ELECTRIC PROPERTIES OF MACRORIIOLECULES. V. THEORY OF I O S I C POLARIZATIOS IS POLYELECTROLYTES BY CHESTERT. O'KOXSKI Department of Chenzistry, Unwerszty of California, Berkeley 4, Californza Receaved N o v e m b e r 10, 1969
Ionic transport phenomena are shown to he important to t,he dielectric constant, as well as the dispersion and conductivity of polgelrctrolyte solutions. The effects of the excess conductivity arising from mobility of ions a t the interface, and-with thin ion atniospheres-the charge transport due to counterions, may be expressed in terms of a two-dimensional conductivit,y. The electrical boundary value problem is formulated in terms of this surface conductivity, the dielectric constant and volume conductivity of tlie medium, and an anisotropic dielectric constant and volume conductivity for the particle. These parameters have important physical significance for macromolecular structures. It is proposed t'hat this model is more appropriate for polyelect,rolytes than either the Debye-Falkenhagen or 1Iaxwell-Wagner models. Isotropic spheres are considered in detail, to show the effects of a surface conductivity on the internal field, the induced polarization, and the dielectric constant, dispersion, and conductivity of dilute systems. Reasonable approximations were found n-hich obviate detailed treatment of the boundary-value problems for the generalized ellipsoid and the limiting forms of a cylinder. Anisometric particles of uniform surface conductivity are shown to be electrically equivalent to particles of anisotropic volume conductivity. Equations are given for the complex dielectric constant, the relaxation times, and the low and high frequency dielectric and conductivity increments of dilute systems of ellipsoidal particles. Numerical calculations of the increments are made for randomly and completely oriented ellipsoids of revolution over a wide range of parameters appropriate to dilut,e aqueous media. Important differences between needle and disk-shaped particles are found. I n oriented systems of anisometric particles, counterion tranqiort prodllces a st,rong anisotropy of the low frequency dielectric increment, and an anisotropy of conductivity apart from the contribution due to anisotropy of the frictional coefficient of the polyion. Further, the internal field depends upon orientation, and this effect on polyion mobility may greatly exceed the frictional coefficient effect discussed by Eigen and Schwarz. The results indicate that essentially all of the known dielectric properties of aqueous proteins, nucleic acids, nucleoproteins, and charged colloids are explicable in terms of this model. It is shown that polarization of the counterion atmosphere reduces the internal field and thus diminishes permanent dipole polarization. Available experiment:tl data, generally incomplete for the present interpretation, are discussed in as far as possible. The need for further experiment's to test, the theory is indicated, and some crucial ones are suggested. It is predicted that biological effect's in intmse high frequency fields may be enhanced by the use of pulsed radiation.
1. Introduction
Of the maiig types of systems for which the frequency-d(>pendentdielectric properties have been investigated, perhaps the least understood are solutions of polyelectrolytes and colloidal electrolytes. These classes include the proteins, nucleic acids, synthetic polyelectrolytes, micelles and many colloidal dkpersions. Formulation of an adequate theory of the dielectric constant and conductivity of these systems requires treatment of the charge transport processes arising from the motion of the ions or other charge carriers in the solvent, in the ion atmosphere, and in the solutes or the suspended phase. Keglect of these effects can lead to seriou5 errois in the interpretation of dielectric constant and conductivity. A recent interpretation evidmtly in error will be discussed below. There oxists no general theory of dielectric constant and conductivity rigorously applicable to p ~ ~ l y e l ~ c t i ~ n Debye l ~ t e ~ and . Falkenhagen' trenlcd the dielectric constant of simple clectrolytes in terms of a niodel involving point charges in a dielecti ic continuum. Several investigators2 have treated tlie conductivity of simple electrolytes and in recent years there have been some significant refinemer tiizlj3 which are in accord n ith experimental result'. w l l above the low concentrations for which the Debye-Huckel limiting law is valid. But the refinements are not adequate for polyelectrolyt CP, where particle dimensions and shape are impoi taiit constants of the system. 11) (a) P Dehjt' and 11 FalkenliaRen. (1928), (b) J E ' l e i t r o c h e m , 34, 562 (1928). f 2 ) (a) L Onsiger. Ph7/sik. Z., 27, 388 (h) R XI. Fiioss a n d I, Onsaper, Proc Aotl rlilc1 0 1 R \ 61, OIB fl”
- C1”)2]AJ/
eJf’Cif’)/(eiZ
(€12
A €lf12)
+
[(t,”€i
+
~1’”) - 1 - e,eif’)~4,’(ei2 f
ei’”)]’
Sumerical calculations of the dispersion may be made from these equations after appropriate dmplification for a given system. Fricke’s equations are for E , = Eb = ec = eZ, and eaN = Eb” = t o N = tZ”, that is, he did not consider anisotropic particles. As in his case there are three relaxation times for a given ellipsoid, corresponding to the three axes. By generalization of Fricke’s equation” 14 for anisotropic particles, the relaxation times in dilute systems are Ti
-
= 1
((3.4)
4XZKJ/W
where k,” i b the true volume conductivity along the j-axis, and K,’ is the surface contribution to K,, the effective conductivity. For ellipsoids, values of K,’ may be obtained from equation 5.1 or 5 . 2 to 5.4. If tlie particles are cylindrical rods or flat disks, ecpttions 5.6 to 5.8 may be employed for the surface contributions, and the value of A , may be taken as hat of a n ellipsoid of revolution of the same axial ratio. The volume fraction should be the actua external volume of the particles, determined (ixperimentally. Random Orientation.-For a dilute suspension of randomly oriented ellipsoidb, equation 6.2 becomes -
611
THEORY OF IONIC POLARIZATION IN POLYELECTROLYTES
May, 1960
+ € l ( l / A j - 1)
‘j
‘$7 K j
f K1(1/.4] - 1)
(6.11)
where K ~ ,in e.5.u.. is given by equation 6.5. In the case of Maxwell-Wagner polarization, that is, zero surface conductivity, these are dependent on the shape of the particle, but independent of size. In our model they depend both upon size and shape, since K,’, the surface contribution to the effective conductivity, clearly depends upon size as well as shape. Thus, randomly oriented polydisperse systems may be expected to show three broadened regions of dielectric dispersion, or two if there is an axis of symmetry. If the frequency is lorn enough so that all dielectric constants are small relative to the corresponding loss factors, the fields are determined by the conductivities, and the equation for the dielectric increment, A q , reduces to ej/ei
“
j=a,b,c
[I
- 1
+
+
- 1)lAj l)Aj]?
( ~ j / ~ i
(Kj/KI
-
(6.12)
The low frequency conductivity increment is AKI_ = _ K1
E
3 j = a , b j c1 + a*
- 1 - 1)Aj
K~/KI (Kj/Kl
(6.13)
At sufficiently high frequencies so that loss factorq are all small relative to the corresponding dielectric constants
and
(0.15)
For the present model and the Maxwell-Wagner model, no allowance is made for the possibility of the applied field causing large modifications of ion distributions a t the boundaries, so the equations are strictly valid only for small fields. Fricke has further restricted application of his high frequency equations to frequencies “above all dispersion region$.” If this is interpreted literally, his equations 22 and 23 and our 6.14 and 6.13 could only be applied above the dipolar dispersion region of polar solvents which would seriously restrict their use, for example, in aqueous systems. In obtain-
CHESTERT. O’KONSKI ing equations 6.14 and 6.15 from 6.9 and 6.10 here, it was seen that this restriction is not necessary. If one stays below the dipolar dispersion region of the solvent, where the dipolar contribution to e l ” is always much less than €1, the equations for A q , and AE,~‘’ are applicable. Thus, they may be applied for dilute aqueous solutions up to around a few hundred megacycles. When the conductivities are so high that ion atmosphere and dipolar dispersion regions overlap significantly, the more exact expressions 6.9 and (3.10 should be used. It should he kept in mind that the equations are for dilute systems, ie., Ae < E; and A K < K ~ . Numerical Results.-To illustrate the various effects of size, shape, magnitude of the surface conductivity, and solvent conductivity upon the dielectric and conductivity increments, numerical calculations were made for a wide range of values of the important parameters of the system. These are directed primarily a t elucidation of the propertics of rigid particles of relatively low dielectric ronstanf and bulk conductivity, in a medium of high dielectric constant and low to moderate conduct ivit y. First, to illustrate the effect of particle shape on the primary quantities appearing in the equations, values of the depolarization factors A, and Ab = A,, and of the surface contributions to the effective conductivity, K ~ ’ and Kb’ = K ~ ’ , are presented in Table I. These apply to ellipsoids of revolution, where 2n is the symmetry axis and 2b = 2c, with axial ratio p = a/b. The surface conductivity X and the particle volume were held constant, and equations 5.la, to -c were used. The K’ values are given in units of K,’, the value for a sphere of the same volume and surface conductivity. It is seen that K&’ increases monotonically with increasing axial ratio, whereas Kb’ goes through a minimum at p slightly greater than one, increasing markedly for axial ratios much less than one, and somewhat less for axial ratios much greater than one. The important depolarizing factors are the smallcr ones, for it is along the corresponding asis that the field esel-ts its greater effect. TABLE I L)EPOLARIZINGFACTORS A N D SURFACE CONTRIBUTIONS 1o THE EFFECTIVE CONDUCTITITY US. AXIAL RATIOOF SPHEROIDS p = o.’h
200 100 50 30 20 In a
’1 R
O.OQO1333 .0004666 ,001466
.003466 .006733
3
,02026 .0.5579 . 1085
1
,2330 ,3333
3/2 2/3 1/3 1/10 1 /30 1/100 1/200
.4459 ,6354 XAO; . 9 197 0844 992 4
h
0,4999 .‘I997 ,4992 .4982 ,4966 .4898 ,4720 ,4457 ,3834
*a’/*B’
5.848 4.642 3.684 3.107 2.714 2.154 1.710
1.442
,3333
1,lG 1.000
,2770 .1822
0.8735 ,6933
OA!l59
.’tci!l 3218 ,2154 ,1710
,
02500 .O O 7 7 W ,003910
*hf/k.)
3.724 2 . 956
2.348 1,983 1 .736 1 ,393 1,144 1.022
0.963 1,000 1.103 1.475 3,002 6.ltil
13.72 21.77
Yol. 64
Low frequency conductivity increments of randomly oriented systems were computed from equation 6.13 for a system of particles with zero volume conductivity, for varying axial ratio, again keeping the particle volume (47ruo3/3) constant, for various values of 2k/UoK1 = K ~ ‘ / K ~ . Results are presented in Table 11. No assumption regarding dielectric constants was required. For K * ‘ / K ~ = 1, the field is not affected by insertion of the sphere for its effective conductivity is equal to that of the medium, and AK = 0. But with highly elongated particles, the combination of small depolarization factor and increased effective conductivity along the longer axis produces positive conductivity increments. These increase rapidly for large valuw of p or l / p , and, of course, increase with K ~ ’ , / K ~ . Thus, very large contributions to the low-frequency conductivity may occur in relatively dilute qyqtems of highly anisometric particles with surface conductivity. The numerical calculation in Part 4 for spheres with Alao = 2 X lov3 ohms-‘ cm.-l in LIP KCl corresponds to Ks‘/hl = 27. For oriented systems, the conductivity iiicrernentq depend upon whether the field is oriented along the a or the b-axis, and these are designated AK, and A h b , respectively. Values for parallel orientation are given in Table 111. They are obtained by taking the appropriate term of the sum of equation 6.13, with the coefficient 82 rather than 6d3. It is apparent that an anisotropy of the conductivity of the solution will be produced by partial orientation of the particles, and its magnitude can be obtained from these values if the degree of orientation is known. High frequency conductivity increments for L: broad range of p and K ~ ’ / K are ~ given in Table IT’. Equation 6.15 was employed, with = 78.54 (&O, 25”) and E, = Eb = e, = 3.00. The increments are only weakly dependent upon the latter Comparison with Table quantity, for small ej/el. I1 shows that AKh is greater than A K ~for large K ~ ’ / K ~ . This is because the local field remains large a t high frequencies, but approaches zero a t low frequencies for high K ~ ’ / K ~ . In oriented systems, AK, and AKb were computed from the appropriate term of the sum in equation 6.15. They are presented in Table 1’. Again, large anisotropies are observed. The high frequency dielectric increment is independent of conductivity ratio, as see11 by equation 6.14. Table VI lists values of Aeh//el& calculatcd for = 78.54 and = € b = e, = 3.00. It is negative and becomes more negative a t very low axial ratios, as expected from considerations of the dielectric constants of i n d a t i n g p a r t i ~ l e s . ’ ~The dielectric increments of oriented systernq a1.o are listed in Table VI. The low frequency dielectric increments were computed for the usual axial ratios, with h g ’ / K 1 values ranging from 1 to 100, el = 78.54, E, = 3.00, K,O = 0 and are shown in Table VII. Additional p-values were computed to delineate the interesting itiflvimum which mas found at the higher hs’/K1 vn111c5 and high p . A t K ~ ’ / K = ~ 1, the low frequency dielectric increment is aln~xysnegai i \ c iii t h e range l’loo ’< p ’< 100. Iiisertioxi of
TABLE I1 IiOW FREQVTENCY~ O S I ~ l l i ~ ' l ~ I VINCREMENTS ITY t's. hXlA1, RA'I'IO I'OIt ~ P I I E R O I L ) ~ ' K'S,X* = in 30 0.1 0.3 1.0 3.0
P
+
100 30 10 i
3/2
1 2/3 1/a 1/10 I /30 1/100
100
44.59
16.07 10.29 6,656 3 , w25 2.350 2,250 2.350 3 ,030 6.702 16.27 '44.32
-0.9032 0.05089 1.871 5.345 - 1.120 - ,3610 I ,137 3.648 -1,258 2.485 - ,6614 0.5959 -1,312 - ,8703 .I555 1,532 - .!)IO0 ,02190 1.248 -1.294 1.200 -1.286 - ,9130 .DO00 1,252 - .9142 ,02195 - 1,300 -1.465 - ,9452 ,1643 1.619 -2.264 -1.174 .8400 3.525 -4.247 -1.567 8.089 2.411 -8,526 -1.907 6.570 20.2:1 trc wl:ttive valiirs pcr unit voliime fraction, ~ t i 1 / ~ , 8 ? .
__
__-.
128.3 51.11 14.71 4 , 350 :3 , 071 2.912 3.058 4,03!) 9.518 25.29 78.80
24.59
10,56 3.913 2.858 2.719 2.850 3.733 8.604 22.15 C,R,(i-l
TABLE I11 r-
-
,.---0
.
-
a5
P
-
I2OW FREQUENCY CONDUCTTVITY ISCREMENTS O F ORIENTED SPHEROIDS
____
- -
=
.'s/n,
10
h
a
ba
. ~ . .-.1,0
_____ b
a
-
-_
----IO a
b
- -
-.~.-100
--b
a
-
1.99 1.87 381 1.60 4 4 . 5 0.536 -1.09 3.64 0.989 12.8 1.09 1.81 149 1.43 27.2 -1 .3:3 2 00 ,660 8.09 - ,891 40.1 2,01 1.76 1.27 1 4 . 5 -1.49 ,330 4.92 1 . 13 - ,797 8.66 2,20 1.80 1.08 5.46 ,0223 2.44 0,422 - ,!I43 -1.50 2.54 2.00 4.14 3.04 1.55 1.10 ,140 - ,0371 - 1.12 -1.38 2.91 2.25 2.!Jl 2.25 1 .20 1.20 -1,2!) -1.29 ,000 .DO0 2.19 3 .4!) 2.66 1.74 1.41 -1.18 - ,134 ,100 0.941 - 1.54 1.54 5.29 1.24 3.92 ,640 2.11 - 2.28 -I.O(i - ,381 .437 13.7 9.61 1.13 0.881 5.14 -0.736 - ,995 1.76 ,293 - 5.32 1 .02 37.4 ,714 2 4 . 1 - ,0358 1 2 . 2 - ,388 -1.91 4.57 -12.0 -26.6 ,371 -3.45 11.6 - ,543 30.6 ,540 6 6 . 2 0.968 118 a Column a contains values of A K ~ / K ~where &, Atis is the conductivity increment for orientation of the a-axis along the field; column h contains A K I , / K ~ Sfor ? orientation of the b-axis along the applied field. 100 30 10 :i 3/2 1 2 /:3 1/3 1/10 1/80 1/100
,
+
TABLE IV FREQI-ENCY COND1:CTIVITY INCREMESTS, ~ . ~ , / ~ , 67',q. ~) AXIALR A T I O
FIIGH
I
--
h)Sjkl
0.1
P
100 30
32 1
213 1/3 1 1/30 1,'100
--0.7770 --1.068 ,305 --1.389 --1.282 -.1.296
_ _2.2.56 -4223 --7 740
=
3.0
1.0
10
100
7.192 24.90 8G,88 883,: 4.256 16.09 5 7 . 4 9 589.9 2.424 39.03 405.1 1"13 6.488 25,30 2 6 7 . 2 0'7316 5'222 20"'' 223'0 ,6679 5.000 20.16 235.1 224,3 ,7370 5,254 21,06 7,197 28.01 295,6 72.16 7 4 8 . 7 4,509 19.54 49.78 180.1 1856 12.54 9 2 . 7 3 335.2 3453 23.45
spheres of the same effective conduct'ivit'y as the solvent does not affect the field distribution, but the dielectric constant decreases because of t'he lower dielectric constant of the sphere. The effect of n lower internal dielectric constant is overcome at sufficiently high values of K a t and K ~ ' , so the increment becomes positive a t higher values of K ~ ' / K ~ , the more so when p # 1. Large positive values at' low and high p are associated wit,h low values of the depolarizing factor, and arise from t,he terms containing K~ in equation 6.12. This is of particular interest regarding the interpretation of the large dielectric increment's of anisometric polyelectrolytes. If K ~ / Q is large enough so tijj/tiI >> 1, the corresponding term of the siini apprnaches
TABLE
P
HIGHFREQUENCY 0.1
b
I
- 0.536 - , G9l - ,797 ,942 -1.11 - 1.28 - 1.53
- -
v
h S D U c T I V I T Y INCREMENTS O F ORIESTED ~ P € I E R O I D S n
-.---1,o a
hli/hl
_~_. ---~-3,0 _
b
=
a
-0.895 3.6j 8.97 12.9 8.38 - 1 . 26 2.12 5.32 -1.46 1.22 3.03 5.70 3 -1,49 0.676 1.33 4.27 3/2 -1.38 ,598 0.798 4.40 1 -1.28 ,668 ,668 5.00 .877 ,667 6.23 -1.18 2/3 1.86 ,946 11.0 - 2.2; -1.01 1/3 1/10 - 5.31 -0.730 8.78 2.37 40.1 1 /30 -11.9 - ,379 26.7 5.44 113 1/100 -24.0 ,370 44.5 12.9 197 The numbers in column a are A K ~ / K , &in, column b, A K I , / K ~ ~ ~ . 100 31) 10
+
_ ~ _ _10_
--
b
a
30.9 19.9 13.0 7.60 5,63 5.00 4.77 5.28 9.27
45.5 30.3 21.4 16.9 17.7 20.2 25.0 43.1 I50
18.4
414
40.8
729
~-
-.
b
108 71.1 47.9 29.5 22.6 20.2 19.1 20.5 33.4
464
63.7
4277 7576
135
ino
a
I:!)
180 215 266 -lLX
~
--. b
to94
311 240 215 204 216 si4 (il6
1392
I00
r--
I
I
I
I
The low frequency dielectric increments, 4e, and of oriented systems are presented in Table VIII. Again, el = 78.51, 6, = 3.00, and kjO = 0 for all axes. The numerical calculations have been presented in the form of tables as the wide range of values encountered would require a n excessive number of curves for interpolations of reasonable accuracy. However, because low frequency dielectric increments have been widely investigated, curves of Aq/el& 2's. p for various K * ' / K ~ are shown in Fig. 4 to portray the general trends and to facilitate interpolations over the wide range of \ d u e s . Aeb,
30 10 N
'0, 5 3.0
I
1. o
0.3 0.1 l/lOC
7. Applications Proteins, Nucleic Acids and Nucleoproteins.P. Fig. 4.--I>ielectric increments us. axial ratio, p , for ellip- Dielectric dispersion has been studied in many soids of revolution. Particle volnme conductivity is zero. protein solutions. The earlier work, which has The surface conductivity and the particle volume are held been reviewed in several places,192o was interpreted constant as p changes. in terms of equations developed for insulating systems. The possibility of ionic polarization TABLEVI HIGHFREQVENCY DIELECTRIC DECREMENTS FOR COMPLETE mechanisms, and variables like solvent conductivity, macromolecular charge and eonfiguration, A N D R A N D OORIENTATION M and counterion composition, which effect their P -- Ars/eidr - Aebjridz - Aeh/cidn contributions, were ignored, and the dielectric 100 0.9622 1.852 1.555 increments were attributed solely to permanent 30 ,9650 1.847 1.553 dipole orientation effects. Dielectric relaxation 1.818 1.539 10 ,9809 times were interpreted in terms of the rotational 3 1,074 1.683 1.480 diffusion constants of rigid hydrodynamic models. 1.429 1.240 I , 524 3,/2 From the above equations and numerical calcula1 1.416 1.4163 1.416 tions, i t is clear that the ion atmosphere polariza1.311 I . 435 2/3 1.684 tion may account for the same effects previously 1.602 2.373 1.166 1/3 attributed to orientation polarization. Thus, the 1/10 5.5863 1.031 2.549 existence of two relaxation times can be c~plained 1/30 11.11 0.98563 4,360 on the basis of a spheroidal model w t h surface 1/100 18.08 0.9690 G.671 conductivity a5 an alternative to one carrying a TABLEVI1 n o n - a d dipole moment. Either idea appears Low FREQUENCY DIELECTRICINCREMENT, A e l / ~ J i ~ , VR. equally acceptable qualitatively and, in general, A X I A L RATIO the possibility of contributions from both must be KS'/KI considered. However, there are several instances 10 1.0 3.0 30 100 P where the permanent dipole orientation theory is 200 --0.009257 0.7073 1.025 2.307 14.08 clearly inapplicable, namely, tobacco mosaic T iim, ,5305 1.162 3.617 23.66 100 -- ,1554 hemocyanin and desoxyribonucleate 50 -- ,3277 ,4502 1.399 5.374 29.04 An early suggestion of this came from n study 30 -- ,4553 .3965 1.681 6.708 27.01 of the Kerr effect iii solutions of t: lnrge rigid 20 -- ,5543 ,3571 1.934 7.217 21.90 macromolecule, tobacco mosaic T i i us (TMV), ,3109 2.265 6.390 12.18 10 -- ,7128 which showed very strong electric orient a t'ion 5 -- ,8435 ,2620 2.206 4.472 6.247 which persisted at frequencies far n1,o.i-e the ex,2133 1.970 3.329 3 -- ,9139 4.142 pected orientational di-spersioii region and which 2.975 1.683 2.568 - - ,9577 .1482 3/2 was dependent upon electrolyte 1 - - ,9618 ,1337 1.628 2.454 2.826 The role of the ion atmosphere waz r1u:dit:itively ,1510 1.690 2.566 2.965 2/3 -- ,9618 suggested on the basis of that study. The ob3.900 -- ,9837 ,2710 2.131 3.322 1/3 served influence of electrolyte concentration caii be --1.427 ,8075 4.420 7.415 1/10 9.088 explained through its effect on t hc parnineiers 1/30 --1.531 9.702 18.20 1.818 23.76 K , / K ~ which strongly affect, the dielectric incremeiit~, 1/100 --2.125 3.821 22.56 49.95 72.05 as shown above. -4 more quantitative treatment of 6.365 35.63 88.25 136.8 lj200 -4.4'73 the orientation effects on the preseiil model inl/Aj, which is a large quantity along the long axes volves additional considerations v-hich will be of highly anisometric particles. The maximum in discused ieparately.?2 A discussion of the limiting A q observed for p > 1 is augmented and shifted to case n hen ~ , / h , i. very large ( h l negligiblc) is conlarger p at greater ~ j / ~ 1 . Thus, of conductivity (19) J. L. Oncle), Cli. 2 2 , in "Proteins, .Imino Acids and Pepassociated with t'he counterions of a polyelectro- tides," by E J. Culm and J. T. Edsall. Reinhold 1'11ld V o r ~ ), XPW lyte is a very important fact.or in determining T u r k , K. Y , 1943. (20) J T Cdssll. Fo71s. chem Torschung, 1, 119 (i(l4D) del-one often ignored in past discussions with (21) C T O'Konski and B H. Zimm, Science. 111, 1 13 (1950). reference to theories developed for insulat,ing ( 2 2 ) C . T O'Konski and S. Kmuw, maniiscript in jraratlon for systems. publication. 1/30 1/10
1/3
1
3
10
30
100
1311
TABLEVI11
Low F R E Q U E N DIELECTRIC CY INCREMENTS FOR ORIENTED SPHEROIDSO ,--~-~
~~~~
,_.
P
.~..~
40 2(? 10
5 :i
312 1
-. 0.!)57 -. ,932 -. ,944 -. .9X3 -. ,921 -. .8%< -. .8fi-l -'
.85li
-. -.
.8!fG .90:!
2'4
-.
J/X
-. 1.39
1/10 1J30 1/100 lj200
-. 3 . 2 3 -. 4 14 -. G.8ti
1.07
-14.7 Coliimns a ant1 b
~
-----3,o
_..__
b
a
200 100 50
a
~-
~
- .. . I , ( )
a
~
-
-
.
K
S
!
/
K
b
l
10
_.
~
a
+0.492 -0.921 1.52 -0.514 ,243 - .873 1.23 - ,000855 - ,0196 - ,790 1.07 ,833 - ,217 - ,682 0.936 + I ,78 - ,371 - ,563 ,817 + 2 . M - . v23 - , "10 ,610 + 3 . 7 8 - ,835 - .0OG74 ,396 + 3 . 7 5 - ,943 . 129 ,256 $3.08 - ,989 ,170 ,137 $2.07 - ,962 , 134 ,134 $1.63 - .907 ,0705 ,191 $1.30 - .780 - ,0777 ,445 +0.941 ,011 - ,526 - ,464 1.44 - ,230 -1.03 3.24 ,384 ,244 -1.97 6.72 ,0765 ,620 -0,330 9.71 - ,159 wfer to AK,JK~&and AKI,/K,&, for orientation of the
+
+
+ + + +
+ +
+ + +
tained in rhc more recent and complete study of the Kerr elfect in TMT' solutions.'j The dielecti ic properties of aqueous solutions of hernocyr ai t i and sodium hyaluronate have been studied ~y .Jacobson and co-workers. 2 4 - - ? 5 They noted that in these systems also, the dielectric relaxation time- ere too short to be explained reasonably on the basis of dipolar reorientation. I n studie Td, where Ti is the ionic as indicnting zero dipole moment along the long relaxation time and T d is the orientational reaxis and a transverse component which was very laxation t,ime. From the Debye-Stokes relnt,ion, large, about 2 X l o 4d. This appeared c ~ n s i s t e n t ~one ~ obtains, for spheres in water a t 23” wit,h the ma,giiitude of the intrinsic Kerr constaiit T d = 2.72 x 10’’ Uo3 (7.1) reported by B e n ~ i t . But ~ ~ it is not consistent Applying equation 6.11 to a’ sphere, with the aswith the negative sign of the b i r e f r i n g e n ~ e . ~ ~ sumption that ~1 is low, taking €1 = 78.5 and Ej = 3 The macromolecule has a negative optical ani~i = 7.1 X ao/X ( A in ohms-’) (7.2) sotropy f:~ctor,~’ so the direction of orientat’ioii is with the long axis along the electric field, contrary Thus, r d , / ~ i = 3.8 X loz3ao2/A. If X is of the to the orientn,tion expected for a large transverse order of ohm-’ it follows that Ti >, ~d only dipole moment. Since the macromolecule is known if a0 2 5 A. This is of the order of toheradius of a to be highly charged, it seems clear that a11 of the small ion with its solvation shell. Thus, unless observed electxical properties can be explained in ionic transport a t surfaces is one or two orders of terms oi ion ,atmosphere polarization. This does mngnit,ude less than we have reason to believe, not rule out the existence of a transverse dipole application of dielectric theories developed for (36) I. AI. Klotz, Science, 128, 815 (1958). insulating systems is never fully j ust,ified for (37) W..Dstwald, Kolloid-Z., 46, 248 (1928). aqueous solutions a t ordinary t’emperatures. The (:38) E. /L. IIauser, J . R h e o l o g y , 2, 5 (1931). s:ime statement applies for non-spherical ions. (99) 13. I:reundlich, “Kapillarchemie,” Aliad. Verlag. m.b.H., The Kirkwood-Shumaker protonic mechnnism13 Leipzig. 19::Z. Tol. 11, p. 108-169, 624. (40) J. L , Russell and E. K. Rideal, Proc. nay. Soc. (London), 164A, will become important if it is the major contribu540 (1936). tion to A, and if K j / K 1 is of order unity or greater. (41) C. 1’. GoodEve. Trans. Forodoy Soc., 36, 342 (1939). Its order of magnitude cannot be ascertained be(42) T. (!, I,nurent, Arkiu Kemi. 11, 503 (19.57): C . A . , 64, 11522~ cause the mobilities of the protons among the ( 1 958). (43) A. Jungner and I. Jungner, Acto Chem. Scand.. 6 , 1391 (1952). proton acceptor sites is not known from eit,her (44) I. Tinoco, Jr., J . A m . Chem. Soc., 77, 4486 (1965). (48) C. T. O’Konski, K. Yoshioka and W. H. Orttung, THIS.JOUR(4.3) H. 13enoit, Ann. Phyn ...6 , 561 (1951). Haltner, Thesis, University of California, Berkeley, 1054. erf and H. A. Scheraga, Chem. Reus., 61, 260 (1951).
R ~ L 63, , 15.58
(IR,?R). (49) I. Tinoco, , T I . , a n d I 1., the local field effect may far exceed t,he frict8ionnl coefficient consideration, in de(61) U. Sohindew-olf, Sa(urzcissenschaften. 40, 435 (1953); 2. phgszk. Cikeni. [.\-.P. 1, 1, 129 (1931); z. E’lekfrochem., 58, 697 (1954). CK2) K, Hwkmann, Z. p h u s i k . Ciiem. (-\‘em Polge), 9 , 318 (1956). ( H 3 ) 11. Ligen a n , l G. Schwarz, i b i d . , 4, 380 (1955). (64) AI. Eigen and 0. Sehnars, J. Colloid Sci., 12, 181 (1957). (05) G . Pc,limnrz, S . P h p i k , 146, 563 (19%).
’
Yol. 01
termining the anisotropy of the mobilit,y of the polyion, defined in the customary way in terms of the external field. This will make the anisot,ropy of the polyion contribution even greater than supposed in Schtvarz’s treatment, but it does not follow that the polyion cont,ribut,ioiiwill exceed the contributions from the relatively mobile counterions which are computed 011 the present model. A complete theory must include both effects. Direct observations of t’he polyion mobility as a function of electric field (or degree of orientation) will probably be required to evaluate contributions of polyions and other ions separately. The electric field orientation problem was treated independently by S c h w a r ~and ~ ~O’Konski and Haltner23in terms of t.he limiting model in which the particles are considered very good conductors relative t o the medium. I n the latter study the absolute magnitude of the Kerr constant w-as studied in a carefully chosen model and it was found t o be significantly below the value calculated for t)he limiting case. The theory has been extended t o the present model and will be presented elsewhere.22 Concentrated Systems : Biological Tissues, Polycrystalline Powders, Soils.-The above discussion on the origin and the magnitude of the surface conductivity effect, and its dependence upon particle size and shape is of interest in the further interpretation of the dielectric properties of tissues and cell suspensions, which have been reviewed recently by 8chwan.l6 As a consequence of a high localized conduct,ivity in interfacial regions, special effects may be produced in biological systems by application of short burst5 of intense high-frequency radiation, e.g., light,ning strokes or high-frequency pulses. At frequencies below the dipolar dispersion region of water, the temperature rise in the ion atmosphere of polyelectrolytic components, e.g., the desoxyribonucleic acid in genes, would be expected to exceed the average temperature rise of the systtem until the heat can he carried away from t’hesurface by thermal conductmion. This may give rise to much greater biochemical changes in pulsed fields than expected on the basis of t’hemean t’emperature rise of the system. The high values of the low frequency dielect’ric constants and the dielectric dispersion observed in crystalline powders containing very small amount,s of adsorbed wat,er vapor are very prob:ibly the result of surface conductivity produced by mobile ions in adsorbed films of ~ a t e r . ~The ~ . presence ~ ~ of mobile ions on the surf:ices of anisometric particles of clay probably is responsible for the large low frequency dielectric constnnts and the broad region of dielectric dispersion observed in soils and for the variations observed with soil type and moisture T.O’Konski, J . Am. Chern. Soc., 73, 3093 (1951). (67) E. J . 3Inrpliy and H. €1. Lowry, TKI* . I n v R N A L , S4, 598 (1930). (68) S. 9. Banerjee and R.D. ,Joshi, Phil. M a g . , 26, 1025 (1938); for a criticism of the method SPE 13. R. I,. I,aniont, ihid., 29, 521 (1940). (69) A . R. von h‘igprl, “Dielect.ric Materials and A~~iilications,” Technology Press, M.I.T., arid John W l e y and Sons. Inc., h-.I?., 1954, (66) C.
1.’ 314.
(70) C. T. O’Konski, unpublished data.
May, 1900
COEFFICIENTS OF EVAFOR ITION
AYT)COXDRNS~\TION
619
Protein Hydration from Dielectric Studies.-- the macromolecular shape may be c~mputed.~O Interpretation of the high frequency dielectric From the values of A E I J E of ~ ~Table ~ VI, it is clear decrements of protein solutions in t’erms of a that the calculated hydration will depend upon the solvated model having a low dielect’ric c o i i ~ t a n t ~ ’ . ’molecular ~ shape, the variation being greater in appears to be :L sound approach to the problem going from spheres to oblate ellipsoids than in for compact molecules, providing the data can be going from spheres to prolate ellipsoids. obtained in a region where the loss contributions Acknowledgments.-It is a pleasure to acfrom ionic processes make only small contribubions knowledge the collaboration of J. J. Hermans on t o the field distribution ( E j ” l/w, where w is the angular frequency at out while the author vas a John Simon Guggenwhich the decrement is observed. Wherever heim Fellow at the University of Leiden. The information regarding molecular asymmetry is assistance of K. Bergmann in reading the available, 1;he dielectric decrement appropriate to manuscript and checking the equations, and T. J. Scheffer in making the numerical calculations is (71) G. H. IIsggis, ‘r. J. Buchannn and J. B. IIasted, .Tolure, 167, also gratefully acknowledged. This work was 007 (1951). supported in part by the Xational Science Founda(72) T.J. Buchanan, G. H. Haggis, .J. B. Hasted and B. J. Robinson, tion. ?’roc. Rou. Soc ( L o n d o n ) ,A213, 379 (1952).
(‘OEI~FICIESTS OF EVAPORATION AND COSDESSATIOS BY J. P. HIRTHAND G. AI. POUND l l c t r r l q RPwnt ch Lnborntory, Carnegze Institute of Technology, Pztlsbzirqh, Pennsylvanza Received November 1 1 . 1959
A model is developed for cryst,al growth and evaporation which invokes the Iiossel concept of a crystal surface being composed of low index faret,s, of low binding energy for adsorbed atoms, separat’ed by ledges of higher binding energy. The treatment indicates that surface diffusion kinet’ics are important in most inst’ances of crystal growth and evaporation. Prior kinetic treatments, such as those involving limitations of cryst,al growth because of entropy effects or diffusion in the vapor phase, are modified to include surface diffusion effects. Values for evaporation and condensation coefficient predicted by t,he kinetic treatment are compared with experimentally determined values.
Introduction Since the relation between the vapor pressure of a substance and the flux of the substance leaving a surface in equilibrium with the vapor was developed by Hertz,’ Knudseq2 and Langmuir3 there has been a vast research effort to investigate the quanti1,y c y j ambiguously called the condensaticn coefficient, the evaporation coefficient, or the accommodation coefficient in the expression J , = =t.,(pe - p ) / ( 2 m k T ) ~ / z
(1)
where the plus sign is used for evaporation, the minus for condensation, J , = J , is the net vaporization flux in molecules per per second, J , = J c is the net condmsation AUY, 1 = 1, 2 , 3, ---, dependent on the process being considered, p , = the equi1it)rium pressure, p = the actual yapor pressure, w = molecular mass, and 1; and 5“ have their usual meaning In general, cy1 in condensation does not equal in evaporation under similar experimental conditions as mas assumed by early investigator.. -41~0, there are often several factors affecting a , which may or may not be mutually independent. Accordingly, the following treatment of tht>various factors affecting J , is presented in order that the interaction of these factors might be clarified. The various factors are considered separately, the simpler cases being treated initially and complexities introduced in sequence. In each case, -he consideration of surface diffusion in the kinetics reprevnts the new contribution of this work. (1) €1. Hertz, A n n Phus 17, 177 (1882). (1) M . Knutisrn, 7 6 1 d , 29, 179 (1909) ( 3 ) I Langrruir, P h y s Rev , 2, 329 (1919).
Theory I. Clean Crystalline Surface with a Monatomic Vapor Phase.-Iii this case. the internal partition functions of an atom in the crystal are vibrational and in the vapor pha-e are translational. Hence, rotational degree.. of freedom are not activated in evaporation or condensation and, as a first apprcximation, it is possible t o use a model of evaporation and growth in which atoms are considered to be bound by near neighbor bond^.^,^ It has been chomn by Burton, Cabrera and Frank6 in the case of growth and by Hirth and Pound7 in the case of evaporation, both following the general ideas of 1701mer,8 that the kinetics involve the steps: (a) adsorption or desorption to the vapor phase from the surface; (b) surface diffusion; and (e) movement into or out of a kink position in a monatomic ledge 011 the crystal surface, a t which the equilibrium concentration of adsorbed atoms is maintained, see Fig. 1. For ideal crystals, ie., large crystals bounded by low indev planes and with no lattice imperfections other than the monatomic ledges and adsorbed atom.., the monatomic ledges arise from crystal edges in evaporation. The adsorbed atoms on the surface develop the distribution shown in Fig. 2, where it is shown that the equilibrium concentration of adsorbed atoms is maintained at ( 4 ) TI- Iiossel ,\ach Ges W zss Gottzngen 135 (1927) ( 5 ) I N Stranski, 2 p h y s z k Chem 136, 269 (1928)
W. K Burton, N Cabrwa and F C. F r a n k , Phil. Trans. Rot,. 243A, 299 (1931). (7) J P Hirth and G. lf Pound, J . Chem P h i i s , 26, 1216 (1957). ( 8 ) ,\I Voliiirr ‘ Kinetih der Phismbildiing,” Rteinkopff, Dresdrn and Leipiig, 1939 (11)
&or
