that the points of entanglement are more closely spaced (as they appear to be in polymethyl methacrylate42compared with polystyrene) or that each coupling point is more effective in reducing the mobility of the chain and hence increasing the average friction coefficient in long-range cooperative motions. I n the latter case, the dependence of steady flow viscosity on molecular weight should involve an exponent higher than the 3.4 which appears to be characteristic of vinyl polymer^.^?*^',^^ Preliminary datazgdo indicate that this is so. The significance of the height of the plateau in @ remains uncertain. It decreases slightly with increasing molecular weight, being 3.0, 1.4 and 1.3 X lo5 dyne/cm.’, respectively, for fractions L, R2b and H. I n concentrated solutions of high molecular weight polyi~obutylenes~ the plateau heights range from 0.G to 1.0 X 1oj dyne/cm.2. The plateaus found in relaxation spectra of undiluted un(44) 11. F. Johnson, W. \V. Evans, I. .Jordan and J. D. Perry, C o l l o i d ,Sc.i., 7, 498 (19.52). (4.7) F, Bueclie, J . A p p l . P h ~ p . 24, , 423 (19.58).
J.
irulcnnized rubbers of high molecular weight4‘I,4’ range from 1.4 to 4 X 1oj dyne/cma2. We suspect that this rather general level is determined by the nature of the random distribution of entanglement points and not on the structure of either backbone or side chains. Acknowledgments.-This work was part of a program of research on the physical structure and properties of cellulose derivatives and ot,her polymers supported by the Allegany Ballistics Labornt,ory, Cumberland, Maryland, an establishment owned by the Unitfed States Navy and opera,ted by the Hercules Powder Company under Contract NOrd 10431. It was also supported in part by a grant from Research Corporation and by the Research Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation. (413) W. K u h n , 0. Ktinzle a n d 9. Preissniann, Helv. Chim. Acto, 30, 307 (1947). (47) L. J . Zapan, 8. L. Sbufler a n d T. TV. DeWitt, J . Polumer S c i . , in pres&
THE ANOhL41dOUS MONOLAYER ADSORPTION OF HELIUM’ BY R L - D RPAL . ~ S I N G HA~N D WILLIAM BAND Depnrltt/e/il sf Phusics, ?’he Stale College of Washington, Pullman, WashingLon R e c e i v e d February 17. 1066
Lennard-Jones d ~ v e l o p e da quarituni mechanical theory of physical adsorption on the hypothesis that adsorption energy is due t o induced dipole-dipole interaction between the adsorbed atom and the adfiorliing surface, the latter being treated as a conducting plane without structure. The present paper reports an attempt, t o understand tjhe anom:tlously high density in the first monolayer of helium adsorbed on solid surfaces in terms of this Lennard-Jones model, by taking into account the fact that the forces causing adsorption perturl) the helium atom wave function and so modify the interaction between neighboring helium atoms. The minimum strength of the forces needed to cause anomalous adsorption is found to be much too large to be explained in terms of induced dipole-dipole effects, and it is suggested that the anomaly may be more profitably discussed in terms of relatively strong permanent (noli-uniform) local crystal fields in the adsorbing surface,
Introduction The anomalously high density of helium adsorbed in the first monolayer on solids was first reported by Schaeffer, Smith and Wendell3 using carbon adsorbents. The surface area was measured by analyzing nitrogen isotherms. The volume of helium adsorbed in the first layer was found to be several t’imes greater than expected. A similar anomaly was found by Long and Meyer4 to occur with Fe203 adsorbers, and by i\,Iastrangelo and Aston6 with TiOz. Frederikse and CrorteP confirmed the results with Fez03. Helium is apparently the only substance exhibiting this effect. When the anomaly was first discovered, there was considerable doubt about its reality: the B.E.T. theory’ of adsorption, by means of which the volume i i i the first layer is calculated, might be inapplicable. The fact that it yields the anomaly ( I ) Supported in part by ONR. (2) O n leave of absenae from t h e University of Allahabad. (3) W. D. Schaeffer, W. R. Smith and C. B. Wendell, J. .4m.Cheln. Soc., 71, 863 (1949). (4) E. Long and L. Meyer, Phys. Rev., 76, 440 (1949); 84, 5 5 1 (1952); 86, 1035 (1952). (5) 8. V. R. RIastrangelo a n d J. G. Aston, J. Chem. Phys., 19, 1370 (1951). (GI H. P. R. Frederikse and C. J . Gorter, Phvsico. 16, 403 (1950). (7) 5. Brunauer, P . H. E m m e t and E. Teller, J. A m . Cliem. Soc., 60, 309 (1488).
might itself invalidate the theory: according to the B.E.T. model, each adsorbed atom becomes a possible site for another adsorbed atom in the next monolayer, but with an anomalously dense first layer, this cannot be true. Band8 modified the B.E.T. model to include this possibility explicitly a t tthe outset, recalculated the volume in the first layer, and obtained essentially the same effect. There seems no reasonable doubt that the interatomic distance in the helium monolayer is no greater than 2.0 X lo-* em., compared with approximately twice this distance in liquid helium. Several qualitative explanations of the anomalous packing in the first monolayer have been suggested. The most promising was in terms of zero-point energy. The interatomic distance in helium for a minimum van der Waals potential is about 2.8 X lop8 cm. ; but by increasing the interatomic discm. as observed in liquid tance to about 4 x helium, the zero-point energy is so greatly reduced that stability is gained in spite of the increase in van der Waals potential. The liquid is “blown up” by its zero-point energy, and the adsorption mechanism is supposed to collapse this pressure. In one sense this explanation leads to more trouble than it resolves. Solid helium under 35 atm. or (8) W. Band, Phys. Rev.. 76, 441 (1949).
RUDRAPALSINGHAND WILLIAM BAND
GG4
more pressure has a greater zero-point energy than the liquid phase, although it is still quite smallabout 7 3 cal./mole as determined from the Debye temperature 32.5"K. The interatomic distance in the anomalous monolayer is much less than that in solid helium, and so implies an even larger zeropoint energy in the monolayer. A crude estimate in terms of the Heisenberg uncertainty relation suggests that the zero-point energy is inversely proportional to the square of the interatomic spacing, and this yields an estimated zero-point energy in the monolayer of the order of 350 cal./mole. The observed energy of adsorption is about 50 cal./mole at high percentage coverage, so the potential responsible for adsorption must have a net depth not less than 400 cal./mole. At the interatomic distance 2 X lo-* cm. in the monolayer, the interatomic potential of two unperturbed helium atoms, using Lennnrd-Jonesg potential turns out lo to be about 1200 cal./mole repu1,sive. To explain the required net negative 400 cal./mole we therefore have to find an adsorption mechanism capable of producing a total negative potential of some 1600 cal./ mole. The straightforward induced dipole-dipole interaction between a helium atom and a conducting plane, the Lennard-Jones model, l 1 is inadequate by a factor of about one hundred to provide so great an energy. The field intensity needed can be estimated as follows: The energy of an atom having polarizability a i n a field of intensity F is ' / N Y F ~ . Taking the polarizability of helium atomI2 as 0.206 X ~ m .the , ~ field intensity required to induce an energy equivalent of l G O O cal./mole works out as F = lo6 dynes/e.s.u., or 0.058 atomic unit. Such fields may be large enough to perturb significantly the van der Waals attraction between two adsorbed helium atoms. Thus, the dipole moment of one helium atom, induced by adsorption, could be opposite in sign to that induced in any neighboring atom, thus adding a direct dipole-dipole attraction between neighboring adsorbed helium atoms. The more strongly polarized the helium atoms are by adsorption, the greater the effect. It was hoped that possibly this increased attraction would permit a smaller adsorption field intensity to produce the required adsorption potential, and so allow the LennardJones picture to be retained. However, the correction turns out to be less than (3% of the energy of adsorption, and no appreciable change in the field intensity results. The perturbation calculations are outlined in the next section. Perturbation Calculations Hass6I3 has given the wave function of a helium atom in a perturbing electric field F in the form u = e-"(?] ?z) (1 ern) { 1 A(zl 2 2 ) B ( r m TZZAI +
+
+
+ +
+
(1) (9) J. E. Lennard-Jones, Proc. Phys. Soc., 43, 471 (1931). (IO) Using the more recent values of the Lennard-Jones constants given b y D. ter Haar, "Eleinents of Statistical Mechanics." Rineliart & Company, Inc , New York, N. Y., 1954, p. 179. (11) J. E. Lennard-Jones, Trans. Faraday Soc., 28, 333 (1932); Proc. R o y . SOC.(London), 1 6 6 A , G , 29 (193G). (12) W. H. Reesoin, "Helium," Elserier PuLlisliing Co., New Pork, N. Y..1042. (13) H. R . Flnw6, Pmr. Camb. Phil. Sor., 26, 543 (1930).
Vol. 59
where rl, r2 are the distances of the two electrons from the nucleus, rI2 the inter-electron distance; 21,2 2 the electron coordinates in the direction of the field F ; c = 0.3G4, n = 1.840, A = 0.3844F1 B = 0.3845F; all the quantities are expressed in Hartree atomic units. We use thin function i n the more approximate form u1 = e-n(rl + 19) (1 f c)'12) { 1 4 x 1 2 2 TITI T2X2) 1
+
+ +
+
(2)
with A = 0.3815F. The adsorbing surface is the y-z plane, and the z-axis joins two neighboring nuclei. The neighboring helium at,om, polnrised in the opposite sense, is represented hy t,he witve function uz = e-n(ra
+ rd)
(1 f
cyI4)
{1
-
:t(x3
+ x4 + r3xa + 7,4.x1) } (3)
where the coordinates are relative to the second nucleus. The coulomb interaction between these two atoms expressed in atomic units is
v = (1/fi%zIz$
+
22124
2253
+
f
- 22123 + 21x4 + 2/1u4 - 222% + 22x4 + 1/22/4 - 22224)
1/11/8
Yl?/3
(4)
where R is the distance between the two nuclei. The first- and second-order perturbation energies due to this interaction are formally AEi = ( l / N ) f ..fu ~ ~ V dU~ ~ i d' ~ d 7 3 d (~54n ) AEp = ( l / N ) f ..f ui2 1' uz2 d ~ di n ( 1 7 3 drc/(Eu
+
Ezi
- Eiz - Ezz)
(5h)
where N = f ..fu12 u2?drl drz dr3 d n ; Ell, Ezi are the negative energies of the two atoms in their unperturbed ground states, and E12,Ez2 are the negative energies of the two atoms in their first excited states. Numerically these energies are E,, Ezl = 2.75 atomic units, E12 E,, = 0.70 atomic unit. All the integrals involved in evaluating eq. 5 have been ~ o m p u t e d and ' ~ we find
+
+
A E l = -0.4128 F2/NRa
AT
+ 2.8110F2 + 2.2181F4)/(2.05NR6) (6) 0.2G71 + 0.3GOGF2 + 0.121GF4 -(0.08256
AEi =
In the absence of polarization, F = 0, the second order perturbation becomes ESo= -G58 X 10-l2 (uo/R)'jerg, which compares favorably with the value given by S l ~ t e r namely, ,~~ -G8 X ( U " / Rerg. ) ~ Taking R = 2 X cm., or about 3.8 atomic units, and assuming the estimated value of F as 0.058 atomic unit, as discussed in the previous section, the first-order dipole-dipole term in atomic unit, eq. G turns out to be -2.67 X while the second order, van der Waals term, is about -50.5 x atomic unit. The increased attraction due to the field F is thus only about GYo, and amounts in effect to no more than 18.6 cal./mole. The possible effect of polarization on the repulsive term in the Lennard-Jones potentia.1, due to overlap, can hardly be estimated with any degree of precision. Some idea of the effect however may be obtained by evaluating the first order dipole(14) R. P. Singh, O N R technical report project 24, 1954. (15) K. C. Slater, Phys. Rev., 32, 355 (1928).
N R 010-603, June
dipole term AEl in terms of the symmetrized wave function LLS
=
(ti(
- U I ( 1,4)~2(2,3)- ~ 1 ( 2 , 3 ) ~14) 2( f
1,2)~2(R,4)
U 1 ( 3 , 4 ) U z (1 , 2 )
The resulting value of AE1 with F = 0.033 turns out to bel4 positive 2.1G X lo5 atomic units. Discussion We have shown that the possible increase in mutual attraction between neighboring adsorbed helium atoms due to their polarization in the adsorbing field is no more than (3%; a very rough indication also has been given that the overlap repulsive component in the interaction between them is increased by an amount that may just about cancel the increased attraction. I n any case it is quite clear that the perturbations in question are far too small to alter the problematic situation appreciably: namely, that it is still necessary to postulate an adsorbing field intensity of roughly 0.058 atomic unit in order to bind the adsorbed atoms against their mutual repulsion, against their high zero-point energy, and still retain 50 cal./mole energy of adsorption. The major difficulty therefore remains: if the adsorbing mechanism is capable of holding helium atoms against their mutual repulsion of 1200 cal./mole a,t full coverage, plus about 350 cal./mole zero-point energy, plus 50 cal./mole energy of adsorption, why does the energy of adsorption increase to no more than
about 100 cal./mole a t 10% coverage, when presumably the repulsive 1200 cal./mole has practically disappeared? There seems no escape from the conclusion that the adsorbing mechanism does not merely balance the mutual repulsion by means of a compensating negative energy, but that the mechanism is such as actually to remove the repulsion almost completely a t all coverages of the monolayer. I t is however clear that the present perturbation theory based on the Lennard-Jones model is quite inadequate to achieve this. It is suggested that the adsorbed monolayer may be regarded as a two-dimensional alloy with the adsorbing surface, the helium atoms entering the surface structure of the adsorbing solid. The forces iiivolved are the local crystalline fields leaking out through irregularities in the surface. Such irregularities are presumably of molecular dimensions, and the fields of a higher order of intensity than those existing between polarized helium atoms. Each adsorbed helium atom may be completely immersed in such a local field, and so become essentially a part of the lattice structure of the adsorbing surface, interactions between neighboring helium atoms being completely swamped by the perturbing effects of such a process. This picture has the merit that it would explain why only the smallest of atoms exhibit the effect; otheix like nitrogen are too large to he suhmerged in the surface irregularities.
A MINIMUM-PRINCIPLE FOR NON-EQUILIBRIUM STEADY STATES BY THORA. BAK Conlribidion f r o m Tibe Dcpartrnents of Cheniislry, The University of Copenhagen, Copcnhacqcn, Denninrk, and Columbia University, New York, New York Received A p d 19, 1966
It is shown that a non-equili1)riuin st,eady state is accompanied by a minilnuin ill a characteristic funct,ion which mag he described as the sum of resistances times fluxes squared. For a steady state new equilibrium t,lie funct,ion is, except for n constant factor, identical with the entropy production. This minimum principle provides the Ilnsis foi, n i l est,ensioii of Le Chat,elier’s principle.
Introduction Prigogine‘ has shown that for an open system in which two irreversible processes take place, the steady state is characterized by the minimum rate of entropy-production consistent with the external constraints which prevent the system from reaching equilibrium. The theorem has been generalized by de Groot2 to systems with an arbitrary number of reactions. The proofs given by both authors are based on the Onsager3 relations, and the nature of the processes is therefore immaterial. Recently Klein and Meijer4have derived the theorem by the methods of statistical mechanics for a particular irreversible process-the flow of matter and energy through a small capillary connecting two containers of an ideal gas maintained a t slightly (1) J. Prigogine, “Etude Theriiiodynamiqiie den PIiCnotn6nes irrhernibles,” Editions Desoer LiBge, 1947, Chal). V. (2) S. R. d e Groot, “Thermodynamics of Irreversible Processes,” Nortli-Holland Pobl. Co., Amsterdam, 1951, C l ~ a g .X. (3) L. Onssger, P h y s . Rev., 37,4 0 5 (1931); 38, 2268 (1931). ( 4 ) hf. J. ICkin a n d P. H. E. Meijer, i b i d . , 96, 2,50 (195.1.).
different temperatlures by two heat baths. Although the general use of the principle of minimum entropy production \.vas initiated by Prigogine’s paper, it is worth mentioning that a special case was proved almost a century ago in a nonmlays rather unknown paper by Helmho1tz.j It is a necessary condition for all the proofs given for the theorem that there be linear relations between the forces and f l ~ x e s . ~This , ~ means t,hatt if we want to apply the theorem to a system in which heat conduction, diffusion or chemical reactions take place, we can only use it for steady states which are near true thermostatic equilihrium. It can be proven for t,he steady states which are characterized by this minimum i n t,he rate of entropy production that they are stabilized in the sense that Le Chatelier’s principle is Tralid. It would be interesting to apply this theorem to ( 5 ) H. Helmholtz, “Wi~senschaftlicli~ .4blinndliingrn y o n H . T T d t n holts,” Leipzig, 1882, I , 223. ( G ) I
