Freeze Casting: A Review of Processing, Microstructure and Properties via the Open Data Repository, FreezeCasting.net

1. Introduction

• Freeze-casting’s tunable porosity supports applications spanning biomaterials, filtration, electrodes, sensors, batteries, pharmaceuticals, foodstuffs, and composite materials.Porosity can be adjusted through suspension characteristics and solidification conditions, but this flexibility makes systematic processing–structure–property analysis difficult.Source: §1 · pp. 1–2
• The FreezeCasting.net repository contains data from over 800 papers, 6,000 experimental specimens, and 10,000 samples linking processing conditions to microstructural, mechanical, and physical properties.The database is publicly available through a web interface and is intended to support dissemination, experimental design, and modeling.Source: §1 · p. 2

1.1.1. Anisotropic vs. isotropic freezing

• Anisotropic freezing directionally rejects particles or solute ahead of a solidification front, producing vertically aligned elongated pores, whereas isotropic freezing yields predominantly non-aligned structures.Anisotropic processing typically involves bottom-up solidification, sublimation, and often sintering; isotropic methods nucleate and grow crystals with random orientations.Source: §1.1.1 · p. 3

1.1.2. Pore structures

• Among reported anisotropic structures, lamellar, dendritic, and honeycomb morphologies comprise 60%, 20%, and 19%, while isotropic structures are predominantly cellular at 92%.Microstructures are also quantified using porosity, wavelength, pore size, wall size, and dendrite spacing where applicable.Source: §1.1.2 · p. 4

1.1.3. Characterization methods

• Two-dimensional characterization can misrepresent three-dimensional freeze-cast structures and cannot readily quantify interconnectivity, percolation, curvature, or tortuosity; optical measurements of pore width were 5–45% higher than 3D reconstruction values.Sublimation and sintering can further modify templated microstructures, motivating database distinctions among solidification, frozen, green, and sintered samples.Source: §1.1.3 · pp. 4–5

2.1. Systematic literature review

• The review targeted papers published between 1998 and 2017 through Google Scholar and Web of Science searches.Search terms covered freeze-casting, ice-templating, ice segregation induced self-assembly, unidirectional freeze-drying, freeze-gelation, gelcasting, and phase separation methods.Source: §2.1 · p. 5
• Citation tracking supplemented database searches by manually checking references from initially identified relevant papers and their associated sources.Source: §2.1 · p. 5
• The combined searches yielded over 2,000 keyword-related studies before eligibility filtering.Source: §2.1 · p. 5
• Studies were excluded when they lacked a solidification step, used a nonsacrificial second phase, or produced fibers or powders instead of porous scaffolds.Source: §2.1 · p. 5
• Non-English papers and studies lacking data in the categories of interest were not included in the extracted dataset.The passage specifies 33 non-English papers among the exclusions.Source: §2.1 · p. 5
• Solely numerical or theoretical studies and review articles were placed in a general literature table without data extraction.This treatment included 12 numerical papers, 12 theoretical papers, and 43 review articles to prevent overrepresentation of data subsets.Source: §2.1 · p. 6

2.2. Data extraction

• Data were manually extracted from text and tables, with plots digitized when values were unavailable and solid loadings converted from wt.% to vol.% using prioritized density sources.Density values came first from the paper, then supplier data, then reasonable estimates from other literature sources.Source: §2.2 · p. 6

2.3. FreezeCasting.net database

• The database uses SQL/MySQL while supporting SQL-compliant systems and downloadable JSON, XML, and CSV formats.The interactive table also supports variable selection, server-side processing, and result export.Source: §2.3 · p. 6, §2.3.2.1 · p. 8

2.3.1. Database architecture

• The hierarchical relational schema separates primary, secondary, property, and supplementary tables, enabling new variables and tables without altering the overall schema.The primary tables contain authors, papers, and samples; secondary tables store experimental data; property tables cover particles, fluids, molds, and bulk materials.Source: §2.3.1 · p. 6

2.3.1.1. Primary tables

• Unique sample IDs link processing-condition-specific results to paper citations and connect each sample with corresponding secondary-table data.This structure accommodates papers reporting multiple outcomes under varying conditions such as sintering temperature, solid loading, or cold-plate temperature.Source: §2.3.1.1 · p. 7

2.3.1.2. Secondary tables

• Secondary tables organize processing–structure–property variables across suspension preparation, solidification, sublimation, sintering, microstructure, and mechanical characterization.Examples include constituent concentrations, viscosity, cooling rate, cold-plate temperature, pore and wall width, and compressive strength.Source: §2.3.1.2 · p. 7

2.3.2. User interface

• The public web interface provides background information, downloadable database formats, and tools for exchanging and exploring freeze-casting data.The repository is accessible at FreezeCasting.net and is intended to facilitate data exchange among researchers.Source: §2.3.2 · p. 7

2.3.2.1. Data exploration and visualization

• The interactive plotting application lets users plot any two of 24 variables, filter datasets, transform visualizations, regress data, and export filtered tables.The variables span suspension characteristics, solidification conditions, microstructure, and mechanical properties; citation information is available through plot-point tooltips.Source: §2.3.2.1 · pp. 7–8

2.4. Statistical analysis

• Python and SciPy supported linear and nonlinear least-squares fitting, logarithmic power-law regression, and Pearson correlation analysis.The analysis code was provided as a Jupyter notebook using the IPython kernel.Source: §2.4 · p. 8
• Means were compared with paired t-tests, while one-way ANOVA followed by Bonferroni-corrected paired t-tests was used for more than two groups.Data were reported as mean ± standard deviation and median, with p < 0.05 defining statistical significance.Source: §2.4 · p. 8

3.1. Templating pore structures using fluids and additives

• Fluid choice and additives regulate pore morphology by altering freezing kinetics, solidified-fluid crystallinity, and particle redistribution.Aqueous processing predominates, but camphene and tert-butyl alcohol are commonly used non-aqueous alternatives.Source: §3.1 · p. 8

3.1.1. Aqueous processing

• Aqueous anisotropic freezing predominantly yields lamellar pores with mean aspect ratio 5.5±3.2 (N=212), while additives or faster solidification can promote dendritic, interconnected structures.Lamellar channels favor fluid flow, whereas greater wall interconnectivity improves compressive response by reducing internal wall buckling.Source: §3.1.1 · pp. 8–9

3.1.1.1. Binders

• Increasing PVA from 0 to 20 wt.% in 25 vol.% Al2O3 suspensions increased secondary branching and reduced pore width from ~25 μm to ~9 μm.Comparable lamellar-to-dendritic transitions were reported with PVA in hydroxyapatite, alumina, yttria-stabilized zirconia, collagen, and PVA systems.Source: §3.1.1.1 · p. 9
• Binder-induced transitions arise through increased viscosity and particle engulfment, while binder rejection from ice can cause constitutional undercooling that promotes secondary-arm coarsening.Sucrose and related sugars can instead drive dendritic-to-cellular transitions by slowing ice growth and producing smaller, more spherical crystals.Source: §3.1.1.1 · p. 10

3.1.1.2. Simple alcohols

• For 10 vol.% ZrO2, 5 vol.% IPA produced aspect ratio 20 with 30 μm pores, compared with ~5 and 10 μm without IPA; higher IPA concentrations reduced these values.In 10 vol.% TiO2, 5 vol.% IPA similarly produced aspect ratio ~10 versus ~3 without IPA.Source: §3.1.1.2 · pp. 10–11

3.1.1.3. Glycerol

• Glycerol increases lamellar-wall bridging and can produce cellular structures, but it consistently decreases pore width regardless of whether suspension viscosity rises or falls.This indicates glycerol’s cryoprotectant effect on ice-crystal size dominates viscosity-dependent explanations; it may also reduce ice-lens cracking.Source: §3.1.1.3 · pp. 11–12

3.1.1.4. Unique pore structures

• Unique aqueous routes produce honeycomb and equiaxed architectures using zirconium acetate, gelatin networks, tumbling, space-holders, or hydrogen peroxide.At 9 vol.% H2O2, the lamellar pattern was lost and a cellular structure resulted.Source: §3.1.1.4 · p. 12

3.1.2. Non-aqueous processing

• Non-aqueous fluids generate material-specific templates: camphene, cyclohexane, and dioxane commonly yield dendritic pores, whereas TBA and DMSO can produce tubular or honeycomb structures.TBA is the second most commonly used freeze-casting fluid after water.Source: §3.1.2 · p. 13

Suspension stabilization

• A stable suspension is the necessary first step because particle interactions control packing and solidified-fluid morphology, whereas instability promotes defects such as ice lenses.The stabilization table contains over 2,000 viscosity and/or zeta-potential samples spanning 20 particle types as of August 1, 2017.Source: §3.2 · p. 13
• For aqueous Al2O3, ζ decreases with increasing pH, becomes highly unstable at the isoelectric point, and indicates broad stability from pH 4–12 when |ζ| ≥ 30 mV, depending on dispersant conditions.Higher |ζ| corresponds to greater stability, and both positive and negative ζ values can stabilize suspensions.Source: §3.2 · p. 13
• Dispersants prevent flocculation by promoting electrostatic repulsion, but excessive pH can sharply increase viscosity and excess dispersant can induce depletion flocculation.For Al2O3, 3–4 wt.% dispersant produced lower |ζ| than 0.5–2 wt.% dispersant in the plotted data.Source: §3.2 · p. 14
• Because stabilization depends on the particle, fluid, and additives, the database supports filtering by material identities, concentrations, particle sizes, and additive roles to guide experimental procedures.Relevant stabilization investigations are reported in refs..Source: §3.2 · p. 14

Database-wide relationship

• Across 2,855 observations, solid fraction explained only ~1% of porosity variation, whereas sublimated unsintered samples reached ~40% explained variance.The overall regression was ϕp = -1.0±0.3·ϕs + 86%±6% with R2 = 0.01; ~82% of the data described sintered samples.Source: §3.3 · pp. 14–15

Sintering effects

• Sintering lowers the regression intercept and increases scatter because volumetric shrinkage varies with sintering conditions.The intercept was 82%±10% for sintered samples versus 92%±3% for unsintered samples, a difference attributed to sintering shrinkage.Source: §3.3 · p. 15

Fluid-specific relationships

• Water and camphene produced stronger solid-fraction–porosity relationships than the pooled database, while TBA data were not statistically significant.Water had R2 = 0.43 and camphene R2 = 0.37; for water, green samples were more linear than sintered samples.Source: §3.3 · p. 15

Material-specific relationships

• Material-specific models differed substantially: polymers had the highest porosity at a given solid fraction, metals the lowest, and sintered aqueous TiO2 explained ~93% of porosity variance.Regression slopes were -1.6±0.3 for metals and -0.8±0.2 for polymers; the TiO2 model had a = -1.7±0.2 and b = 99.1%±0.7%.Source: §3.3 · pp. 15–16

Sintering shrinkage

• Average sintering shrinkage was highest for metals, intermediate for ceramics, and lowest for polymers, with significant differences across all shrinkage measures.One-way ANOVA found significant group differences for diametric, linear, and volumetric shrinkage, with p <0.0001 for each.Source: §3.3 · p. 16

3.4.1. Solidification velocity

• Increasing solidification velocity decreases wavelength, pore width, and wall width overall, with wavelength most strongly dependent on velocity (adj. R2=0.29).For velocity changes from 10 to 100 μm/s, wavelength, pore width, and wall width decrease by 68%, 60%, and 38%, respectively.Source: §3.4.1 · pp. 17–18

3.4.2.1. Constant substrate temperature

• Constant-substrate-temperature cooling controls average velocity but produces gradient microstructures, with pore width increasing by 236% on average from sample bottom to top.The freezing front slows with distance from the cold source, and gradient structures may support tissue-engineering, biomedical-implant, and fuel-cell-electrode applications.Source: §3.4.2.1 · p. 19

3.4.2.2. Linear cooling

• Linear cooling and double-sided cooling do not maintain constant velocity; double-sided cooling reduces but does not eliminate pore-width gradients, averaging 88%±49% top-to-bottom increase.Cooling rate correlates weakly with solidification velocity (r=0.65, N=82), so direct, spatially resolved velocity measurements are more informative.Source: §3.4.2.2 · pp. 20–21

3.4.2.3. Achieving constant freezing velocity

• Exponential and parabolic cooling can achieve near-constant freezing velocities, producing relatively uniform pore widths, but effectiveness depends on material, sample size, and cooling-model assumptions.Exponential cooling achieved 10–50 μm/s and ~18 μm pores at ~20 μm/s, whereas parabolic cooling produced 20 μm pores over 1.3 mm at ~10 μm/s.Source: §3.4.2.3 · pp. 21–22

3.4.3. Undercooling

• The thermodynamic freezing-point model is not confirmed by reported suspension data, although greater freezing depression is expected for smaller particles and experimental reporting is highly ambiguous.The model predicts concentrated-particle effects dynamically, but interface-temperature measurements during solidification are needed for verification; particle size and additive effects complicate predictions.Source: §3.4.3 · pp. 22–23
• Particle volume fraction correlates strongly and linearly with undercooling (N=37, r=-0.9, R2=0.8, p<0.0001), offering indirect support for the freezing-point model.The relationship may reflect particle-assisted nucleation and changing interface conditions, but confounding variables require broader statistical analysis.Source: §3.4.3 · p. 24

Porosity

• Anisotropic structures span mean porosities of 65±14–82±18% when sintered and 80±18–87±17% when green, while lower bounds are approximately 15–28%.These lower bounds correspond to dendritic, lamellar, and honeycomb structures, below which their patterns are often lost or become cellular.Source: §3.5 · p. 24

Pore and wall widths

• Sintering reduces mean lamellar pore width by approximately 40% and wall width by approximately 29%, although distributions are highly skewed and variable.Mean pore widths decrease from 73±90 to 44±85 μm, and mean wall widths from 24±47 to 17±24 μm, for green versus sintered samples.Source: §3.5 · p. 24

Cellular structures

• Sintered cellular structures have a mean pore diameter of 42±87 μm, with reported values below 5 μm to 580 μm depending on the solidification technique.Combined sponge-replication and freezing produces macropores spanning 13–580 μm, whereas traditional isotropic freezing often yields mean pore diameters below 5 μm.Source: §3.5 · p. 25

Porosity–dimension relationships

• At 60% porosity, pore widths range from below 1 to approximately 400 μm, indicating only weak dependence on total porosity.Pore width has r=0.19 and adjusted R2=0.04, while wall width shows moderate relationships with porosity and pore width.Source: §3.5 · p. 25

4. Mechanical properties

• Compressive strength is commonly measured for over 2,000 samples, with parallel loading typically stronger than perpendicular loading in anisotropic freeze-cast structures.The database includes compressive strength, flexural strength, and elastic modulus for porous and selected dense composite materials.Source: §4 · pp. 25–26

4.1. Compressive strength

• The analysis fits Gibson–Ashby open-cell and honeycomb models by regressing normalized compressive strength against relative density, using β1=1 for parallel loads in anisotropic structures.Bulk flexural strength scales ceramics and bulk yield strength scales metals; only sintered samples are included, with C2 fitted when wall and pore dimensions are unavailable.Source: §4.1 · p. 26

4.1.1. Metals

• 67% of variance in normalized metal compressive strength is predicted by the open-cell model versus 57% by the honeycomb model across the full density range.For relative density below 0.5, open-cell fitting is better; above 0.5, honeycomb fitting is better, while impurity data may affect fitted intercepts.Source: §4.1.1 · p. 27

4.1.2. Ceramics

• 53% of variance in normalized ceramic compressive strength is predicted by the open-cell model versus 41% by the honeycomb model.The regression gives C2=0.69±0.10 and β1=2.2±0.1, while anisotropic structures outperform isotropic ones above relative density ≈0.18.Source: §4.1.2 · p. 28

4.1.2.1. Pore structure

• Open-cell fitting is better than honeycomb fitting for each major ceramic pore structure, while dendritic and equiaxed structures most closely follow open-cell behavior.Regression exponents are 2.5±0.3 for lamellar, 1.6±0.3 for dendritic, and 2.3±0.2 for honeycomb anisotropic structures.Source: §4.1.2.1 · p. 29

4.1.2.2. Fluid

• 81% of variance in normalized ceramic compressive strength is predicted for TBA-processed materials, compared with lower predictability for water and camphene.The fitted β1 decreases from 3.1±0.2 for TBA to 2.2±0.2 for water and 1.8±0.3 for camphene.Source: §4.1.2.2 · p. 30

4.1.2.3. Test specimen geometry

• 96% of TBA strength variance is explained when aspect ratio is added to relative density, versus approximately 88% using relative density alone.Aspect ratio decreases predicted strength for TBA and water but increases it for camphene, while specimen-shape effects are difficult to interpret because shapes are unevenly represented.Source: §4.1.2.3 · pp. 31–32

4.1.2.4. Comparison with previous modeling approaches

• The reviewed results contradict prior recommendations favoring the out-of-plane honeycomb model, while elastic-buckling extensions can predict physically unreasonable ceramic strengths.For Al2O3 at relative density 0.4, one buckling relation predicts approximately 144 GPa, about 50 times the fully dense material strength of 2.95 GPa.Source: §4.1.2.4 · pp. 32–33
• The open-cell model explains ~67% of normalized compressive-strength variance in metals and ~53% in ceramics, with no significant anisotropic–isotropic ceramic difference.For ceramics, 78% of data represent anisotropic materials and 22% isotropic structures; ceramic scatter may reflect microporosity, specimen-volume/aspect-ratio variation, and other processing factors.Source: §4.1.2.4 · p. 33
• For ceramics, the open-cell model fits anisotropic dendritic and isotropic equiaxed structures better, whereas neither model fits lamellar, honeycomb, or cellular structures well.The honeycomb model also fails to provide a good fit for lamellar, honeycomb, and cellular ceramic structures.Source: §4.1.2.4 · p. 33

5. Concluding remarks

• The repository contains experimental data from over 800 freeze-casting papers linking processing conditions with microstructure and material properties.Its stated aims are to disseminate relevant data, improve experimental design, encourage industrial use, and accelerate modeling efforts.Source: §5 · p. 33

All materials

• Porosity depends little on suspension fluid but strongly on material type, while low-shrinkage materials show greater dependence on initial solid fraction; pore width is not strongly tied to total porosity.These conclusions distinguish the effects of fluid, material type, sintering shrinkage, and porosity on freeze-cast pore dimensions.Source: All materials · p. 34

Anisotropic materials derived from aqueous suspensions

• Structure wavelength correlates most strongly with solidification velocity for sublimated rather than sintered samples, whereas cooling rate does not significantly correlate with microstructural parameters.Structure wavelength combines the widths of a lamellar pore and its adjacent wall; relationships using pore width or wall width alone are statistically weaker.Source: Anisotropic materials derived from… · p. 34

Metals

• For anisotropic sintered metals, the Gibson–Ashby open-cell model best describes compressive strength below relative density 0.5, while the out-of-plane honeycomb model fits better above 0.5.The preferred model therefore changes with relative density.Source: Metals · p. 34

Ceramics

• For anisotropic and isotropic freeze-cast ceramics, the Gibson–Ashby open-cell model fits experiments better than the honeycomb model, contrary to previous reviews.For anisotropic ceramics from water, camphene, and tert-butyl alcohol suspensions, adding specimen aspect ratio to relative density improves compressive-strength regression, but its predicted effect varies by fluid and needs further testing.Source: Ceramics · p. 34

Table 1: Porosity–volume-fraction regression

• For all samples, volume fraction decreases with total porosity with slope -1.0[-1.3,-0.7]**** and intercept 86[80,92]**** across N=2855 samples.The corresponding correlation and determination coefficients are r=-0.12 and R2=0.02****, with a significant regression fit.Source: Tables · p. 89, Table 1 · p. 89

Table 2: Sintering shrinkage

• Table 2 reports mean sintering shrinkage as changes in diameter, height, and/or volume for metals, ceramics, and polymers.The supplied passage identifies the compared material classes but does not provide the mean numerical values.Source: Table 2 · p. 91

Table 3: Solidification-velocity regression

• For green materials, microstructural parameters show a significant relationship with solidification velocity, with ln(C1)=5.4[2.7,8.1]**, k=-0.9[-1.5,0.2]*, N=6, and R2=0.77*.The fitted relationship is based on references [223, 758, 759]; carbon-polymer composites show ns coefficients for N=4.Source: Tables · p. 92, Table 3 · p. 92

Table 4: Cooling-rate regression

• Table 4 defines least-square regression parameters for relating microstructural parameters to applied cooling rate through ln(λ)=C1+k·ln(v).The table specifies confidence intervals, significance notation, and R2 as the coefficient of determination, but the supplied passage does not include fitted values.Source: Table 4 · p. 93

• Table 5 organizes demonstrated porosity for anisotropic and isotropic freeze-cast microstructures across solidification, green, frozen, and sintered states.The anisotropic classes are lamellar, dendritic, and honeycomb; isotropic classes are cellular and equiaxed.Source: Table 5 · p. 94
• Table 6 summarizes pore and wall widths for anisotropic and isotropic freeze-cast microstructures across solidification, green, frozen, and sintered states.Summary values are reported as means with standard deviations, with N denoting the number of samples.Source: Table 6 · p. 95
• Tables 7–9 describe micromechanical-model and regression parameters linking relative density to compressive strength for freeze-cast metals and ceramics.The ceramic analyses distinguish anisotropic versus isotropic structures and, for anisotropic ceramics, compare fluid types including tert-butyl alcohol, water, and camphene.Source: Table 7 · p. 96, Table 8 · p. 97, Table 9 · p. 98
• The regression analyses report model coefficients, confidence intervals, significance levels, sample counts, and coefficients of determination, while comparing alternative equations for anisotropic ceramic materials.The models use log-transformed relationships involving normalized compressive strength and relative density; one formulation additionally includes an aspect-ratio term.Source: Table 7 · p. 96, Table 8 · p. 97, Table 9 · p. 98
• Bulk flexural-strength values were drawn from CES EduPack or literature, and rule-of-mixtures estimates were used for mixed particle systems.The listed reference values cover ceramic and related materials including Al2O3, hydroxyapatite, yttria-stabilized zirconia, Si3N4, SiC, and mullite.Source: N R2 References C1 k STRUCTURE… · pp. 97–99

• Unidirectional freeze casting drives dendrites along the thermal gradient, rejects particles into interdendritic regions, and leaves elongated pores after sublimation and optional sintering.The schematic shows nucleation, dendritic propagation, particle packing, fluid removal, and scaffold densification as sequential stages.Source: Figure 2 · p. 102
• Isotropic freezing produces randomly oriented crystals and nearly isotropic pores, while freeze-cast materials exhibit lamellar, dendritic, honeycomb, cellular, and equiaxed morphologies.The pore-structure schematic distinguishes anisotropic lamellar, dendritic, and elongated-honeycomb forms from cellular-honeycomb, equiaxed, and cellular structures.Source: Figure 3 · p. 103, Figure 4 · p. 104
• FreezeCasting.net organizes citation, processing, experimental, property, and suspension-stability data and supports plots of zeta potential, porosity, shrinkage, and microstructural dimensions.Repository visualizations categorize measurements by fluid, material, pore structure, solidification velocity, cooling rate, and sample state.Source: Figure 5 · p. 105, Figure 12 · p. 112, Figure 13 · p. 113
• Binders, additives, fluid chemistry, and non-aqueous processing conditions generate substantial variations in anisotropic and non-standard pore structures.The figures show morphology changes with PVA, gelatin, IPA, glycerol, zirconium acetate, space-holders, and fluids including camphene, cyclohexane, dioxane, tert-butyl alcohol, and DMSO.Source: Figure 7 · p. 107, Figure 8 · p. 108, Figure 9 · p. 109
• The database figures examine how structure wavelength, pore width, and wall width depend on solidification velocity and cooling rate across material classes and processing conditions.Additional plots resolve pore width by sample height, cooling function, cooling-rate regime, and constant-temperature or double-sided cooling configurations.Source: Figure 17 · p. 117, Figure 18 · p. 118, Figure 19 · p. 119
• N=1,032 ceramic measurements show normalized compressive strength regressed against relative density with R2=0.59 and p<0.0001, while metal data give N=60, R2=0.67, and p<0.0001.The strength analyses further categorize ceramic relationships by anisotropy, pore structure, and fluid type, with specimen characteristics summarized for Eq. 7.Source: Figure 27 · p. 128, Figure 28 · p. 129, Figure 29 · p. 130

Figure 32

• Figure 32 relates normalized compressive strength to relative density, test specimen volume, and aspect ratio across anisotropic freeze-cast ceramics.The plots use Eq. 8 and were created through the FreezeCasting.net open-data repository.Source: Figure 32 · p. 133
• The data are categorized by tert-butyl alcohol (N=116), water (N=403), and camphene (N=89), with water results further distinguishing lamellar, dendritic, and honeycomb structures.The water categories are shown as blue circles, red symbols, and green symbols, respectively.Source: Figure 32 · p. 133
• Solid regression lines, 95% confidence intervals, and 95% prediction regions summarize the plotted relationships, with coefficients and bulk σ𝑓𝑓 0 values provided in Table 10.The regression lines are black, while confidence and prediction regions are shown with shaded and dashed boundaries, respectively.Source: Figure 32 · p. 133