Course M6 Lecture 9//004 (JAE) Course M6 Lecture 9//004 Polymer solutions and melts Scattering methods Effects of excluded volume and solvents Dr James Elliott Online teaching material reminder Overheads (PDF) for st half M6 available from: www.msm.cam.ac.uk/teaching/online.html#iii www.cus.cam.ac.uk/~jae00/teaching.html#materials
Course M6 Lecture 9//004 (JAE). Introduction Previous lecture considered statistics of isolated polymer chains in vacuo Of course, in practice, polymers interact with solvents and other polymers They also interact with themselves excluded volume In this lecture, we will see how enthalpic interactions modify the entropic statistical behaviour of an ideal polymer chain First, we will review some scattering methods. Particle scattering methods How can we measure R g and C experimentally? SANS, deuterated and H chains have different scattering cross-sections: thermal neutrons: q =0-0 Å SAXS, synchrotron radiation: q = 0-0 Å Light scattering, HeNe laser: q < 0 Å incoming beam θ scattered beam
Course M6 Lecture 9//004 (JAE).. The structure factor Represents interference function between monomer scattering centres ( expiq ij ) I ( q) S( q) = L I N N m i, j ecause monomer density is approximately Gaussian, scattering function is given by: S z ( e + z ) ; z R ( g q z D q) = = Debye function.4 Guinier s law and Zimm plots At low q, the Debye scattering function simplifies to: S ( q) N ( + q D R g / ) Guinier s law So, plotting S versus q yields a measure of R g. This is known as a Zimm plot. Furthermore, plotting ln(s) versus q should also yield a straight line. This is known as a Guinier plot.
Course M6 Lecture 9//004 (JAE).5 ratky plots As q, the Debye function turns into a power law: S D ( q) N q R g Hence, plotting Sq versus q (ratky plot) yields plateau region which contains information about chain structure..6 Evidence for chain swelling From scattering measurements, we know that the size of polymer coils has a concentration dependence which have not so far accounted for in our theory log R G log c w 4
Course M6 Lecture 9//004 (JAE).7. Self-avoiding random walks Let s reconsider the simple model of a random flight or freely jointed chain We will use a mean-field argument for the repulsions between the monomer units, which act to expand the chain. These have the form: N ktn a U kt Na = πr 4πR 4 g They are balanced by an entropic contracting force: S = k ln k R g [ W( Rg, N) ] = Na g.7. Self-avoiding random walks So, writing down the Helmholtz free energy of the chain: N a R g F = U TS = k + T 4πRg Na and minimising F N a = 0 Rg 4π 5 R N a g 4 Rg R + Na g = 0 That is, scaling exponent ν = 0.6 for S.A.W. (c.f. ν = 0.5 for ideal random flight chain) 5
Course M6 Lecture 9//004 (JAE).7. Self-avoiding random walks The preceding approach is known as the Flory-Fischer argument, and gives almost exactly the same results as exact calculations based on renormalisation group theory (which predict ν = 0.588) The fact that mean-field theory agree so well with exact calculations is highly surprising! This is one example of the power of scaling theories in polymer physics (de Gennes).8. Polymer melts What happens now if we consider many polymer chains? Surely the situation will become hopelessly complicated, and the scaling relations will break down? In fact, the situation becomes even more simple In a polymer melt, the monomer concentration should be constant, i.e. independent of R g 6
Course M6 Lecture 9//004 (JAE).8. Polymer melts This means that there are no net expansive forces, in other words it does not matter whether monomers in contact are part of the same chain or on different chains Result is that the scaling exponent for an ideal random walk (ν = 0.5) is recovered!! Statistically speaking, polymer melts are simpler than isolated chains.9 Chain collapse Another simple case is when a polymer experiences a very strong attractive potential This could be either a low temperatures (we have previously ignored attractive part of vdw force) or when polymer is placed in very poor solvent This globular state is very important in biopolymers (proteins, DNA) and is also significant in synthetic polymer processing Simple scaling law R N a i.e. ν = 0. g 7
Course M6 Lecture 9//004 (JAE).0 Effect of solvents Apart from a melt, under what other circumstances can we recover ideal scaling? It turns out that, under certain conditions, dilute polymers can also behave as ideal chains Under these conditions, the polymer is said to be at its theta point (or Θ-point) At this point, the repulsive forces due to excluded volume are exactly balanced by the compressive forces exerted by the surface tension of the solvent In order to understand why this situation arises, we need a simple model of polymers in solution.. Regular solution theory for liquids Use this to calculate the free energy of mixing in a meanfield approximation on a lattice with effective coordination number Z interactions U ε mix = NZΦ Φ Z χ k T,ε, ε ( ε ε ε ) ( ε ε ε ) Umix = NχΦΦ k T per site 8
Course M6 Lecture 9//004 (JAE).. Regular solution theory for liquids If χ < 0, then there is an enthalpic tendency to mix If χ > 0, then there is an enthalpic tendency to demix If χ = 0, the system is termed ideal (athermal) However, it is the free energy of mixing which determines whether the components actually mix For simple liquids (not polymers!): S mix = k Fmix k T N( Φ = Φ lnφ lnφ + Φ lnφ ) + Φ lnφ + χφ Φ per site Flory-Huggins free energy.. RST for solvated polymer The form of the free energy determines the phase diagram, of which more in lecture For polymers, the entropy of mixing is reduced by the degree of polymerisation M S mix [( Φ / ) lnφ + Φ lnφ ] = k N M p As M is often very large, the entropy of mixing is greatly reduced compared to mixing of simple solutions F Φ mix p = lnφ k T M p s p + Φ lnφ s per site s s s + χφ Φ p 9
Course M6 Lecture 9//004 (JAE).. RST for solvated polymer In the context of a binary mixture of polymer and solvent, the χ parameter represents the solvent quality If χ < 0.5, then it is said to be a good solvent If χ > 0.5, then it is said to be a poor solvent If χ = 0.5, then it is said to be a Θ-solvent F Π = V tot Φ V n p = a p F = tot F mix a V ( F mix / Φp) Π = = Φ F p (/ Φ Φ p) p Φ mix p.. Theta conditions Using the Flory-Huggins form of the free energy, and writing Φ p = Φ and Φ s = Φ (binary system), we obtain the following expression for the osmotic pressure: a Π k T = Φ + ln Φ χφ M Φ We can now formulate the Θ-conditions by making power series or virial expansion for osmotic pressure at low Φ ( χ) a Π Φ = + Φ k T M Θ-conditions when χ = 0.5 0
Course M6 Lecture 9//004 (JAE).. Theta conditions The definition of Θ-conditions is when the second virial coefficient disappears One of the consequences of this is that χ = 0.5 Since χ is a characteristic of a particular polymer/solvent system, and χ ~ /T, then it follows that Θ-conditions will only occur at a certain temperature, the Θ-temperature This is analogous to the oyle temperature at which a gas behaves ideally.. Theta conditions Zimm plot from light scattering data for dilute PS solution in cyclohexane under Θ-conditions (T Θ = 4.7 C) Line shows results for ideal random walk chain
Course M6 Lecture 9//004 (JAE)..4 Theta conditions Concentration dependence of osmotic pressure of PS solutions in cyclohexane around the Θ-temperature.4 Good solvents When χ < 0.5, the polymer is said to be in a good solvent This means that the solvent interacts favourably with the polymer, leading to an expanded chain configuration (with scaling exponent ν > 0.5) Expanded chains behave qualitatively differently to ideal chains In particular: Asymptotic r monomer distribution is negative exponential rather than Gaussian Asymptotic r 0 monomer distribution is zero rather than a maximum R g about 0% smaller than expected from R
Course M6 Lecture 9//004 (JAE).5. Scattering data for expanded chains Chain structure dependent on length scale Zimm plots from neutron scattering data for dilute solutions of PS in cyclohexane At temperatures above the theta point (T Θ = 4.7 C) the behaviour at low q becomes non-ideal.5. Scattering data for expanded chains At short distances, chains are ideal, whereas at larger distances they behave like expanded chains Change-over occurs at thermic correlation length
Course M6 Lecture 9//004 (JAE) Lecture summary This lecture, we started by introduced some important scattering methods for measuring the dimensions of polymer coils: Zimm, Guinier and ratky plots We then tried to account for the concentration dependence of coil size in dilute solution via the theory of self-avoiding random walks We found that chains in dilute solution in a good solvent are expanded, but in the melt they are ideal Chains can also be ideal in solution at their Θ-point, where the attractive enthalpic forces exactly balance the repulsive entropic forces Finally, we reviewed scaling laws for expanded chains 4