Lectures on Modern Physics

Jiunn-Ren Roan

21 Dec. 2007

Soft MatterWhat Is Soft Matter?

PolymersFundamental DefinitionsCommon PolymersConfiguration and ConformationThe Ideal ChainNon-ideal ChainsBlock Copolymers

ColloidsFundamental DefinitionsParticle Size and Size DistributionForces between Colloidal Particles

References

Soft Matter

Soft matter, according to Pierre-Gilles de Gennes, the winner of the Nobel Prizein physics in 1991, includes polymers, colloids, surfactants (amphiphiles), andliquid crystals. These materials are “soft” because their mechanical responsesare often, though not always, intermediate between solids and liquids. The term“soft matter” is synonymous with “soft condensed matter” or “complex fluids”.

The building blocks of most soft matter are organic molecules, so for a verylong time physicists showed little interest in soft matter. Most studies werecarried out by chemists, chemical engineers, materials scientists, or even foodscientists. De Gennes switched from “hard” matter to soft matter in mid 1960s.It takes, however, about 30 more years before physicists’ interest in soft matterbegan to surge.

Because of soft matter’s interdisciplinary nature, it will be very helpful to knowa little more about relevant chemistry, especially molecular structure andphysical chemistry.

What Is Soft Matter?

Polymers

A polymer molecule or a macromolecule, according to the IUPAC (InternationalUnion of Pure and Applied Chemistry) definition, is

A molecule of high relative molecular mass, the structure of whichessentially comprises the multiple repetition of units derived, actuallyor conceptually, from molecules of low relative molecular mass.

Related to this definition is the IUPAC definition of the oligomer molecule:

A molecule of intermediate relative molecular mass, the structure ofwhich essentially comprises a small plurality of units derived, actuallyor conceptually, from molecules of lower relative molecular mass.

Thus, whether a multi-unit molecule is an oligomer or a polymer depends onits molecular mass – it is regarded as having an intermediate relative molecularmass if it has properties which do vary significantly with the removal of oneor a few of the units.

A polymer can be synthesized from monomer molecules. Polymerization isthe process of converting monomer molecules into a polymer. The number ofmonomeric units in a polymer is called the degree of polymerization.

Fundamental Definitions

Polymers

A homopolymer is a polymer derived from one species of monomer. Acopolymer is derived from more than one species of monomer.

Copolymers can be further classified according to the number of monomerspecies used in copolymerization: bipolymers are copolymerized from twomonomer species, terpolymers from three monomer species, quaterpolymersfrom four monomer species, etc.

A special type of copolymer is the block copolymer. Especially important arediblock and triblock copolymers, because they have found many applications.

MonomerPolymer

Polymerization

From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995).

Polymerization

Polymerization+

Homopolymer

Copolymer

Polymers

A copolymer of unspecified type is named as poly(A-co-B). Others are namedas follows:

Note that an alternating copolymer poly(A-alt-B) may be considered as ahomopolymer polyAB derived from a hypothetical monomer AB.

In a statistical copolymer the sequential distribution of the monomeric units obeysknown statistical laws. A special case of statistical copolymer is the randomcopolymer, named poly(A-ran-B), in which the probability of finding a givenmonomeric unit at any given site in the chain is independent of the nature of theadjacent units.

From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995).

polyA

polyA-block-polyB

polyA-graft-polyB

poly(A-alt-B)

poly(A-stat-B)

Polymers

Common Polymers

I. W. Hamley, Introduction to Soft Matter (Wiley, 2000).

Polymers

I. W. Hamley, Introduction to Soft Matter (Wiley, 2000).

Polymers

To understand how polymers are named, a few more IUPAC definitions areneeded. A constitutional unit is an atom or group of atoms comprising a partof the essential structure of a polymer. A monomeric unit (or monomer unit)is the largest constitutional unit contributed by a single monomer molecule tothe structure of a polymer. A constitutional repeating unit is the smallestconstitutional unit the repetition of which constitutes a polymer.

Take poly(ethylene) as an example. Its constitutional repeating unit is –CH2–,while its constitutional unit can be any one of the following groups: –CH2–,–CH2CH2–, –CH2CH2CH2–, etc. Since poly(ethylene) is normally synthesizedfrom ethylene, H2C=CH2, the monomeric unit of poly(ethylene) is –CH2CH2–.

Polymers can be named as poly(constitutional repeating unit) or poly(monomerunit). The former is called structure-based and the latter source-based. Thestructure-based names are seldom used in practice.

Finally, note that a polymer can have more than one constitutional repeating unitand, therefore, more than one possible structural-based name. For example, theconstitutional repeating unit for poly(butadiene) can be either –CH=CHCH2CH2–or –CH2CH=CHCH2–. Ambiguities such as this have been resolved in IUPAC’sCompendium of Macromolecular Nomenclature.

Polymers

Configuration and ConformationThe ‘permanent’ stereostructure of a polymer is called its configuration. Theconfiguration of a polymer is permanent in the sense that it is defined whenthe polymer is synthesized and is preserved until the polymer reacts chemically.

A polymer’s configuration is thus defined by its molecular architecture. Majormolecular architecture types are linear, branched, ladder, star, and network:

From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995).

Polymers

However, molecular architecture alone does not completely define a polymer’sconfiguration. A polymer’s configuration is also determined by the way atomsare arranged about double bonds (if any) and chiral centers.

It is well known that about a double bond two arrangements, cis- and trans-, arepossible:

Thus, about each double bond, there will be two distinct configurations.

A chiral center is a carbon atom to which four different groups are attached.The four groups have two different orientations in space and they result inisomers (called stereoisomers) that are mirror images of each other, but arenot superimposable on each other:

From R. T. Morrison and R. N. Boyd, OrganicChemistry, 4th ed., Allyn & Bacon, Boston (1983).

From R. T. Morrison and R. N. Boyd, OrganicChemistry, 4th ed., Allyn & Bacon, Boston (1983).

chiralcarbons

Polymers

Thus, a poly(ethylene) molecule has only one configuration,

whereas a poly(propylene) molecule has isotatic configuration,

syndiotactic configuration,

and atactic configuration.

From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).

From G. Strobl, The Physics of Polymers, 2nd ed., Springer-Verlag, Berlin (1996).

From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).

From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).

Polymers

While configuration defines the ‘permanent’ stereostructure of a polymer,conformation refers to the ‘transient’ stereostructures generated by rotationsabout single bonds. These stereostructures are transient in the sense thatinterconversions among the rotational minima are rapidly executed because thebarrier heights of bond rotational potentials are usually only a few RT, quitesurmountable at room temperature.

From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).

Polymers

Because of these rapid interconversions, polymers are very flexible and can beregarded as a long, flexible piece of string:

P. J. Flory, Statistical Mechanics of Chain Molecules, John Wiley, New York (1969).

From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).

From H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, New York (1971).

Polymers

The simplest model for a flexible polymer is the random walk model. Sincethe model allows the polymer chain to cross itself, it defines is an unrealistic

polymer, i.e. an ideal chain.

Consider a random walk on a square lattice. Letb be the step size (bond length), N the number ofsteps, and rn the displacement vector of the nthstep (bond vector).

On a square lattice, rn can be b1, b2, b3, or b4 withequal probability. Because the walk is random,different steps are not correlated. Therefore,

A convenient way to define the size of a polymermolecule is the end-to-end vector:

The Ideal Chain

From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).

Polymers

Because R and –R occur with equal probability and cancel each other out, giving

the end-to-end vector itself is not a good measure of the polymer size. On theother hand, R2 is immune to this problem, so it has become a standard measureof the polymer size.

For the random walk considered here,

and size of the polymer is given by the end-to-end distance RF (the subscript Fstands for Paul J. Flory, a chemist who pioneered polymer physics)

Note that the size of the polymer is proportional to N1/2 and the above derivationalso holds for a three-dimensional random walk on a cubic lattice:

I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).

b3b4

b2

b1b6

b5

Polymers

We can proceed further and calculate the probability distribution function of Rfor a random walk on a cubic lattice. Let P(R, N) be the probability that anN-step walk results in an end-to-end vector R. From the site at the (N-1)th stepto the final site at the Nth step, there are six equally possible ways:

If the polymer is very long, N 1 and RF b, then we can expand P(R-bi, N-1):

It is easy to show that

Therefore,

and this gives

which is a partial differential equation for P(R, N).

Polymers

For a very long polymer, we expect that large-scale properties such as polymersize will not be affected by small-scale details like number of nearest neighbors.Indeed, it can be shown that for a very long polymer the specific structure of thelattice on which the polymer is modeled makes no difference at all. Therefore,the same partial differential equation holds for all kinds of lattices.

The initial condition for P(R, N) is simply

i.e. the walker remains at the starting point before taking the first step. It canbe verified that the solution to the partial differential equation subject to thisinitial condition is

Thus, the probability distribution of R is a Gaussian (normal) distribution.

Knowing the probability distributionenables us to find all kinds of averagessuch as the end-to-end distance:

which has the same form as before.

Polymers

Since Gaussian distributions are mathematically very amenable, it is convenientto assume that the bond vector rn itself follows a Gaussian distribution:

The ideal chain thus defined is called a Gaussian chain.

Because bond vectors are not correlated, the probability distribution for the entireGaussian chain is given by

Let the position vectors of the “beads” (latticesites) joined by the bond vectors r1, r2, ..., rN beR0, R1, ..., RN. Because rn = Rn-Rn-1, the probabilitydistribution for the Gaussian chain becomes

where . From this distribution function, it can be shownthat for any n and m

From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).

Polymers

The equilibrium state of the Gaussian chain must be described by a distributionfunction proportional to the Boltzmann factor

so if we write

then U can be regarded as the potential energy for a system of springs connectedin series:

Thus, the Gaussian chain model is often called the bead-spring model.

The spring constant for the entire chain is the equivalent spring constant for thesystem of springs in series:

This will be used to find the size of a non-ideal chain.

Polymers

The ideal chain model is obviously incorrect and the fact that a polymer chaincannot cross itself, which is a manifestation of the Pauli exclusion principle,must be taken into account. The resulting effect is usually called the excludedvolume effect and the polymer that cannot cross itself is called an excludedvolume chain.

In models defined on a lattice such as the random walk model, the excludedvolume effect is achieved by prohibiting the same lattice site being stepped onmore than once, thus defining a self-avoiding random walk.

In models defined in a continuous space such as the Gaussian chain model, theconvention is to use an excluded volume parameter to model the repulsiveinteraction between polymer segments.

Non-ideal Chains

From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).

Prohibited!

Polymers

The repulsive interaction comes into effect when two polymer segments collide,so it is proportional to the probability of two segments being at the same point.

Consider a polymer of N segments and size R. If we assumethat the segments are uniformly distributed in the volumeoccupied by the polymer, then the probability of finding asegment within the volume is proportional to N/Rd, whered is the dimension of space. So the repulsive energy at thepoint where the two segments collide is proportional to

where is the excluded volume parameter. Therefore, thetotal repulsive energy is

On the other hand, the elastic energy is assumed to be that of a Gaussian chain:

So the total energy is given by (omitting all numerical coefficients)

From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).

R

I. Teraoka, P

olymer Solutions, John W

iley & S

ons, New

York (2002).

Siz

e (n

m)

Molecular weight (g/mol)

Slope = 0.5936

Polymers

Minimizing the total energy gives the optimumsize, which is identified as the optimumend-to-end distance

The relation between size and molecular weight(or degree of polymerization) is often written as

where the exponent is called the Flory exponent. The exponent for Gaussianchains has been derived to be 1/2 whereas the minimum-energy argument here,devised by Flory himself, gives for theexcluded volume chain

that is, = 3/5 for a polymer in solution(d = 3) and = 3/4 for a polymer adsorbedon a substrate (d = 2).

The exponent = 3/5 agrees with experimentsvery well. In fact, the agreement is so wellthat for a long time it was thought to be exact.

I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).

Polymers

In general, polymers of different types are immiscible. When polymers of twoimmiscible types, A and B, are connected together to form block copolymers,the immiscibility will tend to separate A blocks and B blocks as far away aspossible. Meanwhile, however, the chemical bonds that join the A blocks toneighboring B blocks do not allow complete separation of connected blocks.

In a block copolymer melt, the balance between immiscibility and chemicalconnectedness results in A-rich and B-rich domains. The size of each domainis mainly determined by the length of the block that dominates the domain, sodomains are usually very small, in the order of 10 nm to 500 nm. Theappearance of these small domains in block copolymers is called a microscopicphase separation or microphase separation.

Block Copolymers

From A. Yu Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules, American Institute of Physics, New York (1994).

Polymers

From M. Kleman and O. D. Lavrentovich, Soft Matter Physics, Springer-Verlag, New York (2003).

OBDD = Ordered bicontinuous double diamond

ColloidsFundamental Definitions

The colloidal range (or colloid dimension), according to the IUPAC definition,is roughly between 1 nm and 1 m. A system is called a colloidal system orsimply a colloid if subdivisions or discontinuities in the system occur, at leastin one direction, in the colloidal range. Thus, the solution of gold particlesstudied by Faraday in 1857, porous solids, fibers, thin films, and foams all arecolloidal systems.

A colloidal dispersion is a system in which particles of colloidal size of anynature (e.g. solid, liquid or gas) are dispersed in a dispersion medium, acontinuous phase of a different composition. If the colloidal particles have theproperties of a bulk phase of the same composition, the term dispersed phase(or disperse phase) is used to refer to the particles.

A latex is a fluid colloidal system in which each colloidal particle contains anumber of polymers. The milky sap of many plants are latexes, in which thecolloidal particles are aggregates of biopolymers such as proteins and starches.(Because phase separation probably will occur in bulk aggregates of the samecomposition, for plant latexes the term “dispersed phase” should not be used.)

Colloids

Types of colloidal dispersion

Dispersed phase

Dispersion medium IUPAC Name Examples

Liquid Gas

Aerosol of liquid particles

Fog

Smoke

Hairspray, mist

Solid Gas

Aerosol of solid particles

Smoke

Gas Liquid Foam = froth Fire-extinguisher foam

Liquid Liquid Emulsion Milk, mayonnaise

Solid Liquid

Colloidal suspension Printing ink, paint, toothpaste

Gas Solid Foam = froth Insulating foam

Liquid Solid Ice cream

Solid Solid Opal, pearlA fluid (gas or liquid) colloidal system composed of two or more componentsmay be called a sol. Thus, aerosol, fog, smoke, foam, emulsion, and colloidalsuspension all are sols.

Colloids

Particle Size and Size DistributionCharacterization of particle size and the associated distribution is an importantissue in colloid science. If all the particles in a colloidal system are of (nearly)the same size, the system is called monodisperse; otherwise it is heterodisperse.A heterodisperse system is called paucidisperse if the particles have only a fewdifferent sizes and polydisperse if the particles have many different sizes.

A very important fact to bear in mind is that particle size and size distributionresults should be regarded as relative measurements, so extreme caution shouldbe exercised when comparing results from different instruments. This is becausedifferent instruments are based on different physical principles and even whenthe instruments are based on the same physical principle, they may use differentalgorithms, components, etc. that may cause great variation in the measurements.

Care also should be taken when reading particle size and size distribution databecause they can be presented in various forms and because some instrumentsreport size results as diameters, while some as surface area.

Numerous techniques have been devised for particle size analysis. A usefulguide to some of these techniques was recently issued by the National Instituteof Standards and Technology of the United States.

Colloids

From A. Jillavenkatesa et al., Particle Size Characterization, NIST Special Publication 960-1 (2001).

Colloids

Data for particle size and size distribution can be represented in tabular or graphicforms. Three graphic forms for size distribution in common use are histogram,differential distribution curve, and cumulative distribution curve.

From R. J. Hunter, Foundations of Colloid Science, Vol. 1, Oxford University Press, Oxford (1986).

Colloids

A histogram, ni(di), can be replaced by a differential distribution curve F(d)defined by F(di) di = number of particles in the range di to di+di = ni(di). Ifthe width di is a constant , thenF(di) = ni(di)/, which can be sketcheddirectly from the histogram.

From R. J. Hunter, Foundations of Colloid Science,Vol. 1, Oxford University Press, Oxford (1986).

Modal size

From R. J. Hunter, Foundations of Colloid Science,Vol. 1, Oxford University Press, Oxford (1986).

d10

d90

Mediansize, d50

Colloids

The differential particle size distribution curve F(d) is a sort of probabilitydistribution, because by definition

where N is the total number of particles and fi is the fraction of particles in therange (di, di+di), i.e. the probability of finding a particle of size in this range.

In probability theory the jth moment of a probability distribution f(d) is given by

Consider the second moment. It can be written as

where Ai is the surface area of a particle of diameter di. This suggests that

is an area-averaged diameter, called the area mean diameter. Similarly, thelength mean diameter and volume mean diameter are defined as

respectively.

Colloids

The standard deviation of the size distribution as usual is defined by

It is a measure of the spread of the distribution and is expected to vanish if allthe particles have the same size. Another way to measure the spread is the ratioof the area mean and length mean diameters:

where PDI = polydispersity index. Note that PDI ≥ 1.

Colloids

Forces between Colloidal ParticlesAmong the possible forces between colloidal particles, the most important iselectrostatic forces, followed by the van der Waals forces, and the inertial forcesare the weakest. Forces due to thermal agitation (Brownian forces) and viscosityare equally important, whereas the ubiquitous gravity that dominates macroscopicscales only plays a minor role on the microscopic colloidal scale.

From W. B. Russel et al., Colloidal Dispersions, Cambridge University Press, Cambridge (1989).

References 1. U. W. Gedde, Polymer Physics (Chapman & Hall, 1995). 2. W. V. Metanomski ed., Compendium of Macromolecular Nomenclature (Blackwell

Science, 1991). 3. A. D. Jenkins et al., Pure Appl. Chem. 68, 2287 (1996). 4. I. W. Hamley, Introduction to Soft Matter (Wiley, 2000). 5. P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press,

1979). 6. M. Doi, Introduction to Polymer Physics (Oxford University Press, 1995). 7. D. H. Everett and L. K. Koopal, Definitions, Terminology and Symbols in Colloid

and Surface Chemistry (Division of Physical Chemistry, International Union of Pure and Applied Chemistry, 2001).

8. R. J. Hunter, Foundations of Colloid Science (Oxford University Press, 1986) 2 Vols. 9. A. Jillavenkatesa et al., Particle Size Characterization, NIST Special Publication

960-1 (2001).