Radical polymerization

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Radical_polymerization

In polymer chemistry, free-radical polymerization (FRP) is a method of polymerization by which a polymer forms by the successive addition of free-radical building blocks (repeat units). Free radicals can be formed by a number of different mechanisms, usually involving separate initiator molecules. Following its generation, the initiating free radical adds (nonradical) monomer units, thereby growing the polymer chain.

Free-radical polymerization is a key synthesis route for obtaining a wide variety of different polymers and materials composites. The relatively non-specific nature of free-radical chemical interactions makes this one of the most versatile forms of polymerization available and allows facile reactions of polymeric free-radical chain ends and other chemicals or substrates. In 2001, 40 billion of the 110 billion pounds of polymers produced in the United States were produced by free-radical polymerization.

Free-radical polymerization is a type of chain-growth polymerization, along with anionic, cationic and coordination polymerization.

IUPAC definition

A chain polymerization in which the kinetic-chain carriers are radicals.

Note: Usually, the growing chain end bears an unpaired electron.

Initiation

Initiation is the first step of the polymerization process. During initiation, an active center is created from which a polymer chain is generated. Not all monomers are susceptible to all types of initiators. Radical initiation works best on the carbon–carbon double bond of vinyl monomers and the carbon–oxygen double bond in aldehydes and ketones. Initiation has two steps. In the first step, one or two radicals are created from the initiating molecules. In the second step, radicals are transferred from the initiator molecules to the monomer units present. Several choices are available for these initiators.

Types of initiation and the initiators

Thermal decomposition

The initiator is heated until a bond is homolytically cleaved, producing two radicals (Figure 1). This method is used most often with organic peroxides or azo compounds.

Figure 1: Thermal decomposition of dicumyl peroxide

 

Photolysis

Radiation cleaves a bond homolytically, producing two radicals (Figure 2). This method is used most often with metal iodides, metal alkyls, and azo compounds.

Figure 2: Photolysis of azoisobutylnitrile (AIBN)

 

Photoinitiation can also occur by bi-molecular H abstraction when the radical is in its lowest triplet excited state. An acceptable photoinitiator system should fulfill the following requirements:

• High absorptivity in the 300–400 nm range.
• Efficient generation of radicals capable of attacking the alkene double bond of vinyl monomers.
• Adequate solubility in the binder system (prepolymer + monomer).
• Should not impart yellowing or unpleasant odors to the cured material.
• The photoinitiator and any byproducts resulting from its use should be non-toxic.

Redox reactions

Reduction of hydrogen peroxide or an alkyl hydrogen peroxide by iron (Figure 3). Other reductants such as Cr²⁺, V²⁺, Ti³⁺, Co²⁺, and Cu⁺ can be employed in place of ferrous ion in many instances.

${\displaystyle {\text{H}}_2^{}{\text{O}}_2^{}+{\text{Fe}}^{2+}⟶{\text{Fe}}^{3+}+{\text{HO}}^{-}+{\text{HO}}^{\cdot }}$ Figure 3: Redox reaction of hydrogen peroxide and iron.

Persulfates

The dissociation of a persulfate in the aqueous phase (Figure 4). This method is useful in emulsion polymerizations, in which the radical diffuses into a hydrophobic monomer-containing droplet.

Figure 4: Thermal degradation of a persulfate

 

Ionizing radiation

α-, β-, γ-, or x-rays cause ejection of an electron from the initiating species, followed by dissociation and electron capture to produce a radical (Figure 5).

Figure 5: The three steps involved in ionizing radiation: ejection, dissociation, and electron-capture

 

Electrochemical

Electrolysis of a solution containing both monomer and electrolyte. A monomer molecule will receive an electron at the cathode to become a radical anion, and a monomer molecule will give up an electron at the anode to form a radical cation (Figure 6). The radical ions then initiate free radical (and/or ionic) polymerization. This type of initiation is especially useful for coating metal surfaces with polymer films.

Figure 6: (Top) Formation of radical anion at the cathode; (bottom) formation of radical cation at the anode

Plasma

A gaseous monomer is placed in an electric discharge at low pressures under conditions where a plasma (ionized gaseous molecules) is created. In some cases, the system is heated and/or placed in a radiofrequency field to assist in creating the plasma.

Sonication

High-intensity ultrasound at frequencies beyond the range of human hearing (16 kHz) can be applied to a monomer. Initiation results from the effects of cavitation (the formation and collapse of cavities in the liquid). The collapse of the cavities generates very high local temperatures and pressures. This results in the formation of excited electronic states, which in turn lead to bond breakage and radical formation.

Ternary initiators

A ternary initiator is the combination of several types of initiators into one initiating system. The types of initiators are chosen based on the properties they are known to induce in the polymers they produce. For example, poly(methyl methacrylate) has been synthesized by the ternary system benzoyl peroxide-3,6-bis(o-carboxybenzoyl)-N-isopropylcarbazole-di-η⁵-indenylzicronium dichloride (Figure 7).

Figure 7: benzoyl peroxide-3,6-bis(o-carboxybenzoyl)-N-isopropylcarbazole-di-η⁵-indenylzicronium dichloride

 

This type of initiating system contains a metallocene, an initiator, and a heteroaromatic diketo carboxylic acid. Metallocenes in combination with initiators accelerate polymerization of poly(methyl methacrylate) and produce a polymer with a narrower molecular weight distribution. The example shown here consists of indenylzirconium (a metallocene) and benzoyl peroxide (an initiator). Also, initiating systems containing heteroaromatic diketo carboxylic acids, such as 3,6-bis(o-carboxybenzoyl)-N-isopropylcarbazole in this example, are known to catalyze the decomposition of benzoyl peroxide. Initiating systems with this particular heteroaromatic diket carboxylic acid are also known to have effects on the microstructure of the polymer. The combination of all of these components—a metallocene, an initiator, and a heteroaromatic diketo carboxylic acid—yields a ternary initiating system that was shown to accelerate the polymerization and produce polymers with enhanced heat resistance and regular microstructure.

Initiator efficiency

Due to side reactions and inefficient synthesis of the radical species, chain initiation is not 100%. The efficiency factor f is used to describe the effective radical concentration. The maximal value of f is 1, but typical values range from 0.3 to 0.8. The following is a list of reactions that decrease the efficiency of the initiator.

Primary recombination

Two radicals recombine before initiating a chain (Figure 8). This occurs within the solvent cage, meaning that no solvent has yet come between the new radicals.

Figure 8: Primary recombination of BPO; brackets indicate that the reaction is happening within the solvent cage

Other recombination pathways

Two radical initiators recombine before initiating a chain, but not in the solvent cage (Figure 9).

Figure 9: Recombination of phenyl radicals from the initiation of BPO outside the solvent cage

Side reactions

One radical is produced instead of the three radicals that could be produced (Figure 10).

${\displaystyle \text{R}\cdot +{\text{R}}^{\prime }-\text{O}-\text{O}-{\text{R}}^{\prime }⟶{\text{ROR}}^{\prime }+{\text{R}}^{\prime }{\text{O}}^{\prime }}$ Figure 10: Reaction of polymer chain R with other species in reaction

Propagation

During polymerization, a polymer spends most of its time in increasing its chain length, or propagating. After the radical initiator is formed, it attacks a monomer (Figure 11). In an ethene monomer, one electron pair is held securely between the two carbons in a sigma bond. The other is more loosely held in a pi bond. The free radical uses one electron from the pi bond to form a more stable bond with the carbon atom. The other electron returns to the second carbon atom, turning the whole molecule into another radical. This begins the polymer chain. Figure 12 shows how the orbitals of an ethylene monomer interact with a radical initiator.

Figure 11: Phenyl initiator from benzoyl peroxide (BPO) attacks a styrene molecule to start the polymer chain.

 

Figure 12: An orbital drawing of the initiator attack on ethylene molecule, producing the start of the polyethylene chain.

Once a chain has been initiated, the chain propagates (Figure 13) until there are no more monomers (living polymerization) or until termination occurs. There may be anywhere from a few to thousands of propagation steps depending on several factors such as radical and chain reactivity, the solvent, and temperature. The mechanism of chain propagation is as follows:

Figure 13: Propagation of polystyrene with a phenyl radical initiator.

Termination

Chain termination is inevitable in radical polymerization due to the high reactivity of radicals. Termination can occur by several different mechanisms. If longer chains are desired, the initiator concentration should be kept low; otherwise, many shorter chains will result.

• Combination of two active chain ends: one or both of the following processes may occur.
  • Combination: two chain ends simply couple together to form one long chain (Figure 14). One can determine if this mode of termination is occurring by monitoring the molecular weight of the propagating species: combination will result in doubling of molecular weight. Also, combination will result in a polymer that is C₂ symmetric about the point of the combination.
    

Figure 14: Termination by the combination of two poly(vinyl chloride) (PVC) polymers.

Radical disproportionation: a hydrogen atom from one chain end is abstracted to another, producing a polymer with a terminal unsaturated group and a polymer with a terminal saturated group (Figure 15).

• Figure 15: Termination by disproportionation of poly(methyl methacrylate).

Combination of an active chain end with an initiator radical (Figure 16).

Figure 16: Termination of PVC by reaction with radical initiator.

Interaction with impurities or inhibitors. Oxygen is the common inhibitor. The growing chain will react with molecular oxygen, producing an oxygen radical, which is much less reactive (Figure 17). This significantly slows down the rate of propagation.

Figure 17: Inhibition of polystyrene propagation due to reaction of polymer with molecular oxygen.

 

Nitrobenzene, butylated hydroxyl toluene, and diphenyl picryl hydrazyl (DPPH, Figure 18) are a few other inhibitors. The latter is an especially effective inhibitor because of the resonance stabilization of the radical.

• Figure 18: Inhibition of polymer chain, R, by DPPH.

Chain transfer

Contrary to the other modes of termination, chain transfer results in the destruction of only one radical, but also the creation of another radical. Often, however, this newly created radical is not capable of further propagation. Similar to disproportionation, all chain-transfer mechanisms also involve the abstraction of a hydrogen or other atom. There are several types of chain-transfer mechanisms.

• To solvent: a hydrogen atom is abstracted from a solvent molecule, resulting in the formation of radical on the solvent molecules, which will not propagate further (Figure 19).
  

Figure 19: Chain transfer from polystyrene to solvent.

 

The effectiveness of chain transfer involving solvent molecules depends on the amount of solvent present (more solvent leads to greater probability of transfer), the strength of the bond involved in the abstraction step (weaker bond leads to greater probability of transfer), and the stability of the solvent radical that is formed (greater stability leads to greater probability of transfer). Halogens, except fluorine, are easily transferred.

To monomer: a hydrogen atom is abstracted from a monomer. While this does create a radical on the affected monomer, resonance stabilization of this radical discourages further propagation (Figure 20).

Figure 20: Chain transfer from polypropylene to monomer.

To initiator: a polymer chain reacts with an initiator, which terminates that polymer chain, but creates a new radical initiator (Figure 21). This initiator can then begin new polymer chains. Therefore, contrary to the other forms of chain transfer, chain transfer to the initiator does allow for further propagation. Peroxide initiators are especially sensitive to chain transfer.

Figure 21: Chain transfer from polypropylene to di-t-butyl peroxide initiator.

To polymer: the radical of a polymer chain abstracts a hydrogen atom from somewhere on another polymer chain (Figure 22). This terminates the growth of one polymer chain, but allows the other to branch and resume growing. This reaction step changes neither the number of polymer chains nor the number of monomers which have been polymerized, so that the number-average degree of polymerization is unaffected.

• Figure 22: Chain transfer from polypropylene to backbone of another polypropylene.

Effects of chain transfer: The most obvious effect of chain transfer is a decrease in the polymer chain length. If the rate of transfer is much larger than the rate of propagation, then very small polymers are formed with chain lengths of 2-5 repeating units (telomerization). The Mayo equation estimates the influence of chain transfer on chain length (x_n): ${\displaystyle \frac{1}{x_n}={\left(\frac{1}{x_n}\right)}_o+\frac{k_{tr}[solvent]}{k_p[monomer]}}$. Where k_tr is the rate constant for chain transfer and k_p is the rate constant for propagation. The Mayo equation assumes that transfer to solvent is the major termination pathway.

Methods

There are four industrial methods of radical polymerization:

• Bulk polymerization: reaction mixture contains only initiator and monomer, no solvent.
• Solution polymerization: reaction mixture contains solvent, initiator, and monomer.
• Suspension polymerization: reaction mixture contains an aqueous phase, water-insoluble monomer, and initiator soluble in the monomer droplets (both the monomer and the initiator are hydrophobic).
• Emulsion polymerization: similar to suspension polymerization except that the initiator is soluble in the aqueous phase rather than in the monomer droplets (the monomer is hydrophobic, and the initiator is hydrophilic). An emulsifying agent is also needed.

Other methods of radical polymerization include the following:

• Template polymerization: In this process, polymer chains are allowed to grow along template macromolecules for the greater part of their lifetime. A well-chosen template can affect the rate of polymerization as well as the molar mass and microstructure of the daughter polymer. The molar mass of a daughter polymer can be up to 70 times greater than those of polymers produced in the absence of the template and can be higher in molar mass than the templates themselves. This is because of retardation of the termination for template-associated radicals and by hopping of a radical to the neighboring template after reaching the end of a template polymer.
• Plasma polymerization: The polymerization is initiated with plasma. A variety of organic molecules including alkenes, alkynes, and alkanes undergo polymerization to high molecular weight products under these conditions. The propagation mechanisms appear to involve both ionic and radical species. Plasma polymerization offers a potentially unique method of forming thin polymer films for uses such as thin-film capacitors, antireflection coatings, and various types of thin membranes.
• Sonication: The polymerization is initiated by high-intensity ultrasound. Polymerization to high molecular weight polymer is observed but the conversions are low (<15%). The polymerization is self-limiting because of the high viscosity produced even at low conversion. High viscosity hinders cavitation and radical production.

Reversible deactivation radical polymerization

Main article: Reversible-deactivation radical polymerization

Also known as living radical polymerization, controlled radical polymerization, reversible deactivation radical polymerization (RDRP) relies on completely pure reactions, preventing termination caused by impurities. Because these polymerizations stop only when there is no more monomer, polymerization can continue upon the addition of more monomer. Block copolymers can be made this way. RDRP allows for control of molecular weight and dispersity. However, this is very difficult to achieve and instead a pseudo-living polymerization occurs with only partial control of molecular weight and dispersity. ATRP and RAFT are the main types of complete radical polymerization.

• Atom transfer radical polymerization (ATRP): based on the formation of a carbon-carbon bond by atom transfer radical addition. This method, independently discovered in 1995 by Mitsuo Sawamoto and by Jin-Shan Wang and Krzysztof Matyjaszewski, requires reversible activation of a dormant species (such as an alkyl halide) and a transition metal halide catalyst (to activate dormant species).
• Reversible Addition-Fragmentation Chain-Transfer Polymerization (RAFT): requires a compound that can act as a reversible chain-transfer agent, such as dithio compound.
• Stable Free Radical Polymerization (SFRP): used to synthesize linear or branched polymers with narrow molecular weight distributions and reactive end groups on each polymer chain. The process has also been used to create block co-polymers with unique properties. Conversion rates are about 100% using this process but require temperatures of about 135 °C. This process is most commonly used with acrylates, styrenes, and dienes. The reaction scheme in Figure 23 illustrates the SFRP process.
  

Figure 23: Reaction scheme for SFRP.

• Figure 24: TEMPO molecule used to functionalize the chain ends.
  Because the chain end is functionalized with the TEMPO molecule (Figure 24), premature termination by coupling is reduced. As with all living polymerizations, the polymer chain grows until all of the monomer is consumed.

Kinetics

In typical chain growth polymerizations, the reaction rates for initiation, propagation and termination can be described as follows:

${\displaystyle v_i=\mathrm{d}[M\cdot ]/\mathrm{d}t=2k_{d}f[I]}$

${\displaystyle v_p=k_p[M][M\cdot ]}$

${\displaystyle v_t=-\mathrm{d}[M\cdot ]/\mathrm{d}t=2k_t[M\cdot {]}^2}$

where f is the efficiency of the initiator and k_d, k_p, and k_t are the constants for initiator dissociation, chain propagation and termination, respectively. [I] [M] and [M•] are the concentrations of the initiator, monomer and the active growing chain.

Under the steady-state approximation, the concentration of the active growing chains remains constant, i.e. the rates of initiation and of termination are equal. The concentration of active chain can be derived and expressed in terms of the other known species in the system.

${\displaystyle [M\cdot ]={\left(\frac{k_d[I]f}{k_t}\right)}^{1/2}}$

In this case, the rate of chain propagation can be further described using a function of the initiator and monomer concentrations

${\displaystyle v_p=k_p{\left(\frac{f{k}_d}{k_t}\right)}^{1/2}[I{]}^{1/2}[M]}$

The kinetic chain length v is a measure of the average number of monomer units reacting with an active center during its lifetime and is related to the molecular weight through the mechanism of the termination. Without chain transfer, the kinetic chain length is only a function of propagation rate and initiation rate.

${\displaystyle \nu =\frac{v_p}{v_i}=\frac{k_p[M][M\cdot ]}{2f{k}_d[I]}=\frac{k_p[M]}{2(f{k}_d{k}_t[I]{)}^{1/2}}}$

Assuming no chain-transfer effect occurs in the reaction, the number average degree of polymerization P_n can be correlated with the kinetic chain length. In the case of termination by disproportionation, one polymer molecule is produced per every kinetic chain:

${\displaystyle x_n=\nu }$

Termination by combination leads to one polymer molecule per two kinetic chains:

${\displaystyle x_n=2\nu }$

Any mixture of both these mechanisms can be described by using the value δ, the contribution of disproportionation to the overall termination process:

${\displaystyle x_n=\frac{2}{1+\delta }\nu }$

If chain transfer is considered, the kinetic chain length is not affected by the transfer process because the growing free-radical center generated by the initiation step stays alive after any chain-transfer event, although multiple polymer chains are produced. However, the number average degree of polymerization decreases as the chain transfers, since the growing chains are terminated by the chain-transfer events. Taking into account the chain-transfer reaction towards solvent S, initiator I, polymer P, and added chain-transfer agent T. The equation of P_n will be modified as follows:

${\displaystyle \frac{1}{x_n}=\frac{2k_{t,d}+k_{t,c}}{{k_p}^2[M{]}^2}{v}_p+C_M+C_S\frac{[S]}{[M]}+C_I\frac{[I]}{[M]}+C_P\frac{[P]}{[M]}+C_T\frac{[T]}{[M]}}$

It is usual to define chain-transfer constants C for the different molecules

${\displaystyle C_M=\frac{k_{tr}^M}{k_p}}$, ${\displaystyle C_S=\frac{k_{tr}^S}{k_p}}$, ${\displaystyle C_I=\frac{k_{tr}^I}{k_p}}$, ${\displaystyle C_P=\frac{k_{tr}^P}{k_p}}$, ${\displaystyle C_T=\frac{k_{tr}^T}{k_p}}$

Thermodynamics

In chain growth polymerization, the position of the equilibrium between polymer and monomers can be determined by the thermodynamics of the polymerization. The Gibbs free energy (ΔG_p) of the polymerization is commonly used to quantify the tendency of a polymeric reaction. The polymerization will be favored if ΔG_p < 0; if ΔG_p > 0, the polymer will undergo depolymerization. According to the thermodynamic equation ΔG = ΔH – TΔS, a negative enthalpy and an increasing entropy will shift the equilibrium towards polymerization.

In general, the polymerization is an exothermic process, i.e. negative enthalpy change, since addition of a monomer to the growing polymer chain involves the conversion of π bonds into σ bonds, or a ring–opening reaction that releases the ring tension in a cyclic monomer. Meanwhile, during polymerization, a large amount of small molecules are associated, losing rotation and translational degrees of freedom. As a result, the entropy decreases in the system, ΔS_p < 0 for nearly all polymerization processes. Since depolymerization is almost always entropically favored, the ΔH_p must then be sufficiently negative to compensate for the unfavorable entropic term. Only then will polymerization be thermodynamically favored by the resulting negative ΔG_p.

In practice, polymerization is favored at low temperatures: TΔS_p is small. Depolymerization is favored at high temperatures: TΔS_p is large. As the temperature increases, ΔG_p become less negative. At a certain temperature, the polymerization reaches equilibrium (rate of polymerization = rate of depolymerization). This temperature is called the ceiling temperature (T_c). ΔG_p = 0.

Stereochemistry

The stereochemistry of polymerization is concerned with the difference in atom connectivity and spatial orientation in polymers that has the same chemical composition.

Hermann Staudinger studied the stereoisomerism in chain polymerization of vinyl monomers in the late 1920s, and it took another two decades for people to fully appreciate the idea that each of the propagation steps in the polymer growth could give rise to stereoisomerism. The major milestone in the stereochemistry was established by Ziegler and Natta and their coworkers in 1950s, as they developed metal based catalyst to synthesize stereoregular polymers. The reason why the stereochemistry of the polymer is of particular interest is because the physical behavior of a polymer depends not only on the general chemical composition but also on the more subtle differences in microstructure. Atactic polymers consist of a random arrangement of stereochemistry and are amorphous (noncrystalline), soft materials with lower physical strength. The corresponding isotactic (like substituents all on the same side) and syndiotactic (like substituents of alternate repeating units on the same side) polymers are usually obtained as highly crystalline materials. It is easier for the stereoregular polymers to pack into a crystal lattice since they are more ordered and the resulting crystallinity leads to higher physical strength and increased solvent and chemical resistance as well as differences in other properties that depend on crystallinity. The prime example of the industrial utility of stereoregular polymers is polypropene. Isotactic polypropene is a high-melting (165 °C), strong, crystalline polymer, which is used as both a plastic and fiber. Atactic polypropene is an amorphous material with an oily to waxy soft appearance that finds use in asphalt blends and formulations for lubricants, sealants, and adhesives, but the volumes are minuscule compared to that of isotactic polypropene.

When a monomer adds to a radical chain end, there are two factors to consider regarding its stereochemistry: 1) the interaction between the terminal chain carbon and the approaching monomer molecule and 2) the configuration of the penultimate repeating unit in the polymer chain. The terminal carbon atom has sp² hybridization and is planar. Consider the polymerization of the monomer CH₂=CXY. There are two ways that a monomer molecule can approach the terminal carbon: the mirror approach (with like substituents on the same side) or the non-mirror approach (like substituents on opposite sides). If free rotation does not occur before the next monomer adds, the mirror approach will always lead to an isotactic polymer and the non-mirror approach will always lead to a syndiotactic polymer (Figure 25).

Figure 25: (Top) formation of isotactic polymer; (bottom) formation of syndiotactic polymer.

However, if interactions between the substituents of the penultimate repeating unit and the terminal carbon atom are significant, then conformational factors could cause the monomer to add to the polymer in a way that minimizes steric or electrostatic interaction (Figure 26).

Figure 26: Penultimate unit interactions cause monomer to add in a way that minimizes steric hindrance between substituent groups. (P represents polymer chain.)

Reactivity

Traditionally, the reactivity of monomers and radicals are assessed by the means of copolymerization data. The Q–e scheme, the most widely used tool for the semi-quantitative prediction of monomer reactivity ratios, was first proposed by Alfrey and Price in 1947. The scheme takes into account the intrinsic thermodynamic stability and polar effects in the transition state. A given radical ${\displaystyle M_i^o}$ and a monomer ${\displaystyle M_j}$ are considered to have intrinsic reactivities P_i and Q_j, respectively. The polar effects in the transition state, the supposed permanent electric charge carried by that entity (radical or molecule), is quantified by the factor e, which is a constant for a given monomer, and has the same value for the radical derived from that specific monomer. For addition of monomer 2 to a growing polymer chain whose active end is the radical of monomer 1, the rate constant, k₁₂, is postulated to be related to the four relevant reactivity parameters by

${\displaystyle k_{12}=P_1{Q}_2\exp (-e_1{e}_2)}$

The monomer reactivity ratio for the addition of monomers 1 and 2 to this chain is given by

${\displaystyle r_1=\frac{k_{11}}{k_{12}}=\frac{Q_1}{Q_2}\exp (-e_1(e_1-e_2))}$

For the copolymerization of a given pair of monomers, the two experimental reactivity ratios r₁ and r₂ permit the evaluation of (Q₁/Q₂) and (e₁ – e₂). Values for each monomer can then be assigned relative to a reference monomer, usually chosen as styrene with the arbitrary values Q = 1.0 and e = –0.8.

Applications

Free radical polymerization has found applications including the manufacture of polystyrene, thermoplastic block copolymer elastomers, cardiovascular stents, chemical surfactants and lubricants. Block copolymers are used for a wide variety of applications including adhesives, footwear and toys.

Free radical polymerization has uses in research as well, such as in the functionalization of carbon nanotubes. CNTs intrinsic electronic properties lead them to form large aggregates in solution, precluding useful applications. Adding small chemical groups to the walls of CNT can eliminate this propensity and tune the response to the surrounding environment. The use of polymers instead of smaller molecules can modify CNT properties (and conversely, nanotubes can modify polymer mechanical and electronic properties). For example, researchers coated carbon nanotubes with polystyrene by first polymerizing polystyrene via chain radical polymerization and subsequently mixing it at 130 °C with carbon nanotubes to generate radicals and graft them onto the walls of carbon nanotubes (Figure 27). Chain growth polymerization ("grafting to") synthesizes a polymer with predetermined properties. Purification of the polymer can be used to obtain a more uniform length distribution before grafting. Conversely, “grafting from”, with radical polymerization techniques such as atom transfer radical polymerization (ATRP) or nitroxide-mediated polymerization (NMP), allows rapid growth of high molecular weight polymers.

Figure 27: Grafting of a polystyrene free radical onto a single-walled carbon nanotube.

Radical polymerization also aids synthesis of nanocomposite hydrogels. These gels are made of water-swellable nano-scale clay (especially those classed as smectites) enveloped by a network polymer. They are often biocompatible and have mechanical properties (such as flexibility and strength) that promise applications such as synthetic tissue. Synthesis involves free radical polymerization. The general synthesis procedure is depicted in Figure 28. Clay is dispersed in water, where it forms very small, porous plates. Next the initiator and a catalyst are added, followed by adding the organic monomer, generally an acrylamide or acrylamide derivative. The initiator is chosen to have stronger interaction with the clay than the organic monomer, so it preferentially adsorbs to the clay surface. The mixture and water solvent is heated to initiate polymerization. Polymers grow off the initiators that are in turn bound to the clay. Due to recombination and disproportionation reactions, growing polymer chains bind to one another, forming a strong, cross-linked network polymer, with clay particles acting as branching points for multiple polymer chain segments. Free radical polymerization used in this context allows the synthesis of polymers from a wide variety of substrates (the chemistries of suitable clays vary). Termination reactions unique to chain growth polymerization produce a material with flexibility, mechanical strength and biocompatibility.

Figure 28: General synthesis procedure for a nanocomposite hydrogel.

Electronics

The radical polymer glass PTMA is about 10 times more electrically conductive than common semiconducting polymers. PTMA is in a class of electrically active polymers that could find use in transparent solar cells, antistatic and antiglare coatings for mobile phone displays, antistatic coverings for aircraft to protect against lightning strikes, flexible flash drives, and thermoelectric devices, which convert electricity into heat and the reverse. Widespread practical applications require increasing conductivity another 100 to 1,000 times.

The polymer was created using deprotection, which involves replacing a specific hydrogen atom in the pendant group with an oxygen atom. The resulting oxygen atom in PTMA has one unpaired electron in its outer shell, making it amenable to transporting charge. The deprotection step can lead to four distinct chemical functionalities, two of which are promising for increasing conductivity.

Wheeler–Feynman absorber theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Wheeler%E2%80%93Feynman_absorber_theory

The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assumption that the solutions of the electromagnetic field equations must be invariant under time-reversal transformation, as are the field equations themselves. Indeed, there is no apparent reason for the time-reversal symmetry breaking, which singles out a preferential time direction and thus makes a distinction between past and future. A time-reversal invariant theory is more logical and elegant. Another key principle, resulting from this interpretation and reminiscent of Mach's principle due to Hugo Tetrode, is that elementary particles are not self-interacting. This immediately removes the problem of self-energies.

T-symmetry and causality

The requirement of time-reversal symmetry, in general, is difficult to reconcile with the principle of causality. Maxwell's equations and the equations for electromagnetic waves have, in general, two possible solutions: a retarded (delayed) solution and an advanced one. Accordingly, any charged particle generates waves, say at time ${\displaystyle t_0=0}$ and point ${\displaystyle x_0=0}$, which will arrive at point ${\displaystyle x_1}$ at the instant ${\displaystyle t_1=x_1/c}$ (here ${\displaystyle c}$ is the speed of light), after the emission (retarded solution), and other waves, which will arrive at the same place at the instant ${\displaystyle t_2=-x_1/c}$, before the emission (advanced solution). The latter, however, violates the causality principle: advanced waves could be detected before their emission. Thus the advanced solutions are usually discarded in the interpretation of electromagnetic waves. In the absorber theory, instead charged particles are considered as both emitters and absorbers, and the emission process is connected with the absorption process as follows: Both the retarded waves from emitter to absorber and the advanced waves from absorber to emitter are considered. The sum of the two, however, results in causal waves, although the anti-causal (advanced) solutions are not discarded a priori.

Feynman and Wheeler obtained this result in a very simple and elegant way. They considered all the charged particles (emitters) present in our universe and assumed all of them to generate time-reversal symmetric waves. The resulting field is

${\displaystyle E_{\text{tot}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)+E_n^{\text{adv}}(\mathbf{x},t)}{2}.}$

Then they observed that if the relation

${\displaystyle E_{\text{free}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)-E_n^{\text{adv}}(\mathbf{x},t)}{2}=0}$

holds, then ${\displaystyle E_{\text{free}}}$, being a solution of the homogeneous Maxwell equation, can be used to obtain the total field

${\displaystyle E_{\text{tot}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)+E_n^{\text{adv}}(\mathbf{x},t)}{2}+\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)-E_n^{\text{adv}}(\mathbf{x},t)}{2}=\sum _n{E}_n^{\text{ret}}(\mathbf{x},t).}$

The total field is retarded, and causality is not violated.

The assumption that the free field is identically zero is the core of the absorber idea. It means that the radiation emitted by each particle is completely absorbed by all other particles present in the universe. To better understand this point, it may be useful to consider how the absorption mechanism works in common materials. At the microscopic scale, it results from the sum of the incoming electromagnetic wave and the waves generated from the electrons of the material, which react to the external perturbation. If the incoming wave is absorbed, the result is a zero outgoing field. In the absorber theory the same concept is used, however, in presence of both retarded and advanced waves.

The resulting wave appears to have a preferred time direction, because it respects causality. However, this is only an illusion. Indeed, it is always possible to reverse the time direction by simply exchanging the labels emitter and absorber. Thus, the apparently preferred time direction results from the arbitrary labelling.

Alternatively, the way that Wheeler/Feyman came up with the primary equation is: They assumed that their Lagrangian only interacted when and where the fields for the individual particles were separated by a proper time of zero. So since only massless particles propagate from emission to detection with zero proper time separation, this Lagrangian automatically demands an electromagnetic like interaction.

T-symmetry and self-interaction

One of the major results of the absorber theory is the elegant and clear interpretation of the electromagnetic radiation process. A charged particle that experiences acceleration is known to emit electromagnetic waves, i.e., to lose energy. Thus, the Newtonian equation for the particle (${\displaystyle F=ma}$) must contain a dissipative force (damping term), which takes into account this energy loss. In the causal interpretation of electromagnetism, Lorentz and Abraham proposed that such a force, later called Abraham–Lorentz force, is due to the retarded self-interaction of the particle with its own field. This first interpretation, however, is not completely satisfactory, as it leads to divergences in the theory and needs some assumptions on the structure of charge distribution of the particle. Dirac generalized the formula to make it relativistically invariant. While doing so, he also suggested a different interpretation. He showed that the damping term can be expressed in terms of a free field acting on the particle at its own position:

${\displaystyle E^{\text{damping}}({\mathbf{x}}_j,t)=\frac{E_j^{\text{ret}}({\mathbf{x}}_j,t)-E_j^{\text{adv}}({\mathbf{x}}_j,t)}{2}.}$

However, Dirac did not propose any physical explanation of this interpretation.

A clear and simple explanation can instead be obtained in the framework of absorber theory, starting from the simple idea that each particle does not interact with itself. This is actually the opposite of the first Abraham–Lorentz proposal. The field acting on the particle ${\displaystyle j}$ at its own position (the point ${\displaystyle x_j}$) is then

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)+E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}.}$

If we sum the free-field term of this expression, we obtain

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)+E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}+\sum _n\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)-E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}}$

and, thanks to Dirac's result,

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}{E}_n^{\text{ret}}({\mathbf{x}}_j,t)+E^{\text{damping}}({\mathbf{x}}_j,t).}$

Thus, the damping force is obtained without the need for self-interaction, which is known to lead to divergences, and also giving a physical justification to the expression derived by Dirac.

Criticism

The Abraham–Lorentz force is, however, not free of problems. Written in the non-relativistic limit, it gives

${\displaystyle E^{\text{damping}}({\mathbf{x}}_j,t)=\frac{e}{6\pi c^3}\frac{{\mathrm{d}}^3}{\mathrm{d}{t}^3}x.}$

Written in relativistic form (SI units), the magnitude of the force is the proper time derivative of the proper acceleration:

${\displaystyle F=\frac{e^2}{6\pi {ϵ}_0{c}^3}\frac{\mathrm{d}}{\mathrm{d}\tau }\alpha (\tau ).}$

Since the third derivative with respect to the time (also called the "jerk" or "jolt") enters in the equation of motion, to derive a solution one needs not only the initial position and velocity of the particle, but also its initial acceleration. This apparent problem, however, can be solved in the absorber theory by observing that the equation of motion for the particle has to be solved together with the Maxwell equations for the field. In this case, instead of the initial acceleration, one only needs to specify the initial field and the boundary condition. This interpretation restores the coherence of the physical interpretation of the theory.

Other difficulties may arise trying to solve the equation of motion for a charged particle in the presence of this damping force. It is commonly stated that the Maxwell equations are classical and cannot correctly account for microscopic phenomena, such as the behavior of a point-like particle, where quantum-mechanical effects should appear. Nevertheless, with absorber theory, Wheeler and Feynman were able to create a coherent classical approach to the problem (see also the "paradoxes" section in the Abraham–Lorentz force).

Also, the time-symmetric interpretation of the electromagnetic waves appears to be in contrast with the experimental evidence that time flows in a given direction and, thus, that the T-symmetry is broken in our world. It is commonly believed, however, that this symmetry breaking appears only in the thermodynamical limit (see, for example, the arrow of time). Wheeler himself accepted that the expansion of the universe is not time-symmetric in the thermodynamic limit. This, however, does not imply that the T-symmetry must be broken also at the microscopic level.

Finally, the main drawback of the theory turned out to be the result that particles are not self-interacting. Indeed, as demonstrated by Hans Bethe, the Lamb shift necessitated a self-energy term to be explained. Feynman and Bethe had an intense discussion over that issue, and eventually Feynman himself stated that self-interaction is needed to correctly account for this effect.

Developments since original formulation

Gravity theory

Main article: Hoyle–Narlikar theory of gravity

Inspired by the Machian nature of the Wheeler–Feynman absorber theory for electrodynamics, Fred Hoyle and Jayant Narlikar proposed their own theory of gravity in the context of general relativity. This model still exists in spite of recent astronomical observations that have challenged the theory. Stephen Hawking had criticized the original Hoyle-Narlikar theory believing that the advanced waves going off to infinity would lead to a divergence, as indeed they would, if the universe were only expanding.

Transactional interpretation of quantum mechanics

Main article: Transactional interpretation

Again inspired by the Wheeler–Feynman absorber theory, the transactional interpretation of quantum mechanics (TIQM) first proposed in 1986 by John G. Cramer, describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. Cramer claims it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes, such as quantum nonlocality, quantum entanglement and retrocausality.

Attempted resolution of causality

T. C. Scott and R. A. Moore demonstrated that the apparent acausality suggested by the presence of advanced Liénard–Wiechert potentials could be removed by recasting the theory in terms of retarded potentials only, without the complications of the absorber idea. The Lagrangian describing a particle (${\displaystyle p_1}$) under the influence of the time-symmetric potential generated by another particle (${\displaystyle p_2}$) is

${\displaystyle L_1=T_1-\frac{1}{2}\left((V_R{)}_1^2+(V_A{)}_1^2\right),}$

where ${\displaystyle T_i}$ is the relativistic kinetic energy functional of particle ${\displaystyle p_i}$, and ${\displaystyle (V_R{)}_i^j}$ and ${\displaystyle (V_A{)}_i^j}$ are respectively the retarded and advanced Liénard–Wiechert potentials acting on particle ${\displaystyle p_i}$ and generated by particle ${\displaystyle p_j}$. The corresponding Lagrangian for particle ${\displaystyle p_2}$ is

${\displaystyle L_2=T_2-\frac{1}{2}\left((V_R{)}_2^1+(V_A{)}_2^1\right).}$

It was originally demonstrated with computer algebra and then proven analytically that

${\displaystyle (V_R{)}_j^i-(V_A{)}_i^j}$

is a total time derivative, i.e. a divergence in the calculus of variations, and thus it gives no contribution to the Euler–Lagrange equations. Thanks to this result the advanced potentials can be eliminated; here the total derivative plays the same role as the free field. The Lagrangian for the N-body system is therefore

${\displaystyle L=\sum _{i=1}^N{T}_i-\frac{1}{2}\sum _{i\ne j}^N(V_R{)}_j^i.}$

The resulting Lagrangian is symmetric under the exchange of ${\displaystyle p_i}$ with ${\displaystyle p_j}$. For ${\displaystyle N=2}$ this Lagrangian will generate exactly the same equations of motion of ${\displaystyle L_1}$ and ${\displaystyle L_2}$. Therefore, from the point of view of an outside observer, everything is causal. This formulation reflects particle-particle symmetry with the variational principle applied to the N-particle system as a whole, and thus Tetrode's Machian principle. Only if we isolate the forces acting on a particular body do the advanced potentials make their appearance. This recasting of the problem comes at a price: the N-body Lagrangian depends on all the time derivatives of the curves traced by all particles, i.e. the Lagrangian is infinite-order. However, much progress was made in examining the unresolved issue of quantizing the theory. Also, this formulation recovers the Darwin Lagrangian, from which the Breit equation was originally derived, but without the dissipative terms. This ensures agreement with theory and experiment, up to but not including the Lamb shift. Numerical solutions for the classical problem were also found. Furthermore, Moore showed that a model by Feynman and Hibbs is amenable to the methods of higher than first-order Lagrangians and revealed chaoticlike solutions. Moore and Scott showed that the radiation reaction can be alternatively derived using the notion that, on average, the net dipole moment is zero for a collection of charged particles, thereby avoiding the complications of the absorber theory.

This apparent acausality may be viewed as merely apparent, and this entire problem goes away. An opposing view was held by Einstein.

Alternative Lamb shift calculation

As mentioned previously, a serious criticism against the absorber theory is that its Machian assumption that point particles do not act on themselves does not allow (infinite) self-energies and consequently an explanation for the Lamb shift according to quantum electrodynamics (QED). Ed Jaynes proposed an alternate model where the Lamb-like shift is due instead to the interaction with other particles very much along the same notions of the Wheeler–Feynman absorber theory itself. One simple model is to calculate the motion of an oscillator coupled directly with many other oscillators. Jaynes has shown that it is easy to get both spontaneous emission and Lamb shift behavior in classical mechanics. Furthermore, Jaynes' alternative provides a solution to the process of "addition and subtraction of infinities" associated with renormalization.

This model leads to the same type of Bethe logarithm (an essential part of the Lamb shift calculation), vindicating Jaynes' claim that two different physical models can be mathematically isomorphic to each other and therefore yield the same results, a point also apparently made by Scott and Moore on the issue of causality.

Conclusions

This universal absorber theory is mentioned in the chapter titled "Monster Minds" in Feynman's autobiographical work Surely You're Joking, Mr. Feynman! and in Vol. II of the Feynman Lectures on Physics. It led to the formulation of a framework of quantum mechanics using a Lagrangian and action as starting points, rather than a Hamiltonian, namely the formulation using Feynman path integrals, which proved useful in Feynman's earliest calculations in quantum electrodynamics and quantum field theory in general. Both retarded and advanced fields appear respectively as retarded and advanced propagators and also in the Feynman propagator and the Dyson propagator. In hindsight, the relationship between retarded and advanced potentials shown here is not so surprising in view of the fact that, in field theory, the advanced propagator can be obtained from the retarded propagator by exchanging the roles of field source and test particle (usually within the kernel of a Green's function formalism). In field theory, advanced and retarded fields are simply viewed as mathematical solutions of Maxwell's equations whose combinations are decided by the boundary conditions.