J. Phys. Chem. B 2009, 113, 3941–3946
3941
Depletion Theory and the Precipitation of Protein by Polymer† Theo Odijk‡ Complex Fluids Theory, KluyVer Laboratory of Biotechnology, Delft UniVersity of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands ReceiVed: July 29, 2008; ReVised Manuscript ReceiVed: December 12, 2008
The depletion theory of nanoparticles immersed in a semidilute polymer solution is reinterpreted in terms of depleted chains of polymer segments. Limitations and extensions of mean-field and scaling theories are discussed. An explicit expression for the interaction between two small spheres is also reviewed. The depletion free energy for a particle of general shape is given in terms of the capacitance or effective Stokes radius. This affords a reasonable explanation for the effect of polymer on protein precipitation. Introduction
F1 =
It is difficult to set up theories of the polymer distribution near surfaces because the computed polymer inhomogeneity is sensitive to the nature of the approximations introduced. PierreGilles de Gennes devoted considerable time and effort to trying to understand these problems starting with his survey1 of forty years ago, which is still illuminating to read, and culminating in papers containing now classic ideas like self-similarity2 and the proximal exponent.3 His concise, powerful note4 on polymer depletion by a small sphere was strangely neglected by the colloid community for a long time until the peculiarity of depletion on nanoscales was reassessed merely a decade ago.5,6 There has been a flurry of activity in the statistical physics of nanocolloids immersed in polymer solutions in what has been termed the protein limit (see, e.g., refs 7-11 and references therein). Here, I would like to emphasize simple aspects of polymer depletion by nanoparticles in the spirit of ref 12. Let us recall the argument introduced by de Gennes4 to compute the free energy of depletion involved in immersing a nanosphere into a semidilute polymer solution. The solvent is not just “good” but needs to be really “excellent” (see below); i.e., the excluded-volume β between the Kuhn segments equals A3 where A is the segment length. If the radius a of the sphere is larger than A, it is plausible to assume that a and the polymer correlation length ξ are the only relevant length scales in the problem. The latter is given by13
( ξa )
4/3
kBT
(2)
Here, kB is Boltzmann’s constant and T is the temperature. In the following I discuss several problems concerning the depletion interaction between nanospheres and polymer and its usefulness in explaining the precipitation of proteins by polyethylene glycol. The arguments leading to eq 2 are rather abstract. For instance, it is not clear how the depleted segments are physically involved in the work of depletion. Here, I invoke a different kind of scaling argument to derive the free energy. This has two advantages: we may write down inductive expressions that are independent of the magnitude of the excluded-volume effect, and it becomes possible to derive scaling relations even when there are more than two relevant length scales. For intermediate solvents, the polymer may be regarded as quasi-ideal. A limitation of the resulting selfconsistent field (SCF) picture is here pointed out. Hanke, Eisenriegler and Dietrich14 have computed the interaction between two nanospheres in an ideal semidilute polymer by field theory. This interaction is reevaluated by the method of images and involves a series expansion with interesting properties that are discussed in the Appendix. In the quasi-ideal limit, the depletion energy may be calculated for particles of arbitrary shape. This enables one to gauge the solubility of proteins as a function of added inert polymer which is compared with experiment. Scaling in Terms of Segments Depleted
ξ)A
-5/4
c0-3/4
(1)
where the concentration c0 is the number of polymer segments per unit volume. For a nanosphere dissolved in a semidilute solution of low concentration, one readily has a , ξ. De Gennes then argues that there is a volume of order a3, independent of ξ, surrounding the sphere from which polymer is depleted.4 Hence, the number of segments depleted is of the order of a3c0. Because the free energy of depletion F1 must be proportional to this number, i.e., must be proportional to the concentration c0, one concludes that4 † ‡
Part of the “PGG (Pierre-Gilles de Gennes) Memorial Issue”. E-mail:
[email protected].
On the average, the number of polymer segments depleted from the vicinity of the sphere is a3c0, so one might be tempted to think naively that the free energy of depletion could be something like F1 = a3c0kBT. But this disregards entirely the fact that a particular segment is connected to other segments that are all on one single polymer chain because a , ξ. A depleted test segment is typically connected to h others where a = h3/5A, in view of the excluded-volume effect. Therefore, the number of degrees of freedom is reduced by a factor h
F1 = a3c0h-1kBT
(3)
which agrees with eq 2. Note that this derivation is valid in the mean for the effective number of degrees of freedom is actually
10.1021/jp806722j CCC: $40.75 2009 American Chemical Society Published on Web 02/26/2009
3942 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Odijk
less than unity in eq 3. Moreover, eq 3 is a generalization of eq 2 for the latter only applies when the excluded-volume effect is fully exerted whereas the reasoning leading to the former is independent of excluded volume. Furthermore, the particle does not need to be a sphere: under more general circumstances one may consider Vc0 segments depleted on the average from some volume V and the number h defined appropriately. The application of eq 3) to the following two examples will bear this out. Water is often the solvent of choice in experiments concerning the thermodynamic properties of polymer-nanocolloid or polymer-protein mixtures. Although water-soluble polymers dissolve readily in aqueous solution, the solvent must often be regarded as “intermediate” (β , A3) rather than “excellent”.5 The polymer chain then interacts with a nanoparticle in a quasiideal manner.12 In effect, the excluded-volume parameter z ) h1/2β/A3 may remain smaller than unity if the nanoparticle is not too large. In that case, a string of depleted segments behaves like a Gaussian chain: h = a2/A2 if the particle is a sphere. This would impose the condition12 a < A4/β, which may be easily met in practice. Equation 3 then leads to
( )
a k T ξid B
F1 ) k1A2ac0kBT ) k1
(4)
where k1 is a numerical coefficient and ξid ) 1/(A2c0) is a quasiideal correlation length. The latter relates to a state in which the polymer chains are supposed hypothetically ideal. Equation 4 was previously obtained via the route of introducing a correlation length which, however, is an ambiguous quantity when the solvent is intermediate.15 Here, its derivation is direct and ξid emerges as a derived quantity. Equation 4 may also be obtained within a self-consistent field picture which, in addition, allows us to discuss the segment density c(r b) where b r is the vector distance from the center of the sphere. Within the quasi-ideal approximation, one may argue in favor of dropping excluded-volume terms in the SCF picture b) where at a , ξ, which leads to a Laplace equation12 for Ψ(r b)2 the segment density c(r b) ) c0Ψ(r
∆Ψ(b) r )0
(5)
The solution to eq 5 with Ψ ) 0 at the boundary of the sphere is
Ψ(r) ) 1 -
a r
(6)
This leads to eq 4 again with a value k1 ) 2π/3 for the coefficient.16 At this juncture, it may be useful to point out some limitations of the SCF approach. At some distance r* from the sphere, excluded-volume effects must come into play in eq 5 but then the argumentation for the validity of a mean-field approach also breaks down. Correlations are no longer negligible then. From renormalization theory,6 we know that in the case β ) A3 we have asymptotically
Ψ(r) - 1 ∼
a r
x
()
(7)
with an exponent set equal to 4/3 instead of 1.30,6 in line with an expression quoted below (eq 10). When the solvent is intermediate (β , A3), we again adduce reasoning based on the parameter z above to show that eq 6 is only valid for r < r* ) A4/β. Beyond r*, eq 6 must join smoothly to
Ψ(r) = 1 -
aA4/3 β1/3r4/3
(8)
This reduces to eq 7 if a > r*. In the case of excellent solvent (β ) A3), the SCF theory for the polymer distribution near a surface is incorrect for a number of reasons. First, the correlation length ξ is given by a wrong power law, which de Gennes2 proposed to amend by suitably adjusting the exponent in the excluded-volume term in the SCF equation. Nevertheless, the equation remains purely diffusive so it cannot mimic, for instance, the segment distribution about a sphere expressed by eq 7. A further amendment might be to introduce a fractional SCF equation. In the vicinity of the sphere, this would reduce to
r1-D
d D-1 -Θ dΨ r r )0 dr dr
(
)
(9)
which is a stationary generalized diffusion equation of fractal order D and with a diffusion coefficient r-Θ. (For a discussion of fractional diffusion equations, see ref 17.) Equation 7 imposes the constraint D - Θ ) 10/3 on the exponents D and Θ, which leaves one of them to be suitably chosen. However, a fractional SCF equation is still not entirely satisfactory. Density fluctuations are merely accounted for in a preaveraged sense. Thus, we lack a fully quantitative theory of polymer depletion about a sphere at all distances and at all excluded-volume strengths. The notion of depleted segments also allows us to introduce a relation between the depleted concentration about a sphere c(r) - c0 and the pair correlation function g(r) pertaining to a single chain in the bulk13 (see eqs 6 and 7)
c(r) - c0 = -c0a3h-1g(r)
(10)
This is universal and independent of the strength of the excluded-volume effect. The structure of the depletion hole is analogous to the pair correlation structure of the polymer removed provided the latter is normalized by the effective number of degrees of freedom depleted by the sphere. In the second example of the application of eq 3, I compute the attractive force between two almost touching nanospheres immersed in a semidilute polymer. If the separation between the sphere centers is R so the separation between their surfaces is s ) R - 2a (s , a), their area of interaction is of the order sa ≡ λ2. Thus, the polymer is depleted from an additional volume V = s2a compared with the case when the two spheres actually touch. If the solvent is excellent, a depleted chain of h segments may be viewed as a string of blobs each of size s = m3/5a and consisting of m segments and the string has dimension λ =(h/m)3/4s = h3/4A5/4s-1/4, in line with the excluded-volume effect pertaining to two dimensions. The analogue of eq 3 then yields
∆F =Vc0h-1kBT =A5/3a1/3sc0kBT
(11)
Depletion Theory and the Precipitation of Protein
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3943
TABLE 1: Scaled Depletion Energy between Two Nanospheres as a Function of Their Separation R/2a
1
1.1
1.2
1.3
1.5
2
4
6
10
2 - C2/(4πa)
0.614
0.585
0.558
0.532
0.486
0.395
0.222
0.1538
0.0952
In the quasi-ideal approximation (a < A4/β), similar reasoning but now applied to Gaussian chains leads to
∆F = A2sc0kBT
(12)
In both cases the force of attraction -d∆F/ds is independent of the separation, at least when the latter is small. Interaction between Two Spheres in the Quasi-Ideal Limit It is of interest to consider the interaction between two nanospheres separated at distance R in the quasi-ideal limit. Hanke et al.14 have derived this within a field theory for a Gaussian model at its critical point which is equivalent to the ideal-chain limit. It is useful to consider the case of concentric spheres first and then to employ a particular conformal mapping to evaluate the interaction between nonoverlapping spheres.18 Their result (see eq 3.18 in ref 14) within our context may be written as
F2 1 ) A2c0(C2 - 8πa) kBT 6
(13)
∞
C2 ) 16πa sinh τ
∑
1 1 + exp(2l + 1)τ l)0
2s 1 s cosh 2τ ) 1 + + a 2 a
2
()
(14)
(15)
Here, I have used the fact that the free energy of depletion F2 is directly related to the capacitance C2 of the two particles, as has been shown via Green’s first identity.12 The units of capacitance have been chosen in such a way that C1 ) 4πa for a single sphere. Nevertheless, the interaction between two spheres has never been considered a trivial problem19 and its evaluation at close separations has been a focus of attention in various fields.20 It proves interesting to rederive eq 14 by the method of images.21 One needs to solve eq 5 with boundary conditions Ψ ) 0 at their surfaces and Ψ ) 1 at infinity. In the electrostatic analogy, the two spheres are grounded. A positive test charge is placed at the center of the first sphere. The potential Ψ on the second sphere is brought to zero by alternately adding appropriate image charges on the center line between the two spheres: these are negative within the second sphere but positive in the first. In this case, the capacitance of the first sphere would be ∞
C1 ) 4πa sinh τ
∑ sinh(2n1 - 1)τ
(16)
n)1
cosh τ )
R 2a
(17)
as outlined in a lengthy calculation in ref 21. This expression ultimately arises from a summation of image charges within
the first sphere each denoted by index n. Next, a similar test charge of positive sign is placed at the center of the second sphere. The potential on the first sphere is rendered uniform by again adding image charges on the center line, but the signs are interchanged. The potential on both spheres is now identical and can be set equal to zero by subtracting a constant. The “even” terms absent from eq 16 represent the additional image terms needed to keep both spheres grounded though they are of opposite sign. Thus, the capacitance of the two spheres becomes ∞
C2 ) 8πa sinh τ
n+1
∑ (-1) sinh nτ
(18)
n)1
In terms of a single τ eqs 15 and 17 are indeed identical. But it has not been possible to prove the identity of eqs 14 and 18, though an exhaustive numerical analysis shows they must be (E. Eisenriegler, private communication). The series represented by eq 18 is interesting in its own right and some analysis of it is presented in the Appendix. An expansion of eq 14 at small separations has been given by Hanke et al.14
[
C2 ) 8πa ln 2 +
( 61 ln 2 - 241 ) as ]
(19)
Equations 13 and 19 are consistent with the scaling estimate given by eq 12. The function 2 - C2/(4πa) is very slowly varying (see Table 1) and has a maximum equal to 2 - 2 ln 2 as the spheres touch. Inevitably, eq 11 breaks down at separations R beyond r* as the excluded-volume effect starts to play a role. Nevertheless, the decay of F2 with R is not fast enough for a finite computation of the second virial coefficient. A cutoff at R = ξ must be introduced, as has been discussed by Eisenriegler22 in the excellent-solvent case (β ) A3). The opposite limit of large spheres near ideal polymers has been addressed by Tuinier et al.23 Protein Precipitation by Polymer How well does an expression like eq 4 work? An important phenomenon in bioengineering is the precipitation of proteins by inert polymer. This has been studied for a long time,24-28 but the most thorough quantitative study is that of Atha and Ingham.29 They determined the solubility S of a host of proteins as a function of the polyethylene glycol (PEG) added to the suspension. The concentrations of PEG were well into the semidilute regime. The logarithm of the solubility turns out to be a purely linear function of the PEG concentration and the resulting slopes are a monotone function of the protein radius, the latter being an equivalent quantity derived from the diffusion coefficient via the Stokes-Einstein relation (see Figure 1). The proteins may deviate significantly from an ideally spherical shape, so let us account for this fact. First, we know that for a compact particle of general shape, eq 4 may be generalized to
F1 )
A2 Cck T 6 1 0 B
(20)
3944 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Odijk
Figure 1. Slope of the linear plot of 10log S versus w as a function of the effective Stokes radius of various proteins. The data are taken from ref 29. The proteins are lysozyme (Lys), R-lactalbumin (R-Lac), chymotrypsin (Chy), human serum albumin (HSA), human γ-globulin (IgG), aldolase (Ald), thyroglobulin (Thy), and human fibrinogen (Fib).
in terms of the capacitance C1.12 Next, Hubbard and Douglas have shown by analysis and by simulation30,31 that the Brownian friction coefficient of a particle of general shape is directly proportional to its capacitance to an excellent approximation. (It is also good to recall that the capacitance itself is essentially proportional the particle’s surface area.32,33) Accordingly, I set C1 ) 4πas. The chemical potential µ of a protein in a saturated solution containing inert polymer is given by
µ ) const + kBT ln S + F1
(21)
The protein solution is assumed to be ideal. Because the solution is in equilibrium with protein in the form of a solid (e.g., an aggregate), µ must be a constant. Equation 21 thus predicts that ln S is proportional to the polymer density, which is indeed what is found experimentally.29 In differential form we have
limit as the degree of polymerization reaches 4000. Nevertheless, judging from their Figure 3 and Table 2, a downward shift of about 15% in the magnitude of the numerical coefficient in eq 23 might be needed to account for finite size corrections. (2) At the larger protein radii displayed in Figure 1, one expects the quasi-ideal approximation to break down. The excluded-volume effect exerts itself, so the de Gennes result eq 2 should start to apply. But the data do not bend upward away from the line. Furthermore, at larger radii, the protein limit (as , ξ) should also become tenuous but the solubility curves are remarkably linear29 so we remain far from a colloidal limit that would imply nonlinear plots. (3) Conventional attractive forces have been neglected. Proteins like lysozyme35 and streptavidin36 are known to be attracted to PEG although it is not clear how the results from these experiments impact on the depletion curve shown in Figure 1. A theory of the influence of attraction in an SCF approach has been given in ref 37. A well with a depth of about 0.5 kBT per Kuhn segment would be enough to explain the deviation of lysozyme from the depletion line in Figure 1. Such a number is quite realistic. (4) The current depletion theory is a vast improvement over the conventional approximation in the protein field where the polymer chain is viewed as an impenetrable sphere and the semidilute region is dismissed altogether. The hard-particle curve displayed in ref 29 strongly overestimates the data in Figure 1 and bends upward. (5) The solubility curves are linear, which implies that the interactions between the proteins are either absent or effectively compensated. There are substantial electrostatic forces between them at the concentrations of electrolyte added to the mixtures in the precipitation experiments.29 There are also attractive interactions of the usual type as well as the depletion interaction given by eq 13. Quasi-ideal conditions in terms of the protein interactions could apply as is known to happen in the case without polymer. A theory of the latter has been put forward recently.38 Concluding Remarks
∆ ln S 2π 2 )A as ∆c0 3
( )
(22)
which is rewritten in practical units as
( ) 2
∆10log S 24π Rg a )∆w ln 10 M s )-0.036as
(23)
In eq 23, as is expressed in nm, w in (% g)/mL, and the quantity Rg2/M (the square radius of gyration of the polymer divided by its molar mass) is given in units (nm2 mol)/g. The radius Rg must be measured in the Θ state, and this has been carried out for PEG by Kawaguchi et al.34 who determined Rg2/M to be 0.001 11 (nm2 mol)/g. I have plotted eq 23 in Figure 1. On the whole the simple SCF model works reasonably well though the comparison is subject to the following qualifications. (1) The polymer used by Atha and Ingham,29 PEG4000, in the solubility experiments leading to the results in Figure 1 is not as long as one would ideally want it to be. They did study the dependence of protein solubility on the molar mass of added PEG, and the curves do seem to start to reach an asymptotic
In summary, the quasi-ideal depletion theory leading to eq 23 gives a much better description of the solubility data than one would expect on the basis of the reservations outlined above. Here and in previous work,39 we also conclude that linear depletion laws for thermodynamic quantities are valid up to surprisingly high protein concentrations. The linearity in polymer concentration is in accord with scaling and SCF arguments for nanoparticles. The quasi-ideality with respect to both polymer and protein extends well beyond its range of validity and it will be interesting to see if it can be argued on theoretical grounds in future work. I met Pierre-Gilles de Gennes in 1978 when he became Lorentz professor in the Lorentz Institute for Theoretical Physics at the University of Leiden. I talked to him a lot in the next six years on topics both scientific and personal. Being young, brash and predestined to nonconformity, I was often insecure at the time but he strongly urged me to stick to my own research program, advice that I have heeded ever since. We will miss his elegance and insight. Acknowledgment. I thank Edgar Blokhuis and Peter Prinsen for logistic help. I am grateful to Erich Eisenriegler for pointing out to me that eqs 14 and 18 are actually identical.
Depletion Theory and the Precipitation of Protein
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3945
Appendix
k(n) )
The mathematical physics of two spheres interacting under a variety of fields has been reviewed by Jeffrey and van Dyke.20 An expansion of the interaction at small separations may cause vexing problems. In this respect it is of interest to analyze the series given by eq 18, which arises from the method of image charges. First, it is noted that the expansions expressed by eqs 14 and 18 are quite different in character. In effect, if we define j ) 1 + ) exp τ, we rewrite eq 14 as
I(j) )
∞ C2 1 ) (j - j-1) 2n+1 8πa j +1 n)1
∑
(24)
We may compute the limit of touching spheres rather heuristically by letting tend to zero upon noting that (1 + )2n+1 tends to exp 2n; setting t ) n, we replace the summation by an integral
lim I ) 2 f0
∫0∞ dt e2t 1+ 1 ) ln 2
(25)
On the other hand, the function J stemming from eq 18 should be identical to I ∞
(-1)n+1 J(j) ) (j - j ) n -n n)1 (j - j )
∑
-1
(26)
∞
f0
(32)
It turns out to be convenient to let p be some uneven integer of order τ-1/2. A Taylor expansion of k in terms of τ is then possible to compute J1. We have
1 1 1 1 z(n) ) τ2 + τ2 + O(p2τ4) 6 n n+1 6
(
)
(33)
so that
[
1 1 J1 ) τ2 ln 2 + (p + 1) 6 2
∑
n)p+2
]
( n1 - n +1 1 ) + O(p τ ) 3 2
uneven
(34)
The sum in eq 34 may be shown to be O(τ1/2). Equation 30 is now expressed as an integral with the help of the Euler-Maclaurin formula
J2 )
1 2
∞ dn z(n) + R ∫p+2
(35)
1 1 1 R ) z(p + 2) - z′(p + 2) + z′′′(p + 2) - ... 2 6 90 (36) A Taylor expansion of k(n + 1) is convenient
Letting f 0 in eq 26 yields
lim J )
sinh τ 1 sinh nτ n
(-1)n+1 ) ln 2 n n)1
∑
(27)
1 k(n + 1) ) k(n) + k′(n) + k′′(n) + ... 2
(37)
for the integral in eq 35 is then evaluated immediately In the limit of touching spheres ( f 0), the series given by eq 24 remains absolutely convergent but the expansion eq 26 arising from the image-charge method ultimately becomes conditionally convergent (eq 27). A rigorous analysis is needed to show how the limit eq 27 is attained. Inspection of the series given by eq 26 proves that there are mathematical difficulties both in a first part of the series and in the remaining part. Thus it is expedient to break up the series as follows
J ) ln 2 + J1 + J2
(28)
p
J1 )
∑
z(n)
uneven
These terms are computed by Taylor expanding eq 32 at small nτ
1 1 k(p + 2) ) τ2 - (p + 2) + O(p3τ4) (39) 6 p+2
[
]
[
]
1 1 + 1 + O(τ3) k′(p + 2) ) - τ2 6 (p + 2)2
(40)
k′′(p + 2) ) O(τ5/2)
(41)
which readily gives the terms on the right-hand side of eq 36
∞
∑
(38)
(29)
n)1
J2 )
∞ 1 dn z(n) ) k(p + 2) + k′(p + 2) + ... ∫p+2 2
z(n)
(30)
n)p+2 uneven
with
z(n) ) k(n) - k(n + 1)
(31)
1 z(p + 2) ) τ2 + O(τ3) 6
(42)
z′(p + 2) ) O(τ5/2)
(43)
Equations 28, 34, 35, 36, and 38 thus yield
3946 J. Phys. Chem. B, Vol. 113, No. 12, 2009
1 1 J ) ln 2 + τ2 ln 2 + O(τ5/2) 6 4
(
)
Odijk
(44)
which agrees with eq 19. A term of order τ3/2 is absent in eq 44 because the p constant terms in eq 34 are exactly compensated by terms in the remaining infinite series. References and Notes (1) De Gennes, P. G. Rep. Prog. Phys. 1969, 32, 187. (2) De Gennes, P. G. Macromolecules 1981, 14, 1637. (3) De Gennes, P. G.; Pincus, P. J. Phys. Lett. 1983, 44, 241. (4) De Gennes, P. G. C. R. Acad. Sci. Paris 1979, 288, 232. (5) Odijk, T. Macromolecules 1996, 29, 1842. (6) Eisenriegler, E.; Hanke, A.; Dietrich, S. Phys. ReV. E 1996, 54, 1134. (7) Tuinier, R.; Rieger, J.; de Kruif, C. G. AdV. Colloid Interface Sci. 2003, 103, 1. (8) Eisenriegler, E. J. Chem. Phys. 2006, 125, 204903. (9) Eisenriegler, E. In Soft Matter; Gompper, G., Schick, M., Eds.; Wiley-VCH: New York, 2005; Vol. 2. (10) Hooper, J. B.; Schweizer, K. S.; Desai, T. G.; Koshy, R.; Keblinski, P. J. Chem. Phys. 2004, 121, 6986. (11) Fuchs, M.; Schweizer, K. S. J. Phys. Condens. Matter 2002, 14, R239. (12) Odijk, T. Physica A 2000, 278, 347. (13) De Gennes, P. G. Scaling concepts in polymer physics; Cornell University Press: Ithaca, NY, 1979. (14) Hanke, A.; Eisenriegler, E.; Dietrich, S. Phys. ReV. E 1999, 59, 6853. (15) Brochard, F. J. Physique 1983, 44, 39. (16) Odijk, T. Biophys. J. 2000, 79, 2314.
(17) Hilfer, R. Applications of fractional calculus in Physics; World Scientific: Singapore, 2000. (18) Eisenriegler, E.; Ritschel, U. Phys. ReV. B 1995, 51, 13717. (19) Jeffrey, G. B. Proc. R. Soc. London A 1912, 87, 109. (20) Jeffrey, D. J.; van Dyke, M. J. Inst. Math. Appl. 1978, 22, 337. (21) Smythe, W. R. Static and dynamic electricity; McGraw-Hill: New York, 1968. (22) Eisenriegler, E. J. Chem. Phys. 2000, 113, 5091. (23) Tuinier, R.; Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 2000, 113, 10768. (24) Polson, A.; Potgieter, G. M.; Largier, J. F.; Mears, G. E. F.; Joubert, F. J. Biochim. Biophys. Acta 1964, 82, 463. (25) Juckes, I. R. M. Biochim. Biophys. Acta 1971, 82, 463. (26) Ho¨nig, W.; Kula, M. R. Anal. Biochem. 1976, 72, 502. (27) Middaugh, C. R.; Lawson, E. O.; Litman, G. W.; Tisel, W. A.; Mood, D. A.; Rosenberg, A. J. Biol. Chem. 1980, 255, 6532. (28) McPherson, A. Crystallisation of biological macromolecules; Cold Spring Harbor Laboratory Press: New York, 1999. (29) Atha, D. H.; Ingham, K. C. J. Biol. Chem. 1981, 256, 12108. (30) Hubbard, J. B.; Douglas, J. F. Phys. ReV. E 1993, 47, R2983. (31) Douglas, J. F.; Zhou, H. X.; Hubbard, J. B. Phys. ReV. E 1994, 49, 5319. (32) Russell, A. J. Inst. Elect. Eng. 1916, 55, 1. (33) Chow, Y. L.; Yovanovich, M. M. J. Appl. Phys. 1982, 53, 8470. (34) Kawagucki, S.; Imai, G.; Suzuki, J.; Mivahara, A.; Kitano, T.; Ito, K. Polymer 1997, 38, 2885. (35) Bloustine, J.; Virmani, T.; Thurston, G. M.; Fraden, S. Phys. ReV. Lett. 2006, 96, 087803. (36) Sheth, S. R.; Leckband, D. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 8399. (37) Odijk, T. Langmuir 1997, 13, 3579. (38) Prinsen, P.; Odijk, T. J. Chem. Phys. 2004, 121, 6525. (39) Wang, S.; van Dijk, J. A. P. P.; Odijk, T.; Smit, J. A. M. Biomacromolecules 2001, 2, 1080.
JP806722J
