Presentation on theme: "T. Witten, University of Chicago University of Chicago, October 2014."— Presentation transcript:
T. Witten, University of Chicago University of Chicago, October 2014
Identical, asymmetrical shapes in the microscopic world G. Meng, Arkus, Brenner, Manoharan, Science 2010 S.Sacanna, Irvine, Rossib, Pine, Soft Matter, (2010) http://images.search.yahoo.com/search/images;_ylt=A0PDoV6h0WBRqmk AWlWjzbkF;_ylu=X3oDMTBxbGdlMTFzBHNlYwNmcC1hdHRyaWIEc2xrA3 ZpZXdtb3Jl?p=pawl& Shaped colloidal objects enable massively parallel functionality Probing Self-assembly Orientational disorder limits this functionality. Can we use the asymmetry to reduce the disorder?
Organizing colloidal objects by eg sedimentation Dispersion of randomly oriented colloidal objects How much orientation is possible? How to achieve complete alignment? initial orientations are different common forcing evolves them differently final orientations are the same
Can common forcing organize a system? Force F(t) …generally not!
Why achieve complete alignment? Identically oriented objects respond identically to arbitrary further forcing Everything rotates and translates together Cf. phase-coherent spins in nuclear magnetic resonance (Otherwise, responses are different) We can guide them to probe different environments attach to other objects
Aligning colloidal objects by programmed global forcing How forcing rotates suspended objects –Space and time scales for sedimentation –3x3 “twist matrix” determines motion Constant force –Generic objects are partly self-aligning –Residual disorder: phase angle around rotation axis Non-constant force –Strategy: transient behavior reduces orientational disorder –alternating forces progressively lead to full alignment Benefits and Barriers of full alignment Brian Moths, Jonah Eaton, Tomer Goodfriend, Haim Diamant, TW University of Chicago, Tel Aviv University http://jfi.uchicago.edu/~tten/ChiralSedimentation/
Sedimentation rotation: how strong? eg. cell in water sinks at about 1 diameter/sec centrifuge at 10 6 g rotational diffusion 1000 x slower Onsager mobility matrix M determines motion: velocity v, angular velocity vv A T T T S FF = M V F = Prior work: Happel &Brenner text: Finds M for some specific objects Doi & Makino 2003, 2005: eg. sedimentation + shear for separation Gonzales Graf & Maddocks, 2004: classify types of rotation This work: controlling orientation of generic objects force torque 10 micron “twist matrix” T : F creates rotation F 1mm 1 sec ~.1 g Andreev, Son & Spivak, 2010: hydrodynamics of chiral dispersion If = 0, = T F
Alternatives to gravity: Electrophoresis… Perturbing E field gives proportional rotational response: = T E 10 micron E 10 4 v/m + ++ + + + + D. Long, A. Ajdari, Phys. Rev. Lett., (1998) speed ~ 1 diameter/sec Arbitrary vector perturbation has a T… like sedimentation – – – – – – – Other driving forces Temperature gradients Concentration gradients … behave analogously
Two aspects of object determine twist response T forcing point where F is applied. torque = 0 Displacing forcing point affects antisymmetric part of T R T’ = T + S ||R || 0 R z -R y -R z 0 R x R y -R x 0 ||R || We measure R from “center of twist”, where T is symmetric Affects this symmetric part of T Shape: hydrodynamic drag profile Three orthogonal principal axes Three principal eigenvalues --- may be + or - R… and thus T… can be altered without changing the shape.
Sedimentation with constant F: how much alignment is possible? in body frame, T is fixed, F rotates: dF/dt = F but = T F dF/dt = ( T F) F R F(t) Alignment: many initial F’s one final F symmetric T No alignment nearly symmetric F 2 stable fixed points, (stable orbits) F moves on a sphere Significant alignment strongly asymmetric F 1 stable fixed point axial alignment
Constant sedimentation force F can orient objects, … Sedimentation organizes orientation if R ≠ 0 For large enough R, random orientation F orients with one direction in body. Such objects are “axially-aligning” …BUT: orientation is incomplete Φ undetermined
Time dependence suggests strategy for complete orientation Until steady state is reached, T concentrates points in angle space: transient response favors alignment We can make transient behavior continue by changing F in time aligning (polar) axis Φ ω time
Simplest time-dependent F: change from, to Brian Moths Φ (Φ) Φ angles are bunched Increased alignment! Objects initially axially oriented in direction F Objects reoriented in direction, Φ Ψ
Finding T for a real object: “Fourblob” Requires determining hydrodynamic force and torque under an imposed v and ω HYDROSUB software Represents composite object using many little draggers (Stokeslets) In a 10 4 g centrifuge this object with spheres of 1 micron diameter and specific gravity of 1.1 in water would rotate at 25. rad/sec. T = reduced J.Garc ́ ıa dela Torre and B.Carrasco,Biopolymers 63, 163 (2002).
ΦiΦi Does alignment improve after many switches of F? Φ i+1 F Ψ(Φ0)Ψ(Φ0) Φ0Φ0 random wait α 0 Φ 1 = Ψ(Φ 0 + α 0 ) α1α1 Φ 2 = Ψ(Φ 1 + α 1 ) αiαi Ψ(Φ) … Each slice ΔΦ feels successive bunching and spreading. Is there a net bunching effect on probability p(Φ)? p(Φ) Φ ii+1 varying α Ψ’ Ψ’>1 Ψ’<1 ΔΦ i Δφ i+1 Ψ Φ time
Bunching-spreading randomness is multiplicative ---makes bunching dominate Bunching wins on average Ψ 0 π 2π 1/B 1/S density ρ ρB ρS ϕ Example: equal probability for bunching or spreading Point density ρ after 1 cycle ρ ρS, ρB After 2 cycles, bunching/spreading multiplies ρ ρS 2, ρS B, ρB S, ρB 2 After n cycles, ρ ρS n, …, ρS k B n-k, … ρB n ….broad distribution of densities Median density, ρS n/2 B n/2 ~ ρ [1 + a 2 ] n/2 >> ρ Random wait α a/2
Statistical alignment: alignment entropy must decrease Brian Moths…if Ψ(Φ) is monotonic Entropy H[p(Φ)] – dΦ p(Φ) log p(Φ) quantifies disorder H is maximal when p(Φ) is constant (ie Φ is random) Smaller H p bunched into narrower region(s) After one step of , p(Φ) p(Φ i+1 ) = p(Φ i ) / Ψ’ New H[p] H α ~ H α – H = dΦ p(Φ + α) log Ψ’(Φ) ~ ~ Φ i + α i Change of H is weighted average over “bunching factor” Ψ’ Average change of H for arbitrary random waits α: ~ H α – H α = dΦ p(Φ + α) α log Ψ’(Φ)= (const) dΦ log Ψ’(Φ) ~ negative for monotonic Ψ Thus after many steps, H - ∞ p(Φ) is arbitrarily narrow. Φ p(Φ)
Full alignment without monotonic Ψ(Φ) Suppose Ψ(Φ) is not monotonic: Bunching still occurs when |Ψ’| <1 dΦ log | Ψ’(Φ) | still gives a measure of bunching. Jonah showed: Average change of entropy H for arbitrary random waits α: H α – H α < log | Ψ’(Φ) | Φ J[Ψ(Φ)] ~ If J[Ψ(Φ)] < 0, disorder in phase 0: J[Ψ(Φ)] gives a lower bound on the average ordering rate.
Arbitrary gentle shifts in force give alignment F Ψ(Φ0)Ψ(Φ0) Φ0Φ0 random wait α 0 Φ 1 = Ψ(Φ 0 + α 0 ) α1α1 Φ 2 = Ψ(Φ 1 + α 1 ) … Gentle alternation gives alignment, (above) Repeat the argument with arbitrary shifts F Ψ0(Φ0)Ψ0(Φ0) Φ0Φ0 random wait α 0 Φ 1 = Ψ 1 (Φ 0 + α 0 ) α1α1 Φ 2 = Ψ 2 (Φ 1 + α 1 ) … Change of entropy H at i-th step depends only on i’th Ψ Δ i H α < log | Ψ i ’(Φ) | Φ J[Ψ i (Φ)] If these J i are negative, entropy decreases. Shifts can be random!
Conclusion: alignment of a phase angle happens readily Arbitrary gentle changes of forcing seem sufficient. Example*: Colloids in a chaotic fluid If these objects are in the same flow, v(t), they will tend to align over time These methods seem applicable to controlling any system with a set of random phase angles to be aligned. * Thanks to Kevin Mitchel, UC Merced Tentatively verified with random shifts Shown for weakly rotating force. Gives phase locking Can alignment happen spontaneously?
After alignment: the dance begins Eg. force objects to unstable fixed-point orientation Identically oriented objects respond identically to further forcing Everything rotates and translates together …but is it practical?
Practical limitations look surmountable Can we fully align objects that don’t axially align? YES: one extra step gives an initial axially aligned state What forcing method is optimal? Electrophoresis? Seems most promising, but quantitative prediction of T is harder Do interactions between the objects help or hinder alignment? Hydrodynamic interactions are rich for rotating objects –Haim and Tomer Overall they HINDER alignment: dispersion must be dilute.
Conclusion For colloid-containing fluids Programmed forcing can orient microscopic bodies so that they respond as one coherent, collective motion usually confined to pristine dynamical systems Requirements on the forcing are surprisingly non-restrictive. For phase-coherence phenomena Our random forcing methods seem useable to bring any set oscillating systems into a common phase Comparing with existing methods will give insight. Applicable to generic asymmetric objects of closely similar form.