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High field conduction in native cellulose and its structural implications
High Field Conduction in Native Cellulose and its Structural Implications E.
J.
MURPHY 1
School of Engineering and Applied Science, Columbia University, New York, New York 10027 Received March 7, 1974; accepted May 22, 1974 It is shown that the conductivity of native cellulose in vacuo increases rapidly with voltage gradient; it is given by a sum of two exponential terms. While the conductivity of this material is ionic, as evidenced by the development of H 2 in approximate agreement with Faraday's law, it is proposed that the increase of conductivity with voltage is caused by local electronic conduction in transient thermally-generated defects in the form of voids. These defects have essentially the same basis in statistical mechanics as the defects on which the theory of ionic conduction in crystals is based (Frenkel and Schotky defects); they should be present in a concentration proportional to a Boltzmann factor. They give mobility to the ions generated by the thermal dissociation of the molecules. In order to explain the voltage dependence it is proposed that negative ions tend to collect in small excess on one side of a defect or gap and positive on the other. The unoccupied levels in the positive ions are above the occupied levels in the negative ions, but when an adequate electric field is applied they can be brought into coincidence, and wave-mechanical transport of an electron occurs, as in the Gurney theory of electrolysis. Developing this quantitatively leads to the relation g = (d Inu/dV)kTe-" where g is the width of the gap, u the conductivity, k Boltzmann's constant, T the temperature in OK, and e the electronic charge. The values of g calculated from the data cluster around two mean values: g = 19.5 A and g = 34.9 A, the first corresponding to the higher, the second to the lower part of the range of voltage gradients. This model indicates that H 2 and O2 should be generated in the defects or voids by this type of internal electrolysis. Experimentally H 2 and CO are generated at a rate increasing with voltage, such that at 0.5 X 106 V/cm the yield is about 2 moles/f, i.e., too large for an electrode effect. Thus th€ model explains both the voltage-dependence of conductivity and that of the electrochemical yield (of H 2), using properties already needed for the explanation of conduction and electrolysis. The presence of transient defects (or voids) in thermodynamic equilibrium can also be used in the model for the explanation of the conductivity/water-content relationship. It is proposed that the conductivity is proportional to the probability of simultaneous adsorption of single water molecules at nine discrete sites, characteristic of each transient defect.
INTRODUCTION
A dependence of conductivity on voltage gradient has been observed in a number of materials, both solid and liquid. In some cases it is a small effect, in others very large. We show here that native cellulose exhibits a large increase of conductivity as the applied voltage is increased, and attempt to 1 Present address: 217 East 66th Street, New York, NY 10021.
interpret this in terms derived from other aspects of the conduction behavior. The model developed depends upon the comparative weakness of hydrogen bonds and the consequent largeness of the transient defects (holes) on which the theory of conduction is based, together with an adaptation of the Gurney theory of electrolysis to conduction in the thermally generated gaps. The structure is pictured as having transient, mobile 442
Journal of Colloid and Interface Science. Vol. 49, No.3, December 1974
Copyright © 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
CONDUCTION IN CELLULOSE
gaps about 20-40 A in width and present in a concentration proportional to a Boltzmann factor, i.e., in thermodynamic equilibrium. The temperature dependence of the conductivity of native cellulose has been described elsewhere (1); it is in good agreement with the theory of ionic conduction. The conductivity of this material is very sensitive to its water content; this has also been investigated and the quantitative relationship explained by means of a model depending upon simultaneous occupancy by single water molecules of discrete, contiguous adsorption sites (2). Thus this material exhibits two types of conduction process: (i) the intrinsic (dry state) conduction, and (ii) the conduction contributed by internally adsorbed water molecules. We attempt to show here that at high voltage gradients there is a third type involving electrons in the gaps and ions in the solid. While the temperature dependence, watercontent dependence, and voltage dependence of conductivity could be regarded as independent processes, we attempt here to connect them by deriving them as aspects of a single model. This model depends mainly on the presence of voids or defects, similar in their statistical mechanical origin to the defects (e.g., Frenkel defects) involved in conduction processes in crystals, but taking the form of rather large loops because of the chain-like (and periodic) structure of cellulose molecules, and the comparative weakness of hydrogen bonds between chains. The water molecules are considered to be adsorbed on exposed OR groups during the elementary step in water-induced conduction. This is a model for conduction applicable mainly to a very loose hydrogen-bonded solid structure, rather than to dense crystals. The picture of transport and conduction processes to which this model leads may have a general application to transport and other processes in membranes of biological interest, as well as to some technological processes.
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EXPERIMENTAL ARRANGEMENTS
The material on which the measurements were made (electrical condenser paper) is nearly pure cellulose beaten in water in a way which produces a homogeneous solid of considerable strength. The beating process modifies the surfaces of the fibers so that they are held together by a gel-like structure which presumably depends ultimately upon a system of hydrogen bonds (3). The fibers of cellulose have a degree of crystallinity, e.g., a repeat interval of 10.3 A along the axis of the chain molecule (4). A structure of this kind, combining a degree of crystallinity with gel-like properties, may be representative of a class of materials of biological and technological interest. The term "native cellulose" distinguishes natural products from synthetic cellulosic polymers; the properties of the latter are described, for instance, by Flory (5), those of the former by Frey-Wyssling (6). It has been found, by a great many measurements, that the conduction behavior of native cellulose is regular and reproducible. This is perhaps surprising in view of the physical complexity of the materials used, as compared, fcr instance, with the structural simplicity of a single crystal of an alkali halide. This suggests that the regular features of structure on the molecular scale, rather than irregular features on a larger scale (such as fiber contacts), determine the conduction behavior predominantly. The conductivity measurements were made upon condenser paper of standard composition, held between foil electrodes under mechanical pressure. In most specimens the foil electrodes were composed of Sn or AI, but in addition similar measurements were made Ag, Cu, and Cu-Be foil as electrodes. The specimens were suspended in Pyrex glass tubes by means of the leads used to apply the voltages and measure the currents. The leads entered the tube through tungsten-toglass seals. The tubes enclosing the specimens were sealed to a pumping station and evacuated to pressures less than 10-4 Torr, as mea-
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~.
J.
MURPHY
sured on a McLeod gauge. The cellulose remained thoroughly dry during the measurements, except when water vapor at a known pressure was deliberately introduced by opening a stop cock leading to a bulb containing pure water vapor. The specimen tubes were immersed in a constant temperature bath of oil. The rates of generation of water vapor by thermal decomposition of the cellulose and of permanent gases by electrolysis were monitored by means of the McLeod gauge. The conductivity was determined by applying Ohm's law to data obtained by means of a voltmeter and a Leeds and Northrup HS galvanometer calibrated by means of a standard megohm. The determination of the true value of the conductivity is often complicated by variation of the current (for a constant voltage) with time, but this has been discussed fully in previous papers by this writer, which also give further detail regarding experimental arrangements and methods used in conductivity measurements on this type of material (2). However, high voltage measurements (i.e., measurements at voltage
-
gradients extending up to the breakdown level) present special difficulties. One of these is the tendency for electrical discharges to occur in the ambient air. In the present method this was avoided by two conditions: (a) the specimens were very thin, so that gradients up to about 1.2 X 106 Vjcm could be produced in the material by applying voltages not exceeding 2400 V; and (b) the specimen was enclosed in an evacuated tube where the pressure was low enough that glow discharges did not occur. The agreement of the temperature dependence of conductivity with the requirements of the theory of ionic conduction in solids (1) also gave assurance that no gas-phase conduction was influencing the conduction behavior observed. EXPERIMENTAL RESULTS
The data plotted in Fig. 1 illustrate the dependence of the resistivity of dry native cellulose on the applied voltage gradient for several temperatures of the specimen. In the theoretical discussion which follows we are concerned only with the slope of the 10glT vs
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FIG. 1. Resistivity of native cellulose as a function of applied voltage gradient. (The curves for conductivity have slopes of the same magnitude but opposite sign.) One specimen was exposed to a relative humidity of 32%, another to a water vapor pressure of 9 mm Hg; all other specimens were in vacuo at pressures below 10-4 mm Hg. Journal of Colloid and Interface Science, Vol. 49, No.3, December 1974
445
CONDUCTION IN CELLULOSE
voltage gradient curve; this is the same in magnitude though opposite in sign to the slope of the logp vs voltage gradient curve, where IJ is the conductivity in (n cm)-1 and p (=IJ-1) the resistivity in (n cm). Data are also given for two cases where water vapor was introduced so that the specimen had some conduction due to the presence of adsorbed water molecules. At lOO°C the water had little effect on the slope of the 10gIJ voltage gradient curves, but at 35°C it produced a substantial reduction. This would be expected from the model to be presented here. The influence of electrode material on the slope was also investigated, by using Ag, Cu, and Cu-Be foil in place of Al or Sn. The curves obtained are omitted in order to avoid complication of the figure, but the slopes calculated from them are included in Table 1. No influence of the electrode material is evident. This indicates that the voltagedependence is not likely to be derived from electrode effects for these would vary with the work function of the metal, films of oxide, and other conditions of the surface. These must then be minor influences. The voltage-dependence data show stability and reproducibility in general; however, in one case measurements repeated after an interval of one month during which the specimen was at 120°C showed a significant decrease in slope. This may imply that an annealing process goes on in the cellulose at high temperatures, changing the structure somewhat. 'lilE VOLTAGE-DEPENDENCE OF CONDUCTIVITY IN OTHER MATERIALS
Before discussing in detail the voltagedependence of the conductivity of native cellulose it will be useful to outline briefly the nature of voltage-dependence as observed in other materials. A comparatively small increase of the conductivity with voltage occurs in electrolytic solutions (the Wien effect) (7). This, which is perhaps the most familiar ex-
TABLE I GAP-WIDTHS
(g)
IN ANGSTROM UNITS
(A)
CALCULATED
FROM THE VOLTAGE DEPENDENCE OF CONDUCTIVITY
Temperature ('C)
35.0 75.0 100 112 120 125 130 140 160 160 160
Water vapor pressure (mm.Hg)
(Eq. [8J)
Electrode material
10 12
Sn or Al Sn or AI
19.6 16.6
10-' 10-4 10-4 10-4 10-4 10-4 10-4 10-' 10-'
Sn or AI Sn or Al Sn or AI Sn or AI Sn or Al Sn or Al Ag Cu Cu-Be
21.3 16.7 20.8 18.5 18.3 18.2 21.9 21.2 21.9
30.8 35.9 34.4 36.8 35.0 36.5
Note: The gap-width g2 is derived from the larger slope, g2 from the smaller slope. The slopes used in calculating gap-widths by means of Eq. [8J were taken from curves similar to those illustrated in Fig. 1 but including some other data. The average for gl is 19.5 A, for g2 34.9 A.
ample of voltage-dependence, has been attributed to a decrease in the drag exerted on each ion by its Debye-Hlick.el ionic atmosphere as the drift velocity increases (8). For weak electrolytes Onsager has developed a theory involving an increase in the degree of dissociation into free ions as the voltage is increased (9). Oils often exhibit a very large increase of apparent conductivity with increasing voltage; several hypotheses have been proposed to explain this, among them field emission of electrons from the electrodes (10). A. v. Hippel has found that the conductivity of single crystals of the alkali halides increases with increasing voltage at field strengths approaching the breakdown level, and has attributed it to field emission from the electrodes (11). Beran and Quittner (12) found that the conductivity of rocksalt crystals increases exponentially with applied voltage, but Hochberg (13) later showed -that when the crystal has been annealed (thereby presumably removiIlg incipient cracks) the volt-
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age-dependence vanishes. In measuring the conductivity of crystals it is sometimes found that the conductance of the specimen increases with voltage up to some level such as 600 V and then becomes independent of voltage. This can be traced to poor contact between the dielectric and the electrode, and it is, of course, not properly a voltage-dependent conductivity. For semiconductors, such as germanium, it has been found by Ryder (14) that the conductivity decreases with increasing applied voltage. This has been attributed to the electrons interacting more strongly with the lattice the higher their velocity, so that the mobility is decreased. The application of this result to the present problem is that the possibility that electronic conduction could become more prominent than ionic as the voltage increases, because of an increase of electron mobility with voltage, is apparently excluded. There is also little basis for expecting an increase in ionic mobility (by a substantial amount) as voltage increases because the ions move in thermal equilibrium with the lattice and their kinetic energy is not substantially increased by the applied field. Thus neither ionic conduction, nor electronic, in a homogeneous material would be expected to give an increase of conductivity with voltage gradient as large as that observed in the present experiments. An investigation of the conductivity of mica by Poole (15) leads to results rather closely related to those of the present investigation. He found that the conductivity of mica increases with voltage, and can be expressed as a sum of two exponential terms. MODEL FOR THE VOLTAGE-DEPENDENCE OF THE CONDUCTIVITY OF CELLULOSE
In the model to be developed here the increase of conductivity with voltage is attributed to the presence of voids in which electronic conduction of a special character can occur to an extent increasing with the voltage. The voids are not, as a rule, per-
manently situated at fixed places; a void may disappear at one place when another appears at another place, thereby maintaining a fixed concentration (given by a Boltzmann factor) at a given temperature. The electronic conduction process in the voids (gaps or defects) is modeled on the theory of electrolysis pro~ posed by R. W. Gurney (16). The present model accounts for certain puzzling features of the electrochemical yield of H 2 and CO, as well as the voltage dependence of conductivity. The above is a brief indication of the nature of the model to be developed in more detail below. The experimental relationships upon which it is based are mainly the following: (i) The temperature dependence of the conductivity of native cellulose agrees with the theory of ionic conduction in solid dielectrics. This theory is based to a large extent upon "defects" present in temperature-dependent concentration given by a Boltzmann factor (17, 18). Therefore we expect to find such defects in cellulose, and it is proposed that, because this material is composed of long-chain molecules, the defects are realized as interchain loops which may be rather large. (ii) Cellulose contains crystalline regions (micelles) which according to X-ray diffraction evidence are of limited extent, though there may be a great many of them. Between the crystalline micelles is a region of less orderly intermicellar material. It is proposed that the larger of the thermally-generated defects or holes tend to appear in the intermicellar regions. (iii) A current passing through dry cellulose generates H 2 and CO (with a little C02) at a rate in approximate agreement with Faraday's law (19). However the yield increases with voltage and at the highest voltage gradient used (0.5 X 106 Vjcm) reaches about 2 mol/f. This is somewhat larger than is to be expected from an ordinary electrolytic reaction at electrodes; moreover the yield should be independent of voltage since the applied voltage was much larger than decom-
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position potentials. (Increasing the voltage merely brings ions up to the electrode more rapidly; it does not change the electrode processes.) ELECTRONIC CONDUCTION IN A GAP
From (i) and (iii) we infer that thermal energy causes the dissociation of cellulose to form H 30+ and OH- ions. These ions would tend to become adsorbed at suitable sites on the chain molecules. If the material contains defects or gaps in temperature-dependent concentration there will be ions in the material around the gaps. When a voltage is applied there should be a tendency for positive ions to be present in excess on one side of the gap and negative ions at the opposite side. The adsorbed ions would. tend to be associated with the cellulose molecules in somewhat the same way as ions in aqueous solutions are associated with water molecules. The adsorbed ion should form a complex with the large molecule at the site where it is adsorbed, such that each stationary state of the isolated ion would become spread out into a band of levels. The population of electrons in each level is then proportional to exp( - W;jkT), where Wi is the energy of the ith level, as in atomic equilibria in general. When ions of opposite sign are separated by a void of not too great width, the application of a voltage of suitable magnitude can appreciably change the potential energy distribution, so that resonance tunneling of an electron can occur between the highest occupied level of the negative ion and the lowest unoccupied level of the positive ion. The population of ions with suitable energy for tunneling should then be a function of temperature and voltage, proportional to exp(-W + Veg)/kT, where W = W+ - W_, the energy gap between the lowest unoccupied level of the positive ion and the highest occupied level of the negative ion; V is the voltage gradient and g the width of the gap. (In effect, the applied voltage reduces the potential energy of an electron in the positive ion
447
by Veg/2 and increases that of an electron in the negative ion by the same amount, so that the two levels become the same.) The model is related to the Gurney theory of electrolysis, wherein it was shown that an exponential dependence of current on voltage (at low voltages) could be explained by the tunneling of an electron from electrode to positive ion, or from negative ion to electrode (16). More explicitly, if RH+ is an adsorbed ion at one side of a void of width g and XOHan ion at the opposite side, an applied voltage V acting in the right direction gives W(V) = W - Veg; and when Veg = W, tunneling can occur with substantial probability unless g is too large. This process occurring in a void will increase the conductivity locally by in effect short-circuiting a few unit cells momentarily. (By hypothesis, the voids are not permanent, but appear and disappear.) In this model the main part of the conduction still depends upon diffusional migration caf ions, since voids are present only in a few places at any given time, or only for a small fraction of the time at any given place. No domain is then void free at all times, but all are void free part of the time. The void-free material retains its ohmic conductivity, but the material as a whole (i.e., the material containing a concentration of voids) shows an increase of conductivity with voltage. The activation energy for conduction, as a property of the void-free material, should be essentially independent of the applied voltage; this seems to be the case, at least for a small range of voltages (1). Permanent voids, such as might be expected to be present in some materials, should behave in a way similar to temporary voids except that they would tend to maintain the heating effect of the current in one place, with consequent rise of local temperature, instability of conductance, and ultimate dielectric breakdown by a selfaccelerating process. When the heating is distributed uniformly because of the voids not being permanently present in fixed places
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this self-accelerating process is not nucleated. Moreover, in this material a condition for the permanence of a void may be that its size be substantially greater than that of the voids generated by thermal energy. CALCULATION OF THE WIDTH OF THE VOID FROM VOLTAGE DEPENDENCE
In the model outlined above we have positive adsorbed ions in excess at one side of a gap of width g, and negative ions in excess at the other side. Favorable thermal fluctuations of energy permit resonance tunneling at a given voltage V. The probability of such an event is such that the conductivity u IS given by
u = A exp(Veg - W)/kT,
[lJ
where A is the product of the voltage-independent quantities in the full expression for ionic conduction. The logarithmic derivative of the conductivity with respect to voltage gradient (in e.s.u.) is then given by
dlnu/dV = eg/kT.
[2J
Thus on this model the slope of the (loglT, V)curve depends only on the width, g, of the gap, the absolute temperature T in OK, the Boltzmann constant k, and the electronic charge, e. Converting from natural logarithms to the base 10, and solving for g, we have g
= 2.303 (kT/e)d 10gu/dV.
[3J
The values of g given in Table I were calculated by means of Eq. [3J from the curves in Fig. 1, together with a few other curves of similar type. The values of the gap width, g, given in Table I, appear to be distributed about two average values: g1 = 19.5 A and g2 = 34.9 A. There is a considerable scattering of the points, but since the data were derived from specimens at different temperatures, with different electrode materials, and in some cases with different water content, the scattering is perhaps not larger than ought to be expected. The absence of a definite effect of electrode material on the voltage-dependence indicates
that surface films on the electrodes are not likely to have had an important influence. GENERATION OF GAS BY DISCHARGES WITHIN THE DIELECTRIC
As already indicated [ (iii) above] the passage of a current through dry cellulose leads to the development of hydrogen and carbon monoxide with yield which increases with voltage gradient. It was at :first thought that this was a result of some kind of bombardment process controlled in its rate by the conductivity of the material, but it seemed unlikely that a conductivity-temperature relationship agreeing closely with the requirements of the theory of ionic conduction would be obtained if bombardment processes formed a substantial part of the whole conduction process. Moreover, the model for the voltagedependence of conductivity which has just been described appears also to provide a natural explanation of the voltage-dependence of the electrochemical yield by the processes indicated below. The transference of an electron from RH+ at one side of a gap to XOH- at the opposite side can produce H 2 at one side and O 2 at the other, for the situation is similar to that prevailing in electrode reactions of certain types according to the Gurney theory. In that theory resonance tunneling of electrons between ions in solution and electrons in the metal is responsible for the electrode reaction. It also leads to an exponential dependence of current on voltage, in the range of very low voltages. On this basis it is proposed that a quasielectrolytic process can occur in gaps within a dielectric when the voltage gradient is high enough, by a process resembling those occurring at metallic electrodes. In order to obtain the products of electrolysis which have been observed, it is necessary to introduce secondary reactions. The transfer of four electrons across an internal gap in the dielectric would yield H 2 , O2 , and H 20 accord40H- ~ 2H 2 O2 6H 20. ing to: 4H 30+ By an auxiUary experiment (involving in-
Journal of Colloid and Interface SGiente. Vol. 49. No.3. December 1974
+
+ +
449
CONDUCTION IN CELLULOSE
troducing oxygen gas into the system at a low pressure with the cellulose maintained at temperatures in the range 6o-160°C) it was found that O2 reacts with the cellulose rapidly enough that it is likely that O 2 developed by an electrolytic or quasielectrolytic process would appear as CO, CO 2 , and H 20. (The experiments showed that H 2 and CO are the main products of electrolysis; H 20 would not be detected because it is also evolved by thermal decomposition.) This model indicates that the voltagedependence of conductivity in native cellulose can be considered to be associated with highvoltage electrode-free electrolysis occurring in temporary gaps present in temperature-dependent concentration given by a Boltzmann factor. Although tunneling of electrons is involved here, the process is different from that on which field emission depends. For when a wave-mechanical tunneling process is involved in a conduction process, two extreme conditions can be distinguished: (i) the tunneling process itself is the rate-determining process, and (ii) the time required to set up conditions suitable for rapid resonance tunneling is the rate-determining process. The first of these applies to field emission, for the applied field modifies the shape of the potential energy barrier and the resulting current is given by the Fowler-Nordheim expression (20). This is the condition which presumably would apply to the explanation of the voltage dependence of conductivity in alkali halide crystals near the breakdown level, proposed by v. Hippel, and to the voltage dependence of the conductivity of oils, as proposed by LePage and DuBridge. The second condition, (ii), applies to the theory of protonic conduction in' water, as developed by Wannier (21), to the Gurney theory of electrolysis, and, according to the present' hypothesis, to electrode-free gap conduction occurring in cellulose at high voltage gradients. It may also apply to conduction in mica, because of the presence of easy cleavage planes, and to conduction in rock salt crystals which may con-
tain faults which can be annealed out. Therefore condition (ii) may apply to the results of Poole and those of Beran and Quittner, previously mentioned. VOIDS IN TEMPERATURE-DEPENDENT CONCENTRATION
We return now to some further discussion of the voids or gaps upon which the model is based. Structural defects in concentration proportional to a Boltzmann factor (e.g., Frenkel or Schottky defects) are an important part of the theory of ionic conduction in solid dielectrics. Because cellulose is composed of long chain molecules, we propose that the defects are realized as inter-chain "loops" between nodes where the chains are held together with maximal strength. Diffusion and mobility in solids requires an activation process which can be described in general terms merely as depending upon an activation complex, or a potential energy barrier, or, as in the present case, more explicitly as holes or loops requiring a certain energy for their formation. The voltage-dependence indicates that gap widths of about 19.5 A and about 34.9 A are of frequent occurrence. The smaller of these has about the length of two cellobiose units, the larger between 3 and 4 units. It is to be expected that the larger gaps would be present in lower concentration because more energy is required for their formation. Moreover, a condition for observing the smaller gaps by this type of measurement is that the larger ones be present in much smaller concentration (otherwise the influence of the larger ones would swamp that of the smaller ones). TRANSIENT DEFECTS IN RELATION TO WATER-INDUCED CONDUCTION
The size of the voids (transient loop defects) as indicated by the voltage-dependence data would be expected to bear some relatlon to the model which has already been proposed for the dependence of the conductivity of
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cellulose on its water-content (2). In that model the loop defect provides nine adsorption sites for single water molecules between each pair of ion-generating sites. In a periodic structure the ion-generating sites should be periodically distributed; and, for simplicity, we assume that the ions are generated at the nodes of the loops. In order to accommodate nine adsorption sites, a loop defect would have to be about 2! cellobiose units in length, if we assume that each OR in the cellobiose chain is able to adsorb one water molecule. This is in rough agreement with the gap width gl of Table 1. The longer loops (g2) would not be likely to have an observable effect on low-voltage conduction (i.e., on water-content dependent conduction) because of their comparatively small concentration. From the conduction data for dry native cellulose as a function of temperature it has been calculated that the energy for dissociation into ions is about 40 kcal/mole and the activation energy for mobility about 10.6 kcal/mole (1). The latter would correspond to the simultaneous breaking of two hydrogen bonds. If we assume that pairs of chain molecules are held together strongly at sites separated by the length of one cellobiose unit, and that the chains are bonded together at the ends of this unit by two hydrogen bonds, then the breaking of a pair of hydrogen bonds at this site would open up the structure to form a loop two cellobiose units in length. This loop will have a total of 8 OR groups, at each of which a water molecule can be adsorbed. The breaking of a pair of interchain hydrogen bonds at this site would then also expose two more hydrogen bonds for water adsorption. We can then assume that, effectively, these two bonds are available for adsorption of water molecules only for a fraction of the time, and therefore instead of forming a sequence of ten discrete sites for the adsorption of water molecules, the number is about 9.3 as indicated· by the experimental data; (The meaning of a fractional site is
that it is available for adsorption of a water molecule only for a part of the time.) The tendency for loop defects to form may be modified by the presence of adsorbed water molecules. Therefore loop-defects may form in substantial concentration even at comparatively low temperatures (e.g., room temperature) when the material contains a considerable concentration of adsorbed water. KINETIC ACTIVITY OF THE STRUCTURE
The loop defects of this model appear at random in the material; they are temporary, vanishing at one place and appearing at another, just as the thermal energy is distributed in a solid at equilibrium, or as the Frenkel or Schottky defects are distributed, that is, there is a statistical randomness of distribution in the accessible space. The Schottky and Frenkel defects do not make their presence evident in X-ray diffraction patterns of crystals, first because they are irregularly distributed and secondly because they are present in small concentration. Similarly, it would not be expected that the loop defects of our model would influence X-ray diffraction patterns in a direct way. Also the adsorbed water molecules would not be expected to yield liquid water rings (in diffraction patterns) because the molecules are, by hypothesis, molecularly dispersed in the material. And if the water molecules move about at random in the cellulose in conjunction with (or following) the changing distribution of thermally-generated loop defects, then the distribution of the water will be randomized' so that water molecules held at fixed places by adsorption would not form a sufficiently regular structure to appear in X-ray diffraction patterns. These considerations agree with the observation that the water content of cellulose does not influence X-ray diffraction patterns, except in extreme conditions (22, 23). It seems possible on this model that as loop defects move about in the material a gradual change of structure by an annealing
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CONDUCTION IN CELLULOSE
process could occur, if the structure is not initially at equilibrium. The hydrogen bonds between chains could slowly change their arrangement so as to pull the structure together to form configurations of lower free energy. There is a little evidence from the voltage-dependence data that this occurs when the material is held at a high temperature for a period of the order of a month, for the voltage-dependence changes in a manner suggesting that the larger loop-defects have been annealed out to some degree. When the material contains a suitable amount of adsorbed water the "annealing" process may go on quite rapidly even at room temperature. CONCLUDING DISCUSSION
The above considerations regarding the distribution of water molecules in cellulose may be summarized as follows. The water molecules are uniformly distributed in the material on a macroscopic scale by the occupancy of discrete adsorption sites (OH groups). However, the concentration fluctuates on the molecular scale in essentially the same way as the distribution of thermal energy. As loops develop by thermal excitation, water molecules become concentrated locally at the free sites in the loop defects. The presence of the adsorbed water molecules may facilitate the formation of loops at low temperatures. It is to be noted that it is not necessary that the elementary step in the conduction process be occurring simultaneously at all parts of the dielectric; for, in fact, the denial of this on the ground that it would have a negligible probability of occurring, and the inference that conduction must depend upon randomly occurring independent steps, was one of the main considerations in the development of the Frenkel theory for ionic conduction in crystals (24). It is only necessary then that the elementary step in conduction occurs at a given time only at places where there is a transient defect containing nine adjacent adsorption sites each occupied by a single water molecule, or at a given place only at
times when this condition prevails. This is essentially the same as the process of conduction in crystals, except more complicated in detail. The picture of conduction in native cellulose proposed here in order to explain consistently a variety of aspects of the conduction behavior, involves a rather lively continual shifting about of thermally generated transient defects, together with the adsorbed water molecules associated with them. It is a structure considerably different from that of an ionic crystal, or even that of a synthetic polymer, though it has aspects in common with them. The internal, thermally-produced diffusional movements of this structure are likely to be on a considerably larger scale than those of ordinary crystals or synthetic polymers, because they depend on the breaking of weak bonds (hydrogen bonds); this is particularly likely in the presence of adsorbed water. The model applies not only to migration in an electric field, but to diffusional processes in general. It may be that this type of structure and diffusion process is applicable to the functioning and behavior of biological tissues. The model may also be useful in interpreting the transport and other behavior of gels. The inference that internal, electrode-free generation of electrolytic products can take place in a homogeneous dielectric because of the presence of transient voids in temperaturedependent concentration, proportional to a Boltzmann factor, may have applications to dielectric breakdown in some types of material, as well as to other processes. The annealing process indicated at high temperatures may be of interest in connection with slow changes in the properties of some types of materials, e.g., gels. REFERENCES 1. 2.
MURPHY, MURPHY,
E.
J., J. Phys. Chem. Solids 15,66 (1960).
E. J., J. Phys. Chem. Solids 16, 115
(1960.) 3. CAMPBELL, W. BOYD, Forest Service Bulletin No. 84, Dept. of Interior, Ottawa, Canada, 1933.
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4. MEYER, K. H. AND MISCH, L., Helv. Chim. Acta 20, 232 (1937). 5. FLORY, P. J., Science 124, 53 (1956). 6. FREY-WYSSLING, A., "Submicroscopic Morphology of Protoplasm," Elsevier, New York, 1948. 7. WIEN, M., Phys. Z. 28, 834 (1927). 8. Joos, G. AND BLUMENTRITT, M., Phys. Z. 28, 836 (1927). 9. ON-SAGER, L., J. Chem. Phys. 2, 599 (1934). 10. LEPAGE, W. R. AND DuBRIDGE, L. A., Phys. Rev. 58,61 (1940). 11. VON HIPPEL, A., Phys. Re'"~. 54, 1096 (1938). 12. BERAN, O. AND QUITTNER, F., Z. Pliys. 64, 740 (1930). 13. HOCHBERG, B., Z. Phys. 65, 635 (1931). 14. RYDER, E. J., Phys. Rev. 90, 766 (1953). 15. POOLE, H.H., Phil. Mag. 42, 488 (1921).
16. GUIUaY, R. W., Proc. Roy. Soc., London SeT. A 134, 137 (1931). 17. MOTT, N. H. AND GURNEY, R. W" "Electronic Processes in Ionic Crystals." Clarendon Press, Oxford, 1940. 18. SEITZ, F., "The Modern Theory of Solids," McGraw-Hill, New York, 1940. 19. MURPHY, E. J., Can. J. Res. 41, 1022 (1963). 20. FOWLER, R. H. AND NORDHEIM, L., Proc. Roy. Soc. London Ser. A 119, 173 (1928). 21. WANNIER, G. H., Ann. Physik (Leipzig), 24, 545 (1935). 22. MEYER, K. H. AND MARK, H., Z. Pltys. Chem. 115, 385 (1925). 23. HERMANS, P. H., "Physics of Cellulose Fibers." Elsevier, New York, 1946. 24. FRENKEL, J., Z. Phys. 1, 466 (1933).
Journal of Colloid and Interface Science. Vol. 49, No.3. December 1974
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MURPHY 1
School of Engineering and Applied Science, Columbia University, New York, New York 10027 Received March 7, 1974; accepted May 22, 1974 It is shown that the conductivity of native cellulose in vacuo increases rapidly with voltage gradient; it is given by a sum of two exponential terms. While the conductivity of this material is ionic, as evidenced by the development of H 2 in approximate agreement with Faraday's law, it is proposed that the increase of conductivity with voltage is caused by local electronic conduction in transient thermally-generated defects in the form of voids. These defects have essentially the same basis in statistical mechanics as the defects on which the theory of ionic conduction in crystals is based (Frenkel and Schotky defects); they should be present in a concentration proportional to a Boltzmann factor. They give mobility to the ions generated by the thermal dissociation of the molecules. In order to explain the voltage dependence it is proposed that negative ions tend to collect in small excess on one side of a defect or gap and positive on the other. The unoccupied levels in the positive ions are above the occupied levels in the negative ions, but when an adequate electric field is applied they can be brought into coincidence, and wave-mechanical transport of an electron occurs, as in the Gurney theory of electrolysis. Developing this quantitatively leads to the relation g = (d Inu/dV)kTe-" where g is the width of the gap, u the conductivity, k Boltzmann's constant, T the temperature in OK, and e the electronic charge. The values of g calculated from the data cluster around two mean values: g = 19.5 A and g = 34.9 A, the first corresponding to the higher, the second to the lower part of the range of voltage gradients. This model indicates that H 2 and O2 should be generated in the defects or voids by this type of internal electrolysis. Experimentally H 2 and CO are generated at a rate increasing with voltage, such that at 0.5 X 106 V/cm the yield is about 2 moles/f, i.e., too large for an electrode effect. Thus th€ model explains both the voltage-dependence of conductivity and that of the electrochemical yield (of H 2), using properties already needed for the explanation of conduction and electrolysis. The presence of transient defects (or voids) in thermodynamic equilibrium can also be used in the model for the explanation of the conductivity/water-content relationship. It is proposed that the conductivity is proportional to the probability of simultaneous adsorption of single water molecules at nine discrete sites, characteristic of each transient defect.
INTRODUCTION
A dependence of conductivity on voltage gradient has been observed in a number of materials, both solid and liquid. In some cases it is a small effect, in others very large. We show here that native cellulose exhibits a large increase of conductivity as the applied voltage is increased, and attempt to 1 Present address: 217 East 66th Street, New York, NY 10021.
interpret this in terms derived from other aspects of the conduction behavior. The model developed depends upon the comparative weakness of hydrogen bonds and the consequent largeness of the transient defects (holes) on which the theory of conduction is based, together with an adaptation of the Gurney theory of electrolysis to conduction in the thermally generated gaps. The structure is pictured as having transient, mobile 442
Journal of Colloid and Interface Science. Vol. 49, No.3, December 1974
Copyright © 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
CONDUCTION IN CELLULOSE
gaps about 20-40 A in width and present in a concentration proportional to a Boltzmann factor, i.e., in thermodynamic equilibrium. The temperature dependence of the conductivity of native cellulose has been described elsewhere (1); it is in good agreement with the theory of ionic conduction. The conductivity of this material is very sensitive to its water content; this has also been investigated and the quantitative relationship explained by means of a model depending upon simultaneous occupancy by single water molecules of discrete, contiguous adsorption sites (2). Thus this material exhibits two types of conduction process: (i) the intrinsic (dry state) conduction, and (ii) the conduction contributed by internally adsorbed water molecules. We attempt to show here that at high voltage gradients there is a third type involving electrons in the gaps and ions in the solid. While the temperature dependence, watercontent dependence, and voltage dependence of conductivity could be regarded as independent processes, we attempt here to connect them by deriving them as aspects of a single model. This model depends mainly on the presence of voids or defects, similar in their statistical mechanical origin to the defects (e.g., Frenkel defects) involved in conduction processes in crystals, but taking the form of rather large loops because of the chain-like (and periodic) structure of cellulose molecules, and the comparative weakness of hydrogen bonds between chains. The water molecules are considered to be adsorbed on exposed OR groups during the elementary step in water-induced conduction. This is a model for conduction applicable mainly to a very loose hydrogen-bonded solid structure, rather than to dense crystals. The picture of transport and conduction processes to which this model leads may have a general application to transport and other processes in membranes of biological interest, as well as to some technological processes.
443
EXPERIMENTAL ARRANGEMENTS
The material on which the measurements were made (electrical condenser paper) is nearly pure cellulose beaten in water in a way which produces a homogeneous solid of considerable strength. The beating process modifies the surfaces of the fibers so that they are held together by a gel-like structure which presumably depends ultimately upon a system of hydrogen bonds (3). The fibers of cellulose have a degree of crystallinity, e.g., a repeat interval of 10.3 A along the axis of the chain molecule (4). A structure of this kind, combining a degree of crystallinity with gel-like properties, may be representative of a class of materials of biological and technological interest. The term "native cellulose" distinguishes natural products from synthetic cellulosic polymers; the properties of the latter are described, for instance, by Flory (5), those of the former by Frey-Wyssling (6). It has been found, by a great many measurements, that the conduction behavior of native cellulose is regular and reproducible. This is perhaps surprising in view of the physical complexity of the materials used, as compared, fcr instance, with the structural simplicity of a single crystal of an alkali halide. This suggests that the regular features of structure on the molecular scale, rather than irregular features on a larger scale (such as fiber contacts), determine the conduction behavior predominantly. The conductivity measurements were made upon condenser paper of standard composition, held between foil electrodes under mechanical pressure. In most specimens the foil electrodes were composed of Sn or AI, but in addition similar measurements were made Ag, Cu, and Cu-Be foil as electrodes. The specimens were suspended in Pyrex glass tubes by means of the leads used to apply the voltages and measure the currents. The leads entered the tube through tungsten-toglass seals. The tubes enclosing the specimens were sealed to a pumping station and evacuated to pressures less than 10-4 Torr, as mea-
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sured on a McLeod gauge. The cellulose remained thoroughly dry during the measurements, except when water vapor at a known pressure was deliberately introduced by opening a stop cock leading to a bulb containing pure water vapor. The specimen tubes were immersed in a constant temperature bath of oil. The rates of generation of water vapor by thermal decomposition of the cellulose and of permanent gases by electrolysis were monitored by means of the McLeod gauge. The conductivity was determined by applying Ohm's law to data obtained by means of a voltmeter and a Leeds and Northrup HS galvanometer calibrated by means of a standard megohm. The determination of the true value of the conductivity is often complicated by variation of the current (for a constant voltage) with time, but this has been discussed fully in previous papers by this writer, which also give further detail regarding experimental arrangements and methods used in conductivity measurements on this type of material (2). However, high voltage measurements (i.e., measurements at voltage
-
gradients extending up to the breakdown level) present special difficulties. One of these is the tendency for electrical discharges to occur in the ambient air. In the present method this was avoided by two conditions: (a) the specimens were very thin, so that gradients up to about 1.2 X 106 Vjcm could be produced in the material by applying voltages not exceeding 2400 V; and (b) the specimen was enclosed in an evacuated tube where the pressure was low enough that glow discharges did not occur. The agreement of the temperature dependence of conductivity with the requirements of the theory of ionic conduction in solids (1) also gave assurance that no gas-phase conduction was influencing the conduction behavior observed. EXPERIMENTAL RESULTS
The data plotted in Fig. 1 illustrate the dependence of the resistivity of dry native cellulose on the applied voltage gradient for several temperatures of the specimen. In the theoretical discussion which follows we are concerned only with the slope of the 10glT vs
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200 300 400 SOO VOLTAGE GRADIENT IN KV/CM.
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FIG. 1. Resistivity of native cellulose as a function of applied voltage gradient. (The curves for conductivity have slopes of the same magnitude but opposite sign.) One specimen was exposed to a relative humidity of 32%, another to a water vapor pressure of 9 mm Hg; all other specimens were in vacuo at pressures below 10-4 mm Hg. Journal of Colloid and Interface Science, Vol. 49, No.3, December 1974
445
CONDUCTION IN CELLULOSE
voltage gradient curve; this is the same in magnitude though opposite in sign to the slope of the logp vs voltage gradient curve, where IJ is the conductivity in (n cm)-1 and p (=IJ-1) the resistivity in (n cm). Data are also given for two cases where water vapor was introduced so that the specimen had some conduction due to the presence of adsorbed water molecules. At lOO°C the water had little effect on the slope of the 10gIJ voltage gradient curves, but at 35°C it produced a substantial reduction. This would be expected from the model to be presented here. The influence of electrode material on the slope was also investigated, by using Ag, Cu, and Cu-Be foil in place of Al or Sn. The curves obtained are omitted in order to avoid complication of the figure, but the slopes calculated from them are included in Table 1. No influence of the electrode material is evident. This indicates that the voltagedependence is not likely to be derived from electrode effects for these would vary with the work function of the metal, films of oxide, and other conditions of the surface. These must then be minor influences. The voltage-dependence data show stability and reproducibility in general; however, in one case measurements repeated after an interval of one month during which the specimen was at 120°C showed a significant decrease in slope. This may imply that an annealing process goes on in the cellulose at high temperatures, changing the structure somewhat. 'lilE VOLTAGE-DEPENDENCE OF CONDUCTIVITY IN OTHER MATERIALS
Before discussing in detail the voltagedependence of the conductivity of native cellulose it will be useful to outline briefly the nature of voltage-dependence as observed in other materials. A comparatively small increase of the conductivity with voltage occurs in electrolytic solutions (the Wien effect) (7). This, which is perhaps the most familiar ex-
TABLE I GAP-WIDTHS
(g)
IN ANGSTROM UNITS
(A)
CALCULATED
FROM THE VOLTAGE DEPENDENCE OF CONDUCTIVITY
Temperature ('C)
35.0 75.0 100 112 120 125 130 140 160 160 160
Water vapor pressure (mm.Hg)
(Eq. [8J)
Electrode material
10 12
Sn or Al Sn or AI
19.6 16.6
10-' 10-4 10-4 10-4 10-4 10-4 10-4 10-' 10-'
Sn or AI Sn or Al Sn or AI Sn or AI Sn or Al Sn or Al Ag Cu Cu-Be
21.3 16.7 20.8 18.5 18.3 18.2 21.9 21.2 21.9
30.8 35.9 34.4 36.8 35.0 36.5
Note: The gap-width g2 is derived from the larger slope, g2 from the smaller slope. The slopes used in calculating gap-widths by means of Eq. [8J were taken from curves similar to those illustrated in Fig. 1 but including some other data. The average for gl is 19.5 A, for g2 34.9 A.
ample of voltage-dependence, has been attributed to a decrease in the drag exerted on each ion by its Debye-Hlick.el ionic atmosphere as the drift velocity increases (8). For weak electrolytes Onsager has developed a theory involving an increase in the degree of dissociation into free ions as the voltage is increased (9). Oils often exhibit a very large increase of apparent conductivity with increasing voltage; several hypotheses have been proposed to explain this, among them field emission of electrons from the electrodes (10). A. v. Hippel has found that the conductivity of single crystals of the alkali halides increases with increasing voltage at field strengths approaching the breakdown level, and has attributed it to field emission from the electrodes (11). Beran and Quittner (12) found that the conductivity of rocksalt crystals increases exponentially with applied voltage, but Hochberg (13) later showed -that when the crystal has been annealed (thereby presumably removiIlg incipient cracks) the volt-
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E. J.~MURPHY
age-dependence vanishes. In measuring the conductivity of crystals it is sometimes found that the conductance of the specimen increases with voltage up to some level such as 600 V and then becomes independent of voltage. This can be traced to poor contact between the dielectric and the electrode, and it is, of course, not properly a voltage-dependent conductivity. For semiconductors, such as germanium, it has been found by Ryder (14) that the conductivity decreases with increasing applied voltage. This has been attributed to the electrons interacting more strongly with the lattice the higher their velocity, so that the mobility is decreased. The application of this result to the present problem is that the possibility that electronic conduction could become more prominent than ionic as the voltage increases, because of an increase of electron mobility with voltage, is apparently excluded. There is also little basis for expecting an increase in ionic mobility (by a substantial amount) as voltage increases because the ions move in thermal equilibrium with the lattice and their kinetic energy is not substantially increased by the applied field. Thus neither ionic conduction, nor electronic, in a homogeneous material would be expected to give an increase of conductivity with voltage gradient as large as that observed in the present experiments. An investigation of the conductivity of mica by Poole (15) leads to results rather closely related to those of the present investigation. He found that the conductivity of mica increases with voltage, and can be expressed as a sum of two exponential terms. MODEL FOR THE VOLTAGE-DEPENDENCE OF THE CONDUCTIVITY OF CELLULOSE
In the model to be developed here the increase of conductivity with voltage is attributed to the presence of voids in which electronic conduction of a special character can occur to an extent increasing with the voltage. The voids are not, as a rule, per-
manently situated at fixed places; a void may disappear at one place when another appears at another place, thereby maintaining a fixed concentration (given by a Boltzmann factor) at a given temperature. The electronic conduction process in the voids (gaps or defects) is modeled on the theory of electrolysis pro~ posed by R. W. Gurney (16). The present model accounts for certain puzzling features of the electrochemical yield of H 2 and CO, as well as the voltage dependence of conductivity. The above is a brief indication of the nature of the model to be developed in more detail below. The experimental relationships upon which it is based are mainly the following: (i) The temperature dependence of the conductivity of native cellulose agrees with the theory of ionic conduction in solid dielectrics. This theory is based to a large extent upon "defects" present in temperature-dependent concentration given by a Boltzmann factor (17, 18). Therefore we expect to find such defects in cellulose, and it is proposed that, because this material is composed of long-chain molecules, the defects are realized as interchain loops which may be rather large. (ii) Cellulose contains crystalline regions (micelles) which according to X-ray diffraction evidence are of limited extent, though there may be a great many of them. Between the crystalline micelles is a region of less orderly intermicellar material. It is proposed that the larger of the thermally-generated defects or holes tend to appear in the intermicellar regions. (iii) A current passing through dry cellulose generates H 2 and CO (with a little C02) at a rate in approximate agreement with Faraday's law (19). However the yield increases with voltage and at the highest voltage gradient used (0.5 X 106 Vjcm) reaches about 2 mol/f. This is somewhat larger than is to be expected from an ordinary electrolytic reaction at electrodes; moreover the yield should be independent of voltage since the applied voltage was much larger than decom-
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CONDUCTION~IN CELLULOSE
position potentials. (Increasing the voltage merely brings ions up to the electrode more rapidly; it does not change the electrode processes.) ELECTRONIC CONDUCTION IN A GAP
From (i) and (iii) we infer that thermal energy causes the dissociation of cellulose to form H 30+ and OH- ions. These ions would tend to become adsorbed at suitable sites on the chain molecules. If the material contains defects or gaps in temperature-dependent concentration there will be ions in the material around the gaps. When a voltage is applied there should be a tendency for positive ions to be present in excess on one side of the gap and negative ions at the opposite side. The adsorbed ions would. tend to be associated with the cellulose molecules in somewhat the same way as ions in aqueous solutions are associated with water molecules. The adsorbed ion should form a complex with the large molecule at the site where it is adsorbed, such that each stationary state of the isolated ion would become spread out into a band of levels. The population of electrons in each level is then proportional to exp( - W;jkT), where Wi is the energy of the ith level, as in atomic equilibria in general. When ions of opposite sign are separated by a void of not too great width, the application of a voltage of suitable magnitude can appreciably change the potential energy distribution, so that resonance tunneling of an electron can occur between the highest occupied level of the negative ion and the lowest unoccupied level of the positive ion. The population of ions with suitable energy for tunneling should then be a function of temperature and voltage, proportional to exp(-W + Veg)/kT, where W = W+ - W_, the energy gap between the lowest unoccupied level of the positive ion and the highest occupied level of the negative ion; V is the voltage gradient and g the width of the gap. (In effect, the applied voltage reduces the potential energy of an electron in the positive ion
447
by Veg/2 and increases that of an electron in the negative ion by the same amount, so that the two levels become the same.) The model is related to the Gurney theory of electrolysis, wherein it was shown that an exponential dependence of current on voltage (at low voltages) could be explained by the tunneling of an electron from electrode to positive ion, or from negative ion to electrode (16). More explicitly, if RH+ is an adsorbed ion at one side of a void of width g and XOHan ion at the opposite side, an applied voltage V acting in the right direction gives W(V) = W - Veg; and when Veg = W, tunneling can occur with substantial probability unless g is too large. This process occurring in a void will increase the conductivity locally by in effect short-circuiting a few unit cells momentarily. (By hypothesis, the voids are not permanent, but appear and disappear.) In this model the main part of the conduction still depends upon diffusional migration caf ions, since voids are present only in a few places at any given time, or only for a small fraction of the time at any given place. No domain is then void free at all times, but all are void free part of the time. The void-free material retains its ohmic conductivity, but the material as a whole (i.e., the material containing a concentration of voids) shows an increase of conductivity with voltage. The activation energy for conduction, as a property of the void-free material, should be essentially independent of the applied voltage; this seems to be the case, at least for a small range of voltages (1). Permanent voids, such as might be expected to be present in some materials, should behave in a way similar to temporary voids except that they would tend to maintain the heating effect of the current in one place, with consequent rise of local temperature, instability of conductance, and ultimate dielectric breakdown by a selfaccelerating process. When the heating is distributed uniformly because of the voids not being permanently present in fixed places
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this self-accelerating process is not nucleated. Moreover, in this material a condition for the permanence of a void may be that its size be substantially greater than that of the voids generated by thermal energy. CALCULATION OF THE WIDTH OF THE VOID FROM VOLTAGE DEPENDENCE
In the model outlined above we have positive adsorbed ions in excess at one side of a gap of width g, and negative ions in excess at the other side. Favorable thermal fluctuations of energy permit resonance tunneling at a given voltage V. The probability of such an event is such that the conductivity u IS given by
u = A exp(Veg - W)/kT,
[lJ
where A is the product of the voltage-independent quantities in the full expression for ionic conduction. The logarithmic derivative of the conductivity with respect to voltage gradient (in e.s.u.) is then given by
dlnu/dV = eg/kT.
[2J
Thus on this model the slope of the (loglT, V)curve depends only on the width, g, of the gap, the absolute temperature T in OK, the Boltzmann constant k, and the electronic charge, e. Converting from natural logarithms to the base 10, and solving for g, we have g
= 2.303 (kT/e)d 10gu/dV.
[3J
The values of g given in Table I were calculated by means of Eq. [3J from the curves in Fig. 1, together with a few other curves of similar type. The values of the gap width, g, given in Table I, appear to be distributed about two average values: g1 = 19.5 A and g2 = 34.9 A. There is a considerable scattering of the points, but since the data were derived from specimens at different temperatures, with different electrode materials, and in some cases with different water content, the scattering is perhaps not larger than ought to be expected. The absence of a definite effect of electrode material on the voltage-dependence indicates
that surface films on the electrodes are not likely to have had an important influence. GENERATION OF GAS BY DISCHARGES WITHIN THE DIELECTRIC
As already indicated [ (iii) above] the passage of a current through dry cellulose leads to the development of hydrogen and carbon monoxide with yield which increases with voltage gradient. It was at :first thought that this was a result of some kind of bombardment process controlled in its rate by the conductivity of the material, but it seemed unlikely that a conductivity-temperature relationship agreeing closely with the requirements of the theory of ionic conduction would be obtained if bombardment processes formed a substantial part of the whole conduction process. Moreover, the model for the voltagedependence of conductivity which has just been described appears also to provide a natural explanation of the voltage-dependence of the electrochemical yield by the processes indicated below. The transference of an electron from RH+ at one side of a gap to XOH- at the opposite side can produce H 2 at one side and O 2 at the other, for the situation is similar to that prevailing in electrode reactions of certain types according to the Gurney theory. In that theory resonance tunneling of electrons between ions in solution and electrons in the metal is responsible for the electrode reaction. It also leads to an exponential dependence of current on voltage, in the range of very low voltages. On this basis it is proposed that a quasielectrolytic process can occur in gaps within a dielectric when the voltage gradient is high enough, by a process resembling those occurring at metallic electrodes. In order to obtain the products of electrolysis which have been observed, it is necessary to introduce secondary reactions. The transfer of four electrons across an internal gap in the dielectric would yield H 2 , O2 , and H 20 accord40H- ~ 2H 2 O2 6H 20. ing to: 4H 30+ By an auxiUary experiment (involving in-
Journal of Colloid and Interface SGiente. Vol. 49. No.3. December 1974
+
+ +
449
CONDUCTION IN CELLULOSE
troducing oxygen gas into the system at a low pressure with the cellulose maintained at temperatures in the range 6o-160°C) it was found that O2 reacts with the cellulose rapidly enough that it is likely that O 2 developed by an electrolytic or quasielectrolytic process would appear as CO, CO 2 , and H 20. (The experiments showed that H 2 and CO are the main products of electrolysis; H 20 would not be detected because it is also evolved by thermal decomposition.) This model indicates that the voltagedependence of conductivity in native cellulose can be considered to be associated with highvoltage electrode-free electrolysis occurring in temporary gaps present in temperature-dependent concentration given by a Boltzmann factor. Although tunneling of electrons is involved here, the process is different from that on which field emission depends. For when a wave-mechanical tunneling process is involved in a conduction process, two extreme conditions can be distinguished: (i) the tunneling process itself is the rate-determining process, and (ii) the time required to set up conditions suitable for rapid resonance tunneling is the rate-determining process. The first of these applies to field emission, for the applied field modifies the shape of the potential energy barrier and the resulting current is given by the Fowler-Nordheim expression (20). This is the condition which presumably would apply to the explanation of the voltage dependence of conductivity in alkali halide crystals near the breakdown level, proposed by v. Hippel, and to the voltage dependence of the conductivity of oils, as proposed by LePage and DuBridge. The second condition, (ii), applies to the theory of protonic conduction in' water, as developed by Wannier (21), to the Gurney theory of electrolysis, and, according to the present' hypothesis, to electrode-free gap conduction occurring in cellulose at high voltage gradients. It may also apply to conduction in mica, because of the presence of easy cleavage planes, and to conduction in rock salt crystals which may con-
tain faults which can be annealed out. Therefore condition (ii) may apply to the results of Poole and those of Beran and Quittner, previously mentioned. VOIDS IN TEMPERATURE-DEPENDENT CONCENTRATION
We return now to some further discussion of the voids or gaps upon which the model is based. Structural defects in concentration proportional to a Boltzmann factor (e.g., Frenkel or Schottky defects) are an important part of the theory of ionic conduction in solid dielectrics. Because cellulose is composed of long chain molecules, we propose that the defects are realized as inter-chain "loops" between nodes where the chains are held together with maximal strength. Diffusion and mobility in solids requires an activation process which can be described in general terms merely as depending upon an activation complex, or a potential energy barrier, or, as in the present case, more explicitly as holes or loops requiring a certain energy for their formation. The voltage-dependence indicates that gap widths of about 19.5 A and about 34.9 A are of frequent occurrence. The smaller of these has about the length of two cellobiose units, the larger between 3 and 4 units. It is to be expected that the larger gaps would be present in lower concentration because more energy is required for their formation. Moreover, a condition for observing the smaller gaps by this type of measurement is that the larger ones be present in much smaller concentration (otherwise the influence of the larger ones would swamp that of the smaller ones). TRANSIENT DEFECTS IN RELATION TO WATER-INDUCED CONDUCTION
The size of the voids (transient loop defects) as indicated by the voltage-dependence data would be expected to bear some relatlon to the model which has already been proposed for the dependence of the conductivity of
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MURPHY
cellulose on its water-content (2). In that model the loop defect provides nine adsorption sites for single water molecules between each pair of ion-generating sites. In a periodic structure the ion-generating sites should be periodically distributed; and, for simplicity, we assume that the ions are generated at the nodes of the loops. In order to accommodate nine adsorption sites, a loop defect would have to be about 2! cellobiose units in length, if we assume that each OR in the cellobiose chain is able to adsorb one water molecule. This is in rough agreement with the gap width gl of Table 1. The longer loops (g2) would not be likely to have an observable effect on low-voltage conduction (i.e., on water-content dependent conduction) because of their comparatively small concentration. From the conduction data for dry native cellulose as a function of temperature it has been calculated that the energy for dissociation into ions is about 40 kcal/mole and the activation energy for mobility about 10.6 kcal/mole (1). The latter would correspond to the simultaneous breaking of two hydrogen bonds. If we assume that pairs of chain molecules are held together strongly at sites separated by the length of one cellobiose unit, and that the chains are bonded together at the ends of this unit by two hydrogen bonds, then the breaking of a pair of hydrogen bonds at this site would open up the structure to form a loop two cellobiose units in length. This loop will have a total of 8 OR groups, at each of which a water molecule can be adsorbed. The breaking of a pair of interchain hydrogen bonds at this site would then also expose two more hydrogen bonds for water adsorption. We can then assume that, effectively, these two bonds are available for adsorption of water molecules only for a fraction of the time, and therefore instead of forming a sequence of ten discrete sites for the adsorption of water molecules, the number is about 9.3 as indicated· by the experimental data; (The meaning of a fractional site is
that it is available for adsorption of a water molecule only for a part of the time.) The tendency for loop defects to form may be modified by the presence of adsorbed water molecules. Therefore loop-defects may form in substantial concentration even at comparatively low temperatures (e.g., room temperature) when the material contains a considerable concentration of adsorbed water. KINETIC ACTIVITY OF THE STRUCTURE
The loop defects of this model appear at random in the material; they are temporary, vanishing at one place and appearing at another, just as the thermal energy is distributed in a solid at equilibrium, or as the Frenkel or Schottky defects are distributed, that is, there is a statistical randomness of distribution in the accessible space. The Schottky and Frenkel defects do not make their presence evident in X-ray diffraction patterns of crystals, first because they are irregularly distributed and secondly because they are present in small concentration. Similarly, it would not be expected that the loop defects of our model would influence X-ray diffraction patterns in a direct way. Also the adsorbed water molecules would not be expected to yield liquid water rings (in diffraction patterns) because the molecules are, by hypothesis, molecularly dispersed in the material. And if the water molecules move about at random in the cellulose in conjunction with (or following) the changing distribution of thermally-generated loop defects, then the distribution of the water will be randomized' so that water molecules held at fixed places by adsorption would not form a sufficiently regular structure to appear in X-ray diffraction patterns. These considerations agree with the observation that the water content of cellulose does not influence X-ray diffraction patterns, except in extreme conditions (22, 23). It seems possible on this model that as loop defects move about in the material a gradual change of structure by an annealing
Journal of Colloid and Interface Science, Vol. 49, No.3, December 1974
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process could occur, if the structure is not initially at equilibrium. The hydrogen bonds between chains could slowly change their arrangement so as to pull the structure together to form configurations of lower free energy. There is a little evidence from the voltage-dependence data that this occurs when the material is held at a high temperature for a period of the order of a month, for the voltage-dependence changes in a manner suggesting that the larger loop-defects have been annealed out to some degree. When the material contains a suitable amount of adsorbed water the "annealing" process may go on quite rapidly even at room temperature. CONCLUDING DISCUSSION
The above considerations regarding the distribution of water molecules in cellulose may be summarized as follows. The water molecules are uniformly distributed in the material on a macroscopic scale by the occupancy of discrete adsorption sites (OH groups). However, the concentration fluctuates on the molecular scale in essentially the same way as the distribution of thermal energy. As loops develop by thermal excitation, water molecules become concentrated locally at the free sites in the loop defects. The presence of the adsorbed water molecules may facilitate the formation of loops at low temperatures. It is to be noted that it is not necessary that the elementary step in the conduction process be occurring simultaneously at all parts of the dielectric; for, in fact, the denial of this on the ground that it would have a negligible probability of occurring, and the inference that conduction must depend upon randomly occurring independent steps, was one of the main considerations in the development of the Frenkel theory for ionic conduction in crystals (24). It is only necessary then that the elementary step in conduction occurs at a given time only at places where there is a transient defect containing nine adjacent adsorption sites each occupied by a single water molecule, or at a given place only at
times when this condition prevails. This is essentially the same as the process of conduction in crystals, except more complicated in detail. The picture of conduction in native cellulose proposed here in order to explain consistently a variety of aspects of the conduction behavior, involves a rather lively continual shifting about of thermally generated transient defects, together with the adsorbed water molecules associated with them. It is a structure considerably different from that of an ionic crystal, or even that of a synthetic polymer, though it has aspects in common with them. The internal, thermally-produced diffusional movements of this structure are likely to be on a considerably larger scale than those of ordinary crystals or synthetic polymers, because they depend on the breaking of weak bonds (hydrogen bonds); this is particularly likely in the presence of adsorbed water. The model applies not only to migration in an electric field, but to diffusional processes in general. It may be that this type of structure and diffusion process is applicable to the functioning and behavior of biological tissues. The model may also be useful in interpreting the transport and other behavior of gels. The inference that internal, electrode-free generation of electrolytic products can take place in a homogeneous dielectric because of the presence of transient voids in temperaturedependent concentration, proportional to a Boltzmann factor, may have applications to dielectric breakdown in some types of material, as well as to other processes. The annealing process indicated at high temperatures may be of interest in connection with slow changes in the properties of some types of materials, e.g., gels. REFERENCES 1. 2.
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16. GUIUaY, R. W., Proc. Roy. Soc., London SeT. A 134, 137 (1931). 17. MOTT, N. H. AND GURNEY, R. W" "Electronic Processes in Ionic Crystals." Clarendon Press, Oxford, 1940. 18. SEITZ, F., "The Modern Theory of Solids," McGraw-Hill, New York, 1940. 19. MURPHY, E. J., Can. J. Res. 41, 1022 (1963). 20. FOWLER, R. H. AND NORDHEIM, L., Proc. Roy. Soc. London Ser. A 119, 173 (1928). 21. WANNIER, G. H., Ann. Physik (Leipzig), 24, 545 (1935). 22. MEYER, K. H. AND MARK, H., Z. Pltys. Chem. 115, 385 (1925). 23. HERMANS, P. H., "Physics of Cellulose Fibers." Elsevier, New York, 1946. 24. FRENKEL, J., Z. Phys. 1, 466 (1933).
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