Lecture 23: Lattice Models of Materials; Modeling Polymer Solutions

Transcription

1 Lecture 23: Lattice Mdels f Materials; Mdeling Plymer Slutins Tday: LAST TIME...2 The Bltzmann Factr and Partitin Functin: systems at cnstant temperature...2 A better mdel: The Debye slid...3 EXAMINATION OF HEAT CAPACITIES OF DIFFERENT MATERIALS...6 DEGREES OF FREEDOM IN MOLECULAR MODELS(1)...8 Excitatins in materials...8 Cmplete mlecular partitin functins...9 LATTICE MODELS FOR TRANSLATIONAL DEGREES OF FREEDOM...11 Assumptins in simple lattice mdels...11 FLORY-HUGGINS THEORY OF POLYMER SOLUTIONS...13 The entrpy f plymer slutins...13 REFERENCES...19 Reading: Engel and Reid Dill and Brmberg Ch. 15 Slutins & Mixtures, pp Dill and Brmberg Ch. 31 Plymer Slutins, pp Supplementary Reading: Details f rtatinal, vibratinal, and electrnic partitin functins fr simple mlecules: Engel and Reid Lecture 24 Lattice Mdels f Materials 1 f 19

2 Last time The Bltzmann Factr and Partitin Functin: systems at cnstant temperature Hw d we treat systems at cnstant temperature in statistical mechanics? We needed t determine hw the prbability f mdel micrstates depends n temperature. We fund the answer by minimizing the Helmhltz free energy with respect t the pssible micrstate prbabilities p j. This analysis gave us the Bltzmann factr and the partitin functin: Once we had the cncept f the partitin functin, we began tackling a first example prblem: the Einstein slid. Atms f a crystalline slid are assumed t vibrate in x, y, and z with a single well-defined frequency as quantum mechanical harmnic scillatrs. We started by slving fr the mlecular partitin functin: Frm here we determined the partitin functin fr a system f N nn-interacting, identical, distinguishable scillatrs: % "h# ( 2 ' e * Q = ' e "h# $1* &' )* 3N % h# = e 2kT ' ' h# &' e kt $1 ( * * )* 3N The partitin functin fr this simple mdel allwed calculatins f the internal energy and heat capacity f a crystalline slid: #" C V = 3Nk E b % $ T & ( ' 2 " E T e # " E & % e T )1( $ ' 2 Lecture 24 Lattice Mdels f Materials 2 f 19

3 A better mdel: The Debye slid The Einstein mdel makes the simplificatin f assuming the atms f the slid vibrate at a single, unique frequency: g g Density f states g(w) ω D t v v v ml (a) (b) Frequency distributin g(v) fr crystal. (a) Einstein apprximatin. (b) Debye apprximatin ω/10 13 radians s -1 The Debye distributin f frequencies, with the experimental distributin f frequencies fr cpper. The distributin is shwn as a functin f ω = 2πv. The experimental distributin is bviusly cmplicated enugh that a thery t reprduce such a distributin wuld likely be difficult t prduce. g in Figure 5-4 abve frm Hill is the distributin f vibratinal frequencies present in the crystal. In the Einstein mdel, nly ne vibratinal frequency is assumed fr all atms in the crystal. Hwever, atms sitting n different lattice sites may have difference accessible vibratinal frequencies- which depend n what neighbrs they feel arund them- this is seen in the cmplex distributin f vibratinal frequencies shwn in Figure 22.8 frm Mrtimer fr a real sample f cpper. The Debye mdel apprximates the true frequency distributin by assuming the distributin shwn in Figure 5-4(b): a distributin that is cntinuus up t sme frequency cut-ff (ν m ). The Debye expressin fr heat capacity becmes: Lecture 24 Lattice Mdels f Materials 3 f 19

4 EINSTEIN MODEL DEBYE MODEL #" C V = 3Nk E b % $ T & ( ' 2 " E T e # " E & % e T )1( $ ' 2 C V = k b * h" # h" & k % ( e b T $ k b T ' g h" 2 (" )d" # & kt % e )1( $ ' This apprximatin leads t a heat capacity behavir near zer Kelvin which better captures experimentally-bserved behavir: T " 0, C V " 12Nk b# 4 % T ( ' * 5 & ) $ D % where $ D + h, ( m ' * = Debye temperature & ) k b 3 The Debye mdel perfrms quite well fr predicting the thermal behavir f many slid materials: 25. mle Cv, jules/degree Debye Einstein Al θ D = 385 K T/θ Cmparisn amng the Debye heat capacity, the Einstein heat capacity, and the actual heat capacity f aluminum. Lecture 24 Lattice Mdels f Materials 4 f 19

5 Examinatin f heat capacities f different materials If heat capacities crrelate with mlecular degrees f freedm in a material, we might expect materials that have similar degrees f freedm t have similar heat capacities. This is in fact seen fr many materials. Cnsider first a cmparisn f the heat capacity in 3 different crystalline nn-metals: 2 Cp, J/ ml.k NaCl Ge NiSe 2 9R 6R 3R Temperature, K Mlar heat capacity at cnstant pressure f three crystalline nnmetals. (Ge crystal structure frm Thus in these structurally-related crystals, the heat capacity per N Av atms is very similar, ~3R, r 25 J/mle K. We will shw later in the term that this plateau value can be predicted by treating the atms in the slid as a cllectin f harmnic scillatrs. Lecture 24 Lattice Mdels f Materials 6 f 19

6 c v /3R 1.0 Rbl NaCl FeS2 MgO 0.8 Diamnd Temperature, K Temperature variatin f c v /3R f nnmetals. (1 ml f diamnd, 1 ml f Rbl, NaCl, 2 and MgO; and 1 ml f FeS 3 2.) Lecture 24 Lattice Mdels f Materials 7 f 19

7 Degrees f freedm in mlecular mdels(1) Excitatins in materials We mdeled the atmic vibratins in a crystalline slid using 3 degrees f freedm- harmnic scillatins in X, Y, and Z. We saw that a mdel using nly these 3 degrees f freedm prvides reasnable predictins fr the behavir f the heat capacity f many slids. Other materials may have ther imprtant degrees f freedm that we shuld accunt fr t btain gd statistical mechanics predictins f their behavir. The imprtant mlecular degrees f freedm include: Translatin 1. Translatin 2. Rtatin 3. Vibratin 4. Electrn excitatin Mlecules that have freedm t mve within their cnfining vlume (e.g. cntainer) have translatinal degrees f freedm. Fr example, the mlecules f a gas can ccupy psitins thrughut the vlume in which they are enclsed via diffusin. Rtatin Mlecules can ften stre energy in rtatins abut bnds between atms: Lecture 24 Lattice Mdels f Materials 8 f 19

8 Vibratin The electrnic glue hlding mlecules tgether allws vibratins that stre energy: Symmetric stretch V 1 = 1340 cm 2 Bend V 2 = 007 cm 2 Asymmetric strech V 3 = 2340 cm 2 Symmetric stretch Bend Asymmetric strech V 2 1 = cm V 2 = 519 cm 2 V 3 = 1361 cm 2 Vibratinal mdes f carbn dixide. Electrnic excitatins Orbital Energy (kj / ml) s 3s 2s 4p 3p 2p 4d 3d 4f s 3s 2s 4p 3p 2p 4d 3d 4f s Hydrgen s Multielectrn atms Cmplete mlecular partitin functins A cmplex system may have all f these degrees f freedm. T make calculatins fr a given mdel, we need t knw hw t put these degrees f freedm tgether in the partitin functin. Independent degrees f freedm A cmmn apprximatin is t assume that each degree f freedm in the mlecules f the system is independent, with a unique amunt f energy fr each pssible state f that degree f freedm (let s use DOF as an abbreviatin fr degree f freedm). Thus a mlecule with bth vibratinal and electrnic DOFs has states characterized by ne ttal energy cntaining independent cntributins frm the vibratin and electrnic excitatins: Lecture 24 Lattice Mdels f Materials 9 f 19

9 The subscript j refers t the single state that has the given characteristic vibratinal and electrnic energy. Because we assume they are independent, the value f E j vib des nt depend n the value f E j elec, and vice versa. The partitin functin f this system with independent DOFs is: Where the independent energies have been split ff int partitin functins fr each DOF, q vib and q elec. In general, a cmplete mlecular partitin functin made up f independent degrees f freedm can be written as the prduct f the individual DOF partitin functins: Lecture 24 Lattice Mdels f Materials 10 f 19

10 Lattice mdels fr translatinal degrees f freedm We intrduced statistical mechanics as a set f tls fr calculating macrscpic thermdynamic quantities frm mlecular mdels. These mdels can be derived either frm quantum mechanics (e.g. the Einstein slid) r frm simpler nn-quantum mdels. There are many cases when the quantum nature f available energies in the system f interest d nt dminate and we can use s-called carsegrained lattice mdels t capture the imprtant aspects f the material s behavir. Earlier, we have seen that chemical ptential mdels such as the regular slutin r ideal slutin mimic sme real experimental data reasnably well. Hwever, a questin unanswered is- where did this mdel cme frm? What abut the regular slutin mdel- hw d mlecular interactins give rise t miscibility gaps? Answers t these questins can be fund in simple lattice mdels. Assumptins in simple lattice mdels ASSUME: Lecture 24 Lattice Mdels f Materials 11 f 19

11 In this simple lattice mdel, we assume that nly translatinal degrees f freedm matter in the determinatin f pssible statistical mechanical states- the states f the system are simply defined by the number f unique ways the mlecules can be arranged n the lattice. We take N fr the ttal number f mlecules (N = N A + N B ). T determine the entrpy f mixing, we need the number f distinguishable states W fr the mixed and unmixed cmpnents. Fr the unmixed pure cmpnents, there is nly ne distinguishable state (all lattice sites ccupied by either A r B), s the entrpy is 0 (S = k ln 1 = 0). These assumptins can be used t derive bth the ideal slutin mdel f the chemical ptential and the regular slutin chemical ptential. As an example f the utility f lattice mdels fr materials, we will nw derive the entrpy and enthalpy f mixing using a simple lattice mdel fr plymer slutins, based n the Flry-Huggins thery f plymer slutins. Paul J. Flry s extensive wrk n the statistical thermdynamics f plymers was awarded the nbel prize in chemistry in 1974 Image remved fr cpyright reasns. Phtgraph f P. J. Flry. Lecture 24 Lattice Mdels f Materials 12 f 19

12 3.012 Fundamentals f Materials Science Fall 2005 Flry-Huggins Thery f Plymer Slutins Flry-Huggins thery was develped t prvide a mlecular theretical basis fr the free energy behavir f plymer slutins- t allw predictins f miscibility behavir based n plymer mlecular structure. Thus, ur bjective is t derive a theretical descriptin f the free energy f mixing, which can be used t predict phase diagrams f plymer slutins: "G mix = "H mix # T"S mix =? The entrpy f plymer slutins T mdel a plymer slutin- a cllectin f high mlecular weight plymer chains mixed with a smallmlecule slvent, we! mdel the plymer chains as beads cnnected by unbreakable bnds n a cubic lattice. Slvent mlecules fill single sites f the 3D lattice: Unmixed State Pure Plymer Mixed State Pure Slvent (Hmgeneus Slutin) + vlume fractin f plymer: "p = Vp Nn p = V p + Vs M vlume fractin f slvent: "s =! Vs n = s V p + Vs M! Lecture 24 Lattice Mdels f Materials 13 f 19

13 In this lattice mdel, we are cncerned with ne degree f freedm in the calculatin f the entrpy- the translatinal entrpy: "S mix = S slutin # S unmixed = S slutin # S pure plymer pure slvent ( + S ) We are thus lking t derive expressins fr W slutin and W pure plymer - the number f cnfiguratins pssible fr the plymer + slvent r the plymer alne n a 3D lattice. CONFIGURATIONS OF A SINGLE CHAIN We start by lking at the number f ways t place a single plymer chain n the lattice: What are the number f cnfrmatins fr first bead? With the first bead placed n the lattice, what is the number f pssible lcatins fr the secnd segment f the chain? Lecture 24 Lattice Mdels f Materials 14 f 19

14 Mving n t placement f the third segment f the chain: We repeat this prcess t place all N segments f the chain n the lattice, and arrive at ν 1, the ttal number f cnfiguratins fr a single chain: COUNTING CONFIGURATIONS FOR A COLLECTION OF CHAINS We can fllw the same prcedure shwn abve fr a single chain t btain the number f cnfiguratins pssible fr an entire set f np chains. We start by placing the FIRST SEGMENT OF ALL n p CHAINS. The number f cnfiguratins fr the first segment f all n p chains is ν first : The number f cnfiguratins fr the (N - 1) remaining segments f all n p chains is ν subsequent : Lecture 24 Lattice Mdels f Materials 15 f 19

15 3.012 Fundamentals f Materials Science Fall 2005 Putting these tw cnfiguratin cunts tgether, we have the ttal number f cnfiguratins fr the cllectin f np chains f N segments each: " first" subsequent W = n p!! The factr f np! Crrects fr the ver-cunting since the plymer chains are indistinguishable, and we can t tell the difference between tw cnfiguratins with the same plymer distributins but different chain identities: We are nw ready t calculate the number f unique states fr the unmixed and mixed states: UNMIXED STATE: PURE SOLVENT: PURE POLYMER: Nn p = M W pure plymer # z "1& n p (N "1) M! =% ( $ M ' ( M " Nn p )!n p! S pure plymer = kb lnw pure plymer Lecture 24 Lattice Mdels f Materials 16 f 19

16 Nn p + n s = M # W slutin = z "1 & % ( $ M ' MIXED STATE: n p (N"1) M! ( M " Nn p )!n p! ( ) = )S mix = S slutin " S unmixed = S slutin " S pure slvent pure plymer + S = S slutin pure plymer " S = k b ln W slutin W pure plymer Applying Stirling s apprximatin: ln x!" x ln x # x arriving at a final result: Lecture 24 Lattice Mdels f Materials 17 f 19

17 "#S mix = $k b [ n s ln% s + n p ln% p ] Lecture 24 Lattice Mdels f Materials 18 f 19

18 References 1. Dill, K., and S. Brmberg Mlecular Driving Frces, New Yrk. 2. Mrtimer, R. G Physical Chemistry. Academic Press, New Yrk. 3.. Lecture 24 Lattice Mdels f Materials 19 f 19