Dimensional Changes and Models

III. DIMENSIONAL SWELLING CHARACTERIZATION OF CHEMICALLY

2. Dimensional Changes and Models

much interest in the nature of alkali-stuffed silicate glasses, in particular surface compression magnitude, which is dictated by the molar volume resulting from alkali stuffing and the associated relaxation behavior at the time of stuffing and afterward.^24,25

calculations can be made. In this case, the free end parallel to the direction of invading ion diffusion is observed for perpendicular swelling. An early estimate of this deformation was demonstrated in the finite element analysis of Sane and Cooper²⁹ and later by Jain and Varshneya.³⁰ Laboratory measurements referencing this feature were demonstrated by researchers attempting to improve the flatness of substrates for hard magnetic disks for data storage.³¹

Of these dimensional features, several studies have utilized the step height approach to characterize chemically strengthened glasses. Early work by Burggraaf¹⁴ attempted to measure the step height induced by selective chemical strengthening of sodium aluminosilicate exchanged with potassium. Step height measurements made with a roughness meter were somewhat unreliable, but allowed for an estimated expansion in the diffusion (free) direction of about 10^-4 per mol% K2O in terms of the LNDC. This result had order of magnitude agreement with the LNDC estimated from stress- birefringence measurements, which were about 6x10^-4 per mol% K2O.

Measurements of step height have enabled a materials behavior model to be assembled. Glebov, et al.³² used a step height measurement to study potassium replacing sodium exchange in K8 optical glass (alkali borosilicate³³) as a function of chemical strengthening temperature and time. The researchers assembled a simple model allowing local strain to have elastic and plastic components, using step height observations and optical ellipsometry measurements of stress state as inputs. In the form of Eq. (8), the model simply splits the LNDC B(z) into elastic B_ela(z) and plastic B_pla(z) components.

Allowing the z-axis to be the diffusion direction, then within the x-plane the total (or initial) strain xx^tot

 

^z is equated to the sum of the elastic strain ^elaxx

 

^z and plastic strain

 

^z

pla

xx :³²

  

^z

  

^z xx^pla

 

^z

ela xx tot

xx   

  1  (10)

where ν is Poisson’s ratio and the input elastic strain is of the biaxial plane strain condition, hence the (1- ν) term to convert to linear strain for the summation. Integrating over all of the chemically strengthened layers in the z-direction, the integral of Eq. (10) is:³²

         



^{xxtot z dz 1 xxela z dz xxpla z dz (11)}

which has limits of integration (not shown) from the case depth, i.e. depth of neutral stress point from surface, to the surface. Note, the strain state is assumed equivalent in the x- and y-directions, i.e. _xx^tot

 

z ^tot_yy

 

z . Thus similar equations to Eq. (10) and Eq. (11) can be expressed for strain within the y-plane and in the y-direction. A second equation is constructed by equating the dimensional swelling in the z-direction, called the

“step height” here, to the integral of the free strain in the diffusion direction (z-axis):³²

     

   

 



 

















dz z dz

z

dz z dz

z dz

z z

pla xx tot

xx

pla yy pla

xx tot

xx











2 (12)

where Δz is the step height and, again, the limits of integration (not shown) are from the case depth to the surface. Glebov, et al. invoke volume conservation, so plastic strain is limited to shear flow, thus the factor of two preceding the _xx^pla

 

z integral due to equal plastic strain contributions from the x- and y-dimensions. With Eq. (11) and Eq. (12), the unknowns of integrated total strain and integrated plastic strain can be computed. The contributions to Δz are not known as a function of z-position and prevent initial strain and plastic strain from being determined versus z-position, but the solution to this system of equations is sufficient to allow for estimates of average total strain _xx^tot, average plastic strain _xx^pla, plastic strain-to-total strain ratio R^pla, and, if the average concentration is known, the total maximum strain _xx^totmax or stress _xx^max.

Glebov, et al.³² applied this system of equations to their data set and observed the average plastic strain _xx^pla increased with increasing temperature between 400 and 520 °C, whereas the average total strain _xx^tot (referred to as initial strain by the authors) was nearly constant with increasing temperature, until about 520 °C. That is, the initial potassium accommodation by the glass network displayed a constant average expansion, of which the fraction that relaxed by shear flow increased with increasing temperature.

This lead to the conclusion that for low-temperature ion exchange, plastic strain is temperature-dependent and the initial volume change associated with alkali stuffing causes displacement of constituents about the alkali ion, but does not require thermal activation.

Varshneya³⁴ noted a potential modification to the model of Glebov, et al.³² in which the plastic strain xx^pla

 

^z is split into plastic deviatoric xx^{pla D}

 

^z

  (volume conserving) and plastic hydrostatic xx^{pla H}

 

^z

  (shape conserving) contributions. In this approach, Eq. (11) is modified as follows:

 

^zdz

   

^zdz

 

^zdz xx^{pla H}

 

^zdz

D pla xx ela

xx tot

xx

  



^{  1        (13)}

And Eq. (12) is modified as follows:

       

      

 



  































dz z dz

z dz

z

dz z dz

z dz

z dz

z z

H pla xx D

pla xx tot

xx

H pla xx D

pla yy D

pla xx tot

xx















2 (14)

Note, the plastic hydrostatic strain _xx^plaH

 

z is equivalent in-plane and out-of-plane, allowing densification to alleviate elastic strain, together with shear flow from the original model.

Varshneya^19,34 has also proposed local elastic-plastic yielding behavior during chemical strengthening. Upon the entry of a stuffing alkali ion to a host site, bond bending or stretching occurs consisting of independent hydrostatic and deviatoric elastic limits. Beyond the hydrostatic limit, dilatation no longer contributes to expansion, thus leads to densification. Beyond the deviatoric limit, shear no longer contributes to in- plane expansion, but volume is conserved. Alkali site expansion and stretching is proposed to leave the site in a higher energy configuration than the non-stuffed state, but this configuration resides in a local potential energy minimum from which escape is mediated by limited thermal energy of the ion-exchange process.

A goal of the present work is to examine the applicability of these proposed models and mechanisms where possible.