Kinetic models for polymers with inertial effects

Novel kinetic models for both Dumbbell-like and rigid-rod like polymers are derived, based on the probability distribution function $f(t, x, n, \dot n)$ for a polymer molecule positioned at $x$ to be oriented along direction $n$ while embedded in a $\dot n$ environment created by inertial effects. It is shown that the probability distribution function of the extended model, when converging, will lead to well accepted kinetic models when inertial effects are ignored such as the Doi models for rod like polymers, and the Finitely Extensible Non-linear Elastic (FENE) models for Dumbbell like polymers.


Introduction
In this paper we derive novel kinetic models for both Dumbbell like and rigid-rod like polymers in the presence of inertial forces. The model is to describe dynamics of the probability distribution function embedded in high dimensional configuration space due to inertial effects. We then prove that the limit equation of the new model when inertial force vanishes leads to current models with no inertial effects such as the FENE model and the Doi model, respectively. This illustrates consistency of our kinetic models with existing models.
The wide range of applications of polymer materials has attracted new areas of academic and industrial research. The synthesis of different type of polymers has enlarged the range of applications of polymer materials to areas where mechanical properties are important. Materials made up of macro-molecules such as polymers display properties that completely differ from those made from small molecules. The description of polymer dynamics is often based on large assemblies of molecules, the characteristics could be modeled in terms of their statistical properties.
Most polymers are long chains or branches of repeated chemical units. The full description of each atom in the polymer by molecular dynamics is not feasible for the huge computational effort. Coarse grained models are often expected with macroscopic space and time properties of complex fluids. Typical models such as beadspring chain for flexible polymers and the rigid rod model for liquid crystalline polymers have been established by the pioneers in polymer science.
In general, the flow modeling of polymers has to take into account the internal structure, characterized by both positional and orientational order of phases. Such incorporation is often done by adding new balance equations to those that govern structure-less Newtonian fluids. These new balances must be evaluated from the behavior of polymers. According to the relative size of the bending persistence size and the length of the polymer, two canonical types of polymers are widely studied: the Dumbbell model and the rigid-rod like model. In modeling motion of polymers, it is essential to explore an accurate method of solutions of the Langevin equation for particles undergoing Brownian movement (rotational or translational) under the influence of external fields. A large number of reviews, text books and monographs on the theory, applications and rheology of polymeric materials have appeared in the literature, see e.g. [18,19,20,17,10,11,36,8,33,24,38].
There are three main levels of description of polymeric fluids: atomistic modeling, kinetic modeling [11,20], and the macroscopic approach of continuum mechanics [36]. We shall exploit the kinetic approach. Models of kinetic theory provide a coarse-grained description of molecular configurations wherein atomistic processes are ignored altogether (Doi and Edwards [11], Bird et al [36], and Ottinger [33]). Kinetic theory models for polymer solutions are most naturally exploited numerically by means of stochastic simulation or Brownian dynamics methods [33]. A kinetic theory model when equipped with an expression relating stress to molecular configurations plays an important role in developing micro-macro methods of computational rheology [35,23]. In current kinetic theory models for polymers, the inertia of molecules is often neglected. However, neglect of inertia in some cases leads to incorrect predictions of the behavior of polymers. The forgoing considerations indicate that the inertial effects are of importance in practical applications, e.g., for short time characteristics of materials based on the relevant underlying phenomena.
It is thus the goal of this paper to model dynamics of the density distribution of polymers when the inertial force is no longer ignorable. More precisely we shall be particularly interested in modeling two canonical types of polymers: Dumbbell-like and rod-like polymers, which when inertial forces are not considered have been well understood. We first derive kinetic models including inertial effects from particle dynamics (continuum limit in the Brownian motion), we then show the limit of the augmented models when inertial forces vanish leads to the inertia-free model.
We now summarize our main results for two types of polymers.
1.1. Dumbbell-like polymers. A mcromolecule is idealized as an 'elastic dumbbell' consisting of two 'beads' joined by a spring which can be modeled by an endto-end vector n. Here n is in a bounded ball B(0, n 0 ), which means that the extensibility of the polymers is finite. Let f (t, x, n, p, q) denote the distribution function of Dumbbell-like polymers on the space variables x ∈ R d , d = 2, 3, the translational velocity p ∈ R d , the end-to-end vector n as well as the orientational velocity q ∈ R d . And t is the time. The novel kinetic model to be derived is where ζ is the frictional coefficient for the beads with mass m, k B is the usual Boltzmann constant, T is the absolute temperature, and F is the spring force between beads. The force usually derives from a potential, and has different forms for different models. For the well known FENE potential where H is a spring constant [36].
The above model under the scaling where Our result for Dumbbell like polymers could thus read as follows: Theorem 1.1. The limit ǫ → 0 of f is given by f 0 = ρM where ρ = ρ(t, x, n) ≥ 0 and M are given by Furthermore, ρ(t, x, n) satisfies the following kinetic equation Polymers are idealized as rods of fixed length. The orientation space is n ∈ S d−1 . Let f (t, x, n, p, ω) denote the distribution function of rod-like polymers on the space variables x ∈ R d , the translational velocity p ∈ R d , the orientational vector n as well as the angular velocity ω.
Here ω is on the tangent bundle T n S d−1 . And t is the time. Our rescaled kinetic model can be formulated as where Here ǫ denotes the inertial parameter similar to (1.2), u is the fluid velocity, and U is certain interaction potential of rods. R = n× ∇ n is the rotational gradient operator, and k B , T denote the Boltzmann constant and absolute temperature, respectively. ζ t , ζ r are frictional coefficients in x and n directions. Our result for rod-like polymers then reads as follows: Theorem 1.2. The formal limit ǫ → 0 of f is given by f 0 = ρM where ρ = ρ(t, x, n) ≥ 0 and M are given by Furthermore, ρ(t, x, n) satisfies the kinetic equation Our derivation of these kinetic models is based on establishing motion laws of polymer molecules, followed by a conversion into the kinetic description. The formal limit when inertia vanishes is justified by taking the classical approach for hydrodynamic limits. To this end a rescalling is adopted, so that the collision operator is set on fast scale, and the dissipation of the collision operator drives the states to the unique equilibrium states M . Derivation of models for Dumbbell-like polymers and the formal limit justification are given in §3, and those for rod-like polymers are given in §4. Some concluding remarks are presented in §5.

2.
Kinetic models for Dumbbell polymers 2.1. Equations of motion with non-trivial inertia. We consider a polymer consisting of two-beads connected by one-spring. Each bead as a coarse grained particle represents several chemical units and experiences four kinds of forces in the dilute case where there is no interaction for inter-and intra-dumbbells.
Assume the two beads are positioned at x 1 and x 2 , the Langevin equation of the beads are just balance of different forces expressed as Here W i are independent standard Brownian motions, ζ i are frictional coefficients for the i-th bead with mass m i , σ i = √ 2k B T ζ i by fluctuation-dissipation theorem, where k B is the Boltzmann constant, and T is the temperature. The spring forces F 1 = −F 2 = F by Newton's Third Law depends only on the end-to-end vector n.
A natural configuration for this underlying polymer includes both position of the polymer with these two beads connected to one spring and the end-to-end vector denoted by n = x 2 − x 1 . We assume that m 1 = m 2 = m and ζ 1 = ζ 2 = ζ. Then the difference and average of the above two equations gives The approximation utilizes the equation If the inertia were ignored for this two beads spring system, we would have the following stochastic equations (SDE), The differential operator or the infinitesimal generator corresponding to the process (x(t), n(t)) is given by with , denoting the Euclidean inner product in R d . The configuration of Dumbbell like polymers can be illustrated in Figure 1: When inertial force is significant, one has to take the inertial effects into consideration. We now rewrite the system (2.1)-(2.2) into a first-order systeṁ This system can be considered as a degenerate stochastic differential equation with a singular diffusion matrix of 4d × 4d. The differential operator corresponding to the process (x(t), n(t), p(t), q(t)) is also uniquely defined. In fact for the above dissipative stochastic differential equation with well-defined initial data, a unique solution (x, n, p, q)(t) up to an explosion time is ensured if u(x) and F are smooth functions of the configuration variables.

Kinetic description.
We now formulate the kinetic description of the above motion laws. For a given instant t, the random variables X = (x, n, p, q) can be characterized by a probability density function (PDF) f (t, x, n; p, q) defined by In words, the right-hand side is the probability that the random variableX falls between the sample space values X and X + dX for different realizations of the polymer motion. Thus the above statistical ODE system can be converted into a PDE of the form It is known that in the case with no inertial forces the kinetic equation is also of the Fokker-Planck type. The model follows from the motion law (2.3)-(2.4) as follows: where ρ(t, x, n) denotes the corresponding probability density function.
Remark 2.1. The polymer scale is much smaller than the fluid scale, the diffusion in space is often ignorable. Here we make no such distinction.
It is natural to ask whether the effective dynamics in (2.6) is dominated by (2.7) when inertia vanishes. As a first step we shall identify an effective equation satisfied by the limit of f when m ↓ 0, if the limit exists. In the remainder, we will always suppose that f is as smooth as needed.
2.3. Scaling. We make the following scaling to obtain where The remaining task in this section is to justify the following satisfying Q(M ) = 0 and ρ(t, x, n) solves the kinetic equation (2.7).
As usual the limit equation that is associated with (2.8) does not depend on details of the operator Q. Rather, it depends on Q possessing certain properties related to conservation, dissipation, and equilibria that are stated below. Note the operator Q acts on the augmented variables (p, q) only and leaves other variables (t, x, n) as parameters. Thus we list some properties of Q as an operator acting on functions of (p, q) only, which are essential in the derivation of the limit equation.
2.4. Properties of Q. Note that the operator Q(f ) has the following properties • The operator can be written as a conservative form • The conservation form leads to for every f ∈ L 1 (dpdq). This relation expresses the physical laws of mass conservation during the action of Q. Moreover, this is the only such conservation law. This means that • Dissipation. There is a nonnegative function η(f ) that is an entropy for the operator Q. This means that for η(f ) ∈ L 1 (dpdq) we have whose vanishing characterizes the local equilibrium of Q. For flexible poly- • Existence of equilibria. From the above dissipation property we see that the function f (p, q) such that Q(f ) = 0 forms a space identified through This equilibrium is unique up-to a constant multiplier. We normalize the density function then

2.5.
Asymptotic limit as ǫ ↓ 0. Equipped with these properties we now investigate the asymptotic limit. Define Then integration of (2.8) together with conservation property of Q yields We suppose that as ǫ ↓ 0 the limit of f is identified by f 0 . It follows from the equation . This implies that f 0 = M (p, q)ρ 0 (t, x, n), for the operator Q does not act on (t, x, n). Hence the probability density function f takes the following form Note that pM dpdq = qM dpdq = 0, we thus have (2.12) The limit of these flux functions depends only on the limit of g ǫ . Linearity of the operator Q and Q(M ) = 0 leads to Using the equation (2.8) we obtain Let g ǫ → g 0 as ǫ → 0, then it is clear that Again note that Q is a linear operator, we may assume the ansatz for g 0 as follows.
In comparison with the expression of Q(g 0 ) above, we see that it is sufficient to find a, b, c, d such as Q(a) = pM, These relations are understood in the sense that the operator acts on each component of the underlying vector. In order to identity (a, b, c, d) we state the following lemma: Lemma 2.1. Let Q be the collision operator and g ∈ L 2 (R p × R q ) such that gdpdq = 0. The problem We also note that from It remains to determine only a and b. Let a = αM p we have {a, c} · qdpdq = 0.
These enable us to determine the asymptotic limits of the fluxes.
Now the equation (2.10) divided by M dpdq asymptotically converges to This is exactly the equation (2.7) derived from ignoring inertial forces.
Remark 2.2. With these kinetic equations for Dumbell like polymers, it is natural to understand how these polymers contribute to the macroscopic flow governed by a coupled Navier-Stokes system with the stress determined by Here De is called Deborah number, which is the most important parameter in non-Newtonian fluids. Re and γ are the Reynolds number and viscosity ratio, respectively. The tensor force follows the case when inertial force is ignored, the derivation of the tensor force with inclusion of inertial effects is beyond the scope of this work.

Rigid rod-like polymers
Though many polymers are flexible, there is still a large class of polymers which are not flexible and assume a rod-like structure. Rod-like polymers have some peculiar properties and have attracted a great deal of attention.
We consider rod-like molecules in concentrated regime. Rod-like polymers can have only two kinds of motion, i.e., translation and rotation. The translational Brownian motion is the random motion of the position vector x of the center of mass, and the rotational Brownian motion is the random motion of the unit vector n (|n| = 1) which is parallel to the polymer. We shall build a kinetic model for the probability distribution of orientational motions of rod in every point of phase space (x, p). This serves as a microscopic equation, which is expected to be coupled with the macroscopic equation (the Navier-Stokes equation) for the fluid velocity.

The gradient is
The rotational gradient is

Divergence of a vector
The gradient in terms of ω is defined only for a given n, and will be understood from now on as n · ∇ ω f = 0, ∀f.
To be more specific, we regard the identical liquid crystal molecules as inflexible rods of a thickness b which is much smaller than their length L, as illustrated in Figure 2: where ζ t is the friction coefficient and u(x) is the fluid velocity field. If f is the probability distribution in x, the Brownian force is expressed as F B = −∇ x (k B T lnf ), where −k B T lnf is the chemical potential. For nontrivial inertial force, the distribution needs to be accounted in an extended environment with inclusion of p =ẋ. The corresponding translational Brownian force thus reduces to This can be justified by a similar derivation based on Brownian motions as that in Section 2. The scaled coefficient reflects balances between the friction force and the inertial force. Putting together we have the following translational motion laẇ In what follows we shall conveniently use the chemical potential to describe the Brownian force.

Rotational Brownian motion.
We consider a rod rotating with angular velocity ω, then the rotational motion can be described as where J is the moment of inertia and T is the total torque. 1. Rotational frictional force. Consider a rod of length L placed in a viscous fluid with fluid velocity u(x), where x is the center of mass of the rod.
The rod is parameterized by s, ranging from −L/2 to L/2, then the position vector of the s-point on this rod is written as Let v(s) and F (s) be the velocity of this point, and forces acting on it. The velocity v(s) is expressed by the angular velocity ω The frictional force at x(s) is where ξ(s) is the frictional coefficient, being symmetric ξ(s) = ξ(−s). Note that The frictional force thus reduces to Thus the total torque induced by the frictional force acting on the rod is Symmetry of ξ(s) = ξ(−s) leads to ξ 1 = 0. Then using n · ω = 0 we have where ζ r = ξ 2 denotes the rotational friction coefficient. 2. Thermodynamic potential force.
Let the potential be denoted by U . Then d(E+U ) = 0 gives dE = −dU = −∇ n U ·dn. Note that from |n| = 1 we have ω = n ×ṅ, which leads to ωdt = n × dn. We thus have T · (n × dn) = T · ωdt = dE = −∇ n U · dn. Therefore T × n = −∇ n U , i.e., T = −RU, (3.5) where R is the rotational gradient operator given by (3.1). 3. Rotational Brownian force. The following property will be used on our derivation of forces below. Lemma 3.1. For a fixed vector a, let x = a × y. Then for any function smooth g, ∇ y g = −a × ∇ x g and a · ∇ y g = 0, ∀g.
Proof. Let ǫ ijk as the usual permutation symbol, then x i = j,k ǫ ijk a j y k . We thus have This gives the desired relation. We now show the second claim.
which ensures that a · ∂ y f = 0 holds for any smooth function g For any fixed s the associated Brownian force is calculated by Note that for each fixed s and vector n, the mapping from p(s) to ω is well-defined. The result in Lemma 3.1 gives Thus the corresponding Brownian torque is determined by which when combined with the above calculations leads to Note that the moment of inertia for the rod is Usingω · n = 0, we thus have This together with all forces involved leads to the following motion laẇ n = ω × n, (3.9) where U is the interaction potential.
Here R x and R p is the configuration space for variables x and p. B is some localized kernel.
3.3. Kinetic equations. We shall derive a kinetic equation in the phase space (x, p, n, ω) with (x, p) ∈ R 3 × R 3 and (n, ω) ∈ T S 2 , where T S 2 := {(n, ω)| n ∈ S 2 , ω ∈ T n S 2 }, (3.11) which is usually called the tangent bundle of the manifold S 2 . We start from the continuity equation of a formal form This equation can be simplified when restricted on the tangent bundle: (n, ω) ∈ T S 2 . First we state the following Lemma 3.2. Let (n, ω) be any element of the tangent bundle T S 2 , mapped to another element (m,ṅ) ∈ T S 2 such that m = n,ṅ = ω × n, then for any smooth function f we have Proof. Let ǫ ijk be the usual permutation symbol, thenṅ l = jk ǫ ljk ω j n k . Further we have where δ ij is the Kronecker delta symbol. This gives the first relation in (3.13). The second relation follows from Using similar arguments we have the following: Lemma 3.3. Let g be any smooth vector function, then (3.14) Note from ω = n ×ṅ we haveω = n ×n leading toω · n = 0. Alsȯ n = ω × n,n =ω × n − |ω| 2 n.
Equipped with above lemmas and relations, we are able to reduce the equation (3.12). We only check the last two terms on the left-hand of (3.12). First, where R := n × ∇ n is the rotational gradient operator. For any smooth function g, Lemma 3.2 and Lemma 3.3 imply that ∇ṅg = −(n × ∇ ω )g.
This enables us to simplify the last term in equation (3.12) ∇ṅ · (nf ) = −(n × ∇ ω ) · ((ω × n − |ω| 2 n)f ) Therefore the effective kinetic equation becomes Note here the coefficient L 2 /12 has been absorbed in both the ζ r and the potential U , which is independent of both p and ω.

3.4.
Scaling. We are interested in the solution behavior when inertia vanishes. We now make the following scaling under this scaling the system (3.15) written into the new variables becomes where Our next task is to investigate the formal limit ǫ → 0 of this problem.
Theorem 3.1. The limit ǫ ↓ 0 of the f ǫ is given by satisfying Q(M ) = 0 and ρ(t, x, n) solves the following kinetic equation The above justification procedure remains valid.
3.5. Properties of Q. The operator Q here acts on the augmented variables (p, ω) only and leaves other variables (t, x, n) as parameters. Thus we list some properties of Q as an operator acting on functions of (p, ω) only, which are essential in the derivation of the limit equation. Set with ω satisfying ω · n = 0 for any fixed n ∈ S 2 . This form enables us to conclude the following Lemma 3.4. The operator Q has the following properties: (i) The operator Q can be written as Moreover, this is the only such conservation laws. In other words, for any f ∈ L 1 (dpd n ω), (iii) Dissipation. There is a nonnegative function η(f ) that is an entropy for the operator Q. This means that for η(f ) ∈ L 1 (dpd n ω) we have In the case of rod-like polymers, η ′ (f ) = f /M and the dissipation production is The left-hand side of this relation is non-positive as Thus ψ must satisfy both ∇ p ψ = 0 and ∇ ω ψ = 0. Then one has n · ∇ ω ψ = 0. These ensure that ψ must be independent of (p, ω). 3.6. Asymptotic limit as ǫ ↓ 0. We now investigate the formal limit of the problem, assuming all involved functions are smooth and convergence holds true as needed.
By property (ii) we have f 0 = ρM , with ρ = ρ(t, x, n) ≥ 0 and n ∈ S 2 . Note that Q acts only on (p, ω), ρ are functions of (t, x, n). To find this dependence, we use the generalized collisional invariants. We integrate the equation with respect to (p, ω) to find the continuity equation where the density and fluxes are defined by ρ ǫ (t, x, n) = f dpd n ω, Here the integration over ω is interpreted as for any fixed n with (n, ω) ∈ T S 2 . In the limit ǫ → 0, ρ ǫ → Cρ 0 and J ǫ i → CJ 0 i (i = 1, 2) with C = M dpd n ω, and we obtain (3.20) Thus we may assume that f ǫ = ρ ǫ (t, x, n)M (p, ω) + ǫg ǫ (t, x, p, n, ω), which leads to Q(f ǫ ) = ǫQ(g ǫ ). To determine the limiting flux we need to explore the limit of g ǫ . Using the kinetic equation (3.16) we have Let g ǫ → g 0 as ǫ → 0, then it is clear that Again note that Q(f ) is a linear operator, we may assume the ansatz for g 0 as follows.
In comparison with the expression of Q(g 0 ) above, we see that it is sufficient to find a, b, c, d such as Q(a) = pM, In order to uniquely determine {a, b, c, d} we state the following Lemma 3.5. Let g ∈ L 2 (dpd n ω) such that gdpd n ω = 0. The problem has a unique weak solution in the space H 1 (dpd n ω).
Proof. For each fixed n ∈ S 2 , we apply the Lax-Milgram theorem to the following variational formulation of (3.21): for all φ ∈ H 1 (dpd n ω). The function M is bounded from above and below on R d p × T n (S 2 ), so the bilinear form at the left-hand side is continuous and coercive on H 1 (dpd n ω). The right-hand side is a continuous linear form on H 1 (dpd n ω) due to the zero average of g over R d p × T n (S 2 ). Note also that from It is straightforward to verify that {b, d} · pdpd n ω = 0, {a, c} · ωdpd n ω = 0.
These enable us to determine the asymptotic limits of the fluxes.
In order to further simplify the above fluxes, we state the following lemma: Lemma 3.6. Consider the space p ∈ R 3 and (n, ω) ∈ T S 2 . For any vector A we have M (p · A)pdpd n ω = k B T A M dpd n ω, (3.26) M (ω · A)ωdpd n ω = k B T (Id − n ⊗ n)A M dpd n ω. (3.27) Apply this lemma to the above expressions to obtain A simple calculation shows for arbitrary A that (Id − n ⊗ n)(n × A) = n × A. Therefore Now the limiting equation (3.20) divided by C = M dpd n ω reduces to ∂ t ρ 0 + ∇ x · (u(x)ρ 0 ) + R · ((n × ∇ x u · n − RU ζ r )ρ 0 ) = k B T ζ t ∆ x ρ 0 + k B T ζ r R · Rρ 0 .
Regrouping with ρ 0 replaced by ψ leads to This is exactly the equation derived for rod-like polymers with no inertial force.
Let A = KB, the transformation gives n·ω=0 e −ω 2 /a (A · ω)ωd n ω = Here we have used the fact that

Concluding remarks
In this work we have derived novel kinetic equations for Dumbbell-like polymers as well as rod-like polymers. Inertial forces are taken care of by an augmented environment in an extended configuration space. In the case of rod-like polymers, the augmented space for orientation is just a tangent bundle of the usual sphere. We have also shown that the formal limit of the augmented equation recovers the usual inertia-free kinetic models explored in literature.