# Solid state phase transformation kinetics in Zr-base alloys

A. R. Massih  Quantum Technologies AB, Uppsala Science Park, Uppsala SE-75183,
Sweden  Lars O. Jernkvist  Quantum Technologies AB, Uppsala Science Park,
Uppsala SE-75183, Sweden

###### Abstract

We present a kinetic model for solid state phase transformation
($\alpha\rightleftharpoons\beta$) of common zirconium alloys used as fuel
cladding material in light water reactors. The model computes the relative
amounts of $\beta$ or $\alpha$ phase fraction as a function of time or
temperature in the alloys. The model accounts for the influence of excess
oxygen (due to oxidation) and hydrogen concentration (due to hydrogen pickup)
on phase transformation kinetics. Two variants of the model denoted by A and B
are presented. Model A is suitable for simulation of laboratory experiments in
which the heating/cooling rate is constant and is prescribed. Model B is more
generic. We compare the results of our model computations, for both A and B
variants, with accessible experimental data reported in the literature
covering heating/cooling rates of up to 100 K/s. The results of our comparison
are satisfactory, especially for model A. Our model B is intended for
implementation in fuel rod behavior computer programs, applicable to a reactor
accident situation, in which the Zr-based fuel cladding may go through
$\alpha\rightleftharpoons\beta$ phase transformation.

##  1 Introduction

Alloys based on group IVB metals of the periodic table, comprising titanium
(Ti), zirconium (Zr) and hafnium (Hf), are widely utilized in aerospace and
marine, nuclear energy, and biomedical products [1, 2, 3, 4, 5, 6]. They share
common chemical, physical and electronic attributes; in contrast to disparate
nuclear properties[6], e.g. Zr versus Hf. In particular, in the solid state
they are in two stable crystal structures: low temperature $\alpha$ (hexagonal
closed-packed) and high temperature $\beta$ (body-centered cubic).

Zirconium base alloys are materials of choice for fuel cladding in light water
reactors because of their low neutron absorption cross section and good
corrosion resistance. The alloys comprise Zircaloy-2 (1.5wt%Sn, 0.12wt%Fe,
0.05wt%Ni, 0.1wt%Cr, 0.12wt%O) for boiling water reactor, and Zircaloy-4
(1.5wt%Sn, 0.15wt%Fe, 0.1wt%Cr,0.12wt%O) or Zr1wt%Nb0.1wt%O (or Zr1NbO) among
others for pressurized water reactor.

Zirconium alloys in solid state undergo a phase transformation from $\alpha$
phase (space group P$6_{3}$/mmc) to $\beta$ phase (space group IM3m) at
ambient pressures [4]. Solid state phase equilibria of these alloys basically
comprise three phase domains, namely, the $\alpha$ phase, ($\alpha+\beta$)
coexisting phase, and the $\beta$ phase. The phase equilibria have been
evaluated for Zircaloy-4 in [7] and for the Zr-Nb-O system in [8]. The
starting temperature of the $\alpha\to(\alpha+\beta)$ transition for
Zircaloy-4 is about 1090 K and for the Zr1NbO alloy is around 1040 K [9].
Similarly, the start of the $(\alpha+\beta)\to\beta$ transition is about 1250
K for Zircaloy-4 and 1210 K for Zr1NbO [9]. The phase transition is first
order, meaning that a latent heat is produced during the transition, amounting
to $\Delta H\approx 4$ kJmol-1 [10].

The solid state phase transformation kinetics under nonisothermal conditions
for Zr-alloys has been modeled by means of nonspatial dependent (point
kinetics) differential equation method previously [11, 12, 13]. The method
described in [12, 13] posited that the rate of change for the volume fraction
of the transformed phase is a functional of the fraction of the transformed
phase in thermal equilibrium and the relaxation time for the phase transition
which are both functions of temperature. It fitted the available experimental
data on Zircaloy-4 and Zr1NbO well for constant heating/cooling rates of up to
100 K/s. However, parameters in this model, henceforth referred to as model A,
depend explicitly on the heating/cooling rate $q=dT/dt$, and its applicability
is limited to cases where $q$ is constant and known in advance. Model B,
introduced in the present paper, is free from this constraint.

The rest of the paper is organized as follows. In section 2, we briefly review
experimental data in the literature pertinent to our modeling. In section 3,
we describe the aforementioned model A and model B plus the modified
parameters that account for excess oxygen and hydrogen concentration in the
material. In section 4, we compare our model computation results with the
experimental data surveyed in section 2. In section 5, we analyze our model
and discuss its results and its applicability. A summary and conclusions are
presented in section 6.

##  2 Experimental data

In this section, we provide a brief survey of published experimental data on
solid-to-solid ($\alpha\rightleftharpoons\beta$) phase transformation of
zirconium alloys used widely in the nuclear industry. In particular, we go
over the data that we have utilized in our modeling and analysis. The alloys
discussed here are listed in table 1 with their nominal chemical compositions.
We first recount the phase equilibria properties in regard to the excess
oxygen concentration in the alloy and the hydrogen content of the alloy. Then
we peruse data on phase transformation of the alloys during time varying
temperature, namely continuous heating and cooling.

Alloy | Alloying elements, wt%
---|---
| Sn | Fe | Cr | Nb | O
Zircaloy-4 | 1.5 | 0.21 | 0.1 | … | 0.125
Zircaloy-4L | 1.3 | 0.21 | 0.1 | … | 0.125
Zr1NbO(M5) | … | … | … | 1.0 | 0.125
E110 | … | … | … | 1.0 | 0.05
ZIRLO | 0.94 | 0.09 | … | 0.9 | 0.120
Table 1:  Nominal chemical composition of some Zr-base alloys.

The phase boundary temperature versus excess oxygen concentration for
Zircaloy-4 was determined by Chung and Kassner [14]. These authors studied
Zircaloy/oxygen phase diagram by resistivity measurements and by
metallographic examination of equilibrated specimens. Chung and Kassner
determined the location of phase boundary of oxidized Zircaloy-4 specimens
annealed to form uniform oxygen distribution in the $\alpha$ phase region.
Similar specimens were brought out to equilibrium temperatures in $\beta$
phase and in ($\alpha+\beta$) mixed phase and then quenched to obtain the
corresponding phase boundaries. We have extracted phase boundary data, i.e.
$\alpha/(\alpha+\beta)$ and $\beta/(\alpha+\beta)$ transus, determined by
metallographic examination from table 2 of [14]. These data are displayed here
in Fig. 1 by circles.

The phase boundary temperature versus excess oxygen concentration for certain
ZrNbO alloys for $\beta/(\alpha+\beta)$ transus was determined by Hunt and
Niessen [15]. The authors investigated this transus for ZrNbO alloys
containing up to 5 wt% niobium by metallographic examination. The oxygenated
specimens were heated into the $\beta$ phase region then quenched in water.
Thereafter, they examined the microstructure of the samples for retained
$\alpha$ phase in the transformed $\beta$ matrix. Their data on Zr0.5wt%NbO
were used to our intent for Zr1NbO, which are shown in figure 1 by blue
diamonds. Because oxygen is an alloying element in Zircaloy-4 and Zr1NbO
(table 1), to obtain the excess oxygen, we have subtracted 0.12 w% from the
absolute values of the aforementioned data.

A few data points for low excess oxygen concentrations are available for
Zr1NbO close to the $\alpha/(\alpha+\beta)$ boundary [16]. These data are
shown in Fig. 1 as green diamonds. Toffolon et al. [16] determined these data
by calorimetry starting from a Zr1wt%Nb binary alloy. A more detailed
assessment of the phase equilibria of the Zr-Nb-O ternary system can be found
in [8].

Kinetics of solid state phase transformation in Zr-based alloys has been
investigated experimentally over the years. Early studies include references
[17, 18, 19], which used electrical resistivity versus temperature
measurements to determine the $\alpha/\beta$ phase boundaries in Zircaloy-2.
Corchia and Righini studied phase transformation [17] in the temperature range
of 1000 - 1400 K with heating rates in the range 200 - 1400 K/s and cooling
rates 10 - 60 K/s. They concluded that, in nonequilibrium conditions, the
kinetics of the phase transformation is strongly influenced by the
microstructure of the specimen and by inhomogeneous distribution of the fast
diffusing elements (Cr, Fe, Ni). Arias and coworkers employed much slower
heating/cooling ($\approx 0.05$ K/s) to establish the $\alpha/\beta$ phase
boundaries [19]. They also evaluated [18] the influence of tin, iron and
oxygen on the ($\alpha+\beta$) field. The aforementioned early studies provide
us with some experimental facts and are not in a form that could be used for
direct comparison with our modeling.

A systematic study of $\alpha/\beta$ phase transformation of stress-relieved
annealed (SRA) Zircaloy-4 and recrystallized (RX) Zr1NbO alloy was carried out
by Forgeron et al. [11]. The study includes phase transformation under quasi-
equilibrium conditions, i.e. very low heating/cooling rates, ranging from
0.002 to 0.33 K/s. The researchers used DSC, i.e. Differential Scanning
Calorimetry, to determine the fraction of $\beta$ (or $\alpha$) phase as a
function of temperature in the alloys. They also made direct observation of
microstructure by optical microscopy on samples annealed for a few hours at
different temperatures in the $(\alpha+\beta)$ two-phase domain. The
experimental uncertainty of the phase fraction measurement was estimated to be
around $\pm 0.05$ or less.

In order to study the kinetics of phase transformation, Forgeron et al. [11]
utilized dilatometric technique on small samples (12 mm in length) machined
from as-fabricated cladding tubes. The dilatometric measurements for
Zircaloy-4 samples included data with heating/cooling rates of 10 and 100 K/s
but only with heating rates 10 and 100 K/s for Zr1NbO alloy. As in their DSC
measurements, the uncertainty of the phase fraction measurement is reported to
be $\leq 0.05$ and $\pm 10$ K for measured temperatures.

In a subsequent study, Brachet et al. [20] investigated the influence of
hydrogen content in the alloy on $\alpha/\beta$ phase transformation
temperatures of SRA Zircaloy-4 and to some extent that of RX Zr1NbO. Most of
the data presented in [20] are for Zircaloy-4L (table 1) with some discussion
on the behavior of Zr1NbO alloy. The data include quasi-equilibrium DSC
measurements of the transus temperatures $T_{\alpha/\alpha+\beta}$ and
$T_{\alpha+\beta/\beta}$ as a function of hydrogen concentration in the range
of [H]$\approx 10$ wppm (as-fabricated value) to [H]$\approx 970$ wppm; see
Fig. 2.

Figure 1: Phase boundary temperatures versus metal excess oxygen concentration
for Zircaloy-4$\equiv$Zry4 (circles) [14], Zr-0.5%NbO (blue diamonds) [15],
and Zr-1%Nb (green diamonds) [16]. The lines are empirical correlations for
Zry4 and Zr1NbO alloy. Figure 2: Phase boundary temperatures versus hydrogen
concentration for Zircaloy-4L measured upon heating ($\triangle$
$\triangleright$) and cooling ($\triangledown$ $\triangleleft$) at $\approx
0.17$ K/s. The lines (between heating and cooling) are assumed to represent
the equilibrium temperatures; after Brachet et al. [20].

No continuous set of data on $\beta$ phase volume fraction versus temperature
at different concentrations of hydrogen in Zircaloy-4 is reported in [20].
However, computations with some verifications for $\beta$ phase volume
fraction versus temperature at hydrogen concentrations of $[H]\approx 0$, 500
and 1000 wppm are reported, which we have used to compare the results of our
model calculations in section 4. Regarding kinetic experiments, the
$\alpha/\beta$ phase transformation temperatures were measured by dilatometric
technique for Zircaloy-4L during heating at rates of 10 K/s and 100 K/s as a
function hydrogen content (in the range 10 - 970 wppm) at $\beta$ volume
fractions 0.1, 0.5 and 0.90 [20]. The measurement uncertainty for the volume
fraction and temperature is around $\pm 0.05$ and $\pm 10$ K, respectively.

Frechinet [21] investigated the $\alpha/\beta$ phase transformation of
Zircaloy-4 by electrical resistivity, dilatometry and DSC. His resistivity
measurements comprised heating/cooling ratings of $\pm 5$ K/s and $\pm 0.8$
K/s for which the relative amount of $\beta$ phase fraction as a function of
temperature was determined. Frechinet’s [21] dilatometric measurements were
carried out at a heating rate of 0.8 K/s as a kind of verification of his
resistivity measurements. Some discrepancy between the two methods can be
observed in regard to the evolution of $\beta$ phase. His DSC measurements
covered heating rates of 0.02, 0.08 and 0.33 K/s, which can be considered as
equilibrium or quasi-equilibrium conditions for phase transformation.

A comparative study of the phase transformation behavior of SRA Zircaloy-4 and
a RX Zr1NbO alloy at low heating rates is reported in [9]. Kaddour et al. [9]
used both resistivity and DSC to determine the evolution of $\beta$ phase
during heating as a function of temperature for Zircaloy-4, in which the
heating rates utilized were $\approx 0.8$ K/s (resistivity) and $\approx 0.3$
K/s (DSC). For Zr1NbO alloy, in addition to resistivity and DSC, they used
image analysis, i.e. optical microscopy to determine the volume fraction of
$\beta$ phase as a function of temperature. The results obtained by these
different techniques are in fair agreement. The onset of the $\alpha\to\beta$
transformation temperature determined by resistivity was about 1093 K for
Zircaloy-4 and 1043 K for Zr1NbO alloy.

More recent investigations of $\alpha\to\beta$ phase transformation kinetics
include Nguyen et al’s [22, 23] in-situ synchrotron X-ray diffraction (SXRD)
measurements of cold-worked (CW) Zircaloy-4L subjected to heating rates in the
range of 10 to 100 K/s, and electrical resistivity measurements of CW
Zircaloy-4L and RX Zircaloy-4L at 100 K/s [22, 24], supporting their SXRD
measurements. In a new study, Jailin et al. [25] using dilatometric technique
measured the $\alpha\to\beta$ phase transformation kinetics of SRA Zircaloy-4L
subjected to very high heating rates, i.e. from 50 to 2000 K/s. They found
that from thermal equilibrium to 500 K/s, the temperature at the start of
transformation shifts toward higher values with increasing heating rates.
However, above 500 K/s, the effect of heating rate saturates. Moreover, they
found that the temperature at which the transformation terminates, i.e. when
the alloys is in complete $\beta$ phase, is not much affected by heating rate.

Some phase transformation kinetic measurements on related Zr-Nb alloys have
also been reported in the literature [23, 26]. Nguyen [23] has reported in-
situ SXRD measurements of $\alpha$ to $\beta$ phase transformation of RX ZIRLO
(table 1) at a heating rate of 10 K/s, which indicates that the starting
$\alpha\to\beta$ phase transformation temperature is around 1040 K compared to
$\approx$ 1130 K for CW Zircaloy-4, measured at the same heating rate.
However, at about 1170 K the two alloys’ $\beta$ fractions become roughly
equal, i.e., $\approx 0.4$. Běnes et al. [26] examined the $\alpha/\beta$
transformation of a type of Zr1NbO alloy, namely the E110 alloy, with an
oxygen concentration of about 0.05 wt% (cf. table 1). They measured the
temperature of the transformation at low heating rates in the range of 0.05 to
0.33 K/s by a DSC technique. Their measurements indicate a weak shift
(increase) in the onset of $\alpha$ to ($\alpha+\beta$) transition temperature
with heating rate. The equilibrium onset transition temperature for this alloy
was put as $T_{\alpha/(\alpha+\beta)}=1020$ K.

##  3 Models

In this section, we describe the models that are used to evaluate the
experimental data discussed in the preceding section. The models comprise both
thermal statics (equilibrium) and kinetics (time varying) for
$\alpha\rightleftharpoons\beta$ phase transformation in Zr-alloys (table 1)
for the intend of computing the $\beta$ (or $\alpha$) phase volume fractions
as a function of time and temperature. Portions of the models were presented
in our earlier publications [12, 13, 27].

###  3.1 Thermal equilibrium

We assume that the volume fraction $\beta$-phase material under thermal
equilibrium, denoted as $y_{\mathrm{eq}}$, depends on the Zr-alloy under
consideration, temperature $T$, and excess oxygen and hydrogen concentrations
$\bm{x}=(x_{\mathrm{O}},x_{\mathrm{H}})$ in the metal [12, 13], namely

$y_{\mathrm{eq}}(T,\bm{x})=\frac{1}{2}\left[1+\tanh{\left(\frac{T-T_{\mathrm{mid}}(\bm{x})}{T_{\mathrm{span}}(\bm{x})}\right)}\right],$
(1)

where $T_{\mathrm{mid}}$ and $T_{\mathrm{span}}$ are material specific
functions of the cladding metal layer excess oxygen concentration (>1200
wppm), and hydrogen concentration (> 10 wppm). They are related to the middle
and the span of the mixed-phase temperature region at equilibrium. They are
calculated from the mixed phase lower and upper boundary temperatures,
$T_{\alpha}$ and $T_{\beta}$, through

$T_{\mathrm{mid}}(\bm{x})=\frac{T_{\alpha}(\bm{x})+T_{\beta}(\bm{x})}{2},$ (2)
$T_{\mathrm{span}}(\bm{x})=\frac{T_{\beta}(\bm{x})-T_{\mathrm{mid}}(\bm{x})}{2.3}.$
(3)

For the cladding materials considered (table 1), the phase boundary
temperatures $T_{\alpha}$ and $T_{\beta}$ can be written as

$T_{\alpha/\beta}(\bm{x})=T_{0}+A_{1}\>x_{\mathrm{O}}^{m}-A_{2}x_{\mathrm{H}},$
(4)

where $x_{\mathrm{O}}$ and $x_{\mathrm{H}}$ are the weight fraction of excess
oxygen and hydrogen in the cladding metal respectively, and $A_{1}$, $A_{2}$
and $m$ are constants given in table 2. Equation 4 treats $x_{\mathrm{O}}$ and
$x_{\mathrm{H}}$ as independent variables with no interaction between them.
Furthermore, the relation is appropriate for $x_{\mathrm{O}}<0.01$ and
$x_{\mathrm{H}}<0.001$. The phase boundary temperatures versus excess oxygen
concentration, calculated through Eq. (4), are compared with experimental data
for Zircaloy-4, Zr-0.5%NbO and Zr-1%Nb in Fig. 1.

Alloy | Phase boundary | $T_{0}$ | $A_{1}$ | $A_{2}$ | $m$
---|---|---|---|---|---
type | temperature | (K) | (K) | (K) | (-)
Zircaloy-4 | $T_{\alpha}$ | 1075 | 2.713$\times$105 |  $1.52\times$105 | 1.732
$T_{\beta}$ | 1250 | 3.138$\times$104 |  $2.20\times$104 | 0.929
Zr1NbO | $T_{\alpha}$ | 1016 | 2.713$\times$105 | —– | 1.732
$T_{\beta}$ | 1240 | 3.138$\times$104 | —– | 0.929
Table 2: Constants for calculating phase boundary temperatures of Zircaloy-4
and Zr1NbO cladding through Eq. (4).

###  3.2 Kinetics

Our approach to kinetics of the $\alpha/\beta$ phase transformation in
zirconium alloys is similar to the method used by Leblond and Devaux [28] for
diffusion controlled transformations in steels. We assume that there are only
two phases, $\alpha$ and $\beta$, and we denote the volume fraction of phase
$\beta$ by the variable $y$. We also posit unique phase transformations
$\alpha\rightleftharpoons(\alpha+\beta)\rightleftharpoons\beta$ occurring at
specified temperatures. Furthermore, we assume that the kinetics equation
furnishes the equilibrium proportion $y_{\mathrm{eq}}$ discussed in the
foregoing section.

The general form of the kinetics or the time derivative of $y$,
$\dot{y}=dy/dt$, depends only on $y$ and $T$, namely

$\dot{y}=f(T,y),$ (5)

where $f$ is a function of thermal history $T(t)$ and $y$ is the volume
fraction of $\beta$ phase, and in thermal equilibrium
$f(T,y=y_{\mathrm{eq}})=0$. In modeling the kinetics of the $\alpha/\beta$
transformation, two variants of the model are presented here, which we call
model A and model B. In model A, a key parameter entering the kinetic equation
is temperature rate of change ($q=dT/dt$), i.e. in principle
$\dot{y}=f(q,T,y)$, whereas in model B, $q$ does not appear explicitly in this
equation. In the forthcoming two subsections, we will delineate the equations
used for these two models.

#### Model A

We consider a simplest equation of type (5), by assuming
$y=y_{\mathrm{eq}}(T)$ is a steady-state solution at constant temperature,
i.e. $f(T,y=y_{\mathrm{eq}})=0$ for each $T$. We obtain $f$ by assuming $y$ is
not too far from thermal equilibrium; thereby we Taylor expand $f$ up to the
first power

$f(T,y)\approx f\big{(}T,y_{\mathrm{eq}}(T)\big{)}+\frac{\partial f}{\partial
y}\Big{\arrowvert}_{y=y_{\mathrm{eq}}}\big{[}y-y_{\mathrm{eq}}(T)\big{]}.$ (6)

Since the first term on the right-hand side is null, we can write Eq. (5) as

$\dot{y}=\frac{1}{\tau(T)}\Big{[}y_{\mathrm{eq}}(T)-y\Big{]},$ (7)

where $\tau$ is a relaxation time, which is temperature dependent and formally
is defined as

$\frac{1}{\tau(T)}=-\frac{\partial f}{\partial
y}\Big{\arrowvert}_{y=y_{\mathrm{eq}}}.$ (8)

$\tau(T)$ is a positive quantity and characterizes the duration of phase
transformation. Equation (7) assumes initial values $y(0)=0$ for heating and
$y(0)=1$ for cooling. For model A during heat-up, $\tau$ is calculated through
an empirical relation by [13]

$\tau=\frac{\exp\big{[}16650/T\big{]}}{60457+18129q},$ (9)

where $\tau$ is in s, $T$ is in kelvin, and $q=dT/dt$ is the heating rate in
K/s. Furthermore, $y_{\mathrm{eq}}(T)$ is given by Eq. (1).

During cooling ($\beta\to\alpha$), however, as the temperature is lowered
($q<0$), $\tau$ rapidly increases using this formula. And at sufficiently low
temperatures, it blows up thus rendering artificially $\dot{y}=0$ in Eq. (7).
Surely the formula (9) requires a constraint during cooling to get meaningful
results. Hence, in our computer program, we have modified Eq. (9) according to

$\tau_{cool}=\frac{\exp\big{[}16650/(T+T_{\mathrm{b}})\big{]}}{60457+18129|q|},$
(10)

that is, we have introduced a lower-bound temperature constraint
$T_{\mathrm{b}}$ to avoid unphysically slow phase transformation during rapid
cooling to low temperature. Choosing $T_{\mathrm{b}}=1000$ K, gives reasonable
results for experiments performed with cooling rates from $-5$ to $-100$ K/s;
other choices are also possible. In model A, the heating/cooling rate $q$ also
enters the expressions used for the phase boundary temperatures $T_{\alpha}$
and $T_{\beta}$ [13].

Model A may be used for an experiment or a fabrication process when the
heating/cooling rate $q$ is prescribed in advance. As will be shown in section
4.2, the model describes measured data satisfactorily, i.e. it is engineered
to do so.

#### Model B

As noted in the foregoing subsection, the relaxation time in model A embraces
the temperature rate $q$, thereby renders it ineffective for reactor fuel rod
simulations, where temperature history is not known a priori and may vary
nonuniformly and rapidly during transients. In order to remedy this
shortcoming, in model B we have extended equation (7) to a higher order and
modified the relaxation time $\tau$ in Model A to fit the measured data
suitably. Moreover, we have introduced a cutoff on $\tau$.

The basic equations for the volume fraction of the $\beta$ phase in model B
are

$\frac{dy}{dt}=\frac{1}{\tau(T)}\Big{(}(y_{\mathrm{eq}}(T)-y)\pm\frac{b}{y_{\mathrm{eq}}(T)}\big{(}y_{\mathrm{eq}}(T)-y\big{)}^{2}\Big{)},$
(11)

where the +sign is for heat-up ($y<y_{\mathrm{eq}}$) while the $-$sign is for
cool-down ($y>y_{\mathrm{eq}}$), and $y_{\mathrm{eq}}(T)$ is given by Eq. (1).
The relaxation time $\tau$ (in s) is given as

$\tau=B\exp(E/T),$ (12)

where $B$, $b$ and $E$ are constants; see table 3. During cooling, we have
introduced an upper cutoff limit on time constant $\tau\leq\tau^{\ast}$ to
avoid excessively long relaxation times, namely we take

$\tau=\min(\tau^{\ast},B\,e^{E/T}),$ (13)

where $\tau^{\ast}=6$ s is used in all our computations (table 3). It will be
shown in section 4.2 that model B reproduces measured data fairly
satisfactorily.

Alloy type | $E$ | $B$ | $b$ | $\tau^{\ast}$
---|---|---|---|---
| (K) | (s) | (-) | (s)
Zircaloy-4 | 54000 | $1.05\times 10^{-19}$ | 0.3 | 6.0
Zr1NbO | 54000 | $1.55\times 10^{-19}$ | 0.3 | 6.0
Table 3: Model B parameter constants, Eqs. (11)-(13).

##  4 Results: Computations vs. measurements

In this section, we use the models described in the foregoing section to
evaluate some of the data alluded in section 2. We compare model computations
of the volume fraction of $\beta$ phase with experimental data through a
number of figures.

###  4.1 Thermal equilibrium

In Fig. 3, the temperature dependence of the equilibrium volume fraction of
the $\beta$ phase Zircaloy-4 and Zr1NbO alloy using Eq. (1) is shown against
the experimental data (markers) extracted from ref. [11]. Solid lines are
calculated with the model parameters listed in table 2 whereas the dashed
lines are those from ref. [13]; here $x_{\mathrm{O}}=x_{\mathrm{H}}=0$. As can
be seen, the agreement is quite good. It is noted that the transition
temperatures for Zr1NbO are lower than for Zircaloy-4 and the volume fraction
of $\beta$ phase shifts to a lower temperature because niobium acts as a
$\beta$ phase stabilizer.

The influence of hydrogen content on phase transformation of Zircaloy-4 is
displayed in Fig. 4. It is seen that the results of our computations for
hydrogenated Zircaloy-4 are in fair agreement with experimental data and
analysis in [20]. However, the deviation for as-fabricated Zircaloy-4
(H$\approx 0$ wppm) is salient. This is because, our equilibrium model is
calibrated with the data in [11]; discs in the figure. Note that as hydrogen
concentration is increased, the $\alpha\to(\alpha+\beta)$ transition
temperature shifts to a lower value and the volume fraction of the favored
phase is moved to lower temperatures. We should note that, to our knowledge,
no similar measurements on the effect of excess oxygen on phase transformation
of zirconium alloys have been reported in the literature.

Figure 3: Equilibrium volume fraction of $\beta$ phase as a function of
temperature calculated (lines) vs. measured data (markers) [11] for Zircaloy-4
and Zr-1NbO alloy. Solid lines are calculated with the model parameters listed
in table 2 whereas the dashed lines are those from ref. [13]. Figure 4:
Equilibrium volume fraction of $\beta$ phase as a function of temperature
calculated for Zircaloy-4 with several concentrations of hydrogen. Solid lines
are calculated with model parameters listed in table 2 whereas the dashed
lines are those extracted from figure 12 of ref. [20], which are based on
analysis of experimental data. The disks show measured data for
$\text{H}\approx 0$ wppm from ref. [11].

###  4.2 Kinetics

#### Model A

The results of model A computations were presented earlier [13], however, with
some clarification included here that was missing in that reference; e.g.
usage of Eq. (10), and additional data. We perform our computations using this
model and parameter values as in [13] for the sake of a benchmark with model
B, which is the main contribution of our paper.

We evaluate some of the measurements discussed in section 2, namely data in
[11] with heating/cooling rates $q=dT/dt=\pm 10,\pm 100$ K/s and in [21] with
$q=dT/dt=\pm 5$ K/s. We solve Eq. (7) with initial conditions $y(0)=0$ on
heating and $y(0)=1$ on cooling together with relations (9)-(10) for the
relaxation times (with $T_{\mathrm{b}}=25|q|$) by using an explicit Runge-
Kutta method. The results for the fraction of the transformed phase for
Zircaloy-4 under heating/cooling are shown in Fig. 5. The markers denote
measured data while the lines are calculations according to model A. The
corresponding calculations and data for Zr1NbO [11] subject to heating rates
$q=10$ and $100$ K/s are shown in Fig. 6. It is seen that as $q$ is increased,
the transformed volume fraction shifts to higher/lower temperatures under
heating/cooling, respectively.

---
Figure 5: Volume fraction of $\beta$ phase as a function of temperature
calculated (lines) by using model A vs. measured data (markers) [11, 21] for
Zircaloy-4 at different heating/cooling rates (top/bottom) where $q=dT/dt$.
Figure 6: Volume fraction of $\beta$ phase as a function of temperature
calculated (lines) by using model A vs. measured data (markers) [11] for
Zr1NbO at heating rates $q=dT/dt=10$ and 100 K/s and at thermal equilibrium.

#### Model B

Here, we solve Eq. (11) with initial conditions $y(0)=0$ on heating and
$y(0)=1$ on cooling together with relations (12)-(13) for the relaxation time
by using an explicit Runge-Kutta method. The results of model B computations,
the volume fraction of $\beta$ phase versus temperature, against measured data
for Zircaloy-4 (markers) which were performed at thermal rates of $\pm
0.8(\approx\pm 1)\pm 5,\pm 10,\pm 100$ K/s and at thermal equilibrium (cf.
section 2) are shown in Fig. 7. As can be seen from this figure, the agreement
between calculations and measurements are satisfactory, except for the case of
heating at 100 K/s, at which computations underestimate the measurements at
temperatures $\approx 1230$ and $\approx 1280$ K. However, the trend in
$\alpha\to\beta$ phase transition is qualitatively featured. The corresponding
calculations and data for Zr1NbO [11] subject to heating rates of $+10$,
$+100$ K/s and at thermal equilibrium are plotted in Fig. 8; cf. Fig. 6. As
can be seen, the concordance between computations and measurements is fair for
this alloy.

Figure 7: Volume fraction of $\beta$ phase as a function of temperature
calculated (lines) by using model B vs. measured data (markers) for Zircaloy-4
at different heating/cooling rates (left/right panel) where $q=dT/dt$.
Measured data refs. are For00[11], Fre01[21], Bra02[20], and Ngu17[22]. Figure
8: Volume fraction of $\beta$ phase as a function of temperature calculated
(lines) using model B vs. measured data (markers) [11] for Zr1NbO at heating
rates $q=dT/dt=10$ and 100 K/s and at thermal equilibrium.

To illustrate the phase transformation kinetics expected for reactor fuel
cladding under a postulated loss-of-coolant accident (LOCA) in a pressurized
water reactor (PWR) [29], we apply model B to a cladding temperature history
measured in a test within the QUENCH-LOCA experimental program at the
Karlsruhe Institute of Technology, Germany. This program comprised seven out-
of-reactor LOCA-simulation tests on bundles with 21 electrically heated and
extensively instrumented fuel rod simulators with various Zr-base cladding
materials [30]. The rods in the considered test, QUENCH-L1, were clad with as-
received Zircaloy-4 [31]. More specifically, we consider the cladding
temperature history measured for rod 4 at an elevation 450 mm from the bottom
of the heated section (thermocouple TFS 4/8 [31]), which is shown in the left
panel of Fig. 9. The temperature history is typical for a postulated large-
break LOCA in a PWR: An initial heat-up phase, representing loss of primary
coolant water at full reactor power, is followed by cooling in steam with a
rate around 3-4 Ks-1. At $t$=207 s, the axial steam flow through the fuel
bundle is blocked by liquid water that reaches the lower (cold) part of the
bundle during re-flood of the reactor pressure vessel by the emergency core
cooling system. The fuel cladding temperature then rises to about 870 K until
the water front reaches the thermocouple position and the cladding is rapidly
quenched.

Figure 9: Left panel: Measured temperature history from the QUENCH-L1 LOCA
test considered in calculations. Right panel: Calculated $\beta$-fraction in
Zircaloy-4 cladding for the simulated temperature history, with and without
consideration of cladding high-temperature oxidation. The equilibrium curve
from Fig. 3 is included for comparison.

For illustration, this temperature history is simulated with model B. The
calculated results are shown in the right panel of Fig. 9 for two different
cases: For the ’no oxidation’ case, the effects of high-temperature oxidation
are neglected, keeping $x_{\mathrm{O}}$=$x_{\mathrm{H}}$=0 throughout the
simulated test. For the ’oxidation’ case, the evolution of $x_{\mathrm{O}}$
during the test is calculated through the correlation by Leistikow and Schanz
[32], while hydrogen pick-up during the test is neglected
($x_{\mathrm{H}}$=0). The calculated results in Fig. 9 suggest significant
kinetic effects on the cladding phase composition during a typical PWR large-
break LOCA. The calculated effects of cladding high-temperature oxidation are,
however, moderate. The calculated excess oxygen concentration in the cladding
metal is merely 280 wppm at end of the simulated test. High-temperature
(600–1200∘C) oxidation kinetics and hydrogen pickup of zirconium alloys,
important during LOCA conditions, have been extensively studied in the
literature over the years. Some recent studies include refs. [30, 33, 34].

Next, we consider the case of hydrogenated Zircaloy-4 and rely on experimental
data reported in [20]. As mentioned in section 2, no continuous set of data on
$\beta$ phase volume fraction versus temperature at different concentrations
of hydrogen in Zircaloy-4 have been reported in [20]. Brachet et al. [20],
besides equilibrium data, have presented data on heating at 10 K/s and 100 K/s
for $\beta$ phase volume fractions of 0.1, 0.5 and 0.9 for Zircaloy-4
containing hydrogen concentrations in the range $\approx$ 10 to 970 wppm. We
have extracted these data from their figure 8 by grouping the data to four
levels of hydrogen concentrations, namely, $\approx$ 12 wppm, 188 wppm, 500
wppm, and 920 wppm. The results are depicted as a contour plot in Fig. 10 for
the heating rate of 100 K. In this figure, the $\beta$-fraction values are
interpolated between the measured data points of 0.1, 0.5 and 0.9 for the sake
of contiguity. The corresponding computations with model B are presented in
Fig. 10. It is seen that measurements and computations are in fair agreement,
albeit the complexity of the phenomenon.

In order to recount the aforementioned experimental data on Zircaloy4+H, we
have extended the relaxation time relation of phase transformation, i.e. Eq.
(12), to account for hydrogen concentration as

$\tau=\frac{B\exp(E/T)}{1-0.0003\bar{x}_{\mathrm{H}}\big{(}1-\exp(5000/T)\big{)}}.$
(14)

Here, $\bar{x}_{\mathrm{H}}$ is hydrogen concentration in metal in wppm. This
modification is similar to that of extending the diffusion coefficient of
hydrogen in a metal in the presence of hydrogen traps [35]. Relation (14)
should give plausible results for hydrogen concentrations of up to 1000 wppm
in Zircaloy-4 in $T$-range $\approx 1000-1600$ K.

Figure 10: Contour plots of $\alpha/\beta$ phase transformation temperatures
in Zircaloy-4 as a function of hydrogen concentration. (a) Measured on heating
at 100 K/s, (b) calculated (model B) on heating at 100 K/s. Measurements are
based on the data reported in [20].

In order to get an overview of the influence of hydrogen on $\alpha/\beta$
phase transformation in Zircaloy, we have shown the results of our model B
computations in Fig. 11. The plots in this figure display the volume fraction
of $\beta$ phase as a function of temperature for several values of hydrogen
concentration in Zircaloy-4 at heating rates of 10 K/s and 100 K/s. It is seen
that as hydrogen content in the material is increased, the transition from
$\alpha$ to $\beta$ phase initiates at a lower temperature.

Figure 11: Volume fraction of $\beta$-phase as a function of temperature for
several values of hydrogen concentrations in Zircaloy-4 calculated by using
model B at heating rates: (a) 10 K/s and (b) 100 K/s.

As has been noted the deviation between model B computations and measurements
are most prominent for the case of Zircaloy-4 subjected to a heating rate of
100 K/s (cf. Fig. 7) and even, to some extend, at a cooling rate of $-100$
K/s, cf. Fig. 7. Thereupon, an uncertainty evaluation of measured data and
their deviation from the model calculation are worthy of estimation. As
alluded in section 2, the uncertainty in measured volume fraction of $\beta$
phase as a function of temperature is estimated to be about $\sigma=\pm 0.05$.
Moreover, there is an uncertainty on temperature measurements of around $\pm
10$ K. In Figs. 12 and 12, we have plotted the measured data for Zircaloy-4
for $q=dT/dt=\pm$10 K/s and (b) $q=dT/dt=\pm$100 K/s including uncertainties
(bars) of $\sigma=\pm 0.05$ on $\beta$ phase volume fraction and $\pm 10$ K on
temperature. We have also included in these figures the corresponding model B
computations for comparison.As can be seen the model B computations for the
case of $q=\pm$10 K/s are reasonably satisfactory. However, for the case of
$q=\pm$100 K/s, the model B deviates from measured $\beta$ phase fractions up
to around 0.15 in certain temperature ranges, Fig. 12. For Zr1NbO at heating
rates $q=+10$ and +100 K/s, Fig. 8, the data are reproduced fairly
satisfactory.

Figure 12: Volume fraction of $\beta$ phase as a function of temperature
calculated (lines) by using model B vs. measured data (markers) for
Zircaloy-4. The bars indicate $\pm$0.05 and $\pm 10$ K uncertainties on
measured $\beta$-fraction and temperature, respectively. (a) $q=dT/dt=\pm$10
K/s and (b) $q=\pm$100 K/s.

##  5 Discussion

The phase boundary temperatures versus excess oxygen concentration, calculated
through Eq. (4), were compared with experimental data for Zircaloy-4 and
Zr-0.5%NbO cladding in Fig. 1. The data were obtained from metallographic
analyses of cladding tube samples, quenched from different equilibrium
temperatures. We should note that $T_{\beta}$, calculated through Eq. (4), may
exceed the Zircaloy or Zr1NbO melting temperatures at high oxygen
concentrations. The melting (solidus) temperature of Zircaloy-4 increases from
about 2025 K at zero excess oxygen concentration to about 2320 K for
$x_{\mathrm{O}}\geq$ 0.04 [36].

Turning now to the thermal equilibrium relation (1), we stated this formula
without any theoretical justification or explanation, albeit that it captures
experimental data adequately. Let’s write the hyperbolic tangent part of this
equation (the first term 1/2 is just a constant shift on the ordinate) in a
simplified form

$y_{0}(T)=\frac{1}{2}\tanh{\left(\frac{T-T_{0}}{\sqrt{2}\xi}\right)},$ (15)

where $T_{0}\equiv T_{\mathrm{mid}}$ and $\xi\equiv
T_{\mathrm{span}}/\sqrt{2}$. Indeed, Eq. (15) is a stationary solution of the
Cahn-Hilliard equation of phase field theory in the temperature domain, namely

$\xi^{2}\frac{d^{2}y_{0}}{dT^{2}}=\frac{\delta f[y_{0}(T)]}{\delta y_{0}(T)},$
(16)

where $f(y)=y^{4}-y^{2}/2$ is a double-well function (potential) of $y$; see
e.g. [37, 38]. This can be verified either by direct substitution or by
integrating Eq. (16) with appropriate boundary conditions, e.g.
$y_{0}(T_{0})=0$ and $y_{0}^{\prime}(T_{0})=\sqrt{2}/(4\xi)$. Differentiation
of Eq. (15) with respect to $T$ gives

$\zeta(T)\equiv
y_{0}^{\prime}(T)=\frac{1}{2\sqrt{2}\xi}\mathrm{sech}^{2}{\left(\frac{T-T_{0}}{\sqrt{2}\xi}\right)}.$
(17)

It is seen that the parameter $\xi$ appears both in the argument of the sech-
function and as its coefficient in inverse. In phase field theory parlance,
$\xi$ is the _correlation length_ , here the temperature span of the
coexisting phase. It plays an important role in the analysis of phase
transitions. Figure 13 shows schematic plots of $y_{0}$ and $\zeta$ as a
function of temperature. It is seen that $\zeta$ is highly localized in
temperature with a maximum at the center of the bump, $T=T_{0}$, and falling
rapidly to zero for $|T-T_{0}|>\sqrt{2}\xi$.

Figure 13: Schematic plots of $\alpha/\beta$ phase transition in equilibrium.
Top: the hyperbolic tangent profile of $y_{0}$ corresponding to two coexisting
phases in the span of $\xi$. Bottom: The slope of the profile
$\zeta=dy_{0}/dT$ indicating the sharpness of the transition.

Experimental data indicate that the level of impurities in the material
affects the width of $\xi$. For example, as hydrogen concentration in
Zircaloy-4 is increased, so does $\xi$, namely at $\bar{x}_{\mathrm{H}}\approx
0$ wppm, $\xi\approx 27$ K, whereas at $\bar{x}_{\mathrm{H}}=500$ wppm,
$\xi\approx 37$ K; using relation (3) and data in table 2 for Zircaloy-4 with
$\xi\equiv T_{\mathrm{span}}/\sqrt{2}$. In regard to the influence of oxygen
on the Zr-O phase equilibrium, oxygen is an "$\alpha$ stabilizer". It expands
the $\alpha$ phase region of the phase diagram by formation of an interstitial
solid solution in zirconium [4]. That is, excess oxygen widens $\xi$. For
example, at an excess oxygen concentration of 1wt%, $\xi\approx 80$ K; see
relation (3) and data in table 2 for Zircaloy-4. We may also compare $\xi$ for
Zr1NbO vs. Zircaloy-4. Again using data in table 2, we see that for
$x_{\mathrm{H}}\approx 0$ wppm, $\xi_{\mathrm{Zr1NbO}}\approx 34$ K, i.e.
somewhat wider than that for Zircaloy-4. This is expected, since Nb is a
$\beta$ stabilizer, meaning that it widens the $(\alpha+\beta$) domain in Zr-
based alloys [4].

Regarding the issue of hydrogen in Zircaloy, we should say that in the studied
range of temperature and hydrogen concentration, cf. figures 4, 10, 11,
hydrogen is primarily in solid solution, i.e. not in zirconium hydride phases.
This can be checked from data on the temperature dependence of hydrogen
solubility limit in zirconium alloys [39, 40] and the ensuing phase diagram
[39]. It is, however, worth recalling that the temperature dependence of the
hydrogen solubility limits for hydride dissolution (during heat-up) and
precipitation (during cool-down) are different in Zircaloys [40]; see Fig. 14.

Figure 14: Solid solubility limits for hydrogen in Zircaloy versus inverse
temperature, i.e. hydride dissolution and precipitation. The Arrhenius plots
are based on measurements of Une and Ishimoto [40] in the temperature range of
about 320 to 900 K.

Turning now into the issues in kinetics, the $\alpha\leftrightharpoons\beta$
phase transformation in zirconium base alloys comprise the process of
nucleation and growth. For instance, on heating from $\alpha$ phase, the
transformation involves $\beta$-nucleation, $\beta$-growth and
$\beta$-completion and vice versa; i.e. $\alpha$-nucleation, $\alpha$-growth
and $\alpha$-completion. A commonly used macroscopic model for this process is
the Kolmogorov-Johnson-Mehl-Avrami [41, 42, 43, 44] or KJMA description [45,
46]. The KJMA model assumes that tiny spherical grains nucleate at a constant
rate per unit volume in the metastable phase and grow isotropically at
constant velocity once formed. As first formulated by Kolmogorov [41], the
volume fraction of the favored phase at time $t$ can be expressed as

$\phi(t)=1-\exp\Big{[}-\frac{\pi}{3}\Big{(}\frac{t}{\tau_{d}}\Big{)}^{d+1}\Big{]},$
(18)

where $d$ is the space dimension, $\tau_{d}=(Iv^{d})^{-1/(d+1)}$ is a
characteristic time, $I$ is the nucleation rate per unit volume with dimension
unit $[\mathrm{T^{-1}L^{-d}}]$ and $v$ is the interface velocity (dimension
[$\mathrm{LT^{-1}}$]). In differential form, it reads

$\frac{d\phi}{dt}=\frac{\pi}{3}(d+1)IR_{m}^{d}(1-\phi).$ (19)

where $R_{m}\equiv vt$ is the grain radius obeying an algebraic growth law.
The nucleation process in Kolmogorov’s theory is considered as homogeneous,
meaning that, it occurs uniformly throughout the metastable phase. In most
cases, however, nucleation is heterogeneous. That is, nuclei primarily form at
impurities and defects in the crystal, which are present before the
transformation started. A simple model in the spirit of KJMA has been
discussed in [47], which places nuclei randomly with density $\rho$ throughout
the solid. When the phase transformation is initiated, spherical grains form
at each nuclei and grow isotropically with velocity $v$. Both the KJMA model
and its simplified heterogeneous nucleation variant [47] are a mean-field type
theory. Meaning that, there is no spatial variation in $\phi$ through the
solid, i.e. $\nabla\phi=0$.

In [12], we made a detailed comparison between Eq. (18) and our model A. To
sum up that account, we note that under isothermal condition, by putting
$\phi=y/y_{\mathrm{eq}}$, we can write Eq. (7) in the form

$\frac{d\phi}{dt}=\frac{1}{\tau}\big{(}1-\phi\big{)}.$ (20)

Similarly, the model B kinetics, as described by Eq. (11), can be written as

$\frac{d\phi}{dt}=\frac{1}{\tau}\Big{(}\big{(}1-\phi\big{)}+b\big{(}1-\phi\big{)}^{2}\Big{)}$
(21)

Now a naive comparison of Eq. (19) with (20) yields

$\tau^{-1}=\frac{\pi}{3}(d+1)R_{m}^{d}I.$ (22)

That is, the relaxation time used in our kinetic model is inversely
proportional to the nucleation rate $I$ and $R_{m}^{d}$, cf. Eq. (12),
$R_{m}^{d}I\sim\exp[-E/T(t)]/B_{i}$, which its time dependence is merely
through temperature. In the material science literature, Eq. (18) is often
expressed in the simple form [48]

$\phi(t)=1-\exp[-Kt^{n}],$ (23)

where $K$ is a function of temperature designating the nucleation and growth
rates, and $n$ is a numerical exponent (Avrami exponent) whose value may vary
from $\sim 1$ to 4. According to [48] if there is no change in the nucleation
mechanism, $n$ is independent of temperature. So, as can be inferred from Eq.
(20), our model A corresponds to $n=1$ and $K=1/\tau$.

The behavior of the model B, as described by Eq. (21), is slightly more
complicated. Solution to Eq. (21) subject to $\phi(0)=0$ under isothermal
condition reads

$\phi(t)=1-\frac{e^{-t/\tau}}{1+b(1-e^{-t/\tau})},$ (24)

which is not in the form of the Avrami relation (23), except for $b=0$. For
the sake of illustration, we have plotted this solution in Fig. 15 for several
values of $b$. As can be seen from these plots, the behavior for $b=0$ and
$b=0.3$ is rather close. So, we expect even for $b=0.3$ the Avrami exponent
should be around one.

Figure 15: Plots of the relation (24) for several values of the model
parameter $b$. Model A corresponds to $b=0$, while for model B, we have chosen
$b=0.3$, cf. table 3.

It is worth mentioning here that the space dimension $d$ (or $n\equiv d+1$)
appearing in the foregoing equations depends on genre of nucleation and/or
growth. Avrami [44] attributes $n=1$ to 1-d growth (needles), $n=2$ to 2-d
growth (plates), $n=3$ to 3-d growth (spheres or polyhedra), and $n=4$ to
nucleation and 3-d growth.[49] Whereas Cahn, [50] in a subsequent theoretical
study of the Avrami model, relations of type (23), concludes that after site
saturation, the stage in a reaction at which all potential nucleation sites
are consumed, the reaction proceeds by growth alone. In Cahn’s analysis $n=1$
describes nucleation on grain surfaces, $n=2$ nucleation on grain edges and
$n=3$ nucleation on grain corners.[49]

We should note that our model B is tailored for fuel rod behavior computer
program application, e.g. the FRAPTRAN program [51, 52], where its parameters
are independent of thermal rate $dT/dt$, contrary to model A. Its performance
against experimental data is fairly good, considering the uncertainty in the
measured measured phase fraction and temperature, which is about $\pm 0.05$
and $\pm 10$ K, respectively. The model, despite some theoretical
justification, is basically empirically based with adjustable parameter
constants. Its results perhaps can be improved further by more tune-up of
those constants. The model should be applicable to thermal rating in the range
$dT/dt=-100$ to +100 K/s and hydrogen contents in the alloys up to around 1000
wppm; however, with more uncertainty toward higher thermal rates.

##  6 Summary and conclusions

In this paper we have introduced a new model (model B) for the kinetics of
$\alpha\rightleftharpoons\beta$ transformation in zirconium alloys used in the
nuclear industry as fuel cladding material. We first surveyed pertinent
experimental data for the alloys reported in the literature, section 2. They
constitute an empirical basis for our model. We then presented our model B
together with our previously published model A, the latter for the sake of
comparison and completeness, section 3. The main difference between models A
and B is that model B is formulated such that it can be applied to arbitrary
heating/cooling histories, while model A is restricted to cases with a
constant heating/cooling rate. Moreover, the kinetic equation for model B
extends to a higher order term in Taylor expansion about the equilibrium. Both
models are extended to account for the effect of excess oxygen and hydrogen
content in the alloy. Surplus in oxygen may occur under an accident situation
due to the high temperature metal-water reaction as the cladding temperature
raises say from roughly 650 K to 1470 K. The increase in the hydrogen content
of cladding from its as-fabricated value of around 10 wppm occurs during
normal reactor operation and is enhanced during an accident, again due to the
metal-water reaction. All the experiments which we have evaluated with our
models were performed on unirradiated materials. The effect of irradiation on
$\alpha\rightleftharpoons\beta$ is believed not to be significant, at least
for the rather low heating rates expected in light water reactor LOCA; cf.
Fig. 9. The thermodynamic stability of crystal structures is the dominant
factor for phase transformation. During $\alpha\to\beta$ phase transformation
as the cladding temperature raises, the irradiation-induced defects would
mostly be removed as the recrystallizing front is swept through the lattice
[53].

We compared the results of our model computations with experimental data as
regards the relative amounts of $\beta$ (or $\alpha$) phase fractions as a
function of temperature in the alloys, section 4. Our computations are mainly
on Zircaloy-4, including hydrogenated material, and to a limited degree on
Zr1NbO material because of dearth of available measurements for this alloy.
The agreement between calculations and measurements is fairly satisfactory
considering the complexity in the details and the measurement uncertainties.
Finally, in section 5 we made a theoretical analysis of the thermal
equilibrium expression used and qualified for the transformation. In that
section, we also briefly discussed the kinetics in regard to nucleation and
growth, and delineated the range of applicability of the putative model.

Our model B is basically empirically based, with adjustable parameter
constants. It is intended for implementation in fuel rod behavior computer
programs and reproduces available experimental data fairly satisfactory.

## Acknowledgements

We are thankful to Tatiana Taurines for pointing out the shortcomings in an
earlier version of model B and providing us ref. [21] and Jean-Christophe
Brachet for helpful communications regarding measurements in ref. [11]. The
work was partially supported by the Swedish Radiation Safety Authority (SSM)
under the contract number SSM2018-4296.

## Author contributions statement

A.R.M. and L.O.J. contributed equally to the content of this work. All authors
reviewed the manuscript.

## Additional information

Competing financial interests: The authors declare no competing financial
interests.

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# On uniform Mazur intersection property

Pradipta Bandyopadhyay  Stat–Math Division, Indian Statistical Institute, 203,
B. T. Road, Kolkata 700108, India<EMAIL_ADDRESS>,  Jadav Ganesh
Dept of Mathematics, SLABS, SRM University, Neerukonda, Mangalagiri Mandal,
Guntur Dist (AP), India-522502<EMAIL_ADDRESS>and  Deepak Gothwal
Stat–Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata
700108, India<EMAIL_ADDRESS>Dedicated to Professor Bor-Luh
(Peter) Lin on his 85th birthday!

###### Abstract.

In this paper, we show that a Banach space $X$ has the Uniform Mazur
Intersection Property (UMIP) if and only if every $f\in S(X^{*})$ is uniformly
w*-semidenting point of $B(X^{*})$. We also prove analogous results for
uniform w*-MIP.

###### Key words and phrases:

Mazur Intersection Property, w*-(semi)denting points.

###### 2010 Mathematics Subject Classification:

46B20 _To appear Studia Math._

##  1\. Introduction

The Mazur Intersection Property (MIP) — every closed bounded convex set is the
intersection of closed balls containing it — is an extremely well studied
property in Banach space theory. MathSciNet search with keywords “Mazur(’s)
Intersection Property” returns 53 hits. The paper [8] is an excellent survey
of the MIP, its generalisations and variants. In particular, one should
mention [7], where a complete characterisation was obtained, most well-known
one stating that the w*-denting points of $B(X^{*})$ are norm dense in
$S(X^{*})$. They also considered the property in dual spaces that every
w*-compact convex set is the intersection of balls (w*-MIP). In [4], Chen and
Lin introduced the notion of w*-semidenting points and showed that a Banach
space $X$ has the MIP if and only if every $f\in S(X^{*})$ is a w*-semidenting
point of $B(X^{*})$. Among subsequent papers, we must mention [6].

A much less studied uniform version of the MIP (UMIP or UI) was introduced by
Whitfield and Zizler [9]. MathSciNet finds only one citation to [9], namely,
[8], in the last 30+ years. Characterisations similar to [7] were also
obtained, but an analogue of the w*-denting point characterisation was
missing, which perhaps is a reason for its being less pursued.

In this paper, we show that a Banach space $X$ has the UMIP if and only if
every $f\in S(X^{*})$ is uniformly w*-semidenting point of $B(X^{*})$, thus
filling a long felt gap. In the process, we also present simpler proofs of
some characterisations in [9]. We also introduce a w*-version of UMIP in the
spirit of [7] and obtain similar characterisations.

We hope that our result will also be a small step towards answering the long
standing open question whether the UMIP implies that the space is super-
reflexive.

##  2\. Characterisation of UMIP

We consider real Banach spaces only. Let $X$ be a Banach space. For $x\in X$
and $r>0$, we denote by $B(x,r)$ _the open ball_ $\\{y\in X:\|x-y\|<r\\}$ and
by $B[x,r]$ _the closed ball_ $\\{y\in X:\|x-y\|\leq r\\}$. We denote by
$B(X)$ the _closed unit ball_ $\\{x\in X:\|x\|\leq 1\\}$ and by $S(X)$ the
_unit sphere_ $\\{x\in X:\|x\|=1\\}$.

For $x\in S(X)$, we denote by $D(x)$ the set $\\{f\in S(X^{*}):f(x)=1\\}$. Any
selection of $D$ is called a support mapping.

For a bounded set $C\subseteq X$ and $x\in X$, let

  * $(a)$

$d(x,C)=\inf\\{\|x-z\|:z\in C\\}$, the distance function; and

  * $(b)$

$diam(C)=\sup\\{\|x-y\|:x,y\in C\\}$, the diameter of $C$.

######  Definition 2.1.

[9] We say that a Banach space $X$ has the Uniform Mazur Intersection Property
(UMIP) if for every $\varepsilon>0$ and $M(\varepsilon)\geq 2$, there is
$K(\varepsilon)>0$ such that whenever a closed convex set $C\subseteq X$ and a
point $p\in X$ are such that $diam(C)\leq M(\varepsilon)$ and
$d(p,C)\geq\varepsilon$, there is a closed ball $B\subseteq X$ of radius $\leq
K(\varepsilon)$ such that $C\subseteq B$ and $d(p,B)\geq\varepsilon/2$.

######  Remark 2.2.

In the original definition in [9], $M(\varepsilon)=1/\varepsilon$.

######  Definition 2.3.

A _slice_ of $B(X)$ determined by $f\in S(X^{*})$ is a set of the form

$S(B(X),f,\delta):=\\{x\in B(X):f(x)>1-\delta\\}$

for some $0<\delta<1$. For $x\in S(X)$, $S(B(X^{*}),x,\delta)$ is called a
w*-slice of $B(X^{*})$.

We say that $x\in S(X)$ is a _denting point_ of $B(X)$ if for every
$\varepsilon>0$, $x$ is contained in a slice of $B(X)$ of diameter less than
$\varepsilon$.

A w*-denting point of $B(X^{*})$ is defined similarly.

For w*-semidenting points, we use an equivalent definition.

######  Definition 2.4.

$f\in S(X^{*})$ is said to be a _w*-semidenting point_ of $B(X^{*})$ if for
every $\varepsilon>0$, there exists a w*-slice $S(B(X^{*}),x,\delta)\subseteq
B(f,\varepsilon)$.

A semidenting point of $B(X)$ can be defined similarly.

We begin with a variant of Phelps’ Parallel Hyperplane Lemma [6, Lemma 2.1].
We include the well-known proof as we will use a step of the proof later.

######  Lemma 2.5.

For a normed linear space $X$, if for $f,g\in S(X^{*})$ and $\varepsilon>0$,
$\\{x\in B(X):f(x)>\varepsilon\\}\subseteq\\{x\in X:g(x)>0\\}$, then
$\|f-g\|<2\varepsilon$.

###### Proof.

By the given condition, if $x\in B(X)$ and $g(x)=0$, then
$|f(x)|\leq\varepsilon$. That is $\|f|_{\ker(g)}\|\leq\varepsilon$. By Hahn-
Banach Theorem, there exists $h\in X^{*}$ such that $\|h\|\leq\varepsilon$ and
$h\equiv f$ on $\ker(g)$. It follows that $f-h=tg$ for some $t\in\mathbb{R}$.
Then $\|f-tg\|=\|h\|\leq\varepsilon$. Now, if $y\in\\{x\in
B(X):f(x)>\varepsilon\\}$, then $g(y)>0$ and

$\varepsilon<f(y)\leq(f-tg)(y)+tg(y)\leq\|f-tg\|+tg(y)\leq\varepsilon+tg(y).$

It follows that $t>0$. And

$|1-t|\leq\big{|}\|f\|-\|tg\|\big{|}\leq\|f-tg\|\leq\varepsilon.$

Thus,

$\|f-g\|\leq\|f-tg\|+|1-t|\leq\varepsilon+\varepsilon=2\varepsilon.$

∎

The theorem below is equivalent to [5, Theorem 2.1] with a much simpler proof.

######  Theorem 2.6.

Let $X$ be a Banach space. Let $A\subseteq X^{**}$ be a bounded set. Then
there exists a closed ball $B^{**}\subseteq X^{**}$ with centre in $X$ such
that $A\subseteq B^{**}$ and $0\notin B^{**}$ if and only if $d(0,A)>0$ and
there is a w*-slice of $B(X^{*})$ contained in

$\\{f\in B(X^{*}):x^{**}(f)>0\mbox{ for all }x^{**}\in A\\}.$

###### Proof.

Let $A\subseteq B^{**}[x_{0},r]$ and $0\notin B^{**}[x_{0},r]$. Then
$\|x_{0}\|>r$. Clearly, $d(0,A)\geq\|x_{0}\|-r>0$.

Let $S=\\{f\in B(X^{*}):f(x_{0})>r\\}$. Then $S$ is a w*-slice of $B(X^{*})$.
And if $g\in S$, then for any $x^{**}\in A$,

$g(x_{0}-x^{**})\leq\|x_{0}-x^{**}\|\leq r$

and hence,

$x^{**}(g)\geq g(x_{0})-r>0.$

Thus,

$S\subseteq\\{f\in B(X^{*}):x^{**}(f)>0\mbox{ for all }x^{**}\in A\\}.$

Conversely, let $d=d(0,A)>0$ and let $x_{0}\in S_{X}$ and $0<\varepsilon<1$ be
such that

$\\{f\in B(X^{*}):f(x_{0})>\varepsilon\\}\subseteq\\{f\in
B(X^{*}):x^{**}(f)>0\mbox{ for all }x^{**}\in A\\}.$

Let $M=\sup\\{\|x^{**}\|:x^{**}\in A\\}$. By the proof of Lemma 2.5, for all
$x^{**}\in A$, there exists $t\in\mathbb{R}$ such that $1-\varepsilon\leq
t\leq 1+\varepsilon$ and

$\left\|\frac{tx^{**}}{\|x^{**}\|}-x_{0}\right\|\leq\varepsilon.$

Then for $\lambda\geq M/(1-\varepsilon)$,

$\displaystyle\|x^{**}-\lambda x_{0}\|$ $\displaystyle\leq$
$\displaystyle\left\|x^{**}-\frac{\|x^{**}\|}{t}x_{0}\right\|+\left|\frac{\|x^{**}\|}{t}-\lambda\right|\leq\frac{\varepsilon\|x^{**}\|}{t}+\lambda-\frac{\|x^{**}\|}{t}$
$\displaystyle=$
$\displaystyle\lambda-\frac{\|x^{**}\|}{t}(1-\varepsilon)\leq\lambda-\frac{d(1-\varepsilon)}{1+\varepsilon}.$

Therefore,

$A\subseteq B^{**}\left[\lambda
x_{0},\lambda-\frac{d(1-\varepsilon)}{1+\varepsilon}\right]\mbox{ and clearly,
}0\notin B^{**}\left[\lambda
x_{0},\lambda-\frac{d(1-\varepsilon)}{1+\varepsilon}\right].$

∎

######  Remark 2.7.

From Theorem 2.6, it can be shown that if every $f\in S(X^{*})$ is a
w*-semidenting point of $B(X^{*})$ then $X$ has the MIP. See Theorem 2.11
$(b)\Rightarrow(a)$ below.

######  Definition 2.8.

For $\varepsilon,\delta>0$ and $x\in S(X)$, denote by

$\displaystyle d_{1}(x,\delta)$ $\displaystyle=$
$\displaystyle\sup\limits_{0<\lambda<\delta,\ y\in B(X)}\frac{\|{x+\lambda
y}\|+\|{x-\lambda y}\|-2}{\lambda},$ $\displaystyle d_{2}(x,\delta)$
$\displaystyle=$ $\displaystyle diam(S(B(X^{*}),x,\delta)),$ $\displaystyle
d_{3}(x,\delta)$ $\displaystyle=$ $\displaystyle diam(D(S(X)\cap
B(x,\delta))).$

######  Definition 2.9.

[9] For $\varepsilon,\delta>0$, define

$M_{\varepsilon,\delta}(X)=\\{x\in
S(X):\sup_{0<\|y\|<\delta}\frac{\|{x+y}\|+\|{x-y}\|-2}{\|y\|}<\varepsilon\\}.$

In other words, $M_{\varepsilon,\delta}(X)=\\{x\in
S(X):d_{1}(x,\delta)<\varepsilon\\}$.

The lemma below is quantitatively more precise than [9, Lemma 1]. It follows
from [1, Lemma 2.1]. We include the details for completeness.

######  Lemma 2.10.

For any $\alpha,\delta>0$, we have

  * $($i$)$

$\displaystyle d_{2}(x,\alpha)\leq d_{1}(x,\delta)+\frac{2\alpha}{\delta}$

  * $($ii$)$

$d_{3}(x,\delta)\leq d_{2}(x,\delta)$

  * $($iii$)$

$d_{1}(x,\delta)\leq d_{3}(x,2{\delta})$.

###### Proof.

$(i)$. Fix $\alpha$, $\delta>0$. Let $d_{2}=d_{2}(x,\alpha)$. For each $n\geq
1$, we can choose $f_{n}$, $g_{n}\in S(B(X^{*}),x,\alpha)$ such that
$\|f_{n}-g_{n}\|>d_{2}-\frac{1}{n}$. Choose $y_{n}\in B(X)$ such that
$(f_{n}-g_{n})(y_{n})>d_{2}-\frac{1}{n}$. Then

$\frac{\|x+\delta y_{n}\|+\|x-\delta
y_{n}\|-2}{\delta}\geq\frac{f_{n}(x+\delta y_{n})+g_{n}(x-\delta
y_{n})-2}{\delta}\geq d_{2}-\frac{1}{n}-\frac{2\alpha}{\delta}.$

Thus, $d_{1}(x,\delta)\geq d_{2}-\frac{2\alpha}{\delta}$.

$(ii)$. Let $y\in S(X)\cap B(x,\delta)$ and $f_{y}\in D(y)$. Then,

$0\leq 1-f_{y}(x)=f_{y}(y-x)\leq\|y-x\|<\delta.$

Thus, $f_{y}\in S(B(X^{*}),x,\delta)$. Therefore, $D(S(X)\cap
B(x,\delta))\subseteq S(B(X^{*}),x,\delta)$.

$(iii)$. Let $\lambda\leq\delta$. Observe that for any $y\in B(X)$,

$\displaystyle\left\|\frac{x\pm\lambda y}{\|x\pm\lambda y\|}-x\right\|$
$\displaystyle\leq$ $\displaystyle\left\|\frac{x\pm\lambda y}{\|x\pm\lambda
y\|}-(x\pm\lambda y)\right\|+\lambda=\big{|}1-\|x\pm\lambda
y\|\big{|}+\lambda$ $\displaystyle=$ $\displaystyle\big{|}\|x\|-\|x\pm\lambda
y\|\big{|}+\lambda\leq 2\lambda.$

Let $\displaystyle f\in D\left(\frac{x+\lambda y}{\|x+\lambda y\|}\right)$ and
$\displaystyle g\in D\left(\frac{x-\lambda y}{\|x-\lambda y\|}\right)$. Then

$\displaystyle\frac{\|x+\lambda y\|+\|x-\lambda y\|-2}{\lambda}$
$\displaystyle=$ $\displaystyle\frac{f(x+\lambda y)+g(x-\lambda
y)-2}{\lambda}$ $\displaystyle\leq$ $\displaystyle(f-g)(y)\leq\|f-g\|.$

∎

We now come to our main theorem. The conditions $(c)$–$(e)$ are reformulations
of $(ii)$–$(iv)$ of [9, Theorem 1] in the language of [7, Theorem 2.1]. We
give a self-contained proof of $(a)\Leftrightarrow(b)$, our main result. The
proofs of $(c)\Rightarrow(e)$ and $(c)\Leftrightarrow(d)$ are essentially from
[9, Theorem 1], which we include for the sake of completeness and we will need
them in the next section. For the rest, our arguments are either new or
simpler.

######  Theorem 2.11.

For a Banach space $X$, the following are equivalent :

  * $(a)$

$X$ has the UMIP.

  * $(b)$

Every $f\in S(X^{*})$ is uniformly w*-semidenting, i.e., given
$\varepsilon>0$, there exists $0<\delta<1$ such that for any $f\in S(X^{*})$,
there exists $x\in S(X)$ such that

$S(B(X^{*}),x,\delta)\subseteq B(f,\varepsilon).$

  * $(c)$

The duality map is uniformly quasicontinuous, i.e., given $\varepsilon>0$,
there exists $\delta>0$ such that for any $f\in S(X^{*})$, there exists $x\in
S(X)$ such that $D(S(X)\cap B(x,\delta))\subseteq B(f,\varepsilon)$.

  * $(d)$

Given $\varepsilon>0$ there exists $\delta>0$ such that every support mapping
maps a $\delta$-net in $S(X)$ to an $\varepsilon$-net in $S(X^{*})$.

  * $(e)$

Given $\varepsilon>0$ there exists $\delta>0$ such that for every $f\in
S(X^{*})$, there exists $x\in M_{\varepsilon,\delta}(X)$ such that
$D(x)\subseteq B(f,\varepsilon)$.

###### Proof.

$(a)\Rightarrow(b)$. We use an idea from [6, Lemma 2.2$(ii)$].

Let $0<\varepsilon<1$ and $M(\varepsilon)\geq 2$ be given. Choose
$K=K(\varepsilon)$ as given by $(a)$ for $\varepsilon/3$. We may assume
w.l.o.g. that $K\geq 1$.

Let $f\in S(X^{*})$. Consider $C:=\\{x\in B(X):f(x)\geq\varepsilon/3\\}$.
Then, $d(0,C)\geq\varepsilon/3$. Also, $diam(C)\leq 2\leq M(\varepsilon)$.

So, we have $B=B[x_{0},r]$ containing $C$, $r\leq K$ and
$d(0,B)\geq\varepsilon/6$.

Since $d(0,B[x_{0},r])\geq\varepsilon/6$, $\|x_{0}\|\geq
r+\varepsilon/6>r+\varepsilon/9$.

Claim : $S=:S(B(X^{*}),x_{0}/\|x_{0}\|,1-K/(K+\varepsilon/9))\subseteq
B(f,\varepsilon)$, that is, $(b)$ holds for $x=x_{0}/\|x_{0}\|$ and
$\delta=1-K/(K+\varepsilon/9)$.

Let $g\in S$. Since, $r/(r+\varepsilon/9)\leq K/(K+\varepsilon/9)$,
$g(x_{0}/\|x_{0}\|)>r/(r+\varepsilon/9)$, which implies
$g(x_{0})>r\|x_{0}\|/(r+\varepsilon/9)>r$. That is, $\inf g(B)>0$. So,

$\\{x\in B(X):f(x)>\varepsilon/3\\}\subseteq C\subseteq
B\subseteq\\{x:g(x)>0\\}.$

By Lemma 2.5, $\|f-g/\|g\|\|<2\varepsilon/3$. Also, $g\in S$ implies
$\|g\|\geq g(x_{0}/\|x_{0}\|)>K/(K+\varepsilon/9)$. Hence,

$\|f-g\|\leq\|f-g/\|g\|\|+(1-\|g\|)\leq
2\varepsilon/3+\varepsilon/(9K+\varepsilon)<\varepsilon.$

This proves the claim.

$(b)\Rightarrow(a)$. Given $\varepsilon>0$ and $M(\varepsilon)\geq 2$, it
suffices to show that there is $K(\varepsilon)>0$ such that whenever a closed
convex set $C\subseteq X$ is such that $diam(C)\leq M(\varepsilon)$ and
$d(0,C)\geq\varepsilon$, there is a closed ball $B\subseteq X$ of radius $\leq
K(\varepsilon)$ such that $C\subseteq B$ and $d(0,B)\geq\varepsilon/2$.

Let $\varepsilon>0$ be given. Let $L=M(\varepsilon)+\varepsilon$. Choose
$0<\delta<1$ for $\varepsilon/4L$ obtained from $(b)$. Let $K=L/\delta+1$. We
will show that this $K$ works.

Case I : $C\setminus B[0,M(\varepsilon)+\varepsilon/2]\neq\emptyset$.

Choose $z\in C\setminus B[0,M(\varepsilon)+\varepsilon/2]$. Then $C\subseteq
B[z,M(\varepsilon)]$, $d(0,B[z,M(\varepsilon)])\geq\varepsilon/2$ and
$M(\varepsilon)\leq K$.

Case II : $C\subseteq B[0,M(\varepsilon)+\varepsilon/2]$.

Define $D=\overline{C+\frac{\varepsilon}{2}B(X)}$. Then $D\subseteq B[0,L]$
and $d(0,D)\geq\varepsilon/2$, and hence, $D$ is disjoint from
$B(0,\varepsilon/2)$. By separation theorem, there exists $f\in S(X^{*})$ such
that $\inf f(D)\geq\varepsilon/2$. By choice of $\delta$, there exists
$x_{0}\in S(X)$ such that

$S(B(X^{*}),x_{0},\delta)\subseteq B(f,\varepsilon/4L).$

It follows that if $g\in B(X^{*})\cap B(f,\varepsilon/4L)$ and $z\in D$, then
$g(z)\geq f(z)-\|f-g\|\|z\|\geq\varepsilon/2-\varepsilon/4=\varepsilon/4>0$.
Therefore,

$\displaystyle S(B(X^{*}),x_{0},\delta)$ $\displaystyle\subseteq$
$\displaystyle B(X^{*})\cap B(f,\varepsilon/4L)$ $\displaystyle\subseteq$
$\displaystyle\\{g\in B(X^{*}):g(x)>0\mbox{ for all }x\in D\\}.$

By the proof of Theorem 2.6, for $K\geq\lambda\geq L/\delta$,

$D\subseteq B\left[\lambda
x_{0},\lambda-\frac{d(0,D)\delta}{2-\delta}\right].$

It follows that

$\displaystyle\displaystyle C\subseteq B=B\left[\lambda
x_{0},\lambda-\frac{\varepsilon}{2}-\frac{d(0,D)\delta}{2-\delta}\right],\quad\mbox{and}$
$\displaystyle\displaystyle
d(0,B)=\frac{\varepsilon}{2}+\frac{d(0,D)\delta}{2-\delta}\geq\frac{\varepsilon}{2}.$

This completes the proof.

$(b)\Rightarrow(c)$. As observed in the proof of Lemma 2.10$(ii)$, $D(S(X)\cap
B(x,\delta))\subseteq S(B(X^{*}),x,\delta)$.

$(c)\Rightarrow(e)$. Let $\varepsilon>0$ be given. Then from $(c)$ for
$\varepsilon/3$, there exists $\delta>0$ such that for any $f\in S(X^{*})$,
there exists $x\in S(X)$ such that $D(S(X)\cap B(x,2\delta))\subseteq
B(f,\varepsilon/3)$. This implies $d_{3}(x,2\delta)\leq
2\varepsilon/3<\varepsilon$. By Lemma 2.10$(iii)$,
$d_{1}(x,\delta)<\varepsilon$, _i.e._ , $x\in M_{\varepsilon,\delta}(X)$. And
$D(x)\subseteq B(f,\varepsilon/3)\subseteq B(f,\varepsilon)$.

$(e)\Rightarrow(b)$. Let $\varepsilon>0$ be given. By $(e)$, there exists
$\delta>0$ such that for every $f\in S(X^{*})$, there is an $x\in
M_{\varepsilon/4,\delta}(X)$ such that $D(x)\subseteq B(f,\varepsilon/4)$. So,
$d_{1}(x,\delta)<\varepsilon/4$. By Lemma 2.10$(i)$,
$d_{2}(x,\alpha)<\varepsilon/2$, for $\alpha<\delta\varepsilon/8$. That is,
$diam(S(B(X^{*}),x,\alpha))<\varepsilon/2$. Now, $D(x)\subseteq
S(B(X^{*}),x,\alpha)$. Hence for any $g\in S(B(X^{*}),x,\alpha)$ and $f_{x}\in
D(x)$

$\|f-g\|\leq\|f-f_{x}\|+\|f_{x}-g\|\leq\varepsilon/4+\varepsilon/2<\varepsilon.$

Therefore,

$S(B(X^{*}),x,\alpha)\subseteq B(f,\varepsilon).$

$(c)\Rightarrow(d)$ is obvious.

$(d)\Rightarrow(c)$. If $(c)$ does not hold, there exists $\varepsilon>0$ such
that for all $\delta>0$, there exists $f_{\delta}\in S(X^{*})$ such that for
all $x\in S(X)$, there exists $z_{x}\in S(X)$ and $f_{z_{x}}\in D(z_{x})$ such
that $\|z_{x}-x\|<\delta/2$ and $\|f_{\delta}-f_{z_{x}}\|\geq\varepsilon$.
Then $\\{z_{x}:x\in S(X)\\}$ is a $\delta$-net, but $\\{f_{z_{x}}:x\in
S(X)\\}$ is not an $\varepsilon$-net in $S(X^{*})$. ∎

##  3\. Characterisation of w*-UMIP

In this section, we study the w*-version of UMIP defined above.

######  Definition 3.1.

We say that a dual Banach space $X^{*}$ has the w*-Uniform Mazur Intersection
Property (w*-UMIP) if for every $\varepsilon>0$ and $M(\varepsilon)\geq 2$,
there is $K(\varepsilon)>0$ such that whenever a w*-compact convex set
$C\subseteq X^{*}$ and a point $f\in X^{*}$ are such that $diam(C)\leq
M(\varepsilon)$ and $d(f,C)\geq\varepsilon$, there is a closed ball
$B\subseteq X^{*}$ of radius $\leq K(\varepsilon)$ such that $C\subseteq B$
and $d(f,B)\geq\varepsilon/2$.

For $f\in S(X^{*})$, the inverse duality mapping is defined as

$D^{-1}(f)=\left\\{\begin{array}[]{ll}\\{x\in S(X):f(x)=1\\}&\mbox{ if
\quad}f\in D(S(X))\\\ \emptyset&\mbox{ if \quad}f\notin
D(S(X))\end{array}\right.$

The fact that $D^{-1}(f)=\emptyset$ for some $f\in S(X^{*})$ introduces some
technicalities in the discussion of w*-UMIP.

######  Definition 3.2.

For $\varepsilon,\delta>0$ and $f\in S(X^{*})$, denote by

$\displaystyle d^{*}_{1}(f,\delta)$ $\displaystyle=$
$\displaystyle\sup\limits_{0<\lambda<\delta,~{}g\in
B(X^{*})}\frac{\|{f+\lambda g}\|+\|{f-\lambda g}\|-2}{\lambda},$
$\displaystyle d^{*}_{2}(f,\delta)$ $\displaystyle=$ $\displaystyle
diam(S(B(X),f,\delta)),$ $\displaystyle d^{*}_{3}(f,\delta)$ $\displaystyle=$
$\displaystyle diam(D^{-1}(D(S(X))\cap B(f,\delta))).$

######  Lemma 3.3.

For any $\alpha,\delta>0$, we have,

  1. (i)

$\displaystyle d^{*}_{2}(f,\alpha)\leq
d^{*}_{1}(f,\delta)+\frac{2\alpha}{\delta}$

  2. (ii)

$d^{*}_{3}(f,\delta)\leq d^{*}_{2}(f,\delta)$

  3. (iii)

$\displaystyle
d^{*}_{1}(f,\alpha)\leq\frac{d^{*}_{3}(f,\delta)+2\alpha}{1-\alpha^{2}}$ if
$\alpha<\sqrt{\delta+1}-1$.

###### Proof.

$(i)$ follows from Lemma 2.10$(i)$.

$(ii)$ holds since $D^{-1}(D(S(X))\cap B(f,\delta))\subseteq
S(B(X),f,\delta)$.

$(iii)$. Let $d^{*}_{3}=d^{*}_{3}(f,\delta)$. Choose
$0<\alpha<\sqrt{\delta+1}-1$. Then $\alpha^{2}+2\alpha<\delta$. Let $g\in
S(X^{*})$, $0<\lambda<\alpha$. As before,

$\left\|\frac{f\pm\lambda g}{\|f\pm\lambda g\|}-f\right\|\leq 2\lambda.$

Find $h_{1},h_{2}\in D(S(X))$ such that

$\left\|\frac{f+\lambda g}{\|f+\lambda
g\|}-h_{1}\right\|\leq\lambda^{2}\quad\mbox{and}\quad\left\|\frac{f-\lambda
g}{\|f-\lambda g\|}-h_{2}\right\|\leq\lambda^{2}$

and $x_{1},x_{2}\in S(X)$ such that $h_{i}\in D(x_{i})$, $i=1,2$.

Observe that
$\|h_{i}-f\|\leq\lambda^{2}+2\lambda\leq\alpha^{2}+2\alpha<\delta$, i.e.,
$h_{1},h_{2}\in D(S(X))\cap B(f,\delta)$. Thus $\|x_{1}-x_{2}\|\leq
d^{*}_{3}$. Now,

$0\leq 1-x_{1}\left(\frac{f+\lambda g}{\|f+\lambda
g\|}\right)=x_{1}\left(h_{1}-\frac{f+\lambda g}{\|f+\lambda
g\|}\right)\leq\left\|\frac{f+\lambda g}{\|f+\lambda
g\|}-h_{1}\right\|\leq\lambda^{2}.$

So, $x_{1}(f+\lambda g)\geq(1-\lambda^{2})\|f+\lambda g\|$. Similarly,
$x_{2}(f-\lambda g)\geq(1-\lambda^{2})\|f-\lambda g\|$. So, we have

$\displaystyle\frac{\|f+\lambda g\|+\|f-\lambda g\|-2}{\lambda}$
$\displaystyle\leq$ $\displaystyle\frac{x_{1}(f+\lambda g)+x_{2}(f-\lambda
g)-2(1-\lambda^{2})}{\lambda(1-\lambda^{2})}$ $\displaystyle=$
$\displaystyle\frac{(x_{1}+x_{2})(f)-2+\lambda(x_{1}-x_{2})(g)+2\lambda^{2}}{\lambda(1-\lambda^{2})}$
$\displaystyle\leq$
$\displaystyle\frac{\|x_{1}-x_{2}\|+2\lambda}{1-\lambda^{2}}\leq\frac{d^{*}_{3}+2\lambda}{1-\lambda^{2}}\leq\frac{d^{*}_{3}+2\alpha}{1-\alpha^{2}}$

(since $\displaystyle\frac{d^{*}_{3}+2\lambda}{1-\lambda^{2}}$ is increasing
in $\lambda$). Thus,

$d^{*}_{1}(f,\alpha)\leq\frac{d^{*}_{3}+2\alpha}{1-\alpha^{2}}.$

∎

######  Remark 3.4.

The proof of $(iii)$ is adapted from the proof of [2, Lemma 2]. Notice that
unlike the proof of [7, Lemma 3.1], our proof uses just the Bishop-Phelps
Theorem and not Bollobás’ extensions.

We now come to our characterisation theorem. Though the proof is similar to
that of Theorem 2.11, there are added technicalities as pointed out earlier.

######  Theorem 3.5.

For a Banach space $X$, the following statements are equivalent :

  * $(a)$

$X^{*}$ has w*-UMIP.

  * $(b)$

Every $x\in S(X)$ is uniformly semidenting, i.e., given $\varepsilon>0$, there
exists $\delta>0$ such that for any $x\in S(X)$, there exists $f\in S(X^{*})$
such that

$S(B(X),f,\delta)\subseteq B(x,\varepsilon).$

  * $(c)$

The inverse duality map is uniformly quasicontinuous, i.e., given
$\varepsilon>0$, there exists $\delta>0$ such that for any $x\in S(X)$, there
exists $f\in D(S(X))$ such that $D^{-1}(D(S(X))\cap B(f,\delta))\subseteq
B(x,\varepsilon)$.

  * $(d)$

Given $\varepsilon>0$ there exists $\delta>0$ such that every support mapping
on $X^{*}$ that maps $D(S(X))$ into $S(X)$ maps any $\Delta\subseteq D(S(X))$
that is a $\delta$-net in $S(X^{*})$ to an $\varepsilon$-net in $S(X)$.

  * $(e)$

Given $\varepsilon>0$ there exists $\delta>0$ such that for every $x\in S(X)$,
there exists $f\in M_{\varepsilon,\delta}(X^{*})\cap D(S(X))$ such that
$D^{-1}(f)\subseteq B(x,\varepsilon)$.

###### Proof.

$(a)\Rightarrow(b)$ is similar to Theorem 2.11 $(a)\Rightarrow(b)$.

$(b)\Rightarrow(a)$. Define $D=C+\frac{\varepsilon}{2}B(X^{*})$. Then $D$ is
w*-compact, and $d(0,D)\geq\varepsilon/2$. Therefore, $D$ is disjoint from the
w*-compact convex set $3\varepsilon/8B(X^{*})$. Hence there exists $x\in S(X)$
such that $\inf\hat{x}(D)\geq 3\varepsilon/8$.

Rest of the proof is similar to Theorem 2.11 $(b)\Rightarrow(a)$.

$(b)\Rightarrow(c)$. Let $\varepsilon>0$ be given. By $(b)$, there exists
$\delta>0$ such that for any $x\in S(X)$, there exists $g\in S(X^{*})$ such
that

$S(B(X),g,2\delta)\subseteq B(x,\varepsilon).$

As in [3, Lemma 3.2.6], if we choose $0<\eta<\delta/4$ then for any $h\in
B(g,\eta)$,

$S(B(X),h,\delta)\subseteq S(B(X),g,2\delta).$

By Bishop-Phelps Theorem, there exists $f\in D(S(X))\cap B(g,\eta)$. Then

$S(B(X),f,\delta)\subseteq S(B(X),g,2\delta)\subseteq B(x,\varepsilon).$

And $D^{-1}(D(S(X))\cap B(f,\delta))\subseteq S(B(X),f,\delta)$. Therefore,
$(c)$ holds.

$(c)\Rightarrow(e)$. Let $\varepsilon>0$ be given. Then from $(c)$ for
$\varepsilon/4$, there exists $\delta>0$ such that for any $x\in S(X)$, there
exists $f\in D(S(X))$ such that $D^{-1}(D(S(X))\cap B(f,\delta))\subseteq
B(x,\varepsilon/4)$. This implies $d^{*}_{3}(f,\delta)\leq\varepsilon/2$. By
Lemma 3.3$(iii)$,

$\displaystyle d^{*}_{1}(f,\alpha)$ $\displaystyle\leq$
$\displaystyle\frac{d^{*}_{3}(f,\delta)+2\alpha}{1-\alpha^{2}}\leq\frac{\varepsilon/2+2\alpha}{1-\alpha^{2}}<\varepsilon$
$\displaystyle\mbox{ if }\alpha<\sqrt{\delta+1}-1\mbox{ and
}\frac{\varepsilon/2+2\alpha}{1-\alpha^{2}}<\varepsilon.$

That is, $f\in M_{\varepsilon,\alpha}(X^{*})$. Clearly, the choice of $\alpha$
depends only on $\varepsilon$. And $D^{-1}(f)\subseteq
B(x,\varepsilon/4)\subseteq B(x,\varepsilon)$.

$(e)\Rightarrow(b)$. Let $\varepsilon>0$ be given. Choose $0<\delta<1$
obtained from (e) for $\varepsilon/4$. So, given $x\in S(X)$, there is $f\in
M_{\varepsilon/4,\delta}(X^{*})\cap D(S(X))$ such that $D^{-1}(f)\subseteq
B(x,\varepsilon/4)$. As $f\in M_{\varepsilon/4,\delta}(X^{*})$,
$d_{1}^{*}(f,\delta)<\varepsilon/4$. By Lemma 3.3,
$d_{2}^{*}(f,\alpha)<\varepsilon/2$ for $0<\alpha<\varepsilon\delta/8$. That
is, $diam(S(B(X),f,\alpha))<\varepsilon/2$.

Also, $D^{-1}(f)\subseteq B(x,\varepsilon/4)$ and $f\in D(S(X))$. Hence for
any $y\in S(B(X),f,\alpha)$ and $z\in D^{-1}(f)$

$\|x-y\|\leq\|x-z\|+\|y-z\|\leq\varepsilon/4+\varepsilon/2<\varepsilon.$

Therefore,

$S(B(X),f,\alpha)\subseteq B(x,\varepsilon).$

$(c)\Rightarrow(d)$ is obvious.

$(d)\Rightarrow(c)$. If $(c)$ does not hold, there exists $\varepsilon>0$ such
that for all $\delta>0$, there exists $x_{\delta}\in S(X)$ such that for all
$f\in D(S(X))$, there exists $g_{f}\in D(S(X))$ and $y_{f}\in D^{-1}(g_{f})$
such that $\|f-g_{f}\|<\delta/2$ and $\|x_{\delta}-y_{f}\|\geq\varepsilon$.

Define $\Delta=\\{g_{f}:f\in D(S(X))\\}$. Clearly, $\Delta$ forms a
$\delta/2$-net in $D(S(X))$. Since $D(S(X))$ is dense in $S(X^{*})$, $\Delta$
forms a $\delta$-net in $S(X^{*})$ as well. Define a support mapping
$\Phi:D(S(X))\to S(X)$ as follows :

  * $($i$)$

If $g\in\Delta$, we choose any $f\in D(S(X))$ such that $g=g_{f}$ and define
$\Phi(g)=y_{f}$. Observe that, by definition of $\Delta$, there is at least
one such $f\in D(S(X))$.

  * $($ii$)$

If $g\in D(S(X))\setminus\Delta$, define $\Phi(g)=y$ for some $y\in
D^{-1}(g)$.

Now, as $\|x_{\delta}-\Phi(g)\|\geq\varepsilon$ for all $g\in\Delta$,
$\Phi(\Delta)$ does not form an $\varepsilon$-net in $S(X)$. So, $(d)$ does
not hold. ∎

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