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Crystal growth rates above Tg (Diffusion-controlled growth)

Mark Ediger and Lian Yu (University of Wisconsin-Madison)

Peter Harrowell (University of Sydney)

NSF Chemistry

Based partly upon JCP 128, 034709 (2008) 1

The liquids community claims to have understood many important aspects of the dynamics of supercooled liquids over the last 20 years. Does this new

knowledge have an impact on our understanding of crystallization from

the supercooled liquid?

We say yes! 2

Outline

•  Experimental measurements of crystal growth

•  What controls the growth rate at low T? •  Relationship between D and growth rate •  Absolute growth rates •  Related work by Zanotto and Greer

3

Crystal growth rates in single component supercooled liquids

Thermodynamic control

Kinetic control

TNB

Plazek and Magill, JCP 45, 3038 (1966)

TNB

Tg

4 Aside – some of these measurements were motivated by the need to understand crystal growth in metals (e.g., Turnbull).

Crystal growth rates at low temperatures are kinetically controlled

We can extract the kinetic growth rate ukin from experimental measurements of the growth rate u by removing the contribution of the thermodynamic driving force. Ediger et al, JCP 2008

-11

-10

-9

-8

-7

-6

log

u (m

/s)

1.000.950.900.850.800.750.70T/Tm

TNB (Magill and Plazek) kinetic growth rate ukin growth rate u

5

Plazek and Magill, JCP 45, 3038 (1966)

ukin = u/(1-e-ΔG/RT)

Outline

•  Experimental measurements of crystal growth

•  What controls the growth rate at low T? •  Relationship between D and growth rate •  Absolute growth rates •  Related work by Zanotto and Greer

6

Mapes, Swallen, and Ediger, J. Phys. Chem B (2006); Sillescu and coworkers, Z. Phys. Chem. 1992 (high temp D values)

-18

-16

-14

-12

-10

-8

-6

log(

D)

(cm

2 s-1

)

340320300280260240Temperature (K)

-10

-8

-6

-4

-2

0

2

log

(Tη

-1)

(K

P-1)

Tg

T/η

Wilson (1900) and Frenkel (1932) assume that ukin is governed by D or η; their arguments would not

distinguish between D and η

But we know that D and η have distinct temperature dependences near Tg…

7

We can test relation between crystal growth rate and η (but not D) for a large number of organic systems.

Data from Scherer/Uhlmann, Magill/Plazek, Yu, and Wu/Yu; Note: all single component systems

-12

-10

-8

-6

-4

log

[u/(

1 - e

xp(-Δ

G/RT

))]

(m

/s)

86420log (η/Pa s)

OTP sorbitol α-phenol o-cresol indomethacin δ indomethacin α indomethacin γ TNB slope = -1

Crystal growth rates always have a weaker temperature dependence than η

Three indomethacin polymorphs show same slope (Wu/Yu)

8

We can also test relation between crystal growth rate and η (but not D) for a large number of

inorganic systems.

Data from laboratories of Wagstaff, Weinberg, Zanotto, Uhlmann, Bergeron, and Gutzow. Comparison made by Ediger et al., JCP 2008

Crystal growth rates often have a weaker temperature dependence than η. Note: liquid/crystal compositions are the same.

9

The conventional view of these results (e.g., Uhlmann)

•  The crystal surface explains why ukin does not scale with viscosity •  Only a fraction f(T) of sites at the crystal surface are available for growth •  f(T) has whatever temperature dependence is required to make this explanation work.

Problems with the conventional view: •  f(T) often exhibits a sharp change in slope near 1.25 Tg. Why? •  The three crystal polymorphs of indomethacin have very different surfaces but would require the same f(T) •  In 1967, Plazek and Magill stated for TNB that “mass transport in crystal growth and viscous flow do not have the same temperature dependence” 10

We claim that the relationship between ukin and η is controlled by properties of the

liquid not properties of the crystal surface

1.0

0.8

0.6

0.4

slope

10080604020fragility (m)

inorganics organics

These systems contain Li+

Slope from log ukin vs log η Fragility is defined from η(T) at Tg

11

High fragility glassformers show a non-Arrhenius temperature dependence as Tg

is approached from above.

12

Ediger and Harrowell, J. Chem. Phys. (2012)

m = (d log η/ d Tg/T)|Tg

Possible explanation for ukin ~ η-ξ with��� ξ dependent upon fragility

1)  Assume ukin ~ D for all liquids 2)  Then D ~ η-ξ 3)  Then ξ for diffusion depends

systematically upon fragility

Points 1 and 2 are known to be reasonably accurate for the four liquids where this can be tested. Point 3 would be an important new result, implying that spatially heterogeneous dynamics more strongly influence diffusion in fragile liquids. 13

Outline

•  Experimental measurements of crystal growth

•  What controls the growth rate at low T? •  Relationship between D and growth rate •  Absolute growth rates •  Related work by Zanotoo and Greer

14

The crystal growth rate is more closely connected with diffusion than with viscosity

or structural relaxation

15

IMC

Diffusion: Swallen et al., Soft Matter 2011 Crystal growth rates: Wu and Yu, JPCB 2006

-10

-8

-6

-4

-2

0

2

log

( τ/s

), lo

g (η

) +C

, - lo

g (D

) +

C'

320300280260240T (K)

o-Terphenyl-----------------

τα (DR) τα (NMR) log (η) + C -log (D) + C' -log (ucorr) + C''

Tg

Tm

For OTP, crystal growth rates correlate better with D than η

DR = Richert; NMR = Sillescu; η = Plazek; ukin = Scherer/ Uhlmann

16

Absolute growth rates correlate reasonably with entropy of fusion

ukinτα/a = molecular layers per structural relaxation time; evaluated at η = 104 Pa s. The error bar on the y axis is ± 1 due to uncertainty in τα.

The line shown is the equation given by Burke, Broughton, and Gilmer, JCP 1988. We are not aware of a derivation.

17

Outline

•  Experimental measurements of crystal growth

•  What controls the growth rate at low T? •  Relationship between D and growth rate •  Absolute growth rates •  Related work by Zanotto and Greer

18

19 Nascimento and Zanotto, JCP 133, 174701 (2010)

Nascimento et al. •  Four most

fragile systems: Li2O(2SiO2) PbO(SiO2) cordierite

diopside

20

Chalcogenide-based phase change material (GST)

•  Ge2Sb2Te5

•  Rewritable optical data storage

•  Many assumptions

21 Orava, Greer, Nat. Matls. (2012)

Summary •  Temperature dependence of crystal growth rate

correlates better with D than η. Why? •  Conventional interpretation emphasizing f(T), the

fraction of sites where the crystal can grow, has no predictive power.

•  New interpretation implies that the decoupling of D from η depends systematically on fragility. This is an important insight into diffusion and spatially heterogeneous dynamics.

•  Prediction of growth rates for polymorphic organic liquids: Use ΔSfus and ukin ~ η-0.75

•  New approach re-integrates research on supercooled liquids and crystallization

22