Lecture3Notes - Lecture 3 Models for Polymer Conformation...

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Unformatted text preview: Lecture 3: Models for Polymer Conformation Ideal Chains 1 Monday, September 27, 2010 Recall from last time... • Isomers of polymers exist – Structural Isomerism – Cis/trans (geometric) Isomerism – Stereoisomerism • • • • Isomerism can be characterized using techniques such as NMR spectroscopy Different isomers of polymers behave differently (i.e. crystallization) Isomerism is important in both synthetic polymers and natural polymers Isomerism affects chain conformation • Depending on the side groups on a polymer chain, their arrangement, and the backbone in general, polymers can adopt different conformations • Configuration: permanent geometry that results from the spatial arrangement of the bonds (cis or trans, R or S) Conformation: geometry adopted by a polymer chain as a result of bond rotation • 2 Monday, September 27, 2010 Course Overview 1A 10-10 1 nm -9 10-7 10-8 10 1m -6 10 10-5 10-4 1 mm -3 10 1 cm 10-2 10-1 organ collagen tropohelix red blood cell small blood vessel atoms, e. coli capillaries critical size bone defect bond lengths DNA, virus eyeball tooth 1m 0 10 person Lectures 3 and 4: enthalpic and entropic effects on chain conformation based on interactions of repeat units (not atoms) O HO O H N O N H O HO H N O OH HN H 2N NH Monday, September 27, 2010 NH2 = = 3 Polymer Conformation • The conformation of a polymer chain will be strongly affect by the intraand intermolecular interactions of the chain with itself and its environment • In order to better understand and predict the 3D shape of a polymer chain based on these interactions, we need a starting point • Models of conformation often treat polymers as ideal chains, but the real world (and our experiments) is more complicated DEFINITIONS Ideal chain: no interactions between repeat units far apart along the chain, even if they approach each other in space (do not interact with solvent or itself) Real chain: attractive and/or repulsive interactions between/within repeat units as well as with solvent (interacts with solvent and itself) 4 Monday, September 27, 2010 Ideal Chains • Polymer backbones are flexible due to rotation about sigma bonds • The large number of possible rotations (due to the large number of bonds) means polymers can adopt many conformations (entropically favorable) • Ideal polymer chain conformation can be described as a random walk • The basic relationship between the mean-square end to end distance of a polymer chain and the number of repeat units in the chain is: R 2 = Nb 2 • Polymer size can alternatively (and preferably) be described by a radius of gyration, which is related to the end-to-end distance • Various models of ideal chain behavior take into account geometrical constraints of chemical bonds and atoms 5 Monday, September 27, 2010 Flexibility Mechanisms in Polymers • Consider a single repeat unit of PE; the C atoms adopt a tetrahedral geometry • The bond length between carbon atoms is almost constant at l = 1.54 Å. The angle between neighboring bonds is 68°. • The main source of polymer flexibility is the variation of torsion angles. 6 Monday, September 27, 2010 Flexibility Mechanisms in Polymers The trans state of the torsion angle ϕ is the lowest energy conformation of the consecutive CH2 groups. The changes in the torsion angle lead to energy variations • The gauche (+ and -) interactions are the next lowest energy. The energy difference between gauche and trans Δε determines the relative probability of a torsion angle being in the gauche state in thermal equilibrium; in general, it is also influenced by the values of the torsion angles of neighboring monomers. • The energy barrier ΔΕ determines the dynamics of conformational rearrangements. energy • ΔΕ Δε 7 Monday, September 27, 2010 Flexibility Mechanisms in Polymer Chains • Any section of the chain with consecutive trans states is in a rod-like zig-zag conformation. If all torsion angles of the chain are in the trans state, the chain has the largest possible value of its endto-end distance Rmax. • Rmax is proportional to the number of bonds in the chain skeleton, n, and their projected length (lcos (θ/2) along the contour of the chain. This is referred to as the contour length of the chain. Rmax = nl cos • • • θ 2 Gauche states lead to flexibility in the chain conformation since each gauche state alters the conformation from the all-trans zig zag. Typically, chains are not all trans, but broken up by gauche states; the chain is rod-like on scales smaller than the all trans-length, and flexible on larger length scales Typically all-trans sections comprise fewer than ten main chain bonds Monday, September 27, 2010 8 Possible Chain Conformations • Assume that a chain segment has three local minima (e.g. trans, gauche, gauche seen in PE), and that the chain segment can only exist in one of these three minima (not in the higher energy eclipsing states, for example) • To simplify, we’ll also assume all three minima have the same energy, so we are exactly as likely to find the segment in one conformation as the next • Let’s think about the first two bonds in this segment – how many different possibly conformations do we have, ignoring redundancies? G’ G T’ G’ 3 2! G’ T T’ G’ G’ G T’ G’ Now, if this is just for 2 bonds, imagine a polymer chain! That would be an enormous number of possible conformations. 9 Monday, September 27, 2010 Polymer Chain Conformations • The enormous number of possible conformations allows us to use a statistical approach to gain insight into the nature of polymeric materials and their properties • In order to do this we need a parameter that tells us something about the shape • We use the end to end distance R R 10 Monday, September 27, 2010 Robert Brown (1773-1858) Around 1827 Brown made a systematic study of the “swarming motion” of microscopic particles of pollen. This motion is now referred to as Brownian movement. (Brownian motion). At first, “…I was disposed to believe that the minute spherical particles were in reality elementary units of organic bodies.” Brown then tested plants that had been dead for over a century. He remarks on the “vitality retained by these molecules so long after the death of the plant” Later he tested: “rocks of all ages … including a fragment of the Sphinx” Conclusion: origin of this motion was physical, not biological. His careful experiments showed that motion was not caused by water currents, light, evaporation or vibration. He could not explain the origin of this motion. Many later experiments by others were inconclusive. But by the late 1800’s the idea Brownian movement was caused by collisions with invisible molecules gained some acceptance. 11 Monday, September 27, 2010 Random Walks • For simplicity’s sake, let’s first consider a one dimensional random walk • We can repeat this random walk several times, and end up with a distribution 12 Monday, September 27, 2010 Conformation of Polymer Chain • Imagine a chain of 200 segments • Imagine the polymer chain has a carbon backbone, and that the bond rotational angles limit it to trans and gauche • The occurrence of trans and gauche is weighted according to potential energy J, Mark and B, Erman, Rubberlike Elasticity: A Molecular Primer. John Wiley and Sons, New York (1988) Monday, September 27, 2010 13 Random Walks Now, let’s generalize. Instead of each step having a length of 1, let the length be l : R 2 = Nl 2 R 2 0.5 Freely jointed chain model = N 0.5 l This means that if we have a chain of 10000 bonds each with a length of l, the average end to end distance is 100 bonds! We’d have to stretch it to 100 times its length before the bonds would experience stress There are many many conformations of the chain available, and only one of these is the fully extended chain; thermal motion would cause an extended chain to rearrange to one of these coiled up chains This is the origin of rubber elasticity, and we’ll revisit it later! Monday, September 27, 2010 14 Random Walks • For polymers, we consider steps of equal length, defined by chemical bonds, rather than observing the movement of a particle over an interval of time • Polymer chains have connectivity and volume constraints – they can’t re-occupy an already occupied step, and the connectivity to the previous monomer limits the next step • Steric issues and bond rotations geometrically limit the steps restricted unrestricted 15 Monday, September 27, 2010 Random Walks • In the previous example, we left a few important details out • Bonds angles need to be restricted to reflect reality Freely rotating chain R 1 + cos θ = Nl 1 − cos θ 2 2 C C • We can also account for hindered rotation: Hindered rotation model R 2 θ C ⎛1 + cos θ ⎞⎛1 + ϕ ⎞ = Nl ⎜ ⎟⎜ ⎟ ⎝ 1 − cos θ ⎠⎝ 1 − ϕ ⎠ 2 16 Monday, September 27, 2010 Characteristic Ratio • While the scaling relationship between <R2> and N remains the same, we can add in a prefactor that helps to describe chain conformation better: R 2 = C∞ Nl 2 C∞ = • • R2 Nl 2 This characteristic ratio is the ratio of the actual end to end distance over the random flight model It describes chain stiffness or bond rotational freedom 17 Monday, September 27, 2010 Characteristic Ratio • Depends on the local stiffness of the polymer chain; 7-9 typical for flexible chains • Bulkier side groups tend to lead toward higher C∞, but there are many exceptions • Flexible polymers have many universal properties independent of repeat unit structure 18 Monday, September 27, 2010 Characteristic Ratio Polymer C∞ b(Å) ρ(g/mL) M0(g/mol) 1,4-polyisoprene 4.6 8.2 0.830 113 1,4-polybutadiene 5.3 9.6 0.826 105 polypropylene 5.9 11 0.791 180 PEO 6.7 11 1.064 137 PDMS 6.8 13 0.895 381 PE 7.4 14 0.784 150 PMMA 9.0 17 1.13 655 Atactic polystyrene 9.5 18 0.969 720 19 Monday, September 27, 2010 Kuhn Length • We’ve (somewhat arbitrarily) defined each step as a bond length, but it doesn’t need to be defined this way • Instead of using individual monomers or bonds, we can use segments, that, on average, behave as a freely jointed unit when taken collectively • This is called a Kuhn segment or Kuhn length (b), and it follows the same scaling as the freely jointed chain model 2 N K bK = R 2 = C∞ Nl 2 • Kuhn segments represent a group of monomers; each monomer is not completely freely rotating, but the larger segment is Equivalent freely jointed chain model 20 Monday, September 27, 2010 Wormlike Chain Model (Kratsky-Porod Model) • • • • • A special case of the freely rotating chain model for very small values of the bond angle A good model for very stiff polymers Since the bonds angle is very low, it takes a large number of monomers to make the equivalent freely jointed segment The Kuhn length, b, is therefore very large The characteristic ratio is also very large C∞ = • 1 + cosθ 4 ≅2 1 − cosθ θ Worm like chains are stiff on scales shorter than b, but are not completely rigid and can fluctuate and bend 21 Monday, September 27, 2010 Rotational Isomeric State Model • Most successful model used to calculate the details of conformations of different polymers • Bond lengths l and bond angles θ are fixed • Each molecule is assumed to exist only in discrete torsional states corresponding to the potential energy minima; fluctuations about the minima are ignored • Each of the n-2 torsion angles can be in one of three states - trans, gauche + or gauche - (t, g+, g-) • The whole chain has 3n-2 rotational isomeric states • In this model, the states are NOT equally probable; correlations between neighboring torsional states are included • The relative probabilities of the states of neighboring torsional angles are used to calculate the mean-square end to end distance and characteristic ratio. 22 Monday, September 27, 2010 Summary: Ideal Chain Models FJC FRC HR RIS l fixed fixed fixed fixed θ Free fixed fixed fixed ϕ Free Free Controlled by U (ϕ ) t, g+, g- Next ϕ independent? Yes Yes Yes No C∞ 1 ⎛1 + cos θ ⎞ ⎜ ⎟ ⎝ 1 − cos θ ⎠ ⎛1 + cos θ ⎞⎛1 + cos ϕ ⎞ ⎜ ⎟⎜ ⎟ ⎝ 1 − cos θ ⎠⎝ 1 − cos ϕ ⎠ specific 23 Monday, September 27, 2010 Radius of Gyration • While the end-to-end distance is useful for these models, it’s not something that we can readily measure in the lab • Additionally, end-to-end distance doesn’t work well for branched or cyclic chains • We can measure the radius of gyration of our polymers using a number of techniques (we’ll get specific about techniques in a later lecture) • The mean square radius of gyration, <Rg2>, is defined as the average square distance from all monomers to the center of mass of the polymer • <Rg2> is related to the mean square end to end distance of a polymer chain, and varies with the architecture: Ideal Chains Linear Ring f-arm star <Rg2> = Nb2/6 Nb2/12 [(N/f)b2/6](3-2/f) H polymer (Nb2/6)89/125 24 Monday, September 27, 2010 Radius of Gyration of Rod-like chains • Relationships also exist for rigid polymers • The relationship between Rg2 and end-to end distance is different than that derived for an ideal chain: N 2b 2 L2 R= = 12 12 2 g Rigid Objects Disk Sphere Rod Cylinder R g2 = R2/2 3R2/5 L2/12 (R2/2)+L2/12 25 Monday, September 27, 2010 Self-Avoiding Walk • So, we’ve accounted for the bond angle issue, and in some ways, the hindered rotation • What about the volume issue? Two segments cannot occupy the same space • This is a more difficult problem • The bottom line: R2 A more generalized version: ν tells us specific information about the polymer conformation (shape) Monday, September 27, 2010 0.5 = N 0.6 l R2 0.5 = constant × N ν R 2 0.5 = KM ν R = KM ν Random, self-avoiding walk in 3D Really important Equation 26 A Brief Summary A polymer can adopt many conformations Polymers with no interactions between monomers separated by many bonds along the chain are called IDEAL CHAINS The mean square end to end distance for an ideal (long, linear) chain is: R 2 ≅ C∞ nl 2 C∞ is a characteristic ration that describes the stiffness of the chain We have defined the Kuhn segment, b, for a polymer chain, which is related to the number of monomers n. From the Kuhn segment, we obtain an expression for the mean square end to end distance for ideal chains Several models of ideal chain conformation exist: freely rotating chain, freely jointed chain, worm-like chain, hindered rotation and rotational isomeric state 27 Monday, September 27, 2010 Summary Cont. The mean square radius of gyration, <Rg2>, is defined as the average square distance from all monomers to the center of mass of the polymer It is related to the mean square end to end distance REFERENCES: Essentials, Chpt 8 Fundamentals Chpt 7 Strobl Chpt 2 Rubenstein/Colby Chpt 2 28 Monday, September 27, 2010 Next Time Real chains … review the types of interactions References for next lecture: Strobl Chpt 2-3 Fundamentals Chpt 9 Essentials Chpt 11 Rubenstein/Colby Chpt 3 29 Monday, September 27, 2010 “Structure and chain conformation of a (1->6)-a-D-glucan from the root of Pueraria lobata (WiIId.) Ohwi and the antioxidant activity of its sulfated derivative” H. Ciu, Q. Liu, Y. Tao, H. Zhang, L. Zhang, K. Ding. Carbohydrate Polymers 2008, 74, 771-778 • Investigating the physical properties and structure of a polysaccharide from a medicinal herb, P. lobata • The herb is used in Eastern medicine to manage diabetes, hypertension, heart disease Rg 2 1 2 = 0.0112 M 0.56 ± 0.008 30 Monday, September 27, 2010 ...
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