INFLUENCE OF MOLECULAR WEIGHT AND ARCHITECTURE ON POLYMER DYNAMICS. A Dissertation. Presented to. The Graduate Faculty of The University of Akron

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1 INFLUENCE OF MOLECULAR WEIGHT AND ARCHITECTURE ON POLYMER DYNAMICS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Yifu Ding May, 005

2 INFLUENCE OF MOLECULAR WEIGHT AND ARCHITECTURE ON POLYMER DYNAMICS Yifu Ding Dissertation Approved: Advisor Professor Alexei P. Sokolov Committee Member Professor Mark D. Foster Committee Member Professor Purushottam D. Gujrati Committee Member Professor Gustavo A. Carri Accepted: Department Chair Professor Stephen Z. D. Cheng Dean of the College Professor Frank Kelley Dean of the Graduate School Professor George Newkome Date Committee Member Professor Jutta Luettmer-Strathmann ii

3 ABSTRACT Molecular weight (MW) and architecture are two important parameters of a synthetic polymer. Their roles on polymer properties including dynamics have not been well understood yet. In this thesis, we have used various techniques including light, neutron scattering and dielectric spectroscopy to elucidate their influences on polymer dynamics within a broad time (frequency) range, covering chain, segmental relaxation and fast dynamics. Comparisons between different polymers were made to understand the role of chemical structure in determining MW dependence of the dynamic behavior. Experimental results showed that different physical properties studied appear to have similar molecular weight dependence in the sense that they all saturate when chains approach Gaussian coil behavior. We demonstrate that the difference in the molecular weight dependence for various polymers does not correlate with either the difference in the Kuhn segment length or molecular weight between entanglements. Instead, we propose to introduce an additional parameter, m R (molecular weight associated with each step of the Random walk in the approximation of Gaussian chain) that might be important for characterizing the molecular weight dependence of chain statistics and many physical properties. iii

4 The most intriguing result is that the molecular weight dependence of the fast dynamics, sound velocity and fragility observed in polystyrene is opposite to the one observed in polyisobutylene, although T g increases with molecular weight in both cases. We speculate that difference in symmetry of the monomer structures is responsible for the opposite behavior. Studies of the influence of architectures on fast and segmental relaxation were also carried out. We found that in the case of polybutadiene both of them scale better with total molecular weight instead of the molecular weight of each arm, as suggested by the chain end free volume model. Analysis of the branching effect on the segmental relaxation illustrates similarity to the blending of the same polymer with different molecular weights. iv

5 ACKNOLEDGEMENT This Dissertation is realized under the direction of my advisor Dr. Alexei Sokolov, who not only provided me all the encouragements and inspirations but also showed me the great passions needed to become a great scholar. I am grateful to Dr. Alexander Kisliuk for helping of the experiments and sharing the experiences of a great experimentalist. I would like to thank Dr. Vladimir Novikov for helping with the theories and data interpretations. This research work would not be successful without any of them. I thank all group members for the discussion and companionship, especially former member Dr. Gokhan Caliskan. I want to acknowledge Vivek Sharma, Disha Mehtani and Ryan Hartschuh for correcting my English. I appreciate all the remarks, corrections and suggestions from my committee members Dr. Mark Foster, Dr. Gustavo Carri, Dr. Purushottam Gujrati and Dr. Jutta Luettmer-Strathmann. I thank the funding for my research provided by NSF. I sincerely thank my parents for their love and support and my wife Hongying for her trust and caring through all the years. v

6 TABLE OF CONTENTS Page LIST OF TABLES..xi LIST OF FIGURES...xii CHAPTER I. INTRODUCTION...1 II. HISTORICAL BACKGROUND Chain dynamics Rouse model Reptation model Experimental studies of Rouse dynamics Molecular weight dependence Chain segment: studies on spatial dependence Segmental relaxation Gibbs-DiMarzio theory..3.. Free volume theory 5..3 Coupling model (CM) Secondary relaxation..9 vi

7 .4 Fast dynamics.9.5 Research objectives III. EXPERIMENTAL DETAILS Light scattering theory Continuous medium approach Molecular approach Light scattering components Elastic scattering Quasielastic scattering Inelastic scattering Raman scattering Brillouin scattering Light scattering techniques Laser Optics Samples and cryostats Spectrometers Raman spectrometer Tandem Fabry-Perot interferometer (TFPI) Data treatment Raman TFPI vii

8 3.5 Dielectric spectroscopy Principle Experimental details.. 60 IV. DEPOLARIZED LIGHT SCATTERING (DLS) STUDY OF DYNAMICS IN POLYMERS Introduction Samples and techniques Chain dynamics in DLS: theory Results from DLS Dependence of the DLS spectrum on scattering wave vector MW dependence of DLS spectrum Analysis and discussion Importance of chain statistics Segment size of polystyrene (PS) Steady state compliance J e Dynamic mechanical relaxation data of unentangled PS melts Universal MW dependence of T g Comparison of DLS and viscoelastic measurements Conclusions. 98 V. INFLUENCE OF MW ON FAST DYNAMICS AND FRAGILITY OF POLYMERS Introduction Samples and techniques viii

9 5.3 Results and their analysis Discussion MW dependence of the Brillouin, microscopic and the Boson Peak frequencies Dependence of fragility on MW for different polymers Conclusions VI. INFLUENCE OF ARCHITECTURE ON FAST AND SEGMENTAL DYNAMICS AND THE GLASS TRANSITION IN POLYMERS Introduction Fast and segmental relaxation of polybutadiene (PBD) Samples and techniques Results Discussion Unusual local relaxation in PBD Introduction Samples and techniques Results and discussion Dielectric studies on PS with different architectures Samples and techniques Results and discussion Conclusions...16 VII. SUMMARY.164 REFERENCES ix

10 LIST OF TABLES Table Page 4.1 Relaxation times from DLS spectra and viscosity measurements Sample information for PS and PIB Boson peak and Brillouin peak positions measured at T=80 K for PS and at T=100K for PIB Fragility indexes for different polymers Characterizations of the PBD Characterization of PS with different architectures T g and fragility from dielectric measurements.161 x

11 LIST OF FIGURES Figure Page 1.1 Polymer dynamics at different time scales, represented by the time dependence of the stress relaxation. Secondary relaxation usually merges with segmental relaxation at high temperature..1 Mean square displacement of a chain segment varies with time. The t 1/4 dependence is the fingerprint of reptation behavior Power law dependence of mean square displacement (<r >=- 6ln[S inc (Q,t)]/Q ) measured by neutron spin echo experiments Scaling of the amplitudes of Rouse modes for C 100 H 0 melt at T= MW dependence of self-diffusion coefficient for (a) n-alkanes, different lines represent different temperatures; 6 and (b) PEO melts Steady state compliance of monodisperse polystyrene as a function of MW Self-diffusion coefficients of PDMS melts as a function of MW Dynamic bead size in dilute solution estimated by NSE xi

12 .8 Intermediate scattering function S(Q,t) at different scattering wave vector Q for (a) PDMS and (b)pib melts. Experimental points are fitted with Rouse model (line). The number represents different experimental Q value Q dependence of the elementary Rouse rate W of PIB 51. The Rouse prediction is that W is invariant to Q as indicated by the shaded region Temperature dependence of viscosity (Angell s plot) 58. The slope in this plot at T=T g defines the fragility of the material Temperature dependence of segmental relaxation time of: (a) PDMS 74 and (b) PS 68. Lines represent different MWs...1 T g scales with the arm length for PS stars 81. Lines represent different methods used to measure T g Momentum transfer during the scattering process: scattering wave vector q Different scattering components. Broadening of the central line is quasielastic scattering, and two inelastic peaks (±ν) are Stokes and anti-stokes components Energy transfer for elastic and Raman scattering (Stokes and anti- Stokes) Brillouin spectrum of PS (M n =9100) at 80K in the back scattering geometry (θ=180 ). The central peak is the Rayleigh peak, and the two side peaks are Brillouin peaks (see the text) Basic schemes of the instrument used for Raman and FPI measurements 113. The dash line indicates the part only used for the interferometer Optical diagram for the subtractive mode, see the text for the components xii

13 3.7 Intensity of the transmitted light through interferometer for different wavelengths Tandem Fabry-Perot interferometer, Sandercock model Transmitted intensity of two FP etalons separately (upper two) and in tandem mode Transmission function of the interferometer for FSR 350GHz Example of data treatment for Raman spectrum of polystyrene at 80K. (a) Raman spectrum after subtraction of dark counts, and the straight line represents the fluorescence background. (b) Raman spectrum after subtraction of fluorescence background. (c) Spectral density presentation (see the text for detail) Susceptibility spectra of PBD at different frequency range. (a) FPI spectra at different FSR and Raman spectrum. (b) Joint spectrum, the dash line indicates the joint point A typical dielectric relaxation spectrum for glass forming systems DLS susceptibility spectrum χ (ν) of styrene dimer at T=400K Scattering geometry II: XY plane is the scattering plane. k i, k f is the incoming and the final wave vectors, respectively, and θ is the scattering angle. This geometry is used in connection with the molecular theories (a) DLS susceptibility spectra of PDMS 550 at different scattering angles: ( ) 10, ( ) 90, ( ) 180 ; (b) DLS susceptibility spectra of PDMS 116,500 at different scattering angles: ( ) 90, ( ) DLS spectra of PDMS with various MW at (a) T=96K and (b) T=400K. The spectra are presented as the imaginary part of the susceptibility. The segmental relaxation frequencies at both temperatures are marked with the arrows. The solid lines present slopes characteristic for small molecules χ (ν) ν, and for the Rouse dynamics χ (ν) ν xiii

14 4.6 MW dependence of the exponent a in PDMS at T=400K and T=96K, dashed lines show limits a=1 for small molecules and a=0.5 for the Rouse dynamics MW dependence of the exponent compared to the MW dependence of R g (data from ref.18). The arrow marks M e The MW dependence of the ratio <R g >/M w in PS (data from ref.19). The arrow marks M e The MW dependence of T g and of <R g >/M W (a) for PDMS (T g data from refs.79,80); (b) for PS (T g data from ref.67). Also the MW dependence of density (data from ref.130) in PS is shown in (b). The arrows mark M e Steady-state compliance of PS normalized by MW and density (J e 0 ρ/m) as a function of MW. The dotted line represents the theoretical value calculated from J e 0 =0.4M/ρRT. The data is taken from ref.8-31, as tabulated in ref Dynamic shear modulus G*(ω) of PS with MW 1000, 5000: (a) Loss modulus G (ω) and (b) Storage modulus G (ω). Empty symbols represent experimental data for PS with various MWs: 1000 (square); 5000 (circle). The lines represent the fits: Cole-Davidson equation for 1000 (solid line); Fit for 5000 according to Rouse prediction (dash line), and corresponding one Rouse mode component (dash dot line) Dynamic shear modulus G*(ω) of PS with MW 1000, (a) Loss modulus G (ω) and (b) Storage modulus G (ω). Empty symbols represent experimental data for PS with various MWs: 1000 (square); (circle). The lines represent the fits: Cole-Davidson equation for 1000 (black solid line); Fit for according to Rouse prediction (dash line), and corresponding two Rouse mode components (dash dot line) T g /T g ( ) as a function of MW scaled by the mass of random step m R. m R ~560 for PDMS and m R ~5100 for PS (see the text for definition of m R ) 91 xiv

15 4.14 DLS susceptibility spectra of (a) PDMS 16 ( ); (b) PDMS 550 ( ); (c) PDMS 150 ( ). Thick lines represent the fitting results; the dashed lines represent the component relaxation processes as noted; Thin lines in (b) and (c) represent the spectrum of PDMS 16, used as a comparison Zero-shear-rate viscosities for PDMS with M = 150 ( ) and 550 ( ). The activation energies calculated from the respective slopes are indicated DLS spectra of PS with various molecular weights at (a) T~T g (for different molecular weight: ( ), 550; ( ), 990; ( ), 170; ( ), 1400.) and (b) T=80K. The inset of (b) is the imaginary part of susceptibility spectra χ "( v) = I n ( v) v, which shows the microscopic peak for PS with different molecular weight. (Different symbols in (b) represent PS with different molecular weights: ( ), 550; ( ), 990; ( ), 8000; ( ), 9100; ( ), ; ( ), 00600) Inelastic neutron scattering spectra averaged over all scattering angles for PIB at 100K for various molecular weights: ( ), 370; ( ), 640; ( ), 00; ( ), 3470; ( ), Brillouin scattering spectra of (a) PS at 80K and (b) PIB at 100K with different molecular weights (shown by numbers). Inset of (a) represents a typical Brillouin spectrum with doublet (± ν) at both sides of the elastic line (ν 0 ) Molecular weight dependence of positions of microscopic peak ν micro ( ), Brillouin peak ν B ( ), and Boson peak ν BP ( ) for (a) PS at 80K and (b) PIB at 100K. All the frequencies are normalized by the value of the largest molecular weight ν inf Molecular weight dependence of density for PS at T g. ( ) represents the original measurements from ref.30; ( ) Estimation based on the equation provided with the T g parameter from ref.67. Both data show that density variation is within 1% of the average value xv

16 5.6 T g /T g, inf as a function of molecular weight for PS ( ), from ref.67 and PIB, from ref.145 ( ). T g,inf is the glass transition temperature of the polymer with infinite molecular weight, 373K for PS and 09.5K for PIB Molecular weight dependence of T g /T g,inf and V sound /V sound,inf for linear ( ) and crosslinked PDMS, respectively. The abscissa represents the M n for linear PDMS, and the molecular weight between crosslinking, M c for crosslinked PDMS. Both T g and V s are normalized by the value at infinite MW, data are from refs , Relaxation-to-vibration ratio as a function of molecular weight for (a) PS, and (b) PIB. The ratio is estimated from light scattering spectra for PS at T=80K (Figure 5.1b) and neutron scattering for PIB at T=100K (Figure 5.). Also included in (a) is the fragility index m as a function of molecular weight for PS (data from ref.68) Correlation between fragility and size of cooperative units at T g Temperature dependence of segmental relaxation time for different molecular weight (a) PMMA (data from ref.153) and (b) methyl-groupterminated Polypropylene glycol (data from ref.7) Molecular weight dependence of fragility index m for PS ( ), PDMS ( ), and PIB ( ). Shaded area marks the fragility range expected for monomers and oligomers Conformational energy maps for a meso (left) and racemic (right) dyad of PS Low frequency Raman spectra of PBD samples at different temperatures: 8k star thick solid line, 8k linear thin solid line and k linear symbols (a) Temperature dependence of the quasielastic intensity integrated over the range 5-10 cm -1. The dashed lines present linear approximations for two temperature regimes. The intersection of the lines gives an estimate of T x. (b) The same data plotted as a function of T/T g, where T g is taken from the DSC measurements..137 xvi

17 6.3 Light scattering susceptibility spectra of PBD measured at T=93K in a broader frequency range. Significant variations appear in the tail of segmental relaxation (dominates at ν<5 cm-1) Neutron scattering spectra of PBD molecules with total molecular weights and arm molecular weights far above M e. No experimentally significant differences are seen as architecture is varied Dependencies of the glass transition temperature, T g, and of the temperature marking the onset of the glass transition in the fast dynamics, T x, (a) on total molecular weight and (b) on molecular weight of an arm Dependence of segmental relaxation (susceptibility integrated from 0.03 cm -1 up to 0.3 cm -1, Figure 6.3) (a) on total molecular weight and (b) on molecular weight of an arm Dependencies of T g, in three series of phenolic terminated dendritic poly(benzyl ethers) with different core structures (data from reference 33), (a) on total molecular weight and (b) on the number of ends, f, divided by the total molecular weight Dependence of the T g in star PS (data from reference ) (a) on total molecular weight and (b) on the number of ends, f, divided by the total molecular weight DLS susceptibility spectra of (a) PBD with 7% vinyl content at 35K; (b) PBD with 89% vinyl content at 30K; (c) Polyisoprene at 96K DLS susceptibility spectra of (a) PBD with 7% vinyl content and (b) PBD with 89% vinyl content at different temperatures. Lines show results of the fit Temperature dependence of characteristic relaxation time obtained from the fit: solid circles - PBD with 7% vinyl content; open circles - PBD with 89% vinyl content; the line shows Arrhenius fit of the data for PBD with 7% vinyl content. Segmental relaxation time in 1,4 PBD (star, our light scattering data and open triangles, dielectric data from ref.167) is presented for comparison 154 xvii

18 6.1 Scheme of the structures of the hyperbranched PS, from ref Dielectric loss ε vs frequency ν for 15-end-branch PS at different temperatures. Temperature decrease from right to left (from 11.5 to 8.5 o C) Segmental relaxation for all six samples at T T g +0 o C Temperature dependence of τ α of PS with different architectures. Meanings of the symbols are listed in the figure. T g is defined as the temperature where τ α ~1 s..160 xviii

19 CHAPTER I INTRODUCTION Polymeric materials are widely used in daily life, e.g. fibers, plastics, rubbers and thermosetting resins. They display outstanding mechanical properties: strength and toughness. Most of these important properties come from their large molecular weight (MW), compared with the monomers. Many physical properties show asymptotic MW dependence, e.g. the glass transition temperature (T g ). Large MW is desired when these properties are considered. In another aspect, industrial processing usually becomes more difficult as viscosity increases, which is a direct consequence of increase of MW. The balance between these two considerations determines the optimum degree of polymerization of the final product. With the advance of synthetic chemistry, polymers with different architectures can be designed and produced. Hyperbranched, including dendritic polymers have potential applications such as drug delivery 1 ; star polymers show interesting properties in surface segregation. Ring (or cyclic) polymer attracts interests due to their low processing viscosity and low heat of evolution and volatiles during the potential ring opening polymerization 3. 1

20 Variations of MW and architecture of polymers also provide unique model systems to investigate many fundamental problems. Situations are much more simplified in both cases due to the invariance of the chemical structure. For example, ring polymer is the best model polymer to eliminate the chain end effect that is considered very important in the linear polymers 4. Variation of MW is also used to study complex dielectric relaxation behavior 5-7. Dynamics of polymers is complicated due to the complexity of the chain molecules. In the order of decreasing time scale, polymers display various types of dynamics: chain relaxation (coordinated segmental motion), segmental relaxation (structural or α relaxation), secondary relaxation and fast dynamics (Figure 1.1). Figure 1.1 Polymer dynamics at different time scales, represented by the time dependence of the stress relaxation. Secondary relaxation usually merges with segmental relaxation at high temperature.

21 Studies of the influences of both MW and architecture on polymer dynamics have a long history. The main focus of most investigations has been on segmental and chain dynamics. The latter is responsible for the characteristic viscoelastic behavior of polymers. Chain dynamics has been well understood with the development of Bead- Spring Model 8,9. Some progresses have been made in understanding the MW dependence of T g, but due to the mystery of the glass transition itself, a full understanding of both MW and architecture effect on segmental dynamics is still not reached 10. Regarding fast dynamics, including the fast relaxation and the collective vibrations on the picosecond time scale, almost no conclusions can be drawn on MW and architecture effects due to the lack of experimental data. Generally, the level of understanding of the dynamics of polymers decreases as we go to shorter time scale. In this thesis, we will explore the influence of MW and architecture on these types of dynamics with the help of light scattering, neutron scattering and dielectric spectroscopy. We focus on comparing the experimental results to molecular length scales such as the Kuhn length used to describe the chain statistics. Moreover, a few polymers were compared to explore the importance of the chemical structure. This thesis is organized in the following way: Chapter, historical background: I will briefly review important experimental and theoretical investigations for polymer dynamics with the emphasis on MW and architecture effects in the order of decreasing time scale. 3

22 Chapter 3, experimental details: Introduction of the principle, instrumentation of different light scattering techniques and dielectric spectroscopy. Treatment and interpretation of the data are also presented. Chapter 4, depolarized light scattering (DLS) study of the MW dependence of chain relaxation. The transition of dynamic behavior from small molecule to polymeric behavior is discussed. A comparison between two polymers reveals the deficiency of the traditionally defined Kuhn length in explaining the MW dependence. Mechanism of the chain relaxation in DLS is also discussed. Chapter 5, MW dependences of fast dynamics and fragility are explored. Surprisingly, fast dynamics shows strong dependence on MW. Moreover, opposite trends are observed for polystyrene and polyisobutylene, which is ascribed to their difference in symmetry of the substitution in the chemical structure. Chapter 6, influence of architecture on the fast and segmental relaxation: Fast dynamics and T g are found to scale better with total MW instead of arm length. Branching causes variation in local relaxation environment, which broadens the segmental relaxation, and reduces the fragility of the material compared to linear chains. 4

23 CHAPTER II HISTORICAL BACKGROUND.1 Chain dynamics Chain relaxation is characteristic for polymers. Polymer chains are usually described as random coils in the unperturbed states 11,1. To study their viscoelastic properties, simpler and more intuitive model chains are needed. Different single chain models have been adopted in the literature. For linear flexible chains, the Bead Spring Model (BSM) is the most widely used 8,9. In the BSM, a single chain is visualized as a combination of beads and frictionless springs that connect them 8,9,13. This model chain originates from polymer chain statistics. Consequently, it is a freely jointed chain with oriented segment b (spring between two beads), which is composed of a certain amount of monomers. The effect of the medium is reduced to one parameter, the friction coefficient (ξ). The segment b of the BSM is usually defined as the Kuhn segment. It is known that a real chain is not freely jointed. However, if a chain is long enough, its chain statistics can still be described by a Gaussian distribution 11,1. The unit of this equivalent freely jointed chain is called a Kuhn segment. For a Gaussian chain, < R0 >= NlK = C nl0 5 (.1)

24 where R 0 is the end-to-end distance, N and l K are the corresponding number and length of the Kuhn segment, C is the characteristic ratio of the chain, and n and l 0 are the number and length of the bond of the real chain. Assume that the maximum contour length R max of the real chain is equivalent to that of the Kuhn chain 11,1, R = Nl = (.) K max nl 0 Substitute (.) into (.1); we reach the traditional definition of a Kuhn length, l K = C (.3) l 0 The elastic force of the spring originates from the fluctuation of the end-to-end vector R, following the Gaussian distribution, P( R) ~ e 3R / Nb. The resulting entropic force constant is 3k B T. Nb Starting from this model chain, several theoretical models were developed to describe the viscoelastic properties of the linear flexible polymers. The simplest model is the Rouse model, originally developed for dilute solution. Later the Zimm model was developed to include hydrodynamic interactions. The Reptation model (or tube model) was developed by DeGennes to describe entanglement effects, which appears for long chain polymers in the case of concentrated solutions or melts. Detailed descriptions and comparisons of these models can be found in the books by Ferry (for Rouse/Zimm model) and Doi & Edwards (for Reptation model). In our study, only bulk behavior is considered. Thus, the Rouse and Reptation models are relevant. 6

25 .1.1 Rouse model Two basic assumptions for the Rouse model are Gaussian approximation of the chain statistics and local dissipation of the perturbation 8,9,13. In the melt state, the hydrodynamic interactions are screened out, and the Rouse model is believed to work with the assumption of Gaussian chain for chains without entanglements. Applicable situations includes both unentangled chains and entangled chains at time scales shorter than the onset of entanglement, τ e. In the BSM, each segment undergoes a random walk in time, but the whole chain moves as well. The distance a segment moves will be large with respect to the segment size but small when compared with the whole molecule. The so-called Rouse modes are defined as the coordinated motions of certain numbers of segments. They correspond to motions at different time and length scales. Each Rouse mode is a single exponential decay process 13. The shortest relaxation time of the Rouse mode corresponds to that of the time correlation function of a single segment motion, and the longest relaxation time (τ 1 ) corresponds to that of the whole chain. Correspondingly, the length scale associated with different Rouse modes is between the segment size b and radius of gyration of the chain (R g ). Microscopic information about the Rouse chain can be obtained by observing the average segment diffusion <r > of a chain, 1/ ( t) ~ t r (.4) 7

26 Therefore, compared with a small molecule (Fickian diffusion, <r >~t), each segment is slowed down due to the chain connectivity. This so-called sub-fickian diffusion continues until the displacement reaches R g (or the entanglement constraint), at Rouse time τ 1. When <r(t)> R g, 0 1 N N T k b B = ξ τ (.5) Usually, five parameters can be investigated to compare with theory: τ 1,η, D, R g, and relaxation time spectrum {τ k } k =1,,N 8,9. The predictions of the Rouse model for these macroscopic parameters are: (1) Rouse time: τ 1 ~M ; () Steady state viscosity: η~m 1 ; (3) Diffusion coefficient of the chain: D~M -1 ; (4) Radius of gyration: R g ~M 0.5, and (5) ; 1 ) ( 1 ' N p p M RT G G N p N p p p R + = + = = = τ τ ω τ ω ρ τ ω τ ω (.6a) = = + = + = N p N p p p R p p M RT G G ) ( 1 " τ ω τ ω ρ τ ω ωτ (.6b) 8 Relationships (1) to (4) give the molecular weight (MW) scaling of the different properties, which are only related to the longest Rouse mode. On the other hand, equations (.6a and b) give the description of the dispersion of the relaxation spectrum, which is the summation of contributions from all the Rouse modes. It can be seen from this relationship that the Rouse model is a general Maxwell model with a summation over p single exponential relaxation processes. Other mechanical properties such as

27 compliance can be calculated with the knowledge of modulus. For example, steady state compliance, J e 0, is, 0 J e = J g + Ld lnτ (.7) where J g is the glassy compliance due to segmental relaxations contribution at very short times, and the second part comes from the coordinated motion of the segments, that is, the chain relaxation. Usually, J g is entirely negligible compared with the chain retardation. 3 Therefore, the calculation of J 0 e provides a very good test for the chain relaxation contribution. Rouse theory gives the clear prediction that 8,9, J e 0 M = 0.4 (.8) ρ RT.1. Reptation model When MW is sufficiently large (M>M c ), polymers display a characteristic dynamic behavior: entanglement. Most viscoelastic properties show a dramatic change at a certain MW. A plateau region can be found in mechanical measurements and the viscosity starts to show a much stronger MW dependence 8,9. This effect can be described by the Reptation model, which visualizes the constraint of entanglement as a tube that 3 3 confines the motion of a chain. It predicts that τ 1 ~ M, and η ~ M. Compared to a Rouse chain, the mean square displacement of a segment shows more complicated time variation in the Reptation model, as shown in Figure.1. 9

28 Figure.1 Mean square displacement of a chain segment varies with time. The t 1/4 dependence is the fingerprint of reptation behaviour 9. Besides the traditional rheological experiments, many other experimental data show that the reptation model gives a good description of the dynamics of an entangled chain. As examples, neutron scattering experiments by Richter et al. and simulation work by Binder et al. give strong support for the Reptation model For example, recently, Wischnewski et al. reported the direct observation of the transition from Rouse (t 1/ ) to Reptation (t 1/4 ) behavior using Neutron Spin Echo measurements (Figure.) 0. Despite the success of this model, experimental deviations from the model predictions have been found, such as the well-known M 3.4 scaling of viscosity instead of M 3.0. Modified tube models Constraint Release, Contour Length Fluctuation and Tube Dilation, were developed to explain this deviation. Moreover, this model does not explain the nature of the entanglement. It is still not clear how the molecular weight between entanglement (M e ) depends on the chemical structure. An empirical model was developed 10

29 based on the concept of packing length to understand the origin of the entanglement. Details about the entanglement dynamics can be found in the extensive review by Watanabe 1. Figure. Power law dependence of mean square displacement (<r >=-6ln[S inc (Q,t)]/Q ) measured by neutron spin echo experiments Experimental studies of Rouse dynamics Molecular weight dependence With the increase of MW, the dynamics of the linear chain will change from Rouse to Reptation type with the onset of entanglement on a microscopic time scale. This transition has received much attention and been widely studied both theoretically and experimentally 8,9, The tube diameter (or corresponding molecular weight, M e ) is the most important of the viscoelastic properties. On the other hand, the transition from 11

30 diffusion behavior of a small molecule to the coordinated segmental relaxation of a chain has not received enough attention. It is easy to imagine that the system will deviate from the simple Rouse model if there are only a few statistical segments in the chain. In a series of studies by Paul and co-workers using molecular dynamics simulation and neutron scattering, the chain dynamics of a C 100 H 0 melt were compared with the Rouse model 3-5. Three marked deviations of the chain dynamics from the Rouse model were found: (1) Center of mass diffusion for times smaller than the Rouse time is sublinear instead of linear; () Only the first three modes (p=1,,3) follow the Rouse scaling for the eigenmodes: X p ( 0) 1/ p, higher modes do not follow this scaling as shown in Figure.3; (3) These higher modes are not single exponential. It was speculated that these deviations were due to either non-gaussian behavior of the chain or residual intermolecular interactions. Figure.3 Scaling of the amplitudes of Rouse modes for C 100 H 0 melt at T=

31 This failure of the simple Rouse model is more evident in the viscoelastic properties of short MW polymers. Figure.4(a) shows that the diffusion coefficient of n- alkanes strongly deviates from the Rouse prediction (M 1 ), with the exponent varying from -1.8 to-.7, depending on the temperature 6. This deviation was also found in the case of PEO melts, as shown in Figure.4(b) 7. Figure.4 MW dependence of self-diffusion coefficient for (a) n-alkanes, different lines represent different temperatures; 6 and (b) PEO melts 7. Besides these relatively recent studies, earlier mechanical measurements of unentangled polystyrene (PS) 8-31 also revealed deviations from the Rouse model predictions. These deviations were documented by different authors, and stressed in Ferry s book 9. Recently, Majeste 31 summarized all the measurements on the steady state compliance of short MW PS, as shown in Figure.5. The deviation from the Rouse 13

32 prediction is clearly seen at MW shorter than However, no clear answer has been provided so far. Computer simulations show that the time dependence of mean square displacement of a segment does not reach the Rouse asymptotic value (~t 0.5 ) for chains with several statistic segments either Specifically, this exponent is around 0.6. And the Rouse exponent (0.5) has never been observed in mechanical measurements. Figure.5 Steady state compliance of monodisperse polystyrene as a function of MW 31. On the other hand, poly(dimethylsiloxane) ( PDMS) has been shown by both mechanical and neutron scattering measurements to be the polymer to follow Rouse behavior most closely 17,7, The MW dependence of the self-diffusion coefficient of PDMS also follows Rouse predictions, as shown in Figure

33 Therefore, it is not clear at what chain length the dynamics of a chain molecule start to show the Rouse behavior and how this length depends on the chemical structure. As stated by Paul et al., It is a very challenging and fundamental question to what degree a polymer model which has been shown to quantitatively reproduce experimental static and dynamic properties adheres to the simple Rouse model generally used to analyze the melt 3. Figure.6 Self-diffusion coefficients of PDMS melts as a function of MW Chain segment: studies on spatial dependence Details of chemical structures are neglected in the Rouse model. At large length scales, a real chain can be treated as a Gaussian chain with the assumption of Kuhn segment. However, with the decrease of length scale, local dynamics come into play

34 Dynamics at a short length scale will show deviations from the standard Rouse model. Neutron Spin Echo (NSE) provides the opportunity to investigate this problem. With a wide frequency and Q (momentum transfer) range, NSE can measure the single chain structure factor from its coherent scattering 17. Consequently, deviations from the Rouse model have been found for large Q (corresponding to short length scale), and the segment (or dynamic bead) size can be estimated 40,4. Figure.7 shows the estimation of the bead size for PDMS and PS in solution 4. Surprisingly, the segmental size obtained (~55Å) is much larger than the Kuhn length l K for PS (~0Å). On the other hand, the segment size of PDMS (~16Å) is not much different from its Kuhn length (~11-14 Å) 4,43. Interestingly, as we showed above, PDMS is also the system that follows best the Rouse model (Figure.6), while PS shows strong deviations (Figure.5). Figure.7 Dynamic bead size in dilute solution estimated by NSE 4. 16

35 It is known that local conformational transitions of polymers are usually controlled by two parameters: energy difference between the neighboring states (e.g. gauche and trans), E, and the barrier separating them, ε 11,44. Two different models were developed to describe the deviation from the Rouse model. The first one was developed by Harnau et al. based on the concept of bending elasticity, which is essentially a semi-flexible chain model instead of the BSM The stiffness of the chain is modeled by the concept of persistence length As a result, a persistence length instead of a statistical segment is introduced to describe the end-to-end distance distribution. The second is called the internal viscosity approach, first developed by Allegra, then used by Richter et al. in their analysis of NSE experimental results 36,49,50. The local stiffness is described as resulting from an additional dissipative relaxation process such as jumps over barriers. The difference between these two approaches is that the first one emphasizes the static flexibility ( E) and the latter dynamic flexibility ( ε). Although both approaches were capable of describing the structure factors obtained in the neutron scattering experiment, inconsistencies still existed. Arbe et al. compared the Q dependence of the chain dynamics between PDMS and PIB, and found that static stiffness cannot explain the difference observed between them 36. Specifically, PDMS and PIB have similar static stiffness (C, or Kuhn length), but they have a strong difference in the Q dependence of the chain dynamics: the Rouse model works for PDMS up to Q~0.4Å -1, whereas it fails already at Q~0.15 Å -1 in the case of PIB (Figure.8). In other words, the lower limit of the length scale of Rouse dynamics is -3 times larger for PIB than PDMS. Therefore, the static stiffness or traditional Kuhn segment is not capable 17

36 of explaining the observed difference 36. Detailed analysis on data from PIB melts indicates that at Q~0.15 Å -1, the elemental Rouse rate, W, starts to show a Q dependence (Figure.9) 51. On the other hand, simulation work by Krushev et al. show that there is almost no difference in terms of chain dynamics between a freely rotating chain ( ε=0) and realistic chain, only a shift of the diffusion constant 5, inferring internal viscosity cannot be the only reason for the deviations from Rouse model. Figure.8 Intermediate scattering function S(Q,t) at different scattering wave vector Q for (a) PDMS and (b)pib melts. Experimental points are fitted with Rouse model (line). The number represents different experimental Q value

37 Figure.9 Q dependence of the elementary Rouse rate W of PIB 51. The Rouse prediction is that W is invariant with Q as indicated by the shaded region. Therefore, the reason for the failure of standard Rouse model at short length scale is still not clear. Moreover, the concept of sub-rouse modes was proposed by Plazek et al. to describe the abnormal relaxation modes observed between the chain and segmental relaxation modes Experimental observations of sub-rouse modes were reported for PIB in melt and PS in solution. However, no clear microscopic description for this mode has been given and it is unclear whether it is common for all polymers. Hence, there remain questions, such as, What defines the shortest Rouse mode, and why is the dynamic segment different from the traditionally defined Kuhn segment?. Segmental relaxation Segmental relaxation is the terminology used to refer to the α relaxation in polymeric systems. The glass transition temperature, T g, is one of the most important properties related to the freezing of segmental relaxation. It can be measured by various 19

38 techniques. Usually, it describes the temperature where the segmental relaxation reaches 100 s 10,41. Besides T g, fragility is another important property of a glass forming liquid, first introduced by Angell to classify the glass forming liquids 56,57. Fragility can be defined from the temperature dependence of both dynamic and thermodynamic properties. As shown in Figure.10, the kinetic fragility is defined as the temperature dependence of viscosity η or relaxation time τ α at T g, d logη( T ) m = d( T g / T ) T = T g (.9) Figure.10 Temperature dependence of viscosity (Angell s plot) 58. The slope in this plot at T=T g defines the fragility of the material. 0

39 This parameter describes how fast viscosity (or relaxation time) changes as temperature approaches T g. If the dependence is Arrhenius-like, the system is called a strong liquid. Otherwise, it is fragile. That is, the temperature dependence deviates from Arrhenius behavior. Details about the concept of fragility can be found in a recent review by Ruocco et al 59. It has to be noted that the temperature dependence of viscosity cannot be used as the definition of fragility for polymers. As is shown in the above section, viscosity is a function of the chain length, and usually the contribution from chain dynamics has a different temperature dependence than that of segmental relaxation near T 60,61 g. Therefore, for polymers it is proper to define fragility using the temperature dependence of the segmental relaxation time. Understanding fragility and its correlation with other glass properties is one of the ongoing hot topics in the field of glass research. The behaviors of polymers have often been shown to deviate from the correlations observed for other glasses. For example, according to the Potential Energy Landscape (PEL) approach, the heat capacity jump during the glass transition C p (T g ) should increase with the fragility, because for a fragile system, more states have to be visited during the glass transition 58. This correlation is found in the case of non-polymeric glasses. However, such a correlation does not apply for polymer glasses 6,63. Hence, the applicability of PEL to polymers has been challenged. The fragility of polymers can also be estimated through the empirical WLF equation (or equivalent VFT equation). Specifically, Angell pointed out that from the WLF equation: log a T C1( T Tref ) =, C 1 16 because it is the result of ratio of T ( T C ) ref 1

40 relaxation time at T g (τ α ~100s) to that at very high temperature (τ 0 ~10-14 s); and C is directly related to that of the fragility as, m 1 C / T 64. From the equality between ~ g WLF and VFT equation, C =T*-T 0, and usually T g is used as T*. Therefore, m / ~ 1 C / Tg ~ T0 T g. T0 from VFT equation is believed to be similar to the Kauzmann temperature T 58,65,66 K. Measurements on the fragility of polymers have been performed for different series of systems. It has been shown that fragility is a function of MW 67,68, side chain length 69, blend composition 70, and pressure The nature of all these dependences is not clear and the results seem to be inconsistent. In terms of MW dependence of fragility, Roland et al. showed that for PDMS (Figure.11 (a)), fragility is independent of MW 74, whereas for PS it increases strongly with MW (Figure.11 (b)) 67,68. (a) (b) Figure.11 Temperature dependence of segmental relaxation time of: (a) PDMS 74 and (b) PS 68. Different lines represent different MWs.

41 Different models including both thermodynamic and dynamics approaches have been developed in the past to describe the glass transition phenomenon. Reviews and books on this topic can be easily found. Here, we only consider the theories and experimental effects related to polymeric materials, especially the MW effect...1 Gibbs-DiMarzio theory The Gibbs-DiMarzio (G-D) theory was a thermodynamics approach 10,41, Using statistical mechanics methods based on the Flory-Huggins lattice model, the configurational entropy, S c, of a glass forming system can be calculated. Since all the calculations are based on equilibrium condition, G-D theory predicts that there is a true second order phase transition at the temperature T, where S c (T, P)=0. The resulting glassy state is a true equilibrium state. Consequently, the Kauzmann Paradox is resolved for thermodynamic reasons. In the study of glass transitions, usually the empirical glass transition temperature T g is investigated experimentally, and T g is much above the T used in G-D theory. This difference was solved by assuming that at the experimental time scale, the ratio of T /T g is constant. T g can be determined by S c =0 76. The close relationship between T g and T is also predicted by analyzing the temperature dependence of the relaxations based on the Adam-Gibbs theory 77. Polymer researchers have especially favored the G-D theory because it provides explanations for the polymer-specific problems: variation of T g with MW, chain stiffness, cross-linking and composition of copolymer and blends 10. 3

42 For the MW dependence, G-D theory predicts: x ln v0 ( )( x 3 1 v 0 1+ v ) + ( 1 v 0 0 ( x + 1)(1 v ) 0 ln3( x + 1) )ln ( ) = xv0 x ε ε exp( ) ktg ktg ε ln(1 + exp ) ε kt (1 + exp ) g kt g (.10) Where, x is twice the degree of polymerization; ν 0 is the volume fraction of the holes in the lattice, which is controlled by the intermolecular energy E h ; and ε is the flex energy determining the temperature dependence of the shape (configuration), which is related to the intramolecular stiffness. Therefore, the G-D model is a so-called minimal model that takes into account both intermolecular (E h ) and intramolecular ( ε) effects of polymer chains 10. Experimentally, the former can be estimate from the T g of the infinitely long chain, and latter from the thermal expansion above and below T g. G-D theory also gives an interesting prediction that in contrast to linear polymers the T g of a ring polymer should decrease with increase of MW 78, which was experimentally proved to be qualitatively correct for the case of cyclic PDMS 10,79,80. Many authors have tested the G-D theory. The theory can well describe the experimental data in most cases. However, quantitative disagreements for certain systems such as PDMS exist, and the reason is still not known 10. On the other hand, fundamental questions remains such as whether the true second order phase transition really exist at T, how to reach T, and the equilibrium glassy state. 4

43 .. Free volume theory The free volume concept was first used to describe viscosity of liquids by Dolittle. It provides an interpretation for two famous empirical equations in glass forming liquids: the WLF ( log a T τ ( T ) C1( T T*) = log = ) and VFT equations τ ( T*) C + ( T T*) B ( τ = τ 0 exp( ) ). The main idea is that free volume controls segmental relaxation. As T T 0 temperature decreases, the fraction of free volume decreases. When it reaches a certain value at some temperature, the system is frozen. This temperature is T g and the frozen state is the glassy state 10. The MW dependence of T g can also be explained by the free volume theory. It is assumed that the chain ends contribute more free volume because they have higher mobility. The increase of T g is caused by the decrease of chain end free volume with increase of MW. Assuming that each chain end contributes excess free volume θ, the specific volume contributed by chain ends can be estimated as transition happens at an iso-free-volume state, θn M n A. Because the glass θn M n A α [ T T = f g, g ] (.11) where α f is the thermal expansion coefficient of the free volume and T g, is the glass transition temperature of an infinitely long chain. Rewriting this formula gives T M T θn 1 A g ( n ) = g, (.1) α f M n 5

44 Details about different free volume models can be found in McKenna s review 10. Despite its popularity, this approach has been challenged by many experimental results. For example, the assumption of an iso-free volume state of the glass cannot explain the physical aging and relaxation behavior at the glassy state and it cannot explain the T g dependence of polymers without chain ends, i.e. ring polymers. On the other hand, this approach can also be examined by the study of polymers with different architectures, where the number of chain ends can be altered. It predicts that for star polymers, T g depends on the MW of the arm, T g f B = Tg ( ). (.13) M n where f is the number of chain ends. Figure.1 showed this relationship works for polystyrene (PS) stars with different numbers of arms 81. Figure.1 T g scales with the arm length for PS stars 81. Lines represent different methods used to measure T g. 6

45 However, measurements on large MW polybutadiene stars showed that T g is independent of the number of the chain ends even when the number is as large as hundreds 8. Similarly, measurements of PIP stars did not show the influence of number of chain ends 83. Therefore, the key question here is at what arm length the chain end still remains important...3 Coupling model (CM) CM is a relaxation-based glass transition model developed by Ngai 61,84,85. According to this model, there exists a crossover time t c. At times shorter than t c, segments relax independently and exponentially: φ t ) = exp( t / τ ), t < (.14) ( 0 t c here τ 0 is the primitive segmental relaxation time. At times longer than t c, cooperative segmental relaxation starts to take place due to the intermolecular coupling. As a result, the relaxation function follows the stretched exponential behavior, as described by KWW equation: φ ( τ t (.15) 1 n t ) = exp( t / a ), t > c At t c, combine (.14) and (.15), [ n ] 1/(1 n ) t c 0 τ α = τ (.16) This equation gives the correlation of the primitive and cooperative segmental relaxation times. The so-called coupling parameter n is related to the stretching parameter β, n = 1 β (.17) 7