Langmuir 2007, 23, 12737-12751
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Long-Range Correlations, Geometrical Structure, and Transport Properties of Macromolecular Solutions. The Equivalence of Configurational Statistics and Geometrodynamics of Large Molecules Stefano A. Mezzasalma Department of Biophysics, Biochemistry and Macromolecular Chemistry, Trieste UniVersity, Via Giorgieri 1, 34127 Trieste, Italy ReceiVed June 26, 2007. In Final Form: August 22, 2007 A special theory of Brownian relativity was previously proposed to describe the universal picture arising in ideal polymer solutions. In brief, it redefines a Gaussian macromolecule in a 4-dimensional diffusive spacetime, establishing a (weak) Lorentz-Poincare´ invariance between liquid and polymer Einstein’s laws for Brownian movement. Here, aimed at inquiring into the effect of correlations, we deepen the extension of the special theory to a general formulation. The previous statistical equivalence, for dynamic trajectories of liquid molecules and static configurations of macromolecules, and rather obvious in uncorrelated systems, is enlarged by a more general principle of equivalence, for configurational statistics and geometrodynamics. Accordingly, the three geodesic motion, continuity, and field equations could be rewritten, and a number of scaling behaviors were recovered in a spacetime endowed with general static isotropic metric (i.e., for equilibrium polymer solutions). We also dealt with universality in the volume fraction and, unexpectedly, found that a hyperscaling relation of the form, (average size) × (diffusivity) × (viscosity)1/2 ∼ f(N0, φ0) is fulfilled in several regimes, both in the chain monomer number (N) and polymer volume fraction (φ). Entangled macromolecular dynamics was treated as a geodesic light deflection, entaglements acting in close analogy to the field generated by a spherically symmetric mass source, where length fluctuations of the chain primitive path behave as azimuth fluctuations of its shape. Finally, the general transformation rule for translational and diffusive frames gives a coordinate gauge invariance, suggesting a widened Lorentz-Poincare´ symmetry for Brownian statistics. We expect this approch to find effective applications to solutions of arbitrarily large molecules displaying a variety of structures, where the effect of geometry is more explicit and significant in itself (e.g., surfactants, lipids, proteins).
Introduction Previous papers outlined the fundamentals and some applications of the special theory (SbR) of a recent Brownian relativity (BwR), devised to deal with universality and configurational statistics in polymer solutions by a new approach.1-8 If special relativity9 may be defined as the theory of inertial frames (in uniform relative motion), Brownian relativity starts as the statistical theory of uncorrelated (or unperturbed) random paths, irrespective of the variable by which they may vary or evolve. It is based on the two following postulates:7 From the rest frame of the laboratory, static shapes and dynamical trajectories point out universal Brownian observers for the laws of statistics (BP1); The “diffusive horizon”, the difference between the diffusion coefficients of a Brownian observer and the laboratory, is invariant from any universal frame or random path (BP2). BP1 describes a “universal” principle of relativity, among timelike and shapelike stochastic realizations of an unperturbed Brownian motion. BP2 settles the quantity taking place of the light speed (c) in special relativity. The spacetime which these postulates define is insensitive to interchanging static (shapelike) with dynamic (timelike) random paths. Random realizations, built either on time (t, liquid (1) Mezzasalma, S. A. J. Phys. Chem. B 2000, 104, 4273. (2) Mezzasalma, S. A. J. Stat. Phys. 2001, 102, 1331. (3) Mezzasalma, S. A.; Angioletti, C.; Cesa´ro, A. Macromol. Theory Simul. 2004, 13, 44. (4) Mezzasalma, S. A. Chem. Phys. Lett. 2005, 403, 334. (5) Mezzasalma, S. A. Ann. Phys. 2005, 318, 408. (6) Mezzasalma, S. A. J. Phys. Chem. B 2006, 110, 23507. (7) Mezzasalma, S. A. J. Colloid Interface Sci. 2007, 307, 386 (Feature Article). (8) Mezzasalma, S. A. Chem. Phys. 2007, 334, 232. (9) Dixon, W. G. Special RelatiVity; Cambridge University Press: Cambridge, 1978.
molecules) or repeat units (N, chain molecules), provide “equivalent” (or universal) statistical observations. The word universal should not be confused with covariant (of standard relativity). Brownian relativity lies actually in between an absolute and a covariant description of the (statistical) laws of nature. BP1 and BP2 interpret the universal laws of polymer physics as privileged observations, conducted from the (rest) reference system of the laboratory, but agreeing with any class of Brownian frames. Diffusive horizons are simply the quantities preserved in such observations, and it was shown that including the reference diffusion coefficient (e.g., the laboratory’s) into this definition results into an interesting setting. We stress, to avoid misunderstandings, that BP1 and BP2 were meant initially to redefine the ideal chain (notion), or a polymer solution at the Flory’s Θ point,10 by the spacetime in which it lies. For instance, the meaning of the light speed invariance should not be confused with the constancy concept arisen from BP2. To briefly recall SbR, consider a Minkowski’s spacetime, where any event comprises a time and a mean squared length, both observed at the hydrodynamic limit, where Einstein’s law for Brownian movement applies.11 In particular, at the diffusive horizon linked to the liquid molecules, the frame of reference O′ ) (t′, r′2) will detect1,7
r′2 ) ∆′t′
(1)
where r′(t′) is the stochastic process in (continuous) time t′ for the random position r′, and the motion barycenter is set to r′(0) (10) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, New York, 1979. (11) Boon, J. P.; Yip, S. Molecular Hydrodynamics; Dover Publications: New York, 1991.
10.1021/la701891m CCC: $37.00 © 2007 American Chemical Society Published on Web 11/02/2007
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) 0. The quantity r′2 is a configurational average and may either refer to the distance travelled by a single tagged particle or the endpoint contraction of a polymer snapshot. The diffusion coefficient (D) is replaced by the diffusive horizon (∆′ ) D D0), set in BP2 to take into account the value chosen as reference (D0). The numerical constant accounting for the space dimensionality (2d) will not produce any effect in our theory and is therefore included into ∆′. From another frame O′′ ) (t′′, r′′2), one will similarly have
r′′2 ) ∆′t′′
(2)
The invariance of diffusive horizons preserve the intervals, σ and σ′, between events in O′ and O′′ that are arbitrarily close to each other:
dσ′2 ) dσ′′2
(3)
i.e., with
Figure 1. Geodesic rectangles in (a) Einstein and (b) Brownian relativities.
dσ′2 ) ∆′dt′ - dr′2 2
(4) 2
Brownian-Lorentz transforms (t′, r′ ) f (t′′, r′′ ) were derived with the prescription to leave three quantities invariant, i.e., the diffusive interval, the rules of length contraction and time dilation.7 In matrix notation, they are represented by
( ) () t′′
r′′
2
) ′′L′
t′
r′2
(5)
where ′′L′(D′, D′′) ) I + (′′∆′/D′) ′′P′ sums the identity transform to a perturbation, depending on the O′′ diffusion coefficient, ′′P′ ) ′′P′(D′′), and proportional to the “relative diffusive horizon”, defined as ′′∆′ ) D′ - D′′. Superscripts specify the direction of the transformation, here from O′ (timelike) to O′′ (shapelike), bearing in mind that changing frame equals to revert the spacetime scales at which the observations are performed. Hence, the two molecular dimensionalities (N T 1) and thus their diffusion coefficients (D′ T D′′) are to be exchanged, getting ′L′′(D′′, D′) ) I + (′∆′′/D′′) ′P′′ and ′P′′ ) ′′P′(D′). The cost to be paid for these three prescriptions is obviously a singular matrix transformation, breaking unavoidably covariance. Nonetheless, the scaling picture of an ideal chain with given Kuhn’s step size (l) was achieved.1,4,7 BwR opens an interesting picture for statistics, the equivalence for dynamic and static random paths being one of the central aspects, and can be important in many stochastic processes, where diffusive media are often shared by a couple at least of subsystems. For example, whenever statistics are simpler or more accessible in one of its domains, direct indications on the other spacetime scales would be obtained. In polymer solutions, which identify the most natural BwR application, it would be as chain shapes were “collecting” the statistics of the motion of liquid molecules. This insight was successfully applied to a couple of recent analyses. In the first, the probability distribution function of a “true self-avoiding walk polymer” is modeled as a universal Percus-Yevick hard-sphere solution for the Ornstein-Zernicke function of the host liquid.6 The second tackled the anomalous scaling of passive structure exponents, for turbulent advections, by the anomalous exponents of star polymer partition functions, suggesting a new prospect to research on turbulence in liquids.8 SbR should be extended now to a general-relativistic approach (GbR), accounting explicitly particle correlations for. Actually, GbR was already stated and applied in a first work on BwR1 but
here will be discussed and examined in some more detail. Close to Einstein’s view of gravity, this paper will first proceed with formulating a principle of equivalence for geometry and statistics and then with writing the geodesic and field equations arising in BwR. The Brownian version (EbP) of Einstein’s equivalence principle (EqP), shortly outlined in the next section, will extend BP1 up to including molecular correlations. Afterward, some applications to linear and flexible macromolecules in solution are taken back and completed in a Brownian spacetime with general static isotropic metric. Principles of Equivalence. Einstein’s View and CurVature of Space. The principle of strong equivalence of gravitation and inertia (EqP) is the basic postulate founding general relativity and the Einstein’s field equations. The other, said of weak equivalence, states the equivalence of gravitational and inertial masses, but its delicate relationship with EqP will not be treated here. EqP addresses all laws of nature, irrespective of the laboratory features, while the latter limits itself to make all laws of motion equivalent. According to the former, “one can always find a locally inertial coordinate frame”, in a sufficiently small neighborhood of every spacetime point in a gravitational field, “where all physical laws take the same form as in unaccelerated Cartesian systems in the absence of gravitation”.12 Provided to stay inside small spacetime regions, a gravitational field at rest is equivalent to a reference frame moving with constant acceleration in a spacetime devoid of gravitation. Einstein concluded that, in a freely falling elevator for example, no external static and homogeneous gravitational field could be measured.13 The elevator defines a “local” inertial frame, where gravitation is not felt and each massive body behaves as it were free. He thus indicated a class of observers retaining the special picture of relativity, from which proceeding toward a generalization including gravitational effects. Two of his central insights were that time had to run slower with increasing gravity and, as a consequence, spacetime had to be curved. To convince ourselves by a sketchy example, consider the worldlines portrayed by the rectangle in Figure 1a. Wishing to go in uniform motion from A to B and from A′ to B′, whenever AB ) A′B′, the two times would also be equal, t(AB) ) t′(A′B′). In a gravity field, with (12) Weinberg, S. GraVitation and Cosmology; John Wiley and Sons: New York, 1972. (13) Stephani, H. General RelatiVity; Cambridge University Press: Cambridge, 1985.
Properties of Macromolecular Solutions
intensity decreasing with increasing height, the upper clock will proceed faster. This means that “closing” the worldlines by traveling uniformely along AB and A′B′ for equal time intervals will no longer be possible. In trying to do so, a deficit length (or angle) will be left along the upper path length (A′B′) proportionally to t′(A′B′) - t(AB). Einstein’s standpoint focuses on this time interval as an invalidation of the Pythagorean theorem applied to a pair of inertial frames. He could thereby extend the equivalence for flat and inertial systems into one another for curved and accelerated frames. It is finally instructive stressing that, independent of the reasons behind, to reduce such an ample class of laws to be equivalent might seem at first sight being very naı¨ve and not scientifically sound. However, given the unlimited number of possible options, Einstein’s assumptions were certainly plausible a priori, even if not necessary. Brownian Principle of EquiValence for Geometry and Statistics. When statistics are Gaussian, SbR deals with free Brownian particles and ideal chains as statistically equivalent (universal) systems.4-8 Just as “the laws of physics are the same in all inertial frames” (first postulate of special relativity), “timelike” and “shapelike” reference systems yield in SbR an equivalent description of the laws of statistics (for ideal, or “special” systems). In a spacetime modeled by BP1 and BP2, to distinguish a molecular path (random trajectory in time) from a monomer path (random polymer shape) is no longer meaningful. However, Gaussian chains form a strongly idealized class of macromolecules, with short-range (or null) segment correlations and well described in a “phantom” immaterial spacetime.10 To deal with “real” macromolecules, long-range correlations must be accounted for.14 Thus, as soon as liquid and polymer molecules become carriers of finite (excluded) volumes, or any other interaction form, the flat space hypothesis is clearly abandoned for a more realistic “material” space. Fortunately, including correlations in SbR can be done in close analogy with ordinary relativity. Therefore, when the vacuum is replaced by fixed matter and energy distributions, the equations are to be written in a curved geometry. Here, every chain statistic will induce an equivalent geometry, ascribing the influence of correlations to the same kind of geometrical changes occurring in general relativity, from a (flat) Minkowski’s metric to a curved space (ηµν f gµν). At this point, restating EqP in Brownian language simply requires following the logics of Einstein’s formalism. We can regard any perturbation to the Gaussian statistics as the analogue of a (gravity) field in standard relativity but should careful of a couple of differences. Consider again the (diffusive) worldlines in Figure 1b. They consist now of random paths in the long wavelength regime and with distinct correlation degrees, still depicting a similar rectangle as that in Figure 1a. The average time to cross each side will tend to decrease with increasing correlations, and the former deficit length (or angle) will be positive. Accordingly, a correspondence between field intensity and molecular disorder establishes in BwR. This scheme may both still refer to timelike and shapelike frames. For single linear macromolecules, the “ground” level is set to an ideal Rouse chain, with its (relaxation) times τm (or modes m). They are the correlation times of chain portions of (N/m) repeat units, the longest of which τ1 (∼N2) obviously dominates the dynamics of uncorrelated systems. On varying the extent of correlations, one will obtain the sample paths and times that are typical of other polymer models. (14) Scha¨fer, L. Excluded Volume Effects in Polymer Solutions; SpringerVerlag: Berlin, 1999.
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We can now enunciate a principle of equivalence for geometry and statistics (EbP), basically extending BP1 to any random path endowed with long-ranged (enough) correlations. On the basis of EqP and the above considerations, it may sound like correlated random paths in time, moving in a flat geometry, and unperturbed random shapes, evolving in a curved space, are equivalent (EbP), BwR providing no way to discriminate between these circumnstances. Such as EqP was, for the first relativity postulate in the special formulation, EbP becomes in this way a straight extension to BP1. At first sight, it appears to be obvious, at least qualitatively. Once the unperturbed system is set to embed a background Minkowski spacetime, each worldline will clearly lie in a space with metric tensor corresponding to the arbitrary correlations with which it diffuses. However, just as EqP in ordinary relativity, EbP tells us something more than a bare correspondence. It postulates that the form of universal relationships, for any random path in an ideal system, mantains also in conditions of nonideality. Brownian Metric Tensor and the Equivalence of Correlations and Curvature. Statistical Pseudo-Coordinates. Before proceeding further, one happens to fall across the weakest point of BwR, consisting of certain restrictions to the general form of coordinate transformation laws. SbR was written originally for a particular class of frames, lying at the hydrodynamic limit, with radial symmetry and only detecting second-order statistical moments.15 As in special relativity, it was practically defined in (1 + 1) Cartesian dimensions, a time and a space. The latter takes into account the spatial dimension along which the relative movement establishes. In translational kinematics, it is the uniform motion direction (xk ) Vkt), while SbR is written in a spacetime that is homogeneous and isotropic with respect to diffusion (r2∝Dt). This implies the two spaces are related by xk2 f 2
r2 ) ∑xk , and thus the most natural definition for the radial coordinate is xk f F )
xr2, where
r2 )
∫r2 dPN
(6)
depends on the involved probability measure for a molecular ensemble of N units, PN ≡ P(r, t; N).16 Wishing to deal with F as an independent variable, one may imagine to vary it unconditionally, as a radial or Cartesian coordinate is used to be arbitrarily varied. Otherwise, each of these displacements will be equivalent to a variation of the probability measure. These remarks, though being elementary in themselves, are useful to recall that changing (spatial) coordinates induces a variational problem for the probability density pN(r, t). Moreover, as being statistically representative of whole ensembles, they should rather be regarded as pseudo-coordinates. A given value of F will generally correspond to an ensemble represented by PN. Some details are obviously lost if chain states are described by statistical moments instead of probability measures, but BwR will let the relations among universal features better emerge. In summary, when the issue of changing coordinates through their probability functions (or functionals) is passed by, one enters the universality domain of BwR. In this case, every statistical information is accessed through the behavior of the metric tensor elements (i.e., diffusive and geometrical quantities). Remember that not all coordinate transformations will be feasible at this stage, but only those preserving the symmetry set by the long(15) Brzezniak, Z.; Zastawniak, T. Basic Stochastic Processes; SpringerVerlag: Berlin, 1999. (16) Grosberg, A. Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; Cornell University Press: Ithaca, New York, 1979.
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wavelength (hydrodynamic) regime. Otherwise, wishing to look into statistics, EbP requires to develop the variational problem raised by conceiving coordinates as probability functionals. This issue is left for future work, but a look at some preliminary results will be taken here in the last section. In the general theory of Brownian relativity (GbR), the pseudocoordinate basis in which polymer problems are carried out will be denoted by σµ. In working with it, the consistency of the involved transformation rule, σµ f σ′µ, should be verified every time necessary. For instance, polar and Cartesian representations, σµ ) {τ ) xt,F )
x
x
xr2, θ, φ} and σ′µ ) {x0 ) τ, x1 ) xx2, x2
) y2, x3 ) z2}, clearly transform consistently. Remember also that, throughout this paper, Greek letters will be used to indicate that the coordinates and (tensor) quantities where they are used can be either time- or spacelike, putting off the Latin alphabet only for spatial coordinates alone (e.g., µ ) 0,1, 2, 3 while k ) r, θ, φ). Metric Tensor. Ascribing statistics to the geometry of worldlines, diffusing along a curvilinear path of length σ:
dσ2 ) -γµνdσµdσν
Figure 2. Scheme of spaces of positive (spherical-elliptic), zero (parabolic), and negative (hyperbolic) constant curvatures.
(7)
requires now to deal with the Brownian metric tensor (γµν). According to EbP, it will serve to define in GbR a “locally universal” frame, bringing the perturbed system back to unperturbed. To check how a change in the system statistics may alter geometry, one easily realizes that a connection between Brownian reference frames and kinematics of diffusion can be built in close analogy to Einstein’s arguments. Imagine thus to lay down a random path, still in the long-wavelength regime, and seek the law for the average end-to-end distance as a function of time. When the process is correlated, the second-order statistical moment will deviate, at some point or for a while, from the linear time-dependence of free particles and ideal chains.16 Let this behavior be written as a general diffusion law, r2 ∼ ωR, in some evolutionary variable (ω) and R > 0. Whenever R * 1, from some time in the diffusive regime on, the long-tail statistics will make the random position vector no longer Gaussian and the “anomalous” regime of “subdiffusion” (R < 1) or “enhanced diffusion” (R > 1) will hold.17 In this model, negative or positive deviations from linearity are well discriminated, and deficit or excess lengths will emerge according to the sign of R - 1 (pushing the analogy further, one might also introduce a timelike “acceleration” (d2r2/dt2), and ascertain that the only “unaccelerated” frames are those evidently with R ) 1). On mapping t f N, the same reasoning would obviously concern the arise of deficit or excess angles in path configurations of correlated chains, leading to perfectly dual conclusions for shapelike frames. In general, positive and negative deviations from the law of Brownian motion will take place, respectively, in hyperbolic and spherical frames of reference, otherwise said of positive (kG > 0) and negative (kG < 0) Gaussian curvatures (Figure 2).18 In particular, to say something more, some assumption should be made. Einstein, alongside EqP, supposed that curvature (i.e., the deficit angle) were to be ascribed to the mass (M) enclosed inside the portion of space considered. His suggestive hypothesis, kG (17) Ott, A.; Bouchaud, J. P.; Langevin, D.; Urbach, W. Phys. ReV. Lett. 1990, 65, 2201. (18) Ramsay, A.; Richtmyer, R. R. Introduction to Hyperbolic Geometry; Springer-Verlag: New York, 1995.
Figure 3. Scheme of unperturbed and perturbed Brownian systems embedding a (concentric) horocyclic reference frame.
∝ M, related for the first time matter to geometry and opened to a feasible experimental confirmation of his theory.19 BwR, on the other hand, relies on Einstein’s law for Brownian motion, holding in the long-wavelength domain. This approach moves “backward”, from such “events at infinity” to recovering the effect of correlations at finite wavelengths in the characteristic function.6,20 A notion which lends itself to exemplify a reference frame in the hydrodynamic regime is that of “horocycle.”18 It yields a peculiar reference frame, having both the character of a (“straight”) line and a circle with “center at infinity”, whose simplicity is useful to illustrate EbP. To briefly sketch it out, let A, B be points on a line l and c be a circle of radius AB, centered at B. When, at A fixed, B moves to infinity along l, an Euclidean c tends to the straight line perpendicular to l that is passing to A. In a hyperbolic space, as ABf∞, the circle does not converge to a line but to a horocycle. To fix the ideas, consider thus the couple of schemes in Figure 3. In Figure 3a, spacetime is taken to be flat and, when an unperturbed random path undergoes the effect of (positive) correlations, the random path interval (eq 4) shall lengthen by a certain (positive) amount. EbP affirms that, on letting this path free to “devolve” in a curved spacetime, such an increment is preserved. We may come thus to Figure 3b, embedding perturbed (p) and unperturbed (u) paths in a horocyclic coordinate system in a space of constant curvature. It is known that the arclength ratio of two horocycles is scaling with their radial distance (r) as exp(r/L), where L ) x(2π/|kG|) is the Lobachevskij’s constant and L f ∞ indicates the Euclidean limit. In this special case, a position translating EbP would be σp/σu ) exp(r/L), σi being the diffusive intervals for Gaussian (u) and (long-ranged) correlated (p) paths. As the parallelism angle (R) follows from (19) Misner, C. W.; Thorne, K. S.; Wheeler, J. A. GraVitation; W. H. Freeman and Company: San Francisco, 1973. (20) Mezzasalma, S. A. J. Colloid Interface Sci. 2006, 299, 589.
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the Bolyai-Lobachevskij formula,18 tan(R/2) ) exp(-(r/L)), j /2) would bridge in the simplest the simple relation σu/σp ) tan(R fashion the configuration of a correlated path to a mean R j value (Figure 3b). In the Euclidean case, where paths are ideal and R j ) π/2 (kG ) 0), deficit (or excess) angles disappear. This, of course, was only an example, hardly suitable to model the concrete situations that will next be faced, but already indicates that molecular trajectories and polymer shapes can be set in a variety of arrangements, where geometry, to borrow some words by Bernard Riemann,21 has a direct statistical meaning in its own right. With this in mind, we can continue with the most important, geodesic and field, equations of the general theory. Postulate of Geodesic Motion. We have recalled how Einstein’s EqP allows a description of a free falling system in terms of special relativity. It ensures the existence of a class of locally inertial observers, defined on a scale where spacetime can still be regarded unperturbed. Whatever the nonlocal form of the (curved) spacetime metric, there exists a free falling frame where the motion is still “geodesic” (Appendix I-A). In macromolecular solutions, it is known that molecular hydrodynamics and polymer theory guarantee the existence of unperturbed regimes, in which statistics becomes (back) to Gaussian.10,11 Close to standard relativity, this normally occurs for some (outer) long-wavelength scale, far distant apart from correlation and interaction sources but does not mean that such frames will flatten geometry everywhere. Consider thus a worldline expression of the same form as in eq 7, γµν being the Brownian metric for a given system of statistical coordinates (σµ) and curvilinear length (σ). EbP requires it be related to an ideal frame (e.g., ιµ):
( )
d2ιµ )0 dσ2
(8)
evolving with no correlation in a (flat) spacetime with Brownian Minkowski’s metric η j Rβ:
γµν )
( )( )
∂ιR ∂ιβ η j Rβ ∂σµ ∂σν
(9)
Proceeding as in Appendix I-A, one arrives at
( )( ) ( )( )( )
∂ιµ d2σν ∂2ιµ dσν dσλ + )0 ∂σν dσ2 ∂σν∂σλ dσ dσ
(10)
of course, obtaining again
( ) ( )( )
ν d2ιτ dιλ τ dι + Γ h )0 νλ dσ dσ dσ2
(11)
Γ h τνλ ) lim Γτνλ
(12)
where gRβfγRβ
imply that the final scaling laws will be Gaussian too. Rather, it describes the variation of the ideal behavior, evaluated from a reference frame in which EbP applies. Brownian Einstein’s Equation. Einstein’s field equation of general relativity can be derived from the (Hilbert’s) principle of stationary action, summarized in Appendix I-B. With similar premises and notations, i.e.
∫
{( ) ( ) ( )}
∂γFν ∂γµν γλF ∂γFµ + ν µ 2 ∂σ ∂σ ∂σF
h µν G h µν ) R
γµνR h )T h µν 2
The last two are the Brownian-relativistic geodesic equation and state that, in a locally universal frame, each random path evolves as it were ideal. As already mentioned, this does not (21) Riemann, B. U ¨ ber die Hypothesen, welche der Geometrie zugrunde liegen; University of Go¨ttingen: Go¨ttingen, Germany, 1854.
(15)
The left and right sides are, respectively, the Einstein (G h µν) and energy-momentum (T h µν) Brownian tensors, the former depending on the Ricci (R h µν) and the curvature scalar (R h ).12 The constant k′ is converted into another (unit) coefficient, which may be ignored in scaling calculations like the following. As a short outline, the procedure that can be generally adopted to solve polymer problems comprises (i) a characterization of mass and energy sources, by their time-rates of change (i.e., T h µν), (ii) a derivation of the Brownian metric from the field equation, and (iii) the calculation of a (test) polymer path as a Riemannian geodesics. Said more exactly, point (ii) starts from the spatial geometry and its rate of change at a given time and determines the spacetime geometry at the same instant. Hereinafter, the geometrodynamic equations goes on to predict at any time the 4-dimensional geometry and the mass-energy flowing throughout it. Facing (i-iii) will both take advantage of EbP and the geodesic postulate, but one should not think of them as strictly independent principles. Test polymers take part in fact of the total matter distribution (i.e., the energy-momentum tensor), and it is reasonable that the postulate of geodesic motion could be deduced from the field equation, instead of being axiomatically imposed. This was the line followed by Einstein himself, in cooperation with Grommer, showing that metric singularities cannot be arbitrary and their form is set by the field equation.22 A central reason of this lies clearly into the nonlinearity of eq 15. In a linear theory, one can find further solutions by superimposing one solution with point singularities. Here, the influences of distinct bodies cannot be summed up to get a resultant effect (as, instead, in classical gravity). Any field contains energy and is part of its own source, thus should rather be treated as an inseparable whole. Energy-Momentum and Stress Tensors. Classical Field Theory. The energy-momentum tensor is defined in the Einstein relativity by the variation of the matter action (see eq 14):
(
δx-gLM 1 x-gTRβ ) 2 δgRβ (13)
(14)
the general equation of BwR follows at once from setting gµν f γµν, g ) detgµν f γ ) detγµν, and building all tensor quantities accordingly. We will continue to denote them by overscripting their symbols and write, in covariant form
is the BwR analog of the affine connection,12 i.e.
Γ h λµν )
∫
δ x-γR h d4σ ) -k′ x-γT h µνδγµν d4σ
)
(16)
and fulfills the continuity equation, expressible as the vanishing of the covariant divergence (the semicolon operator):12 Rβ ) T;R
( )
∂TRβ + ΓRRλTλβ + ΓβRλTRλ ) 0 R ∂x
(17)
(22) Landau, L. D.; Lifsits, E. M. Field Theory; Riuniti: Rome, 1982.
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The former condition, however, does not fix TRβ univocally. When the angular momentum is conserved, a further constraint to be regarded is given by the symmetry property
TRβ ) TβR
(18)
ceasing of course to be valid in the presence of torsion. It is useful to report this tensor in explicit form (still in natural units):22
(
Fj ‚‚‚ s ‚‚‚ l l Rβ T ) s ‚‚‚ σij ‚‚‚ l l
)
(19)
where row and column indeces are ordered again as R, β ) {0, k ) 1, 2, 3}, k denoting the spatial components. Here, the proper energy density T00 ) Fj equals the mass density (Fj). The vector Tk0 ) sk is the momentum density and, specularly, T0k denotes the density of the energy flux (across the surface perpendicular to k). The second-rank matrix σij is known as the stress tensor, and quantifies the momentum flux. We assign it the description of all momentum components (i), flowing in the unit time through the unit surface perpendicular to each coordinate (j). Thus, while diagonal terms reflect the hydrostatic pressure (σkk ) p), those off-diagonal are used in ordinary relativity to characterize the viscous behavior. To treat various situations of interest (point particle systems, electromagnetic fields, and so on), a number of (inequivalent) tensor expressions is available. An ideal (or perfect) fluid for instance, without viscosity and heat conduction, will be described by12
2 σRβ ) uR;β + uβ;R - θgRβ 3
(24)
denotes the (shear) stress tensor. Polymeric Fluids and Brownian RelatiVity. In polymeric networks (e.g., fluids and rubbers), the molecular origin of stress is mostly ascribed to intramolecular forces and, without significant concentration gradients, one normally works with25
σ(n) ik ) ηL(κik + κik) + pδik + Πik
(25)
(n) as a function of time, σ(n) ik ) σik (t). Here, ηL is the solvent k viscosity, κik ) (∂Vi/∂x ) is the (spatial) velocity gradient tensor, δik is the covariant Kro¨necker symbol, and
Πik )
ηP
πik
τPF2u
(26)
In nonideal (or imperfect) viscous fluids, it modifies instead by an extra term (τRβ):
is the polymer extra-stress term. It is proportional to a symmetric mean “configuration tensor”, πik ) RiRk, averaged over all chain segments and giving a dynamic state variable for the intramolecular structure.26 Its physical significance is to account for the orientation of the random vector R(s, t) ) (∂r(t)/∂s), tangential to the macromolecule at segment s and time t. The prefactor reports a (single) characteristic time (τP), dominating the relaxation toward equilibrium, the polymer viscosity coefficient for infinitesimal shear (ηP), and the unperturbed root-mean-square extension (Fu). Πik had been essential in explaining the influence of polymer concentration on viscoelasticity. In the very dilute regime, viscoelastic effects are very small and the major contribution is viscous. With increasing concentrations, polymers get more and more entangled, and this term increases rather quickly.25 To understand the way which the previous expressions should be regarded herein, the experiences acquainted with polymer and relativity theories are to be stuck together their main differences from BwR. A first descends from focusing on universal laws, intended here to be Brownianly covariant, and carrying the energy-momentum tensor into a spacetime defined differently (xµ f σµ). Second, the invariant quantity is no longer a velocity but (dimensioned to) a diffusion coefficient. Because of the square root of the temporal coordinate in the unperturbed diffusive interval (ctfxDt), this will demand that our energy-momentum tensor be multiplied by some (characteristic) time. We will continue to work with a natural system of units, now with respect to diffusion (D′ ) 1). Thus, provided to properly rewrite the imperfect energymomentum tensor, the large-wavelength phenomenological expression of T h Rβ can be set to
Rβ TRβ ) TRβ i +τ
T h Rβ ) τjRβ
Rβ + (p + Fj)uRuβ TRβ i ) pg
(20)
where uλ () (dxλ/ds)) is the local value of the λ-th 4-velocity component, with -gRβuRuβ ) 1. Again, p and F are measured from a locally inertial frame that, at the instant of measurement, happens to be moving with the fluid (or “comoving”) at given relative velocity (V). In the absence of gravity, the metric is Minkowskian, gRβ ) ηRβ, while the 0-velocity component equals the Lorentz factor, u0 ) (1 - v2)-1/2, and u ) u0v. Thus, when the fluid is at rest at some position and time, uRuβ ) diag(1, 0), the energy-momentum tensor assumes the (diagonal) form which is typical of spherical symmetry:
j , pI) TRβ i ) diag(F
(21)
(22)
We will limit for now to (linear) viscosity contributions, and ignore further thermodynamic or nonlinear effects.23 A commonly adopted expression is24
τRβ ) -ζθhRβ - ηhRλhιβσλι
(23)
where hRβ ) gRβ + uRuβ is the projector on the rest frame of the observer comoving with the fluid, θ () uF;F) is the volume expansion, ζ and η are, respectively, the bulk and shear viscosity coefficients, and (23) Gron, O. Astroph. Space Sci. 1990, 173, 191. (24) Novello, M.; d’Olival, J. B. S. Acta Phys. Pol. 1980, B11, 3.
(27)
On comparing the equations for polymer and relativistic hydrodynamics, the stress tensor can be separated again in (three) different contributions:
j Rβ(bulk) + σ j Rβ(shear) + Π h Rβ σ j Rβ ) σ
(28)
where the last polymer term is added (to eq 24) to generalize ΠRβ to BwR. We will work, as usual, with incompressible liquids and inextensible chains, for which volume expansions and bulk (or “second”) viscosity coefficients can be neglected,23,26 i.e. (25) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (26) Grmela, M. J. Rheol. 1989, 33, 207.
Properties of Macromolecular Solutions
jτRβ ) -hhRµhhβνσ j µν
Langmuir, Vol. 23, No. 25, 2007 12743
(29)
with σ j Rβ(bulk) ) 0 and hhRβ ) γRβ + (′′∆′)R(′′∆′)β projecting now on the rest (unperturbed) frame of the Brownian observer “codiffusing” with the liquid (see eq 5). At this point, one can take advantage of EbP and discuss the polymer problem in the timelike representation of an observer resting with the laboratory. It turns out σ j Rβ(shear) ) 0, [(′′∆′)R(′′∆′)β ) diag(1, 0)], and
j λµ -τjRβ ) (γRλ + δR0δλ0)(γβµ + δβ0δµ0)σ
(30)
G h RR ) τjRR
or, expliciting each polymer term:
h Rβ + δR0Π h β0 + δβ0Π h R0 + δR0δβ0Π h 00 -τjRβ ) Π
(31)
Concerning the spatial components, a general expression matching the former requirements is
Π h ik ) -E h ikπ j ik
(32)
where E h ik depends on the adopted viscosity model and the geometrical part π j ik ) rirk is made for convenience nondimensional by rescaling Rs ) Furs. Concerning the (pure) temporal component, it turns out τj00 ) -4Π h 00. For equilibrium polymer solutions, the spatial matrix is diagonal, π j ik ) δik, while τj00 is related to the thermal fluctuations of chain configurations.27 Working with one dominant relaxation process, E h ik ) ηPδik, the (phenomenological) relation -Π h 00 ) Fj implies the traceless tensor condition:
h 00 - Π h ii jτRR ) 0 ) 4Π
(33)
getting back, upon N f 1, to a well-known result of the kinetic theory (here, in physical units), η1 ∝ FjD′.28 With this position, the influence of Brownian relativistic cooperative motions is neglected, but a perfectly dual analogy between eq 21 and (in contravariant representation):
τjµν ) diag(Fj, ηPI)
j FF ) a, γ j θθ ) F2, and γ j φφ ) F2 sin2 θ. This with -γ j 00 ) b, γ metric “staticity”, of course, does not prevent the system from having its own dynamics, just as statistical fluctuations do not forbid to attain an equilibrium situation. Rather, it means that the diffusive interval (or proper time) will not be dependent upon time (τ), but on space (i.e., R ) (F, θ, φ)) through the only rotational invariants, R2, dR2, and R‚dR.12 The metric γ j µν lends itself to a suitable description of equilibrium polymer solutions, for which the Einstein tensor is also diagonal (G h ii), leading to a field equation of the form
(34)
is attained, provided any transport coefficient be implicitly dependent on the repeat unit number of the Brownian frame moving across the fluid. Note that, conceptually, the fluctuationdissipation theorem30 has taken the place of the equation of state R ) 0).12 We could also set, in all generality, (i.e., coming from TiR R τjR ) f(Fj, ηP), and observe that in the “pure” relativistic case (corresponding here to a simple liquid), the trace function f = 0. Once the length scale is changed (1 f N), this relation will vary according to the geometrodynamics induced by the polymer molecule. Verifying this point means to solve Einstein’s equation, as shown by the forthcoming section for a relevant class of metric tensors. General Static Isotropic (GSI) Brownian Metric. In a former analysis,1 configurational and transport properties of a concentrated polymer solution were investigated by eq 15. We took the solution of a GSI metric in polar coordinates (γ j µν):
dσ2 ) b(F)dτ2 - a(F)dF2 - F2(dθ2 + sin2 θdφ2) (35) (27) Hinch, E. J. Phys. Fluids 1977, 20, S22. (28) Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill: New York, 1965. (29) Douglas, J. F.; Freed, K. F. Macromolecules 1994, 27, 6088. (30) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: Oxford, 2003.
(36)
for each of the four components G h RR, as worked out in Appendix II. We can promptly note that, on approaching N f ∞, the temporal component gets proportional to the chain density and diffusion coefficient, i.e., can be set to G h 00 f 0. Regarding the spatial components, a model with E h ik ) ηPδFF was considered.1 It h θθ ) G h φφ comprised G h FF ) ηP and two equivalent constraints, G ) 0, assumed to hold at the radial coordinate F. In this description, polymer and solvent move together as a unique spherical body, “freely draining” with each other at the sphere of radius F, which is the average dimension of the chain molecule. Recall that the free draining limit basically corresponds to a Rouse chain, whereas in the Kuhn’s “impermeable sphere model”, strong hydrodynamic interactions are assumed (“nondraining” limit).29 In these terms, one yielded in the end the equation:
ψa F Fψb′ + 1 ) ψab + 2 ψb
(37)
with the short-hand notations, f′ ≡ (df/dF), ψf ≡ (d ln f/dF), and being
lim ψa ) ψτjFF Ff∞
(38)
Its analysis, performed with arbitrary concentrations, produced a modified Stokes-Einstein law (eq 40), which may be re-presented in light of the above discussion. With increasing N (.1), eq 33 will be changing form and a new relation between tensor quantities and statistical coordinates is expected. Specifically, it turns out1
jτ-1/2 FF ∼ FNDN
(39)
the index N referring of course to the unit number of a mean chain molecule of equivalent contour length (Nl). Since τjFF ) ηP, one gets a scaling law that, unless of a dimensional constant prefactor, can be written as
FNDNxηP ∼ N0
(40)
Its universal exponents, for the chain size (ν), diffusion (δ), and viscosity () coefficients, afford a satisfactory description of a solution of linear, homogeneous, neutral, and flexible macromolecules, either in unentangled or entangled regimes. This agreement is also preserved in the polymer relaxation time (τP ∼ Nσ), by which the previous equation rewrites as1 0 F3N τ-1 P xηP ∼ N
(41)
and in the measured values, which raised a famous discrepancy between theory and experiment (-δ ) 2 against = -2.2, ) 3 against = 3.4, with σ = ).30 Keep in mind that a major BwR advantage is not the determination of single universal exponents
12744 Langmuir, Vol. 23, No. 25, 2007
Mezzasalma
but of (new) scaling relations. The last two, being preserved in a number of regimes, provide in fact a good example of Brownian covariant universal law. The analysis of a GSI metric is going now to be completed and proceeded throughout the next paragraphs. Weak and Stationary Limit. A first point left to be faced is the weak and stationary regime, so as to derive the universal behavior corresponding to the Newtonian (nonrelativistic) limit. For nonrelativistic matter, the time-time metric component in ordinary relativity takes the form, -g00 = 1 + V(r), for some (Newtonian) potential function (V). As the Einstein’s equation reduces in this case to
∇2g00 ) -8πGT00
(42)
with T00 = Fj, we step back to a Poisson’s equation for Newtonian gravity (∇2V ) 8πGFj).22 For a GSI Brownian metric, where tfxt and γ j 00 turns out to depend on xηP, a simple dimensional analysis gives the quantity F = (∇2γ j 00)2 the meaning of a (viscous) force. In particular, since
j 00 ) τj00 ∇2γ
(43)
from a reference frame linked to an unperturbed portion of fluid, it obeys
F = [-∇2
2 ∫ τj1/2 FF dF′] ∼
( ) ∂τj1/2 FF ∂F
2
) Fj 2
(44)
Thereby, let x denote the fluid density exponent, Fj ∼ Nx, the scaling eq 44, i.e.
)ν+x 2
(45)
is expected to apply upon x f 0 (fluid density independent of N). In fact, the law ) 2ν is satisfied in the unentangled regime by30 (ν ) 1/2; ) 1) and, correspondingly
F ∼ ηPF-2
(46)
will be the viscous force exerted by a polymer solution flowing in a cavity set by an ideal “phantom” chain size. The weak and stationary limit thus returns a Brinkman’s behavior, arising in colloid and polymer hydrodynamics as a special case of the Navier-Stokes equation.31 Continuity Equation (Bianchi’s Identities). The continuity equation is connected to the Bianchi identities for the Riemann curvature tensor:19
µν jτ;µ )
(
)
∂x-γτjµν +Γ h νµλτjµλ ) 0 µ ∂σ x-γj 1
(48)
Each 4-vector component has been explicited in Appendix II, where the expressions of a(F) and b(F) were regarded in the GSI case, and the main conclusions are the following. Since Γ h 0RR ) µ0 0 (eq 36), the temporal condition (τj;µ ) 0) simply confirms that
F2sinθτj00xa/b3 does not depend explicitly on time. In line with µθ µφ ) 0) and azimuth (τj;µ the metric isotropy, also the zenith (τj;µ ) 0) restrictions produce similar constraints. The radial constraint µF (τj;µ ) 0) yields instead F3(1 + τj00) ∼ xηP and, as G h 00 f 0 (F f ∞):
xηP ∼ N0 F3
(49)
or
ν)
6
(50)
Though coming from a closure relation and derived for entangled systems (Appendix III), this scaling law agrees with a number of regimes too. For a concentrated polymer solution (ν ) 1/2), one would obtain ) 3,which is just the viscosity exponent descending from the reptation theory.10 For a real chain molecule, with average size going from the renormalization group prediction32 to the Flory value,10 0.588 e ν e 3/5, the exponents restricted by the Bianchi’s identities would belong to 3.36 e e 3.53, matching satisfactorily the mean value (=3.45 against 3.4), as well as the experimental and numerical ranges, 3.1 j j 3.6.10,30 Effect of Volume Fraction. A point left open in the former work was the influence of polymer concentration, or volume fraction (φ). To this purpose, it is useful stepping back to the special theory and reconsidering the Brownian-Lorentz factor, γB ) 1 - (′′∆′/∆′), bringing the local mobility change into the rules of length contraction and time dilation. We will start from a low-concentration solution at the Θ point, where SbR is expected to hold, and regard the general expression:
γB )
D′′ - D0 D′ - D0
(51)
and, in Einstein’s theory, express a closure relation for the whole geometrical structure (first equality), forbidding any energymomentum exchange with the field (second equality). In BwR, this conservation law indicates that any transport property can change at the expense of a corresponding (geometrodynamic) perturbation to the configurational polymer statistics. For a GSI metric, with γ j 2RRτjRR ) τjRR and x-γ j ) 2 xabF sinθ, everything is ready to carry out eq 47 that, for a symmetric tensor becomes
i.e., with ∆′ ) D′ - D0 and ′′∆′ ) D′ - D′′.1,7 Note that scaling behaviors in N and φ are in BwR conceptually different. There, approaching universality requires to bring the chain molecule at rest (D0 ) 0). In the present case, where polymer chains spread throughout the liquid, this transformation ceases to be local, and γB should be a function of φ. The most general way to do that is introducing the cooperative (i.e., many-chain) diffusion coefficient, Dc(φ) and linking the laboratory frame to its value in the limit of infinite dilution, Dc(φf0) ) D′′0. Dc comes from the relaxation decay of concentration fluctuations, describing the cooperative diffusion of segments in each of the uncorrelated “blobs” partitioning the chains statistically.10 It appears as the phenomenological coefficient in Fick’s equation for the hydrodynamic description of monomer density fluctuations, and is expressible by a fluctuation-dissipation relation for the concentration gradient of the osmotic pressure (Π) and the friction coefficient (R), Dc ) (1/R)(∂Π/∂c).33 Moreover, Dc ∼ φχ
(31) Kang, K.; Wilk, A.; Buitenhuis, J.; Patkowski, A.; Dhont, J. K. G. J. Chem. Phys. 2006, 124, 044907.
(32) Zinn-Justin, J. Quantum Field Theory and Critical Phenomena; Clarendon Press: Oxford, 1993.
(
R h µν -
)
h γµνR 2
;µ
µν )0)T h ;µ
(47)
Properties of Macromolecular Solutions
Langmuir, Vol. 23, No. 25, 2007 12745
Figure 4. Comparison among (a) light deflected by a mass, (b) path deviated by a polymer entanglement, and (c) EbP.
increases with the volume fraction, the exponent χ (> 0) reflecting the solvent conditions (e.g., χ ) 1 at the Θ point) and, in dilute systems, equates the self-diffusion coefficient.10,34 Hence, we can work with
D′′0 Dc(φ) - D′′0 = w* φ γB(φ) ) D′ D′
(52)
where the condition D′ . D′′0 and the expansion Dc(φ)=D′′'o(1 + w*φ) were exploited.35 The constant w* accounts for the threebody contact interactions on which the osmotic pressure mainly depends at the Θ point, where the osmotic second virial coefficient is vanishing.16,30 Again, for ideal chains, D′′o/D′ ∼ N-1 (friction is additive), and one has synthetically
γ B = φP
(53)
i.e., a linear dependence on the polymer number concentration φP ) φ/N. The first aim is finding the scaling law for the polymer size (in three dimensions); thus, we consider the “pervaded” volume (∼F3) of a chain molecule of average size F. Because φF3 gives the net amount of polymeric material, the universal quantity FV ) (φF3)ν stands in a direct relation with F, i.e.
FV(F, φ) ∼ l3ν-1F(φ)
(φF3)ν ∼ φ1/2F
(57)
i.e., still unless of a dimensional proportionality constant
F(φ) ∼ φ(1-2ν)/[2(3ν-1)]
(58)
The second behavior to be found should nearly express the dual relationship for time intervals, in which polymers diffuse by a distance of the order of the contracted size (F). To point it out, we rewrite eq 54 by a single characteristic dimension, FV ≡ l3ν V (φ), and rearrange eq 56 in light of the time dilation rule corresponding to eq 55:1,7
t ) γ Bτ
(59)
t and τ still referring to a polymer and a rodlike molecule, respectively. Working in the time domain therefore requires the “unit changes” xγBfγB and lV f t, obtaining
t3ν(φ) ∼ t(φ) γB
(60)
from which eq 59 constrains the chain relaxation time to
(54)
(φτ)3ν ∼ φτ φ
(61)
τ(φ) ∼ φ(2-3ν)/(3ν-1)
(62)
or the Kuhn size (l) playing the role in BwR of an invariant step length. Now, remember the general form of the length contraction law, for the Brownian frames of a polymer molecule and a rodlike chain long L1,7
F ) xγBL
(55)
Combining the last two equations at L ∼ F yields the desired contraction rule, i.e.
FV(F, φ) ∼ xγBF(φ)
(56)
which, being γB ∼ φ (eq 53), gives us the first scaling result (33) Csiba, T.; Jannink, G.; Durand, D.; Papoular, R.; Lapp, A.; Auvray, L.; Boue´, F.; Cotton, J. P.; Borsali, R. J. Phys. II 1991, 1, 381. (34) Teraoka, I. Polymer Solutions; Wiley: New York, 2002. (35) Oono, Y.; Baldwin, P. R.; Ohta, T. Phys. ReV. Lett. 1984, 53, 2149.
The diffusion coefficient comes from the derived rules of length contraction and time dilation, (φF3)ν ∼ φ1/2F′ and (φτ)3ν ∼ φ2τ′, rewritten for simplicity in two separate frames. In this way
φ-νD3ν ∼ φ-1D′
(63)
and again, at the common scale, D′ f D:
D(φ) ∼ φ(ν-1)/(3ν-1)
(64)
Before going further, a couple of considerations are in order. The three scaling laws, for the average size, characteristic time, and diffusion constant were determined from a SbR analysis of cooperative diffusion. We expect them to be Brownianly covariant
12746 Langmuir, Vol. 23, No. 25, 2007
Mezzasalma
for a number of regimes, particularly for those in which the role played by chain entanglements is not crucial. Equation 58, in fact, applies to solutions at the Θ temperature and any volume fraction and to semidilute solutions in a good solvent,10 where the average size decreases more weakly as F ∼ φ-0.12 (ν = 0.588). At higher concentrations, excluded volumes are screened out and chain conformations get back to be (nearly) ideal (ν ) 1/2). Both eq 62 and eq 64 are equally describing the semidilute chain relaxation times and (the Zimm) diffusion coefficients, either in Θ or good solvents (for a thorough discussion, see still ref 30). The last law we are interested in, for the viscosity scaling, is more difficult to achieve from simple spatio-temporal considerations, but let us remember the former eqs 40 and 41. We wrote them in the variable N, but the generalized Lorentz factor accounting for the polymer concentration (eq 53) is written in the number of chain molecules per unit volume (φP). Thereby, one can work with a chain ensemble where N is fixed and φ is free to vary. By doing so, using the exponents of eqs 58, 62, and 64 in eq 40, viscosity is predicted to scale as
η(φ) ∼ φ1/(3ν-1)
(65)
This, however, is just the scaling result it had to turn out, giving the polymer contribution to (i.e., the specific) viscosity in unentangled semidilute solutions in Θ and good solvents.30 We thus come to the important conclusion that all such four quantities fulfill the covariant scaling eq 40 also in the variable φ. In formula, let
F ∼ Nνφν′,
τ ∼ Nσφσ′
(66)
D ∼ Nδφδ′,
η ∼ N φ ′
(67)
and
it turns out that1
δ + ν ) 3ν - σ ) 2
(68)
and, equally
δ′ + ν′ ) 3ν′ - σ′ ) -
′ 2
(69)
Within the present discussion, it is legitimate to wonder whether the available exponents for entangled polymer solutions satisfy again eqs 40 and 41, initially derived to model arbitrarily dense systems. Nicely, the values established by the reptation theory10 at the Θ point and in very good (i.e., “athermal”) solvents30
σ′ )
3(1 - ν) ν-2 3 , δ′ ) , ′ ) (athermal) 3ν - 1 3ν - 1 3ν - 1 (70) 7 14 7 (Θ point) σ′ ) , δ′ ) - , ′ ) 3 3 3
(71)
are solving eqs 69, still with ν′(ν ) 1/2) ) 0 (eq 58), i.e., 2δ′ ) -2σ′ ) -′. Throughout the preceding analysis, none of the quantities characterizing the entanglement dynamics could be addressed (e.g., strand relaxation times, Edwards’s tube diameter, correlation lengths, etc.). The next section will take advantage of the present scaling picture, suggesting a BwR interpretation of the entanglement problem in polymer solutions. Entanglement Effects. Path Configuration and (Light) Deflection. In Appendix V-A, the geodesic eq 11 is carried out
for evaluating the shape of Brownian orbits embedding a GSI metric. The final result, which we start from, is a known expression for the azimuth angle:12
∫
φ ) (lφ
x
a(F)
F - F2 - l2φ b(F) 2
d lnF
(72)
where lφ > 0 and g 0 are characteristic constants of the random configurations both in time and shape (i.e., the unit number N). The former is related to the ratio between shapelike and timelike curvilinear abscissas in orbits at fixed average dimension F. The latter reparametrizes the Brownian interval, (dσ/dσ′) ) x, only vanishing for the long-wavelength diffusions of a single liquid molecule and of a monomeric unit within a polymer chain, i.e., when their proper times can be normalized independently. The above relation was exploited in Einstein’s theory to prove that light rays bend toward massive objects (e.g., a planet mass), deflecting by ∆φ/2 ) |φ(rm) - φ∞| - π/2, where φ∞ is the incident direction and rm is the distance of closest approach to the mass.12 Here, it will be applied to model the influence of entanglements in a polymer solution. The new “scattering” phenomenology is depicted in Figure 4b, and visually compared in Figure 4a to a light deflection experiment. A liquid molecule ( ) 0) self-diffuses, from a point at infinity where the Brownian metric is Minkowskian, to the vicinity of a polymer chain. When these reference systems, one timelike and the other shapelike, do not interact at all, the single molecule would proceed its Brownian motion undisturbed, returning again to infinity by the opposite direction. Otherwise, any correlation will have the effect to “distorce” (or “deflect”) the unperturbed free particle trajectory. The outcome will be evidently a random path, bending toward (or away) the correlation source, and the line to be followed in BwR is 2-fold. We can (i) set an entanglement point (or event) along the path and (ii) calculate the azimuth angles, before and after it, through the difference ∆φ as a function of F. From EbP, this (“flat”) molecular trajectory possesses the same statistics of a (“curved”) macromolecular shape, the long-range correlations of which affecting the value taken by ∆φ ) φ(F) - φ∞ (Figure 4c). The last step is expliciting the azimuth difference (under ) 0). As (dF/dφ)Fm ) 0 (Figure 4a), it turns out lφ ) Fm/xb(Fm) (eqs 135-136), when eq 72 becomes
∆φ(F) )
∫F∞
x
a(r) d ln r B(r)
(73)
with B(r) ) B[b(r), r; Fm] ≡ (r2/F2m)(b(Fm)/b(r)) - 1. When the Brownian metric is Minkowskian (a ) b ) 1), configuration paths get back to ideal, with constant azimuth (∆φ ) 0). Note that, at the origin and detection points, γµν is unperturbed and φ is unambiguously defined. Hence, ∆φ should be connected to measurable properties, as will be checked up in the fortcoming subsection. Scaling BehaVior in Semidilute Solutions. Equation 73 is integrated in Appendix V-B from the entangled point Fe (≡ Fm) to the point at infinity of a GSI space, and the result may be approximated to
∆φ ∼ Fe/(2ν)
(74)
The quantity Fe is meant actually to be a (dimensionless) length scale, increasing with the entanglement correlations. Convergence reasons impose < 2ν, a condition which certainly applies in
Properties of Macromolecular Solutions
Langmuir, Vol. 23, No. 25, 2007 12747
the Zimm limit ( ) 3ν - 1; ν < 1), when monomer-solvent (in the chain pervaded volume) and monomer-monomer hydrodynamic interactions are strong enough. Actually, it also holds on approaching the Rouse limit (/2 j ν), where hydrodynamic (as well as excluded volume) interactions are screened out.16 The scaling eqs 40 and 41 were able to model concentrated (Gaussian) systems. We will examine here a semidilute entangled solution and focus on the universal regime at the scale of a chain portion of size somewhat proportional to the (static) correlation length (ξ). This quantity, separating smallscale single-chain from large-scale many-chain conformational statistics, is expected in turn to be proportional to the hydrodynamic screening length (ξH ∼ ξ), discriminating between the Zimm and Rouse limits.36 A suitable reference framework is provided again by the Edwards’s tube and de Gennes’s reptation concepts.25 Accordingly, the topological constraints produced by molecular entanglements would bound the polymer motion to proceed curvilinearly, across a tubelike region of given diameter (d) and contour length (LP) of its “primitive path” (i.e., the shortest “curvilinear axis” of the tube). In Gaussian systems, they connect by the random-walk statistics of N/Ne entanglement strands, each with segment number Ne = Nd/LP = (d/l)2. At the Θ point and in athermal solvents, the relationship d ∼ ξ is also expected.30 On this basis, consider Fe ∼ Ne-z. The exponent z (g0) may be discussed in light of the mean square azimuth deviation, σ2φ≡(∆φ)2, collecting positive and negative deflection contributions together. Equation 74 in fact is a local relation, referring to a single entangled point. To get a scaling behavior for σ2φ, it will suffice dividing its squared value (i) by the “overlap” volume fraction (φ*), at which the solution enters the semidilute regime, and (ii) by another statistical factor, weighting the entanglement formation. We know that the whole concentration domain is organized in four regions (dilute, semidilute unentangled, semidilute entangled, concentrated) by the overlap, entanglement, and semidilute-concentrated crossover volume fractions (0 j φ* j φej φc j 1, respectively). The latter is a constant value that, in athermal solvents, is φc = 1, making highly concentrated systems to be still semidilute. Thus, the other statistical weight can be set to the reciprocal of the concentration ratio giving the width of the semidilute unentangled domain, which is30
{
φe const (athermal) ∼ -1/4 (Θ point) φ* N and, after recalling φ* ∼
(76)
In athermal solvents σ2φ ∼ Ne-z/νN3ν-1 and, since ) 3ν - 1
σ2φ ∼
( ) N Nez/ν
3ν-1
(77)
we may require no other scale be coming into play (i.e., a scale homogeneity), and set (z ) ν)
Fe ∼ N-ν e
x(∆L )
2
P
∼
x
N Ne
(79)
going to zero with (d/LP) f 0. To face instead Θ solutions, the behavior in eq 75 should be dealt with. Precisely, when φe/φ* ∼ N-1/4, scale homogeneity requires a new z value, (z/ν) f z′(ν)/ν ) 1 - 1/[4(3ν - 1)], which is z′(1/2) ) 1/4. Let us turn now to a dynamical, timelike description. We want to express the chain time by the shapelike, configurational properties (Fe, F, σφ), and this can be done by eq 134 of Appendix IV, Feτ = σφF2, yielding
Fe ∼ σφDe
(80)
with diffusion coefficient De ∼ F2/τ. When σφ = constant, eq 80 identifies two universal regimes, in N and φ. The former corresponds again to the (high dilution) limit (d/LP) f 0, for σφ ∼ N0 and De ∼ Fe(Ne f N) ∼ N-ν scales as the Zimm diffusion coefficient. Correspondingly, when σφ ∼ φ0
τe(φ) ∼ Nze(φ)ξ2e (φ)
(81)
i.e., a scaling law for quantities defined at the entanglement spacetime scale (τ ) τe and F ) ξe). In particular, if F ≡ ξ and z ) ν, then τ ∼ ξ2d and Nνe = d ∼ ξ, identifying the strand relaxation time in each correlation volume to τξ ∼ ξ3, as it has to be. Nevertheless, eq 81 goes a little further, on larger length scales = (ξ, d]. Let τe ∼ φσe to denote the entanglement relaxation time of a strand, provided with Ne ∼ φνe, at the tube diameter scale d ∼ φνd. The previous constraint, i.e.
σe ) 2νd + zνe
(82)
is satisfied in both limits, characterized by16,30
3ν ν 1 σe ) ,ν ),ν )3ν - 1 d 3ν - 1 e 3ν - 1 (athermal, z ) ν) 5 2 4 σe ) - , νd ) - , νe ) 3 3 3 1 Θ point, z ) 4
(
)
(83)
(84)
(75)
N1-3ν:10
σ2φ ∼ φeφ*-2(∆φ)2
σφ ∼
(78)
Note that, on approaching the Rouse limit, the random-walk statistics given by F-1 e ∼ xNe∼d would establish. Alongside, azimuth deflection and tube length fluctuations25 would scale simultaneously as (36) Szydlowski, J.; Van, Hook, W. A. Macromolecules 1998, 31, 3266.
As mentioned below eq 81, eq 82 equally holds at the correlation length scale (ξ ∼ d), when strand relaxation time (τξ ∼ τe) and repeat unit number (g ∼ Ne) are referred to a correlation volume (∼ ξ3). Statistical Gauge Transformation and Electromagnetic Analogy. The coordinate transformation rule, between two frames O and O′, is
γ′µν )
( )( )
∂σ ∂σω γω ∂σ′µ ∂σ′ν
(85)
We address the simplest case in which one of the two lies in a Minkowski spacetime, O′ ) {ξµ, ηRν}, and the other is a Brownian system, O ) {σµ,γRν}:
( )( )
ηRβ ) γµν
∂σµ ∂σν ∂ξR ∂ξβ
(86)
We will not care here of the specific form to be assigned the probability density (or Green’s function), which coordinates are defined through. It only matters that a set of real or complexvalued statistical functions, PR ) PR(ξω), generally exists to
12748 Langmuir, Vol. 23, No. 25, 2007
Mezzasalma
well-pose the following definition
∫
σR[PR(ξω)] ) [ PR(ξω) d4 ξ]1/2
(87)
Consider thus a molecule in the local frame O′, endowed with (small) intrinsic fluctuations. The insight is that particles, in a way, “build” their own frames by performing random-walks under Brownian-relativistic constraints. We set out thus to focus on a coordinate change in O′, as detected by O, through the only variations of PR. Each σR will vary by the change of P that is consequent, at given γµν, upon ξµ f ξµ + δξµ:
δσR [PR(ξω)] ≡ 2
∫δPR(ξω)d4ξ
(88)
Accordingly, the variation observed by O turns out to be
( )[( ) ( ) ]
γµν ∂σµ2 δηRF ) µ ν R 2σ σ ∂ξ
ν2
∂δσ ∂ξF
-
ν2
( ) δξ
)
κ
∂σ δσ 2 ∂ξF 2σν
(89)
( )[( ) ( )( )]
γµν ∂σµ2 2σµσν ∂ξR
2
2
δξκ
0)η
(90)
)0
(91)
δξκ
2γµν ∂σµ Fκ 1 ∂2σν2 ∂σν ∂σν ) ν η 2 ∂ξF∂ξκ σ ∂ξR ∂ξF ∂ξκ
(92) or
γµν∂Rσµ0σν ) 0
0[G(ξω) exp(iqRξR)] ) 0
(95)
and coupled to a Lorentz-like condition
∂µ G ) 0
4iq ∂G Vc 0
(96)
(93)
where, for brevity, ∂R ≡ (∂/∂ξR) and, hereinafter, 0 ≡ ηRF∂R∂F 2 ) ∇2 - V-2 c ∂0 will denote the d’Alembertian in Cartesian coordinates, left indicated for some velocity Vc. We do not have room here to examine this relation in detail but use this opportunity to make a couple of remarks and a calculation. A first point is that the arbitrariness in eq 93 is of the same nature of the one shown by the solutions of the Maxwell and Einstein equations and can be eliminated by fixing a particular “gauge”. Thus, while the Lorentz condition is employed in electrodynamics for the 4-vector potential, the ambiguity in the metric tensor of the ordinary relativity can be removed by a particular reference system. A suitable choice is given by the “harmonic” coordinates, µ γRβΓRβ ) 0, meaning that their (invariant) d’Alembertian is vanishing, 0xµ ) 0.22 Hence, the analogy with eq 93 is rather evident. It denotes a “statistical gauge” invariance that, at first sight, seems to widen the class of probability equations for Brownian statistics. To ascertain this point, we address the concrete case of an experimentalist, who measured the scattering law of a system, S ) S(q, ω). It is a double-Fourier-transformed van Hove’s function (G), containing all information about temporal and spatial correlations in the wave vector (q) and frequency (ω) domains.11
(97)
q denoting the wave vector magnitude. We introduce now an intrinsic diffusion coefficient, D* ) ω/(4q2), a heat-diffusion operator, 0D/ ) ∇2 - 1/D*∂0, and a Wick’s rotation, t f it.39 It is an analytic continuation in time, frequently used in field theory and statistical mechanics, motivated here by the complexvalued coordinate definition (eq 94). Symbolically
[0D/ - 0 + ∇2]G ) 0
( ) ( ) [ ( ) ( )( )] δηκR
(94)
with qµξµ ≡ ωt - q ‚ r and d4ξ ) drdt. In this way, PRbecomes the product of a plane wave, solving a hyperbolic differential equation, times a typical solution of the parabolic heat-diffusion equation. To meet eq 93, a restricted gauge transformation38 is thereby adopted in the form
0G )
from which Fκ
∫ exp(iqµξµ)G(ξω)d4ξ
They may be easily joined together for a nondispersive wave (qµqµ ) 0)
Any metric variation will produce an energy-momentum perturbation (say, tµν ∼ δηµν), whose conservation law ((∂tµν /∂ξµ) ) 0) fixes the additional constraint
( )
2
2
∂2σν 1 ∂σν ∂σν 2 F κ F κ ∂ξ ∂ξ 2σν ∂ξ ∂ξ
δηκR
ΣR )
ν2
that, after expanding δPR ) (∂PR/∂ξκ)δξκ, implies
δηRF
The experimentalist can thus proceed with defining a generalized coordinate system (ΣR), from which recovering the second-order description (σR) somewhat upon |q|, ω f 0, i.e.
(98)
so that, under the nonrelativistic limit Vc ≡ c f ∞, then 0 f ∇2, and eq 98 reduces to a heat-diffusion equation, 0DG ) 0, when D* f D.
Conclusions The influence of long-range molecular correlations on universal laws arising in polymer solutions was accounted for by a general theory of Brownian relativity. The agreement witnessed by former analysis is confirmed here in several scaling regimes, both depending on the monomer number and polymer volume fraction. In our formulation, entangled polymer systems were modeled in terms of stronger spacetime distorsions caused by the entanglement points, acting in close analogy to the deflection of a light ray grazing a spherically symmetric matter field. The final calculation points out a family of statistical “gauge” transformations, in which the heat-diffusion equation represents a special case of a wider class of Lorentz-covariant probability equations for Brownian statistics. With regard to feasible issues left for future work, the most noteworthy are likely arising from macromolecular systems where geometry is strongly involved, such as amphiphilic structures (e.g., polymer-like micelles40,41), biomolecules (ranging from small peptides to proteins42), and (37) Gradshteyn, I. S.; Ryzhik, I. M. Tables of Integrals, Series and Products; Academic Press: New York, 1973 (38) Jackson, J. D. Classical Electrodynamics; John Wiley and Sons, Inc.: New York, 1962. (39) Wick, G. C. Phys. ReV. 1954, 96, 1124. (40) McKee, M. G.; Layman, J. M.; Cashion, M. P.; Long. T. E. Science 2006, 311, 353. (41) Mezzasalma, S. A.; Koper, G. J. M.; Shchipunov, Y. A. Langmuir 2000, 16, 10564. (42) Wales, D. J. Energy Landscapes; Cambridge University Press: Cambridge, 2003.
Properties of Macromolecular Solutions
Langmuir, Vol. 23, No. 25, 2007 12749
polymers of various flexibility near interfaces (where confinement geometry effects are determinant43).
(G) contributions, and g ) det gµν. The latter defines the EinsteinHilbert action (SG):
Appendix I
1 k′LG ) x-gR 2
Equation.12
A. Einstein’s Geodesic Consider a free falling reference frame ({ξλ}), where the motion is still geodesic:
( )
d2ξµ )0 ds2
(99)
where R is the so-called curvature scalar and k′ is used to recover the Newton theory in the nonrelativistic limit, G being the gravitational constant. Thus, one obtains
-2k′δSG )
with proper time:
ds2 ) -ηµνdξµdξν
(100)
and Minkowski tensor ηµν ) diag (-1, 1, 1, 1). To proceed, in a sufficiently small neighbourhood of {ξλ}, the motion should connect two spatial points by the shortest distance or, equivalently, two spacetime events by the longest path. In order to write the relation with another arbitrary frame:
ds2 ) -gµνdxµdxν
(108)
(
∫
Rµν -
)
gµνR δgµνx-g d4x 2
(109)
λ η + Γηµλ Γκη - Γµκ Γλλη
(110)
where Rµν is the Ricci tensor:
Rµκ )
( ) ( ) ∂Γλµλ ∂xκ
-
λ ∂Γµκ
∂xλ
the trace of which gives the curvature scalar:
R ≡ RRR ) gµνRµν
(101)
(111)
while the coefficient in the other variation: the free-falling system should be worked out in terms of the new coordinates ({xλ}) and metric (gµν). This is easily done after rewriting the geodesic equation as
( )( ) ( )( )( )
∂ξµ d2xν ∂2ξµ dxν dxλ + )0 ∂xν ds2 ∂xν∂xλ ds ds
(102)
δSM )
( ) ( )( )
( )( )
gµν )
( )( )
∂xτ ∂2ξµ ∂ξµ ∂xν∂xλ
Rµν -
(103)
(104)
Γ h FFF )
{( ) ( ) ( )}
∂gFν ∂gµν gλF ∂gFµ ) + 2 ∂xν ∂xµ ∂xF
(105)
Γ h Fφφ ) Γ h F00 )
(106)
∫ δLGd4x + ∫ δ(x-gLM)d4x ) 0
(107)
the Lagrangean density being a sum of matter (M) and gravity (43) Claessens, M. M. A. E.; Tharmann, R.; Kroy, K.; Bausch, A. R. Nature Phys. 2006, 2, 186.
Fsin θ a(F)
1 b′(F) 2 a(F)
(114)
θ θ Γ h Fθ )Γ h θF ) F-1
B. Einstein’s Equation.12,19 In the principle of stationary action (S) leading to the Einstein field equation, the metric tensor is imagined to undergo an arbitrary infinitesimal variation, gµν f gµν + δgµν, with the prescription that δgµν(|xτ| f ∞) ) 0. Correspondingly, the first-order variation of the action functional is required to be vanishing
δS ) δSM + δSG )
1 dlna 2 d(F)
F Γ h Fθθ ) a(F)
is related to the arbitrary metric through
Γλµν
(113)
A. Affine Connection and Geodesic Equation.12 Consider the diagonal metric with γ j 00 ) -b(F), γ j FF ) a(F), γ j θθ ) F2, and 2 2 γ j φφ ) F sin θ. Since its inverse is specified by γ j µνγ j µν ) 1, the only nonvanishing components are
β
∂ξ ∂ξ ηRβ ∂xµ ∂xν
gµνR ) k′Tµν 2
Appendix II (GSI Metric)
which, since R
(112)
where, in natural units (c ) 1), k′ ) -8πG.
Γτνλ denotes the components of the so-called affine connection:
Γτνλ )
∫ Tµνx-gδgµν d4x
defines the energy-momentum tensor (Tµν). Therefore, from the last and the last but two relations, the Einstein equation is
and some algeabric manipulations, leading to the well-known geodesic equation of motion: ν d2xτ dxλ τ dx + Γ )0 νλ ds ds ds2
1 2
Γ h θφφ ) -
sin 2θ 2
(115)
φ φ Γ h Fφ )Γ h Fφ ) F-1
Γ h φφθ ) Γ h φθφ ) cot θ
(116)
1 dlnb 2 dF
(117)
and τ τ )Γ h Fτ ) Γ h τF
where a prime indicates (d/dF). From them, the geodesic equation turns out to equal the annullment of the following 4-vector (from
12750 Langmuir, Vol. 23, No. 25, 2007
Mezzasalma
( )
top, µ ) τ, F, θ, φ), which we write as
0) 0)
∂Tφ ∼0 ∂φ
2
d τ dlnb dτ dF + dF dσ dσ dσ2
d2F 1 b′(F) dτ 2 1 dlna dF 2 + + 2 dF dσ dσ2 2 a(F) dσ F dθ 2 Fsin 2θ dφ 2 a(F) dσ a(F) dσ
( )
( )
( )
The other two conservation laws need some more elaboration. In the second angular constraint (ν ) θ), the only nonvanishing component of the affine connection is Γ h θφφ, hence
jτφφ
( )
d2θ 2 dF sin 2θ dφ 2 0) 2+ F dσ 2 dσ dσ (118)
B. Einstein’s Equation.12,19 For a GSI metric, the only nonvanishing components of the Einstein tensor are
b(F)
[1 - a(F) - Fψa] F2a(F) 1 G h FF ) 2[a(F) - 1 - Fψb] F F2ψb G h θθ ) gj(a, b; F) 2a(F) G h φφ ) sin2 θ Gθθ
a(F)τjFF
)
(120)
with the notation ψ... ≡ (dln.../dF). Solving the former equation determines the missing metric components, and it turned out1
∫
∫
b(F) ∝ exp(n jτ1/2 τ1/2 FF dF) ∼ 1 + n j FF dF (121)
a(F) ∼ τjFF,
with n , 1. Equations 40 and 41 descend from comparing the first-order variation of b to a corresponding concentration perturbation in a dense system.
As the energy-momentum tensor is diagonal, the continuity equation simplifies into
(
)
∂x-γτjνν +Γ h νµµτj µµ ) 0 (∀ ν) (122) ∂σν x-γj 1
to be worked out for a GIS metric (eq 116), where the affine connection coefficients are the same as in Appendix II-A. Since Γ h 0RR ) 0, the temporal constraint (ν ) 0) returns
( )
∂T0 =0 ∂τ
∂fθ ) fφ - fθ sin2 θ ) 0 ∂θ
(
(126)
)
(
)
ψa ∂ ln TF 1 ψb ) F-3(τjθθ + csc3 τjφφ) jτ00 + τjF ∂F 2 b a (127)
[
F-3ηP(fφ + fθ sin3 θ) ) sin3 θ 2F-1 +
(123)
i.e., T0 ) F2sinθτj00xa/b3∼F2FsinθxηP does not depend explicitly on time. Also the azimuth equation (ν ) φ) points out a similar condition. Let Tφ ) (τjφφ/F4 sin2 θ), where τjφφ = fφηP and fφ is a strictly positive constant coefficient, then
n η (1 + τj00) 2x P (128)
]
which, on averaging again over Ω, reduces to
(9π32 + 1)f F θ
-3
ηP ) 2F-1 +
n η (1 + τj00) 2x P (∀ fθ * 0; θ * 0, π) (129)
and, upon F f ∞
xη
Appendix III
µν ) jτ ;µ
( )
with TF ) F2sinθτjFFxb/a3∼F2sinθηP-1/2. After working out these expressions, one is left with
ψa 1 + F ψab - ψdb/dF ψb 2
(
(125)
since fθ is a (positive) constant too. We should note that the present equations are not well defined at the planes tangent to the surface that is bounding the polymer molecule. Here, in particular, θ * 0, π when F ∼ Nνl. That said, to relate the coefficients fθ, fφ and get rid of angular dependences, the last equation can be averaged over the solid angle (dΩ ) sin θdθdφ) by a uniform probability distribution function, p(Ω) ) (4π)-1, obtaining in the end fφ ) (2/3)fθ. F h FFF, Γ h θθ ,Γ h Fφφ * 0, is certainly The radial condition, with Γ h F00, Γ the most meaningful:
(119)
where the function g is specified by
1 + Fgj(a, b; F) )
) ( )
∂lnx-γ j ∂τjθθ sin 2θ φφ τj + ) ∂θ ∂θ 2
tan θ sin2 θ
d2θ 2 dφ dF dφ dθ + + 2cotθ 2 F dσ dσ dσ dσ dσ
G h 00 )
(
from which, and the position τφφ ) fθηP, it turns out
( )
0)
(124)
P
= C* F3
(130)
with constant prefactor C* ) 9nπ/[2(32 + 9π)]. The above equation requires, ηP . F2 . ηP-1, a condition that normally holds in entangled polymer dynamics.
Appendix IV A. GSI’s Geodesic Equation. Consider the geodesic equation:12,13,19
( ) ( )( )
ν d2σR dσλ R dσ + Γ h )0 νλ du du du2
(131)
generally parametrized by u ) u(σ). It must be solved in each σµ and, for a GSI Brownian metric, the starting equations are those in Appendix II-B. As in classical mechanics, a convenient way to proceed is seeking the integrals of motion. Thus, isotropy implies that limiting the description to the equatorial plane is sufficient, θ ) (π/2). A second integral of motion comes from
Properties of Macromolecular Solutions
Langmuir, Vol. 23, No. 25, 2007 12751
)l (dφ du )
F2
(132)
φ
where, in ordinary relativity, lφ stands for a constant angular momentum density (i.e., per unit mass). Here, it settles a constant ratio between curvilinear abscissas of pure timelike (u) and shapelike (w ) Fjφ) paths (dw/du ) lφ/Fj), what is naturally expected in our geodesic problem. A third originates from choosing the reparametrization as
φ ) (lφ
∫
( )
F - F2 - l2φ b(F)
d ln F
(137)
B. Polymer Path Azimuth Deviation. To work out the integral eq 73, recall eq 121 and set for brevity τjFF ∼ FR (R ) /ν). From
b(Fe) b(F)
du ) b(F) dτ
x
a(F) 2
) exp(-n
∫FFτj1/2 FF dF)
(138)
e
(133) one gets
so that, whenever b is nearly unperturbed, u approximates the Brownian time coordinate. These two relations can be obviously joint together for the azimuth and time coordinates, getting
B[F(k)] ) k2 exp[-ke(kζ+1 - 1)] - 1
dφ ) lφb(F) F dτ
Fe being the distance of closest approach to the Brownian path, while k ≡ F/Fe, ζ ≡ R/2, and ke ≡ n/(ζ + 1)Fζ+1 e . We sum over azimuth contributions in [Fe, ∞) and, since ke , 1, approximate the integral to
( )
2
(134)
Exploiting all such constraints in the geodesic equation yields a last integral of motion:
( )
()
lφ 1 dF ) - + a(F) dσ F b(F) 2
2
e
(135)
where, in relativistic mechanics, j ) (dσ/du) g 0 is a constant (dimensionless) energy per unit mass (equal to zero only for light rays). In BwR, it means a parametrization ratio, between different correlated path, the condition ) 0 only applying to liquid molecules. Now, instead of focusing on the time history of orbits (the explicit trajectory of motion), we will examine their shape. To obtain it, it is necessary to get rid of the temporal coordinate and express φ ) φ(F). It can be derived from eliminating u, yielding in the end:
a(F) dF 2 1 - 2 + F-2 ) - 2 4 dφ F l φb(F) lφ
( )
from which, a quadrature implies
∫F∞
(136)
x
a(F) d ln F = exp(-ke)Fζe B(F)
∫1∞ k 2- 1
(139)
ζ
xk
-1
dk (140)
which, if ζ < 1, converges to37
1 ∆φ ∼ B1/2(1-ζ),1/2 exp(-ke)Fζe 2
(141)
where Bm′,m′′ indicates the β function evaluated in m′ and m′′ and is a constant term. The stretched exponential in ke plays the role instead of a scaling correction, preventing φ from increasing indefinitely. However, for n , 1, the bound it sets lies far away the entanglement point and may be ignored. Acknowledgment. The author thanks Prof. Francisco Baralle (I.C.G.E.B-Trieste) for kind hospitality. LA701891M
