## T

t

Surface separation, [Á]

Surface separation, [Á]

Figure 5 Interactions between biological macromolecules show pronounced universality at close surface-to-surface separations (or equivalently at very large densities). Hydroxypropyl cellulose, schizophillan, different DNA salts, xanthan, and DDP bilayers at small intermolecular separations (given in terms of the separation between effective molecular surfaces of the interacting molecules) all show strong repulsive interactions decaying with about the same characteristic decay length. The log-linear plot is thus more or less a straight line. (Composite data courtesy of D. C. Rau.) See the color insert for a color version of this figure.

the case of DNA, where the ''surface'' structure has a characteristic scale of 1 to 2 A, we realize that the hydration decay length in this case would be almost entirely determined by the surface structure and not the bulk solvent properties. Given the experimentally determined variety of forces between phospholipids (20), it is indeed quite possible that even in the simplest cases the measured decay lengths are not only those of the water solvent itself, but also include the surface properties via the characteristic scale of the surface ordering lH.

The other important facet of this theory is that it predicts that in certain circumstances the hydration forces can become attractive (11). This is particularly important in the case of interacting DNA molecules where this hydration attraction connected with condensing agents can hold DNAs into an ordered array, even though the van der Waals forces themselves would be unable to accomplish that (22). This attraction is always an outcome of nonhomogeneous surface ordering and arises in situations where apposing surfaces have complementary checkerboard-like order (11). Unfortunately, in this situation, many mechanisms can contribute to attractions; therefore, it is difficult to argue for one strongest contribution.

2. Electrostatic Forces

Electrostatic forces between charged colloid bodies are among the key components of the force equilibria in (bio)colloid sys-

Figure 6 The hydration force. Marcelja and Radic (19) introduced an order parameter P that would capture the local condition, or local ordering, of solvent molecules between the surfaces. We represent it as an arrow (that has magnitude and direction) on each water molecule that is trapped between the two apposing surfaces and is being acted upon by the surface fields, depicted schematically with a bold line below each of the three drawings. Minimizing the energy corresponding to a spatial profile of P, leads to a configuration where P points (for example) away from both surfaces, and there is thus mismatch at the midplane (the dotted line below the leftmost drawing). The entropy would favor completely disordered configurations with no net value of P (the dotted line below the rightmost drawing). The free energy strikes a compromise between the two extrema, leading to a smooth profile of P, varying continuously as one goes from one surface to the other (the dotted line below the bottom drawing). As the two surfaces approach the nonmonotonic profile of the order parameter P leads to repulsive forces between them. See the color insert for a color version of this figure.

tems (23). At larger separations, they are the only forces that can counteract van der Waals attractions and thus stabilize colloid assembly. The crucial role of the electrostatic interactions in (bio)colloid systems is well documented and explored, following the seminal realization of Bernal and Fankuchen (24) that electrostatic interaction is the stabilizing force in tobacco mosaic virus (TMV) arrays.

Although the salient features of electrostatic interactions of fixed charges in a sea of mobile countercharges and salt ions are intuitively straightforward to understand, they are difficult to evaluate. These difficulties are clearly displayed by the early ambiguities in the sign of electrostatic interactions between two equally charged bodies that were first claimed to be attractive (Levine), then repulsive (Verwey-Overbeek), and finally that they were usually repulsive except if the coun-terions or the salt ions are of higher valency (25).

Here we introduce the electrostatic interaction on an intuitive footing (Fig. 7). Assume we have two equally charged bodies with counterions in-between. Clearly the minimum of electrostatic energy WE (28), which for the electrostatic field configuration at the spatial position r, E(r), is proportional to the integral of E2(r) over the whole space where one has nonzero electrostatic field, would correspond to adsorption of counterions to the charges leading to their complete neutralization. The equilibrium electrostatic field would thus be entirely concentrated next to the surface. However, at finite temperatures, it is not the electrostatic energy but rather the free energy (26), F = WE - TS, also containing the entropy S of the counterion distribution, that should be minimized. The entropy of the mobile particles with the local density p;(r) [we assume there are more than 1 species of mobile particles, (e.g., counterions and salt ions) tracked through the index i] is taken as an ideal gas entropy (26), which is proportional to the volume integral of 2i[pi(r)ln(pi(r)/io) - (pi(r)-pio)], where is the density of the mobile charges in a reservoir that is in chemical equilibrium with the confined system under investigation. Entropy by itself would clearly lead to a uniform distribution of counterions between the charged bodies, pi(r) = pi0, whereas together with the electrostatic energy it obviously leads to a nonmonotonic profile of the mobile charge distribution between the surfaces, minimizing the total free energy of the mobile ions.

The above discussion, although far from being rigorous, contains the important theoretical underpinnings known as the Poisson-Boltzmann theory (27). To arrive at the central equation corresponding to the core of this theory, one simply has to formally minimize the free energy F = WE - TS, just as in the case of structural interactions, together with the basic electrostatic equation (28) (the Poisson equation) that connects the sources of the electrostatic field with the charge densities of different ionic species. The standard procedure is now to minimize the free energy, take into account the Poisson equation, and what follows is the well-known Poisson-Boltz-mann equation, the solution of which gives the nonuniform profile of the mobile charges between the surfaces with fixed charges. This equation can be solved explicitly for some particularly simple geometries (27). For two charged planar surfaces, the solution gives a screened electrostatic potential that decays exponentially away from the walls. It is thus smallest in the middle of the region between the surfaces and largest at the surfaces. The spatial variation of the electrostatic interaction is just as in the case of structural interactions described with a characteristic decay length, dubbed the Debye length in this case, which for uni-uni valent salts assumes the value

Figure 7 A pictorial exposition of the main ideas behind the Poisson-Boltzmann theory of electrostatic interactions between (bio)colloidal surfaces. Electrostatic energy by itself would favor adsorption of counterions (white circles) to the oppositely charged surfaces (black circles). The equilibrium profile of the counterions in this case is presented by the dotted line below the leftmost drawing. Entropy, to the contrary, favors a completely disordered configuration (i.e., a uniform distribution of counterions between the surfaces), presented by the dotted line below the rightmost drawing. The free energy works a compromise between the two principles leading to a nonmonotonic profile of the counterion density (25), varying smoothly in the intersurface region. As the two surfaces are brought close, the overlapping counterion distributions originating at the fixed charge at the surfaces (the bold line below each drawing) create repulsive forces between them. See the color insert for a color version of this figure.

of \D = 3 A/V7, where I is the ionic strength of the salt in moles per liter. A 0.1 molar solution of uni-uni valent salt, such as NaCl, would thus have the characteristic decay length of about 9.5 A. Beyond this separation, the charged bodies no longer feel each other. By adding or removing salt from the bathing solution, we are thus able to regulate the range of electrostatic interactions.

The exponential decay of the electrostatic field away from the charged surfaces with a characteristic length, independent (to the lowest order) of the surface charge, is one of the most important results of the Poisson-Boltzmann theory.

Obviously, as the surfaces come closer together, their decaying electrostatic potentials begin to interpenetrate (25). The consequence of this interpenetration is a repulsive force between the surfaces that again decays exponentially with the intersurface separation and a characteristic length again equal to the Debye length. For two planar surfaces at a separation D, bearing sufficiently small charges, characterized by the surface charge density ct, so that the ensuing electrostatic potential is never larger than kBT/e, where kB is Boltzmann's constant and e is the elementary electron charge, one can derive (27) for the interaction free energy per unit surface area F(D) the expression F(D) ~ ct2 exp( —D/\d). Obviously the typical magnitude of the electrostatic interaction in different systems depends on the magnitude of the surface charge. It would not be unusual in lipids to have surface charge densities in the range of 1 elementary charge per 50 to 100 A2 surface area (29). For this range of surface charge densitities, the constant prefactor in the expression for the osmotic pressure would be of the order 0.4 to 1.2 x 107 N/m.

The same type of analysis would also apply to two charged cylindrical bodies (e.g., two molecules of DNA) interacting across an electrolyte solution. What one evaluates in this case is the interaction free energy per unit length of the cylinders (30), g(R), where R is the separation between the cylinders that can be obtained in the approximate form g(R) ~ exp( - R/\d). It is actually possible to also get an explicit form (30) of the interaction energy between two cylinders even if they are skewed by an angle 0 between them. In this case, the relevant quantity is the interaction free energy itself (if 0 is nonzero, then the interaction energy does not scale with the length of the molecules) that can be obtained in a closed form as F(R,0) ~ ^2\d R1/2 exp( — R/\D)/sin(0).

The predictions for the forces between charged colloid bodies have been reasonably well borne out for electrolyte solutions of uni-uni valent salts (31). In that case, there is near quantitative agreement between theory and experiment. However, for higher valency salts, the Poisson-Boltzmann theory does not only give the wrong numerical values for the strength of the electrostatic interactions, but also misses their sign. In higher valency salts, the correlations among mobile charges between charged colloid bodies due to thermal fluctuations in their mean concentration lead effectively to attractive interactions (32) that are in many respects similar to the van der Waals forces that are analyzed next.

3. van der Waals Forces van der Waals charge fluctuation forces are special in the sense that they are a consequence of thermodynamic and quantum mechanical fluctuations of the electromagnetic fields (15). They exist even if the average charge, dipole moment, or higher multipole moments on the colloid bodies are zero. This is in stark contrast to electrostatic forces that require a net charge or a net polarization to drive the interaction. This also signifies that the van der Waals forces are much more general and ubiquitous than any other force between colloid bodies (9).

There are many different approaches to the van der Waals forces (15,33). For interacting molecules, one can distinguish different contributions to the van der Waals force, stemming from thermally averaged dipole-dipole potentials (the Kee-som interaction), dipole-induced dipole interactions (the Debye interaction), and induced dipole-induced dipole interactions (the London interaction) (34). They are all attractive and their respective interaction energy decays as the sixth power of the separation between the interacting molecules. The magnitude of the interaction energy depends on the electromagnetic absorption (dispersion) spectrum of interacting bodies, thus also the term dispersion forces.

For large colloidal bodies composed of many molecules, the calculation of the total van der Waals interactions is no trivial matter (15), even if we know the interactions between individual molecules composing the bodies. Hamaker assumed that one can simply add the interactions between composing molecules in a pairwise manner. It turned out that this was a very crude and simplistic approach to van der Waals forces in colloidal systems because it does not take into account the highly nonlinear nature of the van der Waals interactions in condensed media. Molecules in a condensed body interact among themselves, thus changing their properties (c.f. their dispersion spectrum) that in their turn modify the van der Waals forces between them.

Lifshitz, following work of Casimir (9,15), realized how to circumvent this difficulty and formulated the theory of van der Waals forces in a way that already includes all these non-linearities. The main assumption of this theory is that the presence of dielectric discontinuities as in colloid surfaces, modifies the spectrum of electromagnetic field modes between these surfaces (Fig. 8). As the separation between colloid bodies varies, so do the eigenmode frequencies of the electromagnetic field between and within the colloid bodies. It is possible to deduce the change in the free energy of the electromagnetic modes due to the changes in the separation between colloid bodies coupled to their dispersion spectral characteristics (35).

Based on the work of Lifshitz, it is now clear that the van der Waals interaction energy is just the change of the free energy of field harmonic oscillators at a particular eigenmode frequency m as a function of the separation between the interacting bodies D and temperature T, m = m (D,T). With this equivalence in mind, it is quite straightforward to calculate the van der Waals interaction free energy between two planar surfaces at a separation D and temperature T; the dielectric permittivity between the two surfaces, e and within the surfaces, e', must both be known as a function of the frequency of the electromagnetic field (35). This is a consequence of the fact that, in general, the dielectric media comprising the surfaces as well as the space between them are dispersive, meaning that their dielectric permittivities depend on frequency of the electromagnetic field [i.e., e = e(m)]. With this in mind one can derive the interaction free energy per unit surface area of the interacting surfaces in the form F(D) = A/12^D2, where the s.c. Hamaker coefficient A depends on the difference between the dielectric permittivities of the interacting materials at different imaginary frequencies. It can be

Figure 8 A pictorial introduction to the theory of Lifshitz-van der Waals forces between colloid bodies. Empty space is alive with electromagnetic (EM) field modes that are excited by thermal as well as quantum mechanical fluctuations. Their frequency is unconstrained and follows the black body radiation law. Between dielectric bodies, only those EM modes survive that can fit into a confined geometry. As the width of the space between the bodies varies, so do the allowed EM mode frequencies. Every mode can be treated as a separate harmonic oscillator, each contributing to the free energy of the system. Because this free energy depends on the frequency of the modes, that in turn depend on the separation between the bodies, the total free energy of the EM modes depends on the separation between the bodies. This is an intuitive description of the Lifshitz-van der Waals force (15). See the color insert for a color version of this figure.

Figure 8 A pictorial introduction to the theory of Lifshitz-van der Waals forces between colloid bodies. Empty space is alive with electromagnetic (EM) field modes that are excited by thermal as well as quantum mechanical fluctuations. Their frequency is unconstrained and follows the black body radiation law. Between dielectric bodies, only those EM modes survive that can fit into a confined geometry. As the width of the space between the bodies varies, so do the allowed EM mode frequencies. Every mode can be treated as a separate harmonic oscillator, each contributing to the free energy of the system. Because this free energy depends on the frequency of the modes, that in turn depend on the separation between the bodies, the total free energy of the EM modes depends on the separation between the bodies. This is an intuitive description of the Lifshitz-van der Waals force (15). See the color insert for a color version of this figure.

in general split into two terms: the first term in the Hamaker coefficient is due to thermodynamic fluctuations, such as Brownian rotations of the dipoles of the molecules composing the media or the averaged dipole-induced dipole forces and depends on the static (m = 0) dielectric response of the interacting media, whereas the second term is purely quantum mechanical in nature (15). The imaginary argument of the dielectric constants is not that odd because e(i£) is an even function of £, which makes e(i£) also a purely real quantity (35).

To evaluate the magnitude of the van der Waals forces, one thus has to know the dielectric dispersion e(M) of all the media involved. This is no simple task and can be accomplished only for very few materials (34). Experiments seem to be a much more straightforward way to proceed. The values for the Hamaker coefficients of different materials interacting across water are between 0.3 and 2.0 X 10-20 J. Specifically for lipids, the Hamaker constants are quite close to theoretical expectations except for the phosphatidylethanolamines that show much larger attractive interactions probably due to head-group alignment (31). Evidence from direct measurements of attractive contact energies as well as direct force measurements suggest that van der Waals forces are more than adequate to provide attraction between bilayers for them to form multilamellar systems (36).

For cylinders the same type of argument applies, except that due to the geometry the calculations are a bit more tedious (37). Here the relevant quantity is not the free energy per unit area but the interaction free energy per unit length of the two cylinders of radius a, g(R), considered to be parallel at a separation R. The calculation (38) leads to the following form g(R) ~ A a4/R5, where the constant A again depends on the differences between dielectric permittivities Ey, the parallel, and e^ , the perpendicular components of the dielectric permittivity of the dielectric material of the cylinders, and Em, the dielectric permittivity of the bathing medium.

If, however, the 2 interacting cylinders are skewed, then the interaction free energy G(R, 0), this time not per length, is obtained (38) in the form G(R) ~ (A + Bcos20)(a4/R4sin0.) The constants A and B describe the dielectric mismatch between the cylinder and the bathing medium at different imaginary frequencies. The same correspondence between the ther-modynamic and quantum mechanical parts of the interactions as for two parallel cylinders also applies to this case. Clearly, the van der Waals force between two cylinders has a profound angular dependence that in general creates torques between the two interacting molecules.

Taking the numerical values of the dielectric permittivities for two interacting DNA molecules, one can calculate that the van der Waals forces are quite small, typically 1 to 2 orders of magnitude smaller than the electrostatic repulsions between them, and in general cannot hold the DNAs together in an ordered array. Other forces, leading to condensation phenomena in DNA (10) clearly have to be added to the total force balance in order to get a stable array. There is as yet still no consensus on the exact nature of these additional attractions. It seems that they are due to the fluctuations of counterion atmosphere close to the molecules.

### 4. The DLVO Model

The popular Derjaguin-Landau-Verwey-Overbeek (DLVO) (9,25) model assumes that electrostatic double-layer and van der Waals interactions govern colloid stability. Applied with a piety not anticipated by its founders, this model actually does work rather well in surprisingly many cases. Direct osmotic stress measurements of forces between lipid bilayers show that at separations less than ~ 10 A there are qualitative deviations from DLVO thinking (39). For micron-size objects and for macromolecules at greater separations, electrostatic double-layer forces and sometimes van der Waals forces tell us what we need to know about interactions governing movement and packing.

### 5. Geometric Effects

Forces between macromolecular surfaces are most easily analyzed in plane parallel geometry. Because most of the interacting colloid surfaces are not planar, one has either to evaluate molecular interactions for each particular geometry or to devise a way to connect the forces between planar surfaces with forces between surfaces of a more general shape. The Derja-guin approximation (9) assumes that interactions between curved bodies can be decomposed into interactions between small plane-parallel sections of the curved bodies (Fig. 9). The total interaction between curved bodies would be thus equal to a sum where each term corresponds to a partial interaction between quasi-plane-parallel sections of the two bodies. This idea can be given a completely rigorous form and leads to a connection between the interaction free energy per unit area of two interacting planar surfaces, F(D), and the force acting between two spheres at minimal separation D, f(D), 1 with the mean radius of curvature R1 and the other 1 with R2. The formal equivalence can be written as follows, f(D) = 2^ (R1R2/(R1 + R2)) F(D). A similar equation can also be obtained for 2 cylinders in the form, f(D) = 2^(R1R2)1/2 F(D).

These approximate relations clearly make the problem of calculating interactions between bodies of general shape tractable. The only caveat here is that the radii of curvature should be much larger than the proximal separation between the two interacting bodies, effectively limiting the Derjaguin approximation to sufficiently small separations.

Figure 9 The Derjaguin approximation. To formulate forces between oppositely curved bodies (e.g., cylinders, spheres, etc.) is very difficult, but it is often possible to use an approximate procedure. Two curved bodies (two spheres of unequal radii in this case) are approximated by a succession of planar sections, interactions between which can be calculated relatively easy. The total interaction between curved bodies is obtained through a summation over these planar sections.

Figure 9 The Derjaguin approximation. To formulate forces between oppositely curved bodies (e.g., cylinders, spheres, etc.) is very difficult, but it is often possible to use an approximate procedure. Two curved bodies (two spheres of unequal radii in this case) are approximated by a succession of planar sections, interactions between which can be calculated relatively easy. The total interaction between curved bodies is obtained through a summation over these planar sections.

Using the Derjaguin formula or evaluating the interaction energy explicitly for those geometries for which this indeed is not an insurmountable task, one can now obtain a whole zoo of DLVO expressions for different interaction geometries (Fig. 10). The salient features of all these expressions are that the total interaction free energy always has a primary minimum, that can only be eliminated by strong short-range hydration forces, and a secondary minimum due to the compensation of screened electrostatic repulsion and van der Waals-Lifshitz attraction. The position of the secondary minimum depends as much on the parameters of the forces (Ha-maker constant, fixed charges, and ionic strength) as on the interaction geometry. Generally, the range of interaction between the bodies of different shapes is inversely proportional to their radii of curvature.

Thus, the longest-range forces are observed between planar bodies, and the shortest between small (pointlike) bodies.

Figure 10 A menagerie of DLVO interaction expressions for different geometries most commonly encountered in biological milieu. Two small particles, a particle and a wall, 2 parallel cylinders, a cylinder close to a wall, 2 skewed cylinders and 2 walls. The DLVO interaction free energy is always composed of a repulsive electrostatic part (calculated from a linearized Poisson-Bolt-zmann theory) and an attractive van der Waals part. Charge: e, charge per unit length of a cylinder: charge per unit surface area of a wall: ct, C is a geometry-dependent constant, e the dielectric constant, k the inverse Debye length, and p the density of the wall material. The functions K0(x) (the Bessel function K0) and Ei(x) (the exponential integral function) both depend essentially exponentially on their respective argument.

Figure 10 A menagerie of DLVO interaction expressions for different geometries most commonly encountered in biological milieu. Two small particles, a particle and a wall, 2 parallel cylinders, a cylinder close to a wall, 2 skewed cylinders and 2 walls. The DLVO interaction free energy is always composed of a repulsive electrostatic part (calculated from a linearized Poisson-Bolt-zmann theory) and an attractive van der Waals part. Charge: e, charge per unit length of a cylinder: charge per unit surface area of a wall: ct, C is a geometry-dependent constant, e the dielectric constant, k the inverse Debye length, and p the density of the wall material. The functions K0(x) (the Bessel function K0) and Ei(x) (the exponential integral function) both depend essentially exponentially on their respective argument.

What we have not indicated in Fig. 7 is that the interaction energy between two cylindrical bodies, skewed at a general angle 0 and not just for parallel or crossed configurations, can be obtained in an explicit form. It follows simply from these results that the configuration of two interacting rods with minimal interaction energy is the one corresponding to 0 = ^/2 (i.e., corresponding to crossed rods).

### 6. Fluctuation Forces

The term ''fluctuation forces'' is a bit misleading in this context because clearly van der Waals forces are already fluctuation forces. What we have in mind is thus a generalization of the van der Waals forces to situations where the fluctuating quantities are not electromagnetic fields but other quantities subject to thermal fluctuations. No general observation as to the sign of these interactions can be made, they can be either repulsive or attractive and are as a rule of thumb comparable in magnitude to the van der Waals forces.

The most important and ubiquitous force in this category is the undulation or Helfrich force (41). It has a very simple origin and operates among any type of deformable bodies as long as their curvature moduli are small enough (comparable to thermal energies). It was shown to be important for multila-mellar lipid arrays (41) as well as in hexagonal polyelectrolyte arrays (42) (Fig. 11).

The mechanism is simple. The shape of deformable bodies fluctuates because of thermal agitation (Brownian motion) (26). If the bodies are close to each other, the conformational fluctuations of one will be constrained by the fluctuations of its neighbors. Thermal motion makes the bodies bump into each other, which creates spikes of repulsive force between them. The average of this force is smooth and decays continuously with the mean separation between the bodies.

One can estimate this steric interaction for multilamellar lipid systems and for condensed arrays of cylindrical polymers (Fig. 11). The only quantity entering this calculation is the elastic energy of a single bilayer that can be written as the square of the average curvature of the surface, summed over the whole area of the surface, multiplied by the elastic modulus of the membrane, KC. KC is usually between 10 and 50 kBT (43) for different lipid membranes. If the instantaneous deviation of the membrane from its overall planar shape in the plane is now introduced as u, the presence of neighboring membranes introduces a constraint on the fluctuations of u that basically demands, that the average of the square of u must be proportional to D2, where D is the average separation between the membranes in a multilamellar stack. Thus, we should have u2 ~ D2. The free energy associated with this constraint can now be derived in the form (40) F(D) ~ (kBT)2/ (KCD2), and is seen to decay in inverse proportion to the separation between bilayers squared.

It has thus obviously the same dependence on D as the van der Waals force. This is, however, not a general feature of undulation interactions as the next example clearly shows. Also, we only indicated the general proportionality of the interaction energy. Calculation of the prefactors can be a difficult (44), especially because the elastic bodies usually do not

Figure 11 Thermally excited conformational fluctuations in a multilamellar membrane array (small molecules are waters and long-chain molecules are phospholipids) or in a tightly packed polyelectrolyte chain array (the figure represents a hexagonally packed DNA array) lead to collisions between membranes or polyelectrolyte chains. These collisions contribute an additional repulsive contribution to the total osmotic pressure in the array, a repulsion that depends on the average spacing between the fluctuating objects. See the color insert for a color version of this figure.

Figure 11 Thermally excited conformational fluctuations in a multilamellar membrane array (small molecules are waters and long-chain molecules are phospholipids) or in a tightly packed polyelectrolyte chain array (the figure represents a hexagonally packed DNA array) lead to collisions between membranes or polyelectrolyte chains. These collisions contribute an additional repulsive contribution to the total osmotic pressure in the array, a repulsion that depends on the average spacing between the fluctuating objects. See the color insert for a color version of this figure.

the elastic energy can be written similarly to the membrane case as the square of the local curvature of the polymer, multiplied by the elastic modulus of the polymer, integrated over its whole length. The elastic modulus Kc is usually expressed through a persistence length Lp = KJ(kBT). The value of the persistence length tells us how long a polymer can be before the thermal motion forces it to fluctuate wildly. For DNA, this length is about 50 nm. However, it spans the whole range of values between about 10 nmfor hyaluronic acid, all the way to 3 mm for microtubules. Using the same constraint for the average fluctuations of the polymer away from the straight axis, one derives for the free energy change due to this constraint the relationship F(D) ~ (kBT)/(Lp1/3D2/3) (42).

Clearly, the D dependence for this geometry is much different from the one for van der Waals force, which would be D — 5. There is thus no general connection between the van der Waals force and the undulation fluctuation force. Here again, one has to indicate that if the interaction potential between fluctuating bodies is described by a soft potential, with no discernible hard core, the fluctuation interaction can have a profoundly different dependence on the mean separation (42).

Apart from the undulation fluctuation force, there are other fluctuation forces. The most important among them appears to be the monopolar charge fluctuation force (45), recently investigated in the context of DNA condensation. It arises from transient charge fluctuations along the DNA molecule due to constant statistical redistributions of the counterion atmosphere.

The theory of charge fluctuation forces is quite intricate and mathematically demanding (46). Let us just quote a rather interesting result, viz. if two point charges interact via a ''bare'' potential V0(R), where R is the separation between them, then the effect of the thermal fluctuations in the number of counterions surrounding these charges would lead to an effective interaction of the form V(R) ~ — kBTV02(R). The fluctuation interaction in this case would thus be attractive and proportional to the square of the bare interaction.

This simple result already shows one of the salient features of the interaction potential for monopolar charge fluctuation forces, viz. it is screened with half the Debye screening length [because of V02(R)]. If there is no screening, however, the monopolar charge fluctuation force becomes the strongest and longest ranged among all fluctuation forces. It is however much less general than the related van der Waals force, and it is still not clear what the detailed conditions should be for its appearance, the main difficulty being the question of whether charge fluctuations in the counterion atmosphere are constrained.

interact with idealized hard repulsions but rather through soft potentials that have both attractive as well as repulsive regimes.

The same line of thought can now be applied to flexible polymers in a condensed array (42). This system is a 1-dimensional analog of the multilamellar membrane system. For polymers,

### 7. Lessons

Molecular forces apparently convey a variety that is surprising considering that they are all to some extent or another just a variant of electrostatic interactions. Quantum and thermal fluctuations apparently modify the underlying electrostatics, leading to qualitatively novel and unexpected features. The zoo of forces obtained in this way is what one has to deal with and understand when trying to make them work for us.

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