1406
The Journal of Physical Chemistry, Vol. 83,
No. 1 I,
1979
B. Berner and D. Kivelson
The Electron Spin Resonance Line Width Method for Measuring Diffusion. A Critique§ Bret Bernert and Daniel Kivelson*+ Department of Chemistry, University of California, Los Angeles, California 90024 (Received November 20, 1978) Publication costs assisted by the National Science Foundation
The concentration (C) dependent part of the ESR line width (AH) of a paramagnetic solute in a diamagnetic solvent arises from intermolecular interactions between paramagnetic molecules, and these interactions are strongly modulated by the translational motions of these molecules. Thus (dAH/dC) depends upon the diffusion constant ( D ) ,and measurements of (dAH/dC) can be used to determine D. Although this method of measuring D may not be the most favored one for bulk media, it could be one of the few methods applicable to the study of diffusion within bilayers. We have evaluated this procedure for measuring D by studying (dAH/dC) in a number of bulk solvents over a very wide range of ( T / q )where T is the temperature and q is the solvent viscosity. At low viscosity, the spin-exchange interactions dominate, and dAH/dC is linear in D or in ( T / v ) .A t moderate viscosities dAH/dC varies little with ( T / q )because the dipolar interaction which varies as D-I competes with the spin-exchange interaction. In fact, (dAH/dC) remains relatively constant over quite a wide range of ( T / v ) , but at very low T / v the (dA.H/dC) values evidence an upsweep with decreasing ( T / q ) .The low ( T / v )results are not yet understood, but it is clear that (dAH/dC) gives useful diffusion data at low viscosities (high T / v ) only.
Introduction The concentration dependent part of the line widths of the ESR spectra of liquid solutions of paramagnetic species are dependent upon intermolecular spin-spin interactions modulated by the relative translational motions of the paramagnetic molecules. ESR line width measurements can, therefore, be used to determine the translational diffusion constant, D , of the paramagnetic molecules.’-‘ This approach to the diffusional problem should be most effective in low viscosity liquids where motional narrowing (or broadening) effects are important, and less effective for highly viscous systems where the molecular motion is sufficiently slow that translational modulation of spinspin interactions are relatively unimportant. Although one would not usually turn to ESR line width measurements in seeking to determine diffusion constants, in membranes and bilayers most of the alternative techniques are inapplicable and the ESR line width method may provide a useful tool for studying the “fluidity” properties of these systems.s-ll Unfortunately, the interiors of most membranes and bilayers are relatively viscous which clearly limits the applicability of ESR diffusion studies in these media. I t is the purpose of the present study to analyze the ESR line width approach to diffusion and to determine the reliability of the diffusional information which it can yield. Unfortunately, our conclusion is that the method is not very promising. In the next section we summarize the theory which we have used to analyze the problem, and in the third section we present the experimental data along with a discussion of the results together with some conclusions. In the following section the assumptions inherent in the theory and comments on the limitations of theory are presented together with some generalizations and extensions of the results given in the second section. In the fifth section the experimental procedures are presented and discussed, and comparisons with the work of other groups is given. Finally, in the last section, a brief summary, already given in the third section, is included. This format has been used 8 Supported in part by a grant from the National Science Foundation. *Miami Valley Laboratories, The Procter & Gamble Corp., Cincinnati, Ohio 45247.
0022-3654/79/2083-1406$01 .OO/O
to allow the nonspecialist reader to extract the essence of our arguments in the first three sections.
Summary of Theory The ESR technique for studying diffusion has involved the introduction of small paramagnetic molecules (probe molecules) such as di-tert-butyl nitroxide (DTBN) or 2,2,6,6-tetramethylpiperidinyl-l-oxy (Tempo) into various The ESR line width, AH, is measured a t various molar concentrations, C, of probe species, and the quantity, d AHldC, which is dependent only upon intermolecular spin-spin interactions, is determined. Provided the concentrations studied are sufficiently small, only two-body interactions among spins are important, and dAH/dC is a constant independent of C. (Actually, as we shall discuss, there is a slight C dependence a t very low C, a region which we ignore.) dAH/dC as a function of temperature, T , is the experimental information obtained; a careful examination of the theory of diffusion and of the relevant spin-spin interactions enables us to extract diffusional results from the measurements. For a Brownian particle diffusing in a bulk liquid, the Stokes-Einstein relation D = kT/6~qr (1) holds, where k is the Boltzmann constant, r is an effective hydrodynamic radius, and is the coefficient of shear viscosity. For small diffusing molecules this relation does not hold exactly, but it works quite well, and in our studies of bulk fluids we shall assume that eq 1 is valid. In membranes and bilayers, one cannot measure 1 directly, but a measurement of D would be an interesting measure of the fluidity of the membrane or bilayer interiors. Of course, D in these cases depends not only upon solute size as indicated in eq 1 but also upon the ratio of solute size to membrane width; furthermore, D may both be anisotropic and vary from point to point across the membrane. The ESR method yields an average value of the isotropic (trace of tensors) D across the membrane or bilayer.1° We turn now to the relationship between dAHldC and diffusion. Both intermolecular electron spin-spin dipoles and Heisenberg exchange interactions can contribute to dAHldC. The exchange interaction is presumably short range, and we assume that it has a constant value as long 0 1979 American
Chemical Society
ESR Line Width Method for Measuring Diffusion
as two paramagnetic molecules are within a distance 2rex of each other and that it vanishes when the interspin separation is greater than 2rex. Furthermore, in our experiments the exchange interactions are sufficiently strong to cause loss of phase coherence in a time rl, the characteristic time of duration for a radical-radical collision; therefore, exchange occurs on a single collision. Consequently, dAH,,/dC lis proportional to the intermolecular collision frequency v; for a paramagnetic molecule with a nuclear hyperfine structure arising from a nucleus with spin 1:1A5,7 r
’ I
where AH is the peak-to-peak width of a line in the derivative ESR spectrum and y, is the gyromagnetic ratio for the electron. By imeans of diffusion theory the collision frequency v := 8 r N A D r e X10-3)C (
(3) is obtained where N A is Avogadro’s number.2 Below we shall discuss conditions under which these results apply, but here we note that (dAH,,/dC) should be linear in D , and, if the Stokes-E,instein relation (eq 1 ) is valid, linear in T/q. We have yet to consider the dipolar spin-spin interaction as a contributor to dAH/dC. This interaction must be treated in both the motional and static limits. In the motional limit, for the interaction of electron spins at X-band frequencies12J3 dmdip 6 , ” h ’ ~ N ~ ( l o - ~ ) --[9 + 10I][Dr,]-l (4) dC 75(21 1 ) f i
+
where 2r, is the distance of closest approach of the two spin dipoles. If the Stokes-Einstein relation holds, dAHdIp!dC is proportional to (o/T). Equation 4 is valid provided r c / T , > 1 (7) and dnHcIlp- 426,h -(8) dC 9 an expression which is clearly independent of D as well as q. (In eq 8, dAHdip/dC= 49.03 G/M.) We expect (dAHldC) to be the sum of exchange and dipolar contribution^.^ If the Stokes-Einstein relation is applicable, then provided inequality 5 holds1’ dAH/diC = A(T/q) B ( q / T ) (9) where I-,E
+
2y,3h2~2NA(10-3)(9 + 101) B=25(21+ l)&h At low viscosities ( T / q )is large, the first term (the exchange term) in eq 9 dominates, and (dAH/dC) is linear
The Journal of Physical Chemistry, Vol. 83,
No. 11, 1979 1407
in (T/q). At intermediate values of (T/q),the two terms in eq 9 are comparable and (dAH/dC) is relatively insensitive to T/q, or to the value of the diffusion constant. At high viscosities the second term (the dipolar term) in eq 9 dominates and (dAH/dC) is inversely proportional to (T/q). At sufficiently high viscosities the exchange contributions become negligible and the dipolar interactions are virtually static; in this case (dAH/dC) is given by the expression in eq 8. For nitroxides I = 1 and 6, = 1.759 X lo7 d / G ; therefore, A = 4.85 X 10-3(rex/r)P G/M deg and B = 7.63 X 104(r/rc)deg G/M P. The solid line in Figure 1 depicts eq 9 for nitroxides under the assumption that rex= r, = r; the dashed line with double dots depicts the static limit in eq 8. The discussion in the last paragraph suggests that only at very low viscosities would one expect ESR measurements of (dAH/dC) to yield useful diffusional data. In the following sections we shall present experimental data and investigate the theory in more detail.
Experimental Results Experimentally determined values of (dAH/dC) over five orders of magnitude of T/q are given in Figure 1 for DTBN in a number of solvents. At low viscosity (large T/q), (dAH/dC) is reasonably linear in T / q as predicted; i.e., the first term (the exchange term) in eq 9 dominates; however, in this region the values of (dAH/dC) at a given T/q vary appreciably from solvent to solvent, which suggests that ( r e x / r )in eq 10 is solvent dependent. (The spread in these values is somewhat masked in Figure 1by the fact that a log-log plot has been used.) Values of (rex/r) with B = 0 needed to make the theory in eq 9 fit the data for low viscosity solvents are 1.04 for ~ e n t a n e0.83 , ~ for hexane,1° and 0.50-0.717 for water. At intermediate values of (T/q) the experimental data show a flattening out which is in keeping with the theoretical predictions; however, eq 9 predicts a reasonably sharp minimum not compatible with the extremely flat experimental curve in the lo2 deg/P I T/q Ilo3 deg/P region of Figure 1. To illustrate this latter statement, in addition to the solid curve calculated from eq 9 with rex = r, = r, we have drawn two other curves based on eq 9 but with (rex/r)and (r/r,) taken as adjustable parameters; at high T / q these two curves were selected so that they agreed ( r e x / r= 0.65) with the experimental curve for water;34 one of the curves was adjusted ( r / r , = 0.23) so that the minimum value of (dAH/dC) = 1.5 G/M, the minimum experimental value, and the other curve was adjusted ( r / r , = 0.015) so that the T / q value of the minimum corresponded to the ill-specified experimental value of T/q = 700 K / P at the minimum. Quite clearly there are serious discrepancies between the values of (dAH/dC) determined in our experiments and those predicted on the basis of eq 9. The very flat region observed in the experimental (dAH/dC) curve in Figure 1 could be explained by the onset of static behavior, i.e., the exchange contribution becomes negligible and the dipolar contribution is described by the appropriate static expression in eq 8. However, this explanation is not totally satisfactory both because the experimental plateau is considerably lower than the calculated static one and because the experimental (dAH/dC) exhibits an unexplained upsweep at very high viscosities (very low T / q ) (see Figure 1). This upsweep appears to be solvent dependent in that it begins at different T/q values for different solvents. If the Stokes-Einstein relation were to fail at moderate and low values of T / q , then the “anomalous plateau and high viscosity upsweep” in Figure 1 might be a reflection of this “Stokes-Einstein breakdown”. Such is apparently
1408
The Journal of Physical Chemisiry. Vol. 83. No.
10
11, 1979
10'
IO'
T/I,
B.
Berner and D. Kivelson
I
10'
105
t'K/poise)
dAHldCvs. T / q lor DTBN in various Solvents. The three motionally modulated theoreticai curves have the form given in eq 9 with given on the curves. Symbols are for perdeuterated DTBN in degassed ethylene gbcol. and 0 for DTBN in degassed ethylene glycol, A in nondegassed ethylene glycol, in aqueous ethylene glycol, 0 in pentane? 0 in hexane,l6 * in glycerol. and + in water. (Some of the water data are from ref 7.) The value 01 ( r e x l r )= 0.65 was obtained by fitting the curves to the ( T I S ) data for water. Figure 1.
( r e J r ) and ( r l r , )
--8
50 -
E 40-
\
D
-w ' ! I
'
I
, 0
20
no
30
&.y 40
50
w 60Q
:
10-
0
0
70
ab
T ("C)
Figure 2. dAHldC vs. T (OC) for DTBN in MBBA and Tempo in phospholipiz bilayer. The solvents are indicated as follows: e. degassed MBBA: 0,undegassed MBBA:18 X , phospholipid bilayer." The area on the kfl indicates s o l i MBBA (freezing point 16 'C). The dotted area on the right indicates both isotropic MBBA and fluid bilayer (clear point 40-50 "C. and lipid transition temperature is 41 OC. The unmarked region is both nematic MBBA and soli lipk.3 The horizontal line at 49.03 GIM indicates the static limit calculated from eq 8. not the case. Diffusion measurements carried out on similar systems by means of an NMR technique known as paramagnetically enhanced relaxation NMR (PER-' NMR)" suggest that the Stokes-Einstein relation is valid over the entire range of interest to us. Although the PER-NMR measurements do not yield directly the diffusion constant for the probe molecules but a diffusion constant which is the average of the diffusion constants for solvent and paramagnetic probe molecules, enough information can he extracted to indicate that the anomalous hehavior of (dAH/dC) a t low ( T / q )is not a result of Stokes-Einstein breakdown. T o see this we note first that a calculation based on the Stokes-Einstein relation yields a D for DTBN in viscous ethylene glycol comparable to the measured self-diffusion constant for ethylene glycol.'6 The plateau and upswing region in our data could be explained on the basis of the simple motionally modulated dipolar theory only if D for DTBN far exceeded both its Stokes-Einstein value and the value for the
0O0a, 0$3? 008 0 0
y o ~
o
0
0
o
0 20-
0 Ib
m
o 30-
O
0
0
0
ESR Line Width Method for Measuring Diffusion
The Journal
6ot
X
L
---o
ZOOO
’
4000
T/q
I
I
6000 CK/poise)
I
I
8000
I
1 10000
Figure 4. (dAHldC) vs. T I T . +, perdeuterated DTBN (mid-field line) in ethylene glycol. (dAHldC) values for undeuterated DTBN were about 10% higher. 0, DTBN in triphenyl phosphite.” 0,dilute Tempo in aqueous solutions of Mn2+.22X, OH-Tempo in 51:49 aqueous glycerol solution.23 Additional values for OH-Tempo are as follows: lo2(T / v ) : 134, 180, 228, 284,348, 412;dAH/dC: 60,88, 92,92, 141, 217, respectively.
in bilayers below the internal freezing temperature. However, it should be noted that the upswing for solids depicted in Figure 2 aire much steeper than those observed for the viscous normal liquids depicted in Figure 1;furthermore, the values for (dAH/dC) in solids swing up far above the static-limit value given by eq 8. The plateau regions observed in liquids, liquid crystals, and bilayers are all very similar. These phenomena are not yet understood, but the very large values of (dAH/dC) in solid MBBA and in “frozein” lipid may be due to phase separation and the concentration of paramagnetic nitroxides in certain phases. The comparison of theory and experiment suggests that not only is the (dAH/dC) determined by ESR insensitive to ( T / q )at all but high values of ( T / q ) but , also that we still do not have a good understanding of the underlying theory at low T/q. Tlhe similarity between liquid crystal and lipid bilayer data suggests similar structure and dynamics in these systems. A more detailed analysis of the theory and experiment is given in ref 19. The work of several other groups having direct bearing upon our results is discussed below and is plotted in Figure 4 along with some of our data for perdeuterated DTBN in ethylene glycol. (a) Bales and PatroneZ0in their careful studies of DTBN in low viscosity fluids have found it necessary to postulate a high viscosity plateau similar to the one observed in this work. (b) Data for vacuum distilled, degassed triphenyl phosphite are given in Figure 4;the absolute calibration of the line widths is somewhat uncertainaZ1A reasonably distinct minimum occurs at about 2200 K / P , slightly higher than would be expected from the curves and data on Figure 1. ?‘he,incipient upsweep in (dAH/dC) with decreasing T / q below 1700 K / P may be in closer agreement with the theoretical curves in Figure 1 than are our ethylene glycol data in Figure 1. (c) Devaux et al.9 studied large spin labels in egg lecithin liposomes above the internal freezing transition. They reported a minimum in (dAH/dC) vs. T and a very slight low temperature upswing, and they interpreted this result in terms of exchange and dipolar contributions similar to those in eq 9. However, their “upswing” was measured over a very small temperature range and the experimental uncertainties do not permit one to distinguish between an “upswing” and the start of a plateau region. The fact that a t high T the (dAH/dC) increases with increasing temperature suggests that the liposomes studied by Devaux
of Physical Chemistry, Vol. 83, No. 71, 1979 1409
et al. are more fluid than those studied by Dix et al.,loJ1 or else that for the spin labels used by Devaux et al. the value of ( r e x / r )and ( r c / r ) are larger than for DTBN and Tempo, a very unlikely result. (d) Sackmann and co-workers8 also studied the concentration dependence of large spin labels in liposome bilayers and analyzed their data in great detail in terms of both exchange and dipolar interactions. With decreasing temperature they observe an upswing in dAH/dC followed by a plateau, and explain their data largely in terms of dipolar interactions near and in the static limit. Their values of dAH/dC appear to be concentration dependent. (e) Vasserman et alazzstudied the diffusion and ESR spectra of Tempo in various amorphous polymers. (dAH/dC) for these systems is virtually independent of T; for some systems this constant value is the static-limit value of eq 8 while for others it is closer to the plateau value in Figure 1. If their values of D are converted to ( T / q )values by means of the Stokes-Einstein relation, their measurements are all at ( T / q )< 250 K/P.19 (f) Also in Figure 4,diVl/dC is given for dilute Tempo solutions in aqueous glycerol broadened by the presence of appreciable concentrations (C) of paramagnetic ions.23 This is a slightly different experiment than the one we carried out. (g) Finally data are givenz4in Figure 4 for OH-Tempo in aqueous glycerol solutions; although the T / q values are too high to observe a plateau in dAH/dC, the gradual flattening of the curve as T / q decrease suggests the onset of a plateau similar to that assumed by Bales and PatroneZ0 (see paragraph b above). F u r t h e r Theoretical Discussions The theoretical results, the comparison of theory with experiment, and our general conclusions have already been given, but a more careful analysis of the theoretical expressions and their ranges of applicability are required to confirm the results. We carry out such an analysis here. To derive the spin exchange formula, eq 2, one notes that since the molecules under study have nuclear spin (I), due to the hyperfine interaction the unpaired electron has (21 1) different environments; therefore, any given bimolecular collision has a probability (21 1)-l of being between “like” spins and 21(21 l)-l of being between “unlike” spins. Interactions between “unlike” spins are the only ones that lead to spin relaxation. In our sharp-cutoff model of exchange, each collision between unlike molecules leads to spin relaxation (eq 2) provided the colliding molecules are within a distance 2reXof each other and within that distance
+
+
+
>> 1
(12) where Jo is twice the exchange integral of two paramagnetic molecules in c0ntact.j If we assume that T~ is of the order of a diffusional jump time and the jump length is about 1 A, then J02T12
10-16/D (13) if D is described by the Stokes-Einstein relation and r N 3.2 A for DTBN, then T~ N 5 X 10-7q/T, If Jo N 4 X lo1’ s-’, as previously e ~ t i m a t e dthen , ~ condition 1 2 is valid whenever Ti
( T / ~ 4 x 1011 s-1.
1410
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
The relative magnitudes of the exchange and motional dipolar terms in eq 9 can be readily estimated for I = 1 nitroxides if we set rex = r = r,. If we consider either term negligible if it i s less than 10% of the other, we find that the exchange dominates when (T/q) > 1.5 X lo4 deg/P
(15)
and the dipolar term dominates when (T/q)
< 1.5 X
lo3 deg/P
(16)
To derive the expression for the motionally narrowed dipolar line width, eq 4, we again note that since the molecules under study have nuclear spin ( I ) ,any given bimolecular interaction has a probability of (21 + 1)-lof being between like spins and 21(21+ 1)-’of being between unlike spins. Then we can make use of Abragam’s eq 79 and 89 subject to this weighting of “like” and “unlike” interactions and subject to the condition in eq 5. Furthermore, if
>> 1
UB2TC2
(17)
B. Berner and D. Kivelson
not well studied intermediate regime lying between the static and motionally narrowed limits. A t very low (T/q) the upswing could be due to the onset of the true rigid limit. Once again we note that the upswing begins a t different ( T / q )for different solvents (see Figure 1). The theory of spin exchange described above is obviously oversimplified and it might be useful to consider a slight extension of the model. The spin exchange integral J (rij,Qij)is a function of the intermolecular separation (rij) and of the relative orientation (Q,) of the two interacting molecules. The dependence upon orientation is one which we do not, at present, know how to treat. The dependence on rLjis probably short range and approximately of the form
J ( r i j )= K / ( r i j ) n for ri’ > r; K is a constant and n is probably a fairly large numi;, perhaps n > 8. If the exchange radius rex = ro at some Tocorresponding to a low viscosity qo, on the basis of eq 22, we would expect
where w, is the electron Larmor frequency, and a2rC2 > ( T / # >> 107
(19)
where ( T / q )is given in (deg/P). The upper limit is not relevant since, as indicated in eq 15, the exchange interaction dominates long before the upper limit is reached. The “motional” expressions of Abragam are subject to eq 5; since AH for DTBN i s of the order of 1-5 G and AH = (2/d\/3y,)T2-’, eq 5 becomes
(T/#
>> l o 6
(20)
It follows, of course, that the static limit for the dipolar interaction, eq 8, applies when eq 7 holds, Le., when ( T / q ) 2
