Volume 39A, number 4

22 May 1972

PHYSICS LETTERS

VAN DER WAALS ATFRACTION BETWEEN CONDUCTING MOLECULES

P. RICHMOND*, B. DAVIES and B.W. NINHAM Departments of Applied Mathematics, Australian National University, Canberra A. C. T., Australia Received 14 March 1972

We discuss the van der Waals force between two long conducting~chains,due to current-current correlations. An assumption used by other authors is shown to be inappropriate. Further, the effects of finite temperature give rise to forces of longer range than they predict. 2 (l/mXap/an), p = n 0eu, v5 0e, n0 is the equilibrium density of charge carriers of charge e and mass m. J and p are related via the continuity equation. The electric field, E, is derived from a scalar potential, which satisfies Poisson’s equation inside and Laplace’s equation outside the cylinder. For a single thin cylinder the appropriate axially symmetric solutions inside are 2—k2) p(r, 0) += B1 A10(ur) and ~(r,20) = = k2 + —4irAI0(ur)f(u 0(kr) where u w2)/u2. Outside 4(r, 0) = CK 0(kr). The dispersion relation for the normal modes is obtained by applying the usual boundary conditions plus the condition that the radial current is zero at the surface of the conductor. After some algebra we obtain

In a recent letter Chang et al. [1] have applied the Lifshitz theory [2] of van der Waals forces to two conducting cylinders. As a result of what are essentially correlations of current fluctuations they predict an attractive force per unit length across a vacuum between two thin parallel chains a distance R apart which is proportional to R3 (ln(R/a)3/2, rather than the usual R~behaviour. The existence of such a new long range force could well be of much importance for a variety of biological applications. We show in this note that when screening is taken into account, the force is proportional to R3 at zero temperature and to R2 at non-zero temperature. We restrict ourselves to very thin i~ylinders,ignore retardation and perturb about the modes of an isolated cylinder. The conduction processes can be described by a standard hydro dynamic equation* [3, 4J which in linearised form becomes

where 1

~+v~Vp =-~-E

where

(1)

n

~,

—

kK~(ka)— kI~(ka)—7uI1~(ua) K0(ka)

—

(2)

10(ka)—710(ua)

2I~(ua). This gives [ka ~ 1]. y = kw~I1~(ka)/uw w2—u~k2~}(ka)2w~K 0(ka). Fora AD(AD = this yields the result of Changet al. [1]. However we suggest that the limit a ~ AD is more appro~ priate i.e., w v5k. This result can in fact be derived from quantum mechanical calculations for ir electrons interacting via a coulomb potential which is screened by the a electrons [5]. Furthermore it turns out that when a AD the hydrodynamical model exhibits significant radial currents which are inappropriate to the present problem where the current is mainly in the axial direction. For two thin cylinders far apart it can be shown that only the m = 0 mode is important. Solution of the boundary value problem for the potential, now yields the dispersion relation 301 ~-

* **

Queen Elizabeth II fellow. This approach does not incorporate damping due to electron-electron, -phonon or -impurity scattering. We have in mind ~relectrons in a linear chair and the plasma mode which essentially determines the van der Waals force between thin cylinders will in any case be undamped within a quantum mechanical R.P.A. treatment of the first mechanism [5]. The second mechanism is omitted here for simplicity but can be shown for thin cylinders not to affect matters in leading order. The third mechanism will we believe not be important for many biological macromolecules where the backbone is periodic.

~,

PHYSICS LETTERS

Volume 39A, number 4

(3\

w2—v~k2 ±2v~k2K0(kR),

2) is the equilibwheredensity = (q0e/m) andcarriers q0 = (irn0ea rium of charge per unit length of cylinder. For each value of k, eq. (3) gives two normal modes for the coupled pair of cylinders, the frequency of each being slightly perturbed from the isolated cylinder case. The van der Waals free energy may be computed by assigning the free energy ~g(w, T) = ökBTln[2 sinh(hw/2kBT)] to each of the normal modes and summing over k [6, 7], ~ represents the difference between the quantity with given R and R oo. We can readily solve eq. (3) for w; the van der Waals free energy is then /v 0\’~ kBT —~--~---)—i---’ R h~pAD/kBT G(R, I) 8 (4) -~

—~

~‘

R~hwAD/kBT. 2 p 4irv~R The long range finite temperature term is modified by a factor (61/63)2 if the cylinder has dielectric constant e~and the intervening medium has dielectric constant 63. The corresponding factor for the zero temperature term is more complicated. To determine the magnitude of this force for biological systems it is clearly necessary to include an intervening medium such as water on an electrolyte which is strongly dispersive. Moreover anisotropy, finite radii and retardation effects must be included in a general treatment. It is known that these give rise to important quantitative and qualitative effects in planar geometries [8-10]. For the two cylinder problem the —~

302

,

22 May 1972

exact dispersion relation can be derived [11-13] and details of the corresponding effects on the interaction between cylindrical macromolecules will be published shortly. We are grateful to Professor D.P. Craig for bringing to our attention some early work on this problem by Coulson and Davies [141.

References [1] D.B. Chang, R.L. Cooper, J.E. Drummond and A.C. Young, Phys. Letters 37A (1971) 311. [2] E.M. Lifshitz, Soy. Phys. JETP 2 (1956) 73. [3] G. Rickayzen, Theory of superconductivity (Wiley: Interscience, New York, 1965). [4] B. Davies and B.W. Ninham, J. Chem. Phys., to be published. [51 P. Richmond and B. Davies, submitted to mt. J. Quantum Chem. [6] B.W. Ninham, V.A. Parsegian and G.H. Weiss, J. Stat. Phys. 2 (1970) 323 [7] B. Davies, Phys. Letters 37A (1971) 391. [8] B.W. Ninham and V.A. Parsegian, Biophys. J. 10 (1970) 646. [9] V.A. Parsegian and B.W. Ninham, Biophys. J. 10 (1970) 664. [10] V.A. Parsegian and B.W. Ninham, J. Coil. Interface Sci. BI (1971) 405. [11] D.J. Mitchell, B.W. Ninham and P. Richmond, submitted to Biophys. J. [12]D.J. Mitchell, B.W. Ninham and P. Richmond, J. Theoret. Biol., to be published. [13] B. Davies, B.W. Ninham and P. Richmond, submitted to J. Chem. Phys. [14] J. Coulson and B. Davies, Trans. Farad. Soc. 48 (1952) 777.