Extending differential dynamic analysis to dark-field microscopy

Objective

Sample chamber

Condenserlens

Opaque stop

Bright-field

Dark-field

Dark-field microscopy

Optical train Au nanoparticle dispersion

High signal-to-noise ratio in dark-field imaging yields improved DDM statistics.

Extending DDM to dark-field is non-trivial since due to linear space variance.Linear space variance: scatterer intensity varies as scatterer traverses (x,y) imaging plane

0

100

200

Inte

nsity

, i par

t

x0|t x1|t+Δt

X1|t+ΔtX0|t

Standard D(q,Δt) analytical decomposition (i.e. standard analysis method) does not necessarily hold à Linear space variant analysis required

α x1( ) ≈α x0( )+ x1 − x0( ) ⋅∇α x0+

Intensity of individual scatterer

Linear space variant decomposition:

( )( )( )( )

0

2

22

0

, Self SelfV q t D t D tα

χα

∇Δ = Δ = Δ

x

x

( ) ( )2, exp Selfg q t q D tΔ = − Δ

( ) ( ) ( )|Part i i ii iα= −x x x x x

Taylor expansion of LSV

( ) ( )

( ) ( ) ( ) ( )

2ˆ, ,

1 , ,

D t I t

A g t V t B

Δ = Δ Δ

= − Δ + Δ +⎡ ⎤⎣ ⎦

q q

q q q q

Brownian motion:

LSV effects are minimal in realistic experiments à LSI analysis should hold.

3 1~ 3 10 mχ µ− −×

1 10.3 5m q mµ µ− −≤ ≤0.01

qχ≤

Realisticvaluesofχ

Objective: Determine conditions under which DDM can be successfully applied to dark-field imaging.

0.1 1 10 10010

100

1000

Increasing q

Dark-fie

ld

Bright-field

Imag

e st

ruct

ure

func

tion

D(q

, Δt)

(a.u

.)

Time step, Δt (s)

Brownian motion

Applying dark-field DDM to optically dense materials.

0.1 10.1

1

10

100

Rel

axat

ion

time,

τ (s

)

Wavevector, q (µm-1)

Dynamics of optically dilute plasmonic nanoparticle dispersions Summary• DDM can be used with dark-field microscopy to characterize

materials (e.g. plasmonic nanoparticles) that would be impossible in other imaging modes.

• DDM analysis can be used on dark-field images to extract dynamic information (e.g. diffusivity), even in optically dense media.

• We developed an analytical framework to interpret dark-field DDM data, and determine conditions under which standard DDM analysis fails.

• Dark-field DDM complements the existing suite of tools for characterization of soft matter dynamics.

95±13 nm Au nanoparticles diffusing in a Newtonian fluid (sucrose solutions) imaged under dark-field, φ = 5 x 10-

8.

Diffusivities measured MPT and DDM agree, confirming accuracy of LSI analysis in LSV illumination.

Inten

sity

Position

Ex. Coarsening of thermosensitive nanoemulsions (bright-field)2

( , )I t t+Δx( , )I tx ( , ; )I t tΔ Δx ˆ( , ; )I t tΔ Δq

exp)1(exp),(21 ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

β

tta

ttatqg

Differential dynamic microscopy measures the same correlation function as measured in dynamic light scattering.

Fast mode: overdamped fluctuations of fractal network in dense phase

Slow mode: superdiffusive motion (ß > 1) often seen in arrested systems

Applications of DDM:Confocal microscopy Phase-contrast microscopy Polarized microscopy

Suspension hydrodynamics3 Motility of E. coli4 Dynamics of liquid crystals5

Autocorrelation function:

References & Acknowledgements 1. F. Giavazzi, D. Brogioli, V. Trappe, T. Bellini and R. Cerbino. Phys Rev E, 2009, 031403.2. Y. Gao, J. Kim, M.E. Helgeson. Soft Matter, 2015, 6360.3. P. Lu et al. Phys Rev Lett, 2012, 218103.4. V. Martinez et al. Biophysical Journal, 2012, 1637.5. F. Giavazzi et al. Soft Matter, 2014, 3938.6. A.V. Bayles, T.M. Squires, M.E. Helgeson. Soft Matter, 2016, submitted.

Software developed in collaboration with Yongxiang Gao. AVB supported by NSF GFRP No. DGE1144085. AVB and TMS funded in part under Award No. CBET 1438779. MEH funded in part under Award No. CBET 1351371.

Dark-field differential dynamic microscopy Alexandra V. Bayles, Todd M. Squires and Matthew E. HelgesonDepartment of Chemical Engineering, University of California, Santa Barbara

Introduction: probing rheology using fluid motion

Advantages• Real space information• High spatial resolution• Probes heterogeneities

Differential dynamic microscopy (DDM) of complex fluids1

5 µm

Multiple particle tracking microrheology

Several techniques exist to characterize the structure & rheology of complex fluids on optical length scales. Certain techniques are better suited to characterize specific fluids than others.

Disadvantages• Probes provide indirect

measurement of dynamics• Fails for optically dense

materials

hν

Scattering pattern

Inte

nsity

TimeFixed q

Photocorrelation spectroscopy (dynamic light scattering)

Advantages• No probes required • Direct measure of

dynamics • Large sampling volume

Disadvantages• Fourier space information• Data comes pre-averaged• Complicated to measure

dense fluids

r(t +Δt)

r(t)

Interested in using DDM?

FFT of difference imagesDDMCalc

Input: Micrograph series + frame rate + pixel size

MATLAB package for performing DDM analysis available at:http://engineering.ucsb.edu/~helgeson/ddm.html

Ensemble averaging( ),D tΔq

Fitting for LSI decomposition with ( ) ( ), exp / ( )g q t t qτΔ = −Δ

Ensemble + azimuthal averaging( ),D q tΔ

Output: ( ),D q tΔ ( )qτ( ),D tΔq

Dark-field DDM can be used in cases where MPT fails.

95±13 nm Au NPs in 50 wt% sucrose, φ = 3x10-5

DDM has not yet been applied to dark-field microscopy, despite its distinct advantages for strongly scattering materials.

g(q,t)

Inten

sity

Position

0.1 10.01

0.1

1

10

100

1000 30 wt% Sucrose 40 wt% Sucrose 50 wt% Sucrose q-2D-1

self

Rel

axat

ion

time

τ (s)

Wavevector, q (µm-1)0.1 1 10 100

0.1

1

10

100

1000 30 wt% Sucrose 40 wt% Sucrose 50 wt% Sucrose (4λ2 − 4/3D0tE)+4D0Δt

Mea

n sq

uare

d di

spla

cem

ent

r2 2D(Δ

t) (µ

m2 )

Time step, Δt (s)

1 10 1000.1

1

10

100

D(q

,Δt)/

A(q)

Normalized time step, q2DselfΔt

χ/q 0 (LSI) 0.01 0.05 0.1 0.5

Method Diffusivity (μm2/s)d-DDM 0.3115± 0.0006

Stokes-Einstein 0.32± 0.04

Multiple particle tracking (MPT)

Soln Diffusivity (μm2/s)50wt% 0.372 ± 0.00240wt% 0.811± 0.00530wt% 1.58± 0.01

Soln Diffusivity (μm2/s)50wt% 0.373± 0.00140wt% 0.8070± 0.000630wt% 1.5809± 0.0006

Dark-field DDM

( ) ( ) ( ) ( ) ( ) ( )2 2ˆ ˆ, , 2 1 ,D t I t N i S g t BΔ = Δ Δ = − Δ +⎡ ⎤⎣ ⎦q q q q q q

Dynamic image structure function à Analytical decomposition

Structure factor Detection noise

Autocorrelation function

Scattererintensity profile