CROSSOVER FROM UNENTANGLED TO ENTANGLED DYNAMICS: MONTE CARLO SIMULATION OF POLYETHYLENE, SUPPORTED BY NMR EXPERIMENTS. A Dissertation.

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1 CROSSOVER FROM UNENTANGLED TO ENTANGLED DYNAMICS: MONTE CARLO SIMULATION OF POLYETHYLENE, SUPPORTED BY NMR EXPERIMENTS A Dissertation Presented to The Graduate Faculty of the University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Heng Lin May, 006

2 CROSSOVER FROM UNENTANGLED TO ENTANGLED DYNAMICS: MONTE CARLO SIMULATION OF POLYETHYLENE, SUPPORTED BY NMR EXPERIMENTS Heng Lin Dissertation Approved: Accepted: Advisor Dr. Wayne L. Mattice Committee Member Dr. Ernst D. von Meerwall Committee Member Dr. Ali Dhinojwala Committee Member Dr. Gustavo A. Carri Department Chair Dr. Mark D. Foster Dean of the College Dr. Frank N. Kelley Dean of the Graduate School Dr. George R. Newkome Date Committee Member Dr. Richard J. Elliott ii

3 ABSTRACT A two-bead move algorithm for dynamic Monte Carlo simulation has been designed to reflect the true randomness of local torsion dynamics on a high coordination lattice (nnd lattice). All possible configurations of two consecutive beads of a chain on this high coordination lattice have been included in a library by this algorithm. The moves were implemented by randomly choosing one of the possible configurations from the library. Thus there is no artificial rule for the moves. The algorithm is capable of introducing new bond vectors to local configurations without going through chain ends. The chain-cross has been eliminated in this two-bead move algorithm. The algorithm is fast enough for simulating polyethylene (PE) melts ranging from C40 to C34 on a regular desktop computer. The results of our simulation confirmed there were finite chain length effects, e.g. chain length dependent friction coefficients and non-gaussian statistics for short PE chains. A detailed comparison has been made among the experiments, prior simulations by other groups, and the results of our new algorithm. The diffusion coefficients scale with molecular weight (M) to the -1.7 power for short chains and -. for longer chains at 180 C, which coincides very well with experimental results. Due to the finite chain length effect, no pure Rouse scaling in diffusion has been observed. The reptation-like slowdown can be clearly observed when M is above 400 according to the mean square displacements of middle beads. The slope 0.5 predicted by the reptation theory was iii

4 missing for the intermediate regime of diffusion; instead a slope close to 0.4 appeared, indicating that additional relaxation mechanisms exist in this transition region. The relaxation times extracted by fitting the autocorrelation function of the end-to-end vectors scale with M to.5 and.7 power using the reptation model and KWW equation, respectively, for entangled chains. The dynamic Monte Carlo algorithm has also been used for simulating a bimodal mixture of PE with one entangled component (M~4.5k) and one unentangled component (M~1k). Detailed normal mode analyses have been presented. Rouse dynamics have not been observed at the short time and distance scale. The KWW equation generally provides better fits than does the pure exponential decay for the data of normal mode relaxation. The KWW index β varies with the concentration and increases with the decrease of mode numbers until a plateau is reached. Tube dilation was observed before the gradual disappearance of a reptation-like slowdown with increasing short chain content in mixtures. Reasonable agreements have been reached when Hess theory and Pearson-von Meerwall model were used to fit the concentration-dependent diffusion coefficients of PE in the mixture. Additional bimodal mixtures of PE studied by our simulation involved a small amount of short N-alkane in an entangled PE matrix. Most of our data agree with the NMR experimental measurements. Extrapolating the concentration-dependent diffusion coefficients of N-alkanes to the trace limit did not lead to the complete restoration of Rouse scaling of the M dependence of the trace diffusion coefficients (D tr ). Although the M dependence of D tr was found to be obviously weaker than that of the diffusion coefficients in monodisperse melts (D melt ), a residue of the slope still exists. Thus, the commonly seen excessive M dependence of short polymer iv

5 chains may result from two contributions. One is the non-iso-friction environment of the polymers at the different chain lengths, which can largely be eliminated by the trace diffusion measurements. Another one, which is related to the underlying physics of the residue of the slope in trace diffusion, is still mysterious. It may link to the non-gaussian statistics of short polymer chains. v

6 ACKNOWLEDGMENTS First, I would like to express my deepest gratitude to my advisor Prof. Wayne L. Mattice. Wayne was always there to discuss my ideas, proofread and markup my papers and chapters with his extraordinary patience. He never stopped providing me with his continuous support and encouragements. Wayne has showed me the wisdom and the spirit of a true scholar throughout these years. Without his guidance and generosity, this project could not have been possibly finished. A special thanks goes to my co-advisor Prof. Ernst D. von Meerwall with whom I have a lot of extremely constructive discussions. I am also greatly indebted for the great deal of time that Ernst has spent with me, working side-by-side on the NMR measurements of the trace diffusion coefficients of N-alkanes. Without his help, none of the experiments could have been conducted. Learning from him and working with him have been delightful experience. Besides my advisors, I would like to thank the rest of my committee members: Prof. Ali Dhinojwala, Prof. Gustavo Carri from polymer science department, and Prof. Richard Elliott from chemical engineering department for commenting and reviewing my dissertation. I am also grateful for the informative discussions with my current group vi

7 members: Mr. Fatih Erguney, Dr. Numan Waheed and my former group members: Dr. Guoqiang Xu who has helped me understand the previous version of the codes, Dr. Sagar Rane, who inspired me in the writing of new Monte Carlo moves in the current version of the codes. I also want to thank Dr. Xu, Dr. Rane and Dr. Carin Helfer for liberating me, once a FORTRAN ignorant, by generously passing on their programming skills. Last, but not least, I want to thank my wife Qi Li for putting up with me in these years, my one-and-half-year old baby girl Chloe for all the happiness she has brought into my life, my parents Baoqi Lin and Zhilong Li for raising me up and providing me their unconditional love and support, my sister Chen Lin for teaching me being a stronger person. I dedicate this dissertation to them, my lovely family members. vii

8 TABLE OF CONTENTS Page LIST OF TABLES...x LIST OF FIGURES xi CHAPTER I. INTRODUCTION II. THEORETICAL BACKGROUND AND LITERATURE SURVEY Rouse theory The description of entanglements Reptation theory Modifications of the reptation theory Survey of the simulations for entanglement dynamics Bond fluctuation model (BFM) Kolinski and Skolnick s diamond lattice MC Kremer-Grest coarse grained molecular dynamics Theodorou s Atomistic MC and MD simulation Other important simulation models..71 III. SIMULATION METHOD IV. CROSSOVER TO THE ENTANGLED DYNAMICS: MONODISPERSE POLYETHYLENE MELTS 99 viii

9 4.1. Single Chain Static properties of Melts Dynamic properties of Melts Summary..137 V. CROSSOVER TO THE ENTANGLED DYNAMICS: BIDISPERSE POLYETHYLENE MELTS Introduction Simulation Methods Static properties Normal mode analysis Diffusion in bidisperse mixtures Trace Diffusion of N-alkanes in bidisperse mixtures Summary 173 REFERENCES ix

10 LIST OF TABLES Table Page 3.1. The coordinates of the 1 nearest neighbors for bead j shown in Figure 3.1(b). Reproduced from ref Conformations at the two C C bonds to bead j in terms of the vectors i j (rows) and j k (columns)! Reproduced from ref The Average Mayer Functions and the Converted Effective Interaction Parameters for the Indicated Shell Vectors. Reproduced from ref. 84 (Temperature at 370 K) All possible destinations of any two neighboring beads The illustration of seven classes of bead configurations with different numbers of alternative moves. The sequences of 5 numbers under every bead move represent the bond vector connecting nnd beads, in which 3 underlined middle numbers represent 3 changing nnd bond vectors after bead moves. These sequences of bond vectors are followed by the torsional states on the underlying diamond lattice before and after the two bead moves 96 x

11 Figure LIST OF FIGURES Page.1. Short time mean square displacements of short polyethylene chains: the solid line is the theoretical calculation of the CDGLE theory by Guenza 31, the dash line is the theoretical prediction of the Rouse model. The filled circles are center of mass diffusion data, and open squares are middle monomer diffusion data. Symbols from simulations 7 and lines from theoretical calculations 31. Reproduced from ref The rubbery plateau of polymer melts. Reproduced from ref The tube segments at time 0 and at time t. The tube segment AC and DB have been reached by chain ends before the time t. (not shown). As the result, at time t, the tube segment AC and DB are no longer in the original tube The monomer diffusion predicted by the Rouse model and reptation model Common moves in Monte Carlo simulations: a) Verdier-Stockmayer local moves; b) bilocal reptation moves c) pivot moves, reproduced from ref The lock-in conformation for local and reptation moves on a square lattice Sketch of the bond-fluctuation model of polymer chains on the three-dimensional simple cubic lattice. Reproduced from ref The mean square displacements vs. time for N=00. Reproduced from ref The mean square displacements vs. time for N=00. Reproduced from ref xi

12 .10. Perpendicular component and parallel component of mean square displacements along the constructed primitive path versus time t. The solid diamonds shows the data for a single chain in a frozen environment. The open symbols are data from multiple chain simulations. Reproduced from ref Mean square displacements versus time t for different chain length, the inset figure is for the scaling of diffusion coefficient D. Reproduced from ref The relaxation time versus chain length, the inset figure is for the scaling of normal modes. Reproduced from ref Mean square displacements of middle monomer g1 versus time t for different chain length. Reproduced from ref The center of mass diffusion coefficient vs. chain length. Reproduced from ref Diamond lattice and nnd lattice. Reproduced from ref. 87 a) Diamond lattice with sites alternately represented by open and filled spheres. b) nnd lattice by omitting the open spheres from a) Torsion and Collapse on nnd lattice. Reproduced from ref. 87 a). Two left torsion angles determine the angle i-, i-1, i, and right torsion angles determine the angle i-1, i, i+1. b) The triangle depicted the configuration of unrealistic collapses Schematic representation of a section of the coarse-grained chain on the nnd lattice, along with the detailed counterpart on the underlying diamond lattice. The two successive nnd bond vectors i j and j k, defined by the three nnd beads i, j, and k, determine the rotational isomeric state adopted by the two real bonds, j- and j+, to and from the central bead, j, to be γ and δ, respectively. Reproduced from ref Continuous L-J potential (the curve)and discretized L-J potential for nnd lattice (bars) Reproduced from ref Single bead moves and possible two bead moves depicted analog to cubic lattice, modified from ref xii

13 3.6 Single bead moves strains due to the forbidding of the unrealistic collapses, modified from ref The diffusion coefficients D and mean sq. radius of gyration <s> of a single chain plotted against M The relaxation time of end-to-end vectors versus M for single chain simulations The mean square radius of gyration <s> obtained at different cut-off distance of L-J potentials plotted against M. The line is based on the regression of the 5 longest chains simulated with shells The characteristic ratios between <s> and nl plotted against 1/n, where n is carbon number and l is carbon-carbon bond length The ratio <r>/<s> versus M; non-gaussian features shown for short chains Various kinds of mean square displacements (msd) of C60 vs. Monte Carlo Steps (MCS), g1 is msd of the middle bead on the chain, g is msd of the middle bead relative to the center of mass; g3 is the msd of center of mass; g4 is to the msd of end beads and g5 refers the msd of end beads relative to the center of mass. The solid line shown in the bottom of the figure has a slope of Mapping our simulation results of polyethylene onto the results of Bond Fluctuation Model, as reported by Tries et al.80 The color lines are our results, and the black lines are BFM results. From the uppermost to the lowermost: g4, g5, g1, g, g3 for both color lines and black lines respectively Normalized autocorrelation function of the end-to-end vector r vs. normalized time. The numbers inside the box are molecular weights The autocorrelation function of end-to-end vectors of C40 fitted by equation 4. (dotted line) and 4.3(dash line) respectively The autocorrelation function of end-to-end vectors of C34 fitted by equation 4. (dotted line) and 4.3 (dash line) respectively β obtained by fitting the data to the KWW equation plotted against M.117 xiii

14 4.1. g3 of C10, C138, C48 and C34 (from top to bottom) vs. time; the dash lines are just for showing slope The diffusion coefficients D vs M, open squares from the experiments 83, 101 open triangles and closed spades obtained by calibrating MCS with real time using two-shell (330MCS ~ 1ps) and three-shell (530MCS ~ 1ps) simulations. There two regression lines in the figure based on upper left 4 data points and lower right 5 data points from two shell simulations, respectively. The numbers in the figure are slopes of these regression lines g1 of C10 (thick line) vs. time, the thin regression line with an initial slope of 0.545; no slowdown observed before transition to the free diffusive region g1 of C17 (thick line) vs. time, upper thin regression line with an initial slope of 0.54, and lower thin line with a slope of 0.5; intermediate region (0.5 ~ million MCS) with the slope of g1 of C48 (thick line) vs. time, upper thin regression line with an initial slope of 0.5, and lower regression thin line with a slope of 0.5; intermediate region (0.7 ~ 5 million MCS) with a slope of 0.46 by regression. The inner intermediate region (3-4 million) with a slope of 0.4 by regression g1 of C34 (thick line) vs time, upper thin regression line with an initial slope of 0.5, and lower thin regression line with a slope of 0.5; intermediate region (1 ~ 9 million MCS) with a slope of 0.44 by regression. The inner intermediate region (6-7 million) with a slope of 0.39 by regression Comparison between the results from Harmandaris et al 46 and our simulation results in middle bead movements. The dash red line is the same simulation results directly calibrated by C50 shown in46, which is about 1.5 times faster than the results calibrated by the experimental data g1 of C40, C60, C10, C138, C17, C1, C48, C34 (from top to bottom) vs time; the C40 and C60 diffuse much faster than others at short time scale while other chains move at the similar rate at the short time scale 19 xiv

15 4.0. g1 of C40, C60, C10, C138, C17, C1, C48, C34 (from top to bottom) vs time by shifting all curves to iso-friction regime The Diffusion coefficients D vs M after shifting data to iso-friction regime The Rouse mode relaxation of C34 described by KWW equation. Linear fitting of these lines lead to β andτin KWW equation β in KWW equation obtained at different molecular weights vs. N/p The relaxation time τ according to reptation and KWW vs. M Single-chain coherent intermediate scattering function obtained from our simulation at 180 C at various q values (0.04, 0.1, 0.0, 0.8, and 0.36 Å -1 ) for different molecular weights: (a) C (b) C (c) C (d) C The two step relaxation of normal modes predicted by Hess theory. Reproduced from ref The mean square radius of gyration <s> of long chain and short chain components in the mixture against the weight concentration of short chains The example of normal modes fitting of long chain components in the mixture according to KWW equation β extracted from the normal mode fitting of long chain components in the mixture according to KWW equation for all the concentration studied The effective relaxation times from normal modes fitting of long chain components in the mixture and pure components according to KWW equation The effective relaxation times from normal modes fitting of long chain components in the mixture plotted against the concentration of short chain components 160 xv

16 5.7. The example of normal modes analysis of long chain components in the mixture plotted against the time in searching for the two step relaxation in Hess theory An example of normal modes analysis of long chain components in the mixture plotted against the time in linear scale for showing the data fluctuation near zero The example of middle bead diffusion of long chain components in the mixture plotted against the time (red dash lines for illustration) The diffusion coefficients of long chain and short chain components in the mixture plotted against the short chain concentration with the prediction of Hess theory and von Meerall-Pearson model Linear(dash) and FV(solid) extrapolation of the diffusion coefficients of short PE trace in the PE matrix of 4.5K at the different concentration obtained from our simulations Linear(dash) and FV(solid) extrapolation of the diffusion coefficients of short PE trace in the PE matrix of 33K at the different concentration obtained from the experiments Diffusion coefficients of short PE trace from simulation and experimental results via the extrapolation to the trace limit. The trace diffusion coefficients of unentangled polybutadiene(pbd) chains were obtained from reference 118. Dash lines are just for convenience 17 xvi

17 CHAPTER I INTRODUCTION The structure of the simplest molecule was still a mystery when Berzelius coined the word polymeric in in order to describe the substances such as ethylene (C H 4 ) and butene (C 4 H 8 ), both of which consist of /3 of hydrogen and 1/3 of carbon but with different properties. Back then, almost no scientist had the idea of long chain molecules in mind, until Staudinger, a professor of chemistry at the Federal Institute of Technology in Zurich (ETH), first presented this concept in detail in 190. This concept had been very controversial until it was finally accepted in Before this concept was established, chemists had categorized most natural macromolecules, such as Hevea rubber, starch and cellulose, into colloids which are aggregates of small molecules. While the concept of polymers was being established, the development of hydrodynamics and kinetics made the determination of the molecular weight possible for macromolecules. As early as 1845, for particles suspended in viscous media, Stokes showed the external force F=f v=6πηa v for a sphere with the radius a moving through a liquid at a steady state velocity v. f, i.e. 6πηa, is the frictional coefficient, and η is the viscosity of the liquid. In 1906, Einstein 3 suggested that this macroscopic law of hydrodynamics could be applied to the motion of a molecule as long as it behaves like a rigid sphere. Further developments in hydrodynamics, Brownian motion theory, the 1

18 newly invented centrifuge device, and the Stokes law finally formed a solid basis for measuring molecular weights of polymers by the ultracentrifuge method. Brownian motion theory and centrifuge devices also won the Nobel prizes for Perrin in physics and for Svedberg in chemistry respectively, in 196. Another development of early polymer dynamics also involved a method of determining the molecular weight, which was based on an Einstein s work by measuring the solution viscosity of a polymer. The famous Stokes-Einstein 3 relation predicted the diffusion coefficient D = RT / 6πηa N A, in which N A is Avogadro s number and a is the radius of a dissolved molecule. In the process of estimating this radius, Einstein developed a theory and predicted η/η 0 = 1+.5Φ i.e. η sp = (η- η 0 )/η 0 =.5Φ, in which η 0 is the solvent viscosity, and Φ is volume fraction of dissolved molecules. Note that the equation derived by Einstein for rigid spheres is completely independent of particle size. However, there was no doubt that an increasing size of polymer led to higher solution viscosities even if the volume concentration was the same. The discrepancy is due to the effective volume occupied by a polymer chain being bigger than dry volume of polymer segments. In the process of estimating the effective volume of a dissolved polymer molecule, Kuhn recognized in 1930s 4 that local chemistry of polymers did not influence the basic law: <r > = Nb, in which <r > is the mean square end-to-end distance for a polymer chain, b has been named Kuhn length and N is the number of random working segments. However he didn t precisely predict the index a in Mark- Houwink relationship η sp = KM a which was an experimentally observed relationship in measuring molecular weights of polymers by solution viscometry. The discrepancy between his prediction and the experimental results was mainly due to the excluded

19 volume effect, which required that two random walking monomers must not occupy the same space, although Kuhn indeed argued the correction would be <r > ~ N 1+ξ. This problem was unsolved until Flory came into the field. In 1940 s Flory and Fox 5 solved the excluded volume problem and found <r > 1/ ~ N 3/5 for a repulsive interaction and <r > 1/ ~ N 1/3 for an attractive interaction between the segments. Flory 5 also correctly predicted the excluded volume disappears in polymer melts, which would lead back to <r > 1/ ~ N 1/. One additional problem observed experimentally was the puzzling concentration dependence of η sp / Φ, which was unexplained until the extension of Einstein s theory. By counting into hydrodynamic interactions between suspended spheres, the final theory led to η sp / Φ = Φ. The accidental discovery of the vulcanization process by Charles Goodyear in 1839 signified the beginning of the polymer industry. He noticed that a sulfur-coated natural rubber specimen being carelessly brought in contact with a hot stove charred like leather. 1 This fact resulted in the vulcanization process which crosslinks the rubber into the network and virtually led to the takeoff of rubber industry by eliminating the stickiness of rubber. However nobody truly understood the origin of the elasticity of rubbers until Meyer 1 speculated the retraction of rubber is an entropy-driven process after he observed the absorption of heat when rubber retracts. Kuhn 6 solved the major portion of this problem with his rubber elasticity theory in At the same time, the transition from a viscous liquid to a non-equilibrium solid state, glassy state, was found helpful by Ferry and Parks 7 in explaining the fact that unvulcanized rubber is viscous at the high temperature but brittle in the cold. The glass 3

20 transition was also investigated in great detail by Flory and Fox later, 8 who found that glass transition temperature T g is linear in the reciprocal of the chain length (apparently the chains in their study were not long enough) and governed by free volume instead of viscosity. Besides the glass transition, these authors 9 also investigated the behavior of polymer melts and discovered a universal melt viscosity η transition against molecular weight M. The abrupt increase of the slope in the log-log plot of η vs. M was interpreted by Bueche 10 as the emergence of entanglements when chains reach a critical length. Other detailed investigations found that polymers have a spectrum of relaxation times extending over several orders of magnitude. Rouse then developed a theory in 1953 to explain this wide distribution of relaxation times. 11 In his model, a polymer was represented by beads, with friction coefficient ζ, connected by Hookean springs. It has been known that only a part of the relaxation spectrum can be directly obtained in experiments at a given temperature. Ferry 1 suggested that any unknown relaxation time τ i at temperature T could be obtained through a known relaxation time at the temperature T 0 by using τ i (T)=a T τ i (T 0 ), where a T has been called shift factor. Because a T primarily reflects the temperature dependence of the segmental friction coefficient ζ, William, Landel and Ferry related a T to T-T g and discovered that the WLF equation: log a T = (T-T g )/(51.6+ T-T g ) is nearly a universal relationship for many different polymers. Since various interesting aspects were discovered based on the chain length and temperature dependence of polymer dynamics, many theorists have attempted to solve the mysteries, however there still has no universal picture regarding polymer dynamics from the theoretical point of view. Existing theories, which are overwhelmingly 4

21 phenomenological, have been developed either for glass transition or entanglement dynamics but seldom for both. Even for entanglement transition alone, a satisfying theory is not easy to find. This thesis is aimed at providing some critical information for better understanding the chain dynamics of a realistic polymer, especially the crossover of polymer dynamics at the entanglement transition, which is very important for understanding the nature of entanglements, but is often paid little attention in current theories. The current theories of the entanglement dynamics can be divided into two large categories. One category can be called cooperative motion models, which generally feature a segment in a test chain dragging a segment in another chain along within a finite time and distance. However because the exact nature of entanglements is unknown, there has been a great difficulty to faithfully define this cooperative motion. Another category can be categorized as the generalized reptation model Unlike the isotropic motion of chains predicted in the first category, this category features a test chain moving along its own contour, which was assumed as the necessary consequence due to the entanglement effects. However there has been no rigorous derivation to prove this is exactly the consequence due to the entanglements. Furthermore, other chains in this approach were treated as an effective mean field. The original reptation model did not explicitly address the inherent multiple-chain nature of the entanglement. Both of these two categories have left some discrepancies with the existing experiments. The reptation model is more favorable currently because it tends to reconcile more experimental results 1 and its modifications have been relatively effective. Still, there are controversies inside of the framework of the reptation model, which will be covered in the following chapters. 5

22 By appropriately constructing the molecular model for a polymer chain, computer simulation is a unique resource for the exploration of the molecular information of the entanglement dynamics. Simulation usually involves fewer assumptions than theory does. It can provide microscopic information at the short distance and time scale that otherwise may not be accessible from experimental methods. Specifically on the entanglement dynamics, the reptation theory is still at the level of describing generic polymers. For example, the reptation model assumes chains are already long enough to be treated as Gaussian random walks on the length scale of M e, the critical entanglement molecular weight. This is not true in most cases for realistic polymers in experiments and is the least controlled approximation in the theory. Many simulation methods do not require this assumption. The objective of this thesis is to create a faithfully designed Monte Carlo algorithm, which can be used to simulate static properties and dynamics of polyethylene melts. Testing existing theories, acquiring detailed molecular level information and accurately predicting material properties are our goals. Critical comparisons of simulation results with NMR experiments will also be made. This thesis will be arranged as the following order. Two most popular models for polymer dynamics, the Rouse theory and the reptation theory, will be reviewed in detail in Chapter II. Various simulation methods and their results will also be surveyed in Chapter II. Our own simulation algorithms including recently developed two bead moves will be covered in the Chapter III. The detailed simulation results of monodisperse polyethylene melt at 180 C will be presented, discussed and compared with existing theories and experiments in Chapter IV. The Chapter V will focus on bidisperse 6

23 polyethylene melts. Some modifications of the reptation model will be tested against the simulation results as well. 7

24 CHAPTER II THEORETICAL BACKGROUND AND LITERATURE SURVEY.1. Rouse theory The Rouse theory 11 is one of the most popular molecular theories for unentangled polymer dynamics. In Rouse model, polymer chains were treated as freely jointed chains. The repeating units in such chains are coarse grained beads, which generally include a number of repeating chemical units with such a minimal length that the orientation of the subchain with such a length is totally uncorrelated with any others. The coarse graining forms a random walking chain as described by Kuhn. Another very important feature of freely jointed chains is no excluded volume among them, which means that unlike the chemical repeating units, these coarse grained beads can occupy the same space at the same time. Although the concept of zero excluded volume among random walking coarse grained beads is not straightforward, it has been predicted by Flory and examined by experiments. 3 With these characteristics of freely joint chains, the distribution function of the radius of gyration can be shown to be approximately Gaussian as long as the chain is long enough. The most fundamental property of a Gaussian chain probably is <R > = Nb, as Kuhn derived for polymers. Thus Gaussian chains approximation is one of the most fundamental assumptions of Rouse model. However one should be extremely cautious when using Rouse model to examine the real experimental data or simulations 8

25 results based on realistic polymers. In order for chains to be Gaussian, they have to be long enough, while in order for chains to be unentangled, they have to be short enough. Hence, for a specific polymer, it is not guaranteed there must exist a window of intermediate chain length that is neither too short to be Gaussian, nor too long to be unentangled. We will come back to this approximation later. It is interesting that the original Rouse model was designed for a single chain surrounded by a continuum of solvent molecules instead of other identical chains. Hence the surrounding was treated as a mean field Newtonian medium, which has a viscosity independent of shear rate. Each coarse grained bead has a friction coefficient ζ in this medium, and pairs of beads connected by frictionless Hookean springs, with a force constant: 3kT/b, in which b is the root mean square distance between the weight center of two connected beads. The force equation was constructed based on the balance of 3 terms: Brownian motions of beads, spring forces, and the frictions experienced by beads in media using the Langevin equation. 18 dx U ζ = + f (t) (.1) dt x Here, ζ is the friction term, f is the Brownian term, and U captures the connection between beads. We can firstly insert only Hookean interactions in U term and ignore the interbead hydrodynamic interactions and excluded volume interactions. Then we insert Stokes law relationship into the friction term. Thus a Langevin equation for every bead can be constructed as the following equations show, where m is the mass of a bead and b is the bond length. n means the nth repeating unit. 0 and N represent two chain ends respectively. 9

26 r dr dt 0 3k T r B = ( R ζb 0 r R ) + 1 r f 0 (.) r dr dt n 3k B = (Rn Rn 1 Rn+ 1) ζb T r r r + r f n (.3) r dr dt N 3k B = ( RN R ) N 1 ζb T r r + r f N (.4) Here the diffusion coefficient in Einstein relationship D=k B T/ζ has been assumed to be independent of R n. It should be noted here that the Rouse theory considered that every bead has identical local environments, which is not necessary ture. 4 There may exist a significant chain end effect for short polymer chains. Simultaneously solving these coupled equations for all beads involves the normal mode analysis, which is able to solve a linear set of first order differential equations by finding the normal coordinates X p defined by r X p = 1 N pπn dncos( ) R r n N N 0 ( t) with p=0, 1,, (.5) Each of them is capable of independent motion. The introduction of normal modes leads to the decomposition of coupled Langevin equations. Then Langevin equations above can be rewritten as: r X p 6π k BT = p t ζnb r X p + r f p (.6) Since f is the random Brownian force, they are also independent of each other. Thus after the motion of a polymer chain is decomposed into independent modes, the time correlation functions of these normal coordinates then can be calculated using 10

27 < f ( t) f ( t) >= 6Dδ, δ ( t t') (.7) The result shows that n m n m < r X pα r k BT ( t) X qβ (0) >= δ pqδα exp( t / τ p ) β k p (.8) where, p and q =0, 1,,., and α and β are x, y, z. Inside this equation, τ P = τ 1 /p, and τ R = τ 1 = ζn b / 3π k B T. (.9) For relaxation time, by the inverse transformation of equation (.5), one may also give the end-to-end vector of the polymer as the following: r r R = R N r R r pπ N P = X P (( 1) 1) cos( 0 P= 1 ( N + 1) ) (.10) Since the result is dominated by k values which are extremely small compared to N, for r r r N r end to end vector we can have R = R R = (.11) N 0 4 X P P= 1, odd Thus the time correlation function of the end to end vector will be r r < R( t) R(0) >= 16 N < P= 1, odd r r X P X P > (.1) where the assumption that different modes are uncorrelated has been used. By introducing the correlation function of X P, which has been just developed, the final time correlation function of end to end vector is r r N 8b < R( t) R(0) >= ( N + 1) π P= 1, odd 1 P exp( tp / τ ) R (.13) τ R = ζn b / 3π k B T= N /3π w, (.14) where w= ζb / k B T is also called the reorientation rate of the monomers. By the inverse transform of the equation (.5), one may also obtain the 11

28 instantaneous position of a monomer, r R = r N r pπ 1 X 0 + X P cos( ( n + )) ( N + 1) P= 1 (.15) in which, the X 0 is actually the center of mass r R N 1 r G = dnr n N 0 (.16) using the equations above, we write the monomer diffusion as: r r r r N r < ( Rn ( t) Rn (0)) >=< ( X 0 ( t) X 0 (0)) > + 4 < ( X p= 1 p r ( t) X p (0)) pπ 1 > cos ( )( n + ) N + 1 (.17) According to the normal mode analysis, X p can be expressed analytically and lead to: r r N k BT 4b 1 < ( Rn ( t) Rn (0)) >= 6 t + ( N + 1) (1 exp( tp Nζ π P p= 1 When t>>τ R, the first term dominates, yielding / τ )) R pπ 1 > cos ( )( n + ) N + 1 (.18) r r < ( R ( t) R (0)) n n >= 6Dt k BT = 6 t Nζ (.19) where clearly D k T B = (.0) Nζ It is natural to see eventually all monomers of the polymer diffuse as a whole with diffusion coefficient D. While t<<τ R, other terms may dominate, the final result for every monomer can be a little different, if we average over all monomers, we have < r r 4k Tb 3πζ B ( Rn ( t) Rn (0)) >= ( ) t (.1) This is anomalous diffusion at the short time; the averaged result shows that mean square displacement of a typical monomer scales as t

29 The derivation of the relaxation modulus in the Rouse model is not very relevant to our topic, because the original derivation in the Rouse model is for dilute solutions, the resulting intrinsic viscosity scales as N, which is inconsistent with the experiment anyway. It was discovered later that, due to neglecting the hydrodynamic interactions, the Rouse model actually fits polymer melts better instead of polymer dilute solutions. Hence the macroscopic properties such as G(t) and η can be given using the rheological relationships below: ρrt G ( t) = exp( t / τ p ) M p (.) where τ p = ζn b / 3π k B Tp has the molecular level meaning. Known relaxation modulus allows the viscosity to be calculated as ρrt η = G( t) dt = τ p M 0 p τ 1 M ~ N (.3) Now let us summarize the result of the Rouse model, which will be the foundation the reptation theory discussed later. D ~ 1/N τ ~ N η ~ N By the way, it has turned out that neglecting intramolecular hydrodynamic interactions and the assumption of zero excluded volume among all beads along the chain are unacceptable for describing a single polymer chain in solution, although these two terms do disappear for polymer melts above certain distance level. Zimm 5 proposed a model which added intramolecular hydrodynamic interactions and kept most other 13

30 assumptions of the Rouse theory. Briefly, in strong contrast with the Rouse model, the friction coefficient of a whole chain is ξ = η 0 R, in stead of ξ = ζn in the Rouse model at the free draining limit. In other words, Rouse theory is more like a moving "chain without carry solvent molecules with itself, while Zimm theory is more like a moving ball" carrying solvents inside its effective volume. Reasonable agreements with experimental data have been achieved by Zimm theory in real dilute solution, although the chains in good solvents are no way to be Gaussian as Zimm assumed in the first place. Unfortunately, the exact distribution of chain conformation is hard to solve due to the non-markovian nature of the chains with the excluded volume. Fortunately, the major conclusions of the Zimm model are not very sensitive to the improper form of the distribution function of chain conformation. 18 Coming back to what has been discussed earlier in this chapter: although the Rouse theory captured the major interactions inside the polymer melts, the predictions of the Rouse model are not necessarily observable for polymer melts in experiments due to the interference of non-gaussian effects and chain-end effects approaching from lower molecular weights region, where friction coefficient ζ has a molecular weight dependence. This region may be then directly followed by the entangled regime. In fact, the chain end effects have been considered as a major role in smearing the Rouse scaling in the low molecular weight region. 4 It is noticed that the computer simulations for the last 10 years have also challenged the Rouse theory, especially at the short time and distance scale. 6,7,8,9 As we mentioned earlier, no excluded volume effects have been included in Rouse theory. In real polymer melts, they are only screened out only at a certain distance. 30 By including 14

31 the excluded volume effect in dense systems, both molecular dynamics (MD) and dynamic Monte Carlo simulation showed there is an anomalous diffusion for both unentangled and entangled melts at short time and distance scales. The mean square displacement of the center of mass of polymers scales as t at the short time scale in dense solution or melts. 7 Simulation shows that for t > τ R, the center of mass diffusion obeys the Rouse dynamics, and the Fickian diffusion is recovered. In the Rouse theory, local intermolecular interactions are ignored because it is basically a single chain in solvents theory. The center of mass in the Rouse model follows Brownian dynamics and the mean-square displacement scales as t 1 at all time scales. Realistically, the polymer melts show different structures than normal small molecule liquids, such as correlation holes in pair correlation functions. Intermolecular interactions occurring on the length scales shorter than R g lead to the failure of Rouse theory in the satisfying description of the short-time dynamics. By explicitly including both intramolecular and intermolecular forces, Guenza derived a generalized Langevin equation (GLE) for the cooperative dynamics (CDGLE) for interacting flexible polymer fluids. 31 The theory has been found to be in excellent agreement with computer simulations of Kremer and Grest. 7 In Figure.1, Guenza concluded that the introduction of the local intermolecular forces strongly improves the description of the global dynamical properties of polymer melts in the short-time domain. Finally it should be noted that, although the Rouse model is not a perfect theory for polymer dilute solutions, it is generally agreed that overall the Rouse model is suitable for description of unentangled polymer melts, especially for macroscopic 15

32 Figure.1. Short time mean square displacements of short polyethylene chains: the solid line is the theoretical calculation of the CDGLE theory by Guenza 31, the dash line is the theoretical prediction of the Rouse model. The filled circles are the data of center of mass diffusion, and open squares are middle monomer diffusion data from MD simulations 7 The lines are from theoretical calculations 31. Reproduced from ref

33 properties at large time and distance scale. It is well known that the Rouse model breaks down for high molecular weight polymer melts, and the origin of this breakdown has been long attributed to the existence of entanglements, which have been studied for roughly half a century. 3 But people hardly agree on what causes the entanglement and even what is the entanglement. In the next section, the observation of entanglement effects will be briefly revisited, and then followed by the brief review of reptation theory for the description of entanglement effects.. The description of entanglements The dynamics of long chain polymer fluids is not a solved problem largely due to the different conjectures about the real meaning of the entanglements. When polymer chains are long enough to enter the entangled regime, the long relaxation time of the polymer chain conformation and the slowdown of transport processes are nearly universal. The early recognition of entanglement mainly came from the similarity in rheological responses between uncrosslinked polymer and crosslinked rubber. The first signal was the elastic recovery of uncrosslinked polymer. As it is well known that elastic recovery of rubber is due to crosslinks, the observable elastic recovery of uncrosslinked polymer has been naturally accepted as the result of some kind temporary crosslinks, i.e. entanglements. From the same underlying mechanism, both for stress relaxation and dynamic shear modulus, an intermediate plateau region shows up for uncrosslinked polymers with reasonable high molecular weights. This plateau region won't disappear for rubbers, but eventually fades away at the long time for uncrosslinked polymer melts, which indicates the links between the uncrosslinked polymers are temporary in nature. 17

34 . Figure.. The rubbery plateau of polymer melts. Reproduced from ref.. 18

35 This is illustrated as rubber plateau in their rheological responses as Figure. shows: after a small, rapid, step strain, the measurement of the stress relaxation gives a modulus plateau at the intermediate time region, and the length of this plateau depends on the molecular weight. As a natural legacy of rubber theory 3, at the microscopic scale, entanglement has been traditionally considered as the dynamical consequences of chain connectivity and backbone uncrossability due to intermolecular repulsive excluded volume forces. Inspired by rubber theories, physical origin of entanglement has been viewed as the result of a transient network due to topological interactions among the long and interpenetrating polymer chains. The relaxation and transport processes of every individual chain are severely retarded by this kind of knotty network structure. If entanglements can be treated like crosslinks 3, by taking advantage of the existence of rubbery plateau, the distance or the molecular weight M e between entanglements can be calculated employing rubber elasticity theory. It has also been found that entanglement molecular weight M e ~ϕ -1 where ϕ is the volume fraction of polymers in their compatible solvents. Besides the rubbery plateau, which has not been predicted by the Rouse theory, the new scaling law for the molecular weight and concentration dependences of diffusion processes and relaxation processes has been found, along with extended power law frequency dependence of stress, dielectric, and other dynamic response functions. For example, almost universally for polymers above certain critical molecular weight M c, the melt viscosity η~ M 3.4, while the Rouse theory predicts: η~ M 1. Moreover, a link was found between M e and M c : M c M e, 1 although this relationship has been challenged 19

36 recently. 33 Besides the backbone uncrossability or the resemblance of knots, there exist several other pictures for entanglement. One may imagine two chains may met and have some monomers in them stick to each other before they eventually leave each other after some time. Another alternative view may consider entanglement as an effective field. Neither of these crude pictures can explain why some monomers may stick; or why chains have to reach certain length to show entanglement effects and how these entanglements eventually relax. Immediately after realizing the entanglement phenomenon, several theoretical treatments have emerged trying to capture its real microscopic nature. These theories including temporary network model pioneered by Green and Tobolsky 34 and cooperative motion approach pioneered by Bueche 10 in 1950s. The reptation model developed by de Gennes and Edwards has been introduced in 1970s 17,18 and followed by relatively new theories by Ronca, 13 by Fixman, 35 and by Schweizer 15 developed in 1980s. These new theories are somewhat different but all based on generalized Langevin equations (GLE). The slowing down of dynamics may not be unique to linear polymers. For other different macromolecular architectures, such as cyclic polymer rings, star branched polymers, rigid rods, H like molecules and even gels of non-interpenetrating microgel particles, an entanglement-like behavior can also be seen. It seems that a dense enough collection of macromolecules is the only requirement of this dynamic slowing down. 36 Without a decisive nature of entanglements, most of the phenomenological theories have been constructed based on the strikingly different models for real space motion for the long polymer chains. Among them it is commonly acknowledged that the reptation model 0

37 has described or predicted more experimental facts than any other models. 1 Hence the major assumptions and derivation of reptation model will be given in the following section..3. Reptation theory As we just mentioned, one of the central problems in polymer dynamics is to achieve the understanding of the entanglement phenomenon, better at the molecular level. In the last 30 years, the reptation hypothesis 17,18 has drawn large attention on its reconciliation of a wide range of experimental observations, and simulation works. Both experimental and theoretical studies inspired by reptation theory in the last 30 years have been fruitful. 1 In order to further understand and appreciate all these efforts, a review of reptation model and its different versions by different modifications is given here. The following session is organized into 3 sections. First of all, the definition and discussion of entanglement effects in the reptation scheme will be briefly reviewed. The major derivation steps will be illustrated. Secondly, the examinable predictions and fingerprints of reptation will be introduced with the emphasis on transport phenomenon. Finally the comparison between reptation and the experimental evidence for or against it will be concluded. Reptation theory has been constructed for the motion of a single chain in an effectively static tube formed by topological constraints due to other surrounding chains. The chains relax via a stochastic sliding motion along its own contour inside the tube. As a result, it has been also called tube theory. Reptation theory started to gain popularity in late 1980s and especially after the critical review by Lodge et al, 1 who found that the 1

38 reptation model explains a wide variety of phenomena reasonably well, and must be viewed as an extremely successful theory. The theoretical construction of reptation is introduced here in minimal words. The theoretical discussion of the theory formulation is not our intention, but rather for a sketch of rough picture with emphasis on critical postulates and final examinable conclusions. Although the precise description of the entanglement effects probably requires at least chains, the original reptation model, just like the Rouse model, is still a singlechain theory. The major phenomenological assumption of the reptation model is that the major portion of this single polymer except its ends move back and forth following its own contour with a constant contour length L. It is similar to a snake moving in a tube which has similar shape with its own contour. This motion assumes a certain diffusion coefficient, which can be identified as the diffusion constant of the Rouse model, D=k B T/Nζ. The contour length L is a new parameter comparing to the Rouse model. Seemingly there is another new parameter a in reptation theory, which is the step length of creation of a piece of new tube when chain ends continuously choose random directions. However when chains are supposed to be Gaussian, the end to end distance <r >=Nb =La is a natural relationship. Hence, a is not an independent new parameter. Note, most theories including reptation theory treat chains as random walks on the length scale a, which is not necessary true for realistic polymers. Furthermore, unlike some other assumptions in the reptation model, this approximation cannot be removed for infinitely long chains. This is the least well-controlled approximation in the reptation model and may influence generality of the reptation model. In experiments, the a can be treated as the entanglement length, which is the root

39 mean square end-to end distance of a chain with the molecular weight M e =ρrt/g 0 n, 1 where G 0 n is the rubbery plateau value of the relaxation modulus and ρ is the mass density of the polymer. Similar to the derivation of the Rouse model, the time correlation function of R e is the reasonable first step to obtain the characteristic relaxation of the system. First of all, this time correlation function can be shown equal to the time correlation function for the remaining tube segments starting at point c as Figure.3: the tube segment vanishes when it is reached by any of chain ends. As the time passes, the part of the chain CD remains in the original tube, while segment AC and DB are in the new tube. The end to end vector at the time zero: r R r r r = (.4) ( 0) RA(0) C + RCD + RDB(0) And the end to end vector at time t: r r r r R( t) = R + R + R (.5) AC CD Since the segment AC and DB are uncorrelated, DB r r L < R( 0) R( t) >=< R >= a < σ ( t) >= a dsψ ( s, t) CD 0 (.6) where σ is the contour length of the tube segment CD, and ψ(s, t) is the probability that this segment starting from point s remains at time t. If ψ(s, t) is averaged over all possible s, the ψ(t) in the following equation will be the average probability for any remaining tube segment. L 1 ψ ( t) = dsψ ( s, t) (.7) L 0 3

40 Figure.3. The tube segments at time 0 and at time t. The tube segment AC and DB have been reached by chain ends before the time t. (not shown). As the result, at time t, the tube segment AC and DB are no longer in the original tube. 4

41 Let ψ(ξ, t, s) be the probability that the remaining tube segment starting from point s moves a distance ξ at time t. Then = s L s s t d t s ),, ( ), ( ξ ξψ ψ (.8) which means in order to keep this tube segment, the highest displacement should be between s-l(backward motion) and s(forward motion). Also we have: ),, ( ),, ( s t t s t + >= + < ξ ψ ξ ξ ψ (.9) Expanding the left side of this equation, ),, ( ),, ( ),, ( ) ( 1 ),, ( ),, ( ),, ( s t t D s t s t s t s t s t c ξ ψ ξ ξ ψ ξ ψ ξ ξ ξ ψ ξ ξ ξ ψ ξ ξ ψ + = > < + > + < = > + < (.30) Insert the result back to equation (.9), and take the limit as t goes to the zero: ),, ( ),, ( s t D t s t c ξ ψ ξ ξ ψ = (.31) where D c is one dimensional curvilinear diffusion coefficient. Nζ T k D B c =, hence the reptation model is intrinsically connected to the Rouse model. With the boundary conditions ) ( ),0, ( ξ δ ξ ψ = s 0 ),, ( = s t ξ ψ at s = ξ and L s = ξ (.3) the solution of (.31) is the following ) / )exp( ) ( )sin( sin( ),, ( 1 d p t p L s p L s p L s t τ ξ π π ξ ψ = = (.33) 5

42 where τ d = L /D c π, average probability for any remaining tube segment is L 1 1 ψ ( t) = dsψ ( s, t) = ds dξψ ( ξ, s, t) = L L 0 L s 0 s L p: odd 8 p π exp( p t / τ ) d (.34) Thus the end to end vector correlation is r r L < R(0) R( t) >= a < σ ( t) >= a dsψ ( s, t) = alψ ( t) = Nb 0 p: odd 8 p π exp( p t / τ ) d (.35) This has exactly the same form of the Rouse model. The only difference is the relaxation time. Insert the expression of the Rouse diffusion coefficient into D c, and use Nb =La: τ d ζn b = (.36) π k Ta B Here τ d is proportional to N 3, while in the Rouse model, ζn b τ R = (.37) 3π k T B τ R is proportional to N Next, the monomer diffusion is derived as the following. The MSD of monomer s is: r r ϕ ( s, t) =< ( R( s, t) R( s,0)) > (.38) In order to calculate φ(s,t), φ(s,s,t) is introduced: r r ϕ ( s, s', t) =< ( R( s, t) R( s,0)) > (.39) For all s, except s=a, and s=l, we have ϕ ( s, s', t + t) =< ϕ( s + ξ, s', t) > (.40) The right hand side of the equation can also be expanded in the same manner as the one in derivation of autocorrelation function of end-to-end vectors. A similar one dimensional 6

43 diffusion equation can be obtained: ) ',, ( ) ',, ( t s s s D t t s s c ψ ϕ = (.41) The boundary conditions of this equation can be shown as ' ) ',, ( 0 s s a t s s t = = ϕ (.4) a t s s s L s = = ) ',, ( ϕ (.43) a t s s s s = =0 ) ',, ( ϕ (.44) The solution is obtained as the following: ) ' )cos( ))cos( / exp( (1 1 4 ' ) ',, ( 1 L s p L s p t p p La t L a D s s a t s s d p c π π τ π ϕ + + = = (.45) when s=s ) ( ))cos / exp( (1 1 4 ), ( ),, ( 1 L s p t p p La t L a D t s t s s d p c π τ π ϕ ϕ + = = = (.46) There are also two extreme time limits with regarding to τ d, when t<<τ d, the second term dominate, replacing cos (pπs/l) with the average ½, and converting the sum into integral: 1/ 0 ) ( 1 )) / exp( (1 1 4 ), ( π τ π ϕ D t a t p p La t s c d = = (.47) When t>τ d the first term dominates: t N b T k a t L a D t L a D t s B c c = = = ) 3 1 ( 6 ) 3 1 ( 6 ), ( ζ ϕ (.48) Note here D is 3 1 N b T k a B ζ, and it scales as N - in contrast to the N -1 in the Rouse model. 7

44 Also it seems that the time scaling law of monomer diffusion is much like the one of the Rouse model, where MSD~t 1/ at short time, while MSD~t 1 for longer time. However for reptation model, there is additional intermediate scaling for intermediate time. Considering at very short distance scale, i. e. d<<a, polymer don t feel any constraint from the tube, the Rouse diffusion at the short time is recovered. < r r 4k Tb 3πζ B ( Rn ( t) Rn (0)) >= ( ) t (.49) Once the segment has moved a distance a, it feels the tube, a new time regime set in, thus 3 ζ 4 τ e = πa, which is independent on N. Note here mean square displacement 4 k BTb (MSD) don t have N dependence either, while the just derived result as r < ( R t r R n ( ) n (0)) >= Dct a( ) π 1/ for t<<τ d shows N dependence. After the following derivation, we will know that the r < ( R r n ( t) Rn (0)) >= Dct a( ) π 1/ is actually not to describe the behavior for t 0, but for τ R <<t<<τ d! Once the monomer diffuses to longer distance, it feels the constraint imposed by the tube, the chain start to feel the perpendicular constraint from the tube, and only have freedom to move along the tube. This will be a one-dimension analog of the monomer diffusion for the Rouse model at the short time, then 1 3 4k Tb 3πζ B < ( sn ( t) sn (0)) >= ( ) t (.50) However, now s n is no long the coordinates respect to the origin but the position with respect to the tube. 8

45 r < ( R n For τ e <t<τ R, the chain as a whole does not move, then using r ( s) R n ( s')) >= a s s' and assume that 1/ < sn ( t) sn (0) > < ( s( n ( t) sn (0)) > (.51) The following equation can be obtained < r r >= 4 ) 7 k T ( ) ζ 1/ 4 1/ B 1/ 4 1/ 4 ( Rn ( t) Rn (0)) ( ab t (.5) For τ R <t< τ d, in this regime, monomers still feel the tube existence, but then it is already beyond the τ R, which means that we can use the Rouse diffusion result at longer time in stead of short time diffusion used above, however the new equation still should be the one dimensional analog. Thus < ( s ( t) s (0)) >= D t (.53) n n c Again we use the same approximation as the one in the last regime, which assumes that tube doesn t change appreciably, we obtain r < ( R n r ( t) R n (0)) kbt >= ( ) ζ 1/ a( ( t N ) + 1) 1/ (.54) If this equation is compared with the one derived directly from the monomer diffusion of r r Dct reptation model < ( Rn ( t) Rn (0)) >= a( ) π 1/, there is only a difference in numerical constants. The scaling for mean square displacements of monomer is summarized below < r r 1/ 1/ ( Rn ( t) Rn (0)) >= Nb ( ) ~ t τ R t when t τ ( independent of N) (.55) e 9

46 < r r 1/ 1/ 4 1/ 4 ( Rn ( t) Rn (0)) >= Nb ( ) ( ) ~ t L τ R a t whenτ e t τ R (.56) < r r 1/ 1/ ( Rn ( t) Rn (0)) >= Nb ( ) ~ t τ d t when τ R t τ d (.57) < r r 1 ( Rn ( t) Rn (0)) >= Nb ( ) ~ t τ d t when τ t (.58) d The relationships above tells us if the samples of the same polymer are all entangled, at the short time, their MSD curve should merge to one line, and start to split from the τ R of the chain with the smallest molecular weight. For viscoelastic behaviors, it can be shown that c π τ G( t) t N t R 1/ 1/ = k BT ( ) ( ) ~ when t τ e (.59) G = k BT ( ) ~ N t when τ e t τ d 3 c π b a (.60) G( t) = G Ψ( t) = G exp( tp / ) ~ exp( t) τ d π p p: odd when τ t (.61) d b 3 η = G( t) dt = G dt exp( tp / τ ) = H N d ζ (.6) 4 π p a 0 p: odd 0 Finally we should be aware that some assumptions used in deriving these relationships can be too strict. For example, the rigid tube doesn t exist especially for 30

47 chain(rouse) Monomer diffusion Rouse vs. Reptation 3D(Rouse) ln MSD(t) 1/ 1 1/ 1/4 1 short entangled chain 1/ long entangled chain lnt τ e τ R τd chain tube 3D Figure.4. The monomer diffusion predicted by the Rouse model and reptation model. 31

48 polymer melt with molecular weight almost same order of the magnitudes as the entanglement molecular weight N e. Also, all analysis here takes the case to the limiting situation, which means, in real system, there can not be sudden transitions like that seen in following figures. All transitions will be much more gradual than the predictions. Figure.4 summarizes the discussion related to monomer diffusion above, which has been called the fingerprints for reptation motion in polymer melts..4 Modifications of the reptation theory Of course, the strict reptation motion is only the representation for a free chain moving in a network. For polymer melt, all chains are moving, hence a real polymer chain is not nearly as confined by its neighbors as the strict reptation assumes. It is also apparent in reptation theory that η ~ τ ~ N 3, is not consistent with the well known experimental relationship η ~N There were significant discrepancies between the original tube model and experiments on the self diffusion coefficients of long chain polymer fluids, which scale as M -. ~ -.3 powers instead of M predicted in tube theory. 38 The pre-terminal frequency dependence of dielectric and stress loss module do not agree with each other and are both much broader than the pure reptation predictions. These inconsistencies are considered as the result of contour length fluctuation which incorporates the additional relaxation of the probe chain itself, or as the result of constraint release, which accounts for the matrix mobility. 39 Here a brief summary is given for these important modifications. Because the original reptation model considered a chain with tube segments which itself follows Gaussian statistics, thus the distance between centers of the neighboring segments actually is not a fixed but a fluctuating 3

49 number. The average of these fluctuating numbers is tube diameter a. As the result, the contour length L is actually not fixed either. Overall, contour length fluctuation (CLF) induces a faster relaxation near chain ends and effectively reduces the L in the original reptation model to: L f = L( ( a / R)) (.63) and the relaxation time and viscosity respectively to: τ (.64) 1/ d, = τ d ( ( N e / N) ) f 1/ 3 η f = η ( ( N e / N) ) (.65) These two equations resemble the power law relationships τ ~ N 3.4 and η ~ N 3.4 in a wide range of N/N e from The CLF has been frequently presumed to induce no displacement of the center of mass and have no effect on the center of mass diffusion coefficient of the linear chain. CLF has been also interpreted differently 40 which led to: D = D (1 O( N / N) ) 1/ 1 f e (.66) up to N/N e > 300, and this resembles a power law relationship D ~ N -.4. In contrast of CLF based on a single chain approach, constraint release considered the multiple chain effects. Constraint release (CR) considered when a constraint forming matrix chain diffuses away the entanglement segment of the probe chain can jump locally additional to its reptation motion. In the original model of constraint release, by assuming the constant contour length, 41 the relaxation time for constraint release process is: 3 3 τ CR ~ ζ N M N P N e (.67) where N M is the chain length of matrix and N P is the chain length of probe chain. The diffusion coefficient: 33

50 3 1 D = D rep + D CR, where DCR ~ ζn M N P N e. (.71) It has been found that the hydrodynamic interaction becomes important for CR effect only when N p > N M N e > N e 3. Because N e is at least the order of 100 for polymers, N p will have to be at least 10 6 for significant hydrodynamic interaction. Thus the Rouse dynamics is safe for describing most practical systems. These two modifications within the picture of reptation have offered new perspectives on entanglement and served to reconcile the reptation predictions with a broad range of experimental data. However they both have significant challenges from both theoretical and experimental sides. For example, although the original CLF model predicts the same scaling of diffusion coefficient versus molecular weight as a strict reptation model did, the newer generation of CLF model predict that D ~ N -.4. Because CLF is a single chain approach, it should also predict that trace diffusion D tr ~ N -.4 in a matrix consisting of much longer chains. However experiments found that tracer diffusion is proportional to N - instead of N Constraint release (CR) has also assisted reptation theory to obtain the better agreement with experimental data. However the idea constraint release itself, implicitly assuming two chain contacts as an entanglement, which is somewhat contradicting with the reptation model that regards entanglement effects as a continuous effective field (with the length scale of 50 Å). Secondly, the picture of constraint release is very subjective, for example, if the constraint release is uniform along the chain or if the magnitude of constraint release is too significant to invalidate the hypothesis in derivation of reptation theory? Hence the theory built on these subjective conjectures may also be subject to argument. Coming back to the diffusion, if the experimental observation D tr ~ N cannot be explained by CLF, 34 3

51 will current CR model explain it? For monodisperse homopolymer melts, τ CR 3 M P 3 e ~ ζn N N ~ N 5, because N M = N p. This is much longer than the reptation time 3+ where τ ~ N, so it seems that the self-diffusion scaling can not be explained by CR either. Overall many modifications of original tube theory have been made in the last 0 years including contour length fluctuation, reptons, double reptation, constraint release, tube dilation, and nematic coupling. Most modifications have been done to weaken the rigid constraints of the tube and enhance the relaxation for chains of finite length. These correction efforts have been usually based on the educated guesses of the meaning of the entanglements, and sometimes empirical parameters without a clear microscopic meaning were involved for fitting data. However, whether these corrections are physically reasonable, or have explained the behaviors that have not been properly described by pure reptation is still a subjective matter in a certain sense. There are also some experimental evidences, which have formed the significant challenges for the reptation theory. 36 For example, although polymer rings shall not reptate and a special kind of microgels does not interpenetrate with each other, their viscoelastic responses are surprisingly similar to linear chain melts. The similarity of viscoelasticity of melt of linear chains, rings, and microgels appears totally incompatible with the reptation framework. 36 As the same as 30 years ago, reptation theory remains a phenomenological theory, and has still not been derived from the intermolecular forces at the microscopic molecular scale. Moreover, due to its highly phenomenological character, the description of entanglements at the level of the time dependent intermolecular forces cannot be addressed. The prediction of the influence of liquid 35

52 structure on dynamics is not attainable or requires extra assumptions. Computer simulations at present can not reach much higher molecular weight than M e, which is not adequate for decisively testing the reptation hypothesis at the chain trajectory level. Overall quantitative agreement on transport and viscoelastic material properties has not yet been reached. Different modifications of reptation theory have given better predictions on one or more of these properties, but they are still not under a unified frame yet..5 Survey of the simulations for entanglement dynamics Computer simulation is one of the most powerful tools for understanding the intrinsic nature of polymer dynamics and for checking theories because it embraces all the necessary microscopic information, such as various autocorrelation functions and diffusion at the very short time scales, which is very difficult to observe in experiments. It was the simulation that rendered a strong support to the reptation model back in 1988, which demonstrated t ¼ regime clearly for the mean square displacement of middle beads in polymer melts. 43 This signature of reptation was observed by experimentalists only recently 44. By appropriately constructing a molecular model for a polymer chain, simulation can also provide critical information to experimentalists in certain highly sophisticated experimental procedures, however simulation does not always more effortless than experiments. In fact simulating a sufficiently long realistic polymer, residing deeply in the reptation regime, has never been an easy task mainly due to the need of extremely demanding computing power. Consequently, most simulations involving entanglements only dealt with coarse grained polymer chains, which is 36

53 perfectly understandable, because the entanglements phenomena, for instance, η ~ M 3.4, has been found almost universal for flexible polymers. Only recently entangled realistic polymers, such as PE, can be studied by MD 45, 46. This should not be considered as a trivial development because at last the structure-property relationship of entangled realistic polymers can be studied by computer modeling, especially when the local packing and monomer friction may be very important for certain macroscopic properties. 47 On the other hand, the incorporation of chemical details naturally introduces finite excluded volume effect and stiffness into the system, which have not been considered in the classic theories and modeling but can be very important for precise description of chain statistics and dynamics. 48 The stiffness is also deeply correlated with the intriguing packing length problem. 33 The excluded volume effects induced by the L-J potential at the short distance scale can also produce local packing orders probably leading to unforeseen effects beyond the prediction of the reptation model. 49 The developments of a simulation method for a realistic polymer can provide valuable information for the comparison of the experiments, which have been always based on realistic polymers. It is also necessary for practical purposes. For example, predicting diffusive properties or viscosities of a polydisperse PE melt is still beyond the reach by a generic reptation model 18, but can be easily studied by computer simulations. Inside the framework of simulation for realistic polymers, molecular dynamics is a nature choice for studying polymer dynamics because the MD numerical scheme integrates the equations of motion for a polymer chain. However, the traditional MD simulation of polymers involves a large span of time scales from fast bond vibrations to 37

54 slow reorientation of whole polymer chains. Additionally, the dynamic simulation for realistic polymers requires an intermolecular interaction, such as Lennard-Jones potential, in order to obtain the realistic local packing. The evaluation of LJ potentials is a computationally expensive step. As a result, the simulation of long polymer chains by traditional MD is generally a slow process. Many numerical schemes have been applied to conquer this problem. For example, in order to save the compute time for tracking the change of the LJ potential in the system, a multiple step time (MTS) algorithms, namely rrespa developed by Sandia Lab, was applied to reach the longest relaxation time of C In this algorithm, the fast degrees of freedom, such as the stretching, bending and torsions of bonds, are integrated with a small time step, for instance fs, while the change of LJ interactions is included in slow modes, evaluated in a bigger time step, for instance 10 fs. Consequently, the most time consuming LJ tracking process in the simulation is significantly reduced. Even before the serious data collection begins, the traditional MD may still not be efficient enough to obtain an equilibrated structure in a short time. Thanks to a new Monte Carlo scheme for the fast equilibration of the PE chains, 50 A 300 ns-long simulation is now possible in MD simulation for united atom polyethylene melts using the equilibrated conformations obtained from MC as the input of MD. Monte Carlo simulations may provide an alternative choice for the dynamical simulation of realistic polymers. Compared with MD, MC is usually a more flexible algorithm. One can argue that fast motions, such as bond stretching and bending, only serve as the heat bath for the slower motions, e.g. torsions, which make the torsion dynamics a category of random motion without memory of its past history. 51 Thus this 38

55 slow motion can be modeled by Markovian master equation, which virtually opens the door for the dynamic simulation of realistic polymers via MC algorithms. Indeed, in MD simulation, at the length and time scale where a monomer starts feeling the environment, its motion will be eventually averaged by various encounters between itself and surrounding monomers. This averaging process leads to the stochastic nature to the segmental dynamics, which serves as one of most important basis of Rouse theory. The dynamic MC itself is usually built upon this stochastic nature of moves according to the Metropolis criterion naturally containing the detailed balance principle. As a result, including the bond stretching and bending into MC algorithms is no longer necessary, which greatly saves computing time. Furthermore excluding bond stretching and bending results in fixed bond length and bond angle, which, combined with the symmetrical torsion angles of many realistic polymers, enables a lattice MC algorithm to use the fast integer computation. The elemental moves of MC can be very flexible for the sake of computing speed; however there are also some important criteria for moves used in dynamic MC simulations for polymers. First of all, all elemental moves should have local and random nature. 51 It may be important to incorporate elemental moves which are able to introduce new bond vectors to the attempted segment as well. 5 To check a newly developed dynamic Monte Carlo scheme, averaging these stochastic moves over certain distance and time scale should lead to expected dynamics for well-understood systems 53. In other words, before further applications, one should be cautious and compare the results from dynamic MC with those from experiments and MD simulations. Secondly one should realize that the construction of current dynamic MC algorithms has not taken the hydrodynamic effects into account. Thus dynamic MC will only be able to reproduce 39

56 non-realistic Rouse dynamics instead of realistic Zimm dynamics for dilute polymer solutions. Here we would like to cover the principle of dynamic Monte Carlo method in detail. Actually Monte Carlo methods are used as computational tools in many areas of chemical physics. Although this technique has been largely associated with obtaining static, or equilibrium properties of model systems, Monte Carlo methods may also be utilized to study dynamical phenomena. 54 Although molecular dynamics (MD) simulations are able to describe the trajectories of individual atoms or molecules, it is usually a very inefficient process for the equilibration of samples. Polymers are long chain molecules with many repeating monomer units. Thus the characteristic length scales of such kind of molecules range from chemical bond length (a few Å), to the dimension of the whole polymer coil (a few hundred Å). The corresponding relaxation times may vary several orders of magnitudes. An efficient equilibration and the sampling of equilibrium properties by MD are very tedious. Monte Carlo may provide an alternative way to simulate the dynamics of polymer. The fast equilibration can be achieved with nonlocal moves, 55, 56, 57, 58 which may alter the conformation of a large portion of a chain at once. One may ask whether there is any direct relationship between the Monte Carlo dynamics and the true physical time of the experiment, which could be also obtained by a numerical integration of the equation of motion in MD simulations. Indeed, the application of the Monte Carlo method to the study of dynamical phenomena requires a self-consistent dynamical interpretation of the technique and a set of criteria under which this interpretation may be practically extended. In a recent publication, 59 certain 40

57 inconsistencies have been identified which arise when the dynamical interpretation of the Monte Carlo method is loosely applied. These studies have emphasized that, unlike static properties, which must be identical for systems having identical model Hamiltonians, dynamical properties are sensitive to the manner in which the time series of events characterizing the evolution of a system is constructed. These studies have underscored the importance of using a Monte Carlo sampling procedure in which transition probabilities should be based on a realistic model of a particular physical phenomenon under the consideration, in addition to satisfying the usual criteria for thermal equilibrium. Unless transition probabilities can be formulated in this way, a relationship between Monte Carlo time and real time cannot be clearly demonstrated. For polymers, any thermodynamic property can be calculated by averaging an observable A, say, mean square radius of gyration, over all configurations. 1 < A >= Z dxa ( x )exp[ β U ( x )] (.68) In the case of A as the mean square radius of gyration, x may be a set of r ij which is the vector from any one monomer i to another monomer j, where Z is the partition function: Z = exp[ β U( x )] dx, (.69) 1 here the Boltzmann factor Px ( ) = exp[ β U( x)] Z (.70) represents the probability density of one specific x at equlibrium. However generally the integral can not be solved analytically because the number of x approaches infinity and there is no way to solve the multiple integration of such high dimensional configuration space. Monte Carlo may provide a numerical solution this integral by 41

58 generating a random sample of configuration points. The master equation can be approximated by < A > i Ax ( ) exp[ βu( x)] i i exp[ βu( x )] i i (.71) Apparently when m approaches infinity, this is the numerial solution for the master equation (.68). Normally a uniform discretization is used in numerical methods. For example, still taking the radius of gyration as the example, dr j can be transferred into a sinθ j dθ j dφ j, where j is a integer from 1 to n, where 0 θ<π, 0 φ<π. However, if we uniformly discretize the θ and φ, the most of knots are on the surface instead of inside of this n dimensional cube. The number of knots inside this cube will be X=[(p-)(p-)/(p)(p)] n = [(1-1/p)(1-/p)] n =exp{n[ln(1-1/p)(1-/p)]} (. 7) When step p is approaching infinity, ln(1-1/p)=-1/p and ln(1-/p)=-/p, insert these two equations back, we get X=exp(-3n/p) clearly, when n is large, X will be extremely small. Hence the uniform discretization of individual axis leads to a seemingly unexpected nonuniform distribution of the knots in the super dimensional space. As the result, the random choices of X in Monte Carlo simulation are more efficient than uniform discretization for multiple integrations. The simple random choosing of X is called simple sampling method in Monte Carlo. However for polymer chains, not every configuration x has the same weight on calculating the <A> because they don t possess the same Hamilton U(x). In fact most of them are negligible due to their high energy. As a result, the most samples in simple sampling method have been wasted. If we can generate random samples according to some distribution P(x) equal to or close to weight 4

59 distribution of x at the equilibrium state, the discretized master equation then can be simplified to < A > i Ax ( i) exp[ βu( xi)]/ Px ( i) = 1 exp[ βu( x )] / P( x ) M i i i M 1 Ax ( ) i (.73) when P(x) = exp(-βu(x)). Only samples with significant weight are selected, and by directly averaging these samples, the <A> can be obtained. Sampling important configurations with some specific probability distribution is called importance sampling method in Monte Carlo simulations. However, we don't know how to obtain a distribution at least close to P(x) at the equilibrium to achieve the importance sampling. Remember that we cannot generate random samples but have to generate samples according to some rule which often makes favorable states. About 50 years ago, Metropolis and his coworkers 60 solved this problem by constructing a Markov process, which means that the evolution from configuration x1 to x depends on a transitional probability W(x1 x). By definition, in a Markov process the x1 and x should be independent. At the Equilibrium State, the flux out from x1 should equal to the flux into x1. Then equation P(x)W(x x1)=p(x1)w(x1 x) (.74) can satisfy this condition. This is the celebrated condition of detailed balance. Then it is identified by Metropolis and his coworkers that the detailed balance condition is verified if one uses W(x1 x)=min(1, P(x)/P(x1)). (.75) 43

60 The probability density is based on the Boltzmann measure ~exp(βu), and the resulting distribution is guaranteed to be the equilibrium distribution. Then, in the case of polymer, what has to be done is to set up a displacement rule, and record the energy E1 at x1 and E at x, using Metropolis algorithm to compute the transitional probability, W. This algorithm can be implemented by using a single uniformly distributed random number between 0 and 1, and the system moves from x1 to x under the condition that random number < W(x1 x). The only remaining liberty is to adjust the displacement rule. The time-honored procedure will allow about half of the moves are rejected. There is a dynamic interpretation of Markov process. The time evolution of this Markov chain can be characterized by the following master equation for the probability of x1 at time t. At the equilibrium state, clearly the left hand side equal to zero. Then right hand side is naturally detailed balanced condition. dp( x1, t) = [ W( x x1) P( x) W( x1 x) P( x1)] (.76) dt x1 x As a result, the P(x) eq is just the static solution for P(x, t) for this dynamic master equation. Hence an expectation value, say, dimension of a chain, according to their conformation x can be the average of either canonical ensembles or the time evolutions. The basic principle in statistic mechanics tells us that these two averages only equal to each other when a system attains the ergodicity. In simulation, this concept simply means that any configuration x1 can be eventually reached from another configuration x. We can t overemphasize the ergodicity in Monte Carlo simulations. Together with the detailed balance, ergodicity is the fundamental concept which not only insures that the simulation will converge to the correct probability density, but also serves as a 44

61 foundation of dynamic Monte Carlo. In Monte Carlo simulations, ergodicity can be broken in two ways: trivially, when the Monte Carlo dynamics only samples part of configuration space; viciously, ergodicity even can be broken when the algorithm is formally correct, but simply too slow. Ergodicity breaking of this former kind is relatively easy to be fixed, say, by considering a wider range of possible moves. However vicious ergodicity breaking sometimes goes unnoticed for a while, because it may show up clearly only in particular variables. The system can be stuck in some local equilibrium, which does not corresponding to the thermodynamically stable state. It always invalidates the Monte Carlo simulation. For molecular dynamics, the vicious non ergodicity also exists. In fact, Ergodicity has not been proved for any system that has been seriously investigated with molecular dynamics simulations. 61 Even abstract mathematical proof that no trivial accident happens does not protect the simulation from a vicious one. The absolute ergodicity can rarely be evaluated explicitly, and the mathematical analysis is useful only to detect the trivial kind of ergodicity violation. Very often careful data analysis and much physical insight are needed to assure the practical correctness of the algorithm. Several general measures in practice to check the ergodicity include Occasionally do very long runs. Use different starting conditions. For example quench from higher temperature/higher energy states. Shake up the system. Use different algorithms such as MC and MD Compare to experiment. 45

62 Here the conceptual difference between the equilibrium Monte Carlo and dynamic Monte Carlo is realized. In the former case, we essentially have unrestricted choice of algorithm since the only interest there is to generating independent configuration x distributed according to P(x). The temporal correlation is just a nuisance. However in the dynamic Monte Carlo, this correlation becomes the main object of our curiosity. One always can ask if there is any relationship between the Metropolis dynamics and the true physical time. Indeed, no equations of motion have even been seen in the discussions above. However, there has been a lot of discussion of this point and many simulations have been devoted to an elucidation of this question for, say, the hard sphere liquid. All these studies have confirmed there is a direct relationship as long as only local moves are used. The difference between the two approaches corresponds to a renormalization of time. But how the master equation of dynamic Monte Carlo is related to the physical dynamics of polymers? It is concluded 51 that when in a system the degrees of freedom can be split into two sets, a fast one and a slow one, which are only weakly coupled with each other, then even if the fast one possess its own intrinsic dynamics, say, Newtonian, the slow one should not have an intrinsic dynamics of their own. In another word, the fast dynamics just serves as a heat bath that allows thermally activated random jumps of the slow dynamics to occur. When the time scale between jumps is much large than the relaxation time of fast dynamics, the slow dynamics can be modeled by a Markovian master equation as in the master equation of dynamical Monte Carlo, where the variables representing fast dynamics disappear and the master equation only depends on the 46

63 variables representing the slow dynamics. The change of dp(x, t)/dt will only depend on the present state X only, which means there is no memory of the past history of the system. However it is not immediately clear that Monte Carlo simulation is also useful to describe self-diffusion of long flexible polymer chains in polymer melts. One sure still can argue that the fast processes (bond length and bond angle vibrations, etc.) act as a heat bath for the slower processes, such as thermally activated t, g+, g- jumps in the torsion potential. Nevertheless, since the time scale of these vibrations is only to 3 orders of magnitude smaller than the reorientation time of these jumps, it is clear that this argument holds only approximately. Furthermore, because most of Monte Carlo dynamics adopt coarse grained model, there may be more distortion in distance scale of these jumps, for which length scale traveled by atoms in such a conformational transition is of the same order as the smallest length scale in the coarse-grained model, about.5 Å for nnd Lattice in our model and about Å for the thermal version of the bond fluctuation model. Hence one can only hope for an accurate description of phenomena on much larger length scales. In practice, a critical problem is how to construct the moves in Monte Carlo simulation to simulate these random jumps even if they don t have intrinsic Newtonian dynamics. Saying it in a straightforward way, the more realistic these coarse grained moves are, the more accurate the model represents the true dynamics of the system. Like in the case of hard fluids, it is already proved that only local moves roughly correspond to the random local conformational changes. More importantly, all moves adopted in the dynamics Monte Carlo have to be at least almost ergodic. For the standard self avoiding walks with an 47

64 Figure.5. Common moves in Monte Carlo Simulations a) Verdier-Stockmayer local moves b) bilocal: reptation moves c) pivot moves, reproduced from ref

65 excluded volume interaction on cubic lattice, a variety of moves has been proposed (see Figure.5). But there are problems: neither the Verdier-Stockmayer algorithm 63 nor the slithering snake algorithm are strictly ergodic, even in a trivial sense, there are certainlocked-in configurations as shown in Figure.6, however, conversely if one does not start with this conformation, one can never reach this conformation. Hence the sampling of phase space provided by these algorithms is not complete. Although the resulting errors seem to be small in many cases of interest, 64 this problem should be kept in mind. The pivot move has been proved as ergodic algorithm in the trivial sense. 57 The newer version includes highly efficient double pivot moves 65 involving chains. However they are all non local moves, and may not be feasible for dynamic simulations. As a result we go back to local moves, and found another problem with the original version of Verdier-Stockmayer algorithm is an unrealistic slowdown 66, 67, 53 when only end rotations and the kink jump move are included, interpreting kink-jumps as exchange of two neighboring bond vectors, it is clear that new bond vectors are created only at the chain ends and must diffuse slowly into the chain interior to equilibrate the configuration. The modified version of Verdier-Stockmayer 68,69 solved this problem by including rotational moves. Hence even pure local moves can not guarantee the physical dynamics can be reproduced in the simulation once the interaction energy, say, excluded volume effect, is applied. While both the slithering snake and pivot algorithms relax the chain configurations much faster, they clearly do not correspond to any physically realistic 49

66 Figure.6. The lock-in conformation for local and reptation moves on a square lattice. 50

67 chain dynamics happening in polymer systems. Hence they can only be used in the fast equilibration of the system. However, slithering snake moves have been used in the simulations of the primitive path in the reptation model. 70,71 In fact it is known that slithering snake move is highly artificial, because no local rearrangements are allowed and density fluctuations can only occur at chain ends. The idea is that the snake corresponds to the primitive path of the reptation model and each monomer to a coarsegrained tube segment of an originally local model. The mapping of both levels of description is nontrivial. Furthermore, this kind of mapping must involve the density. Baschnagel and his coworkers 71 acknowledged that it is possible that the high densities used in the bond fluctuation model are in fact too large to correspond to any realistic volume densities of an underlying microscopic model under the condition of the nonlocal moves. Indeed as we will see in the following section devoted to the bond fluctuation model, the mapping procedures indicated that 5 C-C bonds corresponds to one BFL bond, which varies from to 10 lattice steps on underlying cubic lattice. These authors also failed to mention that in fact some global moves including slither snake moves violate detailed balance principle, which may make their conclusions tentative although recent work shows that detailed balance principle is way too strict for Monte Carlo simulations. 7 The brief review of practical implementation of different algorithms above is mainly limited to the simple cubic lattice, in which generally coordination number is 6 in three dimensions. Of course, BFM model is an exception, which will be discussed later. On diamond lattices, it is also found that single bead move is not enough to reproduce the Rouse dynamics. Both 3 and 4 bond moves are required. 73 Although dynamic 51

68 simulation can be done using pure local moves on cubic lattice, it has been realized if excluded volume is included, these local moves are too slow. As we mentioned, even if these local moves are ergodic in mathematical principle, slow dynamics may still lead to vicious kind of ergodicity breakdown. By construction of new lattices, the coordination numbers of lattice can be increased to larger number, for example, in nnd lattice, As a result, local moves can be speeded up and non ergodicity or incorrect slow dynamics may disappear due to this additional freedom. Furthermore, according to the series of dynamic Monte Carlo studies on different lattices, including high coordination number face centered lattice, 75 Kovac and coworkers discovered that results of multiple chain simulations on low coordination number lattices, such as diamond lattice by Kolinski 73 or cubic lattice by Verdier, 76 are not totally independent of lattice. In those cases, simple cubic lattice and diamond lattice systems have shown strong deviations from Rouse-like behavior of the chain-length dependence of the relaxation times and the diffusion constant, as the concentration of the system increases. The similar kind of effects has been found in Bond fluctuation model. Kovac and his coworkers 69 have identified that as the concentration of the system increases, the 90 crankshaft moves on simple cubic lattices and 4 bond motions on diamond lattices can be suppressed, while the face centered lattice, only using single bead moves, shows only moderate deviations from Rouse behavior. Hence it should be kept in mind that the different concentration dependence of moves at different length scales might lead to lattice specific dynamics. By far, it seems that there is no detailed analysis available in this aspect for bond fluctuation model. Although only single bead move allowed in bond fluctuation model and coordination number is 5

69 extremely high in this model, the moves still can be separated into different ranges. Whether high concentration specifically suppresses the relatively longer-range moves like they did in cubic and diamond lattice system is unknown. Therefore, sometimes it is essential that simulations be done by using a lattice model in which elementary motions of only a single length scale are needed to see how important the simple suppression of the longer length scale motion is in explaining the concentration dependence of the dynamics. Of course there is a probability that this suppression of long range local moves is physically realistic, and the concentration on different lattices corresponds to different physical densities. Finally it is should be noticed that the lattice simulation by Kovac and his coworkers again was criticized by Verdier upon the lattice constraints and the use of move rules. 77 It should be mentioned that while in most coarse grained Monte Carlo dynamics, the concentration of bead occupancy is an empirical adjustable parameter for the simulations of polymer melts. The Monte Carlo simulation based on realistic polymers, such as nnd model, 74 don t have concentration as an adjustable parameters because every beads stands for a realistic chemical unit. The lattice spacing is defined by the realistic bond length of these realistic chemical unties. The concentration hence is low and fixed corresponding to the realistic density of certain polymer melt at certain temperature. In the next section, several important dynamic Monte Carlo models will be introduced, and the main results of these models will be briefly discussed. 53

70 .5.1. Bond fluctuation model (BFM) Original bond fluctuation model 78 is an athermal model in which no potential for bond lengths and bond angles has been built in. Although a simple cubic lattice is used in BFM, the effective monomer no longer occupies one lattice site as those on other simple cubic lattices do. As Figure.7 shows, the monomer occupies a cubic cell defined by surrounding 8 lattice sites in 3 dimensions (4 in Dimensions). The effective bonds connecting the centers of these cubes can have lengths varying from, 5, 6, 3, 10 lattice spacing. By this combination of bond lengths, an excluded volume interaction is enough to ensure the non crossing of chains. In BFM, a total of 108 possible nextneighbor pairs can be identified, of which about 1 can be occupied simultaneously. For simulating the static and dynamic properties of polymer melts, an empirical volume fraction 0.5 is needed. 79 The thermal version of BFM has also been developed, in which a temperature dependent effective potential corresponding to different bond length and bond angles has been assigned so as the probability distribution of BFM bond angles and bond lengths overlaps the one of the realistic polymer. 80 However, probably because of high computational demands, the BFM s thermal version has not been widely used. Before discussing the result of BFM, several important definitions of mean square displacement are given as: g 1 (t): the mean-square displacement of the monomers in the center of the chains g l (t)=<(r N/ (t) - R N/ (0)) > (.77) 54

71 Figure.7. Sketch of the bond-fluctuation model of polymer chains on the threedimensional simple cubic lattice. Reproduced from ref

72 g (t): the corresponding quantity of g 1 (t) in the center-of-mass system of each chain g (t)=<((r N/ (t) - R N/ (0))- (R CM (t) - R CM (0))) > (.78) As we discussed in the Rouse model, g 1 (t) ~t 1/ when t<τ R and g 1 (t) ~t 1 when t>τ R, while the g (t)=g 1 (t) when t<<τ R and g (t)~ constant~<r g > when t>>τ R Similarly, for chain ends, we have g 4 (t)=<(r end (t) R end (0)) > (.79) g 5 (t)=<(r end (t) R end (0))-( R CM (t) - R CM (0))) > (.80) It can be derived from the Rouse theory, g 4 (t)/g 1 (t)=, when t<<τ R. When t>> τ R, g 5 (t)~4<r g >. 81 Of course, one of the most important quantities is MSD of the center of mass of polymer chains. It is given as g 3 (t)=<(r cm (t) R cm (0)) > (.81) Rouse theory has predicted that g 3 (t) always scale as t 1. The major conclusions of BFM include: Rouse theory is not consistent with the simulation results for the scaling at t<< τ R, which have showed that g 1 (t) ~ t 0.55 instead of t 0.5. From a simple scaling argument, Binder and Paul 51 rationalized this result by the effects of the excluded volume effect which is not screened out at the short distance scale. Once the chain reaches the distance scale where the excluded volume effect has been screened out, the scaling will go back to Rouse scaling as t 0.5. Thus when the concentration φ is only 0.05, which corresponds to the dilute solution, no Rouse scaling is found, however at higher concentration, Rouse scaling is recovered at the longer time or distance scale. This shows that simulation is a good supplement of the Rouse theory, 56

73 which is inadequate to catch all the facts by totally neglecting the excluded volume effects and hydrodynamic effects at all distance scales. Paul and his coworkers also found 79 that the relaxation times turn out scattered by fitting the autocorrelation functions, and by using diffusion plots, the τ R can be indirectly checked from several intersection points. For example, τ 3 is time at the intersection of g and g 3. According to the Rouse theory, τ R is only different from these relaxation time by a numerical constant. Hence, they found the scaling at the low concentration is corresponding to the Rouse model plus the effects of excluded volume effects in dilute solutions. For high molecular weight chains, the scaling moves away from the Rouse prediction with increasing concentration on the lattice. Finally an empirical concentration φ = 0.5 has been decided as the lattice occupancy for the BFM model to reasonably describe the polymer melts instead of polymer solutions mainly based on the scaling of the relaxation time, which is one of the biggest assumptions in BFM model. After the decision of the correct melt density, the immediate following procedure would be simulating rather long chains at φ = 0.5 in order to check reptation theory. The next figure shows the mean square displacement of the center of mass and the middle monomer in much longer simulations for much longer chains, which includes 00 coarse grained monomers. The major conclusions of Figure.8 include: Anomalous diffusion of center of mass at the short time scale has been found. Instead of the g 3 ~t 1, g 3 ~t 0.8 was discovered. As we already discussed in the section of the Rouse model, the local intermolecular interactions in Guenza s cooperative dynamics model 31 are responsible for this anomalous scaling. Probably the more exciting finding is that over a wide intermediate range of time, g 1 shows a scaling close to ¼, which is the prediction 57

74 Figure.8. The mean square displacements vs. time for N=00. Reproduced from ref

75 of the reptation. g 3 also shows a range with ½ slope, which is also predicted by the reptation model. However it seems the length of these ranges for g 1 and g 3 is not totally consistent. This inconsistency is especially pronounced for different τ e found form g 1 and g 3 curves respectively. According to the reptation theory, starting from τ e, g 1 and g 3 should start reptation, and show t 1/4 and t 1/ scaling respectively. Two τ e decided from g 1 and g 3 are different by nearly two orders of magnitudes. No reliable result can be obtained from the longer chains or longer simulation probably due to insufficient computing power at that time. So we think the result is not decisive in this case. The diffusion coefficients obtained in this method are plotted in Figure.9 with other results from the experiments and other simulations. It seems that all data are very close together. For a very short chain, BFM was used to test the anomalous diffusion of g 3 at the short time scale, the polymer in dilute solution doesn t show this anomalous behavior, while this behavior gradually shows up when the concentration increases. This may be consistent with cooperative dynamics approach 31 which tries to capture the multiple chain nature of polymer melts. Recently, since the large increase of the computing power, the BFM was used to examine much longer chains than those in Paul s work. Instead of using g 1, Kreer and his coworkers 8 used the cubic invariant g 6 to reexamine the diffusion results. g 6 is given as (.8) g 6 /t 1/4 is plotted versus time in this case. Of course, if the t 1/4 scaling over the intermediate time range predicted by reptation theory does exist, then for enough long polymer 59

76 Figure.9. The mean square displacements vs. time for N=00. Reproduced from ref

77 chains, g 6 /t 1/4 will reach a plateau. There is a plateau region for longest chains with N=51, which is about 14 times of N e. For N=18, a transition to the plateau shows up and disappears quickly. According to the reptation theory, τ e is the time when g 1 enters into t 1/4 region. Hence a value of τ e can be determined in Figure.1. However, in order to clarify the inconsistency in Paul s work, the characteristic relaxation times in reptation theory have been identified also with other g i curves. Remember what has been discussed in the section of reptation theory, before τ e, we have MSD as < r r 4k Tb 3πζ B ( Rn ( t) Rn (0)) >= ( ) t 3 4 ζ and τ e as τ e = πa 4 k Tb B by inserting τ e back to the t in the expression of the MSD, we obtain g 6 (τ e ) ~ g 1 (τ e ) ~ g (τ e ) = a /3. Then once the τ e is found, the tube diameter can be calculated easily. It is known that the expressions of autocorrelation function of both the Rouse and the reptation model have the exact same mathematical form as 8 < Re ( t) Re (0) >= Nb exp( p t / τ ) ee (.83) p π p: odd The only difference is the value of the relaxation time τ ee. Hence fitting the autocorrelation curve for both reptating or non reptating chains has the same procedure, one easy way is inserting the τ ee into t, the scaled autocorrelation function < Re ( t) Re (0) > 8 = exp( ) = p < R > p π e p: odd (.84) Taking this value, it is easy to find the τ ee for all samples with different molecular weight. Kreer and coworkers 8 also used the exact same data of N=51 to test another major theory for description of entanglement: polymer mode coupling 61

78 theory. It seems that the initial 1-φ(t) (φ(t) as normalized time correlation function of endto-end vectors) scales as t 9/3 and later it scales as t 3/8, which fit into the prediction of the polymer mode coupling theory. However one sample is not enough to decisively make a certain conclusion. Finally if all τ ee are plotted versus N, a transition from scaling N.31 to N 3.3 has been found from BFM. The diffusion coefficients scale as N - at a very long chain limit. All these fitting results are more or less consistent with the experiment results, although N - is not an experimental measured scaling, N -.. Beside the dynamical information, the basic static information, such as dimension of the polymer chains has been given by BFM. Chains approach Gaussian limit until N>100, which is about 4 times of N e in BFM. Some important results from other simulations will be reviewed very briefly next..5.. Kolinski and Skolnick s diamond lattice MC By using a dynamic MC simulation on a dense diamond lattice, Kolinski et al 6 have showed that reptation within a tube is not a necessary condition to observe D~N ~.4 and τ ee ~N 3~3.4. Reptation is observed only when a frozen environment surrounds the chain of interest. In a polymer melt, there is no separation of time scales between the chain of interest and the other surrounding chains; the polymer behaves much more like the classical Rouse chain. The anomalous g 3 ~ t 0.9 has been discovered at the short time scale, which is roughly consistent with BFM. Furthermore, the chain ends are more mobile than is predicted by the Rouse model with a uniform bead friction coefficient. Figure.10 shows that the ratio between perpendicular component and parallel component of chain movement along the constructed primitive path. Only in a frozen environment (shown as 6

79 Figure.10. Perpendicular component and parallel component of mean square displacements along the constructed primitive path versus time t. The solid diamonds shows the data for a single chain in a frozen environment. The open symbols are data from multiple chain MC simulations. Reproduced from ref

80 Figure.11. Mean square displacements versus time t for different chain length in MD, the inset figure is for the scaling of diffusion coefficient D. Reproduced from ref. 4 64

81 the solid diamonds), the anisotropic motion has been detected. Otherwise isotropic motions have been found in melts Kremer-Grest coarse grained molecular dynamics The molecular dynamics method used by Kremer et al 7 is not the traditional one including the chemical details. In this method, the coarse grained monomers are very weakly coupled to a frictional background and to a heat bath. The interaction potential is purely repulsive portion of L-J potential between all monomers with a strong attractive interaction between neighbors along the chain. Earlier, Kremer et al 4 noticed that the mobility of monomers near the end of the chain is much larger than near the center of the chain. Such end effects have not been taken into account in the original tube model. As a result, instead of averaging all monomer motions suggested by reptation theory, only the monomers at the middle of the chain are used to calculate monomer diffusion. From Figure.11, they found that the middle bead diffusion clearly shows the t 1/ scaling at the short time as theories suggested, and more importantly, for the first time, a clear t 1/4 scaling was detected for N>100, which is about 3-5 times larger than N e in such kind of simulation. However this exponent was challenged by the most recent simulations by Theodorous s 46 and our simulation in this dissertation based on realistic polymers. It is unlikely that chains can be confined as perfectly as in the strict reptation model at such a short chain length. It should be noted that in this coarse grained molecular dynamics simulation, the short time monomer diffusion does not depend on the chain length as the Rouse theory suggests, and in reality such a phenomenon means the inner monomers do not feel the 65

82 Figure.1. The relaxation time versus chain length, the inset figure is for the scaling of normal modes. Reproduced from ref. 4 66

83 chain ends. In Figure.11, they also found that the relaxation time shows Rouse scaling t for short chains and reptation scaling t 3 for long chains. Of course, just like earlier simulations using BFM, this result from molecular dynamics did not found t 3.4 obtained from the experiments. This is the intriguing facts of simulations Theodorou s Atomistic MC and MD simulation Because of the significant computational cost, almost none of earlier simulations is capable of dealing with the realistic polymers. The process of coarse graining sometimes leads to ambiguities, such as purely empirical lattice occupancy in BFM and the diamond lattices by Kolinski et al. The empirical intermolecular neighboring potentials used in Kremer-Grest coarse grained molecular dynamics are also subjective to argument. It is sometimes arbitrary to match the properties calculated from the coarse grained model to those obtained from the experiments by using realistic polymers. For example the usual mapping process involving comparisons of the characteristic ratio between the realistic polymer and coarse grained chains in simulation is not appropriate. It has been shown that polymers with the same persistence length may display tremendously different dynamics and possesses different material properties. Thus it is always desirable to directly simulate the dynamics of realistic polymers. Not only the comparison with the experiments will be more convenient, but also it may lead to practical applications of molecular simulations. Theodorou and his coworkers 46 have taken advantage of their newly developed double pivot move algorithm 65 to achieve the fast equilibration of polyethylene long chains in Monte Carlo simulations. The equilibrated conformation can be directly input into the molecular dynamics simulations. 67

84 The dynamical information then can be reliably obtained in a more straightforward way than dynamic Monte Carlo. Their results have showed a possible transition from Rouse to reptation starting from C 00 in the following figure. Clearly more simulation data for longer chains are needed to identify the finger print of the reptation, i.e. g 1 ~ t 1/4. A natural advantage of realistic simulation is that the macroscopic properties obtained can immediately be compared with the experimental results. The simulation shows that a range of D~M -1.7 for oligomers due to the free volume effects of abundant chain ends, which is followed by a range of D~M -1. This can be a Rouse region according to the authors; however this result is not consistent with the pulse gradient NMR experiments. 83 Finally the authors 46 calculated the friction coefficients by reverse of the diffusion coefficients using both the Rouse model and Reptation model according to the following equations: k B T ζ = for the Rouse model, and DN = k B Ta ζ for reptation model. 3DN < R > Between C 100 and C 150, the friction coefficients don t change, which means that they fit the Rouse model very well. Again, it is not consistent with the experimental data. 83 The viscosity was calculated with the following equations: RT < R > η = ρ for Rouse and 36MD RT < R >< R > η = ρ for reptation. 36MDa 68

85 Figure.13. Mean square displacements of middle monomer g 1 versus time t for different chain length. Reproduced from ref

86 Figure.14. The center of mass diffusion coefficient vs. chain length by MD. Reproduced from ref

87 .5.5. Other important simulation models There are many other important models for efficiently simulating polymer chains, for example, a fast algorithm by Smith et al. 106 using discontinuous potentials, a model using the transferable force field 1 which has been proved as an exceptional candidate for simulating asymptotic behaviors of n-alkanes towards high polymers. Although they cannot be reviewed in details here, these methods should be appreciated. 71

88 CHAPTER III SIMULATION METHOD As we mentioned in the Chapter II, the simulation of long chain polymers often involves several orders of magnitude of times and length scales. Without efficient simulation methods, it is almost impossible to do modeling over this vast range of time and length scales. In Monte Carlo (MC) simulations on lattices, by discretizing conformational space into lattice sites, the degree of freedom of polymers, such as a bond stretch at the short times scale, has been discarded or tremendously reduced. Hence MC has been a popular choice for polymer modeling since the 1960's 66. The Gaussian statistics of polymer melts naturally led to the coarse graining of several monomer units into a random walk unit. Although both measures gained tremendous speed advantage in simulations, they might have brought out potential problems too. The reduction of the degree of freedom may lead to ergodicity problems or unrealistic dynamics. The coarse graining of monomer units results in loses of the information of chemical details and may bring ambiguity on the real material properties on lattices. In order to avoid these potential problems, the high coordination number is needed to maintain enough degrees of freedom for achieving ergodicity and representing the realistic dynamics. It is also desired that the ability for assigning the coarse grained beads back to corresponding chemical units is preserved. 7

89 The recently developed nnd lattice, 74 based on an underlying diamond lattice, have 1 first-shell coordination numbers, in comparison with the 4 of the diamond lattice and 6 of the simple cubic lattice. The underlying diamond lattice is an excellent representation of the saturated carbon covalent bonds. The nnd lattice is a distorted cubic lattice with the 60 angle between any pairs of axes. The coarse grained polymer chains on explicit nnd lattices can be reverse-mapped back to the underlying diamond lattice as realistic polymer species. Both short range and long range potentials used on the nnd lattice define the kind of polymer species Hence the corresponding simulation of realistic polymers on a nnd lattice is generally more relevant to practical interests. The first step of constructing a nnd lattice is mapping polymer chains to diamond lattices in Figure 3.1 (a). Taking a polyethylene chain as an example, each diamond lattice site represents a methylene unit; the lattice step length comes out naturally as C-C bond length. If every second methylene unit (shown as opened sites in the following Figure 3.1) is taken out and its mass is merged to neighbors, the rest of the lattice sites forms a distorted cubic lattice in Figure 3.1 (b) with the coordination number of 1. Table 3.1 includes the collection of all these 1 vectors for the 1 coordination numbers of the first shell, and from now on, the vector from x j, y j, z j to x j, y j+1, z j will be abbreviated to 0+0 etc. 73

90 Figure 3.1. Diamond lattice and nnd lattice. Reproduced from ref. 87. a) Diamond lattice with sites alternately represented by open and filled pheres. b) nnd lattice by omitting the open spheres from a). 74

91 Table 3.1. The coordinates of the 1 nearest neighbors for bead j shown in Figure 3.1b. Reproduced from ref

92 Taking 1.53 Å as C-C bond length, the nnd lattice spacing will be.5 Å because the diamond lattice has the bond angle of Every bead on nnd lattices represents a two-carbon unit on the main chain of a polymer. The occupancy of the lattice can be calculated from the realistic density of a polymer. For polyethylene, generally, it is around 18%, in comparison with the 50%-100% occupancy of the abstract coarse-grained polymer chains on various other lattices including BFM, 79 Kovac s Face centered cubic lattice, 75 Kolinski s diamond lattice, 73 Pakula full occupied lattice. 88 Since on the nnd lattice, for polyethylene, every bead is an ethylene monomer, they can no longer be taken as a pure random walk unit. Instead, both intra and intermolecular energy has to be built into the model to achieve a realistic description of the polymer. This built-in realistic energy naturally results in a thermal model which explicitly includes the temperature, while most other lattice models, which implements random walk or self avoiding random walk, are athermal models simulating polymers at an infinite temperature. The simulation speed is a price to pay for the realistic potential in our system. However, low lattice occupancy at least partially compensates the cost of the realistic potential. Low lattice occupancy also minimizes the danger of the concentration dependence of the particular algorithm discovered by Kovac and coworkers 75 on a high occupancy lattice. The intra-molecular (short range) potential can be introduced by direct applying rotational isomeric state model 89,90 on the underlying diamond lattice. For polyethylene, the conformational partition function Z can be expressed by the following equation including all combinations of torsion states for all bonds of a polymer chain. 76

93 (3.1) Where the columns from the left to the right are t, g+, g- of bond i respectively, the rows from the top to the bottom are t, g+, g- of bond i-1 respectively. σ is the statistical weight of the second bond (bond i) in the conformation corresponding to tg+, tg-, g+g+, g-g- of i-1 and i, σω is the one corresponding to g+g-, g-g+. However, if one has to recover the original underlying diamond lattice from the nnd lattice, there is an ambiguity since two underlying diamond lattices can exist for one nnd lattice. Once the underlying lattice is specified, the true conformation of chemical bonds can be determined by identifying the category of the pairs of two consecutive nnd bond vectors as shown in Figure 3. (a). The Table 3. is a typical library used in the program. The reverse of the bond denoted as rev in the table. It has been forbidden in the simulation since the excluded volume interaction has been strictly observed. Not only the excluded volume was implemented on the nnd lattice, it was also done for underlying diamond lattice in order to avoid two methylene units occupying a same site once the coarse grained polymer chain was mapped to united atoms. Indeed, some pairs of nnd bond vectors represent unrealistic collapse of two carbons onto one diamond lattice site. The collapse is denoted as col in the table, and it is shown in Figure 3. (b). 77

94 Figure 3. Torsion and Collapse on nnd lattice. Reproduced from ref. 87 a). Two left torsion angles determine the angle i-, i-1, i, and right torsion angles determine the angle i-1, i, i+1. b) The triangle depicted the configuration of unrealistic collapses. 78

95 Table 3.. Conformations at the two C C bonds to bead j in terms of the vectors i j (rows) and j k (columns). Reproduced from ref

96 By specifying the underlying lattice, the ambiguity on the configuration and the collapse problem has been solved by the library in Table 3.. Then according to the partition function, the probability of bond i at bond state α, p α,i can be calculated using conventional partial derivatives. 91 On the nnd lattice, the real bond is associated with coarse grained beads. Hence instead of using real bond index i in p α,i, the bead index j is used, and there are real bonds connected to this bead, the proceeding real bond is denoted by -, and the following real bond is denoted by +. Then the probability p ± k,j, denotes the probability that the real bond preceding + or following - the jth nnd bead is in state k. Two successive nnd vectors, i j and j k, specify that the two real bonds ( j- and j+) adjacent to the jth nnd bead are in states γ and δ, as depicted in Figure 3.3. ± Similarly, q kl, j denotes the conditional probability that the C C bond j ± is in state l, given that the proceeding C C bond is in state k. Hence, the probability of a conformation may be expressed by (3.) A new conformation is produced by moving one or several beads, the new state due to this movement is denoted as *. Moving the chain end, i.e. the 1st bead, directly change the bond state of - and + adjacent to bead, as a result, the conditional probability of 3- adjacent to bead 3 is modified too. The probability of moves then can be directly calculated according to these changes from P new / P old. Of course, any move resulting in the reverse of nnd bonds or collapse of inexplicit Cs will be immediately rejected. 80

97 Figure 3.3. Schematic representation of a section of the coarse-grained chain on the nnd lattice, along with the detailed counterpart on the underlying diamond lattice. The two successive nnd bond vectors i j and j k, defined by the three nnd beads i, j, and k, determine the rotational isomeric state adopted by the two real bonds, j- and j+, to and from the central bead, j, to be γ and δ, respectively. Reproduced from ref

98 (3.3) Similarly, moving the bead, 3 or further inner bead results the 3 corresponding formulas below (3.4) (3.5) The previous studies 91 found that exclusion of collapses tremendously slows down the simulation. The possible reason will be discussed latter in the section of the developing dynamics Monte Carlo on nnd lattices. In order to simulate the dense system, such as polymer melts, it is necessary to include the long-range intermolecular potential, which produces the correct structure, characterized by pair correlation function of the system. In the construction of this long range potential, the directly bonded bead pairs are ignored because they have been taken care of by RIS based short-range potential discussed above. 84 All other pairs in the system should be identified and the interaction between them can be calculated using conventional Lennard-Jones (L-J) potential. However, only at the ideal limit, all pairs have to be considered, in practice; usually a cutoff distance is used as a compromise between the computational speed and the accurate description of the system. The 8

99 repulsive portion of L-J potential at the distance less than the lattice spacing,.5 Å, is effective modified to the hard potential by not allowing the double occupancy of a same lattice site. The typical L-J potential is depicted in the following figure. The 6-1 potential in the second row of the equation is commonly used to describe L-J potential, σ is the collision diameter, at which the potential equals to 0, and ε is the well depth in Figure 3.4. The x axis of the figure is scaled by the σ. Both σ and ε have to be chosen empirically because theories are not advanced to the level for determining these parameters for monomers by measuring material properties. However for small molecules, such as ethane or ethylene gas, these parameters can be calculated by measurable properties. 9 For the repeating units of polymers, using L-J parameters close to these gases is the common practice in molecular simulations. Here one educated conjecture would tells that the range of L-J parameters is between the ones of ethane (ε/k=30, σ=4.4 Å) and the one of ethylene (ε/k=05, σ=4. Å). The discretization procedure of the continuous L-J potential lead to the 3 way potential on the nnd lattice as equation (3.6) shows. The discretization involves Mayer function f, which is e βu(r) -1. The integration of this function over the space has been commonly used in the evaluation of second viral coefficient for imperfect gases. (3.6) 83

100 Figure 3.4. Continuous L-J potential (the curve) and discretized L-J potential for nnd lattice (bars) Reproduced from ref

101 In order to condense the potential function all over the space onto the lattice sites, the first step is to obtain the cell averaged Mayer function for every lattice site relevant to the origin. The cell averaged Mayer function is defined as (3.7) In calculation of this cell averaged Mayer function, one reference monomer is fixed as the origin of the lattice, another monomer can be sitting anywhere in the targeted unit cell on the nnd lattice. The center of this unit cell is the very lattice site where this monomer occupied in the simulation. Because the nnd lattice is a distorted cubic lattice, not every neighboring site has the same sphere symmetry regarding to the origin, even if they are in the same neighboring shell. Hence for every neighboring shell, sites with the same sphere symmetry are grouped together; the shell averaged Mayer function then can be obtained as the arithmetic mean of the different groups of cell averaged Mayer functions. Once the shell averaged Mayer function is obtained, according to the definition, the shell averaged interaction energy is calculated according the following equation. (3.8) Table 3.3 shows the estimated interaction energy values for the every shell according to this discretization procedure. In Figure 3.4, this discretized L-J potential is shown as the horizontal bars. Generally only 3 shells are used in the nnd simulation because as the figure shows, the magnitude of attractive tail of 4 th and 5 th shells is small. In fact, to produce the correct structure of the liquid, repulsive portion of L-J potential is enough in most cases

102 Table 3.3. The Average Mayer Functions and the Converted Effective Interaction Parameters for the Indicated Shell Vectors. Reproduced from ref. 84 (Temperature at 370 K) 86

103 The attractive portion of L-J is only for cohesive nature of substance. In most nnd simulations, an attractive tail (usually on the 3 rd shell) is usually included because the structure of liquid is not the only interest of nnd simulations. Other important material properties, such as the solubility parameter of a polymer, are also targeted in nnd simulations. This property can also be used for evaluating the correctness of the L-J parameters. Another strategy of the evaluation of the L-J parameter is also based on the very cohesive nature of the polymer melt. By increasing one side of the periodic boundary of a box to the infinity, a polymer film can be obtained. The L-J parameter can be obtained by making the plateau density at the film center approaching to the realistic density. 86 With both short-range interaction and long range interaction selected, Monte Carlo simulation can be started as long as the algorithm for bead movements is constructed. Because of the high coordination number and low lattice occupancy, the nnd lattice naturally becomes a good candidate for the dynamic Monte Carlo algorithms. The single bead move algorithm is a nature choice, which allows a bead moving to any unoccupied site as long as the chain connectivity is maintained and no-collapse rule is implemented. The basic idea is depicted in the following figure in D, a single bead, such as 0, 3, 5, 10 can be moved to 0, 3, 5, 10 respectively. Thanks to the high coordination numbers of nnd lattice, not only the traditional Verdier-Stockmayer moves such as end bead free rotation (0 0 ) and kink jump(3 3 ) are easily reproduced, but the moves similar to 90 crankshaft (10 10 ) and 180 crank-shaft (10 10 ) are also included. However, by forbidding collapses of the implicit C units, many of these possible single 87

104 Figure 3.5. Single bead moves and possible two bead moves depicted analog to cubic lattice, modified from ref

105 moves are not allowed, 91 and it is especially true for a group of crucial relaxation, such as single bead crank-shaft moves. According to Clancy et al, 91 as depicted in Figure 3.6, forbidding collapses results in some locked conformation for crank-shaft moves. One example of this locked crankshaft conformation is depicted in the figure, the bead 4 can t move to 4, 4, or 4. Every these new position leads to a collapse! As a result, Clancy et al developed an inner chain pivot algorithm (note, this should be differentiated from the common pivot moves developed by Sokal et al 57 ), which allow two or more beads to move simultaneously by exchanging the three or more nnd bond vectors respectively. Clancy s idea is very simple. For example, in order to move three beads together at the same time, four bond vectors connected to these 3 beads will be affected. There is a constraint in moving these beads, which is to maintain the connectivity. This constraint can be expressed in the following equation where l is the bond vector: l 1 +l +l 3 +l 4 = l 1 +l +l 3 +l 4 By reversing the sequence of these bond vectors, this constraint requirement is easily fulfilled. The idea can be expressed in the following: l 1 +l +l 3 +l 4 = l 1 +l +l 3 +l 4 =l 4 +l 3 +l +l 1 which means that the relationship between the new bond vectors and old bond vectors is: l 1 = l 4, l = l 3, l 3 = l, l 4 = l 1 This is depicted in Figure After exchanging the bond vectors, bead 4 and 5 move to the positions of 4 and 5. Actually this idea is identical to the Kranbuehl-Verdier algorithm in the simple cubic lattice, which was criticized by Hilhorst et al. 53 Note, in Figure 3.6, the combination from 1-6 can t move (the bond vector is 89

106 1,1,5,9,3,), even with Clancy s rule. From atom 3-6, the bond vector is 5,9,3, the reverse bond vector is 3, 9, 5 which also lead to a collapse. From atom -5, the bond vector is 1,5,9, the reverse bond vector is 9,5,1, which lead to a collapse again. (9,5 is always a collapse combination) For the higher bead number of pivot move, as long as reverse the bond vector, 5->9 will become 9->5, the collapse will refuse the any move unless separating 5 and 9 vector. Indeed, by just changing the order of bond vector sequence, the certain conformation (represented by one or more bond vector) will never reach other conformation until it diffuses to chain ends. Although no ergodicity problem exists in this algorithm because certain conformation still can reach other conformation, there is a direct dynamic consequence. As we mentioned earlier, the correct dynamic Monte Carlo should present the randomness nature of torsion dynamics. The inability of transforming certain conformation into another possible conformation locally doesn t represent this randomness. In other words, the dynamic Monte Carlo moves have to be as real as possible. Although pivot algorithm was always combined with single bead move in our program, the former one completely dominates the latter one due to its larger scale. What was lacking in pivot moves seems to be the rotational moves which are able to introduce new bond vectors locally to relax the polymer configuration. 77 In cubic lattices, it is usually called 90 crankshaft moves. Another problem of pivot move is its length scale motion. In the program, pivot move usually extended from beads to 6 beads, which may not be considered as a local move. In other words, the dynamics up to 6 bead range can be distorted. The larger scale of motion of pivot moves results in one more problem, which is the possible chain crossing. Chain crossing certainly is not physical in reality. 90

107 Figure 3.6 Single bead moves strains due to the forbidding of the unrealistic collapses, modified from ref

108 The problems of using pivot moves in simulating polymer dynamics motivated us to design a new class of move, which has to fulfill several requirements for the proper description of polymer dynamics. First of all, the algorithm should use moves which are random, local and realistic. Otherwise the final results obtained from the simulation may only represent the dynamics of a particular design of moves instead of the realistic dynamics. One bead move involving two bonds is a natural choice of elemental moves in almost all dynamic Monte Carlo simulations. However as we mentioned before, if one bead move excludes collapses on nnd lattice, it is often not fast enough in terms of computing speed. On the other hand, adding two-bead move involving three bonds has been a common practice in other dynamic Monte Carlo simulations. The two-bead move not only improves computing speed, but also has a realistic basis in the cooperative motions of polymer segments. 94 However, introducing two-bead move on the nnd lattice is not as straightforward as on cubic lattice, where combinations of moves are very limited and results can easily be visualized as 180 and 90 crankshaft moves. nnd lattice doubled the coordination number of cubic lattice and introduced more versatile destinations for a single bead move. It was expected that two-bead move on the nnd lattice will introduce much more categories of moves. By mutating all possible vector combinations of any two neighboring beads using a computer, we found there are total 6 categories of two-bead moves on nnd lattice. Additionally two-bead move naturally incorporates single-bead move by its definition because all possible destinations of neighboring beads included. It was found that there are total 948 independent conformations for two neighboring beads on the nnd lattice as shown in Table 3.4. Among the 6 categories of bead moves, 4 of them include out-of-plane (rotational) 9

109 moves. The last category in the table can have as many as five different rotational moves which enable the new bond vector to be introduced. Two bead moves on the nnd lattice correspond to the change in torsion angles at five to eight C-C bonds on the underlying diamond lattice. There was a concern whether both single bead move and two bead move occur at the same time scale. 95 It has been found later that the Monte Carlo dynamics have neither mixture nor lattice dependence of the moves, which strongly suggested that there is no distortion of dynamics due to the incorporation of the two bead moves, as long as the occupancy of the lattice is not too high. 96 On the nnd lattice, the typical occupancy is about 0%, which is way below the concentration that may lead to the deviation of the dynamics. Thus the incorporation of two bead moves is reasonable based on earlier observations. Two-bead move naturally incorporates the out-of-plane motions similar to 90 crankshaft moves on cubic lattice, which easily introduces new bond vectors locally. It provides many more destinations for moving beads than internal pivot moves. 91 This algorithm is also expected to more closely represent the stochastic nature of microscopic motions in polymer melts because there is no artificial rule of motions involved. All that a bead does is to randomly chooses a new location in all of possible neighboring sites. By the way, a number of the moves, which may lead to the nnd collapses i.e., the overlap of implicit methylene units, have been discarded and have not been counted into the possible alternative positions. In Table 3.4, a portion of possible moves have been plotted for a convenient visualization. The categories in the upper row allow a limited number of out-of-plane motions, while the categories in the lower row don t have out-of-plane motions. Generally compact configurations have more alternative moves 93

110 Table 3.4. All possible destinations of any two neighboring beads. Name Total Total number numbers of Examples (bond vectors) of alternative groups in positions this category All trans 1,1,1;. 1 1 trans 1,1,3; 3,1,1; 1,3,1; 3 48 Triclinic 1,5,3; (3!) Rotation Monoclinic + Rotation 3 group rotation 5 group rotation 1,5,9; 5,9,1; 5,1,9; 9,1,5; 3,1,7; 3,7,1; 7,3,1; 1,3,7 1,1,1; 1,1,1; 1,1,1 1, 5, 7 (3!) 3,3,7; 1,1,8; 1,5,11;5,11,1; 11,1,5; 5,1,11; 3,5,9; 5,3,9; 5,9,3; 9,3,5; 3,7,4; 1,7,; 11,7,1; 9,7,10; 1,1,6; 6,1,1; 1,6,1; 5,9,4; 4,5,9; 9,4,5; 5,7,6; 6,7,5; 8 (in groups) 18 9 (in groups) 4 10 (in 3 groups) 4 1 (in 5 groups) 1 SUM

111 available, while the totally extended configuration have no alternative move except chain ends. By checking into every individual category of moves, possible chain crossings can be identified, which involve a first shell neighboring bead of both initial and resulting conformations from another chains. By checking the moves involving this kind of neighboring beads, the chain crossing can be eliminated. In fact, due to the LJ potential, the long range first-shell neighboring pairs are extremely rare due to the highly repulsive interaction in such a short distance. Skipping the check of chain crossings seemingly didn t result in a much different dynamics, but in any case, the following reported data are all from the program the enforces the check. This kind of check can also be visualized in the table. The red bead is a long range first shell neighbor which may lead to chain crossing. In the current simulation, for a single MCS step, there is /3 probability for a single bead move and 1/3 probability for a two bead move. Hence there is a 1:1 opportunity for every bead to be moved by either single bead or two bead moves in every step. The acceptance of single bead moves and two bead moves in our simulation is judged by Metropolis rules explained in Chapter II. Before data could be collected, all simulations have been run long enough to reach the equilibration. The signs of reaching equilibration include: the stabilized average polymer dimensions, the large displacement of the center of mass comparing to the radius of gyration, and the nil of autocorrelation functions of end-to-end vectors. All simulations were conducted on typical desktop computers. 95

112 Table 3.5. The illustration of six classes of two bead configurations with different numbers of alternative moves. The sequences of five numbers under every two bead move represent the bond vector connecting nnd beads, in which three underlined middle numbers represent three changing nnd bond vectors after two bead moves. These sequences of bond vectors are followed by the torsional states on the underlying diamond lattice before and after the two bead moves. 10 alternative positions ttttg + g - tg g - tg + tg + g + tg - 8 alternative positions g - tg + ttg - g - g tg + tg - g + ttt 9 alternative positions tttg + g + g + g - t g - g - g - g - ttg + t 1 alternative positions tttg+g-g+g-g g-tg+g+g-g-g+g- 3 alternative positions ttttg + ttt g + tg - tttg + t 6 alternative positions g - tg - tg - tg + t ttg - tg - tg + t 0 alternative position ttttttg + t ttttttg + t 96

113 A successful algorithm should fulfill the ergodicity requirement. In this aspect, for nnd lattice, the coordination number is 1 compared with 6 of a Cubic lattice. For SAW chains, the lattice with high coordination number has fewer ergodic problems due to frozen conformation which cannot be reached or escaped from if only implementing ordinary local moves. Ergodicity is broken at this point because either other chains can not sample this configuration space or this chain can not sample other configuration spaces. 57 This kind of problem is less relevant to our simulation. By introducing interactions, the polymer chains on nnd lattice actually are not the same as a typical SAW chain. At the typical melting temperature of polyethylene, the short range interaction and long range interaction of polyethylene generate very few first shell contacts on nnd lattice, typically 1 or pairs observed at each MCS. A configuration as compact as the frozen configuration in SAW cubic lattice 57 virtually has nil probability to seen at 453K. However this still doesn t guarantee there is no ergodicity breaking in our simulation. Sometimes even if an algorithm is formally correct, simply being slow may still lead to ergodicity breaking as we discussed earlier. It is known that the absolute ergodicity can rarely be evaluated explicitly by math, no matter for MD or MC. Thus comparing to other algorithms holds significant importance in discussion of the results. Our algorithm based on realistic polymer also enables us to compare our results with the experiments, which we consider is the most important judgment of a newly developed algorithm. 97

114 Summary A new bead move for Monte Carlo simulation has been designed to reflect the true randomness of local torsion dynamics on nnd lattice. The rotational moves are included, which introduce the new bond vectors to the local conformation without going through the chain ends. All possible configurations of two consecutive beads on nnd lattice can be visited locally and there is no artificial rule of the moves. 98

115 CHAPTER IV CROSSOVER TO THE ENTANGLED DYNAMICS: MONODISPERSE POLYETHYLENE MELTS As already discussed in detail in the last chapter, when our simulation calculates short range intramolecular interaction, the beads simply represent the explicit methylene units themselves. However when long range interaction is needed, the mass of implicit methylene units is then merged into their explicit neighbors which are directly connected together on the lattice. 84 In this case, a bead represents an ethylene unit. On nnd lattice, the short range interaction of chains, extending within the range of the nd nearest connecting beads, is based on the torsion energy described by RIS model. The 1st order and nd order energies used in this work are Eσ=.7kJ/mol, and Eω=14.6kJ/mol respectively. In the simulation, not only the nnd lattice site cannot be double occupied by beads, but also the underlying diamond lattice don t allow the double occupancy of implicit methylene units, termed as nnd collapse, 86 which results from the specific arrangement of consecutive torsion states in a short range leading to the overlaps of implicit methylene units. Although it doesn t guarantee there is no overlap among underlying hydrogen atoms, the elimination of nnd collapses indeed makes our model more comparable to the united atom model. For the rest of the intrachain and interchain pairs of beads in the system, the long 99

116 range interaction is described by a discretized Lennard-Jones potential as shown in equation (4.1). This long range interaction is essential to reproduce the local structure of multiple chain systems. U LJ 1 6 σ σ = 4ε (4.1) r r In the current work, ε/k = 185K and σ = 4.4Å. Based on these two parameters, the discretized U L,J can be calculated. Only those of the first neighboring shells, which represent the repulsive part of the LJ potential, are adopted in our simulation as the following: u 1 = and u = kj/mol. The interactions beyond the nd shell are negative numbers, which may help in producing the cohesive nature of polymers. However, for our NVT Monte Carlo simulation, the cohesive nature is automatically realized due to the dimension of the simulation box. Furthermore, besides providing cohesive nature to the material, attractive energy is not an important factor in determination of the structure of a material. 93 It is also good to have a strongly repulsive 1st shell potential, which virtually eliminates the long range collapses, although they are rare between the long range pairs of the implicit methylene units. The box size used in the simulation is steps (50Å 50Å 50Å), as long as its size exceeds twice the root mean square radius of gyration, <s > 1/ 0. Otherwise, the box size increases identically in all directions until it meets times of <s > 1/ 0 for relatively long chains. The temperature used in all simulations is 453K. Once the box size, temperature, and chain length are decided, the number of chains in the box is a fixed number depending on the experimental density. 97 After the chain number is obtained, the bead occupancy can be calculated accordingly. About 17-18% of lattice sites are 100

117 occupied in our simulations for polyethylene depending on the realistic density Single Chain The statistic and dynamic characteristics of a single isolated chain based on our simulation are presented in Figure 4.1. Only the RIS interaction and repulsive portion of L-J potential has been adopted. In the discrete version of L-J potential for our lattice, this corresponds to choosing only the first two shell interactions. Every data point is an average of 3 to 4 replicas. All slopes shown in Figure 1 are based on data regression. The mean square of radius of gyration, <s >, scales with the molecular weight to the power 1.19, very close to the theoretical predication, 1., for the expanded chains in dilute solutions. Because the dynamic Monte Carlo is based on stochastic monomer motions without hydrodynamic interactions, instead of the Zimm dynamics, the Rouse dynamics is expected if the algorithm is correct. 51 However, our results show a finite chain length effect. That D is inversely proportional to M, which is predicted by Rouse theory, can only be realized for longer chains. For chains shorter than C00, D approaches to the slope 1 asymptotically. This finite chain behavior is seldom discussed in the classic theory of dilute solution, although it may not surprise experimentalists and simulators dealing with oligomers in a bulk state. It is known that for oligomers or short polymer chains, the glass transition temperature T g is the function of M. It increases with M until the degree of polymerization reaches certain value. This fact indicates the segments in shorter chains experience less friction than those in longer chains, which naturally leads to an extra chain length dependence for the diffusion coefficient D under certain limiting 101

118 <s > (nm ) ± D (nm /millionmcs) -1.0± M Figure The diffusion coefficients D and mean sq. radius of gyration <s > of a single chain plotted against M. 10

119 chain length. Therefore this result is reasonable considering a realistic polymer is simulated, while the theoretical prediction is based on the hypothesis assuming the segmental friction coefficient is independent of chain length. Although our simulation corresponds to diluted chains in their good solvents without hydrodynamic interactions, the chain length dependence of the friction coefficient appears in our results as well. The underlying reason of this extra dependence includes non-gaussian behaviors 98 or in the melt, the extra free volume at chain ends. 97 This issue will be addressed again later when we discuss polymer melts. In Figure 4., the relaxation time τ ee, is obtained from fitting the auto correlation function of end-to-end vectors with the following equation with τ ee as the only parameter: < Re ( t) Re (0) >= Nb p: odd 8 p π exp( p t / τ ee ) (4.) The equation was derived from the Rouse theory and has been found identical to the one derived from reptation theory. 18 The resulting τ is plotted against chain length M. The regression shows that τ scales with M to the power.05, which is close to the Rouse prediction,, but a little lower than., the theoretical prediction allowing excluded volume effects. Some may still argue that the equation 4. is not valid for longer chains here because the Rouse theory always assumes Gaussian statistics which is absent for a single chain in its good solvent, in which <s > scales with M to 1. instead of 1. Although this is a legitimate argument, approximating a diluted long chain to the Gaussian distribution was acceptable because the final scaling results are not very sensitive to this approximation as seen the Zimm theory. 18 Therefore, our algorithm has yielded the expected dimension scaling law and the Rouse-like dynamics for chains 103

120 10 τ (millionmcs) 1.05± M Figure 4.. The relaxation time of end-to-end vectors versus M for single chain simulations. 104

121 longer than C00; for shorter ones, the extra mobility of repeating units results in chain motions faster than the typical Rouse dynamics. 4.. Static properties of Melts The static properties of polyethylene chains have been studied in the melt state. The most time consuming calculation in dense melts is finding long range interaction pairs. As a result, a short cut-off distance is preferred for the sake of computing speed. As we mentioned before, in our simulation, the L-J potential has been discretized for nnd lattice. The first shells are repulsive and the 3 rd is attractive. The further shells are also attractive but interaction energies are much closer to zero than those of the first 3 shells. It is known that the local structure of a liquid is dominated by repulsive energy The attractive tail of the LJ potential is mainly for generating a cohesive nature of substances. 93 Therefore, by taking advantage of NVT Monte Carlo, it may be possible to use only shells to generate proper dimension of polyethylene. The mean square radius of gyration <s > of PE chain has been computed from the simulations using either shell and 3 shell interactions. In Figure 4.3, the solid symbols were from 3 shell simulations and open symbols were from shell simulations. There is not much difference in terms of dimensions. Later we will show the similar observation in term of dynamics properties as well. Thus most data in the rest of paper were all collected from shell simulations. The expected scaling law of polymer dimensions in the bulk state is that <s > is proportional to M. However, in Figure 4.3, there is a similar interesting feature observed here as seen in dilute conditions. This time, instead of dynamic properties, the static 105

122 10 1.0±0.03 <s > (nm ) 1 shell 3 shell M Figure 4.3. The mean square radius of gyration <s > obtained at different cut-off distance of L-J potentials plotted against M. The line is based on the regression of the 5 longest chains simulated with shells. 106

123 properties, <s >, of short chains approach to the slope 1 asymptotically, which again shows that the packing of short chains generates neither standard static properties nor standard dynamics properties. In Figure 4.4, C s,n, the characteristic ratio, <s > / nl (n is the carbon number, and l is the carbon-carbon bond length), is plotted against 1/n. The slope from the simulation is 18, and the intercept is 1.1, both of which are in excellent agreement with earlier RIS calculations 99. In order to verify the statistics of these short PE chains, the ratios of mean square end-to-end distance and mean square radius of gyration, <r >/<s >, are plotted against M in Figure 4.5. This expected ratio for Gaussian chains equals to 6. From Figure 4.5, short chains are significantly non-gaussian as the ratio indicates. 99 The ratio gradually approaches 6 as the chain is getting longer Dynamic properties of Melts The major purpose of our algorithm is to model the dynamics properties, especially diffusion in polyethylene melts, which can be extracted directly from the simulation without being biased by applying theories. The Figure 4.6 is a popular plot for the mean square displacement of the center of mass, middle segment, and chain ends. 79 Our algorithm captures all characteristics which have been found by other MC or MD algorithms. For example, the initial slope of g3 is 0.83, which coincides with literature findings. This feature wasn t been explained by theory until The initial slope of g1 is 0.55 which is higher than the 0.5 predicted by the Rouse theory, but is identical to earlier findings. 79 Although it has been discussed earlier that short chain diffusion may not be well described by the Rouse theory, Figure 4.6 shows that 107

124 ±0.0 1 Cs,n ± /n Figure 4.4. The characteristic ratios between <s > and nl plotted against 1/n, where n is carbon number and l is carbon-carbon bond length. 108

125 <r> /<s> M Figure 4.5. The ratio <r >/<s > versus M; non-gaussian features shown for short chains. 109

126 msd (nm ) g3 g1 g g4 g MCS Figure 4.6. Various kinds of mean square displacements (msd) of C60 vs. Monte Carlo Steps (MCS), g1 is msd of the middle bead on the chain, g is msd of the middle bead relative to the center of mass; g3 is the msd of center of mass; g4 is to the msd of end beads and g5 refers the msd of end beads relative to the center of mass. The solid line shown in the bottom of the figure has a slope of

127 qualitatively, the displacements of center of mass, the middle bead, and chain ends agree with the prediction of Rouse theory except at short time and distance scale. Perhaps, for polyethylene, the most accessible plot of this kind can be found in the coarse grained simulation using the bond fluctuation model 80. The Figure 4.7 is the result of overlapping our simulation of C10 on the result of C100 by BFM, which has been verified by molecular dynamics simulation. 98 The g1, g and g3 of these two sets of results almost perfectly overlap each other, but g4 and g5 are somewhat different. It is perfectly understandable because an end bead in our simulation represents a methyl unit while in BFM, one end bead may include about 5 carbon atoms and their pendant hydrogen atoms. Naturally the end beads in our case move faster than theirs. By incorporating more inner repeating units, g4 and g5 can approach the value from BFM. In Figure 4.8, the normalized auto correlation functions of end-to-end vector <r r> from various length of polyethylene chains are plotted together against normalized time t/τ. τ is the longest relaxation time of the polymer chain obtained by fitting the <r r> to equation 4.. They overlap each other reasonably well. In spite of non-gaussian behaviors, equation 4. usually gives very good fits for shorter chains as shown in Figure 4.9. The KWW relationship (equation 3) gives a little lower quality fit for short chains in Figure 4.9. β < r r >= exp( ( t / τ ) ) (4.3) However for longer chains, the fittings of <r r> according to equation 4. is usually not perfect, and the quality of fitting is lower than those fitted according to KWW equation as shown in Figure

128 Figure 4.7. Mapping our simulation results of polyethylene onto the results of Bond Fluctuation Model, as reported by Tries et al. 80 The color lines are our results, and the black lines are BFM results. From the uppermost to the lowermost: g4, g5, g1, g, g3 for both color lines and black lines respectively. 11

129 Figure 4.8. Normalized autocorrelation function of the end-to-end vector r vs. normalized time. The numbers inside the box are molecular weights. 113

130 <r r> MCS Figure 4.9. The autocorrelation function of end-to-end vectors of C40 fitted by equation 4. (dotted line) and 4.3(dash line) respectively. 114

131 <r r> MCS Figure The autocorrelation function of end-to-end vectors of C34 fitted by equation 4. (dotted line) and 4.3 (dash line) respectively 115

132 These results qualitatively agree with a recent MD simulation for polybutadiene melts 100 and show that equation 4. is a poor description of dynamics at the Rousereptation transition, although it was universal in both Rouse and reptation region according to the theory. In Figure 4.11, β in equation 4.3 obtained from the fitting is plotted against the molecular weights. The data show a decreasing trend of β with M. In Figure 4.1, the mean square displacements of the center of mass from C10, C138, C48, and C34 are shown. The diffusion coefficients can be calculated from these curves and the results are plotted against the molecular weights in Figure In Figure 4.13, closed squares are simulation results using only two repulsive shells. Open triangles are the simulation results including the 3 rd attractive shell Open spades are experimental results. 83, 101 The MCS has been rescaled to real time according to the displacements in real distance. Our results indicate there is no much difference between two shell and three shell simulations. The only difference is the rescaling coefficient shown in the caption of Figure There are two regression lines in Figure 4.13 obtained respectively from the data points of the 4 shorter chains and 5 longer chains from two shell simulations. These two regression lines cross each other at M The slope for short chains coincides with what has been obtained from MD simulation 46 as well. The slope for longer chains is -., which is lower than MD simulation but perfectly overlaid on the experimental results. It is noticed that no Rouse scaling is found in our simulation, which is contrary to some other simulations. 46,10. Although there is no clear consensus whether a Rouse scaling regime really exist for polyethylene melts, in the light of earlier discussion, we believe that the molecular-weight dependent segmental friction coefficients of short chains could extend into the crossover regime towards the entangled 116

133 β M Figure β obtained by fitting the data to the KWW equation plotted against M. 117

134 10 1 g3 (nm ) MCS Figure 4.1. g3 of C10, C138, C48 and C34 (from top to bottom) vs. time; the dash lines are just for showing slope

135 region based on our observation from static properties of melts and single chain dynamic properties. These properties show that non standard packing may exist which may generate excessive free volume even beyond the molecular weight 000, higher than the believed entanglement length M e. Of course, theoretically it is not known to what extent this non standard packing will influence diffusion scaling. Comparing Figure 4.3 and Figure 4.4, a different asymptotic chain length limit could be seen depending on which property, either <s > or <r >/<s >, is being looked at. This result is expected because theory shows that <r > and <s > have different approach to their asymptotic limits 99. The non standard packing of chain segments creates free volume in the melts, which explains the lower density for shorter chains, which in turn leads to the lower friction coefficients of shorter chains. There is no window for the Rouse scaling to appear because chains are not sufficiently in a normal packing even after they start entering the entanglement regime. Searching for the signature of reptation of the longer chains, the g1 of C10, C17, C48, and C34 are plotted against MCS in the following four figures. In Figure 4.14, for C10 with the molecular weight of 1430, the initial slope of g1 is 0.545, and the slope of g1 gradually increases to the slope 1 without any evidence of a slowdown, which is a close resemblance to the Rouse prediction. In figure 4.15, for C17 with the molecular weight of 410, the initial slope is 0.54, but after the initial stage, this slope drops to 0.5 and this regime continues until it transits to the free diffusion regime. It is hard to say whether this can be considered as a 119

136 10-1.7±0.1 1 D cm /10 6 s ± M Figure The diffusion coefficients D vs M, open squares from the experiments 83, 101 open triangles and closed spades obtained by calibrating MCS with real time using twoshell (330MCS ~ 1ps) and three-shell (530MCS ~ 1ps) simulations. There two regression lines in the figure based on upper left 4 data points and lower right 5 data points from two shell simulations, respectively. The numbers in the figure are slopes of these regression lines. 10

137 slowdown due to the entanglement, because there is a possibility that after a non typical dynamic behavior at short time and distance scale, the polymer moves more like a Rouse chain with a slope exactly equaling to 0.5 before it starts free diffusion. However, at this crossover chain length, it is also probable that the drop of slope is too subtle to be detected. Nevertheless the scaling plot in Figure 4.13 indicates this may be an initialization of entangled regime. In Figure 4.16, for C48 with the molecular weight of 3474, after the initial stage, the slope drops and an intermediate regime appears before the slope rises back to 0.5. If this intermediate regime has a slope 0.5, then it is exactly what reptation theory predicted. The slope is in this intermediate region, which indicates that entanglement effects start emerging but this confinement by the entanglement is not as severe as that in the melts of infinitely long chains. In other words, this relates to the additional relaxation besides the reptating motion of chains. It could be contour length fluctuation or constraint release, although we don t know what are the predictions of them at the low end instead of the high end of chain length. It is also likely to be understood as a sum of the Rouse motion and reptating motion. For C34 with the molecular weight of 4538, the trend is more evident shown in Figure The intermediate slope decreases to 0.44 and the lower slope is 0.39 in the inner regime of the intermediate region. These results also agree with molecular dynamics simulation using united atom model of PE. 46 Some parameters with clear physical meaning can be potentially extracted from Figure 4.16 and Figure 4.17 according to the theory. The characteristic slowdown starts at the region between

138 100 C10 10 g1 (nm ) E+04 1.E+05 1.E+06 1.E+07 MCS Figure g1 of C10 (thick line) vs. time, the thin regression line with an initial slope of 0.545; no slowdown observed before transition to the free diffusive region. 1

139 100 C17 10 g1 (nm ) E+04 1.E+05 1.E+06 1.E+07 1.E+08 MCS Figure g1 of C17 (thick line) vs. time, upper thin regression line with an initial slope of 0.54, and lower thin line with a slope of 0.5; intermediate region (0.5 ~ million MCS) with the slope of

140 100 C48 g1 (nm ) E+05 1.E+06 1.E+07 1.E+08 MCS Figure g1 of C48 (thick line) vs. time, upper thin regression line with an initial slope of 0.5, and lower regression thin line with a slope of 0.5; intermediate region (0.7 ~ 5 million MCS) with a slope of 0.46 by regression. The inner intermediate region (3-4 million) with a slope of 0.4 by regression. 14

141 100 C34 g1 (nm ) E+05 1.E+06 1.E+07 1.E+08 MCS Figure g1 of C34 (thick line) vs time, upper thin regression line with an initial slope of 0.5, and lower thin regression line with a slope of 0.5; intermediate region (1 ~ 9 million MCS) with a slope of 0.44 by regression. The inner intermediate region (6-7 million) with a slope of 0.39 by regression. 15

142 million MCS, where the mean square displacement of the middle bead reaches nm for both C34 and C48. From classic reptation theory, at this point, the chain starts feeling the confinement of tube. However an intriguing fact is that the chains felt the tube much earlier than we would expected from experimental measurements. The entanglement length of PE according to the experiments is around 3.8 nm which is larger than what is seen here. It is possible that our model is still not capable to precisely capture the entanglement length scale; however after carefully reviewing the g1 of long chains in MD simulation results of united atom model of PE, a slowdown length scale similar to our results can be recovered. In Figure 4.18, the mean square displacement of C48 in solid red line in our simulation is plotted against the real time obtained by calibration of MCS using the experimental data. The black/white background is from ref. 46. Clearly our C48 based on the calibrated real time moves slower than C Calibrating our results according to the g1 from the MD simulation 46 results in red dash line in Figure Two calibration results show C50 46 move 1.5 times faster than C48 in our system, nevertheless, the shape of the curves are extremely similar. As we have mentioned, the tube diameter a of PE has believed around 38Å. 18 However according to both MD and MC results in Figure 4.18, the depression of the mean square displacement clearly starts earlier than a /3, not to mention a, which is around 14.4 nm (log g = 3.16Å ) in Figure This earlier slowdown of the segmental diffusion actually has been found by Putz and Kremer 103 when they compared the entanglement length obtained from monomer diffusion with the one obtained from the viscoelastic measurements. However experimental measurement didn t support the observation. 104 Here what we found 16

143 Figure Comparison between the results from Harmandaris et al 46 and our simulation results in middle bead movements. The dash red line is the same simulation results directly calibrated by C50 shown in 46, which is about 1.5 times faster than the results calibrated by the experimental data. 17

144 coincides with many other simulations 45,46,103 and may prove it is not an artifact of simulation at all. The underlying reason, however, is worth further study. When g1 of all chains are plotted together, it is easy to understand what the molecular weight dependent segmental friction coefficient is. It is noted that in theory if the segmental friction coefficient is a constant without a chain length dependence, the g1 at the initial stage should be identical for all chains. An overlapped curve at the short time and distance scales should have been observed. Yet in Figure 4.19, evidently the middle bead in short chains moves much faster in the first place even before this middle bead don t know that it belongs to a small chain. The intrinsic mobility virtually shadowed the classic scaling of Rouse dynamics. In Figure 4.19, it is also clear that g1 of long chains at initial stage are much closer to each other than g1 of the shorter chains. Actually diffusion curves of shorter molecules can be shifted so that the initial portion of the curves can overlap each other as shown in Figure 4.0. The new diffusion coefficients based on this shift can be computed. The results are shown in the Figure 4.1. It seems that there is still a residual effect after shifting to iso-friction. The Rouse scaling, D ~ M -1, still couldn t be obtained Equation.5 in the Chapter II have a discrete version. It is called Rouse mode, and can be calculated directly from the simulation. Although Rouse predicted that normalized time correlation function of these modes should follow: r r r < Xp( t) Xp(0) > / < Xp >= exp( t / τ ) p (4.4) 18

145 . 1.E+03 1.E g1 nm 1.E E+00 1.E-01 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 MCS Figure g1 of C40, C60, C10, C138, C17, C1, C48, C34 (from top to bottom) vs time; the C40 and C60 diffuse much faster than others at short time scale while other chains move at the similar rate at the short time scale. 19

146 1.E+0 1.E+01 g1 nm 1.E+00 1.E-01 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 MCS Figure 4.0. g1 of C40, C60, C10, C138, C17, C1, C48, C34 (from top to bottom) vs time by shifting all curves to iso-friction regime. 130

147 ±0. D cm /10 6 s -.± M Figure 4.1. The Diffusion coefficients D vs M after shifting data to iso-friction regime. 131

148 it is generally not possible to fit these modes with a simple exponential function. Instead, KWW equation again was found to be a better choice, especially for the short time. The normalized auto correlation function of Rouse modes are presented in Figure 4.. The lines can be fitted to extract β and τ in KWW equation, and the effective relaxation of these modes is : β τ τ eff = exp[ ( t / τ ) ] dt = Γ(1/ β ) (4.5) β 0 Where Г(x) is the gamma function: Γ z 1 t ( z) = t e dt 0 (4.6) The results of fitting several different lengths of chains by KWW equation is shown in Figure 4.3. The βfrom those chains above the entangled molecular weight is plotted against N/p, a master curve appears, and shorter unentangled chain C60 seems to fall close to this master curve as well. It is observed that β continues to rise when the N/p is below 100 and levels off when N/p is beyond 100. According to Shaffer s explanation 105, low β value generally signifies the strong constraints of the dynamics. It is natural to consider the low β at the short distance scale as a consequence of dominant local interactions from the stiffness and connectivity. The transition point of β may symbolize the blob size in a dynamic sense. The local constraint is screened out above this scale. It may respond to the entanglement transition. Although C60 shows a similar trend in the initial stage, there is no plateau region observed for C60. However there were different observations in the literature. 13

149 -ln(xp(0)xp(t) ) log( p log(t) Figure 4.. The Rouse mode relaxation of C34 described by KWW equation. Linear fitting of these lines lead to β and τ in KWW equation. 133

150 β C60 C17 C48 C N/p Figure 4.3. β in KWW equation obtained at different molecular weights vs. N/k. 134

151 Shaffer observed no plateau of β but a decreasing trend vs. N/p for entangled chains. He concluded this trend is the result of the entanglement effects. While Padding etc. 45 did observed the plateau region the same as we found here, they also identified an interesting minimum β region at the scale of two blob size. Because the length scale was clearly shorter than the experimental entanglement chain length, they considered this as a new length scale, termed as slow down length N s. This new length scale is not supported from our results. The relaxation time τ obtained by fitting equation 4. and KWW equation from the five entangled chains is plotted again chain length M in Figure 4.4. Note the relaxation time of KWW equation plotted in the figure is the effective relaxation obtained from integrating Gamma function. The intermediate slope.5 and.7 appear. It is interesting that chains are already entangled in term of the scaling of translational diffusion while they are not fully entangled in terms of the scaling of the rotational diffusion. This scaling of the relaxation time could be identical to the scaling of the viscosity in this regime. If it is the case, then this scaling is very close to the experimental results, although few theory can explain this experimental phenomenon very well. 106 At this moment, without further work, not much input can be given here. Finally the single-chain coherent intermediate scattering functions are given at several molecular weights because currently the neutron spin echo (NSE) is one of the main experimental techniques used to study dynamics of polymer chains. The experiments are able to give the single-chain intermediate scattering functions over a 135

152 100 τ (m MCS) 10 reptation.49±0.05 KWW.70± M Figure 4.4. The relaxation time τ according to reptation and KWW vs. M 136

153 wide range of length and time scales, which can be directly compared with simulation results of polymer chain dynamics. The normalized single-chain dynamic structure factor can be obtained from simulations of the isotropic melt utilizing the simple relationship shown in equation 4.7. The scattering centers are nnd beads in our simulations. Singlechain coherent intermediate scattering functions obtained according to the equation , 107 are shown in the rest of Figure 4.5 for 4 different chain lengths respectively. S( q, t) S( q,0) < = < N i, j= 1 N i, j= 1 r r sin( q i ( t) j (0 ) r r > q i ( t) j (0 r r sin( q i (0) j (0 ) r r > q ( t) (0 i j (4.7) In the equation 4.7, i and j are the beads belong to the same chain. The single chain coherent intermediate scattering functions calculated from our simulation are very close to those obtained from MD simulation. 46 However there are no experimental data available currently at these chain lengths and at this temperature. We welcome the direct comparison with the experimental data as soon as they become available Summary A dynamic Monte Carlo algorithm based on random two bead moves has been developed on the nnd lattice. The static and dynamic properties have been studied for polyethylene chains ranging from C40 to C34 in melts and C40-C790 in diluted systems using this algorithm. For diluted systems, Rouse scaling was obtained only after M above The non standard packing of short chains due to the non Gaussian statistics generate extra free volume expressed as low density in melts. This phenomenon results 137

154 C10 <S(q,t)>/<S(q,0)> ns Figure 4.5. Single-chain coherent intermediate scattering function obtained from our simulation at 180 C at various q values (0.04, 0.1, 0.0, 0.8, and 0.36 Å -1 ) for different molecular weights (a) C

155 <S(q,t)>/<S(q,0)> C ns Figure 4.5. Single-chain coherent intermediate scattering function obtained from our simulation at 180 C at various q values (0.04, 0.1, 0.0, 0.8, and 0.36 Å -1 ) for different molecular weights (b) C

156 1 <S(q,t)>/<S(q,0)> C ns 100 Figure 4.5. Single-chain coherent intermediate scattering function obtained from our simulation at 180 C at various q values (0.04, 0.1, 0.0, 0.8, and 0.36 Å -1 ) for different molecular weights (c) C

157 1 <S(q,t)>/<S(q,0)> C ns 100 Figure 4.5. Single-chain coherent intermediate scattering function obtained from our simulation at 180 C at various q values (0.04, 0.1, 0.0, 0.8, and 0.36 Å -1 ) for different molecular weights (d) C

158 in an extra chain length dependence of the diffusion in both diluted systems without hydrodynamic interactions and melts even after the entanglement effect sets in. The chain-length dependent friction coefficient resulting from this phenomenon also closes the window for Rouse scaling regime for diffusion of polyethylene melts. Our simulation results on diffusion coefficients indicate that entanglement starts at around M of 1500, while analysis of g1 shows that the signature of reptation appears only after 400. Many of our results, such as diffusion scaling of short chains and long chains; g1, g, g3 of C10; as well as the distance scale and slopes of g1 for long chains, are supported by either experiments or other simulation algorithms. Our results, supported by an earlier simulation, 46 show that the description of autocorrelation function of end-to-end vector r according to the classic reptation model is not precise for slightly entangled chains, while KWW equation can usually give a better fit of the data. It is not clear if the rotational diffusion and translational diffusion have different dependence on M at the Rousereptation transition due to the inadequate fitting based on the theory. More work is needed in this area to enable an extraction of reliable relaxation times of polymer chains, better without being biased by models. It is intriguing to see that chains feel the entanglement confinement at much shorter distance scale than the tube diameter measured by experiments. 14

159 CHAPTER V CROSSOVER TO THE ENTANGLED DYNAMICS: BIDISPERSE POLYETHYLENE MELTS 5.1 Introduction In the last chapter, the transition from the pseudo-rouse dynamics to reptation dynamics of monodisperse polyethylene melts has been discussed. What has been found is that after certain critical molecular weight M c, the diffusion coefficients decrease more rapidly with molecular weight, which signifies the starting point of entangled dynamics. The extracted M c from our simulation is 1500, which roughly corresponds to mean square radius of gyration nm from our simulation. This result agrees with the extracted entanglement length from the middle bead diffusion, however it apparently is much smaller than the experimental value. It is also known the results from experimental measurements themselves differ depending on the experimental methods. For example, the entanglement length calculated from rubbery plateau using rubber elasticity theory has been called M e, because it is different from the transitional molecular weight M c obtained from viscosity measurements, both of which differ from the M c obtained from diffusion measurements. According to reptation theory, they should be the same length scale, which corresponds to the tube diameter a. The underlying physics of the relationship between M e and M c is still a very intriguing topic. Thus investigating the 143

160 initial formation of the tube and physics behind the tube diameter a itself has become an important issue. There are several theories 13, 15, 36, 37, 108, 109,110, 111,11 which are related to the emergence of the entanglement effect, in other words, these theories should have the ability to cover the transition region. Richter and his coworkers 115 have checked most of them by using their own experiments and by summarizing the experimental data from other groups. Their results clearly favor reptation model instead of other theories which are incompatible with the reptation model. Especially for transitional region, it has been found among these theories, Hess model, which is compatible with reptation model, is very promising. Hence we mainly focus on the Hess theory and other modifications which are inside the reptation frame here. Hess theory has recently been extended and thoroughly checked by Richter and his coworkers. 114, 115 Favorable agreements have been reached according to their studies. The generalized Rouse model by Hess describes the microscopic dynamics of polymer melts. Unlike the Rouse model, Hess introduced excluded volume effects, which appears as an additional memory function of entanglement friction resulted from the surrounding chains other than the test chain in the generalized Langevin equation. Via the excluded volume effects in Hess model, both motions of the test chain and of the surrounding chains contribute to the relaxation of the entanglement friction function, which implicitly incorporates the constraint release effects. The overall center of mass diffusion in the Hess model can be successfully decoupled into the contribution of curvilinear diffusion D along the chain contour and D, which is perpendicular to the chain contour. 144

161 (5.1) φ is the mean excluded-volume energy between two arbitrary segments. D R is the Rouse diffusion coefficients. For monodisperse polymer melts, the analytical solution of the last equation exists: (5.) Clearly above the critical molecular weight N c, the perpendicular motion freezes and only curvilinear motion exists. For a bimodal system, 115 the formula has been made suitable to be used in the transitional region: (5.3) There is another important conclusion in extended Hess theory that predicted the two step function of time correlation function of normal modes for the higher modes as shown in the following figure reproduced from literature. This two step relaxation will be checked in following discussions. Note the step transition happens at the very low value 145

162 Figure 5.1. The two step relaxation of normal modes predicted by Hess theory. Reproduced from ref