The Development of Guidance for Solving Polymer-Penetrant Diffusion Problems in Marine Hardware

.................................................................................................................. ii ACKNOWLEDGEMENTS .......................................................................................... iii DEDICATION .............................................................................................................. iv TABLE OF CONTENTS ............................................................................................... v LIST OF TABLES ....................................................................................................... vii LIST OF FIGURES .................................................................................................... viii CHAPTER 1: INTRODUCTION .................................................................................. 1 CHAPTER 2: DIFFUSION FLOWCHART TOOL .................................................... 12 INTRODUCTION ................................................................................................ 12 FORM THE PROBLEM....................................................................................... 14 IDENTIFY THE MATH & SELECT A SOLUTION METHOD ........................ 27 APPLY THE SOLUTION METHOD & SOLVE FOR WHAT YOU WANT.... 35 GENERAL GUIDANCE ...................................................................................... 38 CHAPTER 3: DIFFUSION RESOURCES ................................................................. 42 INTRODUCTION ................................................................................................ 42 DATABASE RESOURCES ................................................................................. 43 SIMULATION ...................................................................................................... 47 EXPERIMENTATION ......................................................................................... 47 GENERAL DIFFUSION REFERENCE .............................................................. 49 CHAPTER 4: DIFFUSION PROBLEMS ................................................................... 50 INTRODUCTION ................................................................................................ 50 STEADY-STATE, PLANE SHEET, SPECIFIED BOUNDARY CONCENTRATION............................................................................................. 50 UNSTEADY PLANE SHEET, IMPERMEABLE BOUNDARY ....................... 57 vi CONCENTRATION-DEPENDENT DIFFUSION COEFFICIENT ................... 66 HETEROGENEOUS MATERIALS .................................................................... 66 OTHER RELEVANT PROBLEMS ..................................................................... 67 CHAPTER 5: BARRIER POLYMER APPLICATION .............................................. 69 CHAPTER 6: SUMMARY AND OUTLOOK ............................................................ 76 REFERENCES ............................................................................................................. 77 APPENDIX 1: MARINE-HARDWARE-RELEVANT DIFFUSION PROBLEM QUESTIONS ............................................................................................................... 82 APPENDIX 2: MARINE-HARDWARE-RELEVANT DIFFUSION PROBLEM GEOMETRIES ............................................................................................................ 83 APPENDIX 3: MARINE-HARDWARE-RELEVANT DIFFUSION PROBLEM INITIAL & BOUNDARY CONDITIONS .................................................................. 84 APPENDIX 4: DIFFERENTIAL EQUATION SOLUTION METHOD APPLICABILITY CHART ......................................................................................... 85 APPENDIX 5: UNSTEADY-STATE DIFFUSION PROBLEM WORK, UP TO CHOOSING A SOLUTION METHOD. ..................................................................... 87 APPENDIX 6: UNSTEADY-STATE DIFFUSION PROBLEM WORK, SEPARATION OF VARIABLES METHOD ............................................................. 90 APPENDIX 7: CRANK TEXT PROBLEM, LAPLACE TRANSFORM .................. 95 BIBLIOGRAPHY ...................................................................................................... 100

There is a need for the development and trial of an accessible, robust procedure that can be followed to solve polymer-penetrant diffusion problems associated with marine hardware failures, incorporating some of the complexities abounding in these types of systems.
The goals of this guidance are to instruct design engineers in the appropriate methods to apply when confronted with two fundamental questions related to diffusive mass transfer in marine hardware coatings and components: A. How long until X occurs? 'X' can be the concentration or flux of a chemical species at a certain location in the hardware reaching a specific level of interest.
B. What material should I select for my application to avoid diffusive-masstransfer-related issues? Question A may be encountered when dealing with an existing system that may be experiencing a problem, or to assess risk for a potential problem in an existing system. It may also be an important question to answer when finding a solution to question B. Many performance requirements are made of materials used in marine hardware systems besides those related to diffusive mass transfer (e.g. structural strength, acoustics, corrosion resistance), and may dominate the material selection decision of a design engineer. It has been my experience that reliance on materials of historical familiarity is sometimes used as a surrogate to understanding complex phenomena, even if the modern application is different from the historical one.
Motivating examples from work experience with marine hardware: The origin of this thesis topic comes from my experience working at the Naval Undersea Warfare Center (NUWC) in Newport, RI. Problems within the scope of cases, the diffusion of water through multiple layers of different materials led to an electrical failure of the internal hardware. In one case a proposed dip-coating fix was revealed to be insufficient as the diffusion coefficient of water through the dip-coating was too high for it to be an effective barrier. This was shown by analogy in a bruteforce experiment instead of by predictive modeling due to my insufficient knowledge to completely describe the phenomena. In the other investigation the measurement of diffusion coefficient by the gravimetric mass uptake method was confounded by runaway water absorption forming entrained liquid blisters in the samples. Again, my lack of sufficient knowledge led to the inability to address the unexpected behavior and hampered diffusion coefficient calculation.
I believe good engineering design practice for material selection could be used to reduce the occurrence of these types of delayed problems. Through my involvement in the design end of marine hardware I found a lack of such reasonable guidance for material selection which incorporates diffusion behavior and other deleterious chemical interaction. Non-scientific accelerated water bath testing or vague 'compatibility' statements are often the only guidance offered to engineers that address these types of potential problems. These methods are presented as a 'one-size-fits-all' approach, but their results are highly dependent on sample materials, geometries, and temperature response of material property. In addition, the 'acceptable' numbers resulting from such tests reflect the common materials of the time; it is with great peril that one should apply such non-specific evaluation methods to the exotic and novel materials considered for the advanced hardware design of the future.
One application that is dependent upon polymer-penetrant diffusion is the use of environmentally acceptable biocides to prevent marine biofouling on flexible surfaces not accommodating to the typical painted-on anti-fouling solution. Marine biofouling causes noise and drag to a marine vessel (2), and can interfere with the proper operation of sensor arrays. In this case, the biocide is incorporated into the flexible coating material and protects the surface by slowly leaching out over time, discouraging marine organisms from landing and attaching. The success of this mechanism involves the diffusion of the biocide (penetrant) through the coating (polymer) until it can no longer support the minimum release of biocide from the surface necessary to prevent fouling. The time to reach this minimum is a critical design parameter and is dependent upon the material properties of the biocide and coating, the geometry (thickness) of the coating, the degree of loading of biocide in the coating, and the ocean environment the surface is exposed to. A model is needed that considers these parameters and that can predict the efficacy of a biocide-laden coating solution.
To summarize my motivations, I am finding I need to solve polymer-penetrant diffusion problems, but they are occurring in different systems which have their own unique modeling challenges. Examples include the aforementioned biocide efficacy question, material selection for new design, accelerated life testing prediction, failure analysis, and diffusion measurement. Each system may be unique enough that it has its own field of advanced study. Do I have to be an expert in each field in order to solve the diversity of problems I encounter? No! That would take many lifetimes. Instead, I want to use the basic functional components of each of them in order to solve the majority of problems I encounter. More advanced problems would then require collaboration, the necessary resources for which will have already been conveniently compiled in this work. Diffusion is anticipated to be important for future navy systems that incorporate novel and exotic composite materials and additives. Polymers, plasticizers, and leachable components will all be subject to the persistence of random molecular motion, much more so than the ceramic and metallic materials of historic familiarity.
At the end of the day, I am most interested in improving the reliability of marine hardware systems, accurately predicting and extending their service life, and reducing maintenance costs especially those associated with marine biofouling. I also hope to contribute innovative solutions to the incorporation of polymers in new applications and to improve quality control and specification to ensure the U.S. Navy is procuring appropriate materials in its systems. This would most likely manifest as scientifically valid, application-specific test requirements on procurements which rapidly and inexpensively check materials and components for problematic behaviors such as fluid incompatibility, permeation resistance, and incorrect formulation.

Difficulty with subject matter, lacking literature:
I have had difficulty in approaching these types of problems, and this lack of sufficient understanding partly motivated the choice of thesis topic. What seemed lacking in the literature was guidance akin to the TurboTax® tax preparation software concept, i.e. a user-friendly step-by-step approach which introduces the appropriate topics and asks for the right decisions along the way towards the ultimate solution.
Human memory is naturally spatial (3) and works well with a flowchart since it represents a sequence of events as though along a path. Solving mass transport problems requires applying concepts of mass balance, constitutive relationships, differential equations, and evaluation of transcendental functions in their proper order.
Getting the order of operations correct may seem straightforward to an experienced practitioner, however I can attest that it is confusing and frustrating for the novice. One particularly vexing trap is to stray too far from the original problem at hand while delving into the nuance of applying one of the aforementioned concepts, a sort of 'Hansel and Gretel' bewilderment as the breadcrumbs of our thought process fall prey to the complex gingerbread diorama challenge of incorporating new knowledge. The added concept of mole-based or mass-based units leads to mass transport requiring at least twice the generic symbols and equations to handle the permutations of scenarios as compared to heat and momentum transport. Mathematical analogy is made in the literature between heat conduction and mass diffusion (4) (5), however mass diffusion has unique nuances which don't appear in heat transfer such as chemical reaction (1) and an apparent discontinuous concentration driving force between phases at equilibrium (although chemical potential is continuous, but practical transport phenomena equations presented in the literature avoid using chemical potential).
It is clear that much work has been done related to polymer and penetrant diffusion in the last 60 years and advancements continue today. In spite of all this effort, the incorporation of these diverse behaviors into the solution of real-world applications is not straightforward in the literature. Beyond the negation of simplifying assumptions, the hapless engineer is sent down an unfamiliar path delving deeply into the particular science of a unique behavior, with guidance returning the user to the original problem at hand noticeably absent. No user-friendly process seems to exist to pursue (in general) the solution of diffusion problems involving penetrants in polymers; no one resource has combined all tools necessary to approach the diversity of behaviors potentially encountered. Indeed, how does one navigate the dense jungle of polymer-penetrant diffusion knowledge between the polymer-penetrant diffusion problem and its solution?
This thesis work offers to create that guidance to applying modern advances in the polymer-penetrant diffusion field to real-world problems, compiling the state-of-the-art and making it accessible to those not deeply involved in the basic research.

Complexity of the phenomena:
Many of the assumptions simplifying mass transport in gases and liquids do not apply to diffusion through polymers, as shown in the literature. In 1968, Fujita (6) notes that the sorption and permeation of organic vapors is "exceedingly complex" in polymers at temperatures below glass transition (Tg). Crank (5) highlights that the diffusion of vapors in high-polymers, different from most (e.g. dilute) systems, is concentration dependent, and a "very marked, characteristic feature" at that. Crank goes on to note a number of numerical solutions exist for these situations, however these solutions vary and are not uniformly applicable to all polymer-vapor systems. Vieth (7) concisely displays the potential for complexity when a condensable vapor (as opposed to a permanent gas) is diffusing through a rubbery polymer (above Tg); activation energy for diffusion (ED), solubility, and diffusion coefficient can all be strong functions of penetrant concentration, more so if the temperature is near the polymer's Tg or if the penetrant acts as a plasticizer to the polymer. Stastna & De Kee (8) propose that "[g]enerally, the diffusion behavior of polymers cannot be described by Fick's law with constant boundary conditions." Neogi (9) affirms that, although the three assumptions of homogeneity, isotropy, and local equilibrium apply to conventional transport phenomena used to address fluids, "[n]one of these apply to solid polymers uniformly." Cussler (1) asserts that for low-molecular-weight solutes in a polymer solvent, diffusion modeling requires a blend of concepts from diffusion in solids and diffusion in liquids; I found it rather telling that this major work on diffusion phenomena purposefully omits a detailed discussion on diffusion in solids and restricts its scope to fluid systems.
Finally, Kwan (10) summarizes many of these historically observed peculiar behaviors, stating that it is "…essential… to understand the nature of the diffusion of low molecular weight liquid penetrants into a given polymer matrix," and admits that after "several investigations" and "a variety of studies" there are now "various models" which describe the diffusion process.
One of the reasons for the complexity of these types of systems is the range of polymer and penetrant characteristics which can influence the diffusion process. The diffusion coefficient is nearly always temperature dependent, and may be concentration dependent (5). A very marked step-change in diffusion behavior generally occurs around the glass transition temperature of the polymer; over a narrow temperature range the types of accessible motions available to the polymer chain segments change significantly, impacting the mechanism of penetrant diffusion except for very small penetrant molecules, such as monatomic or diatomic gases. The relative time scales of polymer and penetrant molecular motion play a primary role in determining the type of diffusion which occurs in the system; Neogi (9) summarizes different power-law behavior of sorption data versus time in the expression ∞ = with α values reflecting the type of diffusion (e.g. pseudo-Fickian, classical/Fickian, anomalous, case II, super case II). Interaction or the lack thereof between polymer and penetrant molecules can lead to plasticization or penetrant clustering, respectively (7).
Chemical reaction can also occur during the diffusion process, removing and generating chemical species with different diffusion behaviors (11) (12). Finally, diffusion through non-isotropic systems (e.g. inhomogeneous bulk material, composites, and layers) must accommodate different diffusion behaviors in different phases and the transfer between them (1) (5).

How I am trying to solve the problem:
Specifically, the goal is to produce a straightforward step-by-step process flowchart tool for modeling the concentrations of small molecule penetrants (e.g. water, solvents, oils, plasticizers) within glassy and rubbery solid polymers as a function of position and time, making adjustments to include phenomena atypical to that found in the diffusion of gases and liquids.
This tool shall: be usable by the average scientist or engineer not deeply versed in the specifics of polymer-penetrant diffusion; provide guidance to the user to avoid mistakes, sanity check, and assess the expected range of accuracy of the prediction; be sufficient to address most types of polymer-penetrant diffusion problems; and be validated through experimental data both historical and novel.

Summary of the sections that follow:
Summary, Ch.2 Diffusion flowchart tool: The key product of this thesis effort consists of a process flowchart tool which can be followed step-by-step to develop the equations and solutions of mass transport in solids, specifically polymers. The fundamentals of mass transport are reviewed from the literature, and the methods put forth by these authors form the basis of the flowchart process. The resulting flowchart process can be broadly described in three steps: 1, forming the problem; 2, identifying the math; 3, applying a solution method. Each step will be expanded upon in the detailed description of the tool. To facilitate use by the target audience, general guidance and pitfall avoidance is also included to increase these users' chance of success.

Summary, Ch.3 Diffusion Resources:
This chapter summarizes the resources available for those seeking quantitative answers to their diffusion problem. It is intended as the 'next step' for users of the Extensive math notes are included to convey the scope of the effort required in order to develop solutions for mass diffusion problems. Summary, Ch.5 Barrier polymer application: This section describes a method for dramatically reducing diffusion through a barrier polymer by adding nano-thick micron-sized clay platelets. Various models have been proposed to correlate the characteristics of the additive with the apparent diffusion.
This section is included to illustrate complexities of real-world transport problems and connect their origins to the fundamental steps of the flowchart tool.

Future work / looking forward:
Current research in polymer-penetrant diffusion seems to focus in general on the need for computational solutions to the problem; good experimental techniques are critical for end-use application, but the predictive power of computational methods is necessary given the vast number of experimental permutations of polymer-penetrant compound combinations. Bernardo et al.'s (13) review of polymer membrane separation technology emphasized the need for molecular dynamics studies and improved understanding of transport on a molecular level to advance the field, building on the computer simulation necessary to describe amorphous and semicrystalline polymers. Diffusion can be the determining factor of long-term reliability and performance, however purely experimental characterization is impractical; predictive models are necessary for optimizing systems and extrapolating short-term experimental results (14).
Since diffusion seems to be a lesser-well-understood property but with potentially disastrous consequences, it is my hope that this guidance will be a useful addition to the design engineer's toolbox that will improve material selection in marine hardware systems.

INTRODUCTION
The entirety of the flowchart process used to solve polymer-penetrant diffusion problems in marine hardware can be grouped into three parts ( Figure 1  Within each part are a sequence of steps that are to be followed in roughly the given order, with some exceptions noted in the detailed summaries. Table 1 provides an outline to this flowchart process. This flowchart tool was developed through review of the following relevant textbook literature on the subject of diffusion: Stastna & De Kee (8), Vieth (7)

IDENTIFY THE MATH & SELECT A SOLUTION METHOD
Step 7 Identify the type of differential equation.
8 Identify the types of initial & boundary conditions. 9 Select the appropriate solution method.

APPLY THE SOLUTION METHOD & SOLVE FOR WHAT YOU WANT
Step 10 Apply the solution method.
11 Verify the final solution satisfies the conditions of the problem.
12 Solve for what you want. Start by clearly and absolutely defining the desired solution. In the words of the late, great philosopher Yogi Berra, "You've got to be careful if you don't know where you're going 'cause you might not get there!" (18). This is an important first step as there will be many subsequent steps involving different physical and mathematical concepts, likely prompting one or more reference searches; it is easy to get lost on your way to the solution. Having a clearly defined goal will be like a beacon, always allowing you to return to the original problem at hand. This first step, 'Clearly define the desired solution', is asking the user what 'u' want. To borrow notation from Farlow (19), the notation u, ux, and uxx are used to describe the dependent transport variable 'u' (e.g. concentration, temperature, velocity) and its first and second derivatives with respect to an independent variable 'x' (e.g. position, time). The user is asked the fundamental question, "What do you want to know?" Typically sought answers in mass transfer are In addition, the time (or position) at which any of the above achieve a certain value of interest may be the desired solution. This latter goal would then be achieved through either algebraic rearrangement of the solution resulting from this process; or by some 'solver' method such as Newton-Raphson, Goal Seek (Microsoft Excel), or another root-finding algorithm. These methods are not covered in the scope of this work, but may be necessary in practice especially for any complicated or transcendental functions.
A guide for example questions expected to occur for a marine-hardware-relevant diffusion problem can be found in Appendix 1. It is assumed that users are looking to identify how to solve their polymer-penetrant diffusion-related problem in the sense of fixing it practically, not just how to describe it.
Once the desired solution value is identified, it is critical to answer the question, "How is your desired solution related to the concentration profile?" This is important to establish because no matter what the final desired solution is, the precursor solution necessary to get there (which is developed while following this process) is the concentration profile of the system, i.e. cA(x,t). For momentum and heat transport problems the analogous precursor solutions are the velocity and temperature profiles, respectively. If the desired solution is the concentration value at a specific position and time, or the time required for the concentration to reach a certain value at a specific position, then no additional relationship is needed. If instead the flux value is desired (usually from a boundary surface), then the key relationship will be the constitutive relationship between species flux and driving force and will inherently incorporate the concentration profile. An important assumption here is that local equilibrium exists between each side of the interface at the boundary. This is true for a system that is 'diffusion limited', i.e. the diffusion through the bulk phase is the slow process. If instead the system is boundary limited then a mass transfer coefficient and interphase mass transfer solution process is appropriate; this subject is beyond the scope of this work, however more information on it can be found in chapter 8 of Cussler's text on diffusion (1). If the cumulative amount of diffusing species that has moved past a specific position after an elapsed time is desired, then the relationship is the integral of the flux at that position over the cross-sectional area and the elapsed time. For steadystate conditions this will simplify to straightforward multiplication. Finally, the relationship for the average concentration of a region of the system at a particular snapshot in time is the integral over that system dimension at that specific time.

Draw picture(s).
An idealized picture of the problem will help to visualize the scenario and link the physical problem with the abstract math used to describe it. A second picture illustrating the macro-system schematic which the simplified problem exists in can be helpful for knowing your problem's place in the larger picture (16), but for the purposes of solving the problem it can be optional. Here the user is asked the question, "What does your physical system look like?" A guide for example geometries expected to occur for marine-hardware-relevant diffusion problems can be found in Appendix 2.
Drawing the simplified picture will establish the space dimension(s) of the problem. The coordinate system (rectangular, polar, spherical) is chosen based on the simplest shape of the problem relative to what the user wants to know. Polar and spherical coordinate systems are not covered in the scope of this work; however, conversion guidance between coordinate systems can be found in the literature (16). A useful tip here is to choose the coordinate system direction (i.e. the direction of increasing axis values) to coincide with what the user intuitively believes to be the direction of decreasing concentration, i.e. the net flow direction of the species of interest. For example, the position z = 0 should correspond with the boundary location of greatest concentration. This will make the flux positive and the concentration gradient negative in the positive coordinate (z) direction. This approach may not be universally applicable, as shown later on in chapter 4 in one of the worked-through reference solutions from the literature (4).
A good picture will emphasize both the region of interest of the problem and its boundaries. Boundaries and their characteristics are an important part of the final solution; they greatly change its form from one model to the next even if the systems share the same fundamental differential equation. One feature to establish while drawing the picture is to decide if the physical space is an open or closed region, i.e. is one or more direction considered to be of infinite length, or is there finite length to the system? A leaching problem into the infinite expanse of ocean or absorption into a very thick material will likely be semi-infinite and have one space dimension go on to infinity. The behavior of a layer with finite thickness separating two phases will have to be considered as finite length.
These considerations also influence whether a short-time solution or a longtimes solution is necessary for the model to be valid. These limits of interest with space and time factor in to the dimensionless ratio 2 which influences the convergence of the final solution. If the diffusion coefficient of the system can be determined along with the length and time of interest, then the value of this ratio can assist with choosing the appropriate solution method later on. Guidance for the validity of one or the other (e.g. short vs. long times) can be found in (20). For a plane surface the short-time solution is valid for 2 < 10 −2 . The long-time solution is valid for 2 > 0.5

Define initial and boundary conditions.
Here the user is asked the question, "What is known at the boundaries of the system being modeled?" This is referring to what is or is expected to be known at the boundary positions (space and time) of the system in terms of the following possible values:

Known concentration
The concentration value of the penetrant species of interest is typically specified throughout the bulk of the film at time zero as the initial condition; it may be zero, uniform, or vary as a function of position. The concentration value may also be specified at the boundaries. This is usually specified as constant, but can vary as a function of time at the boundaries. The time-varying case is not covered in the scope of this work, although further guidance can be found in the literature (4) (5). Specifying concentration is simpler in theory, but in practice it can be difficult to determine. The following values are more likely be known:

Known exterior phase value & equilibrium relationship
It is typically easier to know through measurement and to control in practice the bulk concentration of the exterior phase to the system of interest. Translating this to the diffusion system requires an assumption of local equilibrium at the phase boundary and an equilibrium relationship. Local equilibrium implies a steady-state mass balance at the phase boundary, i.e. flux out equals transfer in, even if there is a discontinuous concentration; that relationship is given by a partition coefficient equation (e.g. Henry's law, = ) and 'hypothetical' concentrations. More information on this can be found in chapter 8, section 5: 'Mass Transfer across Interfaces' of Cussler's text (1).
This boundary condition may also be a constant value or can vary with time.

Impermeable
The third boundary condition likely to be encountered is that of the impermeable boundary. This condition implies that no flux of the penetrant species occurs at that boundary position. Mathematically this leads to the relationship that the first derivative of concentration with respect to position is zero at that boundary position, A reactive boundary condition may also be specified if a heterogeneous chemical reaction with the penetrant species is occurring. This situation is not explored further in the scope of this work, but example problems can be found in (16) A guide for example initial and boundary conditions expected to occur for marine-hardware-relevant diffusion problems can be found in Appendix 3.
The user, in searching for these known values, will inevitably discover a multitude of different units to describe the concept called 'concentration'. It will be generally helpful (although not critical) to establish these units and a mechanism to enable their conversion so that the final solution can be obtained from this process. In general, values of concentration are described as an amount of material per unit volume of space, with the 'amount' being in terms of moles or mass of penetrant species.
Respective 'fractions' (e.g. mole fraction, mass fraction) may also be encountered, defined as mole/mass of penetrant species divided by the total moles/mass of all material in a given unit volume of space; as a result, the mole/mass fraction term for 'concentration' is dimensionless. Due to experimental convenience, 'partial pressure' values can also be used to describe the 'concentration' of a penetrant species. For practical use of this, an equilibrium relationship is required to convert to other concentration units; the reader is encouraged to refer to the now familiar chapter 8 from Cussler's text (section 5.3) (1).

Define the constitutive equation for species flux.
This is the point at which the relationship between the species flux and the concentration gradient is specified, along with any other mechanisms influencing the motion of matter across a fixed plane in space (or appropriate moving reference frame).
This relationship is defined in accordance with the user's understanding of what is going on in their system of interest. The curious reader is referred to (16), chapter 24 as a brief introduction to fluxes of mass due to gravity, electric, and magnetic influences; these mechanisms are not covered in the scope of this work. Even if the user is just interested in the concentration profile as their final solution, this is still an important step. The species flux equation will be the bridge between the concentration gradient and the next step, the shell mass balance.
During the development of the diffusion flowchart tool, this step was originally created as 'determine the math' and also included the subsequent step; however it was later expanded into the more specific steps shown here, prompting users to separately specify the constitutive equation for species flux and the shell mass balance. This was done to give these related but separate concepts the proper space to be more completely grasped rather than lumping them together, as well as making it more convenient to change to a different constitutive equation should the need arise. It is anticipated that complexities involving non-Fickian diffusion (expected with polymer-penetrant scenarios) might be better handled down the road if initially these concepts are treated separately.
A variety of equations to describe this flux and account for these different mechanisms can be found in the literature. At the very least a molecular flux term is expected, and if the bulk medium is a gas or liquid then a convective flux term may be appropriate. Equations including these terms for a mass and mole basis as well as for fixed coordinates and those relative to the velocity of the fluid can be found in (16) This step is rather automatic and seems to be without decisions to make. The one question to the user that is important to ask at this time is, "Is the diffusion coefficient concentration-dependent?" If unfamiliar with the polymer-penetrant system of interest, the user will have to look up this sort of interactive behavior in the available literature. In the likely event that this information cannot be found in sufficient detail (to find a mathematical function for DAB = f(cA)), the user will have to conduct an experiment to determine the relationship between diffusivity and concentration. See the Experimentation section in chapter 3: Diffusion Resources for additional information on this subject.
As routinely stated in the literature the diffusion of penetrants through polymers is markedly concentration-dependent. Many textbook problems in gases or liquids will have as an assumption a diffusion coefficient that is not a function of time, position, concentration, or otherwise; however the tendency is for this to not be the case in solids, especially polymers (1) (5) (16) (21). This feature may be glossed over in the typical instruction of transport phenomena, where mass transfer is often taught in an analogous fashion following heat transfer. This heat analog approach to teaching mass transfer loses the significance of a dependent-variable-varying coefficient, e.g. thermal conductivity changing with temperature, mass diffusion coefficient changing with concentration. In the first chapter of Carslaw & Jaeger's treatise on modeling heat conduction (4), the authors point out that this characteristic is much more pronounced in mass transport than it is in heat conduction, and subsequently deals no further in the text with such nonlinear behavior. Mathematically, the nonlinear differential equation cannot be solved by superposition of elementary solutions as sums or integrals (22). As will be shown through demonstration of the flowchart tool, incorporating the concentration-dependent diffusion coefficient requires advanced solution methods, usually numerical approaches.
This characteristic is important to establish at this point because it will have an immediate impact in the following steps of the process.

Conduct a shell mass balance.
Chemists think in molecules, mechanical engineers think in continua; chemical engineers think in both. The equations developed in this process reside in the realm of continuum mechanics; the intricacies of the diffusion coefficient must be traced back to molecular interaction.
This step, based on the concept of conservation of matter, builds the relationship which accounts for the mass of penetrant material in the bulk of the system of interest as a function of position and time. The balance is performed on a 'shell' volume section within the bulk, which is then allowed to shrink to differential thickness; this is then applicable to the entire region between the boundaries. The flux of penetrant 'in to' and 'out of' the shell volume comes from the constitutive equation for species flux defined previously.
Going back to the picture drawn in step 2, a thin slice is added consisting of two parallel plates (or planes) in the middle of the bulk region. The plates are oriented such that the thickness (thin) dimension is in parallel with the anticipated mass diffusion direction; in other words, the plates are perpendicular to the principal direction of diffusion. In their text on transport phenomena; Bird, Stewart, and Lightfoot use the phrase: "thin in the axis direction of anticipated species flux" (16). In order to complete the shell mass balance, the user must answer the following two questions:

Steady vs. Unsteady State
Based on their system of interest, the user must decide if the concentration of penetrant is changing with time over the region of the system. If not, then the system is at steady-state and there will be zero accumulation within the shell. Otherwise, the system is unsteady (i.e. dynamic) and a time derivative must be incorporated in the shell balance. A consequence is that the mass accumulation rate within the volume is not zero, i.e. ≠ 0.

Chemical Reaction
If the penetrant species is reacting in this system, then a reaction rate term (gain or loss) must be included in the mass balance. If the reaction is homogeneous (i.e. occurring throughout the bulk region of the system) then this term is included in the balance. If the reaction is heterogeneous (i.e. occurring at the system boundary/phase interface), then the reaction rate is not included here and is instead defined in the boundary conditions. Reactive boundary conditions are not covered in the scope of this work.
As the volume thickness is allowed to approach zero, a derivative relationship with respect to position is developed between the fluxes at the opposing faces. This is where the concentration-dependence of the diffusion coefficient will be important, as is described in the next step.
6. Form the differential equation.
Combining the results of the previous two steps will generate the differential equation that is to be solved in the rest of the process. Plug the constitutive equation for species flux into the flux term from the shell balance and take the derivative, resulting in a differential equation with concentration as the dependent variable. This equation will be analyzed in the next part of the process; re-writing it into standard or general form (19) (23) will facilitate the analysis in the later steps.
For a constant diffusion coefficient, the usual instruction is to bring it outside of the derivative and simplify the overall equation. However, if the diffusion coefficient is expected to vary with concentration then this cannot be done, resulting in a nonlinear differential equation. Table 2 illustrates this difference.

IDENTIFY THE MATH & SELECT A SOLUTION METHOD
The differential equation resulting from the previous section will now be classified. This classification combined with the nature of the boundary conditions and any initial conditions will determine the solution method most appropriate for solving the differential equation. Here, the 'solution' will consist of an expression for species concentration as a function of position and, if applicable, time.
7. Identify the type of differential equation. Certain solution methods will only work for differential equations of a certain type. For example, the method of the integrating factor is only applicable to first-order ordinary linear differential equations with continuous coefficient functions (23). The differential equation must have these characteristics or be transformed into one having them before the integrating factor method can be applied.
Arranging the differential equation into general form is helpful for this evaluation. Differential equation characteristics include the order of the differential equation, the number of independent variables (1-ordinary, ≥ 2-partial), its linearity and homogeneity, and the dependence of coefficients with respect to the independent variables. The additional linearity characteristic found in (24) of quasilinearity describes a PDE with coefficients that are a function of the dependent variable and/or any of its derivatives of lower order than that of the differential equation; nonlinear PDEs therefore have coefficients that are a function of the derivatives of the dependent variable and are the same order as that of the differential equation.
Linear 2 nd -order PDEs can be categorized by basic type: parabolic, hyperbolic, and elliptic. This feature is determined by the resulting value of the following calculation involving the coefficients of the general form of the PDE: …with A, B, and C coefficients from the linear 2 nd -order general equation of dependent variable 'u' (19): The linear unsteady-state diffusion problem (heat and mass) will in general be parabolic, i.e. B 2 -4AC=0 (19) (24). If 2 space dimensions are considered, then an elliptic differential equation will likely result.
It is prudent at this point to also identify the ranges of the independent variables over which this PDE applies, i.e. the bounds of the space and time variables. This is important as it will determine the applicability of some solution methods later on.

Identify the types of initial and boundary conditions.
Prior to this work the author did not fully appreciate how terribly important the initial and boundary conditions are to the final solution of the problem. The addition of a separate step focused on boundary condition identification seeks to make that clear for future users who have a similar under-appreciation. The literature suggests three aspects for classifying the types of boundary conditions in the diffusion problem (19) (24); in the author's opinion the latter reference has a more straightforward breakdown.
First, the overall problem is classified based on the ranges over which the independent variables (e.g. position and time) are allowed to vary. If at least one of the independent variables has an open region (i.e. is allowed to vary to positive or negative infinity), then the overall problem is classified as an initial-value problem. The common example of this is the time variable, which is usually allowed to increase towards positive infinity. This can also be the case for the space variable in the examples of infinite (-∞ < x < ∞) and semi-infinite (-∞ < x ≤ 0] or [0 ≤ x < ∞) media. If instead the independent variables all vary over a closed region, then the overall problem is a boundary-value problem (24). Farlow (19) suggests distinguishing an infinite media problem as an initial-value problem and all others with at least one open independent variable range as an initial-boundary-value problem. The second worked-through problem discussed in chapter 4 is therefore classified as an initial-boundary-value problem because it is physically bounded but occurs at unsteady-state.
Second, the behavior of the dependent variable (e.g. concentration) at the boundaries is classified. Three different behaviors are considered here, resulting in the following named types of boundary conditions of a problem (24). A boundary condition of the 'first kind' is called a Dirichlet condition, and specifies the value of the dependent variable (e.g. concentration) for a particular value of an independent variable (e.g. position). The dependent variable may be constant, or a function of independent variable(s) (e.g. time, position). This also describes the typical kind of initial condition encountered in simplified diffusion problems (i.e. a specified initial concentration profile). Although simpler to model, experimentally determining this value exactly at the boundary region is terribly difficult (1). It is more practical experimentally to specify and control an exterior bulk concentration value instead and use a local equilibrium relationship, e.g. Henry's law, to define the boundary just inside the phase of interest for the problem. The boundary condition of the 'second kind' is called a Neumann condition, and specifies the value of the derivative of the dependent variable (e.g. concentration) with respect to an independent variable (e.g. position) for a particular value of an independent variable (e.g. position). This derivative may be constant or a function of independent variable(s) (e.g. time, position) just like the Dirichlet condition. One example is when an impermeable boundary is specified; in this case, the species flux at the boundary is set to zero. Finally, a boundary condition of the 'third kind' is called a Robin condition, and specifies a relationship between the dependent variable (e.g. concentration) and its derivative with respect to an independent variable (e.g. position) for a particular value of an independent variable (e.g. position).
Examples of this include when flux at the boundary is dictated by a mass-transfer coefficient and the difference between the concentration at the interior surface and the concentration in the exterior bulk, as well as when a concentration-dependent heterogeneous reaction is specified at a boundary surface (16).
Third, linearity and homogeneity are two characteristics used to classify boundary conditions, much as they are used to describe differential equations. The applicability of a solution method may depend on these characteristics of the boundary conditions. See Appendix 4 and relevant math textbooks (19), (23), (24) for some guidance on applicable solution methods.
All of these different types of boundary conditions are expected to be encountered when solving diffusion problems.
9. Select the appropriate solution method.
Having prepared the differential equation from the mass balance and constitutive equation for species flux, and determined the boundary conditions from the picture and problem description, the next general step for the user is to choose a solution method.
There are many different methods by which a differential equation may be solved, some better than others for a particular scenario; a helpful guide on the variety was found in (19). The challenge presented here to the practicing engineer is to choose a solution method that will be successful and will give an accurate answer in a reasonable amount of time without being overly complicated. It was assumed by the author that in order to do this, the characteristics of the differential equation and boundary conditions need to be analyzed. At the very least, the initial goal here is to ensure a solution method is chosen that will ultimately be successful and not doomed to fail from the start.
Two options lie before the flowchart user now that they are armed with the knowledge of the differential equation, initial and boundary conditions, and their classification: search the literature for an already worked-out problem with the same equation and condition (or some analog), or determine the appropriate solution process for solving the problem directly. A brief summary of the advantages and disadvantages of each approach are presented in Table 3.

Look it up
Less work.
Likely validated with links to relevant data.
May not exist. Still need to translate notation, mind assumptions.

Solve it yourself
Better understanding.
Only limited by mathematical prowess.
May take a long time if not familiar with solving differential equations.
Prone to frustration. The example scenarios used to illustrate the flowchart process tool in action will be presented with the latter option chosen, however any reference solution found will also be presented along with the appropriate conversions necessary.
To progress down the path of self-solution, what is desired at this point in the flowchart process is a straightforward continuation of steps to follow leading directly to the appropriate solution. Mathematics being logical, this should be the case at least conceptually, if not theoretically. In practice, however, the diversity of solution methods and their various rules of applicability add a degree of complexity which is outside the current scope of the author's practical mathematical knowledge. The paraphrase "experience will enlighten" seems to be the go-to panacea to address this shortfall; the author finds this frustrating given the aforementioned logical permanence of math and the inability to directly program the fuzzy concept of 'experience' into a computer or translate into a flowchart tool. An attempt to capture this is available in Appendix 4, which seeks to provide a guide for determining the appropriate solution method. Further iterations through the developing flowchart tool will hopefully bring this enlightenment; as a consolation, the various worked-out solutions should prove useful down the road.
When this flowchart process results in a linear ordinary differential equation, a helpful reference for analytical solution methods is (23) (25) and specifically for second order linear ODEs with constant coefficients (26).
When this flowchart process results in a linear partial differential equation, a helpful reference for analytical solution methods is (19). Qualifying examples for this situation are unsteady-state (i.e. dynamic) solutions of the concentration profile. The method of separation of variables can be used to solve linear homogeneous partial differential equations with variable coefficients; the Laplace transform can be used to solve linear inhomogeneous partial differential equations, but requires constant coefficients.
When this process results in a nonlinear ordinary or partial differential equation (which may be most of the time), a helpful reference for numerical solution methods is (24). Expect to encounter this when modeling a scenario with a concentration- The user may decide to look up a worked-through solution. This will likely involve translating notation and converting units. Cussler's text, §3.5 (1) contains a guide for using references in this manner. Crank's text (5)  There will be parameters in this final equation (e.g. thickness or other dimensions, the diffusion coefficient) that will need to be specified or obtained by some method. Refer to Chapter 3: Diffusion Resources for a more detailed description of the following sources of this information.
A database or text reference of diffusion coefficients is a natural first resource, however be prepared not to find data for the specific species pair of interest. Even if there is, the temperature of the data may be different from the application, and the user may be hard pressed to find a detailed model of the diffusion coefficient's temperature dependence for their situation.
Computer simulation of chemical pairs leading to accurate diffusion coefficient prediction of penetrants in polymers is still in development; greater confidence exists in the ability to predict relative diffusion rates of different penetrants in polymers.
Experimentation will likely be necessary to obtain an accurate value for the diffusion coefficient. A wide variety of experimental methods have been developed for the purpose of measuring diffusion coefficients, some of which can be found in §5.6 of (1), organized by utility, accuracy, difficulty of operation, and cost.

General approach pitfalls
The first pitfall to avoid is having inappropriately set expectations at the outset regarding the time required, number of different concepts engaged, and number of steps that must be taken to get a functional solution to a polymer-penetrant diffusion problem.
A user who is used to one-step solutions involving a quick reference look-up will be frustrated when their problem's complexity is one step greater than the solutions found in the reference; the permutations of solutions possible for different scenarios of boundary conditions and geometry are so large that one shouldn't expect to find the right answer to their specific problem (in a timely fashion) simply by scanning examples in the text.
This process will take some time, multiple pieces of paper, and quite a few whiteboard erasure iterations. Multiple steps and mini conclusions along the way preclude keeping the whole problem 'in one's head,' discouraging the use of shortcut strategies. In addition, seemingly small differences in scenarios (e.g. axis direction, short vs. long times) lead to different solutions almost incomparable in form that appear dissimilar and unrelated. As such, the user must attack these types of transport problems with a methodical approach; scanning textbooks grasping at straws for a shortcut solution will inevitably lead to frustration and failure.

Differential equation pitfalls
There are a few pitfalls to be aware of when the user is at the stage of solving the differential equation created in the first portion of the process.
When using a solution method (e.g. separation of variables) ensure the method is valid for the scenario. For example, the method of separation of variables requires homogeneous boundary conditions; if the user does not first transform their nonhomogeneous boundary condition then they will have to re-do the work. Chapter 7 of Farlow's text (19) illustrates this and briefly covers the applicability of some solution methods to different scenarios.
Solving differential equations may involve developing equations then plugging those equations into other differential equations. The pitfall to avoid here is forgetting to conduct the differentiation after substitution. This situation is likely to surface when Crank's treatise on the mathematics of diffusion (5) includes a chapter (8)  Non-dimensionalizing variables is more good practice than pitfall avoidance.
Crank advocates the technique in particular for use with numerical methods, primarily because sets of different solutions can be obtained by scaling, and "…fundamental parameters are often highlighted and analogies with physically different systems become clearer…" (5). Non-dimensionalizing reduces the problem to pure math, so solution attack is not clouded by parameters specific to the field of study (19). In addition, non-dimensionalizing variables can assist with analytical solution methods by enabling math 'tricks' to be used. For example, non-dimensionalizing physical boundaries so that a space region varies from 0 to 1 as opposed to 0 to L enables the use of orthogonality to simplify some series summations of orthogonal functions.
The guidance encourages developing the boundary conditions early, even if initially they are done descriptively without numbers or equations. This emphasis seeks to bridge the gap between the physical situation (grasped by most spatially-minded humans) and the abstract math which will be solved to describe it (not intuitively grasped by most humans, including some nascent chemical engineers). Also, a loose rule of thumb when applying boundary conditions using analytical solution methods is to apply the homogeneous boundary condition first.

Unit pitfalls
Double-check units along the way. If the units are not working out to that which is expected, then there is likely an error. It is good practice to check units at each miniconclusion in the process (e.g. constitutive equation for species flux, shell mass balance, boundary conditions). At the very least, this must be done at the end when getting back to solving for the value of interest for the problem (27).
With many different ways to describe just the value of concentration, unit conversion will almost certainly need to be implemented prior to the successful conclusion of the problem. A comprehensive table of unit conversions can be found in the text by Bird,Stewart,and Lightfoot (16), as well as in texts by other authors on the subject. Finally, some general resources related to diffusion are briefly reviewed, focused on reference material for solving differential equations as well as where to turn to for worked-through solutions of the diffusion problem for various scenarios, geometries, and boundary conditions.

Print resources
Textbooks on the subject of diffusion in polymers have some experimental records throughout them (7) (8) (15), however they are not in general designed to be databanks of experimental results. Larger volumes/compilations designed to tabulate such data exist and tend to be a few editions in at this point (28) (29) (30). The section on permeation in the Polymer Handbook tabulates mostly permeability, but intermittently includes information on solubility, and diffusivity. The McKeen text tabulates permeability and the less-generic vapor transmission rate, with no data on solubility or diffusivity. Both references contain little to no data on polyurethane materials, an important family of polymers to some marine hardware systems.
The references just mentioned as well as those throughout the literature all use a variety of different relatable parameters to report transport phenomena results.
Examples include the transmission rate, the permeation rate, the permeance, the diffusion coefficient, the solubility, and the permeability coefficient. These parameters may be derived from the others using values associated with the experiment that was run (e.g. thickness, temperature and the penetrant partial pressure) and by making assumptions (e.g. permeability coefficient is independent with thickness). Figure 2 is a guide to deriving permeation parameters from those available based on their interrelationships (31). (31) Each parameter may also be reported using a variety of different units, some of which by convention include members of both the English system and the Metric system of units (e.g. permeation rate: gram·mil·m -2 ·day -1 ). The permeability coefficient units can be particularly confusing:

Figure 2 Roadmap of permeation parameter conversions
with a volume, a length, and an area unit all within one parameter and no unit cancellation, and a pressure (but not ambient pressure -it's the penetrant partial pressure driving force across the membrane). Expect to perform unit conversions and to not be able to use literature values directly as they are found.

Digital resources
Due to the sheer number of permutations of polymer-penetrant scenarios, print records are ultimately physically incapable of conveniently containing the majority of tabulated properties of possible polymer-penetrant pairs. Computerized resources with remote, online access are a practical alternative to printed tables, not to mention the search and analytical advantages of digital records. The free website MatWeb (32) (48) which enables the determination of the concentration-dependence of the diffusion coefficient from a single sorption curve measurement. Crank and Park (15) allude to determining this dependence from sorption rate, and time lag measurements. From sorption measurements, a series of experiments run at successively smaller intervals of uniform initial concentration can be used to determine the relationship between the diffusion coefficient and the concentration (49). At the very least, the equations for a constant diffusion coefficient can be used and the resulting value will be an average of the diffusion coefficient values for the range of concentration tested. The dependence of the diffusion coefficient on concentration can be determined from time lag measurements, however the fundamental relationship between D and C must be known a priori (50). Just as a warning, it has been reported that the dependence of D on C will be different depending on the measurement technique chosen (51).

GENERAL DIFFUSION REFERENCE
Textbooks guiding users in the solution of partial and ordinary differential equations exist focused on analytical methods (19) and numerical methods (24). The treatises on mass diffusion (5) and on heat conduction in solids (4) both contain many worked-through simplified problems (although they are by no means simple) for various geometries, scenarios, and boundary conditions in transport phenomena.

INTRODUCTION
Examples of diffusion problems that are likely to be useful to the user are those of the membrane geometry with specified or impermeable boundary conditions. These scenarios can be used to describe diffusion through simplified 'container' structures (cylindrical hoses, layered boxes, etc…) and coatings, respectively. Descriptions of the process to solve these diffusion problems using the flowchart tool are presented in their entirety in order to convey to the user the true scope of the effort involved. Their influence on the development of the flowchart tool is described to explain its origins.
The complexity introduced when considering a concentration-dependent diffusion coefficient is demonstrated in part using the flowchart tool, with guidance to the user for additional references. The fundamental theory thus far has considered the bulk material to be homogeneous, however the practical applications of the user will inevitably require heterogeneous composite materials to be considered. The promising technology of nanoclay platelet filler material will be touched upon here from the standpoint of solving the diffusion problem; a more complete description of the topic will be delved into in the subsequent chapter. Finally, two additional marine-hardwarerelevant scenarios of likely interest will be briefly described but not solved.

Introduction
The first problem to work through was that of steady-state diffusion through a plane sheet. This scenario is found in the experimental determination of the diffusion coefficient through the time-lag method, and it is straightforward to work through with examples found directly in the literature (5). This problem was worked through in two ways: by general but not identical textbook guidance (16), and then by comparison to the identical problem solution in the literature (5). The concept of nondimensionalizing the problem was also introduced (52).
In this first effort the generic constitutive equation for species flux was obtained from the literature (16)     This reference includes the convective flux term as the bounded region is a gas phase.  Should you fail to find the problem or do not have access to these resources, solve the differential equation with the deduced boundary and initial conditions. o This is a second-order ordinary linear homogeneous parabolic differential equation, which means that its solution should be attainable through applying the regular rules of integration twice on the system. o The boundary conditions are both Type 1, Dirichlet, so straightforward substitution and algebraic manipulation should allow for the deduction of the two resulting constants of integration.  The Crank, 1975 reference should be used in this case.

APPLY THE SOLUTION METHOD AND SOLVE FOR WHAT YOU WANT:
11. Apply the solution method.
 To use Crank's method, we must convert notation from the mole basis to a mass basis. Table 4 describes the conversions necessary for the differential equation, and

Influence on the flowchart tool
Working through the problem with analogous but not identical guidance established the initial flowchart steps to solution stated previously. It also established the connection between identifying the parts of the problem (e.g. differential equation, composed of mass balance and constitutive equation; boundary conditions) and their role in forming the final solution (e.g. differential equation is solved for the dependent variable; boundary conditions are used to determine the constants of integration).
Comparing the problem to an identical problem solution in the literature introduced the concept of converting notation (e.g. the initial problem dependent variable was mass fraction; the literature problem dependent variable was molar concentration) and emphasized the importance of double-checking units in equations (27).

Introduction
The second problem that was worked through was that of unsteady diffusion through a plane sheet, with an impermeable boundary condition specified at one surface and a constant concentration specified at the other. This scenario is found in applications involving coatings over materials impervious to the penetrant (e.g. metals, glass, etc.). Even though the underlying material is impervious to the penetrant, there may be some interaction which is dependent on the penetrant concentration in the material presented to the exterior surface (e.g., corrosion, adhesion).
This was a more complicated problem than the first and prompted significant review of math concepts (differential equations), requiring a working through of other example problems in other textbooks.

Influence on the flowchart tool
Working through this problem revealed unexpectedly (to the author) that two valid solutions applicable over different ranges of the independent variables were possible, prompting an addition to the flowchart tool to account for and guide similar encounters in the future. This arises from different solution convergence rates depending on the value of the dimensionless ratio 2 .

Description of the worked-through problem
The math notes for the first phase of the flowchart process can be found in Appendix 5. The numbering of the steps do not match because a prior numbering scheme was retained to show how the flowchart process was developed. Time spent forming the problem further refined the flowchart process, establishing the specific step of clearly defining the initial and boundary conditions of the problem. This first section was completed methodically, with no additional surprises encountered.

Step 1: Clearly define the desired solution
The desired value for this scenario is the concentration of species A at the impermeable boundary face after an elapsed amount of time, i.e. the unsteady-state concentration profile. It is imagined there is some threshold value of concentration at this surface which is crucial for a particular application.
Step 2: Draw a picture (Figure 4)  Convective flux is not anticipated because the bulk polymeric coating material is stationary. The diffusion is assumed to be constant and not concentration-dependent, for simplicity.
Step 5: Conduct a shell mass balance − * = There is no chemical reaction specified, so no generation or consumption term for species A is required. An accumulation term is expected since this is an unsteadystate problem: over the time range of interest the concentration goes from being zero everywhere to a uniform value, i.e. accumulation occurs.
Step 6: Form the differential equation 2

=
The equations from the previous two steps are combined. The diffusion coefficient moves outside the derivative because (in this case) it is assumed constant.
Step 7: Identify the type of differential equation Identification of the characteristics of the formed problem PDE and BCs was guided by (19), with additional reference to (24). The differential equation for this problem is a second order homogeneous linear partial differential equation with constant coefficients. The linear unsteady-state diffusion problem (heat and mass) will in general be parabolic, i.e. B 2 -4AC=0 (19) (24), and this is indeed the case for the second workedthrough problem.

Step 8: Identify the types of initial and boundary conditions
For the second worked-through problem a homogeneous linear Neumann condition (flux specified) is present at the impermeable face of the plane sheet, and a nonhomogeneous linear Dirichlet condition (concentration specified) is present at the other face.
The initial concentration everywhere is zero (homogeneous linear Dirichlet condition). After solving it is possible to confirm that at long times the concentration everywhere will be equal to the concentration of species A in equilibrium with the exterior environment (if the resulting solution is valid at long times).
Step 9: Select the appropriate solution method Having identified the type of PDE and BCs, the next step is to choose an appropriate solution method. The first recommended approach is to review the literature for an identical or analogous problem already worked-through. In the case of this impermeable boundary problem, a number of similar examples were found but none looked directly applicable. The initial search of Crank's text (5)  Clear guidance for making the modification was not readily apparent, and there also appeared to be a typo in the equation from the book (second summation, first sin() term, was 'π' instead of 'L' in the denominator).
The closest match from the initial literature search was a heat transfer solution in (4), for the case in which the end x = 0 is insulated and the end x = L is kept at V and the initial temperature is f(x). Note that the direction of flux is reverse to the length axis here as compared to the boundary conditions established for this problem. Note also that here the summation starting point is at n = 0, whereas the summations in Crank's equation start at n = 1; perhaps this would be changed when modifying Crank's equation for half-symmetry.
Unfamiliarity with the method of converting a heat transfer solution to a mass transfer solution precluded the immediate adoption of the result found in (4). Guidance for this conversion process can be found in a number of references, e.g. (4) (5) (16), however the experience of developing the solution without using a preconceived one was deemed valuable. Therefore, the approach was changed to applying a solution method appropriate for the formed initial-boundary-value problem.
Step 10: Apply the solution method A similar problem found in (19) was solved using the method of separation of variables. This was worked through for the text example, and then the same strategy was applied to the impermeable boundary problem; see Appendix 6 for the math notes.
In this case, applying the method of separation of variables directly to the formed problem was inappropriate as it cannot be used for a problem where one or more of the The theory of the Laplace transform has been covered in a number of textbooks (4) (5) (19) (23) (53). In a nutshell, the goal is to transform a hard problem in original variables into an easy problem in transformed variables, solve it, then apply an inverse transform (via reference tables) to the solution in order to return to the original variables (19). The example problem in the text (19) demonstrating the use of the Laplace transform was worked through, however it turned out to be very difficult at the inversetransform step to return to the original variables of interest; consulting an alternate reference (53) was required in order to finish the problem. The lesson learned here was that not all textbook example problems are good ones. Continuing on, the Laplace transform method was applied to the impermeable boundary problem. However, the inverse-Laplace transform again proved to be a stumbling block, with an even more complex result to invert than the textbook example problem; no transform pair could be found in the literature. One lesson here is that the Laplace transform method is powerful, but loses much of its effectiveness when no table of transform pairs contains the particular transformed solution of interest. A later reference was found (22) with particular advice on dealing with the types of transformed equations resulting from using the Laplace transform on diffusion problems; running this obstacle to ground was postponed when further literature searching resulted in discovering an analogous worked-through problem.
A nearly identical worked-through impermeable boundary problem was found in (5)  This concept of working through a problem and ending up with two viable solutions applicable over different ranges of the independent variables was unexpected, and prompted an addition to the process flowchart tool to account for similar encounters in the future.
Step 11  a second-order quasilinear homogeneous partial differential equation with variable coefficients. Reference searching for a worked through solution did not find any readily available; those that were found were for infinite and semi-infinite media (5). Time did not allow for further work on this problem, but numerical methods are the likely path to success (24).

HETEROGENEOUS MATERIALS
One scenario of diffusion in heterogeneous materials is discussed in the following chapter and concerns primarily impermeable platelet filler material. Barrer's chapter §6 in (15)  It is especially important that the thickness not change as it factors significantly in the diffusion coefficient calculation. This invariance may not be the case if significant penetrant uptake occurs and swells the polymer disk. Fortunately, there is guidance for interpreting data from such an occurrence that can be found in §10.6.5 of (5).
Another diffusion problem of marine hardware relevance is that of leaching.
Desorption of volatile or otherwise mobile material components can change material properties such as glass transition temperature, stiffness, and sound speed. In some cases it may be desired to lose a penetrant over time, such as an anti-fouling biocide or entrained water (e.g. drying out hardware). The boundary conditions associated with leaching phenomena may be different from the scenarios mentioned before. For example, the penetrant diffusivity in the polymer may be high such that a slow surface removal (modeled by an interphase mass transfer equation) controls the rate of penetrant loss.

CHAPTER 5: BARRIER POLYMER APPLICATION
This section is included to illustrate some complexities of real-world transport applications and connect their origins to the fundamental steps of the flowchart tool.

Introduction
The goals of a barrier polymer are to minimize permeation while being minimally present. Permeation should be a non-issue for the effective service life of the product it is protecting and should not dictate the thickness of coatings. At the same time, any barrier additive or structural content should be minimized and not adversely alter the production process, i.e. be a 'facile' process (55). Nano-composite platelet additives offer a solution to this problem, achieving dramatic improvement in permeation reduction with minimal loading; it is expected this can be further enhanced by ensuring the platelets are properly oriented (56).

Barrier mechanism
The primary mechanism by which platelets and polymer crystals retard small molecule permeation is tortuosity (56) (57) (58). The additive itself being nearimpermeable (especially relative to the polymer), the small molecule is forced to diffuse around the structure. This effectively increases the 'thickness' of the coating as permeating molecules must travel a longer overall path through the polymer. A high aspect ratio enhances this mechanism by maximizing the size of the obstacle the molecule must circumnavigate while minimizing thickness, and thereby weight. A cartoon of this mechanism is shown in Figure 6.
(a) (b) Figure 6 Tortuosity cartoon illustrating diffusion (a) normal to platelet orientation, and (b) parallel to platelet orientation. Figure 6-a is an idealized cartoon of barrier platelets forcing a permeating molecule to travel a twisted, tortuous path through the polymer. If properly oriented like this, the platelets act like shingles on a roof (56). Care should be taken to ensure that diffusion parallel to platelet orientation is not a problem, as illustrated in Figure 6b. This extreme alignment may not be as effective a barrier in the transverse direction.
A random orientation may not have the optimal barrier property as in Figure 6-a, but it is more likely to provide a decent barrier in all directions.

Permeation Prediction
Platelet polymer additives and polymer crystals reduce the permeation rate of small molecules through the bulk polymer. Although qualitatively understood, the accurate quantitative prediction of this reduction is a challenging problem. Barrer's chapter in (15) deals extensively with the subject of diffusion in heterogeneous media.
Mittal (59)  Mittal argues that using factorial design methods to include these nonideal behaviors in the model accounts for experimentally observed behaviors such as the reduction in barrier performance when the polymer is saturated with nano-platelets at a particular filler fraction (Φf). Properties such as platelet aspect ratio (α) and percent randomness can be calculated from experimental permeation behavior and filler volume fraction.
These models were derived with the underlying assumption of ideal Fickian diffusion behavior, such that the permeability coefficient is the product of the diffusion coefficient and solubility. In the scope of the flowchart tool, these models took step 4 and added terms to the constitutive equation for species flux to account for the increased resistance to flow due to the dimensional tortuosity of the added platelet flakes.
Assuming a platelet is a solid cylinder of diameter d and thickness t, the equation for surface-area to volume ratio is: The larger A/V ratio means more surface area for adhesion of the nano-platelet to the polymer is available for less volume of filler added, and that more barrier influence can occur with lower loading of additive. This exfoliation typically requires chemical assistance through treatment with organophillic cationic surfactants (e.g. alkyl ammonium salts) (56) or sonication (62). Shear from processing also promotes exfoliation of the sheets, and it is worth noting at this time the different regimes that can be expected of nano-platelet additives in polymers: segregated, exfoliated, and intercalated.
Segregated implies virtually no dispersion, more like a two-phase system.
Exfoliated describes platelet sheets that are well removed from one another.
Intercalated describes separate sheets that are relatively close to each other, less than the order of the platelet diameter. In this case, single polymer chains wind their way between sheets in the space referred to as the 'gallery' (56).

Platelet Orientation & Polymer Processing
It has been noted (54) that dilute concentrations of nano-clay sheets provide some tortuosity and restrict permeation, but semi-dilute concentrations amplify the effect through multiple scatterings of the penetrant molecules between the less-disperse sheets. One difficulty in processing is that higher concentrations of additives increase the polymer melt viscosity such that the material cannot be extruded. The conundrum is therefore a limit on barrier additive loading level due to processing considerations.
One clever approach that gets around this problem is the use of multilayer co-extrusion with post-processing annealing (54).
In multilayer co-extrusion, two feed streams are fed through a series of layer multiplying elements which split, spread, and recombine the streams. The result is an extruded product with any number of alternating layers of material. In Decker et al. (54) this method was used to prepare alternating layers of low-density polyethylene (LDPE) with linear-low-density polyethylene (LLDPE) loaded with MMT nanoclay at a low concentration so as not to interfere with the extrusion process.
During the annealing phase, polymer-polymer interdiffusion of the LDPE and LLDPE resulted in a 'moving-front' of the phase boundary due to the different molecular mobilities of the two polymers. The result was a compaction of the clayloaded layer and an expansion of the other. The end result is regions of high nanoplatelet additive providing enhanced barrier properties. The feat worth noting is that this was achieved without initially high loading that would make processing the polymer by extrusion nigh-impossible.
Platelet orientation can also be influenced by other mechanisms, not just shearing forces in extrusion-type flow. Yucel et al. (62) describe a multilayer solution casting method which results in platelet orientation as the solvent evaporates. The polymer chains collapse as the solvent leaves, compacting the remaining polymer and nano-clay. Large internal stresses from this behavior resulted in bending of the nanoclay at high loading.

Platelets and Polymer Crystals
In addition to acting as barriers, platelet additives can also serve as nucleation sites for polymer crystals. This behavior was described in Girdthep et al. and was cited as the reason for enhanced strength in an unexpected orientation of the final product (63).
After melt blending PLA (poly-lactic acid), PBAT (poly-butylene-adipate-coterephthalate), the combatibilizer TBT (tetrabutyl titanate), and kaolinite with silver; the result was processed through blown film extrusion. The researchers found extended and oriented polymer chains in the machine direction (MD), but also found crystal growth from the clay platelets in the transverse direction (TD). They reported that subsequent tensile testing revealed greater strength in the TD orientation than the MD orientation for the nano-composite sample. However, the data referred to in the article did not obviously corroborate the reported results.
Work done by Chatterjee et al. (64) described the barrier ability and influence of processing on polymer crystals without additives. An interesting result was the existence of a critical draw ratio, below which polymer processing and orientation of crystals actually increases permeation rate.
Forming high-density polyethylene (HDPE) through a blown film process followed by machine direction orientation processing resulted in three different polymer crystal structures, each with different barrier properties. The HDPE from the blown film contained random, spherulite crystals which lent a degree of tortuosity to the polymer. At low draw ratios, the crystals organized into a lamellar structure in the machine direction; this effectively decreased tortuosity and increased measured permeation rates. After a certain point (the 'critical draw ratio') the lamellar crystals transform to a microfibrillar zig-zag pattern with smaller crystals developing around the fibers. This decreased the amorphous fraction and, along with the increased orientation, resulted in significant reduction of permeation rate. The results of this effort are already bearing fruit for the author, as he is using the flowchart process tool to tackle the anti-fouling biocide leaching problem mentioned earlier in the text. Further refinement of the flowchart tool is expected as it continues to be used in this manner.
This body of work is anticipated to be a foundation for future guidance documentation for marine hardware designers to assist them in material selection decisions and avoid diffusion-related problems. Initial dissemination is expected to be to acoustic hardware engineers. integrating factor (23)  integrating factor (23) characteristic equation (23) undetermined coefficients (23) variation of parameters (23) Separation of variables (19) N N o The z axis goes from left to right as that is the anticipated direction of flow (high to low concentration). 3. Roughly define boundary conditions. o Initially the coating has zero concentration of species A. o After immersion, the concentration of A just inside the surface immediately reaches equilibrium with the exterior environment and remains at that value the entire time it is immersed. o No species A leaves the coating at the coating-metal surface, i.e. the flux of A in the positive z direction is zero at z = 0 o At large times, the concentration throughout the coating will be uniform and equal to the concentration of A in equilibrium with the exterior environment. 4. Put what you want to know in terms of the flux or other transport property. o | = = ( , ) o Since the question is concerned with finding the time required to achieve a specific concentration at a specific position, what you want to know is simply an algebraic re-arrangement or simple iterative solution (Newton's method, Goalseek in Excel) of the concentration profile. Since this is developed by 'solving' the differential equations formed in this process, no additional equations are necessary.  … which is solvable by separation of variables. The result U(x,t) is then reintroduced to the original problem, u(x,t) = c A0 + U(x,t). Changing notation to match our problem:  u(x,t) = c A0 + U(x,t)  … becomes  c A (z,t) = c A0 + γ(z,t) Also, non-dimensionalize the space variable z so the range of the problem goes from 0 to 1:  ζ = For 0 ≤ z ≤ L, 0 ≤ ζ ≤ 1 The final problem ready for solution by separation of variables is therefore:  PDE (for 0 ≤ ζ ≤ 1 ; 0 < t < ∞)  … is the concentration profile in the half-sheet. This converges for moderate to large values of Dt/L 2 (i.e. 'long' times).