Repulsion - Steric(entropic)

Introduction

"When a sol of gelatin, for instance, is added to a gold sol prepared by the reduction of a gold salt in an alkaline medium, it appears that the gold sol is strongly protected against the flocculating action of electrolytes."

H.R. Kruyt, Colloids: A textbook; H.S. van Klooster*, Translator; John Wiley & Sons: London; 1927; p. 87. (*Who I met in the late 1960's)

It had been known for a long time that electrolytes would flocculate many sols; gold sols were a common example. These were call lyophobic colloids. The colloids insensitive to electrolyte were, in hindsight, polymeric. They were call lyophilic colloids. Kruyt reports here that some combinations of the lyophilic colloids could "protect" the lyophobic colloids from salt addition. This lyophilic colloids were also called "protective" colloids.

We now know this mechanism to be polymer adsorption; and in the present context, examples of steric stabilization.

From the very beginning, the stability of polymer-stabilized sols has been understood primarily in terms of the solution solubility behavior of the polymer. Polymer-coated sols are stable when the polymer is both adsorbed and soluble; and unstable even when the polymer is adsorbed if it is no longer soluble.

 Stability of a thin film or a dispersion requires a repulsive force. In this case a "steric" or "entropic" barrier.

Simple model of steric stabilization

 The dispersion energy for two spheres increases as two spheres near each other by Brownian motion. $\Delta G_{121}=\frac{-A_{121}d}{24H}$ For the kinetic energy to remain greater than the attractive energy, the distance must be kept greater than H. $kT>\frac{A_{121}d}{24H}$ If polymer layers of thickness 't' around each particle just touch at this distance, 'H': $kT>\frac{A_{121}d}{48t}$ or $t>\frac{A_{121}}{48kT}d$ For example: Reference Polymer thickness for stabilization as a function of particle diameter: Reference

Polymer Size

A polymer increases the viscosity of the solution in a manner dependent on molecular size.

 This polymer size can be calculated from the intrinsic viscosity: $\left[ \eta \right]=\underset{c\to 0}{\mathop{\lim }}\,\frac{1}{c}\left( \frac{\eta _{solution}}{\eta _{solvent}}-1 \right)$ $\left\langle r^{2} \right\rangle ^{1/2}=\left( \frac{2}{5}\frac{MW}{N_{0}}\left[ \eta \right] \right)^{1/3}$ Where MW is molecular weight and N0 is Avogadro’s number. Or from c* where c* is the concentration where the viscosity is not linear in concentration. $\left[ \eta \right]=\frac{1}{c*}$ Or from a theory where l is the “Kuhn” length. $R_{g}=\frac{l\sqrt{n}}{\sqrt{6}}$
• What about the viscosity DURING the polymerization process?

As a monomer is converted to polymer in a homogeneous system, the viscosity can increase rapidly. In a high viscous medium, small monomer molecules can still diffuse readily to growing chains. So the rate of propagation remains relatively constant, but large growing chains cannot diffuse easily toward each other, and therefore the rate of termination can decrease considerably. Also the "degree of polymerization" increases. This sudden increase in rate, called the "autoacceleration" or the "Trommsdorff effect", is pronounced when high-molecular-weight polymer is formed, since the viscosity of the solution increases in proportion to the molecular weight raised to somewhere between the second and fourth power in many cases. The transition from normal kinetics to autoacceleration can be quite sharp and it can be aggravated by the higher heat generation which can raise the temperature and further increase the rate.

A slightly better model

 Take into account the compressibility of the outer reaches of the polymer chain:

Polymers in solution - Phase diagrams

 Sterically stabilized dispersions are stable when the polymer is soluble – the one phase regions. The higher temperature is called the "lower critical temperature" and the lower temperature is called the "upper critical solution temperature. (No kidding!) 141 nm silica particles- with grafted polymer. Pictures were taken at 0 C and 60 C. The particles phase-transfer with the change in polymer solubility. The upper liquid is ethylacetate and the lower, water. Dejin and Zhao, Langmuir, 2001, 23, 2208

Steric Effects and steric repulsion

Steric Effects

Steric effects arise from the fact that each atom within a molecule occupies a certain amount of space. If atoms are brought too close together, there is an associated cost in energy due to overlapping electron clouds (Pauli or Born repulsion), and this may affect the molecule's preferred shape (conformation) and reactivity.

Chemisorbed long-chain thiol can still be physisorbed as well. While S-As bond acting as a lever and steric repulsion of the first CH2 unit acts as a pivot, the NH2 endgroup of the molecule still feels quite strong attraction to the surface.

There are a few types of steric effects:

Steric hindrance or steric resistance occurs when the size of groups within a molecule prevents chemical reactions that are observed in related smaller molecules. Although steric hindrance is sometimes a problem, it can also be a very useful tool, and is often exploited by chemists to change the reactivity pattern of a molecule by stopping unwanted side-reactions (steric protection). Steric hindrance between adjacent groups can also restrict torsional bond angles. However, hyperconjugation has been suggested as an explanation for the preference of the staggered conformation of ethane because the steric hindrance of the small hydrogen atom is far too small.

Steric shielding occurs when a charged group on a molecule is seemingly weakened or spatially shielded by less charged (or oppositely charged) atoms, including counterions in solution (Debye shielding). In some cases, for an atom to interact with sterically shielded atoms, it would have to approach from a vicinity where there is less shielding, thus controlling where and from what direction a molecular interaction can take place.

Steric attraction occurs when molecules have shapes or geometries that are optimized for interaction with one another. In these cases molecules will react with each other most often in specific arrangements.

Chain crossing — A random coil can't change from one conformation to a closely related shape by a small displacement if it would require one polymer chain to pass through another, or through itself.

Steric Repulsion

The origins of steric or electrosteric repulsion lie in both volume restriction and interpenetration effects, although it is unlikely that either effect would occur in isolation to provide a repulsive force. In many industrial processes where the coagulation of colloidal particles would naturally occur, steric repulsion between particles can be induced by the addition of a polymer, to prevent the approach of the particle cores to a separation where their mutual van der Waals attraction would cause flocculation to occur. Complete particle surface coverage by absorbed or anchored polymer at high concentration can produce a steric layer that prevents close approach of the particles. The steric layer also acts as a lubricant to reduce the high frictional forces that occur between particles with large attractive interactions. The time-dependent, displaceable and slow-forming hydrolyzed inorganic layers which lead to repulsive electrosteric forces between mica surfaces in 0.1M Cr(NO3)3 electrolyte have been reported.

The magnitude of the repulsion resulting from steric forces is dependent upon the surface rare of the particle that the polymer occupies and whether the polymer is reversibly or irreversibly attached to the particle’s surface.

Adsorbed and nonadsorbing surfactants and polymers are widely used to induce steric stabilization. The principal advantages of steric stabilization over charge stabilization are:

• 1. Provision of stability in nonpolar media where weak electrical effects occur
• 2. Use of higher levels of electrolyte in aqueous media without causing flocculation
• 3. Reduction of electroviscous effects arising from particle charge by the addition of electrolyte without flocculation.
• 4. Dispersion stabilization can be achieved at higher particle concentrations

Graft or block copolymers commonly used as steric stabilizers are designed to have two groups of different functionality, A and B. A is chosen to be insoluble in the dispersion medium and has strong affinity for the particle surface while B is selected to be soluble but have little or no affinity for the particle surface. Other steric interactions, which give rise to short-range repulsion in aqueous dispersions due to bare size of ions present at the particle solution interface at high ion concentrations, have been observed.

Optimizing Steric Repulsion

In order to optimize the steric repulsion, we consider the steric potential:

$V(h)=2\pi kTR\Gamma ^{2}N_{A}\left( \frac{V_{p}}{V_{s}} \right)(0.5-x)\left( 1-\frac{h}{2\delta } \right)^{2}+V_{elastic}$

Where k is Boltzmann’s constant, T is the absolute temperature, R is the particle radius, $\Gamma$ is the amount adsorbed, $N_{A}$ is Avogadro’s number, $V_{p}$ is the specific partial volume of the polymer, $V$ is the molar volume of the solvent, x is the Flory-Huggins parameter, $\delta$ is the maximum extent of the adsorbed layer and Velastic takes account of the compression of polymer chains on close approach.

From this equation, we can see that:

1. Higher adsorbed amount($\Gamma$) will result in more interaction/repulsion

2. Distinct cases emerge for different Flory-Huggins chain-solvent interaction parameter x. When x<0.5 (good solvent condition), there is maximum interaction on overlap of the stabilizing layers. Osmotic forces cause solvent to move into the highly concentrated overlap zone, forcing the particles apart. If x=0.5, the steric potential goes to zero and for x>0.5 (poor solvent conditions), the steric potential becomes negative and the chains will attract, enhancing flocculation.

3. The steric interaction starts at h= $2\delta$ as the chains begin to overlap and increases as the square of the distance.

4. The final interaction potential is the superposition of the steric potential and the van der Waals attraction.

The following papers by Evans and Napper are helpful in understanding steric stabilization:

Sterically Stabilized Dispersion

For sterically stabilized dispersions, the resulting energy-distance curve often shows a shallow minimum, Vmin, at a particle-particle separation distance h comparable to twice the adsorbed layer thickness delta. For a given material, the depth of this minimum depends on the particle size R and adsorbed layer thickness delta.

Hence Vmin decreases with increase in $\frac{\delta }{R}$, as illustrated in the figure above. This is because as we increase the layer thickness, the van der Waals attraction weakness so the superposition of attraction and repulsion will have a smaller minimum. For very small steric layers, $V_{\min }$ may become deep enough to cause weak flocculation, resulting in a weak attractive gel.

On the other hand, if the layer thickness is too large, the viscosity is also increased due to repulsion. This is due to the much higher effective volume fraction $\Phi _{eff}$ of the dispersion compared with the core volume fraction. We can calculate the effective volume fraction of particles plus dispersant layer by geometry and we see that it depends on the thickness of that adsorbed layer, as illustrated in the figure below.

The effective volume fraction increases with relative increase in the dispersant layer thickness. Even at 10% volume fraction we can soon reach maximum packing ($\Phi =0.67$) with an adsorbed layer comparable to the particle radius. In this case, overlap of the steric layers will result in significant viscosity increases. Such considerations help to explain why the solids loading can be severely limited, especially with small particles. In practice, solids loading curves can be used to characterize the system and will take the form of those illustrated below:

A higher solids loading might be achieved with thinner adsorbed layers but may also lead to interparticle attraction, resulting in particle aggregation. Clearly a compromise is needed: choosing an appropriate steric stabilizer for the particle size of the pigment.

Steric Effects and Protein Structure

Steric effects have profound impacts on chemically bonded structures. Many, many organic chemistry reactions have stereoselectivity that is based on steric hindrances (for example, there is a bulky group blocking a molecule from attacking this area, thus changing the outcome of the reaction). Indeed, steric effects are nearly universal and affect the rates and energies of most chemical reactions to varying degrees. In biochemistry, steric effects are often exploited in naturally occurring molecules such as enzymes, where the catalytic site may be buried within a large protein structure. In pharmacology, steric effects determine how and at what rate a drug will interact with its target bio-molecules.

One of the most important and unanswered questions in modern biology soft-matter is how exactly proteins fold to a stable energetic state. As a professor once told me, “If anyone could understand this…this…this miracle, he would win the Nobel Prize!” While this professor forgot to point out that it could be a woman who figures out the solution to this folding puzzle, he nonetheless made his point very clearly. Steric effects play a big role in the current understanding of how proteins fold. There are many conformational constraints (on the protein) that are due to steric effects. For example, it has been found that central residues in a helical peptide cannot adopt dihedral angles from strand regions without encountering steric collisions. This forces alpha-helical segments to be followed by beta-strand segments via an intervening connecting linker. (For more information, see Protein Science (2004) 13:633-639).

The picture below shows the three main chain torsion angles of a polypeptide. The bonds in the polypeptide are only somewhat free to rotate around each other. One can simulate all the different possible conformations, and only some of them are sterically allowed.

Sterically disallowed (and allowed) conformations are typically displayed by using a Ramachandran Plot. Below is an example of such a plot:

In the diagram above the white areas correspond to conformations where atoms in the polypeptide come closer than the sum of their van der Waals radii. These regions are sterically disallowed for all amino acids except glycine which is unique in that it lacks a side chain. The red regions correspond to conformations where there are no steric clashes, i.e. these are the allowed regions namely the a-helical and a-sheet conformations. The yellow areas show the allowed regions if slightly shorter van der Waals radii are used in the calculation, i.e. the atoms are allowed to come a little closer together. This brings out an additional region which corresponds to the left-handed a-helix. Disallowed regions generally involve steric hindrance between the side chain C methylene group and main chain atoms. One amino acid—Glycine—has no side chain and therefore can adopt phi and psi angles in all four quadrants of the Ramachandran plot. Hence it frequently occurs in turn regions of proteins where any other residue would be sterically hindered.

Understanding of steric effects in protein folding helped lead to the alpha-helix model, as well as discoveries of distortions from alpha-helices.

The opposite of steric repulsion: Depletion Attraction

The image illustrates polymers in a colloid suspension that fail to adsorb on the colloid surface. This might actually produce the opposite than intended effect and flocculate the colloid mixture through a mechanism known as depletion attraction. To understand the concept of depletion, let us approximate the polymer particle volume by a sphere of radius $r_g$ (the polymer's radius of gyration). When the two colloidal particles come close, the polymer particles are excluded from the volume between the two colloids. Therefore the polymers exert a force equivalent to their osmotic pressure on opposite sides of the two colloidal spheres, which pushes them together. The depletion mechanism is strongly dependent on the amount of excluded volume and consequently on the sizes and geometries of the excluding and excluded particles. The free energy gained when the two spheres come into contact was first calculated by Asakura and Oosawa to be:

$\Delta F = (1+\frac{3D}{2d})nk_BT$

where D is the diameter of the large (colloidal) spheres, $d =r_g$ is the diameter of the polymer particles and n is the volume occupied by the polymers.

Far from being simply a streric stabilization accident, depletion is actually an important force in crowded cellular environments where it actively drives the assembly of various molecular structures. This is often referred in literature as 'macromolecular crowding' and constitutes a fairly revolutionary idea: the tight packing of proteins and organelles within the cell is itself the attractive force that drives the assembly of macromolecular structures. The crowding model scores points for universality, since it is an interaction solely dependent on volume which can account for several disparate bio-assembly processes. The following image illustrates exactly that. On it, the depletion agent (small spheres or small polymers) is not depicted. Instead, the image shows particles and processes that can be driven by depletion.

Category A sums up protein-protein assembly, which includes the bundling of protein filaments (iii) and the helical conformation of a previously linear filament (iv). Here's a thought on this first part: Since filaments are bundled by depletion attraction, one could assume that the rigidity and tensile strength of the cytoskeletal filamentous structure in vivo is simply a consequence of tight cellular packing!

Category B on the image alludes to chromatin-chromatin interactions in the nucleus. Chromatin is depicted as a blue line, while the orange beads stand for transcription factors bound on the chromatin thread. Illustration B(i) suggests that the entropic cost of chromatin looping is actually overcome by depletion attraction.

Finally, category C illustrates that confinement of particles generates a type of depletion, since enclosing a sphere in a confined space (such as a pore) generates a large overlap volume.

Images and text adapted from: Marenduzzo D., Finan K. & Cook P.R., 'The depletion attraction: an underappreciated force driving cellular organization, Journal of Cell Biology 175, 681-686, (2006)