Relaxation in Glass: Review of hermodynamics Lecture 11: hermodynamics in the Glass ransition Region
hermodynamic Functions 1 st Derivatives emperature Dependence of the Entropy swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 2
hermodynamic Functions 1 st Derivatives emperature dependence of the Gibb s Free-Energy s G l s G S G g l G S G g S G swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 3
hermodynamic Functions 1 st Derivatives V s < V l < V g g g l l s s V G V G V G swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 4
Homework Exercise for next time: Derive an expression for the following quantity in terms of easily measured quantities and apply it to liquid id B 2 O 3 slightly above its melting point and crystalline B 2 O 3 slightly below its melting point S 1 1 1 V S S V S S Cp Cp Cp S V V V S swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 5
he Enthalpy as a function of temperature, H() he heat capacity Cp measures how much heat it takes to raise the temperature of the system by one degree q H C ( ) he enthalpy can be calculated from the heat capacity for a large change in temperature H ( 2 ) H ( 1 ) 2 H ( ) d 1 1 If the Cp() of solid io 2 is 17.97 + 0.28 x 10-3 - 4.35 x 10-5 / 2 cal/mole-k, the H melt is 16 kcal/mole, and the Cp() for the liquid is 21.4 cal/mole-k, plot the Cp(), H() and calculate how much heat is required to heat 10 lbs. io 2 from room temperature up to 2500 K? 2 C d swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 6
Enthalpy function for glass forming liquids Now consider the temperature dependence of the Enthalpy for a liquid cooled from above its melting point to room temperature along two cooling paths: First assume thermodynamic equilibrium holds and the liquid readily crystallizes at its melting (freezing) point to form the equilibrium crystalline phase and then continues to cool to room temperature Second, assume kinetics holds and the liquid bypasses the equilibrium crystallization and super-cools to the glassy state. swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 7
Enthalpy Changes in the Glass ransition Region H() decreases continuously with cooling Slope of the H() curve is the heat capacity which changes from liquid-like to solid-like values in the transition region Change in heat capacity at the glass transition Cp(g) measures the differences between the liquid and solid (glassy) Cp values Sub-g annealing and relaxation can occur if liquid is given sufficient time to relax to lower enthalpy state Molar enthalpy supercooled liquid glass fast liquid Cp liquid Cp glass H melting slow crystal emperature Cp crystal swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 8
Heat Capacity changes at g: Cp(g) he change in slope in enthalpy at g is a measure of the difference between heat capacity of the liquid and the glass Heat capacity of glasses arises mostly from vibrational contributions rotational ti and translational ti l degrees of freedom have been frozen out Heat capacities of liquids arise from all three contributions rotational, translational, and vibrational He eat Capaci ity Cooling Curves glass g emperature Supercooled liquid (large) Cp(g) (small) swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 9
Heat Capacity changes at g: Cp(g) Exercise: What would the enthalpy curve look like in the region of g if the change in heat capacity at g for some reason vansished, that is Cp(g) 0? He eat Capaci ity Cooling Curves glass g Supercooled liquid (large) Cp(g) (small) (zero) emperature swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 10
Heat Capacity changes at g:cp(g) Glass transition occurs when: hermal energy, heat, being input into the glass has filled all the available thermal degrees of freedom, vibrations, in the glass he vibrational states are essentially filled and at maximum amplitude Additional heat supplied to the glass must be accommodated by other degrees of freedom Rotational and translational degrees of freedom now become available and as such, Cp liquid >> Cp glass swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 11
Heat Capacity Changes at g: Cp(g) Covalently bonded liquids exhibit strong rigidly held structures (SiO 2, for example) Generally exhibit higher h glass transition temperatures t and smaller Cp(g) values Molecular, or ionic salt liquids exhibit fragile, weakly held structures (sucrose, for example) Generally exhibit lower glass transition temperatures and dlarger Cp(g) C ) values Behaviors can be interchanged by chemically changing the liquid Depolymerizing covalent liquids through non-bridging oxygens swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 12
emperature dependence of the Entropy, S() Entropy measures the disorder in the system It is always positive, and always increases with temperature Hot things are always more disordered than cool things Entropy can go to zero at 0 K S( 2 ) S( 1 ) 2 1 S ( ) d 2 1 C d If the Cp() of solid io 2 is 17.97 + 0.28 x 10-3 - 4.35 x 10-5 / 2 cal/mole-k, the H melt is 16kcal/mole, and the Cp() for the liquid is 21.4 cal/mole-k plot the S() and calculate the change in entropy when 10 lbs. io 2 is cooled from 2000 K to room temperature? swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 13
emperature Dependence of the Entropy S() supercooled liquid Mola ar entropy glass slow fast S melting Cp liquid / liquid crystal Cp crystal / emperature m swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 14
Entropy Changes below m Entropy is intimately linked to liquid state behavior Highly disordered liquid being reversibly ordered at the freezing point to the the crystalline phases Corresponding reversible change in entropy, H(m)/m Glass forming liquids exhibit continuously changing entropy that shows no discontinuities Entropy decreases with temperatures At g, continuously changes from liquid-like values to solid like values tal liquid -S cryst S l 0 g Fast Slow Cooling emperature m S m swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 15
Kauzmann aradox If entropy curve continued along meta-stable equilibrium liquid line At some temperature below g, the entropy of the liquids would appear to decrease below that of the crystal How could a liquid, with its inherent structural disorder, have an entropy lower than that of the corresponding crystal he Kauzmann temperature, K, is the temperature where the entropy of the liquid would intersect that of the equilibrium crystal Glass at this temperature is often called an ideal glass tal iquid -S cryst S l Fast K Slow emperature m S f swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 16
Kauzmann aradox Simultaneous to rapidly decreasing entropy Viscosity is increasing Structural relaxation time is rapidly increasing ime required for the liquid to continue to follow the equilibrium line becomes dramatically longer Liquid falls out of equilibrium at a temperature above the Kauzmann temperature because the time required for it to remain in equilibrium simply becomes much longer than the experimental time scale tal iquid -S cryst S l 0 Fast K Slow S f emperature m ity) og(viscos l swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 17
Excess Entropy S excess of the Supercooled Liquid he important quantity is the extra entropy the liquid has above that of the crystal at the same supercooled temperature Equilibrium liquids above the melting point have no excess entropy Supercooled liquids have excess entropy because they have not lost the entropy of melting given to the liquid on melting A supercooled liquid, at maximum has the entropy of melting to lose below m At K the liquid has lost all of this entropy liquid -S crys stal S 0 Fast K Slow S f emperature m log(visc cosity) swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 18
Excess Entropy S excess of the Supercooled Liquid Cpliquid ( ) Cpcrystal ( ) Sexcess( ) S melting d At just above the melting point, S excess ( m ) = 0 At just below the melting point, S excess ( m ) = S m m S liq quid -S crysta al As decreases below m, S excess decreases due to the loss of entropy through cooling. At the Kauzmann temperature, all of the entropy gained by the liquid by not crystallizing has been lost through cooling 0 Fast K Slow S f emperature m sity) log(visco swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 19
S S Vanishing Excess Entropy S excess at K excess melting ( K ( ) 0 S m ) K m Cp melting liquid K m Cp ( ) Cp liquid crystal ( ) Cp ( ) d S crystal S liquid - crystal ( ) d S f Fast log( (viscosity) 0 Slow K swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 20
Gibb s Free-Energy Change at g G = H - S Gibbs Free-Energy change at m is continuous, there is no Latent Free-Energy Change as is the case for the enthalpy and entropy At the melting point G liquid = G crystal Below the melting point G liquid > G crystal and G crystallization < O Above the melting gpoint Gibb s Free-Ene ergy Liquid Crystal G liquid > G crsytal G melting < O G( ) At any point S( ) emperature m swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 21
Gibb s Free-Energy Change at g Glasses then fall off the liquid line at progressively lower temperatures t the slower the cooling rate Gibbs Free-Energy of the glass behaves more like the crystal than the liquid Glass transition range is the range of where the Gibb s Free-Energy changes from liquid-like values to solid-like values Gibb s Free-Ene ergy Liquid Glasses Crystal m emperature swmartin@iastate.edu MI: Relaxation in Glass - Lecture 11 hermodynamics in the Glass ransition Region 22