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Polymer Crystallization: Growth by Secondary Nucleation
Polymer Crystallization: Growth by Secondary Nucleation Crystallization can proceed in three ways that were originally elucidated in a seminal article by Burton et al. (1951). These ways are (i) crystallization by secondary nucleation, (ii) crystallization by surface roughening or surface melting, and (iii) crystallization by the intervention of screw dislocations. Unlike metals and many other atomic solids, polymers, being in the form of chains, have a much more complicated mechanism of growth. This complication is a direct result of their long chain nature and of the necessity for coordinated movements between the atoms making up the chain. The situation is further complicated by the process by which the lowest internal free energy of the chain can be attained, since the attainment of lowest free energy is an integral part of the crystallization process. The attainment of lowest free energy involves the internal free energy of the chain determined by its conformation, or its detailed shape (Boyd and Phillips 1993). Invariably this conformation involves a straightening out of the chain, which in the melt is usually a random coil. The lowest free energy conformation tends to correspond to the form of either a linear zigzag for some polymers, or a helix for many others. So, when compared to the process occurring in other types of materials, there is a complicated internal molecular process, which either precedes crystallization or occurs concurrently with it. Such a process makes crystallization by surface roughening a more unlikely phenomenon than in other materials. The surface melting process involves an equilibrium between atoms in the surface layer and atoms sitting on top of it. Because of the coordinated movements of the chain, atoms cannot simply jump out of the surface for polymers. Roughening must, therefore, be an integral part of the formation of the crystal surface by deposition, making it an unlikely event. Hence, experimental crystallization kinetics have invariably been interpreted using secondary nucleation theory and it has been very successful over the years. In this article the development of secondary nucleation theory for polymers and its experimental verification will be outlined. 1. Requirements of a Crystallization Theory for Polymers From the earliest days (Keller 1968, Phillips 1990, 1994) it has been recognized that a theory of isothermal crystallization in polymers must be able to describe or explain why the growth rate is linear with time, but dependent on the exponential of the negative reciprocal of the supercooling, and also why the thickness of the crystals is dependent on the reciprocal supercooling. Variations on that basic theory must also explain molecular weight and pressure depen-
dencies, and also why the growth rate of crystals in a copolymer decreases strongly with the comonomer content. All of these requirements are satisfied by secondary nucleation approaches for the vast majority of polymers. It must also be recognized that theories of crystallization of polymers can only be tested through the use of linear defect-free polymer chains (such as linear polyethylene). The requirement regarding the behavior of copolymers is in fact a requirement for explanation of the behavior of defective chains, of which copolymers are only one class. All stereoregular polymers, such as polypropylene, contain randomly placed defects in large amounts (usually more than 1% of the chain atoms) which make them behave like copolymers, and, hence, these are inappropriate polymers for the testing of theories of polymer crystallization. 2. Basic Models of Secondary Nucleation The simplest model that can be used involves the placement of a nucleus of undecided shape and size on a flat surface of the growing crystal and then determining its size and shape through the use of lowest free energy considerations. When applied specifically to polymers the model is modified to recognize that the polymer chains will invariably be aligned parallel to one another and transverse the thickness of the crystal. The result of the theoretical formulations is that there are three dimensions to the nucleus, since it is in the form of a box. Each of those dimensions has a critical value determined essentially by the free energy of the surface normal to the dimension under consideration. The principal simplification that can be made to this model is that the nucleus is a monolayer, having only two dimensions to be determined. Both schemes result in equations which predict the first two requirements of a theory as outlined above, namely the dependencies of both the linear growth rate and the crystal thickness on reciprocal supercooling. All of these formulations assume that the critical nucleus formation is the slowest and, hence, the rate-determining step. Once the nucleus is deposited on the surface the remaining chains will deposit against the nucleus and do so very rapidly, thus completing the growth step. They do, however, have the principal defect that they are continuum theories that do not take into account in any specific way the mechanism by which the polymer molecules actually add to the crystal. They are also defective in that they are missing a critical energy term needed to ensure that crystal growth occurs (Hoffman and Miller 1997). 3. The Hoffman Lauritzen Approach The detailed Hoffman Lauritzen approach has evolved over the years and many of its facets are missed or misunderstood by many readers. Essentially, it uses a 1
Polymer Crystallization: Growth by Secondary Nucleation chemical reaction rate theoretical approach involving a series of successive reactions. Since each reaction depends on the prior one the result is a flux rate of polymer chains, in units of individual stems, onto the crystal surface. Because it takes the individual stem unit approach the highest energy step is the first one and the critical nucleus is a single stem. Conceptually, this is the biggest difference compared to the earlier continuum nucleus approaches. The approach calculates the free energy of formation of the single stem nucleus in terms of the volumetric free energy of fusion as the free energy released when the stem is in crystallographic register with the underlying substrate, and the surface free energies of the adsorbed single stem. The deposition process may occur anywhere between the two extremes of a fully amorphous adsorbed state followed by rearrangement to crystallographic register, and of the chain becoming fully aligned in the melt before jumping onto the surface. Reality may be different for different systems. For instance, there could be a mesophase or hexagonal state as an intermediate state, generating a system close to the second extreme. The actual situation is handled through an apportioning of the free energy of fusion between the two stages of the first deposition. For instance, in the case of fully amorphous adsorption there would be no free energy of fusion released in the deposition step and all of it would be released during the rearrangement to full crystallographic register. For all cases, the impediment to formation of the critical nucleus is the surface free energy of the adsorbed single stem, appearing as the lateral surface free energy. (In a later version of the theory the lateral surface free energy was related to the change of entropy from a random coil to an oriented state and computed using the characteristic ratio of the polymer chain.) The apportionment factor is represented by the symbol Ψ which can have values between 0 and 1, but cannot be predicted a priori and for lack of information on the detailed adsorption process, for convenience is taken as one of the extremevalues. A debatable point has always been the question of whether the nucleus should really be a single stem. There has been experimental evidence for 30 years that the critical nucleus should be about three stems in size (Andrews et al. 1971, Lambert and Phillips 1990), but attempts at using a multiple stem approach have proved mathematically intractable. The problem really comes down to one of population dynamics of potential states. Recent approaches using simulations have also suggested that the multiple stem nucleus would be the most favored state. Nevertheless, the Hoffman Lauritzen theory produces equations, which describe experimental behavior in an excellent way. E
G
E
G
U* Kg G l G exp k exp k ! kT ) T ∆T f H R(T _ H F F c c 2
(1a)
where G is growth rate, U* is the activation energy for transport, T_ is the temperature at which the extrapolated melt viscosity is infinite, Kg is the nucleation constant, ∆T is the supercooling and f is a correction factor that accounts for the variation of the enthalpy of fusion ∆Hf with T. The nucleation constant is given by: jb σσ T ! Kg l ! e m k∆Hf
(1b)
where σ and σe are lateral and end surface energies of ! is the crystal, b is the thickness of a crystal stem, T m ! melting temperature of an infinitely the equilibrium thick crystal and k is Boltzmann’s constant. A major extension of the theory involved the recognition that the deposition of a single critical nucleus may not always occur and that multiple nucleation generates a different situation. The situation is handled best in general conceptual terms by considering it to be a competitive situation between the rate at which critical nuclei are deposited on the surface and the rate at which the chains deposit laterally to complete the growth step (Fig. 1). This leads to three distinct situations or regimes; I—the classical situation in which the rate of secondary nucleation is slowest, II—a situation in which the rates of secondary nucleation and lateral spreading are comparable, and III—a situation in which the rate of i g Regime I
Regime II
Regime III
Figure 1 Schematic diagram of the competition between the rate of secondary nucleus deposition and the rate of surface spreading.
logG–log (∆T)+U*/2.3R(Tc–T∞ )
Polymer Crystallization: Growth by Secondary Nucleation
III
II
I 1/Tc(∆T )f
Figure 2 Schematic diagram of the effects of regime transitions on growth kinetics.
secondary nucleation is fastest (Fig. 2). The changes in regime generate changes in Eqn. (1), given by the value of the parameter j, which are manifested as slope changes when experimental data are plotted. These three situations occur naturally in many polymers as the crystallization temperature is reduced. The vast majority of polymers studied show regimes II and III, whereas few show regime I which is the classical situation. Lower molecular weight tends to favor the lower regimes. The effect of microstructural chain defects has been handled in the main by assuming that they are primarily insoluble in the crystal for energetic reasons and, hence, impede the formation of the critical nucleus. A simple, but elegant, theoretical approach is that of Andrews et al. (1971), who used the Flory formulation for random distribution of defects in a random copolymer, and assumed that nothing else changed in the mechanism. This approach essentially assumes that the critical nucleus is of a given size at a given temperature and pressure and must be made up of continuous lengths of crystallizable units. Hence, only sequence lengths greater, or equal to, the critical length present in a critical nucleus will take part in nucleus formation. The relation derived is one, found
experimentally, in that the logarithm of the growth rate is inversely proportional to the fractional defect content. The equation also predicts that the critical nucleus size, in the form of a continuous folded chain, can be derived. In the two extant cases published the critical nucleus was multiple stem, not single stem, except in the extreme case of a highly defective chain where only a single stem nucleus is possible. The Andrews’ equation was derived for a constant crystallization temperature and does not address the effect of microstructural content on regimes. This subject has been researched and reviewed (Phillips 1993) and the basic elements of behavior are established. Bibliography Andrews E H, Owen P J, Singh A 1971 Microkinetics of lamellar crystallization in a long chain polymer. Proc. R. Soc. London 324, 79–97 Boyd R H, Phillips P J 1993 The Science of Polymer Molecules. Cambridge University Press, Cambridge Burton W K, Cabrera N, Frank F C 1951 The growth of crystals and the equilibrium structure of their surfaces. Proc. R. Soc. A 243, 299–358 Hoffman J D 1983 Regime III crystallization in melt-crystallized polymers: the variable cluster model. Polymer 24, 3–26 Hoffman J D, Miller R L 1997 Kinetics of crystallization from the melt with chain folding. Polymer 38, 3151–212 Hoffman J D, Miller R L, Marand H, Roitman D B 1992 Relationship between the lateral surface free energy-sigma and the chain structure of melt-crystallized polymers. Macromolecules 25, 2221–9 Keller A 1968 Crystallization of polymers. Rep. Prog. Phys. 31, 623–75 Lambert W S, Phillips P J 1990 Regime transitions in a nonrepeating polymer: crosslinked linear polyethylene. Macromolecules 23, 2075–81 Phillips P J 1990 Polymer crystals. Rep. Prog. Phys. 53, 549–604 Phillips P J 1993 Effects of chemical microstructure on regimes in polymer crystallization. In: Dosiere M (ed.) Crystallization of Polymers. Kluwer, Dordrecht, The Netherlands, pp. 301–11 Phillips P J 1994 Spherulitic crystallization in macromolecules. In: Hurle D T J (ed.) Handbook of Crystal Growth. Vol. 2. Elsevier, Amsterdam, pp. 1169–215
P. J. Phillips
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 7253–7256 3
dencies, and also why the growth rate of crystals in a copolymer decreases strongly with the comonomer content. All of these requirements are satisfied by secondary nucleation approaches for the vast majority of polymers. It must also be recognized that theories of crystallization of polymers can only be tested through the use of linear defect-free polymer chains (such as linear polyethylene). The requirement regarding the behavior of copolymers is in fact a requirement for explanation of the behavior of defective chains, of which copolymers are only one class. All stereoregular polymers, such as polypropylene, contain randomly placed defects in large amounts (usually more than 1% of the chain atoms) which make them behave like copolymers, and, hence, these are inappropriate polymers for the testing of theories of polymer crystallization. 2. Basic Models of Secondary Nucleation The simplest model that can be used involves the placement of a nucleus of undecided shape and size on a flat surface of the growing crystal and then determining its size and shape through the use of lowest free energy considerations. When applied specifically to polymers the model is modified to recognize that the polymer chains will invariably be aligned parallel to one another and transverse the thickness of the crystal. The result of the theoretical formulations is that there are three dimensions to the nucleus, since it is in the form of a box. Each of those dimensions has a critical value determined essentially by the free energy of the surface normal to the dimension under consideration. The principal simplification that can be made to this model is that the nucleus is a monolayer, having only two dimensions to be determined. Both schemes result in equations which predict the first two requirements of a theory as outlined above, namely the dependencies of both the linear growth rate and the crystal thickness on reciprocal supercooling. All of these formulations assume that the critical nucleus formation is the slowest and, hence, the rate-determining step. Once the nucleus is deposited on the surface the remaining chains will deposit against the nucleus and do so very rapidly, thus completing the growth step. They do, however, have the principal defect that they are continuum theories that do not take into account in any specific way the mechanism by which the polymer molecules actually add to the crystal. They are also defective in that they are missing a critical energy term needed to ensure that crystal growth occurs (Hoffman and Miller 1997). 3. The Hoffman Lauritzen Approach The detailed Hoffman Lauritzen approach has evolved over the years and many of its facets are missed or misunderstood by many readers. Essentially, it uses a 1
Polymer Crystallization: Growth by Secondary Nucleation chemical reaction rate theoretical approach involving a series of successive reactions. Since each reaction depends on the prior one the result is a flux rate of polymer chains, in units of individual stems, onto the crystal surface. Because it takes the individual stem unit approach the highest energy step is the first one and the critical nucleus is a single stem. Conceptually, this is the biggest difference compared to the earlier continuum nucleus approaches. The approach calculates the free energy of formation of the single stem nucleus in terms of the volumetric free energy of fusion as the free energy released when the stem is in crystallographic register with the underlying substrate, and the surface free energies of the adsorbed single stem. The deposition process may occur anywhere between the two extremes of a fully amorphous adsorbed state followed by rearrangement to crystallographic register, and of the chain becoming fully aligned in the melt before jumping onto the surface. Reality may be different for different systems. For instance, there could be a mesophase or hexagonal state as an intermediate state, generating a system close to the second extreme. The actual situation is handled through an apportioning of the free energy of fusion between the two stages of the first deposition. For instance, in the case of fully amorphous adsorption there would be no free energy of fusion released in the deposition step and all of it would be released during the rearrangement to full crystallographic register. For all cases, the impediment to formation of the critical nucleus is the surface free energy of the adsorbed single stem, appearing as the lateral surface free energy. (In a later version of the theory the lateral surface free energy was related to the change of entropy from a random coil to an oriented state and computed using the characteristic ratio of the polymer chain.) The apportionment factor is represented by the symbol Ψ which can have values between 0 and 1, but cannot be predicted a priori and for lack of information on the detailed adsorption process, for convenience is taken as one of the extremevalues. A debatable point has always been the question of whether the nucleus should really be a single stem. There has been experimental evidence for 30 years that the critical nucleus should be about three stems in size (Andrews et al. 1971, Lambert and Phillips 1990), but attempts at using a multiple stem approach have proved mathematically intractable. The problem really comes down to one of population dynamics of potential states. Recent approaches using simulations have also suggested that the multiple stem nucleus would be the most favored state. Nevertheless, the Hoffman Lauritzen theory produces equations, which describe experimental behavior in an excellent way. E
G
E
G
U* Kg G l G exp k exp k ! kT ) T ∆T f H R(T _ H F F c c 2
(1a)
where G is growth rate, U* is the activation energy for transport, T_ is the temperature at which the extrapolated melt viscosity is infinite, Kg is the nucleation constant, ∆T is the supercooling and f is a correction factor that accounts for the variation of the enthalpy of fusion ∆Hf with T. The nucleation constant is given by: jb σσ T ! Kg l ! e m k∆Hf
(1b)
where σ and σe are lateral and end surface energies of ! is the crystal, b is the thickness of a crystal stem, T m ! melting temperature of an infinitely the equilibrium thick crystal and k is Boltzmann’s constant. A major extension of the theory involved the recognition that the deposition of a single critical nucleus may not always occur and that multiple nucleation generates a different situation. The situation is handled best in general conceptual terms by considering it to be a competitive situation between the rate at which critical nuclei are deposited on the surface and the rate at which the chains deposit laterally to complete the growth step (Fig. 1). This leads to three distinct situations or regimes; I—the classical situation in which the rate of secondary nucleation is slowest, II—a situation in which the rates of secondary nucleation and lateral spreading are comparable, and III—a situation in which the rate of i g Regime I
Regime II
Regime III
Figure 1 Schematic diagram of the competition between the rate of secondary nucleus deposition and the rate of surface spreading.
logG–log (∆T)+U*/2.3R(Tc–T∞ )
Polymer Crystallization: Growth by Secondary Nucleation
III
II
I 1/Tc(∆T )f
Figure 2 Schematic diagram of the effects of regime transitions on growth kinetics.
secondary nucleation is fastest (Fig. 2). The changes in regime generate changes in Eqn. (1), given by the value of the parameter j, which are manifested as slope changes when experimental data are plotted. These three situations occur naturally in many polymers as the crystallization temperature is reduced. The vast majority of polymers studied show regimes II and III, whereas few show regime I which is the classical situation. Lower molecular weight tends to favor the lower regimes. The effect of microstructural chain defects has been handled in the main by assuming that they are primarily insoluble in the crystal for energetic reasons and, hence, impede the formation of the critical nucleus. A simple, but elegant, theoretical approach is that of Andrews et al. (1971), who used the Flory formulation for random distribution of defects in a random copolymer, and assumed that nothing else changed in the mechanism. This approach essentially assumes that the critical nucleus is of a given size at a given temperature and pressure and must be made up of continuous lengths of crystallizable units. Hence, only sequence lengths greater, or equal to, the critical length present in a critical nucleus will take part in nucleus formation. The relation derived is one, found
experimentally, in that the logarithm of the growth rate is inversely proportional to the fractional defect content. The equation also predicts that the critical nucleus size, in the form of a continuous folded chain, can be derived. In the two extant cases published the critical nucleus was multiple stem, not single stem, except in the extreme case of a highly defective chain where only a single stem nucleus is possible. The Andrews’ equation was derived for a constant crystallization temperature and does not address the effect of microstructural content on regimes. This subject has been researched and reviewed (Phillips 1993) and the basic elements of behavior are established. Bibliography Andrews E H, Owen P J, Singh A 1971 Microkinetics of lamellar crystallization in a long chain polymer. Proc. R. Soc. London 324, 79–97 Boyd R H, Phillips P J 1993 The Science of Polymer Molecules. Cambridge University Press, Cambridge Burton W K, Cabrera N, Frank F C 1951 The growth of crystals and the equilibrium structure of their surfaces. Proc. R. Soc. A 243, 299–358 Hoffman J D 1983 Regime III crystallization in melt-crystallized polymers: the variable cluster model. Polymer 24, 3–26 Hoffman J D, Miller R L 1997 Kinetics of crystallization from the melt with chain folding. Polymer 38, 3151–212 Hoffman J D, Miller R L, Marand H, Roitman D B 1992 Relationship between the lateral surface free energy-sigma and the chain structure of melt-crystallized polymers. Macromolecules 25, 2221–9 Keller A 1968 Crystallization of polymers. Rep. Prog. Phys. 31, 623–75 Lambert W S, Phillips P J 1990 Regime transitions in a nonrepeating polymer: crosslinked linear polyethylene. Macromolecules 23, 2075–81 Phillips P J 1990 Polymer crystals. Rep. Prog. Phys. 53, 549–604 Phillips P J 1993 Effects of chemical microstructure on regimes in polymer crystallization. In: Dosiere M (ed.) Crystallization of Polymers. Kluwer, Dordrecht, The Netherlands, pp. 301–11 Phillips P J 1994 Spherulitic crystallization in macromolecules. In: Hurle D T J (ed.) Handbook of Crystal Growth. Vol. 2. Elsevier, Amsterdam, pp. 1169–215
P. J. Phillips
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 7253–7256 3