This is because, in the absence of c 2, we can define free energy functions $$ Q^x_r = \left( \fraca_xb_\!x \right)^r-1 , \qquad Q^y_r = \left( \fraca_yb_\!y \right)^r-1 , $$ (A9)which generate the equilibrium distributions $$ c_r^eqx = Q_r^x c_1^r = \fracb_\!xa_x \left(\fraca_x c_1b_\!x \right)^r \;\; > \;\; c_r^eqy = Q_r^y c_1^r = \fracb_\!ya_y \left( \fraca_y c_1b_\!y \right)^r . $$ (A10)If

a x /b x  < a y /b y then the latter (Y) will be the dominant crystal type at equilibrium, whilst X is the less stable morphology at equilibrium. These last two words are vital, since, at early times, the growth rates depend on the relative sizes of the growth rates a x and a y . It is possible for the less stable form to grow first and EVP4593 manufacturer more quickly from solution, and be observed for a significant period of time, since the rate of

convergence to equilibrium also depends on the fragmentation rates and so can be extremely slow (see Wattis 1999 for details). In the presence of grinding, the crystal size distributions also depend upon the strength of dimer interactions, that is, the growth rates α x c 2 + ξ x x 2, α y c 2 + ξ y y 2 and the grinding rates β x , β y . The steady-state size distributions will depend on the relative this website growth ratios due to grinding (α x c 2 + ξ x x 2)/β x and (α y c 2 + ξ y y 2)/β y as well as the more traditional terms due to growth from solution,

namely a x c 1/b x and a y c 1/b y . Such systems with dimer interactions have been analysed previously by Bolton and Wattis (2002). The presence of dimer interactions can alter the size distribution, and in non-symmetric Silibinin systems such as those analysed here, dimer interactions can alter the two distributions differently. Two points are worth noting here: (i) for certain parameter values, the less stable stable form (Y, say, with a y /b y  < a x /b x ) may be promoted to the more stable morphology by grinding (if (α y c 2 + ξ y y 2) / β y is sufficiently greater than (α x c 2 + ξ x x 2) / β x );   (ii) grinding may make a less rapidly nucleating and growing form (Y, say, with a y  < a x ) into a more rapidly growing form if α y c 2 + ξ y y 2 is sufficiently greater than α x c 2 + ξ 2 x 2.   In systems which can crystallise into three or more forms, we may have the case where x is more stable than y and y is more stable than z; thus, at equilibrium x will be observed. Furthermore, if a x  < a y  > a z we may observe type y at early times due to it having faster nucleation and growth rates than x and z.