Interaction Forces between Membrane Surfaces - ACS Publications


obtained from the computer simulations. This solution can be compared .... continuum (with e = 80) enclosed between surfaces that represent the phosph...
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1 Interaction Forces between Membrane Surfaces Downloaded by 205.217.247.57 on February 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0235.ch001

Role of Electrostatic Concepts Max L . Berkowitz and K. Raghavan Department of Chemistry, University of North Carolina, Chapel Hill, N C 27599

We briefly review the role electrostatic concepts play in the theoretical description of forces acting between membrane surfaces. A special emphasis is given to a discussion on the nature of the hydration force.

THE

MOST IMPORTANT FORCES ACTING

BETWEEN

M E M B R A N E SURFACES

are

van d e r Waals, electrostatic, a n d hydration. T h e first two forces are explained by the D e r j a g u i n - L a n d a u - V e r w e y - O v e r b e e k ( D L V O ) theory ( J ) ; the exis­ tence o f the hydration force was anticipated before it was measured ( 2 ) . T h e van d e r Waals force is always attractive a n d displays a p o w e r law distance dependence, whereas the electrostatic a n d hydration forces are repulsive a n d exponentially decay w i t h distance. T h e electrostatic force describes the interaction between charged m e m b r a n e surfaces w h e n the separation b e ­ tween surfaces is above 10 molecular solvent diameters. T h e hydration force acts between charged a n d uncharged m e m b r a n e surfaces a n d at distances b e l o w 10 molecular solvent diameters; its value dominates the values o f van d e r Waals a n d electrostatic forces ( 3 ) . T h e t e r m " h y d r a t i o n " reflects the belief that the force is due to the structure o f water between the surfaces. Electrostatic a n d hydration forces are similar i n some respects: b o t h are exponential a n d repulsive a n d their theoretical description involves c o u p l i n g electrostatic concepts a n d ideas b o r r o w e d f r o m statistical mechanics. A l ­ though the nature o f the electrostatic force is solidly established, this is not the case for the hydration force. T o illustrate the role the electrostatic 0065-2393/94/0235-0003$08.54/0 © 1994 American Chemical Society

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

4

BIOMEMBRANE ELECTROCHEMISTRY

concepts play i n the description o f the interaction between the m e m b r a n e surfaces, w e w i l l describe the theory o f the electrostatic a n d hydration forces. T h e description o f the electrostatic force w i l l b e b r i e f because details can be f o u n d i n the excellent book b y V e r w e y a n d O v e r b e e k (J). T h e m a i n emphasis o f this review w i l l be o n the description o f the hydration force.

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Electrostatic Force Because molecules at an a q u e o u s - m e m b r a n e interface often carry a net charge, electrostatics must be an important component o f the theory that explains the f u n c t i o n i n g o f this interface. A l s o , because a very large n u m b e r o f molecules constitute the interface, it is anticipated that the theory couples the electrostatic concepts w i t h the concepts o f statistical mechanics. A m o n g the theories that illustrate this c o u p l i n g , w e find the rather simple but surprisingly successful G o u y - C h a p m a n theory ( 4 ) . T h e successes a n d failures o f the G o u y - C h a p m a n theory i n explaining the properties o f biological membranes are detailed i n the recent review b y M c L a u g h l i n ( 5 ) .

Gouy-Chapman Theory. T h e first theory that successfully treated the electric double layer present at a phase boundary was the G o u y - C h a p ­ man theory (4). I n its simplest version it considers an interface between a homogeneously charged solid surface a n d an ionic solution. T h e ions i n the solution are m o d e l e d as point charges a n d the solvent is m o d e l e d as a dielectric c o n t i n u u m w i t h dielectric constant €. T o s i m p l i f y the description o f the theory w e consider a symmetric z:z electrolyte solution. Place the surface charge at plane x = 0 a n d let the space charge due to m o b i l e ions extend f r o m x = 0 to infinity. T h e starting equation i n the G o u y - C h a p m a n theory is Poisson's equation, w h i c h couples the space-charge density p w i t h the electrostatic potential eV i|; = - 4 i r p

(1)

2

w h e r e € is the dielectric constant o f the m e d i u m a n d V is the L a p l a c e operator. A t this point the G o u y - C h a p m a n theory assumes that the average concentration o f ions at a given point c a n be obtained f r o m the average value o f the electrostatic potential at the same point through the application o f the B o l t z m a n n distribution; that is, 2

n_= n= +

n exp(ze\\i/ kT)

(2)

n exp(— zety/kT)

(3)

0

0

where n is the b u l k concentration o f ions, n_ ( n ) are local concentrations o f negative (positive) ions, e is the electronic charge, k is the B o l t z m a n n 0

+

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

1.

5

Interaction Forces between Membrane Surfaces

BERKOWITZ A N D RAGHAVAN

constant, T is the temperature o f the system, a n d z is the valency o f the ions. T h e total charge density p is t h e n given b y the expression p = ze(n

— n_)

+

= —2zen

0

(4)

smh(ze\\f/kT)

Substitution o f 4 into e q 1 results i n the P o i s s o n - B o l t z m a n n equation: e V i|i = Sirzen 2

(5)

sinh(zety/kT)

0

E q u a t i o n 5 the fundamental equation o f G o u y - C h a p m a n theory. W h e n

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ze\\f < < kT, e q 5 can b e linearized. T h e resulting equation is 8TrzVn i[f 0

V ^ =



2

ekT where

I|I = TT X

( ) 6

D

X , w h i c h is usually called the D e b y e

length, is given b y the

D

expression X

2

= ekT/8itz

D

V n

(7)

0

F o r a flat interface, i|i is a function o f x only. U s i n g the boundary conditions that i|/ = 0 a n d dty/dx = 0 w h e n x -> °°, the following exact solution for the z\z electrolyte at the flat interface c a n b e obtained: 2kT *

=

l + l n

^

ze

aexp(-x/X ) D

7 1 — a exp( —

7TT x/\ ) D

where exp( zety /2kT) 0

a

— 1 (9)

exp(%e*Ji /2fcT) + 1 0

v|i is the potential at the surface ( x = 0). F o r ze\\f /kT e x p ( - x / X ) 0

(10)

D

E q u a t i o n 10 can be obtained directly f r o m the linearized P o i s s o n - B o l t z m a n n equation. A n important connection between t h e surface potential, surface charge density a , a n d the density o f ions c a n b e established. Because the system is electroneutral, w e c a n write

a

= /

dx

9

•'o

= —

f

(11)

-rrdx=

4TT JQ dx

4Tr\ax/ o x =

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

6

BlOMEMBRANE ELECTROCHEMISTRY

E q u a t i o n 11 is often c a l l e d the contact value t h e o r e m . C o m b i n i n g eqs 8 a n d 11, w e get the desired relationship:

" = y —

s

i

n

M

h

(12)

F o r the b i o m e m b r a n e surfaces the situation ze\\f >> 0

kT

often occurs. I n

this limit, e q 12 is s i m p l i f i e d a n d has the f o r m n ( 0 ) = n exp(ze$ /kT)

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0

0

= 2TTv /ekT

(13)

2

E q u a t i o n 13 predicts that the concentration o f counterions at the surface o f the m e m b r a n e , n(0), is p r o p o r t i o n a l to the square o f the m e m b r a n e charge density a n d is independent o n the valency a n d the b u l k concentration o f the ions. E q u a t i o n 13 together w i t h other predictions f r o m G o u y - C h a p m a n theory are subject to experimental verification. T h e agreement b e t w e e n the G o u y - C h a p m a n theory a n d the experiment is rather nice, considering h o w simple it is. T h e recent review b y M c L a u g h l i n ( 5 ) s h o u l d be consulted for further details

o f the

experimental tests,

successes,

a n d limitations

of

G o u y - C h a p m a n theory. F r o m the rigorous treatment o f the double-layer p r o b l e m o n the m o l e c u ­ lar level, it becomes clear that the G o u y - C h a p m a n theory o f the interface is equivalent to a m e a n field solution o f a simple p r i m i t i v e m o d e l ( P M ) o f electrolytes at the interface ( 6 ) . T o consider the correlation b e t w e e n ions, integral equations that describe the P M are devised a n d solved i n different approximations. A n "exact s o l u t i o n " o f the P M of the electrolyte can be obtained f r o m the c o m p u t e r simulations. T h i s solution can be c o m p a r e d w i t h the solutions o b t a i n e d f r o m different integral equations. F o r detailed discus­ sion o f this topic, refer to the review b y C a r n i e a n d T o r r i e ( 6 ) . I n m a n y cases, the molecular description o f the solvent must be i n t r o d u c e d into the theory to explain the complexity o f the observed p h e n o m e n a . T h e analytical treat­ ment i n such cases is very involved, b u t initial success has already b e e n achieved. Some

o f the theoretical developments along these

fines

were

reviewed by B l u m ( 7 ) .

Double-Layer Interaction.

W h e n two charged surfaces approach

each other their electrical d o u b l e layers start to overlap and, as a result, the surfaces experience a repulsive force. F o r flat surfaces the forces can be explained i n terms o f the osmotic pressure due to the difference i n the i o n i c concentration i n the region b e t w e e n the plates a n d the concentration i n the b u l k region. T h e r e f o r e , w e c a n write ( I ) p = kT(n + +

n_-2n ) 0

= 2n kT[cosh(ze\\t /kT) 0

m

-

l]

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

(14)

1.

BERKOWITZ A N D RAGHAVAN

Interaction Forces between Membrane Surfaces

7

where p is the disjoining pressure a n d i|/ is the potential at the m i d p l a n e between the surfaces separated b y a distance h.lf ze\\i /kT > 1), w e c a n show that t|; ~ 2i|i ( x = h/2) ( I ) . Therefore, at these conditions w e get f r o m eqs 8 a n d 9 that D

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m

= 8a exp( -h/2\ )

ze^JkT

(16)

D

Substitution o f e q 16 into e q 15 produces the final result f o r the disjoining pressure: p = 64n kTa 0

2

exp(-h/\ ) D

(17)

w h e r e a is given b y e q 9. E q u a t i o n 17 is an often q u o t e d result o f the D L V O theory that predicts that at distances greater than the D e b y e length the repulsive pressure between charged membranes decays exponentially, a n d the decay length is given b y the D e b y e length.

Hydration Forces Experimental Results. T h e D L V O theory, w h i c h is based o n a c o n t i n u u m description o f matter, explains the nature o f the forces acting between m e m b r a n e surfaces that are separated b y distances b e y o n d 10 molecular solvent diameters. W h e n the interface distance is b e l o w 10 solvent diameters the c o n t i n u u m picture breaks d o w n a n d the m o l e c u l a r nature o f the matter should b e taken into account. I n d e e d the experiment shows that for these distances the forces acting between the molecularly smooth surfaces (e.g., mica) have a n oscillatory character ( 8 ) . T h e oscillations o f the force are correlated to the size o f the solvent, a n d obviously reflect the molecular nature o f the solvent. I n the case o f the r o u g h surfaces, o r m o r e specifically b i o m e m b r a n e surfaces, the solvation force displays a monotonic behavior. It is the nature o f this solvation force ( i f the solvent is water, t h e n the force is called hydration force) that still remains a p u z z l e . T h e hydration (solvation) forces have b e e n measured b y using the surface force apparatus ( 9 ) a n d b y the osmotic stress m e t h o d ( J O , I I ) . Forces between phosphatidylcholine ( P C ) bilayers have b e e n measured using b o t h methods a n d g o o d agreement was f o u n d . T h e general picture that emerges f r o m the experiments is that u p to distances ~ 3 n m l i p i d bilayers i n water r e p e l each other w i t h an exponen-

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

8

BlOMEMBRANE ELECTROCHEMISTRY

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tially decaying force. T h e decay constant o f the force varies f r o m the value less than 0.1 n m to above 0.3 n m ( 3 ) . I f the lipids are neutral, the repulsion is eventually balanced b y the attractive van der Waals forces o f D L V O theory. I f the lipids are charged, the repulsion continues b u t w i t h a decay constant compatible w i t h the value obtained f r o m the double-layer theory. T h e addition o f ions to water can, i n some cases, provoke the appearance o f a n attractive force ( 3 ) , a p h e n o m e n o n o f obvious relevance to m e m b r a n e fusion. T h e previously described measurements have b e e n p e r f o r m e d o n lipids i n aqueous solutions, b u t l i p i d bilayers also swell i n some other solvents (12) a n d the results o f such measurements compare quite w e l l w i t h the aqueous case. I n addition, hydration (solvation) forces act between D N A polyelectrolytes ( 1 3 ) a n d polysaccharides (14). These facts make the interpretation o f the forces even m o r e c o m p l i c a t e d a n d it is n o w o n d e r that different ap­ proaches to explain the nature o f this solvation force exist. So far n o truly ab initio theory has b e e n proposed. T h e existing theories i n c l u d e models based o n the electrostatic approach, the free energy approach, a n d a n approach based o n the entropic o r protrusion m o d e l .

Electrostatic Models. T h e solvated p h o s p h o l i p i d bilayer c a n be considered as a n electrostatic p r o b l e m , w h e r e water is m o d e l e d as a dielectric c o n t i n u u m ( w i t h e = 80) enclosed between surfaces that represent the p h o s p h o l i p i d membranes. T h e h e a d groups o f the m e m b r a n e molecules c a n be represented b y dipoles a n d the tails c a n b e represented b y a dielectric c o n t i n u u m (e ^ 2). T h e repulsion hydration force is d u e to the interaction o f the zwitterions o f the h e a d groups w i t h the image charges. T h e first analysis o f such an electrostatic m o d e l was p e r f o r m e d b y Jonsson a n d W e n n e r s t r o m ( J W ; 15). Because the h e a d groups o f the m e m b r a n e molecules are m o b i l e , J W represented the zwitterions as a two-dimensional fluid described b y a radial distribution g ( r ) . T o simplify the p r o b l e m , J W assumed that the surface separation was large a n d that there was n o correlation between the surfaces. F r o m their analysis J W c o n c l u d e d that the force d u e to the images strongly d e p e n d e d o n the character o f the pair distribution f u n c t i o n . T h e force h a d an exponential decay w h e n the correlation between the zwitterions was strong. F o r the uncorrelated zwitterions, the force f o l l o w e d the p o w e r law. Because the calculated value o f the force was o f the same magnitude as i n the experiment, J W considered their theory a success. T h e theory p r o ­ posed b y J W was extended b y K j e l l a n d e r ( 1 6 ) , w h o investigated the m o d e l i n greater detail b y r e m o v i n g some overrestrictive assumptions. K j e l l a n d e r c o n ­ c l u d e d that the force was strongly dependent o n the m o d e l used for g ( r ) a n d that the agreement between the J W results a n d the experiment was fortu­ itous. M o r e o v e r , Kjellander showed that a change i n the location o f the dielectric discontinuities w i t h respect to the location o f the zwitterions caused the results to undergo a dramatic change.

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

1.

BERKOWITZ A N D RAGHAVAN

Interaction Forces between Membrane Surfaces

9

T h e electrostatic m o d e l p r o p o s e d b y J W was later thoroughly analyzed i n a series o f works p e r f o r m e d b y the " S w e d i s h - A u s t r a l i a n g r o u p " (17-19), w h e r e exact f o r m a l treatment a n d approximate methods w e r e u s e d to solve the p r o b l e m . T h e authors o f these papers considered two electrostatic models. I n the first m o d e l they investigated the interaction between two planar surfaces (separated b y a distance h) w i t h m o b i l e ions adsorbed onto t h e m (the net surface charge was zero). T h e surfaces w e r e i m m e r s e d i n the dielectric c o n t i n u u m w i t h the dielectric constant e B e h i n d each surface a different dielectric m e d i u m (with the dielectric constant € ) was p l a c e d . I n the second m o d e l the m o b i l e ions w e r e replaced b y m o b i l e dipoles that w e r e oriented p e r p e n d i c u l a r to the surfaces. I n b o t h models the m o t i o n o f the particles was restricted to the w e l l - d e f i n e d plane. F r o m the analytical treat­ ment, w h i c h i n c l u d e d images a n d correlations, the following asymptotic results for the pressure were obtained for the first m o d e l : P

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2

1. W h e n e = e (no image charges are present i n the system), the interaction arises f r o m the charge fluctuations. T h e pres­ sure, P , is attractive a n d displays an asymptotic behavior: x

2

P ~ -T/h

3

(h

-> oo)

(18)

T h e pressure is independent o f the i o n concentration a n d i o n i c radii; it only depends o n the temperature T a n d the distance between the layers. 2. W h e n € # e , image charges are present i n the system. T h e image repulsion between electroneutral planes is exactly can­ c e l e d b y the static van d e r Waals interaction, a n d the leading t e r m for the total pressure is exactly the same as i n the previous case. x

2

I n the second m o d e l (and i n the case that € # e ), the pressure between surfaces arises f r o m d i p o l e correlations a n d dielectric images. Asymptotically the pressure is x

P ~ -A/h

3

+ B/h

4

- C/h

5

(h

2

-> oo)

(19)

T h e van der Waals pressure is not screened a n d it dominates at large distances; the second t e r m is due to image repulsion, a n d the t h i r d t e r m is due to the attraction between dipoles o n different surfaces. These asymptotic results w e r e c o m p a r e d w i t h n u m e r i c a l results obtained b y solving the hypernetted chain equation ( H N C ) for the p a i r distribution function, w i t h M o n t e C a r l o ( M C ) simulations a n d w i t h a m e a n field approxi­ mation. T h e agreement between the n u m e r i c a l results a n d the asymptotic results was fairly good at distances above h — 1.5 n m . Substitution o f the

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

BlOMEMBRANE ELECTROCHEMISTRY

10

parameters that c o r r e s p o n d to the zwitterionic l i p i d lamellae into the forego­ i n g pressure expressions resulted i n the attractive total pressure. W e have c o n c l u d e d that the electrostatic m o d e l p r o p o s e d b y J W is quite an interesting m o d e l b y itself, b u t i t is very sensitive to the fine details o f the system and, therefore, cannot explain the quite generally measured exponen­ tial dependence o f hydration forces o n the surface separation.

Free Energy Approach.

N e a r l y i m m e d i a t e l y after the p u b l i c a t i o n

o f the first data o n the hydration force ( 1 0 ) , M a r c e l j a a n d R a d i c ( M R ) p r o p o s e d a very elegant theory to explain the nature o f the observed strong

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force ( 2 0 ) . A c c o r d i n g to M R theory the force is d u e to the modification o f water structure near the m e m b r a n e - w a t e r interface. T h e water molecules near the interface d i f f e r f r o m the water molecules i n the b u l k : they are m o r e " o r d e r e d . " T o describe this " o r d e r , " o n e c a n introduce a n o r d e r parameter T|(x) a n d p e r f o r m a L a n d a u - t y p e expansion o f the free energy density

g(x);

that is,

g = where g

0

g

o

+ a t , + c(dr\/dxf + -

(20)

2

is the free energy density i n b u l k water, a n d a a n d c are constants.

T h e corresponding m i n i m i z a t i o n p r o b l e m results i n the differential equation

d T\(x)/dx 2

2

-

= 0

(a/c)i\(x)

(21)

A s s u m e n o w that the interfaces are p o s i t i o n e d at x = h/2 a n d that

t\(h/2)

= —j]( — h/2)

a n d x = — h/2

= T| . T h e m i n i m i z a t i o n o f free ()

energy

density given b y equation 2 0 subject to these b o u n d a r y conditions results i n the following f o r m f o r the o r d e r parameter:

T](x) = Tj sinh(Kx)/sinh(Kfe/2)

(

0

w h e r e K = (c/a) .

)

T h e excess free energy p e r unit area, A G , is

l/2

AG

2 2

= f

h/2

(g

-

g o ) dx

=

(flc)

1 / 2

T

1

2

0

coth(/ K/2) l

-h/2

J

T h e pressure, p , is given b y the derivative o f the excess free energy:

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

(23)

1.

BERKOWITZ A N D RAGHAVAN

Interaction Forces between Membrane Surfaces 11

F o r K / I > > 1, the repulsion follows the exponential law:

p = 4 p e x p ( — Kh) 0

= 4 p exp(—/i/X)

(25)

0

w h e r e X = 1 / K . F r o m the derivation described previously, it follows that the value o f p

0

(the pressure w h e n the surfaces are at close contact) is deter­

m i n e d b y the degree the surface orders the water and, therefore, depends o n the properties o f the surface. T h e decay parameter X is d e t e r m i n e d b y the

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degree the o r d e r i n g is propagated through water and, therefore, according to M R theory (20)

characterizes water only.

M a r c e l j a a n d R a d i c (20)

d i d not specify what quantity plays the role o f

the order parameter. M a n y different variables can be considered as c a n d i ­ dates for this role, but the orientational polarization o f water has the greatest appeal. Indeed, i n later publications G r u e n a n d M a r c e l j a (21)

explicitly

considered this quantity as an order parameter i n the p r o b l e m . A n emphasis o n the orientational polarization allows easy construction o f an intuitive picture for the origin o f the hydration force, a n d this picture is given i n F i g u r e 1. A s can be seen f r o m this figure, due to the preferential orientation a n d symmetry, the resulting dipole to the left o f the m i d p l a n e OO' repels the dipole to the right o f the m i d p l a n e . T h e net result is a repulsive force. T h e M R theory predicts that the decay constant is independent o f the nature o f the surface, but the experimental data show that this is not the case and that the decay constant does d e p e n d o n the nature o f the surface (3). Recently, K o r n y s h e v a n d L e i k i n ( K L ; 22)

extended the

theory and demonstrated h o w this dependence replaced the homogeneous

Marcelja-Radic

can be explained. T h e y

boundary conditions used i n M a r c e l j a - R a d i c

theory b y the inhomogeneous boundary conditions. A c c o r d i n g to K o r n y s h e v a n d L e i k i n , the description o f the inhomogeneous character o f the b o u n d -

O'

O Figure 1. Pictorial representation of the Marcelja-Radic

theory.

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

BlOMEMBRANE ELECTROCHEMISTRY

12

aries is measured b y the correlation f u n c t i o n

S ( H ) = |

( ) 2 6

R

label the surfaces. I n reference 2 2

the authors show that o n l y S^iR) contribute t o t h e repulsion force. T h e F o u r i e r transform o f this correlation function, S (Q) (usually c a l l e d t h e U

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structure factor), w h i c h is d e f i n e d as

S (Q)

(27)

= fdRS (R)exp(-iQR)

U

u

determines the inhomogeneous c o n t r i b u t i o n to the hydration pressure

F

i n h

,

w h i c h is given b y t h e expressions

PiM

= P (h) + P (h) y

c i( )

p

h

,oc

= lT-

QSiAQ)

Q

d

2W

V

0

(28)

2

r

1

sinh [^(K 2

I n e q 2 9 , 1 / K is t h e characteristic reference 2 2 p o i n t e d out, S (R)

2

+