Radiotracer Studies of Self-diffusion in the Plastic Solids Norbornylene and Norbornane BY ALAN V. CHADWICK" AND JACQUES W. FORREST 1- University Chemical Laboratory, University of Kent, Canterbury, Kent CT2 7NH Received 28th December, 1977 Radiotracer self-diffusion measurements have been made for norbornane (bicyclo[2,2, llheptane) and norbornylene (bicycIo[2,2,l]hept-2-ene) in their h.c.p. plastic crystalline phases. In the case of norbornane the measurements were limited by the poor quality of the specimens; however, the results are consistent with existing n.m.r. data and support the view that self-diffusion occurs via a mono- vacancy mechanism. More extensive measurements were possibJe for norbornylcne, and in the temperature range 251 to 308 K the atmospheric pressure results fit the equation Measurements on oriented norbornylene specimens showed that any anisotropy of the self-diffusion was within the experimental error.Pressure effect studies for norbornylene at 281 and 301 K yielded an activation volume equal to 0.840.1 times the molar volume. These results are in excellent agreement with existing n.m.r. data and confirm a monovacancy mechanism of self-diffusion. The present work suggests that in cases where there is a disagreement between radiotracer and n.m.r. measurements this cannot be explained simply in terms of the entropy of fusion of the material. Translational self-diffusion in plastic crystals, molecular solids in which the molecules are globular and undergo rapid endospherical reorient ation, has been monitored by radiotracer, nuclear magnetic resonance (n.m.r.) and plastic deformation (creep) measurements.' 9 Trends in the diffusion parameters have been associated with variations in the entropy of fusion, ASf, and the entlialpy of self-diffhion, AH, is usually compared with the latent heat of sublimation, E,.The activation enthalpies from n.m.r. measurements, AHn, exhibit an apparent dependence on ASf ;1-4 AH, increases from 1 L, to 2 L, as ASf increases from 1 PP to 2.5 R, where R is the gas constant. The activation enthalpies from tracer measurements, AH,, are consistent with the values from creep experiments and are usually equal to -2Ls irrespective of AS,.'. Thus for materials with AS, < 1.7 R values of AH, are usually greater than those of AHn. This could be taken as an indication that the two techniques are not monitoring the same process,2 although it has been noted that the differences between the n.m.r.and radiotracer self-diffusion coefficients, D, and D,, are not large. More measurements for low ASf materials are needed to understand the diffusion behaviour in these systems. Since the plastic crystals have close-packed structures, the predominant point defects are expected to be lattice vacancies and therefore self-diffusion is usually assumed to occur by a monovacancy mechanism.l* Detailed calculations of AH are not available for plastic crystals, but to a first approximation it would be expected to be similar to that found for rare gas crystals. In these latter materials theoretical calculations of AH for a monovacancy mechanism yield AH !Y 2 L, and this is in f Present address : Materials Section, Building Research Station, Garston, Herts.2562A . V . CHADWiCK AND J . W. FORREST 2563 agreement with experiment.6 In addition to a simple comparison of AH and L,, three other methods have been employed to obtain more direct information on the diffusion mechanisms in plastic crystals ; (i) measurement of the isotope mass effect on self-diffu~ion.~* * and (iii) measurement of the effect of hydrostatic pressure, P, on diffusion.1°-12 The first two methods make use of the fact that in the solid state the translational diffusion of a particular molecule is not necessarily a random process and successive jumps of the molecule may be ~0rrelated.l~ The correlation factor, f, is a measure of the degree of correlation and can be calculated for a given diffusion mechanism, lattice structure and experi- mental diffusion technique.In method (i) the correlation factor for tracer diffusion, ft, is determined and compared with the theoretical values. The ratio D,/D, is equal toft&, wheref, is the correlation factor for n.m.r. diffusion, and in method (ii) this ratio is compared with the theoretical values. This latter method is less direct and at presentf, is not well-known for all structures. It also requires careful calibration of the temperature scales in the two experiments. Studies of D as a function of P allow evaluation of the activation volume for self-diffusion, A V,. Information on the diffusion mechanism can be obtained by comparing the ratio of AV, and the molar volume, kT,, with theoretical estimates for the various diffusion mechanisms and with values of this paranieter in other systems in which the mechanism has been identified from other types of experiment.In this paper we report the results of radiotracer studies of self-diffusion in norbornam (bicyclo[2,2,l]lieptane) as a functioa of temperature, T, and in norborny- lene (bicyclo[2,2,l]hept-2-eiie) as a function of T and P. Norbornyiene has one plastic phase l4 (129 K-melting point) in which the structure is close to being ideally hexagonal close-packed (h.c.p.) with c/n = 1.61 l 5 and ASI. = 1.22 R.'" Norbornane has two plastic phases and AS, = 1.53 R.14 In the lower phase (131-306 K) the structure is h.c.p. with cia = 1.50 l 5 and in the higher phase (386 K-melting point) the structure is face-centred cubic (f.c.c.).For both these materials n.m.r. spin lattice relaxation times in the rotating frame, Tlp, have been measured as a function of T 4 and P.lo* Since these are low ASf materials the comparison of tracer and n.m.r. diffusion parameters will be important. (ii) Comparison of D, and D, 6 * EXPERIMENTAL MATERIALS The details of purification of the niaterials and preparaiion of the tritium-labelled tracers have been described elsewhere.16 Vapour phase chromatography indicated that norbornylene and norbornane contained < 1000 p.p.m. and < 10 p.p.m. impurity, respec- tively. The major impurity in norbornylene was its isomer nortricyclene (tricyclo[2,2,1 ,03s5]- heptane) which we were unable to remove completely; however, since the norbornane was prepared by hydrogenation of the norbornylene it seems reasonable to assume that the levels of other impurities were -c 10 p.p.m.Good crystals of norbornylene (10 mm diameter x 50 mm long) were gown from the melt by the Bridgman-Stockbarger technique and by sublimation growth. The orientations of the melt-grown crystals were determined by X-ray methods. Samples were cut parallel and perpendicular to the c axis, the maximum angle of misalignment being about 10'. A crystal deliberately doped with 7000 p.p.m. nortricyclene was melt-grown. Norbornane crystals grown from the melt shattered on cooling through the phase transition at 306 K. Good crystals of the h.c.p. phase were obtained by sublimation growth; however, they were sensitive to thermal shock.Cooling these crystals to 243 K or cutting them with a razor blade introduced cracks and although these annealed out after several hours, the samples were considered to be of a poor quality for diffusion experiments.2564 SELF-DIFFUSION IN PLASTIC SOLIDS DIFFUSION PROCEDURE The basic sample preparation, annealing and sectioning procedures were similar to those used in previous work.5 The tracer deposit was applied to the sample in the form of a saturated solution in n-pentane and the sectioning was performed in a deep-freeze maintained at 243 K. During an anneal the diffusion couple was encapsulated in mercury to eliminate evaporation, Diffusion anneals under pressure were performed in an apparatus similar to that described by McKay and Sherwood.12 The essential features of the system were a commercial pressure vessel (Pressure Product Industries), a liquid pressure transmitting system, a manual hydraulic pump (Enerpac P228-7) and a calibrated Budenberg gauge.The temperature and pressure variations of the sample during anneals were < +O.l K and < +2 MN m-2, respectively. Standard procedures were used to correct the anneal times for the heating and cooling, and pressurising and de-pressurising periods. RESULTS NORBORNANE The solution of the diffusion equations for this type of tracer sectioning experiment is C(x) = ____ ‘ exp ( -x2]4D,t). (zDtf)* Here C(x) is the specific activity of tracer that has penetrated to a depth x into the crystal, Q is the total amount of tracer initially deposited on the sample and t is the time of the diffusion anneal.100 I x2/10-’ m2 FIG. 1.-Typical diffusion profile for norbornane : 9, raw data points for a run at T = 303.5 K with t = 233 200 s ; x , the same points after subtraction of the tail (see text). The straight line fit yields Dt = 9.8 x 10-ls m2 s-l. A typical profile for norbornane is shown in fig. 1 and clearly this cannot be fitted to eqn (1) with a unique value of D,. The long tail is typical of pipe diffusion,’. l7 i.e. diffusion along bulk defects such as grain boundaries and dislocations. This wasA . V . CHADWICK AND J . W. FORREST 2565 to be expected from the poor quality of the samples. By subtracting a linear extra- polation of the tail off of the raw data points it was possible to estimate the contribution from lattice diffusion to the profile.The fitting error in D, from this procedure was +30 % ; however, the absolute error was difficult to assess. At worst the values of Dt obtained represent an upper limit to the true values. The poor quality of the sample and the low diffusion coefficients meant that measurements were restricted to a very short temperature range, 293-304 K. In fig. 2 the present results are compared with the n.m.r. data. lo3 KIT FIG. 2.-Self-diffusion coefficients for norbornane plotted as a function of reciprocal temperature : a, radiotracer results ; x , n.m.r. data.4 T.P. is the h.c.p.-f.c.c. transition temperature. NORBORNYLENE Typical diffusion profiles for norbornylene are shown in fig.3. In contrast to norbornane there was only a slight tailing at deep penetration. Diffusion coefficients were evaluated by neglecting these tails and fitting the remainder of the profile to eqn (1). The fitting errors gave an uncertainty in D, of about 10 %. The profiles x2/10-' m2 FIG. 3.-Typical diffusion profiles for norbornylene : the lines are best fits to the points and the arrows indicate 9Dtt. m2 s-' ; 0, T = 251.3 K, t = 240 4320 s, P = atmospheric, Dt = 1.45 x 10-15 m2 s-I ; +, T = 301.2 K, t = 226 440 s, P = 34.5 MN m-2, Dt = 2.67 x x , T = 281.0 K, t = 272 160 s, P = 0 , T = 299.3 K, t = 150 600 s, P = atmospheric, Dt = 6.49 x m2 s-' ; atmospheric, Dt = 1.75 x m2 s-l.2566 SELF-DIFFUSION IN PLASTIC SOLIDS were linear within the region x < 3Jzt which is an indication that the values of D, were representative of lattice diffusi0n.l When it was possible this point was further checked by showing that D, was independent of the anneal time.Unfortunately this was not possible when D, was -10-15 m2 s-I since the arsneal times necessary to obtain the minimum reasonable penetration were around the maximum limit for convenient and accurate measurements, i.e. one month. The anneal times used were similar to those employed by Hampton and Sherwood in their careful tracer study of cyclohexane. Thus precautions were taken to minimize the effect of pipe diffusion. If these procedures were not completely successful then D, would be an over-estimate of the lattice diffusion and the error would increase with decreasing values of D,.Consequently AH, and A V, would be underestimated. 103 KIT FIG. 4.-Self-diffusion coefficients for norbornylene plotted as a function of reciprocal temperature : 0, 8 and + are the radiotracer results for crystals with faces (1 c-axis, I c-axis and with random orientations, respectively ; x , n.m.r. data.4 M.P. is the melting point. The atmospheric pressure results and the corresponding n.m.r. data are shown in fig. 4. The results for oriented crystals show that any anisotropy of self-diffusion is less than the experimental error. Since the structure of norbornylene is close to ideal h.c.p. this result was not unexpected. At 273 K, D, for the nortricyclene doped crystal was the same as that for the nominally pure crystals. Again this is not an unexpected result due to the similarity of the structures of dopant and host and the relatively low doping level employed.A least-squares fit of the tracer results to an Arrhenius equation in the form D, = D,, exp (-&H,/RT) yielded + 3.3 D,, = k . 6 - lo-’ m2 s-l and AH, = 49.2f2 kJ mol-’. Within the limits of the present experiments AH, is independent of 7‘. The results of the pressure experiments are shown in fig. 5. coefficient can be expressed as l 8 The self-diffusionA . V . CHADWICK AND J . W . FORREST 2567 where c1 is the lattice parameter, v is a vibrational frequency (usmlly taken as the Debye frequency), /? is a geometric factor, and AGd is the Gibbs free energy for diffusion. The activation volume, AVd, is defined as = [a(AGd)/ap]T (4) and hence from eqn (3) The second term on the r.h.s.of eqn (5) is negligible and AVd is obtained from the slope of a plot of log D against P. The present results yielded AVd = (82+6) x m3 mol-1 at 301 K and AVd = (75k5) x m3 mol-1 at 281 K. Within the limits of this study AYd is independent of P and T. P/MN m-” FIG. 5.--Self-diffusion coeffcients for norbornylene plotted as a function of pressure : 0 and 9 are the radiotracer results in randomly oriented crystals at 301.2 and 281.0 K, respectively; x , n.m.r. data at 296 K. The lines are best fits to the points. DISCUSSION The available diffusion data for norbornane and norbornylene are collected in table 1. The present tracer results for norbornane are not sufficient in themselves to yield information on the mechanism of self-diffusion.The n.m.r. data are consistent with a monovacancy mechanism of self-diffusion in norbornane. The value of AHn/Ls is I .76 5 0.06 which is only slightly smaller than the theoretically predicted value for this mechanism. However, the prediction is an approximation based on an analogy with the rare gas crystals and the difference may not be significant. The strongest evidence for this mechanism being operative comes from the n.m.r. pressure data. Hard-sphere calculations for close-packed solids (f.c.c. and h.c.p.) predict an activation volume for vacancy formation equal to V, and the corresponding migration volume equal to -0.8 V,. Experimental determinations of AVd/Vm in systems where the vacancy mechanism has been confirmed by other techniques are considerably less than 1.8, e.g.0.7 to 0.95 in metals,19* 2o 0.72 to 1.3 in plastic crystals l2 and 1.2 in naphthalene.21 It is usually accepted that AVd/Vm < 1 is due to some inward relaxation of the molecules surrounding the vacancy and this would explain the result for norbornane. It would be unwise to evaluate AH, from the few data points in this study; however, they roughly parallel the temperature2568 SELF-DIFFUSION I N PLASTIC SOLIDS dependence of the n.m.r. data. The value off,/& for vacancy diffusion in a h.c.p. structure is not known but it would not be expected to be very different from that for a f.c.c. structure, i.e.ft/fn N 1.4. Thus the present estimate of 1.5k0.2 is extra support for monovacancy self-diffusion occurring in norbornane.TABLE 1 .-SELF-DIFFUSION PARAMETERS FOR h.c.p. NORBORNANE AND NORBORNYLENE The present norbornane reference norborny lene 1.53 14 1.22 3 1 f l 4 33+ 1 - - 1.49+ 0.1 1 1.76+_ 0.06 4 1.475 0.06 1.5k0.2 this work, 4 1.5k0.2 100.2 15 95.8 0.99+0.01 (299 K) 11 0.88s40.02 (296 K) 0.86+0.06 (301 K) 0.7840.05 (281 K)} reference 14 4 this work 4 this work, 4 15 10 this work tracer measurements for norbornylene yield AH&, as 1.49 & 0.1 1 which is reasonable evidence for a monovacancy mechanism even though it is smaller than the theoretically predicted value. Again this difference may be due to a weakness in the theoretical model. There is also the possibility that AHt has been under- estimated due to an undetected contribution to D, from pipe diffusion.It is difficult to eliminate this possibility although a comparison with other tracer studies of plastic crystals would suggest that it is unlikely. Unless norbornylene has a different bulk defect nature to other materials of this type then the present experiments should have minimized the effect of pipe-diffusion and AHt should not be significantly smaller than the true activation enthalpy for lattice diffusion. Similarly, the value of A Vd/ V,, -0.8, should be representative of lattice diffusion and, by analogy wiih other materials, support the view that self-diffusion in norbornylene involves a monovacancy which is slightly relaxed. The only directly comparable tracer study of a close-packed plastic crystal has been made for f.c.c. pivalic acid l2 where A Vd/ V, was found to be 1.3.For four f.c.c. plastic crystals, including pivalic acid, A Vd/ V, has been determined by the creep technique l2 and the values are in the range 1 .O to 1 .3. The fact that norbornylene lies outside this range may be a consequence of the difference between f.c.c. and h.c.p. structures although considerably more data for both lattice types would be needed to verify this point. In addition, it should be remembered that the measured AV, is not the sum of the local defect formation and migration volumes. It is these latter volumes that ought to be used in comparing various materials ; however, their evaluation from A Vd requires a knowledge of Poisson's ratio.Ig Since this is not available for the plastic crystals that have been studied it is not meaningful to try to interpret small variations of AV,/V,.For norbornylene there is a good agreement between AHt and AHn, the latter being obtained from data points over a much wider temperature range.4 The n.m.r. data show a region of apparently increased activation energy at the highest tempera- tures, as can be seen in fig. 4. This has been observed in other low ASf plastic crystals. It has been attributed to the use of the Torrey model to evaluate D, from the Tlp data and is not due to any change in the diffusion process.22 As with norbornane, the ratio of DJD, is 1.5+_0.2 and provides additional support for monovacancy self- diffusion. The agreement between the tracer and low pressure n.m.r. values of AV, for norbornylene is excellent. The n.m.r. study lo revealed a region of low AV, at pressures which would be beyond the scope of a tracer study.A .V . CHADWICK AND J . W. FORREST 2569 In summary, norbornane and norbornylene are two low ASf plastic crystals for which tracer and n.m.r. studies of diffusion yield concurrent results; in the case of norbornylene these include D, A H and AVd, which can be interpreted in terms of monovacancy self-diffusion. It may be significant that both these materials are relatively brittle.23 Differences between AH,, and AH, in some soft, low ASf, materials are now well-substantiated, notably in cyclohexane, * although differences in D and AVd from the two techniques are not large. There is still no complete explanation of this effect. Recent work 22 would suggest that the n.m.r.data does not monitor pipe-diffusion and the isotope-mass effect suggests that the lattice diffusion processes monitored by the tracer methods are the same for all plastic crystals irrespective of ASf. One major obstacle in resolving the discrepancy is the fact that the tracer measurements are limited to D, > lO-I5 m2 s-l and therefore cannot be used to investigate the possibly interesting regions at high P and low T. The conclusion of the present work is that the source of the discrepancy is not simply the numerical value of ASf and that the explanation needs to be sought in terms of the nature of the plastic crystalline state. One of us (J. W. F.) thanks the University of Kent for the award of a graduate assistantship. The authors acknowledge valuable discussions held with Drs.N. Boden, R. Folland and J. H. Strange. A. V. Chadwick and J. N. Sherwood, Point Defects in Solids, ed. J. H. Crawford, Jr. and L. M. Slifkin (Plenum Press, New York, 1975), vol. 2, p. 441. J. N. Sherwood, Surface and Defect Properties of Solids (Spec. Period. Rep., Chemical Society, London, 1973), vol. 2, p. 250. P. Bladon, N. C. Lockhart and J. N. Sherwood, Mol. Cry~t. Liq. Cryst., 1973, 19, 315. R. Folland, R. L. Jackson, J. H. Strange and A. V. Chadwick, J. Phys. Chem. Solids, 1973, 34, 1713. A. V. Chadwick, J. M. Chezeau, R. Folland, J. W. Forrest and J. H. Strange, J.C.S. Faraday I, 1975,71,1610. 6A. V. 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Labelled Compounds, 1975,11, 242. l7 J. N. Sherwood and D. J. White, Phil. Mag., 1967, 16,975. l 8 P. G. Shewmon, Diffusion in Solids (McGraw-Hill, New York, 1963). D. Lazarus and N. H. Nachtrieb, Solids under Pressure, ed. W. Paul and D. M. Warschauer (McGraw-Hill, New York, 1963), p. 43. 2o N. L. Peterson, Solid State Physics, ed. F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1968), vol. 22, p. 409. 21 E. Hampton and J. N. Sherwood, J.C.S. Faraday I, 1975, 71, 1392. 22 N. Boden, J. Cohen and R. T. Squires, MoZ. Phys., 1976, 31, 1813. 23 R. M. Hooper and J. N. Sherwood, Surface and Defect Properties of Solids (Spec. Period. Rep., Chemical Society, London, 1977), vol. 6, p. 308. (PAPER 7/2275)