Chapter 7

Colloids

7.1 Colloidal phase behavior

Wigner crystal, to colloidal liquid, to hard-sphere gas electrostatic interactions between particles,qualitative; depletion attraction, simple estimates (case of micelles, of polymer coils; semidilutesolution case; most e↵ective is c*, range set by diameter) colloidal crystals, variation with salt,qualitative argument (main reference to phase diagram?) comments on liquid-gas and gas-solid,with respect to depletion and size of object aggregation – DLA, RLA (qualitative)

Marginal screening length

Depletion attractions

simplest case: dilute hard-sphere micellar solution and colloidal particles [me, TAW 5.1.4] use ofDerjaugin approximation Similar result for dilute solution polymer depletion interactions – semidi-lute solution; range is xi, semidilute pressure acts [TAW p. 121] most e↵ective is short chains atoverlap concentration

173

174 CHAPTER 7. COLLOIDS

7.2 Di↵usion of colloids

Dynamic light scattering

Consider a dilute sample of small particles, small compared to the wavelength of light, illuminatedby a coherent plane wave. Because the scatterers are small (pointlike), they scatter approximatelyisotropically. The scattered intensity I is proportional to the square modulus |A|2 of the scatteredamplitude, a result familiar from our earlier discussion of static small-angle scattering.

Now we are interested in how the scattered intensity varies in time, as a result of the motionof the particles. Roughly speaking, if a particle moves a distance of the order of the wavelengthof light, the phase of the scattered wave arriving at the detector from that particle will change byof order 2⇡. (More precisely, if a particle moves in the direction of the scattering wavevector bysuch a distance, the scattered wave will change phase by this amount; however, random motionof the particles will always lead to some displacements along the scattering wavevector.) As weshall see, the uncorrelated motion of the particles in this way causes the scattered intensity tofluctuate. Measuring the time dependent fluctuations of the scattered intensity at a given scatteringwavevector is called dynamic light scattering (DLS).

The timescale of the fluctuations is su�cient to reveal details about the motion of the particles.In particular, if the particles are di↵using, we can infer their di↵usion constant D, and throughthe Stokes-Einstein relation their hydrodynamic radius a. Thus by DLS we can measure the sizeof dilute colloidal particles, by observing how fast they di↵use.

To analyze this situation, we begin with the scattered amplitude, which is

A /X

i

eiq·ri (7.2.1)

in which ri is the position of the ith scatterer with respect to the origin, taken in the center of theilluminated region. We are interested in the intensity autocorrelation function,

S(q, t) = hI(q, t)I(q, 0)i /X

i,j

X

k,l

heiq·(ri(t)�rj(t))eiq·(rk(0)�rl(0))i (7.2.2)

Assuming the motion of di↵erent particles are uncorrelated, only terms above with i = j andk = l, or i = l and j = k, survive the average. The first of these pairings is what results fromaveraging I(q, t) and I(q, 0) separately; thus we may write

S(q, t) / N2 +N2ieiq·(r(t)�r(0))i2 = N2(1 + g(t)2) (7.2.3)

The first term arises from the square of the average intensity; we recall from our discussion of staticscattering that the average intensity from N uncorrelated particles scales as N .

We focus on computing g(t). Evidently g(0) = 1, while for su�ciently large times, r(t) andr(0) become uncorrelated, and g(t) vanishes. Thus g(t) describes the decay of correlations in thelocation of the particles, on the length scale q�1 (since the particles must move a distance of order2⇡/q to destroy the correlation).

If the particles are di↵using, then the displacement r(t) � r(0) is a Gaussian random variable,in which each component has zero mean and variance 2Dt. Thus we may evaluate the average overrandom trajectories of the particle in time, much as we did previously for similar integrals involvingthe random walk trajectories of polymer chains in space. Following this approach, we find after abit of algebra

g(t) = e�Dq2t (7.2.4)

7.2. DIFFUSION OF COLLOIDS 175

As anticipated, the timescale for correlations to decay is set by the time for a particle to di↵usea distance L2 = Dt such that Lq is of order unity. If we make measurements at di↵erent scatteringangles and hence di↵erent wavenumbers q, we expect to find that the slope �(q) of the decay oflogS(q, t) versus time is 2Dq2. This can be tested by plotting �(q) versus q2. (Recall q = 2q

0

sin ✓/2,where q

0

= 2⇡/� is the incoming wavenumber for incident radiation of wavelength �.) The particledi↵usion coe�cient D can be extracted from the slope of �(q) versus q2.

Speckle

As an aside, we may ask how correlated is the scattered intensity between nearby locations onthe face of the detector, which is to say at nearby wavenumbers. Thus we examine the angularcorrelation

C(�q) = hI(q)I(q + �q)i � hI(q)i2 (7.2.5)

Analyzing this correlation function as before, assuming randomly located pointlike scatterers,we find

C(�q) / N2(1 + he�i�q·ri2)�N2 (7.2.6)

in which the particles r are randomly located throughout the illuminated (scattering) volume, oflinear dimension L.

We see that the intensity at adjacent scattering wavevectors will be correlated if �q is smallenough that �qL is less than unity. Now we work through the geometry to see how large a displace-ment this corresponds to on the face of the detector, some distance R away from the scatteringvolume. The change in wavenumber �q corresponds roughly to an angular displacement d✓ accord-ing to �q ⇡ q

0

d✓ (assuming the scattering angle is reasonably small); then the displacement on thedetector face is

dR = Rd✓ = R�q/q0

= R�/(2⇡L) (7.2.7)

With typical numbers, � = 5000A, R = 100cm, L = 1mm, we have dR ⇡ 0.1mm.The “granular” appearance to the eye of laser light scattered from a small illuminated region is

known as “speckle”; we have estimated the size of a single speckle. It is often desirable to arrangedetectors for dynamic light scattering so that the detector aperture is of order one speckle in area;this maximizes the coherence of the scattered light across the detector, and hence maximizes theamplitude of the intensity fluctuations relative to the average intensity.

Long time tails

In our previous discussions of the random motion of colloidal particles under the action of thermalforces, we have tacitly assumed that the motion was di↵usive, Brownian motion. Now we examinethat assumption a bit more closely, to ask on what timescale a particle with an initial velocity withrespect to the surrounding fluid, is able to “forget” its initial velocity.

Suppose that at time t = 0, a neutrally buoyant particle suspended in quiescent fluid is given a“kick”, resulting in an initial velocity v

0

presumably as the result of random thermal forces (pressurefluctuations on the particle surface). How long does memory of that initial velocity persist?

This question is answered by the velocity autocorrelation function,

C(t) = hv(t) · v(0)i (7.2.8)

which evidently vanishes for long times. At t = 0, C(t) equals the variance in particle velocity,equal to 3kT/m by equipartition. One might naively expect that this function would decay expo-nentially in time, perhaps with the characteristic time ⌧ = a2/⌫, which is the only time that canbe constructed from the particle radius and fluid kinematic viscosity.

176 CHAPTER 7. COLLOIDS

But consider the fate of the momentum carried by the particle after an initial kick; with initialvelocity v

0

, the particle carries a momentum (4⇡/3)a3⇢0

v. Remember that momentum is conserved;where does the momentum go, so that the particle can forget its initial velocity? Evidently, themomentum must be absorbed into the fluid.

Momentum transport in a fluid is likewise governed by a conservation law, of the general form

@gj@t

+ri⇧ij = 0 (7.2.9)

in which ⇧ij is the ith component of the current carrying the jth component of the momentum.For an incompressible viscous fluid, the momentum current takes the form

⇧ij = �⌘(rivj +rjvi) + p�ij (7.2.10)

Viscosity transports momentum in a fluid; if we have a positive gradient @xvy, this results in aleftward current of the y component of momentum, as the faster upward-moving fluid to the rightaccelerates the slower fluid to the left.

The resulting equation for momentum density, in linearized approximation, is

@g

@t� ⌫r2g +rp = 0 (7.2.11)

which is essentially a di↵usion equation, with a pressure term present to enforce incompressibility(which implies that v and thus g must be divergenceless). Here the momentum di↵usivity ⌫ is justthe kinematic viscosity ⌘/⇢.

All of which is to say, momentum spreads out into the fluid surrounding the particle by di↵usion.In a time t, the momentum initially carried by the particle can only di↵use a distance of order

p⌫t,

spreading out more or less uniformly over a volume of fluid of order (⌫t)3/2. The particle andsurrounding fluid then move together, with a velocity determined by momentum conservation, as

a30

⇢0

v(0) ⇠ (⌫t)3/2v(t) (7.2.12)

This implies the initial velocity kick must decay as a power law in time,

v(t) ⇠ v(0)a3/(⌫t)3/2 = v(0)(⌧/t)3/2 (7.2.13)

Thus the velocity autocorrelation function actually decays with a power-law “long time tail”, ratherthan the naively expected exponential decay exp(�t/⌧).

Note that the characteristic time ⌧ may be interpreted as the time for momentum to di↵usein the fluid a distance of order the particle radius a. For a particle of radius 1µ, in a fluid ofviscosity 1cP and density 1g/cm3, this timescale ⌧ is 1µsec. The thermal velocity

p3kT/m of such

a particle, which would be the typical “initial velocity” of interest, is about 0.2cm/sec. The particlethus travels a distance of about 20Aduring the time ⌧ ; the long time tail evidently is only relevantfor rather small random displacements of the particle.

It would seem, then, that there is no way to use dynamic light scattering to look at motion ofcolloidal particles on such short length scales. But as we shall see in the next section, a di↵erentform of dynamic light scattering is much more sensitive to small motions of particles, and has beenused to study such small random displacements in colloidal systems.

7.2. DIFFUSION OF COLLOIDS 177

Di↵using wave spectroscopy

In conventional dynamic light scattering, it is important that the index contrast between theparticles and the surrounding fluid is weak enough that no significant multiple scattering occurs;that is, the light scattered by one particle does not scatter again from a second particle beforereaching the detector.

As a consequence, particles must move a distance of order the wavelength of light before theycan dephase the scattered wave at the detector. A more recently developed scheme for dynamiclight scattering operates instead in the regime of strongly multiply scattered light — so muchmultiple scattering, in fact, that the light paths themselves are random walks within the sample.

A typical light path through the sample in this multiple scattering limit would then involvescattering from not just one but many particles. In such a case, uncorrelated motions of eachparticle along the light path, much smaller than the wavelength of light, result in many small phaseshifts that combine to dephase the light along the path.

In such an experiment, the light follows many di↵erent paths, from an ensemble governed bythe statistics of random walks in the scattering medium. The ensemble of paths can be controlledto some degree by choosing where to send light into the sample and where to observe scatteredlight. For example, if light is injected at one point on the face of a slab sample of thickness h, andscattered light collected from a point on the opposite face, the light paths will all tend to involvemany scatterers. The number of scatterers will be of order n = h2/l2

0

, in which l0

is the mean freepath, the typical distance between successive scattering events, because n is the typical number ofsteps required for a random walker (the light) to traverse the slab in steps of l

0

.

On the other hand, if the light is injected and collected from a point on the same side of the slab,the light paths correspond to a Gambler’s Ruin problem (the random walk light path is injectedabout one mean free path into the sample, and takes random steps until it exits the sample). Theresulting ensemble of paths comprises a broad distribution of path lengths (number of scatterers).

To compute the autocorrelation function for the scattered intensity, we begin by writing thescattered amplitude as

A =X

p

eiqL(p) (7.2.14)

in which L(p) is the total length of the light path p, and q = 2⇡/� is simply the wavenumbercorresponding to the incident light. (There is no “scattering wavevector” in this problem.) Thescattered intensity I is equal to |A|2; we seek the intensity autocorrelation function

C(t) = h�I(t)�I(0)i = |hA ⇤ (t)A(0)i|2 (7.2.15)

In the average hA ⇤ (t)A(0)i, each path contributes coherently only with itself, leading to

hA ⇤ (t)A(0)i =X

p

heiq(L(p,t)�L(p,0))i (7.2.16)

in which the angle brackets denote the average over the motion of the particles, which change thelengths of all the paths.

To find out how the lengths of a given path changes, write the path as a sequence of particlepositions {r

1

(t), r2

(t), r3

(t), . . .}. Let di = ri+1

� ri be the ith step in the path; let �ri(t) =

178 CHAPTER 7. COLLOIDS

ri(t)� ri(0) be the random displacement of the ith particle. The total path length is then

L(t) =X

i

|di + �ri+1

(t)� �ri(t)|

⇡X

i

qd2i + 2di · (�ri+1

(t)� �ri(t))

⇡X

i

(|di|+ ni · (�ri+1

(t)� �ri(t))) (7.2.17)

in which ni is a unit vector along the initial direction of the ith step.Hence �L(t), defined as the di↵erence L(t)� L(0), is simply

�L(t) =X

i

�ni · �ri(t) (7.2.18)

in which �ni is defined as ni�1

� ni; note that �ni is a random vector of lengthp2 (because ni�1

and ni are uncorrelated).Now, observe that �L(t) is a Gaussian distributed random vector (because it is a linear combi-

nation of such vectors, namely the �ri(t) ) of zero mean, and variance

h(�L(t))2i = 2n/3h(�r(t))2i = 4nDt/3 (7.2.19)

in which n is the number of scatterers along the given path. Performing the average over theparticle motions then leads to

hA ⇤ (t)A(0)i =X

p

e�q24n(p)Dt/3 (7.2.20)

in which n(p) denotes the number of scatterers in the path p.Now we are ready to perform the sum over paths, to obtain the amplitude autocorrelation

function:

hA ⇤ (t)A(0)i =Z 1

0

dnP (n)e�q24nDt/3 (7.2.21)

Now we need the distribution of path lengths for the random walk light paths. We shall evaluatethis for the case of light injected and collected from the same spot on the face of a semi-infinitesample.

For this case, the path length distribution is the same as the lifetime distribution for theGambler’s ruin problem. We solved this in an early lecture, using the method of images. Thelifetime distribution is given by the di↵usive flux at the absorbing interface,

P (n) = @n

"e�(z�a)2/(4Dn)

p4⇡Dn

� e�(z+a)2/(4Dn)

p4⇡Dn

#

z=0

(7.2.22)

In the present case, the injection distance a is the mean-free path l0

; the di↵usion constant D isl20

/6 (so that in each direction x, y and z, the mean-square displacement is hx2i = 2Dn = l20

n/3,for a total of hr2i = l2

0

n). Correspondingly, we have

P (n) / n�3/2e�3/(2n) (7.2.23)

where the proportionality constant can be fixed by normalization.

7.2. DIFFUSION OF COLLOIDS 179

Finally, using this expression for P (n) in the expression for hA⇤(t)A(0)i performing the integral,and squaring to find C(t), we obtain

C(t) = e�p

32q2Dt (7.2.24)

The broad distribution of path lengths leads to a “stretched exponential”; the log is not linear intime as a simple exponential decay would be, but decays faster at earlier times (from the contribu-tion of long, easily dephased paths) and more slowly at later times (as the remaining correlationsare increasingly dominated by shorter and shorter paths, which remain correlated for longer times).