DERIVATION OF A SIMPLE POLYMER/SOLVENT SYSTEM VIA THE

HYBRID MIXTURE THEORETIC APPROACH

by

Tessa F. Weinstein

B.S., Metropolitan State College of Denver, 1999

A thesis submitted to the

University of Colorado at Denver

in partial fulfillment

of the requirements for the degree of

Master of Science

Applied Mathematics

2002

. f /* f t'

jrtLl.

This thesis for the Master of Science

degree by

Tessa F. Weinstein

has been approved

by

os/Q^/oS-

Date

Weinstein, Tessa F. (M.S., Applied Mathematics)

Derivation of a Simple Polymer/Solvent System via the Hybrid Mixture Theo-

retic Approach

Thesis directed by Assistant Professor L.S. Bennethum

ABSTRACT

A two-scale theory involving swelling polymeric media is considered. Macro-

molecular relaxation of polymer chains is modeled by considering the solid phase

as viscoelastic and the fluid as viscous. The mesoscopic averaging procedure is

laid out in detail and the mesoscopic field equations for mass, momentum, energy

and entropy balance are given. Constitutive theory is developed. Independent

variables are chosen and constitutive restrictions are derived for a two-phase,

multi-component system with negligible interfacial effects. Non-equilibrium,

equilibrium, and near-equilibrium results are obtained. The system is simplified

to a two-phase, single constituent per phase scenario. An experiment is pro-

posed and relevant governing equations are given. Appropriate assumptions are

made to reduce the system to one which is reasonably solvable.

This abstract accurately represents the content of the candidates thesis. I

recommend its publication.

m

Signed

L.S. Bennethum

IV

DEDICATION

This thesis is dedicated to the memory of John Lord Knight. Without his sup-

port, encouragement, and kind heart this work would not have been possible.

ACKNOWLEDGMENT

I would like to thank my advisor, Lynn Bennethum, for her time, patience, and

advice over the course of the last two years.

CONTENTS

Figures ............................................................... ix

Tables.................................................................. x

Chapter

1. Introduction and Overview of HMT..................................... 1

2. Microscale Averaging and the Governing

Field Equations...............i................................... 9

2.1 Microscale Governing Equation..................................... 10

2.2 Averaging Procedure............................................... 11

2.3 Mesoscale Field Equations......................................... 14

2.3.1 Conservation of Mass............................................ 14

2.3.2 Conservation of Linear Momentum................................. 16

2.3.3 General Case.................................................... 19

2.3.4 Conservation of Angular Momentum................................ 20

2.3.5 Conservation of Energy ......................................... 25

2.3.6 Entropy Balance................................................. 28

3. Assumptions and Constitution........................................ 30

3.1 Standard Assumptions and Resulting

Entropy Inequality................................................ 30

3.2 Constitutive Assumptions.......................................... 32

3.2.1 Entropy Inequality.............................................. 34

3.2.1.1 Non-Equilibrium Results....................................... 40

3.2.1.2 Equilibrium Results........................................... 41

Vll

3.2.1.3 Near-Equilibrium Results..................................... 44

4. Derivation of a Model of a Simple Polymer/

Solvent System..................................................... 48

4.0. 2 Assumptions and Entropy Inequality for a

Simple Polymer/Solvent System.................................. 49

4.0. 3 General Non-Equilibrium Results................................ 50

4.0. 4 Equilibrium Results ........................................... 50

4.0. 5 Near-Equilibrium Results....................................... 51

4.0. 6 Total Stress and Total Heat Flux............................... 53

4.0. 7 Generalized Darcys Law........................................ 53

4.0. 8 A First Approximation.......................................... 54

5. Discussion and Future Work......................................... 62

Appendix

A. Complete Nomenclature............................................. 64

B. Relations between Phase and Species Variables..................... 69

C. Identities Needed to Obtain Entropy Equation (3.4).............. 71

References............................................................ 72

vm

FIGURES

Figure

1.1 Different types of polymer molecules............................ 2

1.2 Scales of Observation........................................... 5

4.1 Proposed Experiment and Corresponding Geometry.................. 59

IX

TABLES

Table

2.1 Quantities for Microscopic Field Equations

x

1. Introduction and Overview of HMT

We develop a two-scale theory for swelling polymer systems, herein called

the microscale and mesoscale. Swelling polymers exhibit a hierarchy of scales

due to their complex porous structure. While the author recognizes that this

is naturally a three scale problem, it is the belief of the author that a full

understanding of the two-scale problem must be attained before the three-scale

problem is considered.

In recent years polymers have become increasingly important in technolog-

ical industries. Swelling polymers have a myriad of applications including but

not limited to: construction, agriculture, drug delivery, and food stuffs. For

ease of exposition we will restrict our discussion to drug delivery. While sev-

eral useful drug delivery systems have been developed, most have been found

by trial and error. Accurate mathematical models are needed to elucidate the

mechanics of drug release and to suggest new alternatives to current method-

ologies. Drug delivery systems are usually classified as diffusion-, swelling-,

or chemically-controlled systems. Of these, swelling-controlled systems are the

least understood because of the complex interactions that occur during release

[37],

In order to develop a model, a basic understanding of polymer terminology,

characteristics, and behavior is needed. Although there is still some debate

on the subject, polymers generally have molecular weights of 25,000 g/mol or

larger, and increased chain length means increased entanglement of polymer

chains. Monomers are the building blocks of polymers, e.g. amino acids or

1

Linear

Branched

Network

Figure 1.1: Different types of polymer molecules

sugars. Monomers link together with other molecules of the same or different

type to form polymers. A homopolymer is made up of a single monomer, whereas

a copolymer is made up of two or more types of monomers. A repeating unit is

a segment of a macromolecule that forms the basic unit of the macromolecule

(excluding the ends). In other words, we can form a complete polymer by linking

an adequate number of repeating units together. A linear polymer is one in which

every repeating unit is attached to exactly two others. A branched polymer is

one in which repeating units are not linked exclusively in linear fashion. See

Figure 1.1 for a schematic representation. A network polymer is formed by

chemically linking together linear or branch polymers; this process is referred to

as crosslinking. This same process is known as vulcanization for rubbers. See

[39, 44, 47, 42],

Polymers can be broadly classified into two groups: amorphous polymers,

which can be further classified as crosslinked or un-crosslinked, and crystalline

polymers. Amorphous portions of a polymer are the areas of the polymer with

long chain length where the polymer tends to coil about itself; this is known as

entanglement. The entanglement of amorphous regions makes polymers flexible,

and helps them hold together under stress. Unvulcanized natural rubber is

an example of an un-crosslinked amorphous polymer. Vulcanization of natural

2

rubber introduces covalent sulfur bonds so that the chains are chemically bonded

to one another. Crystalline polymers exhibit a crystalline structure, however,

most of them contain regions of amorphous material. Because of the complicated

nature of crystalline polymers, the model developed herein would need to be

extended. Thus, we restrict our discussion to polymers of the amorphous type,

[17].

Different polymers exhibit a whole spectrum of behavior dependent on var-

ious properties such as degree of cross-linking, crystallinity, and surrounding

temperature. In particular, some polymers are so highly sensitive to temper-

ature that certain properties change almost discontinuously at a temperature,

Tg, called the glass transition temperature, [17]. Such polymers are called ther-

mosets. Below the glass transition temperature, Tg, polymers are solid, hard,

even brittle and are considered to be in the glassy state. Amorphous polymers in

the glassy state are sometimes called amorphous liquids or supercooled liquids.

It is well known that amorphous polymers below Tg are not in thermodynamic

equilibrium; they still flow, but the time scale for observing creep and flow is

very long. Above Tg polymers enter the glassy transition, where the polymer

softens. As the temperature increases polymers will enter a rubbery plateau,

rubbery flow state, and finally a viscous flow state [42]. Diffusion properties of

a polymer will change drastically as the polymer nears Tg [30].

Figure 1.2 gives a schematic representation of the problem being considered

at various scales. At the microscale we can distinguish between the solid matrix

of the polymer and the fluid (or solvent); this is on the order of microns. At

the mesoscale the solid and fluid appear as a mixture; this is represented by the

3

mesoscale polymer. Figure 1.2 also depicts the the polymer particle immersed

in a bulk fluid. Inclusion of the bulk fluid in our model would constitute a third

scale, which would be considered the macroscale, and is not considered here.

Traditional models of drug delivery systems begin with a concentration form

of Ficks law of diffusion. Depending on the type of system modeled and geom-

etry, Ficks law is modified accordingly. For example, the diffusion coefficient

can be modified by considering it as the mutual diffusion of the drug in the

polymer, the diffusion of the drug alone, or concentration dependent. The pri-

mary problem with these models is that they are of limited applicability since

they only apply to Fickian diffusion phenomena [37]. It is well known that non-

Fickian diffusion occurs when polymers enter the glass-rubber transition [30].

Various authors have expanded the Fickian model to account for anomalous

behavior, but these extensions are largely heuristic [37]. A variation on the tra-

ditional approach is to treat coefficients of Ficks and Darcys law as stochastic

processes using the Karhunen-Loeve expansion, [22]. However, this increases

the dimension of the problem by treating the random aspect of the problem

as a new dimension and it is not entirely clear how the resulting coefficients

relate back to the physical problem. Lustig et al. [30], address the problem

using continuum thermodynamics. However, they do not have a variable that

directly accounts for the moisture content of the polymer. Low [29] found that

the swelling pressure in smectitic clays (specifically montmorillonite), is highly

dependent on moisture content. We find it reasonable to assume that the same

will be true of the swelling of polymers. Hybrid mixture theory (HMT) has

recently been applied to polymeric and biopolymeric systems by Singh et al.

4

Macroscale

bulk fluid

Mesoscale REV

mesoscale polymer

Mesoscale

Microscale

Figure 1.2: Scales of Observation

5

[41]. Hybrid mixture theory has all of the advantages of classical mixture the-

ory plus the added advantage that mesoscale variables can be directly related

to microscale counterparts. This last method is the context in which our model

is systematically developed.

Hassanizadeh and Gray developed hybrid mixture theory (HMT) in a series

of papers, [24, 25, 26] aimed at single phase flow in granular media. In [24] the

microscopic field equations are averaged once to obtain mesoscale analogues.

Bowens continuum theory of mixtures is then applied to obtain constitutive

relationships [14, 15, 16]. Bowen uses the classical mixture theory framework

and exploits of the entropy inequality in the sense of Coleman and Noll, [18],

then linearizes about variables becoming zero at equilibrium to obtain constitu-

tive relationships at the larger scale. Hassanizadeh further extended the theory

to include a multi-component fluid phase in a granular material and obtained

generalized forms of Darcys and Ficks laws [27, 28]. Later, Gray and Has-

sanizadeh applied HMT to a system with three single constituent phases and

included interfacial effects.

Some the aforementioned papers contained the incorrect result that, at equi-

librium, the chemical potential of two phases are equal to each other. This con-

tradicts the classical thermodynamic result that, at equilibrium, the chemical

potential of a single constituent in two phases is equal. In 1994, Achanta et al.

correctly employed HMT with the additional axiom of equipresence [2], which

requires that before exploiting the entropy inequality it is assumed that each

phase is composed of the same N constituents. Only after the entropy inequal-

ity has been exploited can the concentration of certain constituents be set to

6

zero. In [2], the authors derive the macroscale field equations for each phase and

interface of a three-phase, multi-constituent media.

From 1994 to present HMT has been successfully employed to model swelling

and shrinking behavior in gels, food stuffs, and colloidal systems where phase

interactions play an important roll in the mesoscopic and macroscopic behavior,

[3, 6, 7, 10, 33, 34, 35, 36]. All of these works assumed an elastic solid phase at

the microscale. Thus, they are applicable to systems exhibiting macroscopic vis-

coelastic behavior, but not to systems in which viscoelastic behavior is observed

in much smaller scales, such as the solid phase of polymers. In [41], Singh et

al. propose a constitutive theory to model a two-phase polymeric system with a

viscoelastic solid phase, a viscous fluid phase, and obtain novel forms of Darcys

and Ficks laws.

Despite the clear progress that has been made with the application of HMT

to the study of polymers it is the opinion of the author that no one has yet

correctly simplified the problems of diffusion and absorption of a solvent to a

satisfactory degree. These systems are simply too complicated to solve numeri-

cally. It is the aim of this work to reduce the latter problem to one that is both

tractable and comparable with current literature in polymer science.

In Chapter 2 we give the microscale governing equations, review the averag-

ing procedure, give the definitions of mesoscale quantities in terms of their mi-

croscale counterparts, and give a detailed derivation of all of the mesoscale field

equations (mass, momentum, etc.). In Chapter 3 we discuss the simplifying as-

sumptions and constitution of the system being modeled. We obtain an entropy

inequality that is valid for systems with a visco-elastic solid phase and viscous

7

liquid phase, then obtain non-equilibrium, equilibrium, and near-equilibrium

results for that system, as well as a generalized Darcys law. In Chapter 4,

we reduce the entropy inequality previously obtained for a two-phase system

composed of a single constituent per phase. Again, we obtain non-equilibrium,

equilibrium, near-equilibrium results, and a generalized Darcys law. We then

derive a system of equations that we believe is tractable. Finally, in Chap-

ter 5 we discuss avenues for further research both with this model, and more

sophisticated ones.

8

2. Microscale Averaging and the Governing

Field Equations

The purpose of this section is to lay out in detail the averaging procedure

used in HMT for any two-scale, multi-constituent, multi-phase material. The

two scales are herein called the microscale and mesoscale. At the microscale one

can distinguish between phases. It is at this scale that the field equations (mass,

momentum, etc.) are known to hold and properties such as density, velocity, and

mass are clearly defined. The mesoscale is order of magnitudes larger than the

microscale. At the mesoscale one can no longer distinguish between individual

phases. Because an averaging process is performed to obtain an analogue on this

larger scale, properties such as density and velocity are now viewed as somewhat

blurred. Herein we develop one such technique.

Before continuing we need to mention a few basic underlying assumptions.

First, we assume that the material we are modeling has negligible interfacial

effects. That is, the interface has no thermodynamic properties and is massless.

Thus, no constituent present gains or loses mass, momentum, or energy when

crossing an interface. This will place special restrictions on each of the field

equations and will be discussed in further detail in subsequent sections. Second,

the material we are modeling has a representative elementary volume (REV);

that is, a volume for which averaged properties will remain the same if the REV

is made slightly larger or smaller. In addition we require that the REV size and

shape remain the same for all space and time. Such an REV does not exist if the

material under question is too heterogeneous. For a more detailed discussion of

9

the existence of such an REV see [5, 19]. It is important to remember that for

the current discussion, the following theory is applicable to any material meeting

these requirements.

2.1 Microscale Governing Equation

Each phase is denoted by small Greek letters (a, /3, 7), and species (or

constituents) are denoted by j, j 1,..., N. For the purposes of simplicity and

brevity we assume that all interfaces are massless and have no thermodynamic

properties. Interfacial effects can be included by following the approach provided

by [23].

Additionally, we assume that there are no internal surface discontinuities,

meaning that each phase is the union of several isolated simply connected vol-

umes, [24]. Using the notation of Eringen, [21], the constituent, microscopic

field equations for a given phase, a, can be stated as

^-(fPip3) + V (p>v3iP3) V V p>fj = p>G3 + (Pi)3 (2.1)

at

where p3 is the mass density, ip3 is the mass average (over the phase) thermody-

namic property of constituent j, v3 is the mass average velocity vector, i3 is the

flux vector, f3 is the body source, G3 is the net production, and ip accounts for

the influx of ip from all other constituents (e.g. due to chemical reactions). If the

medium consists of only one constituent then ip is equal to zero. This equation

holds on the microscale for mass, linear and angular momentum, energy, and

entropy. Table 2.1 lists the quantities used for each field equation.

In Table 2.1, t is the second-order stress tensor, g is the external supply of

momentum (gravity), r is the position vector referenced to a fixed coordinate

10

Table 2.1: Quantities for Microscopic Field Equations

Quantity $ i f $ G

Mass 1 0 0 r 0

Linear Momentum V t 9 i + rv 0

Angular Momentum r x v r xt r xg r x (i + rv) m 0

Energy e + \v2 tv + q g v + h Q + i v + r(e + \v2) 0

Entropy V 4> b r) + rv A

system, e is the internal energy density function, h is the external supply of

energy, q is the heat flux, 77 is the entropy density, is the entropy flux, b is the

external supply of entropy, and A is the entropy production.

2.2 Averaging Procedure

The averaging procedure used in HMT is based on works of various au-

thors and was developed at approximately the same time [4, 32, 45, 46]. While

many different methods are available [19], we choose to use the one which is

computationally the simplest where field equations are averaged via weighted

integration. Here we use the indicator function of the a-phase as the weight,

and treat the averaged quantities resulting from the weighted integration as dis-

tributions. This allows us to bypass the difficulties of defining the derivative of

averaged quantities that result from the weighted integration [40, 38]. Addition-

ally, the weighting function used here may result in averaged quantities that do

not correspond to physical quantities measured. This problem can be overcome

by choosing a weighting function that represents the experimental apparatus

used to measure physical properties [19].

11

Let SV denote the REV, SVa denote the portion of the a-phase within SV,

and 5Aap denote the portion of the a/3 interface within 5V. It is assumed that

SVa and SAap are isolated simply connected regions. Express the magnitude of

SV by |5V|, then the volume fraction can be written as

e(M) = M (2'2)

so that

5> = 1- (2.3)

a

Letting r and x denote the position vector and the centroid of the REV, respec-

tively, r can be written

r x + Â£,

(2.4)

where Â£ is the local coordinate referenced to the centroid of the REV and varies

over all of SV. The indicator function for the cx-phase is given by

7a (**>*)

1 if r e SVa

0 if re 5V@,

Then

\5Va\{x,t) = J7a{x + Â£,t) dv(g) (2.5)

SV

represents the magnitude of the volume SV in the cx-phase. Following Has-

sanizadeh and Gray [24], we make the following definitions:

p>a(x,t) = j p>{r,t)ja{r,t) dv{Â£) (2.6)

sv

is the average mass over |<7Va|,

W){x,t) = J dv(i) (2.7)

sv

12

is the volume average property of 'ip3, and

= c,1 [P>(r,t)ipi(r,t)ja(r,i) dv(() (2.8)

H> \sva\Jv

I

is the mass average property of 'ip3. We would like the mesoscale field equations

to be the analogues of the microscale field equations. To ensure this we will apply

the following theorem that allows us to interchange the order of differentiation

and integration. This result is due to Cushman [20].

Theorem 2.1 If w01^ is the microscopic velocity of the interface a/3 and na

is the outward unit normal vector of dVa indicating the integrand should be

evaluated in the limit as the a/3-interface is approached from the aside then

We now have everything we need to average equation (2.1) from the mi-

croscale to the mesoscale. We begin with the conservation of mass, then linear

momentum, angular momentum, energy, and lastly entropy. Corresponding bulk

equations are derived for each field equation and restrictions that result from

the assumption that the interface has negligible thermodynamic properties are

given.

13

2.3 Mesoscale Field Equations

2.3.1 Conservation of Mass

Substituting the appropriate quantities from Table 2.1 into equation (2.1)

the conservation of mass at the microscale is

-(^) + V-(^V) = ^P. (2.11)

Formally, we multiply by the indicator function, integrate over the mesoscopic

REV, and divide by the magnitude of the REV, |5V|. Using Theorem 2.1,

relationships (2.6)-(2.8), and substituting back into equation (2.11), we have

+ V (eVV*)

^W\ / ^

*** 5Aa,

waPj v3) nada + e01^^

(2.12)

Now, using the material time derivative, which is given by

Dai d a.

-- =---\-v 3 V,

Dtdt

(2.13)

we obtain the mesoscopic mass balance for constituent j in phase a

Dai (gQpQJ) + eapQi (v vaj) = e*/ + . (2.14)

L/t ,

In the equations above, v3 is the microscopic velocity of constituent j, wa^i is

the velocity of the jth constituent in the aj3 interface, and na is the outward

unit normal vector of 5Va. Complete nomenclature is given in Appendix A. Our

motivation in defining the mesoscopic variables is twofold. First, we would like

the mesoscopic variables to coincide with actual physical variables that can be

measured via mesoscale experiments. Second, we would like the definition of the

14

mesoscopic variables to be as consistent as possible with their microscale coun-

terparts. Unless otherwise stated, the mesoscopic variables have the following

definitions:

(2.15)

is the average mass over 5Va,

(2.16)

is the mass averaged velocity,

(2.17)

is the net rate of mass gained by constituent j in phase a from phase /?, and

is the rate of mass gain due to interaction with other species within the same

phase.

Bulk phase variable definitions are not always intuitive and are defined so

as to preserve the form and interpretation of the mesoscale equations. To obtain

the bulk phase counterpart for the conservation of mass we make the following

definitions:

(2.18)

N

(2.19)

is the mass density of phase a, and

(2.20)

15

is the concentration of the jth constituent in the a-phase. All other definitions

of bulk phase variables are given in Appendix B. Using equations (2.19) and

(2.20) in equation (2.12) we have:

A

?-{eapaCa!) + V (eapaCaivai) = ^e% (2.21)

Summing over j = 1,..., N we obtain:

4 (V) + V(eV) = Â£ ej. (2.22)

Defining ^ as ^ + va V, we obtain the bulk phase counterpart for the con-

servation of mass,

)

r\a (
L +ep(V-v) = Y,n, (2.23)

where the following restrictions apply:

N

Vr0' = 0 Va and

3=i

e?' +ej' = 0 i\ II

(2.24)

(2.25)

The meaning of equation (2.24) is that the gain of mass of the bulk phase due

to chemical reactions alone must be zero. Equation (2.25) says that the rate of

mass gained by phase a from phase /3 is equal to the rate of mass gained by

phase j5 from phase a, i.e; no mass is lost in the interface.

2.3.2 Conservation of Linear Momentum

Substituting the appropriate quantities from Table 2.1 into equation (2.1)

the conservation of linear momentum at the microscale is

^-{ppyi) + V (/^W) V V pigi = fPi + pPr^vi. (2.26)

C/ 0

16

Using the same method as for the conservation of mass equation we obtain:

d_

dt

{eapaivai) V {ea{(tj)a + p^v^v0* p^v^vo01))

+ V {eapa*vaiva-#) Â£apaiga>

= 7^ [ [tj + pPvj(waPi vj)] na da

tZ\W\ J

6Aap

+ ea paj (t +rJvi ),

(2.27)

--~Ct

where gaj = g3 is the mass average external supply of momentum and all other

variables retain the meaning they were given in the previous section. Addition-

ally, we define the mesoscopic stress tensor, also known as the Cauchy stress

tensor, as:

taj = (tj')01 + paiva*vas pajvjyja. (2.28)

Now, subtract vai times equation (2.12) from equation (2.27) to obtain

r\

Â£apaj~(va^) + Eap0ljvaj V (') V (eatai) eapajgaj

C/6

V - [ \fj + pjyjfw01^ vj)] na da

tfAmJ [

&a 5 A,

a0

-ml'"

waPj vj) na da

+ Â£apaj (i rvj) .

(2.29)

Using (2.13) we obtain the mesoscopic linear momentum balance for constituent

j in the ct-phase:

Â£apaj^r(va^) V (eta>) e0paig0lj = TS (2.30)

17

where we have made the following definitions:

To =

. . f [tf + p>v^(wa^j v*)] na da

P Va\ L J

5Aap

- va> J fp (wa^i v5) na da

5Aap

(2.31)

represents the effect constituent j of phase /? has on the rate of change of me-

chanical momentum of the same constituent in phase a, and

i 3 = eapai (i + r3v^ v^r3 )

(2.32)

is an exchange term that takes into account all gain of momenta due to the

presence of other species but not due to chemical reactions.

Again, we would like to obtain the bulk phase counterpart. Beginning with

equation (2.27) and using (2.20) we have

r\

^(Â£apaCaiva3) V (Â£a(ta paiva>'a tt^>a)) Â£apaCajga>

L/C

+2V(eapa*Caivai va) V(Â£apaCajva va)

= Y 7777-7 [ W + pPvHw^i uJ)l na da

& |n%

+Â£apw(t + Â¥&*). (2.33)

Subtracting va times equation (2.21) we get

r\ r\

^-(eQpaCaJvai) va-~-(Â£apaC^) (2.34)

-V (Â£a(tai paiVai'a V01*01))

Â£apaCajgaj + V (2Â£apaj Caj vaj va)

-V(Â£apaCa*va va) vaV (Â£apaca*vai)

= Y Tap + Y Cp vaj + ^ v*j Y v ^v* (2-35)

P P^ot P^ct

18

Summing from j = l,...,iVwe obtain

(2.36)

where the following restrictions apply:

N

+rCjvaja) = 0 Va

(2.37)

(2.38)

Equation (2.37) says that linear momentum can only be lost due to interactions

with other phases. Equation (2.38) says that the interface can hold no linear

momentum.

2.3.3 General Case

Equation (2.1) is the constituent, microscopic general field equation. We

will find it useful to average this equation up to the mesoscale. The details

of the averaging are much the same as they are for the conservation of linear

momentum, so we will not repeat them here. Let us just say that the general

case, after averaging from the microscale to the mesoscale, can be written as

^-(Â£apajipia) V (ea((i:)a + paj vaj p^viipi*))

ot

+ V {eap^va^a) sapaJjia

=F& W3 J pyw^i -

6Aap

pPj -iP)] na da

(2.39)

19

We will find it convenient to substitute the quantities from Table 2.1 directly

into equation (2.39) as the averaging procedure has already been performed.

2.3.4 Conservation of Angular Momentum

Conservation of angular momentum is probably the most difficult field equa-

tion to upscale simply because the calculations are tedious. Thus, we will go

into more detail in this section than in others so that the reader may more eas-

ily reproduce these results. We begin by substituting the appropriate terms for

the conservation of angular momentum from Table 2.1 into equation (2.39) and

obtain

r\

(eapajr x via) + V (Â£apajvajr x via) V

(JL

(ea(r x tj)a

+ Â£apajvajr x via Â£apajv:>(r x Â£apajr x gia

= ^ | J [r x & + pP{r x vi)(waPi u-7)] na da

a

T5"TWr / - Ji n" da

k to ,i

+ Â£apajrx(i +rvi) Â£apajfriJf

(2.40)

Next we subtract x crossed with equation (2.27) (the conservation of linear

momentum at the microscale), where x is the macroscale field variable. To do

this we will need several identities:

r x via = x x vaj + Â£ x via,

vir x via = x x x via, and

r x gia = xx gaJ + Â£x gia,

20

among others similar to these. These identities can be verified by combining

(2.4) with the fact that (2.6) (2.8) are integrals with respect to Â£ only, and not

with respect to x. Starting with the left hand side, for the 4r terms we have

d

d

{eapajx x va*) + (epQj'Â£ x vi ) x x {eapajv^)

x *)

(2.41)

For the V- terms we have:

V [e paj vaj x x vaj + Â£apajvajÂ£i x iPQ eax x (t^)a

- x tj)a Eapaivaix x vaJ eapa*vaiÂ£ x via

+ Eapajx x viyi + EapajviÂ£ x vi\

+ X X V [Ea(tj)a + Eap^vaiv^ Eapa>VHP Â£apaiVaJVa>]

= V (sapaivaix x Va*) X x V {eapa*vaJva>)

- V (sax x (tj)a) + a; x V (Â£a(tj)a) V (eQ(Â£ x tj)a)

- V (EapajVaiX XVai) + X X V- (Â£apaivaiva3)

+ V (Â£apajx x vivia) -ix V' (Â£apaj vivia)

+ V (EapaviÂ£ x J'). (2.42)

The first and third lines after the equality sign in equation (2.42) cancel. Note

that if we switch to indicial notation,

-V (Â£ax x (tj)a) + x x V (ea(tj)a)

(s %itjkÂ£ijl),k + (^ tjk),kÂ£ijl

= 'C XifetjkEijl Â£ %itjk,k&ijl T Â£ ^i^jk,kÂ£ijl

= Â£ ^ik^jkSijl = Â£ tj{Eiji, (2.43)

21

where the last equivalence is left in indicial notation because there is no equiv-

alent in direct notation, we have

V (Â£apajx x vivia) x x V (Â£apajv^v^a)

= (eap0ixivjvkeiji)ik Xi^p^vjv^jSiji

= ea Paj XijkVjVkEiji + Â£apajXl(vjVk)ikÂ£lji

- Â£ap^Xl(vJVk)>jÂ£lji

= eQ paj SikVjVkEiji = 0, (2.44)

and

V (Â£apajVaj X X Vai)-XX V (Â£a paj Vaj Vaj)

= {Â£apajViXiÂ£ijk)ti Xl(Â£apajViVj)jÂ£ijk

= Â£CpajXiiiViVjÂ£ijk + Â£a pai Xi(ViVj) ,iÂ£ijk

- Â£apaj Xi (V{ Vj ) yjÂ£ijk

= Â£a Paj 5uViVjÂ£ijk = 0. (2.45)

If we use identities (2.43) (2.45) and treat the external supply of momentum

-------rCC

term, r x gi similarly, the left hand side becomes

d_

dt

(Â£a^jx7) + V (Â£apajV^ X via -Â£a{$, x tj)a)

Â£atjiÂ£iji ~ Â£apaji x gi .

(2.46)

22

Turning our attention to the right hand side of the equation, we examine

the terms with no sum:

^a,- : a ---~o ------:----:---a

eapajx x (i +r3vi ) Â£apajm3 +eapaiÂ£ x (i3 +r3vi)

Â£Â£ja-:-----a

xx Â£apaj (i r3 + vi )

= Â£apajrn3 + eapajÂ£> x (i +r3vi) . (2.47)

Now, examining the terms on the right hand side which involve a sum, we have

E

. f . [ \x x t3 + pP{x x v3)(wal3i uJ)l na da

J

^^ct/3

+E

P^OL

jfiyj /[

6Aap

-E

Â£'

Wc

a r

/ [t3 + piyi(wa^i v3) na da]

* Ot J

SA,

a(3

=E

p?a

S'

]5Vc

f [Â£ x tj + p>(Â£ x v3){waPj uJ)J na da

* Ol\ J

(2.48)

Combining the left and right hand side, we now have the conservation of angular

momentum at the mesoscale:

d

dt

{Â£apaÂ£ xvi ) + V {Â£apa*viÂ£ x vi ea{Â£ x t3)a)

Â£aI X (t3)a Â£apaiÂ£ X gi

Â£apajm3 +Â£apajÂ£x(i +r3vi)

+E

P<*

J [Â£ x tj + pP(^ x vj)(waPj vj)] na da

8Aap

(2.49)

23

Equation (2.49) simplifies to

~Â£at^Â£iji = maj + Maj + y^(mgj), (2.50)

where we have replaced (iJ) by taj (we can do so because of equation (2.28))

and we have made the following definitions:

mjC* = Â£apajrnaj (2.51)

is the rate of gain of angular momentum due to interaction with other species

within the same phase,

r\

Mai = -^(e>ijx?) V (eap^v^ x ia) + V ea(Â£ x tj)a

C/ 6

______ ______________:-------a

+ EapajÂ£ x gja + Â£apajÂ£ x (iJ + r^vi) (2.52)

is the rate of angular momentum gain by constituent j in phase a due to the

microscale angular momentum terms, and

m

= |^T| J [Â£ x t3 + X vJ)(wa^ vJ)] na da (2.53)

is the rate of angular momentum gain by constituent j in phase a due to inter-

action with phase /?.

Summing over j = 1,..., N yields the conservation of angular momentum

for the bulk phase

= Ma + Y^ "ip, (2.54)

where the following restrictions apply:

N maj = 0 Va, (2.55)

3=1 rrip' + rrfj = 0 (2.56)

24

Equation (2.54) implies that, in general, the mesoscale stress tensor ta is not

symmetric for multi-phase systems. However, if taj is symmetric then the right

hand side of (2.54) must be zero. Restriction (2.55) says that bulk-phase angular

momentum can only be lost due to transfer to other phases, restriction (2.56)

holds because we assume that the interface is massless.

2.3.5 Conservation of Energy

Beginning as we did for the conservation of angular momentum, by substi-

tuting the appropriate terms for the conservation of energy from Table 2.1 into

equation (2.39), we obtain

d

1^-

(Â£apaj(ei + -vi vi )) V (sa((t3 v3 + q3)a + pajvaj (ei + -v^ vi )

C/ b Zj Z

,tOl 1

-paivi(ei + -vi vi) )) + V (Â£apajvaj (eJ -1vi vi ))

2 2

Eapaj (gi via + hia)

Â£i*i[/l-"

5Aap

+ q3 + p3(e3 + ~v3 v3)(wal3j v3)

z

na da

1 f

+ -vi vi) / p3 (waPj v3) na da

5 A

+sOLpOLj (Qj + V v3 + ?3(e-? + -v3 v3) ).

2

(2.57)

Now, from equation (2.57), subtracting equation (2.29) times vai and equation

(2.12) times (eia + v^a), performing massive algebraic manipulations, and

again using equation (2.13), we obtain the conservation of energy equation at

the mesoscale:

eapaj^{ea3) V (eaqai) Â£ata> : - eapa3ha3

= Qai+Qf, (2-58)

25

where the colon indicates the tensor dot product (a : b = Yhij aijhj)- Here we

have made the following definitions:

hai = + g3 via gai va3 (2.59)

is the external supply of energy,

1-,---= 1

eUj = e3 Hv3 v3-vai vai

(2.60)

is the energy density,

q<*j = + (ti vjy t-i . + paivai (eaJ + -V<*J v<*j)

2

i ; _Q

pajv3(e3 + -v3 v3)

(2.61)

is the partial heat flux vector for the jth component of phase a,

Qaj = Â£apaj (Q3 + i3 v3 (i + r3v3 v^r3 ) vaj + r3(e3 Hv3 v3)

2

r3 (e3 Hv3 v3 ))

2

(2.62)

is the rate of energy gain due to interaction with other species within the same

phase not due to mass or momentum transfer, and

q3 +13 v3 4- p3(e3 Hv3 v3)(wa^i v3)

2

na da

v0lj) J fj{waPj v3) na da

5Aap

v

J [t3 + fPv3{wa^3 V3)]

na da

J4p

(2.63)

is the rate of energy transfer from phase /3 to phase a not due to mass or

momentum transfer.

26

The bulk phase conservation of energy equation is found in an analogous

manner to the bulk phase conservation of linear momentum by using (2.20) and

subtracting va times (2.33) and (e + \va va) times (2.21) and then summing

over constituents. This yields

eapa

Daea

Dt

V (eaqa) eata : Vva sapaha = ]rQÂ£,

/S^Q

where the following restrictions hold:

Nr /

E

3=1 L

Qa* +ij- va*a + raj leai + ^(va^a)2

= 0 Vck, and (2.65)

+

QiJ + v* + eÂ£ (e^ + i(u^)2

0 j = l,...,N.

(2.66)

The first restriction says that bulk-phase energy can only be lost due to transfer

with other phases, and the second restriction states that the interface retains no

energy.

27

2.3.6 Entropy Balance

After substituting the appropriate quantities from Table 2.1 into (2.39) the

entropy balance at the mesoscale for constituent j in phase a is

Â£apajrjja) V (Â£a((ft}a + paiva>fta

- p^voft )) + V ) Â£apajV

= Â£aPaj \ a ixt/ i [ft + ft ft (^Q/3j ft) ri da

Â£Â£ '-Pa\oVa\ J

a0

^da

$ Ad 0

+ Â£apaj Ai + eapaj rji + r3 vi

(2.67)

Subtracting rfi times (2.12) and using (2.13) we obtain

Dai

Â£api

Dt

fa') V (Â£a
= Â£$?+5' + A>,

where we have made the following definitions:

Aaj = Ai

(2.68)

(2.69)

is the entropy production,

,v. rOC

rf3 =

is the entropy of the jth constituent in the ophase,

4>aj = (ft )a + ftvair\ai paivift

(2.70)

(2.71)

28

is the partial entropy flux vector for the jth component of the a-phase,

Wa = ba3

(2.72)

is the external entropy source,

$

P

[
eariai r

J

pi (waPi yi) na da

(2.73)

(3

is the entropy transfer to the jth component in the a-phase through mechanical

interactions with the same component in the /3-phase, and

rf3

Â£apajr]ia + ripi hr^apaj

(2.74)

is the net entropy gain due to interaction with other species within the same

phase

The bulk-phase counterpart is given by

Dap

eapa 1

Dt

where the following restrictions apply:

N

i=i

3

paba =Â£8/+X. (2.75)

0 Va, (2.76)

0 (2.77)

The first restriction states that entropy can only be lost due to interactions with

another phase, and the second restriction states that the interface can hold no

entropy.

29

3. Assumptions and Constitution

Thus far, the theory introduced applies to all media meeting the require-

ments laid out in the beginning of Chapter 2. Obviously, one of the strengths of

the theory is that it has a broad scope of application. However, as outlined in

the introduction, we are interested in the theory as it applies to polymeric and

biopolymeric systems. First, we will make a few standard simplifying assump-

tions, and give the resulting form of the entropy inequality. Next, we will make

constitutive assumptions as they apply to polymeric and biopolymeric systems

and give definitions and identities needed to compute the entropy inequality

for the system defined by our specific constitution. Lastly, we will obtain non-

equilibrium, equilibrium, and near equilibrium results.

3.1 Standard Assumptions and Resulting

Entropy Inequality

Henceforth we assume that the stress tensor, ta, is symmetric. We also

assume that the system is in local thermal equilibrium. That is, because the

phases are viewed as overlaying continua, we assume the temperature of all

constituents in all phases at a single point is the same for all phases. This

assumption can be expressed as

T = Tai = Vo, V/3,Vj. (3.1)

Note that this does not mean that we are assuming that the temperature is

constant; T is still a function of time and space.

30

Next, we assume that the system is thermodynamically simple in the sense

of Eringen [21]. This means that entropy flux and external supplies of entropy

are due to heat flux and external supplies of heat alone, respectively. This

assumption can be expressed as:

foaj

ba> = ~jT- (3.3)

Entropy is a mathematically useful quantity. However, experimentally it cannot

be measured directly. Thus, we choose to perform a Legendre transformation to

convert the internal energy ea> into the Helmoltz free energy Aai,

Aaj = eaj + Trfi, (3.4)

allowing us to choose temperature instead of entropy as an independent variable.

The second law of thermodynamics states that the total entropy generated

by the system must be non-negative, and is maximum only when the system is

in equilibrium. This statement can be expressed as:

N

a=Ea = EEa^' <3-5)

Q: Ot j1

We begin by solving equation (2.68) for subtract ^ times the conser-

vation of energy equation (2.58), to eliminate the heat source variables,

perform the Legendre transformation with (3.4), use the identities given in Ap-

31

pendix C, then sum over all constituents and phases to obtain

. v-^ ( Â£aPa (DaAa nDaT\

N

+ T I ) : d

\j=l

a f N -

+^(VT) \ 9 Â£

l 3=1 L

- palVa3'a V- + {v

. vaia

J=1

4e*5-

/3^a

iv

[e? +?>]

J-l

+ I >o,

P&*

(3.6)

where uaj, = vai va, and in general a comma in the superscript denotes a

difference in the superscripted quantity.

3.2 Constitutive Assumptions

First, we assume that we have a two-phase system consisting of a solid and

a liquid phase, denoted a = s,l, respectively. The solid phase (the polymer) is

assumed to be viscoelastic and the liquid phase is assumed to be viscous. The

interactions of the two phases results in viscoelastic behavior. The viscoelastic

solid is modeled as a Kelvin-Voigt element, which means that the constitutive

variables will depend on the time rate of change of the strain tensor. The model

can be extended to a general Kelvin-Voigt type by including higher derivatives

32

of the strain tensor, but initially we include only one derivative for simplicity.

The unknowns of the system include:

eapa\ va\ T,

Tj', Tj, Qap\ Qa*, Tp Vaj, Vj-

(3.7)

(3.8)

(3.9)

Conservation of mass (2.14), momentum (2.30) and energy (2.58) correspond to

the unknowns on line (3.8), respectively. Additional equations are obtained by

considering the variables on line (3.9) as constitutive. Thus, every variable on

lines (3.7)-(3.9) has a corresponding equation except for sa. This is known as

the closure issue [13]. Here, we choose to follow Bowen [16] and postulate an

additional dependent variable, ea, so that we assume the evolution of the time

rate of change of volume is a constitutive variable.

Constitutive variables are considered functions of the following set of inde-

pendent (or constitutive independent) variables:

el, eQp% V(eapa0, vas, va*a, T, VT, Es, VES, Es, dl, Vvl>'1. (3.10)

Here we include only el since equation (2.3) implies that either el or es can vary

independently but not both. In our conservation equation ea and paj always

appear together and account for the actual physical make-up of the material.

The variable 'Veaplj accounts for non-local effects of the material make-up.

Including additional gradients of any variable will increase the non-locality of

the model and make it more accurate with respect to that variable. However,

the more gradients we include the more complicated the model becomes. Thus,

we are careful only to include gradients that we feel are relevant to our system.

33

We include dl to account for viscous effects of the fluid. Es and V.E5 account

for elastic behavior in the solid. Again, accounts for non-local effects. Es

allows for viscoelastic behavior as evidenced by the Kelvin-Voigt model [31].

By including additional time derivatives of Es, we could model Kelvin-Voigt

elements in series which would allow for a better fit to data, but this would

also unnecessarily complicate our model. We include T because amorphous

polymers exhibit a high degree of dependency on temperature as discussed in

the introduction. If we do not include VT in our list of independent variables,

exploiting the resulting entropy inequality produces the result that the heat

flux is zero for all time, but this is only true of non-heat-conducting materials.

On the other hand, if we include VT in our list of independent variables, we

recover Fouriers law of heat conduction. Including va,s gives us information

about the exchange of momentum between phases. Lastly, vaj,a and VuJ,Q are

directly related to viscous diffusion and give us information about the exchange

of momentum within a phase.

3.2.1 Entropy Inequality

To arrive at a form of the entropy inequality that is useful we will need

several identities and assumptions. Next, we make the appropriate simplifying

assumptions and give identities needed to obtain the entropy inequality for our

system.

The axiom of equipresence requires that initially all constitutive variables

be functions of the complete set of independent variables, even, for example,

if one phase lacks a certain constituent. Only after the entropy inequality is

34

exploited can the concentration of a species be set to zero. However, it can

be demonstrated that the Helmholtz free energy is not a function of all the

variables in list (3.9) [26]. To simplify the following manipulations, we postulate

the dependence of the Helholtz free energy as follows:

A1 = Al(el,elpl\T,Es,Es) (3.11)

As = As{el,esps\T, ES,ES). (3.12)

(See Bennethum [7] for a more generalized formulation.) Otherwise we adhere

to the axiom of equipresence: all other constitutive variables are considered

a function of list (3.10). Material time derivatives of A1 and As appear in the

entropy inequality (3.4). Using the chain rule, they may be calculated as follows:

1v

DlAl _dAlDlel dAl D(elplJ) dAl DlT

- del Dt +^d(Â£lplJ) Dt + dT Dt

Dt

dAl DlEs dAl DlEs

+ ^---h

DSAS

Dt

dEs

dAs Dlel N

Dt dEs

+Â£

+

del Dt d(sspsi) Dt

dAa DSES dAs DSES

+

Dt

dAs D{espSj) dAs DST

(3.13)

dT Dt

(3.14)

dEs Dt dEs Dt

We will find it useful to be able to convert a material time derivative with

respect to one phase (or species within a phase) into a material time derivative

with respect to another phase (or species within a phase). This can be done

using either of the following two identities:

g(0 g(0

Dt Dt

Â£>*;(.) Â£)(.)

Dt

Dt

+ vlsV{-) (3.15)

+ va'V(-). (3.16)

35

{

After using the identities and techniques discussed above, performing massive

algebraic calculations and grouping terms for ease of exploitation we arrive at

the following form of the entropy inequality:

A=-?5>v(ir+')

a x '

DST

~Dt

el

+T

F~l .

dA1

Â£lPl^rs

: d!

dA

N

dEs

+s>'^+siE K'+'-v

dAs

dEs ^ V r r dpsi

3=1 N

F-1 > : Es

1

T

U ( dAl

ep

del

+ rf VT + ep

,j(SA

BE1

dA1

V.E H-----VÂ£

dEs

J jdAlj

+Â£pWVÂ£+T*

v

l,s

1 { i i dA1 dAs .

Â£lpl + e p :E

T V dEs 8ES

+E

eVT

N r

^2

<
E

j=i L

ta3 vaia pa3 yaja tAaJ q.

1

T

N r

EE

a j=1 L

N r

-~Qi -~a1-

I1/ +1 J + V(eapajAaj) paV(eVj)

Va3a

-?EE

a: j=l

N

f)Aa 1 1

^ + p^T + 5Ki')2 + 5(*-'I)2

4ee

a J=1

N

dAa 1

dpai 2

+-(vQJQ)2

2

1 .dAa

r 3

1 1 ( f)Aa \

+tEEsavr+t'>o, (3.i7)

a j=l \ r /

where d = |(V va + (V u)T- The last term in the square brackets of line

four in equation (3.17) comes from converting ds into the independent variable

36

Es via

ds = F~T Es F~\

(3.18)

where F is the deformation gradient of the solid phase and is defined by (in

indicial notation).

F FkK XktK

dxk

dX

K

(3.19)

where x is the Eulerian coordinate and X is the Lagrangian coordinate. Note

that in the entropy inequality (3.17) the terms involving vaj,a and 'Vvaj,a have

only N 1 independent variables and one dependent variable. This is because

N

J2caj = i> (3-2)

3=1

and

N

J2Caivaia = 0. (3.21)

3=1

The difficulty arises from using intensive variables (variables that do not change

with a change in quantity, e.g. concentrations) as opposed to extensive variables

(variables that change with a change in quantity, e.g. mole numbers). While

upscaling is possible on extensive variables the results are not physically applica-

ble to open systems. To remove the Nth component dependence from vaj,a and

Vri,a in equation (3.17) we will use the following identities due to Bennethum

et al., [11]:

N N-l

PaJ Va3a =

j=1 3=1

Fai

pai

naN

FaN ] .

(3.22)

37

N

Nl

EGaj Vvai'a = ( Gaj ^-GaN ) Vvaja

< V oaU /

3=1 3=1 V F

N-1

-Ga" ^ V

/9aJ \

^ 1

(3.23)

j'=i

where Faj and Gaj are the coefficients of vaja and VuJ,a in (3.17), respec-

tively. Hence, these terms of the entropy inequality become:

dAa dAa

N r

EVE

a j=1 L

Pap0i

dpai dpaN

+ti taN paHAai -AaN)I

pON r V '

: (Vva')

N-l r

?ee Â£ (sr+H (5? +^)

a j=l

-eataNV ( ) + I pa^ pa^- ) V(eV0

\pajv / \ dpai dpN J

V (eapQj (A* AQJV))] va*a.

. dAa y dAa

(3.24)

Before we obtain non-equilibrium, equilibrium, and near-equilibrium results

we make the following definitions for ease of exposition:

Pi As

t,e = pF -Â§&*' (3-25)

and

*=p'F-^i- pT> <3-26)

where tse and tsh are the effective stress, and hydration stress, respectively. The

effective stress accounts for the elastic component of the stress in the solid phase.

The hydration stress accounts for physio-chemical forces between the liquid and

solid phases.

38

Whenever possible we will use the following thermodynamic definitions in

the following sections to simplify notation. The chemical potential, is de-

fined as

pr3 =

dAa

dMai

_ d{eapaAa)

va,T d{eapai)

= Aa + pa^

ea,T dpai

sa,T

(3.27)

and represents the amount of energy required to insert a molecule of constituent

j. Here M*3 is the mass of the jth constituent in the a-phase. The classical

pressure, pa, is

Pa = ~

dAa

dva

= ~{Pa)2

ea,Cai V '

dAa

dp0

N

dAa

= Ypp^

e,Cai ^ r)na.

3=1

dp0

(3.28)

where va is the specific volume of phase a. In the case of a single isotropic solid

phase pa represents the physical pressure of the solid phase. However, for our

system the classical pressure is distinct from the thermodynamic pressure, pa

r = -

dAa

dVa

= ~eapa^~

Mai ,t dea

eapaj ,T

(3.29)

which is the change in energy (extensive) of phase a with respect to a change in

volume of phase a keeping the mass and temperature fixed. This coincides with

the classical definition of pressure in Gibbsian thermodynamics. For a single

phase, single constituent system the previous two definitions are equivalent.

However, for swelling systems these pressures are distinct and related to each

other through

pa = pa + tra,

(3.30)

where 7r, the swelling potential, is

7rQ = eQpa

dAa

dsa

pa,Ca3

(3.31)

39

The derivation of (3.12) will be given in a subsequent section.

3.2.1.1 Non-Equilibrium Results

The following variables are neither constitutive nor independent:

T, VdSjs, Es . (3.32)

This means that they can vary arbitrarily, and because these variables appear

linearly, to avoid violating entropy inequality (3.17), the coefficients of these vari-

ables must be identically zero. This yields the following set of non-equilibrium

results:

y>V(~ + >)) =0, (3.33)

rv \ '

, dAl s s dAs n

S P -;--h Â£ p ---- 0,

8ES dEs

(3.34)

psi

tSj tSN = pSj (ASj ASN pSj + fj,SN) I.

pSN r \ r r /

(3.35)

Equation (3.33), the classical result for one phase, says that T and rj are dual

variables so that + rj = 0. We will make the simplifying assumption that

dAa

~8T

-va,

Vcn = l, s.

(3.36)

In equation (3.34) the partial derivatives are taken while holding the volume

fraction times density terms fixed, which means that these terms can be moved

inside the partial. Defining the total Helmholtz free energy as AT = elplAl +

espsAs, we can write (3.34) as

dAx

dEs

= 0.

(3.37)

40

This means that the total Helmholtz free energy is not a function of Es. Equa-

tion (3.35) contains an Nth component dependence and will be used later to

obtain an equilibrium result for pSj.

3.2.1.2 Equilibrium Results

We choose to define equilibrium to be when the following variables are zero:

il, dl, Es, vls, va>Q, e^', VT. (3.38)

At equilibrium the total entropy of the system is maximum and the net gener-

ation of entropy is minimum. Thus = 0 and > o, where x and y are

variables in the list (3.38). Hence, we obtain the following results:

Pl = P, (3.39)

t = -p'l, (3.40)

Â£l

ts = -psI + tse + -tsh, (3.41)

es

Â£>V = 0, (3.42)

a

t, = p(Vel) Â£y (+ Hv#) , (3.43)

(jm01* -paN)V(e Vj) V (eapai {Aa* AQJV)), (3.44)

41

(3.45)

(3.46)

Equation (3.39) says that at equilibrium the thermodynamic pressures in the

liquid and solid phases are equal. The next equation says that the stress in the

stress in the solid phase is a scalar multiple of the identity and contains the

of the system is zero at equilibrium. In equation (3.43) the last term in round

will be discussed shortly. Finally, we used restriction (2.25) to obtain equation

(3.46), which is the classical thermodynamic result that, at equilibrium, the

chemical potential of a species in two different phases is equal.

To obtain an equilibrium equation for //>, first note that the non-equilibrium

equation (3.35) contains an Nth component dependence. This implies that the

way in which the species are ordered matters. To remove this dependence, sum

(3.35) from j = 1 to IV and use equations (3.27) and (3.28) to obtain

Substituting the equilibrium expression (3.41) for ts into (3.47) and noting that

at equilibrium vSj,s = 0 yields

liquid phase is a scalar multiple of the identity. Equation (3.41) says that the

effective and hydration stresses. Equation (3.42) states that the total heat flux

brackets is new and is due to including Es as a constitutive independent variable.

Equations (3.44) and (3.47) contain an Nth component dependence and which

(3.47)

(3.48)

42

Substituting (3.48) into equation (3.35) results in an equilibrium expression for

/isi:

j.Sj j.se

Ha*I = Asil + +

psi p:

i.sh

Â£SPS

(3.49)

Similarly, we want to remove the Nth component dependence from equation

(3.45) to obtain an equilibrium relation for p,1*. First, sum (3.45) from j 1 to

N, divide by pl, then use (3.27), (3.28) and (3.40) to obtain

iJil = Ahl (3.50)

ph

This equation implies that the shear components of tlj must be zero. This makes

sense when we recall that equation (3.40) implies that the stress in the liquid

phase is a scalar multiple of the identity.

Equation (3.44) also contains an Nth component dependence. Because

vctj,a o at equilibrium, equation (2.37) becomes ^ i 3 =0, and (B.13) re-

3=i

N

ys. ct * ctj

duces to = Tp Thus, when we sum equation (3.44) from j = 1 to N we

3=1

obtain

N

= 5^/zaiV(eVJ) fiaNV{eapa) sataNV

0aN

5 = 1

-V(eapa(Aa AaN))

(3.51)

Now, taking a = l and /? = s, using (3.27), (3.43) and

VAl =

0Â£l

N dAl

T^)v<Â£V') + WVT+Â§Â£ VE+% (3-52)

43

which comes from using equation (3.11), we obtain the following equilibrium

result:

Ths +tj = fiV{elp1*) V(elpljAlj). (3.53)

The above equation is an equilibrium relation for the right hand side. We will

not be using this equation, thus a discussion of it is beyond the scope of this

paper, but we include it here for completeness.

3.2.1.3 Near-Equilibrium Results

The coefficients of the variables that are zero at equilibrium (equilibrium

variables) are a function of these variables. In other words, if zj are the equi-

librium variables and /* are the corresponding coefficients, then for each i we

have fi(zj). We linearize the coefficients to form positive quadratic terms by

using a Taylor series expansion about the equilibrium variables. We truncate all

second order and higher terms. For this reason the results hold only near equi-

librium. Additionally, we can perform a Taylor series expansion of each of the

coefficients about one or more equilibrium variables. This method of obtaining

near-equilibrium results within the context of HMT was clarified by Singh et al.

in [41].

We choose to perform a one variable expansion for all of the equilibrium

variables in (3.38) except for VT, Es, and d!, for which we will perform a two

variable expansion.

Performing a single variable expansion for el, vl,s, v1*'1, V v1}1, and e^3 we

obtain the following near-equilibrium relationships:

plps = r]Â£l, (3.54)

44

T

S

R vls +pl(Vel)

T3s +t = -W vli'1 + ^V{elpli) V(Â£lplAlj), (3.56)

(3.57)

(3.58)

In the above equations R, ffl, Clj, and SJ are the linearization coefficients.

In equation (3.54) 77 is a scalar variable. This equation states that the rate at

which the polymer takes on liquid is governed (at least near equilibrium) by

the difference in the thermodynamic pressures of the two phases. When the

time rate of change of the volume fraction of the liquid phase is zero we recover

the equilibrium relation (3.39) as expected. Equations (3.55) through (3.57) are

constitutive relationships and interpretation of them is beyond the scope of this

paper. The last term in brackets in equation (3.55) becomes zero after applying

the non-equilibrium result (3.36). Equation (3.58) says that mass transfer of

a constituent is governed by the differences in the chemical potentials of that

constituent in each phase. Solid phase diffusion is assumed to be negligible, so

there is no result for vsi,s.

Next we perform a two-variable expansion of the coefficients of VT and Es

about and the coefficient of VT. Set the coefficient of VT for the solid phase

Both R and R? are second order tensors, C j is fourth order, and S-7 is a scalar.

45

equal to Qs and consider the standard decomposition

neq

diss

(3.59)

where the subscripts eq, neq, and diss, mean equilibrium, non-equilibrium, and

dissipative, respectively. Substituting Qs and equation (3.59) into the corre-

sponding terms of the entropy inequality we find

Gs is a fourth order tensor representing the effect of strain rate on stress due

to relaxation processes within the solid phase. Hs is a third order tensor that

causes the thermal gradients to affect the stress inside the anisotropic solid

phase, Js is a third order tensor that represents heat flux in the anisotropic

solid due to strain rate, and Ks is a second order tensor representing heat flux

in the solid due to the thermal gradient. Symmetry relationships follow due to

the symmetry of the stress tensor and strain rate. Substituting (3.36) and (3.61)

into (3.59), and (3.62) into Qs, respectively, we obtain

diss

We now linearize to obtain

S S

e-F~l F~t : Es + Qs VT > 0. (3.60)

T 1*

diss

(3.61)

Qs = KS VT + F Js : Es.

(3.62)

N

+F Gs Ft : Es + Hs VT ^ ps^vs^svs^s, (3.63)

46

Ks VT + F Js : Es

tSj vSj,s pSjvSi,s + ^(vSi,a)2

(3.64)

Similarly, we perform a two-variable expansion of the coefficients of VT and

d! about these variables to obtain

N

tl = -plI + Hl -VT + L1 :dl -^2 plivli'lv{i'\ (3.65)

j=i

ql = Kl -VT + M1 : dl

+ tlj Vlj1 pljVlj1 (,Ali + ~(vlj1)2

7=1 L ^ 2

(3.66)

where Hl is a third order tensor representing the stress in the liquid phase due to

the thermal gradient, Ll is a fourth order tensor representing the effect on stress

due to the rate of deformation, Kl is a second order tensor representing the heat

flux due to the thermal gradient, and Ml is a third order tensor representing the

heat flux due to rate of deformation. As before, symmetry relationships follow.

To obtain a near-equilibrium result for pSj, substitute (3.63) into (3.47) to

get

tsj

p,Sj I = ASj I------

psi

+ p + + F G FT : & + Hs VT^ . (3.67)

This form of the solid-phase chemical potential was first reported by Singh et

al. [41]. The term containing the time rate of strain of the solid phase, Es, is

new. This implies in polymeric systems p,Sj is dependent on the strain rate of

the solid phase.

47

4. Derivation of a Model of a Simple Polymer/

Solvent System

We are interested in further simplifying our system to a two-phase, single

constituent per phase system, so that we might gain insight to the applicability

of this theory. The reader may ask if such systems are of interest in the scien-

tific community. As an example of one such system consider the Poly (methyl

methacrylate)- methanol (PMMA) which has a wide variety of applications.

PMMA is used as a glass replacement (Plexiglas), it is found in acrylic latex

paints and lubricating oils for machinery. Technically, PMMA is an amorphous

polymer. As mentioned in the introduction, amorphous polymers have long

chains of molecules that tend to coil about themselves. For this reason amor-

phous polymers tend to be flexible and hold together well when stressed. The

chemical resistance of PMMA to acids is good, to alkalis is excellent, and to

solvents, such as methanol, is poor. This means that acids have a difficult

time penetrating PMMA, while solvents penetrate PMMA quite easily. The

entangled nature of PMMA produces swelling behavior when put in a solvent

like methanol. Additionally, there are existing models on the PMMA-methanol

system, [43].

The remainder of this section will be organized in the following manner.

First we will make the appropriate assumptions for our entropy inequality to

be consistent with the system described above. This will greatly reduce the

complexity of the entropy inequality (3.17). Next, we will exploit the resulting

entropy inequality to obtain non-equilibrium, equilibrium, and near-equilibrium

48

results, as we did in the previous chapter. Following this, we propose an exper-

iment, list the relevant equations, and reduce the system to one that should be

solvable.

4.0.2 Assumptions and Entropy Inequality for a

Simple Polymer/Solvent System

A species is a compound that cannot be broken down into smaller pieces by

the system. We assume there is exactly one species per phase where the solid

phase is the polymer. This results in a greatly simplified system. First, the

diffusive velocity of the jth component in the o-phase is zero. This is because

there is only one component per phase and it constitutes the entire phase. For

the same reason the gradients of the diffusive velocities, Vv^',a, are also zero.

Both e| and e^J are zero because there will be no mass transfer between species

or phases. Additionally, rj is also zero because chemical reactions within a

phase with only one constituent cannot produce more or less of that constituent.

Having made these assumption we arrive at the following entropy inequality:

a

~fF~1' [(1 ^ Â£ltsh + Â£S*S)] f~T : &

(4.1)

49

4.0.3 General Non-Equilibrium Results

The following variables are neither constitutive nor independent and can

vary arbitrarily.

Ea

DST

~Dt

(4.2)

If the entropy inequality is to hold for all possible processes their coefficients

must be identically equal to zero. This results in the following set of non-

equilibrium relations:

(4.3)

s sdAs xldAl n Â£ P + Â£lpl = 0 dEs dEs (4.4)

Both of these equations are identical to those obtained in Section 3.2.1.1.

4.0.4 Equilibrium Results

The equilibrium variables for this system are the variables in (3.38) that

still appear in equation (4.1). They are:

el, Es, d!, Vls, VT. (4.5)

Using the same argument given in.Section 3.2.1.2 we obtain the following results:

r=& (4.6)

Fl

ts = -psI + tse + tsh (4.7)

Â£s

tl = -plI (4.8)

?1 = p'(Ve') ep (f^(V13') + (4.9)

Y^eaqa = 0 (4.10)

a

50

These are the same equations that appear in Section 3.2.1.2 without the equa-

tions in that section that give results for //>', p,s>, and Ts + i Because there is

only one constituent per phase it does not make sense to talk about the chemi-

cal potential of species. However, it does make sense to talk about the chemical

potential of a phase. This will be discussed in greater detail in later sections.

4.0.5 Near-Equilibrium Results

Near-equilibrium results are obtain as described in Section 3.2.1.3. Per-

forming a single variable expansion for es and vl,s, we obtain the following

near-equilibrium relationships:

plps = T]sl (4-11)

f[ = -it v1* +p{Vel) ep (|^(ViSs) + fg;(V.E))

-sV + (VT). (4.12)

Both equations (4.11) and (4.12) appear exactly as they do in Section 3.2.1.3.

Thomas and Windle [43] investigate a similar system as the one under con-

sideration here in and empirically obtain p as pQS-~Me> for polymer solvent

systems at a fixed temperature below its glassy-transition temperature, Tg, and

refer to it as the viscosity coefficient. In this equation p is the viscosity of the

polymer when the volume fraction of the liquid phase is zero, M is a constant,

and elm is the maximum value that el can attain and occurs when the polymer

is saturated. Achanta [1] suggests that both p and M are highly dependent

on temperature because amorphous polymers exhibit a wide range of behavior-

dependent on temperature. For our purposes we will adopt the assumptions of

51

Thomas and Windle described above. In equation (4.12), R is a second order

tensor. As before, the last term in round brackets will vanish when we employ

(3.36).

Next we perform a two-variable expansion of the coefficients of VT and Es

about these variables and obtain

Â£l

ts = -psI + tse + tsh -F Gs FT :ES -Hs VT. (4.13)

Â£s

qs = F Js : Es + Ks VT, (4.14)

In the above equations Gs, Hs, Js and Ks retain the same meaning they were

given in Section 3.2.1.3. In fact, equations (4.13) and (4.14) differ from equations

(3.63) and (3.64), respectively, only in that they contain no terms with diffusive

velocities.

Performing a two-variable expansion of the coefficients of VT and dl about

these variables we obtain

tl = -plI + Hl VT + Ll : dl (4.15)

ql = ql + Kl VT + Ml : dl, (4.16)

where Hl, Kl, Ml, and Ll also retain the meanings that they were given pre-

viously. Again, the only way that they differ from their previously obtained

counterparts is that they contain no diffusive velocities.

52

4.0.6 Total Stress and Total Heat Flux

Following [10] we define the total stress and particle pressure as

t = ests + Â£ltl, and (4.17)

P = Â£SpS + Â£lpl, (4.18)

respectively.

Substituting (4.13) and (4.15) into (4.17) and using (4.18) we obtain

t = -pi- s (pl + f + - G: - H vr)

+ el (Hl -VT + L1 : dl) (4.19)

for the total stress in a particle. Similarly, we define the total heat flux of a

particle to be

q = Â£sqs + Â£lql. (4.20)

Substituting (4.14) and (4.16) into (4.20) and using (4.10) we obtain

q = el (Kl VT + Ml : dl) + Â£s (ks VT + Js : 23s) . (4.21)

4.0.7 Generalized Darcys Law

To obtain Darcys law, we begin with the conservation of momentum for

the bulk phase, equation (2.36), neglect the inertial term, and substitute in

equations (4.12) and (4.15). We then have

R. vls = V(eV) V(Â£l(Hl VT)) V(Â£l(Ll : d1)) Â£lplgl

+p?(Vs) Â£y (Hr(VE-) + ^[(vi1)) . (4.22)

53

This equation was originally obtained by Singh et al. [41]. R is called the

resistivity tensor, and we assume it is diagonalizable and invertible. Further-

more, R~1 = K, where K is the conductivity tenor. We will be simplifying

this equation in a subsequent section, and so we leave physical interpretation

for later.

4.0.8 A First Approximation

In this section we make further simplifying assumptions with the goal of

arriving at a system that is solvable. To do this we will need several tools. In

Section 3.2.1 we listed several definitions, including chemical potential, classical

pressure, thermodynamic pressure, and swelling potential. We restate them

here, as they relate to the problem under investigation for ease of exposition

and clarity. The chemical potential is given by

(4.23)

the classical pressure, pa, is

N

(4.24)

the thermodynamic pressure, pa, is

(4.25)

and the swelling potential, na, is

(4.26)

54

The last three variables are related through the following equation originally

obtained by Bennethum and Weinstein, [12]:

pa = pa + 7r. (4.27)

This relationship can be derived in the following manner. First note that we

can write the Helmholtz potential of the liquid phase as

A1 = Al(Â£l,Â£lpl\T,...) = Al{Â£l,pl\T,...), (4.28)

that is, our two formulations coincide regardless of what combination of inde-

pendent variables we use. Here the over line on the right hand side emphasized

that a different combination of independent variables is being used. The total

differential of A1 is given by

DAl =

dAl

dÂ£l

dA1

J dAl

d,Â£ T

dpli

dplj

(4.29)

del

dA1

dÂ£l

Elplj

Â£l pi

dÂ£l +

(fe +

alt

dÂ£lpli

dAl

,d(Â£lPlj)

dp1-

dplj +

l

TJ

Â£l 8pli

pli DA

de.

(4.30)

Now take the partial of equations (4.29) and (4.30) with respect to Â£l while

holding pli fixed to obtain

dAl

dÂ£l

dA1 dÂ£l dA1 dpli

p'i dÂ£l pi dÂ£l pli dpli el dÂ£l

plj

dA1 dÂ£l dA1 dpli pli dA1 dÂ£l

dÂ£l elpli dÂ£l pli dpli e1 dÂ£l pli Â£l dpli sl dÂ£l p'i

(4.31)

After canceling the appropriate terms we have

dAl

dÂ£l

pli

dX_

dÂ£l

e1 pli

+

pli dA

Â£l dpli

(4.32)

55

Multiplying through by elpl we recover (4.28).

Consider the following experiment. A thin polymer film is affixed to a sta-

tionary object and surrounded by solvent. Assume that changes in the volume of

the polymer are due primarily to changes in the height of the polymer. Because

we are interested in predicting the swelling behavior of this system, we will need

Darcys law. However, equation (4.22) is far too complicated for our present

purpose. As a first approximation, we neglect gravity and all second order and

higher terms. Thus, the relevant equations for the proposed experiment are

pl = ps (4.33)

pl =pl + 7Tl (4.34)

Pl ~pS = T]Â£l (4.35)

Rvls =pl(VÂ£l) V(eV) (4.36)

Â£* +Â£*W = 0 (4.38)

Â£l + Â£s = 1, (4.39)

where equations (4.37) and (4.38) come from the mass balance for each bulk

phase, (2.23) and the assumption that the density of each phase is constant.

The unknowns of the system include

pl, ps, tt1, il, vl>s, sl, ss, pl. (4.40)

56

Thus, we have six equations and eight unknowns.

As mentioned previously, equation (4.35) says that time rate of change of

the volume fraction of the liquid phase is governed by the difference in the ther-

modynamic pressures of the two phases. Here, p is not considered an unknown

of the system because it can be approximated by [43]

p oe[m*74], (4.41)

where M is a constant, elm is the maximum volume fraction of the liquid phase,

and p is the viscosity of the dry polymer. It is this variable that accounts

for the polymer structure. Consider a polymer submerged in solvent. At first,

regardless of the polymer structure, the polymer is in a glassy state and polymer

chains are entangled. As the polymer takes on liquid the polymer swells. If the

polymer is not crosslinked, as may be the case with linear polymers, the solid

phase will be unable to support tension and the pressure in the solid phase will

be equal to the pressure in the liquid phase, i.e. ps = pl. On the other hand,

if the polymer is crosslinked, as in the case of network polymers, then the solid

phase can support tension. The degree of crosslinking or entanglement is taken

into account by p. Initially, 77 770, which means polymers with small rj0 like to

swell, and are therefore not terribly entangled nor highly crosslinked. Polymers

with large rf have more difficulty swelling, and are thus either very entangled

or highly crosslinked. This can be seen by comparing the two two values of el

that result from using p greater than one, p less then one, and the same left

hand side in equation (4.35). Using (4.41) and calculating M at equilibrium

we find that M = ln(^), and because p < p, we know that M is always

positive. According to [43] M controls the sharpness of the diffusion front, with

57

low values representing little to no diffusion front and high values representing

a sharp diffusion front.

It has been argued in various ways [1, 41] that ps 0, or at least that

ps 7Tl. If we adopt this assumption, and use (4.34) and (4.35) in (4.36) we

arrive at the following form of Darcys law:

R vls = -V(eV) elV(r]il), < (4.42)

which resembles the form of Darcys Law used by Achanta et al. [1]. An impor-

tant feature of this equation is that fluid flow is dependent on both the gradient

of the volume fraction of the liquid phase and the time rate of change of the

same variable. Additionally, rj is a function of the volume fraction of the liquid

phase. In other words, the system is highly dependent on el.

If we think of the swelling potential, irl, as a function of el only, then

V(eV) = -^eT"(Ve;). Replacing R~l with K, and using the approximation

for rj given above we can rewrite (4.42) as

vl's = -K +elV{r)e(-Mel/e)il)SJ .

Add (4.37) and (4.38) and using (4.39) we get

V-vs = -V- (elvl's). (4.44)

Substituting this back into (4.37) we obtain

il + (1 el)V {elvls) = 0 (4.45)

We will need to transform our system of equations from Eulerian to La-

grangian coordinates and back again because it is computationally easier to

58

solvent

1 V/,

'/z

polymer

(exagerated)

stationary

object

solvent

wet polymer

dry polymer

stationary

object

Figure 4.1: Proposed Experiment and Corresponding Geometry

59

model solids in Lagrangian coordinates. See Figure 4.1 for a schematic repre-

sentation of the experiment and proposed change of coordinates. Here, z is the

Eulerian coordinate and Z is the Lagrangian coordinate. Suppose that V is the

Eulerian REV and that Vs is the solid in the REV; then

. U A(AZ)

V A(Az)

where A is the cross-sectional area. Taking the limit of this equation as A

approaches zero, we have

r)7

~ = = l Â£l. (4.47)

This equation defines the transformation between coordinate systems.

In terms of the Eulerian coordinates (4.43) and (4.45) are

vls = -K

d(eW) dsl 0 ;g(e(-^7

del dz ^ Â£ dz

(4.48)

Â£* + (!-Â£*)

d{elvl>s)

dz

= 0.

(4.49)

Because of (4.47) we can change from Eulerian to Lagragian coordinates using

d(-) d(-) dz = {l Â£l)d(-)

dz dZ dz

dZ

(4.50)

Thus, in Lagrangian coordinates (4.48) and (4.49) become

is r, (, i.d(elTTl) del 0 u l d(e(-~Ms!/s!m')Â£l

v1'8 = -# (1- - e1)-

del dZ

dZ

and (4.51)

Sl + (1 Â£l)

2d(Â£lVl,s)

dZ

= 0.

(4.52)

60

The primary stumbling block in this analysis is that we do not have an explicit

expression for irl in terms of el. In [12] the authors rigorously derive an empirical

relationship originally obtained by Low [29], allowing them to derive an equation

in which n1 is explicitly dependent on the thickness of the water layers in a well-

ordered clay. One of the primary assumptions that leads to this expression is

that the sample of clay is assumed to be well-ordered, and therefore there is

a balance of forces. This is not the case in most polymeric and biopolymeric

systems. In particular, polymer gels are cross-linked, and therefore permanently

entangled. Additionally, uncross-linked polymers can behave as if they are cross-

linked when swelling occurs quickly. Thus, the well-ordered assumption does not

apply to these systems and there is no balance of forces between the solid and

liquid phases. The author still believes that it is reasonable to assume that nl

is a strong function of el. However, it is not clear how to measure this variable

experimentally independent of the other pressures. Achanta, [1], assumes that

we can use the elastic modulus of the dry polymer to approximate ^, but

does not give any formal derivation.

61

5. Discussion and Future Work

A numerical solution to this problem that can be compared to current lit-

erature is needed. However, it is not always clear how the variables presented

in this analysis, as well as others like it, are related to quantities measured in

physical experiments. A clear connection needs to be made between the phys-

ical properties measured experimentally and the variables that appear in the

governing equations, (e.g. tt1 in equation (4.42)) before a comparison can be

made in a meaningful way. The geometry proposed in the previous chapter was

chosen because it is one-dimensional and therefore el is easily measured. Once

an equation of state is obtained for the swelling pressure, that is not a function

of the equilibrium variables, the problem can be non-dimensionalized, and ap-

propriate initial and boundary conditions can be imposed, and the problem can

be solved numerically. The problem is similar to a class of moving boundary

value problems known a Stefan and Stefan-Neumann boundary value problems.

In addition, there is a plethora of future work to be done with respect to

diffusion, temperature dependence, phase change, and electroquasistatic prob-

lems. While Darcys law governs fluid flow, modeling drug delivery requires a

Fickian type equation to predict diffusion of species. A generalized Ficks law

of diffusion can be obtained by substituting (3.56) and tl* from (3.57) into the

momentum balance (2.30). However, a detailed analysis of such an equation is

beyond the scope of this work, and is left for future study. As mentioned before,

polymers are highly sensitive to temperature. A model including the effects of

temperature that could accurately predict the change in diffusive and swelling

62

behavior associated with the polymer nearing Tg would constitute a major ad-

vance in polymer modeling. To this end, we would include the VT term (second

term) of (4.22), and any other relevant equations. Entangled polymers that are

not crosslinked and swollen at a reasonably slow rate will eventually disentangle

and dissolve into the solvent. This is a phase change problem and is usually

modeled by considering a jump in coefficients. Finally, swelling is affected by

the pH of water (i.e. concentrations of hydrogen in the liquid phase) indicating

that the solid phase is chargee. The foundations of the application of electroqua-

sistatics to swelling porous media is laid out in [9, 8]. Future research in these

areas is required to gain a comprehensive understanding of polymer behavior.

63

Appendix A. Complete Nomenclature

Superscripts, Subscripts, and Other Notations

aj jth component of a-phase on mesoscale

o-phase on mesoscale

" denotes exchange from other interface or phase

k'1 difference of the two quantities, i.e. -k -l

\aj microscopic property of constituent j in phase [subscript] (non-averaged)

Latin Symbols

8Aai3: Portion of a/5-interface in representative elementary volume (REV)

Aaj, Aa: Helmholtz free energy density [J/Kg]

baj,ba\ External entropy source [J/(Kg-s-K)]

Caj: Mass fraction of component [-]

da: Rate of deformation tensor, approximately equal to the symmetric part of

va [1/s]

eaj,ea: energy density [J/Kg]

e^J: Rate of mass transfer from phase [subscript] to phase [superscript] per

unit mass density [1/s]

64

Es: Strain tensor of the solid phase [-]

F: Deformation Gradient of the solid phase [-]

gaj ,ga\ External supply of momentum (gravity) [m/s2]

Gs: Fourth order tensor, effect of strain rate on stress due to relaxation

processes in the solid phase

ha*, ha: External supply of energy [J/ (Kg-s)]

Hl,Hs: Third order tensors representing the effect of thermal gradients on

the stress

^OLj

i : Rate of momentum gain due to interaction with other species within the

same phase per unit mass density [N/Kg]

i : Rate of momentum gain due to interaction with other species within the

same phase per unit mass density [N/Kg]

Js: Third order tensor representing the heat flux in the anisotropic solid due

to strain rate

Kl, Ks: Second order tensors representing heat flux in the liquid and solid

phases, respectively, due to the thermal gradient

K: Second order tensor representing the permeability of the solid matrix

Ll\ Fourth order tensor representing the effect on stress due to the rate of

deformation [-]

65

maj: Rate of angular momentum gain due to interaction with other species

within the same phase per unit mass density [N-m/Kg]

TUp : Rate of angular momentum gain by constituent j in phase a due to

interaction with phase /? [N-m/Kg]

Maj: Rate of angular momentum gain due to the microscale angular momen-

tum terms see Appendix C [N-m/Kg]

Ml: Third order tensor representing the heat flux due to the rate of defor-

mation

n: Unit normal vector pointing out of a-phase within mesoscopic REV [-]

rnaj: Rate of gain of angular momentum of constituent j from other con-

stituents in phase a [m2/s2]

qai \ Partial heat flux vector for the component of phase [J/(m2-s)]

qa: Heat flux vector for phase a [J/(m2-s)]

Qaj: Rate of energy gain due to interaction with other species within the

same phase per unit mass density not due to mass or momentum transfer

[J/(Kg-s)]

Qp Qp\ Energy transfer rate from phase [subscript] to phase [superscript] per

unit mass density not due to mass or momentum transfer [J/(Kg-s)]

r: Microscale spatial variable [m]

66

raj: Rate of mass gain due to interaction with other species within the same

phase per unit mass density [1/s]

R: Second order tensor called the resistivity tensor

t: Time [s]

T: Temperature [K]

taj: Partial stress tensor for the component for phase [N/m2]

ta-. Total stress tensor for the phase [N/m2]

Tp }Tpi Rate of momentum transfer through mechanical interactions from

phase [subscript] to phase [superscript] per unit mass density [N/Kg]

vai,va: Velocity [m/s]

SV: Representative elementary volume (REV)

6Va: Portion of a-phase in REV

wa/3i: Velocity of constituent j at interface between phases a and /3 [m/s]

x: Macroscale spatial variable [m]

Greek Symbols

7a: Indicator function which is 1 if in mesoscopic region a and zero otherwise

ea: Volume fraction of a-phase in mesoscale REV [-]

Aai, A: Entropy production per unit mass density [J/(Kg-s-K)]

67

Â£: Microscale spatial variable which varies over REV for fixed x: r x + Â£

[m]

rjai,r]a\ Entropy [J/(Kg-K]

rjaj: Entropy gain due to interaction with other species within the same

phase/interface per unit mass density [J/(Kg-s-K)]

0: Total entropy flux vector for the phase [J/(m2-s-K)]

Entropy transfer through mechanical interactions from phase [sub-

script] to phase [superscript] per unit mass [J/(kg-s-K)]

pai: Partial mass density of component of a-phase [Kg/m3] so that eapaj

is the total mass of j^ constituent in phase a divided by the volume of

REV

pa: Mass density of a-phase averaged over /a-phase [Kg/m3]

68

Appendix B. Relations between Phase and Species Variables

N

= CajAai

3=1

N

ba = ^ Caj baj

3=1

Cri

Ci = (

pa

N 1

ea = Y^ caj (ej -vaja v
3=1

3=1

N

g = J2Caj9aj

3=1

N

ha = J2Cat(hai +gaiVaia))

3 =1

----------

E-^a.

"f

j=i

N

V MJ

j=i

(B.1)

(B-2)

(B.3)

(B.4)

(B.5)

(B.6)

(B.7)

(B.8)

(B.9)

69

N 1 qa = 5^[gaj' + tai vaia pai(eaj + -Va^a Vaja)vaia] 3=1 2 (B.10)

N 1 Qp = Y)Qp+t7 yai,a +r^ea^-v^a v01^)} 5=1 (B.ll)

N ta = ^(taj palvaiaVaja) 3=1 (B.12)

j=i (B-13)

N va = Y^, Ca*va* 3=1 (B.14)

N rf = Cajr}ai 3=1 (B.15)

'~3 III a < (B.16)

N pa = 'Epai 3=1 (B.17)

N ai ~ p^V^rf*) 3=1 (B.18)

i=i (B.19)

70

Appendix C. Identities Needed to Obtain Entropy Equation (3.4)

N

Â£

3=1

Eapai Dai Aai EapDaAa ( Aa^

~T Dt ~ ~T dT + ~T~eP

N

Aai

3=1

r3

Sa

Aa (V v ")

(C.l)

Zc t m t n+ z_^ t 1 v '

T '' Dt T Dt z' T

j=i j=i

N N

Yl j;taj Vva^' = ^2
i=i i=1

(C.3)

N

N

Â£Â£8? = -Â£'

j=l /3^a j=1 /3^q

(C.4)

AT

j=l /3/a L

T v + r [A"1 +TlCXJv'"f

(C.5)

N

Â£Â£? = -Â£Â£ ^ *+

j=l /3^a a /3^a

N

+ Â£ [t? + i^(o^)2l ) + Â£Â£ ej? (Aa*+T?7a*) (C.6)

j-i

/?A* j=l

71

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