Moisture Storage and Transport in Concrete - Lutz H. Franke - E-Book

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Lutz H. Franke

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Beschreibung

Comprehensive insight on moisture transport in cement-based materials by means of experimental investigations and computer simulations

Moisture Storage and Transport in Concrete explores how moisture moves through cementitious materials, focusing on its absorption, storage, and distribution with the help of experimental investigations and computer simulations. The text discusses the different ways moisture moves, such as through vapor or capillary action, as well as how it affects the properties of cement-based materials, offering new insights and models to help understand and predict moisture behavior in these materials, which can be important for construction and maintenance.

After a short introduction to the topic, the text is split into five chapters. Chapter 1 covers surface energetic principles for moisture storage in porous materials. Chapter 2 explores real pore structure and calculation methods for composition parameters. Chapter 3 explains basic equations for the description of moisture transport. Chapter 4 discusses experimental investigation results with regard to the modeling of moisture transport in concrete materials. Chapter 5 showcases modeling of moisture transport, taking into account sorption hysteresis and time-dependent material changes.

Written by a highly qualified author, Moisture Storage and Transport in Concrete also includes discussion on:

  • Dependence of surface energy of water on temperature, on relative humidity of air, and for aqueous salt solutions
  • Calculation of the pore size dependent distribution of inner surfaces using the moisture storage function
  • Temperature influence on the capillary transport coefficients and differences between capillary pressure and hydraulic external pressure
  • Adsorption and desorption isotherms of the CEMI reference material and causes of differences between adsorption and desorption isotherms
  • Sorption isotherms and scanning isotherms of hardened cement paste and concrete
  • Modeling of vapor transport and drying by evaporation of concrete

Moisture Storage and Transport in Concrete is an essential reference to help researchers and professionals to make informed decisions for the construction of concrete-based infrastructure, enabling them to avoid common issues such as corrosion of reinforcement steel, deterioration of concrete strength, and the growth of mold and mildew.

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Table of Contents

Cover

Table of Contents

Title Page

Copyright

Preface

1 Surface Energetic Principles for Moisture Storage in Porous Materials

1.1 Introduction

1.2 Surface Energy and Spreading of Liquids on Solid Surfaces

1.3 Basic Equations for Liquid Absorption in Material Pores

1.4 Sorptive Storage on Material Surfaces and on the Inner Surface of Pore Systems

References

2 Real Pore Structure and Calculation Methods for Composition Parameters

2.1 Illustration of the Pore Structure of Selected Materials

2.2 Calculations on Porosity, Degree of Hydration, and Material Densities

References

3 Basic Equations for the Description of Moisture Transport

3.1 Moisture Flows at the Volume Element

3.2 Base Modeling of Moisture Transport

3.3 Structure of the Simulation Program

References

4 Experimental Investigations with Regard to the Modeling of Moisture Transport in Mortars and Concrete

4.1 Preliminary Remarks on Moisture Storage

4.2 Concrete Data for the Experimental Investigations

4.3 Data on Porosity of the Considered Materials and Influence of Treatments on Porosity

4.4 Hysteretic Moisture Storage Behavior as Important Issue with Respect to Modeling

4.5 Water Storage Behavior Under Changing Moisture Boundary Conditions with Consideration of the Air‐Pore Content

4.6 Adsorption and Desorption Isotherms

4.7 Results on Capillary Water Absorption Depending on Initial Water Content and Time

References

5 Modeling of Moisture Transport Taking into Account Sorption Hysteresis and Time‐Dependent Material Changes

5.1 Preliminaries

5.2 Modeling of Capillary Transport

5.3 Modeling of Vapor Transport and Drying by Evaporation of Concrete

5.4 Realistic Modeling of Drying by Evaporation for Ceramic Bricks, Calcium Silicate Products, and Porous Concrete

References

Bibliography

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1 Comparison of pore‐volumes and densities of a mortar and a concr...

Chapter 3

Table 3.1

Diffusivities

Dw() and Dw() [/s] of mortars (cement contents 51...

Chapter 4

Table 4.1 Composition of the cement bonded materials used for reported test...

Table 4.2 Air voids content of mortar REF made without air‐entraining agent...

Table 4.3 Hardened cement paste HCPIII (mixture see

Table 4.1

), prestorage ...

Table 4.4 Salt solutions to generate defined relative air humidities for is...

Table 4.5 Measured water content and water adsorption values of mortar REF ...

Chapter 5

Table 5.1 Results of vapor diffusion measurements using

pre‐drying conditio

...

List of Illustrations

Chapter 1

Figure 1.1 (a) Model of the water‐storage function for cement‐bound material...

Figure 1.2 (a) Adsorption and desorption isotherms and scanning loops measur...

Figure 1.3 Orientation of water molecules and schematic representation of th...

Figure 1.4 Testing Model : Measuring the surface energy and surface tension ...

Figure 1.5 Measuring of the surface tension by the bracket‐method

Figure 1.6 Measuring surface tension and corresponding liquid surface shapes...

Figure 1.7 Measurement results for the surface energy of water as a functio...

Figure 1.8 Surface tension of aqueous sodium chloride solution at 15–35 C....

Figure 1.9 Incremental progress of a water droplet () on a solid surface ...

Figure 1.10 Formation and definition of capillary pressure in a cylindrical ...

Figure 1.11 Possible equilibrium suction heights for three model capillaries...

Figure 1.12 Meniscus shape in cylindrical pores as a function of pore radius...

Figure 1.13 Temperature dependence of the Henry parameter 1/ of  and , an...

Figure 1.14 Vapor molecule physisorption on solid surfaces. Designations of ...

Figure 1.15 Sorption isotherms for water vapor with pore‐free surface. (a) F...

Figure 1.16 (a) Measured points and regression curve of the adsorption isoth...

Figure 1.17 BET sorption curves according to Eq. (1.53) as a function of par...

Figure 1.18 Comparison of the BET modeled sorption curve according to Eq. (1...

Figure 1.19 Sorption isotherm for the REF material according to

Figure 1.16

...

Figure 1.20 Excess surface work curves according to Eq. (1.63) for the adsor...

Figure 1.21 Comparison of the curve calculated according to Adolphs/Setzer...

Figure 1.22 Schematic representation of a cylindrical pore with increasing r...

Figure 1.23 Measured sorption isotherm of the sample material REF and its ...

Figure 1.24 Course of the inner surface of the reference material REF above ...

Figure 1.25 (a) Cylindrical capillary pore with meniscus and sorbed layer th...

Figure 1.26 Course of surface tension or surface energy in sorption films up...

Figure 1.27 Comparison of sorption isotherms for plane surfaces of water vap...

Figure 1.28 (a) Cylindrical capillary pore with meniscus and sorbed layer th...

Figure 1.29 (a)

Influence of surface curvature resp. radial tensile stress o

...

Figure 1.30 (a) Relative air humidities at which

filling of cylindrical po

...

Figure 1.31 Comparison of relative film thicknesses on the inside of cylindr...

Figure 1.32 The adsorption isotherm of the material REF, the derived net sto...

Figure 1.33 Schematic representation of capillary condensation in two differ...

Figure 1.34 Calculated film thicknesses on the inside of water‐filled cylind...

Figure 1.35 Adsorption isotherm of the material REF and net storage function...

Figure 1.36 Schematic representation of the principle of an AFM (atomic forc...

Figure 1.37 Results of D force scanning of the muscovite mica material inte...

Figure 1.38 Three‐dimensional AFM‐images of muscovite mica/electrolyte solut...

Figure 1.39 Measurement results of G. Zhao et al. from [53]

on water vapor so

...

Figure 1.40 Figures and results of modeling from Kuchin et al. [71] and Kuch...

Figure 1.41 Modeling of the force transmission from the meniscus edge into t...

Figure 1.42 Schematic illustrated behavior of the shape of the menisci and t...

Chapter 2

Figure 2.1 (a) TEM image of Reichinger of synthetic silicate MCM‐41 with par...

Figure 2.2 Microscopic pictures from fractured surfaces of partially sintere...

Figure 2.3 (a) Light microscope image of the polished section of a ceramic b...

Figure 2.4 Two groups of MIP cumulative pore volume curves, relative to thei...

Figure 2.5 Microstructure of hardened cement paste HCP schematical.

Figure 2.6 Microscopic picture of Portland cement clinker structure before g...

Figure 2.7 SEM pictures of alite grains in HCP of Portland cement with a ‐r...

Figure 2.8 Sample of hardened S paste, , hydrated for eight years at 20 C...

Figure 2.9 Op‐CSH in a sample of hardened ‐S paste, , hydrated for three ...

Figure 2.10 TEM pictures to characterize the different microstructure of CSH...

Figure 2.11 Hydration of HCP of

ordinary Portland cement CEM I 52.5

.(a) ...

Figure 2.12 SEM overview images of the UHPC used; UHPC Composition: Portland...

Figure 2.13 STEM high angle annular dark field (HAADF) images from a UHPC sa...

Figure 2.14 STEM high angle annular dark field (HAADF) image from a point wi...

Figure 2.15 r 0.14 as measure for the biggest mean effective capillary p...

Figure 2.16 Identification of the radius

r

Entry

to concrete C with  [nm] u...

Figure 2.17 Illustration of the development of the internal (empty) volume (...

Figure 2.18 (a) Calculated maximum chemical shrinkage of Portland cement sa...

Figure 2.19 Calculated relation of the total pore volume with and without ...

Figure 2.20 Calculated degrees of possible hydration of Portland cement samp...

Figure 2.21 Development of the degree of hydration of HCP prisms made of Por...

Figure 2.22 Model diagram for the determination of the mathematical function...

Figure 2.23 Evolution of the degree of hydration of HCP prisms made of blast...

Figure 2.24 Measurements by Adam [39] of the hydration course of concrete sl...

Figure 2.25 Development by volume of the components of hydrating white Portl...

Chapter 3

Figure 3.1 Determination of the hydraulic conductivity, compare Todd [1].

Figure 3.2 Hydraulic conductivity [m/s].

Figure 3.3 Temperature effect on the viscosity of pure water.

Figure 3.4 Measurements of the influence of external hydraulic pressure on ...

Figure 3.5 Viscosity dependence of solutions of calcium chloride on temperat...

Figure 3.6 Moisture distributions of a ceramic brick material according to t...

Figure 3.7 Effect of value on hydraulic conductivity of HCP. Measured valu...

Figure 3.8 Definition of the relevant pore radii (determined via the MIP mea...

Figure 3.9 NMR measuring device from Krus [11] for measuring the water conte...

Figure 3.10 Section of a 1D net discretization with capillary moisture flow ...

Figure 3.11 Single volume element with inflow and outflow and water contents...

Figure 3.12  Basic water storage function of mortar REF

and derived net st

...

Figure 3.13  Water storage function of mortar REF, showing the dependence ...

Figure 3.14 Elements of the solution system.

Chapter 4

Figure 4.1 Comparison of MIP‐results after standard‐drying at C/1 mbar and

Figure 4.2 Comparison of MIP‐results after C‐drying of concrete. (a) Pore s...

Figure 4.3 Carbonation of the edge of a REF‐Mortar cube  [] after long‐tim...

Figure 4.4 MIP pore distribution of the non‐carbonated part and the carbonat...

Figure 4.5 Part from a

prism

(REF series after 84 days of initial water stor...

Figure 4.6 Mortar REF, Slice (series S2), thickness  [mm]. (a) Water conten...

Figure 4.7 Mortar REF, Slice (series S3), thickness 7 [mm]. (a) Test with ...

Figure 4.8 Slice (thickness  [mm]), cut from the bottom of a prism at the e...

Figure 4.9 Thin‐sections with air voids filled with blue resin, materials wi...

Figure 4.10 Air voids content of mortar MA (made without air‐entraining agen...

Figure 4.11 Size distribution of air voids in concrete samples from a Michig...

Figure 4.12 Size distribution of air voids in lab concrete mixtures with air...

Figure 4.13 SEM images of inner surfaces of air voids in concrete mixtures w...

Figure 4.14 CEMIII‐Mortar MIII, prestorage 12 weeks at 98 RH/20 C: Compari...

Figure 4.15 CEMI‐Mortar REF, prestorage 12 weeks at 98%C: Influence on the ...

Figure 4.16 CEMI‐Mortar REF, 84d water prestorage, slices  []: After C/1 ...

Figure 4.17 Influence of drying procedure on imbibition tests of CEMI‐Mortar...

Figure 4.18 CEMI‐Mortar (REF), 84d water prestorage: Influence of drying pro...

Figure 4.19 CEMI‐Mortar REF, 84d water prestorage, : Measured basic‐isother...

Figure 4.20 REF mortar slices 40407.5 [], 84d water prestorage, measuring...

Figure 4.21 Water‐vapor adsorption and desorption curves measured on synthet...

Figure 4.22 Schematic illustration of the cause of the difference between wa...

Figure 4.23 Schematic illustration from Zandavi and Ward [18] and Zandavi [1...

Figure 4.24 Illustration of the volumes of the pore types of mortar REF, cal...

Figure 4.25 REF mortar‐slices 7.5 [mm] (series C1 and C3): 240d water presto...

Figure 4.26 Water content curves measured on REF mortar‐slices 7.5 [mm] (ser...

Figure 4.27 Water content curves measured on REF mortar‐slices 7.5 [mm] : ...

Figure 4.28 Water content curves measured on REF mortar‐slices 7.5 [mm] : ...

Figure 4.29 Cross section got by breaking a ] slice of REF mortar, stored 5...

Figure 4.30 TGA tests of thin HCP‐slices (size 1 [mm]) on possible dilution ...

Figure 4.31 Schematic figure of the precipitation of a thin calcium carbonat...

Figure 4.32 Schematic illustration of the water suction into capillary pores...

Figure 4.33 Absolute pressure  [Pa] in capillary pores dependent on capilla...

Figure 4.34 Dissolution of air‐pores (air bubbles) with initial size in su...

Figure 4.35 Comparing tests of water absorption on mortar‐REF‐samples. (a)...

Figure 4.36 Water adsorption curves measured on mortar REF prisms 404014...

Figure 4.37 Water adsorption curves measured on mortar REF prisms  [] afte...

Figure 4.38 Water adsorption curves measured on mortar REF prisms 404080...

Figure 4.39 Water adsorption curves measured on mortar REF prisms 404015...

Figure 4.40 Water adsorption curves measured on mortar REF prisms 4040150 ...

Figure 4.41 Schematic diagram about the dependence of possible water content...

Figure 4.42 Results of adsorption/desorption measurements

on mortar REF sli

...

Figure 4.43 Results of adsorption/desorption measurements on mortar REF sl...

Figure 4.44 Desorption isotherm with special look at the sharp water content...

Figure 4.45 Experimental control of the desorption behavior of water‐stored ...

Figure 4.46 Measured water storage function by combining the results of pr...

Figure 4.47 Measured water storage function by combining the results of pres...

Figure 4.48 Water content measured on a series of prisms and cubes of mortar...

Figure 4.49 Comparison of cumulative pore volume curves (a) and correspondin...

Figure 4.50 Comparison of measured water storage curves of 3 REF mortar‐se...

Figure 4.51 Mortar‐REF, Tests on influence of prestorage methods on 1D capil...

Figure 4.52 View of adsorption and desorption curves of selected building ma...

Figure 4.53 Vapor sorption and over‐hygroscopic storage behavior of brick Wi...

Figure 4.54 Vapor sorption and storage behavior of brick Joens (Germany), t...

Figure 4.55 Adsorption and desorption behavior of sand‐lime bricks.

(a)

Test...

Figure 4.56 Measured adsorption and desorption isotherms on

Hardened Cement

...

Figure 4.57 Measured adsorption and desorption isotherms on

Hardened Cement

...

Figure 4.58 Measured adsorption and desorption isotherms on

mortar MIII

( a...

Figure 4.59 Scannings measured on

HCPI and HCPIII

/TUHH.

(a)

 Main Isotherms a...

Figure 4.60 Adsorption isotherms and adsorption scannings,

measured on concr

...

Figure 4.61 Measured adsorption and desorption isotherms and scannings.

(

...

Figure 4.62 Phenomenon of crossing of primary desorption isotherms by subseq...

Figure 4.63 Sketch describing the dependencies or parameters used for modeli...

Figure 4.64 Course of adsorption and desorption scannings for the water‐satu...

Figure 4.65 Sketch describing the effective inner water surface at the vap...

Figure 4.66 Comparison of measured and calculated adsorption scannings of a ...

Figure 4.67 Observed course of adsorption scannings and desorption scannings...

Figure 4.68 Sorption isotherms and scannings determined on HCPI. Comparison ...

Figure 4.69 Comparison of sorption isotherms of mortar REF ()

after sealed

...

Figure 4.70 Schematic diagram with labels for the Eqs. (4.17) to (4.20) for

Figure 4.71 Sorption isotherms of mortar REF ()

after water storage

and aft...

Figure 4.72 Scannings calculated based on the procedures in Sections 4.6.4 a...

Figure 4.73 Comparison of corresponding calculated adsorption scannings (das...

Figure 4.74 DVS measured adsorption and desorption isotherms and scanning is...

Figure 4.75 Dependence of the shape of desorption isotherms on the value, ...

Figure 4.76 DVS measured adsorption and desorption isotherms and scanning is...

Figure 4.77 Comparison of measured values (

b

) from Zhang et al. [40]/with pe...

Figure 4.78 Sorption isotherms for mortar MIII deduced by calculation from t...

Figure 4.79 Sorption isotherms of HCPIII and mortar MIII measured after

15 m

...

Figure 4.80 Comparison of measured and calculated adsorption isotherms as a ...

Figure 4.81 Relative humidity RH measured in closed containers above crushed...

Figure 4.82 Schematic illustration of the moisture exchange up to transport ...

Figure 4.83 Desorption isotherms of HCP PI from Portland cement CEM I 52.5 R...

Figure 4.84 Desorption isotherms of HCP PIII (Blast furnace cement CEM III/A...

Figure 4.85 Boiling point and vapor pressure curves, respectively, of water ...

Figure 4.86 Effective water‐pressure curve in the pores (of a hardened cemen...

Figure 4.87 Results of desorption studies on the influence of cavitation on ...

Figure 4.88 Results of

desorption tests

using the desiccator method at eleva...

Figure 4.89 Tests on the temperature influence on the desorption and absorpt...

Figure 4.90

Desorption isotherms

at elevated temperatures measured on HCPI (o...

Figure 4.91 Adsorption isotherms for different temperatures for CEM I (mater...

Figure 4.92 The computational modeling results presented are based on the ma...

Figure 4.93 Desorption isotherms at elevated temperatures describing measure...

Figure 4.94 Calculated curves based on the material and the curves from

Figu

...

Figure 4.95 Measured values of scanning isotherms of HCPI (CEMI+ limestone ...

Figure 4.96 Comparison of applicability of MIP‐measurements as bordering “de...

Figure 4.97 Desorption isotherms of sandstone material, previously doped wit...

Figure 4.98 Capillary water uptake curves measured in two independent projec...

Figure 4.99 Plot of the shape of experimental water uptake curves as a funct...

Figure 4.100 Water content curves in [kg/] of mortar REF derived from mea...

Figure 4.101 Capillary water uptake coefficients, respectively, slope of the...

Figure 4.102 Capillary water uptake after standard drying of 2 UHPC products...

Figure 4.103 Base curve (echo train intensity decay) of an NMR measurement o...

Figure 4.104 Results of NMR analyses on HCP of white cement (), 28d of wate...

Figure 4.105 Evidence of change in water structure with decreasing pore size...

Figure 4.106 Schematic description of water storage in HCP material at an in...

Figure 4.107 Adsorption isotherm of the material REF and net storage functio...

Chapter 5

Figure 5.1 Water content distribution and penetration depth curves after one...

Figure 5.2 Definition of water transport and vapor transport properties as a...

Figure 5.3 Schematic picture of the dependence of the liquid water transport...

Figure 5.4 Calculated capillary

water uptake curves

(red) into longitudinall...

Figure 5.5 Schematic figure about the impact of drying and wetting on the mo...

Figure 5.6 Diagram explaining the modeling of the possible time‐dependent de...

Figure 5.7 Diagram to illustrate the modeling of the possible time‐dependent...

Figure 5.8 Calculated capillary water uptake curves (blue) into longitudinal...

Figure 5.9 Distribution curves of capillary‐absorbed water into prisms of le...

Figure 5.10 Modeling the slope of the scanning isotherms for the mortar REF ...

Figure 5.11 Illustration of the possible scanning behavior of differently wa...

Figure 5.12 Control calculation for reverse sorption experiments on the cour...

Figure 5.13 Illustration of the influence of the

hysteretic storage behavior

Figure 5.14

Resulting water distributions in the test body of Figure

 5.13

a, i

...

Figure 5.15 Influence of the

hysteretic storage behavior

within the test bod...

Figure 5.16 Influence of the

hysteretic storage behavior

within the test bod...

Figure 5.17 Influence of hysteretic storage behavior within the test body co...

Figure 5.18 Results of the vapor diffusion measurements according to DIN EN ...

Figure 5.19 Dependence of the water vapor diffusion coefficient on the deg...

Figure 5.20

(a)

Vapor transport through a 10 [cm]‐thick (initially water‐satu...

Figure 5.21 Configuration of the volume elements, here related to the evapor...

Figure 5.22 Comparison between experimental evaporation curve and curve calc...

Figure 5.23 Experimental and calculated evaporation curves with boundary con...

Figure 5.24 Characteristic pore‐ resp. water saturation parameters of a poro...

Figure 5.25 Comparison between (a) the experimental evaporation and (b) the...

Figure 5.26 Comparison between the experimental evaporation curves (black di...

Figure 5.27

(a)

Varying moisture storage stages of the prism series of

Figure

...

Figure 5.28 Results of two‐sided evaporation measurements over the two faces...

Figure 5.29 Time‐dependent water content decrease and associated water conte...

Figure 5.30 Typical images of sawn surfaces after prolonged water storage (l...

Figure 5.31 Influence of surface salt precipitation on the evaporation behav...

Figure 5.32 Measured evaporation curves on water‐stored prisms made of morta...

Figure 5.33 IR spectroscopic analysis of the substance removed from the sur...

Figure 5.34 Measurement of the two‐sided evaporation behavior of test specim...

Figure 5.35 Influence of surface salt precipitation and subsequent abrasion ...

Figure 5.36 Influence of the formation of a surface precipitate on the evapo...

Figure 5.37 Comparison of the one‐sided evaporation for two series of test s...

Figure 5.38 Comparison of the one‐sided evaporation of a test specimen of gi...

Figure 5.39 Results of one‐sided evaporation tests on samples of ceramic bri...

Figure 5.40 (a) Measured adsorption isotherms of two different batches of ce...

Figure 5.41

(a)

 Calculated water content distributions in a 100 [mm] thick ce...

Figure 5.42

(a)

Sorption isotherms of ceramic bricks with  [/]; function f...

Figure 5.43 Calculated one‐sided drying by evaporation of wall panels of dif...

Figure 5.44 Calculated one‐sided drying by evaporation of wall panels or tes...

Guide

Cover

Table of Contents

Title Page

Copyright

Preface

Begin Reading

Bibliography

Index

End User License Agreement

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Moisture Storage and Transport in Concrete

Experimental Investigations and Computational Modeling

 

Lutz H. Franke

 

 

 

 

 

Author

Prof. Dr. Lutz H. FrankeHamburg University of Technology (TUHH)Institute of Building MaterialsBuilding Physics and BuildingChemistryHamburg, Germany

Cover Image: © Westend61/Getty Images

All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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Print ISBN: 978‐3‐527‐35378‐1ePDF ISBN: 978‐3‐527‐84686‐3ePub ISBN: 978‐3‐527‐84685‐6oBook ISBN: 978‐3‐527‐84687‐0

Preface

This book deals with the modeling and computational simulation of moisture absorption, moisture storage, and moisture distribution mainly in concretes or cement mortars, based on our own experimental investigations and numerous national and international publications on the subject. New aspects of moisture transport and its modeling are presented. For comparison, transferable aspects of moisture transport in inert materials such as ceramic bricks and materials with comparable porosity are considered.

One aim of the book is to present the research results as comprehensibly as possible in such a way that the presented content does justice to the title of the book. In doing so, it is attempted to reproduce the presentations as simply as possible, but in a physically and mathematically correct manner.

The results of targeted experiments carried out at the TUHH (Hamburg University of Technology) on moisture storage and transport serve as an important basis. The materials used were cement mortars and concretes, mainly made of Portland cement. The computational simulations were carried out with the help of a separate computational program additionally developed for this problem area. The models used and, where appropriate, mathematical formulations of interest or required for understanding and comparison with previous approaches are given.

Investigation results on the various moisture transport mechanisms such as vapor transport, capillary transport, and effusion are presented. Water‐molecule adsorption in the pore system is considered in more detail and the influences on the transport coefficients, sorption isotherms, and moisture storage functions are highlighted. Special effects such as hysteretic behavior with scanning isotherms, water content distribution, and self‐sealing are considered and modeled, and the influence is highlighted.

Furthermore, not only an alternative modeling of drying, especially of concrete, is presented, but also the drying of ceramic bricks and materials of similar pore size distribution is considered for comparison.

Capillary pressure (and vapor pressure) is used as the driving potential for moisture transport, since this mechanism allows a broader consideration of influences, especially the hysteresis of the moisture storage function. Other authors also suggest using preferably capillary pressure as driving potential for the modeling of transport processes in concrete materials.

The simulation program developed and used in the present project to calculate and verify the results is written in Fortran source code. A number of additional calculations were performed using the program MathCad. The Windows interface to the program was developed by Dr. G. Deckelmann at the TUHH and exists at the institute.

 

January 2024

Prof. Dr. Lutz H. FrankeEmeritus Head of the Institute of Building MaterialsBuilding Physics and Building ChemistryHamburg University of Technology (TUHH)Germany

1Surface Energetic Principles for Moisture Storage in Porous Materials

1.1 Introduction

Most natural mineral materials, with the exception of crystals, have a pore system whose pores can range from very fine nanometer (nm)‐sized pores to the millimeter (mm) range. A recognized classification of pore sizes has been made by International Union of Pure and Applied Chemistry (IUPAC). In the 2015 update of the 1985 report [1]. In this paper, the pores are classified into macropores, mesopores, and micropores.

Building materials such as natural stone, brick and especially concrete cover the full pore size range mentioned. Concrete materials usually contain a substantial concentration of particularly fine pores, which are classified in the group of nanopores.

These porous materials can therefore store liquids in the pore system, especially water, which can be carried in vapor form or in liquid form via surface forces.

The capillary absorbed liquid content is measured in [kg/ or in [, the velocity usually with good approximation as in [kg/ with the material coefficient in [kg/( when constant fluid supply is ensured.

To describe the storage of liquid from vapor uptake, the resulting water content is presented in the form of sorption isotherms as a function of the external relative humidity or after converting the relative humidity to the corresponding capillary pressure in the material.

In a number of (building) materials, such as brick products, a portion of the pores is so large that it is no longer filled by vapor adsorption, even at about 100% relative humidity. These pores can then only be filled capillary by external liquid‐water supply. This water fraction is called the superhygroscopic range of total water uptake. In such cases, the total moisture storage is described by the so‐called moisture storage function.

As indicated in the schematic moisture storage function in Figure1.1a, many authors allow the hygroscopic range of the moisture storage function to extend only to about relative humidity, when in fact it must be defined to about 100% RH. The reason for this is the difficulty of precisely setting the moisture and measuring it accurately in this ‐near range. If the material also contains large pores that cannot be filled by capillary action – for example, air pores – the associated pore volume, which can usually only be filled under pressure, is assigned to the overhygroscopic range.

Figure 1.1 (a) Model of the water‐storage function for cement‐bound material.

Source: Adapted from Fagerlund [2] and Eriksson et al. [3].

(b) Sample moisture adsorption and desorption storage functions for building materials as a function of capillary pressure .

Source: Carmeliet and Roels [4]/Sage Publications.

The curve region above RH can be determined using the pressure plate experiment [5, 6], and Espinosa–Franke [7] as a so‐called suction stress curve depending on the applied capillary pressure. The mutual conversion of is done with Eq. (1.1). In this way, moisture‐storage functions can also be represented completely as a function of instead as, for example, by Carmeliet in Figure 1.1. At relative humidity, the associated capillary pressure is [Pa].

This means that in Figure 1.1a,b only the lower section of the curves (in the hygroscopic region) was determined by sorption measurements. The overhygroscopic ranges thus concern additional capillary water absorption as well as further water absorption under pressure also with (partial) filling of the processing‐related or artificially inserted air pores.

Using the Eq. (1.39) explained in more detail in Section 1.3.2, the vapor pressure dependence of the adsorption and desorption curves can be converted to the corresponding dependence on the associated capillary pressure in [Pa] as follows:

(1.1)

Sorption tests on building materials, in particular cement‐bound materials, yield desorption curves that deviate significantly from the adsorption curves or moisture storage functions during water absorption. The reason for this behavior will be discussed in more detail in Section 4.6. Measurements by, for example, Feldman and Serada [8] or Ahlgren [9] have already made this clear in 1968 and 1972, see Figure 1.2a,b. The measurements also show that a transition between an absorption and desorption curve, or vice versa, occurs on a “short path,” referred to as scanning loops or scanning isotherms. The main focus of Chapters 2, 3, and 4 will be to show the effects of this behavior on the moisture transport and the moisture household of corresponding material bodies.

Figure 1.2 (a) Adsorption and desorption isotherms and scanning loops measured on HCP of Portland cement .

Source: Feldman and Sereda [8]/Springer Nature.

(b) Principle course of moisture storage functions including scanning isotherms of building materials.

Source: Ahlgren [9]/Lund Institute of Technology/CC BY 4.0.

It will be shown that the individual moisture storage functions may have a fundamental importance for the moisture balance of porous materials and the modeling of moisture transport.

1.2 Surface Energy and Spreading of Liquids on Solid Surfaces

Since pore water within moist porous bodies is transported by capillary pressure (and vapor pressure) in the presence of sufficiently fine pores, and therefore capillary pressure is a crucial quantity with respect to moisture transport, the origin of capillary pressure within the pore system is first addressed. This first requires explanations of the role of the surface energy of the substances involved.

1.2.1 Explanations on Surface Energy and Surface Tension

The  molecular arrangement on a water surface surrounded by air is shown schematically in Figure 1.3. In contrast to the interior of water, where the molecules are surrounded by similar molecules in all spatial directions and therefore force effects between the molecules cancel each other out in the summation, the surface lacks balancing molecular partners on the air side.

Therefore, a molecular arrangement is formed at the surface, which leads to inwardly directed cohesive forces (hydrogen bonds) and force effects in the surface  plane.

Figure 1.3 Orientation of water molecules and schematic representation of the attractive forces at the liquid surface, by D. Drummer, Erlangen‐Nürnberg, Germany.

To increase the liquid surface area, work must be done to overcome the cohesive forces of a considered amount of water while the volume remains unchanged. The work to be done per unit area to increase this surface area A is called the surface energy  [LV means liquid versus air], here abbreviated as, corresponding to Eq. (1.2) in differential formulation:

(1.2)
(1.3)

UsingFigure 1.4, it can be shown how surface energy can be determined by surface enlargement in a model experiment:

A water membrane (producible by addition of surfactant) of dimension  is stretched by  with force

F

. The surface (front and back) increases by . The work done to increase the surface  is given in

Figure 1.4

using Eq. (

1.2

).

The displacement work is  ( and F measured).

Figure 1.4 Testing Model : Measuring the surface energy and surface tension by the work of displacement and the boundary force on a (soap)–watermembrane.

Figure 1.5 Measuring of the surface tension by the bracket‐method

The formulations of the work describe the same change in the sample and therefore must be equal in magnitude. Thus,  Eq. (1.3) yields the magnitude of the (specific) surface energy of the surfactant‐added water. It can be seen that the special molecular orientation or the resulting surface cohesion in surface plane of the water membrane can introduce an edge force leading to an increase of the surface area, which is called the surface stress. From Eq. (1.3), it can be derived at the same time that the boundary force F related to the unit of the boundary length in [m] corresponds to the surface energy  of the liquid.

The true value for non‐surfactant water can be determined fairly accurately with the experiment shown in Figure 1.5, in which a wire stretched in a stirrup structure is lifted out of a water surface via a precision balance. The water surface around the wire is lifted until it breaks off when the maximum force F is reached. The surface energy of water is 0.07275 [Nm/ = N/m] at 20 C, correspondingly 72.75 [mN/m].

Figure 1.6 Measuring surface tension and corresponding liquid surface shapes by two different methods [10]. Left: Situation during a Wilhelmy‐plate test. Right: Three states of the film surface during a bracket test (compare Figure 1.5) or with a ring‐shaped wire during a Noy test.

The  examples of surface stress measurements shown in Figure 1.6 illustrate the shapes that water surfaces can attain largely due to surface stress alone. Instead of the bracket test, the Noy method or the Wilhelmy method are predominantly used; compare Figure 1.6 and Welcome to DataPhysics‐Instruments [10].

In the Noy method, a ring‐shaped wire is immersed and then drawn. A high‐speed camera shows that in this process, the water surface takes on the shapes sketched on the right in Figure 1.6 as the pulling force F increases. From the maximum tensile force , the surface energy of the liquid is then calculated according to Eq. (1.3), corresponding to the Wilhelmy method.

1.2.2 Dependence of Surface Energy of Water on Temperature, on Relative Humidity of Air, and for Aqueous Salt Solutions

Results on the dependence of the surface energy of water on the relative humidity of the surrounding air were apparently first reliably determined in 2012 by Pérez‐Díaz et al. [11], supplemented in 2017 by investigation results by Portuguez et al. [12]. Up to now, obviously, no corresponding results could be determined with capillary suction tests or on drops on flat material surfaces due to the mutual influence of liquid and solid or the resulting influences by simultaneous water evaporation.

As a new measuring method, Pérez‐Díaz et al. and Portuguez et al. developed the method of hanging drops in climatic chambers combined with precise drop shape measurements via microscopy and image analysis.

The following Figure 1.7a shows the variation of the surface energy at 100% RH as a function of temperature in comparison to the already known behavior as a confirmation of the measurement methodology used. Figure 1.7b contains the results for different relative humidities at different temperatures. This shows, for example, that at 20 C and 20% RH, the value of is about 5% larger than at 100% RH.

Figure 1.7  Measurement results for the surface energy of water as a function of ambient relative humidity and temperature, basic diagrams from Portuguez et al. [12] with data from Pérez‐Díaz et al. [11] (a) Dependence of the surface energy at 100% RH as a function of temperature, comparison of the measured values of Potuguez (red circles) and of Pérez‐Díaz (blue circles ) with previous tabulated measured values (black signs). (b) Dependence of surface energy on relative humidity and temperature.

The dependence of surface energy on temperature at 100% RH can be calculated by the following Eq. (1.4). At 60 C, = 0.067 [Nm/].

(1.4)

According to American National Institute of Standards and Technology (NIST), at 100% RH and 100 C: and at C:  [Nm/]. The surface energy becomes zero at the critical point of water at 373 C.

Figure 1.8 Surface tension of aqueous sodium chloride solution at 15–35 C.

Source: Chen et al. [13]/with permission of Elsevier.

Furthermore, it is necessary to ask to what extent the surface energies change for aqueous salt solutions or electrolytes. For  limited‐concentration solutions, H. Chen et al. [13] presents a “Gibbs phenomenological surface‐phase” method for numerous salts. Figure 1.8 shows results from this publication for sodium chloride and sodium sulfate solutions. The relatively limited influence up to salt contents of 1 [mol/kg] can be seen.

1.2.3 Spreading of Liquids on a Solid Surface

The  ability of a liquid to wet a solid depends largely on the surface energies of the substances involved. Therefore, the surface energies of solids are also important.

These surface energies can only be measured indirectly at room temperature using drops of test liquids whose surface energies are known. Contact angle measurement is usually used, in which the angle of inclination or the edge angle to the solid surface is determined and the surface energy is derived from this, compare for example Kinloch [14]. Also, so‐called test inks allow an approximate determination of the energetic surface states.

For the evaluation of the results, the following considerations are important:

Is defined first

specific surface energy of the liquid

in the environment air plus vapor.

specific surface energy of the solid

in the environment air plus vapor, as well as

specific interfacial energy

, which can exert a separating effect in the interface between the liquid in contact and the solid due to the different molecular  nature.

For an understanding of the following relationships, we refer to Figure 1.9 of a partially contacting droplet with total surface area of and a given volume.

The spreading of the droplet (L) on the solid surface (S) occurs, if the preconditions are met, due to the effect of attractive forces between the liquid and the solid, which are able to overcome the surface tension of the liquid when the contact area is increased.

The spreading comes to a stop when a minimum of the total energy of the ongoing process is reached. Figure 1.9 represents the situation just before this standstill:

The already existing contact area between (

L

) and (

S

) still increases by the fraction

dA

. On the air side of the contacting droplet, the surface area is thus approximately increased by the value . In total, the droplet surface area thus increases by . For this, the following work must be done on the drop side:

(1.5)

At the same time, adhesion energy is released in the area of contact area increasing dA:

(1.6)

Here expresses that there is not a full saturation of the surface energy in contact but is reduced by the fraction . At equilibrium:

(1.7)

It follows

(1.8)

and for the boundary angle  the well‐known Young equation follows after the transformation:

(1.9)

At complete wetting or dissolution of the drop takes place.

Figure 1.9 Incremental progress of a water droplet () on a solid surface () shortly before reaching the energetic balance of the surface energies involved.

The surface energies present now allow statements about the dispersion behavior of different liquids on different surfaces. This also determines whether a liquid can penetrate a pore system. The measurement of the contact angle in the drop test provides at least approximate information on this. If the contact angle is about 90°, . This results is a “neutral” behavior of the given liquid on the given solid surface. No capillary water absorption takes place in this case. In order that capillary takes place, the angle  must be appreciably less than 90°.

According to the following equation by van Honschoten et al. [15], spreading occurs for a solid–liquid combination when the value S is positive:

(1.10)

1.2.4 Determining the Surface Energies of Solid Surfaces

For   mineral materials used in construction, the contact angle is usually less than 90°, unless the surface has been modified, for example, by a hydrophobizing measure. Kaolin has a surface energy of about 500–600 [mN/m]; HPC concretes with aggregate from granodiorite and from granite with a compressive strength of about 130 [MPa] have, according to Barnat‐Hunek to 1800 [mN/m] [16] and a contact angle of about 10° to 30° when wetted with water at 22 C, resulting in capillary water absorption into a pore system.

Calcium carbonate has a surface energy of only –80 [mN/m], so that largely standing water droplets can be expected on such surfaces.

Plastics, paints, and waxes have a  of 25–40 [mN/m], so that no spreading of water can take place on these surfaces, which is very well seen, for example, on waxed car bodies. Teflon exhibits a  of about 20 [mN/m].

In contrast, ethanol or isopropanol, for example, with surface energies of 22 or 23 [mN/m] on mineral surfaces always show a contact angle close to 0°, i.e. complete spreading and a strong readiness for capillary penetration into capillary pores.

Metal alloys also usually have very high surface energy, ‐values of metals can be taken from Kumikov and Khokonov [17]. Nevertheless, with respect to adhesion, adequate surface pretreatment is especially important for metals.

If, in the case of “unknown” solid surfaces, their surface energy is needed, however, it cannot be determined only by determining a boundary angle with a known liquid. A more accurate determination of the surface energy can be made by at least two edge angle measurements with two different test liquids with different, known polar interaction fractions (for example, from hydrogen bonding) and dispersive fractions  (from van der Waals forces).

The surface energy of the solid as well as the interfacial energy  can then be determined from the results of the boundary angle measurements according to the accepted OWRK method of Owens, Wendt, Rabel, and Kaelble, compare Yuan and Lee [18], Lauth and Kowalczyk [19], and Barnat‐Smarzewski and Smarzewski [16].

In Literature [20], measurement and calculation results from two institutes are reported by Cwikel et al., in which the performance of five different computational models for determining the surface energy of solids is investigated. For this purpose, the contact angles of selected test liquids on 42 different solid surfaces are measured and compared with the predictions of the computational models.

1.3 Basic Equations for Liquid Absorption in Material Pores

1.3.1 Liquid Absorption in Pores by Effect of Surface Energies

Liquid   is also drawn into the inner surface of material pores by the effect of surface tension. The finer the pores, the greater the depth of penetration or rise relative to gravity. In such pores, the liquid is transported as in a tube. This is traditionally shown by the liquid rise height in a cylindrical capillary pore.

1.3.1.1 Derivation of the Capillary Rise via the Adhesion Works and the Potential Energy in Capillaries

The  adhesion works in a standing cylindrical capillary are for the rise height h:

(1.11)
(1.12)

The potential energy to be overcome is:

(1.13)

At equilibrium is

(1.14)

From this follows, shortened by  :

(1.15)

Using Eq. (1.9), this gives the relationship for the achievable height due to adhesion work in a cylindrical capillary:

(1.16)

If instead slit pores with a constant spacing of the pore surfaces ofare present, the result for the rise height is

(1.17)

If gives . At this point, we refer the reader to Section 1.3.1.3, where a more general derivation of the fluid uptake in pores due to adhesion work is  described.

1.3.1.2 Capillary Pressure in Cylindrical Pores and in Slit Pores

The   transport in capillary pores or slit pores caused by the surface energy creates a tensile stress below the menisci in the pores, which is called capillary pressure. As a result of the previous explanations, the resulting capillary pressure can be simply represented according to Figure 1.10. The water column of the pores resp. pore filling is further pulled by the effect of the surface energies resp. the corresponding surface tension at the wall of the pores. In dependence of the existing edge angle  from Eq. (1.9), the force component  in [N/m] is now generated there in the pore direction.

(1.18)

For a cylindrical capillary pore with a given perimeter, the resulting total force follows to

(1.19)

The resulting capillary pressure  cannot be exceeded by the surface energy alone because of the limitation of the surface tension to . The capillary pressure is then (when the surrounding air pressure is not taken into account)

(1.20)

In the presence of slit pores with d = surface distance of the pore walls, the following results instead

(1.21)

When ,  is only half as large as for a cylinder pore, according to Eq. (1.20).

The influence of changing temperatures and possible salinity on  has been given previously.

Figure 1.10 Formation and definition of capillary pressure in a cylindrical pore during water absorption.

1.3.1.3 Capillary Pressure as a Cause of Fluid Transport and Rise Height in a Capillary Pore

From    Extrand and Moon [21], using simple water absorption experiments on glass capillaries, it is shown that the derivation of the rise height in capillary pores in terms of surface work and potential energy presented in Section 1.3.1.1 leads to an unjustified formal restriction on the applicability of the Eq. (1.16).

According to this equation, varying ratios of surface energies along the capillary pore and a varying pore cross section, for example, a larger pore radius in the lower part of the capillary, should also lead to the same riser height (1.16). Extrand and Moon [21] concludes that the capillary pressure at the head of the water column in the capillary is responsible for the increase, independent of the other parameters. In fact, this can be shown and somewhat specified in the following way:

Instead of (1.11) and (1.12), let be written only the local increase of the surface energy fractions or work fractions of the liquid surface and the solid surface in the region of the pore radius

(1.22)
(1.23)

The resulting energy sum does the work of raising the corresponding liquid level. The corresponding mechanical work portion force displacement results at the considered radius from the product of the pore cross‐sectional area and the hydraulic stress acting in the cross‐section times the displacement . For example, the surface of the advancing fluid has the shape of a meniscus. The associated mechanical work is then, with the hydraulic (tensile) stress

(1.24)

At equilibrium is

(1.25)

From this follows, shortened by  and , with  the relation

(1.26)

With the suction stress , the fluid bulk density , and the acceleration due to gravity g, the hydraulic potential or the fluid pressure head is given by

(1.27)

Figure 1.11 Possible equilibrium suction heights for three model capillaries with identical inner radius in the upper region.

Figure 1.11 shall schematically illustrate the relationship between the effect of capillary pressure and gravity in different capillaries. is the rise height in capillary 1 with the associated capillary pore radius according to Eq. (1.16) resp. (1.27).

If the capillary has a radius extension, for example, in the form of a spherical pore, the liquid uptake stops when entering the spherical pore according to the associated radius extension, resp. the lower capillary pressure (capillary 2 in 1.11). If, on the other hand, the sphere pore is filled with water by some action in the sense of a continuous pressure connection to the surface, the rise height reaches the suction height h1 for a capillary radius .

It is to be noted that only the adhesion work in the area of the front meniscus and the capillary pressure there (as suction force) lead to the progress of the water column.

If capillary condensation takes place at the front of the capillary pore due to the boundary conditions (pore radius and vapor pressure, compare Sections 1.3.2 and 1.4.5), liquid is added there accordingly, but the capillary head is not increased.

When the capillaries are horizontal or there is no gravity (but water contact at the base), the capillary is completely filled by the adhesion work up to the head, regardless of its  length.

1.3.2 Pore Filling by Capillary Condensation

In an empty pore system of a dried material, if the external vapor pressure is increased from very small pressures to saturation pressure , the vapor diffusing into the pores leads to sorption layer thicknesses at the pore walls corresponding to the associated “base isotherm.” For water vapor, this is the Eq. (1.50) resp. Figure 1.15a. These sorption liquid layers are also called liquid films.

Practical experience confirms that fine‐porous materials have moisture contents that correlate with the moisture content of the surrounding air. In the following, it will be shown again why this must be so.

As a reminder, the general gas equation for ideal gases is written   first.

(1.28)

There is  Gas pressure in [Pa], [Nm/(Kmol)] = General gas constant, and n = Number of moles of gas in

If  follows  = Molar volume = 24.46 [l/mol] under thermodynamic standard conditions 25 C and 101.3 [kPa] and 22.414 [l/mol] under standard conditions 0 C and 101.3 [kPa].

Equation (1.28) is also valid for the partial pressures of individual gas components in gas mixtures, for example also for the water vapor fraction in air.

Let the saturation concentration for water vapor in air be . At 20 C,  [g/. In contrast, air weighs about 1170 [g/] under standard conditions. The partial pressure for water vapor in air is also:

(1.29)

Herein, 0.462 is the specific gas constant (independent of boundary conditions) for water vapor in [Pa]. Let be the saturation partial pressure of water vapor in air. The temperature dependence is  C.

(1.30)

The vapor concentration is  in equilibrium with a (flat) water surface at a given temperature .

If the water surface is curved concavely, this leads to a higher relative concentration  with respect to the curved water surface . Expressed as relative concentration  is now, which is especially clear when imaging the inner surface of a sphere.

To maintain thermodynamic equilibrium, the liquid surface must absorb vapor molecules from the air in an effort  to reduce. In the initial plane state, the equilibrium between the phases of water and vapor in contact in confined space is as follows:

Since must be, it follows (at unchanged temperature and pressure) .

When there is a curvature of the fluid surface, there is (due to the relationship between surface curvature and pressure ) a change in pressure in the system. Here, first of all, the fluid pressure is meant.

On the part of the liquid, the change in chemical potential (or free enthalpy) to be compensated is compared to the initial state at a flat surface and at unchanged temperature, withmolar volume of the liquid or water and the corresponding saturation pressure is

(1.31)
(1.32)

The fluid pressure can be . This depends on whether the pressure generated in the fluid is negative or positive. In the case of a cylindrical pore with a progressing meniscus of fluid filling in Figure 1.10, the fluid pressure should be assumed to be negative as a tensile stress.

In Eq. (1.20), is the radius of the cylindrical pore. In a circular– capillary, the pressure‐transmitting meniscus has two main radii of curvature

(1.33)

Inserted into Eq. (1.20), the Young–Laplace relation for a circular–cylindrical pore follows.

(1.34)

In the plane initial state,  and thus  in Eq. (1.32). In the present case, and hence  is negative. From this follows for the fluid in the region of the meniscus

(1.35)

The enthalpy change of the liquid is followed by the corresponding reaction of the vapor phase V: From the saturation initial state  as the saturation pressure of the plane  surface.

(1.36)

where is the equilibrium saturation vapor pressure in the concave surface region.

From the requirement  follows with Eq. (1.35) and = mole volume of condensed liquid (18 [] or 18/ [/mol] for water), the Kelvin equation given by in the following formulation for concave surfaces:

(1.37)

Therein, is the external relative humidity. The summary term (constant for given T) corresponds to Eq. (1.68).

Equation (1.37) indicates up to which pore radius, at a given vapor partial pressure  a cylindrical pore is completely filled by condensation from the vapor entering the pore, denoted capillary condensation. Here, for cylindrical pores with constant cross‐section, it is assumed that a meniscus supposed to be stationary is already present due to an interrupted water supply at the base of the capillaries, or in the case of an equilibrium situation as indicated in Figure 1.10. When exposed to an external relative vapor pressure of at least , in the curvature region of this meniscus, the external vapor pressure becomes the saturation vapor pressure, with the consequence of condensation of the vapor present and pore filling.

If, for a given radius , the critical relative vapor pressure or the external relative humidity above which capillary condensation takes place is sought, the result is (Eq. (1.37) transformed):

(1.38)

For materials with pore size distributions “small to large,” the entire material is filled only up to the pore size according to Eq. (1.37) by the vapor diffusing in from the outside.

The Kelvin Eq. (1.37) or the resulting Eq. (1.38) are generally valid also for curved concave liquid surfaces characterized by different principal radii of curvature, as well as for pores with noncircular cross section.

Combining the relations (1.37) and (1.20), we immediately obtain the relation between and

(1.39)

1.3.2.1 Extent of Validity of the Kelvin Equation

An important question is up to which lower pore radius the Kelvin equation is valid. The applicability is considerably limited by the charge distribution and the relation of the pore radius to the molecule size of the fluid under consideration. Numerous authors have commented on this issue in the past. Matsuoka et al. [22] perform atomic force microscopy (AFM) studies on liquid films between curved muscovite‐mica surfaces at different relative humidities. They give a lower pore radius of  [nm] for (the polar) water, e.g.  [nm] for   cyclopentanes.

Fifteen years later, Kim et al. [23] remark, based on measurements with more advanced atomic force microscope (AFM) technology, that a much lower boundary pore radius  [nm] for water can be assumed. They measure in the AFM apparatus at the curved contact surface of the samples capillary condensation down to the mentioned radius and evaluate the results based on the accepted Kelvin–Tolman theory, which gives a correction to Laplace's equation for very small pore radii, compare  also [24].

They emphasize explicitly that this result (also based on AFM studies) refers to the classical Kelvin equation according to Eq. (1.37).

If one includes the relation of Eq. (1.79) according to the representation in Figure 1.22 or Section 1.4.5 and Figure 1.15a, then yields the corresponding real radius [nm], which could then be called as the lower physically detectable pore radius for the applicability of the Kelvin relation.

1.3.3 Saturation Vapor Pressure at the Surface of Convex Shapes

In contrast to concave liquid surfaces, for example, at a meniscus in cylindrical pores, the saturation vapor pressure at convex external surfaces is increased by , so that condensation of vapor occurs only at relative humidities .

It is then valid as liquefaction vapor pressure over an external surface (with the main curvature radii r). Instead of Eq. (1.38), the corresponding equation with reversed sign is then valid.

Since the relative vapor pressure is lower on such surfaces, faster evaporation or drying also takes place there.

In Section 1.4.6.2, the layer thicknesses of water molecules that can form by adsorption on concave or convex surfaces as a function of relative humidity are investigated.

1.3.4 Explanations of the Young–Laplace Equation for Stress on Curved Fluid Surfaces

The relation presented at the same time by Young and Laplace is as follows:

(1.40)

Equation (1.40) gives the mechanical relationship between the curvature of a nonplanar membrane‐like surface, given by the two principal radii of curvature and , and the stress acting in the membrane plane in [N/m] and the pressure  in [N/ acting on the membrane surface (orthogonal).

This pressure can act from a liquid or a gas. Since a “water‐membrane” is formed at the water surface, the surface can be treated with the Young Laplace relation (1.40). Since the membrane stress acts largely as a constant for water surfaces, Eq. (1.40) provides the relationship between the sum of inside and possibly outside orthogonal pressure components on the surface and the curvature of the  surface.

In the area of the concave meniscus in a cylindrical capillary pore the  water pressure acts on the water side as a two‐dimensional tensile stress, on the air side the air pressure (including vapor partial pressure). However, since the air pressure acts on the entire system, it is also present as a component on the “water side” of the meniscus and can therefore be disregarded (except in special cases).