3.012 Fundamentals of Materials Science Fall 2005 Lecture 23: 12.05.05 Lattice Models of Materials; Modeling Polymer Solutions Today: LAST TIME .........................................................................................................................................................................................2 The Boltzmann Factor and Partition Function: systems at constant temperature..................................................................2 A better model: The Debye solid.................................................................................................................................................3 EXAMINATION OF HEAT CAPACITIES OF DIFFERENT MATERIALS....................................................................................................6 DEGREES OF FREEDOM IN MOLECULAR MODELS(1) ........................................................................................................................8 Excitations in materials...............................................................................................................................................................8 Complete molecular partition functions.....................................................................................................................................9 LATTICE MODELS FOR TRANSLATIONAL DEGREES OF FREEDOM ..................................................................................................11 Assumptions in simple lattice models .......................................................................................................................................11 FLORY-HUGGINS THEORY OF POLYMER SOLUTIONS ...................................................................................................................13 The entropy of polymer solutions..............................................................................................................................................13 REFERENCES ...................................................................................................................................................................................19 Reading: Engel and Reid 32.3-32.4 Dill and Bromberg Ch. 15 ‘Solutions & Mixtures,’ pp. 267-273 Dill and Bromberg Ch. 31 ‘Polymer Solutions,’ pp. 593-605. Supplementary Reading: Details of rotational, vibrational, and electronic partition functions for simple molecules: Engel and Reid 32.5-32-9 Lecture 24 – Lattice Models of Materials 1 of 19 12/5/053.012 Fundamentals of Materials Science Fall 2005 Last time The Boltzmann Factor and Partition Function: systems at constant temperature How do we treat systems at constant temperature in statistical mechanics? We needed to determine how the probability of model microstates depends on temperature. We found the answer by minimizing the Helmholtz free energy with respect to the possible microstate probabilities pj. This analysis gave us the Boltzmann factor and the partition function: Once we had the concept of the partition function, we began tackling a first example problem: the Einstein solid. Atoms of a crystalline solid are assumed to vibrate in x, y, and z with a single well-defined frequency as quantum mechanical harmonic oscillators. We started by solving for the molecular partition function: From here we determined the partition function for a system of N non-interacting, identical, distinguishable oscillators: ! Q =e"h#2e"h#$1% & ' ' ' ( ) * * * 3N=eh#2kTeh#kT$1% & ' ' ' ( ) * * * 3N The partition function for this simple model allowed calculations of the internal energy and heat capacity of a crystalline solid: ! CV= 3Nkb"ET# $ % & ' ( 2e"ETe"ET)1# $ % & ' ( 2Lecture 24 – Lattice Models of Materials 2 of 19 12/5/053.012 Fundamentals of Materials Science Fall 2005 Figure by MIT OCW. A better model: The Debye solid The Einstein model makes the simplification of assuming the atoms of the solid vibrate at a single, unique frequency: g g v v t (a) (b) Frequency distribution g(v) for crystal. (a) Einstein approximation. (b) Debye approximation. vml Figure by MIT OCW. Density of states g(w) 0 1 2 3 4 5 �D �/1013 radians s-1 The Debye distribution of frequencies, with the experimental distribution of frequencies function of � = 2�is obviously complicated enough that a theory to reproduce such a distribution would likely for copper. The distribution is shown as a v. The experimental distribution be difficult to produce. Figure by MIT OCW. ‘g’ in Figure 5-4 above from Hill is the distribution of vibrational frequencies present in the crystal. In the Einstein model, only one vibrational frequency is assumed for all atoms in the crystal. However, atoms sitting on different lattice sites may have difference accessible vibrational frequencies-which depend on what neighbors they ‘feel’ around them-this is seen in the complex distribution of vibrational frequencies shown in Figure 22.8 from Mortimer for a real sample of copper. The Debye model approximates the true frequency distribution by assuming the distribution shown in Figure 5-4(b): a distribution that is continuous up to some frequency cut-off (νm). The Debye expression for heat capacity becomes: Lecture 24 – Lattice Models of Materials 3 of 19 12/5/053.012 Fundamentals of Materials Science Fall 2005 EINSTEIN MODEL DEBYE MODEL ! CV= 3Nkb"ET# $ % & ' ( 2e"ETe"ET)1# $ % & ' ( 2! CV= kbh"kbT# $ % & ' ( 2eh"kbTeh"kT)1# $ % & ' ( 2g"( )d"0*+ This approximation leads to a heat capacity behavior near zero Kelvin which better captures experimentally-observed behavior: ! T " 0, CV"12Nkb#45T$D% & ' ( ) * 3where $D+h,mkb% & ' ( ) * = Debye temperature The Debye model performs quite well for predicting the thermal behavior of many solid materials: Debye Einstein Al �D = 385 K 25 20 15 10 5 0 0 .4 .8 1.2 1.6 2.0 . mole T/� Comparison among the Debye heat capacity, the Einstein heat capacity, and the actual heat capacity of aluminum. Cv, joules/degreeFigure by MIT OCW. Lecture 24 – Lattice Models of Materials 4 of 19 12/5/053.012 Fundamentals of Materials Science Fall 2005 Examination of heat capacities of different materials • If heat capacities correlate with molecular degrees of freedom in a material, we might expect materials that have similar degrees of freedom to have similar heat capacities. This is in fact seen for many materials. Consider first
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