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## Lectures on Modern Physics

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**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 ora 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.**Polymerization**Polymerization Polymerization From U. W. Gedde, Polymer Physics, Chapman & Hall, London (1995). Monomer Polymer Homopolymer + Copolymer Polymers A homopolymer is a polymer derived from one species of monomer. A copolymer is derived from more than one species of monomer. 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.**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). Polymers A copolymer of unspecified type is named as poly(A-co-B). Others are named as follows: 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.**From U. W. Gedde, Polymer Physics, Chapman & Hall, London**(1995). 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 R. T. Morrison and R. N. Boyd, Organic**Chemistry, 4th ed., Allyn & Bacon, Boston (1983). chiral carbons From R. T. Morrison and R. N. Boyd, Organic Chemistry, 4th ed., Allyn & Bacon, Boston (1983). 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: 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:**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). 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, 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). 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.**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). Polymers 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). From H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, New York (1971). 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). 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:**b5**b2 b3 b4 b1 b6 I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002). 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 distanceRF (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:**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 RFb, 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.**From M. Doi, Introduction to Polymer Physics, Oxford**University Press, New York (1996). 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 where . From this distribution function, it can be shown that for any n and m**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.**Prohibited!**From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996). I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002). 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. 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.**R**From M. Doi, Introduction to Polymer Physics, Oxford University Press, New York (1996). 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 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 where is the excluded volume parameter. Therefore, the total 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)**I. Teraoka, Polymer Solutions, John Wiley & Sons, New York**(2002). Slope n = 0.5936 Size (nm) I. Teraoka, Polymer Solutions, John Wiley & Sons, New York (2002). Molecular weight (g/mol) 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 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 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). 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.**From A. Yu Grosberg and A. R. Khokhlov, Statistical Physics**of Macromolecules, American Institute of Physics, New York (1994). 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. 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.**From M. Kleman and O. D. Lavrentovich, Soft Matter Physics,**Springer-Verlag, New York (2003). Polymers 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**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.**From A. Jillavenkatesa et al., Particle Size**Characterization, NIST Special Publication 960-1 (2001). Colloids**From R. J. Hunter, Foundations of Colloid Science, Vol. 1,**Oxford University Press, Oxford (1986). 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.**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 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.**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.**From W. B. Russel et al., Colloidal Dispersions, Cambridge**University Press, Cambridge (1989). 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.**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).