How to understand nucleation in crystallizing polymer melts under real processing conditions

Abstract

As has been shown experimentally in our laboratory, the number of athermal nuclei, as found in unnucleated quiescent melts, increases tremendously with decreasing temperatures of crystallization, down to severe degrees of undercooling. One cannot assume that the presence of heterogeneous nuclei can explain this horrible temperature dependence. Moreover, one can conclude that the number fraction of macromolecules participating in these athermal nuclei is extremely low. Macroscopically, these nuclei seem to form a number of spots in a sea of homogeneous undercooled liquid.In the present paper the proposal is made that this number can be estimated from the probability of the occurrence of local molecular arrangements of varying quality, which preexist by accidence in a so-called living equilibrium in the stable melt, i.e. above the equilibrium melting point. During a rapid quench, realistic for processing conditions, these local arrangements are successively stabilized and serve as precursors for the start of crystallization. Dependent on their quality, this stabilization occurs over a broad range of crystallization temperatures. A selection rule for their effectiveness is derived from thermodynamics. In addition, reasons are discussed for the observed strong influence of flow on the formation of nuclei. From the "short-term" creep experiments, which are successful even at low degrees of undercooling, the impression has been obtained that during flow an unimaginable long-distance mechanical interaction becomes effective between the nuclei of crystallization. However, a more convincing explanation has been found recently: it is described at the end of this paper.

Appendices

Appendix 1. Number density of athermal nuclei, as derived from probability considerations

The first question is about the overlap of any coil molecule with the origin of the coordinate system, where a reference sequence of s repeating units of a regular conformation, as needed for a permanent association, will be placed later. If one has a macromolecule of contour length l, one has to find several parameters which are needed for the calculations. From the well-known average square radius of gyration 〈s ²〉=N _k A ²/6 (with N _k being the number of Kuhn segments of length A and l=N _k A) one can derive with N _k=l/A the following value for the (average) radius, r, of the coil:

$$ {r\; = \;{{l^{{1/2}} \;A^{{1/2}} } \over {6^{{1/2}} }}.} $$

(3)

The volume of the coil is then given by

$$ {V_{l} \; = \;{{4\pi } \over 3}\;{{l^{{3/2}} \,A^{{3/2}} } \over {6^{{3/2}} }}.} $$

(4)

An important point of Kuhn's statistics is that the segment density of the coil decreases with increasing contour length. In fact, a coil contains l/l ₀ repeating units ("monomers") of length l ₀ , whereas its volume increases with (l/l ₀)^3/2, as Eq. (4) shows. As a consequence, one obtains for this density ν₀(l):

$$ {\nu _{0} {\left( l \right)}\; = \;{{6^{{3/2}} \,3} \over {4\pi }}\;{1 \over {l^{{1/2}} \;A^{{3/2}} \;l_{0} }}.} $$

(5)

A coil overlaps with the origin if its center of mass lies within the radius r of the coil from the origin. Within this sphere, however, there also lie a lot of repeating units (of length l ₀), which belong to coils with their centers of mass outside this sphere! Only exactly at the origin are all repeating units from centers of mass lying within the said sphere. The corresponding number, Z ₀, of repeating units can be calculated as follows:

$$ {Z_{0} = \nu _{0} {\left( l \right)}{{4\pi } \over 3}r^{3} Z_{l} ,} $$

(6)

where Z _l is the number of coil molecules of contour length l per unit volume of the melt. (For simplicity it is assumed that in this calculation the density of the repeating units which belong to this coil is constant within the coil and falls abruptly to zero only at the radius r.) (4π/3)r ³ is here the volume of the sphere surrounding the origin. For Z _l one has

$$ {Z_{l} = {{\rho N_{{\rm{A}}} l_{0} } \over {M_{0} l}} = {{\rho N_{{\rm{A}}} l_{0} } \over {M_{0} }}{A \over {6r^{2} }},} $$

(7)

where ρ is the density of the melt (in kilograms per cubic meter), N _A is Avogadro's number and M ₀ is the molar mass of the repeating unit (in kilograms per mole). The second expression on the right side of this equation is obtained when Eq. (3) is used for the relation between l and r . If nowEqs. (3), (5) and (7) are inserted into Eq. (6), one obtains the following almost trivial relation for Z ₀:

$$ {Z_{0} = {{\rho N_{{\rm{A}}} } \over {M_{0} }}.} $$

(8)

In this equation every influence of an overlap has disappeared and, with it, the influence of the contour length. In fact, Eq. (8) can be derived directly from the molar mass M ₀ of the repeating unit without any special assumption.

However, only a fraction of these repeating units, as contained in the unit volume of the melt, will be capable of association with a repeating unit of another macromolecule: both units must be part of properly oriented regular sequences, as will be seen later. The influence of the contour length seems to reappear if the probability is considered that the second sequence is part of a macromolecule of a certain length l. In fact, the longer such a macromolecule is, the higher the probability is that the mentioned sequence is part of it. In this way one obtains an expression for the number of pairs, z _1,2, of macromolecules (per unit volume), capable of association, if the restriction is applied that each of them takes part only once. One has

$$ {z_{{1,2}} = {{P^{*} } \over 2}{l \over {l_{0} }}{\left( {{{\rho N_{{\rm{A}}} } \over {M_{0} }}v_{0} } \right)}Z_{l} ,} $$

(9)

where v ₀ is the volume of the repeating unit (i.e. the sphere of influence of this unit). However, the contour length l of the reference macromolecules is only formally contained in this equation, as will be seen quickly. First, it should be observed that the expression in parentheses is dimensionless, as it should be. In fact, Z _l is a number per unit volume, like z _1,2. One can also easily show that the expression in parentheses is of the order of unity. P ^* is a probability. For the first time in this treatment the comment must be made that this probability will strongly depend on the chosen length of the interacting regular sequences. Anyway, P ^* must be very small. As a consequence one is actually permitted to assume that every macromolecule is engaged only once. The factor 1/2 is introduced because of the fact that every repeating unit can be part of the reference sequence at the origin, but also of the participating other regular sequence. If now Eq. (9) is compared with the first expression on the right side of Eq. (7), one notices that z _1,2 is actually independent of l. This means that the first subscript can be omitted. One simply has

$$ {z_{2} = {{P^{*} } \over 2}{\left( {{{\rho N_{{\rm{A}}} } \over {M_{0} }}} \right)}^{2} v_{0} .} $$

(10)

In this equation the occurrence of P ^* reminds us that these repeating units must be parts of regular sequences ready for an attachment. Equation (10) shows that there is no obvious reason why polymers of a higher molar mass should contain a larger number of athermal nuclei than polymers of a lower molar mass. Polydispersity, however, may play a role, but this seems a difficult matter.

Whereas so far the excluded volume has been disregarded, such a simplification will not lead to realistic results when more than two sequences are to be associated in their nearest neighborhood. Actually, already in Eqs. (9) and (10) one should have used instead of v ₀ something like 6v ₀ . In fact, the central strand, as formed by the reference sequence, has about six open places for nearest neighbors in a cross-section perpendicular to its anticipated axis, in a quasi hexagonal packing. This c=6 is a coordination number, which eventually can be adjusted to the crystal lattice which is formed in reality. With each strand added one of these places is filled, but two other neighboring places are created. This means that 12 neighboring places are created, when the first shell of six is filled, as it should be for hexagonal packing.

Before the development of the equations can be continued, one must realize that P ^* is built up of two factors, i.e. P ^*=P _r P, where P _r is the probability that a reference sequence of s repeating units is preformed in the melt, and P is the probability that a second (regular) sequence is formed and can dock under the given circumstances at the reference sequence. If N _c regular sequences (N _c "strands") are to be associated, one has to form the product of (N _c−1) probabilities P, of which every one has first to be corrected for the number of docking places, increasing with N. In this way one can obtain—in a somewhat heuristic way—the number of bundles per unit volume containing N _c strands (Fig. 3), where N _c is independent of the sequence length, as was argued in the Preliminary argumentation.

$$ {\Phi _{N} = {1 \over 2}\;P_{r} P^{{N - 1}} {\left( {{{\rho N_{{\rm{A}}} v_{0} } \over {M_{0} }}} \right)}^{{N - 1}} {{{\left( {N + c - 2} \right)}!\;} \over {{\left( {c - 1} \right)}!}}{{\rho N_{{\rm{A}}} } \over {M_{0} }}.} $$

(11)

In this equation N _c is replaced simply by N for typographic reasons. The front factor 1/2 may probably be replaced by 1/N, because every strand is eligible as the reference sequence; however, such a change by a factor of the order of 0.1 is irrelevant in the "power (of ten) game" to be played with the probabilities. In Eq. (11) the factor containing the faculties can easily be explained. For this purpose it is assumed that N=10 and c=6. With these assumptions one obtains for the product containing v ₀ 6v ₀7v ₀8v ₀...14v ₀ . It is clear that this product contains N−1=9 factors. If one takes the v ₀ together, one can incorporate them in the expression in prantheses, as occurred in Eq. (11). The numerical factor is indeed 14!/5!, which shows the importance of the last but one factor of Eq. (11). One has for this factor the value ≅7.26×10⁸ , which certainly cannot be disregarded.

For an evaluation of Eq. (11) one needs additional assumptions, for example, ρ=0.7×10³ kg m⁻³, M ₀=5×10⁻² kg mol⁻¹, v ₀=(5×10⁻¹⁰)³ m³=1.25×10⁻²⁸ m³ .With these values one has (ρN _A/M ₀)=0.8722×10²⁸ m⁻³ and (ρN _A/M ₀)v ₀≅1.0902. A logarithmic version of Eq. (11) reads

$$ {\log \Phi _{N} = {\left( {N - 1} \right)}\log P\, + \;\log P_{{\rm{r}}} + {\kern 1pt} {\left( {N - 1} \right)}\log {{\rho N_{{\rm{A}}} v_{0} } \over {M_{0} }} + \log \;{{{\left( {N + c - 2} \right)}\;!} \over {{\left( {c - 1} \right)}\;!}}\, + \,\log {{\;\rho N_{{\rm{A}}} } \over {2M_{0} }}.} $$

(12)

With N=10, c=6 and P=P _r/s (where the last expression is an anticipation of Appendix 2) with s=10 , one has logΦ₁₀=10logP _r−9+0.338+8.861+27,758≅10logP_r+27.96. For i-PP we know two extreme values for logΦ, namely 9, as obtained at 150 °C, and 16, as found at 90 °C [1]. For these two temperatures one obtains values for logP _r of ≅−2 and of −1, respectively. These values do not seem unreasonable, as will be explained immediately in the next paragraph. It does not seem superfluous, however, to consider the shape of the curve defined by the rather intractable Eq. (12) with respect to varying values of N. For this purpose the following rough simplifications are introduced: P=P _r/10, c=6, lnN!≅NlnN—N (Stirling). If the modified Eq. (12) is solved for logP and differentiated with respect to N, one obtains a single maximum, which lies for logΦ=9 at N≅16 and for logΦ=16 at N≅12. But this means that the values for logP _r, as estimated earlier, are close to the maximum values, both of ≅−1! However, this argumentation can also be inverted. In order to obtain the experimental values of logΦ with bundles of the order of 10–15 strands one needs probabilities P _r of the order calculated earlier. In order to obtain a feeling for the influence of lower values of N, one must calculate the probability values for the lowest conceivable value of N, namely N=2, by using Eq. (11). In this way one obtains logP _r≅−9.2 for 150 °C and logP _r≅−6.4 for 90 °C. This means that extremely low probabilities would suffice in this case (of N=2) for the experimentally obtained numbers of nuclei. But it is also clear that with the intuitive requirement of an increased number N the demand for a higher probability P _r will become stringent. Fortunately, nature comes our way, as explained later.

In connection with the said intuition the remark made at the end of the Preliminary argumentation with respect to ΔH _f provides us with a consolation. In fact, with two strands not only the tension at the ends of the strands will be negligible but also the lateral cohesion will be insufficient for the initiation of a somewhat more extended ordered state which can serve as a nucleus.

Now we come to the promised argumentation in favor of higher values of P _r characteristic for a polymer melt even in its equilibrium state before the quench to the temperatures of crystallization. In this connection some insights must be recalled which were promoted quite a long time ago. In fact, finite pieces of helical (or zigzag) conformations must be present a priori in every relaxed molecular coil because of the energetic preference of those pieces. In this respect the reader is reminded of the improved theory of rubber elasticity, as described in Treloar's book [45]. One can deduce this fact also from the optical anisotropy of the random links [46, 47].

Appendix 2. A closer look at conditions of effective association

When Eq. (11) was constructed, the numbers of available neighboring places were counted for repeating units in a cross-section perpendicular to the reference sequence, i.e. to the anticipated direction of a bundle of sequences. For the following argumentation some local alignment is assumed, which means that neighboring macromolecules are arranged as in a meal of cooked spaghetti. In this respect the reader is reminded of Pechold's meander model [48]. For the three-dimensional space this means that the conformations of neighboring macromolecules cannot be completely independent of each other because of the requirement that the density of the condensed amorphous matter cannot be an order of magnitude lower than that of the crystallized matter. A favorable argument for local alignment is also the positive influence of elevated pressure on the speed of crystallization [49].

It seems that one can give an estimate for the docking probability P. For this purpose a volume element 2v ₀ s is considered. The longitudinal direction of this (oblong) volume element is assumed to be parallel to the reference sequence (of s repeating units). In such a volume element one has to find the center of mass of a regular sequence of s repeating units, if it is ready for attachment to the reference sequence. The number of those centers of mass is equal to the said volume element multiplied by the number of regular sequences of s repeating units per unit volume of the melt (as a whole). This number is equal to P _r(s)Z ₀/s. The probability for the said center of mass to be found at any location within the volume element 2v ₀ s is therefore 2P _r(s)v ₀ Z ₀. This means that the probability for a complete match, i.e. for the location of this center of mass exactly opposite the center of mass of the reference sequence, is equal to (1/2s)2P _r(s)v ₀ Z ₀=P _r(s)/s, because of v ₀ Z ₀ being equal to 1.

In fact, the absence of (major) stresses at the ends of regular sequences, as indicated in the right part of Fig. 2 for thin enough bundles, seems to be possible only if the ends of all regular strands which form the bundle are at the same "height". It seems impossible that such bundles contain regular sequences which do not match completely and, as a consequence, contain disordered regions in their interior. Or to formulate it in another way, if some units at the end of a regular sequence of, say, s units do not match completely, at best nuclei can be formed with fewer than s effective units. If, however, irregularities already exist at the onset, a useful nucleus cannot be expected anyway.

So, after all, it seems quite easy to formulate the docking probability P. For an assembly of regular sequences of s units one simply has

$$ P{\text{ = }}{P_{{\text{r}}} {\left( s \right)}} \mathord{\left/ {\vphantom {{P_{{\text{r}}} {\left( s \right)}} s}} \right. \kern-\nulldelimiterspace} s{\text{,}} $$

(13)

But this means that for the factor in Φ _N =Φ _N (s) containing the probabilities (see Eq. 11), one has

$$ P_{{\text{r}}} P^{{N{\text{ - 1}}}} {\text{ = }}{{\left[ {P_{{\text{r}}} {\left( s \right)}} \right]}^{N} } \mathord{\left/ {\vphantom {{{\left[ {P_{{\text{r}}} {\left( s \right)}} \right]}^{N} } {s^{{N{\text{ - 1}}}} }}} \right. \kern-\nulldelimiterspace} {s^{{N{\text{ - 1}}}} }. $$

(14)

This argumentation makes an estimate of the total number of athermal nuclei per unit volume of the melt as a function of temperature feasible. According to Eq. (2) the number n ^*=s determines the possibility that a "dormant" athermal nucleus can grow out into a detectable particle: the lower the temperature, the smaller s can be. So, it depends on the chosen crystallization temperature T _k how many dormant nuclei can become active. The lower this temperature, the more terms, as given by Eq. (11), etc., have to be added up. Of course, for a corresponding integration one needs P _r(s). In principle one has

$$ {\Phi _{N} \cong {\int\limits_{n^{*} {\left( {T_{{\rm{k}}} } \right)}}^\infty {\Phi _{N} {\left( s \right)}{\rm{d}}s} }.} $$

(15)

The change of the density of the melt which accompanies the quench is disregarded in this consideration; however, an adequate correction can always be made.

As already mentioned, when the idea of the local alignment was introduced, the molecules, which have to offer the needed regular sequences, are not exactly parallel to each other. However, one can admit a certain average space angle, as a small fraction of 2π. If this fraction is guessed, one can modify Eq. (13) adequately. This can also lead to an estimate for P _r(s) if the numbers of nuclei obtained experimentally at various crystallization temperatures are taken as a point of departure. Of course, there are still many unknown parameters, as there are σ_e, N _c, s, etc. Also crystallographic knowledge will be needed urgently. At the moment, the present author does not feel capable of performing this type of theoretical work. Nevertheless, he feels that the door is opened for realistic calculations in this field. Those calculations could be of importance if the number of nuclei are derived from the chemical structure of the macromolecules. However, such a goal, as desirable for the chemical industry, still seems very hard to attain.

For i-PP the number of nuclei is comparatively low at, say, 150 °C (≅10⁹ m⁻³), but the number increases tremendously with decreasing temperature (see the left side of Fig. 1), so a logarithmic scale is adequate. This means that exact integration of Eq. (15) is not strictly necessary, as it covers a range of higher temperatures where the numbers of nuclei are extremely small. This almost logarithmic dependence on ΔT (see the logarithmic scale for Φ in Fig. 1) is in line with Strobl's consideration [20], according to which the probability for the presence of a regular sequence of s repeating units decreases exponentially with increasing s.