Lecture 7: Boltzmann distribution & Thermodynamics of mixing

Transcription

1 Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters 10, 15 Boltzmann dstrbuton Before contnung wth the thermodynamcs of polymer solutons, we wll brefly dscuss the Boltzmann dstrbuton that we have used occassonally already n the course and wll contnue to assume a workng knowledge of for future dervatons. We ntroduced statstcal mechancs earler n the course, showng how a probablstc vew of systems can predct macroscale observable behavor from the structures of the atoms and molecules that compose them. At the core, we model probablty dstrbutons and relate them to thermodynamc equlbra. We dscussed macrostates, mcrostates, ensembles, ergodcty, and the mcrocanoncal ensemble and used these to predct the drvng forces of systems as they move toward equlbrum or maxmum entropy. Up to Lecture 5, we had only dscussed deal behavor meanng that we assumed there was no energetc contrbuton from nteractons between atoms and molecules n our systems. Here, we buld on the base dscusson of statstcal mechancs as presented n Lecture and used snce to consder how we can develop mathematcal tools that account for energetc nteractons that may occur n real (non-deal systems. The central result s the Boltzmann dstrbuton law, whch provdes probablty dstrbutons based on underlyng energy levels of the system. In ths approach, we now want to consder smple models (often lattce models that allow us to count mcrostates but also assgn an energy E for each mcrostate. We then use ths nformaton to calculate a probablty dstrbuton for the lkelhood that the macrostate assumes the proposed mrcostates at equlbrum. As an example we could consder the example below of a model, four-monomer polymer chan constraned to two-dmensonal space that has four open conformatons and one compact conformaton. To start, we must defne the system and the assocated energy levels. The system s a four-monomer chan wth two energy levels, where each energy level coresponds to the number of monomer monomer nteractons that the chan can make. We wll often take zero as the lowest energy state and refer to ths as the ground state. Dsruptng the contact ncreases the energy of the system by ɛ = ɛ 0 > 0. As an asde, n ths case t s qute clear but we wll use ɛ for ntermolecular energes and E for macrostate energes, whch are equvalent n ths case but ths s not the case as we consder more complex systems. We refer to states other than the ground state as excted states. The ndvdual conformaton refer to mcrostates and each ground or excted state can be acheved by one or more mcrostates. We thus seek to compute the set of proabltes {p } = p 1, p,..., p 5 1

2 Prof. Tbbtt Lecture 7 etworks & Gels that the model chan adopts ts varous conformatons. We wll come back to ths example brefly after we derve the Boltzmann dstrbuton law below. For ths dervaton we wll operate n the Canoncal Ensemble, whch assumes constant temperature T, volume V, and components { }. As before each system has many accessble mcrostates and observable macrostates. The number of mcrostates n macrostate s gven as n and wth total mcrostates we can wrte the probablty p = n. From Lecture Eq. 8, we know that: S k B = p ln p. (1 As before, we are nterested n predctng the macrostate at equlbrum, whch we wll determne when the entropy s maxmzed. Statng ths mathematcally, we want to fnd the set of probabltes {p } such that ds = 0 wth the followng constrants: p = 1 ( p E = E = Ē (3 We can then use the method of Lagrange multplers to solve for ds = 0 from Eq. 1 and the constrants n Eqs. and 3. ds k B = [ln p + 1] dp = [ln p ] dp, (4 dp = 0, (5 [p de + E dp ] = 0. (6 In Eq. 6 the frst term goes away as de 0 as energy levels have lttle dependence on T, V, { }. Therefore, we can solve the set of equatons wth the Lagrange multpler approach as follows: [ ln p α βe ] = 0. (7 At the global maxmum, each term n the sum n Eq. 7 must be equal to zero so we have: ln p = α βe, (8 p = e α e βe. (9 ext, we can determne α wth a normalzaton constrant by dvdng by 1 = p : e α e βe p = e α (10 e βe = e βe e βe = e βe Q, (11 where Q that p E = Ē. e βe s defned as the partton functon. Addtonally, we can defne β wth the constrant

3 Prof. Tbbtt Lecture 7 etworks & Gels dē = [E dp + p de ] (1 ds = k B ln p dp (13 = k B dp [ βe ln Q] (14 = βk B E dp (15 = βk B dē (16 Therfore, we can connect wth what we have shown about thermodynamc drvng forces, namely: ( S = 1 ds U T,V,{ }} T dē = βk B β = 1 k B T. (17 Ths brngs us to the common expresson of the Boltzmann dstrbuton law: p = e E/k BT Q. (18 As before, ths dstrbuton was calculated by maxmzng entropy! We return brefly to the model chan presented before. We can calulate the partton functon as Q = 1 + 4e ɛ0/kbt and the probabltes as p c = 1 Q and p o = 4e ɛ 0 /k B T Q. What s expected to occur at hgh and low T? 3 Thermodynamcs of soluton As we move toward consderng materals constructed from polymer chans, ncludng solvent swollen networks or gels, we need to consder how sngle polymer chans beleve n soluton. Ths can be polymer chans n a soluton of polymer chans or a more classc vew of soluton wth polymer chans n a solvent. The approach, as before n ths course, wll be to develop smple models that allow us to calculate the thermodynamcs of polymer solutons and, today, we start wth smple solutons as we buld toward ths end. To do ths, we wll use the Canoncal Ensemble dscussed above (constant T, V, { } and calculate the Helmholtz free energy F = U T S on a lattce model where S s the entropy of soluton and U accounts for the nteracton energes between molecules on the lattce. Here, we wll consder a lattce wth stes and n A molecules of type A and n B molecules of type B. As before molecules A and B are equvalent n sze and each occupes one lattce ste. We can vsualze the problem as below: Frst, we can calculate the entropy of soluton as before by calculatng the multplcty as the total number of spatal arrangements of the system W =! n A!n B!. Therefore, we can use the Boltzmann equaton (Lecture and Strlng s approxmaton: 3

4 Prof. Tbbtt Lecture 7 etworks & Gels S sol = k B ( ln n A ln n A + n A n B ln n B + n B (19 = k B (n A ln + n B ln n A ln n A n B ln n B (0 [ na = k B ln n A + n B ln n ] B (1 = k B (n A ln x A + n B ln x B. ( where x A = n A / and x B = n B /. Ths can also be wrtten wth the mole fracton of one speces, say x = x A : S sol k B = x ln x (1 x ln(1 x. (3 We can then calculate the energy of soluton. In our lattce model, ths nvolves countng the sum of the contact nteractons between all of the nearest neghbors n the soluton. In the system descrbed above, we can consder three types of nteractons: AA, BB, and AB. The total energy of the system s then: U = m AA w AA + m BB w BB + m AB w AB, (4 where m AA, m BB, and m AB are the number of AA, BB, and AB nteractons, respectvely, and w AA, w BB, and w AB are the nteracton energes wth the assocated contacts. For ths, we take the nteracton energes to be negatve. We then need to count the number of AA, BB, and AB nteractons, whch are not known n general. A convenent approach s to express these n terms of n A and n B. Addtonally, each lattce ste has z contacts avalable. Therefore, and smlarly zn A = m AA + m AB, (5 We can rearrange Eqs. 5 and 6 to solve for m AA and m BB : zn B = m BB + m AB. (6 4

5 Prof. Tbbtt Lecture 7 etworks & Gels We can combne these results wth Eq. 4 to arrve at: ( zna m AB U = = n A + m AA = zn A m AB, (7 m BB = zn B m AB. (8 w AA + ( zwbb ( znb m AB w BB + m AB w AB (9 ( n B + w AB w AA + w BB m AB (30 To compute U, we stll need to calculate m AB and to do ths we wll employ a mean-feld approxmaton as before. We make the assumpton that for any numbers of molecules the psystem s random and unformly dspersed, ths allows us to more easly estmate m AB. To do ths, we consder a specfc ste on the lattc next to an A molecule. The probablty that a B molecule occupes the neghborng ste can be calculated by assumng that B molecules are dstrbuted randomly across the lattce. Therefore, the probablty that a gven ste s occuped by a B molecule p B s gven by: p B = n B = x B = 1 x. (31 On ths lattce, there are z nearest-neghbors so on average for each A molecule so the average number of AB contacts on each A molecule s zn B / = z(1 x. There are a total of n A A molecules, so m AB zn An B = zx(1 x. (3 Therefore the total contact energy of the soluton n terms of n A and n B can be wrtten as: ( zwbb ( U = n A + n B + z w AB w AA + w BB x(1 x (33 ( zwbb = n A + n B + k B T χ AB x(1 x (34 ( where we defne a dmensonless parameter χ AB = z k B T wab w AA+w BB called the nteracton parameter. Then we can compute the full free energy of soluton: F (n A, n B k B T = n A ln n A + n B ln n B + zw AA k B T n A + zw BB k B T n B + χ AB x(1 x. (35 In practce, we are most nterested n computng a free energy dfference between the soluton and the ntal pure states of A and B: F sol = F (n A, n B F (n A, 0 F (0, n B. (36 Thus, we arrve at the full free energy dfference as a functon of the mole fracton x and the nteracton parameter χ AB : F sol k B T = x ln x + (1 x ln(1 x + χ ABx(1 x. (37 Ths s the regular soluton model orgnally developed by Hldebrand n 199 and demonstrates that whle deal solutons are drven purely by entropy, we must also consder the nteractons of molecules when computng the free energes of real solutons. We wll buld on ths n the next lecture as we work toward an understandng of the thermodynamcs of a polymer n soluton, where one speces n many tmes larger than the other speces around t. 5