phenomenology of crystalline polymorphism: overall

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B is endothermic (+3.6 kJ mol-‐1 or +8.834 J g-‐1) and ... the higher melting enthalpy, which differs by +22.8 K ..... representing one of the six two-phase equilibria (line intersecting O1-O2: B-vapor, O1-O3: liquid-vapor, O1-O4: B-liquid, O2-.

PHENOMENOLOGY  OF  CRYSTALLINE  POLYMORPHISM:   OVERALL  MONOTROPIC  BEHAVIOR  OF  THE  CARDIOTONIC   AGENT  FK664  FORMS  A  AND  B     Ivo  B.  Rietveld*,  René  Céolin   Caractérisation  des  Matériaux  Moléculaires  à  Activité  Thérapeutique  (CAMMAT)   Faculté  de  pharmacie,  4  avenue  de  l’observatoire,  75006  Paris     *Corresponding  author:  Ivo  B.  Rietveld,  e-­‐mail:  [email protected],  tel.  +33  1  53739675  

    ABSTRACT   The   stability   hierarchy   of   crystalline   polymorphs   is   often   determined   on   the   basis   of   limited   calorimetric   data   even  when  other  useful  data  such  as  specific  volumes  may  be  available.  This  may  lead  to  wrong  conclusions  or   an  incomplete  picture  of  the  stability  behavior  of  a  compound  and  may  be  problematic  in  many  applications,  for   example   for   pharmaceuticals.   Therefore,   the   topological   approach   has   been   applied   to   the   pharmaceutical   FK664,   which   exhibits   crystalline   dimorphism,   using   heat   and   work   related   thermodynamic   data   to   obtain   a   pressure  –  temperature  phase  diagram  and  elucidate  its  phase  behavior.  The  approach  leads  to  the  conclusion   that  FK664  is  overall  monotropic  with  form  B  the  most  stable  solid  phase.  In  other  words  form  A  does  not  have  a   stable   domain   for   any   pressure-­‐temperature   coordinate.   The   case   of   FK664   demonstrates   that   making   use   of   the  full  available  thermodynamic  data  set  in  combination  with  statistical  information  of  the  phase  behavior  of   small  organic  molecules  and  classical  thermodynamics  leads  to  a  sound  evaluation  of  the  stability  hierarchy  for   crystalline  dimorphism.     Keywords:   monotropy,   phase   diagram,   topological   pressure-­‐temperature   diagram,   crystalline   polymorphism,   phase   relationship,   stability,   (E)-­‐6-­‐(3,4-­‐dimethoxyphenyl)-­‐1-­‐ethyl-­‐4-­‐mesitylimino-­‐3-­‐methyl-­‐3,4-­‐dihydro-­‐2(1H)-­‐ pyrimidinone



INTRODUCTION   In   1901,   Bakhuis-­‐Roozeboom   published   four   pressure-­‐temperature   (P-­‐T)   phase   diagrams   representing   the   four   possible   topological   layouts   of   the  four  phases  (two  solids,  the  liquid,  and  the  vapor)   present   in   the   system   in   the   case   of   crystalline   dimorphism[1].   These   diagrams,   whose   original   drawings   have   been   reproduced   recently   [2],   demonstrate   the   effect   of   pressure   and   make   it   possible   to   evaluate   the   phase   behavior   of   two   polymorphs   over   the   entire   pressure   –   temperature   domain.     Although  Nagasako  evoked  the  fact  that  the  influence   of  pressure  on  phase  equilibria  needs  to  be  taken  into   account   [3],   at   present   it   is   not   the   case   when   describing   the   stability   hierarchy   of   pharmaceutical   polymorphs.   This   may   be   due   to   the   difficulty   to   measure   the   pressure   (the   vapor   pressure   and   the   hydrostatic  pressure);  However,  this  is  not  necessary,   because   only   the   inequality   between   the   vapor   pressures  is  required  to  establish  a  stability  hierarchy   between  polymorphs,  as  shown  in  Figure  1.   In   Figure   1,   the   lines   represent   the   pressure   and   temperature   coordinates,   where   a   condensed   phase   (form   I,   form   II,   or   the   melt   (L   =   liquid))   is   in   equilibrium   with   its   vapor   phase.   Point   1   is   the   intersection  of  the  curves  representing  the  equilibria   between  form  I  and  the  vapor  and  between  the  melt   and   the   vapor.   Thus,   at   the   intersection,   form   I,   the   liquid,   and   the   vapor   (which   has   the   same   pressure   for   both   phases   at   the   temperature   of   intersection)   are  in  equilibrium  defining  the  intersection  as  a  triple   point   with   its   coordinates   being   temperature  T1   and   the   vapor   pressure   at   the   intersection.   Thus,   if   a   melting  point  is  obtained  while  the  condensed  phases   are  in  equilibrium  with  their  vapor  phase,  which  is  the   case   under   ordinary   conditions1,   it   is   equivalent   to                                                                                                                           1  “Ordinary  conditions”  signifies  that  the  system  is  in   thermal  equilibrium  with  its  surroundings  and  that  a   vapor  phase  is  present  which  is  in  equilibrium  with   the  condensed  phase,  thus  ordinary  in  the  sense  of    

the  triple  point  and  the  temperature  of  triple  point  1   is   equivalent   to   the   melting   temperature   T1.   The   same   is   true   for   triple   point   2,   at   which   form   II,   the   liquid  and  the  vapor  are  in  equilibrium  and  for  which   T2   is   the   melting   point   under   ordinary   conditions.   Finally,  point  3  is  the  triple  point  where  form  I,  form   II,   and   the   vapor   are   in   equilibrium,   but   it   also   represents   the   solid-­‐solid   transition   temperature   T3   between  forms  I  and  II  under  ordinary  conditions.  

  Figure 1. The two possible cases for the inequality in the vapor pressures of two polymorphs (a) enantiotropy (b) monotropy (see text for an explanation of the figures) In   Figure   1a,   below   T3,   form   II   is   more   stable   than   form   I.   The   most   stable   form   has   the   lowest   vapor   pressure   for   a   given   temperature,   which   is   clearly   the   case   for   form   II   below   T3.   Form   I   is   the   more   stable   form   from   temperature   T3   up   to   temperature   T1,   above  which  the  liquid  is  the  more  stable  phase.  Such   stability   behavior   between   two   solid   phases   has   been   named  enantiotropy  by  Lehmann  [4,  5].  In  contrast,  in   Figure  1b,  form  I  has  the  lowest  vapor  pressure  and  is   thus   the   more   stable   phase   for   the   entire   temperature  domain  below  the  melting  temperature                                                                                                                                                                                                         how  a  drug  may  be  stored.  It  implies  that  the   thermodynamic  pressure  of  the  system  is  equivalent   to  its  partial  vapor  pressure,  even  if  an  inert  gas  is   present.   2  

T1,   while   form   II   is   less   stable.   Lehmann   called   such   behavior   monotropy,   after   he   observed   cases   for   which  form  II  transformed  non-­‐reversibly  into  form  I   on   heating   at   ordinary   pressure   [4,   5].   “Non-­‐ reversibly”   indicates   that   form   I   does   not   transform   back   into   form   II   on   cooling.   Lehmann   neglected  the   influence  of  pressure  on  phase  changes;  however,  by   including   pressure,   Bakhuis-­‐Roozeboom   arrived   at   the   general   description   of   the   four   possible   P-­‐T   diagrams  for  crystalline  dimorphism.   The   dimorphism   of   FK664   is   a   further   example   in   a   series   of   theoretical   and   experimental   studies   on   polymorphism   through   topological   pressure   –   temperature   phase   diagrams   [2,   6-­‐23].   FK664   is   a   2-­‐ pyrimidone   derivative   (Figure   2)   developed   in   1988-­‐ 1989  by  Fujisawa  Pharmaceutical  Company  as  a  drug   against   heart   failure   [24,   25].   The   stability   hierarchy   between   its   two   polymorphs   is   re-­‐examined   using   heat-­‐  and  work-­‐related  data  at  ordinary  pressure.  

comparing  the  calculated  densities  (Dcal).  The  authors   also   calculated   with   results   from   DSC   measurements   that  the  heat  of  transition  (ΔA→BH)  of  form  A  to  form   B   is   endothermic   (+3.6   kJ   mol-­‐1   or   +8.834   J   g-­‐1)   and   they   concluded   that   “Form   B   is   at   least   more   stable   than   form   A”,   although   no   interconversion   between   the   two   crystal   forms   had   experimentally   been   observed.   Table   1.   Crystallographic   and   calorimetric   data   at   T  =  295   K   for   forms   A   and   B   of   FK664,   C24H29N3O3   with  M  =  407.505  g  mol-­‐1  [26]a     Form  A   Form  B   Crystal  system,   Monoclinic,   Monoclinic,   space  group   P21/c   P21/c   a/  Å   13.504   8.067   b/  Å   6.733   15.128   c/  Å   24.910   18.657   96.55   102.34   β /°   Vunit-­‐cell/  Å3   2250.09   2224.252   Z   4   4   Dcal/g  cm-­‐3   1.20293   1.216909   vspec/  cm3  g-­‐1   0.83130   0.821754   391.0   413.8   Ti→ L/  K   Δi→ LH/  kJ  mol-­‐1  (J  g-­‐1)   33.2  (81.5)   36.8  (90.3)   a

  Crystallographic   data   based   on   values   reported   in   the   CIF   files   (Cambridge   Structural   Database   codes   VUSZEO   and   VUSZEO01),   Dcal:  calculated  density  from  CIF  data,  vspec:  specific  volume  from   CIF  data,  Ti→L:  melting  point  of  phase  i  (=  form  A  or  form  B),  Δi→LH:   heat  of  fusion  of  phase  i.  


  Figure 2. Chemical formula of FK664, (E)-6-(3,4dimethoxyphenyl)-1-ethyl-4-mesitylimino-3-methyl-3,4dihydro-2(1H)-pyrimidinone.





LITERATURE   Structural   studies   and   physicochemical   characterization  of  the  two  forms  of  FK664  have  been   published   in   1991   by   Miyamae   et   al.   [26]   and   the   results   are   compiled   in   Table   1.   The   crystal   structures   were  solved  at  the  same  temperature  (295  K)  and  the   authors  reported  that  form  B  is  the  denser  one  after    

HEAT-­‐RELATED  DATA   Form   B   exhibits   the   higher   melting   temperature   and   the  higher  melting  enthalpy,  which  differs  by  +22.8  K   and   +3.6   kJ   mol-­‐1   (8.8   J   g-­‐1)   from   those   of   form   A,   respectively.  Neglecting  the  difference  in  the  specific   heats   of   the   two   forms,   the   enthalpy   of   transition   from  A  to  B  is  calculated  to  be  −3.6  kJ  mol-­‐1  (−8.8  J  g-­‐ 1 ),   according   to   the   thermodynamic   cycle   shown   in   Figure   3.   Thus   ΔA→BH   is   negative,   opposite   to   the   conclusion   reported   by   Miyamae   et   al   [26],   and   this   implies   that   the   difference   in   Gibbs   energy   between   the   forms   will   decrease   with   increasing   temperature   and   that   form   B   should   eventually   spontaneously   3  

transform   into   form   A   on   heating   following   the   Le   Chatelier  principle.  

potential)   of   the   condensed   phase.   The   vapor   pressure   as   a   function   of   the   temperature   may   be   approximated  by  the  following  expression:   Ln  (Pβ)  =  −Δβ→vaporH  /(RT)  +  Bβ→vapor  

  Figure 3. Hess’ law for the heats of transition between form A, form B, and the liquid; the algebraic sum of the heats of transition over the cycle must be equal to 0. It follows that the heat of transition from A to B must be equal to −3.6 kJ mol-1. Although   the   A-­‐B   equilibrium   has   never   been   observed,  its  temperature  (TB→A)  at  ordinary  pressure   can   be   calculated   using   the   procedure   described   by   Yu  [27]  leading  to  the  following  equation:  

TB→ A =

Δ A→ L H − Δ B→ L H   Δ A→ L H Δ B→L H − TA→ L TB→ L




Using  the  values  provided  in  Table  1,  TB→A  =  895.22  K   is   found,   a   temperature   for   the   B→A   transition   far   higher   than   the   temperatures   of   fusion   for   both   forms.   As   explained   above   (Figure   1),   the   transition   temperature  corresponds  to  the  invariant  equilibrium   of  the  two  solid  forms  and  their  vapor  also  called  the   triple   point   A-­‐B-­‐vapor   (in   equilibrium,   the   vapor   pressures   of   the   two   solid   phases   must   be   equal).   It   implies  that  the  A-­‐B-­‐vapor  triple  point  is  metastable,   because   it   is   located   in   a   domain   of   the  P-­‐T   diagram   where   the   solids   are   not   stable   anymore,   as   the   highest  melting  form  melts  at  413.8  K.   As   the   triple   point   lies   on   the   condensed   phase-­‐vapor   equilibrium   curves,   the   vapor   pressure   of   the   triple   point  can  be  calculated.  The  vapor  pressure  is  a  direct   reflection   of   the   Gibbs   energy   of   the   vapor   phase.   Because   the   vapor   phase   is   in   equilibrium   with   the   condensed   phase   (in   other   words   the   chemical   potentials  are  equal),  the  saturated  vapor  pressure  is   also   a   reflection   of   the   Gibbs   energy   (or   chemical    



in   which   Pβ   is   the   vapor   pressure   (in   Pa)   of   the   condensed   phase   β   (forms   A,   B,   or   the   liquid)   in   equilibrium  with  the  vapor,  Δ  β→vaporH  is  the  enthalpy   (in   J   mol-­‐1)   of   either   vaporization   (liquid)   or   sublimation   (solid),   R   is   the   gas   constant   (8.31446   J   mol-­‐1K-­‐1),  T  is  the  absolute  temperature  and   Bβ→vapor  is   a  constant.   The   boiling   point   at   a   vapor   pressure   of   760   Torr   (1.0133×105   Pa),   identical   to   the   atmospheric   pressure,  and  the  enthalpy  of  evaporation  of  organic   compounds   was   evaluated   with   ACD/Labs   software   [28].   Using   the   boiling   point,   TL→vapor,   of   558.5°C   (831.65  K)  and  the  vaporization  enthalpy,  ΔL→vaporH,  of   84060  J  mol-­‐1,  the  constant  BL→vapor  for  FK664  can  be   calculated   and   is   found   to   be   23.68.   Now,   the   vapor   pressure   (in   Pa)   of   liquid   FK664   as   a   function   of   temperature  (in  K)  can  be  expressed  by:   Ln(PL)  =  −84060/(RT)  +  23.68  




Similar  expressions  for  the  vapor  pressure  of  the  two   solid  phases  can  be  derived  from  eq.  3.   At   the   melting   temperature   of   form   B,   TB→L   =   413.8   K,   the   vapor   pressure   of   the   liquid   can   be   calculated   with   eq.   3   (PL   =   0.4726   Pa)   because   the   vapor   pressure   of   form   B   equals   that   of   the   liquid   at   the   melting  point  under  ordinary  pressure.  The  difference   in   enthalpy   between   form   B   and   the   liquid   is   the   melting   enthalpy.   Therefore,   the   sublimation   enthalpy  of  form  B  is  given  by  ΔB→vaporH  =  ΔL→vaporH  +   ΔB→LH   leading   to   120860   J   mol-­‐1.   As   the   vapor   pressure   is   known   at   TB→L,   constant   BB→vapor   can   be   calculated   leading   to   34.38.   The   vapor   pressure   (in   Pa)  of  form  B  can  now  be  expressed  by:   Ln(PB)  =  −120860/(RT)  +  34.38    



The  dependence  of  the  vapor  pressure  of  form  A  with   the  temperature  can  be  obtained  in  the  same  way:  (i)   4  

ΔB→vaporH   is   obtained   by   adding   ΔL→vaporH   and   ΔB→LH   and   equals   117260   J   mol-­‐1.   (ii)   Because   form   A   and   the   liquid   are   both   in   equilibrium   with   the   vapor   phase   at   the   triple   point   A-­‐L-­‐vapor   (i.e.   at   TA→L=   391   K),   the   pressure   obtained   with   eq.   3   at   this   temperature   (equal   to   0.1137   Pa)   can   be   used   to   calculate  constant  BA→vapor  leading  to  33.90.  The  vapor   pressure  (in  Pa)  of  form  A  can  now  be  given  by:   Ln(PA)  =  −117260/(RT)  +  33.90    



The   two   curves   defined   by   eqs.   4   and   5   intersect   at   the   temperature   where   both   solid   phases   have   the   same  vapor  pressure,  the  triple  point  A-­‐B-­‐vapor.  The   temperature   can   be   calculated   by   setting   the   pressure  equal  in  eqs.  4  and  5  leading  to  895.22  K.   Because   this   approach   is   identical   to   eq.   1,   any   difference   in   the   obtained   transition   temperature   is   due  to  rounding  errors.  In  addition  to  the  metastable   transition   temperature,   this   approach   allows   the   calculation   of   the   vapor   pressures   of   the   condensed   phases  as  they  are  given  by  eqs.  3-­‐5.  At  the  A-­‐B-­‐vapor   triple   point   the   vapor   pressure   of   the   solid   phases   equals   76   MPa.   Instead,   at   the   same   temperature,   the  liquid  has  a  vapor  pressure  of  0.24  MPa,  which  is   far  smaller.  It  indicates  that  the  liquid  is  more  stable   at   this   temperature   and   that   the   A-­‐B-­‐vapor   triple   point   is   metastable   because   it   is   located   in   a   P-­‐T   domain,   where   the   liquid   has   a   lower   vapor   pressure.   Nonetheless,  if  one  were  to  work  at  1  atm  pressure,   and  the  liquid  would  be  present  in  a  container  freely   expanding   against   the   atmosphere,   even   the   liquid   would   completely   evaporate   as   its   0.24   MPa   vapor   pressure   would   only   be   counterbalanced   by   a   mere   0.1  MPa  external  atmospheric  pressure.   Following   the   discussion   of   the   two   scenarios   in   Figure   1,   the   dimorphism   of   FK664   is   clearly   reflected   by  the  case  illustrated  in  Figure  1b.  In  principle,  form   B  would  transform  into  A  on  heating  following  the  Le   Chatelier   principle;   however   this   transition   point   is   metastable.   Nonetheless,   using   only   heat-­‐related   data   limits   the   stability   analysis   to   the   temperature   domain   and   no   inferences   can   be   drawn   concerning   the  phase  behavior  under  increasing  pressure.  It  may    

be  interesting  to  know  whether  the  metastable  phase   at   ordinary   pressure   (here   FK664   form   A)   becomes   stable   on   increasing   the   pressure.   This   is   carried   out   using   work-­‐related   data,   as   shown   in   the   following   section.  

WORK-­‐RELATED  DATA   The  finite  change  ΔU  of  the  internal  energy  of  a  one-­‐ component  system  that  undergoes  a  change  of  state   is  the  algebraic  sum  of  the  heat  received,  Q,  and  the   work   done,   W,   during   the   change.   This   is   written   as   ΔU   =   Q   +   W   and,   according   to   Gibbs,   leads   to   the   fundamental   equation   of   (classical)   thermodynamics   ΔU   =   TΔS   –   PΔV.   P   and   T   are   the   pressure   and   the   temperature   of   the   system   at   which   the   change   occurs,   which   is   accompanied   by   changes   in   entropy   (ΔS)   and   volume   (ΔV).   The   first   principle   states   that   ΔS  =  Sfinal  −  Sinitial  and  ΔV  =  Vfinal  −  Vinitial,  and  the  second   principle   states   that   ΔS   =   Q/T   if   the   change   occurs   “reversibly“,   i.e.   at   an   infinitely   slow   rate.   In   particular,  because  the  parameters  P  and  T  are  more   accessible,   and   because   they   are   intensive,   the   fundamental  equation  of  classical  thermodynamics  is   rewritten  through  the  Legendre  transformation  in  the   form   of   a   variation   of   the   Gibbs   energy   ΔG   =   −SΔT   +   VΔP.   The   pressure   –   temperature   phase   diagrams   are   a   direct   reflection,   or   one   should   say   projection,   of   the   use   of   the   Gibbs   energy   to   determine   phase   stability.  Unfortunately,  work-­‐related  data,  i.e.  P  and   v,  are  virtually  never  taken  into  account,  because  the   pressure   is   seldom   measured   and   work   is   assumed   to   be   negligibly   small,   thus   thermodynamics   is   often   reduced  to  calorimetry.   However,  it  is  nowadays  fairly  easy  to  determine  the   unit  cell  volumes  of  crystal  structures.  In  addition,  in   the   Helmholtz   function,   which   is   also   obtained   by   the   Legendre   transformation   of   the   internal   energy,   F  =  U    −   TS   and   ΔF   =   −SΔT   −   PΔV,   V   and   T   are   the   independent  variables  instead  of  S  and  V  (or  P  and  T   in  the  case  of  the  Gibbs  energy).  v-­‐T  diagrams  can  be   of   practical   interest   in   the   study   of   polymorphism,   because   temperature   and   specific   volumes   are   more   easily   obtained   than   pressure.   v-­‐T   diagrams   are   projections  of  the  function  F(v,T)  on  the  v-­‐T  plane  and   5  

the  specific  volume  v  is  used,  rendering  an  extensive   variable   intensive   by   dividing   by   the   quantity   of   material  present  in  the  system.   For  a  relevant  use  of  the  variables  T  and  v,  the  system   needs  to  be  in  equilibrium  with  its  vapor  phase.  This   implies   that   thermal   expansion   measurements   have   to   be   carried   out   under   saturating   vapor   pressure.   If   the   solid   for   which   the   volume   is   being   measured   is   not  placed  in  a  closed  container  with  a  fixed  volume,   the   surroundings   that   have   to   be   saturated   by   the   vapor   pressure   are   virtually   infinite   and   the   saturation   condition   does   not   seem   to   be   fulfilled.   Nevertheless,   because   the   saturating   vapor   pressure   of   pharmaceuticals   is   in   most   cases   very   small,   the   assumption   that   vapor   pressure   locally   saturates   the   surroundings  of  the  specimen  is  generally  correct.   In   the   case   of   FK664,   the   specific   volumes   obtained   from   the   two   crystal   structures   solved   at   the   same   temperature   have   enough   precision   to   judge   that   vA   is  greater  than  vB.  From  the  inequality  ΔA→Bv  =  vB  –  vA   <  0,  it  can  be  inferred  that  form  A  should  eventually   transform   into   form   B   on   increasing   the   pressure,   following   the   Le   Chatelier   principle.   In   combination   with   the   previous   conclusion   involving   the   calorimetric   data,   it   can   be   inferred   that   the   slope   dP/dT   of   the   equilibrium   curve   A-­‐B   should   be   positive.  To  confirm  the  inference,  the  triple  point  A-­‐ B-­‐L   will   need   to   be   placed   in   the   P-­‐T   diagram.   It   is   located  at  the  crossing  point  of  the  P-­‐T  melting  curves   A-­‐L  and  B-­‐L.  The  melting  curves  are  approximated  by   straight   lines,   which   is   the   simplest   case   of   a   monotonous  curve  [29].   As   the   slopes   of   the   equilibrium   curves   (dP/dT)   are   given  by  the  Clapeyron  equation,  

dP Δs Δh     = = dT Δv T Δv





the  inequalities  in  the  specific  entropy  (Δs  =  Δh/T;  the   equality   is   valid   at   equilibrium,   h   is   the   specific   enthalpy   (in   J   g-­‐1))   and   in   the   specific   volume   (Δv)   between   the   phases   in   equilibrium   with   each   other   need  to  be  established.  In  the  case  that  they  are  not   available   experimentally,   they   can   be   determined    

based   on   the   following   generally   accepted   approximations:   (1)   the   inequality   in   the   specific   volumes   between   two   solid   forms   is   virtually   independent   of   the   temperature.   (2)  Gavezzotti  has  reported  that  molecular  solids  with   expansivities   (or   volumetric   thermal   expansion   coefficients   αv)  of  about  2  ×  10-­‐4  K-­‐1  expand  by  about   5  to  6  %  from  0  K  to  their  melting  temperature  ([30],   p  277).  This  value  is  close  to  the  mean  value  found  for   several   drugs,   as   shown   in   Table   2.   (3)   A   reasonable   estimate   of   the   volume   change   on   melting   can   be   found   in   several   references.   Gavezzotti   reported   ([30],   p   24)   values   from   Ubbelohde   for   volume   changes   on   the   melting   of   organic   compounds   [31,   32].   The   mean   value   in   the   increase   of   the   volume   that   accompanies   melting   is   11.5  %  with  only  2  values  out  of  the  7  –  16  %  range.   This   is   close   to   the   value   of   12   %   reported   by   Goodman   et   al.   [33].   It   is   also   near   the   10   %   value   found   for   several   molecular   pharmaceuticals   (see   compilation  in  Table  2).  Thus  an  increase  of  11  %  will   be   used   in   the   following   for   the   estimate   on   the   volume  change  on  melting.  







One   step   in   the   construction   of   a   topological   P-­‐T   phase   diagram   involving   two   polymorphs   is   to   determine  the  slopes  dP/dT  of  the  melting  curves  and   to   infer   whether   they   cross   at   high   pressure   or   at   negative   pressure,   i.e.   whether   they   diverge   or   converge  with  increasing  pressure.  In  either  case,  the   intersection   is   the   triple   point   A-­‐B-­‐L.   If   it   occurs   at   elevated   pressure,   the   triple   point   is   stable;   at   negative  pressure,  the  triple  point  will  be  metastable   as  the  system  is  in  an  expanded  state.   For   the   slopes   of   the   melting   equilibria,   the   volume   difference   between   the   solid   form   and   the   melt   has   to   be   determined   at   the   temperature   of   fusion.   A   value  of  2  ×  10-­‐4  K-­‐1  for  the  volume  thermal  expansion   6  

αv   for   each   polymorph   is   taken   and   a   linear   change   in   the   specific   volume   as   a   function   of   temperature   is   assumed:   v(T)  =  v0  (1  +  αv  T),  




(7)   -­‐1

with  v0  at  T  =  0  K  and  v(T)  in  cm  g ,   αv  in  K  and  T  in   K  (cf.  Figure  4).   The   values   of   the   specific   volumes   at   the   melting   temperatures   of   forms   A   and   B   are   found   to   be   0.846372   cm3   g-­‐1   at   391   K   and   0.8401914   cm3   g-­‐1   at   413.8   K,   respectively,   using   the   values   obtained   at   295  K  and  reported  in  Table  1  (v0  is  0.784986  cm3  g-­‐1   and   0.7759720   cm3   g-­‐1   for   respectively   form   A   and   form  B).   It   is   generally   accepted   that   the   thermal   expansion   of   organic   molecular   liquids   is   larger   than   that   of   their   related   molecular   solids,   and   the   expansivity   αV   of   their   melts   on   average   is   found   to   be   approximately   1.0  ×  10-­‐3  K-­‐1  [34].  In  the  following,  a  value  of  1.25  ×   10-­‐3  K-­‐1  will  be  used,  as  it  is  close  to  the  average  of  the   values  obtained  from  the  pharmaceuticals  in  Table  2.  

an   expansivity   of   1.25   ×   10-­‐3   K-­‐1   for   the   melt,   the   specific   volume   of   liquid,   vL   /cm3g-­‐1,   as   a   function   of   temperature   (K)   can   be   approximated   with   the   following  linear  equation:   vL  =  0.619195  +  7.73993  ×10-­‐4  T    



It   results   in   vL   =   0.921826   cm3   g-­‐1   at   TA→L   =   391   K.   It   follows  that  ΔB→Lv  =  0.099281  cm3  g-­‐1  at  413.8  K  and   ΔA→Lv=  0.075454  cm3  g-­‐1  at  391  K.   Schematically   this   is   shown   in   Figure   4.   It   can   be   seen   that   the   specific   volume   of   the   metastable   melt   is   larger  than  the  crystalline  solid  A  at  the  temperature   of  the  observed  glass  transition,  Tg  =  327  K  [26].  Along   the   same   line,   the   Kauzmann   temperature[38]   at   which   the   curves   of   the   specific   volume   of   the   melt   and   of   form   A   cross,   can   be   found   below   the   experimental   glass   transition   temperature.   Thus   even   though   the   relationships   of   the   specific   volume   with   the  temperature  have  been  obtained  from  statistical   information,   they   coincide   with   the   available   additional  experimental  and  theoretical  information.  

Table   2.   Thermal   expansion   of   a   number   of   active   pharmaceutical   ingredients   (API)   from   the   literature.;   α V,L:   isobaric   thermal   expansion   coefficient   (or   expansivity)   of   the   liquid   defined   by   equation   (7),   α V,S:   expansivity   of   the   solid,   and  

v(liq)/v(solid):the  ratio  between  the  specific  volumes  of   the  liquid  and  the  solid  at  the  melting  point.   API  

α V,L   3 -­‐1 ×10  (K )   1.26   1.49  

biclotymol   progesterone   a (I/II)     Rimonabant   1.38   a (I/II)   lidocaine   1.30   prilocaine   1.01   ternidazole   0.81   Mean/median   1.21/1.28   a  crystalline  polymorphs  

α V,S   4 -­‐1 ×10  (K )   1.72   1.71/1.63  

v(liq)/v(solid)   at  Tfus   1.13   1.09/1.10  





3.99   2.18   1.80   2.12/1.76  

1.06   1.13   1.11   1.10/1.10  

[35]   [36]   [37]    

[2]   [15]  

The   specific   volume   of   the   liquid   can   be   estimated,   based   on   the   statistical   difference   mentioned   above   of  11%  and  the  specific  volume  of  the  highest  melting   form   at   the   melting   point.   This   leads   to   1.11   ×   vB(413.8  K)  =  0.939473  cm3  g-­‐1  at  TB→L.  Then,  assuming    

  Figure 4. Schematic thermal expansion of FK 664 form A (double trace line), form B (black line), and the liquid (black line turning into broken and subsequently dotted line). Tg: glass transition (ca. 327 K), Tk: Kauzmann temperature.  


To  determine  the  slope  of  the  melting  equilibria,  the   Clapeyron   equation   (6)   is   used.   The   results   on   the   specific   volume   and   the   calorimetric   data   on   the   fusion   in   Table   1   can   now   be   used   to   find   the   following  slopes  dP/dTA-­‐L  =  2.76  MPa  K-­‐1  and  dP/dTB-­‐L   =   2.20   MPa   K-­‐1.   Because   form   B   has   the   higher   melting   point   in   combination   with   the   more   gradual   slope   and   hence   form   A   has   a   lower   melting   point   with   a   steeper   slope   for   the   equilibrium,   the   equilibrium   curves   must   intersect   at   negative   pressure  and  thus  the  triple  point  A-­‐B-­‐L  is  metastable.   The   melting   curves   can   be   approximated   by   straight   lines   that   pass   at   ordinary   pressure   through   the   triple   points   A-­‐L-­‐vapor   and   B-­‐L-­‐vapor,   which   are   equal   to   the   respective   melting   points.   The   pressure   at   these   triple   points   is   given   by   eqs   5   and   4   and   equals   0.11   Pa   and   0.47   Pa   respectively.   Because   the   slopes   of   the  solid  –  liquid  equilibria  are  in  the  order  of  2  to  3   MPa   K-­‐1,   the   pressure   coordinate   of   the   triple   points   can   be   taken   as   0   MPa.   This   leads   to   the   following   expressions   using   slope   and   triple   point   for   the   pressure  (in  MPa)  of  the  two  solid  –  liquid  equilibrium   lines  as  a  function  of  the  temperature  (in  K):   A-­‐L:    

PA-­‐L  =  2.76  T  –  1080  





PB-­‐L  =  2.20  T  –  910  




With  these  equations,  the  coordinates  of  metastable   triple   point   A-­‐B-­‐L   are   found   to   be   P,   T   =   -­‐246   MPa,   302  K.  Although  the  temperature  can  be  found  close   to   room   temperature,   the   pressure   is   clearly   negative.   The  last  two-­‐phase  equilibrium  curve  that  remains  to   be   located   is   the   one   where   A   and   B   are   in   equilibrium   with   each   other.   It   passes   through   triple   points   A-­‐B-­‐vapor   (T,   P   =   895   K,   76   MPa,   see   the   section   on   heat   related   data)   and   A-­‐B-­‐L   whose   P-­‐T   coordinates  have  been  determined  just  above.  Solid-­‐ solid   equilibrium   curves   are   usually   straight   lines,   at   least   up   to   a   few   hundred   megapascals;   thus,   using   the  triple  point  coordinates,  the  equation  is  obtained   in  a  straightforward  way:   A-­‐B:      

PA-­‐B  =  0.54  T  –  409  




It  can  now  be  concluded  that  the  slope  dP/dT  for  the   A-­‐B   equilibrium   curve   is   smaller   (ca.   0.54   MPa   K-­‐1   from  eq.  11)  than  those  of  the  two  melting  curves.   The   purpose   of   these   calculations   is   not   to   obtain   accurate  values  for  the  expressions  of  the  two-­‐phase   equilibria   and   of   the   coordinates   of   the   triple   points   but   to   study   with   a   high   level   of   precision   their   inequalities  and  relative  positions  to  each  other  in  the   phase  diagram.  To  demonstrate  the  robustness  of  the   topological  placement  of  the  phases,  even  though  the   accurate   position   of   the   phase   equilibria   is   not   known,   a   sensitivity   parameter   analysis   has   been   carried  out,  which  can  be  found  in  the  supplementary   information.  






Because  the  coordinates  of  the  four  triple  points  that   exist   in   the   case   of   dimorphism   have   been   determined  (Table  3),  the  P-­‐T  diagram  can  be  drawn   by   using   straight   lines   for   the   two-­‐phase   equilibria   passing  through  the  triple  points,  as  shown  in  Figure   5a.   Table  3.  Coordinates  of  the  four  triple  points  for  the   dimorphism  of  FK664   Triple  point   P  /Pa   T  /K   O1:  B-­‐L-­‐vapor   0.47   413.8   6 O2:  A-­‐B-­‐vapor   76  ×10   895   O3:  A-­‐L-­‐vapor   0.11   391   O4:  A-­‐B-­‐L   -­‐246  ×106   302     However,  the  resulting  phase  regions  are  not  of  equal   stability   ranking,   because   the   P-­‐T   representation   of   phase   equilibria   in   a   one-­‐component   system   is   the   projection   of   the   intersections   of   monotonous   Gibbs   energy   surfaces,   each   of   which   represents   the   Gibbs   energy  minimum  as  a  function  of  P  and  T  for  a  given   phase.  To  determine  which  of  these  surfaces  has  the   deepest   minimum   and   to   apply   a   stability   ranking   to   the  different  phases,  the  alternation  rule  can  be  used.   This   rule   imposes   the   alternation   of   stable   equilibrium   lines   and   metastable   lines   around   a   triple   point,   which   is   a   direct   result   of   the   Gibbs   energy   surfaces   intersecting   and   replacing   one   minimum   8  

surface   (or   stable   phase)   by   another   [29].   Metastability   simply   means   that   the   Gibbs   energy   surface   of   the   phase   is   not   the   lowest   one.   One   metastable   surface   can   intersect   another   and   therefore   the   stability   ranking   projected   on   the   P-­‐T  

plane   can   also   be   extended   into   metastable   triple   points   surrounded   by   alternating   metastable   and   supermetastable   equilibrium   lines   (which   translates   into   even   higher   placed   Gibbs   energy   surfaces   in   relation   to   the   stable   and   metastable   phases).

  Figure 5. Construction of the topological phase diagram of the dimorphism of FK664; P and T are not to scale (a) Placing the triple point coordinates (numbers can be found in Table 3) and intersecting each set of two triple points by a straight line representing one of the six two-phase equilibria (line intersecting O1-O2: B-vapor, O1-O3: liquid-vapor, O1-O4: B-liquid, O2O3: A-vapor, O2-O4: A-B, and O3-O4: A-liquid) (b) The lines and triple points have been ranked according to their stability hierarchy (c) Stable domain of form B (shaded region) (d) Metastable domain of form A (shaded region), where it can occur as a solid even if it will eventually turn into form B. Because   the   highest   melting   solid   is   the   most   stable   phase  just  below  its  melting  point,  it  is  clear  that  the   triple   point   B-­‐L-­‐vapor   is   stable.   Three   stable   two-­‐ phase  equilibria  must  meet  in  this  point  and  turn  into   metastable   on   the   opposite   site   of   the   triple   point.   These   are   the   liquid-­‐vapor   curve   (down   from   the   critical   point),   the   sublimation   curve   of   form   B,   and   the  melting  curve  of  form  B.  This  can  be  seen  around    

triple   point   O1   in   Figure   5b.   It   can   also   be   seen   that   the   metastable   extensions   of   these   two-­‐phase   equilibria   curves   intersect   metastable   triple   points.   The   metastable   extension   of   the   B-­‐vapor   curve   intersects  O2,  which  is  the  triple  point  between  form   A,   form   B,   and   the   vapor   at   a   location   where   the   liquid  is  stable.  This  means  that  the  triple  point  itself   is   metastable   (as   has   already   been   discussed   in   9  

previous   sections)   and   therefore   any   metastable   equilibrium   intersecting   it   must   turn   supermetastable.  The  same  is  true  for  triple  point  O3,   the   melting   point   of   form   A,   which   occurs   in   the   region   where   B   is   stable   (Figure   5c),   and   triple   point   O4,   where   the   two   solids   and   the   liquid   are   in   equilibrium.   The   latter   triple   point   is   located   in   the   region   where   the   system   is   expanded   and   the   only   possible   phase   that   can   exist   in   such   a   state   is   the   vapor.  Thus  O1  is  the  only  stable  triple  point  and  the   intersection   of   the   only   three   stable   two-­‐phase   equilibria  (Figure  5b).  

  Figure 6. Gibbs energy-temperature isobaric sections of the P-T diagram for an overall monotropic behavior (form A less stable than form B for any P and T). Left-hand side: section at pressure P1, right-hand side: section at pressure P2. The inset shows how form A becomes less metastable than form B as T increases, at constant pressure. However at point 5, the liquid has the lowest Gibbs energy. Points 1 to 6 are sections of two-phase equilibrium curves; 1: A-liquid, 2: A-vapor, 3: B-liquid, 4: B-vapor, 5: A-B, 6: liquid-vapor. The   phase   diagram   possesses   the   same   topology   as   Bakhuis-­‐Roozeboom’s   case   4,   for   which   Bakhuis-­‐ Roozeboom   gave   no   example.   Two   other   cases   of   dimorphism   among   APIs   have   been   found   to   exhibit   this   so-­‐called   “overall   monotropic   behavior”:   biclotymol  [2]  and  rimonabant  [22].   In  the  case  of  overall  monotropic  behavior,  the   Gibbs   energy   of   the   less   stable   polymorph   remains   larger   than   that   of   the   other   phases   at   any  P  and   T.  This   is    

illustrated   in   Figure   6   by   depicting   two   G-­‐T   isobaric   sections  of  the  G(P,T)  diagram.  

CONCLUDING  REMARKS   The   preceding   analysis   demonstrates   that   Miyamae   et  al.  were  correct  when  they  concluded  that  form  B   was   the   more   stable   one   of   the   two   known   solid   forms.   It   also   places   this   conclusion   on   firm   thermodynamic   footing.   Finally,   it   is   clear   from   the   analysis   that   form   A   does   not   have   a   stable   domain   under   any   pressure   and   temperature   coordinates   in   relation  to  form  B,  in  other  words  that  the  system  is   overall  monotropic.   In   1901,   when   Bakhuis-­‐Roozeboom   published   the   four   possible   cases   of   P-­‐T   diagrams   for   crystalline   dimorphism,   no   example   was   provided   of   the   fourth   case,  overall  monotropic  behavior.  It  may  have  been   the  lack  of  an  example  that  caused  this  case  to  be  all   but   forgotten,   as   citations   have   been   scarce   in   the   literature.  However,  systems  with  overall  monotropic   behavior   do   exist   as   compounds   with   such   phase   behavior   can   be   found   in   the   literature:   trichloroacetic   acid   [7],   hydrazine   monohydrate   [7],   sulfanilamide   [8,   10],   ferrocene   [11],   and   more   recently   two   pharmaceuticals:   biclotymol   [2]   and   rimonabant  [22].   Supplementary   information:   Sensitivity   parameter   analysis   of   the   inequality   in   the   dP/dT   slopes   of   the   melting  curves  

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