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Polymer Solutions

The article provides a brief general introduction into the concepts of scaling, universality, and crossover scaling, plus the blob concept that provides an intuitive picture of crossover phenomena. We present the most important static and dynamic scaling laws for unentangled uncharged polymer solutions, together with their test and refinement by careful computer simulations. A hoard of simulation methods has been developed for these systems, and these will be briefly discussed as well.


Introduction
Polymers exist in a variety of states and situations. They may appear as bulk systems or in confined geometries (like films) and they may be solid (semi-crystalline, rubbery, or glassy) or liquid. In the liquid state the macromolecules may form a dense melt, or they are dissolved in a solvent of good or poor quality. Finally, for a solution it makes a huge difference if the molecules are charged or not; in the former case one has a polyelectrolyte system.
Theoretical physics is mainly interested in the universal properties of polymer systems, i. e. those properties that do not depend on the details of the chemistry that defines the monomer. Lots of insight has been gained by deliberately discarding these details and rather focusing on the effects that result from the physics of macromolecules as such. The most important aspects here are (i) connectivity, i. e. the macromolecular architecture, which may be linear chains, rings, stars, combs, networks, etc. (in other words, the topology arising from bonded interactions); (ii) non-bonded interactions (here in particular the excluded-volume interaction, sol-vent quality effects, and possibly long-range electrostatics); (iii) (possibly) geometric restrictions; and (iv) (for solutions) the dynamic coupling between the motion of macromolecules and the flow of the surrounding solvent (the so-called "hydrodynamic interaction"). This is altogether the huge field of theoretical polymer physics, for which excellent textbooks (De Gennes 1979;Doi and Edwards 1988;Grosberg and Khokhlov 1994;Rubinstein and Colby 2003) exist. Even though the complicated chemistry has been replaced by simplified or coarse-grained models, understanding the physics is still a challenging and complicated problem, which one cannot simply "solve" by straightforward pencil-and-paper analytical theory. Rather, one has to rely on a combination of intuitive insight, theoretical approximations, experiments in the laboratory, as well as careful numerical studies of welldefined models.
The most successful computer models in the "universal" regime of phenomena are (i) simple lattice models and (ii) bead-spring models in the continuum. Both types of models can faithfully model connectivity and interactions (aspects (i) and (ii) of the previous paragraph); however, they have different strengths and weaknesses when it comes to further aspects. Lattice models are particularly well-suited for Monte Carlo studies of static properties, while bead-spring models are particularly amenable for studying the dynamics of systems with hydrodynamic interactions, and for applying Molecular Dynamics and similar methods, which are much easier to parallelize than Monte Carlo algorithms.
The present article attempts to provide some overview of the physics (statics and Brownian dynamics) of neutral polymer solutions in the bulk, and computer simulations that have provided confirmation and/or refinement of the underlying ideas. For polyelectrolyte solutions and dense melts, please see the contributions by C. Holm and G. S. Grest in this volume, respectively.

Scaling laws
A good deal of theoretical polymer physics is concerned with so-called "scaling laws". This is a very general concept, which has proven extremely useful not only for polymers, but also, e. g., in the theory of critical phenomena, or the study of turbulence. From an abstract point of view, scaling laws are nothing but a special case of the general observation that the mathematical structure of a physical theory is strongly restricted or perhaps even fully determined by the underlying symmetries. Here we deal with a special symmetry, which is the invariance under the rescaling of parameters. Suppose we consider a certain physical quantity Q and we are interested in its dependence on another physical quantity P. As an example, let us think about the dependence of the average size R of a polymer coil on the contour length L of the (linear) molecule. Let us further assume that we pick a certain value of P, P = p, as the basic unit for P. Let q be the value of Q for this particular P value. Then we may write the relation in dimensionless form, where F is a dimensionless function with F(1) = 1. Of course, we could also use a different unit system, by picking a different value p ′ for the P units, and the corresponding value q ′ for the Q units, such that we also have with (in general) another function G, G(1) = 1, where the rescaling factor is given by λ = q ′ /q. On the right-hand side we can similarly introduce φ = p ′ /p. Obviously φ will depend on λ , φ = φ (λ ). Now, the system is scale invariant if φ depends only on λ but not on the point from where the rescaling started (i. e. not on p, q), and if also G = F. In other words, scale-invariant systems are those that are characterized by a lack of intrinsic scale, or those where the physics does not provide some most natural unit system. In our polymer coil example this means that it does not matter how long the subchains are into which the full molecule is decomposed (as long as these subchains are long compared to the size of a chemical monomer, and short compared to the overall contour length). Since we can combine two rescaling transformations with factors λ and µ into a single one with factor λ µ, we have, for scale-invariant systems, and of course φ (1) = 1. By mapping this relation onto a differential equation, it is trivial to show that its solution is a power law, with an undetermined exponent α. Insertion into Eq. 2 yields or, with x = P/p, By picking the special value λ = x α , we see that F is also a power law, In other words, scale invariance automatically gives rise to power laws. For our polymer coil example this means where R 0 is the coil size for contour length L = L 0 , and the Flory exponent ν depends on the physical situation. For flexible (i. e. sufficiently long) polymers in three dimensions ν takes the value 1/2 for random-walk (RW) statistics, while it is roughly 0.59 for self-avoiding-walk (SAW) statistics, which applies in good-solvent conditions. Finally, for a chain that forms a collapsed globule due to attractive interactions, ν = 1/3. It is important to realize that the exponents of scaling laws are typically universal (and this is certainly true for ν). This is so because scale invariance means that the system "looks the same" after proper rescaling. Now, the idea of the renormalization group for polymers (Des Cloizeaux and Jannink 1991;Schäfer 1999) is that one should start from an original system and then subject it to a coarse-graining procedure, where several original monomers are lumped into new effective monomers. Iterating this, the chain more and more "forgets" its chemical details, while only the asymptotic scale-invariant structure remains -and this is the same for all original systems within a so-called "universality class". For polymers, all chains with relevant excluded-volume interactions belong to the universality class of SAWs, while those with turned-off excluded volume to the RW universality class. For simulations, the concept of universality implies that any model can in principle be used, as long as it falls into the universality class that one wishes to study. This in turn means that the construction of models is mainly guided by considerations of conceptual simplicity, computational efficiency, and convenience in general.
Further important universal quantities are amplitude ratios and crossover scaling functions. The latter will be discussed in the next section; the former are simply the ratios of prefactors of scaling laws in dimensionless form. For example, one can study various measures of the size of a polymer coil, i. e. the end-to-end-distance R E ≡ R 2 E 1/2 , the gyration radius R G ≡ R 2 G 1/2 , and the hydrodynamic radius Here N is the number of monomers of the chain, whose coordinates are denoted with r i . In the asymptotic long-chain limit the ratios R E /R G and R G /R H are universal numbers, taking the values √ 6 and 8/(3 √ π) for three-dimensional RWs.

Crossover scaling
There are many situations where one needs to consider the dependence of a quantity on more than a single variable. For example, in polymer solutions one is interested in the dependence of the coil size on degree of polymerization N, concentration c (total number of monomers per unit volume), and solvent quality. In such a situation, scale invariance is expressed by a straightforward generalization of Eq. 6: A particularly important case occurs if there are just two arguments, in which case we have such that apart from the power law x α 1 1 we also have a dependence on the "crossover scaling variable" x 2 ) is then called a "crossover scaling function". Typically the behavior becomes simple in the asymptotic limits In such a case, the crossover scaling function must itself asymptotically behave like an appropriate power law. Finally, if F describes the behavior of a universal ratio, it must itself be universal in the limit of long chains.

Blobs
Crossover phenomena in polymer solutions can conveniently be described in terms of so-called "blobs". A blob is a portion of the polymer chain that is composed of g monomers and has a typical extension (the "blob size") ξ . This length scale marks the crossover between two different behaviors, and typically an energy of k B T (thermal energy) is associated with it. The blob concept provides a nice pictorial description of crossover phenomena and is hence a very useful tool for deriving crossover scaling laws. The most important crossovers in the statics of polymer solutions are those between RW and SAW behavior, driven by (i) attractive effective interactions and (ii) concentration.
For a single isolated chain, the quality of solvent can be measured in terms of an effective interaction energy ε(T ), which measures the temperature-dependent degree of attraction between two monomers. At the temperature of the Theta transition (Lifshitz et al 1978;Schäfer 1999) (T = Θ ) the repulsive and attractive parts of the interaction cancel out, such that effectively the chain behaves as a RW. In the vicinity of T = Θ we may write ε(T ) = ε 0 (1 − Θ /T), which gives rise to a dimensionless interaction parameter z ⋆ = (k B T ) −1 ε 0 (1 − Θ /T ). For any z ⋆ > 0 the chain structure is asymptotically a SAW. However, if z ⋆ is small, the amount of repulsion is too small to disturb the RW statistics on small length scales. This gives rise to a thermal blob size ξ T corresponding to g T monomers, of which each has a size b, T . The blob size is found by equating the total energy in the blob with k B T , i. e. g 1/2 T z ⋆ ∼ 1, or ξ T ∼ b/z ⋆ . Visualizing a very long chain as a SAW composed of RW blobs, one finds R ∼ ξ T (N/g T ) ν or from which the relevant crossover scaling variable z = N 1/2 z ⋆ is read off. For chains that violate the condition N ≫ g T , Eq. 15 is generalized to where the crossover scaling function Next, let us consider the concentration-driven crossover from SAW to RW behavior in perfectly good solvent, as a result of Flory screening of excluded-volume interactions (De Gennes 1979;Rubinstein and Colby 2003). The overlap concentration c ⋆ is the concentration where an arrangement of perfectly swollen chains is just space-filling, i. e. c ⋆ ∼ N/R 3 ∼ b −3 N −(3ν−1) . At concentrations c that significantly exceed c ⋆ , the chains overlap. This gives rise to concentration blobs of size ξ c , containing g c monomers. Since on length scales below ξ c there is no overlap, the SAW structure is unperturbed in this regime. Conversely, on scales above ξ c we have RW behavior. Therefore, ξ c ∼ bg ν c and c from which the natural crossover scaling variable c/c ⋆ is read off. The generalization in terms of a crossover scaling function is while for c/c ⋆ ≫ 1 the power law Eq. 17 is recovered. A solution whose monomer concentration is small but whose chains are so long that there is nevertheless a strong overlap is called "semidilute". The semidilute regime ends at a concentration c ⋆⋆ where the blob size has shrunken to the monomer size, such that no SAW regime is left.
In the general case, the structure is determined by the interplay between concentration and solvent quality effects, or the competition between ξ T and ξ c . For the ideal picture of a semidilute solution, we have b ≪ ξ T ≪ ξ c ≪ R. On length scales below ξ T and above ξ c we have RW behavior. This is due to attraction on the small scales and due to Flory screening on the large scales. For length scales between ξ T and ξ c we have SAW behavior; this regime shrinks more and more upon deteriorating the solvent quality or upon increasing the concentration. This picture gives rise to the generic phase diagram shown in Fig. 1. For more details on the derivation, see the textbook literature, or the Supplemental Material of (Jain et al 2012a). It is important to note that all universal ratios can, in the asymptotic limit of infinite chain length, be expressed in terms of just the two crossover scaling variables z and c/c ⋆ .

Dynamic scaling
Polymer statics provides us with two important length scales, the monomer size b and the coil size R. In case the system needs to be described in terms of a blob picture, there may be a blob size, or possibly even more blob sizes, as additional important length scales. In a dense melt of long chains, there may also be entanglements, which give rise to a "tube diameter" as yet another important length scale. In the present article, we will only consider non-entangled systems, where the chains are either too short or too dilute to develop entanglements.
The idea of dynamic scaling for the Brownian motion of polymers may then be understood as follows: For each length l, there is an associated time τ(l). This time may be viewed as the time that a sub-chain of extension l (in real space, not along the contour) needs to move diffusively by its own size. Alternatively, we may also pick a single monomer and study the time dependence of its root mean square displacement, (∆ r) 2 1/2 . The time τ(l) would then be given by the time that passes until (∆ r) 2 1/2 = l. The underlying concept is here that the single-monomer motion must be consistent with the motion of the object as a whole. Now suppose we consider l values that are well within a regime that is bounded by two of the important scales mentioned in the previous paragraph, with no further important scale in between. Then we again have a lack of a naturally provided unit system, and this applies not only to lengths but also to times. Therefore, the dynamics in such a regime must be described by a power law which defines the dynamic exponent z (not to be confused with the crossover scaling variable of the previous chapter!), applicable to the regime under consideration. Let us first consider a system where no blobs occur. This can either be a dilute solution (c ≪ c ⋆ ) or a dense melt (c ≫ c ⋆⋆ ). The longest relaxation time τ R is associated with the coil size, Dynamic scaling then implies that the mean square displacement for times t with Furthermore, if D denotes the center-of-mass diffusion constant of the chain as a whole, we have For an isolated chain in solvent (dilute limit), the intra-chain hydrodynamic interaction (i. e. the strong coupling of the monomer motion to the flow of the surrounding solvent) gives rise to so-called Zimm dynamics (Doi and Edwards 1988). A characteristic feature of Zimm dynamics is that the diffusive behavior of the chain is essentially that of a Stokes sphere, D ∼ (k B T )/(ηR), where η is the solvent viscosity. More precisely, the approximate Kirkwood theory (Doi and Edwards 1988) predicts D = (k B T )/(6πηR H ). From this, one reads off z = 3.
In case the chain is in a dense melt, the hydrodynamic interactions are screened (more about this below). Therefore there are no hydrodynamic correlations in the monomer displacements, and if entanglements play no role (which is the case if the chains are not too long), then the relevant theory for the dynamics is the Rouse model (Doi and Edwards 1988). Here one simply assumes a homogeneous viscous background and each monomer has a friction constant ζ . The friction coefficients of these monomers simply add up to yield the total friction coefficient of the chain. Therefore D = (k B T )/(Nζ ). Since R ∼ bN 1/2 (in a dense melt we have RW statistics), D ∝ R −2 or z = 4.
In a semidilute solution, the length scale that governs the crossover from Zimm to Rouse dynamics is again the blob size ξ c . Up to this length scale, one has unscreened excluded-volume and hydrodynamic interactions, i. e. SAW statistics and Zimm dynamics. The corresponding time scale is the blob relaxation time τ ξ ∼ ηξ 3 c /(k B T ). The blob as a whole has a Stokes friction coefficient ∼ ηξ c . Computer simulations (Ahlrichs et al 2001) have shown that hydrodynamic interactions are unscreened as long as the time scale is significantly below τ ξ , regardless of length scales. This is reasonable, since on these short time scales all monomers just move with the flow, such that correlations exist even on length scales significantly beyond ξ c . Conversely, for times significantly above τ ξ , the blobs "feel" that they are not free to move (or that chain-chain collisions occur). Therefore, the blobs then dampen the hydrodynamic flow velocity u with a friction force per unit volume of (De Gennes 1976) ∼ ηξ c ξ −3 c u, which is a term that should be added to the Stokes equation for u. For such a situation, the flow field generated by a point force does no longer decay like 1/r (r: distance from the point force) but rather like exp(−r/ξ H )/r, where the hydrodynamic screening length ξ H ∼ ξ c . Hydrodynamic screening is thus understood as a randomization of hydrodynamic correlations, induced in essence by chain-chain collisions.
It should also be mentioned that melts do not exhibit strict Rouse dynamics in the dense limit, even if the chains are quite short. The reason is dynamic coupling of the chain motion to the viscoelastic modes of the "matrix", which gives rise to subtle corrections (Farago et al 2012b,a). Similarly, there are also subtle corrections to the RW statistics of polymer chains in a melt (Wittmer et al 2004). Both results have been obtained by careful computer simulations.

Simulations I: Statics
To study universal static single-chain properties, the method of choice is clearly Monte Carlo (MC) of lattice models, where chains are simply walks on a lattice. Mostly simple-cubic lattices are studied, but other lattice structures are permitted as well. The so-called pivot algorithm (Madras and Sokal 1988) is presently the most efficient method known. Here one randomly selects a sub-chain and rotates it by a random angle around a random axis. This is an MC trial move, which is accepted or rejected by the standard Metropolis criterion. Presently the fastest-known implementation is that by N. Clisby (Clisby 2010), where the information about the chain conformation is stored in a somewhat unconventional manner: Firstly, one defines a bounding box about the chain as a whole. Associated with it are global properties like number of monomers, end-to-end-vector, gyration radius, centerof-mass coordinate. Then one subdivides the chain into two sub-chains, for each of which the analogous information is stored. This is done recursively all the way to the monomer level, such that one obtains a binary tree. The coordinates of each box and its contents are stored relative to the coordinates of the coarser level. This makes it possible to move one such "container" as a whole without the need to ever touch the data of the finer levels that it contains. Similarly, overlap checks are done by checking the overlap between bounding boxes: If they do not overlap, then their contents will surely not overlap either. With such tricks it is possible to reduce the computational complexity of one pivot move to ln N, such that very long chains are accessible. A recent study (Clisby and Dünweg 2016) has thus been able to find for three-dimensional SAWs: ν = 0.58759700(40), R G /R H = 1.5803940(45), R 2 E /R 2 G = 6.253531(10). Similarly, accurate MC calculations have been able to study the Θ transition in three dimensions (Grassberger 1997) and in particular investigate the subtle logrithmic corrections to scaling that occur there. For this study the so-called "PERM" (prune-enriched Rosenbluth method) algorithm was used, where chains are grown step by step, and statistical criteria decide at each step if a chain is terminated, continued, or even branched to generate yet another chain. In this way, unbiased samples of long chains may be generated. Universal crossover scaling functions for the Θ transition were studied as well (Kumar and Prakash 2003), using the methodology of Brownian Dynamics (BD; we will discuss this method briefly below). This study emphasized the importance of appropriate extrapolation procedures: In order to find the crossover scaling function, one should work at a constant value of the crossover scaling variable (here z = N 1/2 z ⋆ ) and study the residual dependence of a universal ratio (like R G (T )/R G (T = Θ )) on the chain length. This residual dependence is a correction to scaling; therefore the asymptotic universal behavior is obtained after extrapolation N → ∞.
The crossover scaling for Flory screening in good solvent was studied utilizing a lattice model and MC simulation (Paul et al 1991). However, here the model was the so-called bond fluctuation model (Carmesin and Kremer 1988), where monomers do not occupy single sites but rather elementary cubes, while the connecting bonds may vary within limits. This allows to implement a MC dynamics that involves simply a random displacement of an elementary cube on the lattice. Indeed it was found that a crossover from SAW to RW statistics occurs, with a crossover length scale ξ c that exhibits the concentration dependence predicted by blob theory. Later, the same model was also used to reveal the corrections to purely Gaussian behavior in a melt (Wittmer et al 2004).
Finally, the double crossover that results from the competition between the Theta blobs and the Flory screening blobs has recently been studied by BD (Jain et al 2012a). The internal blob structure could not be resolved, for lack of sufficiently long chains; however the dependence of total-chain properties like the coil size on concentration and solvent quality was in perfect agreement with blob theory. The same was true for dynamic properties like the diffusion constant.

Simulations II: Dynamics
For studies of dynamics, one needs algorithms that faithfully reproduce the motion of the monomers, at least on the (typically long) time scales that one is interested in. Obviously, brute-force Molecular Dynamics (MD) will satisfy this condition, if it involves all particles in the system. This approach has been highly successful for the studies of melts (see contribution by G. S. Grest in this volume), and can in principle also be applied to the dynamics of solutions, where the hydrodynamic interactions are taken into account by explicit solvent particles. Zimm dynamics for a single chain could thus be successfully established (Pierleoni and Ryckaert 1992;Dünweg and Kremer 1993). For such studies of a single macromolecule in a simulation box one has to take into account that the latter is typically not much larger than the chain itself. Therefore, one has to deal with strong finite-size effects, which scale as R G /L, where L is the linear box size. This is a direct consequence of the long-range nature of the hydrodynamic interactions: Since the flow field generated by a point force decays like 1/r (r: distance from the force center), the correlations between the stochastic displacements of two distinct monomers are proportional to the inverse interparticle distance. The theory thus provides detailed quantitative predictions about the magnitude of such finite-size effects, and this in turn makes it possible to quantitatively check Zimm theory even in a finite-box situation.
However, for solutions MD is nearly always unnecessarily expensive and can rather be replaced by cheaper algorithms that simulate the solvent degrees of freedom in a simplified fashion. The only situations where this is not true are either fairly concentrated solutions, where the solvent contribution to the computational effort is only moderate, or investigations of local atomistic dynamics, where local packing and similar phenomena are of specific interest. In all other cases, the effect of the solvent can be summarized by (i) its quality, which may be modeled by just a suitable effective momomer-monomer interaction, and (ii) the hydrodynamic interactions, which give rise to dynamic correlations between monomers as a result of momentum transport through the solvent. The crucial observation is here that the solute-solvent system is characterized by a large separation of time scales: The slowest degree of freedom in the solvent is diffusive momentum transport, characterized by the "kinematic viscosity" η kin = η/ρ, i. e. the ratio between shear viscosity and mass density, which has the dimension of a diffusion constant. The dimensionless "Schmidt number" Sc = η kin /D then relates this to the diffusion constant D of an immersed particle, or, more generally, to the diffusion constant of some immersed soft-matter object of size R. Typically, in dense fluids Sc ≫ 1 even for solvent particles, due to a sizeable viscosity value -note η kin ∝ η but D ∝ 1/η. For large (isolated) macromolecules, the corresponding Schmidt number is yet much larger, as Sc ∝ 1/D ∝ R. For these reasons, we may either replace the solvent by some sort of "generalized hydrodynamics solver", i. e. a set of more or less artificial degrees of freedom that exhibit the correct hydrodynamic behavior on large length and time scales, or dispose of the solvent altogether, by assuming that the flow field follows the configuration of monomers instantaneously, such that it becomes completely enslaved to the latter, and thus no longer appears as an explicit degree of feedom.
Let us begin with the latter approach. This is the realm of Brownian Dynamics (Öttinger 1995). Here one solves a discretized stochastic differential equation for the monomer coordinates r i , using a finite time step h. The update rule can then be written as Here Greek letters indicate Cartesian indexes with Einstein summation convention. F j is the force acting on particle j, while ↔ µ i j is the mobility tensor that describes the hydrodynamic correlations between the monomers i and j. Typically, the Rotne-Prager tensor (Öttinger 1995) is used. In case one is not interested in correct solution dynamics, one may simply turn the hydrodynamic interactions off and replace ↔ µ i j with a multiple of the unit tensor. In this case, the method will produce Rouselike dynamics for a single-chain simulation. The last term of Eq. 21 denotes the stochastic displacements, where q iα are random variables with while the matrix σ iα, jβ satisfies ∑ k σ iα,kγ σ jβ ,kγ = µ iα, jβ .
This approach was pioneered by a seminal paper nearly forty years ago (Ermak and McCammon 1978) and has seen many refinements since then. The main difficulties are the evaluation of the mobility tensor, which couples all monomers in the system, and the calculation of a suitable square root. These problems have remained a computational challenge for decades. Standard Ewald sums for multi-chain systems (Jain et al 2012b) have met moderate success, but only recently has a method been published (Fiore et al 2017) whose computational effort scales strictly linearly with the number of involved monomers. The generalized hydrodynamics solvers are technically much easier and also much more easy to parallelize. They also scale linearly with the number of monomers, however at the expense of an additional large set of explicit solvent degrees of freedom. These solvers all include thermal fluctuations in some way or another. This is necessary because in soft-matter physics we are dealing with length and time scales that are so small that fluctuations play a role. Obviously, Brownian motion of polymer chains could not be studied if fluctuations were absent. Therefore such methods are not fully macroscopic but are rather frequently called "mesoscale" methods.
One can distinguish two classes of mesoscale methods, depending on the way how thermal fluctuations are treated. The first class, which one may call "MD-like", are particle methods where the amount of thermal fluctuations per degree of freedom is similar to what one would get in an MD simulation. Peculiar to these methods is the impossibility to adjust the degree of thermal fluctuations independently of the macroscopic fluid properties that are relevant for hydrodynamics. Conversely, in the second class, which one may call "hydrodynamics-like", the degree of thermal fluctuations can be adjusted independently of the macroscopic properties. The degree of thermal fluctuations is here a reflection of the degree of coarse-graining: The more atomistic particles are lumped into one mesoscale degree of freedom, the smaller is the amount of thermal fluctuations per mesoscale degree of freedom -simply as a result of Gaussian statistics and the law of large numbers.
As this aspect is typically under-emphasized in the literature, let us illustrate this by a very simple example, a one-dimensional ideal gas, which we simulate by MD, augmented by a Lowe-Andersen thermostat (Lowe 1999) to bring the system to thermal equilibrium. This thermostat simply picks, from time to time, a pair of nearby particles at random. The center-of-mass velocity of that pair then remains unchanged, while the relative velocity is chosen at random, using the appropriate equilibrium Maxwell-Boltzmann distribution, such that the total momentum is conserved. The thermal (root mean square) velocity of a particle is then (k B T /m) 1/2 , where m is its mass. This has macroscopic relevance, since this is also the speed of sound. Now let us assume that we lump M adjacent particles into a new mesoscale particle. The new system is then again an ideal gas, which we wish to simulate with the same method. We then have two choices concerning the question of the mass of the mesoscale particles: Either we can assign the value Mm, which is naively the correct choice, since the bigger particle should indeed exhibit more inertia. Moreover, the thermal velocity (i. e. the amount of thermal fluctuations) is indeed correctly reduced by the factor M −1/2 . However, this comes at the price of also reducing the speed of sound by the same factor -and this is a value that we would prefer to keep constant, in order to maintain the time-scale separation between immersed soft-matter objects and the sound waves. Therefore one typically chooses the value m, thus keeping the macroscopic properties intact, but overestimating the degree of thermal fluctuations. In other words, MD-like methods are typically too restrictive to permit a fully consistent coarse-graining. This dilemma is solved by the hydrodynamics-like methods, where thermal fluctuations are an add-on with adjustable strength to a method that would also work in the strict macroscopic limit with no fluctuations whatsoever.
Dissipative Particle Dynamics (DPD) (Espanol and Warren 1995) is directly derived from MD, which is just augmented by a momentum-conserving Langevin thermostat. Similarly to the Lowe-Andersen method, DPD is based upon pairs of nearby particles, which are however not chosen at random but rather considered in their totality, at every single time step. The projection of the relative velocity onto the interparticle axis is dampened by a Langevin friction. This is compensated by stochastic Langevin forces on the two partcles that also act along the interparticle axis and add up to zero. The total momentum is conserved, and the fluctuationdissipation theorem (FDT) is satisfied. A generalized version also thermalizes the velocity components perpendicular to the axis (Junghans et al 2008), however, it is presently not yet fully understood what effects the implied violation of angularmomentum conservation has on the hydrodynamics.
Quite often, DPD simulations are run with particles that have fairly soft interaction potentials. This is done in the spirit of coarse-graining, which in general leads to such softening of interactions. It also has a practical implication, since softer potentials also allow to use a larger time step. The most radical implementation of that idea is to simply run DPD of an ideal gas as a solvent for soft-matter objects (Smiatek et al 2008). Using an ideal gas has a huge advantage: The solvent degrees of freedom are reduced to their prime function, which is to transmit momentum through the system, and the equlibrium structure of the immersed objects is unaltered compared to immersion in vacuum. The viscosity can nevertheless be adjusted to reflect dense-fluid conditions, by choosing a sufficiently strong friction.
A yet simpler variant is Multi-Particle Collision Dynamics (MPCD) (Gompper et al 2009). Here the ideal-gas particles are sorted into cubic cells. In each cell the algorithm determines its local center-of-mass velocity, and the relative velocities of the particles with respect to it. The latter are then subjected to a random rotation. This "collision step", which conserves both the momentum and the kinetic energy, serves to thermalize the ideal gas and is followed by a standard MD "streaming step".
Both DPD and MPCD are "MD-like", with a coupling of the monomers to the solvent that arises naturally from the setup of the respective algorithms. We will now turn to the "hydrodynamics-like" methods.
Smoothed Dissipative Particle Dynamics (SDPD) (Espanol and Revenga 2003) has been developed to cure the abovementioned deficiencies of DPD. The name suggests a closer proximity to DPD than the method actually exhibits. While DPD comes in spirit fron MD, as essentially a bottom-up approach, SDPD rather is a top-down method: Here the starting point is Smoothed Particle Hydrodynamics (SPH) (Monaghan 2005), which is nothing but a discretization of the Navier-Stokes equations in terms of particles. This looks deceptively similar to MD but is fundamentally different: Firstly, in MD both the equation of state and also the transport coefficients like the viscosity are an output of the atomistic model, and must be determined by simulation. Conversely, in SPH they are input parameters. Secondly, MD particles have as only properties their coordinates and momenta (and possibly their orientations and angular momenta). SPH particles, on the other hand, have additional properties "on board" that one could not even define for MD particles beause their nature is genuinely thermodynamic -volume and entropy, which both change in the course of time as a result of the dynamics. SDPD adds Langevin noise to the SPH equations of motion such that the FDT is satisfied. Although the SDPD particles are thermodynamic objects, it is nevertheless possible to simply connect a set of them via springs and thus obtain an immersed polymer chain with the correct large-scale properties (Litvinov et al 2008). The polymer-solvent coupling is therefore as straightforward as for DPD and MPCD.
Instead of discretizing the Navier-Stokes equations in terms of particles, one may also discretize them via a lattice. One therefore arrives at standard finite-difference or finite-volume schemes (Donev et al 2010;Balboa-Usabiaga et al 2012). Again, one may add thermal fluctuations to the equations to satisfy the FDT.
Finally, one may also simulate hydrodynamics via the Lattice Boltzmann (LB) (Dünweg and Ladd 2009) method. Here one solves a linearized and fully discretized version of the Boltzmann equation known from the kinetic theory of gases. Space and time are discretized in terms of a lattice spacing a and time step h, respectively. Velocity space is also discretized and reduced to a small discrete set of velocities c i . Each lattice site contains a set of real-valued positive variables n i , which are interpreted as the mass density corresponding to velocity c i . The mass density ρ and the momentum density j are then obtained as zeroth and first velocity moment of the populations, The procedure then begins with a collision step, i. e. a re-arrangement of the populations on the site such that mass and momentum are conserved: where the collision "operator" ∆ i satisfies This is followed by a streaming to the adjacent lattice sites, such that the total procedure can be written in terms of the Lattice Boltzmann equation (LBE): This implies that the discrete velocities must be chosen commensurate with the lattice. For example, the popular D3Q19 model (Dünweg and Ladd 2009) which lives on the three-dimensional simple-cubic lattice, involves nineteen velocities, which correspond to the six nearest and twelve next-nearest neighbors, plus the zero velocity.
The method involves lots of adjustable parameters, like the set of velocity shells, associated weight coefficients, and various details of the collision operator. All of these are tuned in order to obtain the correct Navier-Stokes behavior in the continuum limit, which is found from the algorithm by subjecting the LBE to an asymptotic (Chapman-Enskog) analysis. The LBE can therefore be used as a Navier-Stokes equation solver in its own right. Thermal fluctuations are introduced by adding a suitably chosen stochastic collision operator to ∆ i . For further details, see Dünweg and Ladd (2009).
It should be emphasized that in all of the abovementioned "hydrodynamics-like" methods it is very important to make sure that the FDT is not only satisfied in the asymptotic continuum limit, but rather for the algorithm as such. Substantial effort has gone into the development of methods that do satisfy this condition.
In contrast to particle methods, hybrid methods that involve MD for the polymer chains and a lattice algorithm for the solvent need special care for the fluid-particle coupling. A particularly simple approach is a frictional coupling (Dünweg and Ladd 2009), where each monomer is assigned a Stokes friction coefficient. Therefore each monomer is not only subject to the conservative forces coming from other monomers (and possibly yet other sources) but also to a friction force and a Langevin stochastic force. The former dampens the relative velocity of the particle with respect to the local flow field, which is obtained via interpolation from adjacent lattice sites. The latter is just standard Langevin noise that is needed to satisfy the FDT. Back-coupling is obtained by interpolating the thus-resulting momentum transfer back to the lattice and enforcing momentum conservation. Another possibility is to enforce a stick boundary condition, either on the surface of an extended particle (Dünweg and Ladd 2009) or based upon a point-particle picture (Usabiaga et al 2013).
At the end of this section, we briefly wish to mention a few studies that have focused on polymer solution dynamics. Zimm dynamics of a single chain has been studied by BD by many authors, e. g. Fixman (1986); Liu and Dünweg (2003); Sunthar and Prakash (2006), where the last study also investigated the solventquality driven crossover behavior. Single-chain Zimm dynamics was also studied by LB/MD (Ahlrichs and Dünweg 1999), MPCD (Mussawisade et al 2005), and SDPD (Litvinov et al 2008). Not surprisingly, all these studies yield essentially the same results, and it is even possible to quantitatively map them onto each otherthis has explicitly been done for LB/MD vs. pure MD (Ahlrichs and Dünweg 1999) as well as for LB/MD vs. BD (Pham et al 2009).
Detailed studies of the concentration-driven crossover from Zimm to Rouse dynamics have been done by both LB/MD (Ahlrichs et al 2001) and MPCD (Huang et al 2010). Both confirmed the picture of hydrodynamic screening as outlined in Sec. 5, and the latter paper went even beyond to also study non-equilibrium behavior.

Summary
Polymer solution statics and dynamics is a beautiful piece of physics where progress has been made by analytical theory (in particular scaling considerations), experiments, and computer simulations. Improved physical and mathematical insight led to the development of computer simulation methods that went from simple and fairly brute-force to more and more sophisticated and problem-oriented, focussing on the essence of the phenomena one wishes to study. The author hopes that the present contribution has given the reader a glimpse on how fruitfully theory and simulations have worked together in this field. For reasons of both space and also expertise of the author, the present article has only focused on the most basic equilibrium phenomena and completely left out the highly important field of non-equilibrium physics, i. e. nonlinear polymer solution rheology, which would be worth yet another article in this series.