Entropic forces and friction
(c)2017 van Putten 1 Today
Gibbs’ principle
Entropic forces and applications to DNA stretching
DNA
The ultimate data-storage medium storage medium
Friction
Mobility from the van ’t Hoff equation
Macroscopic friction with air and solids
(c)2017 van Putten 2 Gibbs’ principle
First law of thermodynamics: dW = −PdV = dE − TdS
Exchange associated with mechanical work
Gibbs’ principle: compare neighbouring states at same E
−dE = PdV − TdS = 0
E : total mass-energy as measured at infinity
(c)2017 van Putten 3 To heat what is cold …
Two equivalent ways
Warm the cold by heat conduction
Bring cold material into a heat bath
(c)2017 van Putten 4 Heat the cold (I): heat quanta jumping to cold
hot cold
contact: heat quanta jump to cold from hot
J. Lienhard, 2013, “Entropy” - Concept Vignettes (YouTube)
2 ⎛ 6 ⎞ ⎝⎜ 3 ⎠⎟ 625 ΔS1 + ΔS2 = kB ln = kB ln ≅ 0.5798kB > 0 ⎛ 5 ⎞ ⎛ 7 ⎞ 350 ⎝⎜ 2 ⎠⎟ ⎝⎜ 4 ⎠⎟
(c)2017 van Putten 5 Heat the cold (II): pulling in cold material
heat bath
f Entropically m favorable m
(van Putten & Levinson 2012 Cambridge University Press, Ch.10)
f : entropic force, calculate using Gibbs’ principle
6 Entropic force in a rope
Length of rope outside the heat bath: x dW = −Fdx : − Fdx − TdS = dE Entropically dS favorable dE = 0 : F = −T m dx m
(van Putten & Levinson 2012 Cambridge University Press, Ch.10) Isothermal process when the heat bath is arbitrarily large
Rope has a certain #DOF per unit length (2 x #molecules / length)
Rope inside the heat bath is thermalized at kBT
Thermalized #DOF in the rope is proportional to length inside heat bath
(c)2017 van Putten 7 Gibbs’ principle applied to DNA
(c)2017 van Putten 8 Length of Freely Jointed Chains (FJC)
b
Macroscopic length by theory of random walks:
R = b N
Richard Berry, 2013, http://biologicalphysics.iop.org/cws/article/lectures/48662
(c)2017 van Putten 9 Entropic force Ideal gas approximation: 2 mvrms ≈ 3kBT, L = Nb energy in heat 3k T f = = B 0 length b Dimensionless strain: change in macroscopic length z ε = = length L
Entropic force (increase S by curling up): f ~ ε f0
(c)2017 van Putten Hooke’s law ective spring force (pN) force spring ff ective e
Linear extension (x/L)
Richard Berry, 2013, http://biologicalphysics.iop.org/cws/article/lectures/48662
(c)2017 van Putten 11 Hooke’s law R = b N : ⎛ k T ⎞ ⎛ z ⎞ z f (z) = 3 B = 3k T ⎝⎜ b ⎠⎟ ⎝⎜ Nb⎠⎟ B R2 ( ~ pN)
Spring energy: 2 3 ⎛ z ⎞ W ≈ k T 2 B ⎝⎜ R⎠⎟
Richard Berry, 2013, Hooke’s law with k=k(T) http://biologicalphysics.iop.org/cws/article/lectures/48662
(c)2017 van Putten 12 DNA: the ultimate computer storage medium?
13 Bits in a rope: DNA
Organic basis
Deoxyribonucleic acid
Phosphate group glues nucleotides (joining sugar molecules) into single string
Pairs of such strings join to form a helix upon joining individual base pairs
ORIGIN 1930s: deoxyribonucleic from a blend of deoxyribose and nucleic acid nucleotide: nucleoside linked to a phosphate group
(c)2017 van Putten 14 DNA: Gbit information storage
L = 3 billion bp in human DNA
Physical length DNA: o few ×109 × few Α ≈ few m
http://www.chemguide.co.uk/organicprops/aminoacids/dna1.html
(c)2017 van Putten 15 Write & read by synthesizing and sequencing
1 g = 700 TB “TG=1” “AC=0”
Sebastian Anthony, 2012 http://www.extremetech.com/extreme/134672-harvard-cracks-dna-storage-crams-700-terabytes-of-data-into-a-single-gram
(c)2017 van Putten 16 https://www.youtube.com/watch?v=6IJgtMMn7G0
17 Stable: in 700,000 yr old bones
Tested 83 kbyte 2000 year-equivalent storage at 10 oC by one-week aging at 70 oC
Price: 20 MUSD/GByte
120 MUSD in each human cell encoding of about 6 GByte
Grass, R., et al., 2015, Angew. Chem, “Robust Chemical Preservation of Digital Information on DNA in Silica with error correcting codes,” 54, 2555
http://www.newscientist.com/article/mg22530084.300-glassedin-dna-makes-the-ultimate-time-capsule.html#.VSnlRlazvjJ
Jacob Aron, 2015
(c)2017 van Putten 18 Molecular diffusion
(c)2017 van Putten 19 Mobility Drift velocity in a random walk due to an external force:
f : vd = µ f
Pressure gradient (force per unit volume): Px = cf van ’t Hoff’s partial pressure difference: [P] = [c]kBT :
Px = cxkBT
Px cf cx = = kBT kBT
(c)2017 van Putten 20 Diffusion
Diffusion equation: ∂t c + ∂x (vdc) = Dcxx
local time rate-of-change + gradient(concentration current) = molecular diffusion
Stationary solutions (drift balanced by diffusion):
∂x (vc) = Dcxx : vdc = Dcx
⎛ cf ⎞ (µ f )c = D⎜ ⎟ : D = µkBT Einstein relation ⎝ kBT ⎠
(c)2017 van Putten 21 Friction
An emergent property when interacting with a medium consisting of a large number of particles, i.e., a solid surface, air, etc.
(c)2017 van Putten 22 Entropy creation by friction with increase in temperature
Gibbs’ entropic force by increase in (thermalized) phase space
(c)2017 van Putten 23 Frictional heating is… everywhere
https://www.youtube.com/watch?v=RJzyB_qEWyU
(c)2017 van Putten 24 Friction in daily life - useful and inevitable
(c)2017 van Putten 25 Parachutes
Leonardy da Vinci (around 1485) http://www.davincilife.com/davincis-parachute.html
(c)2017 van Putten 26 Contact force
N: normal contact force N F f
(c)2017 van Putten 27 Onset to slip
f (tangential friction resistance force) f = µS N
f = µk N
F > f : acceleration f = F
applied tangential force F
(c)2017 van Putten 28 Static and kinetic friction forces F F = F cosθ, N = F sinθ : = tanθ g g N
f N ⎛ F ⎞ tanθS = ⎜ ⎟ = µS ⎝ N ⎠ S F
⎛ µk ⎞ ⎛ µk ⎞ N ma = ΔF = (µS − µk )N = ⎜1− ⎟ FS = ⎜1− ⎟ Fg cosθ ⎝ µS ⎠ ⎝ µS ⎠ θ F θ Initial acceleration
Fg=mg ⎛ µk ⎞ a = ⎜1− ⎟ gcosθ ⎝ µS ⎠
(c)2017 van Putten 29 Braking distance 1 E = mv2 k 2
W f = Ek
W f = fd +Wair
m = 2000 kg, v = 120 km/hr, µS = 0.6
Ek Neglecting energy dissipation in air: W f ≅ fd : d = µS mg 120000 1 v = = 33.33 ms−1 : E = mv2 = 1.11 MJ 3600 k 2 mg = 2000 kg × 9.8 m s−2 = 19.6 kN 1.11×106 d = m = 94.48 m 0.6 ×19600
(c)2017 van Putten 30 Speed of descent
1 f = C ρv2 A 2 D
2mg v = 2 πCD ρr m = 100 kg, CD = 0.75
r = 5 m : v = 5.3 ms−1
(c)2017 van Putten 31