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Entropic Forces and Friction

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Entropic Forces and Friction 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 ective ff 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.
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