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Master Thesis Defense
Thermophoresis in Liquids and its Connection to Equilibrium Quantities
Benjamin F. Maier
23 June 2014
06/23/14 2
Thermophoresis
● directed movement of particles
● Induced by heat
● in gaseous or liquid systems
here: gas
● solute moves to cold
http://bit.ly/1qy7Gs7, 06/20/14
06/23/14 3
Thermophoresis - Liquids
● Different picture for liquids
● Hot region sometimes preferred
● No coherent prediction possible
http://i.imgur.com/FKLjmUV.jpg, 06/20/14
?
06/23/14 4
Thermophoresis – A Powerful Tool!
Powerful addition to electrophoresis
Braun, et. al. (2006)
Problem: Unpredictable!
DNA migration
06/23/14 5
Thermophoresis in Formulae
phenomenological flux
diffusion thermal flux
Soret equilibrium
TemperatureDensityDiffusion coefficientThermal diffusive mobility
Soret coefficient
measured from Soret equilibrium density
06/23/14 6
Soret Coefficient
● sign of dictates direction of movement
Piazza, et. al. (2008)
prefers cold region
prefers hot region
06/23/14 9
Local Equilibrium
● assumptions
– length scale of solute-solvent interaction
– small heat flux
● Würger (2014)
● Braun/Dhont (2007)
force on solute
solvation free energy
solvation enthalpy
solvation entropy
Which one is true?
06/23/14 10
Open Problems
● local equilibrium applicable?
● if yes, is driving force the entropy or the enthalpy?
● connection of and
● sign change of reproducible?
06/23/14 11
Theory – Brownian Motion
● Overdamped Langevin equation (an SDE)
● interpretation: Ito or Stratonovich?
position change
external force interaction force
Gaussianstochastic process
local diffusion coefficient (not necessarily Einstein's relation)
ItoStratonovich
– friction
06/23/14 12
Ideal Gas Soret Equilibrium Density
● ideal gas without external potential
● interpretation of
– Einstein's relation follows for spatially invariant diffusion coefficient
– may be wrong (Astumian, 2008)
– Will assume
ideal gasequation of state
06/23/14 13
BD Simulation for 1D Ideal Gas
cold hotlinearreflective boundary conditions
06/23/14 14
BD Simulation for 1D Ideal Gas
cold hotlinearreflective boundary conditions
06/23/14 15
1D Ideal Gas with External Potential
cold hot
naïve Boltzmann expectation
−1.0 −0.5 0.0 0.5 1.00.0
0.2
0.4
0.6
0.8
1.0
T
−1.0 −0.5 0.0 0.5 1.00.0
0.2
0.4
0.6
0.8
1.0
T
06/23/14 16
1D Ideal Gas with External Potential
cold hot
naïve Boltzmann expectation
−1.0 −0.5 0.0 0.5 1.00.0
0.2
0.4
0.6
0.8
1.0
T
−1.0 −0.5 0.0 0.5 1.00.0
0.2
0.4
0.6
0.8
1.0
T
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Dynamical Density Functional Theory (DDFT)
● local equilibrium assumption: use equilibrium functional
● traditional functional, dimensionless
● new scaling, adapted ideal gas functional
● interacting particles: how to scale excess functional?
06/23/14 18
Interacting Particles – Solvent and Dilute Solute
two possibilities to scale the excess functional
solute densities
Soret coefficient
Würger, '14 Braun/Dhont, '07
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0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0
0.2
0.4
0.6
0.8
1.0
Gaussian Solute in Ideal Gas Solvent (1D)
equilibrium from thermodynamic integration
Significant difference in predictions from enthalpy and entropyapproaches
periodic boundary conditions
fit function
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Thermophoretic BD Sim. – Gaussian Solute
hotcold
looks like the prediction from the enthalpy
−0.5 0.0 0.5
0.6
0.8
1.0
1.2
−0.5 0.0 0.5
0.6
0.8
1.0
1.2
06/23/14 21
Thermophoretic BD Sim. – Gaussian Solute
hotcold
looks like the prediction from the enthalpy
−0.5 0.0 0.5
0.6
0.8
1.0
1.2
−0.5 0.0 0.5
0.6
0.8
1.0
1.2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4 Soret coefficient
Sign change!
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Homogeneous 1D System of Gaussian Particles
Soret equilibrium density from enthalpy connected to equation of state
hotcold
const.pressuresecond order
virial eq. of state
second virial coefficient
−0.5 0.0 0.5
20
25
30
35
40
45
−0.5 0.0 0.5
20
25
30
35
40
45
T
06/23/14 23
Homogeneous 1D System of Gaussian Particles
Soret equilibrium density from enthalpy connected to equation of state
hotcold
const.pressuresecond order
virial eq. of state
second virial coefficient
−0.5 0.0 0.5
20
25
30
35
40
45
−0.5 0.0 0.5
20
25
30
35
40
45
T
Thermophoretic system seems to follow the equation of state!
06/23/14 24
Summary So Far
● achieved
– local equilibrium assumption seems to be appropriate
– Soret coefficient connected to enthalpy
– sign change of reproducible
– crucial assumption:
● open
– does it describe real systems?
– choice of and its origin
MD simulations
06/23/14 25
MD Setup
● modified SPC/E thermostats (Nosé-Hoover)
● SPC/E solvent● Lennard-Jones noble gas solute
● periodic boundary conditions in z● reflective boundary conditions in xy
● NPT equilibration● NVT runs● measuring the Soret equilibrium
density
● SPC/E: water model(extended simple point charge)
06/23/14 26
First: Solvation Free Energy
fit function
300 350 400 450 500
T / K300 350 400 450 500
T / K
Method: Widom Insertion in thermodynamic equilibrium bulk SPC/E for Ar, Kr and Xe
example: xenon
06/23/14 27
Thermophoretic simulations - Solvent
● 7 simulations of Ar, Xe, Kr with run times t > 200 ns in temperature range 300 K < T < 450 K● measured the Soret equilibrium density and temperature profile of solvent and solute
example: krypton solvent density
following the equation of state!
linear profile
06/23/14 28
Thermophoretic simulations - Solute
● 7 simulations of Ar, Xe, Kr with run times t > 200 ns in temperature range 300 K < T < 450 K● measured the Soret equilibrium density and temperature profile of solvent and solute
example: krypton solute density
06/23/14 29
Soret Coefficient from Density – Krypton
Calculate Soret coefficient as
derivative – how to model ?
Soret eq. density Soret coefficient
Kr, 300 K < T < 350 K
06/23/14 30
Soret Coefficient from Density – Krypton
Calculate Soret coefficient as
derivative – how to model ?
Soret eq. density Soret coefficient
Kr, 300 K < T < 350 K
06/23/14 31
Soret Coefficient from Density – Argon
Calculate Soret coefficient as
Soret eq. density Soret coefficient
Ar, 300 K < T < 350 K
06/23/14 32
Soret Coefficient from Density – Argon
Calculate Soret coefficient as
Soret eq. density Soret coefficient
Ar, 300 K < T < 350 K
For every simulation: Soret coefficient isregion where all fits coincide
06/23/14 33
Soret Coefficient from all Simulations
no significant agreement/results in MD simulations
no sign change
06/23/14 34
Summary & Outlook
● theoretical derivation of Soret equilibrium density for various systems via DDFT
● connection between Soret coefficient and solvation enthalpy/equation of state found in BD simulations
● no significant agreement between Soret coefficient and solvation enthalpy in MD simulations
● agreement between Soret eq. density and eq. of state in MD simulations
● Ito/Stratonovich
● more realistic BD simulations / difference between BD and MD, extract meaningful data from MD
● connection between local diffusion coefficient and local temperature
● consideration of electrostatics and hydrodynamics
06/23/14 35
1D Hard Rods
equation of state
06/23/14 36
2D Ideal Gas in Harmonic Potential
06/23/14 37
2D Ideal Gas in Harmonic Potential
06/23/14 38
Soret Coefficient
● sign of dictates direction of movement
Piazza, et. al. (2008)
prefers cold region
prefers hot region Solute size
dependenceBraun, et. al. (2006)
Polystyrene beads
06/23/14 39
Hypothetical Influence of Friction
Stokes' law consider SPC/E viscosityadditional factor in densityadditional term in Soret coeff.
● 8 hypotheses● sign change in
