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QUARTERLY REVIEWS THE APPLICATION OF MONTE CARL0 METHODS TO PHYSICOCHEMICAL PROBLEMS by M. A. D. FLUENDY~ and E. B. SMITH (PHYSICAL CHEMISTRY LABORATORY OXFORD UNIVERSITY) WITH the advent of fast electronic computers the Monte Carlo method has become a powerful tool in the hands of the theoretical chemist. Based on random sampling it can be used to solve complex problems for which analytical methods are powerless with an accuracy limited only by statis- tical considerations. In many physical problems the basic elements of the calculation may be fully understood but the general complexity prevents solution by standard means. For such problems the Monte Carlo method is particularly appropriate whereas for simple calculations it is generally inefficient. As yet no general review of the applications of the method to problems of chemical interest has been available.Monte Carlo techniques have been used with some success in three fields of particular interest to chemists and these will be reviewed separately (1) the properties of dense gases liquids and solids ; (2) the configuration of macromolecules ; and (3) order-disorder phenomena. In each of these the appropriate analytical theories will be surveyed and their conclusions compared with the results obtained by Monte Carlo methods. The Monte Carlo methods to be described generally involve the simulation of physical systems (in a purely mathematical sense) and the method is not markedly different in principle from the physical simulation of such systems. There have been frequent attempts to use physical models as molecular analogues in order to obtain a better understanding of molecular processes and the structure of matter.Ball bearings seeds and similar objects have all been used to study the distribution of molecules in liquids and metals though because of the practical difficulties much of the work has been confined to two dimen- sions. Morrell and Hildebrand,’ however constructed a model liquid by means of gelatine balls hardened by chemical treatment in a matrix of liquid gelatine of the same density. The “molecular” distribution in this liquid was very similar to that observed in real liquids by X-ray diffraction studies. t Present address Department of Chemistry University of California Berkeley 4 California. W. E. Morrell and J. H. Hildebrand J. Chem. Phys. 1936 4 224. 241 242 QUARTERLY REVIEWS The Monte CarIo Method2# 3 The difficulty of devising appropriate mechanical models has severely limited the value of physical simulations of molecular systems.However the availability of fast electronic computers has made it possible to design calculations involving essentially the same approach. The Monte Carlo technique is in its essential and simplest form an experimental study of a mathematical model of some physical process. Thus if we wished at a game of chance to estimate our probable losses due to the “house percent- age,” we could proceed in several ways. The simplest but probably most expensive would be a direct appeal to experiment; as an alternative we could attempt to analyse the statistics of the game and perform a direct calculation. For all but the simplest games this would be prohibitively difficult.In such a case the statistical approach of the Monte Carlo method may be applied. From our knowledge of the rules of the game we can construct a directly analogous mathematical scheme using random numbers to represent the fall of dice or the distribution of playing cards. We can then “play” a large number of games until the average house winnings converge as closely as desired to a steady value the house percentage. With the aid of a fast electronic computer this apparently in- efficient method may produce a satisfactory result in a comparatively short time. The Monte Carlo method involving such sampling procedures can be applied to a very wide range of problems. A trivial but instructive example is the integration j j = f(x)dx.By choosing N specific values of x (xl x2 . . . xi . . . x,,) at random we can compute 1 i = l The accuracy of the estimate j of 7 will depend on both the number of samples of xt considered and on the nature of the function f(x). For example if we wish to estimate i) = 7 5 E with 95 % certainty the number of samples required ( N ) is given (when N is large) by N = 3-84 var (:)/e2. The variance var (j) is defined as A 1 var (9) = N - [ I f(x)2dx - 721. 0 Monte Carlo Method U.S. Dept. of Commerce National Bureau of Standards Symposium on Monte Carlo Methods (1954) University of Florida John Wiley Applied Maths. Vol. XII. and Sons Inc. New York. FLUENDY AND SMITH MONTE CARL0 METHODS 243 The efficiency of the method in the solution of any given problem will depend on the variance and the key to the use of the Monte Carlo method is the devising of sampling schemes which will produce a small variance and so converge rapidly.Basically most methods of variance reduction presuppose some approximate knowledge of the final answer so that each sample may be classed according to the size of its contribution. Samples are chosen not at random but from some other probability function which is nowhere zero and varies so that the probability of picking a sample is greatest in the region of most importance. Samples selected in this manner are weighted (by the inverse probability of selection) to obtain an unbiased final value. An example of both the Monte Carlo method and of the power of variance-reduction techniques in a very simple case is the determination of the area of a quadrant of a circle (Fig.1). The simplest way would be to FIG. 1. Quadrant of a circle of which the area can be determined by both Monte Carlo and variance-reduction techniques. sample pairs of xi and yf randomly from 0 to 1 estimating the fraction of total points within the quadrant by using the condition xi2 + yi2 < 1. The ratio of such samples to the total number would give the area. In this case the variance may be shown to be 0.17/N. A superior method would involve sampling only x randomly from 0 to 1 to evaluate the integral The variance is now 244 QUARTERLY REVIEWS This method would be three times as efficient as the previous one if the computing time per iteration were unchanged-an assumption which is no more than approximately true. Had only those points lying above the 1,l diagonal been considered and the area of the segment been estimated a variance as low as 0-015/N could be obtained.In more complicated problems even greater savings can sometimes be attained. The evaluation of a simple integral of this type would never be carried out by a Monte Carlo method since even if an analytical procedure were unavailable a systematic numerical quadrature would be very much more efficient. However for multiple integrations of four or more dimensions the Monte Carlo method is usually more powerful. The Monte Carlo method has been used in two somewhat different types of problem. The first class comprises those problems to which in principle a numerical method could be applied but is impracticable owing to com- putational difficulties.High-order multiple integrations are a typical example. The second type of application is in the solution of statistical problems for which no concise mathematical formulation is available but only a set of rules for forming a population from which samples may be drawn. In the physical wor!d there are many problems for which a good conceptual model of the individual events is available but in which the complex statistical nature of their interactions precludes any simple solution. This situation is particularly common in chemistry where the behaviour of dense gases and liquids the structure of polymers and order- disorder phenomena in the solid state and many similar problems have defied any quantitative understanding. The Monte Carlo method can therefoi e be expected to find considerable application in chemical problems.The remainder of this Review is concerned with the several attempts that have already been made to use Monte Carlo techniques to solve such problems. The three main areas are discussed separately. The recent applications of the method to the theory of absolute reaction rates are also reviewed. The States of Matter One of the most general problems which occur in physical chemistry is the evaluation of bulk properties of matter in terms of the forces acting between molecules (and vice versa). The problem in general cannot be solved; thus no adequate theoretical prediction of the properties of liquids or dense gases is possible. In the first place we lack precise knowledge of the forces between molecules. The Lennard-Jones intermolecular potential is generally considered adequate for the inert gases but it is an over- simplification when applied to more complicated molecules.Nevertheless even if an accurate knowledge of the forces between molecules were available the major difficulty would still remain that is the purely statistical problem of accounting for the mutual interactions of large numbers of molecules. The properties of a molecular system can be FLUENDY AND SMITH MONTE CARL0 METHODS 245 expressed in terms of the partition function which for a classical system is given by where p and q are the conjugate momenta and co-ordinates. The contribu- tion of the kinetic energy can be evaluated by direct integration but the contribution of the potential energy is not so easily evaluated. The so- called configurational integral may be written where U(q) is a function of all co-ordinates.As in general this integration cannot be performed most theories attempt to simplify it by approximating the relation between the potential energy of the system and the co- ordinates of the molecules. Thus the potential energy might be expressed as a function of the co-ordinates of a few molecules only. The derived thermodynamic properties are valid only so far as the initial assumptions are valid. Two such approaches are of particular interest. At low densities the equation of state of a gas can be expressed in terms of a series of virial coefficients PVIRT = 1 + B(T)/V + C(T)/V2 + . . . . . . The second virial coefficient B(T) is related to the relative co-ordinates of two (spherically symmetrical) molecules by the equation co [exp (- U(r)/RT) - l]r2dr where r is the separation of the molecular centres.If the variation of the intermolecular potential energy U(r) with r is known B(T) can be evalu- ated. At high densities the cell (or free-volume) theory has been used extensively with moderate success. A reference molecule is considered to move in a potential field produced by its neighbours which are assumed to be fixed on lattice sites. The method is not very satisfactory for liquids but it has been the basis of a large amount of research. The Monte Carlo method differs fundamentally from theories of this type as no attempt is made to simplify the configurational integral.4 if the forces acting between molecules are known it is possible to sample the configurations of a large number of molecules and so evaluate the con- figurational integral with an accuracy limited only by statistical considera- tions.The assumption of the pairwise additive nature of the intermolecular I. Z. Fisher Soviet Physics Review 1960 2 783. 246 QUARTERLY REVIEWS energy is an approximation common to most analytical and Monte Carlo methods. The simplest type of Monte Carlo procedure would sample the configurations generated by a number of molecules placed at random in a regular v01ume.~ The total energy of the system could be calculated from a knowledge of the form of the intermolecular potential enabling each overall configuration to be weighted by the appropriate Boltzmann factor. The value of any equilibrium property can then be obtained by averaging over sufficient configurations.The limitation of this approach is that except at the lowest densities any randomly constructed molecular distribution is likely to be an extremely improbable one. Thus in the case of hard-sphere molecules an acceptable non-overlapping configuration is almost impossible to generate by this method at densities greater than those appropriate to the dilute gas. The Monte Carlo method of Metropolis Rosenbluth Rosenbluth Teller and Teller.6-A more powerful method was devised by Metropolis et aZ.6 The molecules are confined to a rectangular or cubic space by periodic boundary conditions which in effect give rise to an infinite space made up of unit translations of the basic cell. It is considered that at least 32 or 64 molecules in the basic cell are required to give an adequate picture of the condensed states of matter.Initially the molecules are distributed on a lattice and new configurations are generated by moving one molecule at a time. In the case of hard-sphere molecules the theory takes a particularly simple form. If a move were such as to cause an overlap the potential energy would be infinite and the new configuration of the system would have zero probability and need not be considered. In such a case the system is returned to its original configuration. However all configurations which do not involve overlapping molecules are equally probable and contribute to the final average of any property. Each acceptable configuration is analysed in terms of a pair distribution function g(r) of the molecules. This function defines the number density of molecular centres at a distance r from a reference molecule.Thus the number of molecular pairs which are separated by a distance r is (N2/2 Y)g(r)4m2dr. After many configurations have been analysed the average pair distribution function is used to cal- culate the thermodynamic properties in the usual manner. Thus for hard spheres PV/RT = 1 - (27~/3V) [g(~)o3] where 0 is the diameter of the molecules and g(u) is the pair distribution function evaluated at the point of contact. This method represents a great advance on the simple technique first described in which molecules are introduced randomly into an appro- priate space. In place of a series of independent samples each weighted B. J. Alder S. P. Frankel and V. A. Lewinson J. Chem. Phys. 1955,23,417. Metropolis Rosenbluth Rosenbluth Teller and Teller J.Chem. Phys. 1953 21 1087. FLUENDY AND SMITH MONTE CARL0 METHODS 247 by the appropriate Boltzmann factor this method generates new configura- tions in such a way that the probability of their occurrence is proportional to the Boltzmann factor and then weights all configurations equally. This corresponds to the generation of a Markov chain with constant transition probabilities.’ Since the outcome of any trial is not independent but de- pends on the outcome of the trial immediately before it the probability of configuration E k is no longer associated with a fixed probability P k but to every pair E j E k there corresponds a conditional probability P j k . Thus if Ej occurs then the probability that E k follows is P j k . The various probabili- ties form a matrix of transition probabilities.A series of trials obeying statistics of this form is called a Markov chain. The absolute probability a,(”) of state E k after n steps starting from Ej can be expressed where a j is the probability that Ej occurs in the initial trial. a k @ ) should be independent of a if n is large. This is the case if p j k ( n ) converges to a finite limit independent ofj. This is usually so if no periodicities occur. States which have a mean finite recurrence time but are not periodic are called ergodic states and for a stationary distribution to occur all states must be ergodic. All such states can be attained from every other state. For such a system there is a stationary distribution which is unique and tends as n gets large to become independent of the initial state.In a classical molecular system each state occurs with a frequency proportional to the Boltzmann factor for that state. The commonly used transition probabilities are given by :8 k#j; 1 P j k = A i k u k < uj uk > uf = A j k exP[- (uk - U.)/kT]; P j j = - S j k 7 j%fk A j k is only non-zero when the two states differ only in the co-ordinates of one molecule. A further restriction is that A f k is zero if the move is such as to take the molecule a distance greater than that defined by an arbitrary fixed parameter a condition which is included to increase the probability that moves will be acceptable. This parameter determines the rate of con- vergence of the method but not the final average. In the case of hard spheres the above conditions reduce to 7 W. Feller “Probability Theory and Its Applications,” John Wiley and Sons Inc.8 W. W. Wood and F. R. Parker J. Chem. Phys. 1957,27,720. New York 1950 Ch. 15. 248 QUARTERLY REVIEWS The theory together with many computational details has been given by Wood et aL0sg Molecular Dynamics.-The Monte Carlo method of Metropolis et al. is concerned only with the equilibrium properties of molecular systems and the moves which the molecules undergo are not associated with real time but only with an arbitrary computer time. However it is possible to solve with any required accuracy the simultaneous classical equations of motion of a number of molecules-this explicit simulation technique has been called molecular dynamics.lO,ll The molecules are again confined to cubic or rectangular boxes by periodic boundary conditions.To start with the molecules are given equal kinetic energies with velocities in random directions and after each collision the new velocities of the interacting molecules are calculated. The number of collisions as a function of time and the sum of the momentum changes are recorded to enable the colli- sion rate and the pressure to be evaluated. This extension to transport properties represents a great advance in simulation techniques and enables a wide range of problems of a kinetic nature to be tackled. Hard-sphere Systems.-Hard spheres exerting no attraction on each other are the simplest type of molecular model; consequently the proper- ties of systems of such molecules are of particular interest. The potential is defined U(r) = 00; r <a; U(r) = 0; r > 0; where a is the collision diameter of the molecule.Before simulation calculations were undertaken it was a matter of controversy whether such systems could undergo phase transitions. The majority of studies have involved hard sphere molecules in two or three dimension~.*l~-~~ Fig. 2 illustrates the Monte Carlo equation of state for three-dimensional hard- sphere systems and those predicted by various theories. Not unexpectedly the virial expansion provides an adequate model for the low-density results whereas the cell theory is in poor agreement at all except the very highest densities. The most interesting feature of the Monte Carlo results is the phase transition which is observed in hard-sphere systems. The transition is between the fluid and the solid phase and despite the absence of attractive forces both phases are well defined.The Plate illustrates the change in the molecular motions in a system of hard spheres that has W. W. Wood and J. D. Jacobson Proc. Western Joint Computer Conference San * “Two-dimensional hard spheres” are more properly called hard discs. lo T. Wainwright and B. J. Alder NUOVO Cimento Suppl.. 1958 6. 116. 1’ B. J. Alder and T. W Wainwright 3. Cfiern. Pfiys. 1959. 31. 459. 1% M. N. Rosenbluth and A. W. Kosenbluth J Chem Phys. 1954 22 881. 13 W. W. Wood and J. D. Jacobaon J. Chem. Phys. 1957 27 1207. 1‘ B. .I. Alder and T. Wainwright 1. Chem. Plivs. 1957 27 1208. l6 B. J. Alder and T. Wainwright J. Chem. Phys. 1960 33 1439. Francisco California 1959 p. 261. FLUENDY AND SMITH MONTE CARLO METHODS 249 spontaneously undergone a transition from the fluid to the solid phase with no change in density.The photographs were obtained by following the molecular trajectories for 3000 collisions and using the method of molecular dynamics. Hard-sphere systems undergo no gas-liquid transition and behave like substances above their critical temperature. The simplest form of intermolecular potential to give rise to all three states of matter is the square-well intermolecular potential which allows for molecular at traction. I0 5 + FIG. 2. The equation of state for hard-spherical molecules. The unlabelled solid curves represent the results obtained by molecular dynamics for system of 108 molecules. V, i s the volume at close packing. The discontinuity at V/V % 1-55 is the fluid-solid phase transition. The predictions of various analytical methods are labelled F free- volume theory; V virial expansion; s.superposition theory. Other symbols + mo- lecular dynamics for 32 molecules; 0 Monte Carlo results of Wood and Jacobson;Ia 0 Monte Carlo results of Rosenbluth and Rosenbluth.l* (Reproduced by permission from ref. 13.) 250 QUARTERLY REVIEWS Monte Carlo calculations have also been made on binary mixtures of hard-sphere molecules with a radius ratio 5 :3.l6 Preliminary results showed that such systems undergo a contraction on mixing and a feature of the results for such mixtures is that the cell model is found to be a somewhat better approximation when used to predict changes on mixing than when applied to the pure components. Further work is under way. Up to moderate densities the initial configuration of the mixture can be generated by distributing the two species randomly in a lattice.At high densities the problem of generating initial configurations is severe and special techniques are required involving a gradual expansion of the radii of the molecules.17 At these densities the results indicate that transitions between certain types of configuration become very improbable and the total number of states may be effectively divided into more than one class between which transi- tions are extremely improbable. Wood who has called this the quasi- ergodic problem has pointed out that under these conditions spurious results will be attained. The method of molecular dynamics has been applied to hard-sphere systems and the equilibrium properties obtained agree well with the more recent Monte Carlo calculations.The phase transition was studied and found to be absent in systems of four or less molecules. Even with 500 molecules the two phases did not exist in the system simultaneously. The fact that more than four molecules must be involved makes it unlikely that the transition can be studied by analytical means. A full investigation of the dependence of the general thermodynamic properties on the number of molecules in the system has been made.15 Thirty-two molecules give a reasonably adequate prediction of the equation of state in the fluid region whereas at least 64 molecules are required at higher densities. An interesting feature of the results was the speed with which the Maxwellian velocity distribution was attained. Whereas configurational distributions are modified only very slowly the equilibrium velocity distribution is reached after 2-4 collisions per molecule.The equilibrium collision rate is given by the Enskog theory with considerable accuracy at all densities,’* but this theory gives the self-diffusion coefficient correctly only in the fluid region. The exact results which have become available for this simple molecular model have already provided a stimulus to the development of analytical theories. Recently a new theory has been developed19 which enables the equation of state of hard-sphere systems to be predicted with an accuracy never before attained. The use of the exact hard-sphere results as a basis for perturbation treatments of more realistic forms of intermolecular potential has also proved valuable.20 La E.B. Smith and K. R. Lea Nature 1960 186 714. l7 W. W. Wood personal communication. 1 0 H. Reiss H L. Frisch and J. L. Lebowitz J . Chem. Phys. 1959 31 369. J. 0. Hirschfelder C. F. Curtiss and R. B. Bird “Molecular Theory of Gases and E. B. Smith and B. J. Alder J. Chem. Phys. 1959,30 1190. Liquids,” John Wiley and Sons Inc. New York 1954 p. 634. FLUENDY AND SMITH MONTE CARL0 METHODS 25 1 The phase transition in hard-sphere systems. The traces were made by a succession of positions in a 32-particle hard-sphere system as determined by the method of molec- ular dynamics. 3000 collisions (a) in the liquid phase and (b) in the solid region. The solidification has taken place spontaneously as both systems are under identical conditions (V/Vo = 1.525). (Reproduced by permission from Alder and Wainwright.ll) Leunard-Jones Molecules.-The success of the method of Metropol is et al.with simple molecular systems led to its application to a more 252 QUARTERLY REVIEWS realistic molecular modeP The Lennard-Jones intermolecular potential U(r) = 4e[(a/r)12 - ( ~ / r ) ~ ] has been shown to be an adequate basis for the interpretation of the properties of the inert-gas molecules. The details of the calculation are somewhat different from that designed for hard-sphere systems. Any con- figuration j is changed to configuration k by moving one molecule. The potential energies Uj and Uk are then evaluated; if UR < Uj the new con- figuration is accepted. If Uk > Uj then exp [ - ( Uk - Ul)/kT] is compared with a random number between 0 and 1 and if the exponential is larger the new configuration is accepted.If the reverse is true the original con- figuration is retained. These conditions make the probability of any con- figuration proportional to its Boltzmann factor. Because of the compara- tively long-range nature of the Lennard-Jones forces the effect of distant neighbours was approximated in two ways first by assuming a lattice distribution and secondly by assuming a uniform distribution. The latter method proved satisfactory in most cases. The total potential energy of any configuration is estimated (for pairwise additive potentials) by N N i-I 1-1 t w The average potential energy over the Markov chain length gives the excess internal energy of the system. The excess heat capacity is calculated from the relation C,EIR = N[<(U/NkT)'>, - (U/NkT)2] and the equation of state from computations of the pair distribution func- tion g(r) a3 The thermodynamic properties were calculated for Lennard-Jones molecules at the reduced temperature kT/E = 2.74 and over a wide range of density to enable a comparison to be made with the experimental results of Michels et aL21 and Bridgman22 for argon at 55"c.Systems containing 32 and 108 molecules were studied and the results for both numbers were in good agreement. By using the potential parameters E and 0 determined by Michels from a study of the second virial coefficient of argon a com- parison of the Monte Carlo and the experimental results can be made without the introduction of any adjustable parameters. As illustrated by Fig. 3 the Monte Carlo results and those of Michels from 150 to 2000 atm.21 A. Michels H. Wijker and H. Wijker Physica 1949 15 627. 22 P. W. Bridgman Proc. Amer. Acad. Arts Sci. 1935 70 1. FLUENDY AND SMITH MONTE CARL0 METHODS 253 ‘ O 8 e! RT 4 are in excellent agreement but a discrepancy is evident between the calculations and Bridgman’s results from 2000 to 15,OOO atm. This may be due to the inadequacy of the Lennard-Jones intermolecular potential at i f. 6 t 0 8 IS I 06 I ! . I S 2 \‘;--- 3 4 6 8 1 0 3 - !\ 2 high densities or to the fact that the pairwise additive potential approxima- tion is no longer valid. However the facts that the Monte Carlo results provide a better continuation of Michels’s results (whereas there is a dis- continuity between these and Bridgman’s) and that Bridgman’s results for nitrogen reported in the same paper have disagreed with more recent equation-of-state determinations suggest that the Monte Carlo results may be more accurate than the experimental results at high densities.Order-Disorder Problems. Many physical and chemical problems involve order-disorder processes on lattices. Superlattice transitions in alloys and the various co-operative transitions in many solid compounds are common examples. A theory of regular solutions has also been developed by using a lattice model. The general theory of order-disorder processes is appropriate to many similar phenomena.23 aa E. A. Guggenheim “Mixtures,” Oxford University Press London 1952. 254 QUARTERLY REVIEWS The simplest case involves two species A and B distributed on a lattice. The total interaction energy can be written U = N u w u + NBB WBB + NAB WAB where Nap is the number of ct/3 neighbour pairs and wag is the interaction energy of a and species.An interchange energy w is defined as w = WAB - QWAA - &JBB which represents the energy of formation of an AB pair. The total energy is clearly dependent on the distribution of A and B on the lattice that is the degree of order present. The simplest treatment of this problem is to assume that the distribution is random despite the different energies of interaction. This has been called the zero-th approximation and was used by Bragg and Williams in their theory of superlattice tran~itions.~~ In the case of most liquid mixtures w is positive and the tendency is towards phase separation but for many metal alloys w is negative and the tendency to form unlike pairs leads to a regular array called a superlattice in which the number of such pairs is maximum.Both tendencies are overcome by high temperatures which favour random mixing and definite transition temperatures exist at which randomness overcomes the ordering tendency. Such transitions are associated with a large rise in heat capacity. The transition temperature (Tc) is given by Tc = ~ ~ / 2 k for the zero-th approximation where z is the number of nearest neighbours. More sophisticated approximations make some allowance for the fact that the energy differences involved in the formation of various pair leads to a non-random distribution. Thus Guggenheim suggested that a parallel with chemical equilibrium could be drawn and proposed This equation is the basis of the quasi-chemical theory which is a first-order approximation.This theory gives Tc = w/{ k In [z/(z - 2)]}. Higher approximations are prohibitively difficult to evaluate by normal methods but the problem is most suitable for Monte Carlo study. Salsburg Jacobson Fickett and Wood25 applied the Monte Carlo method to a two- dimensional triangular lattice-gas (a lattice whose sites are only partly filled by interacting particles) for which an exact solution is available. The co-ordination number of the lattice is 6 and the molecules interact only 24 W. L. Bragg and J. E. Williams Proc. Roy. SOC. 1934 A 145 699; ibid. 1935 25 Z. W. Salsburg J. D. Jacobson W. Fickett and W. W. Wood J. Chem. Phys. .4 151 540. 1959 30 65. FLUENDY AND SMITH MONTE CARL0 METHODS 255 with nearest neighbours the interaction giving rise to a pair energy E .N such particles are considered distributed on M lattice sites. New con- figurations are generated by moving a “molecule” to an unoccupied site selected at random and the system is contained by periodic boundary con- ditions. The details of the method are similar to that devised by Metropolis et al. If the new configuration has a lower energy it is accepted. If not the exponential of the energy difference is compared with a random number between 0 and 1. An exponential greater than the random number leads to the acceptance of the new configuration otherwise the system is returned to its original state. Systems of from 8 to 98 molecules were studied on lattices for which 50% of the sites were occupied (N/M = 0.5) at a reduced temperature defined by exp(- EIkT) = 2.The results were extrapolated to give values corresponding to an infinite number of molecules. The ratio U/E that is the ratio of the average internal energy per molecule to that of a single pair interaction (or the average number of nearest-neighbour bonds per molecule) was found to be 1.896. The value given by exact treatment26 was 1.8938. About 500,000 configura- tions were considered in each case. A further study was made of the varia- tion of the properties of the system with the fraction of sites occupied. The results show the power of the Monte Carlo method in dealing with problems of this type. In a similar study Fosdick2’ considered a two-dimensional lattice and for a 20 x 20 lattice rapid convergence was reported except near the Curie point but as yet only a preliminary notification of the research is available.The Monte Carlo method has also been applied to more realistic lattices of interest to metallurgists. Guttman28 investigated order-disorder phenomena in a body-cenrted cubic lattice containing two types of atom. He considered systems of from 256 to 1024 molecules at various composi- tion ratios. With OM = OBB = 0 and w = WAB the total energy is given by U = - NAB w where NAB is the number of A-B interactions. The results for an infinite crystal are summarised in the annexed Table together with the predictions of various theoretical approximations. pc represents the value of the short- range order parameter at the transition temperature. p is defined by NAB - NAB (random) = NAB (pertect order) - NAB (random) rs G.F. Newelland E. W. Montrose Rev. Mod.Phys. 1953,25,353. 27 L. D. Fosdick Bull. Arner. Phys. Soc. 1957 2 239. L. Guttman J. Chem. Phys. 1961,34 1024. 256 QUARTERLY REVIEWS TABLE. cubic lattice at the critical temperature.2a Bragg-W ill iams (zero-th order) 0.5 0.693 1 -50 0 Quasi-chemical (1st order) 0-435 0.652 1 a78 0144 Comparison of the results for a 1 :1 superlattice in a body-centred Method kT,/zo S,” CVINk P c Domba 0-385 0.548 (4 0-31 1 Monte Carlo 0.379 0.495 (4 0.4 1 8 dynamique union intern. phys. Paris 1952 p. 177. aDomb Changements de phases Comp. rend. reunion ann. avec comm. thermo- The heat-capacity maximum in the region of order-disorder transition reaches a maximum in an asymmetric manner. The Monte Carlo results are in quite good agreement with the experimental heat capacities for /3cu-zn considering the primitive nature of the model used.Guttman concludes that the power of such calciilations lies not so much in the fact that they can be adapted to give exact numerical solutions of these problems as in the insight they give towards the construction of a simple but adequate theoretical model. F o ~ d i c k ~ ~ investigated order-disorder phenomena in a A3B alloy on a face-centred cubic lattice using a rather more sophisticated model. He defined interaction parameters ~ ( n ) = Qmfi(a) + 8 wBB(~) - WAB(n) where n refers to the order of the neighbour. Considering only first- and second-nearest neighbours two parameters are defined dl) and h = d2)/d1) the latter representing the ratio of the interchange energies of the first and the second series of neighbours.Using systems of 500 lattice sites in 5 x 5 x 5 unit-cell arrays and with periodic boundary conditions he studied the system when X = 0 - 0.25 and - 0.50. The transition temperature at these values of X is given by kTc/d1) = 1.01 1-61 and 2.25 respectively. Taking X = - 0.25 for the Cu,Au system dl) is calculated to be 816 cal./mole compared with values of 802 and 71 1 suggested by previous workers. The Properties of Macromolecular Systems The properties of high polymers of both natural and synthetic origin have been of interest for many years. Their anomalous physical properties such as viscosity which is dependent on the shear rate light scattering and long- range elasticity have provided a challenge to theory. The aim of all work in this field is to explain these macroscopic properties in terms of the detailed molecular geometry of the long chains involved.Unfortunately a know- ‘II L. D. Fosdick Phys. Rev. 1959,116 565. FLUENDY AND SMITH MONTE CARL0 METHODS 251 ledge of the molecular formula and geometry of a polymer in the normal chemical sense is not sufficient since the properties of such molecules do depend directly not upon the formal chemical structure but upon the actual configurations adopted by the polymer chains. In most work the geometric properties of these chain configurations have been reported in terms of the average square distance between the ends of a chain of N links <RN2>av. This quantity furnishes an indication of the size of the molecule and can be used in the prediction of its viscous properties.For example it has been suggested30 for polymer solutions that where yo is the intrinsic viscosity which is obtained by extrapolation to zero concentration and zero rate of shear M is the molecular weight and K a constant. Thus the first step in any rigorous treatment of the properties of a macromolecule must be the evaluation of the “average configuration,” ideally in terms of a probability density function (W) which defines the probability of finding a chain segment of the macromolecule in a given region of space. This function being given the problem is reduced to one in mechanics albeit a very complex one. It is with the calculation of W that we shall be concerned in the remainder of this discussion. In principle it should be possible to calculate Wexactly from a knowledge of the segment-segment interactions and the molecular geometry but in practice the statistical complexity is too great.However many approxi- mate methods have been devised. *The simplest m o d e l ~ ~ ~ t ~ ~ consider a polymer molecule as a long freely jointed chain or random walk of equal steps a model directly analogous to Brownian motion and for which the geometric properties can be determined exactly. Attempts have been made to improve this model by allowing for the effect of restricted rotation and fixed bond angles on the mean chain configuration. In the first such attempt K ~ h n ~ ~ showed that the variations in end-end length due to these molecular factors over a chain of many links could be dealt with by grouping several molecular links together to form a “statistical length.” The actual molecular chain could then be replaced in the calculation by a chain made up of an equivalent number of freely jointed statistical lengths.The exact physical relation of the statistical length to the link length depends principally upon the intramolecular forces assumed in the cal- culation of Boltzmann weighting factors for each configuration. An alternative method (e.g. ref. 34) confines the chain segments to a lattice the tetrahedral diamond lattice being particularly suitable for use when discussing quadrivalent carbon chains. The relative probability for 30 M. L. Huggins J. Phys. Chern. 1938,42 911. 31 E. Guth and H. Mark Akad. Wiss. Wien. Sitzungber. IIb 1934 143 445. 32 W. Kuhn Kolloid Z. 1934 68 2. 33 W . Kuhn KolloidZ. 1936 76 258.34 R. P. Smith J . Chem. Phys. 1960 33 876. 2 258 QUARTERLY REVIEWS each possible step at every stage of the chains’s growth is computed as in the Kuhn model by considering the Boltzmann weighting factor for the step. The random walk on a lattice is known as a Polya walk and it is possible as in the case of a walk in continuous space to calculate the average configuration of the chain. In general the theoretical calculations predict that a limit of the form lim<RR2>av = N12A N-+W exists where I = the individual link length and A is a constant depending on the molecular properties of the chain. Unfortunately all calculations based upon these types of model are deficient in one important respect. No account is taken of the impossibility that two chain segments occupy the same region of space.In other words no allowance is made for the “excluded volume”. The Problem of Excluded Volume.-Despite many attempts to in- corporate this factor in calculations only one exact result has been obtained (on the limiting-step entropy for walks on a lattice).* In general the theoretical treatments have given rise to widely divergent predictions. Indeed it would be little exaggeration to say that all possible predictions as to the effect of excluded volume on chain configuration have been made at one time or another. The difficulty seems to lie mainly in the mathe- matical complexities associated with problems such as this in which the probability of any step depends directly not only upon the immediately prior step but upon all the previoussteps R ~ b i n ~ ~ in an ambitious attempt to secure a solution to the problem of self-avoiding random walks in continuous space was able to show that it was a bad approximation to neglect even high-order interactions (i.e.interactions between segments separated by many chain links) and that no useful approximate solution was likely to be found; however he was able to set an upper bound of 0.5 on E in the empirical equation lim <RN2>av = KN(l+€). in a less fundamental approach assumed that the density distribution of the excluded volume chain about a central reference point was Gaussian in form but broadened by segment-segment interaction. Thus <RN2>av = <RN2>,” A2 where <RN2>at refers to the simple model involving no excluded volume corrections and A is an “expansion coefficient.” From thermodynamic considerations they deduced that A5 - A3 = BN+ and that <RN2>av/N increased with N.Unfortunately dSoo = k log lim N 4 W Flory and * The limiting step entropy dS for walks on a lattice is defined as ) Number of allowed n step walks N+ * (Number of allowed (n- 1) step walks 35 R. J. Rubin J . Chem. Phys. 1952 20 1940. 3a P. J. Flory and T. G. Fox jun. J. Arner. Chem. Soc. 1951 73 1904. FLUENDY AND SMITH MONTE CARL0 METHODS 259 the approximations are of doubtful validity and it has been shown recently3’ that (A5 - A3)/N* is not constant but is an increasing function of A. Other have used the simple unrestricted random walk as a first approximation and have attempted to allow for the effect of the excluded volume by incorporating additional “volume perturbation” terms in equations such as the differential Fokker-Planck equation which describes the distribution of end-end distances for simple random walks.However the solution is very sensitive to the nature of the terms included and con- flicting conclusions have been obtained. Work on the other model of a polymer-the self-avoiding random walk on a lattice (SARWL)-has followed a more purely mathematical course; use has been made of a method in which rigorous upper and lower bounds on E in the relation lim<RN2> av = KN(l+“) are calculated thus enabling the true value to be bracketed.39 Unfortunately it is not possible to produce a very accurate estimate of E in this way since the calculations become pro- hibitively difficult if any but drastic simplifications are made. A method of direct enumeration has also been used in which the number of allowed configurations is counted The self-avoiding random walk has provided the only exact theoretical result in the field.Hammersley and Morton41 were able to show that the limit p = lim CN/CN-l exists for certain lattices where CN and CN-l are the total numbers of N and N - 1 step self-avoiding walks possible on the given lattice. The actual value of p cannot yet be deduced mathematically but must be found experimentally. The constant p is often known as the attrition constant since it determines the,probability that the SARWL will be disallowed at its next step owing to an intersection. It is in fact related to the limiting-step entropy. Monte Carlo Calculations.-The disagreement between the various approximate theoretical treatments of the chain-configuration problem suggests that none of them is entirely satisfactory.Unfortunately it is difficult to use experimental evidence to resolve this conflict since all the experimentally accessible quantities depend upon some bulk property which as yet can only be related to the polymer configuration in a rather arbitrary fashion. In tbis situation it would obviously be valuable to have some means of checking the accuracy of the theoretical prediction at some intermediate stage. A Monte Carlo method is well suited to this purpose since it can supply “experimental” information about chain configuration that would not otherwise be available. An electronic computer can readily be programmed to generate a lattice model of a macromolecule by selecting at random steps along the 87 M.Kurata W. H. Stockmayer and A. Roig J . Phys. Chem. 1960,33 151. 38 E. W. Montroll J. Chern. Phys. 1950,18,734; J. J. Hermans M. S. Klamkin and 3s M. E. Fisher and M. F. Sykes Phys. Rev. 1959 114,45. 40 M. E. Fisher and B. J. Hiley J. Chem. Phys. 1961,34 1253; 1961,34 1531. J. M. Hammersley and K. W. Morton J. Roy. Stat. Soc. 1954 By 16 1 23. N + W N + m R. IJllman J . Chern. Phys. 1952,20,1360; H. M. James ibid. 1953,21,1629. 260 QUARTERLY REVIEWS lattice vectors disregarding all chains which intersect. The “population” of random walk self-avoiding chains so generated can then be used to determine the various averages such as end-end length of theoretical and practical interest. The method used differs in one important respect from that applied in the simulation of systems of hard-sphere molecules at high densities (as in the equation of state investigations) where the new configurations were achieved by the random motion of the constituent molecules over com- paratively short distances.Thus there was a strong correlation between the configurations at successive iterations and the approach to equilibrium was in some sense at least kinetic. In the generation of model polymer chains the packing density is much smaller (though this might not apply to a polymer in a very unfavourable solvent or with strong intersegment at- tractive forces) and it is consequently possible to generate entirely new random configurations of the chain at each iteration (though we shall see later that in some cases greater efficiency can be achieved by retaining a certain amount of the previous configuration).Thus the averaging is carried out over a spatial or Gibbsian ensemble. This method of averaging has several advantages and in particular it should be noted that if this method were not used severe complications would arise due to the existence of certain classes ofState-corresponding perhaps to knotted configurations of the polymer chain-between which the probability of transition while not zero would be very low. Thus the probability of the molecule’s “escaping” from this state during the “time” over which the averaging is carried out may be effectively zero. This type of behaviour is effectively non-ergodic and convergence of the average to the true value for systems showing this behaviour cannot be expected in any reasonable computing time.The first attempts to perform this type of calculation were made by King on simple punched-card equipment,42 but most of the subsequent work has been due to Wall and his co-w~rkers.~~ All the investigations have been concerned with self-avoiding walks on various lattices the tetrahedral being the most intensively studied. Interest has been focused mainly on the variation of the mean-square end-end length as a function of the number of steps in the chain although the radius of gyration ring-closure probability and limiting-step entropy have all been considered. It was found that for most two- and four-dimensional lattices the ratio 4a G. W. King Monte Carlo Method U.S. Dept. of Commerce Nat. Bureau of Standards Appl. Maths. Vol. XII. 43 F. T. Wall L.A. Hiller and D. J. Wheeler J . Chem. Phys. 1954 22 1036; F. T. Wall and L. A. Hiller Ann. Rev. Phys. Chem. 1954 5 267; M. N. Rosenbluth and A. W. Rosenbluth J. Chem. Phys. 1956 23 356; F. T. Wall L. A. Hiller and W. F. Atchison ibid. 1955 23 913; F. T. Wall L. A. Hiller and W. F. Atchison ibid. p. 2314; F . A. Cotton and F. E. Harris J. Phys. Chem. 1956 60 1451; F. T. Wall L. A. Hiller and W. F. Atchison J. Chem. Phys. 1957 26 1742; F. T. Wall R. J. Rubin and I. M. Isaacson ibid. 1957 27 186; F. T. Wall and J. J. Erpenbeck ibid. 1959,30 634; G. S. Rushbrooke and J. Eve ibid. 1959 31,1333; P. J. Marcer D.Phi1. thesis Oxford 1960. FLUENDY AND SMITH MONTE CARL0 METHODS 26 1 < R N ~ > av/N diverged and converged respectively as N increased. But for three-dimensional lattices of practical interest the trend of this ratio is less clear and chains of greater length must be sampled.This is not easy since as the chain becomes longer the probability of successfully adding a further step decreases. The probability that a step will be allowed was foundg3 to decrease exponentially with increasing chain-length i.e. if N = number of s step walks successful and No = number of walks started then N = No exp (- As) where X is called the attrition coefficient. To counter this difficulty various “enrichment” procedures are used and much recent research has been concerned with developing satisfactory schemes for extending the chain lengths that can be sampled. One method due to Wall and Erpenbe~k,~~ utilises allowed chains of a certain length (say s steps) as a basis for the development of a number of longer chains so that a tree of such chains is produced (see Fig.4). It can be seen that r -s stops-- r /Fails / t-S stops --.- -3’steps--* Fails FIG. 4. Chain enrichment by branching. If j is the number of levels at which branching occurs and p the number of branches started at each level wasteful attrition or excess branching is minimised if p = exp (As) where A is a constant and s the number of steps at each level. wasteful branching or attrition is minimised when the number of walks started at each level is equal to the number of walks expected to fail within s steps. A more sophisticated technique devised by Mar~er,*~ uses the branching procedure of Wall and Erpenbeck but instead of adding steps singly whole sections of chain (40 step sections were in fact used) are added.These sections of self-avoiding chain are generated without difficulty by a straightforward Monte Carlo method. This technique is more efficient than the original Wall and Erpenbeck method since intersection within the 262 QUARTERLY REVIEWS individual lengths is prevented and the overall chain attrition much re- duced. With this method reasonably large samples of chains up to 2000 links in length can be obtained. By these techniques a good knowledge of the properties of self-avoiding chains has been built up. The results for tetrahedral chains have been fitted to the relation lim<Rfi2>av = KN(l*) (by analogy with the simple random-walk problem). As longer chains have been sampled it has been found necessary to change the parameters e.g.for N z 200 <RN2>av = BN1-22 was suggested; for N z 800 <Rhi2>av = B'N1-ls gave a better fit; for Most recently of all Marcer has suggested that a relation of the form lim<RN2>av = KNIogN gives a better fit both to his values for chains up to 2000 links in length and to the previous data due to Wall. The typical behaviour of some SARWL's is illustrated in Fig. 5. In the case of the three-dimensional lattice at ieast the limiting behaviour has not been reached and the plot is curved towards the log N axis. Fig. 6 shows a comparison between the Monte Carlo results of Wall et al. and of Marcer and some A factors determined by viscosity measure- N+03 N z 2000 <RN2>av = B"N1.13. N-+m 1-0 2.0 3.0 lo9,0N FIG. 5 . Typical behaviour of (RN2)&, for various types of lattice.( 1 ) Self-avoiding random walk in one dimension (gradient = 2). (2) Self-avoiding random walk on a two-dimensional square lattice. (3) Self-avoiding random walk on a three-dimensional diamond lattice. (4) Self-avoiding random walk on a typical four-dimensional lattice. (5) Unrestricted random walk in three dimensions (gradient = 1). (This Figure is based upon one given by Wall Hiller and Atchis~n.'~) FLUENDY AND SMITH MONTE CARL0 METHODS 263 ments;44 as can be seen the Monte Carlo results conform quite well to the experimental values over the range of “A” values available and indeed suggest that even the comparatively primitive lattice model used in these FIG. 6 . obtained by viscosity measurement^.^' at 30”c. Comparison of “A” factors calculated from Monte Carlo results with .factors “A” factors from viscosity data for polyisobutenes 0 at 20”c; 0 in cyclohexane Limiting dependence of A on N,from Monte Carlo results A Wall Hiller and Wheeler;43 B Wall and E r p e n b ~ k ; ~ ~ C M a r ~ e r .~ ~ Curve F shows the value of A as calculated from the Flory relation A5 - A3 = BN+ The A factors from viscosity measurements were computed by using the relation While for the Monte Carlo results A was calculated as A2 = (R.vz)aV/(RNz) a; B being taken arbitrarily as 1 0 0 for this plot. A’ = .[77o]/KM+. where (RNB) ,” was found by Tobolsky’s method34 using E = 0.8 kcal. mole-l. calculations may form a reliable guide to the configuration of a polymer molecule. A more detailed comparision of theoretical and Monte Carlo results has been made by Kurata et aL3’ Random Walks in Continuous Space.-The self-avoiding walk on a lattice is of course at best only an approximate model of the polymer molecule.A more realistic simulation would allow the walk to occur in continuous space and would incorporate as accurately as possible the various intramolecular-force fields. The application of a Monte Carlo method to a model of this type would correspond quite closely to the solution of the multidimensional integral formulated by Rubin. Naturally the greater mathematical complexity of this type of model will limit the 44 P. J. Flory and W. R. Krigbaum J. PoZymer Sci. 1953,11 37. 264 QUARTERLY REVIEWS chain lengths accessible to computation but it is possible to compute end-to-end length distributions and other characteristic properties which can be compared with experimental observations of for example the transition-state enthalpy in cyclisation reaction^.^^ Computation of Absolute Reaction Rate The theoretical calculation of absolute reaction rates has long been the subject of investigation but at present progress is severely limited by the mathematical difficulties of the transition-state theory.Briefly this theory involves the calculation of a multidimensional surface describing the potential energy of the reacting system as a function of the position of all the nuclei. This of course in itself is an immensely difficult task but in simple cases such as the H2-H reaction a reasonable approximation to the potential-energy surface can be constructed. If equilibrium is assumed between the reactants and the so-called “activated complex” which exists in the neighbourhood of the saddle point in the potential-energy surface it is possible to calculate the concen- tration of the complex in the system.The activated complex can suffer one of two fates; it may decompose to yield the original reactants or reaction may occur with the formation of products. The fraction of such fruitful collisions is known as the “transmission coefficient.” Thus if the potential- energy surface and the concentration of activated complex are known and the transmission coefficient can be calculated our understanding of the reaction rate is complete. The most elegant approach is the purely quantum-mechanical one in which the problem is regarded as the transmission of a wave packet through a potential-energy surface and a solution obtained by solving a time-dependent Schrodinger equation ; unfortunately the mathematical difficulties of this method are at the moment almost pr~hibitive.~~ An alternative procedure possible in some cases is to treat the assembly classically and after computation of the potential-energy surface from quantum-mechanical considerations to formulate the Hamiltonian equations of motion for the system of reacting nuclei.The track of all the nuclei can then be followed by solving these equations at successive short intervals of time and hence the outcome of the collision namely reaction or no reaction can be determined. Wall Hiller and Mazurg7 have used this procedure in an attempt to study the reaction H + H -+ H2 + H. In their method the positions of all the nuclei throughout the interaction are “plotted” on a cathode-ray screen and photographed at short intervals plots similar to Fig.7 being obtained. By repeating this procedure many times for different initial 46 M. A. D. Fluendy unpublished results. E. M. Mortensen and K. S. Pitzer in “The Transition State,” Chem. SOC. Special 4’ F. T. Wall L. A. Hiller and J. Mazur,J. Chem. Phys. 1958,29,255; 1961,35,1284. Publ. No. 16 1962. FLUENDY AND SMITH MONTE CARL0 METHODS 265 conditions of translational vibrational and rotational energy it was possible to determine the relation between these quantities and the probability of reaction. Though the results published so far are very - 4 P z 2 u Y 0 v .t 0 L 0 u ' - 2 - 4 1 t 4 2 0 -2 -4 16-41 I FIG. 7. The figures show the tracks of all the reacting nuclei in the reaction H + H2+ A hydrogen atom is shown coming from the bottom of the diagram to collide with a vibrating hydrogen molecule (the top two tracks).For clarity collinear collisions are illustrated (a) Reaction (b) No reaction (Reproduced with permission from ref. 47.) H2 + H. limited it is apparent that the vibrational energy plays only a small part in activating the complex and that molecules with little or no rotational energy are more likely to react with the colliding hydrogen atom. Conclusion From the preceding discussion it is clear that the Monte Carlo tech- nique can be applied to a wide range of physicochemical problems where it can be used not only to provide numerical results but also to guide the intuition in the construction of new theories. It must be remembered however that it is expensive of computer time and therefore a method of the last resort which should only be used for tackling those problems for which analytical or conventional numerical methods are useless.What- ever the application of the method it is essential that the fullest use be made of any information available concerning the expected result so that an efficient sampling scheme may be devised since in some cases the computing time required to achieve a given accuracy may be reduced by several orders of magnitude by good sampling. Indeed so important are the savings available that it may prove worthwhile to attempt to design a programme that would automatically optimise its sampling as the com- putation proceeded. In fields outside chemistry the method has been shown to have wide applications. Successful applications range from scattering problems in nuclear physics to models of economic behaviour and the reorganisation of industrial stock piles. One thing seems certain; as electronic computers 266 QUARTERLY REVIEWS of the present generation are replaced by new models (at least 100 times faster) the scope of the Monte Carlo method must increase. In particular the properties of systems involving long-range interactions such as plasmas and electrolytic solutions may be studied. The method may also find application in the investigation of extended gravitational systems surface-catalysis effects the probability of knotting and similar topological problems in long-chain polymers and in quantum mechanics. The authors thank Mr. R. P. Bell F.R.S. for advice and encouragement and Dr. D. Handscomb for helpful comments on the manuscript.

ISSN:0009-2681    

DOI:10.1039/QR9621600241

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

ADDITION POLYMERISATION AT HIGH PRESSURES By K. E. WEALE COLLEGE OF SCIENCE AND TECHNOLOGY LONDON S.W.7) (DEPARTMENT OF CHEMICAL ENGMEERING AND CHEMICAL TECHNOLOGY IMPERIAL 1. Introduction.-Pressure is potentially as significant a variable as temperature in chemical reactions and its effects are being more widely investigated as the techniques for developing controlled pressures up to 100,000 atm. become better known. Many gas-phase reactions are considerably influenced by pressures of a few hundred atmospheres because of the increased concentrations of the reactants. Higher pressures of a few thousand atmospheres and upwards produce large chahges in reaction rates in liquids. These are not primarily mass-action effects but are determined by the contraction or expansion accompanying the formation of the transition state from the reacting molecules and they are intimately related to the reaction mechanisms.Measurements of the pressure-dependence of rate constants have provided fundamental information about such processes in many types of reaction including that of addition polymerisation. The first general studies of polymerisation at high pressures made over thirty years ago were prompted partly by the consideration that polymers have higher densities than the monomers from which they are produced. Conant and his collaborators1 and Starkweather2 subjected nearly fifty substances to pressures of 3000-12,000 atm. and found that some which formed no polymer or small amounts at 1 atm. gave substantial yields at the highest pressures. These were principally vinyl compounds and conjugated dienes but included the cyclic hydrocarbon indene the sym- metrical ester diethyl fumarate and higher aldehydes that form soft polymers unstable at ordinary pressure.Another seven olefinic substances were studied by Sapiro Linstead and N e ~ i t t ~ who found a-methyl- styrene to be the most responsive to pressure; and in 1933 Fawcett and Gibson discovered4 the high-pressure polymerisation of ethylene which has become of great industrial importance. At the time of these experiments the physical chemistry of polymerisa- tion was in an early stage of development and it was not possible to inter- pret the effects of pressure in any detail. Much more is now known about the changes caused in the separate reaction steps largely as the result of studies of the polymerisation of styrene.Bridgman and Conant Proc. Nut. Acad. Sci. U.S.A. 1929 15 680; Conant and Tongberg J. Amer. Chem. Soc. 1930 52 1659; Conant and Peterson ibid. 1932,54 628. Starkweather J. Amer. Chern. Soc. 1934,56 1870. Sapiro Linstead and Newitt J. 1937 1784. See Pemn Research 1953 6 1 11. 267 268 QUARTERLY REVIEWS 2. The Polymerisation of Styrene under Pressure.-An early attempt6 to measure the acceleration of the polymerisation of styrene gave the huge factor of 2.24 x lo6 at 140"/1500 atm. A re-investigation by Gillham6 showed that it is considerably less than 10 and that experimental short- comings particularly failure to carry out the reaction isothermally had led to the error. Gillham measured the polymerisation rate for styrene without initiator up to 4000 atm. at 100".At 3000 atm. the rate was 10.1 times faster but the molecular weight of the product was only 1.55 times greater than at 1 atm. The difference between the two factors indicates that the effect of pressure is complex. A simple acceleration of the rate of initia- tion of chains would not increase the polymer molecular weight while alternatively an increased rate of chain growth or a decreased rate of chain termination would cause the overall rate and the polymer molecular weight each to increase by a similar factor. Further measurements on styrene were made by Merrett and Norrish' up to 5000 atm. at 60° using benzoyl peroxide as initiator. With improved techniques they obtained reproducible kinetic results which have been confirmed by later workers. Their findings are illustrated in Fig.1 which shows the logarithm of the overall rate to be nearly a linear function of pressure (with an acceleration factor at 3000 atm. of 7.2). The polymer molecular weight is approximately tripled between 1 and 3000 atm. but increases only slightly at higher pressures. The reaction order with respect to initiator concentration is 0.5 at 1 atm. but decreases a little with increas- 1.0 I Pressure (at m.) FIG. 1. The effect of pressure on the overall rate of polynierisation of styrene (A) at 60" and on tire polymer molecular weight (B) (right-hand scale). Tamman and Pape 2. anorg. Chem. 193 1 200 1 13. Gillham Trans. Faraduy SOC. 1950 46 497. i Merrett and Norrish Proc. Roy. SOC. 1951 A 206 309. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 269 ing pressure An explanation of the changes in rate and molecular weight was suggested in terms of the effect of pressure on the separate steps of the reaction.For the free-radical polymerisation the rate equations are usually written Rate of production of radicals by initiator decomposition Rd = 2fkd[I] (1) Rate of initiation Ri = ki [Z*l[ M] (2) Rate of chain growth R P = kdM1 [R*l (3) Rate of chain transfer Rtr == ktr[R*][Y] (4) Rate of termination Rt = kt[R*I2 ( 5 ) Equation 1 refers to the production of two radicals Z* from each molecule of initiator I by homolysis,fbeing the efficiency of the radicals in starting chains. Chain initiation is by addition of Z* to the monomer M and chain growth proceeds by successive addition of monomer molecules to the chain radical R*. Chain transfer is a radical-displacement reaction between R* and a molecule Y (monomer initiator solvent or polymer) producing a polymer molecule and the radical Y* which initiates a new chain.The kinetic chains are terminated by collisions between polymeric radicals (equation 5) which result in disproportionation or combination the latter being strongly preferred in polymerisation of styrene. The rate constants of liquid-phase reactions vary with pressure according to the equation :* in which d V* is the volume change which occurs when the transition state is formed from the reacting molecules. The decomposition of benzoyl peroxide involves the stretching and breaking of the 0-0 bond so that dV* is likely to be positive for this reaction and k d will be decreased by pressure. Chain growth (and initiation) are bimolecular addition processes for which negative values of dV* and a consequent acceleration by pressure may be expected.Termination by combination should also have a negative dV* but Merrett and Norrish considered that this reaction between two large radicals is diffusion-controlled and is retarded at high pressure because of the increased viscosity of the medium. The transfer reactions are believed to occur through the abstraction of an atom from the molecule Y by the chain radical. Transfer does not generally affect the overall rate of polymerisation but may sometimes determine the polymer molecular weight. As transfer involves both bond-formation and bond- breaking the sign of dV* is uncertain but it was assumed that the rate of transfer is increased by pressure and later evidence supports this.a (In k)/W = - A V*/RT . . . . . . . (6) t~ (a) Evans and Polanyi Trans. Faraday Soc. 1935 31 875; 1936 32 1333; (b) Hamann,‘ ‘Physidhemical Effects of Pressure,” Buttenvorths Scientific hbls. Lon- don 1957. 3 270 QUARTERLY REVIEWS From the general considerations the increased molecular weight and rate of polymerisation are attributed mainly to the acceleration of chain growth by pressure while the levelling-off of polymer molecular weight is due to an alteration in the balance between the termination and the transfer reactions. At 1 atm. most molecular chains are terminated by combination and at first an increase in pressure causes an increase in molecular weight by slowing down this reaction. At higher pressures the accelerated transfer reactions determine the molecular weight although kinetic termination still mainly follows equation 5.The well-known combination of equations 1,2,4 and 5 gives the overall rate of polymerisation Roy in the steady state as . . . . . . (7) where 6 = kp/kt*. The slight decrease in the exponent of [I] at higher pressures may indicate some participation of primary radicals Z* in termination reactions with R*. Many organic reactions are polar in nature and the sign of dV* for these is often determined by differences in the solvation of the reacting molecules and the transition state. The implicit assumptions that solvation changes are unimportant in the various stages of radical polymerisation and that d V* is simply related to alterations in the intermolecular distances and interatomic bond lengths of the reactants are reasonable but require strengthening by more direct evidence.In subsequent work Nicholson and Norrish9 have isolated the effects of pressure on kp kt and kd for styrene and have confirmed most of the inferences made from the earlier results. (a) The pressure-dependence of kp and kt. Measurements of the steady- state rate of polymerisation (equation 7) can determine only the ratio k,/kt and additional information is needed if the two constants are to be separately evaluated. This was obtained from experiments on the photo- chemical polymerisation at high pressures under conditions of intermittent illumination (rotating-sector method) with the results shown in Fig. 2. kt decreases rapidly up to lo00 atm. and then more slowly while kp increases exponentially with pressure.d V* for the chain-growth reaction is calculated to be -13.4 c.c./mole which may be compared with a volume difference of 22.7 c.c./mole between monomer and polymer for the polymerisation at 1 atm. The decrease in kt is probably due to the suggested viscosity effect but this has not been independently established. (b) The pressure-dependence of k d . The thermal decomposition of benzoyl peroxide in carbon tetrachloride solution was studiedg over the same pressure range at 60" and 70". Two mechanisms are distinguishable. The first is a radical-induced chain decomposition which is accelerated by pressure but is ineffective in initiating polymerisation and is probably Nicholson and Norrish Discuss. F'ruduy SOC. 1956 22 97 104. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 271 suppressed in the presence of monomer.The unimolecular decomposition corresponding to equation 1 is retarded by pressure and kd at 3000 atm. is about half of its initial value (Fig. 2). Results have been obtained for the effect of pressure on other homolytic dissociations by EwaldlO and by Walling et a/." The values of d V* calculated from equation 6 are given in Table 1. 0.8 - 0.6 - C -0.4 - u FIG. 2. The efect of pressure on kp (A) and kt (C) for styrene at 30"; and on kg @)for the homolytic dissociation of benzoyl peroxide in carbon tetrachloride at 70" (Nicholson and Norrish*). All the dissociations are retarded by pressure so that AY* is always positive but there are significant variations with the solvent the values in benzene and carbon tetrachloride being roughly twice those in toluene and cyclohexene.The lower values are in agreement with approximate calcula- TABLE 1. A V* for initiator dissociation. Initiator Benzoyl peroxide Benzoyl peroxide Benzoyl peroxide Azoisobutyronitrile Azoisobutyronitrile t-Butyl peroxide t-Butyl peroxide t-Butyl peroxide t-Butyl peroxide Solvent CC14 Acetophenone Toluene Toluene Toluene Cyclohexene Benzene CCl ca4 Temp. 60" 70 80 62.5 62.5 120 120 1 20 120 d V*(c.c./mole) 9.7 8.6 4.8 3.ga 9-4b 5.4 6.7 12-6 13.3 a Initiator disappearance followed photometrically. Rate of radical formation by lo Ewald Disciiss. Faraday Soc. 1956 22 138. l1 Walling and Pellon J. Amer. Chem. Soc. 1957 79 4786; Walling and Metzger iodine-scavenger technique. ibid. 1959 81 5365. 272 QUARTERLY REVIEWS tions of the volume effect which assume a 10% stretching of the 0-0 or C-N bond in the transition state and a constant molecular cross-section (estimated from the van der Waals radii of the bonded atoms).An explanation of the larger values found in the other solvents has been suggestedll but is not discussed here. (c) The pressure-dependence of ktr. The effects of pressure on k d kp and kt postulated by Merrett and Norrish are therefore confirmed and the acceleration of transfer reactions by pressure has also been verified. Walling and Pellon12 examined the polymerisation of styrene in carbon tetrachloride solution up to 6OOO atm. at 60° and found that ktr for the reaction between the polymer radical and the solvent is increased to nearly the same extent as kp. d V* for this transfer is about -1 1 c.c./mole.Toohey and Weale13 found that transfer between polystyryl radicals and triethylamine is also accelerated but that d V* is apparently much smaller (- - 1 or - 2 c.c./mole) than for the chain growth. The abstraction of an atom from a neutral molecule presumably occurs via a bimolecular transi- tion state formed with a contraction in volume. The contraction should be less than in the chain-growth reaction because bond-breaking occurs simultaneously; but if with some substrates the polar structures suggested by Walling1* contribute to the transition state an additional contraction due to solvation may be expected. A variation of dV* for transfer with the structure of the attacked molecule if confirmed could be important in the behaviour of allylic monomers (Section 4). In addition Ewaldl6 has shown that transfer reactions between diphenylpicrylhydrazyl and thiols are strongly accelerated at 10,OOO atm.3. The Relation of the Pressure Effect to the Structure of the Monomer.- Kinetic data for monomers other than styrene are limited to overall rates of polymerisation under pressure. dV* calculated from these is a com- posite quantity (= LIV*~ + AY*& - dY*J2 if equation 7 is valid) and its values are slightly high if as here they are uncorrected for increases in [MI and [I] due to compressibility. However the compressibility cor- rections are generally similar in magnitude and as d V*p is much the largest term the overall dV* is a useful approximate guide to its variation with monomer structure. Values for styrene,13 ally1 acetate,16 methyl meth- acry1ate,l7 a-methylstyrene,l* and a~enaphthylenel~ are shown in Table 2.Walling and Pellon J. Amer. Chem. Soc. 1957 79,4776. Is Toohey and Weale Trans. Faruduy SOC. in the press l4 Walling “Free Radicals in Solution,” Wiley New York 1957 p. 158. l6 Ewald Trans. Furaduy SOC. 1959,55 792. la Walling and Pellon J. Amer. Chem. Soc. 1957 79,4782. I8 Elroy Ph.D. Thesis London University 1961. Lamb and Weale .Roc. Symp. on Phys. and Chem. High Pressures 1962. Romani unpublished results. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 273 TABLE 2. A V* for various monomers from the overall rate of polymerisa- tion. Monomer Styrene Styrene (in toluene S/M = 4-3) Styrene (in triethylamine S/M = 3.3) Ally1 acetate Methyl methacrylate 01- Methylstyrene Acenaphthylene (in toluene S/M = 4.6) Pressure range (atm.) 1-3000 1-4400 1-8500 1-3000 3000-4500 1-2000 i-3aoo Temp.60” 60 60 80 40 60 60 AV* (c.c./mole) - 18 - 17 - 17 - 14 - 19 - 17 - -10 (S/M denotes the mole iatio of solvent to monomer.) For monomers containing the structure CH,=C the substituent groups appear not to affect dV* and ally1 acetate is not necessarily exceptional because the peculiar characteristics of the reaction (Section 4) may affect the apparent d V*. Acenaphthylene contains the cyclic structure (I) and for this monomer d Y* is definitely lower. Calculations suggest20 that the separation between radical and C=C group 4 Hc;c in the transition state is approximately the mean of the van der Waals and the covalent distance while the C=C bond length is almost unchanged. d Y* for chain growth should therefore be about half the volume change for the reaction M-+ (l/n)M, which is roughly true for styrene.21 The close similarity between d V* for styrene with and without a solvent is also of interest.In many organic reactions a large negative dV* is associated with a large negative entropy of activation AS* and both result mainly from the solvation of partial ionic charges in the transition state.8b The considerable negative dY* for styrene chain growth is also linked with a large negative AS* (-35 e.u. at 1 atm.22) but neither can be due to charge-solvation. Conant and Peterson1 suggested that an “align- ment” of monomer molecules is favoured by pressure (i.e. that compres- sion assists formation of the transition state by reducing the translational and rotational entropy of monomer molecules) but in this case the pres- sure-acceleration should decrease with dilution.4. Allylic Monomers at High Pressures.-The study of styrene has yielded much fundamental information about high-pressure polymerisation but this is restricted in some ways by the nature of the monomer. In the temper- ature and pressure range investigated polymerisation of styrene is strongly favoured thermodynamically so that depolymerisation does not occur; and chain transfer to monomer is relatively unimportant compared with chain growth. The influence of pressure on the thermodynamics of polym- (I) 1o Evans Gergeley and Seaman J. Polymer Sci. 1948,3 866. 11 Whalley Discuss. Faraday SOC. 1956,22 146. aa Bamford and Dewar Proc. Roy. SOC. 1948 A 192,309. 274 QUARTERLY REVIEWS erisation and the balance between chain transfer and chain growth is shown by recent work on monomers containing the allyl group CH2:CCH2-.(a) a-Methylstyrene. There is good evidence that at 1 atm. the ceiling temperature for polymerisation of a-methylstyrene CH,:CPhMe is about 60-61 ". At higher temperatures depolymerisation is faster than chain growth and polymer formation is impo~sible.~~ The results of Sapiro Linstead and Newitt3 imply that the ceiling temperature varies with pressure and Kilroe and Weale24 found that it increases linearly from 61" at 1 atm. to 171" at 6480 atm. If an equation is assumed of the same form as the Clausius-Clapeyron relation dTc/dP = T A V / AH and AH has a value25 of -8.4 kcal./mole then the volume change for polymer- monomer equilibrium in the liquid is -14.7 c.c./mole.This is of the approximate magnitude to be expected for the algebraic difference between the estimated values of A Y* for chain growth and depropagation. A sharp maximum in the rate of polymerisation as a function of pressure was also observed. Elroyl* has shown that this is due to solidification of the monomer the freezing point of which increases linearly with pressure from -23.2" at 1 atm. to 60" at 4860 atm. The region in which polymerisation of a-methylstyrene is possible is therefore limited by the ceiling temperatures and the freezing points which both increase markedly with pressure. The molecular weight of the polymer increases with pressure at constant temperature. At 3000 atm. the molecular weight is constant (-28,000) over much of the temperature range in which polymerisation can occur but it rises appreciably near the freezing temperature and falls abruptly as the ceiling temperature is approached.At pressures very close to the freezing pressure polymers with molecular weights of 200,000-600,000 were obtained. Some unsaturated liquid dimer is produced and above the ceiling temperature this is the only reaction product. The dimer probably results from transfer reactions involving abstraction of an allylic hydrogen atom from monomer and the formation of the resonance-stabilised radical CH,:CPh.CH,-. The rate of d;mer formation is accelerated by pressure but to a smaller extent than the chain growth. (b) AU'Z acetate. The polymerisation of allyl acetate (CH2 =CH.CH,. OAc) has been studiedls at 80" and pressures up to 8500 atm. The molecular weight of the polymer is low (-2000) and increases only slightly with pressure but the reaction rate increases exponentially being 50 times greater at the highest pressure than at 1 atm.Degradative transfer with allylic hydrogen to yield an unreactive radical CH,:CH.CH(OAc)* is apparently much faster relative to chain growth in this monomer than in a-methylstyrene. It is considered that ktr/kp is largely independent of es Dainton and Ivin Quart. Rev. 1958 12 61. * Jessup J. Chem. Phys. 1948 16 661. Kilroe and Weale J. 1960 3849. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 275 pressure (so that the molecular weight is nearly constant) but a part of the increase in rate is attributed to an enhanced reactivity of the allylic radicals which may restart chains at high pressures.(c) Isopropenyl acetate. Bywater and Whalley26 have investigated the polymerisation of a-methylvinyl acetate CH :CMe.OAc at 50-120" and up to 10,OOO atm. Products of low molecular weight are formed up to 6000 atm. but at higher pressures the reaction is reasonably fast and yields solid polymers (M - 30,000). The polymerisation resembles that of a-methylstyrene rather than that of ally1 acetate. Allylic transfer appears relatively unimportant and the effect of pressure on rate and molecular weight is probably due to the acceleration of a slow chain growth and a decrease in chain termination; but the possibility of a ceiling-temperature effect at the lower pressures is not excluded. The behaviour of these allylic monomers illustrates the effects of pressure when chain growth is relatively slow and transfer depolymerisation or termination competes strongly.Before monomers with other types of structure are discussed the application of the concepts used in the preceding sections to the high-pressure polymerisation of ethylene will be considered 5. The Polymerisation of Ethylene.-The production of solid polymer (M>20,000) from ethylene by the free-radical process is generally carried out at 1500-3000 atrn. and 160-200". Oxygen or a peroxide is the usual initiator. Many measurements of the effect of pressure temperature and initiator concentration on the reaction rate and polymer molecular weight have been made but only one or two can be referred to here. Difficulties are encountered because of the sensitivity to traces of oxygen (a few p.p.m. may initiate reaction under some conditions) the possibility of phase separation between polymer and monomer and the large varia- tion in the density of the system with pressure.A detailed investigation,' between 400 and 2000 atm. with a variety of initiators showed the rate of reaction to increase rapidly with pressure but the results could not be well expressed in a form such as equation 7. In liquid polymerisations the concentrations are readily corrected for the compression of the liquid (about 10-20% at several thousand atmospheres) but in ethylene the density may double over the experimental pressure range and pressure density and fugacity each failed to represent monomer concentration satisfactorily in the rate equation. An extensive study of the oxygen- initiated polymerisation by Ehrlich and Pittilo28 revealed features (e.g.induction periods and a critical polymerisation boundary) which cannot be represented in terms of the usual steady-state kinetics. However using propane as a diluent to ensure phase homogeneity and t-butyl peroxide as Bywater and Whalley h e r . Chem. SOC. Reprints (Division of Polymer Chem- istry) 1960 Vol. 1 143. *' Laird Morrell and Seed Discuss. Faraday SOC. 1956 22 126. 28 Ehrlich and Pittilo J . Polymer Sci. 1960 43 389. 276 QUARTERLY REVIEWS initiator Symcox and E h r l i ~ h ~ ~ obtained reproducible kinetic behaviour at 130" and 1000-2500 atm. which resembles that found for liquid poly- merisation. The order with respect to initiator is between 0.5 and 0.6 indicating a close approach to binary termination (equation 5). The rate of polymerisa- tion increases exponentially with pressure and the order with respect to monomer is between 1 and 2.The ratio 6 calculated from equation 7 increases with pressure so that k,/kt is larger. Although the ethylene- propane mixture is supercritical its density is high (- 15 moles/l.) and the transition-state equation (6) may reasonably be applied to the system.29 d V* is thus calculated to be -23 c.c./mole which is a feasible value though rather close to the average difference between the molar volumes of monomer and polymer over the range 1000-2500 atm. Therefore when an initiator other than oxygen is used and phase inhomogeneity avoided the effect of pressure appears similar in dense ethylene and in liquid olefins. Polyethylene of low molecular weight (-930) has been obtained30 from the liquid monomer at 0" and only 40 atm.As the order of decreasing temperature-dependence is usually ktr> kp> kt the lower molecular weight accords with the view that at higher temperatures and pressures the chief effect is an increase in the ratio k,/kt. 6. Attempts to Polymerise Polysubstituted Olefins under Pressure.-The monomers so far discussed have all been vinyl or vinylidene compounds containing the structure CH,=C:. Tri- tetra- and 1 2-di-substituted ethylenes (except CF,=CF,) polymerise much less readily if at all; but a few attempts to produce polymers at high pressure have succeeded. Holmes-Walker and Weale31 examined a number of these monomers up to 10,000 atm. generally using benzoyl peroxide as initiator. Several esters of cinnamic acid (PhCH :CHCO,R) give good yields of polymer containing 20-30 monomer units in contrast to their low reactivity at 1 atm.32 The formation of low polymer from diethyl fumarate was confirmed but diethyl maleate is unreactive.Maleic anhy- dride and stilbene did not polymerise nor did 1 ,Zdibromoethylene although 1 ,Zdichloroethylene forms a solid polymer of unknown molecu- lar weight at high pressure.33 Cyclic olefins which are formally included in this section display varying response. Cyclohexene is ~nreactive,~' indene gives low polymers,1~31 while the formation of high polymers from acenaphthylene is accelerated at high pressure.lg The factors governing the reactivity of monomers of this type and their dependence on pressure still remain uncertain. (a) 1,2- Disubstituted ethylenes. Symcox and Ehrlich J.Amer. Chem. SOC. 1962.84 531. ao Padgett and Perry. J. Polymer Sci. 1959 37 543. I1 Holmes-Walker and Weale J. 1955 2295. sa Marvel1 and McCain J. Amer. Chem. SOC. 1953 75 3272. Weale J. 1952 2223. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 277 (b) Tri- and tetra-substituted ethylenes. Attempts to polymerise these monomers under pressure have usually been unsuccessful. Neither iso- pentene Me,C :CHMe nor a-methylstyrene dimer PhMe,CCH :CMePh reacts and ap-dimethylstyrene yields only small amounts of dime^.^ At higher pressures and temperatures (up to 30,000 atm. and 300") Gonikberg and Z h ~ l i n ~ ~ obtained rearranged products with up to six carbon atoms per molecule from tri- and tetra-chloroethylene; and a tetramer or pentamer possibly of cyclic structure from 2,3-dimethylbut-2- ene.The obvious explanation is steric hindrance between the bulky substituent groups (which is absent in the reactive polyfluorinated ethyl- enes). This could be so even though the dimethylbutene forms a trans- parent at -80" by an ionic mechanism since there may be less steric hindrance for the planar carbonium ion than for a radical. There is not yet sufficient evidence to exclude the possibility of radical polymerisa- tion under pressure in at least some cases. (c) 1,l- Disubstituted ethylenes. Although many of these monomers polymerise readily some are less reactive. 1,l Diphenylethylene gave only dimer3 at 5000 atm. as did methyl a-t-b~tylacrylate~~ at 10,OOO atm. But-3-enylbenzene did not react at 5000 atm.3 but 1-isopropenylnaphtha- lene yielded 54 % of low polymer at 10,000 atm.and 125". 7. The Pressure Polymerisation of Non-olefinic Monomers.-The polymerisation under pressure of substances without a C=C bond has received little attention but the scattered results available suggest that this field would repay further exploration. The early work on aldehydes1 has recently been extended. Polyacetaldehyde (M 500,000) is formed at - 180" but an attempt to produce it at 9OOO atm. and 40" was unsuccessfu1.36 Novak and Whalley3' have recently obtained high polymers of acetalde- hyde propionaldehyde n- and iso-butyraldehyde valeraldehyde and chloral at 8000-9000 atm. and room temperature and showed that they have the polyoxymethylene structure [ -CHR.O -In. All the polymers revert spontaneously to monomer at room temperature and 1 atm.and this is probably another instance of the ceiling-temperature effect discussed in connection with a-methylstyrene. A slow reversion at room temperature (to dimer) also occurs with the high-pressure polymer of dimethylaminoborine discovered by Dewing3*. This substance (Me,N-BH, an analogue of isobutene) forms a high- melting polymer at 1 50"/3OOO atm. but diethyl- and dipropyl-amino- borine do not. The black polymer formed from carbon disulphide at s4 Gonikberg and Zhulin Izvest. Akad. Nauk S.S.S.R. Otdel. khim. Nauk 1957,510; 1958 1254; 1959 916; cf. Gonikberg Butuzov and Zhulin Dokludy Akad. Nuuk S.S.S.R. 1954 97 1023. 36 Ashikari Kogyo Kagaku Zasshi 1956,59 1204. s6 Fawcett and Gibson J. 1934 386. 87 Novak and Whalley Trans. Faraday SOC. 1959 55 1490; Canad. J. Chem. 1959 8a Dewing Chem.SOC. Symposium on Inorganic Polymers Nottingham 1%1. 37 1710 1718. 278 QUARTERLY REVIEWS 45,000 atm. above 175" which decomposes to carbon and sulphur if heated to 200" at ordinary pressure probably has the structure [-c -s -1 ,.*9 4 Other polymerisable monomers in this class include epoxycyclohexane which yields a hard transparent product at 12,000 atm.,l and phenyl- acetylene which gives polymers40 at 1000-6000 atm. Although Bengels- dorf41 obtained a yellow solid from acetone at 320"/50,000 atm. this appears to be the result of a series of ketol condensations rather than addition polymerisation. 8. Copolymerisation at High Pressures.-The study of copolymerisation at 1 atm. has given much information about the relative reactivities of free radicals and provided a theoretical understanding of the manufacture of industrially important copolymers.The development of the high- pressure polyethylene process has been followed by many reports of the copolymerisation of ethylene with other monomers but chiefly in patents which are uninformative about the mechanism and kinetics of the reactions. An interesting copolymer of this type is formed42 from ethylene and carbon monoxide at 1000 atm. with t-butyl peroxide as initiator. The liquid or crystalline products (M < 2000) are polyketones and those with the highest carbonyl content correspond to the structure [-COC,H,-] n. Another unusual copolymer is obtained at 2000-8000 atm. from carbon monoxide an olefin and an The product from carbon monoxide ethylene and methanol contains a-hydroxy-ester groups and at the highest pressure the structure tends to [ -CMe(CO,Me)CH,CH,-] n.Lamb and Weale" have recently studied the copolymerisation of some liquid monomers up to 4000 atm. The overall rates of the copolymerisa- tions increase exponentially with pressure the acceleration being about the same as for styrene and the molecular weights of the copolymers increase more slowly and level off at the highest pressures. In the styrene- methyl methacrylate system the reactivity ratios Ta and rb were found not to change with pressure. These are the ratios of the rate constants for chain growth defined as ra = kpaa/kpab (where kpaa is the rate constant for the addition of monomer A to a chain radical ending in monomer A and kpab is for the reaction between monomer B and a radical ending with A); and correspondingly rb = kpbb/kpba.The constancy of ra and rb indicates that in each pair the two rate constants are similarly increased by pressure and probably that d V* is the same for all four reactions in agreement with the conclusion of section 3. 9e Whalley Canad. J. Chem. 1960,38,2105. 40 Korshak Polyakova and Suchkova Vysokomol. Soedineniya 1960,2 1246. 41 Bengelsdorf in Hamann's book ref. 86 p. 187. (4 Coffman Pinkney Wall Wood and Young J. Amet. Chem. Soc. 1952,74,3391. Cairns Coffman Cramer Larchar and McKusick 1. Amer. Gem. Soc. 1954,76 3024. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 279 A third important parameter is defined by 4 = ktab/(ktaa*ktbb)+ in which the k(s are the rate constants for the three possible bimolecular termination reactions between radical chains ending in A and B monomer units.In equimolar mixtures of styrene and methyl methacrylate at 1 atm. 4 x 25 so that the cross-termination reaction is strongly preferred and in the same system 4 appears to vary little with pressure up to 3000 atm. The persisting preference for cross-termination seems at variance with the assumption of diffusion-control for these reactions at high pressure since both types of radical might be expected to diffuse at similar rates and 4 to approach unity. However it is apparent from the work of North and Reed44 that the question of diffusion-control is complicated even in the case of a single monomer at ordinary pressure and no detailed study of the effect in copolymerisation has appeared. 9. Some Further Aspects of High-pressure Polymerisation.-A few other features of high-pressure polymerisation merit a short discussion although they have not been very thoroughly investigated.Most existing studies relate to radical polymerisation and little information is available for ionic reactions. The rate of formation of poly-a-methylstyrene with trichloroacetic acid as initiator is increased by pressure to about the same extent as the radical process,24 and the iodine-initiated polymerisations of isobutyl vinyl ether45 and isopentyl vinyl ether46 are also accelerated to a comparable degree. Ionic initiation depends on the concentration and reactivity of charged complexes which may be at least as dependent on pressure as the chain- growth and the relative importance of the two effects has not been deter- mined for any system.The aldehyde polymerisations may also occur by an ionic mechanism but have not been studied kinetically. (ii) The structure of high-pressure polymers. Polymer molecules formed by the radical mechanism often have long chain branches which have developed from the site of a transfer reaction between a radical and “dead” polymer. There is some evidence that pressure has little effect on branching in polymerisation of styrene. Kobeko et aZ.*’ found the viscosity- concentration curves for solutions of polystyrenes made at 1 and 6000 atm. to be very similar and no difference is detectable by light-scattering rnea~urernents.~~ The resemblance between the increase of molecular weight with pressure for reactions taken to 100% conversiona and to <20% conversion7 also suggests that pressure causes no large increase in branch- ing even with high concentrations of polymer.These results may reflect a difference between the effects of pressure on transfer to polymer and on transfer to monomer and solvent but sufficiently detailed evidence is not ( i ) Ionic pozymerisation under pressure. I4 North and Reed Trans. Faraaby SOC. 1961,57,859. 4s Nahumy Ph.D. Thesis London IJniversity 1961. Hamann and Teplitsky J. Phys. Chem. 1961,65 1654. Kobeko Kuvshinskii and Semenova Zhur. fiz. Khim.. 1950,24,345. 48 Trementozzi and Buchdahl J. Polymer Scf. 1954 12 149. 280 QUARTERLY REVIEWS available on this point. Another type of structural variation which occurs with polydienes appears also to be unaffected by pressure. RichardsonaQ made an infrared analysis of polyisoprenes and polybutadienes produced at pressures between 5700 and 10,000 atm.and detected no important change in the proportions of trans-1,4- and cis-1,4- or of 1,2- and 1,4- linkages present. There is however a very considerable effect of pressure on the structure of polyethylene. The lower density and crystallinity of the ordinary high- pressure polymer compared with the low-pressure (Ziegler) polymer are attributed to short “pendant” side chains occurring at frequent inter- vals along the main chain. RoedelS0 has suggested that these arise from a cyclic internal transfer reaction (annexed) after which further chain growth continues from the secondary radical. At 7000 atm. and 50-60” the side chains are not formed and the polymers have the high density and crystallinity characteristic of the low-pressure product.6f It is well known that polymers of many substituted ethylenes can exist in stereoregular forms in which the configurations of successive asym- metric carbon atoms are either all identical (isotactic polymer) or alternate in some regular sequence (e.g.syndiotactic polymer). If A V* differs for the various modes of chain growth pressure will affect the type of stereo- regularity exhibited by the polymer. A study of poly(methy1 methacrylates) produced between 1 and 7500 atm. shows the effect to exist and the ratio of the propagation constants kiso/ksun to increase from 0.33 to 0.54 in this pressure range.62 Isotactic growth requires a helical conformation of the radical and a strictly oriented approach by the monomer molecule. The larger (negative) d V* for isotactic propagation is therefore probably associated with a greater decrease in entropy in the transition state.(iii) Fast radical polymerisation under pressure. Many monomers polymerise with great rapidity if ionic initiators are used but the products are often of low molecular weight .63 The pressure-acceleration of radical polymerisations is accompanied by an increase in molecular weight so that fairly high polymers might be obtained in rapid reactions under pressure. Sometimes a fast uncontrolled reaction yields decomposition products (e.g. ethylene,64 cyclopentadieneYs5 dichloro- and trichloro- Richardson J. Polymer. Sci. 1954 13 321. Roedel J . Amer. Chem. Soc. 1953 75 6110. 61 Hines Bryant Larchar and Pease Ind. Eng. Chern. 1957,49 1071. 62 Zubov Kabanov Kargin and Shchetinin Vysokomol.Soedineniya 1960,2 1722. 63 Swarc J . Amer. Chem. SOC. 1956,78 1122; Hayes and Pepper Proc. Roy. Soc. 54 Hunter Chem. and Ind. 1955 396. 66 Raistrick Sapiro and Newitt J. 1939 1761. 1961 A 263,63. WEALE ADDITION POLYMERISATION AT HIGH PRESSURES 281 ethylenea3J4) but Klaassens and Gis01fs6 found that when styrene is compressed to 10,OOO atm. at 75" the temperature rises to over 300" in a few seconds and a block of polymer is formed. Molecular weights were not reported but in more recent experiments at 2000-3000 atm. with initial temperatures of 90-100" (in which the temperature increased a little more slowly) polystyrenes with molecular weights up to -500,000 were ~btained.~' Transfer reaction rates increase with temperature more rapidly than chain growth so that higher molecular weights might be attained if the temperature rise was controlled.(iv) Pressure and cyclisation. Under some conditions polymerisable monomers yield cyclic products e.g. tetrafluoroethylene CF2=CF2 can give the perfluorocyclobutane C4F8. It is possible that in some cases pressure as well as temperature may determine which reaction path is followed. This has not been demonstrated experimentally though Walling and P e i ~ a c h ~ ~ found that when the polymerisation of isoprene is suppressed by adding trinitrobenzene the dimerisation to cyclic products such as 4-isopropenyl- 1-methylcyclohexene (11) is strongly accelerated by pressure with d V* about -25 c.c./mole. When no inhibitor is present the rate of 1 polymerisation is also greatly increased. In contrast cyclo- pentadiene is not known to form linear polymers by a free- radical mechanism but the formation of bicyclic dimer is much faster under pressures6 ( d V* = -30 c.c./mole).It has already been noted (section 6) that 2,3-dimethylbut-2-ene yields cyclic products rather than a linear polymer under pressure and at 150" under pressure butadiene which does normally undergo a pressure- accelerated linear polymerisation yields viscous liquid products which are thought to be cyclic tetrarner~.~~ Conclusion.-High pressures produce large effects in addition poly- merisation by promoting processes which involve local decreases in volume and entropy and by opposing others. The results on the polymerisation of a few derivatives of ethylene are now fairly well understood but some important aspects have received little attention and the behaviour of non- olefinic monomers under pressure remains almost unexplored. The extension of detailed studies in these directions should yield novel polymers and reveal new chemical effects. 64 Klaassens and Gisolf J. Polymer. Sci. 1953 10 149. i7 Rabbetts unpublished results. 5* Walling and Peisach J. Amer. Chem. SOC. 1958,80 5819. (n) AY- Slobodin and Rachinskii DokMy Akad. Nauk S.S.S.R. 1947,58,69.

ISSN:0009-2681    

DOI:10.1039/QR9621600267

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

HALOGEN CATIONS By J. AROTSKY (DEPARTMENT OF CHEMISTRY AND BIOLOGY THE HARRIS COLLEGE PRESTON) and M. C. R. SYMONS 1. Introduction MUCH confusion has arisen in the literature over the terms “positive halogen” and “halogen cations”. This confusion appears to arise from the fact that elements of Group VII apart from fluorine form covalently bonded compounds in which the halogen is in the +1 oxidation state. Thus positively-charged halogen compounds where the halogen will migrate to the cathode in electrolysis are common. At an early stage in chemical history1 these compounds were confused with the free halogen cation. Since the alkali metals do not readily form covalently bonded complexes no such confusion has arisen in this case and whereas C1+ Br+ and I+ are often postulated in systems where their presence is chemi- cally improbable references to the equally improbable Li- Na- and K- are unknown to the authors.(a) Definitions and Scope.-Since fluorine does not readily form com- pounds in which it is normally considered to be in the +1 oxidation state the term “halogens” in this review will refer to chlorine bromine and iodine. For convenience we shall employ the following conventions when referring to their oxidation states The total sum of the oxidation states of the elements forming a molecule or ionic species is equal to the charge on the species. Elements bonded to a more electronegative element are in a positive oxidation state and vice versa. Halogen compounds in which the element is in the + 1 oxidation state can be divided into three classes ( i ) Neutral or negatively-charged species.These consist principally of the hypohalous acids and their anions and binuclear interhalogen com- pounds such as iodine monochloride. (ii) Species which have a unit positive charge. These include protonated hypohalous acids such as H 20Br+ and aromatic heterocyclic nitrogen complexes such as pyI+N03- (py = pyridine). These compounds can be referred to as “cationic halogen compounds” or “positive halogen com- pounds”. The latter nomenclature will be used since this prevents confusion with (iii). These will have an electronic configuration ns2np4. We will refer to them as halogen cations and it is with solutions of these ions that this Review is principally concerned. See e.g. Caven and Lander “Systematic Inorganic Chemistry,” Blackie and Son London 1906 p.301. (DEPARTMENT OF CHEMISTRY THE UNIVERSITY LEICESTER) (Z) The simple cations. 282 AROTSKY AND SYMONS HALOGEN CATIONS 283 (b) Historical Significance of this Topic.-One of the first trends noted in the Periodic Table was the increase in the electropositive character of succeeding elements in a given group. The desire to illustrate this trend stimulated a search for iodine species in which the element had an electro- positive character. This in turn led to an almost obsessive desire to prove the existence and comparative stability of the iodine cation in chemical systems. This obsession appears to have been mainly responsible for the confusion which is still apparent in current literature regarding the chemical nature of halogen cations. The discovery2 that iodine and iodine monochloride formed conducting solutions in ethanol and later,3 that iodine itself conducted in the vapour phase and in the molten state led to the conclusion that the iodine cation was present in these systems.Since the cation has only six electrons in its outermost shell its existence could not be accounted for by the octet theory. This contributed to the concept that it was electron-pairing and not octet formation that resulted in stable configurations.* Recently investigations of aromatic nitration have shown that the nitronium ion is a comparatively stable entity and is the reactive inter- mediate in many electrophilic nitration^.^ As a result of this work a rash of postulates arose concerning the existence of positively-charged inter- mediates involved in electrophilic attack of aromatic compounds.Amongst these intermediates halogen cations were prominent. A paper by Bell and Gelless concerned with thermodynamic aspects of the formation of halo- gen cations in aqueous solution has tended to curb enthusiasm for reac- tion mechanisms involving their participation. (c) Reviews.-Many Reviews which purport to be directly concerned with halogen cations as well as halogen compounds in the +1 oxidation state have appeared both in periodicals’ and in text-books.8 Unfortunately in most of these confusion between halogen cations and positive-halogen compounds is apparent. For example the phrase “solvated iodine cation” is often used when referring to protonated hypoiodous acid. Also the phrase “stabilised halogen cation” may be found in reference to cations such as the N-iodopyridinium ion.These criticisms do not apply to a recent extensive review of aromatic halogenation and nitrati~n.~ !a Walden 2. phys. Chem. 1903,43 385. a Thomson “The Corpuscular Theory of Matter,” Constable London 1907 pp. 130 Lewis J Amer. Chem. SOC. 1916 38 762. Ingold “Structure and Mechanism in Organic Chemistry,” Bell London 1953 Bell and Gelles J. 1951 2734. See e.g. Mischenko Zhur. priklad. Khim. 1957 30 665. See e.g. Heslop and Robinson “Inorganic Chemistry,” Elsevier Amsterdam 1960 p. 384; Kleinberg Argersinger and Griswold “Inorganic Chemistry,” Heath and Co. Boston 1960 p. 483. de la Mare and Ridd “Aromatic Nitration and Halogenation,” Academic Press London 1959. 131. 274. 284 QUARTERLY REVIEWS 2. Properties Halogen cations particularly that of iodine have been clearly identified in the gas phase by emission and mass spectroscopy.The spark spectrum of iodinefo shows lines attributable to electronically-excited iodine cations. from which it is possible to show that the ground state of the cation is 3P2 of the configuration 5s25p4 as expected. We can quite safely assume from this that there is also no departure from Hund’s rule of maximum multiplicity in the case of bromine and chlorine cations. The cations will thus have two unpaired electrons and we will see later that this fact is of importance when considering methods of identification. Because there are only four p-electrons in the outer shell halogen cations must be extremely powerful electrophiles though electrophilic activity will be partly governed by the energy required to pair the two unpaired p - electrons.11 However this energy which (from data for iodine cations in the gas phaselo) must be of the order of 10 kcal.mole-l will not be sufficient to lower their reactivity to a value below that of such covalently bonded species as H201+. Two other properties that are relevant to the study of halogen cations are their unit positive charge and the fact that the halogen is in the +1 oxidation state. Detection of a species having these two properties together with high electrophilicity is often used as proof of the presence of halogen cations in the system. It should be emphasised that these properties are shared with most positive halogen compounds. In fact there would only be a difference in electrophilic power a property that it is very difficult to measure quantitatively.Further since halogen cations are such power- ful electrophiles their presence in nucleophilic solvents such as water,B ethanoP2 and nitrobenzene2 is most unlikely. Fortunately another property of halogen cations namely their para- magnetism is more diagnostic of their presence in solution than those listed above since solutions which might conceivably contain halogen cations are unlikely to contain other unidentifiable paramagnetic entities. The magnitude of the magnetic moment for these ions in solution will be difficult to predict since it will surely depend considerably upon the mode of interaction with nearest-neighbour solvent molecules. However a strongly paramagnetic halogen species having the properties listed above will almost certainly be the corresponding cation.The electronic absorption spectrum of halogen cations will not be so useful for purposes of identification. Analysis of the spectrum of the iodine cation in the vapour phaselo shows that there are no permitted electronic transitions above 200 mp. By extrapolation this will also be the case for bromine and chlorine cations. However in any interacting medium the lo Murakawa 2. Physik. 1948 SC 84. 11 Millen quoted in ref. 5. Is Brasset and Kikindai Compt. rend. 1951 232 1840; Kikindai and Cassel ibid. 11 10. AROTSKY AND SYMONS HALOGEN CATIONS 285 configuration of the cations will be more or less perturbed. The situation is closer to that of solvated transition-metal ions than that of ions with a rare-gas configuration. Thus the mediurn may perturb the energy levels in the cation in such a manner that transitions similar to those between the d-levels of transition-metal complexes can occur.Since halogen cations will not by definition be strongly bonded to other species there should be no infrared or Raman absorption bands attribut- able to them. (a) Thermodynamic Considerations.-In 195 1 an important paper by Bell and Gelless appeared concerning the probable existence of halogen cations in aqueous media. Equilibrium constants for reactions shown in Table 1 were estimated from free-energy data computed by means of typical thermodynamic cycles (Table 2). Inevitably numerical values for TABLE 1. Free energies and equilibrium constants for some reactions of the halogens (from Bell and Gelless). X (aq.) + X+ (aq.) + X- (aq.) . .. . . . . (1 ) X c1 Br r AGO (kcal. mole-l) 84(56) 67(39) 5 5 w K 10-y 1040) 10"0( 1030) 1 0-40( 1 0-21) The values in parentheses incorporate our approximate ligand field correction of 28 kcal. moleAL. X (aq.) + H,O + H+ + HOX + X- . . . . . . (2) X c1 dGo (kcal. mole-l) 4.5 K lo3 Br I 11.2 17 lo4 10-13 Xs (aq.) + H,O f H,OX+ + X- . . . . . . . . . (3) X c1 Br dG" (kcal. mole-l) 42 30 K 1 0 3 O 10-20 I 14 1 O-'O TABLE 2. The Born-Haber cycle for the determination of the free energy of X (aq.) = X+ (aq.) + X- (aq.) . . . . . . (4) xs ( g 4 - = (gas) - x+ (gas) + x- (84 t J- J- x2 (a+) x+ (as.) + x- (as.) ionic radii and solvation energies of the cations under consideration have been derived from only partly justifiable extrapolations and hence may be considerably in error. This treatment of the equilibria has been subjected to some criticism.In particular Bell and Gelles made no allowance for the fact that since halogen cations have an incomplete outer p-level some "ligand field 4 286 QUARTERLY REVIEWS stabilisation” must occur giving higher solvation energies for the cations than was originally estimated. Secondly there is strong evidence that HzOX+ (X = Br Cl) is an intermediate in aromatic halogenation^,^ but the equilibrium constants estimated by Bell and Gelles for reaction (3) of Table 1 lead to concentrations of protonated hypohalous acids which are so small that they could hardly act as important intermediates in the relevant halogenations. A considerable body of evidence has now been advancedlsf6 for the existence of iodine cations in oleum solutions.From the spectra of these solutions we estimate a value of about 28 kcal. mole-l for the “ligand field stabilisation” of the iodine cation due to the oriented field of solvent molecules.17 It is unlikely that this stabilisation energy will vary extensively from iodine to chlorine since ligand-field stabilisation energies of transi- tion elements appear to depend more on their configuration and charge than on their size. We have therefore included this figure with the data of Bell and Gelles (Table 1). As a result of this extra solution energy the iodine cation appears to be sufficiently stable to be a reasonable intermediate in chemical reactions. The participation of chlorine and bromine cations in halogenations still appears unlikely. On the question of the reaction described by equation (3) Bell and Gelles had to make rather gross approximations both for the free-energy of reaction between halogen cations and water and for the solvation energy of hypohalous acids.One or both of these approximations appear to us to be somewhat in error for the following reasons. First kinetic evidence for the participation of protonated hypohalous acids in electrophilic substitutions is strong. Secondly it seems possible to obtain approximate values for the acidities of all the protonated hypohalous acids by relating the known value for hypoiodous acid with the differences between the pK values for the acidities of the neutral acids and the results are very different from those which can be derived from the data of Bell and Gelles.6 Thus the difference in the pK values for hypoiodous and hypochlorous acid is of the order of -5 and since this reflects the difference in electro- negativities of iodine and chlorine we suggest that a similar difference in the pK values for the protonated acids is probable.The pK value for hypoiodous acid [equation (25)] is about 1*3.18 Hence the pK value for protonated hypochlorous acid is probably in the region of from -3 to -4. This is in marked contrast with the value of -26 implicit in the data of Bell and Gelles,6 Many other common intermediates in reactions such Symons J. 1957 387. l4 Symons J. 1957 2186. lS Arotsky Mishra and Symons J. 1961 12. le Arotsky Mishra and Symons J. 1962 2582. l7 Orgel “An Introduction to Transition-Metal Chemistry,” Methuen London l8 Buckles and Mills J. Amer. Chem. Soc. 1953 75 552; Bowers and Scott ibid.1961 p. 72. 3582. AROTSKY AND SYMONS HALOGEN CATIONS 287 as the t-butyl carbonium have pK values in this region and thus our calculations support the postulate that protonated hypohalous acids could be reactive intermediates in electrophilic hologenations. This reasoning in no way reflects upon calculations relating to the formation of free halogen cations as has previously been intimated.g Even when one adds an ex- tremely large free-energy term for the “ligand field” stabilisation of free chlorine cations their significant existence in chemical systems is energetic- ally unlikely. Finally even if free halogen cations were formed in aqueous systems the free energy of reaction (1) in Table 1 would require the im- mediate formation of the corresponding protonated hypohalous acid.The argument that the energy of activation of reaction (1) would be large resulting in the stabilisation of free halogen cations in aqueous systems,ll is not compelling since this effect would merely lower the electrophilic reactivity of the halogen cations generally as the p-electrons must pair for the halogen cation to react as an electrophile with any system. Besides the quantitative reasoning given in Section 2 suggests that this barrier is not large. We conclude therefore that although protonated hypohalous acids may well be reactive species involved in electrophilic substitution reactions the participation of halogen cations except possibly iodine cations is improbable in any reaction in aqueous media. It is perhaps significant that we have failed to detect chlorine or bromine cations in sulphuric acid and oleum solutions15 [Section 3(b)] and that if our conclusions regarding the electronic absorption spectrum of solvated iodine cations [Section 3(b)] are correct we can put a limiting value of 1 0 “ ‘ ~ to the concentration of iodine cations in acidified aqueous solutions of hypoiodous acid and related compounds.This conclusion is based upon the fact that no absorption in the 640 mp region could be found for such solutions. By using the data of Bell and Gelles,’? and this value for the concentration of the iodine cation we calculate that the concentration of chlorine cations in comparable solutions is about 10-53~. This result since it rests upon differences in energies rather than absolute values cannot be in error by many powers of ten.3. Preparation The preparation of halogen cations can be discussed in terms of the hypothetical equilibrium HalX + Hal+ + X- . . . . . . . . . . . . . (5) where X- might be halide hydroxide or other suitable anions. In view of the thermodynamic considerations already discussed physical constraints are unlikely to influence such equilibria sufficiently to give detectable amounts of halogen cations. However the equilibrium could be moved to the right if there is either a suitable nucleophilic reagent present to react preferentially with the halogen cations or an electrophile to react Rosenbaum and Symons Mol. Phys. 1960,3,205. 288 QUARTERLY REVIEWS with the anions. The first alternative is trivial since our aim is to prepare halogen catioiis; we mention it here since it seems that such reactions have often occurred in attempts to make halogen cations and failuie to recognise this mode of reaction has led to many erroneous conclusions (Section 4).The latter alternative must therefore be used and it will also be a necessary requirement that no nucleophiles be present which can react with the halogen cations formed. For an electrophile Y the sequence X-+Y+XY- . . . . . . . . . . . . . (6) XY- 3 Further products . . . . . . . . . . (7) is suggested step (7) being necessary if (6) is reversible or if halogen cations can react with XY- to re-form HalX. In the reactions that we have studied a second step of this type appears to be necessary for the formation of appreciable concentrations of halogen cations. (a) Suitable Media.-Halogen cations are very powerful electrophiles and will exist in media containing only the weakest of nucleophiles.Another necessary prerequisite is that the ionizing power of the solvent be high. There should be no reducing agents capable of converting the cations into the halogen or oxidizing agents of sufficient power to oxidize them to the ter- or quinque-valent state. This latter condition is particularly important in the case of iodine. One medium which appears to fill the above requirements is sulphuric acid and the solution chemistry of the halogens in this solvent has received considerable attention. However much of this work has been on iodine compounds in the +3 and +5 oxidation states with which we are not at present concerned. (b) Solutions of Iodine and its Compounds in Sulphuric Acid and 01ems.- When potassium iodate is mired with an excess of iodine in sulphuric acid a solution is formed having a brown-red colour which contains a powerful electrophilic iodinating agent.This is thought to be the cation 13+ formed by the reaction HIO + 71 + 8H.$04 -+ 51,+ + 3HsO+ + 8HS0,- Masson found that when nitrobenzene or chlorobenzene is shaken with such a solution high yields of the rn-iodo-compounds are formed together with molecular iodine according to reactions of the type . . . . . (8) C6H,N0 + I,+ 3 I + C6H41*N0 + H+ (9) C,H,NO + I,++ 21 + C6H41*N0 + H+ . . . . . . . (40) . . . . . . . . The presence of the cations I$ and I,+ is also indicated by cryoscopic spectrophotometric and conductometric rneasurement~.~~~~~ The formation and comparative stability of such ions is considered in detail in Section 6.Masson J. 1938 1708. AFLOTSKY AND SYMONS HALOGEN CATIONS 289 Iodine dissolves in sulphuric acid to give pink solutions having an absorption spectrum similar to that of iodine vapourl* (see however ref. 15). The solubility of iodine is increased from about 1 0 - 3 ~ to 0 . 5 ~ by addition of silver sulphate the solubility of which is concurrently en- hanced. The resulting solutions contain a powerful electrophilic iodinating agent which reacts in a manner comparable with Is+ except that silver iodide is precipitated instead of iodine.21 &I,+ + C6HIN0,+ C6H41.NOs + Agl + H+ . . . . . (11) Cryoscopic and conductimetric measurements indicate that a 1 :1 complex is formed between silver ions and iodine molecules.le These facts indicate that the iodine is entirely converted into Ag12+ and hence that these complexes have considerable stability.Although the geometry of such complexes is unknown it is noteworthy that silver ions readily form com- plexes with olefins and aromatic hydrocarbons and hence we feel that a similar form of bonding giving rise to a structure in which the silver is symmetrically placed with respect to the two iodine atoms may be favoured. Alternatively the structure may be more comparable to that of 13+ in whose formation the role of molecular iodine is again that of a donor rather than an acceptor. In accord with results for other complexes of silver22 the spectrum of “iodine” in Ag12+ is shifted to low energies relative to that of “free” iodine and there is a new intense absorption in the near-ultraviolet region probably associated with an intramolecular charge-transfer transition.l* A possible role of halogen complexes with silver ions in aromatic halogenations is discussed in Section 7.None of these solutions contains an appreciable concentration of para- magnetic species and therefore we conclude that iodine cations are not an important constituent. When sulphur trioxide is added to sulphuric acid the medium is changed in two ways; first the acidity is increased and secondly there is an in- creasing concentration of powerful oxidant. These effects are reflected in the behaviour of iodine in such The apparent solubility of iodine in dilute oleums (30”/,)* is about 0 . 5 ~ . Spectrophotometric studies15 of these brown solutions show that they contain 1$ and sulphur dioxide.If potassium iodate in the stoicheio- metric amount required by reaction (12) is added to the brown solutions HIO + 21,+ + 8H,S04 -+ 71+ + 3H,O+ + 8H,SO,- . . . . (12) the colour changes to a deep blue. This change can also be effected by other oxidizing agents such as potassium persulphate or hydrogen perox- * The concentration of oleums will be expressed in terms of the percentage by weight *l Derbyshire and Waters J. 1950 3694. 2s Tetlow 2. p h p . Chem. 1938 B 40,397; 1939 B 43 198. of sulphur trioxide in 100 ”/ sulphuric acid. 290 QUARTERLY REVIEWS ide. The visible absorption spectra of the blue solutions are characterised by three bands with maxima at 645 500 and 410 mp. Wavelength (mp) FIG. Electronic spctrum assigned to the iodine cation. (Broken curve was derived as described in ref.15.) Similar blue solutions are obtained by direct dissolution of iodine in 65 % 01eum.l~ The apparent solubility of iodine in this medium is extremely large being greater than 1 0 ~ . These solutions contain an exceptionally powerful iodinating agent such inert compounds as pyridine,z3 picoline,24 and phthalic anhydride26 being readily iodinated in positions expected for electrophilic attack. The spectrum of these blue solutions is very similar to that described above and is quite different from that of blue solutions of sulphur in oleum,26 with which these solutions might be confused. An early analysis13 has now been modified16 to give the following conclusions. A band at 280 mp is ascribed to sulphur dioxide since addition of sulphur dioxide merely increases its intensity whilst addition of an oxidiz- ing agent such as potassium persulphate decreases its intensity the rest of the spectrum remaining unaltered.Also when iodine monochloride is dissolved in this medium the spectrum is identical with that of solutions of iodine except that there is no absorption band at 280 mp.16 From the value of the extinction coefficient of sulphur dioxide in this medium it has been shown that one molecule of sulphur dioxide is formed per molecule of iodine This is in accord with the postulate that all the iodine is in the +1 oxidation state in agreement with chemical ana1y~is.I~ A careful study of the visible spectrum of these solutions under a variety of conditions suggests strongly that all three bands in the visible region are a property of a single substance.ls The blue solutions in both dilute and concentrated oleums contain a paramagnetic species14J6 having a magnetic moment corresponding to 1 -44 B.M.per iodine atom as estimated by the Gouy technique,16 and 1.5 B.M. aa Plazek and Rodewald Roczniki Chetn. 1947,21 150. M Plazek and Rodewald Ber. 1937,70 1159. 16 Allen Homer Cressman and Johnson Org. Sjmth. 1947,27 78. u Symons J. 1947,2440. AROTSKY AND SYMONS HALOGEN CATIONS 29 1 by using nuclear magnetic resonance.27 The latter measurements showed that the paramagnetic entity is not prot~nated.~~ Conductometric measurements of both dilute2* and concentrated oleumsl6 and conductometric titrations of these solutions with appropriate solutions of boric a ~ i d ~ ~ s ~ ~ ~ ~ showed that there is one positive charge per iodine atom.Similar results were obtained for solutions of iodine mono- chloride. l6 4. Evidence for the Iodine Cation In summary it has been shown that there is a single coloured species containing iodine in the +1 oxidation state in the blue solutions having the spectrum shown in the Figure. This species has one positive charge per iodine atom is not protonated and is paramagnetic having a moment of about 1.5 B.M. per iodine atom at room temperature. These results are consistent with the concept that the blue species is the cation I+ formed by reactions such as I + ZH,S,07 + ZSO -+ 21+ + HS207- + SO . . . . . . (13) . . . . . . ICI + H,SaO7 + SO,+ I+ + HS,07- + HSOSCI (14) However since this ion has two unpaired electrons an explanation must be found for the relatively small magnetic moment which is much smaller than the “spin-only” value of 2.8.Also since there are no allowed transi- tions for iodine cations in the gas phase above 200 mp,l0 the visible spectrum also requires interpretation. Furthermore in view of the dis- cussion in Section 2(a) it will be necessary to show that this postulate is not contrary to thermodynamic requirements. (a) Solvation of the Iodine Cation.-To a first approximation the solvent stabilisation energy of the iodine cation will be comparable with that of a rare-gas cation having the same radius. Solvation energies in sulphuric acid are not known but must surely be large. In addition there will be a “crystal field” stabilisation since the cation has a 3P ground state. This is expected to give rise to a distorted solvation in which the solvent molecules specifically avoid the filled p-orbital.Comparison with theory and results for transition-metal complexes suggests that this extra stabilisation may be as large as 28 kcal. mole-’. The simplest reaction for iodine cations in oleum is and this must be compared with the reaction I+(aq.) + HaO + IOH,+ of Bell and Gelles? In addition to the “crystal field” stabilisation which was not taken into account by Bell and Gelles reaction (15) is inherently less probable than (16) since two strongly solvated ions are thereby destroyed and HS,O,- ions are far weaker nucleophiles than water. I+(so~v.) + HS,O,-(SOIV.) + IHSgO7 . . . . . . . . . . (15) . . . . . . . . (1 6) a7 Connor and Symons J. 1959,963. a8 Mishra and Symons unpublished results. lo Arotsky and Symons Trans.Faraday Soc. 1960,56 1426. 292 QUARTERLY REVIEWS (b) Effect of Solvation on Spectra.-As has been pointed outis an asymmetric solvent-field of the type depicted which resuIts in a splitting of the p-orbitals into a lower singlet and an upper doublet creates a situation which is comparable with that for transition-metal complexes. Hence p p transitions must be considered and for simplicity this problem was treated as one involving a linear Stark field? Thus using the symbol- ism of linear molecules the ground state of the solvated ion is 3C- and there are three low lying excited states. The transitions under considera- tion are then 317 i f- C- where i = 0 1 or 2. These transitions are formally forbidden since the model has inversion symmetry. However in com- parable systems involving second- or third-row transition-metal complexes relatively intense transitions are often observed such factors as mixing with higher-energy orbitals of opposite parity such as the 5d level or “borrowing” from charge-transfer transitions involving orientated solvent molecules tending to increase the intensity.15 The treatment by Jev0ns3~ shows that the energy difference between these transitions should be equal and about half that between the 3P and 3P1 levels for the ion in the gas phase.Furthermore the lowest energy-transition should be more intense than the others. We therefore postulate that the bands at 640 500 and 410 mp are due to these transitions the band at 640 mp is indeed about three times as intense as the others and the equal spacing of 4400 cm-1 is close to the value of 3600 ern? derived from the data for the ion in the gas phase.lo (c) Effect of Solvation upon Magnetic Susceptibility.-To continue our analogy with compounds of transition metals we find that experimental susceptibilities are often far smaller than the value predicted by using the “spin-only” approximation.A possible approach to this question was given15 using the theory of K~tani.~l Alternatively one can postulate that as a result of coupling with excited singlet states the 3Z- state is split so that the MA = 0 level lies slightly below the M8 = &l level. This would have the effect of lowering the measured susceptibility. A similar argu- ment could be used to account for the very weak paramagnetism attributed to the ion I0+.l8 (a) Alternative Postulates.-A variety of alternatives were considered and rejected because of known properties of the blue solutions.l5 However the possibility that IS03+ was the coloured paramagnetic species could not be eliminated,15 although qualitative considerations made it seem less likely than the original theory.The ion lSO,+ could be paramagnetic for the same reasons as lo+ in that it has local axial symmetry. Also other things being equal it might have a visible spectrum qualitatively similar to that postulated above for solvated iodine cations. sn Jevons “Report on The Band Spectra of Diatomic MolecuIes,” Physical Society 91 Kotani J . Phys. SOC. Japan 1949,4,293. London 1932 p. 100. AROTSKY AND SYMONS HALOGEN CATIONS 293 Since in our view the cation IS03+ is the only important alternative to the iodine cation an experiment was devised which would enable us to draw a clear distinction between them.The conductivities of solutions of sodium chloride and iodine monochloride in dilute oleums have been measured,= and the results provide the required distinction. The conductivities of the latter solutions which have the same visible spectrum as that of solutions of iodine monochloride in 65% oleum vary markedly as the medium is changed and hence an uptake of sulphur trioxide greater than that required by reactions (17) and (18) would be readily detected. In fact the ICI + ZH,S,O 4 I+ + HS0,CI + HS,C),- + H,SO . . . . (17) H,SO + H,S,O f=r H,SO,+ + H&O,- . . . . . . . . (18) conductivity of equimolar solutions of sodium chloride and iodine mono- chloride were the same within experimental error.28 This result whilst being in accord with the iodine cation theory eliminates the concept that ISO$ is an important constituent of these solutions.5. Chlorine and Bromine Cations in Strong Acids In view of the apparently successful preparation of concentrated solu- tions of iodine cations attempts have been made to detect cations of chlorine and bromine under similar conditions.l6 These elements dissolve unchanged in 65% and other oleums and on oxidation give tervalent derivatives in the case of bromine via the cation Br3+. No indication of a stable species containing halogen in the + I oxidation state was detected. Most significantly no paramagnetic species could be detected which result means that the concentration of solvated chlorine or bromine cations was always less than 10-2~.Again no trace of an absorption band which could be assigned to the ion was detected. Since had they been formed they would surely be solvated in a manner comparable to that of the iodine cation we would expect to find bands of low energy. However especially for the chlorine cation the intensities would be expected to be greatly reduced and thus it is difficult to predict a detection limit. This result that the iodine cation is considerably more stable than those of chlorine and bromine is in accord with expectation. 6. Trihalogen Cations These ions have the general formula Hal,+ and known examples are given in Table 3. A consideration of electronic structure32 suggests that TABLE 3. TrihaIogen cations. Cation I3+ IC12f BrF,+ Br3+ Anion HS0,- Sbcls- BrF,- HS,O,- Reference 20 33 * 28 * See e.g.Woolf and Emel6us J. 1949 2865. ** Walsh J. 1953 2,260. 294 QUARTERLY REWWS these ions should be bent and have considerable stability since as was stressed by MassonYao they have a “closed-shell” configuration. Recent structural studies of salts such as (I(&+) (SbC16-)= have shown that these ions have a bond angle of about 90’. So far as we are aware the first ion in this class to be discovered was Is+ which Masson identified in solution in sulphuric acid.20 They are of significance to the subject matter of this Review in that they are powerful electrophilic halogenating agents and have thus often been confused with simple halogen cations. 7. Snmmary of Other Work In this section we shall consider reports other than those discussed above purporting to demonstrate the presence of halogen cations in various systems.There is a considerable mass of work in the literature on this topic much of it a result of loose nomenclature as discussed in Section 1. (a) Kinetic Evidence-In the Reviewers’ opinion the thermodynamic calculations of Bell and GellesYg even when allowance is made for such effects as increased solvation energy of the halogen cations arising from their incompleted outer shellg [see Section 2(a)] make the participation of chlorine cations in reactions in aqueous media appear extremely im- probable . Nevertheless the kinetic evidence in favour of chlorination by the chlorine cation in certain reactions of hypochlorous acid is so strong’ that many workers simply assume that this ion is the usual agent in aqueous acidic media.= This evidence has been summarised by de la Mare and Ridd,O and rests on the significant observation that for relatively unreactive aromatic compounds the rate of chlorination is of the form 4[HOClJ/dt = k,[HOCI] + kJHOCI] [H+] This is interpreted in terms of a rate-determining heterolysis of the oxygen-chlorine bond followed by a rapid attack of the resulting chlorine cations upon the aromatic compound.In accord with this is the observa- tion that when deuterium oxide is used as solvent there is a large increase in rate.s6 This has been accepted as good evidence for the transient formation of chlorine cations in these solutions. However we think it is significant that in order to suppress reactions such as (19) a large excess of silver perchlorate was present in the solutions under con~ideration.~~~~ However reaction rates were insensitive to small changes in the concentra- CI- + H,OCI+ + CI + H,O .. . . . . . . (19) sa Vonk and Wiebenga Acra Cryst. 1959 12,867. * Swain and Ketley J. Amer. Chem. Soc. 1955 TI 3410. a6 de la Mare Ketley and Vernon J. 1954 1290. Harvey and Norman J. 1961 3604. AROTSKY AND SYMONS HALOGEN CATIONS 295 tion of silver perchlorate and thus it was concluded that silver ions were not involved in the reaction whose rate was measured.S0 There is strong evidence that the complex A@$ has considerable stability [Section 3(b)] and that it is a powerful iodinating agent.21 There is also evidence from spectrophotometric solubility and conductometric studies for the formation of AgBr,+ and AgC12+,28 and hence the possible participation of AgC12+ in the halogenations studied by & la Mare and his ~ ~ - w o r k e r s ~ ~ should be considered.Thus an alternative mechanism for chlorination in acidified aqueous solution of hypochlorous acid containing a large excess of silver perchlorate and some solid silver chloride is kl. s1 (20) (21) AgCl(so1id) + H,OCI+ + AgCb+ + H,O . . . . . . . . AgCI,+ + ArH __* ArCl + AgCl(solid) + H+ . . . This will be in accord with the kinetic data if k, is larger than kzo and the concentration of AgCl$ is large compared with that of Cl,. This would make the forward part of reaction (20) ratedetermining and hence the overall rate would be independent of the concentration of silver ions in solution and of the aromatic substrate. We suspect that several other mechanisms could be found for these complex systems which would adequately accommodate the kinetic data.(b) Molten Iodine and Iodine Monochloride.-Molten iodine is a moder- ately good conductor of electricity’ and this has been explained by the equilibrium It is inconceivable that in the presence of a large excess of iodine the free cation would not be completely converted into I$ and hence the equi- librium 31 P I,+ + I,- is surely a better description of the seV-ionisation of iodine. Similarly the electrical conductivity of molten iodine mon~chloride~~ which has been attributed to the equilibrium is better depicted as . . . . . . . . . 21 + I+ + I,- (22) . . . . . . . . . (23) 2lCl PI+ + El,- . . . . . . . . . (24) 31cI P I,cI++ IcI,- (W . . . . . . . . (c) Reaction of Iodine with Ethyl AlcohoL-Iodine dissolves in ethyl alcohol to give brown solutions which have a small electrical conductivity.2 These solutions are diamagnetic and the diamagnetism increases linearly with added iodine;- this indicates that appreciable concentrations of O7 Emel6us and Greenwood J.1950,987. $0 Couty Bull. Soc. chim. France 1938,5 84. 296 QUARTERLY REVIEWS iodine cations are not formed in these solutions. The addition of an alco- holic solution of silver nitrate to one of iodine results in the deposition of 69-74% of the iodine as silver iodide.sB This was interpreted as being indicative of the formation of the free iodine cation together with a species containing iodine in the +3 oxidation Solutions of silver perchlor- ate40 or silver nitrate41 with iodine in phenol remain unchanged but the addition of ethyl alcohol results in the immediate formation of tri-iodo- phenol and silver i~dide.~O,~l Solutions of iodine and silver nitrate in ethyl alcohol cooled to 2" to prevent oxidation of the alcohol when treated with the required amount of iodine to bring the overall oxidation state to +1 were found to displace hydrogen ions from a cation-exchange resin the iodine being absorbed.70% could be recovered as the element from the resin on treatment with aqueous potassium iodide. This was interpreted in terms of the reaction and quoted as evidence for the existence of free iodine cations in the original In view of the chemical evidence given earlier and the thermodynamic considerations of Section 2(a) it appears most unlikely that free iodine cations are present in these solutions.A possible explanation of the results is that the species containing iodine in the +1 oxidation state in these solutions is ethyl hypoiodite or its conjugate acid. All the reactions described above can be accommodated in terms of this postulate although others are possible. These arguments illustrate the danger of relying on purely chemical evidence to establish the presence of free halogen cations in a system. H,O + Resin-I+ + I- + I + Resin-H+ + OH- . . . . (26) (d) Solutions of Iodine Monochloride in Acetonitrile.-The electrical conductance of solutions of iodine monochloride in acetonitrile is con- sistent with the fact that some ions are formed in these solutions. Reaction (24) was postulated in e~planation.~~ Acetonitrile is like ethanol a basic solvent and is readily protonated by acids dissolved in it.43 Nitrogen- containing compounds readily bond to iodine in the +1 oxidation state to form stable species such as the N-iodopyridinium ions.We consider it much more likely that the electrical conductivity of these solutions is due to the equilibrium particularly since no optical absorption attributable to the positively- charged iodine species was detected.42 MeCN + 2ICI + MeCNI+ + ICI,- . . . . . . . . (27) Ushakov J. Gen. Chem. (U.S.S.R.) 1931,1 1258. 40 Birkenbach and Goubeau Ber. 1932 65 395. 41 Fiakov and Gengrinovitch. Inst. Khim. Akad. Nauk. (U.S.S.R.) 1940,7 125. 42 Poyov and Deskin J. Amer. Chem. Soc. 1958 80 2049; Popov and Geske ibid. 43 Kolthoff Bruckenstein and Chantooni J. Arner. Chem. Soc. 1961 83 3927.. p. 2976. AROTSKY AND SYMONS HALOGEN CATIONS 297 (e) Iodine Cations as Catalysts in Polymerisation Reactions.-Cationic vinyl polymerisation can be induced by iodine,44 and equilibrium (5) has been suggested as an explanation. In view of the arguments given above this seems unlikely one alternative being that 13+ ions are the active species. Thanks are offered to Drs. J. J. Betts J. L. Latham and H. C. Mishra for helpful discussion. Eley and Pepper Trans. Fwuday Soc. 1947,43 112.

ISSN:0009-2681    

DOI:10.1039/QR9621600282

出版商:RSC    

年代:1962

数据来源: RSC