Salt solution and crystals

Equations (3.58) and (3.59) are solved simultaneously, taking into account the geometry as well as initial and boundary conditions.

ions. In function of the salt concentration the density of the liquid phase ρ_l (kg/m³) will change (eq.(3.112)).

The index α is used in a summation to refer to all the ions of species from which the solid crystal forms. The gas phase is wet air, which is considered to be an ideal mixture of the ideal gasses dry air (a) and water vapour (v)

The crystallization reaction is given by

2 2

A^α H O A_ν^α_α H O

α α

ν ν ⎛ ⎞ν

+ ↔⎜ ⎟⋅

⎝ ⎠

∑

^{a w}

∏

^{w (3.60)}

As an example, the crystallization reaction for Na2S04 yields:

2

4 2 2 4 2

2Na⁻+S0⁻+10 H O↔Na SO 10 H O⋅ (3.61)

Depending on the conditions, the liquid phase may transform into water vapour and additionally salt crystals may appear according to equation (3.61). The concentration of crystallized salts is expressed by C_x in molcrystal/m³mat.

Analogous to the liquid degree of saturation the volume of crystallized salts can also be expressed in relation to the volume of open pores by S_x (m³crystal/m³open pores). Therefore Cx, the porosity φ₀ and the molar volume υ_C(m³crystal/molcrystal) of the salt are needed.

0 x

x C

S C υ

= φ ⋅ (3.62)

The crystal degree of saturation S_x can be related to the crystal pressure P_x, describing the equilibrium between salt crystals and the liquid phase. It is called the salt retention curve Sx

( )

Px which will be discussed later in detail.

In presence of salt the moisture retention curve also changes. There can be a horizontal shift (Fig. 2.12) due to changes in the capillary pressure and a vertical shift due to changes in density (eq. (3.112)).

Next to the definitions given in this paragraph we additionally refer to the general definition list concerning salt solutions.

3.3.1.1.2 Thermodynamical approach of crystallization (according to Coussy [COU-06])

Thermodynamic equilibrium between co-existing phases of the same substance requires the equality of their chemical potential, or

α α

μ^x=

∑

ν μ^a +ν μ^{w w (3.63)}

μ_v =μ_w (3.64)

The Gibbs-Duhem equation applied separately to the solid crystal phase and aqueous mixture gives

d d d

l Pl n_{α α} nw w

α

φ =

∑

μ + μ ^(3.65)

dP d

φ_{x x} =n_x μ_x (3.66)

The Gibbs-Duhem equation applied to pure water gives

d d *

φ_w P_l =n_w μ_w (3.67)

a) Derivation of the crystallization condition

Rewriting eqs. ((3.65)-(3.67)) in molar volumes

* * * *

d d d d d

l Pl xw w x_{α α} xw w x _{α α}

α α

υ = μ +

∑

μ = μ +

∑

ν μ ^(3.68)

d d

x Px x

υ = μ (3.69)

d d *

w Pl w

υ = μ (3.70)

Introducing ((3.63)) in ((3.69)) gives

d d d

x Px w w _{α α}

α

υ =ν μ +

∑

ν μ ^(3.71)

Eliminating _αd _α

α

∑

ν μ from eqs. ((3.68)) and ((3.71)) yields

*

* *

1 1

d _x ^L d _l ^{w w} d _w

x x x

P P x

x x

ν

υ μ

υ υ υ

⎛ ⎞

= +⎜ − ⎟

⎝ ⎠ ^(3.72)

The liquid water is assumed to act as an ideal solvent, so that its chemical potential μw

satisfies

* *

dμ_w=dμ_w+R Td lnx_w (3.73)

or using eqs. (3.70) and (DEF5)

*

* *

* *

d d d ln

d d d d

1

w w l w

w l w w l

w

P R T x

RT RT N

P x P x

x N x

μ υ

υ υ

= +

= + = −

−

(3.74)

Introducing (3.74) in (3.72) we get

*

* * *

d 1 d 1 d

1

w L

x l

x x

P P R T N x

x x N x

ν υ

υ υ

⎛ Δ ⎞ ⎛ ⎞

= +⎜⎜⎝ ⎟⎟⎠ + ⎜⎜⎝ − − ⎟⎟⎠ ^(3.75)

with the molar volume difference between the aqueous solution and the pure components given by:

( )

* 1 *

l l x x N w x w

υ υ υ ⎡ υ ⎤υ

Δ = − − −⎣ + ⎦ ^(3.76)

According to Coussy [COU-06], the molar volume difference can be written as:

*

l x x

υ δ υ

Δ = (3.77)

where δ is a dimensionless and constant dilatation coefficient, accounting for the volume change resulting from the crystallization process.

Introducing eq. (3.77) in eq. (3.75), we get

( )

¹_{* *} ^*

d 1 d d

1

w

x l

x

P P RT N x

x N x

δ ν

υ

⎛ ⎞

= + + ⎜⎜⎝ − − ⎟⎟⎠ ^(3.78)

( )( )

^*_* ^*_*

0 0

1 ln 1

1

N w

x atm l atm

x

R T x N x

P P P P

x N x

ν

δ υ

⎡⎛ ⎞ ⎛ − ⎞ ⎤

⎢ ⎥

− = + − + ⎢⎣⎜⎜⎝ ⎟ ⎜⎟ ⎜⎠ ⎝ − ⎟⎟⎠ ⎥⎦

(3.79)

The reference state is taken as the solution saturated state (S_l =1), with P_atm the atmospheric pressure and where the molar fraction is equal to the solubility x^*₀.

Because the water vapour is assumed to be an ideal gas forming an ideal mixture with dry air, the variation of the chemical potential is:

dμ_v =R Td lnΦ (3.80)

Using eqs. (3.64) and (3.74), we get

dμ_v =dμ_w=υ_wdP_l+R Td lnx*_w=R Td lnΦ (3.81) or

d d ln *

w l

w

P R T

υ = x^Φ (3.82)

Integrating from the reference state (solution is pure water at atmospheric pressure), we get the modified Kelvin liquid-vapour equilibrium condition

* *

ln ln

l atm 1

w w w

R T R T

P P

x N x

υ υ

Φ Φ

− = =

− ^(3.83)

Combining eqs. (3.79) and (3.83)

* *

* * *

0 0

ln 1

1 1

x w

w

N

x l

x

R T x N x

P P

N x x N x

δ υ ν

υ

υ

⎡ ⎤

⎛ ⎞ ⎛ ⎞

⎛ ⎞

⎢ Φ − ⎥

− = ⎢⎢⎜⎜⎝ − ⎟⎟⎠ ⎜⎜⎝ ⎟ ⎜⎟ ⎜⎠ ⎝ − ⎟⎟⎠ ⎥⎥

⎣ ⎦

(3.84)

During drying from the saturated reference state, large pores will first be invaded by wet air creating an air-solution interface with a curvature radius denoted r_φ. Meanwhile crystals may grow and invade these pores creating a solid crystal-solution interface with curvature radius r_x. The radii r_φ and r_x are governed by the Laplace equation

( )

2 _gl 2 _gl cos

g l

P P

r_φ r_φ

γ σ ⋅ θ

− = = (3.85)

( )

2 cos

2 xl xl

x l

x x

P P

r r

σ φ

γ ^⋅

− = = (3.86)

The mechanical equilibrium of the interfaces between two distinct constituents, liquid/air and crystal/liquid, involves surface energy effects σ_gl respectively σ_xl [COU-06]. Fig.

3.6 shows the schematical view of the interfaces crystal/liquid and liquid/air present in a salt loaded sample.

Fig. 3.6 Schematical view of the interfaces crystal/liquid and liquid/ air present in a salt loaded pore (based on [RIJ-04]).

Assuming equilibrium in a (cylindrical) pore, r=r_φ =r_x or using eqs. (3.83) and (3.84) 1

2 2

atm l x l

gl xl

P P P P

r γ γ

− −

= = (3.87)

Supersaturation Capillary condensation

Px

Px

liquid Crystal θ

pc

pc

liquid

r _Vapour φ

P_l

P_l r

2 σ cos θ

− =

^xl

x l

x

P P

r

2 cos

φ

σ φ

− =

^gl

g l

P P

r

pc

pc

liquid

φ _Vapour

P_l

P_l r

Px

Px

Crystal θ

where we assumed the gas phase to be at reference pressure P_g =P_atm. Introducing eqs. (3.85) and (3.86) into (3.87) we get

* *

* * *

0 0

1 1

2 ln 1 1

x w

w

N

x xl

R T x N x

r N x x N x

δ υ ν

υ

υ γ

⎡ ⎤

⎛ ⎞ ⎛ ⎞

⎛ ⎞

⎢ Φ − ⎥

= ⎢⎢⎜⎜⎝ − ⎟⎟⎠ ⎜⎜⎝ ⎟ ⎜⎟ ⎜⎠ ⎝ − ⎟⎟⎠ ⎥⎥

⎣ ⎦

(3.88)

*

1 ln

2 1

w gl

R T

r υ γ N x

= − Φ

− ^(3.89)

or considering the equality between eqs. (3.88) and (3.89), we get

* *

* * * *

0 0

ln 1 ln 0

1 1 1

C

w w

N

x xl w gl

x N x

N x x N x N x

δ υ ν

υ υ γ

υ γ

⎡ ⎤

⎛ ⎞ ⎛ ⎞

⎛ ⎞ ⎡ ⎤

⎢ Φ − ⎥ Φ

+ =

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎢ ⎥

⎢⎜⎝ − ⎟⎠ ⎜⎝ ⎟ ⎜⎠ ⎝ − ⎟⎠ ⎥ ⎢⎣ − ⎥⎦

⎢ ⎥

⎣ ⎦

(3.90)

which leads to the condition of stabilized crystallization in the general form

* *

* * *

0 0

1 1

1 1

x xl

w

w gl

N

x N x

N x x N x

υ γ δ ν

υ γ

⎛ ⎞

⎜ + ⎟

⎜ ⎟

⎝ ⎠⎛ ⎞ ⎛ ⎞

⎛ Φ ⎞ −

⎜ ⎟ ⎜ ⎟ =

⎜ ⎟

⎜ − ⎟ ⎜ ⎟ ⎜ − ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ^(3.91)

This relation gives the molar fraction x^* for crystallization at a relative humidity Φ. At Φ =1, x^*=x₀^* and with decreasing relative humidity the molar fraction, at which crystallization occurs, increases. This phenomenon is often referred to as

‘supersaturation’.

b) Mole balance of the crystallization reaction

Resulting from the stochiometry of the crystallization reaction, the mole balance associated with the crystallization

n n C

t t

α

ν^→α ^→

⎛ ⎞ ∂

∂ ⎜ ⎟= −

∂ ⎝ ⎠ ∂ ^(3.92)

or using eqs. (DEF2) and (DEF3)

* 0

C

w

n x n n

t t _α ^α

→ ⎡ ⎛ ⎞⎤

∂∂ +∂ ⎢∂ ⎢⎣ ⎜⎜⎝

∑

+ ⎟⎟⎠⎥⎥⎦= ^(3.93)

Using eqs. (DEF6) and (DEF7) and assuming the porosity almost does not change, we get

0

x l

x l

S x S

t υ t υ

⎛ ⎞ ⎛ ⎞

∂ ∂

+ =

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

∂ ⎝ ⎠ ∂ ⎝ ⎠ ^(3.94)

From eq. (3.76) the molar volume of the liquid phase υ_l in eq. (3.94) is

( )

^*

1 (1 )

l N vw x w x x

υ = −^⎡⎣ + ^⎤⎦υ + +δ υ ^(3.95)

or

* *(1 ) *

l xw w x x x vw w

υ = υ + +δ υ − υ (3.96)

The second and third term refer to the molar volume of the crystals when removing the contribution of the hydration water molecules, since this contribution is already accounted for by the first term.

Introducing eq. (3.96) into (3.87) we get

( )

* *

* *

0 0

1 1

ln 1

2

N w

x xl gl

R T x N x

r x N x

ν

υ γ δγ

⎡⎛ ⎞ ⎛ − ⎞ ⎤

⎢ ⎥

= + ⎢⎣⎜⎜⎝ ⎟ ⎜⎟ ⎜⎠ ⎝ − ⎟⎟⎠ ⎥⎦

(3.97)

This relation describes at which molar fraction x^* crystallization can occur in a pore with radius r. The value x^* is lower for larger pores than for fine pores, which means that during drying crystallization will first occur in the largest pores and than with decreasing relative humidity further in the finer pores.

Let the function S r( ) characterize the cumulative pore radius distribution. Then 1−S r( ) represents the cumulative pore volume fraction of pores having a pore entry radius greater than r.

Summarizing, Fig. 3.7 represents the crystallization conditions (3.91) and (3.97).

Remark that for the large values of r_max the value of x^* remains generally close to the value of the solubility x₀^*. This can be the case for many building materials.

0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.7 0.75 0.8 0.85 0.9 0.95

Φ(-)

x* (-)

0.001 0.01 0.1 1 10 100 1000

r (µm)

r

Fig. 3.7 Molar fraction x^* and pore access radius rrelated to the crystallizing pores plotted against the RH Φ [COU-06]

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