Polymer Science. Prof. Dr. J.P. Rabe and Prof. Dr. I. Sokolov

Transcription

1 Polymer Science Prof. Dr. J.P. Rabe and Prof. Dr. I. Sokolov June 30, 2004

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3 Contents 1 Polymer Physics Polymer Physics in a nutshell Ideal chains Polymer chains: from chemistry to physics Ideal chain model The number of configurations and thermodynamics Entropic elasticity of the chain Scaling considerations A mathematical approach to random walks Ideal chain in an external field Excluded volume effects Introduction The Flory approach The lattice model for an excluded-volume chain The structure functions Interaction of excluded-volume chains The notion of blobs Melts and Solutions Polymers in melts Polymer solutions: a Flory-Huggius approach The osmotic pressure of polymer solutions Phase diagram of a polymer solution Polymer blends and block-copolymers Single molecule behavior in bad solvents Polymer Networks The rubber elasticity Polymer networks and percolation Classical approach to polymer networks Dynamics of polymer systems The Rouse model

4 2 CONTENTS Hydrodynamic interactions Concentrated solutions and melts: the reptation model 96 2 Polymer Characterization Introduction Molar mass determination Introduction Colligative methods Osmometry Ultracentrifuge Light scattering Mass spectroscopy Chromatography Electron microscopy Scanning probe microscopy Optical spectroscopy Basics of photophysics Selected (one-photon) photo physical processes Energy transfer processes Infrared spectroscopy Molecular modelling Molecular mechanics potentials Molecular dynamics potentials X-ray and neutron scattering Introduction Basics of x-ray scattering Basics of neutron scanning X-ray and neutron reflectivity Mechanical testing Introduction Viscous fluids Elastic solids Viscoelasticity Creep, stress-strain measurements, dynamic mechanical testing Appendix Symbols Chemical formulae Literature Information about the script

5 Chapter 1 Polymer Physics 3

6 4 CHAPTER 1. POLYMER PHYSICS 1.1 Polymer Physics in a nutshell What is really important to know about properties of single polymer chains? 1. Polymers are long chain molecules consisting of many repeating units segments (chemically: monomers). All polymeric molecules are flexible at larger scale. This property is described by a characteristic persistence length of the molecule. This is a length along the contour of the molecule after which its initial orientation is forgotten. At larger scales the configurations of the chains can be considered as rather random. 2. If we consider the configurations as fully random and neglect all possible interactions between different parts of the chain (phantom chains), we arrive to random-walk-like models. The simplest of them is a simple random walk on a lattice. Each segment of the chain (monomer, or a superunit of the size of persistence length) can be considered as a step in a random direction on a lattice. 3. The RW model gives the distribution of the end-to-end distance of the chain as the function of the number of segments. Both, the method and the result, are worth knowing. ( N ) 2 R N 2 = r i = ri rj = r 2 i + 2 i=1 i=1 i j N r 2 i + 2 ri r j N r i 2 = Na 2 i=1 i j } {{ } 0 4. If we consider the end-to-end distance distribution in an ideal chain, it is Gaussian. There is a deep mathematical theorem (the central limit

7 1.1. POLYMER PHYSICS IN A NUTSHELL 5 theorem) which states that it is always the case if one has to do with the sums of independent, identically distribution random variables: p (x) = 1 2πσ 2 e x2 /2σ 2 σ = (x x) 2 5. The overall number of configurations of a chain on a lattice with coordination number c is c N. The number of configurations of a chain with fixed end-to-end distance is W (x) = C N p (x) (essentially, per definition of the probability). (a) There is a deep connection between the thermodynamical property called ENTROPY, S, and a number of possible configurations (ways) in which a state of a system under given values of external parameters (constrains): S = S 0 + k b ln W. This relation (due to Boltzmann) gives us a possibility to calculate many important, thermodynamically relevant quantities without really going into details of microscopic calculations. The number of configurations W can often be found via combinatorical considerations. (b) In experiment we typically have to do with isothermic conditions. The appropriate thermodynamic potential to describe such conditions is the FREE ENERGEY F. F = U T S U: internal energy, the mean value of all mechanical energies in the system (kinetic, potential, interactions with external fields) T : Temperature, S: Entropy For an ideal chain model (no interaction, U is absolutely independent on the conformation ) F = T S The whole behavior of the chain has an ENTROPIC NATURE: it is not due to the interactions but fully due to thermal motion. A polymer chain at T = 0 does not show any elasticity and hangs as a piece of a flaccid rope. Note: It is not always easy to grasp the statistical mechanical recipes. Nevertheless, be grateful that they exist.

8 6 CHAPTER 1. POLYMER PHYSICS 6. The elasticity of the chain can be calculated using f = df dx = T x cn p (x), where c is the coordination number. We get: f = 3k b T R Na 2 = 3k bt R R 2 0 THE RULE OF THUMB. In discrete models like our random-walk chain the entropy is the order of k b per a degree of freedom Each constrain decreases the entropy by an amount of an order of k b. This increases the free energy. Since the difference in a free energy between the two states, 1 and 2, is just the work necessary to bring the system from state 1 into the state 2, imposing external constrains ALWAYS IMPLIES DOING WORK! 7. A polymer chain in a pore exercises pressure on the pore s walls. It is not easy to imprison the chain: this reduces its entropy (i.e. freedom) and thus implies performing work. Interestingly enough, the pressure on the walls of an anisotropic pore is anisotropic. There is a huge difference between an ideal gas and a polymer! 8. A very important method used for explaining the properties of polymer chains is based on SCALING or DIMENSION considerations. It typically gives correct results up to numerical factors but needs good understanding of the physics: It starts from determining what parameters can be relevant for the description of the corresponding phenomenon. For example, for a single polymer chain only two such parameters exist: these are the k b T and the unperturbed chain radius R 0 = Na 2. The real art is to combine them in a correct way to obtain the physically meaningful results. Note: Such rough consideration have always to be checked by EX- PLICIT CALCULATIONS, by numerical SIMULATIONS or by EX- PERIMENTS. 9. A model based on ideal chains is not always relevant. For example, it is not sufficient for chains in good solvents. The monomers of a real chain effectively repell each other the molecule swells. The equilibrium configuration of the molecule is due to the equilibrium of this repelling force and the molecule s elasticity. Based on this rough

9 1.1. POLYMER PHYSICS IN A NUTSHELL 7 argument, Flory (Nobel prize 1974) had found that R an ν ν 3. This approximation for ν is remarkably good. 5 with 10. Two excluded-volume molecules repell (although not to strong). The same is true for two parts of the same molecule. The repulsion of the parts of the molecule leads to the concept of blobs: it is often helpful to consider the molecule as a string of blobs. What is really important to know about polymer melts and solutions? 1. These are only very dilute solutions in which the chains behave as independent. Since the typical chain size is R an ν, the concentration of chains n < n 1 R 3 1 a 3 N 9/5 in such dilute solution is very low. The volume concentration of monomers Φ = Nn N 4/5 a 3 is also very low. (n = c/n, c: concentration of monomers) 2. If the concentration of chains n > n, they overlap: we enter first the regime of a semi-dilute solution (n > n, but still Φ << 1), and then one of a concentrated solution (Φ 1) and melt (no solvent at all). 3. The chain in a good solvent swells due to the effective repulsion between monomers. In a melt, the monomers are both inside and outside of the spatial region occupied by the chain, the corresponding forces acting inwards and outwards cancel (Flory, 1949). Chains in a melt behave as ideal ones! Note: This picture is pertinent to three dimensional case. 4. A simple model describing behavior of polymer solutions was proposed by Flory and Huggins. This is a lattice model, in which the sites of a cubic lattice can either be occupied by monomers forming a chain, or

10 8 CHAPTER 1. POLYMER PHYSICS by independent solvent molecules. The free energy in this model (up to uninteresting constants and to terms which are linear in Φ) is given by: F = F 0 + N 0 k b T χφ 2 + F 0 = O (0) + O (Φ) Φ N ln Φ }{{} translational entropy of chains + (1 Φ) ln (1 Φ) }{{} entropy contribution of the solvent χ: a Flory-Huggins parameter A Flory-Huggins parameter characterizes the interaction between the monomers and the solvent molecules, and between the solvent molecules among themselves. It equals to: χ = 1 k b T [ E MS E MM + E SS 2 ] The entropy contribution of chains is N times smaller than in the case when the monomers were not connected. This is a so-called translational entropy of the chain: if the first monomer is fixed, all other give essentially a volume-independent (= uninteresting) contribution. 5. The value of χis the only parameter describing interactions within the Flory-Huggins approach. For small Φ(dilute regime) one has F k b T N 0 Φ N ln Φ (1 2χ) Φ Φ The case χ = 0 corresponds to the so-called athermic situation (no interactions at all, ideal solvent); the case χ < 1/2corresponds to good solvents and the case χ > 1/2to bad solvents. The case χ = 1/2 corresponds to a so called θ-situation. Since χ is an effective parameter depending, say, on temperature, one can speak about a θ-temperature, etc. 6. If we consider a single chain in a dilute solution, it swells in good solvents, and takes its ideal conformation at θ-point. In bad solvents the chain shrinks and forms a droplet-like, globular state. For soft, flexible chains (for which the Flory-Huggins theory is a reasonable description) this transition corresponds to an sudden collapse.

11 1.1. POLYMER PHYSICS IN A NUTSHELL 9 7. The osmotic pressure of polymer solutions within a Flory-Huggins theory is given by p osm = k [ ] bt Φ a 3 N ln 1 1 Φ Φ χφ2 The special cases are: (a) dilute solution, Φ << 1/N p osm = k bt Φ a 3 N k bt c N k bt n, an ideal gas of chains (b) semi dilute case, 1/N << Φ << 1 p osm k bt (1 2χ) Φ2 2a3 χ < 1/2 Note: p osm 0 The case χ > 1/2 (bad solvent) corresponds to the separation of the homogeneous solution into a polymer-rich and a solvent-rich phases. (c) concentrated solutions p osm = k bt a 3 ln 1 1 Φ The accuracy of the Flory-Huggins theory in situations (a). and (c). is pretty good, while for the case (b). of semi-dilute solutions it is not enough. 8. Scaling consideration for the osmotic pressure in a semi dilute solution (athermic solvent) is based on the following scaling assumption: p osm = k b T c ( c ) N f N R2 F If the chains overlap strongly, they do not know, whether the monomers are their own or belong to other chains the N-dependence must cancel. Noting that R F an 3/5 we get then p k bt a 3 Φ9/4

12 10 CHAPTER 1. POLYMER PHYSICS (and not Φ 2 as the Flory-Huggins theory proposes). 9. In bad solvents the polymer solution separates into a polymer-rich and a solvent-rich phases. Such phase separation starts at χ min = N and for Φ c = 1 N In the vicinity of the critical point the system can be considered as a dense packing of practically ideal chains. 10. Flory-Huggins theory can also be used to describe the behavior of polymer blends (a mixture of chains of length N A and N B, interaction parameter χ). In this case F N 0 k b T Φ A ln Φ A + Φ B ln Φ B N A N }{{ B } translational entropies + χφ A Φ B Φ B = 1 Φ B Polymer blends demix easier than solutions, the critical point corresponds to χ c = Φ (A) c ( ) 2 N 1/2 A + N 1/2 B 2N A N B N 1/2 B N 1/2 A + N 1/2 B The case N B = 1 cooresponds to the polymer solution. For N B N A = N, χ c 2/N << 1: The blend starts to demix as soon as the interaction gets to be repulsive. 11. A special case of blends corresponds to block-copolymers where immiscible polymers A and B form the two parts of the same chain and

13 1.1. POLYMER PHYSICS IN A NUTSHELL 11 can t really separate. In this case one encounters the effect of microphase separation, the system consists of A-rich and B-rich domains of well-defined size and form. Typical situations (depending on the relative length of the chain) correspond to micelles, cylindrical domains or lamellae. What is really important to know about polymer networks? 1. Cross-linking the chains in a polymer melt or concentrated solution (vulcanization) leads to emergence of rubber-like properties. The theory of rubber elasticity (Flory!) starts from the two postulates: (a) The chains in a melt are ideal (b) The melt is a liquid, its volume does not change strongly under the deformation. Assuming the affine deformation (i.e. that each subchain is deformed in the same way as the whole piece) we get that the stress caused by elongation λ is σ = k b T υ (λ υ/λ 2 ), where υis the concentration of cross-links. The Young s modulus for small elongations is E = 3k b T υ. The Flory theory of rubber elasticity is reasonable up to λ 5, υhas to be not too small. 2. For small υmost of chains are not connected to the elastic skeleton of the network and are either free or form dangling ends. The infinite cluster of connected chains is formed at some υ = υ c. Similar situations take place in many other cases of gelation. Many of such cases are well-described by a lattice percolation theory. The theory (operating with sets of nodes or sites, which are connected by bonds which may be present or absent with probability p) shows that for p < p c only finite clusters of connected sites are present, while for p > p c an infinite cluster exists. p c is lattice dependent, Cp c 1 (C is the coordination number), often Cp c d/ (d 1)(Scher-Zallen invariant). The behavior of the clustermass ( degree of polymerization ), cluster size ξor elastic moduli near the percolation threshold is described by universal power laws. M (p c p) γ, ξ p p c ν for p < p c Note: this ν is not the Flory exponent!

14 12 CHAPTER 1. POLYMER PHYSICS S (p p c ) β (fraction of monomers belonging to inf. cluster) for p > p c E (p p c ) t for p > p c The exponents γ, ν, β, t do not depend on a lattice type and depend only on the dimension of space. 3. A classical picture of gelation due to Flory and Stockmeyer considers this process as a growth of tree-like clusters (dendrimers), it neglects the possibility of forming the loops. Within this picture the percolation problem is exactly solvable, the estimate for p c 1/ (C 1) is not very bad, but the description of M, ξ, S and E in vicinity of the transition point is rather poor. What is really important to know about polymer dynamics? 1. Dynamics of a single chains determines the diffusion coefficient of single chains. Simplest model: the Rouse chain (1953), an ideal phantom chain (no interactions between the monomers) in an immobile solvent assumes independent motion of different monomers. This model describes the chain as consisting of beads and springs (which may symbolize monomers and chemical bonds or larger subunits). Beads immersed in a solvent experience friction and uncorrelated thermal forces (Brownian motion). The overall behavior of the chain can be described in terms of superposition of modes (Fourier representation). The predictions of the Rouse model are often at variance with the experimental findings. The analysis shows that it is the assumption of the immobile solvent, which is crucial. 2. The Zimm model (1956): The hydrodynamic interaction is taken into account. The motion of each monomer causes the solvent motion which acts an all other monomers. This interaction is described by a so-called Oseen tensor. This model leads to better predictions. 3. Concentrated solutions and melts: a reptation model. The main property of polymer fluids is viscoelasticity (e.g. silicon flows under small constant force, but a silicon or latex ball bounces when thrown on the floor!). The first theory of polymer fluid dynamics was developed by P.G. De-Gennes (Nobel prize 1991), M. Doi and S.F. Edwards in 1970s. The chain s motion is strongly hindered by other

15 1.1. POLYMER PHYSICS IN A NUTSHELL 13 chains restricting its lateral motion. The motion along the contour of the chain is unrestricted, as if the chain moved in a tube. Stages of reptation: (a) The chain crowls out of the tube and reorients. (b) The old configuration is forgotten

16 14 CHAPTER 1. POLYMER PHYSICS 1.2 Ideal chains Polymer chains: from chemistry to physics Polymers are long chain molecules consisting of many repeating units-segments (chemically: monomers). All polymeric molecules are flexible at larger scale. This propertyis described by a characteristic persistence length of the molecule. This is a length along the contour of the molecule after which its initial orientation is forgotten. At larger scales the configurations of the chains can be considered as rather random G. Staudinger discovers the linear structure of polymers and introduces the notion of macromolecules. Examples: polyethylene, polystyrol Not necessary the carbon chain: polymethylsiloxan The chains are very long N >> 1 (for synthetic polymers N ), N is a large parameter of the polymer theory. Homopolymers and heteropolymers Previous examples were homopolymers. All structural units were the same. In many cases these units are different. Examples: Biopolymers DNA (4 types of monomers) and proteins (20 types of monomers). synthetic heteropolymers, e.g. block-copolymers like AAABBAAA... Note: Not all polymers are simple linear chains. The chains can branch forming stars, brushes and regular or irregular trees (dendrimers). Physics of polymers The physically most interesting properties of polymeric materials are due

17 1.2. IDEAL CHAINS 15 to the large-scale properties of polymer molecules, which can be considered as flexible, one-dimensional objects. Observation: All long macromolecules are flexible. The mechanism of the flexibility is different for different chains. 1. normal elasticity (Hook s law) 2. rotational isomers (e.g. in polyethylene) The valent angle γ is practically fixed, γ = for typical compounds. The azimuthal angle ϕ may take different values, typically in vicinity of local energy minima. The probability to find a particular value of ϕ is given by the Boltzmann factor ( p (ϕ) exp E ) k b T

18 16 CHAPTER 1. POLYMER PHYSICS Thermal activation leads to jumps between different probable ϕ -values. Characteristic time of such jumps is ( t τ 0 exp E ) b, with τ s k b T If E b k b T we have to do with dynamical flexibility (typical case). A rough picture of a polymer molecule: a coil Scales in polymer physics At small scales the atomic structure of the molecule is resolved. This scale domain belongs to the realm of chemistry. At intermediate scales a molecule can be described as a curved elastic rod. Its typical radius of curvature is given by the persistence length l. If is a contour length along the chain (a chemical distance), and θ is the angle showing the change in orientation (see the Fig.) then: ( cos θ (s) = exp s ) l

19 1.2. IDEAL CHAINS 17 At scales much larger than l the adequate picture of a macromolecule is given by a random coil. At large scales the properties of a macromolecule do not depend on particular details of chemical sequence simplest models are the best. A model: The freely rotating chain: here a two dimensional model ϕ i [0, 2π],ϕ i are independent, a i = a, analog in 3D The mean end-to-end distance R = R 2 = = = N a i i=1 ( N ) 2 a i i=1 N ( a i ) i=1 N ( ai ) i=1 N N i=1 j=i+1 N i=2 i j=1 a i a j a i a j }{{} 0 = Na 2 the typical radius of the molecule R N 1/2. Note: L: a contour length N = l/a R 2 = Na 2 = La

20 18 CHAPTER 1. POLYMER PHYSICS For all models one can introduce an effective segment length l (analog to a) so that R 2 = Ll l - the Kuhn s length (1934). For persistent chains l = l. Example: The behavior of a persistent ( worm-like ) chain. Parallel to our discrete consideration ( L ) 2 L L R 2 = ds U (s) = ds U (s) ds U (s ) U(s) = d r ds L L : a unit vector in the chain direction L s L = dsds t = s s U (s) U (s ) 0 0 cos θ(s s ) }{{} = 2 ds dt cos θ (t) 0 0 L L s ( = 2 ds dt exp t l ) [ ( )] L = 2 l exp 0 0 { L l l R 2 L = 2 for L << l 2L l for L >> l for L << l the molecule behaves as a rigid bar for L >> l the molecule is flexible; its Kuhn segment length is 2 l. In general: The properties of all models at scales L >> l are similar. Further links no links available yet Ideal chain model If we consider the configurations as fully random (phantom chains) and neglect all possible interactions between different parts of the chain, we arrive

21 1.2. IDEAL CHAINS 19 to random-walk-like models. The simplest of them is a simple random walk on a lattice. Each segment of the chain (monomer, or a superunit of the size of persistence length) can be considered as a step in a random direction on a lattice. The random walk (RW) model (Orr, 1947) [Mathematical background: the RW model is statistics, Rayleigh 1880, Bachelier 1900, Pearson 1905]. A coarse-gained model, a close relative of valent-angle-models. 1. the simplest model for estimating the number of possible configurations calculation of entropy. 2. the simplest model for numerical simulations of thermodynamical properties of polymers. Two ways of discussion: Direct enumeration. On our square lattice there are C = 4 ways to choose the next step. (C coordination number) the overall number of configurations of a chain of N steps is C N For the sake of simplicity, let us first consider a 1d-model (C = 2).

22 20 CHAPTER 1. POLYMER PHYSICS How many ways are there to come from beginning (B) to end (E) in N steps (How many configurations with end-to-end distance R)? The chain makes N R = N 2 + R 2 steps to the left and N L = N 2 R 2 the right. The number of ways to arrange this is steps to W (R) = C N R N = CN L N = N! N R! (N N R )! CN K : binominal coefficient The probability to find a way ending at R P N (R) = CN R N C = N! ( N N 2 R ) ( N! R 2 a binominal distribution ) 2 N! A Stirling formula: ( ln (N!) = N + 1 ) ( ) ln (N) N + ln 2π 2 Why? Here is a very rough explanation. ln (N!) = ln ( N) = N ln (k) N k=1 1 ln (k) dk = N ln (N) N + 1

23 1.2. IDEAL CHAINS 21 in comparison with one above, the terms of the order O (N) are lost. Let us use the full form (*) in Eq (1): n n! ln n S(n) n! appr = exp S (n) δ, % (( N ln (P N (R)) = N ln (2) + ln (N!) ln 2 R ) ) (( N! ln R ) )! 2 = N ln (2) + N ln (N) + 1 ln (N) 2 N + ln ( N 2π 2 R ) ( N ln 2 2 R ) 2 N 2 R 2 ln ( N 2π 2 + R ) ( N ln R ) + N R 2 ) P N (R) = exp (ln (p)) 1 = exp ( R2 2πN 2N (a Gaussian distribution) If we wouldn t take a = 1 we would get ) P N (R) 1 = exp ( R2 2πNa 2Na 2 In d-dimensions ( 4 possibilities in 2d or 6 possibilities in 3d) MULTIDIMENSIONAL Gaussian distributions. The overall result reads ( ) d/2 ) d P N (R) = exp ( dr2 2πNa 2Na 2 For example in three dimensions: R = X e x + Y e y + Z e z (all lengths in units of a). P N (R) = (2d) }{{} C N = N! N 1! (X N 1 )!N 2! (Y N 2 )!N 3! (Z N 3 )!

24 22 CHAPTER 1. POLYMER PHYSICS N 1 : number of steps to the right, (X N 1 ): number of steps to the left N 2 : number of steps up, (Y N 2 ): number of steps down N 3 : number of steps forward, (Z N 3 ): number of steps backward With an additional condition 2N 1 X + 2N 2 Y + 2N 3 Z = N N 3 = X + Y + Z + N 2 (N 1 + N 2 ) N: the overall number of steps Using the Stirling formula for the factorials we get: 3 ( ) P N R = 1 2π N exp d a X2 2 N exp Y 2 d a2 2 N exp Z2 d a2 2 N d a2 = P N d (X) P N d (Y ) P N d (Z) Now X,Y,Zare the coordinates of the end of the molecule On the average N steps in each direction d Further links no links available yet The number of configurations and thermodynamics There is a deep connection between the thermodynamical property called EN- TROPY, S, and a number of possible configurations (ways) in which a state of a system under given values of external parameters (constrains), can be realized: S = S 0 + k b ln W This relation (due to Boltzmann) gives us a possibility to calculate many

25 1.2. IDEAL CHAINS 23 important, thermodynamically relevant quantities without really going into details of microscopic calculations. The number of configurations can often be obtained via combinatoric considerations. In experiment we typically have to do with isothermic conditions. The appropriate thermodynamic potential to describe such conditions id the FREE ENERGY F. F = U T S U: internal energy, the mean value of all mechanical energies in the system (kinetic, potential, interaction with external fields) T : Temperature, S: Entropy. The change F in the free energy is equal to work performed under isothermic conditions. For an ideal chain model (no interaction, U is independent on the conformation and can be set to zero) F = T S To see how it works, consider a similar model: a lattice gas. The particles do not interact, so that U = 0 again. number of sites M = L 3 V

26 24 CHAPTER 1. POLYMER PHYSICS number of configurations, N molecules distributed over M sites W = M N V N (different part) S = k b ln W = k b N ln V F = T S = k b T N ln V W = M N /N!(identical part) S = k b ln W k b N(ln V ln }{{} N ) Stirling F = T S = k b T N(ln V ln N) p = F V = Nk bt = RNT V V This is the well-known equation of state of the ideal gas. Note: This result stresses that the behavior of the ideal gas has an EN- TROPIC NATURE: it is not due to the molecular interaction, but fully due to the thermal motion! If the energies of different configurations differ, the useful (equivalent) form of the free energy F is F = k b T ln (Z) Z: a partition function Z = i ( exp ε ) i k b T The( sum goes over all states (i.e. configurations), ε i is an energy of state i, exp ε ) i is a Boltzmann factor (occupation probability for a state i). k b T How to see this? U = F + T S entropy: S = F T

27 1.2. IDEAL CHAINS 25 calculate: F + T S = F T F T = k b T ln (Z) + T k b ln (Z) T k b T ( i E i exp ε ) i i exp ( ε i k b T k b T ) ε i = U ( i ε ) ( i exp ε i k b T 2 k b T Z ) Note: All independent variables of F except for temperature, are extensive, i.e. proportional to the size of the system. This fact will be repeatedly used in the following. The work of external forces under isothermic conditions (T = const.) is equal to the change in free energy F. For example, if we increase R from R to R + R we have F = A = f R. f: the tension force of a molecule f = F (the force with which it acts on a nail is f = F R ) R Further links no links available yet

28 26 CHAPTER 1. POLYMER PHYSICS Entropic elasticity of the chain In some respect the behavior of a polymer chain resembles one of an ideal gas: The whole behavior of the chain has an ENTROPIC NATURE, it is not due to the interactions but fully to thermal motion. A polymer chain at T = 0 does not show any elasticity and hangs as a piece of a flaccid rope. For a RW chain all configurations have the same energy (say 0) Z i 1 W, the number of configurations, and F = k bt ln (W ). In the case of the one-dimensional chain model (1D RW) with a = 1, W (R) = C N R N = N! ( N 2 R )! 2 ( N 2 + R 2 )! 2 N P N (R) Let us return to our expression for ln (p N (R)) (( N ln W (R) = N ln 2 + ln P N (R) = ln (N!) ln 2 R 2 N ln (N) ln (N) N + ln 2π }{{} uninteresting since not R-dependent N 2 R 2 ln 2π ( F = k b T N 2 R ( N 2 + R ) ( N ln 2 R ) + 2 ) )! ( N 2 R (( N ln ) ln ) ( N ln 2 + R ) + N R 2 other terms {( }}{ N 2 + R ) ln ( N 2 + R R ) )! 2 ( N 2 R ) 2 ) +const and f (R) = k bt 2 [ ( ) ( )] N + R N R ln ln 2 2 Small R: f (R) k b T R N (Hook s law for R << N) The force diverges when R approaches N, and the chain is fully stretched.

29 1.2. IDEAL CHAINS 27 Note: Our derivation was for d = 1. For a three-dimensional chain we shall get for R small: f (R) 3k b T R (R- displacement in the units of a) N If we restore a, we shall get: f (R) 3k b T R Na 2 = 3k bt R R 2 0 The same can be obtained from a Gaussian distribution (as an approximation). In 3D: F = k b T ln (Q) k b T ln ( 6 N p N (R) ) ( k b T N ln (6) 3 ( ) ) 3 2 ln 3R2 2πNa 2Na 2 R 2 = F 0 + k b T 3 2 Na, (F 0: R-independent) 2 f = F R = 3k bt R Na = 3k bt R 2 R0 2 Note: The description based on the Gaussian distribution does not predict the force divergence at large elongations it works only for R << R 0. This consideration is an important step in explaining the entropic mechanism of rubber elasticity (see in detail in Section 3). Further links no links available yet

30 28 CHAPTER 1. POLYMER PHYSICS Scaling considerations A very important method used for explaining the properties of polymer chains is based on SCALING or DIMENSION analysis. It typically gives correct results up to numerical factors but needs a good understanding of the physics. It starts from understanding what parameters can be relevant for the description of the corresponding phenomenon. For single polymer chains only two such parameters exist: these are the k b T and the unperturbed chain radius R 0 = Na 2. The real art is to combine them in a correct way to obtain physically meaningful results. As an example consider again the elastic force. Let us calculate the elongation as a function of f. What can x depend on? Clearly on f, on T and on R 0 as the only characteristic of an unperturbed chain (R 0 = N 1/2 a, but both characteristics Nand a themselves are microscopic irrelevant) Let us consider x as a function of f. ( x X of the end!) ( ) fr0 x R }{{} 0 g }{{} k b T }{{} : the only relevant characteristics of the dimension of length : some function : the only dimensionless combination of the relevant parameters ( ) fn x N 1/2 1/2 a ag k b T Note: An argument of the function has always to be dimensionless (unless we know that the function is a power-law). There are no exp (kg) or ln (s). k b T has the dimension of work, and so has fr 0. The second step is to apply additional physical knowledge in order to define the function g. In our case we know that if we connect two chains of the same length, the elongation will be twice the elongation of a single chain. If the elongation of each part is x, the whole elongation is 2 x; note that the

31 1.2. IDEAL CHAINS 29 forces acting on both parts are the same. We know that x N 1 g (ξ) ξ (x N 1/2 g (..N 1/2) N only if x N 1/2 AN 1/2, so that the function g is a linear one) x }{{} Na 2 f k b T R0 2 up to a numerical prefactor. We can revert this scaling relation and say that f k b T x R 2 0 (comparing this with our previous rigorous derivation we see that the factor 3 is lost) A more complex application of scaling analysis: A polymer chain in a pore A polymer chain in a pore exercises pressure on the pore s walls. It is not easy to imprison the chain: this reduces its entropy (i.e. freedom) and thus implies performing work. Interestingly enough, the pressure on the walls of

32 30 CHAPTER 1. POLYMER PHYSICS an anisotropic pore is anisotropic. ideal gas and a polymer! There is a huge difference between an Let us calculate the free energy of the chain. Relevant parameters: k b T : the only parameter of the dimension of energy R 0 = Na L } the two characteristic lengths F k b T f ( a ) N L k b T - the only energy scale f - some function (...) - the only combination of other parameters What is f (...)? We know that the function is an extensive, additive function, F N 1 = f (x) x 2 F k b T N a2 L 2 Note: Changing N by same factor of y changes the value of F f( N) by a factor of y. f(x y) = yf(x) Taking the derivative of both sides of this equation with respect to This identity holds for any value of y. Let us take y = 1 d dy f(x y) = f(x) x 2 y f (x y) = f(x) f (x) = 2 x f(x)

33 1.2. IDEAL CHAINS 31 This is a simple differential equation with separating variables which can readily be integrated: df f = 2 x dx d ln f = 2d ln x ln f = 2 ln x + ln A f = Ax 2 We recognize here Na2 L 2 = K, the number of independent fragments of the chain. The growth of F when decreasing L is due to the loss of entropy due to confinement. The loss is of the order of k b per each K segments (each reflection from the wall takes away one degree of freedom ). THE RULE OF THUMB. In discrete models like our chain the entropy is the order of k b per a degree of freedom each constraint decreases the entropy by an order of k b. This increases the free energy. Since the difference in a free energy between the two states, 1 and 2, is just the work necessary to bring the system from state 1 into the state 2, imposing of external constraints ALWAYS IMPLIES DO- ING WORK! Note: The result can be rewritten as F k b T R 0 L 2 What is the force acting an the wall of a pore?

34 32 CHAPTER 1. POLYMER PHYSICS f [E] [L] k bt L g ( ) R0 L Since the parts of the molecule (no interaction) don t fill each other, for N >> 1 f k ( ) bt an 1/2 L g N L = g (z) z 2 = f k bt L 3 a2 N Note: Such rough consideration have always to be checked by: EXPLICIT CAL- CULATIONS, by numerical SIMULATIONS or by EXPERIMENTS. The explicit calculations are given here (for mathematical oriented students) unveil another interesting property of the chains: if the pore does not have a form of a cube, the pressure exercised by molecules on its different walls are not equal! Further links no links available yet A mathematical approach to random walks

35 1.2. IDEAL CHAINS 33 A one dimensional model (simple RW) X N }{{} = X N 1 }{{} + s N }{{} * : position of the Nth monomer ** : position of the (N 1)th monomer *** : gives us a possible position of the N th monomer relative to the position of the (N 1)th one Example: A simple random walk p (s) = 1 (δ (s a) + δ (s + a)) 2 where δ (x) = { 0, x 0, x = 0 δ (x) dx = 1

36 34 CHAPTER 1. POLYMER PHYSICS The characteristic function of the position of the end of the N-segment chain f N (k) = f N. 1 (k) f s (k) In our example f s (k) = [ 1 2 δ (s a) + 1 ] δ (s + a) exp (ik) ds 2 = 1 (exp (ika) + exp ( ika)) cos (ka) 2 For small k one has cos (ka) 1 k2 a 2 2 ) f N (k) = fn 1 (k) (1 k2 a 2 2 f N (k) f N 1 (k) = a2 2 k2 f N 1 (k) Let us consider N as a continuous variable: N f N (k) = a2 2 k2 f N 1 (k) in a continuous approximation back Fourier transform p (x, N) N = a2 2 2 p (x, N) x2 This is an equation of the type of diffusion equation Check: use the fact that p (x) = 1 exp ( ikx) f (k) dk 2π 2 x p (x) = 1 2 exp ( ikx) f (k) dk 2 2π x 2 = 1 ( ) 2 exp ( ikx) f (k) dk 2π x2 1 exp ( ikx) [ k 2 f (k) ] dk 2π

37 1.2. IDEAL CHAINS 35 Example: a free chain A pdf of the position of the chain s end is Gaussian. It is given by a solution of the equation [I] with an initial condition p (x, 0) = δ (x)(the first monomer at the origin). Solution: p (x, N) = ) 1 exp ( x2 2πNa 2Na 2 Check: ) N p (x, N) = exp ( x2 2 2πa N 2/3 2Na ) x ( 2 2π N 5/2 a exp x2 3 2Na ) 2 ( x2 = 1 ) x ( N 2Na 2 2πNa Na exp x2 2 2Na ) ( ) 2 ( 2 x2 1 = N 2 2Na 2 2πa3 N exp x2 3/2 2Na ( ) πa5 N exp x2 5/2 2Na 2 a2 2 p p (x, N) = N 2 x 2 (x, N) Initial condition: p (x) dx = 1 The width σ 2 = Na 2 0 when N 0 lim N 0 p (x, N) = δ (x) Analogue: RW in 3D

38 36 CHAPTER 1. POLYMER PHYSICS p ( s) = 1 6 k k ( ) f k For RW: = δ ( s (a, 0, 0)) + δ ( s ( a, 0, 0)) +δ ( s (0, a, 0)) + δ ( s (0, a, 0)) +δ ( s (0, 0, a)) + δ ( s (0, 0, a)) ( exp i ) k s p ( s) ds exp (ik x x) exp (ik y y) exp (ik z z) p (x, y, z) dxdydz f 1 (k) = 1 6 [exp (ik xa) + exp (ik y a) + exp (ik z a)] [exp ( ik xa) + exp ( ik y a) + exp ( ik z a)] = 1 3 (cos (k xa) + cos (k y a) + cos (k z a)) k2 xa 2 + k 2 ya 2 + k 2 za 2 = k2 a 2 Analogue to 1D-situation one obtains a 3D diffusion equation p ( r, N) N = a2 p ( r, N) }{{} 6 =:D

39 1.2. IDEAL CHAINS 37 is a Laplace Operator, 2 x y z 2 * : a parameter D plays a role of a diffusion coefficient. The solution for a free chain is again a Gaussian, p ( r, N) = The entropy ( ) 3/2 ) 3 exp ( 3 r2 2πNa 2 2Na 2 S = k b ln W k b ln ( C N p ( r, N) ) k b N ln (C) + k }{{} b ln p (r, N) does not play any role if Nis fixed with fixed ends S = k b ln (p ( r, N)) + S 0 F = U T S = const.k b T ln p ( r, N) Knowing F lets us calculate physical characteristics, say the forces ( [ ( f x = F χ = k b T ) 3/2 ( 3 ln exp 3 ) ]) (x2 + y 2 + z 2 ) x 2πNa 2 2Na 2 3k bt Na 2 x Note: An approach based on a diffusion equation (equivalent to an assumption that chain is Gaussian), does not let for describing a fully stretched chain (this needs a microscopic knowledge of a local bond length!). Example: A rigorous treatment of a chain in a pore A more complex situation: A polymer chain in a pore of size L x L y L z.

40 38 CHAPTER 1. POLYMER PHYSICS R i : beginning of the chain, R f : end of the chain The Green s function ( ) ( ) G R, R0, N P N R R: position of the N-th segment P N ( R ): provided the initial segment is at R i ( ) N G R, R0, N = a2 6 }{{} D ( ) R G R, R0, N ( ) + δ R R0 δ (N) }{{} represent initial condition ( ) δ R R0 δ (N): - this way of writing the equation nothing else as stating the initial condition, i.e. the fact that the initial segment is situated at R 0. Boundary conditions: ( ) No segments outside of the pore G R, R0, N boundaries. = 0 for R0 at the Solution: Separation of variables in Cartesian coordinates: ( ) G R, R0, N = g x (x, x 0, N) g y (y, y 0, N) g z (z, z 0, N) }{{} the behavior in x,y and z -drection corresponds to the independent RW s in the corresponding directions

41 1.2. IDEAL CHAINS 39 The general solution for x-component reads: ( ) g x R, R0, N = 2 ( ) ( ) ) πpx0 πpx sin sin exp ( π2 p 2 Na 2 L x p=1 L x L x 6L 2 x Check: Each term is a solution of the corresponding equation: g x = g x (p) ( ) ) πpx γ (x, x 0, N) = a (x 0 ) sin exp ( π2 p 2 Na 2 L x 6L 2 x ( ) ) πpx g x (p) = a p sin exp ( π2 p 2 Na 2 L x 6L 2 x ( ) ) ) N g(p) a2 2 πpx 6 x 2 g(p) = a p sin ( π2 p 2 a 2 exp ( π2 p 2 Na 2 L x 6L 2 x 6L 2 x ( ) ) + a2 6 a π 2 p 2 πpx p sin exp ( π2 p 2 Na 2 L 2 x L x 6L 2 x = 0 Initial condition: g x (x, x 0, 0) = δ (x x 0 ) g x (x, x 0, 0) = ( ) πp a p (x 0 ) sin x 1 L p=1 x }{{} for N = 0all exponential are equal to 1 δ (x x 0 ) For N = 0 : a p is a sinus-transform of the initial condition. Result : a p (x 0 ) = 2 ( ) πp sin x 0 L x L x a p (x 0 ) = 2 ( ) πp sin x 0 L x L x g x (x, x 0, N) = 2 ( ) ( ) ) πpx πpx0 sin sin exp ( π2 p 2 Na 2 L x p=1 L x Let us return to physics. Both ends of a chain move freely within the pore. The overall number of configurations is, as always, C N p N (R), so that the partition function reads: ( ) W = C N d 3 R d 3 R0 G R, R0, N = Z x Z y Z z C N L x 6L 2 x

42 40 CHAPTER 1. POLYMER PHYSICS with Z x = Lx 0 Lx dx dx 0 g x (x, x 0, N) 0 Performing integration we get Z x = 8L ) x 1 ( π 2 p exp π2 p 2 Na 2 2 p=1,3,5 6L 2 x (integrals for even p over a full period of trigonometric functions vanish). The free energy of a chain in the pore F = k b T ln W = with F i = k b T ln Z i N}{{ ln C} + F x + F y + F z uninteresting Application: forces acting on the walls of the pore. f x = F x 1) large pore. R 0 = Na << L x = πn 2 a the order of unity Z x 8L x π 2 p=1,3,5 }{{} a known series equal to π 2 /8 F y and F z do not depend on x f x = F L x = F x L x = 6L 2 x 1 p 2 = L x k b T ln L x = k bt L x L x The pressure of the chain on the wall is p = f x s = l = k bt L y L z L x = k bt V << 1 all exponentials are of

43 1.2. IDEAL CHAINS 41 This corresponds to the equation of state of an ideal gas. The pressure force is isotropic and the same everywhere. 2) small pore R 0 = Na >> L x πn 2 a 6L 2 x >> 1 Only the first term (with the largest exponential) has to be taken into account Z x = 8L ) x ( π exp π2 Na 2 2 6L x f x = k b T ln Z x = k bt π 2 Na 2 + k b T L x L x L x 6L 2 x = k ] bt [1 + π2 Na 2 k b T π2 Na 2 L x 6L 2 x The pressure on the wall is then p x = The pressure is anisotropic! in a cubic pore L x = L y = L z = L 6L 2 x f x = k bt π 2 Na 2 L y L z V 6L 2 x and F = k b T (ln Z x + ln Z y + ln Z z ) k bt 2L 2 π2 Na 2 p k bt π 2 Na 2 L 3 6L 2 Another example of using the random walk formalism: Ideal chain in an external field

44 42 CHAPTER 1. POLYMER PHYSICS The external field E introduces the asymmetry in transition rates, so that the probabilities p + to go to the right and p to go to the left differ. One has Ea {}}{ p + = exp p U k b T }{{} a Boltzmann factor 1 + Ea k b T Using of this rule guaranties the overall Boltzmann distribution. p + 1 ( 1 + Ea ) 6 2k b T p 1 ( 1 Ea ) 6 2k b T In analogy with simple RW: f 1 (x) = 1 ( 1 + Ea 6 2k b T ) exp (ik x a) ( 1 Ea ) exp ( ik x a) 2k b T (exp (ik ya) + exp ( ik y a)) (exp (ik za) + exp ( ik z a)) F a 6 k b T (ik xa) 1 6 k2 a Result in the coordinate space ( ) p N R = a2 N 6k b T F ( ) ( ) p N R + a2 6 p N R Further links no links available yet

45 1.2. IDEAL CHAINS Ideal chain in an external field Configurations which a chain takes in an external field corresponds to minimization of its free energy. They are always a compromise between the internal energy gain and the entropy loss. Example: Week absorbtion of a single chain One can obtain an exact solution using our rigorous equation. In what follows we use a qualitative method, starting from the energy balance. Suppose the overall number of the monomers confined at the surface is xn(x < 1). Then F k b T R2 0 d 2 }{{} xn ε }{{} * : increase in F due to entropy loss as a consequence of confinement ** : energy gain due to absorption at each xth monomer The monomers are distributed within a layer of thickness d (more or less homogeneously) x a d

46 44 CHAPTER 1. POLYMER PHYSICS Minimizing F : k b T R2 0 d a2 N ε = min 2 d2 2k b T R2 0 d 3 + a2 d 2 N ε = 0 d k bt ε a Further links no links available yet

47 1.3. EXCLUDED VOLUME EFFECTS Excluded volume effects Introduction A model based on ideal chains is not always relevant. For example, it is not sufficient for chains in good solvents. The monomers of a real chain effectively repel each other. The molecule swells. The equilibrium configuration of the molecule is due to the equilibrium of this repelling and the molecule s elasticity. Based on this rough argument, P.J. Flory (Nobel prize 1974) had found that R an ν with ν 3/5. This approximation for ν is remarkably good. Interactions between the parts of the chain Important: Repelling of segments which come too near to each other ( U (r) k b T υ Rn R ) m υ : energy in units of k b T

48 46 CHAPTER 1. POLYMER PHYSICS ( The interaction υ Rn R ) m is short-range, so that one can approximately take ( k b T υδ Rn R ) m The overall interaction energy (the internal energy) U 1 2 k bt 1 2 υk bt N N n m N 0 dn ( υδ Ri R ) j N 0 ( dmδ Rn R ) m This can be expressed through the density of segments: c ( r) = n δ ( r R ) N n p (R n ) dnδ ( r ) R n p (R n ) 0 U 1 2 υk bt d r [c ( r)] 2 Note: As is clear from this equation, the prefactor υ has a dimension of a volume and can be considered as a typical volume of iteration. In spacial dimension d 3 the dimension of υ is [ L d]. In a real situation υ can be a function of T and depends on the properties of a solvent (if any). We take υ > 0 (repulsion), which corresponds to good solvents. Further links no links available yet The Flory approach Flory theory considers a molecule as a cloud of independent particles Flory approximation: { N/R d c (r) 0

49 1.3. EXCLUDED VOLUME EFFECTS 47 U (R) υk b T N 2 Rd R2d The internal energy contribution due to the interaction decays with the growth of R The relevant thermodynamical potential for T = const. is the free energy F U (R) T S (R) With the increasing of the chain s size R internal energy decreases free energy decreases Entropy decreases free energy grows (More and more forbidden configurations) The two contributions have to be balanced. Let us take the entropic contribution like in a Gaussian chain T S k b T R2 Na 2 k b T : the only energetic parameter R 2 : for a scaling approximation it does not play an great role whether is the end-to-end distance, or the overall size. The overall free energy F = υk b T N 2 R d + k bt R2 Na 2

50 48 CHAPTER 1. POLYMER PHYSICS minimization over R dυ N 2 2R + Rd+1 Na = 0 2 R d+2 = 1 2 dυa2 N 3 }{{} non universal Note that the prefactor υa 2 guaranties the correct dimension of the r.h.s. 3 R N d + 2, or R 2 N 2ν with ν = }{{} 1, 3/4, 3/5 }{{}}{{} d=1 d=2 d=3 The approximation for ν is very good, (difference 1%compared to the best numerical values) and in practical applications can be considered as exact. The great success of the Flory method is due to the compensation of the 2 errors. a) Neglecting correlations along the chain (i.e. taking U c 2 ) leads to considerable overestimation of U b) F entr is also considerably overestimated (it is taken for an ideal chain, which has the denser conformation) trying to improve each of the terms typically leads to worse results. Interesting: for d = 4, ν = 1/2: the chains in d = 4 are ideal! (essentially, the chains in all d 4 are ideal). The repelling can only increase the radius of the chain compared with the ideal chain R > R 0. The corresponding contribution to the free energy is: F repelling U (R) υk b T N 2 so that R d υk bt N 2 R d 0 F repelling U (R) υk b T 1 a d N 2 d/2 = υk b T N 2 N d/2 a d }{{} ( Na) d

51 1.3. EXCLUDED VOLUME EFFECTS 49 For d > 4 this energy decreases when N grows, and is thus unimportant for long chains which try to minimize only the elastic energy. For d = 4 F repelling is at most constant ( unimportant contribution) Theoretical idea, giving rise to the so-called ε-expansion. Expansion in parameter ε = 4 d, starting from an ideal chain, ν 1 2 ε Note: In the case when the interaction potential between two monomers is a very strong, short-range repelling (the excluded volume interaction), the overall repelling contribution to the free energy can be considered as purely entropic (the configurations in which the monomer volumes overlap are forbidden the configurations are constrained). { } a d N The number of such constraints number of touches N, where a d N is the total volume occupied by the monomers, so that ad N R d R d is of the order of the probability that another particle is just at the site where the first one is. As we have seen, an entropy loss is per constraint (c.f. with a chain in a pore!) is of the order of k b per touch S }{{} S }{{} 0 k b N 2 /R d * : chain with excluded volume ** : ideal chain F repelling ST = k b T a d N 2 /R d i.e. υ 1 Further links no links available yet

52 50 CHAPTER 1. POLYMER PHYSICS The lattice model for an excluded-volume chain Self-avoiding-walks (SAW). Numerical methods are: Exact enumeration for short chains (different algorithms) + extrapolation Monte Carlo simulations (generate a small amount of configurations of very long chains, enough for getting the probability density function s (pdf)) e.g. using so-called pivot algorithms. Trivial qualitative result W N < C N (essentially W N goes as C N with C < C, e.g. in 3d simple cubic lattice one has C = 4.68 instead of C = 6). Rigorously speaking, for N >> 1 one has W N = const. C N N γ 1 (asymptotically exact for N ). C (just like C) is non-universal and depends on the type of the lattice (i.e. on our rather arbitrary assumptions about the microscopic structure of the monomers). On the other hand, γ depends only on the dimension of space d = 2: γ 2 = 4/3 for all lattices (exact) d = 3: γ 3 = 7/6 for all lattices (good approximation) Note: In d = 1, W N = 2 (go left or go right!) C = 1 and γ = 1 an ν (ν is a Flory expo- The mean end-to-end distance of a chain R F nent) ν 3 3/5, ν 2 3/4, ν 1 = 1

53 1.3. EXCLUDED VOLUME EFFECTS 51 The pdf of the end-to-end distance scales as p N ( r) = 1 ( ) r f RF d p R F and looks as follows (analogue to a Gaussian chain for which δ = 2) Theoretical results (scaling, RG) g = (γ 1) /ν (i.e. for d = 3, g 5/8) and δ = 1/ (1 ν). (i.e. in d = 3, δ = 5/2 the exponential tail decays faster than for a Gaussian chain; for d 4, ν = 2 and thus δ = 2; ideal Gaussian chains!). Further links no links available yet The structure functions How do we learn about the conformations of polymer chains? Because we can measure their structure factor in the experiments on light-,

54 52 CHAPTER 1. POLYMER PHYSICS x-ray-, or neutron scattering. The intensity of the scattered wave is proportional to ( ) G k = 1 N ( exp i N ) 2 k x n n=1 = 1 N exp ( i N ) N ( k x n exp i ) kx m 1 N = 1 N n=1 N N n=1 m=1 N N n=1 m=1 m=1 ( exp i k ( x n x m ) exp ) ( i ) k ( x n x m ) Averaging over the configurations of the chain. ( ) Here 4π k is a scattering vector, k = sin θ λ 2. Explanation: Phase shift ϕ = 2π (s 1 + s 2 ) λ ( = r k1 k ) 2 = r k with k = k 1 k 4π 2 = λ sin θ 2 and k 1 = k 2 = 2π/λ Small k (essentially k a 1):

55 1.3. EXCLUDED VOLUME EFFECTS 53 Universal behavior: ( ) G k 1 [ ( 1 i N k x n x m 1 k [( xn x m )] 2 )] [ = 1 N N ] 2 k 2 ( xn x m ) N 6 }{{} * ) = N (1 k2 3 R2 g * Here we use the fact that the direction of a vector R = x n x m (with respect to the direction of k) is random, so that [ k R ] 2 = 1 R 3 k2 2 To see this, let us calculate the mean square value of the projection of a randomly oriented vector of length l on the x-axis. Take x-direction as a polar axis of a system of spherical coordinates and get x 2 = 1 4π = 1 2 = 1 3 l2 1 1 (l cos θ) 2 sin θdθdϕ l 2 cos 2 θd cos θ

56 54 CHAPTER 1. POLYMER PHYSICS R 2 g: Gyration radius. R 2 g = 1 N N ( rn r s ) 2 n=1 with r s = 1 N r m, r s : coordinates of the center of mass. N m=1 Rg 2 = 1 r 2 N n 2 r n r s + r s 2 n ( = 1 r n 2 2 r n N N ) r m + 1 r N 2 m r n n m m,k ( ) = 1 r 2 2 N n r N 2 n r m r N N 2 m r n n n,m n m,k }{{} 1 = 1 r 2 1 N n r N 2 n r m n m,n 1 = ( rn r 2N 2 m ) 2 m,n Example: A Gaussian chain ( exp i ) k ( x i x j ) ( = P i j ( r) exp i ) k r d 3 r }{{} Fourier transform of a Gaussian ) = exp ( i j k2 a 2 (in 3d) 6 A Gaussian chain ( ) N N G (k) exp n m (ka)2 6 n=1 m=1 ( ) dn dm exp n m (ka)2 6 { [ ( )]} 12 6 = (ka) 2 1 N (ka) 2 1 exp N (ka)2 6

57 1.3. EXCLUDED VOLUME EFFECTS 55 i.e. G (k) = Nf ( k 2 R 2 g) with Rg 2 = 1 2N 2 dn dm n m = a2 N 6 R2 0 6 f (x) = 2 (x 1 + exp ( x)) is a Debye-function (1944) x2 { 1 x/3, for x << 1 f (x) 2/x, for x >> 1 Limiting cases and explanation: k 0, long waves, x << 1, i.e. k Rg 2 << 1 G (k) N (1 R2 gk 2 ) 3 This expression is essentially universal and applicable for any polymer chain (also real). very short waves (x-rays, neutrons) G (k) 12 (ka) 2 λ 2 a 2 sin 2 θ 2 Note: this result is N-independent and gives the typical segment / monomer size. It is the intermediate k-range that probes the chain s structure. Scaling theory of scattering by excluded-volume chains G (k) = N f ( k R 0 ) The asymptotic of f (x) for small x is known: f (x) 1 for x 0 (essentially, f (x) 1 Ak 2 R 2 0) Interesting is the asymptotic of short wavelengths, probing the local structure. In this case G (k) is N-independent (G (k) is the sum of rapidly oscillating functions, which decays very fast with (n m)) Nf ( k an) N 0 f (x) = x 1/ν G (k) ( k a) 1/ν A method measuring ν. Further links no links available yet

58 56 CHAPTER 1. POLYMER PHYSICS Interaction of excluded-volume chains Two excluded-volume molecules repel (although not to strong). The same is true for two parts of the same molecule. The repulsion of the parts of the molecule leads to the concept of blobs: it is often helpful to consider the molecule as a string of blobs. We discuss the interaction of two chains put close to each other. According to our entropy-loss consideration, the free energy contribution due to interaction is F int k b T K K: number of collisions of the segments of two molecules. According to Flory (i.e. taking the coils as the clouds of independent segments) we get K N a3 N R 3 F F int k b T a3 N 2 R 3 F k b T N 1/5 The coils repulse. In this Flory-like approximation the repulsion is strongly overestimated; in reality the repulsion is F int k b T (but there is still some repulsion energy, which is considerable on the molecular scales). Reason for the overestimation: the wrong estimate for the probability of a contact since the segments are not independent. Further links no links available yet

59 1.3. EXCLUDED VOLUME EFFECTS The notion of blobs Model problem: a real chain in a capillary or in a narrow flat cavity ag ν D g (D/a) 1/ν gives us the number of monomers per blob. Since the blobs repel each other (although weakly) the overall system can be considered as a 1-dimensional (pore) or 2-dimensional (flat geometry) real chain of blobs. R tube N (a/d) 1/ν D Na 1/ν D 1 1/ν Na 5/3 D 2/3 D: blob size, i.e. the new segment length, n = N/ (D/a) 1/ν : the number of blobs R cavity DN ν 2 (a/d) ν 2/ν 3 D N ν 2 a ν 2/ν 3 D 1 ν 2/ν 3 N 3/4 a 5/4 D 1/4 The size of the chain decreases when increasing D; for D R F we get R R F ; the chain then consists of the only one blob, and its size ceases to decrease. Free energy per blob: k b T the only parameter of the dimension of energy. Moreover, the coil is characterized by the only one characteristic length, R F an ν ( ) D F k b T ϕ an ν F : free energy per blob ϕ (x), not yet known.

60 58 CHAPTER 1. POLYMER PHYSICS For D << R F the chain consists of many independent blobs, the number of blobs for given D grows N ( a ) 1/ν ϕ (x) x 1/ν and F k b T N D The free energy per blob is F n ( a ) 1/ν N = k D bt ( a ) 1/ν = k b T N D This is essentially a definition of blob! Elasticity of a real polymer chain: Scaling considerations: We know that the elasticity of a polymer chain is of entropic nature and is independent on its microscopic structure a and N can enter the expressions only through R 0 an ν (we had ν = 1/2 for an ideal chain; now we have ν 3/5). Other parameters are f and k b T, from which one can get another characteristic length, k bt f. Analogue to our considerations for ideal chains: ( ) fan ν X R 0 g k b T g: some function (For ideal chain, we got this function via direct calculation.) How does the function g look like for a real chain? linear regime very small linear response X f ( g (z) z 1 )(z 1) f X an ν k b T an ν a 2 N 2ν ( ) kb T f = X a 2 N 2ν κ = k bt : Hook s coefficient an ν f k b T

61 1.3. EXCLUDED VOLUME EFFECTS 59 Ideal chain: ν = 1/2; κ I N 1 Real chain: ν = 3/5; κ R N 6/5 < κ I ( Real chain is softer, due to the repulsion between the segments.) For large elongations we encounter a nonlinear regime. z 1, i.e. X > R 0 For X >> R 0 one can consider the chain as an effective chain of blobs. n is taken so that the elongation per blob is of the order of the unperturbed blob size. Then, within a blob, a conformation of a chain corresponds to the behavior for small elongations. The overall chain of blobs is practically fully elongated and oriented in the force direction: X n (f) = R (n). This defines the blob size: a2 n 2ν f an ν = R 0 (n). Thus, k b T ( ) 1/ν kb T n = af Blob radius R (f) an ν and number of blobs: K = N/n The overall elongation: x = KR (f) an ν N/n = ann ν 1 an (k b T/af) ν 1 ν (note that ν < 1) For an ideal chain (ν = 1/2), this is still a linear dependence, but for real chains we have X f (1 ν)/ν f 2/3, so that strong nonlinearity is encountered. The results hold if the chain is still far from being fully elongated; fa << k b T. Further links no links available yet

62 60 CHAPTER 1. POLYMER PHYSICS 1.4 Melts and Solutions Polymers in melts A chain in a good solvent swells due to the effective repulsion between monomers. In a melt, the monomers are both inside and outside of the spatial region occupied by the chain, the corresponding forces acting inwards and outwards cancel. Chains in a melt behave as ideal ones! This fact, first discovered by Flory (1949), is so unusual, that the people at the beginning didn t believe him. To discuss the situation we note that polymer coils are extremely loose objects. The density (part of the volume taken up by the monomers) of an ideal chain is Φ : concentration of monomer Φ = Nv m R 3 F or R3 0 Φ N V v m Nv m (N 1/2 a) 3 N 1/2 ( vm a 3 ) << 1 The number concentration of monomers is given by c = Φ/v m and it is even less for the excluded-volume chains (v m is a volume of a monomer). On the other hand, polymer melts are macroscopically dense (they are normal fluids) there are many segments within the chain s volume. A conformation of the single chain is due to the fact that interaction of its segments leads to effective repulsion with the repulsion energy U 1 2 υk bt d r [c ( r)] 2 This energy is minimized when reducing density, i.e. stretching the molecule, and is in competition with the growing elastic energy (entropy contribution). In general, the energy per monomer is proportional to the local concentration of other monomers, U υk b T c ( r) where c ( r) is the concentration of other monomers.

63 1.4. MELTS AND SOLUTIONS 61 In a melt the chains are strongly interpenetrating; the local density of the monomers is a constant everywhere. The overall concentration is constant, the effective force to the outside produced by the monomers of the same chains is compensated by the forces to the inside produced by the monomers of other chains effective force is ZERO, and the conformation of the chain is ideal.

64 62 CHAPTER 1. POLYMER PHYSICS This result of Flory was proved (Cotton 1976) in experiments with partly deuterized polystyrene (amplitudes of neutron scattering for D are much larger than for H one can practically see only the tagged molecules, on the other hand the physico-chemical properties of the chains are very similar). Note: The 3d-picture of a polymer melt is like this The situation in 2d-systems is vastly different: even a single ideal chain leads to a monomer concentration c N a 2 which is large and N - R0 2 independent the chains are to some extent separated. Moreover, the typical 2d-configurations of SAWs are closed. Further links no links available yet

65 1.4. MELTS AND SOLUTIONS Polymer solutions: a Flory-Huggius approach These are only very dilute solutions in which the chains behave as independent. Since the typical chain size is R an ν, the concentration of chains n < n 1 R 3 1 a 3 N 9/5 in such dilute solution is very small. The volume concentration of monomers C = Nn N 4/5 a 3 is also very small. (n = c,c: concentration of monomers) N If the concentration of chains n > n, they overlap: we enter first the regime of a semi-dilute solution (n > n,but still C << 1), and then one of a concentrated solution (C 1) and melt (no solvent at all). Increasing the concentration of the chains. The overlapping threshold: The solution can be considered as a dense packing of swollen coils.

66 64 CHAPTER 1. POLYMER PHYSICS c N R 3 F N a 3 N 3ν a 3 N 1 3ν a 3 N 4/5 Volume concentration Φ N 4/5 A Flory-Huggius theory (1942): a lattice model for a polymer solution.

67 1.4. MELTS AND SOLUTIONS 65 Note: Flory Huggius theory belongs to a class of so called mean-field theories. Such theories are never very good (but never very bad). Many experimental results were explained within the picture. Each site can be occupied either by a monomer or by the solvent molecule F = U T S Mean-field a molecule is considered as a cloud of monomers (the only role of the chain is to keep the monomers together). The portion of the sites occupied by the monomers is Φ. Each pair of neighboring sites gives a contribution into the internal energy U. The internal energy corresponds to the mixture of N M and N S = N 0 (1 Φ) solvent molecules: = N 0 Φ monomers U = N SS ɛ SS + N MM ɛ MM + N MS ɛ MS with N SS the number of neighboring sites occupied by solvent molecules, N MM the number of corresponding sites occupied by monomers and N MS the number of pairs of neighbors if different types. Let N S be the total number of sites occupied by solvent molecules and C be the coordinationnumber of the lattice. Then CN S is the number of all bonds starting at sites occupied by solvent CN S = 2N SS + N SM (The factor 2 corresponds to the fact, that each SS-bond is counted twice, as one starting at each of its ends). Analogously, CN M = 2N MM + N SM Thus N SS = C 2 N S 1 2 N SM N MM = C 2 N M 1 2 N SM and U = C (N Mɛ MM + N S ɛ SS ) ( + N SM ɛ MS ɛ SS }{{ 2 } 2 ɛ ) MM }{{ 2 } (1) (2)

68 66 CHAPTER 1. POLYMER PHYSICS (1) - The sum of internal energies of pure solvent and pure monomers (melt). (2) - Change in internal energy compared to separated solvent/melt. For a well-mixed situation N SM = N bonds Φ(1 Φ) = CN 0 Φ(1 Φ) N 0 - the overall number of sites ( U = N 0 C ɛ MS ɛ ) MM + ɛ SS Φ(1 Φ) + U pure 2 }{{} k b T χ with U pure the reference internal energy of unmixed solvent/melt. The Flory-Huggins parameter χ characterizes the quality of the solvent: The solvent with χ = 0 is athermic (no effective interaction, a very good solvent). The case χ = 1/2 corresponds to the so-called θ-solvent, χ < 1/2 good solvents, χ > 1/2 to bad solvents. The entropy: here the meanfield approximation gives S(Φ) = k b N 0 [ Φ N ln Φ N + (1 Φ) ln(1 Φ) ] The free energy is F = F pure + k b T N 0 [ χφ(1 Φ) + Φ N ln Φ N + (1 Φ) ln(1 Φ) ] } {{ } F Explanation for the form of the entropy Compare this with the simple case when N = 1 and the situation corresponds only to the possibility of place the molecules of two types (monomers M and solvent molecules S) on a lattice of N 0 sites. ΦN 0 of the sites are occupied by M and by (1 Φ) N 0. The overall number of possibilities is N 0! (ΦN 0 )! [(1 Φ) N 0 ]! Use the Stirling-formula.

69 1.4. MELTS AND SOLUTIONS 67 The connected chain of N monomers allows for considerably less possibilities than N free monomers. This is approximately described by considering the chains as particle like solid objects: this corresponds to a translational entropy of chains (Φ/N is the concentration of chains). Explanation (very rough) N independent particles (ideal gas) An excluded-volume chain. The first monomer can be neglect excluded volume W = V N placed in V ways, the second in C ways, each next one at most in C 1 ways W < V C (C 1) N 1 S N ln V S ln V + ln C + (N 1) ln (C 1) The free energy per site reads: f = F = N k b T [const 1 + const 2 Φ + ΦN ] ln Φ + (1 Φ) ln (1 Φ) χφ2 0 const 1 : Additive constants in thermodynamical potentials can be set to zero const 2 : the same as when no chains where in play, it does not correspond to interactions between monomers and does not contribute to physically interesting properties. Small concentrations: expand in f series in Φ F k b T N 0 const 2 Φ + Φ N ln Φ (1 2χ) Φ Φ3 Note: the term Φ 2 can be interpreted as an effective monomer-monomer contribution into the free energy. The prefactor (1 2χ) corresponds to an

70 68 CHAPTER 1. POLYMER PHYSICS effective interaction at neighboring sites. χ = 1/2 corresponds to the absence of interaction. In application to real situations one finds that χ depends on temperature. One can anticipate that in a θ-solvent the chains are ideal. This is really the case! Further links no links available yet The osmotic pressure of polymer solutions P osm = F under the constant number of monomers N V mon = const. We use the following identities N mon = N 0 Φ = V a 3 Φ N 0 = N mon Φ and V = N mona 3 Φ. It follows: F V = V (N 0f(Φ)) = Φ (N 0f(Φ)) Φ V = ( ) Nmon Φ Φ f(φ) = ( ) Nmon Φ Φ f(φ) = 1 f(φ) Φ2 a3 Φ. 1 ( ) V Φ Φ 1 ( ) Nmon a 3 Using the Flory-Huggins equations one gets: P osm = k bt a 3 Φ [ Φ N + ln 1 1 Φ ( Φ χφ 2) ] Special cases: Dilute solution, Φ 0 : P osm = k bt Φ a 3 N k bt c N with c being the number of N chains per unit volume. This corresponds to an equation of state of an ideal gas (generally valid for all dilute solutions); this regime is well-described by

71 1.4. MELTS AND SOLUTIONS 69 the theory. Semi dilute solution, N 4/5 << Φ << 1: Expansion up to the second order P osm k bt a 3 [ Φ N + 1 ] (1 2χ) Φ2 1 (1 2χ) Φ2 2 2 The accuracy of the Flory-Huggius theory in this situation is not sufficient, vide infra. Concentrated solution, Φ 1 : P = k bt a ln Φ The logarithmic growth of the pressure. The accuracy of the F-H-theory in this regime is pretty good. Further links no links available yet Phase diagram of a polymer solution Now we discuss the two situations: a) What happens in the regime of intermediate concentrations [in good solvents we have an effectively one-chain behavior at small concentrations Φ and an ideal-chain-like behavior, like in a melt for Φ 1]. b) What happens in bad solvents (and how the corresponding physical effects can be used) Phasediagram of a polymer solution (according to the Flory-Huggius theory). χ = χ (T ).

72 70 CHAPTER 1. POLYMER PHYSICS For Φ 1 F RT N 0 Φ N ln Φ N + (χ 1)Φ (1 2χ) Φ Φ Good solvent: dominance of the steric repulsion 1 2 (1 2χ) Φ2 > 1 6 Φ3 Φ < 3 (1 2χ) (A crossover line between ideal and swollen chains). χ > 1/2: bad solvent, phase separation curve.

73 1.4. MELTS AND SOLUTIONS 71 The extremal point C is a critical point. In bad solvents the polymer solution separates into a polymer-rich and a solvent-rich phases. Such phase separation starts at χ C = N and for Φ c = 1 N In the vicinity of the critical point C the system can be considered as a dense packing of practically ideal chains. Explanation: The free energy per site in a homogeneous phase is [ F (Φ, χ) Φ f(φ, χ) = = k B N 0 N ln Φ N (1 2χ)Φ2 + 1 ] 6 Φ Starting from this expression we explain the form of the spinodal and of the coexistence curve. The spinodal: According to thermodynamics, the system is stable if its free energy attains its minimum. Let us consider the stability of the system against separating into two phases with different monomer concentrations. Let us make the following thought experiment: We separate the whole volume of the system into two parts (say of N 0 /2 sites each) and consider the monomers or solvent molecules crossing the imaginary boundary. This will lead to small changes in the monomer concentrations in each half of the system (say to Φ + and Φ ) and therefore to small changes in the free energy. The total free energy of the whole system, so that the overall free energy will be F = (N/2)f(Φ +, χ) + (N/2)f(Φ +, χ). If this free energy is larger than F (Φ, χ), the external work is needed to put the system in such an inhomogeneous state, if this value is smaller, then such a transition would take place spontaneously.

74 72 CHAPTER 1. POLYMER PHYSICS Assuming to be small we write the stability criterion: f(φ, χ) 1 [f(φ +, χ) + f(φ, χ)]. 2 Expanding f(φ +, χ) and f(φ, χ) up to the second term in we get that the solution is stable whenever 2 f(φ, χ) 0. Φ2 The solution of the equation 2 f(φ, χ) = 0 Φ2 gives us the spinodal, the curve bounding the region of thermodynamical stability of the homogeneous solution. For the free energy per site given by Eq.( ) the spinodal (for small Φ) possesses the form χ(φ) 1 ( 1 + Φ + 1 ). 2 NΦ The critical point: For the free energy per site given by Eq.( ) the spinodal χ(φ) possesses a minimal value χ c. This means that for χ < χ c the homogeneous solution is unstable, independently on what the concentration Φ is. This value of χ is called the critical point. It is given by the solution of the equation d χ(φ) = 0. dφ From Eq.( ) we get χ c = N. The coexistence curve: The equilibrium state of the system is the one in which the free energy of the system attains its minimum. We have to compare the free energies of a system with N 0 sites in the state with homogeneous monomer concentration with the free energy of the twophase system. Let us assume that in the phase-separated system the number xn 0 of the sites is occupied by the solvent-rich phase with the concentration of monomers Φ 1 and the number (1 x)n 0 of the sites is occupied by the polymer-rich phase with the concentration of monomers Φ 2 > Φ 1. Note that

75 1.4. MELTS AND SOLUTIONS 73 since the overall number of monomers is N 0 Φ we have xn 0 Φ 1 +(1 x)n 0 Φ 2 = N 0 Φ, so that the total free energy in the phase-separated system is F 2 (x, Φ, χ) = xn 0 f(φ 1, χ) + (1 x)n 0 f [(Φ xφ 1 ) /(1 x), χ]. The value of x for fixed Φ and χ has to be chosen so that the overall free energy in phase-separated state is minimal: i.e. x is the solution of the equation x F 2(x, Φ, χ) = 0. The coexistence curve corresponds to the equality of the free energies in these homogeneous state and in the phase-separated state, and is given by the solution of the system of two equations: Eq.( ) and the equation F (Φ, χ) = F 2 (x, Φ, χ). This can only be done numerically. The typical position of the spinodal and the coexistence curve are shown in the picture. The critical point is the common point of the spinodal and of the coexistence curve. The Flory-Huggius theory is not very exact near the critical point C (with respect to the form of the coexistence curve). The states in vicinity of c are characterized by the dense packing of practically ideal chains R 0 N 1/2 a (since χ 1/2 ) φ = c N R 3 0 = N 1/2 a 3 a 3 Φ c For Φ << Φ c and χ > 1/2 the solution corresponds to non-overlapping globules In a semi-dilute solution the chains overlap strongly, but the overall concentration of the monomers is still very low (Φ << 1). Further links no links available yet

76 74 CHAPTER 1. POLYMER PHYSICS Polymer blends and block-copolymers The theory of polymer blends within the Flory-Huggins approximation can be built in a way similar to those for solutions. Difference: Solvent entropy term has to be changed for the translational entropy of the other polymer F N 0 = k b T { ΦA N A ln Φ A + Φ B N B ln Φ B + χφ A Φ B (One can again forget about the terms of the first order in Φ.) The overall entropic term for a blend is considerably smaller than for the solution the overall system is less stable against phase separation (demixing). Spinodal: Boundary of thermodynamical stability 2 F Φ = N A Φ + 1 N B (1 Φ) 2χ = 0 Critical point: Minimal value of χ at the spinodal: χ = 1 ( ) 1 2 N A Φ + 1 N B (1 Φ) χ Φ = 0 Φ C = ΦC N 1/2 B N 1/2 A + N 1/2 B and χ C = } ( ) 2 N 1/2 A + N 1/2 B 2N A N B Limiting cases: N B = 1: a solution χ C = 1/2 N B << N A : χ = 1/2N B : The incompatibility is typical if the effective interaction is repulsive. N B = N A : χ C 2/N A << 1 An interesting situation: immiscible polymers A and B form the two parts of the same chain: a block-copolymer χ AB > χ C For χ < χ C the domains form which contain mainly A- or B- blocks: The parts of the chain separate, but cannot form macroscopic phases since they are connected to each other.

77 1.4. MELTS AND SOLUTIONS 75 Result: MICROPHASE SEPARATION. The overall structure depends on the relative length (number of segments) of the parts of the chain. Further links no links available yet Single molecule behavior in bad solvents If we consider a single chain (in a dilute solution), it swells in good solvents, and takes its ideal conformation at θ-point. In bad solvents the chain shrinks and forms a droplet-like, globular state. For soft, flexible chains (for which the Flory-Huggins theory is a reasonable description) this transition corresponds to an sudden collapse: A coil-globule transition. Fully parallel to the Flory theory of swelling: A competition of the interaction energy between the monomers and entropic contributions. α: Swelling coefficient R = αan 1/2 }{{} R 0

78 76 CHAPTER 1. POLYMER PHYSICS The internal energy of the chain has the same form as in the Flory-Huggius theory; now we express it as a function of α: U (α) = }{{} R 3 k b T volume k b T k b T B }{{} 1 2χ c 2 + C }{{} const. c 3 [BR 3 ( NR3 ) 2 + CR 3 ( NR3 ) ] [ BN 1/2 + C ] α 3 a 3 α 6 a 6 The entropic contribution is considered to be one of an ideal chain. If the size of the chain is larger than R 0, we take an equation for a chain pulled at the end, if it is smaller than R 0, the one for a chain in a pore. Thus: T S k b T R2 R 2 0 Simple interpolation: a 2 k b T for the chain pulled at the end k b T R2 0 R 2 k bt/α 2 for the chain in a pore T S = k b T ( α 2 + α 2) All in all: [ BN 1/2 F k b T + C ] α 3 a 3 α 6 a + 6 α2 + α 2 Minimization in equilibrium: F/ α = 0 1/2 3BN α 4 6C α 7 + 2α 2α 3 = 0 } a{{ 3 }}{{} a 6 2x 2y Multiplied by α 4 this gives α 5 α = x + yα 3, y > 0, x { > 0 in good solvent < 0 in bad solvent y: This parameter essentially describes the rigidity of the chain. In a Flory- Huggins lattice model it was 1/6.

79 1.4. MELTS AND SOLUTIONS 77 For very soft chains y 1, for stiff ones y << 1. In a very good solvent x 1 (since B a 3 (1 2χ) and N 1) x = 0: approx. Gaussian coil, α 1 x < 0: The situation depends on whether y > y cr or y < y cr In a very bad solvent x >> 1 (for the same reason). y < y cr Strong attraction: α << 1, terms with positive powers of α irrelevant: }{{} x + yα 3 0 <0 [ ] C 1/3 1/3 C α ( B) 1/3 N 1/6 C i.e. R αan 1/2 α N 1/3 B x 1 : the vicinity of the θ-point. Changing x (i.e. χ, i.e. T Note that in vicinity of the θ-point χ const. (T T θ )) ( ) BN 1/2 1/5 ( ) 1/5 B x > 0: swelling, α >> 1, α, i.e. R αan 1/2 a N 3/5 a 3 n N R B : a droplet of constant density, globular structure. 3 C The transition occurs at x cr 1, i.e. very close to a θ-point. For y > y cr we do not have an abrupt transition but a gradual change of the regime (continuous decrease in size). However, although it is not a transition, the whole changes take place within the interval of x /y 1, (y = 1) and x N 1/2 a 3 The whole transformation takes place within a temperature interval T θ T N 1/2 << 1 (an almost phase-transition). T θ Explanation: Equation for equilibrium swelling coefficient α in a coil-globule transition: x = α 5 α yα 3 }{{} f(α,y) (0 < α < ) Note that due to large positive/negative power in α, the change in f(α, y) stretches over many orders of magnitude.

80 78 CHAPTER 1. POLYMER PHYSICS If f(α, y) is monotonously growing, then for each x there exists only one solution, which is the actual radius of the molecule. If f(α, y) is nonmonotonous, there might be three solutions. To see whether f(α, y) is monotonous calculate df(α, y)/dα = 5α 4 1+3yα 4. Denote α 4 = z: 5z 1 + 3y/z = 1 This is a quadratic equation, which has: no solution for y > 1/60 one solution z = 1/10 for y = 1/60 two solutions z = 1/10 ± 1/ y for y < 1/60 (nonmonotonous f with two extrema)

81 1.4. MELTS AND SOLUTIONS 79 The value of y = 1/60 corresponds to a critical value (change of behavior). Further links no links available yet

82 80 CHAPTER 1. POLYMER PHYSICS 1.5 Polymer Networks The rubber elasticity Cross-linking the chains in a polymer melt or concentrated solution (vulcanization) (Charles Goodyear, ) leads to emergence of rubber-like properties. The theory of rubber elasticity (Flory!) starts from the two postulates: 1. The chains in a melt are ideal. 2. The melt is a liquid, its volume does not change strongly under the deformation. These properties are retained also after introducing crosslinks. The high elasticity of rubber Cross-linking of chains in a melt produces highly elastic, interconnected network. Note that: Chains in the melt are ideal. Entropic elasticity of the ideal chain: ( ) 3kb T f = Na 2 ( R, S = 3 2 k b Rx 2 + Ry 2 + Rz 2 ) R0 2 A rubber network consists of subchains connected at the network nodes.

83 1.5. POLYMER NETWORKS 81 Pull along the x-axis: x λ x x Associated change of the cross-section in y- and z-directions y λ y y, z λ z z The overall piece of rubber behaves essentially as a melt (i.e. as a fluid): its volume hardly changes. V = λ x λ y λ z x 0 y 0 z 0 = λ x λ y λ z V 0 V 0 λ x λ y λ z 1 symmetry: λ y = λ z = λ 1/2 x Assumption: Affine deformation: each subchain is deformed in the same way as the whole network. Entropy change under elongation: ( ) ( ) S = S R S R0 = 3 k b [( ) ( ) ( )] R 2 2 na 2 x Rx R 2 y Ry R 2 z Rz 2 0 = 3 k b [( λ 2 2 na 2 x 1 ) Rx ( λ 2 y 1 ) Ry ( λ 2 z 1 ) ] Rz 2 0 = 1 2 k ( b λ 2 x + λ 2 y + λ 2 z 3 ) n: subchain length (# of monomers per subchain) R0 2 x = R0 2 y = R0 2 z = R2 0 3 = na2 3 Summing over all subchains (whose number is of the order of bv, where b is the concentration of crosslinks) we get: S = k bbv 2 ( λ 2 x + λ 2 y + λ 2 z 3 )

84 82 CHAPTER 1. POLYMER PHYSICS λ x = λ, λ y = λ z = λ 1/2 S = k b bv (λ2 + 2) / (λ 3) 2 f = T S x = T S x 0 λ f stress σ = surface = f = T S z 0 y 0 V λ = k bt b (λ 1λ ) 2 For λ around 1, (λ 1λ ) is approximately 3 (λ 1) 2 The Young s modulus: σ = E x x E = 3k b T b E turns to be the same as the pressure of an ideal gas with molecular concentration 3b. The elastic moduli of polymer networks are not specific for chemical structure of a polymer, but vary dramatically with the density of cross-links. The theory does not take into account:

85 1.5. POLYMER NETWORKS 83 finite contour length of chains which cannot be stretched longer than to Na. mutual entanglements of chains in addition to chemical cross-links non-affine nature of deformation Further links no links available yet Polymer networks and percolation For too small concentrations of nodes cross-links the continuous network is not formed. If the number of cross-links is too small, not each chain is connected to the elastic frame of the system and cannot contribute considerably to the elasticity: one has to have at least 2 nodes per chain in order to form an elastic frame. There exists some critical value of b = b c above which the rubber elasticity appears; at lower concentrations the system is fluid, although it contains a considerable amount of large clusters of interconnected chains. Note: From a chemical point of view a cluster is a single molecule. Many other situations correspond to similar sol-gel transitions in polymer systems. All these models share a reasonable degree of universality; some of their properties are described by percolation theory.

86 84 CHAPTER 1. POLYMER PHYSICS Two different situations can be considered: Growth of clusters in chemical reactions (e.g. in a solution of functional units, stars ), and introduction of cross-links into a preexisting system of chains. Both are to large extent equivalent. Model: Percolation on a lattice If the reaction took place, we consider the bond as present or intact (otherwise it is absent or broken). The molecules (connected units) are now called clusters (Broadbent & Hammersley, 1957). The concentration of intact bonds is p. The cross-linking problem is projected on a so-called site problem of the percolation theory, where the node is present with probability p or absent. p 0: only monomers p 1: the whole system is polymerized p c, at which an infinite cluster (molecule) appears. p c depends on the details of the lattice. Examples:

87 1.5. POLYMER NETWORKS 85 2D lattices p c site p c bond honeycomb square triangular D lattices diamond sc bcc fcc For a bond problem Cp c d d 1 d: the dimension of space, C: coordination number In general, an inequality Cp c 1 holds. Many other properties are universal: mean polymerization degree N w (p c p) γ γ 1.80 in 3d (γ = 43/18 in 2d) for p < p c. Typical cluster size (correlation length) ξ (p p c ) ν ν 0.88 in 3d (ν = 4/3 in 2d) Note: This critical exponent ν is not the Flory exponent! An infinite cluster appears for p > p c. The fraction of monomers belonging to an infinite cluster goes as S (p p c ) β with β 0.41 in 3d and β = 5/36 in 2d. The Lamé coefficients and the Young s module grow as E (p p c ) t. t > β (i.e. the elastic moduli grow slower than S since at transition the major part of the infinite cluster consists of dangling bonds). t 1.7 (exp.) Theoretical model of an infinite cluster:

88 86 CHAPTER 1. POLYMER PHYSICS Further links no links available yet Classical approach to polymer networks Classical picture: Flory, Stockmeyer 1943 Percolation on a tree Bild P : probability that there is an infinite way starting at the root P = 1 (1 pq) C p: probability, that the outgoing bond is intact Q: probability that there is an infinite way starting from a site different from the root An infinite system: Q are the same for the bonds in all shells Q }{{} = 1 (1 pq) C 1 }{{}

89 1.5. POLYMER NETWORKS 87 : probability, that a bond starts an infinite way : probability, that none of the outgoing bonds starts an infinite way Trivial solution: Q = 0 Nontrivial solution: Close to the critical point on can linearize the equation 2 (C 1) (C 2) Q p (C 1) Q p Q Nontrivial solution possible for p > p C = 1 C 1 close to p = p C P grows linear in p. Reaction of z-functional units. The probability that a functionality reacts is p i, the probability that the reaction is terminated (say by a parasitic reaction) is 1 p. The solution of the problem was given by De-Gennes, following Gordon (1962) and uses a rather formal approach; part of the calculations for a simpler approach are proposed as a homework. This approach is based on the so-called generating functions (good for physicists). Let us neglect the possibility of forming loops: the growing clusters have than the form of trees:

90 88 CHAPTER 1. POLYMER PHYSICS Let us take a monomer at random and calculate the probability ω n that it belongs to a cluster consisting of n + 1 monomers. Let us consider a monomer of interest as a root. Given a root we can impose directions on all the bonds. Note that the root has z outgoing bonds, and each other site has one incoming and z 1 outgoing bonds. We can calculate all ω n s using a mathematical trick called GENERATING FUNCTIONS. We introduce two of them: F 0 (θ) = n ω n θ n

91 1.5. POLYMER NETWORKS 89 n: number of monomers other than the root (number of bonds) Note: F 0 (θ) contains all information about the cluster sizes: d n dθ n F 0 (θ) = n!ω n. θ is a formal variable which is doing nothing else as counting the bonds: Each reacted incoming bond brings a θ! This branch is a cluster of n sites. If we add an additional monomer (we have to increase the power of θ by 1. Imagine we know F 1 (θ) = ω nθ n where ω n is the probability that the branch (neglecting the incoming bond) has n monomers. Imagine we connect the two branches characterized by F 1 (θ) and F 2 (θ). We get F (θ) = F 1 (θ) F 2 (θ) = ω n ω m θ n+m n m In particular, if we are interested in the probability to find a cluster of k sites, we get it from the θ k -th term which is m ω mω k m θ k, i.e. really corresponds to counting all the possibilities.

92 90 CHAPTER 1. POLYMER PHYSICS One more advantage is that the calculating of the generating functions can be done using a graphical help, a diagrammatic expansion. The same procedure can be applied for F 1. This is a closed equation, and we have solved the problem! F 1 through F 0, and you get ω n! Express For example: F 0 θ=1 = n=1 ω n (p) for p < p c, n=1 ω n = 1 sum over all infinite n for p > p c there exists an infinite cluster so that F 0 (1) = n=1 ω n = 1 P (p) the probability that a root belongs to an infinite cluster (i.e. gel-fraction). to the Solution for F 1 (1)

93 1.5. POLYMER NETWORKS 91 F 1 = (1 p + pf 1 ) z 1 p = 1 F 1/(z 1) 1 1 F 1 This solution exists for p > p c = 1/ (z 1). For p > p c, F = 1. The behavior of F 0 is also very similar to this picture, S (p p c ) 1 compare with 3d β = 1 β = 0.41 ν = 1/2 ν = 0.85 t = 3 t 1.7 Not a very good model to close to p c. However, the estimates for p c and the behavior far from p c may still be fairly well reproduced! Further links no links available yet

94 92 CHAPTER 1. POLYMER PHYSICS 1.6 Dynamics of polymer systems The Rouse model Dynamics of single chains (determines the diffusion coefficient of single chains) Simplest model: the Rouse chain (1953) ideal phantom chain (no interactions between the monomers) immobile solvent, independent motion of different monomer Model describes the chain as consisting of beads and springs (which may symbolize monomers and chemical bonds or larger subunits). Beads immersed in a solvent experience the friction force of the solvent and uncorrelated thermal forces (Brownian motion). m 2 r n = f t fr 2 n + f n ch + f n th = ξ r n t + κ [( r n+1 r n ) + ( r n 1 r n )] + f ch where f n fr is the friction force acting on the n-th monomer, f n ch is the elastic force acting from the neighboring monomers, and f th is a force due to the thermal motion of solvent molecules. κ is the elastic constant of a spring. f th is a Gaussian, δ-correlated force with zero mean, f th n (t) = 0 f th n,α fm,β th = 2ξδnm δ αβ δ (t t ) Typically, the motion is slow, and the accelerations are small m 0: We get the first order equation ξ r n t =... Equation for normal modes Introducing normal coordinates X p (t) 1 N N 0 dn cos ( pπn ) Rn (t) N

95 1.6. DYNAMICS OF POLYMER SYSTEMS 93 (p = 0, 1, 2,...) (essentially, Fourier-modes of our continuous equation in n-space) one gets following system of equations for X p (t): ( ) ξ p = t X p (t) = k p Xp (t) + fp }{{} a random force with: ξ 0 = Nξ, ξ p = 2Nξ for p > 0 k p = 2π 2 kp 2 /N 6π2 k b T Na 2 p 2 (p = 0, 1,...) and random forces f p having the following properties: f pα = 0 f pα (t)f qβ (t ) = 2δ pq ξ p k b T δ(t t ) The forces with p q are independent the motion is described by a picture of independent modes, each of them corresponding to an Orustein- Uhlenbeck-process (exactly Eqn. (*)). The solution for PDFs of OU-process are known (since 1931)! ξ r n t = 3k b T a 2 κ }{{} for ideal chain ( r n+1 2 r n + r n 1 ) + fn th (t) At larger scales the chain can be considered as a continuous object. In a continuous approximation r (t, n) ξ t = 3k bt 2 r (t, n) a 2 n 2 + fn th (t) (a Rouse equation) This is a stochastic partial differential equation. Boundary conditions: the last monomers are free, no elastic forces from outside the chain r (t, 0) n = r (t, N) n = 0 The overall behavior of the chain can be described in terms of superposition of modes (i.e. in Fourier representation in n), see Grosberg, Khokhlov or Doi and Edwards (*). Result for example for a mean squared displacement of e bead number n (x (t, n) x (0, n)) 2 = 6k bt Nξ t + 4Na2 π 2 p=1 ( )) 1 πpn cos2 1 exp ( tp2 p2 N τ R

96 94 CHAPTER 1. POLYMER PHYSICS with τ R = N 2 a 2 ξ/ (3π 2 k b T ) being the longest relaxation time (Rouse time). This is the most important characteristic time of a problem corresponding to full rearrangement of the chain s conformation. For t >> τ R all monomers (and the whole chain) move in a diffusive manner (x (t, n) x (0, n)) 2 6k bt Nξ the diffusion coefficient x 2 Dt t a a diffusive behavior D = k bt Nξ (N times less then for each monomer; the last one is D = k bt/ξ, according to the Einstein s relation). τ R R2 D Na2 k b T/Nξ N 2 a 2 ξ k b T For t << τ R, the displacement of each segment goes as (x (t, n) x (0, n)) 2 ( ) 12 k b T a 2 1/2 t π ξ i.e. x 2 t 1/2, x 2 t 1/4 Explanation: At short times the overall conformation don t change strongly. The possible displacements are bounded to small subchains, for which τ R t, i.e. to the fragments of the size ( N t k ) 1/2 bt a 2 ξ the overall displacement is of the order of the size of such subchain: ( ) x 2 Na 2 kb T a 2 1/2 ( ) 1/2 t a 2 kb T t 1/2 ξ a 2 ξ The motion of a phantom chain in an immobile solvent follows the predictions of the Rouse theory, independently on what particular chain model of the chain is applied. The predictions of the Rouse model are often at variance with the experimental findings. The analysis shows that it is the assumption of the immobile solvent, which is crucial.

97 1.6. DYNAMICS OF POLYMER SYSTEMS 95 Further links no links available yet Hydrodynamic interactions The Zimm model (1956): the hydrodynamic interaction: The motion of each monomer causes the solvent motion which acts on all other monomers. This interaction is described by the so-called Oseen tensor v ( r 2 ) = Ĥ ( r) v ( r 1) Ĥ: Oseen tensor (2. rank) H αβ = 1 ) (δ αβ + ( e r ) 8πη s r α ( e r ) β (H αβ : Cartesian components, η: solvent viscosity) H (r) r 1 : very long-range interaction The Zimm model, takes into account this interaction and leads to an equation for r (t, n) in the form r (t, n) t = N 0 dmĥnm [ ] 3kb T 2 r (m, t) + f th (m, t) a 2 m 2 The results from the Zimm model are: τ 1 N 3/2, D N 1/2 These findings are confirmed experimentally (θ-solutions). The physical

98 96 CHAPTER 1. POLYMER PHYSICS meaning of these results is as follows: The solvent moves together with a coil. The coil behaves as an impenetrable sphere of radius R 0 = N 1/2 a. The overall friction force (Stokes law) corresponds to a friction coefficient ξ eff = 6πη s R 6πη s an 1/2 ; according to the Einstein s relation D = k b T/ξ k b T/η s an 1/2 Now, τ z R2 D a k b T η s N 3/2 Also an excluded-volume coil behaves as an impenetrable sphere! In experiments on the viscosity of dilute polymer solutions, a solution behaves as a suspension of spheres of the same concentration n and of radius R F or R 0 (good and θ-solvent). Further links no links available yet Concentrated solutions and melts: the reptation model The main property of polymer fluids is viscoelasticity (e.g. silicon flows under small constant force, but a silicon or latex ball bounces when thrown on the floor!). The first theory of polymer fluid dynamics was developed by P.G. De-Gennes (Nobel prize 1991), M. Doi and S.F. Edwards in 1970s. Consider the motion of one tagged chain in a temporary network formed by other chains:

99 1.6. DYNAMICS OF POLYMER SYSTEMS 97 The lateral motion of the chain is strongly restricted, the motion along its contour is however free: this is as if the chain moved in a tube. The form of the tube itself is a random walk s path:

100 98 CHAPTER 1. POLYMER PHYSICS Stages of reptation: The chain crowls out of the tube and reorients. The old configuration is forgotten The disappearance of the tube due to the collective motion of other chains is typically a slower process and can be neglected. τ : a typical relaxation time in a system N e : the typical number of entanglements (effective links) per chain. N e >> 1: in typical situations N e << N e << N: tube diameter d an 1/2 e The ( chain) is a sequence of blobs of size d the tubes length N Λ d an, (N e : number of blobs, oriented along the tube) N e N 1/2 e Note: difference with the case of the pore D t : diffusion coefficient along the tube, like in a Rouse case it is proportional to kt/nξ: The motions of a solvent (if any) are strongly weakened by the network of other chains. τ Λ2 D a2 N 2 Nξ N e k b T N 3

101 1.6. DYNAMICS OF POLYMER SYSTEMS 99 At times t << τ the solution or melt behaves as a solid (is rubber elastic). The entanglements act as effective cross-links. Experiments on viscosity: l (t) l = σl (t) (σ: stress) (for a solid l (t) = const. = 1/E) Experiments give typically τ N 3.4 and η N 3.4. The difference is attributed to finite chain lengths: N e is not to large. The segment motion in the reptation model shows 4 subsequent regimes.

102 100 CHAPTER 1. POLYMER PHYSICS Further links no links available yet

103 Chapter 2 Polymer Characterization 101

104 102 CHAPTER 2. POLYMER CHARACTERIZATION 2.1 Introduction text

105 2.2. MOLAR MASS DETERMINATION Molar mass determination Introduction M 0 molar mass of a repeating unit P degree of polymerization M molar mass of a macromolecule M i distribution of molar masses M i = P i M 0 n i number of macromolecules with P i m i mass of a macromolecule with P i m i = M i N A w i : mass of all macromolecules with P i w i = n i m i = M in i N A w = i w i n = i n i Synthetic polymers are usually not monodisperse, but have a distribution of polymerization degrees P i. The distributions can be characterized by their moments µ l : µ l = 1 n i Mi l k i with l = 1, 2, 3,... and k: normalization Number averaged molar mass M n is given by i µ 1 = n im i wi N A i n = =: M n i n Weight averaged molar mass M w is given by µ 2 = i n imi 2 wi M µ 1 i n = i im i i w =: M w i z-averaged molar mass is given by µ 3 = i n imi 3 wi Mi 2 µ 2 i n = imi 2 i w =: M z im i

106 104 CHAPTER 2. POLYMER CHARACTERIZATION Polydispersity P D(or Uneinheitlichkeit U) P D = U = M w M n 1 For monodisperse polymers all averages are identically M and P D = 0. For polydisperse polymers M w > M n P D > 0 Example: Consider a system of 1000 molecules with the following characteristics: FLASH- TABLE Table 2.1: Methods Overview M n M w M z M η M-range g/mol Absolute Methods Osmometry Elevation of < 10 4 boiling point Lowering of < 10 4 boiling point Electron > 10 5 Microscopy Endgroup 104 Analysis Sedimentation Light Scattering > 10 2 SAXS & SANS > 10 2 Ultracentrifuge > 10 3 Relative Methods Viscometry Very high Distribution of molar masses Gel Permatation Chromatography and Electron microscopy Colligative methods Colligative properties are properties which depend only on the number of particles in a solution. They are independent from the shape of the particles.

107 2.2. MOLAR MASS DETERMINATION 105 Elevation of the boiling point by T b : T b lim = R T 2 1 c c ρ H b M n with : H b : boiling enthalpy Lowering of the freezing point by T f : T f lim = R T 2 1 c c ρ H f M n with : H f : freezing enthalpy Lowering of the vapor pressure by p: p p = X 2 with : X 2 : concentration of solute Further links no links available yet Osmometry Initially the chemical potential µ of a solvent in a solution is lower than the pure solvent. Therefore the solvent molecules tend to pass through the membrane into the solution in order to reach equilibrium. This causes the buildup of pressure in the solution compartment until, at equilibrium, the pressure exactly counteracts the tendency for further solvent flow. Experimental setup:

108 106 CHAPTER 2. POLYMER CHARACTERIZATION Π = RT (1 + A 2 cm n +...) c M n with : A 2 : second virial coefficient A 2 depends on the solute and solvent and therefore describes the net interaction. All three interactions (solute-solute, solute-solvent, and solventsolvent) depend differently on T, therefore A 2 may be positive or negative. It can be also zero for a certain temperature, called θ -temperature T θ. The virial coefficient A 2 may be considered also as deviation from ideality. Limitation is the time required to establish the equilibrium which may take days.

109 2.2. MOLAR MASS DETERMINATION 107 Further links no links available yet Ultracentrifuge Consider a particle of mass m and volume v, i.e. a specific volume v s = v m = 1 ρ in a liquid of density ρ 0 Force on particle F g = mg v s ρ 0 g = (1 v }{{} s ρ 0 ) mg buoyancy causes motion in x-direction Frictional force F f = f dx dt with f : friction coefficient limits acceleration. There are two cases to consider. 1. Steady state Constant sedimentation velocity dx dt if (1 v s ρ 0 ) mg = f dx dt g due to gravity on earth rather small small dx dt larger g in (ultra)centrifuge (Svedberg 1924) (hard to measure)

110 108 CHAPTER 2. POLYMER CHARACTERIZATION due to centrifugal acceleration ω 2 x (1 v s ρ 0 ) mω 2 x = f dx dt Experiment: sedimentation constant s = (1 v s ρ 0 ) M = fs N }{{} A =m dx dt ω 2 x M = N Afs 1 v s ρ 0 Determination of M requires f! (or for spheres: f = 6πη 0 R) with D: Diffu- Use Einstein relation D = k bt f sion coefficient Svedberg equation: M = RT s D (1 v s ρ 0 ) spheres = N A6πη 0 Rs (1 v s ρ 0 )

111 2.2. MOLAR MASS DETERMINATION 109 Besides s, D, v s also T and s 0 (or R, v s, ρ 0 and η 0 ) need to be known [which molar mass average can one determine from the sedimentation velocity?] Experiment: ω = rpm (up to g) of sector shaped cell (small quality, little convection) in vacuum (avoids friction) Sedimentation velocity from optical detection of front : Schlieren-Optics, interference or absorption. 2. Equilibrium Total potential µ = sum of mechanical and chemical potential: µ = 1 µ N A 2 mω2 r 2 = µ 1 2 Mω2 r 2 µ r = µ r Mω2 r = 0 Linear response Flux J of moles of macromolecules per area and time: [ ] µ J = L r Mω2 r with mass conductivity L = c N A f with c: concentration, f: frictional coefficient per mole For T = const. µ r = µ p + r T,c }{{}}{{} r v sm pω 2 r RT c µ c T,p }{{} γ 1+c c c r with γ = 0 for ideal solution, γ: activity coefficient c Ideal solution: µ = Mv s ρω 2 r + RT c r [ c r J = L Mv s ρω 2 r Mω 2 r + RT ] c c r J = sω 2 rc D c r

112 110 CHAPTER 2. POLYMER CHARACTERIZATION Flux is difference between sedimentation rate and back diffusion! Equilibrium: J = 0 1 c c r = ω2 rm (1 v s ρ 0 ) RT Integration between menisus r m and any point r ln c (r) c (r m ) = ω2 M (1 v s ρ 0 ) (r 2 r 2 m) RT Measure ln c (r) versus r 2 experimentally slope gives M Integration between menisus r m and cell bottom r b rb r m c r dr = ω2 M (1 v s ρ 0 ) rb rc (r) dr RT r m c (r b ) c (r m ) = ω2 M (1 v s ρ 0 ) (r 2 b r2 m) c 0 2RT with c 0 : final concentration due to mass conservation Further links no links available yet Light scattering Elastic Scattering: ω in = ω out Inelastic Scattering: ω in ω out With elastic scattering one can measure M w, the shape of the polymer and the virial coefficient. Necessary additional information: concentration and increment in the refractive index Experimental setup

113 2.2. MOLAR MASS DETERMINATION 111 When a beam of light strikes the molecules of a medium, the electrons are displaced and oscillate about their equilibrium positions with the same frequency as the exciting beam. This induces transient dipoles in the atoms or molecules, which act as secondary scattering centers by re-emitting the absorbed energy in all. µ = α E = α E 0 exp (iωt) with µ: dipole moment and α: atom polarizability Oscillating dipole emits light with intensity I Θ = 8π4 α 2 ( I r cos 2 Θ ) λ 4 0 Approximation d << λ 0 I Θ = 8π4 α 2 ( I r cos 2 Θ ) λ 4 0 Scattering by volume (normalized by I 0, r) R Θ = I Θ I 0 r 2 N R Θ = 8π4 r 2 α 2 λ 4 NI 0 ( 1 + cos 2 Θ )

114 112 CHAPTER 2. POLYMER CHARACTERIZATION Scattering characteristic p = χ E χ = ε 1 = n 2 1 ε = ε solution ε solvent = 4πNα ε = n 2 n 2 0 = (n + n 0 ) (n n 0 ) n n 0 2n 0 4πN = α α = n 0 dn c 2π dc N R Θ = R Θsolution R Θsolvent = 2π2 n 2 K = 2π2 n 2 λ 4 0N A ( ) 2 dn dc λ 4 0N A K c ( 1 cos 2 Θ ) = 1 + 2A 2 c R Θ M w ( ) 2 dn cm ( 1 cos 2 Θ ) dc For d >> λ/20, interference from light scattered from different parts of the same molecule has to be taken into account and therefore we introduce the correction factor P (Θ). P (Θ) = 1 16π2 3λ 2 R 2 g sin 2 Θ 2

115 2.2. MOLAR MASS DETERMINATION 113 K c ( 1 cos 2 Θ ) = R Θ M w P (Θ) + 2A 2 c for 1 1 x = 1 + x + x2 + x K c ( 1 cos 2 Θ ) = 1 R Θ M w if Θ 0, which means for small angles 16π2 lim P (Θ) = 1 R 2 Θ 0 3λ 2 g sin 2 Θ 2 K c ( 1 + cos 2 Θ ) = 1 + 2A 2 c R Θ M w Zimm-plot (1 + 16π2 R 2 3λ 2 g sin 2 Θ ) A 2 c Plot of Kc/ R Θ as a function of c for fixed Θ slope gives A 2 Plot of Kc/ R Θ as a function of Θ for fixed c slope gives Rg 2 Intersection of c = 0 and Θ = 0 gives M w Further links no links available yet

116 114 CHAPTER 2. POLYMER CHARACTERIZATION Mass spectroscopy Chromatography Electron microscopy Preparation: Deposit single molecules (isolated from each other) on a substrate with contour in two dimensions Measurement of the contour length: In transmission electron microscopy, resolution is very high (monomer length) but high energy of electron beam can cause the destruction of the sample. High contrast in polymers by staining with heavy atoms. Further links no links available yet Scanning probe microscopy Introduction Atomic Force Microscopy (AFM), also known as Scanning Force Microscopy (SFM) Invented in 1986 by G. Binnig, H. Rohrer and co-workers (Binnig 1986, Rugar 1990) Measured physical quantity: force between a sharp conical tip and a sample surface allows investigations on a wide range of materials including polymers Fig. 1

117 2.2. MOLAR MASS DETERMINATION 115 SFM probes the surface of a sample using a sharp tip (terminal radius 10nm), located at the free end of a cantilever (100µm long, elastic modulus: a few 0.1N/m) Forces of a few pn between tip and sample surface vertical deflections of the cantilever on the Å-scale. Laser beam bounces off the back of the cantilever onto a position-sensitive detector (PSD) (Fig. 1), measuring the displacements of light beams as small as 1nm a mechanical amplification is produced by the length-ratio (cantilever-detector and cantilever itself) detection of sub-å vertical movements of the cantilever tip a computer is used to generate a map of the surface topography (Fig1) measurement-accuracy strongly depends on the tip s movement-accuracy relative to the sample accurate placement of the tip is carried out by a piezoelectric scanner tube (positioning with sub-nm precision) Piezoelectric scanner Actuators (scanner) are typically made of a single radially polarized piezoceramic tube on the in- and outside the tube is plated with an electrode material (usually nickel) the outer electrode is divided along the tube axis into four equal segments (Fig 2) due to the piezoelectric transverse effect, the tube bends when a voltage is applied between an outer and the inner electrode when opposite segments are driven by a signal of the same magnitude but opposite sign, the tube is bent twice as much the probe can be moveed in two dimensions (approximately on a sphere) the inner electrode is usually driven by the signal adjusting the sample-

118 116 CHAPTER 2. POLYMER CHARACTERIZATION probe-distance Fig. 2 electric field E and strain S are (without mechanical stress) related by S = d E with d: piezoelectric strain constant d adopts three different values (on the order of several nm/v ) depending on the direction of the applied electrical field relative to the polar axis and the observed direction the displacement l is given by (including the geometry of the tube): l x = s x U x l y = s y U y l z = s z U z the sensitivities s x, s y and s z for the displacement are related to the piezoelectric strain constant d Fig. 3

119 2.2. MOLAR MASS DETERMINATION 117 d is temperature dependent and not truly constant for large E also the sensitivities for the tube s x-, y- and z- direction are not constant the piezoelectric actuators follow a hysteresis curve and the displacement of an actuator is time dependent (creep) (Fig. 3) all these problems make it difficult to calibrate a piezoelectric tube for accurate SFM operation Forces in SFM Interaction forces between tip and sample are often quite complex due to several factors: the number of atoms from the tip involved in the interaction is not only one, due to the contribution of rather long range forces forces are dependent on the environment (gas, liquid or vacuum) the scan is a dynamic process velocity dependent forces need to be considered the tip can deform the sample Classification of forces Important to distinguish type of forces (tip sample) in order to separate the contributions and correctly interpret the experimental results Long range forces 1. van der Waals forces: due to atomic polarizability and therefore very general

120 118 CHAPTER 2. POLYMER CHARACTERIZATION role of these forces have been discussed by Moiseev and Hartmann ( 1/r 6, spatial extent: one to tens of nm) 2. Electrostatic forces: due to coulombic interactions can occur between an electrostatically charged tip and a charged area of an insulating surface (spatial extent: one to thousands of Ångstrom) 3. Capillary forces: curvature at the contact between tip and sample causes condensation of vapor from the ambient including water from the air surfaces exposed to air environment are typically coated by a layer of water (thickness depends on relative humidity (RH) of the atmosphere and the physico-chemical nature of the object) strong attractive capillary forces (about 10 8 N) result that hold the tip in contact with the surface to avoid capillary forces RH=0% Thundat and co-workers demonstrated that below RH=10% they could not detect decays any further of the capillary forces (Thundat 1993) two simple experimental procedures can minimize the effect of this kind of forces: (a) flood a sealed chamber for the measurements with a dry inert gas such as N 2, He or Ar (b) make use of a fluid cell, that means to perform measurements with both the tip and the sample immersed in a liquid medium Short range forces 1. Repulsive forces: ( 1/r n with n > 8) interatomic repulsion forces have two origins: (a) repulsion between nuclei: overlap of two electronic clouds gives rise to an incomplete screening of the nuclear charges coulombic repulsions

121 2.2. MOLAR MASS DETERMINATION 119 (b) Pauli repulsion: according to the exclusion principles of Pauli, two electrons with the same spin can not occupy the same orbital electrons can only overlap when the energy of one electron is increased 2. Forces of covalent bonds: they originate from the overlap of the wave functions of two or more atoms density of electronic charges is concentrated between the two nuclei this force decreases abruptly for a separation over a few Å the type of interaction can be also called chemisorption 3. Metallic adhesion: derives from the interaction between strongly delocalized electronic clouds, which cause strong interactions ( e r ) important when two metallic surfaces approach to the extent that the electronic wave functions overlap this case can be called physisorption 4. Friction: during the scan, there is a force component parallel to the surface, since the tip is not always oriented exactly perpendicular to the surface this friction tends to twist the cantilever (torsion angle depends on the composition of the surface) measurement of the twist provides chemical information it was also shown that this kind of friction force can be detected at the atomic scale First approximation: forces contributing to the deflection of an SFM cantilever (van der Waals and the interatomic repulsive forces) can be derived from the Lennard - Jones potential [ (σ ) 6 ( σ ) ] 12 ε (r) = 4ε 0 r r ε 0 : potential energy at the minimum σ: effective molecular diameter

122 120 CHAPTER 2. POLYMER CHARACTERIZATION r: the interatomic distance f = grad ε: force (plotted in Fig. 4) two distance regimes are highlighted: 1. the contact regime the cantilever is held less than a few Å from the sample surface repulsive interatomic force between cantilever and sample 2. the non-contact regime the cantilever is held on the order of tens to hundreds of Å from the sample surface Fig. 4 attractive interatomic force between cantilever and sample (largely a result of the long-range van der Waals interactions) Contact mode SFM tip makes soft physical contact with the sample cantilever has a lower spring constant than the effective spring constant holding the atoms of the sample together scanner gently scans the tip across the sample (or the sample under the

123 2.2. MOLAR MASS DETERMINATION 121 tip) the contact force causes the cantilever to bend in order to follow the topographic profile using very stiff cantilevers it is possible to exert large forces on the sample the sample surface is likely to get deformed (may be used in nanolithography ) the total force that the tip exerts on the sample is the sum of capillary plus cantilever forces must be balanced by the repulsive van der Waals force force magnitude varies from 10 8 N to 10 6 N (typically 10 7 N) (with the cantilever pulling away from the sample almost as hard as the water is pulling down the tip) Contact mode SFM can operate in two modes: 1. constant height mode height of the scanner (consequently also the distance sample surface - tip holder) is fixed as it scans spatial variation of the cantilever deflection can be used directly to provide the topographic data set cantilever deflections and thus variations in applied force are small often used for recording atomic-scale images of atomically flat surfaces essential for monitoring fast processes in real-time, where high scan speed is essential 2. constant force mode the deflection of the cantilever is used as input to a feedback loop moving the scanner up and down in z-direction keeping the cantilever deflection constant total force applied to the sample is constant image is generated from the scanner s motion

124 122 CHAPTER 2. POLYMER CHARACTERIZATION scanning-speed is limited by the feedback loop s response-time total force exerted on the sample by the tip can be controlled this mode gives a real topographic map of the sample surface preferred for most applications Non - contact modes alternative modes - to overcome the problem of the friction during scanning in contact mode - to minimize the forces exerted from the tip on the sample - to minimize the effect of the capillary forces SFM cantilever vibrates near (on the order of tens to hundreds of Å) the surface of a sample (Fig. 5) a stiff cantilever is forced to oscillate near its resonant frequency (typically from 200 to 400 khz) with an amplitude of a few hundreds of Å while the tip scans over the sample, the system detects the amplitude of the cantilever s swing and keeps it constant with the aid of a feedback system that moves the scanner up and down keeping the amplitude constant also the average tip-to-sample distance is constant sensitivity of this detection scheme provides sub-å vertical resolution in the image due to the elimination of the shear forces (tip sample) these modes are particularly useful for studying soft or elastic materials (i.e. biological and organic films) as a consequence of the reduction of the overall interaction forces (tip sample), these modes do not suffer from tip or sample degradation effects the lateral resolution that can be reached is a few nm (lower than in the contact mode) Fig. 5

125 2.2. MOLAR MASS DETERMINATION 123 Non Contact SFM (NC-SFM) tip sample interactions are indicated on the force-distance curve (Fig. 4) as the non-contact regime force (tip sample) in this regime is low ( N) force measurement is more difficult than in the contact regime (where it can be several orders of magnitude larger) cantilevers used for NC-SFM must be stiffer than those used for contact SFM because soft cantilevers can be pulled into contact with the sample surface small force values in the non-contact regime and the greater stiffness of the cantilevers used for NC-SFM limit the force resolution, and consequently the lateral resolution, that can be achieved. Tapping Mode(TM) (TM-SFM) or intermittent-contact atomic force microscopy (IC-SFM) is similar to NC-SFM, except that for TM-SFM the vibrating cantilever-tip is brought closer to the sample at the bottom of its travel it just barely hits, or taps the sample notably increase in the lateral resolution the intermittent-contact operating region is indicated on the curve in (Fig. 4) TM-SFM is more effective than NC-SFM both for: imaging larger scan sizes that may include greater variation in sample topography slightly higher resolution that can be achieved due to the stronger tip-sample interaction forces that are sampled

126 124 CHAPTER 2. POLYMER CHARACTERIZATION making use of the Phase Imaging mode, detecting the shift in phase of the vibration further increase in spatial resolution in this imaging mode the contrast is originated from differences in surface adhesion and viscoelasticity very helpful for detecting different phases that coat the sample surface Fig. 6 Further links Tapping Mode(TM)

127 2.3. OPTICAL SPECTROSCOPY Optical spectroscopy Basics of photophysics Selected (one-photon) photo physical processes Energy transfer processes Infrared spectroscopy

128 126 CHAPTER 2. POLYMER CHARACTERIZATION 2.4 Molecular modelling Molecular mechanics potentials Molecular dynamics potentials

129 2.5. X-RAY AND NEUTRON SCATTERING X-ray and neutron scattering Introduction Advantages of X-rays and neutrons: wavelengths on atomic length scale no charging of insulating polymers good penetration into polymers buried interfaces provide contrast which is specific to certain repeat unit nondestructive X-rays have rest mass 0, while neutrons have a finite mass. So the physics of them is different. We treat them both as travelling waves and the media in which they travel (polymer, vacuum) are described by a refractive index n. For both x-rays and neutrons n is very near to Basics of x-ray scattering Basics of neutron scanning X-ray and neutron reflectivity Surface is a particular interface (polymer-vacuum) Most polymers are isolators X-Ray Reflection n 1 cos θ 1 = n 2 cos θ 2 n 2 is smaller than n 1, therefore θ 2 is smaller than θ 1. reflection if θ 1 goes very small angle, then θ 2 will be 0. We can have total n 1 cos θ 1 = n 2 θ = arccos n 2 n 1 Amplitude of reflected and transmitted light is given by: r s = sin (Θ 1 Θ 2 ) sin (Θ 1 + Θ 2 ) t s = 2 cos Θ 1 sin Θ 2 sin (Θ 1 + Θ 2 )

130 128 CHAPTER 2. POLYMER CHARACTERIZATION Brewster angle is angle at which the refractive beam is perpendicular to the reflected beam. Reflected beam and transmitted intensity: r 2 and t 2. Determination of n n = 1 δ = 1 λ2 2π r eρ e ρ e = Nf r e : classical radius of an electron ρ e : x-ray scattering density N: number of atoms per volume f: atom form factor (roughly equal to number of electrons in atom) Neutron Reflection Neutrons are described as particle waves rather than electromagnetic waves for particles: p = mv E = p2 2m = 1 2 mv2 for waves E = hν = ωwhere ω = 2πν = h 2π c = λν = ω k k = 2π λ p = k de Broglie wavelength Schrödinger Equation 2m Ψ + V Ψ = EΨ = Ψ + 2m 2 (E V ) Ψ = 0 ( x 2 + y 2 + ) z 2

131 2.5. X-RAY AND NEUTRON SCATTERING 129 analogous to wave equation for light where 2m 2 n = (E V ) εω2 c 2 ( 1 V ) 1/2 = 1 λ2 E 2π ρ n N: density of the nucle??? b: scattering length, characteristic for each type of the nucleus, e.g. is very different for H and D As a result we have the possibility for labelling polymers at certain positions by exchanging D for H. intramolecular contrast (copolymers) and intermolecular contrast (blends). Further links no links available yet

132 130 CHAPTER 2. POLYMER CHARACTERIZATION 2.6 Mechanical testing Introduction Mechanical properties of polymers depend on time scales: Silly putty is viscous on long and elastic on short time scales. Therefore, these extreme cases shall be considered first. Then combinations of them (viscoelasticity) are treated Viscous fluids A Newtonian liquid is characterized by its viscosity η Viscosity η of a solution depends on concentration c Limiting viscosity number [η] = lim c 0 η η 0 cη 0 with η 0 : solvent viscosity ([η]: sometimes called intrinsic viscosity) characterizes capacity of a polymer molecule to enhance the viscosity of a solution. It depends on size and shape of the polymer. Within a given series of polymer homologs, [η] increases with molar mass M ( method to measure M) Fit by Mark-Houwink-Sakurada equation: [η] = km a linear, flexible polymers at θ-conditions: [η] = k e M 0.50 Newton s Law: F A = σ = η dε dt with σ: stress, η: viscosity, ε: strain σ (ε) time dependent dε = const. (constant deformation) dt σ (ε) = η dε dt = const.

133 2.6. MECHANICAL TESTING 131 Figure 2.1: Dashpot Energy is dissipated Dashpot describes retardations fluid Table 2.2: Viscosities of various fluids η in Ns conditions in C m 2 pentane acetone heptane trichloromethane toluene methanol benzene water ethanol poly(ethylene) ca (400P a) Further links no links available yet

134 132 CHAPTER 2. POLYMER CHARACTERIZATION Figure 2.2: σ = σ(ε) Elastic solids Spring Hooke s Law F A = σ = Eε = E l l 0 with E: Young s modulus, l 0 : length of sample without external stress (σ = 0 ), l: length of sample at σ

135 2.6. MECHANICAL TESTING 133 σ (ε) independent of t! Analog for shear: σ S = Gε S Energy is stored and can be recovered Elastic solid becomes thinner upon elongation Poisson ration: ν p Typical values for ν p : Without volume change: E = 2 (1 + ν p ) G Examples: material E in GN ν m 2 p G in GN m 2 steel granite poly(styrene) natural rubber Further links no links available yet Viscoelasticity Simplest models with good approximation for small perturbations are Maxwelland Voigt-Kelvin-Model Maxwell-Model

136 134 CHAPTER 2. POLYMER CHARACTERIZATION ε = ε 1 + ε 2 σ = σ 1 + σ 2 dε dt = 1 E dσ dt + σ η dσ dt dε dt = σ η dε dt = 0 (constant stress) = 0 (constant strain) dσ dt + E η σ = 0 σ (t) = σ (t = 0) exp ( Eη ) t Exponential relaxation of stress with time. At time τ = η E initial stress is reduced to 1 e τ is called relaxation time. = of the original value. Voigt-Kelvin-Model

137 2.6. MECHANICAL TESTING 135 σ 1 = Eε σ 2 = η dε 2 dt ε = ε 1 = ε 2 σ = σ 1 = σ 2 σ = Eε + η dε 2 dt dσ = 0 constant strain dt Further links no links available yet Creep, stress-strain measurements, dynamic mechanical testing Maxwell- and Voigt-Kelvin-Model still too simple to fit typical mechanical properties like toughness, strength etc. Creep: dε ( ) dσ > 0 for constant stress, e.g. due to weight dt dt = 0 Experiment: Maxwell- and Voigt-Kelvin-Modell in series

138 136 CHAPTER 2. POLYMER CHARACTERIZATION Boltzmann superposition principle: If a system is subjected to a series of stresses σ i at times t i, creep is given by the sum of the identical responses Example

139 2.6. MECHANICAL TESTING 137 Stress-Strain-Measurements Practically important for determination of Young s modulus E brittleness yield strength Tensile stress at uniform rate σ L : stress for brittle fracture Y : yield point B: break point

140 138 CHAPTER 2. POLYMER CHARACTERIZATION Temperature Dependence Example for brittle materials with failure before Y : Poly(methylmethacrylate) at room temperature Example for tough materials: celluloseacetate

141 2.6. MECHANICAL TESTING 139 Rigidity and yield strength decrease with T Maximum elongation increases with T Glass transition temperature T g for cellulose acetate: 273K poly(methylmethacrylate): 320K Fracture energy goes through maximum with increasing T Stress Relaxation σ (t) for dε dt = 0 Force measurement experimentally more demanding than creep σ (t) E (t)

142 140 CHAPTER 2. POLYMER CHARACTERIZATION Rapid change over narrow temperature range glass transition Dynamic mechanical analysis Important for short time scales Sinusoidal stress phase (φ)-shifted sinusoidal response with amplitude A: ε = ε 0 exp (iωt)

143 2.6. MECHANICAL TESTING 141 Response function E (ω) with σ (ω) = ε E (ω) E (ω) = E (ω) + ie (ω) Further links no links available yet

144 142 CHAPTER 2. POLYMER CHARACTERIZATION

145 Chapter 3 Appendix 143

146 144 CHAPTER 3. APPENDIX 3.1 Symbols A: area A 2 : second virial coefficient b: neutron scattering length c: concentration / velocity of light D: diffusion coefficient E: energy / Young s modulus f: friction coefficient / atom form factor F : force G: shear modulus h: Planck s constant : = h/2π H: enthalpy I: intensity J: flux k b : Boltzmann constant l: length L: mass conductivity m: mass M: molar mass M n : number averaged molar mass M w : weight averaged molar mass M z : z-averaged molar mass n: refractive index / number of particles N: number of atoms per volume N A : Avogadro constant

147 3.1. SYMBOLS 145 p: pressure P : degree of polymerization r: distance r e : classical radius of an electron R: universal gas constant R g : radius of gyration R θ : Rayleigh ratio s: sedimentation constant S: entropy t: time T : temperature T g : glass transition temperature v: volume v s : specific volume V : potential energy x: spatial coordinate y: spatial coordinate z: spatial coordinate α: atom polarizability γ: activity coefficient ɛ: strain / dielectric function ɛ s : shear strain η: viscosity [η]: limiting viscosity number µ: chemical potential / dipole moment / moment of a distribution µ: sum of mechanical and chemical potential

148 146 CHAPTER 3. APPENDIX ν P : Poisson ration Π: osmotic pressure ρ: density ρ e : x-ray scattering density σ: stress χ: dielectric permeability ω: circular frequency ω: circular frequency

149 3.2. CHEMICAL FORMULAE Chemical formulae acetate acetone celluloseacetate cellulose ethane ethanol heptane pentane poly(dimethylsiloxane)

150 148 CHAPTER 3. APPENDIX poly(ethylene) poly(methylmethacrylate) poly(methylsiloxan) poly(styrene) toluene