The crystallization of linear polymers is strongly dependent on cooling rate or thermal history and chain length. Longer chains have higher crystallization temperatures and slower crystallization processes with larger temperature windows of transition. Slower cooling also leads to higher crystallization temperatures but with smaller temperature windows of transition. In general, longer chains have slower dynamics which leads to higher degrees of nonequilibrium processes, similar to that of faster cooling. Such general crystallization behavior can be demonstrated by two samples from our simulations with N=10 and 100 under different cooling rates, as shown in Fig. 2a,b.

Similar to experiments, the phase transition of crystallization can be identified by the peaks of exothermal curves. The values obtained from simulations can validate the method’s feasibility through comparison with experimental data. Meyer and Müller-Plathe [18,38] discussed the variations in enthalpy (or volume per monomer) under different conditions during cooling and heating cycles in detail. Regarding the structural characteristics of the crystals, our previous work addressed the structure factor of the crystalline structure during the cooling process [36]. In MD simulations, the enthalpy (H) and specific heat (Cp ) can be calculated by H=E+pV and Cp=dE/dT+pdV/dT, where E represents the internal energy, p is pressure and V denotes the volume of the simulation box. The measured values of H and Cp depend on temperature and cooling rates. The phase transition temperature can be estimated under slow cooling. However, under the fastest cooling conditions for long polymers (e.g., N=100 at qA=2×105, equivalently ∼3.14×109 K/s), the crystallization behavior resembles a glass transition, which makes the crystallization temperature difficult to identify. We note that even at such a fast cooling rate, a small fraction of crystalline structures is still found. Due to the fast cooling rate, which results in low crystallinity, defining the crystalline regions is challenging, and a new method may provide an effective solution to this problem [40].

The phase transitions in simulations are rather slow due to the limited simulation time compared to experiments. Therefore, it is not easy to precisely estimate the Tc when the changes of exothermal curves at the phase transition are not as sharp as in experiments. To better estimate the Tc, we propose a fitting formula for Cp by (2)Cp=h/wsech2(t)+Cp0+CpΔtanh⁡(t),where t=(T−Tc)/w is the reduced temperature around Tc. The first term, h/wsech2(t), comes from the phase transition (structural change), and the prefactor, h/w, is the peak height in specific heat due to this phase transition. The second and third terms are from steady mixing of melt and crystalline phases (we assume that there are only two phases for simplicity). The parameters 2h and 2w represent the enthalpy change and the width of the temperature window of the phase transition. The parameters, Cp0+CpΔ, and Cp0−CpΔ, are the specific heats before and after phase transition. The meanings of these parameters are sketched in Fig. 2c. We have CpΔ=(Cpm∗−Cpc∗)ϕc/2 and Cp0=Cpm∗−CpΔ. Where Cpm∗ and Cpc∗ are the specific heats of a pure melt and a perfect crystal, and ϕc is the final crystallinity after the phase transition. Integration of Cp with temperature gives the enthalpy as (3)H=H0+htanh⁡(t)+wtCp0+wCpΔln⁡[cosh⁡(t)],where H0 is the enthalpy at T=Tc.

These two formulae fit the simulation data well, as shown in Fig. 2d for an example of the sample of N=50. The errors are slightly larger near the beginning and the end of the phase transition, especially during very slow cooling. We can observe that the first term of Eq. (2) is symmetric, while the calculated Cp is slightly asymmetric (with a more gradual left shoulder and a steeper right one). We attribute the main reason of these errors to the simple assumptions of two phases mixing. For polymers, there are precursor states before crystallization and significant reorganization after crystallization [41]. Therefore, the simple assumptions lead to larger errors at the beginning and end of crystallization, particularly under slow cooling rates. However, the behavior near T=Tc is fitted quite well in both H and Cp. The Tc and w obtained by fitting are in good agreement with the values obtained from the simulations, and they are insensitive to the changes of other parameters such as Cp0 and CpΔ.

In simulations, and even in experiments, the crystallization of polymers is a rather slow and continuous process, and the measured Tc and w are strongly dependent on cooling rates. For polymers with high molar mass, the temperature window of transition, w, is always large even under very slow cooling rates. By fitting Eqs. (2) and (3) to the simulation results, we can estimate the two key parameters Tc and w quite well, which describe the temperature location and window of the phase transition. We can then compare the effect of different chain lengths and cooling rates on the crystallization behavior.

It is interesting whether we can approach the limit of w→0 by extrapolating the data at different cooling rates (as q→0), or if we can determine how slow a cooling rate should be to satisfy a phase transition with a certain value of w. We show the relations of q and w for monodisperse samples in Fig. 3a. It is found that there is a power law:(4)w≅αqβ,β=0.68,where the parameter α is a constant for a specific sample. The value of α for longer polymers is larger than that for shorter ones, indicating that the crystallization rates of longer polymers are slower than those of shorter ones. This observation is consistent with experiments. It is noted that the scaling relation is almost length independent, i.e., the exponent β is almost the same for all different chain lengths. However, the physical mechanisms behind this power law and the exponent remain unclear.

In Fig. 3b, we present the results of measured Tc and w. It is observed that the data can be fitted by (5)Tc=Tc∗+a[exp⁡(−w/b)−1],where Tc∗, a, and b are fitting parameters. In the limit of w→0 (or q→0), the measured value of Tc is expected to be Tc∗. Thus we can obtain the theoretical equilibrium crystallization temperature Tc∗ at the limit of infinitely slow cooling, by fitting the data at different cooling rates with Eq. (5). This extrapolation is particularly useful in simulations because polymer crystallization is a slow process, and computer simulations can only be conducted under very rapid cooling conditions.

The crystallization temperature Tc increases with the chain length N, as illustrated in Fig. 3c. We find that there is an empirical relation between Tc and N as (6)Tc=Tc∞−KN−N0,where the N0, K, and Tc∞ are parameters. Tc∞ represents the crystallization temperature of infinitely long chains. This formula is adapted from the Flory-Fox equation, used to fit the glass transition temperature (Tg) of polymers [42,43]. It is observed to fit our simulation data well. In experimental settings, an empirical relation between Tc and Tg is found as Tc=γTg,γ=0.5−0.8. This similarity suggests a strong connection between the crystallization and glass transition of linear polymers.

It is noteworthy that Eq. (6) is also similar to the Gibbs-Thomson equation for the melting temperature Tm=Tm∗(1−2σ/lρcΔh) of crystalline polymers [2]. This analogy arises when we consider l=(N−N0)/f as the effective crystalline thickness and set K=2σf/ρcΔh. Here, Tm∗ represents the equilibrium melting temperature, l denotes the lamella thickness, σ is the surface free energy per unit area, Δh is the increase in enthalpy per unit mass upon melting for an infinitely thick crystal, ρc is the density of the crystalline state and f is the average folding number. It is observed that N0 increases with the rise in cooling rate, aligning with the increase in the amorphous fraction under faster cooling conditions. However, the effective K increases with the cooling rate, as indicated in the caption of Fig. 3c. This suggests that the average folding numbers of crystalline chains f decreases with the cooling rate. This phenomenon can be attributed to the fact that the formation of chain folds requires sufficien time, while the chains do not have enough time to form well folded structures under fast cooling.

The chain lengths range from N=10 to 500, crossing the entanglement length of the melt systems of long chains, estimated to be Ne≅30−50 depending on the temperature and methods of PPA or Z1, as demonstrated in our previous studies [44]. The crystallization temperatures Tc reaches a rough constant at the chain lengths of N>100, and the deviations between measured Tc and Eq. (6) are observable, see Fig. 3c. We note that there are lager fluctuations in Tc-N curves near N=40−50, particularly under fast cooling conditions. During fast cooling in the crystallization process, polymer chains lack adequate time to slide and rearrange. This limitation can lead to a reduction in crystallinity, which subsequently results in fluctuations in Tc. Furthermore, this restricted movement is compounded by entanglement effects. These observations indicate entanglement effects in the crystallization of linear polymers under fast cooling.

As mentioned above in the Introduction section, bidisperse and polydisperse samples also hold significant value in industry, and it is interesting that whether the above empirical formulae found for monodisperse samples stand for polydisperse samples.

We carry out MD simulations using the same parameters for the polydisperse samples with chain lengths following a Schulz distribution, as well as bidisperse samples with N=10 and 100, as detailed in the Model and Simulation Details section. In Fig. 4a, we present the relationship between w and q for both the bidisperse samples and polydisperse samples. We find that the power law described by Eq. (4) remains consistent across most samples, with the exponent β remaining unchanged. This can be attributed to the fact that β is independent of chain length. Consequently, there appears to be no distinction between bidisperse, polydisperse, and monodisperse samples. However, the parameter α is dependent on N, and it appears that the behavior of α is more likely dependent on Nw rather than Nn comparing with the results of monodisperse systems, as the data are more close to that of N=100, as indicated by the red solid triangles in Fig. 4a.

The crystallization temperatures of different chain lengths at various cooling rates, Tc(Ni,q), are obtained from Fig. 3c. Consequently, a simple expectation of the crystallization temperature for polydisperse samples, Tcpd(N∗,q), might be the weight-averaged value as (7)Tcpd(N∗,q)=∑iniNi2Tc(Ni,q)∑iniNi.

Here, ni is the number of polymer chains with a length of Ni, where Ni∈10,100 for the bidisperse samples. The ni values for the polydisperse samples satisfy the discrete Schulz distribution shown in Fig. 1b, derived from Eq. (1). N∗ is the effective chain length of the polydisperse systems. The simulated data and fitting curves based on Eq. (7) are shown in Fig. 4b. For the bidisperse samples, we can see that Eq. (7) fits the data fairly well under cooling rates of qC, qD, and qE. However, for samples with most probable distribution of chain lengths, Eq. (7) only fits the data well at slow cooling (qE). For the fast cooling rates such as qA and qB, neither the binary nor the polydisperse samples satisfy Eq. (7), indicating that the impact of distribution of chain lengths on crystallization becomes more pronounced under rapid cooling conditions.

It is noted again that significant fluctuations in the Tc-N curves under fast cooling conditions are concentrated in the N=40−50 range, coinciding with the entanglement length region of the CG-PVA model systems in the molten state (Ne≅30−50). Unlike the monodisperse samples with N=30−50, both the bidisperse samples with N=10 and 100 and the polydisperse samples, contain long chains exceeding Ne. The mixing of long chains with N>Ne and short chains N<Ne leads to a more complex crystallization behavior, as disentanglement processes may occur but are heavily reliant on relaxation times and exhibit greater stochasticity. Therefore the effects of entanglement restriction should be considered for long chains, particularly under fast cooling, which is consistent with our previous studies of the crystallization of entangled polymers [37,45].

Interestingly, although the fitting formula of Eq. (7) implies a weight averaged assumption, the effective chain lengths N∗ of the polydisperse systems are found to be number-average values, i.e., the effective chain length would be N∗=Nn rather than Nw. The difference between the fitting using Nn and Nw is illustrated in Fig. 4b by the solid and dashed lines.

We display the snapshots at the final cooled state in Fig. 5. Specifically, we select three samples with Nn≅50 following fast cooling at qA and slow cooling at qF. Observing the results, we notice that all three samples exhibit the formation of small crystallites after the rapid cooling rate of qA, while they all develop lamellar structures following the slower cooling of qF. As the number-averaged chain lengths of the three samples are nearly the same, the differences between the crystalline structures give a visualization of the effect of chain length distribution on the crystallization. It is obvious that the lamella thicknesses are different after the slow cooling of qF, while their Tc are almost the same, as shown in Fig. 4b. This distribution dependent crystallization at almost the same Nn and Tc demonstrates a contradiction to the classical theory and needs further investigations.