Looking at your string, I see parameter value pairs in the pattern of Param{Value}Param{Value}. If that is the case, then a regular expression can pull these out using the code below (note that I added a closing curly bracket to your string):
>> pvRawString = 'ternaryMassDensity{Xmole(1)=0.0Xmole(2)=1.0rho(1)=2343rho(2)=2343}ThermalConductivity{Xmole(1)=0.0Xmole(2)=1.0kappa(1)=3,2,13,...,2kappa(2)=2,1,11,...,3Tref=...}ElectricalConductivity{sigma0=1e5Tref=300}Mobility{Xmole(1)}';
>> pvs = regexp(pvRawString, '(?<Parameter>\w+){(?<Value>.*?)}','names');
>> pvs(1)
ans =
Parameter: 'ternaryMassDensity'
Value: 'Xmole(1)=0.0Xmole(2)=1.0rho(1)=2343rho(2)=2343'
The "pvs" variable above is a struct array of length 4, one for each parameter/value pair found. Note that the regular expression above will fail if curly brackets are nested within the parameter value.
