Microbead Rheology Theory and Applica4ons Ian Seim - PowerPoint PPT Presentation

Microbead Rheology Theory and Applica4ons Ian Seim iseim@live.unc.edu 2/2/17

Path Data • Each path is a 4me series with x and y posi4ons recorded at regular 4me intervals – The x-posi4ons can be denoted: X(0 = t 0 ), X(t 1 ), X(t 2 ), … , X(t i ), … , X(T = t N ) • t N = T = total recording 4me of path data – Typically T = 30 seconds for experiments in the Hill lab • τ = lag 4me (inter-observa4on 4me) – The first lag 4me is t i+1 – t i = 1/frame rate (typically 60fps -> τ = 1/60s), but we are oXen interested in mul4ples of this first lag 4me (2/60, 3/60, etc.) – Νumber of data points = N = T/τ

Path Data • We are interested in the proper4es of the increments , ΔX i = X(t i +τ) – Χ(t i ) – t i for i = 0, 1, 2, … , N-τ – some4mes for mul4ples of τ, i.e. 2/frame rate, 3/ frame rate, 4/frame rate, etc. • For a given path and τ = 1/frame rate, we then have N-1 increments in each coordinate • What does the distribu4on of the x- increments look like for this salt water path…?

Mean-Squared-Displacement (MSD) • For a 4me series of posi4ons, X(t i ) for i = 0, 1, 2, … , N, at a given lag 4me τ, the MSD is defined as: N − τ 1 2 ∑ Δ r 2 ( τ ) = [ ] X ( t i + τ ) − X ( t i ) N − τ + 1 i = 0 • At each lag 4me, τ, the MSD is the variance of the corresponding van hove correla4on func4on

Brownian Mo4on • A simple con4nuous-4me stochas4c process • In our applica4ons, we are interested in the brownian mo4on as a random walk of a par4cle – Think of a drunken sailor who stumbles out of the bar with nowhere to go: he takes a sequence of steps, but randomly chooses an angle for each one. How far away from the bar is he aXer some amount of 4me? (on average)

Brownian Mo4on • Regard each increment, ΔX i = X(t i + τ) – X(t i ), as a random variable – ΔX i ~ N(0, 2Dτ) – The increments are independent (the par4cle doesn’t “remember” where it was) MSD ( τ ) = 2 dD τ , d = dimensionality (Einstein, 1905) D = k B T , k = Boltzmann’s constant, η = 6 πη r viscosity, r = radius (Stokes-Einstein) • Fluctua4on-Dissipa4on: allows us to infer dissipa4on (macro) proper4es from observed fluctua4ons (micro) and vice versa

Frac4onal Brownian Mo4on • A generaliza4on of brownian mo4on – The distribu4on of increments is s4ll Gaussian with mean 0, but they are no longer independent, i.e. they are correlated and have “memory” • We introduce a parameter, α, that captures the linear, but not necessarily = 1 scaling of the MSD: MSD ( τ ) = 2 dD τ α • If α < 1, this is called sub-diffusion • If α > 1, this is called super-diffusion

Proper4es of Variance • For N random variables X i we have: ⎛ ⎞ N N ∑ ∑ ∑ ( ) ( ) + Var X i Var X i Cov X i , X j ⎟ = ⎜ ⎝ ⎠ i = 1 i = 1 i ≠ j • In our serng, the N (+1) random variables are increments, ΔX i in each coordinate, and their sum is our observed path • Lets look at what happens to the variance of this sum in two cases …

Brownian Mo4on Δ X i = X ( t i + τ ) − X ( t i ) = N (0,2 D τ ) ⎛ ⎞ N N N ∑ ∑ ∑ ( ) ∑ ( ) + Var Δ X i Var Δ X i Cov Δ X i , Δ X j 2 D τ = 2 ND τ ⎟ = = ⎜ ⎝ ⎠ i = 1 i = 1 i ≠ j i = 1 ( ) = 2 DT = 2 ND τ Var X ( t N ) − X ( t 0 ) • Since the variance of the sum of the smallest increments is equal to the variance of the largest increment, we see that the increments are independent, i.e. covariance = 0

Frac4onal Brownian Mo4on 2 D τ α N i (0, V ij ) Δ X i = , where V ij is a covariance matrix (defined later) ⎛ ⎞ N N ∑ ∑ ∑ V ij = 2 DT α N 1 − α Var Δ X i Var ( Δ X i ) + ⎟ = ⎜ ⎝ ⎠ i = 1 i = 1 i ≠ j ) = 2 DT α ( Var X ( t N ) − X ( t 0 ) • Here we see that the two variances are no longer equal and we see that the increments are now correlated, and covariance is non-zero.

Covariance for fBm 2 D τ α N i (0, V ij ) Δ X i = V ij = 1 α + i − j − 1 α − 2 i − j ⎡ ⎤ α 2 i − j + 1 ⎣ ⎦

Experimental DriX ? We should probably look at the par4cle paths, per movie…

DriX • Due to fluid flow, microscope movement, other external forces, etc. • For now, we assume it is linear (constant in 4me), and we update our fBm model: 2 D τ α N i (0, V ij ) Δ X i = µ τ + • So we see that paths are a “superposi4on” of a stochas4c process, fBm, and a determinis4c, linear driX, with velocity μ

Firng fBm to data using MLE • Using our model of fBm + driX, we can simultaneously fit the parameters D, α, and μ to each experimental path using Maximum Likelihood Es4ma4on • This allows us to accurately recover the diffusive parameters, despite driX (if it is linear), and effec4vely characterize the viscoelas4c fluid and re-construct a “true” MSD

Maximum Likelihood Es4ma4on • Suppose there are N i.i.d. observa4ons, X 1 , … , X N from an unknown probability density func4on with parameters, θ – We want to find an es4mate of θ • We define the joint density func4on for these observa4ons as N ∏ f ( X 1 ,..., X N | θ ) = f ( X i | θ ) i = 1 • Now, we regard the observa4ons as parameters of this func4on, and θ as a variable (free to vary), and we end up with the likelihood L ( θ ; X 1 ,..., X N ) = f ( X 1 ,..., X N | θ )

Maximum Likelihood Es4ma4on • In prac4ce, we mostly use the log-likelihood , which is the natural logarithm of the N likelihood func4on ∑ ln L ( θ ; X 1 ,..., X N ) = ln f ( X i | θ ) i = 1 • Using calculus (if we are lucky), we can take par4al deriva4ves of the log-likelihood func4on and find maximum values of the parameters we are interested in. Otherwise, we can use a minimiza4on rou4ne (fminsearch() in matlab) on the parameter of interest on the nega:ve log-likelihood func4on

Simula4ng fBm Paths 1 α ) α + s a − t − s • Form the covariance matrix, 2 ( t • Compute the square root matrix, Σ (eigenvalue decomposi4on is what I use) • Generate a vector, v, of numbers drawn from a standard Gaussian distribu4on • Let ,this is our 1-dimensional u = 2 D Σ v fBm path • Can easily add a linear driX by choosing a velocity, μ, mul4plying by the corresponding 4me vector, and adding it element-wise to u

Assignments • 1) write code that simulates Brownian paths in 2-d (will have 3 inputs (D, τ, T) and should output a 4me series of posi4ons in x and y • 2) write code to calculate MSD (input is a 2-d path, output is a vector of lag 4mes and a vector of corresponding mean-squared- displacements) • 3) write code that finds the maximum likelihood es4mate of D for a brownian path (input is a 2-d brownian path, and τ; output is an es4mate of D)