Diffusion of a sphere in a dilute solution of polymer coils
We calculate the short time and the long time diffusion coefficient of a spherical tracer particle in a polymer solution in the low density limit by solving the Smoluchowski equation for a two-particle system and applying a generalized Einstein relation (fluctuation dissipation theorem). The tracer particle as well as the polymer coils are idealized as hard spheres with a no-slip boundary condition for the solvent but the hydrodynamic radius of the polymer coils is allowed to be smaller than the direct-interaction radius. We take hydrodynamic interactions up to 11th order in the particle distance into account. For the limit of small polymers, the expected generalized Stokes-Einstein relation is found. The long time diffusion coefficient also roughly obeys the generalized Stokes-Einstein relation for larger polymers whereas the short time coefficient does not. We find good qualitative and quantitative agreement to experiments.
Transport properties of Brownian particles in suspensions are of great interest for all technological applications involving complex fluids such as food technology or oil recovery and they have been studied extensively experimentally (see, e.g., Dunstan and Stokes (2000); Brown and Rymden (1988)) and theoretically (see, e.g., Harris (1976); Felderhof (1978); Batchelor (1983); Medina-Noyla (1988); Dean and Lefèvre (2004)). The diffusion constant of a spherical tracer particle with radius in a simple solvent is to a good approximation given by the Stokes-Einstein relation
with the solvent viscosity and the thermal energy . As demonstrated, e.g., for the case of a tracer sphere in solution of polymers Ullmann et al. (1985); Phillies et al. (1985, 1997); Phillies (1985), the naive approach to replace the pure solvent viscosity with the macroscopic shear viscosity of the polymer solution (as measured in a viscosimeter) in general fails. The polymer solution in the vicinity of the moving sphere is not homogeneous. Even in equilibrium one observes depletion layers or density oscillations (depending on the interaction potentials between the polymers and between the polymer and the particle). In the vicinity of a moving particle, the flowing solvent rearranges the polymers leading to an enhanced polymer density in-front and a reduced polymer density behind the particle Penna et al. (2003), which leads to an enhanced friction Squires and Brady (2005); Rauscher et al. (2007); Gutsche et al. (2008), and to long-ranged solvent mediated effective interactions Dzubiella et al. (2003); Krüger and Rauscher (2007). The time scale for the build-up of these inhomogeneities in the solution in the vicinity of the moving particle is given by the diffusivity of the polymers and the particle size. For most systems this time scale is well separated from the corresponding microscopic time scale of the solvent Bedeaux and Mazur (1974); Pagitsas et al. (1979); Cukier et al. (1980), but rather close to the time scale of the particle diffusion (given by the particle size and its diffusion constant in the pure solvent).
For such systems, the mean square displacement is not linear in time and for the tracer particle one defines a time dependent diffusion coefficient via Dhont (1996)
with the particle position at time . indicates the equilibrium ensemble average. The short and the long time limit of are called the short and long time diffusion coefficients
respectively. The diffusion coefficient is related to the time-dependent mobility coefficient via the fluctuation-dissipation theorem, i.e., the generalized Einstein relation,
with . is the linear response mobility defined by the ratio of the average velocity of the particle and a small and constant external force that starts to act on the particle at . Therefore, at time , the distribution of polymers around the sphere is still in equilibrium, and the short time mobility is solely determined by the hydrodynamic forces on the tracer. Once the sphere is in motion, the distribution of polymers in the vicinity of the tracer becomes anisotropic: it is more likely to find a polymer in front of the sphere than behind it, which reduces the mobility of the sphere. After a sufficiently long time, the polymer distribution becomes stationary and the velocity is related to the force via the long time mobility . In general .
Most of the theoretical work focuses on the semi-dilute regime with effective or mean field models Ogston et al. (1973); Cukier (1984); Altenberger and Tirrell (1983); Langevin and Rondelez (1978), see Masaro and Zhu (1999) for a summary. The depletion of polymers near the surface of the colloid was considered in Ref. Odijk (2000). Recently, a model based on the reduced viscosity of the polymer solution near the colloid due to depletion was introduced Tuinier et al. (2006); Tuinier and Fan (2007). However, up to now, the distinction between short and long time diffusion coefficient has been considered only for suspensions of hard spheres Batchelor (1972); Felderhof and Jones (1983); Hanna et al. (1982); Batchelor (1983); Zhang and Nägele (2002). One main distinction between hard spheres and polymers is, that for hard spheres the hydrodynamic radius is equal to the particles radius, while for polymers, the effective hydrodynamic radius is in general smaller than the radius of gyration. A different hydrodynamic and sphere radius has been used as a model for charged colloids Cichocki and Felderhof (1991), but only the case of equal tracer and bath particles was considered. In this paper, we focus on the regime of a dilute polymer solution but we distinguish between the short and long time diffusivity. We idealize the tracer particle and the polymer coils as hard spheres concerning both, the direct and hydrodynamic interactions, but with a hydrodynamic radius which can be smaller than the interaction radius in the case of the polymers.
We model the tracer particle as well as the polymer coils as spherical overdamped Brownian particles with radii and , respectively, as shown in Fig. 1. The bare diffusion coefficients of the sphere and the polymers in the pure solvent are and (Indices and will denote the sphere and the polymer, respectively, throughout the paper.), respectively, which are calculated via the Stokes-Einstein relation in the pure solvent with viscosity . The hydrodynamic radius which enters the Stokes-Einstein relation is equal to for the tracer and for the polymers, see Fig. 2. We idealize the direct interaction between the tracer and the polymers as a hard-sphere interaction
denotes the distance of the centers of the tracer and a polymer. As a consequence, the center of mass of the polymers can approach the center of the tracer only up to a distance . In the dilute limit, we neglect the mutual interactions of the polymer particles, which reduces the problem effectively to a two-particles system of one tracer sphere and one polymer coil.
The dynamics of the system is described by the Smoluchowski equation for the probability density for finding the tracer sphere at position and the polymer coil at . The Smoluchowski equation is a continuity equation and with the corresponding probability currents we can define the velocity operators such that :
with the external force . and denote the gradient with respect to the position of the sphere and the polymer, respectively. The components of the symmetric diffusivity matrix
have the form
The projector is given by , with . The -dependent coefficients , , , , , and can be expanded in a power series in . We use the coefficients for spheres with no-slip hydrodynamic boundary conditions on their surfaces up to order in the distance according to Ref. Jeffrey and Onishi (1984). We quote the coefficients for two unequal spheres of radii and for the tracer and the polymer, respectively in appendix A. We therefore assume that the polymer interacts with the solvent like a solid sphere with radius , see Fig. 2. For the case of hard sphere suspensions, also lubrication forces have been taken into account Batchelor (1983). In the case considered here, the radius of gyration is always larger than , i.e., the hydrodynamically interacting spheres never come into contact and the far field expansion converges well. The diffusion matrices (9) are valid on the Brownian time scale and for small Reynolds numbers Dhont (1996).
Eq. (II) is translationally invariant since depends only on . As a consequence, is a function of only, and the hard interaction potential in Eq. (6) can be translated into a no-flux boundary condition on a sphere of radius
In thermal equilibrium (for which is a necessary condition) detailed balance holds and all components of the probability currents are zero. The equilibrium distribution and therefore also the initial condition for the dynamical problem Eq. (II) is therefore given by
with the average number density of polymer molecules . Far from the tracer sphere, the polymer distribution should be unaffected by the presence of the sphere and therefore equal to the corresponding equilibrium distribution, which yields the boundary condition
In the linear response regime, the average velocity of the tracer particle is given by ( denotes the time dependent non-equilibrium average)
such that we can calculate the short and long term diffusion coefficients from the solution of the Smoluchowski equation (II): once is known, Eq. (13) yields the mobility coefficient , from which we calculate the diffusion coefficients through the generalized Einstein relation Eq. (5). Since has to be finite for , the integration constant which appears when solving Eq. (5) for has to be zero.
Since only depends on , we have replaced the gradients with respect to the positions of the tracer and the polymer with . and are the result of the direct interactions and the Brownian force, respectively Dhont (1996).
In the short time limit, i.e., at , is given by the initial condition, i.e., by . By symmetry, both and are zero in this limit.
In the long time limit reaches a steady state for , which is given by the solution of the stationary version of Eq. (II) (for )
Since we are interested in the linear response regime, we expand in powers of up to linear order, i.e., we seek a solution of the form Dhont (1996)
Inserting Eq. (18) into Eq. (17) and keeping only terms linear in yields a second order linear differential equation for the coefficient . Since we expand the mobility matrix in Eq. (II) in a power series in up to order , we choose the following ansatz for
which turns this differential equation into an algebraic equation for the coefficients . It can be solved for the in terms of . Note that . finally is determined by the boundary condition (10) at short distances, which reads in terms of ,
We solve the above equations using the computer algebra system Mathematica.
Iii Short and Long-time diffusion coefficient
In the following we evaluate the three contributions to the average velocity of the sphere, i.e., , and from Eqs. (14) with . Since is explicitly multiplied by , in the linear response regime the coefficient in the first contribution has to be averaged with respect to the equilibrium distribution for all times . Therefore the short time diffusion coefficient of the sphere equals the first contribution to the long time diffusion coefficient
For evaluating the interaction velocity defined in Eq. (15) in the stationary limit, we note that the gradient of the potential is only nonzero at and points in direction . This leads to
With a partial integration with respect to the stationary Brownian velocity defined in Eq. (LABEL:eq:bro) is given by
Combining the three contributions, the long time diffusion coefficient of the sphere is finally given by
Eqs. (21) and (25) can also be stated as functions of mass density (which is often used in the experimental literature) of polymer coils rather than the number density by using , with Avogadro’s number and the molecular mass of the polymer coils . The coefficients and relating the mass density to the short and long time diffusion constants are then given by and , respectively.
The coefficients and for the short and long time coefficient are functions of , and and they can be written as the product of and a function that depends only on and . For hard spheres . For large polymers in good solvent conditions, approaches a universal value of , see Dünweg et al. (2002) and references therein. In order to be able to compare our results to the case of hard spheres Batchelor (1983), we introduce the polymer packing fraction and define
Fig. 3 shows and as function of for different values of between one and two.
For , i.e., for tracer particles large as compared to the polymer coils (this limit is often referred to as the colloid limit), the polymer solution as seen from the colloid behaves like a continuum, the probability distribution approaches and both, and converge to the continuum result
This leads to the generalized (sometimes also called effective) Stokes-Einstein relation for the diffusion coefficients
with the Einstein result for the zero-shear high-frequency limiting viscosity of the polymer solution
is the Newtonian viscosity of the (polymer free) solvent. Note that the next term in is of , while the difference between and is of .
For , i.e., in the so called protein limit in which the tracer particle is small as compared to the polymer coils, and the short time diffusion coefficient approaches the value in the pure solvent from below. The long time diffusion coefficient reaches a finite value which is smaller than :
It is important to note that our model is limited to the case were the colloid does not enter the polymer, i.e., the colloid must remain larger then the “mesh-size” of the polymer. In the protein limit, the diffusion coefficients are independent of hydrodynamic interactions since the small tracer does not perturb the solvent significantly. Both short and long time coefficient approach the value of the corresponding calculation neglecting hydrodynamic interactions. This simplifies Eq. (17) significantly and we get the analytic results and , with for .
For a very long polymer in good solvent conditions . The corresponding values of and as functions of are shown in Fig. 4. In contrast to the short time coefficient which decreases monotonically as a function of the long time coefficient has a maximum at . That means that for (), larger spheres are less hindered in their motion by the polymer coils than smaller spheres, while the situation is reverse for (). This is in agreement with experiments as demonstrated in Sec. V.1. The variation of over all is nevertheless rather weak (about %), which means that the generalized Stokes-Einstein relation is an acceptable approximation for all values of . However, this is only the case for , for which the limits of for and for are not too different. In contrast to the long time diffusion coefficient, the short time diffusion coefficient varies strongly as a function of , such that the generalized Stokes-Einstein relation for holds only in the limit . The maximum of at is the result of two competing effects. With decreasing (increasing ) the solvent flow field generated by the moving tracer is weaker and as a consequence the short time diffusivity increases, i.e., and, according to Eq. (25), decrease. On the other hand, the distribution of bath particles around the tracer gets more disturbed since the weaker flow field cannot transport the bath particles around the tracer such that these accumulate infront of the tracer, which reduces the tracer mobility Rauscher et al. (2007); Gutsche et al. (2008). For very large the decreasing short time coefficient dominates but at intermediate the accumulation of bath particles leads to a local maximum of . This mechanism only leads to a local maximum of for intermediate values of .
The long time diffusion coefficient for a tracer particle in a suspension of equal hard spheres is expected to be , see, e.g., Dhont (1996). Our theory yields a slightly larger value since we neglect lubrication forces at small particle distances (which are less important if the interaction radius is larger than the hydrodynamic radius, i.e., for ). For this reason for , while still beeing monotonic, shows the onset of a local maximum in our theory (see Fig. 3), which is in contrast to the results obtained in Refs. Batchelor (1983); Nägele (2003). For polymers with the same interaction radius as the tracer sphere we find . Therefore a tracer particle is much less hindered in its motion by a suspension of equal sized polymers than by a suspension of equal sized spheres. Because the polymer hydrodynamic radius is smaller than the hydrodynamic radius of the hard spheres, the tracer and the polymer interact less strongly via the solvent.
In order to test the accuracy of our results obtained with hydrodynamic tensors up to order , we repeated the calculation with hydrodynamic tensors of the next lower order (). The relative deviation of the two results are (), (), () and () for both long and short time results. vanishes for both and . As expected, the order of the expansion is less critical for , i.e., for small , and taking into account even higher orders should not change the results for significantly. Calculations with lower order approximations to the diffusion tensors ( or less) do not yield the correct generalized Stokes-Einstein relation, Eq. (28).
V Comparison to experiments
v.1 Stretched exponential and scaling exponents
Experimental values of the long time diffusion
with a dimensional constant . It has been noticed that the form (31) has unphysical limits for both and for and , because one expects a finite value for any . Rescaled versions have been suggested Tuinier and Fan (2007); Langevin and Rondelez (1978), where only the difference between the limits and are described by a stretched exponential. This rescaling makes a direct comparison of to experiments difficult, since experimental values are usually extracted from un-rescaled data. Apart from this, the general experimental findings for the protein limit () are (see (Odijk, 2000, Table 1)), i.e., decreases with . In the colloid limit (), was found to be almost independent of with a negative Phillies et al. (1985). The maximum of at (see Fig. 4) is therefore in qualitative agreement with experimental findings. Since is non-monotonic as a function of in our calculation, we conclude that cannot be described by a stretched exponential in over the whole range of size ratios. This is in disagreement with Ref. Tuinier and Fan (2007), in which a universal value and a monotonic behavior was found. This difference might be due to the fact that our prediction is valid in the dilute limit, while the model in Tuinier and Fan (2007) is based on a depletion layer which is more pronounced at higher densities. We also emphasize that the short time diffusion coefficient decreases monotonically with . In Ref. Tuinier and Fan (2007), the distortion of which gives rise to the difference between short and long time diffusion was not taken into account.
In terms of mass density , our result for the long term diffusion constant in Eq. (25) reads
As illustrated in Fig. 4, the dependence of on , i.e., on the ratio of the polymer radius of gyration to the size of the tracer particle is rather weak. We therefore neglect this dependence. Using the scaling of the polymer size with its mass, Doi and Edwards (1986), we find by comparing Eqs. (32) and (33)
In a good solvent, self avoiding walk statistics for a Gaussian chain lead to for the average size of the polymer Doi and Edwards (1986). Using this value, we find
with good agreement to the experimental value found in Ref. Phillies et al. (1985). Note that the generalized Stokes-Einstein relation also leads to the expression in Eq. (34) for . The exponent of the concentration is in our linear theory equal to unity by construction. In summary, all exponents compare well to the experimental findings, see Tab. 1. For polymers at -conditions with we predict .
|Experiment Phillies et al. (1985)|
|Eq. (33)||weak dependence|
|Experiment Odijk (2000)||–|
The phenomenological law (31) is best valid at semi-dilute polymer concentrations as stated in Ref. Phillies et al. (1985). It is in fact non-analytic in for . Despite this, the experimental data summarized in Phillies et al. (1985) also covers the dilute regime and shows similar behavior there, see Sec. V.2. However, we found only very few experiments Phillies et al. (1997) focusing on the dilute regime.
v.2 Quantitative comparison
Our model does neither include long ranged forces between the tracer particle and the polymer coils nor adsorption of the polymer on the tracer. This situation is realized in Ref. Ullmann et al. (1985): poly(ethylene oxide) (PEO) is a neutral polymer, so electrostatic interactions between the sphere and the polymers are absent, and polymer adsorption on the sphere is suppressed by a surfactant.
Neither the radius of gyration nor the hydrodynamic radius of the PEO polymers were measured in the experiments and we calculate them according to Ref. Devanand and Selser (1991) via
Fig. 5 compares the diffusion constants measured by light scattering (Ullmann et al., 1985, Fig. 3c) with our results, which are by construction linear in the polymer mass density . The tracer sphere (polystyrene) has a radius of nm and the polymers have molecular masses of 18500, and amu. The experimental data for the smallest polymer with amu in (Ullmann et al., 1985, Fig. 3c) were not useful at small concentrations due to large scatter. According to Ullmann et al. (1985) the diffusion constant of the tracer particle in the pure solvent was cms. Given these values, there is no fit parameter for the initial slopes in Fig. 5. The overlap concentration is 25.5, 7.1 and 3.2 g/L for the three polymers, respectively. Our theory, which is linear in the polymer concentration, is valid only for . In Fig. 5, the experimental points start to deviate from the straight line for smaller and smaller as increases. For amu, they deviate considerably for g/L. The agreement for concentrations much smaller than is good in all three cases. Note that according to our theory the short time diffusion constant is almost identical to since the polymer coils are small as compared to the tracer particle, i.e., , for the cases shown.
Fig. 6 shows the normalized diffusion coefficient for two different sphere sizes (322 and 51.7 nm with cms Ullmann et al. (1985)) in a solution with a polymer mass of amu. For small polymer concentrations the normalized diffusion coefficient depends only weakly on the sphere size. For the smaller sphere, the ratio of to is given by , such that a continuum theory is not expected to hold. For this value, long and short time coefficient should differ appreciably but the experimental values lie between our predictions for the short and long time diffusion constants. From Ref. Ullmann et al. (1985) it is not clear weather the experiments probe the long or the short time coefficient. Note that at higher concentrations, the larger sphere is less hindered in the diffusion by the polymers, which is consistent with a negative value of .
We developed expressions for the short time and the long time diffusion coefficient of a tracer sphere in a dilute solution of particles with a hydrodynamic radius which can be smaller than the hard sphere radius for the interaction with the tracer particle, e.g., polymers. Solvent mediated hydrodynamic interactions are taken into account up to 11th order in reciprocal distance, therefore neglecting lubrication forces at close distances. Calculating the diffusion coefficients is reduced to the solution of a system of 11 algebraic equations.
The results are in good agreement with experiments in the dilute regime, in which the diffusion constant depends in an affine way on the polymer density. While the short time diffusion coefficient decreases monotonically as a function of the size of the tracer particle, polymers with hydrodynamic radius comparable to the tracer size seem to be most efficient in decreasing the long time diffusion coefficient.
Our current model is limited to low polymer densities, but direct polymer-polymer interactions can be taken into account in the framework of a dynamic density functional theory Penna et al. (2003). Hydrodynamic interactions between the polymers and the tracer sphere can be taken into account in the same way as in this paper Rauscher et al. (2007) and recently dynamic density functional theory has been extended in order to take into account hydrodynamic interactions among the polymer coils Rex and Löwen (2008).
Acknowledgements.We thank R. Tuinier, G. Nägele and J.K.G. Dhont for discussions. M. K. was supported by the Deutsche Forschungsgemeinschaft in IRTG 667. M. R. acknowledges financial support from the priority program SPP 1164 “Micro and Nano Fluidics” of the Deutsche Forschungsgemeinschaft.
The coefficients , and , of the diffusivity matrix for a polymer (hydrodynamic radius ) and a spherical tracer particle of hydrodynamic radius read up to order (Ref. Jeffrey and Onishi (1984))
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