Introduction to Soft Matter Physics(Tfy-3.363)

Lecture 5Hydrophobic Effect and Colloids

Topics today

Lecture 5

Summary of the last lecture: solutions and osmosis (not included in this file)

Hydrophobic effect

a) Small solutes

b) Large solutes

c) Self-assembly

Introduction to colloids

Important factors in colloidal dispersions

Einstein equation for diffusion

Langevin equation

Sedimentation

Hydrogen bonding in H2O

Liquid Ice

The unusual properties of water arise mainlyfrom two factors:

1) Water molecules readily form H-bonds(remember that these are ~ 8 kBT)

2) Water molecules form four H-bonds in a tetrahedral geometry

3D hydrogen bond structures

Whereas in liquid water a typicalhydrogen bond lifetime is ~ 1 ps, in the solid phase (ice) hydrogenbonding leads to stable, opentetrahedral networks.

Hydrophobic effect: small solutes

Provided the solutes are small enough (high surfacecurvature), hydrogen bonding networks may still beformed around them.

Contrary to a common misconception, microscopicallyspeaking hydrophobic solutes are not repelled bywater molecules. In fact, hydrocarbon molecules, for example, often interact more strongly with watermolecules (van der Waals interactions) thanmolecules of their own type. It is just that watermolecules have even stronger mutual interactions.

Instead non-polar solutes (or parts thereof) may beviewed as cavities in water, where hydrogen bondingcannot occur.

This leads to a layer of water with significantorientational correlations around the solute(i.e., decrease in configurational entropy)

∆G ~ volume

Hydrophobic effect: large solutes

For large non-polar solutes, an intact hydrogenbonding network around the solute cannot bemaintained (too low curvature).

To compensate, about one hydrogen bond per molecule near the surface is sacrificed and the water-solute interface is shifted slightly awayfrom the solute surface.

Due to the fact that at standard conditions the free energy cost for forming a water liquid-vaporinterface is small enough (compared to kBT), there is a thin layer of vapor-like water aroundthe solute.

The free energy of solvation, ∆G, for this case is mainly due to enthalpy, instead of entropy. In addition, the effect is proportional to the surfacearea of the solute, not the volume.

∆G ~ area

Free energy of solvation

Entropy dominates

Correlations in the positions the water molecules around thesolute. ΔG proportional to the volume occupied by the solute.

Enthalpy dominates

Formation of a waterinterface away from the solute surface

ΔG proportional to the surface are of the solute.

Surface tension γ

Equilibrium radial distribution of water

Without van der Waals forces

With van der Waals forces

Driving force for assembly

We will return to this topic later on when we discuss self-assembly in more detail.

In principle, one needs to take into account both the entropic (volume) and enthalpic (surface) contributions to the free energy (which are not additive...)

Higher T

Lower T

Colloids

Foams

Paints

Fog, smoke

Aerogel

Milk

Blood

Examples of ”biocolloids”

Hyd

roph

obic

Hyd

roph

ilic

Micelles, self-assembled colloidalparticles consisting of amphiphilicfatty acid or lipid molecules

HDL, High-Density Lipoprotein, 8–11 nm sized biomoleculeaggregates containing cholesterol, phospholipids, apolipoproteins A etc. (”good cholesterol”)

LDL, Low-Density Lipoprotein, 18–25 nm sized similar aggregates(”bad cholesterol”)

Some tidbits from the history of colloids

Thomas Graham (1805 - 1869)

1861, Thomas Graham separates components of a solutionusing a semi-permeable membrane. He calls the componentspermeating the membrane crystalloids, and the ones that do notpermeate the membrane colloids, which is derived from the Greek word κωλλα, ”glue”.

1905, Albert Einstein and William Sutherland formulate the theory of the Brownianmotion. Paul Langevin publishes an alternate approach in 1908.

1857, Michael Faraday experiments on colloidal dispersions of gold; salt-induced coagulation of the colloidal particles.

1937, H.C. Hamaker develops a theory of van der Waals forces between surfaces.

1941, Boris Derjaguin and Lev Landau formulate a theory of colloidal stability.

1948, Evert Verwey and Theo Overbeek improve the forementioned theory; DLVO-theory.

1910, Louis Gouy (and David Chapman 1913): theory of surface charge screening.

1827, Robert Brown publishes his famous study on the jitteringmotion of small clay particles found in pollen grains.

1910, Jean-Baptiste Perrin: sedimentation equilibrium and Avogadro’s number determined.

Classification of colloids

Dispersed phase

External phase

Gas Liquid Solid

Gas

Liquid

Solid

Emulsion

Sol,colloidal dispersion

Colloids with a liquid external phase can also be classified on the tendency of the dispersed particles to aggregate:

Lyophobic: dispersed particles tend to spontaneously aggregateLyophilic: dispersed particles tend to stay dispersed throughout the external phase

(Liquid) Aerosol

(Solid) Aerosol

- Solid foamFoam

(Solid) Emulsion

Solid suspension

Colloids consist of one or more phases, with some characteristic dimensionsbetween ~ 10 nm – 1 μm, dispersed in an external phase.

Surface matters

1 cm 10 nm

10 nm

N

A1 (cm2)

Atot (cm2)

1 106 1012 1018

Colloidal systems are ”all surface”

6

6

2 6 x 10-12

2 x 106

4 x 10-6

4 x 106 6 x 106

Michael Faraday and gold colloids

In 1857, Michael Faraday observed the properties of a dilute solution of colloidal gold. Under normalconditions, the such a colloidal dispersion has a clearred color. However, adding some NaCl into the solution made it turn blue.

Faraday realized that the change in color had to dowith the sizes of the dispersed particles:

In a dispersion, gold colloid particles (< 100 nm) absorb on the green and blue parts of the spectrum, but transmit on the red part. Furthermore, the gold particles are all negativelycharged, thus electrostatically repelling each other.

Adding salt screened these repulsive interactions, and the gold particles started to coagulate. The formation of largeraggregates then resulted in the scattering of blue light fromthe dispersion.

Stability of colloids

Aggregation, coalescence

Dispersion

Creaming

Sedimentation

Effect of colloidal interactions

Effect ofgravity

Important factors in colloidal dispersions

Depletion interactions

Steric stabilization (e.g., by polymer grafting)

Electrostatics

van der Waals interactions

Gravity

Brownian motionConstant, random motion of particles due to collisions with the other molecules in the solution. Displacement of particles is given by the Einstein relation.

Density differences between the solute particles and the external phase lead to sedimentation or creaming of the solutes.

Lyophilic molecules chemically or physically attached to the solute surface prevent aggregation of colloidalparticles. Overlap of the stabilizing molecules results in an osmotic pressure in the overlap region and the stabilized solutes are pushed apart.

Depletion of other solutes (intermediate in size with respect the colloidal particles and the solvent molecules) in a region between two colloidal particles results in an (osmotic) pressure difference. The pressure difference in the depletion region and bulk solventresults in an effective attraction between the colloidal particles.

Will be covered in detail in the next lecture.

Scribes of the ancient Egypt

1 2 3

1) Traditionally, ink was made by dissolvingparticles of carbon black in water.

2) However, within a day or so, the particlesstarted to flocculate and sedimented at the bottom of the container.

3) Adding some gum arabic ( ) into the solution sterically stabilized the colloidalparticles, making the solution more stable.

Brownian motion

Robert Brown (1773-1858)

Incidentally, Robert Brown was also the first to note the ubiquitousnature of a part of eukaryotic cells which he named the ”cell nucleus”.

In 1827, the botanist Robert Brown published a study”A brief account of microscopical observations on the particles contained in the pollen of plants...”, where wereported his observations of irregular, jittery motion ofsmall (clay) particles in pollen grains.

He repeated the same experiment with particles of dust, showing that the motion could not be due to the pollen particles being alive.

Although several people worked on thisphenomenon over the years, a proper physicalexplanation of it had to wait for almost 80 years.

An example of Brownian motion of a particle, recorded for three different resolutionsin time (time steps).

The relation between the diffusion coefficient D and the displacement of a particleundergoing Brownian motion is

Einstein relation

In 1905, Albert Einstein published his PhD thesis on osmotic pressure. Developingthe ideas therein further, later that year he published one of his ground-breakingpapers of that year: the theory of Brownian motion.

Deriving a result, which nowadays is called a fluctuation-dissipation theorem, Einstein showed that the diffusion coefficient of a particle undergoing Brownian motion is

Friction factor of the particle; the frictional force is given by Fdrag = - ξ v

Specifically,

For a spherical particle much larger than the sovent molecules (Stokes-Einstein equation)

whence, for long enough times t

Stokes-Einstein-Sutherland equation?

As a historical sidenote, Einstein did not, in fact, have the precedenceon the result above. Earlier in 1905 (March), a Scotsman/Australian named William Sutherland published a very similar derivation of the Stokes-Einstein equation (which he had publicly presented in a conference already in 1904).

It is not known, why the Stokes-Einstein equation is not known todayas the Stokes-Einstein-Sutherland equation instead (although someauthors have recently suggested it).

William Sutherland(1859-1911)

Sutherland’s article was published in the Philosophical Magazine, a prestigious and wellknown journal. In addition, in 1905 he was already quite famous. For example, he was oneof the two people outside Europe (the other one was J. Willard Gibbs) who were invited to a conference held in honor of Ludwig Boltzmann in 1906. (Einstein was not).

You can get Sutherland’s paper, in addition to other Brownian-motion-related articles, from Peter Hänggi’s web page: http://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html

Dynamics of colloidal particles

Instantaneous force on the colloidalparticle fluctuates wildly...

f(t)

t

f*(t)

t

... But observing the forces oversufficiently long periods of time, wemight see the effective force as:

Langevin equation (1)

Paul Langevin and Albert Einstein, two friends who illuminated the physics of the same phenomenon in two quite different ways.

Instantaneous random force

Let us consider the dynamics of a single colloidal particle under continuousbombardment by the solvent molecules

Langevin equation

The random force f(t) satisfies the following conditions:

Obtained through a fluctuation-dissipation theorem for the problem at hand.

Langevin equation (2)

For the displacement (x,y,z) of a Brownian particle

However, for the mean-square displacements we have

And for the 3D-displacement we have the relation

Since the total displacement of a Brownian particle is obtained with any of the one-dimensional displacements, let us write the Langevin equation in the x-direction as

(L1)

Using the relations

and

We can multiply eq. (L1) with x and rewrite it as

(L2)

Langevin equation (3)

(L3)

We then take the average of eq. (L2), employing the well-known resulst from theequipartition theorem

and further using the notation

we obtain a first-order differential equation

for which the general solution is

Finally, for times , integrating eq. (L4) over time we obtain the result

or

(L4)

Gravitational force affecting the particle V = volume

Bouancy due to the solvent g = acceleration of gravity

Drag force due to the sovent viscosity ρ = mass density

Sedimentation (1)

Charged colloidal particles carried by rivers are neutralized upon the river flow reaching salty sea water. The resulting electrostaticneutralization, flocculation and sedimentation leads to the deposition of the silt at the river delta.

Sedimentation means the (downward) drift of particles in a liquid due to gravity. The essential factors in this process are:

Where the positive sign of force refers to the direction downward. (Note thedifference for the opposite process, creaming.) The sedimentation speed, or theterminal velocity, is obtained by balancing the forces above. For example, in the case of a spherical particle it is given by

where ∆ρ = ρ – ρsol is the difference of the particle and solventmass densities.

Sedimentation (2)

Remember the not-so-evident net motion downward fromthe animation shown at the lecture.

[Simulation and animation by Esa Kuusela]

Some literature on colloids

D. Fennel Evans and Håkan Wennerström, Colloidal domain - Where Physics, Chemistry, Biology, and

Technology Meet (John Wiley & Sons)

Robert J. Hunter, Foundations of colloid science(Oxford University Press)

Ian W. Hamley, Introduction to Soft Matter(John Wiley & Sons)

Denis Weaire, Stefan Hutzler,The Physics of Foams(Oxford University Press)

James W. Goodwin, Jim W. Goodwin,Colloids and Interfaces with Surfactants and Polymers: An

Introduction (Oxford University Press)