Transcript 投影片 1 - NCHU

Lectures on Modern Physics
Jiunn-Ren Roan
21 Dec. 2007
Soft Matter
What Is Soft Matter?
Polymers
Fundamental Definitions
Common Polymers
Configuration and Conformation
The Ideal Chain
Non-ideal Chains
Block Copolymers
Colloids
Fundamental Definitions
Particle Size and Size Distribution
Forces between Colloidal Particles
Soft Matter
References
What Is Soft Matter?
Soft matter, according to Pierre-Gilles de Gennes, the winner of the Nobel Prize
in physics in 1991, includes polymers, colloids, surfactants (amphiphiles), and
liquid crystals. These materials are “soft” because their mechanical responses
are 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 very
long time physicists showed little interest in soft matter. Most studies were
carried out by chemists, chemical engineers, materials scientists, or even food
scientists. De Gennes switched from “hard” matter to soft matter in mid 1960s.
It takes, however, about 30 more years before physicists’ interest in soft matter
began to surge.
Because of soft matter’s interdisciplinary nature, it will be very helpful to know
a little more about relevant chemistry, especially molecular structure and
physical chemistry.
Polymers
Fundamental Definitions
A polymer molecule or a macromolecule, according to the IUPAC (International
Union of Pure and Applied Chemistry) definition, is
A molecule of high relative molecular mass, the structure of which
essentially comprises the multiple repetition of units derived, actually
or 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 of
which essentially comprises a small plurality of units derived, actually
or conceptually, from molecules of lower relative molecular mass.
Thus, whether a multi-unit molecule is an oligomer or a polymer depends on
its molecular mass – it is regarded as having an intermediate relative molecular
mass if it has properties which do vary significantly with the removal of one
or a few of the units.
A polymer can be synthesized from monomer molecules. Polymerization is
the process of converting monomer molecules into a polymer. The number of
monomeric units in a polymer is called the degree of polymerization.
Polymers
Polymerization
From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995).
Monomer
Polymer
A homopolymer is a polymer derived from one species of monomer. A
copolymer is derived from more than one species of monomer.
Polymerization
Homopolymer
+
Polymerization
Copolymer
Copolymers can be further classified according to the number of monomer
species used in copolymerization: bipolymers are copolymerized from two
monomer species, terpolymers from three monomer species, quaterpolymers
from four monomer species, etc.
A special type of copolymer is the block copolymer. Especially important are
diblock and triblock copolymers, because they have found many applications.
Polymers
A copolymer of unspecified type is named as poly(A-co-B). Others are named
as follows:
polyA
polyA-block-polyB
polyA-graft-polyB
poly(A-alt-B)
poly(A-stat-B)
From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995).
Note that an alternating copolymer poly(A-alt-B) may be considered as a
homopolymer polyAB derived from a hypothetical monomer AB.
In a statistical copolymer the sequential distribution of the monomeric units obeys
known statistical laws. A special case of statistical copolymer is the random
copolymer, named poly(A-ran-B), in which the probability of finding a given
monomeric unit at any given site in the chain is independent of the nature of the
adjacent units.
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 are
needed. A constitutional unit is an atom or group of atoms comprising a part
of the essential structure of a polymer. A monomeric unit (or monomer unit)
is the largest constitutional unit contributed by a single monomer molecule to
the structure of a polymer. A constitutional repeating unit is the smallest
constitutional 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 synthesized
from ethylene, H2C=CH2, the monomeric unit of poly(ethylene) is –CH2CH2–.
Polymers can be named as poly(constitutional repeating unit) or poly(monomer
unit). The former is called structure-based and the latter source-based. The
structure-based names are seldom used in practice.
Finally, note that a polymer can have more than one constitutional repeating unit
and, therefore, more than one possible structural-based name. For example, the
constitutional repeating unit for poly(butadiene) can be either –CH=CHCH2CH2–
or –CH2CH=CHCH2–. Ambiguities such as this have been resolved in IUPAC’s
Compendium of Macromolecular Nomenclature.
Polymers
Configuration and Conformation
The ‘permanent’ stereostructure of a polymer is called its configuration. The
configuration of a polymer is permanent in the sense that it is defined when
the polymer is synthesized and is preserved until the polymer reacts chemically.
A polymer’s configuration is thus defined by its molecular architecture. Major
molecular 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’s
configuration. A polymer’s configuration is also determined by the way atoms
are arranged about double bonds (if any) and chiral centers.
It is well known that about a double bond two arrangements, cis- and trans-, are
possible:
From R. T. Morrison and R. N. Boyd, Organic
Chemistry, 4th ed., Allyn & Bacon, Boston (1983).
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 in
isomers (called stereoisomers) that are mirror images of each other, but are
not superimposable on each other:
chiral
carbons
From R. T. Morrison and R. N. Boyd, Organic
Chemistry, 4th ed., Allyn & Bacon, Boston (1983).
Polymers
Thus, a poly(ethylene) molecule has only one configuration,
From G. Strobl, The Physics of Polymers, 2nd ed., Springer-Verlag, Berlin (1996).
whereas a poly(propylene) molecule has isotatic configuration,
From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).
syndiotactic configuration,
From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).
and atactic configuration.
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 rotations
about single bonds. These stereostructures are transient in the sense that
interconversions among the rotational minima are rapidly executed because the
barrier heights of bond rotational potentials are usually only a few RT, quite
surmountable at room temperature.
From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983).
Polymers
P. J. Flory, Statistical Mechanics of Chain Molecules, John Wiley, New York (1969).
Because of these rapid interconversions, polymers are very flexible and can be
regarded as a long, flexible piece of string:
From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).
Polymers
The Ideal Chain
The simplest model for a flexible polymer is the random walk model. Since
the 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. Let
b be the step size (bond length), N the number of
steps, and rn the displacement vector of the nth
step (bond vector).
From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).
On a square lattice, rn can be b1, b2, b3, or b4 with
equal probability. Because the walk is random,
different steps are not correlated. Therefore,
A convenient way to define the size of a polymer
molecule is the end-to-end vector:
From H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, New York (1971).
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 the
other hand, R2 is immune to this problem, so it has become a standard measure
of 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 F
stands 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 derivation
also holds for a three-dimensional random walk on a cubic lattice:
b5
b2
b3
b4
b1
I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).
b6
Polymers
We can proceed further and calculate the probability distribution function of R
for a random walk on a cubic lattice. Let P(R, N) be the probability that an
N-step walk results in an end-to-end vector R. From the site at the (N-1)th step
to 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 polymer
size 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 the
lattice 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 can
be verified that the solution to the partial differential equation subject to this
initial condition is
Thus, the probability distribution of R is a Gaussian (normal) distribution.
Knowing the probability distribution
enables us to find all kinds of averages
such as the end-to-end distance:
which has the same form as before.
Polymers
Since Gaussian distributions are mathematically very amenable, it is convenient
to 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 entire
Gaussian chain is given by
Let the position vectors of the “beads” (lattice
sites) joined by the bond vectors r1, r2, ..., rN be
R0, R1, ..., RN. Because rn = Rn-Rn-1, the probability
distribution for the Gaussian chain becomes
From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).
where
that for any n and m
. From this distribution function, it can be shown
Polymers
The equilibrium state of the Gaussian chain must be described by a distribution
function proportional to the Boltzmann factor
so if we write
then U can be regarded as the potential energy for a system of springs connected
in 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 the
system of springs in series:
This will be used to find the size of a non-ideal chain.
Polymers
Non-ideal Chains
The ideal chain model is obviously incorrect and the fact that a polymer chain
cannot cross itself, which is a manifestation of the Pauli exclusion principle,
must be taken into account. The resulting effect is usually called the excluded
volume effect and the polymer that cannot cross itself is called an excluded
volume chain.
Prohibited!
I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).
From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).
In models defined on a lattice such as the random walk model, the excluded
volume effect is achieved by prohibiting the same lattice site being stepped on
more than once, thus defining a self-avoiding random walk.
In models defined in a continuous space such as the Gaussian chain model, the
convention is to use an excluded volume parameter to model the repulsive
interaction between polymer segments.
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 assume
that the segments are uniformly distributed in the volume
occupied by the polymer, then the probability of finding a
R
segment within the volume is proportional to N/Rd, where
d is the dimension of space. So the repulsive energy at the
point where the two segments collide is proportional to
From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996).
where
total repulsive energy is
is the excluded volume parameter. Therefore, the
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)
Polymers
Minimizing the total energy gives the optimum
size, which is identified as the optimum
end-to-end distance
The relation between size and molecular weight
(or degree of polymerization) is often written as
I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).
The exponent n = 3/5 agrees with experiments
very well. In fact, the agreement is so well
that for a long time it was thought to be exact.
Size (nm)
that is, n = 3/5 for a polymer in solution
(d = 3) and n = 3/4 for a polymer adsorbed
on a substrate (d = 2).
Molecular weight (g/mol)
I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002).
where the exponent n is called the Flory exponent. The exponent for Gaussian
chains has been derived to be 1/2 whereas the minimum-energy argument here,
devised by Flory himself, gives for the
excluded volume chain
Polymers
Block Copolymers
In general, polymers of different types are immiscible. When polymers of two
immiscible 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 as
possible. Meanwhile, however, the chemical bonds that join the A blocks to
neighboring B blocks do not allow complete separation of connected blocks.
From A. Yu Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules, American Institute of Physics, New York (1994).
In a block copolymer melt, the balance between immiscibility and chemical
connectedness results in A-rich and B-rich domains. The size of each domain
is mainly determined by the length of the block that dominates the domain, so
domains are usually very small, in the order of 10 nm to 500 nm. The
appearance of these small domains in block copolymers is called a microscopic
phase separation or microphase separation.
Polymers
From M. Kleman and O. D. Lavrentovich, Soft Matter Physics, Springer-Verlag, New York (2003).
OBDD = Ordered bicontinuous double diamond
Colloids
Fundamental Definitions
The colloidal range (or colloid dimension), according to the IUPAC definition,
is roughly between 1 nm and 1 mm. A system is called a colloidal system or
simply a colloid if subdivisions or discontinuities in the system occur, at least
in one direction, in the colloidal range. Thus, the solution of gold particles
studied by Faraday in 1857, porous solids, fibers, thin films, and foams all are
colloidal systems.
A colloidal dispersion is a system in which particles of colloidal size of any
nature (e.g. solid, liquid or gas) are dispersed in a dispersion medium, a
continuous phase of a different composition. If the colloidal particles have the
properties 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 a
number of polymers. The milky sap of many plants are latexes, in which the
colloidal particles are aggregates of biopolymers such as proteins and starches.
(Because phase separation probably will occur in bulk aggregates of the same
composition, for plant latexes the term “dispersed phase” should not be used.)
Colloids
Types of colloidal dispersion
Dispersed Dispersion
phase
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, pearl
A fluid (gas or liquid) colloidal system composed of two or more components
may be called a sol. Thus, aerosol, fog, smoke, foam, emulsion, and colloidal
suspension all are sols.
Colloids
Particle Size and Size Distribution
Characterization of particle size and the associated distribution is an important
issue 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 few
different sizes and polydisperse if the particles have many different sizes.
A very important fact to bear in mind is that particle size and size distribution
results should be regarded as relative measurements, so extreme caution should
be exercised when comparing results from different instruments. This is because
different instruments are based on different physical principles and even when
the instruments are based on the same physical principle, they may use different
algorithms, components, etc. that may cause great variation in the measurements.
Care also should be taken when reading particle size and size distribution data
because they can be presented in various forms and because some instruments
report size results as diameters, while some as surface area.
Numerous techniques have been devised for particle size analysis. A useful
guide to some of these techniques was recently issued by the National Institute
of 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 graphic
forms. 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) ddi = number of particles in the range di to di+ddi = ni(di). If
the width ddi is a constant D, then
F(di) = ni(di)/D, which can be sketched
directly from the histogram.
Modal size
d90
Median
size, d50
From R. J. Hunter, Foundations of Colloid Science,
Vol. 1, Oxford University Press, Oxford (1986).
d10
From R. J. Hunter, Foundations of Colloid Science,
Vol. 1, Oxford University Press, Oxford (1986).
Colloids
The differential particle size distribution curve F(d) is a sort of probability
distribution, because by definition
where N is the total number of particles and fi is the fraction of particles in the
range (di, di+ddi), 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, the
length mean diameter and volume mean diameter are defined as
respectively.
Colloids
The standard deviation s 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 all
the particles have the same size. Another way to measure the spread is the ratio
of the area mean and length mean diameters:
where PDI = polydispersity index. Note that PDI ≥ 1.
Colloids
Forces between Colloidal Particles
Among the possible forces between colloidal particles, the most important is
electrostatic forces, followed by the van der Waals forces, and the inertial forces
are the weakest. Forces due to thermal agitation (Brownian forces) and viscosity
are equally important, whereas the ubiquitous gravity that dominates macroscopic
scales 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).