Fick's Law of Diffusion

Physical Principles

Fick’s Law describes the diffusive transport of water vapor through porous building materials under a concentration gradient. This fundamental relationship governs vapor diffusion analysis in building envelopes and forms the basis for permeance calculations, condensation risk assessment, and hygrothermal modeling.

The law applies when vapor transport occurs by molecular diffusion rather than bulk air movement, which is the case for most building materials under typical pressure differentials.

Molecular Diffusion Mechanism

Water vapor molecules move through the pore structure of building materials by random thermal motion. The net flux occurs from regions of high vapor concentration to regions of low vapor concentration. The driving force is the concentration gradient, not the pressure gradient directly, though vapor pressure and concentration are related through the ideal gas law.

The diffusion process involves:

• Random molecular motion (Brownian motion)
• Collision with material pore surfaces
• Passage through interconnected pore network
• Adsorption and desorption at pore surfaces
• Concentration-driven net transport

Fick’s First Law - Steady State Diffusion

The first law describes steady-state diffusion where vapor concentration at any point does not change with time.

Mathematical Formulation

The one-dimensional form:

J = -D (dC/dx)

Where:

• J = vapor flux density (kg/m²·s)
• D = diffusion coefficient (m²/s)
• dC/dx = concentration gradient (kg/m⁴)
• Negative sign indicates flow from high to low concentration

Alternative Forms for Building Applications

Vapor Pressure Driving Force:

g = -δ (dp/dx)

Where:

• g = vapor flux density (kg/m²·s)
• δ = vapor permeability (kg/m·s·Pa)
• dp/dx = vapor pressure gradient (Pa/m)

Partial Pressure Form:

g = -δₚ (dπ/dx)

Where:

• δₚ = water vapor permeability (kg/m·s·Pa)
• π = partial pressure of water vapor (Pa)

Relationship to Permeance

For a material layer of thickness L:

g = M Δp

Where:

• M = permeance (kg/m²·s·Pa) = δ/L
• Δp = vapor pressure difference across layer (Pa)

Diffusion Coefficient

The diffusion coefficient quantifies the material’s ability to transmit vapor under a concentration gradient.

Temperature Dependence

The diffusion coefficient varies with temperature according to:

D(T) = D₀ exp(-Ea/RT)

Where:

• D₀ = pre-exponential factor (m²/s)
• Ea = activation energy (J/mol)
• R = universal gas constant = 8.314 J/mol·K
• T = absolute temperature (K)

For water vapor in air at atmospheric pressure:

Dₐᵢᵣ = 2.306 × 10⁻⁵ (T/273.15)^1.81 m²/s

At 20°C: Dₐᵢᵣ ≈ 2.5 × 10⁻⁵ m²/s

Material Diffusion Coefficient

For porous materials, the effective diffusion coefficient is reduced from the air value:

Dₑff = (Dₐᵢᵣ × ε × τ) / ζ

Where:

• ε = porosity (fraction of void volume)
• τ = tortuosity factor (accounts for indirect path)
• ζ = constriction factor (accounts for pore narrowing)

Diffusion Resistance Factor

Building materials are commonly characterized by the water vapor diffusion resistance factor (μ):

μ = Dₐᵢᵣ / Dₘₐₜ

This dimensionless factor indicates how many times more resistant the material is compared to still air of the same thickness.

Vapor Permeability

Vapor permeability relates vapor flux to partial pressure gradient and is the key material property for building envelope analysis.

Definition and Units

δ = vapor permeability

SI Units: kg/m·s·Pa US Units: grain/ft·hr·in.Hg

Conversion: 1 grain/ft·hr·in.Hg = 1.459 × 10⁻¹⁰ kg/m·s·Pa

Relationship to Diffusion Coefficient

δ = D × Mᵥ / (R × T)

Where:

• Mᵥ = molecular weight of water = 18.015 kg/kmol
• R = universal gas constant = 8314 J/kmol·K
• T = absolute temperature (K)

At 20°C for air: δₐᵢᵣ ≈ 1.85 × 10⁻¹⁰ kg/m·s·Pa

Material Vapor Permeability

δₘₐₜ = δₐᵢᵣ / μ

The resistance factor μ is the inverse of the relative permeability.

Vapor Permeance

Permeance is the property of a specific thickness of material, analogous to thermal conductance.

Definition

M = δ / L

Where:

• M = permeance (kg/m²·s·Pa)
• δ = permeability (kg/m·s·Pa)
• L = material thickness (m)

US Units - Perms

Permeance is commonly expressed in perms:

• 1 perm (US) = 1 grain/ft²·hr·in.Hg = 5.721 × 10⁻¹¹ kg/m²·s·Pa
• 1 metric perm = 1 grain/m²·s·Pa = 5.745 × 10⁻¹¹ kg/m²·s·Pa

Vapor Retarder Classification (IBC/IRC)

Vapor Flux Calculation

Single Layer

For a homogeneous material layer:

g = M (pᵢ - pₒ)

g = (δ/L) (pᵢ - pₒ)

Where:

• pᵢ = interior vapor pressure (Pa)
• pₒ = exterior vapor pressure (Pa)

Multilayer Assembly

For multiple layers in series (analogous to thermal resistance):

Total vapor resistance: Rᵥ = Σ(Rᵥ,ᵢ) = Σ(Lᵢ/δᵢ) = Σ(1/Mᵢ)

Total permeance: M_total = 1 / Rᵥ

Vapor flux: g = M_total × Δp_total

Including Surface Resistances

For complete assembly analysis including surface vapor resistances:

Rᵥ,total = Rᵥ,ᵢ + Σ(Lᵢ/δᵢ) + Rᵥ,ₒ

Surface vapor resistances are typically negligible compared to material resistances except for highly permeable materials.

Example Calculation: Wood Frame Wall

Consider a wall assembly (interior to exterior):

1. Gypsum board: 12.7 mm, μ = 8.3
2. Kraft paper: 0.2 mm, μ = 30
3. Fiberglass insulation: 89 mm, μ = 1.0
4. OSB sheathing: 11 mm, μ = 200
5. Building paper: 0.5 mm, μ = 50

Using δₐᵢᵣ = 1.85 × 10⁻¹⁰ kg/m·s·Pa at 20°C:

R_v,total = 1.31 × 10¹⁰ m²·s·Pa/kg

M_total = 7.6 × 10⁻¹¹ kg/m²·s·Pa = 1.3 perms

This assembly qualifies as Class II vapor retarder, with the OSB sheathing providing the dominant resistance.

Concentration Gradient

The concentration gradient is the spatial rate of change of vapor concentration.

Vapor Concentration

Mass concentration of water vapor:

C = ρᵥ = p × Mᵥ / (R × T)

Where:

• ρᵥ = vapor density (kg/m³)
• p = vapor pressure (Pa)
• Mᵥ = 18.015 kg/kmol
• R = 8314 J/kmol·K
• T = absolute temperature (K)

At 20°C and 1000 Pa vapor pressure: C = 1000 × 18.015 / (8314 × 293.15) = 0.0074 kg/m³

Linear Gradient Assumption

For steady-state conditions through a homogeneous material, the concentration profile is linear:

C(x) = Cᵢ - (Cᵢ - Cₒ) × (x/L)

Where:

• x = distance from interior surface (m)
• L = total thickness (m)
• Cᵢ = interior surface concentration
• Cₒ = exterior surface concentration

Gradient Calculation

dC/dx = -(Cᵢ - Cₒ) / L = -ΔC / L

For vapor pressure gradient:

dp/dx = -(pᵢ - pₒ) / L = -Δp / L

Fick’s Second Law - Transient Diffusion

The second law describes time-dependent diffusion where vapor concentration changes with time.

Differential Equation

∂C/∂t = D (∂²C/∂x²)

For one-dimensional diffusion with constant diffusion coefficient.

∂C/∂t = ∂/∂x (D ∂C/∂x)

For variable diffusion coefficient (moisture-dependent permeability).

Hygroscopic Materials

For materials with significant moisture storage capacity:

∂u/∂t = ∂/∂x (D_w ∂u/∂x)

Where:

• u = moisture content (kg/kg)
• D_w = moisture diffusivity (m²/s)

The moisture diffusivity relates to vapor permeability through:

D_w = (δ/ρ_dry) × (dp/du)

Where:

• ρ_dry = dry density of material (kg/m³)
• dp/du = slope of sorption isotherm (Pa·kg/kg)

Characteristic Diffusion Time

The time scale for diffusion through a material:

t_char = L² / (4D)

These time scales indicate how quickly materials respond to changes in boundary conditions.

Mass Transfer Coefficient

The mass transfer coefficient relates vapor flux to a vapor pressure difference, analogous to the convective heat transfer coefficient.

Surface Mass Transfer

At material surfaces, vapor transfer occurs by convection:

g = h_m (p_air - p_surface)

Where:

• h_m = mass transfer coefficient (kg/m²·s·Pa)
• p_air = vapor pressure in bulk air
• p_surface = vapor pressure at surface

Relationship to Heat Transfer Coefficient

Lewis relation for air-water vapor mixtures:

h_m / h_c = 1 / (c_p × Le^(2/3))

Where:

• h_c = convective heat transfer coefficient (W/m²·K)
• c_p = specific heat of air = 1006 J/kg·K
• Le = Lewis number ≈ 0.87 for air-water vapor

For typical conditions: h_m ≈ h_c / (1006 × 0.93) ≈ h_c / 935

Typical Surface Mass Transfer Coefficients

Surface Vapor Resistance

R_v,surface = 1 / h_m

Interior surface: R_v,i ≈ 1.1 × 10⁸ m²·s·Pa/kg Exterior surface: R_v,o ≈ 2.5 × 10⁷ m²·s·Pa/kg

These resistances are significant only for highly permeable materials (M > 100 perms).

Temperature Effects on Vapor Diffusion

Permeability Temperature Dependence

Most building materials show temperature-dependent permeability:

δ(T) = δ₀ exp[a(T - T₀)]

Where:

• δ₀ = permeability at reference temperature T₀
• a = temperature coefficient (1/K)
• Typical range: a = 0.01 to 0.04 K⁻¹

For moisture content also affects permeability:

δ(T, u) = δ₀(u) exp[a(T - T₀)]

Temperature Gradient Effects

In building envelopes with temperature gradients, the vapor pressure gradient depends on both partial pressure and temperature:

g = -δ(T) × dp/dx

Where both δ and p vary with position.

Design Considerations

ASHRAE Standard 160 Analysis

ASHRAE Standard 160 requires evaluation of moisture accumulation:

1. Calculate hourly vapor flux through each layer
2. Determine condensation potential at material interfaces
3. Check moisture accumulation against material limits
4. Verify adequate drying capacity

Critical Interface Identification

Condensation occurs where temperature drops below dewpoint. The critical location is typically:

• Cold side of insulation in heating climates
• Warm side of insulation in cooling climates
• At low-permeability layers (vapor retarders)

Vapor Retarder Placement

Heating-dominated climates: Place vapor retarder on warm (interior) side

Cooling-dominated climates: Either no interior vapor retarder or use permeable materials

Mixed climates: Use “smart” vapor retarders that adjust permeability with humidity

Drying Potential

Assemblies must be able to dry to at least one side:

Inward drying: Permeable interior finish (M > 10 perms)

Outward drying: Permeable exterior sheathing and cladding

Bidirectional: No vapor retarders, all materials relatively permeable

Material Property Data Sources

• ASHRAE Handbook - Fundamentals, Chapter 26 (Heat, Air, and Moisture Control)
• ASHRAE Standard 160 - Design Criteria for Moisture Control
• Building Science Corporation: buildingscience.com
• Oak Ridge National Laboratory (ORNL) materials database
• Material manufacturer technical data sheets
• ASTM E96 test data (water vapor transmission)

Engineering Applications

Condensation Risk Assessment

Compare actual interface conditions to dewpoint:

If T_interface < T_dewpoint → Condensation occurs

Calculate accumulated moisture: Δm = ∫(g_in - g_out) dt

Where:

• g_in = vapor flux arriving at interface
• g_out = vapor flux leaving interface

Hygrothermal Modeling

Advanced analysis using coupled heat and moisture transport:

WUFI, DELPHIN, COMSOL, MOISTURE-EXPERT

These tools solve coupled equations:

• Heat conduction with latent heat effects
• Vapor diffusion (Fick’s Law)
• Capillary liquid transport
• Air infiltration effects
• Solar radiation absorption
• Rainfall absorption

Vapor Pressure Profiling

Calculate vapor pressure at each interface in steady-state:

p_interface,n = p_i - g × Σ(R_v,1 to R_v,n)

Plot vapor pressure profile and compare to saturation pressure (dewpoint) profile to identify condensation planes.

Limitations and Practical Considerations

When Fick’s Law Applies

Valid assumptions:

• Diffusion-dominated transport (negligible air leakage)
• Dilute vapor concentration
• Isothermal or slowly varying temperature
• Homogeneous material properties
• No chemical reactions or phase changes

When Fick’s Law is Insufficient

Fick’s Law alone does not account for:

• Capillary liquid transport - dominant in high humidity
• Air leakage - often exceeds diffusion by 10-100×
• Moisture storage - important for transient analysis
• Sorption hysteresis - different wetting/drying paths
• Rain penetration - exterior moisture source

Air Leakage Dominance

The vapor transport by air leakage at pressure difference Δp_air:

g_air = ρ_v × Q_air = ρ_v × (ELA/1000) × Δp_air^0.6

Where:

• ELA = effective leakage area (cm²)
• Δp_air = air pressure difference (Pa)

Even small air leakage paths transport far more moisture than diffusion through the same area.

Critical design principle: Air barriers are more important than vapor retarders for moisture control.

ASHRAE and Code References

ASHRAE Handbooks

• ASHRAE Handbook - Fundamentals, Chapter 26: Heat, Air, and Moisture Control in Building Assemblies - Material Properties
• ASHRAE Handbook - Fundamentals, Chapter 1: Psychrometrics - Vapor pressure relationships
• ASHRAE Handbook - Fundamentals, Chapter 25: Thermal and Water Vapor Transmission Data

ASHRAE Standards

• ASHRAE Standard 160-2021: Criteria for Moisture-Control Design Analysis in Buildings
• ASHRAE Standard 55-2020: Thermal Environmental Conditions for Human Occupancy
• ASHRAE Standard 62.1-2022: Ventilation for Acceptable Indoor Air Quality

Building Codes

• International Building Code (IBC): Section 1405 - Vapor retarders
• International Residential Code (IRC): Section R702.7 - Vapor retarders
• International Energy Conservation Code (IECC): Air barrier and vapor retarder requirements

Test Methods

• ASTM E96/E96M: Standard Test Methods for Water Vapor Transmission of Materials
• ASTM C1498: Standard Test Method for Hygroscopic Sorption Isotherms of Building Materials
• ASTM C1794: Standard Test Method for Determination of the Water Absorption Coefficient by Partial Immersion

Advanced Topics

Moisture-Dependent Permeability

Many materials show significant permeability variation with moisture content:

δ(u) = δ_dry × f(u)

For wood and wood products: f(u) ≈ exp(b × u)

Where b is material-specific coefficient (typically 5-15).

This nonlinearity requires iterative solution methods for accurate predictions.

Capillary Condensation

At high relative humidity (> 95%), capillary condensation in material pores creates liquid moisture before bulk condensation occurs. This accelerates moisture transport beyond Fick’s Law predictions.

Salt Transport and Efflorescence

In masonry and concrete, dissolved salts affect vapor diffusion and can crystallize at evaporative surfaces, potentially causing deterioration.

Summary

Fick’s Law provides the fundamental framework for analyzing vapor diffusion in building envelopes:

1. First Law governs steady-state vapor flux driven by concentration/pressure gradients
2. Second Law describes transient moisture accumulation and storage
3. Material permeability is the key property, varying with temperature and moisture content
4. Multilayer analysis uses series resistance calculation analogous to thermal analysis
5. Critical design applications include vapor retarder selection, condensation risk assessment, and drying potential evaluation
6. Limitations include neglecting air leakage (often dominant) and liquid transport
7. Modern practice uses hygrothermal modeling that couples Fick’s Law with heat transfer, capillary transport, and air movement for comprehensive moisture analysis

Proper application of diffusion theory, combined with effective air sealing, enables durable building envelope design that controls moisture accumulation and prevents material degradation.