The size of critical secondary nuclei of polymer crystals does not depend on supersaturation

Abstract

It is still a great challenge to determine the size of critical nuclei, which is crucial for a comprehensive understanding of crystallization and for testing the controversial crystallization theories. Here, we propose a method to determine the size of critical secondary nuclei on growth faces of poly(butylene succinate) single crystals in solution, basing on the probability of statistically selecting crystallizable units in random copolymers. In a dilute solution and for a given crystallization temperature, we reveal that the size of critical secondary nuclei was independent of supersaturation, contrary to the well-accepted prediction of existing theories which expect that the size of the critical nucleus increases with decreasing supersaturation. Accounting correctly for the dilution-caused change in the steady-state concentration of clusters of various sizes, we remedy inconsistencies of existing theoretical approaches in deriving the correct size of critical secondary nuclei in solution being independent of supersaturation.

Determining the size of critical nuclei remains challenging. Here the authors propose a method to determine the size of critical secondary nuclei on growth faces of poly(butylene succinate) single crystals in solution, using the probability of statistically selecting crystallizable units in random copolymers.

Introduction

Among the various types of phase transitions, crystallization is a process where amorphous motifs pack orderly to form crystals, which is one of the most widespread and basic ordering processes in nature and industry¹. It is generally accepted that crystallization is initiated by nucleation, which involves an induction period and can be related to surpassing a free energy barrier. While nuclei below a critical size will tend to dissolve again, those larger than the critical size will grow further and form crystals. The formation of critical nuclei signifies usually the rate-limiting step of crystallization.

Though nucleation is the key step of crystallization, the detailed molecular mechanism of nucleation is still hotly debated². Various theories of nucleation have been proposed, which differ in the dependence of the size of the critical nucleus on various thermodynamics-related process parameters. Determining the size of critical nuclei will help to reveal the molecular mechanism and to test nucleation theories. The classical nucleation theory assumes a direct transformation from the amorphous to the crystalline phase^3–6, while multi-step nucleation theory suggests an intermediate phase between the starting amorphous and the final crystalline phase⁷. In addition, the interfacial free energy of nuclei may be different from those of the mature crystal due to size effects⁸ or the presence of an intermediate crystalline phase⁹. So, it is difficult to calculate the exact size of critical nuclei based on nucleation theories without knowing the actual interfacial free energy of critical nuclei.

In textbooks and reviews on crystal nucleation in solution^6,10–12, for the typically assumed spherical shape of the nucleus, the size of the critical primary nucleus in a supersaturated solution is derived as follows

$$r^{*}=\frac{2\gamma v}{RT\ln S}$$ 1

where $r^{*},\gamma ,v,R,T$ and $S$ represent the radius of the critical nucleus, interfacial free energy, molar volume of the crystal motifs (units), gas constant, absolute temperature, and degree of supersaturation of the solution, respectively. $S$ is defined as the actual concentration divided by the equilibrium concentration at the solubility limit. It is well known that when the solution is diluted (i.e., when the solution concentration decreases), the motifs and the formed clusters of various sizes (including the critical nuclei) are simultaneously diluted. However, in deriving Eq. (1), only the dilution of motifs was considered, while the dilution of formed clusters was not taken into account, which sheds some doubts on the validity of Eq. (1).

Recently, advancements in real-time observations of nucleation of small molecules^13–15 and macromolecules¹⁶ have revealed a wealth of information never seen before. However, it is still a challenging task to obtain the size of critical nuclei from real-time observations within a limited explored volume since it is difficult to obtain statistical information on the distribution of the size of various crystal nuclei. Wang et al.¹⁷ developed an ingenious method allowing to probe the critical nucleus size for ice formation with graphene oxide nanosheets of limited size. However, the focus of this method was on the promotion of primary nucleation on heterogeneous surfaces. Thus, the size of the critical nucleus without the impact of a nucleating agent is still difficult to obtain. Determining the exact size of critical nuclei is crucial for both, the testing of nucleation theories and the preparation of nanomaterials of controlled size. Due to the unknown mechanism of nucleation, determining the size of critical nuclei via a theory, which does not rely on severe thermodynamics-related assumptions (such as the capillarity approximation) and detailed experimental parameters is rather appealing.

Nucleation is a stochastic process^18–20, i.e., molecules or sequences of a polymer chain are randomly selected. Recently, we propose a method to determine the size of critical nuclei for either, primary or secondary nucleation, based on the probability of selecting crystallizable units of random copolymer²¹. Details of the theoretical model are presented in Methods. The nucleation rate of random copolymers is derived as follows

$$i^{\prime }=i\cdot p_{\mathrm{A}}^{m+1}$$ 2

where $i$ and $i^{\prime }$ are the nucleation rate of homopolymer and random copolymer, respectively. $m$ is the number of crystalline motifs (here, repeating units of the polymer investigated) contained within the critical nucleus. $p_{\mathrm{A}}$ is the proportion of crystallizable units randomly distributed within the copolymer chains. The germ of this idea was proposed as early as 1971 by Andrews et al.²², although the size of the critical nucleus was incorrectly derived (Supplementary Discussion 1).

In this work, we determine the size of critical secondary nuclei in solution via our approach based on measuring the two-dimensional epitaxial growth rate of single crystals of homopolymer and the random copolymers (Fig. 1). We obtain the size of critical secondary nuclei for single crystals in solution, crystallized at various temperatures, for different solution concentrations and on different crystal faces. The size of critical secondary nuclei is revealed independent of the supersaturation of the solution.

Fig. 1. Determination of the size of the critical secondary nucleus based on the probability of statistically selecting crystallizable units randomly distributed along the chain of random copolymers.

The secondary nucleation rate is obtained from the growth rate on a given crystal face. The decrease in the nucleation rate $i^{\prime }$ of the random copolymer relative to $i$, the one of the homopolymer, can be attributed to the product of the probability of continuous selection of the crystallizable units. $p_{\mathrm{A}}$ is the probability of finding a crystallizable unit along the copolymer chains before the formation of the secondary nucleus, which equals the proportion of crystallizable units in random copolymers. m is the number of crystalline units within a critical secondary nucleus.

Results

Method execution

To determine the size of critical secondary nuclei on the growth face of single crystals formed in solution, poly(butylene succinate) (PBS) was chosen as an example (Supplementary Fig. 1). The crystallizable butylene succinate (BS) units were diluted via random copolymerization. PBS and its random copolymer poly(butylene succinate-ran-butylene 2-methylsuccinate) (PBSM) were synthesized with similar molecular weight and similar molecular weight distribution²³. The fraction of non-crystallizable butylene 2-methyl succinate (BM) units ranges from 1 to 4%. These BM units were randomly distributed within the copolymer chains. More details of the samples are shown in Supplementary Table 1. The concentration of the supersaturated polymer solution was 0.1 $\mathrm{mg}{\mathrm{mL}}^{-1}$, much lower than the overlap concentration of 21 $\mathrm{mg}{\mathrm{mL}}^{-1}$ (see Supplementary Fig. 2 for more details). Therefore, the studied solution can be considered as an ideal dilute solution. PBS single crystals cultured at 69 °C by a self-seeding method²⁴ were used as seeds to provide a thick enough growth front for epitaxial growth of PBS homopolymer and PBSM random copolymers in the whole range of crystallization temperature T_C studied (52 °C ≤ T_C ≤ 69 °C). The PBS seed crystals had uniform size to assure the accurate measurement of the (epitaxial) growth rate of the PBSM single crystals (Fig. 2a). When PBS single crystal seeds were injected into a supersaturated solution of a PBSM copolymer, epitaxial growth of PBSM chains was initiated from the crystal faces surrounding the PBS single crystals. A typical image is shown in Fig. 2b. The growth rate of crystal faces was obtained by measuring the epitaxial growth distance at a specific time. The hexagonal PBS single crystals contained two sets of crystal faces, namely four shorter (110) crystal faces and two longer (020) crystal faces^25,26, as marked in Fig. 2b.

Fig. 2. The size of the critical secondary nucleus at different temperatures.

a Height image of poly(butylene succinate) (PBS) single crystal seeds, obtained by tapping mode atomic force microscopy (TM-AFM). The crystallization temperature T_C is 69 °C. Scale bar, 2 μm. b A typical TM-AFM phase image of a poly(butylene succinate-ran-butylene 2-methylsuccinate) (PBSM) single crystal, epitaxially grown on a PBS seed single crystal. Scale bar, 0.5 μm. a, b The circled number 1 indicates PBS single crystal seeds. The circled number 2 indicates the epitaxially grown PBSM crystals. The circled number 3 and the white dashed box indicate the unwashed crystal rim. The blue double arrow and green dashed double arrow indicate the growth distances of (110) and (020) crystal faces, respectively. c Growth rate $G^{\prime }$ as a function of the proportion of crystallizable units $p_{\mathrm{A}}$, measured at different temperatures T_C. At T_C of 68 °C and 69 °C, $G^{\prime }$ was extremely low, as shown in the inset. The growth rates $G^{\prime }$ are derived from the fitted slopes of the epitaxial growth distances $D$ (each obtained from 12 independent measurements) on (110) crystal faces versus 4 sequential sampling times $t$ and the error bars of $G^{\prime }$ in c represent the standard errors of the fitted slopes of $D$ versus $t$ as presented in Supplementary Fig. 6. d $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at different $T_{\mathrm{C}}$. The numbers next to the dashed lines (in the same colors as the dashed lines) represent the slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at corresponding $T_{\mathrm{C}}$, which should be equal to $m/2$. c, d Blue triangles, orange triangles, purple diamonds, green pentagons, navy-blue circles and pink squares indicate the results at crystallization temperatures of 52 °C, 56 °C, 60 °C, 64 °C, 68 °C and 69 °C, respectively. e The number of crystalline units $m$ within a critical secondary nucleus at different $T_{\mathrm{C}}$. f The number of stems $s$ contained within a critical secondary nucleus at different $T_{\mathrm{C}}$. The error bars in e, f come from the standard errors of the fitted slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ in d. g Schematic diagrams showing the change in the size of critical secondary nuclei with $T_{\mathrm{C}}$.

Our experimental results show that the growth rate of polymer single crystals is proportional to the square root of the polymer concentration in solution, i.e., $c^{1/2}$, as presented in Supplementary Fig. 3. This evidence indicates that the growth of single crystals occurred within Regime II and a critical secondary nucleus consisted of the stems from a single polymer chain (Eq. (18), see Methods for details). In Regime II, the number of crystalline units $m$ within the critical secondary nucleus can be obtained from the fitted slope of the double logarithmic plot of $G^{\prime }$ versus $p_{\mathrm{A}}$,

$$G^{\prime }=G\cdot p_{\mathrm{A}}^{\frac{m}{2}}$$ 3

where $G^{\prime }$ and $G$ represent the (epitaxial) growth rate of the crystalline face of the single crystals of random copolymer and homopolymer at the same concentration of solutions, respectively, and where $p_{\mathrm{A}}$ is the fraction of crystallizable units in random copolymers.

The size of critical secondary nuclei at different temperatures

The variation of the growth rate $G^{\prime }$ of random copolymers with the fraction of crystallizable units $p_{\mathrm{A}}$ at different temperatures $T_{\mathrm{C}}$ is presented in Fig. 2c. The concentration of the supersaturated PBSM solution was fixed at 0.1 $\mathrm{mg}{\mathrm{mL}}^{-1}$. $G^{\prime }$ decreased visibly with decreasing $p_{\mathrm{A}}$. According to Eq. (3), the number of crystalline units $m$ within a secondary nucleus can be obtained from the fitted slope of the double logarithmic plot of $G^{\prime }$ versus $p_{\mathrm{A}}$, as plotted in Fig. 2d. For all the studied $T_{\mathrm{C}}$ (52 °C ≤ T_C ≤ 69 °C) the results showed good linearity. The number of crystalline units $m$ contained within a secondary critical nucleus at different $T_{\mathrm{C}}$ are summarized in Fig. 2e. The thickness of PBS single crystals cultured at different $T_{\mathrm{C}}$ by the self-seeding method is summarized in Supplementary Fig. 4c. The thickness of PBS and PBSM (up to 4% non-crystallizable units) single crystals was almost the same at the fixed $T_{\mathrm{C}}$ (Supplementary Fig. 4d). More details on the measurement of the thickness of single crystals are given in Supplementary Note 3. Provided that the thickness of the amorphous layer of PBS single crystals was relatively negligible compared with that of the crystalline core²⁷, we can calculate the number of crystalline units $r$ within each stem from the lamellar thickness $h$ and crystal lattice parameters of PBS single crystals. We assume that a secondary critical nucleus has a rectangular shape with $s$ stems and each stem consisted of $r$ crystalline (repeating) units. Then, the number of stems $s$ contained in the secondary nuclei can be obtained from dividing $m$ by $r$ and the results are summarized in Fig. 2f. With the increase of $T_{\mathrm{C}}$, the number of stems $s$ and the number of crystalline units m contained within a secondary nucleus increased monotonously from 17 stems and 91 units at T_C = 52 °C to 25 stems and 163 units at T_C = 69 °C (Fig. 2g), respectively.

A secondary critical nucleus containing 163 crystallized BS units at T_C = 69 °C represents a polymer chain sequence with a molecular weight of at least 28 $\mathrm{kg}{\mathrm{mol}}^{-1}$. Considering the non-crystallizable units and a modicum of some crystallizable units in the amorphous fold surface layer of the single crystals, almost an entire polymer chain was required to form the critical nucleus at T_C = 69 °C (Supplementary Table 1), which is proof of the validity of our method for determining the size of critical nuclei. In fact,T_C = 69 °C represented almost the upper limit in temperature for secondary nucleation in solution. At T_C ≥ 70 °C, no growth was detected.

The size of critical secondary nuclei on different growth faces

The relations of the size of critical secondary nuclei with $T_{\mathrm{C}}$ shown in Fig. 2 were derived from the growth of the (110) crystal face. In addition, we determined the size of critical secondary nuclei on the (020) crystal face (Fig. 3a–c). At T_C = 52 °C, the size of critical secondary nuclei on (020) crystal face was almost identical to that on (110) crystal face, but then increased more rapidly for higher T_C. Eventually, at T_C = 68 °C and T_C = 69 °C, the size of critical secondary nuclei on (020) crystal face became so large that no reliable value of the growth rate could be measured within the measurement time. Typical images of epitaxially grown single crystals at T_C=68 °C and T_C = 69 °C are displayed in Fig. 3d. The determination of the size of critical secondary nuclei on different crystal faces of polymer single crystals provides a kinetic perspective for explaining the various shapes and symmetries of polymer single crystals.

Fig. 3. The size of the critical secondary nucleus at different crystal faces.

a The relationship of the growth rate $G^{\prime }$ with the proportion of crystallizable units $p_{\mathrm{A}}$ at different crystallization temperatures $T_{\mathrm{C}}$, measured from the advancement of (020) crystal faces. The growth rates $G^{\prime }$ are derived from the fitted slopes of the epitaxial growth distances $D$ (each obtained from 6 independent measurements) on (020) crystal faces versus 4 sequential sampling times $t$ and the error bars of $G^{\prime }$ in a represent the standard errors of the fitted slopes of $D$ versus $t$ as presented in Supplementary Fig. 8. b $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at different $T_{\mathrm{C}}$, measured from the advancement of (020) crystal faces. The numbers above the dashed lines (in the same colors as the dashed lines) represent the slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at corresponding $T_{\mathrm{C}}$, which should be equal to $m/2$. a, b Blue triangles, orange triangles, green circles and pink squares indicate the results at crystallization temperatures of 52 °C, 56 °C, 60 °C and 64 °C, respectively. c The number of crystalline units $m$ within the critical secondary nuclei on (110) and (020) crystal faces at different $T_{\mathrm{C}}$. The green hexagons and orange dashed curve indicate the results for (020) crystal faces. The pink diamonds and cyan dashed curve indicate the results for (110) crystal faces. The error bars in c come from the standard errors of the fitted slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ in Fig. 2d for (110) crystal faces and those in Fig. 3b for (020) crystal faces. d Typical tapping mode atomic force microscopy (TM-AFM) phase images of a poly(butylene succinate-ran-butylene 2-methylsuccinate) (PBSM) single crystal, epitaxially grown on a PBS seed single crystal at $T_{\mathrm{C}}=68^{\circ }\text{C}$ and $T_{\mathrm{C}}=69^{\circ }\text{C}$, respectively. Scale bar, 0.5 μm. The text below the four images indicates the corresponding crystallization temperature $T_{\mathrm{C}}$ and the proportion of crystallizable units $p_{\mathrm{A}}$ in the random copolymers PBSM. These images demonstrate that growth advanced almost exclusively normal to the (110) crystal faces and barely normal to the (020) crystal faces.

The size of critical secondary nuclei at different solution concentrations

To investigate the variation of the size of critical secondary nuclei as a function of the degree of supersaturation $S$ of the solution, we chose a fixed crystallization temperature of T_C = 60 °C. The concentration of the supersaturated PBSM solution of 0.1 $\mathrm{mg}{\mathrm{mL}}^{-1}$ was increased by a factor of 3, 6, and 10, respectively. The other parameters of the experimental protocol remained the same. The variation of the growth rate $G^{\prime }$ with the proportion of crystallizable units $p_{\mathrm{A}}$ at different concentrations and its presentation in a double logarithmic plot are demonstrated in Fig. 4a-b. The growth rate of single crystal increased significantly when the concentration increased from 0.1 to 1.0 $\mathrm{mg}{\mathrm{mL}}^{-1}$ (Fig. 4a). This indicates that the growth front was not saturated by a layer of adsorbed solute. Otherwise, for a saturated layer of adsorbed solute of constant thickness, the growth rate of single crystals should be independent of solute concentration. The results for the number of crystalline units $m$ within a critical secondary nucleus for different concentrations are summarized in Fig. 4c. It is unveiled that the size of critical secondary nuclei was almost independent of the degree of supersaturation. This surprising result contests the well-accepted prediction of existing theories that the size of critical nuclei depends on the degree of supersaturation of the solution. A striking comparison of our experimental results with the calculated size of critical nuclei according to existing theories allowing to derive the size of critical nuclei is shown in Fig. 4d. The latter is expressed as the relative value $n^{*}$ with respect to the size of critical nuclei at the studied maximum supersaturation, which is set to 1. Details of determining the supersaturation S of secondary nucleation are presented in Supplementary Fig. 11. Details of calculating the theoretical size of critical secondary nuclei according to existing theories are presented in Supplementary Eqs. (6)–(12) and Supplementary Figs. 10–12. Obviously, the size of critical nuclei predicted by existing theories, especially for low supersaturation, is much larger than our experimental results, far beyond uncertainties represented by the error bars.

Fig. 4. The size of the critical secondary nucleus for different supersaturations.

a Growth rate $G^{\prime }$ as a function of the proportion of crystallizable units $p_{\mathrm{A}}$ at different concentrations. The growth rates $G^{\prime }$ are derived from the fitted slopes of the epitaxial growth distances $D$ (each obtained from 12 independent measurements) on (110) crystal faces versus 4 sequential sampling times $t$ and the error bars of $G^{\prime }$ in a represent the standard errors of the fitted slopes of $D$ versus $t$ as presented in Supplementary Fig. 9. b $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at different concentrations. The numbers above the dashed lines (in the same colors as the dashed lines) represent the slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ at corresponding concentrations, which should be equal to $m/2$. a, b Blue triangles, orange triangles, green circles, and pink squares indicate the results in solutions with concentrations of 1.0 mg mL⁻¹, 0.6 mg mL⁻¹, 0.3 mg mL⁻¹ and 0.1 mg mL⁻¹, respectively. c The number of crystalline units m within a critical secondary nucleus at different concentrations. The text next to the data points of pink circles indicates the concentration of the solution $c$ and the corresponding number of crystalline units $m$ within the critical secondary nuclei. d Experimental results and predicted sizes of secondary critical nuclei according to existing theories. The pink hexagons and sky-blue lines indicate the experimental results of secondary nuclei sizes. The navy-blue diamonds and black dashed curve indicate the predicted secondary nuclei sizes based on existing theories, corresponding to the value of the right axis indicated by the black arrow. The predicted sizes based on existing theories are expressed as the relative values $n^{*}$ with respect to the size of critical nuclei at the studied maximum supersaturation, which is set to 1. The error bars in c and d come from the standard errors of the fitted slopes of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$ in b.

Discussion

Next, we will identify the mistake in existing theories, allowing us to derive the size of critical nuclei for crystals grown in solution. In existing theories^6,10–12, the Gibbs free energy change for the formation of a mole of nuclei of size $n$, randomly distributed in solution, is expressed as

$$\mathrm{\Delta }G\left(n\right)=-n\mathrm{\Delta }\mu +\gamma A_{\mathrm{s}}\left(n\right)$$ 4

where $\mathrm{\Delta }\mu$ is the change of Gibbs free energy per mole (chemical potential) of crystal motifs after the phase transformation. $\gamma$ is the interfacial free energy of the nuclei. $A_{\mathrm{s}}\left(n\right)$ is the surface area of one mole of nuclei of size $n$. When $\mathrm{d}\mathrm{\Delta }G/\mathrm{d}n=0$, the change of Gibbs free energy for nucleation reaches its maximum, where the size of critical nuclei $n^{*}$ is obtained. For the typically assumed spherical shape of the nucleus, the radius of the critical nucleus r^∗ is derived as

$$r^{*}=\frac{2\gamma v}{\mathrm{\Delta }\mu }$$ 5

$v$ is the molar volume of the crystal motifs (units).

In solution, $\mathrm{\Delta }\mu$ is defined according to existing theories as^6,10–12

$$\mathrm{\Delta }\mu ={\mu }_{\mathrm{s}}-{\mu }_{\mathrm{c}}=RT\ln S$$ 6

where ${\mu }_{\mathrm{s}}$ and ${\mu }_{\mathrm{c}}$ are the chemical potentials of solute molecules in solution and in the bulk crystal, respectively. The degree of supersaturation $S$ is defined as^6,10–12

$$S=\frac{c_{\mathrm{A}}}{c_{\mathrm{A}}^{\mathrm{eq}}}$$ 7

where $c_{\mathrm{A}}$ $\mathrm{and}$ $c_{\mathrm{A}}^{\mathrm{eq}}$ are the actual concentration and the equilibrium concentration of solute molecule at its solubility limit. By substituting the value of $\mathrm{\Delta }\mu$ (Eq. (6)) into Eq. (5), the radius of the critical nucleus according to existing theories is obtained, as given in Eq. (1). According to the above derivation, the size of critical nuclei is obviously expected to vary with the concentration of the solution.

The derivation of Eqs. (4) and (5) of existing theories followed the classical thermodynamic framework. However, the derivation of the size of critical nuclei (Eq. (6)) is doubtful, which only considers the change in the chemical potential of monomeric motifs and implies the assumption that the chemical potential of the clusters of a certain size is constant upon dilution (at lower supersaturation of the solution). On the contrary, it is apparent that the chemical potential of the clusters of a certain size in solution is related to its concentration. In a more dilute solution (i.e., when the solution concentration decreases), the equilibrium concentration (number density) of the clusters of a certain size should be lower, thus the chemical potential of the clusters with various sizes, including the critical nuclei, should vary with the solution concentration, which is ignored by existing theories in deriving the size of critical nuclei.

As the reference concentration, we consider the simplest but representative case of a dilute concentration $c_0$. Starting from a pure melt, $c_0$ is derived by strong dilution with solvent molecules up to the point until motifs (polymer chains here) are so highly dispersed that the enthalpic effect disappeared. Accordingly, the reference solution is an ideal dilute solution in which the motifs and clusters of different sizes do not interact with each other. In this work, we were concerned with the size of critical nuclei in (ideal) dilute solutions of different concentrations. When the reference solution is further diluted, the added solvent molecules only lead to the change of entropy while the enthalpy of the crystallizable motifs and clusters remains the same during further dilution. The further diluted solution has a concentration of $c_1$ and is also an ideal dilute solution. For brevity, in the following, we will refer to the solution with concentration of $c_0$ and $c_1$ as the reference solution and the diluted solution, respectively.

For nucleation in dilute solutions (in the early stages of crystallization), only a small fraction of motifs has crystallized, so the concentration of motifs can be considered constant and is exactly the concentration of the solution ($c_0$ or $c_1$). In comparison to the reference state, the equilibrium concentration of clusters of the specific size n in the diluted solution of concentration $c_1$ is given as

$$c_{\mathrm{cluster},1}\left(n\right)=c_{\mathrm{cluster},0}\left(n\right)\cdot \frac{c_1^n}{c_0^n}$$ 8

$c_{\mathrm{cluster},0}\left(n\right)$ and $c_{\mathrm{cluster},1}\left(n\right)$ represent the equilibrium concentration of clusters of size n in the reference and diluted solution, respectively. Equation (8) follows the basic expectation of thermodynamic equilibrium and is further verified by the numerical simulation in Supplementary Fig. 15. According to the laws of thermodynamics, the chemical potential of motifs in the diluted solution relative to the reference solution is derived as

$${\mu }_1-{\mu }_0=RT\ln \frac{c_1}{c_0}$$ 9

where ${\mu }_0$ and ${\mu }_1$ represent the chemical potential of motifs in the reference and diluted solutions, respectively. The effect of dilution (the concentration decreases) on the chemical potential of clusters of size n is thus

$${\mu }_{\mathrm{cluster},1}-{\mu }_{\mathrm{cluster},0}=RT\ln \frac{c_{\mathrm{cluster},1}\left(n\right)}{c_{\mathrm{cluster},0}\left(n\right)}=nRT\ln \frac{c_1}{c_0}$$ 10

Here, ${\mu }_{\mathrm{cluster},0}$ and ${\mu }_{\mathrm{cluster},1}$ represent the chemical potential of clusters of size n in the reference and diluted solution, respectively. In the reference solution of concentration $c_0$, for the process of forming one mole of clusters of size n starting from n mole of monomeric motifs, the change in Gibbs free energy is given as

$$\mathrm{\Delta }{G}_0\left(n\right)=\left({\mu }_{\mathrm{cluster},0}-n{\mu }_0\right)+\gamma A_{\mathrm{s}}\left(n\right)$$ 11

with $\gamma$ and $A_{\mathrm{s}}\left(n\right)$ are interfacial free energy and surface area of the clusters of size n, respectively. In the diluted solution of concentration $c_1$, the change in Gibbs free energy $\mathrm{\Delta }{G}_1\left(n\right)$ of the same process (forming one mole of nuclei with size n) is given as

$$\mathrm{\Delta }{G}_1\left(n\right)=\left({\mu }_{\mathrm{cluster},1}-n{\mu }_1\right)+\gamma A_{\mathrm{s}}\left(n\right)$$ 12

Since both the reference solution and the diluted solution are ideal (dilute) solutions, the motifs and clusters do not interact with each other, so the interfacial free energy is the same for clusters of a given size, independent of solution concentration. Therefore, we have

$$\mathrm{\Delta }{G}_1\left(n\right)-\mathrm{\Delta }{G}_0\left(n\right)=\left({\mu }_{\mathrm{cluster},1}-{\mu }_{\mathrm{cluster},0}\right)-n\left({\mu }_1-{\mu }_0\right)=0$$ 13

Namely,

$$\mathrm{\Delta }{G}_1\left(n\right)=\mathrm{\Delta }{G}_0\left(n\right)$$ 14

This means that the change in Gibbs free energy as a result of the formation of one mole of clusters of size n in an ideal dilute solution does not vary with the solution concentration. Equation (14) applies to clusters of any size, given that it is not larger than the size of the critical nuclei. Thus, for $\mathrm{d}\mathrm{\Delta }G/\mathrm{d}n=0$, $n^{*}$ (the size of critical nuclei) should be independent of the concentration of the solution (i.e., independent of supersaturation).

In contrast, by only considering the effect of dilution on the chemical potential of monomeric motifs (Eq. (6)), existing theories draw an erroneous conclusion due to ignoring the effect of dilution on the equilibrium concentration (and, therefore, on the chemical potential) of clusters of various sizes (Fig. 5). Indeed, in solution, since both, the monomeric motifs and the clusters of various sizes (including the critical nuclei), are diluted, the difference of the chemical potential between them does not change upon dilution. Consequently, the size of critical nuclei in a dilute solution should be constant with varying supersaturation.

Fig. 5. Schematic diagram of the effect of the change in concentration of the solution on the size of critical nuclei.

When the solution is diluted (i.e., the concentration decreases), both the motifs and clusters of different sizes (including the critical nuclei) are diluted (with different ratios of dilution, which depend on the corresponding concentrations related to cluster size). The change in Gibbs free energy upon the formation of one mole of critical nuclei is the same in solutions of different concentrations. Thus, the size of critical nuclei does not vary with solution concentration. The the right side in the figure, $\mathrm{\Delta }S$ (orange) reflects the change in translational entropy of clusters upon dilution, which was ignored by existing theories. $\mathrm{\Delta }H$ and $\mathrm{\Delta }G$ represent the enthalpic change and Gibbs free energy change of the corresponding process, respectively. The subscripts 0 and 1 indicate the solution concentrations of $c_0$ and $c_1$, respectively. $R$ is the gas constant. When an ideal dilute solution is diluted from the concentration of $c_0$ to the concentration of $c_1$ ($\mathrm{\Delta }H=0$), the change in Gibbs free energy upon formation of one mole of clusters remains unaffected ($\mathrm{\Delta }{G}_0=\mathrm{\Delta }{G}_1$). Thus, the size of critical nuclei does not change upon dilution. As existing theories only consider the change in translational entropy $\mathrm{\Delta }S$ upon dilution of the monomeric motifs (blue) but ignores $\mathrm{\Delta }S$ upon dilution of clusters (orange), existing theories draw an erroneous conclusion.

The above thermodynamic derivations, implying that the size of critical nuclei does not vary with supersaturation in dilute solutions, are valid for both, small molecules and polymers. The connectivity of crystallizable units distinguishes polymers from small molecules. However, connectivity only affects the dynamic rate constants of forming clusters of various sizes. In the thermodynamic equilibrium state, connectivity has no effect on the difference in Gibbs free energy for the clusters of different sizes.

From the kinetic viewpoint of nucleation, when deriving the size of critical nuclei, the existing theories only consider the effect of a change in polymer concentration on the translational entropy of crystallizable units with size 1 (i.e., the monomeric motifs). However, the corresponding change in translational entropy attributed to the (formed) clusters of various sizes was ignored. Taking all contributions to translational entropy into account, from both, the monomeric motifs and the clusters of various sizes, our microscopic kinetic model²⁸ on nucleation describes correctly the steady-state concentration (number density) of clusters of various sizes (including the critical nuclei) in solutions as a function of the concentration of the solution. All steps of the kinetic analysis are presented in detail in the Supplementary Information. For a simple case of a system consisting of separated solute molecules and solvent, a numerical simulation of solution nucleation yielding the varying steady-state number densities of clusters of a certain size as a function of solution concentration is presented in Supplementary Figs. 15–17. Mathematical derivations (Supplementary Eqs. (18)–(41)) verify the simulation results. The above dynamic analysis of nucleation in solution can describe the situation in which polymer chains are randomly selected from their mixture with solvent molecules. For nucleation of random copolymers, the connectivity of repeating units (including crystallizable and non-crystallizable units) along polymer backbones is reflected in Supplementary Eq. (42) by the decrease of $k_i^{+}$ with increasing number $i$ (e.g., in the Sadler-Gilmer model^29,30, the decrease of attachment rate due to the entropic energy barrier is proportional to the stem length $L$, expressed through an exponential term exp($-KL$) or ${\left[\exp \left(-K\right)\right]}^L$, in which K is the entropic energy barrier of nucleation per stem length divided by RT). By contrast, for the mixture of separated solute molecules (small molecules or polymer chains) and solvent molecules, $k^{+}$ (the rate for attaching a new molecule or a segment from another chain) is constant. Mathematical derivations (Supplementary Eqs. (42)–(44)) demonstrate via Supplementary Eq. (45) how the nucleation rate of random copolymers varies with dilution by non-crystallizable repeating units along the polymer backbones, which is consistent with Eq. (2) of the probability-based model. The kinetic and thermodynamic analyses discussed above, which take into account all the contributions to translational entropy, both of the monomeric motifs and the formed clusters of various sizes, confirm our method for determining the size of critical secondary nuclei (Supplementary Eq. (45)) and supports our conclusion that the size of critical secondary nuclei does not vary with supersaturation in dilute solutions (Eq. (14)).

Under our framework of correctly considering the change in translational entropy of both the monomeric motifs and the clusters of various sizes upon dilution (as a consequence of the dilution-caused change in the steady-state concentration of clusters of various sizes being properly considered), some long-standing problems about nucleation can be reasonably explained. Based on existing theories, the critical nuclei size of polymers should increase with the decreased concentration of crystallizable units (the decreased number of crystallizable units per unit volume), either in melt or in solution crystallization. For instance, Sanchez et al.³¹ deduced that the lamellar crystalline core thickness of random copolymers should increase with the proportion of the non-crystallizable units (in the case that all non-crystallizable units were excluded from the crystalline region). However, the deductions of Sanchez et al. contradict the experimental results that the lamellar crystalline core thickness of random copolymers remains constant with the proportion of non-crystallizable units at a given temperature in melt nucleation reported by Strobl et al.³², Xu et al.²¹, and solution nucleation of Fischer et al.³³. These experimental results can be reasonably explained by the aforementioned theoretical analysis that the size of critical secondary nuclei does not vary with the number of crystallizable units of polymer per unit volume, either in melt or in solution. In fact, the decrease in nucleation rate of random copolymers relative to homopolymer is due to the decrease in the steady-state concentrations of motifs (with a coefficient of $p_{\mathrm{A}}$) and the clusters with various sizes (with a coefficient of $p_{\mathrm{A}}^i$). $p_{\mathrm{A}}$ is the proportion of crystallizable units in the copolymer and $i$ is the size of clusters, as detailed in Supplementary Eqs. (42)–(45).

In summary, we presented a method to determine the size of critical secondary nuclei on the growth face of polymer single crystals in solution. Our approach is based only on the probability of selecting crystallizable units within random copolymer chains but does not rely on prior knowledge of the detailed nucleation mechanism or thermodynamics-related parameters. The number of crystalline stems and crystalline units contained within the critical secondary nuclei, and thus their size, increases monotonously with temperature $T_{\mathrm{C}}$ in the whole studied temperature range. The size of critical secondary nuclei on (110) and (020) crystal faces differs more progressively with increasing $T_{\mathrm{C}}$. Most intriguingly, the size of critical secondary nuclei is revealed to be independent of the concentration of the dilute solution, which contests the well-accepted prediction of existing theories that the size of critical nuclei size varies with the degree of supersaturation of the solution. The existing theories for deriving the size of critical nuclei erroneously assumed that the chemical potential of clusters of a certain size is constant in solutions of different concentrations. To remove this erroneous assumption, we account for the decrease in steady-state concentration (number density) of clusters of a certain size (including the critical nuclei) upon dilution and emphasize that the corresponding increase in translational entropy of clusters of various sizes is significant and thus cannot be ignored. Accordingly, our work provides a more comprehensive understanding of the nucleation mechanism, which removes the discrepancy between experimental results and existing theories. Our experimental findings and the corresponding theoretical interpretation are beneficial for the structural control of nanocrystals, quantum dots, and colloids and may even allow for prohibiting diseases caused by the crystallization of bio-macromolecules in a liquid environment.

Methods

Materials

Succinic acid was purchased from J&K Scientific. 2-methylsuccinic acid, 1,4-butanediol, tetrabutyl titanate, 1,2-dichlorobenzene was purchased from Aldrich. Chloroform and methanol were purchased from Sinopharm. All materials were used as received. PBS and poly(butylene succinate-ran-butylene 2-methylsuccinate) (PBSM) were synthesized via a two-step reaction of esterification and polycondensation in the melt. Products with similar molecular weight and molecular weight distribution were obtained with consistent experimental procedures. (See more details in Supplementary Fig. 1 and Supplementary Methods).

Characterization

The number-average molecular weight of polymers was determined by a ¹H NMR spectrometer (JEOL, ECA-600M) and gel permeation chromatography (Waters-1515). The molecular weight distribution was determined by gel permeation chromatography (Waters-1515). The results are shown in Supplementary Table 1. The molar ratios of the two types of repeating units, BS and butylene 2-methyl succinate (BM), in the copolymers, agreed well with the feeding ratios of the two types of diacids. The names of the samples (PBS, PBSM0.99, PBSM0.98, PBSM0.97, and PBSM0.96) were labeled with the molar proportions of crystallizable BS units in copolymer. The two diacid monomers have similar chemical structure and tended to form random copolymers during polycondensation. The degree of randomness of all copolymers was nearly 1, derived from the results of ¹³C NMR spectrometer (JEOL, ECA-600M). This means that BS units and butylene 2-methylsuccinate (BM) units were truly randomly distributed within copolymer chains.

Preparation of PBS single crystal seeds

PBS single crystals were prepared by a self-seeding method and used as seeds for initiating crystallization of PBSM. The details of the self-seeding procedure are shown in Supplementary Methods. PBS single crystals grew isothermally at 69 °C in an 1,2-dichlorobenzene solution. The lamellar thickness of the resulting seeds was thick enough to provide an epitaxial growth front for secondary nucleation of PBSM within the studied temperature range (52 °C ≤ T_C ≤ 69 °C).

Cultivation and observation of PBSM single crystals

PBSM copolymers were dissolved in 1,2-dichlorobenzene solution (c = 0.1 mg mL⁻¹) at 130 °C. After transfer to an oil bath at the crystallization temperature (52 °C ≤ T_C ≤ 69 °C), a homogeneous but supersaturated solution was established. Within this range of $T_{\mathrm{C}}$, no primary nucleation occurred in this solution during the experiment, as confirmed by dynamic light scattering. Subsequently, a small amount of a solution containing PBS single-crystal seeds was injected into the supersaturated PBSM solution. These seeds induced epitaxial growth of PBSM crystals by attaching molecules on the crystalline faces of PBS seed crystals. A small volume of the solution containing PBSM single crystals was sucked out after different times $t$ of crystallization at $T_{\mathrm{C}}$. The distance, which the PBSM crystals grew on the crystal faces, was determined as a function of $t$. (Supplementary Figs. 6–9) The growth distance and the lamellar thickness of single crystals were determined by atomic force microscopy (Shimadzu, SPM-9600, equipped with NSG-30 probes from TipsNano).

Theoretical model

Nucleation is a stochastic process, in which molecules or sequences of a polymer chain are randomly selected, with only those meeting several criteria being packed into crystalline nuclei. Both diffusion and selection of motifs with the right orientation and conformation affect the nucleation rate. To keep diffusion the same, we consider here only random copolymers and homopolymers with similar chain lengths. Thus, the only remaining difference between nucleation in systems of homopolymer chains and random copolymer chains, which contain crystallizable and non-crystallizable units, is the probability of selecting the crystallizable units on a random copolymer.

The proportion of crystallizable units in random copolymers is defined as $p_{\mathrm{A}}$ and the required number of units in a critical nucleus is defined as $m$. The probability-based model focuses on the ratio of nucleation rates in random copolymers and the corresponding homopolymer. For homopolymer chains, the probability that a selected (attached) unit is crystallizable is 1. For random copolymer chains, the probability that a selected (attached) unit is crystallizable is $p_{\mathrm{A}}$, so the probability that all m units attached to form a critical nucleus are crystallizable is $p_{\mathrm{A}}^m$, compared to the homopolymer. Since the flux of nucleation is determined by the association rate of the critical nucleus and the next crystallizable motif, the nucleation rate (for whatever, primary or secondary nuclei) of random copolymer is given as follows

$$i^{\prime }=i\cdot p_{\mathrm{A}}^{m+1}$$ 15

$i$ and $i^{\prime }$ are the nucleation rates of homopolymer and random copolymer, respectively. The central assumption of this model for determining the size of secondary critical nuclei is based on the case where the random copolymer and homopolymer have the same enthalpic energy barrier for nucleation, and the reduction of the nucleation rate of random copolymer relative to the corresponding homopolymer is dominated by the entropic effect, which is related to the successful selection of randomly distributed crystallizable units.

The validity of this model has been confirmed for determining the size of critical secondary nuclei on the growth front of lamellar crystals of spherulite in melt. However, there are some limitations in the melt. The crystal face of the growth front of spherulite cannot be completely defined. Besides, dilution of the melt (through introducing non-interacting components) does not eliminate the diffusion-limiting behavior of highly viscous molecules, which hinders the study of more critical problems. It should be noted that the previous work used the simplified formula $i^{\prime }=i\cdot p_{\mathrm{A}}^m$, which has a negligible effect on the measured size of critical nuclei. In this paper, the formula for determining the size of critical nuclei is derived in detail (Supplementary Eqs. (18)–(45)) and summarized in Eq. (2).

It should be noted that the connectivity of repeating units along polymer backbones has been considered in this probability-based model, which focuses on the ratio of nucleation rates in random copolymers and the corresponding homopolymer (as detailed in Supplementary Figs. 18 and 19). There, for both copolymers and homopolymers, connectivity between monomers exerts the same restrictions on the movements of molecular segments or chains as a whole. Thus, by determining the ratio of $i^{\prime }$ to $i$, this influence of connectivity is eliminated (Supplementary Eqs. (42)–(45)). Consequently, any difference in nucleation rates between copolymers and homopolymers has to be entirely due to an entropic energy barrier contributed by the corresponding product of probabilities that determines whether a randomly chosen unit is crystallizable or non-crystallizable. A more detailed explanation is given in Supplementary Discussion 5.

In this work, this model was applied to determine the size of critical secondary nuclei on the growth face of polymer single crystals in solution. The secondary nucleation rate $i$ was obtained from the growth rate of a certain crystal face. Depending on the degree of supercooling, different relations exist between the growth rate $G$ of a certain crystal face of a homopolymer and the corresponding secondary nucleation rate $i$^34,35:

$$\mathrm{For}\mathrm{Regime}\mathrm{I\; and}\mathrm{Regime}\mathrm{III}:G\propto i$$ 16

$$\mathrm{For}\mathrm{Regime}\mathrm{II}:G={b\left(2ig\right)}^{\frac{1}{2}}$$ 17

There, $g$ represents the lateral spreading rate of crystalline stems of homopolymers on the crystal growth face after formation of a secondary nucleus. $b$ is the width of a stem normal to the growth face.

Nucleation is a dynamic process. There, the probability of forming a cluster with a size that does not exceed the size of the critical nucleus size is much smaller than the probability of dissolving such a (small) cluster. In our experiments, to form a critical secondary nucleus on the existing growth face of polymer single crystal in solution, polymer chains are randomly selected from a mixture of polymer chains and solvent molecules. The probability to find and choose a polymer chain is proportional to its concentration $c$. If a secondary nucleus contains stems from $n$ different chains, total $n$ times of selection occurred, and the rate of secondary nucleation can be derived as follows

$$i^{\prime }=i\cdot c^n$$ 18

In dilute solutions, the lateral spreading of the crystalline phase will use segments of the chains already adsorbed on the growth face, so dilution will not alter the rate of lateral spreading. Our experimental results, as presented in Supplementary Fig. 3, showed that the growth rate of single crystals is proportional to $c^{1/2}$. This evidence indicates that the growth of single crystals occurred within Regime II, and a critical secondary nucleus consisted of the stems from a single polymer chain. We assume that a critical secondary nucleus has a rectangular shape with $s$ stems, and each stem consists of $r$ crystalline repeating units. The number of crystalline units $r$ within one stem can be derived from the lamellar thickness and the lattice parameters of single crystals. Therefore, the number of stems within a critical secondary nucleus can be obtained by $s=m/r$, with $m$ being the total number of crystalline units within a critical secondary nucleus. In addition, since the lateral spreading rate is determined by the adsorption of polymers and the crystallization rate of each stem, we have

$$g^{\prime }=g\cdot p_{\mathrm{A}}^{r+1}$$ 19

$g^{\prime }$ represents the lateral spreading rate of crystalline stems of copolymers on the crystal growth face after the formation of a secondary nucleus. A detailed discussion of the effect of non-crystallizable units on the lateral spreading rate can be found in Supplementary Eqs. (13)–(17). According to the above derivations, the growth rate $G^{\prime }$ of single crystals of random copolymers can be described as

$$G^{\prime }=G\cdot p_{\mathrm{A}}^{\frac{m+r+2}{2}}=G\cdot p_{\mathrm{A}}^{\frac{sr+r+2}{2}}$$ 20

When s is much larger than 1, Eq. (20) can be simplified as Eq. (3). It should be noted that the slope of $\ln G^{\prime }$ versus $\ln p_{\mathrm{A}}$, which is $\frac{m+r+2}{2}$ essentially reflects the total effect of secondary nucleation rate and lateral spreading rate (including tertiary nucleation rate) on the growth rate of the crystal face. However, in this work $\mathrm{s}\ge 17$, thus $m\gg r+2$. This means the size of the critical secondary nucleus calculated according to Eq. (20) differs little from that calculated according to Eq. (3). The size of critical secondary nuclei has an essentially dominant effect on the growth rate of the crystal face. In this way, the number of crystalline units $m$ within a critical secondary nucleus can be obtained from the fitted slope of the double logarithmic plot of the epitaxial growth rate $G^{\prime }$ versus the proportion of crystallizable units $p_{\mathrm{A}}$ existing along the chain of random copolymers.

Author contributions

Y.L. and J.X. conceived the idea for this work and designed the experiments. Y.L. performed the experiments. J.X. contributed to the theoretical derivation. Y.Z. and T.Z. contributed to the numerical simulation. Y.L., Z.W., T.W., B.G., G.R., and J.X. wrote and revised the paper. All authors discussed the results and commented on the paper.

Peer review

Peer review information

Nature Communications thanks Alicyn Rhoades, who co-reviewed with Xiaoshi Zhang, Dario Cavallo and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All the data supporting the findings of this study are provided in the paper and the supplementary information. The raw data on crystal growth rates and thicknesses are available at 10.5281/zenodo.15171708. All data are available from the corresponding author upon request.

Code availability

Code for the microscopic kinetic model of nucleation is available at https://github.com/markliuy/Microscopic-kinetic-model-of-nucleation or 10.5281/zenodo.15070377.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-58962-5.