15th World Congress on Computational Mechanics (WCCM-XV)
8th Asian Pacific Congress on Computational Mechanics (APCOM-VIII)
Virtual Congress: 31 July – 5 August 2022
S. Koshizuka (Ed.)

SPHERULITE MICROSTRUCTURE FORMATION SIMULATIONBASED ON EFFECT OF MOLDING CONDITIONS ON
POLYLACTIC ACID
LIQIN HONG1, KOICHI TATSUNO2 AND YOSHITERU AOYAGI3
1

Department of Finemechanics, Tohoku University
6-6-01 Aoba, Aramaki, Aoba-ku, Sendai, 980-8579, JAPAN
hong.liqin.p7@dc.tohoku.ac.jp
2

Department of Finemechanics, Tohoku University
6-6-01 Aoba, Aramaki, Aoba-ku, Sendai, 980-8579, JAPAN
k-tatsuno@dc.tohoku.ac.jp
3

Department of Finemechanics, Tohoku University
6-6-01 Aoba, Aramaki, Aoba-ku, Sendai, 980-8579, JAPANaoyagi@tohoku.ac.jp (https://web.tohoku.ac.jp/aoyagi/)

Key words: Crystalline Polymers, Polylactic Acid, Spherulite, Mechanical Properties, Monte
Carlo Simulation
Abstract. This study investigates a simulation model predicting spherulite microstructureformation based on experimental observations. The crystallization process of polylactic acid isobserved at different molding temperatures by in-situ observation, and the rate of spherulite
formation and growth is estimated. Furthermore, according to the Turnbull-Fisher primary
nucleation model and the Lauritzen-Hoffman secondary nucleation model, Monte Carlosimulations of spherulite microstructure formation are performed based on the obtained
experimental results. The validity of the method is verified by a comparison of the simulated
and the experimental results.
1 INTRODUCTION
Thermoplastic polymers are increasingly being used in a wide range of fields because of
their lightweight, low cost, and excellent formability[1][2][3]. Crystalline polymers, which means
thermoplastic polymers with a semicrystalline phase, have a mixture of a crystalline phase witha stable folding structure of molecular chains and an amorphous phase with a disorderedstructure on a microscopic scale. These two layers are alternately stacked to form a structure:lamellar crystal[4][5]. In addition, it is known that lamellar crystals grow radially to produce a
microstructure, which is called a spherulite[6][7]. The mechanical properties of crystalline
polymers are closely related to the dimensions of the spherulite and the crystallinity, and themorphology of spherulite depends on the molding conditions such as temperature andpressure[8][9][10].
Evaluation of the effect of the spherulite microstructure on the mechanical properties of

L. Hong, K. Tatsuno and Y. Aoyagi

crystalline polymers can improve the accuracy of structural analysis. Various studies have been
conducted on microstructure and crystallization models[11][12]. However, since the spherulite isa complicated structure composed of amorphous and crystalline phases, the effects of actual
molding conditions on spherulite microstructure formation and changes in mechanicalproperties depending on spherulite microstructure remain to be clarified.
Various models have been studied to predict the microstructure from the molding conditions.
For example, a prediction model called the phase-field model, which focuses on the structureformation inside spherulite crystals, has been proposed[13]. However, this phase-field model is
only applicable to isothermal crystallization, not to complex temperature conditions. A cellular
automaton model based on microscopic nucleation has also been proposed[14][15]. This model
does not sufficiently consider the effect of temperature on nucleation. Therefore, there are still
many unknowns regarding the effects of temperature, pressure, and material flow on theformation of spherulite structure during the actual forming process.
In this study, we conducted in-situ observation tests at various molding temperatures toconstruct a simulation method for spherulite structure formation of polylactic acid (PLA) basedon experimental evaluations. First, in-situ observation tests were conducted at various molding
temperatures to understand the effect of molding conditions on spherulite formation. Next, atime evolution simulation method of spherulite formation was developed, and its validity was
examined. The parameters used in the numerical simulation were identified based on theobtained experimental results.
2 IN-SITU OBSERVATION TEST OF CRYSTALLIZATION PROCESS
2.1 Specimen preparation for in-situ observation tests
Figure 1 shows a schematic diagram of the specimen. PLA (IngeoRT Biopolymer 4032D,
Nature Works LLC) was used as the material of the specimen. In this study, to evaluate thebehavior of the spherulite structure in bulk material as much as possible, a mold made with 0.45mm thick aluminum foil was placed on a microscope slide to prevent excessive thinning of thepolylactic acid during melting and to maintain a certain thickness. Polylactic acid pellets wereplaced in the hole of the aluminum foil mold and heated at 200 ℃ for 60 minutes using a forced
convection constant-temperature dryer (SOFW-300SB, As one Corporation). By placing acover glass on top of the melted PLA and keeping it at 200 ℃ for further ten minutes, thespecimens used for the in-situ observation test were prepared.

Figure 1 Details of in-situ observation test piece2

L. Hong, K. Tatsuno and Y. Aoyagi

2.2 In-situ observation test using polarized light microscope
A polarizing microscope (KEYENCE Digital Microscope VHX-6000) was used for an insitu observation. The crystalline phase has an optical anisotropy so that the spherulites can beobserved with the polarized light as the bright field of view. Figure 2 shows a schematic diagramof the in-situ observation of the crystallization process of polylactic acid on a heat stage. The
specimens were held at 200 ℃ for 5 minutes to stabilize the temperature, then slowly cooled
for ten minutes until the forming temperature, which was from 80 ℃ to 160 ℃. When it reached
the forming temperature, an observation started, and photographs were taken every 15 seconds.
Since it was difficult to precisely control the molding temperature in each test with the heatstage, temperatures were measured respectively for 28 specimens.
2.3 Spherulite growth rate
The spherulite growth rate is measured by observing the time evolution of the diameter ofspherulites. It is assumed that there is no difference in the growth rate of each spherulite
regardless of the crystallization progress in a specimen. Spherulites that had little contact withthe surroundings were selected from the observed images for analysis. The average value of thechange in diameter with time was obtained for the spherulite growth rate d [m/s]. Since thenumber of spherulites differs in each test because of the forming temperature, 3 to 15 spherulites
were measured in each test.
2.4 Local nucleation rate
Figure 3 shows the schematic diagram of the spherulite number density developing with time.
Note that the nucleation rate is not constant with time because the number of spherulites perunit area decreases as the amorphous area decreases. The graph of spherulite number density isapproximated by

Figure 2 Change in diameter and number of spherulites

Figure 3 Change of the number density of spherulites in an observation area3

L. Hong, K. Tatsuno and Y. Aoyagi n (0)  n(ti )  ns 1  exp  ti  
 ns  

(1)where ns is the spherulite number density at saturation, n is the local nucleation rate, ti is thetime when spherulite formation beginning is thought to be zero, and n (0) is the initial
nucleation rate corresponding to the gradient at ti  0 in Fig. 3. The following nucleation rate
was derived by differentiating Eq. (1).

 n (0) n (ti )  n (0) exp  ti  ns 

(2)

The local nucleation rate at ti  0 , which is independent of the number of spherulite in an
observation area, corresponds to the initial nucleation rate. It is not easy and inefficient to finda moment when nucleation starts from all sequential images. Therefore, we assumed that thenucleation rate is constant for a short period of time at the early stage of spherulite generationand estimated the local nucleation rate from a single image at the early stage. The maximum
diameter of the spherulite d max , the minimum diameter of the observable spherulite d min , andthe number of spherulites Q whose diameter is between d max and d min are measured in theobservation area Vt of the image at the early stage of nucleation. Set the time for the spherulite
diameter reaching d max from d min as to . The difference between the maximum and minimum
spherulite diameters can be expressed by

d max  d min  to d

(3)

The relationship between the number of spherulites and the initial nucleation rate as follows
Q Vt  n (0)to(4)

Therefore, by removing to from Eqs (3) and (4), the initial nucleation rate, i.e., the local
nucleation rate, is obtained.
n (0) Qd(d max  d min )Vt(5)

2.5 Spherulitic structure formation model
The Turnbull-Fisher model[16] and the Lauritzen-Hoffman model[17], which describe primary
nucleation in the liquid phase and secondary nucleation on existing crystal surfaces, were usedin this research. The primary nucleation and the secondary nucleation model are given by EK1Tmo 2n  n0 exp  1 2o RT RT (Tm  T ) 

(6)

 E

K 2Tmod  d0 exp  2 (7)o RT RT (Tm  T ) 
The local nucleation rate n and the spherulite growth rate d obtained from the in-situ tests

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L. Hong, K. Tatsuno and Y. Aoyagi

were compared with the nucleation rate in the Turnbull-Fisher’s primary nucleation model andthe growth rate in the Lauritzen-Hoffman’s secondary nucleation model, as shown in Eqs (6)and (7), respectively. The equilibrium melting temperature Tmo was set to 170 ℃ based on thephysical property data of the raw PLA. K1 is the primary nucleation factor and K 2 is thesecondary nucleation factor. E1 is the energy of molecular chain transportation. E2 is theenergy of molecular chain diffusion. n0 is the reference nucleation rate. d0 is the reference
growth rate. Figure 4 shows a schematic diagram of the temperature dependence of spherulitic
structure formation models. The primary nucleation rate reaches a peak at lower temperatures
than that for secondary nucleation. This is because the critical crystal nucleus size and the Gibbs
energy of the critical nucleus to the liquid are large in the temperature region where crystal
growth proceeds, making nucleation practically impossible, and the critical Gibbs energy
decreases to a value where primary nucleation is possible only in the temperature range withlarge undercooling[18]. The parameters of the spherulite formation model were identified byfitting.
2.6 Results and discussion of spherulite growth rate
In Fig. 5, the spherulite growth rate obtained by the in-situ test was compared with thesecondary nucleation model. It was confirmed that the spherulite growth rate reached a peak at
about 138 °C. There was some variation, but the trend was similar to that of the secondary
nucleation model in the temperature range from 80 °C to 160 °C. The parameters were identified
by curve fitting of the test results while setting the peak temperature to 138 ℃, as shown in

Figure 4 Temperature dependence of nucleation models

Figure 5 In-situ observation results of growth rate and identified secondary nucleation model5

L. Hong, K. Tatsuno and Y. Aoyagi

Table 1.
2.7 Results and discussion of local nucleation rate
Figure 6 compares the local nucleation rate determined by the in-situ test with the primary
nucleation model. The results in the low-temperature range showed a larger variation than thosein the high-temperature range. One of the possible reasons for this is the influence of thetemperature conditions of the in-situ observation test. Because of the slow cooling, nucleation
may occurs during the great range of temperature before reaching the expected value. Also, thearea of the observed region may be another factor. Because the smaller spherulites diameter atlower temperature requires a magnified observation, the region actually goes smaller.
Considering 103 °C as the peak, the parameters of the primary nucleation model were identified
by curve fitting using the experimental results from 103 °C to 160 °C, avoiding the variation
parts. The parameters are shown in Table 2. The nucleation rate’s peak at 103 °C is lower than
the growth rate’s peak at 138 °C in Section 2.6. The position of the peak is considered to bereasonable.

Table 1 Identified parameters of rate dGrowth rate d [m/s]d0 [m/s]

2.05  103

E2 [J]

1.21 10 19

K 2 [J]

9.82  10 22

Table 2 Identified parameters of rate n
Nucleation rate n [m/s]n0 [m-2 s-1]

1.14  1064

E1 [J]

6.16  10 19

K 2 [J]

1.38  1021

Figure 6 In-situ observation results of growth rate and identified primary nucleation model6

L. Hong, K. Tatsuno and Y. Aoyagi

3 DEVELOPMENT AND DISCUSSION OF A SIMULATION METHOD FOR
SPHERULITE FORMATION BASED ON IN-SITU OBSERVATION RESULTS
3.1 Simulation method for spherulite formation
The Turnbull-Fisher primary nucleation model and the Lauritzen-Hoffman secondary
nucleation model described in Chapter 2 were used for the simulation. The parameters wereused from Tables 1 and 2.
Figure 7 shows a computational model for the simulation. A planar analysis area consistingof cells of 100 nm per side was set up. Each cell can be in either the glassy or crystalline state,
and all cells are in the glassy state at the start. A random number R [0, 1] given to all cells at
each computation step determines the state of the cell using
R  k1k1  R  k1  k2

(8)

k1  k2  R
where k1 is the primary nucleation probability and k2 is the secondary nucleation probability.
When R  k1 , the cell transits to crystal state by the primary nucleation. When k1  R  k1  k2 ,the cell transits to crystal state by the second nucleation. When k1  k2  R , the cell stays in theamorphous phase. The probability k1 and k2 are determined as follows:

 E
 2
K1Tmo 2
k1  n0 exp  1 
 a t2o RT RT (Tm  T ) 

(9)

 E

K 2Tmok2  d0 exp  2  mt ao
 RT RT (Tm  T ) (10)Here a is the side length of a cell, 1 10 7 m. t is the step of computation, 0.1s. m is thenumber of adjacent crystal points in the four neighborhoods around the target cell, ranging from
1 to 4.

Figure 7 Monte Carlo method is used to simulate the spherulite formation and growth in a 300 m square area.
The crystalline phase is randomly generated, and the probability is influenced by the situation in the neighborregion.

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L. Hong, K. Tatsuno and Y. Aoyagi

3.2 Discussion of the effect of the slow cooling stage on the local nucleation rate
Here the simulation method developed in section 3.1 is used to evaluate the numerical model
described in section 2.7. The analysis area is 300 m square. Forming temperatures wereanalyzed in 1 °C increments in the range from 80 °C to 120 °C. Figure 8(a) shows the simulated
local nucleation rate and the primary nucleation rate calculated using Eq. (6). Figure 8(b) shows
the simulated image of local nucleation of two minutes after reaching the expected molding
temperature of 117 °C. The microcells that transitioned to the crystalline state by the primary
nucleation were represented by different colors, while the cells that transitioned by thesecondary nucleation were the same color as the adjacent crystalline state cells.
In Fig. 9, the local nucleation rate in the simulation results, including the slow cooling step,
follows the primary nucleation model, which corresponds to the experimental results, in therange of molding temperatures from 110 °C to 120 °C. However, the simulation results from
90 °C to 110 °C was much lower than that of the model. This may be due to the fact that d max
is larger than expected because the formation starts earlier than expected, as described in section
2.7. In fact, in the in-situ observation at 98 ℃, it can be confirmed that spherulites had alreadyformed before the temperature reached 98 ℃.
It was considered that spherulite was affected in the temperature range below 103 °C, which
was considered the peak. However, there was also a possibility that the influence occurs evenFigure 8 (a) Comparison between nucleation rate from simulation results and identified primary nucleation
model and (b) simulation image for 300 m square

Figure 9 Comparison between nucleation rate from simulation with cooling process and typical in-situ
observation results, and identified primary nucleation model8

L. Hong, K. Tatsuno and Y. Aoyagi

around 110 °C. In the range from 80 °C to 90 °C, the simulation results were slightly higher
than the model. Considering that the cooling time was set to 10 minutes regardless of themolding temperature, the cooling rate is higher in the range from 80 °C to 90 °C. Since the local
nucleation rate is considered to be low at these temperatures, the d max from 80 °C to 90 °C wasstrongly affected by the increasing number of generation of spherulites, rather than themeasurement of d max as a larger value than expected. Therefore, it was confirmed that the local
nucleation rate is significantly affected by the slow cooling process.In Fig. 9, the in-situ
observation results at 86, 93, 113, 123, and 128 °C are also shown. Since the primary nucleation
model was fitted from the in-situ test results, they are certainly close. However, the simulation
results tended to differ significantly from observation results. In order to get the local nucleation
rate of simulation considering the slow cooling process close to the value of the in-situ
observation results, the parameters of the primary nucleation model should be re-identified.
Table 3 shows the new parameters. Since there is a large variation at temperatures below
103 °C, the parameters of the primary nucleation model were re-identified by preferentially
fitting the results above 103 °C. Figure 10 shows some in-situ observation results which areclose to simulation results and the curve given by the primary nucleation model with the reidentified parameters. The re-identified curve tends to be higher than the in-situ observation
results. The observation just measured the apparent nucleation rate. We find that the actual local
nucleation rate is larger than that measured by the observation.
3.3 Comparison of in-situ observation results and spherulite formation simulations
Figure 11 shows the simulated microstructures and the in-situ observation results at theTable 3 Re-identified parameters of rate n
Nucleation rate n [m/s]N0 [m-2 s-1]
5.29  1035

E1 [J]

3.05  10 19

K 2 [J]

6.82  10 22

Figure 10 Comparison between nucleation rate from simulation with cooling process and typical in-situ
observation results, and re-identified primary nucleation model9

L. Hong, K. Tatsuno and Y. Aoyagi

Figure 11 In-situ observations and simulation results at
117 °C

Figure 12 In-situ observations and simulation results
at 93 °C

molding temperature of 117 °C, also the time after reaching the expected molding temperature.
In Fig. 11, the primary nuclei generated in the glassy region grow spherically and collide witheach other to form spherulite boundaries, reproducing the characteristic spherulite structure.
The simulation results cannot be directly compared with the in-situ observation test results
regarding the number of spherulite crystals because some are too small to be observed in thein-situ observation test results. However, the simulation results show similar trends in the sizeof spherulites and the percentage of glassy regions at each time to the in-situ observation test
results, which is considered to be a good representation of spherulite formation over time.
Figure 12 shows the comparison at 93 °C. Similar to Fig. 11, the size of the spherulites and thepercentage of glassy regions show similar trends to the in-situ observation test results,
confirming the simulation method's validity.
4 CONCLUSION
The primary and secondary nucleation rates were calculated based on the in-situ observation
tests. A simulation method for spherulite structure formation was developed, representing the
local nucleation rate and spherulite growth rate in a planar region. The conclusions are
summarized as follows.
(1) The local nucleation rates obtained from the in-situ tests scatter widely in the range of
forming temperatures 80 - 103 °C, making it difficult to represent them with a primary
nucleation model.

10L. Hong, K. Tatsuno and Y. Aoyagi(2) The spherulite formation simulation results show that the slow cooling stage affects the
local nucleation rate significantly.
(3) The simulation method, which considered the slow cooling stage, could represent
spherulite formation numerically.
ACKNOWLEDGMENTS
This work was performed with the support of JSPS KAKENHI Grant Number 21K03745.
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