Derivation of Two-fluid Model Based on Onsager Principle

We first need to identify the state variables that characterize the non-equilibrium state of flowing polymer solutions. We choose the state variables as follows:

• •

  Volume fraction of the polymer $\phi$.

  The corresponding “velocity” variable is $\mathbf{v}^{(p)}$, the polymer velocity. The polymer volume fraction and the polymer velocity are related by the conservation law

  $$\dot{\phi}=-\bm{\nabla}\cdot\Big{(}\phi\mathbf{v}^{(p)}\Big{)}=-\nabla_{\alpha}\Big{(}\phi{v}^{(p)}_{\alpha}\Big{)}.$$

  (3)

  Here $\dot{\phi}=\partial\phi/\partial t$ and $\nabla_{\alpha}=\partial/\partial x_{\alpha}$.

• •

  Conformation tensor $\bm{\mathsfit{c}}$.

  $\bm{\mathsfit{c}}$ is a non-dimensional tensor to characterize the microscopic state of the polymer chain. The $\bm{\mathsfit{c}}$-tensor is equal to unit tensor $\bm{\mathsfit{I}}$ when the polymer is at equilibrium, and deviates from $\bm{\mathsfit{I}}$ when the polymer is deformed. Late we will introduce the dumbbell model that presents the polymer chain as a dumbbell consisting of two beads at positions $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$. These two beads are connected by an elastic spring that has an end-to-end vector $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ and a spring constant $k$. The conformation of the dumbbell is then specified by the $\bm{\mathsfit{c}}$-tensor defined by $\bm{\mathsfit{c}}=\frac{k}{k_{B}T}\langle\mathbf{r}\mathbf{r}\rangle$.

  The corresponding “velocity” variable is the material time derivative of $\bm{\mathsfit{c}}$ defined by

  $$D_{t}^{(p)}\bm{\mathsfit{c}}=\frac{\partial\bm{\mathsfit{c}}}{\partial t}+\mathbf{v}^{(p)}\cdot\bm{\nabla}\bm{\mathsfit{c}},\quad{\rm or}\,\,\,D_{t}^{(p)}c_{\alpha\beta}=\frac{\partial c_{\alpha\beta}}{\partial t}+{v}^{(p)}_{\gamma}\nabla_{\gamma}c_{\alpha\beta}.$$

  (4)

  Notice that here we are using the polymer velocity $\mathbf{v}^{(p)}$ to define the material time derivative, not the medium velocity $\mathbf{v}$ which will be introduced next.

• •

  In order to discuss the phenomena of diffusion or migration of polymers, we need to introduce another “velocity” variable representing the velocity of the surrounding. This can be represented by the solvent velocity $\mathbf{v}^{(s)}$, or the medium velocity (volume-average velocity) defined by

  $$\mathbf{v}=\phi\mathbf{v}^{(p)}+(1-\phi)\mathbf{v}^{(s)}.$$

  (5)

  Here we will use $\mathbf{v}$ following rheology convention.

The local flow condition is characterized by the velocity gradient tensor $\bm{\nabla}\mathbf{v}$

and the related rate-of-strain tensor

The general form of the free energy of polymer solution can be written as

where $a(\phi,\bm{\mathsfit{c}})$ is the free energy density. We assume that $a(\phi,\bm{\mathsfit{c}})$ has the following form

The first term $f(\phi)$ is the free energy density of polymer solutions at equilibrium. This term includes the entropic term $\phi\ln\phi$, the interaction term of Flory-Huggins form $\chi\phi(1-\phi)$, and the interfacial energy which depends on the concentration gradient $|\nabla\phi|^{2}$. The second term includes $g(\bm{\mathsfit{c}})$ that represents the elastic energy of deformed polymer chains and is a function of the conformation tensor $\bm{\mathsfit{c}}$. The second term is proportional to the polymer volume fraction $\phi$.

The change rate of the free energy is given by

where $\Pi$ stands for the osmotic pressure and is defined by

Notice that the elastic energy term $g(\bm{\mathsfit{c}})$ has no contribution to the osmotic pressure

The dissipation function also includes several terms. The first one accounts the relative motion of the center-of-mass of the polymer with respect to the medium velocity

where the friction coefficient $\xi=\xi(\phi)$ is in general concentration-dependent. Notice that by use of Eq. (5), the integrand of Eq. (13) is written as $\xi(1-\phi)^{2}(\mathbf{v}^{(p)}-\mathbf{v}^{(s)})^{2}$. Therefor $\Phi_{pv}$ can be understood as the dissipation due to the relative motion between polymer and solvent.

The second term represents the coupling between $D_{t}^{(p)}\bm{\mathsfit{c}}$ and $\bm{\nabla}\mathbf{v}$. We write it in a very general form

The first term is the inter-coupling of $D_{t}^{(p)}\bm{\mathsfit{c}}$, the second term is the cross-coupling between $D_{t}^{(p)}\bm{\mathsfit{c}}$ and $\bm{\nabla}\mathbf{v}$, and the last term is related to the solvent viscosity. Since $D_{t}^{(p)}\bm{\mathsfit{c}}$ and $\bm{\nabla}\mathbf{v}$ are tensors of rank 2, the frictional coefficients $\zeta$ are tensors of rank 4. These coefficients must be positive-definite to ensure that the dissipation function is non-negative.

From the change rate of the free energy (10) and the dissipation functions (13) and (14), the Rayleighian can be written as

where the last term accounts for the incompressibility condition $\bm{\nabla}\cdot\mathbf{v}=0$.

By the variational calculation with respect to the three “velocity” variables, we get the following set of equations

Equation (16) and the conservation equation (3) lead to the time-evolution equation for $\phi$. Equations (16) and (17) give the following force balance equation

from which we can identify the stress tensor as the terms in the big brackets. Equation (18) gives the constitutive equation

These results are quite general, but their usefulness is limited because many phenomenological parameters are introduced in the model. It is not clear how to assign values to these parameters for a practical polymer model. In the following we shall study two existing theories from the view point of this general formulation.

Doi and Onuki Doi and Onuki (1992), and Mavrantzas and Beris Mavrantzas and Beris (1992) proposed two-fluid model for polymer solutions. Their theories are not equivalent to each other, but are based on the same idea that the elastic stress created in the polymer will contribute to the motion of the polymer relative to the solvent if the stress is not uniform. Here we will discuss the two-fluid model focusing on the Doi-Onuki theory. They started with a free energy change rate of the form

The first term $\dot{F}_{\rm mix}$ is the change rate of the mixing free energy

with $\Pi$ is the osmotic pressure given by Eq. (11). The second term $\dot{F}_{\rm el}$ is the change rate of the elastic free energy (Eq. (2.26) of Ref. Doi and Onuki (1992)) and defines the stress exerted on the polymer $\bm{\mathsfit{\sigma}}^{(p)}$ (called the “network stress” $\bm{\sigma}^{(n)}$ in Ref. Doi and Onuki (1992)).

Some comments on this term are in order:

• •

  This derivation is different to the standard derivation based on Onsager principle. We normally start with a free energy as a function of the state variables and then calculate the change rate by performing the time derivative.

• •

  Here the polymer stress $\bm{\mathsfit{\sigma}}^{(p)}$ is put in by hand, therefore we still need a constitutive equation to relate the stress to the state variables.

• •

  Here the polymer velocity is used. In Ref. Doi and Onuki (1992), it was noted “since it is the deformation of the polymer which causes the change of the free energy.” It turns out it is very important to specify which velocity is coupled to the polymer stress.

The dissipation function is given by

due to the relative motion of the polymer chain to the solvent.

The Rayleighian is

which gives

These are Eqs. (2.28) and (2.29) in Ref. Doi and Onuki (1992). Here we have only two time-evolution equations. To complete the formulation, a constitutive equation needs to be specified. This is slightly different from the previous derivation, where the constitutive equation (20) is derived from the condition $\delta\mathscr{R}/\delta D_{t}^{(p)}c_{\alpha\beta}=0$.

For binary mixture of the polymer melt, Doi and Onuki suggested to use the “tube velocity” (or friction-average velocity, also see Tanaka’s works Tanaka (1997, 2000)) in Eq. (24)

where L and S stand for the long and short polymers.

The Rayleighian is written as

which gives

These are Eqs. (4.3) and (4.4) of Ref. Doi and Onuki (1992).

For dilute solutions, the free energy function is given by

where $v$ is the volume of one single dumbbell. The change rate of the free energy is given by Eq. (10)

The variation of $\dot{A}$ with respect to the velocity variables are

The dissipation function related to the $\bm{\mathsfit{c}}$-tensor is given in Ref. Zhou and Doi (2018), with $\dot{\bm{\mathsfit{c}}}$ replaced by $D_{t}^{(p)}\bm{\mathsfit{c}}$,

where ${\rm Tr}$ denotes the trace operation.

Here comes the main question: Which velocity should we use in the velocity gradient $\bm{\mathsfit{\kappa}}$? We have a few options here:

• •

  using the polymer velocity, $\bm{\mathsfit{\kappa}}=\bm{\mathsfit{\kappa}}^{(p)}=(\bm{\nabla}\mathbf{v}^{(p)})^{t}$. This will give the same results of Doi-Onuki.

• •

  using the solvent velocity, $\bm{\mathsfit{\kappa}}=\bm{\mathsfit{\kappa}}^{(s)}=(\bm{\nabla}\mathbf{v}^{(s)})^{t}$. This will lead to a different dynamics.

• •

  using a combination of the polymer and solvent velocities, $\bm{\mathsfit{\kappa}}=(\bm{\nabla}\mathbf{V})^{t}$, with

  $$\mathbf{V}=\alpha_{1}\mathbf{v}^{(p)}+\alpha_{2}\mathbf{v}^{(s)},\quad\alpha_{1}+\alpha_{2}=1.$$

  (40)

  This includes a special case of volume-average velocity $\mathbf{V}=\mathbf{v}$, with $\alpha_{1}=\phi$ and $\alpha_{2}=1-\phi$.

We will continue the derivation using the combination velocity $\mathbf{V}$. In general one should expect $\alpha_{i}$ to be a function of the concentration $\phi$ so it will be position-dependent.

The total dissipation function is given by

The variation with respect to the dynamic variables are

The Rayleighian is written as

The variation of the Rayleighian with respect to $D_{t}^{(p)}\bm{\mathsfit{c}}$ gives

This is the constitutive equation of the Oldroyd-B fluid model, with a small modification that the velocity gradient is given by $\mathbf{V}$.

The variation with respect to $\mathbf{v}^{(p)}$ gives

The variation with respect to $\mathbf{v}^{(s)}$ gives

Equations (48) and (49) give

where $\bm{\mathsfit{\sigma}}^{(p)}$ is defined by

Equation (50) indicates that the tensor $\bm{\mathsfit{\sigma}}^{(p)}-(\Pi+p)\bm{\mathsfit{I}}$ is the total stress tensor and $\bm{\mathsfit{\sigma}}^{(p)}$ is the polymer contribution to the stress tensor. This definition does not depend on the choice of $\alpha_{i}$. For homogeneous solution, the polymer number density is given by $n_{p}=\phi/v$, then Eq. (51) becomes the standard form $\bm{\mathsfit{\sigma}}^{(p)}=n_{p}k_{B}T(\bm{\mathsfit{c}}-\bm{\mathsfit{I}})=G(\bm{\mathsfit{c}}-\bm{\mathsfit{I}})$ with the shear modulus $G=n_{p}k_{B}T$.

Using the expression (51), Eqs. (48) and (49) can then be written as

With the setting $\alpha_{1}=1$ and $\alpha_{2}=0$, we recover the Doi-Onuki results for the polymer solutions (27) and (28). With the setting $\alpha_{1}=\zeta_{\rm L}/(\zeta_{\rm L}+\zeta_{\rm S})$ and $\alpha_{2}=\zeta_{\rm S}/(\zeta_{\rm L}+\zeta_{\rm S})$, we recover the Doi-Onuki results for the polymer blends (31) and (32).

Combining the above two equations, we can obtain

where the last term only appears if $\alpha_{1}$ is position-dependent.

The above equation shows that the relative motion of the polymers with respect to the solvent has two origins: One is the gradient of the osmotic pressure $\bm{\nabla}\Pi$, which corresponds to the usual diffusion due to the concentration gradient. The other one is the gradient of the polymer contribution to the stress tensor $\bm{\nabla}\cdot\bm{\mathsfit{\sigma}}^{(p)}$. This is the essence of the two-fluid model: the polymer contribution to the stress tensor should induce the polymer diffusion, i.e., the stress and the diffusion are coupled. Using the conservation equation, we can see that the time-derivative of the volume fraction has contributions from $\nabla^{2}\Pi$ and $\bm{\nabla}\bm{\nabla}:\bm{\mathsfit{\sigma}}^{(p)}$. These two terms are consistent with previous two-fluid models Beris and Mavrantzas (1994). Existing models have used $\alpha_{i}$ that are independent of the position, therefore the last term in Eq. (54) vanishes.

The magnitude of the stress contribution to the diffusion depends on the choice of $\alpha_{i}$. For $\alpha_{1}=1$ and $\alpha_{2}=0$, the stress contribution is on the order of ${O}(1-\phi)$, the same order of the contribution from the osmotic pressure. This is the original Doi-Onuki result Doi and Onuki (1992). For $\alpha_{1}=0$ and $\alpha_{2}=1$, the stress contribution is reduced to order ${O}(\phi)$, which is small in comparison to the osmotic pressure term. For $\alpha_{1}=\phi$ and $\alpha_{2}=1-\phi$, the stress contribution vanishes.

For homogeneous solutions, the polymer concentration is uniform in space, therefore the polymer velocity $\mathbf{v}^{(p)}$ and the solvent velocity $\mathbf{v}^{(s)}$ are the same, $\mathbf{v}^{(p)}=\mathbf{v}^{(s)}$. Different choices in the velocity gradient $\boldsymbol{\kappa}$ lead to the same time-evolution equations.