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CHAPTER 3: ELECTRICAL MEASUREMENTS IN CONCRETE MATERIALS
This chapter provides a review of the history and application of electrical
measurements in concrete. The definition of electrical conductivity and the principles of
direct (DC) and alternating (AC) conductivity measurements are presented.
The
theoretical principles relating the conductivity of a porous media to its microstructural
properties and humidity condition is discussed.
3.1. A Brief Introduction to Electrical Measurements in Concrete
In concrete materials research, electrical measurements (EM) measure the opposition
of the material to flow of electricity. For example, one can apply a direct (DC) voltage to
the ends of a concrete sample, and measure the sample’s electrical resistance. Most early
researchers, however, recognized that DC measurements may not be accurate due to
polarization effects. For example, Hammond and Robson (1955) observed a change in
the measured resistance when they reversed the direction of electrical current. For this
reason, EM methods which use alternating current (AC) have become more applicable.
Hundreds of research studies have used electrical measurements to investigate
different phenomena in concrete materials. Christensen and coworkers (1994) used EM
to perform an in depth investigation on microstructural changes in hydrating cement
concrete. They observed an increase in the measured resistance with age of the sample
and concluded that this increase is mainly attributed to the consumption of the liquid
phase of the matrix and production of large volumes of solid products during hydration
process. They showed that EM can provide valuable information about concrete’s porous
27
microstructure. Also measurements can be related to factors such as water permeability
and ionic diffusivity of the matrix.
Electrical measurements have been used to monitor moisture transport in concrete.
McCarter and coworkers observed a change in the resistance value when a wet sample is
oven-dried (McCarter and Garvin 1989). Several researchers have studied water and
ionic penetration in concrete’s cover zone and used EM to develop and assess moisture
profiles in drying concrete (McCarter et al. 1996, McCarter 1996, McCarter and Watson
1997, Weiss et al. 1999, and Schieβl et al. 2000). EM has also been widely used to
measure corrosion potentials in reinforced concrete (examples include ASTM C876 and
the study by Schieβl and Breit 1995).
Electrical methods provide several advantages for accurate in-situ testing of materials.
These methods are non-invasive and non-destructive and do not necessitate water
removal from samples prior to testing. Yet, there are also some disadvantages involved.
It is still unclear how to differentiate between separate underlying factors responsible for
electrical conduction in concrete: Factors such as different ions dissolved in the pore
solution, their concentrations, water content, and geometry of the microstructure. Also a
widely accepted standard test procedure does not exist. Each person tends to develop
his/her own testing protocols, and as a consequence, it is difficult to compare results or
replicate tests from different laboratories (McCarter, 2002).
3.2. Resistivity and Conductivity
Ohm’s law in its simplest form relates a DC electrical current (I), passing through a
conductive sample, to the voltage applied to the sample’s ends; V = RI . In this equation
R is electrical resistance which is a function of a material property (so called resistivity),
geometry, and dimensions of the sample. Figure 3.1 shows a cylindrical specimen under
constant DC voltage. Measuring resistance of the samples with different dimensions
shows that electrical resistance is proportional to the sample’s length (L) and inverse to
the surface area (Equation 3.1):
28
R = mρ
L
A
(3.1)
when ρ is the material’s resistivity and m is a geometric factor which equals to “one”
(m=1) for a cylindrical sample with disc shape electrodes at the ends (Figure 3.1). As a
result, resistivity of the material can be determined by measuring the sample’s electrical
resistance and multiplying the measured value by a factor representing the dimensions
and geometry of the test setup (
ρ=
A
R
mL
or
ρ=
1
in Equation 3.2):
k
1
R
k
(3.2)
geometry factor
Conductivity (σ) is defined as the inverse of resistivity (Equation 3.3):
σ=
1
ρ
=
k
R
(3.3)
Conductivity (or resistivity) is a geometry free parameter that represents the
opposition of the material to flow of electricity. Conductivity of concrete, as will be
discussed later in this chapter, is a function of the microstructure’s geometry, moisture
content of the sample, and conductivity of the pore solution.
A
Fig. 3.1. A cylindrical sample under constant DC voltage can be used to see
how dimension and geometry influence the measured electrical resistance
29
The procedure of measuring the material’s conductivity is slightly different when
alternating (AC) currents are used. The equivalent form of Ohm’s law for AC currents is
V = ZI , in which Z is electrical impedance and is again a form of opposition to flow of
electricity. Electrical impedance is a complex number composed of real and imaginary
components; Z = Z ′ + jZ ′′. The real part of impedance ( Z ′ ) is called resistance, while
the imaginary part ( Z ′′ ) is called reactance.
In AC measurements, the sign and magnitude voltage is not constant. For example
the voltage can oscillate between positive and negative peaks in a sinusoidal manner with
constant frequency. The real and imaginary components of electrical impedance are both
functions of this frequency.
In fact, resistance and reactance of a sample can be
measured at different frequencies and plotted against each other. Such a graph is called
Nyquist plot (Figure 3.2a). Each point in this graph corresponds to a distinct frequency.
The impedance components ( Z ′ and Z ′′ ) can be combined to represent the total (or
absolute) impedance, Z , and the impedance argument (or phase angle), θ :
Z = ( Z ′) 2 + ( Z ′′) 2
(3.4)
⎛ Z ′′ 180 o ⎞
⎟⎟
φ = tan ⎜⎜ − ⋅
⎝ Z′ π ⎠
(3.5)
−1
The values of Z and θ for each point can be plotted against the voltage’s frequency to
get Bode plots (Figure 3.2b).
A typical Nyquist plot for concrete is shown in Figure 3.3. This graph is composed of
two half arcs, a larger arc, which is typically in the millihertz to kilohertz range and is
attributed to the passive oxide film on the surface of steel electrodes, and a smaller arc,
typically measured in the kilohertz to megahertz range and is attributed to the sample’s
bulk properties.
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-100000
Z''
increase in frequency
-50000
|Z|
θ
0
-50000
0
50000
100000
150000
Z'
(a)
106
|Z|
105
104
103
102
103
104
105
106
107
106
107
Frequency (Hz)
-100
theta
-75
-50
-25
0
102
103
104
105
Frequency (Hz)
(b)
Fig. 3.2. (a) Nyquist plot, and (b) Bode plots of electrical impedance
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Fig. 3.3. Typical Nyquist plot for hardened concrete (Christensen et al. 1994)
One of the most important parameters that can be obtained from the Nyquist plot is
the “bulk resistance” ( Rb ) which is the resistance ( Z ′ ) at the intersection of the two arcs.
At this point the reactance ( Z ′′ ) of the system is theoretically zero. The frequency
corresponds to this point is called the “cut-off” frequency. The measured bulk resistance
is converted to concrete’s conductivity using Equation 3.3.
In the past, single frequency AC measurements were made at a frequency in the order
of kilohertz to measure the bulk resistance. Christensen and coworkers (1994) showed
that that the cut-off frequency can vary over two orders of magnitude as a function of
hydration time, cement type, and w/c ratio.
Weiss et al. (1999) showed that this
frequency also varies as a function of the moisture content of concrete. As a result, in
order to accurately measure the bulk resistance, it is necessary to measure impedance
over a wide range of frequencies. Such a process is called impedance spectroscopy (IS).
Impedance spectroscopy can be performed by employing a gain-phase analyzer and a
personal computer for data acquisition.
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3.3. Conductivity of Cementitious Matrices
According to the Powers-Brownyard model (Taylor 1990), cement paste can be
envisioned as parallel layers of different components (i.e., capillary porosity, hydration
products, unhydrated cement, etc.) that extend from one side of a sample to the other
(Figure 3.4). The conductivity of a set of n parallel layers can be calculated using the
followings approach. The complete electrical resistance of the set of n layers (Rt) is
related to the resistance of each ith layer (as represented by Ri):
1
1
1
=
+"+
Rt R1
Rn
(3.6)
Rewriting this expression in terms of conductivity (si and Ai are respectively
conductivity and cross sectional area of the ith layer, and m is the geometry factor that is
the same for all layers) results in:
Ri =
σ i Ai
(3.7)
mL
σ t At = σ 1 A1 + " + σ n An
Direction
of current
(3.8)
1
Surface
area= A1
2
I
.
.
.
I
n
Length= L
Fig. 3.4. Powers-Brownyard model of envisioning cement paste as parallel layers
33
Multiplying both sides by the length (L) gives:
σ tVt = σ 1V1 + " + σ nVn
(3.9)
when Vi is the volume of layer i. When the volume fraction of layer i is defined as
follows:
φi =
Vi
Vt
(3.10)
Equation 3.9 can be rewritten in terms of volume fraction of layers:
σ t = σ 1φ1 + " + σ nφ n
(3.11)
If the layers were in series (Figure 3.5), Equation 3.11 would get the following form:
σt =
σ
σ1
+" + n
φ1
φn
(3.12)
Since in an actual concrete sample, the constituting components form a complex system
of serial and parallel layers, none of the Equations 3.11 and 3.12 can be exclusively used.
To consider this fact, Christensen (1993) used a modified equation in the following form:
σ t = σ 1φ1 β 1 + " + σ nφ n β n
(3.13)
Length= L1
I
1
Surface
area= A
2
…
n
Length= L
Fig. 3.5. Envisioning cement paste as serial layers
I
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In Equation 3.13 b is called the structure or the connectivity factor and represents the
tortuosity (twistedness) of the layers.
In a concrete matrix, conductivity of the saturated capillary porosity is several
degrees of magnitude higher than the conductivity of other components. As a result, the
total conductivity of the matrix is practically equal to the conductivity of the saturated
capillary porosity:
σ t ≈ σ oφ cap β
(3.14)
In this equation st is the overall conductivity of the matrix, so is the conductivity of pore
solution, fcap is the volume fraction of capillary porosity, and b is a connectivity factor.
Christensen (1993) showed that in a hydrating cement microstructure each of these
parameters (so, fcap, and b) is changing with time and as a result, an overall change can
be observed in the sample’s conductivity (Figure 3.6 (a) and (b)).
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0 .6
14
10
0 .5
0 .4
8
0 .3
6
0 .2
1
s (s/m)
10
fcap
so (s/m)
12
0.1
4
Series2
so (s/m)
fcap
Series1
2
0 .1
0
1
10
10 0
0
10 00
0.01
1
Age (hr)
10
100
1000
A ge (hr)
(a)
(b)
Fig. 3.6. Variations in (a) pore solution conductivity (so) and volume fraction of capillary
porosity (fcap) and (b) the complete sample’s conductivity (s) as a function of age
(Christensen 1993)
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While Christensen performed his tests on samples maintained at 100% relative
humidity, McCarter and coworkers (1995 and 1996) studied conductivity variations in
samples experiencing drying and rewetting. They used concrete slabs that were exposed
to drying from one surface for a period of 16 weeks. After this period, the samples were
ponded at the drying face with water or a 2 molar NaCl solution. Conductivity profiles of
slabs were obtained before ponding (sB) and after 24 hours of ponding with water (sW)
or NaCl solution (sN), (Figure 3.7).
As it’s shown in the figure a significant increase in conductivity was observed after
rewetting. The increase was more pronounced for NaCl ponded samples in comparison
to water ponded samples due to dissolution of sodium and chloride ions in the pore
solution and increasing its conductivity.
. The research team proposed to link the ratio between conductivities before ponding
and after water ponding (sB/sW) to the degree of saturation of concrete before rewetting
via the following equation:
σB
= (S r ) m
σW
(3.15-a)
when m is independent from porosity and moisture content of the sample and is in the
region of 1.5 to 3.0 for cement pastes and in the range of 1.2 to 2.5 for concretes and
mortars (McCarter et al. 1995). The research team also proposed to use the ratio between
the conductivity of the water ponded sample to the conductivity of the NaCl ponded
sample (sW/sN) to determine the concentration of NaCl dissolved in the pore solution:
σ W σ oW
=
σ N σ oN
(3.15-b)
Using Equation 3.15-b, assuming that soW is known; soN can be calculated and be used to
determine the concentration of chlorides in the pore solution of the NaCl ponded sample.
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100
Bef ore ponding (B)
Water ponding (W)
NaCl ponding (N)
80
s (s/km)
60
40
20
0
0
10
20
30
40
50
Depth from ponding s urface (m m )
Fig. 3.7. The significant increase in the conductivity of dried concrete slabs due to rewetting with
water and NaCl solution (McCarter et al. 1995)
The current research intends to reframe and extend the previous findings in order to
develop a method that accurately accounts for the humidity of concrete and to propose a
procedure that is able to determine the internal humidity (or moisture content) of concrete
based on the measured conductivity.
For this purpose the relationships used by
Christensen to relate conductivity to microstructural parameters were extended to also
account for internal humidity of the microstructure. The procedure is as follows.
Equation 3.14 is derived based on the assumption that all the capillary pores are
saturated with pore solution. In a real life scenario a portion of pores are empty of liquid
phase due to drying or self desiccation. In this case, only the portion of porosity that
contains pore solution is conductive. To reflect this fact, the term fcap in Equation 3.14
should be substituted with fcap-con which represents the volume fraction of capillary
porosity that is capable of conducting electrical current:
σ t = σ oφ cap −con β
(3.16)
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When the system experiences a change in the internal moisture content, each of the
three terms so, fcap-con, and b varies. For example, when a sample loses moisture, the
concentration of ions in the pore solution increases which in turn causes an increase in
the value of so. On the other hand, the volume of the liquid phase and consequently the
volume of conductive porosity decreases (i.e., decrease in fcap-con).
In addition,
connectivity of pores changes which is reflected in a change in the value of b.
Since simultaneous monitoring each of these parameters (so, fcap-con, and b) as a
function of the sample’s internal humidity is difficult, in this study it is proposed that the
conductivity changes caused by humidity variations are separated from the hydration
effects through implementing a humidity factor (fH) in Equation 3.16. The procedure is
to select a reference point at which the sample’s humidity is known and assign the
humidity factor of “one” to this reference point (i.e., fH =1 at reference point). Then, the
sample’s conductivity at other humidities is:
σ t = (σ oφ cap −con β ) ref ⋅ f H
(3.17)
when fH is a function of the difference between the current humidity and the reference
humidity:
f H = f H (∆H = H ref − H cur )
(3.18)
As an example a mature concrete can be considered in which changes in so, fcap-con,
and b due to hydration is negligible. If one selects the 90% relative humidity as the
reference point, conductivity of the sample at 85%RH can be obtained as follows:
σ RH =85% = (σ oφ cap −con β ) RH =90% ⋅ f H ( ∆RH = +5%)
(3.19)
Figure 3.8 illustrates the idea. The humidity factor for a given microstructure can be
obtained by measuring conductivity of the system at known humidities and establishing
the humidity factor function (Equation 3.18).
Then, when the internal humidity is
unknown, measuring the microstructure’s conductivity can be used along with the
obtained humidity factor function to back calculate the internal humidity (Equations 3.17
and 3.18).
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reference state
fH =1
less humidity
fH <1
more humidity
fH >1
Fig 3.8. The humidity factor represents the internal humidity of the system
This is a valid approach as long as the microstructure is invariant with time. A
cementitious microstructure experiences continuous hydrational developments and as a
result the porosity and connectivity of the pores are changing as a function of time. Also,
in a cementitious system the hydration rate and microstructural development are related
to the microstructure’s humidity. However, for simplifying the process of calibration, in
this study, the two following assumptions were made to enables application of the
aforementioned approach to cementitious microstructures:
1- The microstructural development is independent of the moisture history and is
only a function of age.
2- The humidity factor function for a given concrete mixture is independent of the
specimens’ age. This means that one can obtain the “fH” function at a certain age
and apply this function to the samples from the same mixture at different ages.
Using these assumptions, the conductivity of two similar concrete samples (from the
same mixture) cured at different relative humidities (for example one at 100%RH and the
other at 85%RH) at all ages can be related via the following expression:
σ RHcur = (σ oφ cap −con β ) RHref ⋅ f H ( ∆RH = RHref − RHcur )
(3.20)
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As a result the function fH (DRH) can be obtained using a set of calibration specimens and
be used to determine the humidity of similar concrete samples. The comprehensive
details about obtaining and implementing the humidity factor function are described in
Chapter 5.
3.4. Summary
This chapter provided a review of the history and application of electrical
measurements in concrete. The definition of electrical conductivity and the principles of
direct (DC) and alternating (AC) conductivity measurements were presented.
Conductivity of a cementitious microstructure is a function of the pore solution
conductivity (so), the volume fraction of conductive porosity (fcap-con) and the
connectivity of the pores (b). These parameters vary with the sample’s age and internal
humidity. It is proposed in this study to differentiate between the changes caused by
hydration and the changes caused by variations in internal humidity, through introducing
a humidity factor to Equation 3.16. This approach can be used to determine the humidity
of a sample based on its measured conductivity.
approach are described in Chapter 5.
The details of implementing this