Determination of non-isothermal crystallization rate constant of a rotational molding grade LLDPE

W2068120905 · 수집 2026-10-04 18:12:19

T10389 Injection Molding Process and Properties

CoT 라벨링

라벨링 전

PDF 파일

자동 다운로드한 PDF PDF 보기

기본 정보

DOI 제목 발행년도 저널 URL 페이지 열기 PDF URL PDF 열기 발행기관 피인용수 라이선스

추가 정보 초록

The purpose of the present paper was to test the validity of the nonlinear regression method for calculating the non-isothermal crystallization rate constant of the Nakamura's model of a rotational molding grade LLDPE directly from non-isothermal crystallization experiments carried out in a single cell DSC. Cooling rates of 50, 40, 30, 20, 10 and 5 oC/min were used with samples of 3.0 mg under nitrogen atmosphere. Here, good agreement was observed between the experimental relative crystallinity curves and the simulated ones using the calculated parameters by nonlinear regression. It shows that this method can be used to determine the Nakamura's non isothermal rate constant for using in simulation of the cooling phase of rotational molding. In this paper it was used 10-3, 10-4 and 10-14 as the initial crystallinity in the Nakamura's model. However the best average results for all cooling rates was obtained when 10-4 was used. Average spherulitic dimensions of LLDPE studied in this paper did not change significantly with different cooling conditions.

본문

ISSN 1517-7076 artigo 11589, pp.203-211, 2015

# Determination of non-isothermal crystallization rate constant of a rotational molding grade LLDPE

Carlos Alberto Silva de Lima 1, Marcel Andrey de Goes 2,

Luís Antonio Pinheiro 3, Benjamim de Melo Carvalho 4

1,2 Graduate Program in Materials Science and Engineering – State University of Ponta Grossa - UEPG
e-mail: carlos_de_lima@gmail.com; marcel.pg@hotmail.com
3,4 Materials Engineering Department – UEPG, Av.Carlos Cavalcanti 4748, Uvaranas, CEP 84030-900, Ponta Grossa -PR.
[e-mail: lapinheiro@uepg.br benjamim@uepg.br](mailto:lapinheiro@uepg.br)

ABSTRACT
The purpose of the present paper was to test the validity of the nonlinear regression method for calculating
the non-isothermal crystallization rate constant of the Nakamura’s model of a rotational molding grade
LLDPE directly from non-isothermal crystallization experiments carried out in a single cell DSC. Cooling
rates of 50, 40, 30, 20, 10 and 5 oC/min were used with samples of 3.0 mg under nitrogen atmosphere. Here,
good agreement was observed between the experimental relative crystallinity curves and the simulated ones
using the calculated parameters by nonlinear regression. It shows that this method can be used to determine
the Nakamura’s non isothermal rate constant for using in simulation of the cooling phase of rotational molding. In this paper it was used 10-3, 10-4 and 10-14 as the initial crystallinity in the Nakamura’s model. However the best average results for all cooling rates was obtained when 10-4 was used. Average spherulitic dimensions of LLDPE studied in this paper did not change significantly with different cooling conditions.

Keywords: Nonlinear regression method, non-isothermal crystallization, Nakamura’s model, linear low
density polyethylene, rotational molding.

1. INTRODUCTION

One of the difficulties for incorporation of the crystallization phenomenon in softwares for simulation of polymer processing is the necessity of easy and reliable methods for determining the parameters of a given kinetic model chosen to describe the crystallization process.

The nonlinear regression method can be used for obtaining the parameters of kinetic crystallization
models, such as Nakamura’s [1,2], Kamal and Chu’s [3], Dietz [4] and Malkin’s [5], directly from the nonisothermal crystallization data obtained by Differential Scanning Calorimeters. Results of this method are
usually considered good. However, literature mentions unsatisfactory results related to nonlinear regression,
but this behavior is more likely associated to limitations of the crystallization model in describing the nonisothermal crystallization process of the polymer than by the nonlinear regression method itself [4,7]. One
advantage of nonlinear regression is that its procedure for calculating the crystallization kinetic constant is
faster than the master curve approach [8].

The rotational molding is highly dependent on the raw material and it could not exist without adequate polymers for typical conditions of this process. To be rotomolded, a polymer must have heat resistance
in order to avoid chemical oxidative degradation due to long residence time in the oven. The most common
material used in the process is polyethylene (PE), and in this class stands Linear Low Density Polyethylene
(LLDPE) [9]. This polymer is semicrystalline and its crystallization during the cooling phase of rotational
molding affects both properties and processing conditions.

Therefore, the purpose of the present study was to test the validity of the nonlinear regression method for calculating the non-isothermal crystallization rate constant of the Nakamura’s model of a rotational
molding grade LLDPE directly from non-isothermal crystallization experiments carried out in a single cell

Corresponding Author: Carlos Alberto Silva de Lima Received on: 11/09/2014 Accepted on: 29/09/2014

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

Hot Stage-DSC. This equipment reaches controlled cooling rates up to 100 °C/min and allows the in-situ
recording of the developed crystalline microstructure during cooling. Therefore, it is an interesting alternative
for studying non-isothermal crystallization process of polymers.

2. MATERIALS AND METHODS

_Materials_

A specific Linear low density polyethylene (LLDPE) to be used in rotational molding process was used in
this study. This polymer, kindly supplied by Braskem S.A, has melting point of 125 °C and melt flow index
equal to 4.2g/10min [9].

_Methods_

A Differential Scanning Calorimeter DSC 600 Hot Stage from Linkam was used to study the non-isothermal
crystallization kinetics of LLDPE. About 3.0 mg of the material, sealed in an aluminum pan, was heated to
200°C and held at this temperature for 5 minutes before cooling at different rates (50, 40, 30, 20, 10 and 5
°C/min) to 30°C under nitrogen atmosphere. According the procedure recommended by Isayev et al. [10-12]
the experiments were carried out with the same sample for all the six cooling rates in order to obtain good
repeatability for the heat transfer conditions between DSC furnace and the aluminum pan.

The normal procedure for obtaining the non-isothermal crystallization rate constant was applied
using the _experimental data_ . Therefore, it was necessary to define the temperature of the onset of crystallization for each cooling rate. The relative crystallinity (θ) vs. temperature curve was determined (as well as the
corresponding θ vs. time curve) for each cooling rate by partial area calculation in the experimental DSC
curve. In this way, the experimental derivative curve dθ/dt vs. temperature was obtained.

Nonlinear regression method was used to obtain the parameters (1/t1/2)0 and Kg in equation 1 by fitting the differential form of Nakamura’s model, given by equation 2, to the experimental dθ/dt vs. T:

(1)

(2)

where (1/t1/2)0 is a pre-exponential factor that includes all terms independent of temperature; U is the activation energy for the transport of crystallizing units across the phase boundary; Kg is the nucleation exponent;
T∞ = Tg – 30 K is the temperature below which molecular transport ceases; R is the universal gas constant;
ΔT=Tm 0 -T is the degree of supercooling, f=2T/(Tm 0 +T) is a correction factor accounting for the reduction in
the latent heat of fusion as the temperature is decreased, and Tm 0 is the equilibrium melting temperature.

3. RESULTS AND DISCUSSION

The θ vs. temperature curve was determined by applying the method of partial area calculation. Fig. 1 shows
the experimental crystallization rate dθ/dt as a function of temperature for the six different cooling rates used
in this study.

204

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

Figure 1: Crystallization rates dθ/dt as a function of temperature for LLDPE sample cooled at different cooling rates.

As expected, higher cooling rates led to larger and broader peaks and lower onset and peak temperatures, as well as higher crystallization rates. The non-isothermal crystallization data was used to obtain the
parameters Kg and (1/t1/2)0 by nonlinear regression method. Table 1 shows that the Kg obtained from the
nonlinear regression are different for all six cooling rates. The (1/t1/2)0 for the six cooling conditions are different too, mainly for 50 °C/min. For this last cooling rate, the kinetic parameters are much higher than for
the other cooling conditions. Using the calculated K(T) and assuming n equal to 3, the differential form of
the Nakamura equation, given by Equation 2, could be used to simulate the dθ/dt vs. T curves, which are integrated to obtain the θ vs. T curves. These simulated data were compared to the experimental ones to check
the quality of the kinetic parameters calculated by non-linear regression.

Table 1: Crystallization kinetic parameters of LLDPE obtained by the nonlinear regression.

Cooling Rate (1/t1/2)0 (s -1 ) Kg (K 2 )

5 °C/min 1,65 2,49x10 -5

10 °C/min 1,46 9,9x10 -7

20 °C/min 1,13 4,4x10 -7

30 °C/min 0,98 4,5x10 -8

40 °C/min 0,75 2,2x10 -8

50 °C/min 15,38 4,1x10 4

The interval between 5% and 75% of relative crystallinity on each non-isothermal crystallization
curve was used to determinate the kinetic parameters by nonlinear regression. Fig. 2 and 3 show the comparison between experimental and simulated curves at different cooling rates.

205

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

Fig. 4 shows all curves of relative crystallinity as a function of temperature. In general, simulated
curves presented good agreement with experimental ones (mainly at intermediate rates of 40, 30, 20 and 10
°C/min). In each case, the major divergence between experimental and simulated curves appears at the final
stages of the crystallization process. It is not an unexpected behavior once Nakamura’s model usually has
difficulties to describe the end of crystallization process [6].

The differential form of the Nakamura’s Model, equation 2, shows that for θ equal to zero, the rate
of crystallization dθ/dt is zero. In this case, a negligible θt = 0 must be used in equation 2 to obtain the simulated curves, albeit different from zero. Otherwise, (dθ/dt)t = 0 will be zero and θt + Δt will always be zero, as
clearly demonstrated by Galera et al [8]. They showed that θt = 0 equal to 10 -3 was a good value to be used
instead of 10 -14 suggested by Chan [10]. Therefore in the present study it was tested 10 -3, 10 -4 and 10 -14 as the
initial crystallinity in the Nakamura’s model. However the best average results for the set of cooling rates
was obtained when 10 -4 was used as the initial relative crystallinity and this value was applied for calculating
the simulated curves shown in Figures 2, 3 and 4.

As mentioned by Galera et al [8], the Nakamura’s Model presents an extremely slow initial crystallization rate. Therefore, in the initial stages of crystallization, it takes a considerable amount of time to go
from θ = 10 –14 to θ = 10 –3 . In this case, the use of θinitial equal to 10 –3 or 10 -4 is more convenient because it
can be used with a Tic (temperature at the beginning of the crystallization process in DSC experiments) that
is much easier to define than the one coherent with θ = 10 –14 .

(A)

(B)

206

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

(C)

Figure 2: Comparison between experimental and simulated relative crystallinity curves as a function of temperature gen

erated by the Nakamura’s model: (A) 5 °C/min; (B) 10 °C/min and (C) 20 °C/min.

(A)

(B)

207

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

(C)

Figure 3: Comparison between experimental and simulated relative crystallinity curves as a function of temperature gen

erated by the Nakamura’s model: (A) 30 °C/min; (B) 40 °C/min and (C) 50 °C/min.

Figure 4: Comparison between experimental and simulated relative crystallinity curves for all cooling rates.

Figure 5 shows a sequence of images taken during cooling at 30 ºC/min. It is possible to note the
formation of the crystalline phase along the sample as a function of temperature and it is interesting to compare this sequence with crystallization evolution in Figure 3A. Figure 5A shows that at 133 o C the crystallization process has not started yet, fact coherent with Figure 3A. At 107 o C it is clear that a significant degree of crystallinity is already developed in the sample (Figure 3A) and this hypothesis was confirmed by
real-time observation on microscope at this temperature. In contrast, the visualization of crystalline phase is
not so clear in Figure 5B because the loss of resolution in micrograph compared to the image observed on

208

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

microscope and due to the small crystalline structures developed in this sample. However, an increase in the
depolarized light intensity (not shown here) also confirmed that the crystallization process had already started
at 107 o C. The final crystalline morphology for this sample cooled at 30 o C/min is shown in
Figure 5C.

Figure 6 shows the morphology at 30 o C for LLDPE cooled at 5 o C/min from the melt state. It is
possible to observe that nucleation density is still high and the average spherulitic size did not increase significantly when compared to that developed under cooling of 30 o C/min, as shown on Figure 5C.

(A)

(B)

(C)

209

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

Figure 5: Development of microstructure of LLDPE for the cooling rate of 30 ° C/ min at: (A) 133 ° C; (B) 107 ° C and

Figure 6 : Microstructure at 30 ° C for LLDPE  cooled at of 5 °C/min.

4. CONCLUSIONS

A good agreement was obtained between the experimental and simulated relative crystallinity curves using
calculated parameters by nonlinear regression. It shows that this method can be used to determine the Nakamura’s non isothermal rate constant for using in simulation of the cooling phase of rotational molding. In the
present work it was tested 10 -3, 10 -4 and 10 -14 as the initial crystallinity in the Nakamura’s model. However
the best average results for the set of cooling rates was obtained when 10 -4 was used. For the grade of LLDPE
used in the present work spherulitic dimensions did not change significantly with different cooling conditions. The single cell Hot Stage-DSC used in this paper showed to be an interesting alternative for studying
non-isothermal crystallization process of polymers.

5. ACKNOWLEDGMENTS

The authors would like to thank Braskem S.A. by supplying the LLDPE and Fundação Araucária and
CAPES for the financial support.

6. BIBLIOGRAPHY

[1] NAKAMURA, K., KATAYAMA, K., AMANO, T. “Some aspects of nonisothermal crystallization of
polymers. II. Consideration of the isokinetic condition”, _Journal of Applied Polymer Science_, v.17, n.4,
pp.1031-1041, April. 1973.

[2] PATEL, R.M, SPRUIELL, L.J.E. “Crystallization kinetics during polymer processing: analysis of available approaches for process modeling”, _Polymer Engineering and Science_, v. 31, n.10, pp.730-738. May. 1991.

[3] PÉREZ, C. J., ALVAREZ, V.A., STEFANI, P. M., et al., “Non-isothermal crystallization of materbiz/clay nanocomposites”, _Journal of Thermal Analysis_, v.88, n.3, pp. 825-832, July. 2007.

[4] ALVAREZ, V. A., STEFANI, P.M., VÁZQUEZ, A. “Non-isothermal crystallization of polyvinylalcoholco-ethylene”, _Journal of Thermal Analysis_, v. 79, n.1, pp.187-193, Feb. 2005.

[5] FRIEDL, C.F., MCCAFFEREY, N.J. “Crystallization prediction in injection molding”, In: SPE ANTEC
Technical Papers, pp.330-332, Montreal, May. 1991.

[6] CARVALHO, B., BRETAS, R.E.S., “Crystallization kinetics of a PEEK/LCP blend”, _J. Appl. Polym. Sci_ .,
v.55, n.2, p.233. Jan. 1995.

[7] MUBARAK, Y, HARKIN-JONES, EMA, MARTIN, P. J, et al., “Modeling of non-isothermal crystallization kinetics of isotactic polypropylene”, _Polymer_, v.42, n.7, pp.3171-3182, March. 2001.

[8] GALERA, V. V., LUCAS, A.L., CARVALHO, B. M., “Determination of Non-Isothermal Crystallization
Rate Constant for Pseudo-Experimental Calorimetric Data”, _Materials Research_, v.12, n.2, pp. 151-157, June.
2009.

210

LIMA, C. A.S.; GOES, M. A.; PINHEIRO, L. A.; CARVALHO, B. M. revista Matéria, v.20, n.1, pp. 203 – 211, 2015.

[9] COMISSO, T. B., LIMA, C.A.S., CARVALHO, B. M., “Estudo Experimental do Processo de Rotomoldagem de PELBD: Efeitos sobre a Morfologia e Estabilidade Dimensional”, _Polímeros_, v.23, n.1, pp. 97-107,
Feb. 2013.

[10] CHAN, T.V., SHYU, G.D., ISAYEV A.I. “Master curve approach to polymer crystallization kinetics”,
_Polymer Engineering and Science_, v.35, n.9, pp.733-740, May. 1995.

[11] CARVALHO, B. M, BRETAS R. E. S. “Determinação da constante cinética de cristalização não isotérmica de polipropilenos modificados com ácido acrílico e anidrido maleico”, _Polímeros_, v.16, n.4, pp.305311, October. 2006.

[12] MARINELLI, A. L, CARVALHO, B. M, BRETAS, R.E.S. “Evaluation of the master curve approach
for the non-isothermal crystallization of PP/EPR”, _UEPG Ciências Exatas e da Terra, Ciências Agrárias e_
_Engenharia_, v.10, n.3, pp.13-17, Dez. 2004.

211

본문 출처 PDF: PDF 보기 · 다른 PDF는 위 [PDF 업로드]로 올릴 수 있습니다.

활용가능성

이력 0건

아직 편집 이력이 없습니다. 저장할 때마다 이전 내용이 여기에 백업됩니다.