Experimental Colloids I (and I)
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1 Experimental Colloids I (and I) Dave Weitz Harvard Boulder Summer School 7/24/17
2 Experimental Colloids I (and I) Dave Weitz Harvard Boulder Summer School 7/24/17
3 Colloidal particles Colloidal particles are ubiquitous Biology Viruses, macromolecules, organelles Probe particles for bioassays Quantum dots for fluorescent assays Spores, bacteria Processing Paints, coatings, materials control Ceramics
4 Colloidal particles Key control rheology Solid particles behave like continuous fluid Process solids, while flow like fluids eg Paints and coatings Spread paint like a fluid Solidify into a solid coating
5 Colloidal particles Properties set by particle density Concentration of particles low compared to normal material Typical solid: ~10 27 atoms/ m 3 (1 / nm 3 ) Colloids: ~10 18 particles/ m 3 (1 / m 3 ) Latent heat of phase transitions too small to measure Very low pressure: = nk B T ~ x 1.4x10-23 x 300 = 4x10-3 Pa Gas: 3x10 25 molecules/m Pa = 1 atm
6 Soft Solids Easily deformable Low Elastic Constant: Energy Volume Hard Materials ev 3 GPa Soft Materials kt B m 3 1 Pa Soft materials invariably have a larger length scale
7 Continuous phase fluid Thermalization with fluid Equilibrates particles Brownian motion
8 Continuous phase fluid x x Hydrodynamic interactions
9 Colloidal particles Ignore hydrodynamic interactions Thermalize system Important only for dynamics No effect on static properties Consider just two-body interactions between particles
10 Colloidal Interactions U Hard-sphere interactions U = 0 r 2a U = r 2a 2a r Only excluded volume
11 Colloidal Interactions van der Waals interactions Dispersion interactions Dipole-induced dipole interactions Depend on polarizability of material Require different materials Always present for particles in a fluid
12 van der Waals interactions Repulsive Short-ranged Dipole-dipole 1/r 6
13 Stabilizing interactions Particle Fluid Particle
14 Colloidal Interactions - Stabilization Screened Coulomb interaction Inverse screening length Surface potential Ion density
15 Stabilizing interactions Disjoining pressure: Can t compress ions
16 Colloidal interactions stabilizing E b Short range attraction Long range repulsion Sum: Stabilizing barrier V S E b > k B T Colloid stable against aggregation
17 Repulsive Spheres Repulsive interactions
18 Screen charges No Salt Salt
19 Experimental Techniques Light scattering Static light scattering Dynamic light scattering Ultra-small angle dynamic light scattering Diffusing-wave spectroscopy Microscopy Rheology microrheology
20 Dynamic Light Scattering
21 Structure and Dyanmics: Light Scattering = q -1 q =4 n/ sin /2 SLS: < I > vs. q DLS: < I(q,t)I(q,t+ ) > probe structure probe dynamics f(q, )
22 Light Scattering Probes characteristic sizes of colloidal particles q 4 n sin 2 Speckle: Coherence area DYNAMIC LIGHT SCATTERING: Single speckle STATIC LIGHT SCATTERING: Many speckles
23 Dynamic Light Scattering Measure temporal intensity fluctuations Intensity [khz] T im e [s] Obtain an intensity autocorrelation function I() t I( t ) I Lag time ( ) [s]
24 Dynamic Light Scattering Measure temporal correlation function of scattered light: Intermediate structure factor f qt, ~ E 0 E t 2 iq r 0 r t E E t A e 0 m n mn, Time average over all particles 2 2 ~ e q r t 2 ~ e qdt Correlations only between the same particles Cumulant expansion: r 2 (t) ~ Dt Physics: How to change the phase of the field by Each particle must move by ~
25 Ultra Small Angle Light Scattering Probe Structure 2 deg. L. Cipelletti
26 Multispeckle Detection Average over constant q: 0.07 deg to 5.0 deg 100 cm -1 < q < 7000 cm -1 non-ergodic samples avoid excessive time averaging
27 Diffusing Wave Spectroscopy: Very strong scattering TRANSMISSION L MOST LIGHT IS SCATTERED BACK MILK IS WHITE!! D. Pine, P. Chaikin, E. Herbolzheimer
28 P(s): DIFFUSION EQUATION t = 0 I (t) PRL 60, 1134 (88) z 0 * I(t) # PATHS OF LENGTH s = ct P(s)
29 SINGLE PATH [MARET & WOLFE] t=0 t= n SCATTERING EVENTS s = n PATH LENGTH
30 EXTENDED SOURCE g 1 t L * 4 3 6t 1 sinh cosh 8t L 6t 4 6t L 6t 3 * * CHARACTERISTIC TIME SCALE: 0 L * 2
31 TRANSMISSION d 0.6 m = 2%
32 DWS PROBES MOTION ON SHORT LENGTH SCALES PHASE OF PATH CHANGES WHEN PATH LENGTH CHANGES BY ~1 WAVELENGTH ~5000 Å BUT: LIGHT IS SCATTERED FROM MANY PARTICLES L 10 * 10 Estimate: 10 4 MOTION OF EACH INDIVIDUAL PARTICLE CAN BE MUCH LESS CAN MEASURE PARTICLE MOTION ON SCALE OF ~ 5 Å
33 Confocal Microscopy rotating mirrors laser screen with pinhole detector (PMT) microscope sample screen
34 Confocal microscopy for 3D pictures 0.2 m Scan many slices, reconstruct 3D image
35 Rheology sin( t) o sin( t ) o E Elasticity Viscosity o( cos sin t sin cos t) G G
36 Mechanical Properties of Soft Materials: Viscoelasticity Solid: G e i t 0 G ig Fluid: Elastic Viscous
37 Rheology of soft materials G'( )* b, G"( )* b % CB = % CB = % CB = % CB = % CB = Dispersant Series G"( ) 1.6% CB = % CB = G'( ) * a Scaling plot
38 Hard Sphere Phase Diagram Volume Fraction Controls Phase Behavior liquid liquid - crystal coexistence crystal 49% 54% 63% 74% maximum packing RCP 0.63 Increase Decrease Temperature maximum packing HCP = F = U - TS
39 Entropy Drives Crystallization Entropy => Free Volume 0 F = U - TS Disordered: Higher configurational entropy Lower local entropy Higher Energy maximum packing RCP 0.63 Ordered: Lower configurational entropy Higher local entropy Lower Energy maximum packing HCP =0.74
40 Metastable Hard Sphere Phases liquid liquid - crystal coexistence crystal 49% 54% 58% 63% 74% supercooled glass liquid 0.48 xtal 0.54 Maximum packing RCP 0.63 Maximum packing HCP =0.74
41 State diagram for colloidal particles Jammed States: Nonequilibrium solid states Equilibrium Hard spheres
42 Controlled Attraction of Colloidal Particles Depletion attraction fluid fluid + polymer attractive fluid gel Polystyrene polymer, R g =37 nm + PMMA spheres, r c =350 nm T. Dinsmore
43 State diagram for colloidal particles Fractal gels Φ ~ 10-5 Non-equilibrium Strong attractions low Network formation Weakly attractive gels at intermediate volume fractions Jammed States: Nonequilibrium solid states Glasses Weak attractions high Local caging Equilibrium Hard spheres Quasi-equilibrium
44 State diagram for colloidal particles Fractal gels Φ ~ 10-5 Non-equilibrium Strong attractions low Network formation Weakly attractive gels at intermediate volume fractions Jammed States: Nonequilibrium solid states Glasses Weak attractions high Local caging Equilibrium Hard spheres Quasi-equilibrium
45 Hard spheres: -dependent structure factor
46 -dependent relaxation Short-time and long time relaxation processes Inverse-relaxation follows structure factor
47 -dependent relaxation Slopes give relaxation rates effective diffusion coefficients
48 -dependence of viscosity Comparison of frequency dependent data
49 -dependence of short-time diffusion coefficient Correlates with viscosity
50 -dependence of long-time diffusion coefficient Correlates with viscosity
51 Increasing : Approach to the glass transition: -dependent relaxation q-dependent relaxation What is the nature of the relaxation?
52 Viscoelasticity of Hard Spheres ~ Low frequency relaxation a High frequency response Fluid contribution Glassy plateau b
53 Mean square displacement Confocal Microscopy x 2 m 2 3D data -relaxation -relaxation 2D data Volume fraction =0.53 supercooled fluid lag time t (s)
54 Mean-squared displacement = supercooled fluid x 2 m 2 3D =0.56, 100 min (supercooled fluid) 2D Cage trapping: lag time t (s) Short times: particles stuck in cages Long times: cages rearrange
55 Trajectories of fast particles, = micron shading indicates depth
56 Displacement distribution function x 2 m 2 3D t = 1000 s 2D lag time t (s) = 0.53: supercooled fluid Nongaussian Parameter 2 4 x x 2
57 Choose Time with Maximum Non-Gaussian Parameter
58 Time Scale and Length Scale top 5% = tails of x distribution 95% Time scale: t* when nongaussian parameter 2 largest Length scale: r* on average, 5% of particles have r( t*) > r* cage rearrangements =0.53, supercooled fluid
59 Structural Relaxations in a Supercooled Fluid time Number of relaxing particles Relaxing particles are highly correlated spatially
60 Non-Gaussian parameter for glasses glasses No well-defined peak
61 Structural Relaxations in a Glass time Number of relaxing particles Relaxing particles are NOT correlated spatially
62 Fluctuations of fast particles Supercooled fluid = 0.56 Glass = 0.61
63 Cluster Properties Number N f of fast neighbors to a fast particle: Fractal dimension: = 0.56 supercooled fluid
64 Relaxation events are spatially correlated average cluster size volume fraction Cluster size grows as glass transition is approached
65 Dynamical Heterogeneity: possible dynamic length scale Adam & Gibbs: cooperatively rearranging regions (1965) Simulations: Photobleaching: NMR experiments: Glotzer, Kob, Donati, et al (1997, Lennard-Jones) Cicerone & Ediger (1995, o-terphenyl) Schmidt-Rohr & Spiess (1991, polymers)
66 Boulder summer school experiments
67 DWS from hard spheres (a) t (sec)
68 Mean-squared displacement (b) t (sec)
69 Microrheology Measure mean square displacement of probe particles: Light scattering: Dynamic light scattering Motion over larger lengths - lower frequencies Diffusing Wave Spectroscopy Motion over smaller lengths - higher frequencies Calculate Modulus Generalized Stokes-Einstein equation G s as kt r B 2 s max Transform to storage and loss moduli Analytic continuation: s = i G G
70 Complex modulus 2 ) dyne/cm 10 5 G (s)( s (sec 1 )
71 Light scattering rheology G' 10 1 G'' [rad/s]
72 State diagram for colloidal particles Fractal gels Φ ~ 10-5 Non-equilibrium Strong attractions low Network formation Jammed States: Nonequilibrium solid states Glasses Weak attractions high Local caging Equilibrium Hard spheres Quasi-equilibrium
73 Colloidal Stability Partial Stability Many collisions to stick No Stability Sticks every collision
74 Dilute, stable suspension C C C C C C C C R D kt 6 R Destablize C s ~ c 13 2 s ~ ~ D 1 DC 23 C s C C C C WRONG! Irreversible aggregation
75
76
77
78
79 d f ~ 1.70
80 FRACTAL: SELF-SIMILAR NO CHARACTERISTIC LENGTH SCALE M d f ~ R d f : FRACTAL DIMENSION NON-INTEGRAL log M 3 d f d f < d log L
81 DENSITY: DECREASES WITH SIZE f M L d V L d L d f d log FRACTAL log L
82 MASS CORRELATIONS: 3d: cr ( ) ~ 2d: cr ( ) ~ r r 1 3-d 1 2-d f 3 f
83 d =.25 d f ~ 1.75
84 Colloidal Aggregation Colloidal Gold
85 DIFFUSION REACTION LIMITED AGGREGATION DIFFUSE MOTION DIFFUSION-LIMITED STICKS WHERE IT FIRST TOUCHES REACTION-LIMITED MUST COLLIDE MANY TIMES DIFFUSIVE MOTION NOT IMPORTANT.
86 DIMENSIONS: d: Euclidean dimension of space d = 3 real space d = 2 surface d f : d t : Fractal dimension Amount of volume occupied by a space filling object is M ~ R d f Trajectory dimension Fractal dimension of trajectory Random walk: d t = 2 Ballistic motion: d t = 1 No motion d t = 0
87 DIFFUSION-LIMITED CLUSTER AGGREGATION d d d f f t d d d f f t NO INTERPENETRATION BUT CLUSTERS STICK WITH OTHER CLUSTERS d f ~ 1.8 in 3-d.
88 Brown bag calculation
89 Gelation of fractal clusters
90 Gelation of fractal clusters
91 Gelation of fractal clusters
92 Gelation of fractal clusters R c ~ a -1
93 State diagram for colloidal particles Fractal gels Φ ~ 10-5 Non-equilibrium Strong attractions low Network formation Weakly attractive gels at intermediate volume fractions Jammed States: Nonequilibrium solid states Glasses Weak attractions high Local caging Equilibrium Hard spheres Quasi-equilibrium
94 Gelation of Attractive Particles Carbon Black in Oil U
95 spacer 23 m Effect of Volume Fraction Carbon Black in S150N U ~ kt 25 o C 100 m w%
96 Determination of Volume Fraction Shear breaks aggregates into 0.5 m units Concentration Series 25 o C Volume fraction determined from -dependence of Concentration Series 25 o C [Pa s] % CB 3% CB % CB 1.6% CB 10 0 [Pa s] % CB 3% CB 2% CB 1.6% CB ' [s -1 ] ' [s -1 ]
97 spacer 23 m Effect of Volume Fraction Carbon Black in S150N U ~ kt 25 o C 100 m % w%
![[Pa] 10 0 10-1 10-2 10 0 10-1 [Pa s] G' ( ), G"( ) [Pa] 10 0 10-1 10-2 10 0 10-1 [Pa s] 10-3 10-1 10](/docs-images/76/74086051/images/98-2.jpg "0 10 1 10 2 [rad s -1 ] 10-2 10-3 10-1 10 0 10 1 10 2 [rad s -1 ] 10-2 10-3 10-1 10 0 10 1 10 2 [rad")
98 Fluid-Like Behavior Carbon Black in S150N T=25 o C 1.2% CB = % CB = % CB = G' ( ), G"( ) [Pa] [Pa s] G' ( ), G"( ) [Pa] [Pa s] G' ( ), G"( ) [Pa] [Pa s] [rad s -1 ] [rad s -1 ] [rad s -1 ] 10-2
99 Solid-Like Behavior CB in S150N T=25 o C 4% CB = % CB = % CB = G' ( ), G"( ) [Pa] [Pa s] G' ( ), G"( ) [Pa] [Pa s] G' ( ), G"( ) [Pa] [Pa s] [rad s -1 ] [rad s -1 ] [rad s -1 ] 10-1
100 Scaling -Dependence Carbon Black in S150N U ~ kt 25 o C G'( )* b, G"( )* b % CB = % CB = % CB = % CB = % CB = Dispersant Series G"( ) 1.6% CB = % CB = G'( ) * a
101 Scaling along the background-viscosity experimentally accessible range G'( ), G"( ) b a /
102 -Dependence of Fluid-Solid Transition * / * / * / c c c G'p [Pa] G'p [Pa] G'p [Pa]
103 Effect of Dispersant Decrease Interaction Energy Decrease Aggregation Amount of Dispersant controls Interaction Energy
![100 m ~ U [kt] 3.1 5.](/docs-images/76/74086051/images/104-2.jpg "1 8.0 12.")
104 Effect of Dispersant Concentration Carbon Black in Oil = o C spacer 6 m 100 m ~ U [kt] [Disp] w %
105 Scaling: U-Dependence Carbon Black in S150N ~ 0.14 T=25 o C G'( )* b, G"( )* b % Disp U ~ 12.8kT 1.5% Disp U ~ 10.0kT 2.0% Disp U ~ 8.0kT 2.5% Disp U ~ 6.4kT 3.0% Disp U ~ 5.1kT concentration series G"( ) G'( ) * a
106 Fluid-Solid Transition U-Dependence * / 10 1 U c U G'p [Pa] * / U-U c U-U c G'p [Pa]
107 Fluid-Solid Transition Weakly Attractive Systems U [kt] 10 5 Solid Fluid U
108 Fluid-Solid Transitions Carbon Black in Oil U
109 Fluidization through Shear % 10 3 [Pa s] % 2% 1.4% 0.8% 0.4% [Pa] o [s -1 ] o [s -1 ] CB in S150N 25oC
110 Dispersant-Shear Equivalence [Pa] x a(disp) [Pa s] % disp. 1% disp. 2% disp % disp. 4% disp o a ( a(disp) )[ssec -1 ] x a( disp) sec
111 Shear Induced Fluid-Solid-Transition 10 0 y [Pa] fluid solid y = ( - c ) y [Pa] fluid solid y = u (U-U c ) U [kt]
![Phase Boundary in U- Plane 20 15 U [kt] 10 5](/docs-images/76/74086051/images/112-2.jpg "Solid Fluid 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.")
112 Phase Boundary in U- Plane U [kt] 10 5 Solid Fluid U
113 Yield Stress as Phase Boundary y = o ( c ) kt/u [Pa] 1/ yield [Pa] PMMA - PS 5 kt 8.7 kt 16.5 kt Carbon Black in Oil ~ kt Charged Stabilized PS - Salt > 20 kt
114 Phase Boundary in U Plane = const. y = o (U U c ) kt/u [Pa] 10 1 PMMA - PS Data = / yield [Pa] = 0.15 = 0.20 Extrapolation = U/kT Carbon Black = 0.14
115 Jamming Phase Diagram for Attractive Systems U Proposed by: Andrea Liu, Sid Nagel Nature 386, 21 (1998) Completely new way to look at viscosity of soot in oil
116 Dependence on range of interaction
117 Attractive colloidal particles Short Range
118 Attractive colloidal particles Intermediate Range
119 Attractive colloidal particles Long Range
120 Long-range attraction Long Range
121 Reduce gravity
122 Spinodal Decomposition of Colloid Polymer 35 hours
123 ~3 cm Time Evolution of Phase Separation
124 Comparison with Furukawa Theory Long-time evolution of spinodal decomposition
125 Short-range attraction Short Range How does gelation occur? What are interparticle interactions?
126 Structure at gelation Fluid
127 Structure at gelation Gel
128 Weak attractive interaction
129 Cluster distribution: Determine interaction
130 Cluster distribution: Independent of potential
131 Interaction energy at gelation
132 Interaction energy at gelation Same behavior for all, all
133 Gelation is proceeded by spinodal decomposition Implications?
134 S(q) spinodal decomposition Increasing U Increasing t
135 Gelation spinodal decomposition
136 Schematic Phase Behavior for Colloidal Gels Attractive glass U Attractive interaction Spinodal decomposition attractive glass gelation
137 Gelation phase diagram Dynamic arrest of phase separating system
138 How do crystals melt?
139 Colloidal crystals melt at grain boundaries Yodh, Science
140 What if there are no interfaces? Born melting: Elastic catastophe Elastic modulus goes to zero, and crystal melts But how does this actually occur??
141 Volume fraction controls melting Lower Lowest High
142 Phase diagram of Wigner Crystals
143
144 Mean squared displacement
145 Mean squared displacement Lindemann parameter Fraction of lattice parameter
146 Hot particles are highly spatially correlated =
147 Hot particles are highly spatially correlated = 0.101
148 Hot particles are highly spatially correlated = 0.066
149 Hot particles are highly spatially correlated = Melted
150 Hot particles are strongly correlated in space
151 Hot clusters are fractal d f = 1.70
152 Cluster-mass distribution 2 Power-law with exponential cut off
153 Scaling behavior of cluster size
154 Scaling behavior of cluster volume fraction
155 Scaling behavior of elasticity Born (1939)
156 Scaling behavior 1. Cluster size 2. Cluster volume fraction 3. Elasticity
157 Scaling behavior 1. Cluster size 2. Cluster volume fraction Second-order character of 3D melting 3. Elasticity
158 Hot particles lead to non-affine motion Breaks force balance weakens lattice
159 Non-affine motion increases as decreases Calculate elastic modulus including non-affine motion
160 Behavior of elasticity C 44 remains approximately constant with Non-affine modulus does vanish
161 2 nd order behavior for melting 1D melting is always 2 nd order 2D melting is through hexatic 2 nd order 3D melting has 2 nd order character if the lattice is perfect Non-affine shear modulus: Provides weakened regions Mechanical stability is lost Generalizes Born melting 3D melting of Wigner lattice is weakly 1 st order
162 The End
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