Thermodynamics of Melting: Tutorial and Applications (Part II)

T-p paths, Melting, Adiabats and Clapyrons

• T variations in mantle controlled by reversible and irreversible transformation of mechanical energy into internal energy (via pressure and viscous forces) along with heat diffusion and chemical reactions (phase transformations and melting)

• Decompression usually assumed carried out reversibly and adiabatically. I.e., isentropic process

• For isentropic process (no phase change)

⎛ ∂S⎞ ⎛ ∂S⎞ dS = dT+ dp ⎝⎜ ⎠⎟ ⎜ ⎟ ∂T p ⎝ ∂p⎠ T C dS = p dT− αVdP = 0 T ⎛ ∂T⎞ αT ⎜ ⎟ = ⎝ ∂p ⎠ S ρCp p = ρgz ⎛ ∂T⎞ αTg ⎜ ⎟ = ⎝ ∂z ⎠ S Cp T = T exp[αg(z − z ) C ] zo o p • Typical magnitude: α = 2x10−5 K−1, T = 2500K, ρ = 4000kg m−3,Cp =1500 J kg−1 K−1

⎛∂T ⎞ 1 then ⎜ ⎟ = 8.3KGPa− or ~ 0.3 K/km ⎝ ∂p ⎠S

1 • More general model is based on Joule-Thompson (JT) isenthalpic process. This allows for irreversible effects (e.g., heat transfer & friction)

• Recognize that magma ascent requires buoyancy force which is opposed by frictional forces degrading mechanical energy into heat and that heat may be transferred between rising magma and surroundings (irreversible processes hence non isentropic)

• Model for decompression of melt in a crack or pipe under steady flow conditions. Variation of T with depth (z) accounting for enthalpy dependence on p, T, friction, KE and heat transfer under steady flow conditions • Equations express conservation of mass, momentum and energy for melt in a pipe

• Note that quantity modifying pressure gradient is the ⎛ ∂T⎞ isenthalpic gradient from the JT expansion= ⎜ ⎟ ⎝ ∂p ⎠ H

• Easy to show if we neglect inertia (ΔKE, friction (Cf=0) and assume adiabatic (B=0) then isenthalpe simplifies to isentrope

2 • To examine melt fraction dependence on depth useful to examine energy equation in form considering a parcel of peridotite undergoing isentropic decompression

1 2 Δh + 2 Δu − gΔz = q + w' Ø q is heat absorbed (-ve) , w’ is mechanical energy degraded via friction to heat (+ve), h (p,T,f) is specific enthalpy. Assume Δu=0, adiabatic and reversible and get relationship between depth (z), melt fraction f and T. Energy equation becomes

Δh − gΔz = 0 Use thermo to get Δh(p,T,f); then integrate

⎛ 1− αT⎞ C (T− T ) + (p − p ) + f Δh − g(z − z ) = 0 p m ⎝⎜ ρ ⎠⎟ m melting m solve for fraction melt f C (T − T)+ αTg(z − z ) f = p m m

Δhmelting Ø T is along solidus defined by trajectory in Tz space ⎛ ∂T⎞ defined by ⎜ ⎟ ⎝ ∂p ⎠ G

Ø Tm, pm , zm is T, p and depth where Clapeyron slope intersects isentrope; can explicitly solve for f vs z

Ø Physically, enthalpy decrease of material upon decompression is balanced by increase in potential energy and phase change (melt fraction increases)

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⎛ ∂T⎞ ⎛ ∂T⎞ Relationship between melt generation, ⎜ ⎟ and ⎜ ⎟ ? ⎝ ∂p ⎠ G ⎝ ∂p ⎠ S • Shallow mantle has Clapeyron slope (~4 K/km) > isentrope (0.4K/km) hence parcels melt upon decompression • Extent or fraction of melting is proportional to ‘overshoot’

4 • Case when solidus/ isentrope approaches solidus/Clapeyron leads to thick region of low melt fraction melt • Case where Clapeyron slope is negative; might be relevant to deep melting on super earth exoplanet; stability of deep liquid layers is insured

How does Tm depend on depth? • Melting curves for pure substances tend to flatten out in T vs p space; that is, melting curves approach the isentrope ⎛ ∂T⎞ ΔV ⎛ ∂ln T⎞ γ • Recall ⎜ ⎟ = and ⎜ ⎟ = ⎝ ∂p ⎠ G ΔS ⎝ ∂p ⎠ S Ks • MD results show that in liquids γ increases as p increases (isentrope becomes steeper) BUT typical melting curves flatten since ΔV → 0 whereas ΔS (s → liq ) >0 always.

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• This ‘pressure flattening’ is common (silicates, oxides, elements). Some elements have maxima

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ref 15=Wang, 1999

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‘Heuristic models’ for melting & flattening of Tm vs p curves Simon-Glatzel (1929) 1/b T (p) T 1 (p p ) m = m,pr ( + − r π) ' • π and b are constants related to K T andK T

• Some materials show a maximum! o E.g., sodium metal exhibits a maximum! o What happens when if Clapeyron slope turns over? Would we ever see these melts?

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• Extension to SG eqn to acct. for maximum 1/b T (p) T 1 (p p ) exp (p p ) c m = m,pr ( + − r π) (− − r )

o c is related to entropy and volume of melting

Lindemann Criterion (1910) • Melting occurs because of vibrational instability: crystals melt when ave. amplitude of thermal vibration is high fraction of interatomic distance 1/2 2 > δR • s with  = atomic displacement, δ ≈ 1/4 and Rs = one-half of typical bond length

Effects of H2O on Melting

9 • Well known that a little H2O goes a long way in lowering solidus. Two reasons: low molecular mass of water (small mass but lots of particles) & H2O breaks Si-O-Si bonds efficiently

• Dry molten CaMgSi2O6 o Consider the fraction of BO (bridging oxygen) and NBO in dry melt (MD studies show)

Si around O 0.70 0.60 0.50 0 0.40 1 0.30 2 3 0.20

0.10 Proportion 0.00 0 20 40 60 80 100 120 GPa

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Stoichiometry is H2O• CaO• MgO• 2SiO2 • XNBO= 6/7; XBO=1/7; this is dramatic change in melt structure requires many GPa to ‘undo’!

11 • H2O plays a REMARKABLE role in lowering the solidus because it dramatically breaks Si-O-Si bonds at constant p&T by increasing the concentration of [1]O at expense of [2]O • Lowering of solidus in macroscopic terms

ΔH Tdry T m = dry ΔH− RTm ln(1− xH2O ) FINAL SPECULATION Icosahedral packing of oxygen in silicate liquids in the high p limit?

12 • Consider MgSiO3 liquid. Frequency distribution of Oxygen polyhedra. OO[n] is abundance of central oxygens with [n] other oxygens as nearest neighbors

• Note the average O around O increases up to ~ 80-100 GPa followed by ‘collapse’ of structure to oxygen icosahedral packing with ~OO[12]

• This is seen other silicate liquid compositions studied by classical MD and FPMD

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