Olle Inganäs: Polymers structure and dynamics. Polymer physics

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1 Polymer physics Polymers are macromolecules formed by many identical monomers, connected through covalent bonds, to make a linear chain of mers a polymer. The length of the chain specifies the weight of the macromolecule, and one and the same polymer may have chains of different lengths and therefore weights. This was one of the reasons that a stubborn resistance was mounted to the concept of macromolecules; they did not have a specific molecular weight and therefore could not count as chemical species, maybe not even exist. It is only with the success of the macromolecular hypothesis, advocated by Staudinger in the early 20 th century, that the existence of macromolecules of this kind was accepted. The growth of polymer physics and polymer chemistry is strongly intertwined; developments of new materials have had a strong impact on the extension of the frontier of knowledge. Still, natural polymers are abundant, and the essential molecules of life are mainly polymers. The genetic code is enshrined in a sequence of four nucleotides cytosine, guanine, adenosine, thymine - in a linear chain of DA; translation of the genetic code is mediated through another polymer, RA; the genetic information is finally transformed into synthesis of proteins, linear chains of monomers from a library of ca 20 monomers, capable of building all the functional protein structures. These may be active in energy transduction in biological systems, as enzymes for catalysis, they may be used for signalling, in receptors, ion channels and ion pumps. The level of synthesis of biopolymers is so refined and advanced, compared to what we can do in the polymer synthesis plant, that properties of synthetic polymers rarely approach what is available through biological materials. Still, it is the properties of synthetic polymers that have largely been influential on the formation of concepts to describe polymers, and it is mainly as structural materials that polymers are being produced, as plastics for coatings and structural elements. The mechanical properties of polymers have therefore been in focus during the first centennium of polymer physics. The first and foremost physical analysis of polymers is therefore that of polymer geometry. That is done with the help of concepts from statistical physics and thermodynamics. The random coil model of single polymer chains in solution We will start our foray of polymer physics by considering a highly idealised object, a model of macromolecules. Consider a chain of mers, a polymer, connected by covalent bonds between mers, of length + mers, and thus incorporating bonds between mers. We may picture this as a string of beads(point masses) connected by small links of length a. What is the length of this chain? Clearly one answer is the total length of bonds between repeating units L=a, which is named the contour length. But suppose that we would like to calculate the distance from mer to mer +. With the help of vectors of unit length a we can construct a vector R = r i [2.]

Figure 2. Reducing a real polymer chain to a mathematical object Consider now that the square of this length is calculated R 2 = ( Error!)2 = Error!ri rj = r i + 2, ri rj [2.

2 Figure 2. Reducing a real polymer chain to a mathematical object Consider now that the square of this length is calculated R 2 = ( Error!)2 = Error!ri rj = r i + 2, ri rj [2.2] i j Should be a large number, and should the direction of vectors ri be completely random, the second term in [2.2] will approach zero, and be negligible by comparison to the first term. This is because the product ri rj will have values between a 2 and a 2, and all equally probable, therefore leading to a sum of zero. For a random walk of this kind we note that the distance from start to end of the chain R = (R 2 ) /2 = a /2 [2.3] This is the dimension of the random coil, an important concept of polymer physics. This is the geometry of the polymer chain, in a first (and second) approximation. Clearly, the assumptions in this derivation may be less realistic, but as often in physics, brute simplifications can be rewarding. ote that the contour length grows as but the end-to-end distance grows as. The number can easily reach 04 in synthetic polymers. The size of the random coil i.e. the distribution of mers in space- can also be evaluated by calculating the first moment of mass distribution. This can be expressed as the radius of gyration, and is another common measure of the size of the coil. It is related to the end-to-end distance by a simple geometrical correction; we can therefore use these measures interchangeably. We noted that the product ri rj is equal to a 2 for i=j, and anything between a 2 and a 2 if i is different from j. This is an example of a stochastic variable, and we know that the central limit theorem of mathematical statistics give a definitive form to the distribution of these if the

3 number of stochastic variables is sufficiently large. This is the case here. If we ignore constants we find that the probability of finding an end-to-end distance of length r is p(r) exp (- r 2 /a 2 ) [2.4] What does this distribution function tell us? It gives us the probability that in an ensemble of polymer chains of length, a fraction p(r) is characterised by having their starting and ending mers located at a distance r. As we now attempt to calculate thermodynamic properties of the single polymer chain, we will need to calculate the number of ways of arranging the polymer chain, that is, how many paths can the polymer make between start and end? What distinguishes the polymer of length + units from the collection of + mers without any covalent bonds is the number of possible arrangements of these objects in space. We see intuitively that the number of ways of arranging a number of mers is very much larger than that of arranging a single polymer chain, however large that number may be. As the theory of polymer physics is basically a special case of statistical physics, we need to be able to evaluate the partition function of polymeric chains, in order to calculate the entropy and free energy of these objects. The random coil model is very simplified, as it does not forbid the chain to cross itself an unlimited number of times, and as it assumes that the sequence of vectors r has no correlation whatsoever. This is not true for real polymers, where the chemical nature of bonding will reduce the number of possibilities of polymer geometry from that of a random walk to something more limited. Still, a very simple relation between the length of the polymer chain and the dimensions of the not-completely random walk can be obtained, in a so called scaling relationship R = (R 2 ) /2 = a, [2.5] Where the is a coefficient between /2 (for ideal solutions) to 3/5 (for good solutions) where the polymer is swollen. The interaction between chains acts to force them apart by repulsion, leading to expansion of the random coil. To count the number of ways of arranging a polymer chain, we may use yet another brute approximation, in modelling the chain geometry as a random walk on a lattice of some dimensionality d, giving the possibility to choose any of z=2d directions. Consider the random walk illustrated in Fig At each node of the lattice we can let the polymer chain continue in four different directions. There are therefore z + number of chain walks on the Fig. 2.2 A random walk on a 2D lattice

4 lattice. If we consider the subclass of self-avoiding random walks (SAWs) on the lattice avoiding lattice nodes which have already been visited we find that the number of SAWs is less than z(z-) -, which is the number of walks where the chain will not cross at closely spaced positions along the chain. At long distances crossing may occur; therefore this is an overestimate. From knowledge of the distribution function for the random walk we can calculate the number of walks taking us from the start to the end of the polymer. This is the number of paths that is allowed for a chain of length + mers. We have already argued the probability that the chain will start at and arrive at at a distance r, being an instance of application of the central limit theorem. We now notice that, the number of paths going from, is the product of the number of walks on the lattice and the probability that a walk will start at and go exp (- r 2 /a 2 )z(z-) - [2.6] We can now calculate the entropy S of the random coil polymer chain, as we know to be S= k B ln( ) according to statistical physics. We estimate the entropy of the polymer chain as S= S 0 k B (r 2 / a 2) [2.7] where the S0 constant now includes all the non-exponential terms. Therefore the Gibbs free energy is thus found to be G=H-TS= G 0 + k B T(r 2 / a 2) [2.8] The funny consequence is that the free energy of the polymer chain is very much similar to the energy of the harmonic oscillator, something that can be a helpful picture when considering the dynamics of polymer chains. Polymer statistics measures of polymer chain length Synthetic polymers are produced by polymerisation, which does not lead to one and the same length of the polymer chain. The variation of chain lengths in a polymer causes variation of many properties, as well as the modes of forming solid materials. Important measures of this chain weight are the number and weight average molecular weights M n and M w. By definition, in a collection of molecules of length I, with the molecular weight of the chain of length I equal to M i, the number average and the weight average molecular weights are M n = Mi i / i [2.9]

5 M w = MiMi i / Mi i [2.0] The M w is therefore the average molecular weight of the sample, with respect to the weight fraction of chains of length i and weight M i The M n is the average molecular weight with respect to the number fraction of chains of length i. The ratio between these numbers, (M n /M w ), is called the polydispersity ratio, and is one of the measures often used to characterise a polymer sample. The polydispersity is always larger than, and can be correlated to the mode of polymerisation used to produce the sample. Polymers in solution There are very direct ways of measuring the average molecular weight of a polymer sample using the properties of polymer solutions. The viscosity and the osmotic pressure of a polymer solution give direct measures of these parameters; light scattering of polymer solutions can also be used to obtain molecular weights. The theoretical background to these studies is the observation that polymer chains in dilute solution are similar to the atoms in kinetic gas theory, used to analyse the behaviour of ideal gases. This is the point of analogy to the theory of ideal gases; consider the polymer chains as balls of an ideal gas. Chains form (almost) random coils of dimensions R= a b, and mass M 0 and these chains are far apart in the dilute solution (on the average). The viscosity of the solution is increased because of the friction with polymer coils, and is in the first approximation linearly proportional to the concentration of these coils, as originally analysed by Einstein. The viscosity of a polymer solution can then be expressed as = 0 (+ +...) [2.] Where the volume fraction of polymer coils is, and where the concentration of polymer coils is c/m polymer chains of dimension ( 4 /3)( a ) 3 = ( 4 /3)( a(m/m 0 ) ) 3 Therefore we find the viscosity to be related to the weight average molecular weight as lim c 0 ( 0 )/c = K M w (3 -) [2.2] This relation allow us to evaluate the character of the solvent, whether leading to expanded ((3 -) =0.8) or ideal chains ((3 -) =0.5). The osmotic pressure is also related to the concentration of polymer coils, but now weighted by another measure in the relation of the osmotic pressure to the concentration c of polymer of number average molecular weight R / = = lim c 0 /c

6 It is therefore possible to obtain the averaged molecular weights by simple measurements on polymer solutions. Polymer solids - structure The formation of solid state materials from polymer chains is in many ways different from the formation of crystalline or amorphous solids from elemental compounds, such as Al or Si forming crystals, such as glass formed from SiO2. The main difference resides in the fact that collections of polymer chains are normally collections of different polymer chains some of them short and some of them long, some of them branched and some crosslinked ( meaning that two long chains are somehow bonded), some of them carrying chemical or structural deviations. There is therefore not the simple element of repetition that forms the lattice in a crystal, or at best, there is a lot more besides this repeating element. This has a decisive influence on the structure of polymeric solids, which invariably are very disordered, as compared to crystalline materials. This disorder is crucial to understand electronic transport and optical processes in polymer electronics. The limitations and possibilities of polymers are due to the consequences of disorder and solubility; melt and solution processing of polymer materials are the normal methods, and are also possible for materials suitable for polymer electronics. If a polymer sample is sufficiently regular that is that by all chemical and geometrical measures the chains are (almost) identical- then there is a possibility of forming polymer crystals. Such crystals consist of polymer chains forming thin lamellae, where polymer are aligned in parallel for short distances, but then switch directions to once more go back in the lamellae(fig). Chain imperfections such as ends, chemical defects accumulate outside the polymer crystal, and form part of the amorphous disorded phase that is (almost) invariably found close to the polymer crystals. One and the same polymer chains can therefore be part of the crystal and the amorphous phase. In highly regular polyethylene it is possible to have crystallinity approaching 98 % (High Density PolyEthylene, or HDPE for short as the crystallinity and density are directly correlated), in Low Density PE (LDPE) the crystallinity is lower. Mechanical properties vary much between these two different grades.

7 Fig. 2.3 Lamellaes are found in semicrystalline polymers, and contain polymer chains which fold back to form a highly ordered structure. Supramolecular ordering of crystalline and amorphous domains are found in spherulites, where ordering of the polymer chains in lamellae (30-50 nm thickness, microns of side) and segregation between crystalline and amorphous domains give rise to structures -00 μm large, causing rotation of plane polarised light and therefore easily visible in the optical (polarising) microscope.

8 Fig. 2.4 Organisation of polymer chains in both crystalline and amorphous regions leads to structuring of a polymer solid. The mechanical properties of a polymer are much influenced by temperature. At low temperatures, lower than the T g of the polymer, all polymers are rather brittle and glass like. Above the glass transition temperature T g, at a point where the polymer chains found in amorphous regions are somewhat more free to move, to respond to mechanical and thermal energy, the polymer will be softer and flexible. Above the melting temperature range of polymers indeed a range, as all polymers are polydisperse in molecular weight and the melting temperature depends on molecular weight the polymer will flow on stress, and behave like a fluid. It is in this temperature range that melt processing of polymers is typically done. If polymers are crosslinked, by chemical or physical means, the behaviour above the melting range is somewhat more complex- we then have a rubber, that will pull back after stretching.

9 Web resources Some references to polymer materials science Some references to polymer physics L.H:Sperling, Introduction to Physical Polymer Science, John Wiley second edition 992 U. W. Gedde : Polymer Physics Chapman&Hall, second ed Daniel I. Bower: An Introduction to Polymer Physics, Cambridge University Press, 2002 More advanced is P.G.de Gennes: Scaling concepts of polymer physics, Cornell University Press, 99 G. Strobl: The Physics of Polymers. Concepts for understanding their structures and behaviour, Springer, second ed. 997

Proteins polymer molecules, folded in complex structures. Konstantin Popov Department of Biochemistry and Biophysics

Proteins polymer molecules, folded in complex structures Konstantin Popov Department of Biochemistry and Biophysics Outline General aspects of polymer theory Size and persistent length of ideal linear