U. S. Department of CommerceNational Bureau of Standards

Research Paper RP1914Volume 41, August 1948

Part of the Journal of Research of the National Bureau of Standards

Applications of Magnetochemistry to Polymers andPolymerizationl

By Pierce W. Selwood 2

After a brief review of fundamental definitions, experimental methods, theoretical andsemiempirical results on diamagnetic substances, two topics are considered: First, studiesof the diamagnetic anisotropy of crystalline and oriented materials. Using solid naphtha-lene as an illustration, it is shown how to obtain the principal molecular susceptibilities fromthe measured parameters of the single crystal and the molecular orientations in the unit cellas derived from X-ray studies. Conversely, the direction cosines for solid diphenyl arecalculated from the measured macroscopic susceptibilities and the susceptibilities of themolecule. Results recently obtained on the anisotropy of cellulose, protein fibers, andstretched rubber indicate the value of such methods, particularly for oriented polymerscontaining aromatic groups, because of the large effects of the latter. Next, attention isdevoted to the paramagnetism of oxygen and free radicals that can be utilized as a measureof the concentration of the species. Radical concentrations as low as 10~6 mole per liter maybe estimated by means of a modification of the Gouy-balance. It has been employed in astudy of the thermal polymerization of styrene at 66° C. From the changes in diamag-netism in the course of the reaction, the rate of consumption of oxygen present is determined.Assuming this to be due to combination with styrene radicals, the rate of thermal initiationis estimated to be 3.8X10-14 mole-1 liter sec-1. However, in order to obtain by magneticmethods directly the free radical concentration in ordinary chain polymerization processes,the sensitivity would have to be improved by several orders of magnitude.

I. Introduction

There are two principal types of magneticmeasurements from which polymer chemistry maygain information. These are (1) the study ofdiamagnetic anisotropy, and (2) the in situ studyof free radicals and other paramagnetic com-ponents, such as oxygen. Neither of thesemethods has so far received much attention frompolymer chemistry. There is reason to believe,however, that they will both make importantcontributions to our knowledge of polymers andpolymerization in the not too distant future.

The numerical quantity with which magne-tochemists are chiefly concerned is the magneticsusceptibility, x, This quantity is defined asfollows: Let H be the intensity of a magneticfield and B be the magnetic induction in any

1 This paper was presented as part of the 1946-47 series of lectures on theProperties of High Polymers given at the National Bureau of Standards.

2 Professor of chemistry. Northwestern University.

substance placed in the field. We then have5 = ^ + 4 x 5 , where d is the intensity of magnetiza-tion, and d/iJ is the magnetic susceptibility perunit volume, K. Taking d as the density of thesubstance: K/d=x, which is the magnetic suscep-tibility per unit mass.

For nearly all organic compounds the magneticsusceptibility is negative. Such substances aresaid to be diamagnetic; they are repelled by amagnet. The order of magnitude of the suscep-tibility for organic compounds is — 0.3X10"6 to— 0.8X10"8. The susceptibility of water is—0.720X10"6. For diamagnetic substances thesusceptibility is independent of field strength andvirtually independent of temperature.

For many transition group elements and theircompounds, and for organic and inorganic freeradicals, the magnetic susceptibility is positive.These are said to be paramagnetic. The suscep-tibility of paramagnetic substances is often 10 to

Magnetochemistry and Polymers 151

100 times as large numerically as that of diamag-netic substances. Paramagnetic substances areattracted to a magnet. Positive magnetic suscep-tibilities are substantially independent of fieldstrength, but they frequently vary approximatelyinversely as the absolute temperature. Moreoften, they follow the Curie-Weiss law, x=C/-(T+A), where C and A are constants and Tis theabsolute temperature.

Magnetic susceptibilities are most frequentlymeasured by the method of Gouy (fig. 1). A

/ \

FIGURE 1. Principle of the Gouy method.

cylindrical sample is suspended from a balance sothat one end of the sample is in a region of highmagnetic field intensity, and the other end is in afield of negligible intensity. The fields commonlyused are produced by large electromagnets and areof the order of 5,000 to 20,000 gauss. On applica-tion of the field the sample will, if diamagnetic,seem to lose in weight, or if paramagnetic, seem togain in weight. The changes so observed are ofthe order of milligrams.

If the sample is a liquid or a powder, it may becontained in a calibrated glass tube. For most

measurements on organic compounds, it is neces-sary to use a microbalance to gain adequatesensitivity. Temperature control is generallynecessary, and for paramagnetic substances it isfrequently necessary to make measurements overa wide range of temperature. With all refine-ments, a sensitivity of four significant figures isattained without too much difficulty.

There have been innumerable modifications ofthe Gouy method and many other methods oper-ating on somewhat different principles. Thus,the sample may be suspended horizontally ratherthan vertically, the force may be measured bysprings, or by hydrostatic pressure. In allmethods related to the Gouy method the force,/ , exerted on application of the field is given by

where K is the magnetic susceptibility of the sub-stance per unit volume, K0)is the volume suscepti-bility of the atmosphere surrounding the sample,A is the cross-sectional area of the sample, andH is the field strength. The susceptibility ofhydrogen as an atmosphere is generally negligible,but air is appreciably paramagnetic.

The Faraday method for measuring suscepti-bilities measures the force exerted on a smallsample by a field having a fairly high gradientin the direction of motion of the sample. TheQuincke method is often used for liquids; itmeasures the capillary rise, or depression, pro-duced when a strong field is applied to the liquid[1 to 5]1.

Some 30 or 40 years ago, efforts were made toapply standard induction techniques to the mea-surement of magnetic susceptibilities. Thesemethods all failed because the susceptibilities areso very small. But recently Broersma [6] has de-veloped an induction method that is as sensitiveas the best Gouy determinations. At present themethod is considerably more complicated than theGouy method, but further development may makeit the method of choice for routine susceptibilitymeasurements.

II. Atomic and Molecular Diamagnetism

The classical theory of diamagnetism shows thatthe susceptibility per gram atom, %A, is givenby the expression

i Figures in brackets indicate the literature references at the end of thispaper.

152 Journal of Research

where r5 is the mean square radius of the electronicorbits. The same result is obtained in the quan-tum-mechanical theory of magnetism, from whichdiamagnetic atomic susceptibilities may in theorybe directly calculated [7]. For hydrogen

where n and I are principal and subordinatequantum numbers, respectively, Z is the atomicnumber, h is Planck's constant, and m is theelectron mass. Unfortunately, atomic hydrogenis not only difficult to study, but is paramagnetic.This paramagnetism is far larger than the under-lying diamagnetism could possibly be. However,indirect methods for estimating the atomicdiamagnetism give results in good agreement withtheory. Similar calculations give fair agreementwith experiment for the inert gases and for mercuryvapor.

Unfortunately, theoretical calculations of sus-ceptibility for polyatomic molecules are in arudimentary state. For the calculation of mole-cular susceptibilities our only recourse, at present,is the use of empirical constants derived from theconsideration of experimentally found suscepti-bilities for a large number of organic compounds.These constants, known as Pascal's constants, maybe used in much the same way that molecularrefractivities are used in calculating molar refrac-tion. The molar susceptibility is in general givenby

where nA is the number of atoms of susceptibility,XA is the molecule, and X is a constitutive cor-rection depending on the nature of the bondsbetween the atoms. Tables of Pascal's constantsand constitutive corrections are available inworks on magnetochemistry [2]. With their aidsome surprisingly accurate estimations of sus-ceptibilities have been made.

These considerations suggest that the processof polymerization should bring about a change ofmagnetic susceptibility. Farquharson [8] hasexamined the polymerization of 2,3-dimethyl-

butadiene. Consider a dimerization 2 B->B2.The molar susceptibility of the dimer must beXiw=2xB+X, where X is the constitutive correctionfor formation of the new bond. If n molecules ofB polymerize to form Bn, then X M = ^ X B + (fl — 1) \and the relationship between molar suceptibilityand n will be a straight line. The susceptibilityof the polymer per unit mass will be

_ nMB

where MB is the molecular weight of the monomer.The change of x with n will then be representedby a hyperbola. Farquharson has found thatsuch is the case for 2,3-dimethylbutadieiie, andfor certain other polymerizations. Whether thesusceptibility increases or decreases during poly-merization depends on the nature of the bondsbroken and formed. Both cases are known.

From the above it would appear that measure-ment of magnetic susceptibility may afford ameasure of the extent of polymerization and alsoan estimate of the molecular weight. Applicationsof the latter have been made by Farquharson [9]to the polyoxymethylenes. So far, however,there is no evidence to show that the magneticmethod can compete in these respects with anyof the standard procedures in polymer chemistry.It might possibly be adapted to certain in situtypes of studies where other methods are awkward.But the small differences of susceptibilitiesencountered and the effect of isomerism, hydrogenbonding, and impurities, is such as to discouragefurther attempts along this line.

III. Diamagnetic Anisotropy

We turn now to a much more promising appli-cation of magnetism to high polymers, namely,the diamagnetic anisotropy.

A substance, such as a gas, in which the mole-cules are all arranged at random, is truly isotropic.Effective isotropy may be shown by a solid, themicrocrystals of which have completely randomorientation. In crystals of low symmetry thereare three mutually perpendicular directions knownas the axes of principal magnetism. Along theseaxes the direction of magnetism correspondswith the direction of the applied field. The mag-netic susceptibilities along these axes are calledprincipal susceptibilities. Their values are often

Magnetochemistry and Polymers 153

unequal. If such an anisotropic substance ispowdered, the average susceptibility of thepowder is the mean of the three principal suscepti-bilities,

X=l/3(xi+X2+X3)'

Cubic crystalline matter is isotropic, but if apiece of noncubic matter is supended in a mag-netic field it will tend to orient itself so that theaxis of maximum (algebraic) susceptibility in theplane of rotation lies along the lines of force. Inorthogonal crystals the axes of principal suscepti-bility coincide with the crystal axes. In themonoclinic system one principal magnetic axiscoincides with the symmetry axis of the crystals.Many organic compounds crystallize in themonoclinic system.

The additivity of average suceptibilities oforganic compounds shows that even in the solidstate the mutual influence of neighboring moleculeson the magnetic susceptibility must be neglibible.(This is not true for paramagnetic compounds.)It follows that the anisotropy of a single crystalmust be due to the anisotropy of the unit cell, andthat this, in turn, depends only on the resultantanisotropy of the individual molecules in the cell.If the molecule is essentially magnetically iso-tropic, then the crystal will show little or noanisotropy. But if the molecule is anisotropicthen the resultant anisotropy of the crystaldepends only on the relative orientation of themolecules. If the molecules are arranged inlayers, then the resultant crystal anisotropy willbe large, but if the molecules are arranged hap-hazardly, or so that their anisotropies canceleach other, then the crystal will be isotropic.

An obvious way to measure magnetic anisotropyis to orient a single crystal so that one axis lies inthe field gradient in one of the conventionalmethods for measuring susceptibilities. ButKrishnan [10] has developed a far more elegantmethod. Krishnan's method gives the differencebetween any two principal susceptibilities.

If an isotropic sample is cut to spherical shapeit will suffer no orientation in a nonuniform field.Alternatively, an isotropic sample of any shapewill suffer no orientation in a uniform field. Thisis the basis of the Krishnan method.

The sample is suspended by a fine torsion fiberin a homogeneous field (fig. 2). A field of suffi-cient homogeneity may be found in a small region

FIGURE 2. Principle of th& Krishnan anisotropy balance.

between relatively large plane pole pieces. Thefield may be 5,000 to 10,000 gauss. For reasonsof economy a permanent magnet, mounted on asliding table, has some advantages for thesemeasurements. In general one magnetic axis ofthe sample is in the axis of the torsion fiber. Ifthe crystal is isotropic it will suffer no orientationon application of the field, but if it is anisotropic itwill turn until the algebraically largest suscepti-bility approaches the direction of the lines of force.The torsion head may now be turned until thecrystal suffers no orientation on application of thefield. The largest algebraic susceptibility is nowparallel to the field.

If the crystal is made to oscillate, the period ofoscillation is related to the molar magneticanisotropy as follows:

J 7 _=H2m t2

where t0 and t are oscillation periods with the fieldoff and on, respectively, C is the torsional constantof the fiber, m is the mass of the crystal, M is themolecular (or formula) weight, H is the field, andAx is the difference between the two principalsusceptibilities in the plane of rotation.

The anisotropy so obtained is that which existsin the plane of oscillation. The third principalsusceptibilit}^ may be investigated by reorientingthe crystal. Absolute principal susceptibilitiesmay be obtained by measuring one susceptibility

154 Journal of Research

by a direct method; or by using the average (pow-der) susceptibility.

Krishnan [11] has also developed a modificationof the above method. This second procedure hassome practical advantages over the first, especiallyfor very small crystals. The crystal orientation inthe field is adjusted as before so that the largestalgebraic susceptibility in the plane of rotationlies in the direction of the field. Now if thetorsion head is turned through an angle «, thecrystal will turn through a smaller angle <f>. Therelation between a and <t> is such that

where the terms have the same significance as be-fore. If the torsion head rotation is continueduntil 0=TT/4 the crystal will suddenly flip aroundto a new equilibrium position. For this to occurthe torsion head must be turned through an angleac, and we have:

or

The largest diamagnetic anisotropies are shownby graphite, and by aromatic compounds, themolecules of which contain benzene, cyanuric,or phthalocyanic rings. Carbon in the cubic(diamond) form is isotropic with a molar suscepti-bility of —6 X 10~6. But the principal molarsusceptibility of graphite parallel to the hexagonaxis is —264 X 10~6, which is over forty times nu-merically as great as the susceptibility normal tothe hexagonal axis. For naphthalene and anthra-cene the principal susceptibilities are givenbelow [12]:

Principal susceptibilities for single crystal XI06

NaphthaleneAnthracene

— xi

56.075. 5

— X2

146. 4211.8

— X3

76.6* 102. 9

tion studies, and the orientations of the moleculesin the.unit cell are known. It is possible to deducethe principal susceptibilities of the individualmolecules from the principal susceptibilities of thecrystal and the molecular directional cosines.Principal molar susceptibilities for individualmolecules of two compounds are shown below.

Principal susceptibilities for individual molecules X106

Compound

NaphthaleneAnthracene

56. 175.8

- K 2

53.962.6

- K 3

169.0251. 8

For many organic compounds the completestructure has been determined by X-ray diffrac-

Conversely, if the principal susceptibilities ofthe individual molecules are known or can beestimated, and if the anisotropy of the crystal ismeasured, it is then possible in favorable cases tofind the arrangement of molecules in the unit cell.In other words, the magnetic anisotropy may beused for a complete structure determination.More frequently, the anisotropy is of use in simpli-fying structural determinations by X-rays. Ex-amples of these calculations will be given below.

Various suggestions have been made to explainthe abnormal diamagnetism shown along one axisby aromatic compounds. The most popular ex-planation is that the aromatic, or resonance, elec-trons occupy orbits of molecular instead of atomicsize. A satisfactory general theory of anomalousdiamagnetic anisotropy in aromatic compoundshas been developed by London [13]. London'stheory corresponds to the method of molecularorbitals in the theory of chemical binding*

We shall now show how the observed principalsusceptibilities in the crystal may be used to findthe principal susceptibilities for the individualmolecule. Naphthalene is chosen as the example.

Naphthalene belongs to the monoclinic class.The crystalline and magnetic axes of a monocliniccrystal are shown in figure 3, arid figure 4 showsthe relation of a naphthalene molecule to theseaxes. Two of the principal magnetic axes lie inthe (010) or ac plane; the magnetic susceptibilitiesalong these axes are xi and X2- The angle #, whichthe xi axis makes with the c axis of the crystal,taken as positive toward the obtuse angle P be-tween the c and a axes, determines the positions ofthe two magnetic axes. The angle 0 is connected

Magnetochemistry and Polymers 155

FIGURE 3. Principal crystal (a, b, c) and magnetic (1, 2> 3)axes in a monoclinic crystal.

FIGURE 4. Position of naphthalene molecule relative tocrystal and magnetic axes.

with f by the relation 9O°+0+^=obtuse p. Thethird magnetic axis is along the b axis, and the sus-ceptibility along it is X3-

X-ray studies show that the unit cell of naphtha-lene contains two molecules, for each of which thestructure is plane. Both molecules have theirlengths, the lines that join centers of constituentrings, almost in the (010) plane, making an angle

of 12° with the c axis. Considerations of symmetry show that the long axis of the naphthalenemolecule must be one of its magnetic axis. Weshould, therefore, expect the above direction inthe crystal to be one of the magnetic axes of thecrystal. Direct observation of the magnetic axesconfirms this view, and the angle \f/ is actuallyfound to be 12°. The other two magnetic axesare along the breadth of the molecule in the planeof the rings, and along the normal to the plane ofthe rings, respectively.

X-ray studies show that the planes of themolecules are inclined at +65° and —65°, re-spectively, to the (010) plane. We have, there-fore, all the information necessary to find themagnetic susceptibilities of the individual molecule

Xi=Klf

X2=K2 cos2 65°+K3 sin2 65°,

X3=K2 sin2 65°+K3 cos2 65°,

where x and K are principal molar susceptibilitiesof crystal and of molecule, respectively. Sub-stituting the experimentally determined valuesfor xi, X2, X3, we obtain the numerical values forKi, K2, K3 given above.

Biphenyl will be chosen as an example to illus-trate the type of calculation in which the aniso-tropy is used to find the orientation of moleculesin the unit cell. We shall assume that the rings inbiphenyl have the same structure as in benzeneand that the molecule is planar.

The average (powder) molar susceptibility ofbiphenyl is —102.9 X 10~6, which is numerically lessthan twice that of benzene by 7.7X10"6. Thisdifference is obviously the contribution of the twohydrogen atoms that have been dropped. As afirst approximation, we assume that this diminu-tion is the same along the three principal axes ofthe molecule. The principal susceptibilities forthe benzene molecule are Ki = — 37.3, K2=—37.3K3= -^91.2(X 10"6). We then obtain for the prin-cipal molar susceptibilities of the biphenyl mole-cule, all X10"6,

K1 = K 2 =- (2X37 .3 ) -7 .7=-66 .9

K8=— (2X91.2) —7.7= —174.7.

The principal molar susceptibilities for thebiphenyl crystal are

Xi = -63.4X10"6

156 Journal of Research

X 2 =-146.5

X 3 =-98 .9 ,

and the angle \f/ is 20.1°.

There are two molecules in the unit cell. Placeboth molecules with their planes parallel to (100)and their lengths along the c axis. To bring themolecules to their actual orientations, they aregiven the following rotations (fig. 5):

First, a rotation about the c axis of one of themolecules through an angle X, and of the otherthrough an angle —X.

Second, a rotation of both molecules about theb axis through an angle 5, the positive direction ofthe rotation being defined as from the c axis tothe a axis, through the obtuse angle £.

Third, a rotation of the molecules through -\-vand —v, respectively, about the normal to theplane that contains the b axis and the directionof lengths of the molecules after the second rota-tion has been performed.

It is clear that 8=\//= +20.1°, and X and v canbe obtained from the following relationships:

Xi=Ki cos2 v+(K2 cos2 X+K3 sin2 X) sin2 v

X2=K2 sin2 X+K3 cos2 X

X3=Ki sin2 v-\- (K2 cos2 X+K3 sin2 X) cos2 v.

It has already been assumed that xi + X2+X3=Ki + K2+K3; therefore, only two of the above rela-tionships are independent.

Solving, we get

X=31°, v=0°.

The lengths of the molecules lie in the (010) planein the obtuse angle f$ at 20.1° to the c axis, andthe planes of the molecules are inclined at plusand minus 31°, respectively, to the b axis. Thisstructure determination is confirmed by X-rayanalysis. (^=20°, X=32°, *>=0°).

In the above analysis we have evaluated theprincipal susceptibilities of the molecule fromstructure considerations that may not be appli-cable for molecules more complicated than bi-phenyl. In such cases it is possible to obtain themolecular susceptibilities from measurements ofmagneto-optical rotation, according to methodsdescribed by Raman and Krishnan [14].

The structure determination given for biphenyl

FIGURE 5. Position of biphenyl molecule relative to crystaland magnetic axes.

is unfortunately not generally applicable to organicsolids. The simplicity of the derivation cornerfrom the special case of /*«u. For other classesof substances the information obtainable frommagnetic anisotropy is often less, but any aid fromsuch a source is welcome if it lessens the labor ofcomplete structure determinations from X-raydata. The applicability of the magnetic methodis described by Lonsdale and Krishnan [12]. I tmay be summarized as follows:

In the triclinic system, magnetic anisotropymeasurements give directly the molecular orienta-tions.

In the monoclinic and orthorhombic systemsthe magnetic method is of aid in structure deter-minations and is very valuable in a few special cases.

In crystals of high symmetry the magneticmeasurements are of little or no use [15].

The literature does not reveal many studies onthe magnetic anisotropy of polymeric material.A few studies have been made that show the exist-ence of the property in high polymers. But theobvious applicability of the method in the studyof polymers containing aromatic groups seems tohave escaped attention.

Mme. Cotton-Feytis [16, 17] has recently dem-

Magnetochemistry and Polymers 157

onstrated the anisotropy of several natural typesof fibers and of crude rubber. These substancescan, to a first approximation, be considered asuniaxial crystals. The largest numerical suscep-tibility is sometimes longitudinal, sometimes per-pendicular to the long axis. Various types ofcellulose (cotton, sisal, etc.), silk fiber, keratin,and collagen all show varying degrees of anisot-ropy. The magnitude of the anisotropy roughlyparallels the degree of molecular orientation asrevealed by X-ray studies. For example, a normalsample of collagen showing a high degree of molec-ular orientation has an anisotropy ten times aslarge as the same sample shrunk and renderedamorphous by treatment with formaldehyde.

Nilakantan has investigated the anisotropyof wood and cellulose. Wood is anisotropic,apparently because of a molecular anisotropy inthe cellulose molecule and the more or less regularorientation of these molecules along the fiber axis.Lignin and the hemicelluloses are amorphous.

The greatest diamagnetism of cellulose is alongthe fiber axis, which is presumably along the lengthof the molecule. This anisotropy in ^-cellulosederived from teakwood is about xn — X_L——O.lX10~6. The average susceptibility of a-cellulose inpowder form is — 0.508X 10"6. The methodappears to have some utility in confirming themolecular orientations of cellulose shown by X-raystudies.

Crude rubber normally has a certain degree ofanisotropy, and this is altered by hot and coldworking, and by compression and tension. Plexi-glass is said by Cotton-Feytis to yield somewhatsimilar results, but no details are given.

When crude rubber is stretched, the anisotropyincreases but seems to tend toward a limit. Thisparallels the effects observed with X-ray methodsand suggests more or less complete molecularorientation in one direction at high elongations.The anisotropy observed in stretched rubber is,incidentally, quite large. The figures reported byCotton-Feytis for oscillation periods with themagnetic field off and on, respectively, are 85seconds and about 5 seconds. There is no doubtthat the method is a sensitive one for detectingmolecular orientation in such systems.

The effects observed will doubtless be muchmore valuable in the study of high polymers con-taining aromatic groups. It seems probable thatdegrees of molecular orientation quite beyond

detection by other methods will be readily esti-mated by the anisotropy measurements. Thereare obvious applications of such studies to changesoccurring under compression, tension, extrusion,and hot and cold working. The establishment ofrelationships between anisotropy and physicalproperties such as hardness, tensile strength,optical properties, and second-order transitions, isalso a possibility.

IV. Atomic and Molecular Paramagnetism

We turn now to applications in which theparamagnetism of certain molecules or groups maybe used for the elucidation of certain problems inpolymerization. First, however, we shall presentsome general information on atomic and molecularparamagnetism.

The classical theory of Langevin develops anexpression for paramagnetic susceptibility on theassumption that each atom is a small permanentmagnet. These atomic magnets tend to alinethemselves with an applied field, but the aline-ment is resisted by thermal agitation. In modernterminology we identify the atomic magnets withthe magnetic moments produced by orbital andspin movements of the electrons.

Langevin's expression for the molar paramagnet-ism is

where N is Avogadro's number, /x is the permanentmagnetic moment, k the Boltzmann constant, andT the absolute temperature. The correspondingquantum mechanical expression derived by VanVleck [7] is

where 7Z2 is the square of the low frequency part ofthe magnetic moment vector, averaged over time,and this average being itself average over thevarious normal states appropriately weightedaccording to the Boltzmann factor. The quantitya is a combination of high frequency elements ofthe magnetic moment, and of the diamagneticpart of the susceptibility.

In general, the paramagnetic moment consistsof a part derived from an orbital contribution anda part derived from a spin contribution. Theonly major cases in which the orbital contribution

158 Journal of Research

is important is for isolated paramagnetic atomsand for rare earth ions. For those substancesin which only the spin contribution is importantthe paramagnetism is approximately representedby

where S is the resultant spin moment, and 0 is theBohr magneton, which is given by

fherg gauss"

where e is the electronic charge, h is Planck's con-stant, m is the mass of the electron, and c is thevelocity of light.

The effective magnetic moment in such casesis given by

where S may be found from the spectral multi-plicity, or, if the number, n, of unpaired electronsis known,

Finally, the magnetic moment may be foundexperimentally from

or more accurately from

These expressions are of great value in connectionwith studies of transition group ions, such asFe+3, Cr+3, Ni+2, and their respective complexes.However, for all but a very few cases of paramag-netism in molecules there is only one unpairedelectron, and the "spin only" formula applies.Furthermore, in almost all such cases the quantityA is not far from zero. Hence, for paramagneticmolecules such as triphenylmethyl, we find thatthe molar paramagnetic susceptibility at 20° Cis about l,270X10~6, and that,this varies inverselyas the absolute temperature.

Among paramagnetic molecules of interest wemay mention NO, NO2, C1O3, the triarylmethyls,semiquinones, and metal ketyls. All these sub-stances are characterized by having an odd num-ber of electrons. The existence of an unpairedelectron is essential for paramagnetism. Molecular

oxygen is paramagnetic, although it contains aneven number of electrons. The paramagnetismin this case results from the peculiar electronicstate in which two electrons remain unpaired. Asimilar situation exists in a few complex organiccompounds related to the Chichibabin hydro-carbons. Such substances are called biradicals.

It will be clear from the above that a majorapplication of magnetochemistry is in the detec-tion and estimation of molecules containing anunpaired electron spin, that is to say, of freeradicals.

V. Free Radicals

There have been many magnetochemical studiesof free radicals reported in the past 10 years,during which this has become the leading physicalmethod in free-radical chemistry. The generalnature of the results obtainable will be surveyedbriefly [2, 19].

Inasmuch as the most characteristic propertyof a free radical is its i^npaired electron, the mag-netic susceptibility is a most direct measure offree radical existence. Whereas molecular weights,colors, and chemical reactivity may depend uponsecondary factors, it is difficult to see how amolecule that does not contain a transition groupelement can be paramagnetic unless it is a freeradical. We choose to define a free radical as achemical entity, neutral or ionic, which containsone or more unpaired electrons, transition ele-ments being, of course, excluded.

In this way the free radical nature has beenestablished for such substances as diaryl nitricoxide, Fremy's salt, a great variety of hexaaryle-thanes, the hydrazyls, semiquinones, Wurster'ssalts, the metal ketyls, certain diaryl peroxides,diaryl disulfides, and of certain materials in thephosphorescent state. On the other hand, certainorganometallic compounds of which hexaphenyl-dilead is an example, have been shown to bediamagnetic, although molecular weight deter-minations seem to indicate dissociation to thefree radical form.

The magnetic method has a further advantagein its flexibility. Measurements may convenientlybe made on free radicals in solution at varioustemperatures and concentrations, and in varioussolvents. The importance of this type of measure-ment lies in the possibility of obtaining equilibriumconstants at several temperatures, and thus.

Magnetochemistry and Polymers 159»

opening the way for calculations of heats of disso-ciation, free energies, and entropies. A consider-able number of hexaarylethanes have been studiedin this way with a view to finding the effect of-different substituents on the heats of dissociation.Such data are of interest* in connection with theresonance theory of free radical stability.

Other information obtainable from magneticmeasurements on hexaarylethanes includes therate and activation energy of the disproportiona-tion reaction undergone by some free radicals, therelationships between color and free radical con-centration, and rate studies on certain photo-chemical reactions shown by these substances.One of the most valuable features of the magneticmethod is the possibility of studying, in situ,reactions as they take place, provided the con-centration of a paramagnetic component changeswith time, as is the case with free radical reactions121, 22].

I t should be mentioned that the magneticmethod is not infallible. Short-lived radicalscannot be studied with the magnet unless thesteady state concentration is sufficiently high.With the most refined apparatus a free radicalconcentration of 10~6 mole per liter might bejust detectable. Furthermore, there is one casereported in the literature in which the substance•appears to be a biradical but is diamagnetic. Thisis the simple Chichibabin hydrocarbon, to deriv-atives of which reference was made above. Thesimple hydrocarbon appears to catalyze theortho-para hydrogen conversion, which is evidenceof a paramagnetic component, but the compoundshows no trace of paramagnetism as measuredin the usual way [23].

"VI. Magnetic Measurements on ReactingSystems

As applied to reacting systems, magnetic sus-ceptibilities have been used as described aboveto study the rate of disproportionation of hexaar-ylethanes [24]. The transitory existence of semi-quinone free radicals has been demonstratedmagnetically by Michaelis [25], who studiedthe concentration of these substances as a functionof time, and in some cases by a process thatmight be called " magnetic titration," that is,the change of susceptibility during the quanti-tative addition of an oxidizing or reducing agent.A typical example of a reaction that may be

studied in situ magnetically is the slow reductionof dichromate with glucose in dilute solution.This reaction produces a large calculable changeof susceptibility owing to the change from thepractically nonmagnetic dichromate ion to thestrongly paramagnetic chromic ion. No changeof phase complicates this reaction, which hasbeen used to calibrate the very sensitive magneticbalance described below.

The possibility of studying reacting systemsin situ has suggested the use of magnetic measure-ments in polymerization reactions. The infor-mation obtainable in this way would appear tobe of two kinds. First, it would certainly bepossible to detect very small concentrations qfmolecular oxygen and to estimate their changeswith time. And second, the free radical theoryof polymerization suggests the attractive possi-bility of obtaining quantitative information onfree radical concentration during polymerization.

The magnetic study of reactions such as theserequires a degree of sensitivity far beyond thatnormally obtainable with the classical Gouy bal-ance. An apparatus using the Gouy principle,but with several refinements leading to greatly in-creased sensitivity, will be described (fig. 6). This

SAMPLE TUBE

THERMOREGULATOR

FIGURE 6. Horizontal Gouy balance.

uses a horizontal, rather than a vertical, mountingof the sample tube. The horizontal mounting hasactually been used for many years, and has re-cently been adopted by Theorell and by Calvinfor high sensitivity measurements.

The sample tube is suspended horizontallyfrom long delicate fibers. One end of the tube isbetween the poles of a magnet. A permanentmagnet has a great advantage in that the field isquite steady over long periods of time. This is acondition very difficult to attain with an electro-magnet. The substance under investigation is

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placed in the sample tube, which then moveshorizontally as the susceptibility increases or de-creases with time as the reaction proceeds. Thewhole apparatus is, of course, mounted so as toreduce the effects of vibration and of drafts to aminimum.

Displacements of the sample tube are observedfor approximate readings with the aid of a microm-eter microscope focussed on a fiduciary point.Kefined measurements are made with a Fabryand Perot type of interferometer, one plate of whichis mounted on the end of the sample tube. Withthis apparatus the sensitivity is such that 10"~6

mole of free radicals may just be detected [26].Some of the results obtained with this apparatus

are shown in figure 7. These results are for the

TIME IN HOURS

FIGURE 7. Changes of diamagnetic susceptibility duringpolymerization of styrene.

Thermal polymerization of styrene. O, stock room styrene, vacuumdistilled (+ degassed); D, styrene prepared from cinnamic acid, and A, de-gassed.

thermal polymerization of styrene. It will benoted that the magnetic susceptibility suffersrelatively large changes during the first few hoursof reaction. Under the experimental conditionsthis corresponds to a few percent of polymerization.The changes in susceptibility generally take theform of a large increase in diamagnetism followedby a much slower increase. The changes aremarkedly influenced by the purity of the styreneand the method of handling the monomer priorto its introduction into the apparatus.

The initial increase of diamagnetism is oftenmuch greater than would be found for completepolymerization. This "anomalous" diamagneticincrease is believed to be due to the consumption ofdissolved molecular oxygen. The inhibiting actionof oxygen on vinyl polymerization is well known.Assuming that no normal polymerization takesplace during the induction period (a small amount

of polymerization would have a negligible effeqt onthe susceptibility), then the displacement of thesample tube from zero time to,, the time at whichthe curve levels out, approximately represents theinitial concentration of oxygen present. Forcurve 1, figure 7, the initial concentration ofoxygen thus found is 2.9 X10~~4 mole per liter, andthis concentration steadily falls. At 2 hours theconcentration is 0.99 X10"4 mole per liter.

The rate of oxygen consumption is shown bythese results to be zero order. Reasonably con-cordant results are obtained on different samplesof styrene, although the initial oxygen concentra-tion may be quite different.

These results may be used to calculate the rateconstant for initiation. If the initiation processis represented as forming a biradical, oxygen re-acts with these free radicals very rapidly.

CeHs

2 C H 2 = C H

CeHs

CeHs CeHsI I

-CH2— CH— CH2— C H -

C«Hs

CH2-CH-CH2-CH-+O2CeHs

H2C

CH2\

CH—CeHs

\0

Such a process is represented by the equations

2M

R+O2 •RO2,

where R is a free radical and represents the con-centration of free radicals in the kinetic equationsand M is the monomer and its concentration.

— (1)

(2)

Since the rate of consumption of oxygen is experi-mentally zero order, it follows that eq 1 is therate-determining step. That is, the free radicalsare consumed as rapidly as they are formed owingto the rapid action of the oxygen. A steady stateis therefore reached, free radicals being used up

Magnetochemistry and Polymers 161

at the same rate at which they are formed anddR/dt=Oj therefore

andd[O2] 1

l~ dt M2'

—d[O2]/dt may be found from the rate of oxygenloss, and is equal in one of the runs to 2.8X10"8

mole per liter per second, and M (for pure styrenemonomer) is 8.56 moles per liter.

Hence

=3.8X 10~u mole per liter per sec.

This figure is not unreasonable. Bartlett [27]gives a rate constant for initiation in the photo-polymerization of vinyl acetate (which should bemuch higher than for thermal polymerization at66° C) as -3X10- 8 .

The above results are preliminary and are givento indicate the probable direction of magneto-chemical research on polymerization rather thanwith the idea of presenting a completed program.

If vinyl polymerization actually goes througha free radical mechanism there is presented theattractive possibility of determining the free radi-cal concentration by magnetic measurements onreacting systems. There have been some reportsin which this possibility is claimed to have beenrealized [28]. However, the best estimates offree radical concentrations give values of theorder of 10~9 moles per liter. These estimatesare made indirectly and are possibly subject tofairly large error. Nevertheless, opinion at thepresent time is that the sensitivity of the bestmagnetic balances is still several orders from thedetection of free radicals in this way. It may bethat an appropriate choice of reacting systemwould yield free radical concentration muchhigher than that normally believed to be present.But it must be admitted that such a possibilityseems to be somewhat remote.

VII. References

[1] E. C. Stoner, Magnetism and matter (Methuen andCo., Ltd., London, 1934).

[2] P. W. Selwood, Magnetochemistry (IntersciencePublishers, Inc., New York, N. Y., 1943).

[3] W. Klemm, Magnetochemie (Akademische Verlags-gesellchaft, Leipzig, 1936).

[4] S. S. Bhatnagar and R. N. Mathur, Physical principlesand applications of magnetochemistry (Macm'illanand Co., Ltd., London, 1935).

[5] R. W. Asmussen, Magnetokemiske Unders0gelser overUorganiske Kompleksforbindelser (Jul. Gjel-lurups Forlag, Copenhagen, 1944).

[6] S. Broersma, Magnetic measurements on organiccompounds (Martinus Nijhoff, The Hague,1947).

[7] J. H. Van Vleck, The theory of electric and magneticsusceptibilities (Oxford University Press, Oxford,1932).

[8] J. Farquharson, Trans. Faraday Soc. 32, 219 (1936).[9] J. Farquharson, Trans. Faraday Soc. 33, 824 (1937).

[10] K. S. Krishnan, B. C. Guha, and S. Banerjee, Phil.Trans. Roy. Soc. (London) [A] 231, 235 (1933).

[11] K. S. Krishnan and S. Banerjee, Phil. Trans. Roy.Soc. (London) [A]234, 265 (1935).

[12] K. Lonsdale and K. S. Krishnan, Proc. Roy. Soc.(London) [A]156, 597 (1936). This paper correctscertain erroneous data given in one of Krishnan'searlier papers. The incorrect data were unfortu-nately repeated in a review article by P. W. Selwoodand J. Parodi, J. Chem. Education 23, 574 (1946).

[13] F. W. London, J. phys. radium 8, 397 (1937).[14] C. V. Raman and K. S. Krishnan, Proc. Roy. Soc.

[A]113, 511 (1927).[15] W. A. Wooster, Physics of crystals (Cambridge

University Press, Cambridge, Eng.).[16] E. Cotton-Feytis, Compt. rend. 214, 485, 996 (1942);

215,299(1942).[17] E. Cotton-Feytis, Rev. gen. Caoutchouc 21, 26 (1944).[18] P. Nilakantan, Proc. Indian Acad. Sci. 7A, 38 (1938).[19] W. A. Waters, The chemistry of free radicals (Oxford

University Press, Oxford, 1946).[20] R. F. Preckel and P. W. Selwood, J. Am. Chem. Soc.

63,3,397(1941).[21] M. F. Roy and C. S. Marvel, J. Am. Chem. Soc. 59,

2,622 (1937).[22] H. G. Muller, Z. Elektrochem. 45, 593 (1939).[23] G. M. Schwab and N. Agliardi, Ber. 73B, 95 (1940).[24] P. W. Selwood and R. F. Preckel, J. Am. Chem. Soc.

65,895(1943).[25] L. Michaelis, G. F. Boeker, and R. K. Reber, J. Am.

Chem. Soc. 60, 202 (1938).[26] H. Boardman and P. W. Selwood, not yet published.[27] P. O. Bartlett, J. Am. Chem. Soc. 68, 2,381 (1946).[28] J. Farquharson and P. Ady, Nature 143, 1,067 (1939).

O

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