Fluctuation pressure of membrane between walls

Volume 117, number 7

PHYSICS LETTERS A

1 September 1986

F L U C T U A T I O N P R E S S U R E O F M E M B R A N E B E T W E E N WALLS W. J A N K E t Institut ]'dtr Theorie der Elementarteilchen, Freie UniversitSt Berlin, Arnimallee 14, 1000 Berlin 33, Germany

and H. K L E I N E R T 1.2 University of California, San Diego, La Jolla, CA 92093, USA Received 9 May 1986; accepted for publication 20 June 1986

We prove that a tensionless membrane, fluctuating harmonically with free ends between two parallel plates of spacing 2d, generates a pressure of the functional form p = (T/Ad)~'a(.Q, where r is the dimensionless variable ~---- (T//I¢)A//d 2, ~ is the curvature elastic constant, and A the area of the plates. For large A, a(~') becomes a constant which we determine by Monte Carlo simulation to be aoo = 0.060 4- 0.003.

Membrane layers are attracted to each other by van der Waals forces [1] which decrease at intermediate distances d (20 < d < 100 ,~) like 1 / d 3 ,1 The most important repulsive force to keep them apart is provided by thermal out-of-plane fluctuations [3,4]. In the absence of areal tension, these are so violent that they can be seen in an ordinary microscope [5]. The reason for this is that they are controlled only by the curvature energy [6] e = ½~¢(c1 +

¢2) 2,

(1)

where c t = I / R t , c2= 1 / R 2 are the principal curvatures of the membrane and r is the elastic modulus ( ~ (2.3 ~- 0.3) × 10 -12 erg for egg lecithin I Supported in part by Deutsche Forschungsgemeinschaft under grant no. K1256 and U C S D / D O E contract DEAT-0381 ER40029. 2 O n sabbatical leave from Institut fur Theorie der Elementarteilchen, Freie Universit~t Berlin, Arnimallee 14, 1000 Berlin 33, Germany. *.1 The l / d 3 law Pvdw ------1.4 × 1 0 - 1 s e r g / d 3 is valid for intermediate distances (see ref. [1]) as long as retardation effects are negligible. These change the faUoff to 1 / d 5 for larger distances. For very short distances of order 2 - 2 0 there are also repulsive hydration forces and electrostatic forces which overwhelm the van der Walls attraction but which drop of exponentially (see ref. [2]).

membranes at room temperatures). If the membrane is parametrized by the vertical displacement u(x, y) over an (x, y)-plane, this has the lowest 1 2 + ~2)u]2. The two-diapproximation e = ~K[(3j mensional laplacian ~2 = Ox2 + Oy 2 leaves all harmonic field configurations without energy and this causes the violence of the fluctuations. We are thus confronted with the interesting problem of finding the size of the repulsive force between membranes. For a first estimate we shall study two simplified idealizations of the physical system, as proposed by Helfrich [3,5]. The first is to imagine that the undulating neighboring membranes constrain the fluctuations of each membrane on the average, in a similar way as a harmonic potential would do. Then the partition function of each membrane is Z = e -(A/T)f oo

du(x)

×exp(--~TSd2x[(~2u)2*m4u2]),

(2)

where A denotes the area of the membrane at

0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

353

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T = 0. The integral can easily be done giving the free energy density 1

/=

1 c d2k

j

log(

4

+ m 4) = ira2 q- const.

The displacement variable has a gaussian distribution P(u) - e -u:/2°2 with a squared width 0"2 =

(U 2) =

f (2,rr)2 d2k 1 k4+ m4

T 8~:m 2"

brane between two rigid plates. This model has a partition function Z =

(3)

1 September 1986

H f_ du(x) exp ( - " fd2x(O2u):) d

d

also discussed first by Helfrich [3,5]. He resorts to what m a y be called an " i n d e p e n d e n t m e m b r a n e piece a p p r o x i m a t i o n " [5]. Observing that the mean displacement

(4)

(.2) =

d2k

a _

(2,rr) 2 k 4 C o m b i n i n g (3) and (4), we find that the energy changes with average width as follows f=

T2 - 64x(u 2)

(5)

F o r not too small distances d from the neighboring membranes, we m a y follow Helfrich [3] and expect ( u 2) to satisfy ( u 2) = p d 2,

(6)

with p varying only slowly with temperature, in which case we obtain t h e free energy density T2 f = aos~ r d 2 ,

(7)

1

a°~ = 6 4 p '

(8)

resulting in a pressure ~f O(2d)

T2 a°~ r d 3 "

(9)

The size of p is unknown. If we assume somewhat arbitrarily that the membranes at u = + d coincide with the 20 interval of the u distribution, implying that they enclose 95.45% of all possible configurations, we estimate

p = (uZ)/d ~= ~

(10)

and aos~ _ 1 = 0.0625.

(11)

Let us n o w turn to the second idealization which consists of a harmonically fluctuating mere354

r

(13)

4vK ,ff2

diverges with the size of the area, he mentally decomposes the m e m b r a n e into pieces of size AA and argues that between plates at u = _+d, the average size of u z should be (in contrast to our estimate (10)) ( u 2) = +d 2.

(14)

This is compatible with (13) if we imagine the m e m b r a n e to consist of a set of independent pieces of area AA = ~n.r3xd2/T. If these behave like an ideal gas, they exert a pressure

P

T 2 d AA

3 T 2 4,rr 3 xd 3

(15)

u p o n the walls thus resulting in the same law as (9) but with

with

P

(12)

a = 0.0242,

(16)

which is much smaller than (11). We are able to make decisive progress over these estimates. First we observe that by going to the reduced quantities Ured = u/d, Xred = x/vrA, the partition function is seen to have necessarily the functional form (~/2~TA/rd 2 )u2~(z), where z is the dimensionless variable z - ( T / x ) A / d 2, and N the n u m b e r of fluctuating degrees of freedom in the m e m b r a n e ( - n u m b e r of molecules). As a consequence the free energy density has the form

/=

NT

[ 2,nTA ~

T ( T A) ,

(17)

giving rise to a general pressure "law, due to the walls, 3Af T2 a(r), (18) P 3(2d) jcd 3

Volume 117, number 7

PHYSICS LETTERS A

w h e r e A f is the s e c o n d term in (17) a n d a ( r ) = dg(~-)/d~'. O u r s e c o n d p o i n t is that the existence of a non-trivial t h e r m o d y n a m i c limit A ~ ~ implies that the function g ( r ) has the limiting b e h a v i o r g ( ~ ) ~ a ~ " for • ~ ~ . M o r e o v e r , since the m e m b r a n e b e c o m e s free for d ~ ~ , the function g(~') c a n n o t c o n t a i n a n y p o w e r s 1/~" in this limit. Thus, we expect the o n l y corrections to b e of the exp o n e n t i a l t y p e (for instance e-C°nst×'). O u r third a n d m a i n result is a M o n t e C a r l o d e t e r m i n a t i o n of the limiting value a ~ . W e were f o r t u n a t e to have recently finished a s t u d y of the p a r t i t i o n function (12) on a lattice for the p u r p o s e of u n d e r s t a n d i n g a c o m p l e t e l y different p h y s i c a l process, n a m e l y that of defect m e l t i n g in two-dim e n s i o n a l crystals. T h e f u n d a m e n t a l defects are d i s c l i n a t i o n s [7] a n d these i n t e r a c t with each o t h e r elastically b y a p o t e n t i a l

f

d2k

1 September 1986

~ = - ( 3 log Z / 3 T - 1 ) / A shown in fig. l a (x = 1). R i g h t a b o v e the r o u g h e n i n g ( ~ freezing) transition, it d i s p l a y s the D u l o n g - P e t i t law ~ = ½T characteristic for th/e h a r m o n i c elastic fluctuations (in the crystalline state) thus d e m o n s t r a t i n g the irrelevance of the discreteness of u. F u r t h e r , a b o v e the 10

l

I

I

~ .

//" Loplociun RoughModel

///

322SquareLotfice..h-5 (250 + 500) sweeps + : heating. o : cooling.

T~ / / / ' ~'~//// //// //~

1 elk. x

(2~) 2 k 4

i.e. with the s a m e 1 / k 4 c o r r e l a t i o n s as in the p a r t i t i o n f u n c t i o n (12). T h e elastic fluctuations are therefore d e s c r i b e d b y the m o d e l (12). T h e disclinations are b r o u g h t in b y restricting the values o f u to integer n u m b e r s [8]. W e h a d s t u d i e d such a m o d e l on a square lattice of unit spacing, letting u r u n over integer values from - h to h with h = 5, T h u s o u r d a t a c o n t a i n e d the effect of walls at d i s t a n c e 2 d = 10. F o r increasing t e m p e r a t u r e , the m o d e l shows a first-order r o u g h e n i n g transition (TR/K = 1.63) at which the variables u overc o m e their discrete energy barriers. This relates via d u a l i t y to the t r a n s i t i o n of defect m e l t i n g (in the freezing direction). A b o v e this transition, the discreteness of u no longer matters. This is w h y we can use the m o d e l with discrete u ' s as an a p p r o x i m a t i o n to the m e m b r a n e p a r t i t i o n function (12). W e u p d a t e the c o n f i g u r a t i o n s b y sweeping t h r o u g h the entire lattice a c c o r d i n g to a p e r m u t a tion chosen r a n d o m l y after each sweep, a n d a p p l y the s t a n d a r d h e a t - b a t h a l g o r i t h m to each v a r i a b l e u(x) [9]. U s i n g at each t e m p e r a t u r e 250 sweeps for e q u i l i b r a t i o n a n d 500 sweeps for measurements, we f o u n d for a 322 square lattice with p e r i o d i c b o u n d a r y c o n d i t i o n s the i n t e r n a l energy

C J - - T.- 1.63 5

to l

i

, lO T

, ~.5

i

i

-A~

1 20

~

o o

522 Lultice,h=5 (250+500) sweeps lheat, cool. square + o

0

~6o

2~o T2

/

~,

~"

% %~

3~o

40

Fig. 1. (a) The internal energy 0= - ( 0 logZ/OT-1)/area of the roughening model (12) on a 32×32 square lattice with periodic boundary conditions (K = 1). For T = 1.63 it shows the first-order [8] roughening transition dual to the 2D model of disclination melting. For larger T it goes over into the Dulong-Petit law, # = T/2, and for T > 5 we begin seeing the effect of the "walls" at u = _+5 confining the membrane. The deviations from the Dulong-Petit law A# = 7"/2- E are shown in (b). From these we extract &E= -(0.0024-t-0.0001)T 2 such that a~ = 0.060±0.003. The independence of this number on the lattice structure is demonstrated by the data points of a similar run on a triangular lattice which are fitted very well by the same straight line. 355

Volume 117, number 7

PHYSICS LETTERS A

transition, the fluctuations begin "feeling" the restrictions of the "walls" at u = + 5 and the internal energy begins deviating from the Dul o n g - P e t i t law. The deviations A~ = ~ - T/2 are plotted against T 2 in fig. l b for • = 1. They are well-fitted by the straight line A~ = - (0.0024 + 0.0001)T 2.

1 September 1986

l.

0.2

I

t

I

16 a Lair., T = 2 0

(19) Z

This is to be c o m p a r e d with the general form A0

f

8(1/T)

0.I

- ½T

= -(T/A)~'a('r) = -(T2/KdZ)a(r),

(20)

following f r o m eq. (17). Inserting d = h = 5, this implies a(~') = a ~ with a ~ = 0.060 5:0.003

x

3

356

(2

6

Y] [u(x)-u(x+i)]

i=1

I

. . . .

I

-5

. . . .

0

I 5

u

(21)

and confirms that the free energy and the pressure have indeed the forms (17), (18) with g ( r ) = a ~ ' . We have checked that our lattice was large enough for this calculation by doing the same run on a 162 lattice which gave the same result. The M o n t e Carlo n u m b e r for a ~ is m u c h larger than Helfrich's theoretical estimate (16) and in g o o d agreement with our h a r m o n i c oscillator value (11). It is interesting to see that the distribution of u ' s between the walls is very close to being gaussian just as assumed in the h a r m o n i c oscillator approximation. F o r T = 20 (K = 1) this is demonstrated in fig. 2a. The squared width 02, however, is smaller than what is estimated in eq. (10). It is more like d2/5 than the assumed dZ/4. If we plot how ( u ~) varies with temperature in fig. 2b, we find that even though the curve saturates reasonably fast, it is not at all constant in the range T ~ (2, 20) where the pressure has the limiting functional f o r m (9). A n important point is to make sure that our n u m b e r for a ~ is a universal result and does not depend on the lattice structure. This was done by repeating the study for a triangular lattice which gave, indeed, the same n u m b e r as before (see fig. lb). In this comparison, we have to use the energy

2T4~

0.0

)2,

....

I ' "' ' I .... I ' ' ' 16 z Latt., h = ~ x = 5

t~A 4 j,

X,+: h e a t . . o,o: cool. (1000+20000)

(b) ....

I ....

5

I ....

10

I ....

15

20

T Fig. 2. (a) The discrete distribution of u ' s of the model (12) on a 162 square lattice for T = 20 (~ = 1) in comparison with the continuous gaussian distribution of the harmonic oscillator model (with 0 2 = d 2 / 4 = 2 5 / 4 such as to include 95.45% of the data within [ - 5 , 5]). (b) The squared width a 2 -= (u2> of the model (12) on a 162 square lattice against temperature (upper data). The center-of-mass movement of the membrane is included since it contributes to the pressure. Without this movement, we find the lower data points. The squared width of the center-of-mass distribution is given by the difference. The data points are taken in a thermal cycle with 1000 sweeps for equilibration and 20000 sweeps for measurement. The curves are eye-ball fits.

Volume 117, number 7

PHYSICS LETTERS A

w i t h x + i d e n o t i n g t h e n e x t - n e i g h b o r s s u c h as to h a v e the s a m e c o n t i n u u m l i m i t [2]

~T f d2k(O2u)2. F i n a l l y , let us r e m a r k t h a t o u r s t u d y was s t r i c t e d to t h e h a r m o n i c a p p r o x i m a t i o n (12) o f c u r v a t u r e e n e r g y (1). F o r p h y s i c a l m e m b r a n e s , n o n - l i n e a r p i e c e s will h a v e to b e c o n s i d e r e d well. T h i s will b e d o n e in a s e p a r a t e w o r k .

rethe the as

[3] [4]

[5]

[6] T h e a u t h o r s are g r a t e f u l to P r o f e s s o r s N . K r o l l a n d J. K u t i for t h e i r k i n d h o s p i t a l i t y at U C S D a n d to P r o f e s s o r W. H e l f r i c h f o r discussions.

[7] [8]

References [1] B.W. Ninhamand V.A. Parse~an, J. Chem. Phys. 53 (1970) 3398; D.M. LeNeveu, R.P. Rand and V.A. Parse~an, Nature 259 (1976) 601;

[9]

1 September 1986

R.P. Rand, Ann. Rev. Biophys. Bioeng. 10 (1981) 277; S. Nir, Prog. Surf. Sci. 8 (1976) 1. S. Mar~elja and N. Radic, Chem. Phys. Lett. 42 (1976) 129; A.C. Cowley, N. Fuller, R.P. Rand and V.A. Parsegian, Biochemistry 17 (1978) 3163. W. Helfrich, Z. Naturforsch. A 33 (1978) 305. I. Bivas and A.G. Petrov, J. Theor. Biol. 88 (1981) 4591; D. Sornette and N. Ostrowski, J. Phys. (Paris) 45 (1984) 265. W. Helfrich and R.M. Servuss, Nuovo Cimento D 3 (1984) 137; F. Browicz, Zentralbl. Med. Wiss. 28 (1890) 625. P.B. Canham, J. Theor. Biol. 26 (1970) 61. W. Helfrich, Z. Naturforsch. C 28 (1974) 693. H. Kleinert, Phys. Lett. A 95 (1983) 381; W. Janke and H. Kleinert, Phys. Lett. A 105 (1984) 134. D.R. Nelson, Phys. Rev. B 26 (1982) 269; K.J. Strandburg, S.A. Solla and G.V. Chester, Phys. Rev. B 28 (1983) 2717; W. Janke and H. Kleinert, Phys. Lett. A 114 (1986) 255; W. Janke and D. Toussaint, Phys. Lett. A 116 (1986) 387; H. Kleinert, Gauge theory of stresses and defects (World Scientific, Singapore, 1986). M. Creutz, L. Jacobs and C. Rebbi, Phys. Rep. 95 (1983) 201.

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