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Chemistry C3102-2005: Polymers Section

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Chemistry C3102-2005: Polymers Section Chemistry C3102-2005: Polymers Section Dr. Edie Sevick, Research School of Chemistry, ANU 2.0 Thermodynamics of an Ideal Chain: Stretching and Squashing In this section, we begin to investigate the thermodynamics of polymers, starting first, with an isolated ideal chain. We begin with the general relation ∆F = ∆U T ∆S (1) − where the first term on the RHS is the change in the free energy, ∆U is the change in the internal energy of the chain, and ∆S is the change in entropy. Examples of interactions which contribute to ∆U range from Lennard-Jones interactions to complex interactions which scientists might arbitrarily propose, all of which are of the form U = f(rij) where rij is a vector of distances between monomers, solvent, or other particle or surface which might be included in the system. By definition, an ideal chain suffers NO interactions and contributes nothing to the internal energy. Consequently, if we take an ideal chain and end-tether it to a phantom wall, or change the solvent, then the change in the internal energy of the system is 0. On the other hand, the chain has considerable disorder, measured by the number of configurations that it can adopt and quantified by the thermodynamic quantity entropy. If we were to hold the two ends of the chain a distance Na apart so that the chain was taut, we would considerably decrease the number of configurations that the chain could adopt. If S1 is the entropy of the ideal chain in its relaxed, or natural state, and S2 the entropy of the chain in its taut state, then S1 >> S2 or ∆S = S2 S1 would be negative, and by the above equation, ∆F , would be positive. That is, − in stretching the chain to its taut state, we would be performing work on the system to increase the free energy of the system. If we were to release the ends of the chain, the chain would spontaneously relax to its natural state that maximises the number of available configurations; that is the system maximises its entropy and minimises its free energy. Thus, in order to describe the thermodynamics of a chain, we need only quantify the entropy of the chain. In this section, we will investigate the thermodynamics of an ideal chain when it is stretched and when it is squashed. The force of stretching and squashing is determined solely by entropy- such forces are referred to as entropic or elastic forces. Boltzmann’s Principle (after Ludwig Boltzmann (1844-1906)) states that the entropy of the system is S = kB ln[p(Σ)], where p(Σ) is the probability distribution of configurations. Imagine that we could actually count the number of configurations of a chain in its natural and fully- stretched states, then the change in entropy would be ∆S = kB ln [Γ/Γ0] where Γ is the number of taut configurations and Γ0 is the larger number of configurations in the chain’s natural state. 1 EXAMPLE PROBLEM What is the change of entropy associated with an ideal chain of N monomers •• forming a ring? From Boltzmann’s principle, we find that the entropy change for ring formation is Γring ∆S = Sring S0 = kB ln ( ) (2) − Γ0 where the subscript 0 indicates an FJC chain possessing the same number of monomers, of the same size, but whose ends are not constrained. Γring ad Γ0 are the total number of configurations of the ring and the unconstrained chain, respectively. Following section 1.2, we can express Γ in terms of the number of possible random walks that arrive at a destination m. In 1-dimension, the total number of configurations available to the random walker is 2N , as at each step the walker has 2 possible directions. In order for the walker to form a ring, he must return to his starting point, or in the language of section 1.2, m = 0 and n+ = n−. The number of such ring configurations is N! N N (3) ( 2 )!( 2 )! Therefore the probability of observing a random walker returning to his starting point (in 1-dimension) is 1−D 1 N! 2 1/2 pring = N N N =( ) (4) 2 × ( 2 )!( 2 )! πN where the last term on the right hand side resulted from application of Stirling’s approximation, which is valid in the limit of large N. You should see that this formula also results from eq. 29 in the notes, by replacing m with 0. Likewise, you can transform from discrete destinations to a continuous, end-to-edn distance in three dimensions as achieved on the top of page 7 of the class notes. The result is that the probability of observing a ring in three dimensions is − 3 p3 D =( )3/2. (5) ring 2πNa2 Therefore, the entropy change upon ring formation of a chain of N statistical monomers, each of size a, is 3k 3 ∆S = k ln p3−D = B ln ( ) B ring 2 2πNa2 3k = B [ln (3) ln(2πNa2)]. 2 − As N is large, ln(3) << ln(2πNa2), and ∆S is negative, that is, entropy is lost in the process of ring formation. EXAMPLE PROBLEM 2 2.1 Stretching a chain Take an ideal chain, grab each of the free ends, and then extend or stretch the chain by pulling the ends to progressively larger values of R, the end-to-end distance. We want to know the force needed to stretch the chain to half of its contour length. First, lets express the above equation in differential terms: dF = TdS, where dU = 0 by definition. Then, since work, w, − is equal to the force, f, integrated over the distance, R dRf, we can write R0 dF dS f = = T (6) −dR dR We already have an expression of the entropy in terms of the end-to-end distance from the Gaussian distribution of end-to-end distance, p(R): 3 3/2 3R2 S = k ln (p(R)) = k ln ( ) exp ( ) (7) B B 2πNa2 −2Na2 We can simplify this to 3 3R2 S = k ln( )3/2 k ( ) (8) B 2πNa2 − B 2Na2 2 Taking the derivative of S with respect to R gives 3kBR/(Na ) as the first term on the RHS vanishes as it has no R dependence. Thus, the entropic or elastic force needed to stretch a chain to an end-to-end distance R is R f 3k T (9) ∼ − B Na2 Notice that f is a vector quantity, and in the direction opposite to the end-to-end vector. In other words, f is a restoring force. EXAMPLE PROBLEMS An ideal polymer chain is end-tethered to a fixed wall and a weight of mass • m is suspended from the free-end of the chain, as pictured below. The weight is sufficiently small that the elongation of the chain, x, is between the natural size of the chain and its contour length, (N)a<x<Na. Derive an expression for the p chain elongation, x, as a function of the mass of the weight, m First, we assume that the force of extension or stretch- ing force is entirely entropic. We can derive the force, f, versus extension, x, relation starting from simple thermodynamics: E = U T S, − where the elastic force, f, is x dE dS f = = T − dx dx 3 The entropy is given by Boltzmanns principle and the Gaussian probability distribution SkB ln (P (x)) dS d 3 3/2 3x2 = k ln exp . dx B dx h2πNa2 − 2Na2 i Using the identity ln (ab) = ln a + ln b, the last equa- tion becomes dS d 3 3/2 3x2 = k dx B dxh2πNa2 − 2Na2 i , or dS 3x = k dx − B Na2 . Therefore, the equation for the force versus extension is 3x f = k T − B Na2 Now the force on the chain is simply the mass of the weight times gravitational acceleration, or f = mg, so the sought expression is 3x mg = k T − B Na2 . Continuing from above, with the weight attached to the free-end of the chain, the • ambient temperature is increased. What happens to the extension of the chain, and why? From the above equation, you see that the LHS is constant the weight on the free-end of the chain does not change. Thus, if the temperature is increased, then the extension must proportionally decrease. Thus this above result indicates that a stretched chain will contract upon heating. Why? It is primarily an effect due to entropy. You know that entropy rather than interactions dominate because (1) when temperature is increased, the importance of interactions decrease as the magnitude of the interaction, ε, becomes smaller compared with thermal energy,i.e., ε/(k T ) < 1; (2) from E = U TS, you see that increasing the temperature increases the B − role of entropy; (3) the f versus x relation is purely entropic. When temperature is increased, the effect on each statistical monomer is to increase its thermal energy. In other words each statistical monomer becomes more randomised with increased temperature or the entropy of the chain is increased. We know that the conformational entropy of an ideal chain is decreased as its ends are separated apart: or that the ideal chains entropy increases as its end-to-end distance decreases. Thus, with an increase in temperature, there is a randomisation of the statistical monomers, an increase in the chains entropy and a decrease in the end-to-end distance (the chain shortens). Having established that this is an entropic effect, one can analyse this in terms of the exchange of heat and work.
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