# Polymer solutions

## Solvent effects

The equation of state of a gas is commonly expressed as a virial equation:

$\frac{pV}{RT}=1+B\frac{1}{V}+O\left( \frac{1}{V} \right)^{2}\ldots$

The virial coefficients depend on the interactions between the gas molecule.

Similarly, the equation of state for the osmotic pressure of a solution

is commonly expressed as a virial equation: $\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots$

These virial coefficents depend on the interactions between the solute molecules, in our case, polymer molecules.

Polymer overlaps also create “osmotic pressures”.

This osmotic pressure changes the polymer configuration, hence its size in solution.

## Polymer self-interaction and solvent quality

The virial coefficients are a function of polymer pair potentials in solution:

$b_{2}\equiv \frac{1}{2}\int{\left( 1-\exp \left( -\frac{U\left( r \right)}{kT} \right) \right)d^{3}r}$
 When the integral is positive, a good solvent, then the Flory scaling applies. If the integral is zero, the system is said to be at a theta condition. If this integral is negative, i.e. net attractive interactions, higher order terms become important.

The second virial coefficient is difficult to calculate for polymers. In good solvents, the coefficient is essentially the excluded volume. If the size of the polymer is taken as the radius of gyration (from scattering):

$\frac{b_{2}}{R_{g}^{3}}=4.75\pm 0.5$

## Solvent-polymer effects - The Flory 'Chi' parameter

The first model of a self-avoiding polymer is a balance of excluded volume driving the polymer apart with a decrease in entropy with expansion. This led to the Flory scaling:

$r\sim aN^{{3}/{5}\;}$

A second model is to balance excluded volume driving the polymer apart with a measure of the polymer-solvent interactions.

$kT\chi =\frac{1}{2}z\left( 2\varepsilon _{ps}-\varepsilon _{pp}-\varepsilon _{ss} \right)$
$\chi$ is the Flory chi parameter.
 The excluded volume energy is: $F_{rep}=kTv\frac{N^{2}}{2r^{3}}$ The $\chi$ term is: $F_{\text{interaction}}=-2kT\chi v\frac{N^{2}}{2r^{3}}+\text{constant}$ The total is: $F_{rep}+F_{\text{interaction}}=kTv\left( 1-2\chi \right)\frac{N^{2}}{2r^{3}}+\text{constant}$
 $\chi <{1}/{2}\;$ the polymer chain is expanded with a radius $r\sim N^{{3}/{5}\;}$ $\chi ={1}/{2}\;$ the repulsion and attraction exactly cancel; the polymer coil is an ideal random walk, with a radius $r\sim N^{{1}/{2}\;}$. This is known as the theta condition. $\chi >{1}/{2}\;$ the polymer chain collapses to form a compact globule.

## Electrostatic interactions - Bjerrum length and Debye length

Polymer chains, with charge moieties along its length of the same sign, expand. A reasonable question is how strong is the repulsion? The first answer is to consider the scale, that is, length of the electrostatic repulsion.

For Coulombic repulsion the characteristic length is the "Bjerrum length": This is a useful length scale for determining when electrostatic interactions are on the same order as thermal energy.

$\frac{\left( 1.602\times 10^{-19}\text{ }C \right)^{2}}{4\pi \cdot 80\cdot 8.854\times 10^{-12}C^{2}N^{-1}m^{-2}\cdot 1.381\times 10^{-23}JK^{^{-1}}\cdot 300K}=0.7nm\text{ in water at 25C}$

As a first guess we would estimate that the nearest neighbor interactions are sufficient - this is not the case.

When we consider the interaction between larger objects, such as plates and spheres, the characteristic length is the Debye length of electrical double layers:

\begin{align}  & \kappa ^{-1}=\left( 8\pi lc \right)^{-{1}/{2}\;} \\ & c(1mM)=10^{-3}\cdot 6.023\times 10^{23}\frac{ions}{10^{-3}m^{3}}=6.023\times 10^{-4}\frac{ions}{nm^{3}} \\ & \kappa ^{-1}\simeq 10nm\text{ for water with 1mM 1-1 electrolyte} \\  \end{align}

## Repulsion between charges down a chain - 1st model

Suppose all the monomers be charged; estimate the repulsion of one half of the polymer for the other:

 The Coulomb repulsion between sides is: $E\sim \frac{\left( {en}/{2}\; \right)^{2}}{4\pi \varepsilon R}\sim kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}$ If the chain were a random walk: $R\sim n^{{1}/{2}\;}$ This stretch is unphysical. The energy would increase with the 3/2ths power of the length.

Suppose the electrostatic repulsion was balanced by the increase in the elastic energy:

$U_{elastic}\sim kT\frac{R^{2}}{a^{2}n}$

This gives: $kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}\sim kT\frac{R^{2}}{a^{2}n}\text{ or }R\sim n$

That is, a fully extended chain, a rigid rod.

## Repulsion from charges down a chain - 2nd model

Polymer electrolyte chain; like charges evenly spaced between random coils:

Consider the electrostatic force on the center of the chain:

 From position “1” towards the right: $F_{1-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}}}$ From position “2” towards the right: $F_{2-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}}}$ From position “3” towards the right: $F_{3-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}}}$ For all positions: $F_{total}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{j}{j^{2}}}\simeq kTl\left( \frac{n}{R} \right)^{2}\ln n$ The elastic force is work/extension: $F_{elastic}=3kT\frac{R}{a^{2}n}$ The balance is: $kTl\left( \frac{n}{R} \right)^{2}\ln n\simeq \frac{3kT}{a^{2}}\frac{R}{n}\text{ }$ The scaling is: $\frac{R}{n}\simeq a\left( {l\ln n}/{a}\; \right)^{{1}/{3}\;}$ Only sparse charges possible! $\frac{R}{n}\ll a$

## Overlap concentration - c*

$\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots$

The osmotic pressure is nearly idea until the second order term becomes equal to one.

$c*=\frac{1}{b_{2}}$

c* is called the overlap concentration. Even at this concentration the solution is almost all solvent.

c* is also evident in the viscosity as a function of concentration:

Just above c*, called the semi-dilute regime, the polymers in solution overlap but only interact tenuously,

## Semi-dilute regime - The interaction of blobs

Above c* the polymer chains are in contact - albeit with little interaction. As the concentration is increased, the polymers form into chains of random coils, called 'blobs'.

The concentration of polymer in each blob is the same as the overall concentration of polymer in the container - since the blobs fill space.

The blobs from one chain distribute randomly so that the blobs on a chain are as likely to interact with blobs for other chains as with blobs on the same chain.

The simple model is:

• The interaction of blobs on the same chain are unimportant.
• Blobs interact weakly with each other.
• The dimension of a blob is: $\xi _{\varphi }^{{}}$. The size depends on concentration.
• Within a blob, the chain is a self-avoiding chain, D = 5/3
• The blobs on a chain have an unperturbed random walk, D = 2

Therefore the blobs for an ideal solution and the osmotic pressure is:

$\Pi \simeq kT\xi _{\varphi }^{-3}$

## Correlation lengths and blobs

Let $c_{\max }$ be the concentration (density) of polymer when all solvent is gone.

The overlap volume fraction is: $\varphi *\equiv \frac{c*}{c_{\max }}$. Which is typically of order 1%.

The idea is that about c* the polymers start to crowd at their outer edges. The short range order is preserved, but not the long range order. The polymer starts to form blobs.

The (average) local volume is $\left\langle \varphi \left( r \right) \right\rangle _{0}\sim \left( {A}/{r}\; \right)^{d-D}\text{ }r\ll R$

A depends on polymer and solvent.

The local density is highest at r = 0 and decreases from that.

At some r a distance called the correlation length, $\xi _{\varphi }$ :

$\left\langle \varphi \left( \xi _{\varphi } \right) \right\rangle _{0}=\varphi$
$\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}$

Less than $\xi _{\varphi }$ , the polymer is little affected by other chains.

## Osmotic effects and volume fraction

 The solution osmotic pressure is: $\Pi \simeq kT\xi _{\varphi }^{-3}$ The correlation length is: $\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}$ Since D = 5/3 $\varphi =\left( \frac{A}{\xi _{\varphi }} \right)^{d-D}=\left( \frac{A}{\xi _{\varphi }} \right)^{3-{5}/{3}\;}=\left( \frac{A}{\xi _{\varphi }} \right)^{-{4}/{3}\;}$ Or: $\Pi =kTA^{-3}\varphi ^{{9}/{4}\;}$ Light scattering data gives: $\Pi =kT\left( 3.2A \right)^{-3}\varphi ^{{9}/{4}\;}$

Witten Fig. 4.3

## Polymers at surfaces

Polymer chains bound to surfaces can alter interactions between surfaces. For example, they can stabilize colloidal dispersions.

Jones, Fig. 5.4

This image shows two ways that polymers may bind to surfaces. In figure a, the polymers are adsorbed onto the surface, meaning that they are connected to the surface at several points along the polymer's surface. The b figure show polymers that physically or chemically attached to the end. Polymers attached this way form a polymer brush. Often such brushes contain relatively high densities of polymer chains. Interactions between the separate chains cause each chain to extend.

This link contains an interesting simulation of the movement of individual chains in a polymer brush. Note the degree to which individual chains are extended. http://www.lassp.cornell.edu/marko/thinlayer.html

One of the current research goals for polymer brushes is creating surfaces that have switchable properties. [Polymer Brushes: Synthesis, Characterization, Applications Rigoberto C. Advincula (Editor)]

The height of a dense polymer brush may easily be calculated theoretically. Assume a polymer of degree of polymerization, N; link size a; excluded monomer volume, v, and surface density (chains per unit area):* $\frac{\sigma }{a^{2}}$. $\sigma$ is the fraction of surface covered by anchor groups.

 The volume per chain is: $\frac{ha^{2}}{\sigma }$ The stretching energy per chain is: $F_{elastic}=kT\frac{h^{2}}{a^{2}N}$ Combined excluded volume and interaction energy is: $F_{repulsive}+U_{int}=kTv\left( 1-2\chi \right)\frac{\sigma N^{2}}{2ha^{2}}$ The brush height (minimizing energy) is: $h\sim \left[ \sigma v\left( 1-2\chi \right) \right]^{{1}/{3}\;}N$

For high enough densities, the chains in the polymer brush will be strongly stretched with the length going as N instead of $N^{3/2}$.

## Hydrodynamic screening

Assume a flow of liquid through a bed of stationary spheres.

 The viscous-stress, top to bottom, per unit volume is: $\eta _{s}\frac{dv(z+b)-dv\left( z) \right)}{bdz}$ And in the limit of small b, the stress is: $\eta _{s}\frac{d^{2}v(z)}{dz^{2}}$ In a small volume, $b^{3}$, the drag is: $-\frac{kTb^{3}\rho _{2}}{6\pi \eta r}v\left( z \right)$ At steady state the two are equal: $\eta _{s}\frac{d^{2}v}{dz^{2}}=\frac{kT\rho _{2}}{6\pi \eta r}v$ The solution is: $v\left( z \right)=v\left( 0 \right)\exp \left( -\frac{z}{\xi _{h}} \right)$ the screening length is: $\xi _{h}=\sqrt{\frac{2r}{9\Phi }}$

## Diffusion in the semidilute regime

 Diffusion arises from a density gradient: $\frac{\partial \varphi }{\partial t}=\zeta _{c}\nabla ^{2}\varphi$ For independent particles we have the Einstein equation:* $\zeta _{sphere}=\frac{kT}{6\pi \eta _{s}R}$ A dilute solution comprises weakly interacting blobs with radius: $\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}$ For d = 3; D = 5/3 $\xi _{\varphi }\simeq \varphi ^{{-3}/{4}\;}$ And, if we assume the flow inside a “blob” is hydrodynamically screened: $\zeta _{c}\simeq \frac{kT}{\eta _{s}}\varphi ^{{3}/{4}\;}$
n.b. No dependence on MW!
• For a beautiful derivation see Witten, p. 90f: The Brownian motion of a sphere.

## A comparison of diffusion processes

 Brownian motion: Objects move because of thermal fluctuations. The motions are random over time. Long-time behavior only depends on the mean-squared displacement for individual steps. $\zeta =\frac{kT}{6\pi \eta r}$ Viscous flow: The diffusion of momentum perpendicular to the flow. $\frac{\partial v_{x}}{\partial t}=\frac{\eta _{s}}{\rho _{s}}\frac{\partial ^{2}v_{x}}{\partial z^{2}}$ Random-walk polymers: “Similar” to Brownian motion and Viscous flow. $\frac{dp}{dn}=\frac{1}{6}\left\langle r^{2} \right\rangle _{1}\nabla _{{}}^{2}p\left( n,r \right)$

(*Section 4.3.1 in Witten is a terrific, detailed description of Brownian motion. We need to work it in sometime. Next semester?)

## Three Types of diffusion in d dimensions

Quantity Random-walk polymer Diffusing particles Diffusing momentum
Dependent variable Probability: Particle density: Momentum density:
$p\left( n,r \right)$ $\rho \left( r \right)$ $\rho _{s}\vec{\nu }$
Independent variable Monomer number, n Time, t Time, t
Material constant Diffusion constant: Kinematic viscosity:
$\frac{\left\langle r^{2} \right\rangle _{1}}{6}$ $\zeta$ $\frac{\eta }{\rho }$
Equation $\frac{\partial p}{\partial n}=\frac{\left\langle r^{2} \right\rangle _{1}}{6}\nabla ^{2}p$ $\frac{\partial \rho }{\partial t}=\zeta \nabla ^{2}\rho$ $\frac{\partial v_{z}}{\partial t}=\frac{\eta }{\rho }\nabla ^{2}v_{z}$
Mean distance from point source $\left\langle r^{2} \right\rangle _{1}n$ $6\zeta t$ $6\frac{\eta }{\rho }t$

(Witten, Table 4.1)

## How Polymers Behave in Dilute Solutions

Until now we have seen the factors by which the solubility of macromolecules is affected, from both physical, chemical and thermodynamic points of view. Now what happens to these macromolecules when they are dissolved? Due to their large number of carbon atoms bonded together forming a long chain, polymers can generally adopt a lot of conformations. These conformations arise from the numerous internal rotations that can occur through simple C-C bonds, originating a number of rotational isomers. Nevertheless, although the rotation of each bond is able to originate different conformations, due to energy restrictions not all of them have the same probability of occurrence. In such a case, the most stable conformations predominate in solution, like proteins and nucleic acids, that is in biopolymers mainly. However, synthetic polymers particularly, can display a large number of possible conformations, and even though these conformations have not the same energy, the differences are small enough so that the chains can change from one conformation to another. This particularity gives a big flexibility to the macromolecules, and due to this flexibility, the chains do not adopt a linear form in solution, but a very characteristic conformation, known as random coil.

The left image is the Random Coil Model. The right image depicts a C-C simple-bonded chain and its spacial representation

Let's assume C1, C2, and C3 are carbon atoms in the same plane. According to this, the atom C4 can occupy any place throughout the circle, which represents the base of a cone originated by the rotation of the bond E3. The angles of such bonds are symbolized by ω, whereas the location of atom C4 is specified by the internal angle of rotation λ.

For a macromolecule in the solid state, the angle λ has a fixed value due to the restrictions of the network packing. That is why the possible rotational isomers do not occur. Nevertheless when this macromolecule is dissolved, the packing disappears and angle λ can vary widely, originating maximums and minimums of energy. Thus, the probability of reaching diverse stable conformations with each minimum of energy is high. On the other hand, the variation of the internal angle of rotation is associated to an energy change that, at minimums, is small. Hence, the chains can move freely to adopt such stable conformations. The fact that the chains are changing from one conformation to another is also favored, due to the low potential energy of the system. All these factors define, therefore, a flexible macromolecule and from these concepts, the typical random coil form arises.

You might ask if the "shape" or magnitude of the random coil would remain the same once the polymer has been dissolved. You will find that the answer is absolutely negative and that the situation will depend not only on the kind of solvent employed, but also on the temperature, and the molecular weight. The polymer-solvent interactions play an important role in this case, and its magnitude, from a thermodynamic point of view, will be given by the solvent quality. Thus, in a "good" solvent, that is to say that one whose solubility parameter is similar to that of the polymer, the attraction forces between chain segments are smaller than the polymer-solvent interactions; the random coil adopts then, an unfolded conformation. In a "poor" solvent, the polymer-solvent interactions are not favored, and therefore attraction forces between chains predominate, hence the random coil adopts a tight and contracted conformation.

In extremely "poor" solvents, polymer-solvent interactions are eliminated thoroughly, and the random coil remains so contracted that eventually precipitates. We say in this case, that the macromolecule is in the presence of a "non-solvent".

The particular behavior that a polymer displays in different solvents, allows the employ of a useful purification method, known as fractional precipitation. For a better understanding about how this process takes place, let's imagine a polymer dissolved in a "good" solvent. If a non-solvent is added to this solution, the attractive forces between polymer segments will become higher than the polymer-solvent interactions. At some point, before precipitation, an equilibrium will be reached, in which ΔG = 0, and therefore ΔH = TΔS, where ΔS reaches its minimum value. This point, where polymer-solvent and polymer-polymer interactions are of the same magnitude, is known as θ state and depends on: the temperature, the polymer-solvent system (where ΔH is mainly affected) and the molecular weight of the polymer (where ΔS is mainly affected).

It may be inferred then, that lowering the temperature or the solvent quality, the separation of the polymer in decreasing molecular weight fractions is obtained. Any polymer can reach its θ state, either choosing the appropriate solvent (named θ solvent) at constant temperature or adjusting the temperature (named θ temperature, or Flory temperature) in a given solvent.

The θ temperature is a parameter arisen from Flory-Krigbaum theory. It is used to calculate the free energy of mixing of a polymer solution in terms of the chemical potentials of the species. We will further study the θ temperature relationship with other important parameters that characterize dissolved polymers.

### Viscocity and Power Dissipation in the Sphere Model

So far we have analyzed the influence of the solvent and the temperature in the dimensions of the random coil. However is equally important to know what happens to the viscosity of the macromolecular solution as the solvent becomes poorer. Considering the chain molecules as rigid spheres, when a change from a "good" solvent to a "poor" solvent takes place, the spheres become contracted.

We may calculate the effect on viscosity η and power dissipation $\dot{\omega}$ as follows. Say that the pure solvent has viscosity ηs, so that with a shear rate $\dot{\gamma}^2_0$ we have $\dot{\omega}=\eta_s \dot{\gamma}^2_0$. Now add a single sphere of radius R into the solvent. There will be some velocity field around the sphere from the flowing solvent, giving rise to a position-dependent dissipation $\dot{\omega}(r)$. If we integrate over this dissipation for a liquid of volume $\Omega$, we should get the total dissipation:

$\Omega\langle\dot{\omega}\rangle = \int \dot{\omega}(r)\;d^3r = \dot{\omega}_0+\int (\dot{\omega}(r)-\dot{\omega}_0)\;d^3r = \dot{\omega}_0\left[\Omega+\int \left(\frac{\dot{\omega}(r)}{\dot{\omega}_0}-1\right)\;d^3 r\right] = \dot{\omega}_0\left(\Omega+\frac{5}{2}V\right)$

Where V is the volume of the sphere, $V=4\pi R^3/3$. Since power dissipation is proportional to viscosity, we can conclude

$\frac{\eta}{\eta_s} = 1+\frac{5}{2}\frac{V}{\Omega} = 1+\frac{10\pi}{3}\frac{R^3}{\Omega}$

If the volume contains N of these spheres, then then dissipation (and therefore viscosity) is the sum of each sphere:

$\frac{\eta}{\eta_s} = 1+\frac{10\pi}{3}\frac{NR^3}{\Omega}$

From the equation it can be noticed that hs is directly proportional to the volume fraction ф that these spheres occupy. Since, with the necessary considerations, this reasoning can be transferred to macromolecules, which are not rigid spheres, it may be inferred that if the segments are contracted in a "poor" solvent, the viscosity of the solution will be smaller. Therefore, viscosity can be adjusted according to the solvent quality.

Temperature, however, will not affect the viscosity of a polymer solution in a relatively "poor" solvent. In this case, it should be considered that as the temperature increases, the viscosity of the solvent (ηs) decreases. However, on the other hand, when the temperature is raised, a greater thermal energy will be granted to molecules. Consequently, these molecules will tend to expand themselves, increasing their volume fraction (ф). Thus both effects are compensated, and for this reason the change of viscosity due to the increase of the temperature, is not significant.

The measurement of viscosity in dilute macromolecular solutions has a fundamental importance not only in the determination of molecular weights, but also, as we will discuss later, in the evaluation of key parameters for the understanding of the conformational characteristics of polymer solutions.