling rate (°C/min or °C/s)

The Avrami equation is the most common macro kinetic model used to evaluate isothermal crystallization data. For non-isothermal data, the Avrami equation does yield useful empirical information when comparing the rate constant and geometric exponent of different samples. The Avrami parameter ‘k_A’ is a rate constant describing the rate of crystallization, the parameter ‘n_A’ describes the nucleation geometry and typically ranges from 1 to 4– where 1-2 describes rod-like structure, 2-3 describe planar structure, and 3-4 describe sphere-like structure. There is much in the literature regarding interpreting the Avrami geometric exponent.

The Avrami equation is shown in Equation 4:

where:
X(t) = fraction crystallized as a function of time
k_A = Avrami Rate Constant (function of nucleation and crystal growth rate)
n_A = Avrami Exponent (function of growth geometry)
t = time

The Avrami equation can be linearized to form Equation 5:

A plot of the log (-ln(1-X(t)) versus log t is linear (typically 0.2 ≤X(t)<= 0.8) and yields the Avrami parameters k_A (antilog of intercept) and n_A (slope).

The rate constant k_A can be corrected using a method suggested by Jeziorny [4] to account for the differences in heating rate by Equation 6.

Where log k_C is the log of the corrected rate constant and β is the cooling rate.

Crystallization half time (t½) can be calculated from the Avrami data and is expressed as Equation 7:

Sometimes it is more convenient to express a crystallization rate (t) (min^-1), which is simply the reciprocal half-time (Equation 8):

Ozawa modified the Avrami equation to model the non-isothermal kinetics experiment [5]. The relative crystallinity as a function of temperature can be expressed as Equation 9:

The Ozawa equation can be linearized, as shown in Equation 10:

where:
β is the heating rate (°C / min)
K_O is the Ozawa rate constant (min^-1)
n_O is the Ozawa geometric exponent

The crystallization activation energy can be determined by using the obtained rate data in the Arrhenius equation (Equation 11):

where:
ΔE = Crystallization Activation Energy (kJ/mol)
R = Gas Constant (8.314 J / mol K)
T = Crystallization Temperature (K)

k_A^1/nA = Avrami Rate Constant (min^-1)
n_A = Avrami Exponent
τ = Reciprocal half-time (min^-1)

(Friedman isoconversional method)

The differential method is the isoconversional one described by Friedman [7, 8], which follows the change in activation energy as a function of conversion and is described by Equation 12. Taking the log of this equation yields the linearized version, shown in Equation 13. The instantaneous crystallization rate is obtained from DSC data or can be easily generated using Excel™.

The linear growth rate (G) is dependent on temperature (T) and is described by Equation 14.

where:
G is the crystallization growth rate as a function of temperature
G₀ is the preexponential
U* is the activation energy for molecular transport across the melt/solid interfacial boundary and is given by 6.276 kJ /mol
T∞ is the temperature where long range molecular motion ceases (estimated at 50K below the glass transition temperature in this work)
f is a correction factor (Equation 15):

where T⁰_M is the equilibrium melting temperature
K_G is a combined factor related to crystal growth, calculated using Equation 16:

where:
n = 4 for crystallization regimes I and III, and 2 for regime II
b = single layer thickness
σ = lateral surface free energy
σ_e = fold surface free energy
ΔH_F = heat of fusion per unit volume
k_B = the Boltzmann constant

The growth rate G is equivalent to the instantaneous crystallization (dX(t)/dT) rate calculated from the DSC data using Equation 17 [6]:

For the non-isothermal experiment, Vyazovkin [7] proposed a method to calculate the Lauritzen-Hoffman parameters by utilizing the isoconversional method of Friedman to obtain the activation energy as function of conversion at different cooling rates (Equation 18).

Where ΔE_X(T) is the activation energy at extent of conversion and T is the average temperature of where the specific extent of conversion occurs at the cooling rates.

Sample

A commercially available machine screw made from PEEK was used for these experiments. A previous TGA investigation [8] determined that it contained 21.9% filler.

Instrument and Sample Preparation

The DSC experiment was performed using a TA Instruments™ Discovery™ DSC 2500 using nitrogen gas at a purge rate of 50 mL /min. Samples of 3 mg nominal mass were encapsulated using TZero™ Aluminum Pans. Each sample was used once to minimize any effects of thermal degradation.

Samples were heated to 400 °C and held isothermally to destroy process thermal history and any retained structural memory. Samples were cooled to 0 °C using cooling rates from 20 to 1 °C/min.

Data Reduction

The crystallization exotherms are integrated using TRIOS™ Software and data exported to Microsoft™ Excel™ Software. Calculations for all analyses are determined from the DSC data. Typically, the conversion temperature is 50% converted or halftime. In the non-isothermal experiment, the peak temperature is not a reliable estimation of 50% converted as the exotherm is seldom symmetrical. An accurate value can be determined from the raw data utilizing the TRIOS software running integral tool, or from the raw data. Tools in Excel Software can be utilized to automate the process. All calculations, including data-fitting, were done using Excel Software.

Figure 2 shows an overlay of the non-isothermal crystallization exotherms at cooling rates from 1 to 20 °C/min.

Avrami Analysis

Figure 3 shows the fraction crystallized (Equation 2) as a function of temperature. The crystallization data can be converted to the time domain using Equation 3.

The time-based data is then plotted using the linearized form of the Avrami equation (Equation 5) and the rate constant (k_a) and geometric constant (n_a) can be determined from the intercept and slope of the line.

The data is plotted in Figure 4. The data is reasonably linear particularly at the higher cooling rates, but toward the end of the crystallization process, the slope of the line decreases. This is more evident at the slower cooling rates. For this reason, we will treat the crystallization process as two different crystallization regimes (primary and secondary) and will use the Avrami equation to fit each independently. An example of the data treatment for the sample run at a cooling rate of 5 °C/min is shown in Figure 5.

Typically, the linear form of the Avrami equation fits well in the limits of 0.2 <=X(t)<=0.8. Figure 5 shows an example of fitting each crystallization regime separately. Table 1 shows estimated values of X(t) where the primary crystallization regime changes to predominantly secondary indicated by the change in slope of the line. On average, this change of slope occurs at X(t) = 0.67.

A summary of the Avrami non-isothermal kinetics data is shown in Table 2.

Table 1. Fraction crystallized during primary and secondary crystallization

Table 2. Avrami constants n_A and k_A as function of cooling rate

The PEEK sample shows two distinct Avrami exponents in the primary and secondary crystallization regimes indicative of two crystallization mechanisms. The Avrami exponent n_A is approximately 2.5 during the primary crystallization regime indicating a spherulitic morphology and approximately 1.4 during secondary crystallization indicating a rod-like to planar morphology. Both the primary and secondary regimes show little cooling rate dependence on values of n_A (Figure 6).

To obtain the proper units of reciprocal time, the rate constant must be corrected by raising to the 1/n power. Corrected values are summarized in Table 3.

Figure 6. Avrami exponent n_A as function of cooling rate

Table 3. Corrected Avrami rate constants

Alternatively, the corrected rate constant can be obtained in a single step by fitting the following modified Avrami equation [9] shown as Equation 19.

The Avrami rate constant k_A shows cooling rate dependence in both the primary and secondary crystallization regimes attaining higher values in the primary compared with the secondary. This is not unexpected as significant impingement occurs in the secondary regime (Figure 7).

Crystallization half times were calculated using the Avrami parameters and are shown in Table 4 and Figure 8. Crystallization half-times show an apparent asymptotic decay as cooling rate increases. This asymptotic decay indicates that higher cooling rates in the non-isothermal experiment may be a reasonable approximation of processing conditions in correlating the Avrami parameters with end-use properties for our sample.

Figure 7. Avrami rate constant k_A^1/n as function of cooling rate

Table 4. Summary of crystallization half-times

Ozawa Analysis

The Ozawa analysis was carried out in the temperature range of 286 to 308 °C, shown in Figure 9. The results are shown in Figure 10 and summarized in Table 5. The geometric exponent increases n_A from ca. 4 to 6 ending at 3, suggesting a temperaturedependent complex combination of nucleation and growthcontrolled processes. This likely includes competitive athermal nucleation processes dependent on the degree of undercooling [10]. The filler may be a contributing factor to the heterogeneous nucleation process, as may residual un-melted crystal structure which contributes to athermal nucleation. We demonstrated the importance of heating the sample to a temperature significantly higher than the equilibrium melting temperature in a previous application note [1], and in this work we heated to 400 °C, which is above the estimated equilibrium melting temperature of 380 °C [11], but may not be enough to completely eradicate residual structure. Similar results on polytrimethylene terephthalate (PTT) were obtained in another work [12].

As expected, K_O increases with undercooling but K_O^1/No remains fairly constant as shown in Figure 10.

Table 5. Summary of Ozawa data

Figure 10. Temperature dependence of K_O and n_O

Calculation of Activation Energy

Avrami Rate Constant k_A
Activation energy calculated using the Avrami rate constant k_A^1/nA using Equation 11 is -248 kJ/mol, data is plotted in Figure 11.

Peak crystallization rates appear to occur between X(t) = 0.4 to 0.5. (Figure 12).

Figure 11. Activation energy using k_A

Activation Energy Using Isoconversional Method
The activation energy as a function of conversion is shown Figure 13. Energy increases as conversion proceeds until a sharp drop at approximately X(t) = 0.7 corresponding to the primary to secondary crystallization regime change.

Plotting the data from X(t) 0.75 to 0.85 clearly shows the activation energy follows the primary to secondary regime change.

Table 6. Summary of activation energy calculations

Calculation of Lauritzen-Hoffman Parameters
The Lauritzen-Hoffman parameters were calculated using Equation 18 and are shown in Table 7. Data is plotted in Figure 15. Calculations were limited to the primary crystallization regime. In Analysis 1, the value of U* was set to the common value of 6.276 kJ / mol, in Analysis 2, the value of U* was calculated by allowing the software to find a solution¹.

¹Data fitting was done using Solver tool in Excel Software.

Table 7. Lauritzen-Hoffman parameters for PEEK

The crystallization kinetics of PEEK were investigated using the Discovery DSC 2500. Advanced TZero technology results in a stable baseline that is necessary for precise integration of exotherms, especially at lower degrees of conversion.

The Avrami equation is useful in non-isothermal kinetics analyses for empirical data that could potentially be used for assessing effects of additives, fillers, or direct comparison between polymer formulations. For our data set, the activation energy determined from the Avrami rate constant compares well to that generated using Friedman’s isoconversional method.

The Ozawa method fit our PEEK data set well through a broad temperature range with good correlation coefficients. It shows the temperature dependence of the geometric exponent with indications of complex nucleation and potential residual structure. The Friedman isoconversional method for evaluating activation energy as a function of conversion also fit our data well through a broad range of conversion including a significant drop in activation energy during the change from primary to secondary crystallization. The Lauritzen and Hoffman parameters were determined using a non-isothermal approach but only fit the primary crystallization well. The asymptotic decay of the half-time with increasing cooling rate indicates that the non-isothermal experiment may be a reasonable approximation of processing conditions and potential effects of fillers, pigments, and polymer blends by assessing nucleation geometry, rate constants, and activation energy. Correlation with physical properties such as modulus are the logical extension of this study.

The ease of experimental design, automation, and fast turnaround make non-isothermal analyses attractive to a busy analytical laboratory. Data is easily reduced using a spreadsheet, which is an attractive alternative to expensive kinetics software.