Lattice-based simulations of chain conformations in semi-crystalline polymers with application to flow-induced crystallization

J. Non-Newtonian Fluid Mech. 82 (1999) 331±366

Lattice-based simulations of chain conformations in semi-crystalline polymers with application to flow-induced crystallization Jaydeep A. Kulkarni, Antony N. Beris* Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Received 15 June 1998; received in revised form 6 August 1998

Abstract Polymer fiber processes, such as high-speed spinning of nylon and PET, are highly complex involving a complicated interplay between an evolving internal molecular microstructure, macroscopic transport phenomena, such as fluid mechanics and heat transfer, and also non-equilibrium thermodynamics and kinetics affecting nucleation and subsequent crystal growth. All of the above processes are important in determining the final product's semi-crystalline morphology which is primarily responsible for its mechanical properties. Our approach to attack this problem is a hierarchical one: A macroscopic (continuum) model is developed based on the Hamiltonian formalism of non-equilibrium thermodynamics for flowing systems [Beris and Edwards, Oxford University Press, Oxford, 1994]. This approach allows us to follow the dynamic evolution of several macroscopic (continuum) variables involving both kinematic and structural parameters. To implement this approach successfully, an accurate modeling of the (extended) free energy (Hamiltonian) of the system under consideration and the dissipation therein is necessary. While the later is, at the moment, phenomenological, we are developing a first principles approach for the former based on a microscopic modeling of chain conformations using a lattice model. Lattice models have been used extensively before for the analysis of chain conformations in both purely amorphous and semi-crystalline polymers. We have reinterpreted some of the earlier lattice models by systematically deriving the relevant statistics of polymer chains and by outlining the a priori approximations in those models which are necessary to arrive at closed-form expressions. Although we do make use of such earlier work, we have also extended it through a computer-aided analysis. This analysis has enabled us to generate from first principles free energy surfaces for a system consisting of polymer chains, represented as multiple self- and mutually-avoiding random walks, on a 2-D fully populated semi-crystalline lattice. It is shown that the numerical results for dense semi-crystalline systems can be fitted with low order polynomials that provide closed-form approximations for the configurational entropy in terms of non-equilibrium structural parameters, such as the orientation and stretching of the polymer chains. Finally, chain statistics for bulk amorphous polymers have been validated against their theoretical predictions. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Lattice models; Viscoelasticity; Flow-induced crystallization; Dense polymers; Non-equilibrium thermodynamics; High-speed fiber spinning

* Corresponding author. Tel.: +1-302-831-8018; fax: +1-302-831-1048; e-mail: [email protected] 0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 1 7 1 - 2

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Fig. 1. Effect of the fiber spinning conditions on the morphology and the mechanical properties of PET (polyethylene terephthalate) fibers. The birefringence, indicative of the degree of orientation and the crystallinity, and the modulus, indicative of the mechanical strength, of the spun fibers are seen to increase dramatically with increasing spinning speeds. [Reprinted with permission from Perez [76]. Copyright by John Wiley and Sons, New York.]

1. Introduction Polymer fibers represent an annual multi-billion dollar worldwide business thanks to their excellent mechanical and physical properties. These properties (especially the mechanical ones and dyability) depend critically on the material microstructure. The unique properties are obtained when fibrillar crystallites of optimal dimensions are formed within the semi-crystalline morphology of the fibers, and polymer chains of optimal orientational distribution are present within the intervening amorphous regions. This can be usually achieved only under specific (often proprietary) processing conditions such as those encountered in high-speed fiber spinning (Fig. 1). This figure clearly demonstrates the dramatic effect of the processing conditions on, and the strong correlation between, the microstructure and the mechanical properties of polymer fibers. However, technical know-how about these important industrial processes has remained largely empirical, mainly due to the complexity in analyzing the development of microstructure under the highly non-equilibrium conditions realized in high-speed fiber spinning due to macroscopic flow deformation and heat transfer. Moreover, a detailed analysis of the fiber spinning processes spans a wide range of length and time scales (Table 1) making it a true computational challenge with the derivation of quantitative information for the microstructure Table 1 Typical length and time scales associated with the macroscopic flow deformation and the microscopic motions of polymer chains in a fiber spinning process. Note the enormous difference in the scale between the two. Macroscopic flow deformation Microscopic motions

Length scale

Time scale

1m 5 nm

10 ms 10ÿ9 ÿ 1 s

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remaining the barrier for further progress. Note that the extraction of such information characterizing the morphology of a material is a generic issue within material science, especially when this morphology is a result of non-equilibrium (flow) forces. The objective of this study is to lay out a systematic hierarchical approach to model complex transport phenomena encountered in polymer processing, such as high-speed fiber spinning, through the use of a theoretical framework, based on non-equilibrium thermodynamics, developed by Beris and Edwards [1]. Thus, we begin by describing in Section 2 the bracket formulation of non-equilibrium thermodynamics which utilizes microscopic information of a system to predict its macroscopic behavior. Next, the macroscopic modeling of the high-speed fiber spinning process is described in Section 3. The structural model proposed by the present authors [2] is described in more detail. Though the connection between `neck' formation and stress-induced crystallization in high-speed fiber spinning has long been established, this model shows, for the first time, that the neck can result from the ability of the crystalline phase to take up high stresses rapidly. In Section 4, we review the theoretical advances made in the microscopic modeling of semi-crystalline polymers. More specifically, analytical approaches based on a mean field description of lattice models are described in Section 4.1 leading to the well-known Flory [3] and Gaylord [4] expressions. The limitations of these models eventually lead to a new computational approach [5] to model dense polymer systems. This approach is capable of generating efficiently the polymer chain conformations on a fully populated lattice, where excluded volume constraints are taken into account. What distinguishes this approach from the rest is the fact that it leads to the evaluation of absolute thermodynamic quantities, such as the configurational entropy, for small lattices and to their estimation for large lattices. Also, it can incorporate the effect of flow deformation on chain segment distribution and, thus, gives nonequilibrium thermodynamic information (based on microstructure) about the system which can then be used in the hierarchical approach mentioned above to model polymer processes involving evolution of microstructure. The utility of this computational approach is illustrated in model calculations using 2-D lattices of semi-crystalline and bulk amorphous polymers. Finally, the conclusions of the work follow in Section 5. 2. Bracket formulation of non-equilibrium transport phenomena The bracket formulation of the dynamics of structured continua [1] consists of a generalization and extension of equilibrium thermodynamics to non-equilibrium (flow) conditions. It has originated from the pioneering work of Grmela [6], and it is an alternative formulation of the GENERIC description of non-equilibrium thermodynamics [7]. Under the assumption of local partial thermodynamic equilibrium, it considers an extended set of continuum variables which consist of, together with the traditional variables of equilibrium thermodynamics (e.g., density, temperature, etc.), variables characterizing the non-equilibrium structure of the system under investigation (such as, for example, the conformation tensor characterizing the second moment of the end-to-end distribution function of polymer chains) and the velocity vector. This extended set of variables allows for the description of the Hamiltonian functional, which is viewed as the total energy of the system consisting of thermodynamic contributions from the internal and the kinetic energy of the system. The theory also provides a set of evolution equations for all the variables characterizing the system. These equations, which are derived systematically from the bracket formulation of continuum dissipative mechanics (which parallels the development of Hamiltonian bracket equations), describe the time evolution of the variables due to

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Table 2 Comparison between equilibrium thermodynamics and Hamiltonian bracket dynamics; , û represent non-equilibrium variables, and f a nonlinear differential operator Equilibrium thermodynamics

Hamiltonian bracket dynamics

Free energy: F ˆ F …; T † ˆ H…; T ; eq ; eq † Extremum for  ˆ eq : @H @ ˆ 0

Hamiltonian: H ˆ H…; Tÿ; ; †
@H Bracket equations: @ @t ˆ f ; T; ; ; @

conservative and dissipative effects solely in terms of the Hamiltonian. The structure of the dynamic equations depends only on the mathematical nature of the variables that it describes and the material objectivity properties. It is primarily through the description of the Hamiltonian that information about the (microscopic) physics of the system is transmitted to the coupled set of evolution equations for the system variables. One of the advantages of the theory is that it allows for a smooth transition between equilibrium and non-equilibrium descriptions. Under equilibrium conditions, the Hamiltonian reduces to the (equilibrium) free energy and the dynamic equations reduce to the extremum principles of classical equilibrium thermodynamics according to which the internal parameters of the system assume those values for which the free energy of the system is minimized. Thus, under equilibrium, the partial derivatives of the Hamiltonian with respect to all the internal non-equilibrium parameters are zero. See the monograph by Beris and Edwards [1] or the recent paper [7] for further details. These extremum principles, when expressed in terms of the non-equilibrium variables, consistently lead to the equivalent equilibrium expressions leaving only the equilibrium variables as the independent variables characterizing the system. In contrast, these partial derivatives, are, in general, non-zero under flow conditions and they assume values dictated by the evolution equations for the velocity and the internal parameters, both of which are derived in a straightforward fashion from the bracket (or the GENERIC) formulation. The key components of the Hamiltonian bracket theory are contrasted against the equivalent concepts of equilibrium thermodynamics in Table 2. It is this close relationship of the bracket theory with equilibrium thermodynamics that is responsible for the perfect suitability of the former in order to describe such important effects as flow-induced phase transitions. In turn, the theory allows one to naturally describe the effects of structure on a variety of transport phenomena. However, for the theory to be effective, the important physics characterizing the relevant microstructure of the problem needs to be faithfully represented in the system's Hamiltonian (extended free energy) which necessitates a quantitative modeling of the system's microstructure. An example of the application of this approach to the fiber spinning problem based on a simple phenomenological description of the material microstructure is illustrated in Section 3. 3. Macroscopic problem: high-speed fiber spinning The schematic of a typical high-speed fiber spinning process is shown in Fig. 2. Spinning melt is extruded through a spinneret device. Upon extrusion, the melt, due to its viscoelastic nature, exhibits a die swell where the diameter of the fiber increases away from the spinneret. The die swell is usually neglected from the modeling of the spinning process, since it is limited to a distance of approximately

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Fig. 2. Schematic of the high-speed fiber spinning process.

only two die diameters from the spinneret exit. After the die swell, the melt deforms under the action of a tensile force. Along the spinning path, the deforming melt is cooled, solidified, and transformed into a filament with a supramolecular structure. The solidified fiber is received at the other end by a take-up device. As shown in Fig. 2, the spinline is usually associated with a sudden reduction in the fiber diameter referred to as necking. Necking has been attributed to the microstructural changes taking place upon the onset of stress-induced crystallization. The theoretical investigation of fiber spinning processes has been initiated more than 30 years ago with the pioneering work of Matovich and Pearson [8] and Kase and Matsuo [9] based on the analysis of the steady and isothermal melt spinning process of a Newtonian fluid using a slender body onedimensional approximation. Their analysis has been later extended to viscoelastic fluids [10] and to non-isothermal conditions [11,12]. Non-isothermal crystallization kinetics has been studied by Nakamura et al. [13] for slow speed spinning of a Newtonian fluid, but they have not accounted for the presence of the crystalline phase into the stress. The above work on melt spinning and other pre-1982 work has been reviewed in Ref. [14]. The effects of surface tension, gravity, and air drag on the spinline have also been studied [15,16]. The mathematical character of the one-dimensional equations was first studied by Beris and Liu [17,18] who have also studied the behavior of several numerical methods of solution. Later, the one-dimensional approximation has been put on a sound mathematical basis by Bechtel et al. [19]. Recently, Forest et al. [20] have considered the one-dimensional approximation of the isothermal fiber spinning of liquid crystalline polymers whereas, even more recently, the strong

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coupling between the radial and axial heat transfer in axisymmetric non-isothermal fiber spinning processes has also been analyzed [21]. Earlier models, proposed to explain the phenomenon of necking [22,23], are based upon phenomenological expressions for the viscosity of the crystallizing polymer as a function of the relative degree of crystallinity and temperature. The amorphous phase is often treated as a Newtonian fluid and relaxation effects are neglected. Thus, although these models can fit specific experimental results, they lack predictive capability when the processing conditions are changed. Moreover, these models do not provide any information on the conformation of the polymer chains in the two phases. Even when they are used to predict necking in specific experiments, they require a sudden reduction in the apparent extensional viscosity, the nature of which has remained a subject of controversy. For example, it has been proposed [24] that it is due to the liberation of heat during crystallization. Zahorski [25] has studied necking in non-isothermal high-speed fiber spinning with viscoelastic or inertial mechanisms as a problem of sensitivity to external disturbances. Moreover, Ziabicki has reported [16] that non-isothermal conditions coupled with a high Deborah number viscoelastic response also have the potential to lead to the necking phenomenon. However, as a recent analysis by Devereux and Denn [26] indicates, consideration of viscoelasticity alone is not sufficient to provide an adequate understanding of the melt spinning process. In their conclusions, they suggest that flow-induced crystallization needs to be taken into account in future works. Recently, we have proposed a structural model [2] for highspeed fiber spinning based on the framework of non-equilibrium thermodynamics. This model is briefly described next. 3.1. A structural model for high-speed fiber spinning In a recent publication [2], we have considered a one-dimensional (cross-section averaged) approximation for isothermal high-speed fiber spinning with no surface tension and air drag. The fiber is modeled as an inhomogeneous medium with two separate (meso) phases, one semi-crystalline and the other amorphous. The amorphous phase, before and after the onset of crystallization, is modeled as an Extended White Metzner EWM viscoelastic fluid [27]; and the semi-crystalline phase is modelled as an elastic solid [28]. The two phases are considered to act parallel to each other while distributing the applied load. Before the onset of crystallization, the dimensionless momentum balance equation is given as czz ÿ crr ˆ F  vz

(1)

and solved along with the dimensionless constitutive equations for the amorphous phase: vz

 @czz 1  @vz ÿ 2c ˆ ‰ÿczz ‡ 1Š; zz   @z @z De‰1=3…2crr ‡ czz †Šk

vz

 @crr 1  @vz ‡ c ˆ ‰ÿcrr ‡ 1Š: rr   @z @z De‰1=3…2crr ‡ czz †Šk

(2)

Here c*, F*, and v* are the properly dimensionalized conformation tensor of the amorphous phase, tension (constant) in the spinline, and velocity, respectively. The subscripts zz, rr, and z represent components of the corresponding quantities. De is the Deborah number, and k is a characteristic

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constant of the EWM model. The onset of crystallization is determined according to the Flory criterion [3]:    2  hf 1 1 2l N  1 1 ˆ p ÿ : (3) ÿ ‡  kB Tm0 Tm 2  N Crystallization is assumed to setpitwhen the incipient crystallization temperature, Tm, modified by the polymer chain extension, … czz †, exceeds the processing temperature, T0
Tm0 is the equilibrium crystallization temperature, kB is the Boltzmann's constant, hf is the heat of fusion per chain segment, l 1=2 is the length of each chain segment, N is the number of such segments in a chain, and ˆ …3=2Nl2 † . After the onset of crystallization, the dimensionless momentum balance equation becomes …1 ÿ †…czz ÿ crr † ‡ Gr …ezz ÿ err † ˆ F  vz ;

(4)

where  is the relative degree of crystallinity (i.e.,  ˆ 1 corresponds to the final semi-crystalline fiber with an absolute degree of crystallinity determined by the material and processing conditions, typically 10±20%), e is the elastic strain tensor of the semi-crystalline phase, and Gr ˆ Gc/Ga is the ratio of the modulus of the semi-crystalline phase to that of the amorphous phase. This momentum balance equation corresponds to a parallel load transfer between the two phases. This, in turn, corresponds to a system Hamiltonian given by a linear superposition of the Hamiltonians of the two phases. The dimensionless constitutive equations for the amorphous phase can be written as    @ …1 ÿ †  @czz  @vz  (5) …1 ÿ † vz  ÿ 2czz  ÿ czz vz  ˆ  k ‰ÿczz ‡ 1Š;   @z @z @z De 1=3…2crr ‡ czz †    @ …1 ÿ †  @crr  @vz  …1 ÿ † vz  ‡ crr  ÿ crr vz  ˆ  k ‰ÿcrr ‡ 1Š; @z @z @z De 1=3…2crr ‡ czz † and those for the semi-crystalline phase can be written as   @vz @  @ezz  vz  ÿ 2ezz  ‡ ezz vz  ˆ 0; @z @z @z   @v @err @  vz  ‡ err z ‡ err vz  ˆ 0: @z @z @z

(6)

An Avrami-type equation, with an Avrami coefficient of 2, is used in the Lagrangian frame to express the evolution of the degree of crystallinity s        p @ 1 2  2  exp T  ‡ ÿ 3 vz  ˆ 2 Kcr …1 ÿ † ln @z 1ÿ  where Kcr is the properly dimensionalized rate of crystallization, and T is a dimensionless parameter. The above set of equations needs to be solved with appropriate boundary conditions and reasonable values for the various parameters involved in the modeling. For a detailed discussion on dimensionalization, boundary conditions, and parameter values, see Ref. [2].

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Fig. 3. Simulation results for the high-speed fiber spinning process. [Reprinted by permission from Kulkarni and Beris [2]. Copyright by the Society of Rheology.] d* represents a dimensionless diameter, z* the dimensionless distance from the spineret and * dimensionless elongation viscosity.

Despite the simplicity of the microstructural modeling, this model is capable of demonstrating neck formation for a variety of processing conditions (Fig. 3(a)) and material property values consistent with those encountered in practice. Moreover, as can be seen from Fig. 3(b), the initial decrease in the elongational viscosity arises naturally in our model and it is attributed to the combined mechanical response of the amorphous phase and the semi-crystalline phase in the fiber which evolves under the given flow deformation. The ability of the semi-crystalline phase to take-up high stresses rapidly, which we have proposed as the primary cause for neck formation in high-speed fiber spinning, is indicated in Fig. 4 which depicts the contributions of the amorphous and the semi-crystalline phases to the total stress as a function of the distance from the spinneret. However, though this model is capable of explaining the experimental observations qualitatively, it needs a fitting of the various phenomenological parameters to experiments for quantitative comparisons. The origins of these parameters lie in the microstructural details of the polymer. Moreover, the mechanical coupling between the oriented amorphous phase and the crystallites is introduced phenomenologically in this model and as such does not allow us to deduce any information on the system's internal microstructure. To extract this type of information, it is necessary to describe the material microstructure in much more detail. In particular, a microscopic model needs to be used in order to extract thermodynamic (and statistical) information. Recent developments in this direction are mentioned next.

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Fig. 4. Simulation results for the high-speed fiber spinning process. Dimensionless total (axial-radial) stress in the amorphous and the semi-crystalline phases as a function of the dimensionless distance from the spinneret, z*. `Neck' is formed due to ability of the semi-crystalline phase to take-up high stresses rapidly.

4. Microscopic modeling of dense semi-crystalline polymers Dense polymers, under quiescent crystallization conditions (no flow), usually form randomly nucleated spherulitic crystallites [29] as shown in Fig. 5(c) [78]. In contrast, flow-induced crystallization of polymer melts usually gives a highly oriented, row-nucleated crystalline morphology [30]. For example, polyethylene films stretched during crystallization form crystallites which exhibit a shish-kebab structure [30] (Fig. 5(b) [77]) whereas PET and nylon exhibit a fibrillar morphology [31] (Fig. 5(d)). A schematic of a single lamellar crystal obtained by quiescent crystallization of a dilute polymer solution [32] is also shown in Fig. 5(a). Experimental investigations about the development of morphology and kinetics of flow-induced crystallization have been made in detail by Schultz [33±38], McHugh [39±41], and by Ziabicki [42±45]. The theoretical work devoted to studying the development of the microstructure associated with stress-induced crystallization can be broadly classified into three categories: Statistical mechanical models devoted to the equilibrium properties of stretched polymer networks [3,4], statistical mechanical and classical thermodynamic models devoted to the kinetic properties of the crystallization process [42±45], and models connecting the evolution of morphology with flow and transport phenomena [38]. The first statistical mechanical model expressing the equilibrium degree of crystallinity and the incipient crystallization temperature as a function of stretching was proposed by Flory [3]. His results were extended by Gaylord [4] who took into account finite extensibility of the chains, crystallite orientation, and chain folds. He also applied the concepts of irreversible thermodynamics similar to those used by Roe and Krigbaum [46] to obtain rates of crystallization. A rigorous statistical evaluation of these models is given in Appendix A. Ziabicki, in a series of papers [42±45], has proposed a generalized theory of nucleation kinetics which takes into account the internal microstructure of the system. Hay et al. [47] have developed a phenomenological model for row nucleation in polymers that requires knowledge only of the macroscopic parameters of the polymers melt such as stress, heat of fusion, and modulus. Recent advances in connecting the morphology with the macroscopic properties have been made by Schultz [38] who has treated the heat transfer away from the growing crystal front as the rate determining step. The diameter of the fibrils was predicted satisfactorily as a function of the operating conditions for high-speed fiber spinning of PET.

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Fig. 5. Morphology of polymer crystals under various crystallization conditions.

The above discussion shows that the development of the microstructure and the rheological and the mechanical behavior of the medium are intimately connected, and they need to be considered together. The overall problem involving structure, nucleation, and crystallization kinetics is prohibitively

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complex at the moment. Thus, in this work, we provide a first approach to study a small part of it, namely the non-equilibrium semi-crystalline structure based on a lattice model for polymer chain conformations. After a short discussion and analysis in Section 4.1 of some of the analytical techniques which are based on a mean field approach, we described in Section 4.2 a new computational technique for generating chain conformations on fully populated lattices. 4.1. Non-interacting (ideal) chains When an isotopic amorphous polymer melt is subjected to an external stress, the polymer chains get more or less oriented along the direction of stress. This lowers the total number of possible conformations each chain can take and decreases the entropy of the amorphous phase. Since there is less entropy to be sacrificed in going to the crystalline state, the crystallization temperature of the system increases as the externally imposed stress increases. Also, the crystallization rates increase by several orders of magnitude. The size of the primary and the secondary nuclei, the nucleation rate, and the average size of the crystallites are also affected [48]. As noted in the previous section, several models for stress-induced crystallization in elongated polymers having network structures have been proposed in the past to explain these observations. In developing a theoretical framework based on the statistical mechanics of random chains on regular lattices, it is important to distinguish between the realizations corresponding to the same conformation of a single chain (characterized, for example, by its end-to-end distance) and the distribution of various chain conformations in the statistical ensemble of the investigated sample. These differences are summarized in Tables 3 and 4. In these models, it is considered implicitly that crystallization occurs after the polymer has been elongated to the final relative length. Experimentally, this condition would be fulfilled if the sample were elongated at an elevated temperature at which crystallization would not occur and then cooled gradually. Thus, the final step is to be achieved in two separate and distinct steps, i.e. stretching and crystallization, which occurs in that order. We wish to evaluate the entropy of the stretched semicrystalline network with respect to the hypothetical perfectly crystalline polymer. We achieve this in three steps, by evaluating the absolute entropies of the stretched semi-crystalline network, the equilibrium semi-crystalline network, and the relative entropy of the equilibrium semi-crystalline Table 3 Descriptions of a system composed of a single chain or an ensemble of chains in a discretized space Single chain Ensemble of K chains

Macroscopic system (conformation)

Microscopic picture (realizations)

…nxi ; nyi nzi † P…nxi ; nyi nzi † ˆ kidiscrete =K

Zdiscrete …n Pxi ; nyi ; nzi † Z discrete K  S ˆ kB i kidiscrete ln ki discrete i

Table 4 Descriptions of a system composed of a single chain or an ensemble of chains in a continuous space Single chain Ensemble of K chains

Macroscopic system (conformation)

Microscopic picture (realizations)

(x,y,z) g(x,y,z)

f…x; y; z†Z   R total RR S ˆ kB K gln f :
Zgtotal dxdydz

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network with respect to the perfect crystal. Our analysis, which leads to expressions similar to the ones obtained by Flory [3] and Gaylord [4], is presented in Appendix A. Typical results are presented in Fig. 6. In Fig. 6, the dimensionless free energy of a stretched semi-crystalline network consisting of K chains, F…; nf †=KRTm0 (Eq. (A.39)), is plotted as a function of the degree of crystallinity, , for various integral values of the number of folds, nf. The following parameter values corresponding to crosslinked Ê , l ˆ 4.25 A Ê, polyethylene [4] were used: em ˆ 3294 cal molÿ1, e ˆ 3162 cal molÿ1, a ˆ 1.5 A ÿ1 hf ˆ 950 cal mol , ˆ 4, Tm0 ˆ 141.78C, and N ˆ 250. This figure clearly shows that at lower elongations higher number of folds is preferred, whereas at higher extensions a lower number of folds is preferred. Extended chain crystallization is obtained for large strains. These models have the following approximations: 1. A Gaussian distribution is used to represent the isotropic amorphous phase. This distribution is accurate for small extensions of long chains. 2. The various chains behave independently of each other, i.e., the effect of formation of chain bundles is neglected. Hence, strictly speaking, these are nucleation models and they should be used to predict the incipient crystallization temperature rather than the equilibrium degree of crystallinity. 3. All chains undergo the same degree of crystallization. 4. The crystals are fibrillar, i.e., not involving folds; and they are oriented along the direction of stretch. 5. In deriving expressions for the incipient crystallization temperature and the equilibrium degree of crystallinity, crystallization is assumed to occur in a thermodynamically most favorable manner, i.e., a state of equilibrium is assumed. 6. Possible small entropy changes associated with the formation of nuclei are neglected. Thus, for example, the tendency for chains having comparatively large z components in the undeformed state to crystallize selectively is discarded. The Flory and Gaylord models assume isotropicity and complete lack of intra- and inter-chain interactions. As such, they provide a severely oversimplified picture of the internal microstructure which is typically characterized by mesoscopic crystalline assemblies in the form spherulites, lamellae and/or fibrils, as discussed above and shown in Fig. 5. To be able to distinguish between these structures and quantitatively describe the effects of crystallization, a much more detailed and anisotropic microscopic description needs to be employed. Of particular interest are the modifications in the amorphous phase in the adjacency of the crystalline interface. The structure of the amorphous phase in lamellar semi-crystalline polymers was first studied quantitatively by Yoon and Flory [49] using lattice models. Since then, lattice models have been extensively used to study these systems. The application of lattice models in this regard falls into three categories: Gambler's ruin models [50,51,52], mean-field lattice models [53,54±59], and Monte Carlo methods [60,61]. Among these, only the Monte Carlo methods allow for chain interactions and an exact accounting of excluded volume and chain connectivity effects. The Gambler's ruin models involve the analysis of uncorrelated, independently selected, individual random walks as they develop on a lattice. In particular, random walks emanating from one of the lattice boundaries, representing a crystal±amorphous polymer interphase, are examined. Those walks are followed on the lattice, representing the amorphous region, till they either return back to the same boundary from where they had started, thus forming a loop, or reach the opposite lattice boundary, representing the next crystal±amorphous interphase, forming a bridge. Given the simplicity

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Fig. 6. Effect of deformation on polymer microstructure. Each plot shows the dimensionless free energy Eq. (A.39), F(,nf),/ (KRTm0), of a semi-crystalline system of K chains as a function of the degree of crystallinity, , for various integral values of the number of folds, nf, and a specific extension, . The equilibrium number of folds corresponding to the minimum of the free energy is seen to decrease with an increasing deformation (extension) of the polymer chains.

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of the (self-intersecting) random walks as a model, analytical results can be obtained for the Gambler's model chain statistics [50,51,52]. An overview of these results is given in Appendix B. However, it has been shown [5] that the oversimplified assumptions of the Gambler's ruin model place severe limitations as far as the quantitative, and sometimes even qualitative, validity of the resulting predictions is concerned (see also Fig. 10). Moreover, modifications (modified Gambler's ruin) which have been proposed to partially take into account the chain interactions [51,52] have had limited (if any) success [5]. Mean field lattice models [53,55] allow for chain segment orientation probabilities to vary depending on the location and/or the neighbor segment orientation. However, the specific chain layout on the lattice is still not taken into account; and as a result, the predictions from mean field lattice models are sometimes contradictory to each other and have been received with skepticism. Thus emerges a need for a more rigorous approach for microscopic modeling which is discussed next. 4.2. Interacting chains The availability of analytical results is severely limited when polymer chains are allowed to interact with each other on a fully populated lattice, such as the one corresponding to an amorphous region in a semi-crystalline polymer, even when only the excluded volume effects are taken into account. In fact, as outlined in a very recent monograph on the subject [62], we are aware only of some statistical exponents for chain length correlations that can be obtained by analytical approaches for certain types of lattices in the dense (fully populated) limit. Therefore, one has to resort to computer simulations to obtain any detailed information about dense polymers when chain interactions are taken into account. Traditional Monte Carlo methods have been used extensively for this purpose. In this section, we shall elaborate on a new computational approach which has been shown to be useful [5] to obtain such information exactly for small lattices and which can be later coupled with a Monte Carlo scheme to test the ergodicity and improve the efficiency of traditional Monte Carlo schemes. The proposed computational approach has the following features that can be distinguished from a traditional Monte Carlo scheme: 1. The formalism allows for an exact evaluation of absolute thermodynamic potentials, and it can be efficiently implemented in a computer, at least for small lattice sizes. 2. It can incorporate the orientational constraints imposed by flow. Most of the existing Monte Carlo methods can be implemented efficiently only in the case of partially filled lattices where global reallocation moves of the chain segments can be made (involving recursive enrichment [63] and/or chain motions [64,65,66] such as kink or reptation), thus ensuring ergodicity of the Monte Carlo schemes. Unfortunately, these schemes tend to become very inefficient as the dense (fully populated) limit is approached [67]. Recently, however, a new method called configurationbiased Monte Carlo has been developed [68,69] which seems to be efficient in the dense limit. On the other hand, Monte Carlo methods have been developed for fully populated lattices as well [60,70]. However, due to the limited possibilities for chain segment rearrangements on a fully populated lattice, ergodicity of the Monte Carlo algorithms used therein remains questionable [70]. Another issue with traditional Monte Carlo approaches is their difficulty to generate full thermodynamic potentials, such as free energy and entropy. Also, they cannot handle the constraints that are needed to enable the use of

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Monte Carlo techniques under non-equilibrium (flow) conditions. However, there is at least one atomistic Monte-Carlo simulation that the authors know of [71] where such constraints have been successfully handled based on the use of generalized forces developed within the generalized bracket framework. Moreover, in that same work [71], the extended free energy of the system was obtained through an integration based on individual Monte-Carlo realizations corresponding to different values of the generalized forces (identified as the derivative of the free energy with respect to an internal conformation tensor). Therefore, it is clear from the above that there is a substantial need for improving the existing Monte Carlo techniques applied to fully populated lattices. In an effort to contribute towards that goal, we have opted here to develop an alternative approach based on a direct, exact enumeration of the latticerealizable microstates. Even though the computational implementation of the proposed direct technique is very efficient [5], it is understood that, due to the exponential increase in the number of microstates with lattice size, such a technique can be used on its own potentially only for small lattice sizes, as are the particular examples supplied here. Nevertheless, it is believed that it can find more extensive use in the formulation of hybrid direct-Monte Carlo method as the use of the direct method allows for the development of random initial guesses, even in large size lattices, with a high computational efficiency. In any case, use of the direct method in small lattices allows one to avoid the ergodicity ambiguity and also allows for a straightforward application of the flow-induced chain orientation constraints, necessary to study non-equilibrium effects. The new, direct computational algorithm [5] is based upon an enumeration of all possible microscopic configurations in dense polymer systems. In this approach, the macromolecular chains are represented as self- and mutually-avoiding random walks on a fully populated lattice corresponding to the amorphous regions of a lamellar semi-crystalline morphology. The significance of considering mutually-avoiding random walks in dense polymers is that, though all the self-avoiding conformations of a given chain are still possible, they are no longer equally probable and their likelihood depends upon the interactions of that chain with the other chains present in the system [67]. The novelty of the proposed computational algorithm is that the information about chain connectivity is generated in terms of the permutations of the vertical and the horizontal bonds connecting various lattice sites instead of the conventional procedures to generate self-avoiding walks of individual chains. This information can be generated very rapidly through the use of look-up tables. This enables an efficient generation of new microstates. The approach has been able to carry out an exact enumeration of all possible microstates (corresponding to a given distribution of the vertical bonds) for a lattice of size up to 16  16 with realistic memory and unable CPU requirements [5]. These results are extremely encouraging, since as mentioned in Ref. [72] and we quote ``We were simply to generate even a single Hamiltonian chain on any larger lattice in our method. While we estimated that L ˆ 6 for d ˆ 3 and L ˆ 11 for d ˆ 2 should be feasible with unduly large amounts of CPU time, beyond that should be unaccessible with (variants of) our algorithm, and with present day computers.'' A Hamiltonian chain is one which visits all the lattice sites only once. Here L denotes the lattice size, d indicates the dimensions and the algorithm used in Ref. [72] is a recursive and randomized implementation of a conventional enrichment method. The transfer matrix method used in our approach does not automatically exclude the microstates containing closed loops (rings) to allow for the evaluation of thermodynamic potentials and statistics of systems containing only linear chains. This correction for the ring structures to the entropy calculations is always applied a posteriori based on the multiplication of the total number of microstates, obtained by the transfer matrix method, by a correcting factor representing the proportion of the microstates without

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Fig. 7. A microstate corresponding to a 2-D semi-crystalline lattice. There are N0 ˆ 8 lattice sites in a row, and the separation between the two parallel crystalline boundaries is L ˆ 5 rows [5]. The distribution of vertical bonds is v ˆ [844448]. Note that the lattice is fully populated, and the connectivity requirements are also satisfied. We wish to obtain thermodynamic information on a macrostate which comprises a large number of such microstates.

rings in the microstates generated through the transfer matrix method. This factor is determined statistically by generating a large number of microstates and then examining whether they contain ring structures or not. Of course, exclusion of the ring structures in the evaluation of chain statistics is automatic by focusing our attention explicitly on those microstates that do not contain such structures. In order to motivate the reader, we briefly elaborate on a 2-D implementation of the new approach, details can be found elsewhere [5]. In two dimensions, the lamellar semi-crystalline morphology can be modeled by a square lattice (co-ordination number ˆ 4) with N0 sites in a row and a depth of L rows. The first and the last rows are considered adjacent to crystalline lattices occupied by macromolecular chains perfectly parallel to each other along the y direction (see Fig. 7). The chains from the crystalline region continue in the amorphous region so that the lattice is completely filled. This means that all the lattice sites must be occupied by the chain segments and each lattice site must be connected to two and only two (out of four) neighboring lattice sites. Periodic boundary conditions are imposed across the other two boundaries of the lattice which simulate a lamella infinite in the lateral (along x) direction. A connection between two adjacent lattice sites in the same row is referred to as a horizontal bond whereas a connection between two lattice sites in the same column but in adjacent rows is referred to as a vertical bond. We refer to a feasible arrangement of vertical and horizontal bonds which satisfies the space-filling constraints mentioned above as microstate. One such microstate is shown in Fig. 7. In any particular microstate, a chain connecting the two crystalline boundaries is said to form a bridge whereas a chain starting from one crystalline boundary and returning to the same crystalline boundary is said to form a loop. Our aim is to determine the total number of distinguishable microstates, denoted as , for a given distribution of vertical bonds, v, where v(k) is the number of vertical bonds entering the kth row. We are also interested in obtaining statistical information on the loops and the bridges. In this treatment, v is imposed as a constraint and is used to identify the macrostate of the system. This macrostate can be obtained at a specific flow deformation which results in an average segment orientation distribution represented by v. The correlation between the flow and the chain segment

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orientation has already been dealt within the context of a macroscopic non-equilibrium thermodynamic theory [1]. Alternatively, one can see v as an internal parameter with respect to which the free energy needs to be minimized in order to investigate the corresponding macrostate at equilibrium. The configurational entropy Scr of the corresponding macrostate is given by the Boltzmann's fundamental entropy law as Scr …v; N0 ; L† ˆ kB ln… …v; N0 ; L††;

(8)

where kB is the Boltzmann's constant and (v; N0, L) indicates the total number of microstates for a given distribution of vertical bonds, v, corresponding to a semi-crystalline lattice of size N0  L. , as evaluated by the transfer matrix method, is corrected a posteriori for the exclusion of ring structures, as mentioned above. The subscript cr refers to a semi-crystalline system. To enable a comparison between different lattice sizes, the entropy Scr is scaled per semi-crystalline site by dividing it by the total number of sites, N0L. This gives rise to what we wall the `reduced' entropy, scr. Occasionally, we also want to examine the difference between the reduced entropies of a semi-crystalline system and a completely amorphous system at equilibrium. The reduced entropy of an amorphous system can be obtained by using a direct enumeration of the microstates employing periodic boundary conditions across all the four boundaries of the lattice (see Section 4.3 below). This entropy also needs to be corrected for the ring structures in the same way as that for semi-crystalline system. We refer to this entropy difference as the `relative reduced' entropy, scr .

Fig. 8. Relative reduced entropy (scr ) surface for a 2-D semi-crystalline system with N0 ˆ 16 and L ˆ 5. 2 and 3 indicate the fraction of horizontal bonds in the second and the third row, respectively. Thus, 2 ˆ 0.5 indicates that there are 8 horizontal bonds in the second row. The number of vertical (and, thus, horizonal) bonds entering the fourth and the fifth rows is chosen to be equal (symmetry) to that entering the third and the second rows, respectively.

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Fig. 9. Reduced entropy for a 2-D semi-crystalline system with N0 ˆ 16 and L ˆ 5. The figure illustrates the idea that the exact (obtained from simulations) entropy of such systems can be fitted with semi-analytic approximations, such as Approximation 1 (Eq. (9)) and Approximation 2 (Eq. (10) with " ˆ 0.7).

Configurational entropy calculations for the semi-crystalline system are shown in Figs. 8 and 9. It should be noted that the exclusion of the microstates containing ring structures has a relatively small effect on the entropy of semi-crystalline systems [5], at least for the lattice sizes examined in that reference. With this occasion, it is instructive to show that the computer-generated results can be utilized profitably to obtain analytic closed-form approximations which can be potentially used in conjunction with macroscopic dynamic simulations. Two such expressions have been used to fit our

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2-D lattice results (not corrected for ring chains) with the comparison shown in Fig. 9. These are: Approximation 1: 0 " L Y 1 Scr ˆ ln@ N0 L kˆ1

N0 ÿ



2

h…k† N0

N0

!!

N0 ÿ

v…k†



2

v…k† N0

h…k†

! ! #…1=2† 1 N0 A;

(9)

and Approximation 2: 0 " L Y 1 Scr ˆ ln@ N0 L kˆ1

 h  i   2 !! h…k† h…k† N0 ÿ " 1 ÿ h…k† ‡ 1 N0 N0 N0 N0 v…k†

N0 ÿ



v…k† N0

h…k†

2

!!#…1=2† 1 N0 A; (10)

where h(k) is the number of horizontal bonds in the row k and  is a dimensionless adjustable parameter. Though there is certain arbitrariness in the form of these approximations, Fig. 9 indicates that such analytic approximations have the potential to predict the thermodynamic quantities of the system quite closely. Finally, as shown in Fig. 10, significant differences are observed in the statistical analysis of the chain conformations in semi-crystalline systems as obtained by the proposed computational approach [5] and the Gambler's ruin models [50±52]. Fig. 10(a) shows that the average bridge lengths as predicted by the Gambler's ruin models are higher than those predicted by the simulations. This is to be expected since the chains are allowed to fold back on themselves in the Gambler's ruin models. As shown in Fig. 10(b), the Gambler's ruin model [50] predicts the probability for a walk to be bridge, PB, to be independent of the fraction of the horizontal bonds in a row, , whereas the simulations indicate a strong dependence on . Again, this dependence is to be expected since, as  increases, it is more probable for a chain to form a loop than a bridge. Fig. 10(c) shows the bridge distribution, i.e., the fraction of bridges of length LB as a function of LB, for various values of . The simulation results are seen to distinguish clearly between even and odd bridge lengths. In order to compare the simulation results with the Gambler's ruin model [50], Schumann filtering, bn ˆ …1=4†…anÿ1 ‡ 2an ‡ an‡1 †, was applied to obtain coarse-grained simulation averages. The simulation average compares qualitatively with trends exhibited by the Gambler's ruin model. Similar remarks hold true for the probability distribution of loops shown in Fig. 10(d). 4.3. Interacting chains in dense amorphous polymers The proposed direct computational approach can also be used to study chain statistics and conformational thermodynamics in dense amorphous polymers. A completely amorphous state is represented by a square lattice (L ˆ N0) with periodic boundary conditions across all the four boundaries. The approach has enabled us to calculate the reduced entropy of the amorphous lattice in the thermodynamic limit (i.e., as N0 ! 1). It is found to be 0.3668 before the correction for the rings and 0.3255 after the correction. These values compare favorably with the reduced entropy for a single Hamiltonian walk on a 2-D Manhattan lattice (a square lattice with horizontal and vertical alternating

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Fig. 10. Statistical information for a 2-D semi-crystalline lattice with N0 ˆ 16 and L ˆ 5 [5]. It is plotted as a function of the fraction of the horizontal bonds in each row. Thus,  ˆ 0.5 indicates that there are eight horizontal bonds in each row. Note the quantitative and also the qualitative differences between the simulations results of the proposed computational approach [5] and the mean-field Gambler's ruin models [50,51]. The statistical information is plotted with two standard deviations about the mean value, the errorbars are not visible where they are smaller than the symbols used. Note that the results reported here are not corrected for rings.

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orientations) given as G/ (approx. 0.29156...) [73], where G is the Catalan's constant Gˆ1ÿ

1 1 ‡ 2 ÿ


: 2 3 5

(11)

The literature result provides an absolute lower bound for the reported entropy given its restriction to Manhattan lattices. The close agreement is encouraging for the validity of our approach which, in fact, is the only approach providing such information for general lattices for which analytical results are not available. As exact results for Hamiltonian walks are confined to the Manhattan lattice [73,74], the behavior of such walks on non-oriented lattices and the precise scaling of two-dimensional polymers remain unclear [73,75]. Fig. 11 presents indicative results associated with chain statistics on a fully periodic lattice of size N0 ˆ L ˆ 16 containing only linear chains and no rings. As Fig. 11(a) indicates, the critical exponent, v, appears to be 1/2 as conjectured by Duplantier and Saleur [73]. The deviation from this scaling law at small chain lengths arises due to short range correlations between the chain segments where the excluded volume effect is not yet screened. The deviation at high chain lengths arises due to the finite size of the lattice. Fig. 11(b) shows the scaled probability distribution for the end-to-end distance of a chain comprising 60 chain segments. The distribution is Gaussian at large end-to-end distances, as expected. 4.4. Concluding remarks The proposed computational approach has several advantages. The macromolecular chains in a dense polymer system are constrained not only by themselves but also by the neighboring chains. Due to the dense packing, the phase behavior of these systems is very complex since it is affected by the chain connectivity requirements (giving rise to entropic penalties) and by the interactions among the various chains (giving rise to energetic penalties). This explains why accounting only for the intra-chain connectivity, i.e., self-excluded volume interactions using self-avoiding random walks, an approach used popularly to represent isolated chains in dilute polymer solutions, is not sufficient to represent dense polymer systems. The proposed approach treats exactly the chain connectivity and the excluded volume issues for fully populated lattices. It can also include vacant lattice sites and chain ends. The proposed algorithm computes the configurational entropies for small lattice sizes through an exhaustive enumeration and estimates them with tight statistical bounds for larger ones. To the best of our knowledge, this is the first time that the fundamental thermodynamic quantities such as the configurational entropy and the partition function have been obtained for the amorphous regions of dense semi-crystalline systems through a direct computation. Correction for the ring structures is applied a posteriori to the partition function obtained by the transfer matrix method. The proposed approach can incorporate chain-segment orientation constraints, such as the ones imposed by a flow field. The approach can efficiently collect statistical information such as proportions of loops and bridges, their average lengths, the distribution of their lengths, etc. It should be emphasized here that these statistics influence critically the mechanical properties of the semi-crystalline polymers, and thus we believe that the proposed algorithm will give us the much needed capability to predict those properties. Preliminary results involving 3-D lattices (Fig. 12) have also demonstrated the feasibility of the proposed approach for the investigation of 3-D structures although in that case there is a need to

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Fig. 11. Chain statistics for a 2-D amorphous phase. The amorphous lattice consists only of infinite chains (no chain ends) and the `chain length' corresponds to a portion of these chains.

amplify the computational efficiency through the use of stochastic methods. The thermodynamic information obtained using the proposed computational approach can be used to develop accurate closed form expressions for the free energy of the semi-crystalline polymers. These expressions can then be used in the Hamiltonian formalism based on non-equilibrium thermodynamics [1] to study the macroscopic behavior of dense polymer systems under flow deformation.

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Fig. 12. A microstate for a 3-D semi-crystalline lattice. The amorphous phase is constrained between the two planar crystalline boundaries from the top and the bottom with periodic boundary conditions (not shown) across the lateral directions.

5. Conclusions In this work, we have described a systematic hierarchical approach for the quantitative modeling of the high-speed spinning process. The key for a successful implementation of this approach is an accurate understanding of the microstructure from first principles. Towards this end, we have proposed a new computational approach which incorporates chain orientation constraints imposed by flow and is capable of evaluating absolute thermodynamic potentials for small lattice sizes. Although the presented results are not final and considerable amount of work needs to be done for 3-D lattices (which will probably require the use of stochastic Monte Carlo methods), we believe that the framework outlined in the present work provides significant guidance for the establishment of a definite theoretical linkage between the underlying microstructure and the processing conditions that can eventually allow for a more rational design of complex and industrially important polymer processes. Acknowledgements The authors will like to acknowledge very helpful discussions with Dr. Chet Miller from Central Research, DuPont Experimental Station and Dr. Jerold Schultz, Department of Chemical Engineering, University of Delware. This work has been partially supported through a grant from DuPont Company with matching funds provided by the State of Delaware.

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Appendix A A. Statistical analysis of non-interacting chains on a lattice A.1. Conformation realizations for a single Guassian amorphous chain Consider a chain consisting of N directed statistical segments each of length of l. Assuming that each segment can randomly orient in one of the six possible directions (on a cubic lattice), the total number of distinguishable realizations that a single chain can take is Ztotal ˆ 6N :

(A.1)

Now, we wish to calculate the total number of distinguishable realizations if the chain is in a particular conformation, i.e., has a specific end-to-end distance. In the discrete sense, let this end-to-end distance be nx
l; ny
l; nz
l where nx ; ny ; nz are integers chosen such that they give a physically realizable conformation. Let Nx‡ ; Ny‡ ; Nz‡ and Nxÿ ; Nyÿ ; Nzÿ be the number of directed segments in the positive and negative x, y, z axes respectively. Thus, we have the following set of constraints: Nx‡ ÿ Nxÿ ˆ nx ; Ny‡ ÿ Nyÿ ˆ ny ; Nz‡ ÿ NZÿ ˆ nz ; Nx‡ ‡ Nxÿ ‡ Ny‡ ‡ Nyÿ ‡ Nz‡ ‡ Nzÿ ˆ N:

(A.2)

We note that there are six unknowns and only four equations. Consequently, the total number of distinguishable realizations for a chain with an end-to-end vector …nx
l; ny
l; nz
l† is given by the double summation Z…nx ; ny ; nz ; N† ˆ

Nÿny ÿnz ‡nx Nÿ2Nx‡ ‡nx ‡ny ÿnz 2 2

X

Nx‡ ˆnx

X

N!

Ny‡ ˆny

Nx‡ !Nxÿ !Ny‡ !Nyÿ Nz‡ !Nzÿ !

:

(A.3)

The lower bound for Nx‡ is obtained by putting Nxÿ ˆ 0 in Eq. (A.2). The upper bound is obtained by putting Nyÿ ˆ 0 and Nzÿ ˆ 0 in Eq. (A.2) and solving the set of constraints for Nx‡. The lower bound for Ny‡ is obtained by putting Nyÿ ˆ 0 in Eq. (A.2). The upper bound is obtained by putting Nzÿ ˆ 0 in Eq. (A.2) and solving the set of constraints for Ny‡. We can now make a simplifying assumption by treating the three directions as independent of each other and assuming that there are N/3 directed segments in each direction. Then the set of constraints reduces to Nx‡ ‡ Nxÿ ˆ N=3; Nx‡ ÿ Nxÿ ˆ nx ; Ny‡ ‡ Nyÿ ˆ N=3;

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Ny‡ ‡ Nyÿ ˆ ny ; Nz‡ ‡ Nzÿ ˆ N=3; Nz‡ ‡ Nzÿ ˆ nz :

(A.4)

This set of equations can be readily solved for the six unknowns. The required number of realizations is then given as Z…nx ; ny ; nz ; N† ˆ

N! Nx‡ !Nxÿ !Ny‡ !Nyÿ !Nz‡ !Nzÿ !

:

(A.5)

Eq. (A.5) can be simplified using Stirling's approximation. This gives Z discrete …nx ; ny ; nz ; N† ˆ

6N

2

…3=2†

…2N=3†

2

2

eÿ…3=2N†…nx ‡ny ‡nz † ;

(A.6)

where the superscript `discrete' indicates that nx, ny, nz are discrete integer variables. These integer variables can be changed to real variables x, y, z by the substitution x ˆ nx
l; y ˆ ny
l; z ˆ nz
l. Then we can define a probability density f, such that fdxdydz represents the probability that the single polymer chain has a conformation between (x, y, z) and (x ‡ dx, y ‡ dy, z ‡ dz), as f …x; y; z; N† ˆ

1

2

…2Nl2 =3†…3=2†

2

2

2

eÿ…3=2Nl †…x ‡y ‡z † :

(A.7)

Let Zdiscrete(x, y, z; N) represent the total number of possible realizations for a polymer chain with conformations between (x, y, z) and (x ‡ l, y ‡ l, z ‡ l). Then, Z discrete …x; y; z; N†  6N

xZ‡l y‡l z‡l Z Z

f …x; y; z; N†dxdydz; y

x

z

Z discrete …x; y; z; N†  f
l3 ; Z discrete …x; y; z; N† 

6N

2

…3=2†

…2N=3†

2

2

2

eÿ…3=2Nl †…x ‡y ‡z † :

(A.8)

Thus, in this section we have demonstrate a way to calculate the total number of distinguishable realizations possible for a polymer chain having a specified conformation, i.e., a specified end-to-end distance. This has direct relevance in calculating the absolute entropy of a state using Boltzmann's law. The following assumptions were made in deriving Eq. (A.6): 1. The x, y, and z directions are discretized, in the sense that a particular segment can be placed only along one of the x, y, or z axes. This assumption is not expected to introduce a significant error as the length of each directed segment (l) is small, and the number of directed segments (N) is very large. 2. The effect of excluded volume is neglected.

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3. The three directions are treated as independent of each other. This assumption leads to a Gaussian probability distribution. A.2. Statistical ensemble of K chains Consider an assembly of K chains or a statistical ensemble of K chains, or K independent configurations of a single chain in time. Suppose the system is in a particular state wherein there are kidiscrete number of chains with a conformation (nxi ; nyi ; nzi ) such that X i

kidiscrete ˆ K:

(A.9)

The total number of distinguishable realizations possible for the system in this particular state is

ˆ Qi

K! kidiscrete !

i  Y kdiscrete Z discrete …nxi ; nyi ; nzi ; N† i :

(A.10)

Stirling's approximation can be employed to yield the following very important expression ln… † ˆ

X i

kidiscrete ln



 Zidiscrete K ; kidiscrete

(A.11)

where Zidiscrete  Z discrete …nxi ; nyi ; nzi ; N†: The discrete probability distribution of the chains at equilibrium can be obtained by maximizing the entropy (ˆkBln ) of the system of K chains subject to the constraint represented by Eq. (A.9). The result is kidiscrete Zidiscrete : ˆ K Ztotal

(A.12)

For example, for amorphous Gaussian chains kidiscrete 1 ÿ…3=2N†…n2xi ‡n2yi ‡n2zi † e : ˆ K …2N=3†…3=2†

(A.13)

We can derive a continuous probability density g, such that gdxdydz represents the probability that the polymer chains in the ensemble have conformations between (x, y, z) and (x ‡ dx, y ‡ dy, z ‡ dz), as g…x; y; z; N† ˆ

1 …2Nl2 =3†…3=2†

2

2

2

2

eÿ…3=2Nl †…x ‡y ‡z † :

(A.14)

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Then, kidiscrete  K

357

xZi ‡l yZi ‡l zZ i ‡l

g…x; y; z; N†dxdydz; xi

yi

zi

kidiscrete  g…xi ; yi ; zi ; N†
l3 ; K kidiscrete 1 2 2 2 2 eÿ…3=2Nl †…xi ‡yi ‡zi † : …3=2† K …2N=3†

(A.15)

The continuous analog of Eq. (A.11) can be written as   Z Z Z f
Ztotal dxdydz: ln… † ˆ K gln g

(A.16)

In the following sections, we always work with continuous probability distributions. A.3. Polymer systems and Gaussian amorphous phase A.3.1. Equilibrium configuration For equilibrium amorphous phase, we have fZtotal ˆ Ztotal ˆ 6N : g

(A.17)

Substituting Eqs. (A.14) and (A.17) in Eq. (A.16) and applying Boltzmann's law, we get the entropy of the equilibrium amorphous phase as Sae ˆ kB …KN ln 6†:

(A.18)

A.3.2. Stretched amorphous phase Suppose the initially isotropic amorphous network is subjected to a homogeneous strain with components x, y, and z along the x, y, and z directions, respectively. The coordinates of mean position of any junction in the network relative to any other must change by the same factors. Consequently, a chain i characterized by an end-to-end vector (xi, yi, zi) after deformation must have had components (xi/x, yi/y, zi/z) before deformation, provided that the deformation is sufficiently fast so as to freeze the equilibrium distribution. This reasoning is due to Flory [3] and is applicable to solid (rubber-like) polymeric networks. For liquids (e.g., polymer melts), the same argument is valid, according to the central limit theorem, if the polymer chains are sufficiently long. Hence, after deformation, we have gas …x; y; z; x ; y ; z † ˆ

1 …2Nl2 =3†…3=2†

eÿ…3=2Nl †……x =x †‡…y =y †‡…z =y †† : 2

2

2

2

2

2

2

(A.19)

The total number of distinguishable realizations possible for polymer chains with conformations between ri and ri‡dr is still given by Eq. (A.8). Hence, using Eq. (A.16), we get the entropy of the

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stretched amorphous phase as " Sas …x ; y ; z † ˆ kB

# K…2x ‡ 2y ‡ 2z ÿ 3† : ln…x y z † ‡ KN ln6 ÿ 2

(A.20)

Volume changes associated with the stretching process are neglected. A.4. Stress-induced crystallization for unfolded Guassian chains A.4.1. Stretched semi-crystalline network At the instant when the stretched amorphous network is quenched below crystallization temperature resulting in a stretched semi-crystalline network, the network does not have sufficient time to relax, and the configurational probability distribution for the network is still given by Eq. (A.19). In particular, if we assume that the network is stretched only along the z axis by a factor , then the volume remaining constant, we have   1 1 gzs …x; y; z; † ˆ gas x; y; z; x ˆ p ; y ˆ p ; z ˆ  ;   gzs …x; y; z; † ˆ

1

eÿ…3=…2Nl ††…x ‡y ‡…z = †† : 2

…3=2†

…2Nl2 =3†

2

2

2

2

(A.21)

As the individual chains begin to crystallize, the total number of distinguishable realizations possible for the chains changes instantaneously. For simplicity, we assume that all the chains begin to crystallize simultaneously and independently; and that, at any particular instant of time, all the chains have been crystallized to the same extent denoted by ! number of segments per chain. Thus, ! enters as an additional parameter to identify the total number of possible realizations of a chain. We further assume that the crystallized segments are oriented along the z direction, which is the stretch direction, and there is no folding. Given these limitations, the total number of distinguishable realizations possible for chains with conformations between ri and ri ‡ dr is given by fsc
Zsc, total where fsc …x; y; z; !† ˆ and

1 ‰…2…N ÿ

w†l2 =3†Š…3=2†

2

2

2

2

eÿ…3=…2…Nÿ!†l ††…x ‡y ‡…jzjÿ!l† †

(A.22)

Zsc;total ˆ 6…Nÿ!† 2…N ÿ !†:

(A.23)

Using Eq. (A.16), the entropy is given by Z1 Z1 Z1 Sscs ˆ kB ÿ1 ÿ1 ÿ1



 6…Nÿ!† 2…N ÿ !†fsc dzdydz; Kgzs ln gzs

2

Sscs ˆ kB K 4…N ÿ !†ln6 ‡ ln‰2…N ÿ !†Š ‡

Z1 Z1 Z1

Z1 Z1 Z1 gzs lnfsc dxdydz ÿ

ÿ1 ÿ1 ÿ1

3 gzs lngzs dxdydz5:

ÿ1 ÿ1 ÿ1

(A.24)

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359

A.4.2. Equilibrium semi-crystalline network This hypothetical phase can be obtained if the stretched semi-crystalline network at a particular instant of time were capable of relaxing instantaneously. In that case, the configurational probability distribution for the chains is given by gsce ˆ fsc :

(A.25)

The total number of possible realizations for polymer chains with conformations between ri and ri ‡ dr is still given by Eqs. (A.22) and (A.23). Thus, the entropy of the equilibrium semi-crystalline state is Z1 Z1 Z1 Ssce ˆ kB ÿ1 ÿ1 ÿ1

 …Nÿ!†  6 2…N ÿ !†fsc Kfsc ln dzdydz; fsc

Ssce ˆ kB K‰…N ÿ !†ln6 ‡ ln‰2…N ÿ !†ŠŠ:

(A.26)

A.4.3. Equilibrium crystalline phase The entropy difference between an equilibrium semi-crystalline system and a perfectly crystalline system corresponds to the entropy difference associated with the melting of N ÿ ! number of segments per chain. If sf is the entropy of fusion per segment [3], then Ssce ÿ Sce ˆ K…N ÿ !†sf

(A.27)

sf is a thermodynamic quantity which represents that portion of the entropy of fusion arising from the randomness of arrangement of the segments in space, but not including entropy contributions from internal changes within the segments during melting. A.4.4. Stretched semi-crystalline phase relative to the equilibrium crystalline phase Using the expressions developed in the above sections, we can now calculate the entropy of the stretched semi-crystalline phase with respect to the equilibrium crystalline phase. The result is 21 1 1 3 Z Z Z Z1 Z1 Z1 gzs lnfsc dxdydz ÿ gzs lngzs dxdydz5 ‡ K…N ÿ !†sf ; Sscs ÿ Sce ˆ kB K 4 ÿ1 ÿ1 ÿ1

ÿ1 ÿ1 ÿ1

"  # …3=2†    2  N N ! l N  1 N 3 ‡ 2 p ÿ ‡ Sscs ÿ Sce ˆ kB K ln ÿ…! l†2 ‡ N ÿ! Nÿ!  Nÿ! 2  Nÿ! 2 ‡ K…N ÿ !†sf ; where ˆ



3 2Nl2

…1=2†

(A.28)

:

This result is the same as that given by Flory [3] except for the logarithmic term on the right hand side. However, at nucleation, where this theory is strictly valid, ! ! 0; and the effect of this term is negligible.

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For small degrees of crystallinity and low elongations, entropy of the stretched semi-crystalline phase with respect to the equilibrium semi-crystalline phase according to Eqs. (A.24) and (A.26) has a small positive value. This is clearly inadmissible, since the final state may not have higher entropy than the most probable state. Flory [3] has noted that this discrepancy arises mainly because of the assumption about the crystallite orientation. Next step is to calculate the free energy of the stretched semi-crystalline system with respect to the equilibrium crystalline system, If hf represents the heat of fusion per segment, then the heat change accompanying the fusion of N ÿ ! segments per chain is Khf …N ÿ !†. All the structural rearrangements can be considered to occur without a change in internal energy. Hence, the free energy of the stretched semi-crystalline phase with respect to the equilibrium crystalline system becomes F…!† ˆ U ÿ TS; "  …3=2†   N N ! l N 2 ‡ 2 p ÿ…! l† F…!† ˆ Khf …N ÿ !† ÿ kB KT ln Nÿ! Nÿ!  N ÿ!  2    1 N 3 ÿ ‡ ÿ KT…N ÿ !†sf : ‡ 2  N ÿ! 2

(A.29)

Equilibrium degree of crystallinity can be found out readily by minimizing free energy F(!) with respect of !. To simplify the analysis, we can substitute (N ÿ !)/N  , such that 1 ÿ  represents the degree of crystallinity. Also, hf =sf ˆ Tm0 is the equilibrium melting temperature. We get,      2  Nhf 1 1 2l  1 ÿ ‡ 2eq 2 l2 N 2 ÿ ‡ 1:5eq ‡ p ÿ ˆ 0: kB Tm0 T 2  

(A.30)

It should be noted that the term linear in eq is negligible in comparison with the other two terms for realistic values of the various parameters. Incipient temperature of crystallization, Tm, can be found out by setting  ˆ 1 in Eq. (A.30). The result is    2  hf 1 1 3 2l N  1 1 ‡ p ÿ : ÿ ‡ ˆ 2N kB Tm0 Tm 2  N 

(A.31)

At  ˆ 1, Tm < Tm0 which is a direct result of the small positive entropic residuals at low elongations, as discussed in the above paragraphs. A.5. Stress-induced crystallization for folded Gaussian chains Now, we can incorporate the effect of chain folding in stress-induced crystallization. We use the terminology similar to [4]. Thus, each chain is assumed to crystallize to the same extent with nf number of folds, each fold being of width a.  is the number of segments traversing the crystal and l represents the crystal thickness. It is further assumed that the end-to-end vector for the crystallized part lies along the direction of stretch. If lcr is the magnitude of this vector, then ÿ
…1=2† lcr ˆ n2f a2 ‡  2 l2 ;

(A.32)

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361

where  ˆ 1 for even number of folds and  ˆ 0 for odd number of folds. If is the number of segments in each fold, then the number of segments in the crystallized part of the chain is …nf ‡ 1† ‡ nf . Based on the discussion above, we obtain the following results. A.5.1. Stretched semi-crystalline network gzs ˆ

1

2

…3=2†

…2Nl2 =3†

fsc …x; y; z; !; nf † ˆ

2

2

2

2

eÿ…3=2Nl †…x ‡ay ‡…z = †† ;

(A.33)

1 ‰…2…N ÿ …nf ‡ 1† ÿ nf †l2 =3†Š

…3=2†

e…3=2…Nÿ…nf ‡1†ÿnf

†l2 †…x2 ‡y2 ‡…jzjÿlcr †2 †

; (A.34)

and Zsc;total ˆ 6…Nÿ…nf ‡1†ÿnf † 4‰N ÿ …nf ‡ 1† ÿ nf Š:

(A.35)

A.5.2. Equilibrium semi-crystalline network gsce ˆ fsc :

(A.36)

The total number of distinguishable realizations possible for polymer chains with conformations between ri and ri ‡ dr is given by Eqs. (A.34) and (A.35). A.5.3. Equilibrium crystalline phase Taking the reference state of a fully extended and crystallized system at equilibrium, we have Ssce ÿ Sce ˆ K‰N ÿ …nf ‡ 1† ÿ nf Šsf :

(A.37)

A.5.4. Stretched semi-crystalline phase relative to equilibrium crystalline phase Finally, the entropy of the stretched semi-crystalline phase with respect to the fully extended equilibrium crystalline phase is given by "  …3=2† N N ÿ 2 …n2f a2 ‡  2 l2 † Sscs ÿ Sce ˆ kB K ln N ÿ …nf ‡ 1† ÿ nf N ÿ …nf ‡ 1† ÿ nf  2   2 2 2 N  1 N 3 2 2 …1=2† ÿ ‡ ‡ ‡ p …nf a ‡ l †  N ÿ…nf ‡ 1† ÿ nf 2 2  N ÿ …nf ‡ 1† ÿ nf ‡ K‰N ÿ …nf ‡ 1† ÿ nf Šsf :

(A.38)

Since we have ! ˆ …nf ‡ 1† ‡ nf (here ! represents the total number of segments in the crystal and the folds), the free energy of the system can be written as

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"  …3=2† N N F…!; nf † ˆ Kem ‡ Knf e ‡ hf K‰N ÿ !Š ÿ kB KT ln ÿ 2 …n2f a2 ‡  2 l2 † Nÿ! Nÿ!  2   2 2 2 N  1 N 3 2 2 …1=2† ÿ ‡ ÿ KT‰N ÿ !Šsf ; (A.39) ‡ ‡ p …nf a ‡  l † Nÿ! 2  Nÿ! 2  where em and e are the surface energies of the crystallites. Since ! and nf are chosen to be independent variables,  is no longer independent. The degree of crystallization is  ˆ …nf ‡ 1†=N ˆ …! ÿ nf †=N. The degree of crystallization and number of folds at equilibrium can be determined by solving the following two equations simultaneously @F…; nf † @F…; nf † ˆ 0; ˆ 0: @ @nf

(A.40)

However, since nf is an integer, one can plot F(,nf) as a function of  for various integral values of nf and locate the global minimum. Appendix B B. Gambler's ruin analysis Here, we present the available analytical results on the statistics of the lamellar semi-crystalline morphology based on a random walk analysis of the chain conformation subject to the appropriate boundary conditions. This analysis is the basis of the so-called `Gambler's ruin' model [50,51,52]. In the early Gambler's ruin model [50], the two parallel crystalline boundaries are represented as absorbing barriers at z ˆ 0 and z ˆ M ‡ 1 with an amorphous region in between them. A macromolecular chain begins a random walk either at z ˆ 1 or z ˆ M and then continues in the amorphous region until it is finally absorbed by one of the absorbing barriers. The lattice is associated with a probability of p to step in the ‡z direction, q in the ÿz direction, and  in any of the horizontal directions irrespective of the local environment, i.e., the positions of the previous chain segments. However, based upon this simplified description of the amorphous phase, it is possible to develop analytical expressions for the quantities of interest such as average lengths of loops and ties, etc. Here we state results [50] for the case p ˆ q: Probability for a random walk to be a loop …PL † ˆ

M ; M‡1

Probability for a random walk to be a bridge …PB † ˆ Average length of a loop …< LL >† ˆ

1 ; M‡1

2M ‡ 1 ‡ 1; 3…1 ÿ †

Average length of a bridge …< LB >† ˆ

M 2 ‡ 2M ‡ 1: 3…1 ÿ †

(B.1)

J.A. Kulkarni, A.N. Beris / J. Non-Newtonian Fluid Mech. 82 (1999) 331±366

Also the fraction of the loops having length n, gL(n), is given as (     …nÿ2† ) M 1 ÿ X v v for n  2; gL …n† ˆ …1 ÿ †cos ‡ sin2 M vˆ1 M‡1 M‡1

363

(B.2)

and the fraction of bridges of length n, gB(n), is given as (     …nÿ2† ) M X v v ÿcos…v†sin2 for n  M ‡ 1: gB …n† ˆ …1 ÿ † …1 ÿ †cos ‡ M‡1 M‡1 vˆ1 (B.3) By comparing the lattice described in Section 4.2 with the Gambler's ruin model, we note that L ˆ M ‡ 2. Also the loops and bridges, as described in Section 4.2, include the two chain segments, one at the beginning and the other at the end of the loop or the bridge, which belongs to the crystalline phase. These segments are not included in the Gambler's ruin model described above. Thus, a loop of n segments generated by the simulations is equivalent to a loop of n ÿ 2 segments of the Gambler's ruin model. Recently, a modified Gambler's ruin model has been developed [51,52] where the chains undergo random walks with one-step memory, i.e., they avoid retracing the last step; the walks are otherwise unbiased. The results developed there are for a cubic lattice. Corresponding results for the average quantities for a two-dimensional lattice can also be derived and are as follows: PL ˆ

M ; M‡2

PB ˆ

2 ; M‡2

hLL i ˆ

8M 2 ‡ 24M ‡ 13 ‡ 1; 6…M ‡ 2†

hLB i ˆ

M…2M ‡ 5†…2M ‡ 7† ‡ 1: 12…M ‡ 2†

(B.4)

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